Physics Reports 362 (2002) 303 – 407 www.elsevier.com/locate/physrep The role of the in nuclear physics Giorgio Cattapan b; c , L dia S. Ferreira a ; ∗ a Centro de F sica das Interacc ˜ oes Fundamentais (CFIF), Departamento de F sica Instituto Superior T ecnico, Avenida Rovisco Pais, 1096 Lisboa, Portugal b Dipartimento di Fisica dell’ Universit a, I-35131 Padova, Italy c Istituto Nazionale di Fisica Nucleare, I-35131 Padova, Italy Received September 2001; editor : G:E: Brown Contents 1. Introduction 305 2. The free 307 2.1. The as a resonance 307 2.2. Isospin mass splittings in the multiplet and hadron structure 314 2.3. Deviation from spherical symmetry: the E2=M 1 ratio 318 3. Theoretical models for the N –interaction 326 3.1. Basic meson-exchange models for the N interaction 327 3.2. Meson–baryon couplings 330 3.3. The coupled-channel approach to the N system 352 3.4. Relativistic and unitarity corrections in the N coupled-channel approach 357 3.5. QCD-inspired models 360 4. Contents of in nuclei 363 4.1. Light nuclei: the percentage of in the nuclear wave function 363 4.2. Mesonic-exchange currents and 365 4.3. propagation in nuclei 370 4.4. The in nuclear matter 384 5. Conclusions and outlook 388 Appendix A. Dispersion relation constraints 390 A.1. Preliminaries 390 A.2. Fixed-variable dispersion relations 391 Appendix B. The N scattering matrix and electromagnetic corrections 397 References 401 Abstract We review the properties of , and the role it plays in Nuclear Physics. We try to assess what has been ascertained up to now, and what remains to be claried about this hadron, which represents the outstanding structure in the medium-energy pion–nucleon interaction. We consider the free rst, with particular emphasis on those aspects which may give information on the internal hadron structure, such as the isospin mass splittings in the multiplet, and the electromagnetic ngerprints of deviation from spherical symmetry. ∗ Corresponding author. E-mail address: [email protected] (L.S. Ferreira). 0370-1573/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S0370-1573(01)00093-X
Appendix B. The �N scattering matrix andelectromagnetic corrections 397
References 401
Abstract
We review the properties of �, and the role it plays in Nuclear Physics. We try to assess what has beenascertained up to now, and what remains to be clari>ed about this hadron, which represents the outstandingstructure in the medium-energy pion–nucleon interaction. We consider the free � >rst, with particular emphasison those aspects which may give information on the internal hadron structure, such as the isospin masssplittings in the � multiplet, and the electromagnetic >ngerprints of deviation from spherical symmetry.
The excitation of the � resonance is the most striking phenomenon in �N scattering for pionkinetic energies below 300 MeV. This is clearly exhibited by the phase shift behavior, with theP33 partial wave in the J = T = 3
2 channel playing by far the dominant role, and a weak s-waveinteraction even near threshold. The cross section for forming this resonance is at the unitarity limitof 200 mb, testifying its very strong coupling to the �N system. When regarded as an excited state inthe baryon spectrum, the � has the smallest mass gap with respect to the nucleon, about 300 MeV.Because of these features, one may expect that the � can have a prominent role in clarifying itemsin low-energy Hadron Physics and in highlighting non-nucleonic degrees of freedom in NuclearPhysics.
In low-energy hadron physics, the N–� mass diEerence is one of the basic observables quark mod-els are required to reproduce [1]. Moreover, the analysis of the mass splittings within the � multiplet,and of observables such as the � quadrupole moment and the E1=M2 ratio in pion photo-productioncan give information about quantum chromo dynamics (QCD) in the low-energy regime, since theseexperimental data are strictly connected to quark–quark tensor color interactions and=or the eEectivecoupling of constituent quarks with the Goldstone bosons associated with the breaking of chiralsymmetry [2,3].
The � resonance, considered as a baryon with its own spin and isospin quantum numbers, hasby now a long history as an “exotic” degree of freedom in Nuclear Physics. Its coupling to theother hadronic degrees of freedom, i.e. nucleons and mesons, can be described through eEec-tive Lagrangians, whose structure is dictated by symmetry considerations. As a matter of fact, inlow-energy nuclear physics it has become customary to mimic the usual meson-theoretic approachto the nuclear-force problem. One looks at the N → � transition or at the interaction between two�’s as due to the exchange of mesons; a non-relativistic reduction of the corresponding Feynmannamplitudes then provides the required baryon–baryon potentials. In so doing, the meson–baryoncoupling strengths come into play as free parameters, to be determined in reproducing the exper-imental data. This means that the task of extracting signals of non-nucleonic degrees of freedomfrom experiments is unavoidably intertwined with the determination of the “best” values for thevarious meson–baryon coupling constants. Hopefully, one would like to get more information onthe strengths from >rst-principle arguments, namely from a QCD description of hadron structure.As is well-known, in spite of great progresses in QCD lattice calculations, this is still beyond ourpresent possibilities. Even in the framework of more phenomenological quark models, a clear andunquestionable calculation of the hadron–hadron interaction is still missing, since this entails thesolution of a non-trivial few-body problem. It is not surprising therefore, that for many years onehas limited oneself to look for relations between the various coupling strengths based essentiallyon symmetry assumptions. A noteworthy example is the SU (4) quark model, where nucleon and �are members of the same multiplet, the latter being a spin–isospin excitation of the former with nochange in the orbital motion of the constituent quarks; as a consequence, the coupling strengths of �and � mesons to nucleon and � can only diEer by purely geometric factors. Similar considerationsapply to the strong-coupling model, a generalization of the celebrated Chew–Low model for the �Nsystem. Baryons are described as static sources interacting through meson exchange, with a commonscale parameter � for their mutual interaction; for � → ∞, a dynamical symmetry emerges, whichdictates well-de>ned relations among the coupling strengths.
When looking for signals of � excitation in nuclear phenomena, it is natural to start from thetwo-nucleon system. Thus, one may try to see whether an explicit inclusion of � degrees of freedomimproves our understanding of the nuclear force. It is by now well known that intermediate � prop-agation can substantially reduce contributions from the eEective � meson in One-Boson-Exchangemodels of the NN interaction. More generally, to achieve a sounded description of NN bound-stateand scattering observables, one needs a non-perturbative approach. This leads to the NN–N�–��coupled channel problem, which has been the subject of many investigations since the 1970s, withincreasing progress in the number of coupled channels taken into account, and in the realistic natureof the employed baryon–baryon potentials [4,5]. From a fundamental point of view, one has to as-certain to what extent this essentially non-relativistic scheme catches the essential features of whatis in itself a relativistic problem, the description of exchange forces mediated by meson >elds.
If the � is taken as a constituent of the atomic nucleus, one may wonder how much it contributesto the total nuclear wave function. Warnings of caution has been raised in the past about thisissue [6]. As a matter of fact, the � “particle” is in a sense an artifact, which mimics a resonantstructure in the �N continuum. If it is described as a Rarita–Schwinger >eld through the directproduct of a spin 1 vector state and a spin 1
2 state, one has to deal with non-physical spin- 12components [7]. Their contributions can be dismissed on the energy shell by imposing additionalconstraints on the Rarita–Schwinger >eld. In nuclei, however, � propagation occurs oE-shell, andunphysical degrees of freedom are still possible. These ambiguities are masked, but not in principlesolved, by the non-relativistic approximations necessary in order to match � physics with the wavefunctions of nuclear-structure theory. In spite of these problems, the idea of the resonance as apossible constituent of the nucleus is a robust one, as testi>ed by the success of the �-hole model inreproducing photon- or pion-induced scattering cross sections [8]. To reduce the ambiguities inherentto any model-dependent analysis, it is natural to look for the � in few-nucleon systems, and=or inprocesses where this degree of freedom is excited through the simple and well-known electromagneticor weak interactions. In the former case, indeed, one can exploit the presently available Faddeev orvariational techniques to get realistic nuclear wave functions; in the latter, one is naturally led toconsider the electromagnetic and weak currents which in the nuclear system couple to the externalelectroweak >eld. The � can contribute both to the one-body current through direct coupling to thephoton, and to the two-body currents, much in the same way as it comes into play in the intermediateNN interaction.
In nuclear matter, or in heavier nuclei, more eEective descriptions are required, in order to keepcomputations to a tractable level. In particular, short-range NN and N� correlations are described bymeans of eEective forces, in the spirit of Landau theory of quantum Fermi liquids. Great eEorts havebeen devoted to gain information on the proper values to be assigned to the corresponding eEectivestrength parameters [9]. This can be done either by looking at nuclear-structure and reaction data, or,on the theoretical side, through many-body calculations. The main issue is to disentangle contributionsarising from � excitation in the nuclear medium from eEects due to the presence of higher-order“ordinary” con>gurations in the nuclear wave function.
The detailed structure of the N� force can be probed exploiting the diEerent ways by whichdiEerent probes couple to nuclear targets. Pions and photons interact with nucleons and �’s throughlongitudinal and transverse couplings [8], respectively, whereas both types of interactions intervenein charge-exchange reactions [10]. The formation, propagation and >nal decay of the resonance canbe thus explored through diEerent windows, which emphasize diEerent facets of the same basic
dynamical mechanism. The most striking diEerence between nuclear and electromagnetic probes isthat with the former one observes a systematic downward shift of the resonance position with respectto the proton target, whereas the latter only lead to a broadening of the � peak. To ascertain thephysical meaning of this fact, the speci>city of the experimental probe is crucial. Under this point ofview, charge-exchange reactions turned out to have a privileged role. For the same reason, exclusivereactions are expected to convey more information than simpler inclusive experiments, even if theyare obviously more demanding for the theoretical analysis.
The excitation of �-hole states can be regarded as a polarization eEect of the nuclear medium.Together with N -hole excitations, it has been advocated to explain why several observable quantitiesare quenched with respect to the value they ought to have in a simple single-particle description ofthe nuclear response. Well-known examples are the quenching of the nuclear magnetic moment andof the axial coupling constant in nuclei with a closed-shell core plus (or minus) one nucleon [9,11],and the missing strength in the giant Gamow–Teller resonance, excited through intermediate-energycharge-exchange reactions [9,10]. Here also the main problem is to clarify the respective role playedby �-hole excitations, and by the mixing of higher-order con>gurations involving nucleonic degreesof freedom only.
In this paper, we try to give an overall insight into the present understanding of the � undera nuclear physicist point of view, updating what can be found in some previous excellent reviews[12,13]. Needless to say, it is practically impossible in a single paper to cover all the items whichwould deserve attention. For instance, we shall not consider � production in high-energy heavy-ionreactions, nor shall we dwell on � production reactions due to hadron probes, or on � absorp-tion processes in the resonance region. We shall rather follow the scheme outlined above. Thus, inSection 2 we discuss the free �, with particular emphasis on the determination of the resonanceparameters within the baryon spectrum, and on the relation between its isospin mass splittings anddeviation from spherical symmetry and our present views of hadron structure. Theoretical modelsfor the N–� interaction are discussed in Section 3. There, we devote considerable attention to thecoupling strengths among the eEective hadron >elds playing the main role in nuclear physics, namelyamong nucleons, �’s, and the � and � mesons. The coupled-channel N–� problem is also considered,both from a non-relativistic and a relativistic point of view. Finally, modern approaches based uponQCD-motivated quark models are brie?y discussed. The � particle as a constituent of the nucleusis the subject of Section 4, both for light and heavier nuclei. In addition to isobaric contributionsto electromagnetic currents, “classical” topics such as Landau parameters, medium eEects in nuclearreactions, and quenching phenomena are updated. At the end, the merging of the coupled N–�system in nuclear matter is discussed. Conclusions and outlook are given in Section 5.
2. The free �
2.1. The � as a resonance
Under an empirical point of view the � baryon can be perceived as a strong, isolated resonancein �N scattering. In elastic �±p, �+n, and charge-exchange �−p going into �0n cross sections,one can see the presence of a pronounced maximum for a pion lab energy around 190 MeV,which may be assigned to an intermediate state with mass M� ∼ 1230 MeV. In Fig. 1, the total
Fig. 1. Total �−p and elastic �+p cross sections as functions of the pion lab kinetic energy T�. Data from Ref. [14].
cross sections for �∓p scattering are plotted as functions of the beam energy; they show explicitly anexcited state with charge 2e and 0, respectively. If charge independence is assumed, these two statescan be assigned to an isospin multiplet. The isospin value is determined from the ratio of the totalcross sections for isospin-related reactions, such as elastic �±p scattering and the charge-exchangereaction �−p → �0n. As is well-known near the resonance one has
which is consistent with a total isospin T = 32 . In fact four charge states have been identi>ed for the
isobar, that is �−; �0; �+; �++.These considerations can be put on a more quantitative level through a partial-wave analysis of
the scattering data. In principle, the quantum numbers of the � resonance can be guessed throughrather simple quantum–mechanical considerations of �–N scattering. Assuming that the range R ofthe �N interaction is >xed by the pion mass (R ∼ 1=m�), for pion energies in the resonance region(T� ¡ 300 MeV) the usual semi-classical relation l ∼ qR between the angular and linear momentumof the pion gives that the s and p waves must dominate the scattering process. If the spin-free andspin-?ip amplitudes f(q; �) and g(q; �) are expanded into partial waves up to l = 1 one has (seeEqs. (B.13), and (B.15)–(B.16)),
Fig. 2. DiEerential cross sections for �N charge exchange at the � resonance, E�=190 MeV, from diEerent measurements.Figure from Ref. [15].
Here, we have labeled the various partial-wave amplitudes by means of the orbital and total angularmomentum quantum numbers l and j, respectively, and assumed purely elastic scattering, so thatEq. (B.17) is written
fl;j =12iq
(exp 2i�l; j − 1) =1qexp i�l; j sin �l; j ≡ 1
qal; j : (2.3)
Using the standard expression for the unpolarized diEerential cross section
d�d
= |f(q; �)|2 + |g(q; �)|2 ;
it is a trivial task to show that the angular distribution can be written in this approximation as asecond-degree polynomial in cos �,
d�d
� 1q2
(A0 + A1 cos �+ A2 cos2 �) ; (2.4)
where the coeScients A0; A1 and A2 depend upon the partial-wave amplitudes al; j according to
A0 = |a0;1=2|2 + |a1;3=2 − a1;1=2|2 ; (2.5)
A1 = 2Re[a?0;1=2(2a1;3=2 + a1;1=2)] ; (2.6)
A2 = 3|a1;3=2|2 + 6Re[a?1;3=2a1;1=2] ; (2.7)
so that A2 only contains contributions from the p wave. At resonance the angular distribution in theCM system has a parabolic dependence upon cos �, with a remarkable consistency among diEerentdata sets. This is clearly seen for example in Fig. 2, where data for �–N charge exchange at 190 MeVare reported. The diEerential cross sections at resonance can be reproduced with A0 ∼ 1; A1 ∼ 0,and A2 ∼ 3, a result consistent with a dominant l = 1; j = 3
2 contribution, with the correspondingphase shift �1;3=2 passing through �=2. A con>rmation of this assignment can be obtained by looking
at the value of the total cross section at resonance, where the dominant contribution ought to begiven by the jth component
�jT ∼ �
q2(2j + 1)(#j + 1) ;
which, for j= 32 and a purely elastic process (#j=1) gives �T � 8�=q2. This estimate is in reasonable
agreement with the experimental values shown in Fig. 1.The actual, quantitative extraction of the resonance parameters, energy and width, is obviously
far less simple than the above illustrative considerations. This is even more true, when the couplingto the possible reaction channels is taken into account, and a simultaneous identi>cation of thevarious baryon resonances is attempted, in order to map the excitation spectrum of the nucleon.The determination of resonance parameters is generally a two-step process. One has >rst to expresscross section and polarization data in terms of partial-wave amplitudes. For elastic scattering thisis accomplished through partial-wave analysis; for inelastic data one has to resort to isobar models,where the reaction process is described as a coherent superposition of quasi-two-body channels. Thesecond step involves the extraction of resonance parameters from the partial wave amplitudes, andhas been accomplished through several approaches, which diEer in the way of handling the couplingamong channels, the implementation of unitarity, and the very parameterization of the scatteringmatrix.
Benchmarks in partial wave analysis of �N scattering have been the solutions provided by theKarlsrhue–Helsinki (KH) [16–18] and Carnegie–Mellon–Berkeley (CMB) [19–21] groups. Thesestudies emphasized the crucial role played by dispersion relations in constraining the partial waveamplitudes extracted from the data. As the quality of the data base improved in time, new >ts tothe elastic �N data have been pursued by the Virginia Polytechnic Institute and State Universitygroup (the VPI group) [22,23]. They also advocated both forward dispersion relations and dispersionrelations at >xed momentum transfer t to constrain the solution. We shall give some hints about this,when dealing with the much debated question of the optimal value for the �NN coupling constant.
A general prescription which has to be satis>ed in the extraction of the resonance parameters isunitarity. The KH and VPI approaches focused on the elastic �N channel. The former accountedfor all inelasticity through absorption parameters #l; j. In the analysis, one has to take into accountelectromagnetic eEects, so as to de>ne purely hadronic partial wave amplitudes. How this has beenaccomplished by KH is summarized in Appendix B.
In the VPI approach, one accounts for the overall inelasticity introducing an eEective �� chan-nel, and unitarity is implemented through a coupled-channel K-matrix formalism [14]. The elasticscattering amplitude in each partial wave is then given by
Te =ImCeK
1− CeK(2.8)
with K expressed in terms of the 2× 2 K-matrix according to
K = Ke +CiK2
0
(1− CiKi): (2.9)
The functions Ce and Ci are the elastic and inelastic Chew–Mandelstam functions, obtained byintegrating phase-space factors over the appropriate unitarity cuts. Extensive energy-dependent, and
energy-independent (i.e. single-energy) analysis have been performed by VPI, whose results areupdated and can be found in the VPI repository [23].
Other approaches exploited formalisms which allow for many channels. Thus, Manley and Saleski[24] factorized the S-matrix into background and resonant contributions SB and SR, respectively,
S = STR SBSR (2.10)
with SB expressed in terms of a background K-matrix in the usual way, namely
SB =1 + iKB
1− iKB; (2.11)
and the resonant component written as the product of N factors, where N is the number of resonancesbeing parameterized,
SR = S1=21 S1=2
2 : : : S1=2N : (2.12)
Each factor S1=2k is constructed in such a way that its square can be written in terms of a transition
matrix, namely Sk=1+2iTk , which is parameterized in the standard multichannel Breit–Wigner form.As a consequence, near an isolated resonance one simply has S ∼ SR, if background contributionsare negligible (SB � 1). This approach allows for a direct extraction of resonance parameters, andhas been used starting from partial-wave amplitudes for �N → �N scattering and from isobar-modelamplitudes for �N → ��N reactions [24].
A multichannel model, devised to subsume both unitarity and analyticity requirements, has beendeveloped several years ago by the Carnegie Mellon–Berkeley group [19–21], and recently revived byT.-S.H. Lee and collaborators [25]. The basic assumption of this model is that a physical (asymp-totic) channel a can be mapped into another asymptotic channel b through a set of intermediateresonant states i=1; 2; : : : ; R. Both physical and intermediate states can be of two-body (�N; #N; : : :)or quasi-two-body (��; �N; : : :) nature, i.e. one of the two particles in a given channel can be itselfa resonance. The coupling of resonance i to an asymptotic state a is described by a strength pa-rameter +ai and a form factor fa(s), depending upon the total center-of-mass energy
√s, so that the
multichannel T -matrix can be written
Tab =R∑
i; j=1
fa(s)√
�a(s)+aiGij(s)+jb√
�b(s)fb(s); a; b= 1; : : : ; M : (2.13)
Here, �a is the phase-space factor for channel a. For a two-body channel this means that one simplyhas �a = pa=
√s, whereas for quasi-two-body channels a more complex recipe has to be employed,
in order to satisfy the unitarity condition
Im(Tab) =M∑c=1
T?acTcb : (2.14)
Finally, Gij is the full (dressed) propagator, satisfying the Schwinger–Dyson equation
Gij = G0ij +
∑kl
G0ik.klGlj ; (2.15)
where the bare propagator G0ij=�ij(s− s0; i)−1 is diagonal, with non-vanishing elements associated to
the propagation of the ith resonance. The integral equation (2.15) can be reduced to a set of R× R
algebraic equations assuming a factorized form for the self-energy .ij,
.ij =M∑c=1
+ci/c(s)+cj : (2.16)
The physical meaning of (2.16) is obvious. Resonance i can decay into an asymptotic channel c,whose particles can coalesce again to form resonance j. The channel propagators /c have to beparameterized so that the unitarity prescription (2.14) is satis>ed. Here, we shall limit ourselves tooutline how this is accomplished for stable two-particle channels, in order to show how analyticitycan be explicitly introduced in the formalism. For more details the reader is referred to the relevantliterature [25]. If scth is the threshold for channel c, one writes
/c =1�
∫ ∞
scth
f?c (s
′)gc(s′; s)fc(s) ds ; (2.17)
with
gc(s′; s) =1�
[�c(s′)
s′ − s+ i0− P
�c(s′)s′ − s0
]; (2.18)
P the principal-value part of the propagator, and s0 plays the role of a subtraction point. Insertionof Eq. (2.18) into (2.17) immediately gives
Im(/c(s)) = f2c(s)�c(s) ; (2.19)
Re(/c(s)) = Re(/c(s0)) +s− scth
�
∫ ∞
scth
Im(/c(s′))(s′ − s)(s′ − s0)
ds′ : (2.20)
The former of these relations clearly sets the discontinuity of the propagator across the right-hand,unitarity cut. The latter represents a subtracted dispersion relation very similar in form to what can beestablished for �N scattering on the ground of axiomatic >eld-theoretic or S-matrix considerations.The subtraction point s0 can be chosen such that the resulting scattering matrix has convenientanalyticity properties. Thus, for �N scattering one can choose s0 so that the scattering amplitude inthe complex s plane has a pole on the left-hand side, and a branch cut from s�Nth = (m� + M)2 to+∞, as it should be (see Appendix A).This type of parameterization allows also to simulate the left-hand singularities due to t- and
u-channel mechanisms. That these singularities must be there is dictated by crossing symmetry; inpotential-scattering language, they arise because the interactions at work can be always associatedto the exchange of quanta in the crossed channels. These singularities can be easily simulatedintroducing bare states with mass s0; i below the �N threshold, and additional free propagators G0
ij.This prescription introduces non-resonant contributions into the scattering amplitude. One advantageof the CMB approach, is that the separable structure of the transition amplitude, and its dependenceupon s only, make it easy to determine the complex poles of the scattering amplitude in the complexenergy plane.
To >nalize our discussion on the identi>cation of resonance parameters, we would like to mentionthat one can evaluate the K matrix in terms of some eEective Lagrangian, thus reducing drasticallythe number of involved parameters, since background and resonance amplitudes are obtained fromthe same set of Feynmann graphs [26]. Moreover, aspects of the hadron dynamics, such as chiral
Table 1�(1232) Breit–Wigner mass and width, and pole position, in MeV. To be consistent with the Breit–Wigner parameterizationE� =M� − (i=2)2�, the pole position is given as Re(Epole) and −2 Im(Epole). The values recommended by PDG are givenin boldface
symmetry, can be taken into account in a straightforward way. The main drawback of a K-matrix,eEective-Lagrangian approach is that analyticity of the resulting scattering amplitude is no longerguaranteed. At the same time, because of the complicated functional form of the driving potentialV , to restore analyticity by imposing dispersion-relation constraints is far from trivial. As we haveseen, analyticity constraints can be rather easily introduced in a T -matrix parameterization of theCMB type.
The overall >t of the scattering data may show large discrepancies in diEerent resonances, de-pending upon the particular ansatz which has been employed. A remarkable example is given bythe S11(1535) resonance, seen in the T = 1=2; J = 1=2 s-wave channel of the �N system, whoseparameters largely diEer in the various analysis [25], in spite of its high-con>dence (4?) status inthe PDG tables [27]. For strongly excited and well-de>ned resonances, such the P33(1232) � isobar,on the other hand, there is a substantial consensus about the value of the identifying parameters.For the reader’s convenience, we report in Table 1 the � Breit–Wigner mass and width for mixedcharges, given by Refs. [16,20,24] and [23; 28], together with the PDG recommended values [27].In addition, we provide also the position of the corresponding pole of the scattering amplitude in thecomplex energy plane. This information has become more and more popular in recent years [27],since the P33 �N partial wave gets large background contributions from the nucleon pole term(a point to which we shall come back again when dealing with the determination of the �NN cou-pling constant). Whereas the conventional Breit–Wigner parameters strictly speaking pertain to theresonance and to the large background in �N scattering, the pole position is strictly associated tothe resonant state.
As it can be seen from this table, the pole position remains stable near 1210− i50 MeV in all theanalysis. This is true even when the � mass assignment deviates from the PDG recommended value,ranging in the literature from 1210 up to 1241 MeV [29]. This can be understood in the light ofthe background eEects mentioned before, and may be particularly relevant when data coming fromdiEerent experiments are analyzed. Indeed, even for an isolated, narrow multichannel resonance theS-matrix has to contain a (Breit–Wigner) resonance factor and a slowly varying background term,S(E)=SB(E)TSR(E)SB(E). If data from a production experiment are analyzed, the initial factor SB(E)ought to be replaced by other energy-dependent factors, thereby introducing some model dependence.That the pole position and not the Breit–Wigner mass is the quantity which can be most rigorouslyassociated with a resonance has been stressed by HUohler some years ago [28].
Fig. 3. Data on �+d and �−d total cross sections as functions of the pion lab kinetic energy T� from Ref. [31]. The fullline represents the >t obtained with Eq. (2.23) for the �-mass splitting.
2.2. Isospin mass splittings in the � multiplet and hadron structure
The mass values for the various charge states of the � resonance have been extracted fromdiEerent data sets. The �++ and �0 masses have been determined from partial-wave analysis of �Nscattering data [18,30,31], and have the most precise values, whereas for the �+ it has been obtainedfrom pion photoproduction data [27]. The �− mass, on the other hand, has never been determined.Various relations can be found involving the masses within the T = 3
2 � multiplet; they may helpeither in determining the �− mass from the experimentally known masses, or in testing the degreeof charge-independence violation in the � system. A classical example is provided by � scatteringon the deuteron. Since the deuteron is an isosinglet (Td = 0), the neutral and charged pion crosssections must be equal in the limit of charge-independent interactions. This symmetry is reducedto the invariance with respect to the replacement T3 → −T3 (charge symmetry) by the p–n and�±–�0 mass diEerences. Even with this less restrictive assumption, one has that the �+–d total crosssection ��+d ought to be equal to the �−–d one ��−d. Needless to say, electromagnetic eEects haveto be removed before any meaningful comparison with the experimental data may be done. Thisis a feasible although non-trivial task, as we outline in Appendix B. The deuteron being a looselybound system, one can expect that the total cross section is dominated by the contributions due toscattering from a free neutron and a free proton. In the resonance region, therefore, one can writewith an obvious notation
��+d(!) = ��++(!) + 13��+(!) ; (2.21)
��−d(!) = 13��0(!) + ��−(!) : (2.22)
These expressions show that, if the masses associated to the various � charge states are diEerent, thetotal �±d cross sections cannot be the same any longer. This must be re?ected in ��−d(!)−��+d(!),which diEers from zero across the resonance region. This is shown in Fig. 3, where the result ofa classical experiment by Pedroni et al. is exhibited [31]. The solid curve corresponds to a >t with
Table 2Additive quantum numbers of the light and strange quarks
Quark d u s
Q − 13 + 2
3 − 13
Tz − 12 + 1
2 0
S 0 0 −1
Table 3Quark structure of nucleon and �
p n �++ �+ �0 �−
(uud) (udd) (uuu) (uud) (udd) (ddd)
respect to the parameter
D =M�− −M�++ + 13(M�0 −M�+) : (2.23)
Once rescattering eEects, as well as corrections due to the target nucleon motion have been takeninto account, one >nds
D = 4:6± 0:2 MeV : (2.24)
Eqs. (2.23) and (2.24) allow one to >x the �− mass from the already known � masses. In a sense,they allow one to study the �− charge state through the scattering of negative pions from the neutronbound in the deuteron.
The mass splitting within an isospin multiplet can be described by a simple mass formula quadraticin Tz, which has been established many years ago by Weinberg and Treiman [32]:
M = a− bTz + cT 2z : (2.25)
Substituting Tz = − 12 and 1
2 in this relation, one can >x the linear coeScient bN for the nucleonisodoublet, given the n–p mass diEerence
Mn −Mp = bN = 1:29 MeV : (2.26)
Similarly for the �0 (Tz =−1=2) and �+ (Tz = 1=2) states one has
M 0� −M+
� = b� � 1:38 MeV : (2.27)
If, on the other hand, one uses the Weinberg–Treiman relation in Eq. (2.23), one gets D=(10=3)b�=4:6 MeV, in good agreement with the result (2.24).
The equality between the linear coeScients bN and b� in Eq. (2.25) >nds a natural explanationin the constituent quark model of baryons. According to this model, both the nucleon and the �are bound systems of three u and d quarks, which have eEective (constituent) masses mu � md ∼350 MeV. For the reader’s convenience we report in Table 2 the spin-?avor assignments of theu, d, and s quarks, and in Table 3 the quark structure of the nucleon and �.
In order to ful>ll the Fermi statistics for �++, the color degree of freedom was assigned to quarks.The established baryons are then color-singlet (qqq) states [33], and their state functions must beantisymmetric under interchange of any two equal-mass quarks (such as the u and d quarks in thelimit of isospin symmetry). As a consequence, one may write
where the subscripts A and S denote antisymmetry or symmetry with respect to interchange of anytwo of the equal mass quarks. For “ordinary” baryons, only the isospin and strangeness ?avors arerelevant; these three ?avors enjoy an approximate SU (3) symmetry, which can be combined with theSU (2) spin symmetry to give an overall SU (6) spin-?avor symmetry. In this limit the six basic statesare u ↑, u ↓; : : : ; s ↓, where the symbols ↑ and ↓ denote spin up and spin down states, respectively.Group theory then shows that baryons can be classi>ed into various SU (6) and SU (3) multiplets.The four � charge states belong to a J � = 3
2+ spin decuplet, with strangeness S = 0, together with
the .±, .0, S =−1 states, the ;± with S =−2, and the −, S =−3 particle. Looking at Table 3,one sees that the �0 and the neutron have the same (udd) quark structure, apart the diEerent waythe quark spins are coupled. The same applies to the (uud) �+ state and to the proton. In the SU (6)limit, therefore, one expects the n–p mass splitting to be the same as the mass splitting betweenthe �+ and �0 states. This is indeed approximately veri>ed in Eqs. (2.26) and (2.27). Actually,in this limit, the mass splittings can be simply ascribed to the diEerent masses of the down and upquarks, i.e.
M�0 −M�+ =Mn −Mp = md − mu : (2.29)
The observed isospin splitting in the nucleon and � systems, therefore, may provide indicationsabout the corresponding property at the quark level, and the degree of accuracy the quark modelcan reach in describing hadron properties.
Simple quark model considerations can be used to derive a variety of relations for baryon iso-multiplet splittings within the same SU (3) multiplet. In particular, one can >nd relations whichconstrain the unknown �− mass, and allows one to understand the electromagnetic mass splittingsof baryons as a result of quark dynamics. A >rst, basic result is suggested by the Weinberg–Treimanrelation (2.25), which states that the mass splitting within an isomultiplet is given by a mass formulaquadratic in Tz; as a consequence, the third derivative of M with respect to Tz must vanish. Applyingstandard >nite-diEerence formulas [34] one >nds
D3 ≡ M�++ − 3M�+ + 3M�0 −M�− = 0 : (2.30)
This result is actually corroborated by the quark model. To understand how this comes out, onehas >rst to identify the terms in the quark-model Hamiltonian where quark masses may come intoplay [35]. There is obviously the kinetic energy of the quarks. There will then be a one-bodycontribution Kqi , plus a two-body term Kqiqj taking into account the quark–quark central componentsof the interaction, such as an eEective con>ning potential. This contributions can be subsumed intothe kinetic energies, a fact justi>ed on the ground of the virial theorem 〈T 〉= 〈(r=2) dV=dr〉. One hasthen to consider the isospin breaking eEects. These can be classi>ed into (a) the mass diEerence �q
between the up and down quarks; (b) the quark–quark Coulomb interaction; (c) the strong hyper>nesplitting due to the gluon-exchange interaction acting dominantly on pairs in an S-wave state; and(d) the spin-dependent hyper>ne contribution of electromagnetic origin. For all these contributions
one can derive simple scaling laws with the quark masses, on the (reasonable) assumption thatthe relevant quantities do not vary from one baryon to another within the same multiplet. Let usconsider, for instance, the Coulomb interaction energy
VEemij = <QiQj
⟨1rij
⟩; (2.31)
where < ∼ 1137 is the >ne structure constant, and Qi the quark charge. If 〈1=rij〉 does not vary
appreciably within a multiplet one can parameterize this quantity as VEemij =aQiQj, with a a universal
constant. As for the hyper>ne strong and electromagnetic interaction energies, they both can bewritten in the form
VEHFxij = >x × |?ij(0)|2〈�i · �j〉
mimj; (2.32)
where >x depends upon the strong coupling constant for VEHFstrongij , and is given by >=−2�<QiQj=3
for the electromagnetic splitting VEHFemij . If the quark–quark wave function |?ij(0)|2 at the origin is
the same for all quark–quark pairs in the same multiplet, one can parameterize these contributionssimply by
VEHFstrongij = b
〈�i · �j〉mimj
; VEHFemij = cQiQj
〈�i · �j〉mimj
; (2.33)
where b and c are constants. The estimate of the hyper>ne contributions reduces itself to the eval-uation of the mean value 〈�i · �j〉. This can be accomplished through simple considerations. First,one can use the value of the total spin squared S2 to show that⟨∑
i¡j
�i · �j
⟩=∓3 (2.34)
for J = 12 and 3
2 , respectively. From Eq. (2.28) one sees that two like quarks in the S-wave orbitalof a ground-state baryon must be in a spin-symmetric state, namely, they have to couple to spin 1,which implies 〈� · �〉= 1. The same is true for quarks in the ?avor decuplet with J = 3
2 , since theirspins must be aligned together. Finally, in octet baryons one has a pair of like quarks, and two qiqj
pairs, with i �= j and 〈� · �〉=−2.In the light of the above considerations, the various baryon isospin violating mass shifts can be
evaluated almost straightforwardly. For instance, the p–n mass diEerence turns out to be
Mp −Mn = �q + Ku − Kd + Kuu − Kdd +a3+ b
(1m2
u− 1
m2d
)+
c9
(4m2
u− 1
m2d
): (2.35)
Notice that, if isospin violating interactions are ignored, one simply gets Mp − Mn � mu − md.A similar calculation shows that Eq. (2.30) is actually satis>ed. The accuracy of this relation hasbeen investigated by Jenkins and Lebed [36], combining a perturbative treatment of ?avor-breakingeEects with the 1=Nc expansion. In this approach, the completely symmetric spin-?avor SU (2F)representation (see Eq. (2.28)) for the lowest-lying baryons is decomposed into a tower of baryonstates of increasing angular momentum J=1
2 ;32 ; : : : ; Nc=2 for arbitrary Nc. The physical baryons can be
identi>ed with states at the top of the ?avor representations, since in this case the number of strangequarks is of order 1, and not O(Nc). Jenkins and Lebed estimated the accuracy of Eq. (2.30) to be
of order 0′′0′=N 3c , where 0′ and 0′′ � 0′ represent the strong and electromagnetic isospin-symmetry
violating parameter, respectively. With their parameters this implies D3 ∼ 10−3 MeV.A check of the quality of the above constraints on the � masses can be obtained by combining
the linear relation (2.23) with Eq. (2.30) to get
M�0 −M�+ =3D10
; M�− −M�++ =9D10
: (2.36)
Using the value (2.24) for D, and the Breit–Wigner values 1233:6 and 1230:9 for the �0 and �++
masses [18,27], one has
M�+ � 1232:2 MeV; M�− � 1235:0 MeV : (2.37)
Overall, this value of the �+ mass compares fairly well with the values given in PDG(M�+ = 1231:2–1231:8).
The above considerations can be extended to include the masses of strange particles. For instance,one can consider the constraint [36]
with .? ≡ .(1385). This relation is expected to be accurate up to 3× 10−5 [36]. The consistencyof this relation constraining the � masses with Eqs. (2.23) and (2.30), as well as with the knownBreit–Wigner masses, has been investigated in [29]. Since the value of .?
2 extracted from datais not accurate enough, this quantity has been estimated using the experimental value of .2 ≡M.+ + M.− − 2M.0 , and a dynamical quark model calculation of the diEerence .?
2 − .2. Theset of � masses obtained in such a way turned out to be inconsistent with the values given byEqs. (2.23), (2.24), and (2.30). Even if such a conclusion has to be taken with some caution,because of the essential role played by dynamical, quark model calculations, it is worthwhile tonote that this diSculty is overcome if the pole positions of the resonances are used, instead of theBreit–Wigner masses.
2.3. Deviation from spherical symmetry: the E2=M1 ratio
A wealth of information on the � can be gained from its excitation through electromagneticprobes. This is quite understandable in view of the small value of the coupling strength < � 1
137 .The isobar excitation can be obtained either by means of real photons or by exchange of virtualphotons in electron scattering. In the latter case, to a good degree of approximation one can think ofthe electron interacting with the target nucleon through exchange of a single photon of four momenta(!; q) with ! �= |q|, as depicted in Fig. 4, while, for real photons, one obviously has != |q|.
It would be impossible here to consider the subject in all its details; as a matter of fact thereader can >nd excellent reviews in the literature [2]. Here, we limit ourselves to some basic factsillustrating how the existence and features of the � isobar are seen through the electromagneticwindow; we then consider in some detail a topic which is receiving a lot of attention by severalresearch groups, the electric–quadrupole=magnetic-dipole ratio E2=M1.The excitation of the � is clearly perceived in pion photoproduction processes, which complement
what one can learn in elastic pion–nucleon scattering. Thus, both +p → �0p and +p → �+n reactionsexhibit a pronounced resonance behavior for photon energies around 300 MeV, as seen in Fig. 5.
Fig. 4. � electroproduction through exchange of a virtual photon of four momenta Q = k ′1 − k1 ≡ (!; q) between thescattered electron and the nuclear target.
Fig. 5. Total pion photoproduction cross sections from Ref. [8].
The general structure of the photoproduction transition amplitude can be >xed on the ground ofgeneral symmetry arguments. Let us consider a photon of four momenta (!; k) impinging on anucleon, leading to a pion of four momenta (q0 =
√m2 + q2; q) and isospin state b in the >nal state.
Since both the initial particles and the >nal nucleon have two spin states, there are eight degrees offreedom to characterize the transition amplitude. Parity conservation reduces these eight amplitudesto four complex ones. Therefore the number of physical independent observables to be measured atany photon energy and pion scattering angle is seven, once a common phase factor has been >xed.Lorentz and Gauge invariance dictate the structure of the S-matrix to be [37]
Here, p and p′ denote the nucleon four momenta in the initial and >nal state, respectively, and u(p)and Xu(p′) represent the corresponding spinors in the spin and isospin space. The coeScients M�
are purely geometric quantities, to be constructed with the Dirac matrices so as to satisfy invariancerequirements; the whole information on the dynamics of the process is contained therefore in theinvariant amplitudes A(b)
� (s; t), whose dependence upon the external kinematics is best expressed interms of the Mandelstam variables s = (p + k)2 and t = (q − k)2. Note that Eq. (2.39) states thatfor each pion isotopic state there are four scalar functions of energy and angle, hence 12 functionsto describe the four physical processes
+p → �+n; +p → �0p ; (2.40)
+n → �−p; +n → �0n : (2.41)
There are diEerent possible sets of amplitudes [38–40] to describe the above photoproduction pro-cesses. The photoproduction amplitudes can be analyzed under the point of view of their isospin
and spin structure. As for the former, one can proceed in analogy to the pion–nucleon case, therebyobtaining in terms of the Pauli operators for the nucleon isospin Bi
A(b)� = A(+)
� �b3 + 12A
(−)� [Bb; B3] + A(0)
� Bb : (2.42)
The physical amplitudes for the four photoproduction charge channels can be written as linearcombinations of the A(±) and A(0) amplitudes [37]. More interesting on physical grounds is thede>nition of invariant amplitudes referring to the isospin states of the >nal �N system, namely
As for the spin structure of the scattering matrix, in a relativistic framework it is best exhibitedthrough the helicity formalism of Jacob and Wick, which allows for an elegant expression of theobservables in single pion photoproduction [38]. To introduce a decomposition into electric andmagnetic multipoles, however, it is maybe more convenient to re-write the S-matrix in terms oftwo-component Pauli spinors Ci and C†f for the target and >nal nucleon [39]. This can be accomplishedby de>ning the operator F according to
M4�
√s〈�(q)|u(p′)
[4∑
�=1
A�M�
]u(p)|+(k)〉= C†fFCi ; (2.44)
where M is the nucleon mass, and isospin dependence has been omitted for the sake of simplicity.The overall F can be written again in the �N center-of-mass frame in terms of four complexamplitudes Fi, depending on total energy, scattering angle, and the invariants constructed withnucleon spin �, photon polarization vector �, and k ≡ k=|k| and q ≡ q=|q|, which de>ne thescattering plane. One has
Eq. (2.45) is the classical Chew–Goldberger–Low–Nambu (CGLN) representation for the photo-production scattering matrix [40]. In terms of F the diEerential photoproduction cross sections aregiven by
d�d
=|q|!
| C†fFCi|2 : (2.46)
Finally, rotational invariance leads to a decomposition of the four quantities Fi(s; x) (x ≡ cos � ≡k · q) in terms of derivatives of the Legendre polynomials,
Here, the usual notation l± is employed to denote the total angular momentum J = |l ± 1=2| ofthe �N system. The coeScients El± are the electric multipole amplitudes, while the quantities Ml±represent the magnetic multipole contributions. All the observable quantities can be expressed interms of these amplitudes. For instance, inserting Eqs. (2.47) into the CGLN representation (2.45),using (2.46) and integrating over the scattering angle one obtains for the total cross section
One may ask at this point what information can be extracted from the data about the role playedby the various multipole amplitudes. As discussed in Ref. [8], it is well established by now that thephotoproduction cross sections of Fig. 5 are dominated by the E0+ and M1+ dipole contributions.The former produces s-wave pions with the >nal �N system in a J = 1
2 state, the latter is responsiblefor the production of p-wave pions. In this regime Eq. (2.48) becomes,
�tot � 4�|q|!
[|E0+|2 + 2|M1+|2] ≡ �(E0+) + �(M1+) ; (2.49)
and the two dipole contributions are enough to reproduce the experimental cross sections to areasonable degree of accuracy. The non-resonant E0+ component is non-negligible across the wholeresonance region for the +p → �+n reaction, whereas the +p → �0p cross section is dominated bythe magnetic dipole amplitude. This can be nicely explained in simple classical terms [8], consideringthe classical dipole moment −(m�=M)eD, where D is the distance between the neutral pion and the�N center of mass. One can expect at threshold and in the resonance region that �(E0+) scaleslike (m�=M)2, representing a simple center-of-mass correction. In the limit (m�=M) → 0, the electricdipole contributions to neutral pion photoproduction are completely quenched.
The basic theoretical framework for the description of � photoproduction can be obtained on theground of simple isobaric models. We shall not go into the details here, since they can be found inseveral excellent textbooks [8] and review papers [2]. We limit ourselves to recall that the resonant32 ;
32 channel can be described in terms of the excitation, propagation and subsequent decay of an
intermediate � particle, much in the same way as in �N scattering processes.Schematic models in the dipole approximation focus their attention on the leading features of
photoproduction processes with outgoing s- and p-wave pions. Here, we consider in more detailwhat can be ascertained about the role played by smaller multipole amplitudes, and in particular bythe electric quadrupole amplitude E3=2
1+ , because of its relevance to the physics of the � isobar in thecontext of quark models of hadrons. To understand this point, we recall that SU (6) quark modelspredict this quantity to be zero, since the nucleon and � wave functions factorize into a spin anda purely s-wave space part, so that the matrix element of the electric quadrupole operator vanishesidentically and the +N → � excitation is a pure M1 transition, the well-known Becchi–Morpurgoselection rule [41]). Since the mid 1970s, however, de Rujula, Georgi and Glashow recognizedthat gluon-exchange could lead to hyper>ne interquark interactions, with the subsequent presenceof a d-state admixture in the baryon ground state wave function. These tensor eEects, which areresponsible also of the hyper>ne contributions to the baryon interaction energy of Eq. (2.32), inducea small violation of the Becchi–Morpurgo selection rule. In chiral quark models of the nucleonmost of the E2 strength comes from tensor correlations between the pion cloud and the quark bag,or from meson exchange currents between the quarks. In Skyrme models, both the nucleon and
� states are obtained from the same soliton con>guration via an adiabatic SU (2) rotation, and arespherically symmetric. The charge distribution in the spin–isospin stretched state, however, may havea non-vanishing electric quadrupole component, leading to an E2=M1 ratio diEerent from zero. Thesmall value of this observable can be expected on the ground of power-counting arguments in thenumber of colors Nc. As a matter of fact, it can be shown that E2=M1 scales as N−2
c [42].Theoretical calculations of the E2=M1 ratio performed with diEerent models of the nucleon have
led to rather diEerent predictions. Constituent quark models have given −2%¡E2=M1¡ 0, depend-ing upon the strength of the hyper>ne interaction and the bag radius [43–46]. Larger negative valuesin the range −6%¡E2=M1¡−2:5% have been obtained with Skyrme models [42], while chiral bagmodels have led to −3%¡E2=M1¡− 2% [47]. Preliminary results have been also obtained withlattice QCD calculations [48], even if with a large uncertainty, that is E2=M1= (+3± 9)%. Finally,values around −3:5% have been predicted in a quark model with exchange currents [3]. A precisemeasurement of the E2=M1 ratio, therefore, represents an important testing ground for microscopicmodels of the nucleon, and gives information about small but physically important components ofthe baryon wave function.
The determination of the electric and magnetic multipole amplitudes is a demanding task underthe experimental point of view. As we have seen before, one has to deal with eight degrees offreedom, i.e. the real and imaginary parts of the four helicity or CGLN amplitudes of Eq. (2.45).A kinematically complete experiment would require therefore at least eight independent observablesto specify the multipole amplitudes to any order in l [49]. Such an ambitious goal is still beyondour present possibilities, and one has to rely on the diEerential cross section, and the three sin-gle polarization observables ., Pf, and Pi, which are usually referred to as photon asymmetry,recoil nucleon polarization, and target asymmetry, respectively. The photon (or beam) polarizationis de>ned for linearly polarized photons, whose polarization can be perpendicular or parallel to theproduction plane. If the corresponding cross sections are denoted by �⊥ and �‖, respectively, itsoperative de>nition is
.=d�⊥ − d�‖
d�⊥ + d�‖ : (2.50)
For linearly polarized photons on unpolarized targets the diEerential cross section is expressed interms of the unpolarized cross section d�0(�)=d and . by the simple formula
d�(�; /)d
=d�0(�)d
[1− .(�) cos(2/)] ; (2.51)
where / is the azimuthal angle of the emerging pion with respect to the beam direction.Similarly, the emerging or incoming nucleon can have two polarization states Pnf=±1 and Pni=±1
along the perpendicular to the production plane, and one can de>ne
Pf =d�Pnf=1 − d�Pnf=−1
d�Pnf=1 + d�Pnf=−1; Pi =
d�Pni=1 − d�Pni=−1
d�Pni=1 + d�Pni=−1: (2.52)
The extraction of the T=32 M1 and E2 components from a >t to the multipole expansion of the above
observables is prone to the Donnachie’s ambiguity [50], that is the presence of higher partial-wave(l¿ 2) strength in lower partial waves. One can try to overcome this problem by representingthe higher partial waves by the corresponding Born contributions, and by constraining the analysisthrough as many observables as possible, coming from the simultaneous measurement of diEerent
reaction data. Information coming from �N scattering is essential in these analysis. In particular,below the two-pion production threshold one can resort to the Fermi–Watson theorem [51], whichexpresses the phases of the complex multipole amplitudes directly in terms of the �N scatteringphase shifts �T
l±,
ETl± = |ET
l±| exp i�Tl± + n�; MT
l± = |MTl±| exp i�T
l± + n� ; (2.53)
where T = 12 ;
32 .
The multipole amplitudes can be extracted from the experimental database by means of twodiEerent, basic methods, the energy-independent and the energy dependent approach. In the former,one investigates each given energy through a standard C2 procedure, where the >tting parameters arethe real and imaginary parts of the multipole amplitudes. Below the two-pion production thresholdthe Fermi–Watson theorem (2.53) allows one to reduce the number of >tting parameters by a factorof two, since only the absolute value of the amplitudes needs to be determined from the data.In the energy-dependent method, on the other hand, one simultaneously considers the data at allenergies. In this case, the energy dependence of the data must be taken into account either througha suitable parameterization, or by means of dispersion relations. Which one of the two approacheshas to be preferred has been the subject of a long debate in the literature, as we shall see whendealing with the determination of the pion–nucleon coupling constant. It is fair to say, here, thatmaybe the right answer depends upon the situation one is considering. If the data refer to both crosssections and polarization observables at each energy, and are closely spaced in energy, as it happensin a complete, dedicated experiment, one may regard the energy-independent approach as the bestone. For widely spaced data, on the other hand, and when only a few polarization observables areavailable, the energy-dependent approach may be preferable, because continuity is built in from thevery beginning, and systematic errors tend to cancel out. The same approach can be also useful whenthe general structure of the resonances is already known, and one wants to extract small partial-waveamplitudes.
The determination of the E2=M1 ratio requires the measurement of the small E3=21+ amplitude with
respect to the dominant magnetic dipole amplitude, in presence of a large background contributionto the former. High-quality data coming from coincidence experiments are therefore welcome.Up to a few years ago medium-energy electron accelerators were pulsed, with limited photon ?uxesfor these experiments. The situation has drastically changed with the advent of cw-beam facilities,such as the Laser Electron Gamma Source (LEGS) at BNL, or the Mainz Microtron MAMI. Thisexplains the ?ourishing of new determinations of the E2=M1 ratio in the last years.The LEGS collaboration performed high-precision measurements of p(+; �0), p(+; �+) and p(+; +)
cross sections and beam asymmetries in the energy region 209¡E+ ¡ 333 MeV [52]. The reac-tions have been analyzed simultaneously, with a dispersion calculation of Compton scattering as aconstraint on the photopion amplitudes. For each isospin channel the various multipole amplitudesMl± (M= E;M) have been parameterized in the form
Ml± = AB(1 + iT l±�N ) + ART l±
�N ; (2.54)
where T l±�N is the corresponding partial-wave transition amplitude for �N scattering, and the back-
ground term AB contains the contributions of the Born graphs. The physical meaning of this pa-rameterization can be easily understood below the 2� production threshold. In this case, writing the
�N T -matrix in terms of the pion–nucleon scattering phase shifts, T l±�N = sin �l± exp i�l±, Eq. (2.54)
becomes
Ml± = (AB cos �l± + AR sin �l±) exp i�l± ; (2.55)
and the Fermi–Watson theorem Eq. (2.53) is satis>ed. Eq. (2.54) represents a K-matrix-like unita-rization to parameterize the photopion multipoles. Note that at the K-matrix pole associated with the� resonance the corresponding phase is obviously equal to �=2; ReE3=2
1+ = ReM 3=21+ = 0, and one has
E2M1
=E3=21+
M 3=21+
=Im E3=2
1+
ImM 3=21+
=AR
A′R
: (2.56)
Thus, the ratio Im E3=21+ =ImM 3=2
1+ , which is usually compared to theoretical calculations, is essentiallygiven by the ratio of the coeScients AR obtained in the >tting procedure.The LEGS collaboration quoted a percentage,
E2M1
= [− 3:0± 0:3stat+syst ± 0:2model]% ; (2.57)
where the last error takes into account the uncertainties arising from multipole truncation, the choiceof the �N phase-shift solution, and the evaluation of the Compton dispersion integrals.
A diEerent approach has been followed by the MAMI collaboration [53,54]. They determinedthe diEerential cross sections and beam asymmetries for the reaction p(+; p)�0 and exploited theparallel part of the diEerential cross section d�‖=d , when the pion is detected in the plane de>nedby the photon polarization and the beam momentum, to extract E2=M1. Indeed, in the s- and p-waveapproximation, this quantity in analogy to the Eq. (2.4) is given by
d�‖d
=q![A‖ + B‖ cos �+ C‖ cos2 �] (2.58)
with the coeScients A‖; B‖, and C‖ given by
A‖ = |E0+|2 + |3E1+ −M1+ +M1−|2 ;
B‖ = 2Re[E0+(3E1+ +M1+ −M1−)?] ;
C‖ = 12Re[E1+(M1+ −M1−)?] : (2.59)
Central to the analysis of the MAMI collaboration is the ratio of the third and >rst coeScient in theparameterization (2.58) of the cross section,
R ≡ 112
C‖A‖
=Re[E1+(M1+ −M1−)?]
|E0+|2 + |3E1+ −M1+ +M1−|2 : (2.60)
At resonance, where Re(M 3=21+ −M 3=2
1− )=0, neglecting |E0+|2; E1+ and M1− with respect to M1+, andthe isospin 1
On the ground of a mild energy dependence of R, the MAMI collaboration extracted the E2=M1ratio from a >t of cross section and beam asymmetry data for the reaction p(+; p)�0 between 270and 420 MeV; obtaining [53]
E2M1
= (−2:5± 0:2stat ± 0:2sys)% : (2.62)
The basic approximation Eq. (2.61) has been questioned in Ref. [55] and by the VPI group [56].The former used the Mainz data set to extract the E2=M1 ratio without the above approximationby means of their eEective Lagrangian approach [57], obtaining E2=M1 = (3:2 ± 0:25)% as therecommended value. The latter, used their multipole amplitudes, obtained through a combinationof energy-dependent and energy-independent analysis [58], getting the much lower value E2=M1 =(−1:5 ± 0:5)%. A short summary of the debate among the three groups can be found in [54].Here, we limit ourselves to note that a new, more careful evaluation of the E2=M1 ratio throughEq. (2.60), with inclusion of data from a simultaneous measurement of the p(+; �+)n reaction, hasbeen recently published by the MAMI collaboration [54]. They performed both energy-dependentand energy-independent multipole analysis of their data set, the energy dependence being constrainedthrough >xed-t dispersion relations [59]. The >nal result of this new analysis is E2=M1 = (−2:5 ±0:1stat ± 0:2sys), substantially con>rming their previous assignment.
The dependence of the E2=M1 ratio upon the four momenta transfer can be studied through elec-troproduction experiments. Recently, high-precision measurements have been made at the MIT-BatesLinear Accelerator at Q2 = 0:126 GeV=c2 for the reaction p(e; e′p)�0 [60], yielding the value
E2M1
= (−2:1± 0:2stat+sys ± 2:0model)% : (2.63)
It is worthwhile to stress the large uncertainty in the >nal result due to some unavoidable modeldependence in the analysis of data.
This uncertain state of aEairs is quite understandable, in view of the many uncertainties implicit inthe determination of E2=M1. In the >rst place, the intensive experimental activity in the last severalyears has shown that the result is sensitive to the structure of the database being considered.On the other hand, in absence of a >rm dynamical scheme for the �N interaction, one is forced toimplement unitarity through some phenomenological method, such as the K-matrix approach, whichunavoidably introduces some model dependence. Various dynamical models of the �N system havebeen proposed in the past to remove or at least to reduce the ambiguities in the extraction of the+N ↔ � transition amplitudes from photoproduction or electroproduction data [61], with particularattention to a clearer separation between background and resonance contributions to the amplitudes.Due to our limited understanding of the �N dynamics starting from fundamental principles, however,some degree of model dependence is always present. From this point of view, an interesting andoriginal analysis has been performed in Ref. [62]. The essential point of this paper is the separationof a resonant contribution from the total transition amplitude, that implies the introduction of a ��component into the �N scattering state, which vanishes in the asymptotic region. The explicit formof the wave function, therefore, depends upon the model used to de>ne the isobar contribution, andcan be always changed by means of a unitary transformation, much in the same way as one canalways pass from one representation to another in quantum mechanics. The argument is illustratedthrough a model calculation, where the pion production multipoles consist of three terms, a back-ground contribution, a “bare” � resonant multipole, and a vertex-renormalization term, where the
produced pion interacts with the nucleon to give rise to resonance propagation before being emittedin the >nal state. This is diagrammatically illustrated in Fig. 6. One can show that, once the modelis speci>ed, one can construct a whole class of unitary equivalent models which, while providing thesame >t to the data, predict however diEerent resonance amplitudes. Had these conclusions a generalvalidity, they would be a strong challenge to theoreticians. Indeed, a meaningful comparison of theE2=M1 ratio with the results of QCD-inspired models would require a unique, consistent treatmentof the dynamics. Needless to say, these considerations are also relevant with reference to the oldquestion of the role played by isobar components in nuclei.
In principle, information on the deviation from spherical symmetry can be also obtained by lookingat other observable quantities. A noticeable example is the quadrupole moment Q. For the nucleon,angular momentum selection rules imply that it has to be zero even in presence of d-state admixturesdue to inter-quark tensor forces. This is not true for the �, because of its higher spin, and actuallyquark models lead to the prediction Q� ∼ −0:09 fm2 [44]. Unfortunately, the experimental determi-nation of the � quadrupole moment would require the elastic scattering of photons on this unstableparticle. The situation is less unsatisfactory as far as the � magnetic moment F� is concerned.In this case, one can estimate its value from radiative � scattering on proton targets. However,a model-independent analysis of the data is again impossible, since a dynamical model of the �Nscattering amplitude is required. This explains the large uncertainty in our present knowledge ofthis quantity (3:7¡F� ¡ 7:5 nuclear magnetons according to PDG), so that a vis-[a-vis comparisonbetween experiments and quark-model predictions is not possible.
3. Theoretical models for the N–� interaction
The embedding of the � isobar into the nuclear environment requires the meson–baryon–baryon(�N�, �N�, ��� : : :) vertices as basic building blocks. Once these coupling Lagrangians have beengiven, standard non-relativistic reduction techniques provide the various transition potentials oneneeds in nuclear-structure and nuclear-reaction models. Whereas the basic structure of the couplingscan be determined on the ground of general invariance requirements, one is left with the problemof providing information about their strengths, and with the oE-shell freedom associated with thevertex form factors. A direct experimental determination of the strengths is obviously hamperedby the unstable nature of the intervening particles. One is therefore forced to rely on relationsbetween meson–NN and meson–N� strengths following from simple quark-model or strong-couplingconsiderations. In spite of the ?ourishing of quark-model approaches to meson and baryon structure,these relations still retain their value, and we shall review them, stressing the essential role playedin their derivation by general algebraic properties rather than by detailed dynamical assumptions. Inthis perspective, the knowledge of the �NN and �NN coupling strengths is a necessary prerequisite,apart from the obvious basic nature of these quantities. For this reason, we discuss the present
status of their determination >rst, devoting particular attention to the �NN coupling constant, whichhas been the subject of great debate in the nineties. In so doing, we emphasize the role played bydispersion relation constraints in the analysis of the experimental data. We then touch the problem ofthe oE-shell behavior of the meson–baryon–baryon vertices, where more explicit dynamical modelsare required, whether one starts from an eEective approach in terms of hadron degrees of freedom,or from more microscopic quark models.
The comparison of the predictions based on the assumed model N� interactions with nuclearphenomenology entails the solution of a coupled-channel (CC) problem. One is then confrontedwith two questions. On the one hand, one has to ascertain to what extent the solution of theseCC equations—which are essentially non-relativistic—can simulate higher-order contributions of co-variant >eld-theoretic descriptions of hadron dynamics. This is just one aspect of the much moregeneral problem of the role played by relativistic eEects in nuclear physics. On the other, one hasto embed the CC N� system in the nuclear environment. The latter question has received the moresatisfactory, albeit not conclusive, answer in two limiting situations, for few-nucleon (A6 3) nuclei,and for nuclear matter. In the second half of this chapter we consider some aspects of these complextopics, deferring the analysis of isobar excitations in nuclear matter to the last part of the paper.
3.1. Basic meson-exchange models for the N� interaction
An eEective description of the � as an elementary particle can be given in quantum >eld the-ory through the Rarita–Schwinger formalism. As is well-known, this can be accomplished withoutambiguities only for an on-shell �, since its oE-shell propagation and coupling to other hadrons isaEected by the presence of unphysical spin 1
2 degrees of freedom [64,65]. Fortunately, in most ap-plications of low- and medium-energy nuclear physics one can limit to a non-relativistic description,which mimics the meson-theoretic approach to the NN problem. To see how this comes out let usrecall that pion–nucleon coupling can be described either through the pseudoscalar (PS) interactionLagrangian
LPS =−g X (x)i+5� (x) · �(x) ; (3.1)
or by the pseudovector (PV) coupling Lagrangian
LPV =f�NN
m�
X (x)+F+5� (x) · 9F�(x) : (3.2)
Here, m� is the pion mass, (x) is the spinor >eld for the nucleon, and �(x) represents the isovectorpion >eld, with � the usual Pauli vector in isospin space. In the non-relativistic limit one gets fromthese interaction Lagrangians the eEective coupling Hamiltonian
H�NN =−f�NN
m�(� ·∇)(� · �) ; (3.3)
provided that the PS and PV coupling constants are related by
Fig. 7. Processes with �NN , �N� and ��� vertices. The full and double lines represent the nucleon and �, respectively,and the dashed lines the exchange of diEerent mesons.
with � the usual Pauli matrix-vector, and M the nucleon mass. In the static approximation theeEective Hamiltonian (3.3) leads to the well-known NN potential in momentum space [8]
V s�(NN ) =−f2
�NN
m2�
(�1 · q)(�2 · q)q2 + m2
��1 · �2 : (3.5)
Time-retardation eEects can be taken into account considering the full Feynmann propagator for theexchanged pion, bringing the energy-dependence ! of the pion in the potential, that is
V�(NN ) =f2
�NN
m2�
(�1 · q)(�2 · q)!2 − q2 − m2
��1 · �2 : (3.6)
The �N� coupling can be described in a similar way, starting from the Lagrangian
L�N� =f�N�
m�(i F
� T N · 9F�+ h:c:) ; (3.7)
where F� is the Rarita–Schwinger >eld for the �, and T a transition operator connecting isospin 1
2and 3
2 states [12,66]. According to the Wigner–Eckart theorem, its matrix elements between isospineigenstates can be simply de>ned as the Clebsch–Gordan coeScients
〈 32 t�z |T�| 12 tNz 〉= 〈112�t
Nz )| 32 t�z 〉 : (3.8)
Lagrangian (3.7) in the non-relativistic limit leads to the eEective coupling Hamiltonian
H�N� =−f�N�
m�S+ ·∇T
+ · � (3.9)
with S a transition spin operator de>ned in the same way as T. The various coupling and transitionpotentials between NN and N� states can be now written down on the ground of straightforwardperturbation theory. Thus, the NN → VN transition corresponding to Fig. 7 is given by [8,12]
This expression can be immediately obtained from the NN potential (3.6) by appropriately replacingthe coupling constants, and substituting the operators �1 and �1 with S+
1 and T+1 , respectively.
Similarly, the direct and exchange interactions V�(N� → N�) and V�(N� → VN ) become
V�(N� → N�) =f�NNf���
m2�
(�1 · q)(�2 · q)!2 − q2 − m2
��1 · �2 (3.11)
and
V�(N� → VN ) =f2
�N�
m2�
(S+1 · q)(S2 · q)
!2 − q2 − m2�T+1 · T2 ; (3.12)
where the spin–isospin transition operators � and � in V�(N� → N�) refer to the ��� vertex. Theyare uniquely de>ned once their reduced matrix elements between spin or isospin 3
2 states are >xed.One can take [8] 〈 32‖�‖ 3
2〉= 〈 32‖�‖ 32〉= 2
√15.
The potentials given above describe the simplest one-pion exchange interaction between thehadrons. At the next level of complexity in the number of exchanged mesons one has the two-pionexchange contributions [8,67]. For the NN case, symmetry considerations severely restrict the ex-changed quantum numbers. Indeed one can have only the scalar–isoscalar exchange with J =0+ andT = 0 spin and isospin quantum numbers, or the vector–isovector exchange with J = 1−; T = 1.In the vector–isovector channel a prominent role is played by the vector meson � meson, a propertwo-meson resonance with a physical decay width. It can be described in an eEective way by therelativistic Lagrangian
L� =−gVX (x)+F (x)� · F(x) + gT
2MX (x)�FI (x)9I� · F(x) (3.13)
with F(x) a four-vector in con>guration space and a vector in isospin space associated, to the �meson. In the non-relativistic limit one gets the spin- and isospin-dependent �-exchange potential
V� =f2
�NN
m2�
(�1 × q) · (�2 × q)!2 − q2 − m2
��1 · �2 ; (3.14)
with its characteristic transverse vector coupling �× q with respect to the longitudinal coupling � · qappearing in the scalar �-exchange interaction. The �NN coupling constant in (3.14) is related tothe vector and tensor coupling strengths gV and gT by
which shows that the Pauli operators �i and Bj enter into the spin–isospin-dependent �-exchangepotential in the same way as they come into play in the �-exchange NN interaction. By the way, thisresult exhibits the well-known fact that �-exchange produces a tensor NN potential which partiallycounteracts the �-exchange contribution.
These considerations can be immediately extended to the N� system. Owing to the spin andisospin quantum numbers of the intervening hadrons one has that �-exchange can contribute alsoto the N� interaction. Thus, replacing the Pauli spin and isospin operators � and � with S and T,respectively, one gets for the exchange N� → VN transition
V�(N� → VN ) =f2
�N�
m2�
(S+1 × q) · (S2 × q)!2 − q2 − m2
�T+1 · T2 : (3.18)
Similarly, the direct interaction V�(N� → N�) can be written as
V�(N� → N�) =f�NNf���
m2�
(�1 × q) · (�2 × q)!2 − q2 − m2
��1 · �2 : (3.19)
3.2. Meson–baryon couplings
In the Lagrangians and potentials described in the previous section masses and coupling constantsappear as phenomenological parameters. Here, we shall brie?y consider the coupling strengths, asthey can be determined on the ground of experimental analysis or simple theoretical models, startingfrom the pion–nucleon coupling, which, because of its basic nature in nuclear physics, has beenthe subject of very many investigations in the last forty years. This quantity can be determinedthrough diEerent routes, starting from �N scattering data, NN forward dispersion relations, and=orNN partial-wave analysis. This topic has been the subject of great debate in recent years, thereforeit is worthwhile to consider it in some detail.
3.2.1. The �NN coupling constantIt is impossible to do full justice to the enormous amount of work which has been devoted to the
determination of the �NN coupling constant. The >rst estimate of this parameter has been presum-ably given in 1950, on the ground of photoproduction data for charged mesons [68], when �N andNN scattering experiments were still in their infancy, and far less accurate than photoproduction ex-periments. The Kroll–Ruderman Theorem [69], which states that the pion photoproduction amplitudeat threshold is just proportional to the �NN coupling strength, allowed Bernardini and Goldwasser[70] to produce the value f2
�NN =4�= 0:065 in the mid-1950s.Under a >eld-theoretic point of view, a major achievement at those days has been represented
by the static Chew–Low model [71–73]. There, the interaction between a recoilless nucleon andthe pion >eld is described by the pseudovector eEective Hamiltonian (3.3). The whole dynamicalcontent of the theory can be subsumed into the Low equation, which expresses the �N transitionamplitude in terms of the scattering and production amplitudes starting from the same initial stateor ending on the same >nal one. Under a dispersion-theoretic point of view, the Low equationessentially expresses the requirements dictated by unitarity, crossing and analyticity. Note, however,that in general the scattering amplitudes appear in the Low equation both on- and oE-the energyshell. As a consequence of the static limit, in the Chew model this equation can be written only interms of physical scattering amplitudes. In a fully relativistic theory, this is not possible, and properdispersion relation techniques have to be employed. In spite of the drastic approximations inherentto the Chew–Low model, it turned out to be quite successful in describing the low-energy P-wavepion–nucleon scattering, providing the value f2
The very >rst application of forward dispersion relation to �N data to determine the pion–nucleoncoupling is due to Haber–Schaim [75] and Davidon and Goldberger [76]. As explained in moredetail in Appendix A, the essential ingredient of these analysis is the dispersion relation that relatesthe total cross section �± for �±N scattering to the forward �N isospin odd transition amplitudeT (−)(!), evaluated at incoming-pion kinetic energy !, namely[
ReT (−)(!)!
− !2
4�2P∫ ∞
m�
k(!′)!′2
�−(!′)− �+(!′)!′2 − !2 d!′
](!2
N − !2)
= − 4f2 +!2
N − !2
4�2
∫ ∞
m�
k(!′)!′2 (�−(!′)− �+(!′)) d!′ ; (3.20)
where k(!) =√
!2 − m2�. The pole corresponding to the nucleon propagation in the intermediate
state is located at !N ≡ −m2�=2M . The use of dispersion relations represented a great achievement,
since at those times a >rm dynamical scheme for strong interactions was lacking, and one hadto look for constraints which were independent upon detailed assumptions about the dynamics ofstrongly interacting particles. Under this point of view, dispersion relations represented an idealtool, based as they were upon general requirements only, such as analyticity hopefully related tocausality, unitarity and crossing symmetry. Under a practical point of view, the above relation statesthat the quantity on the left-hand side must be a linear function of the pion energy squared, so thatan extrapolation to the nucleon pole !N could give the �N coupling strength. The result of thesepioneering applications of dispersion relation constraints (DRC) was f2
�±NN =4�=0:08± 0:01 for thecoupling in its pseudovector form.
At the beginning of the 1980s Koch and Pietarinen [18], and Kroll [77] obtained what has beenconsidered for several years the textbook values of the �±N and �0N coupling constants, respectively.The former used dispersion relations at >xed momentum transfer t, as functions of the variable
I ≡ 14M
(s− u) =14M
(2s− 2m2� − 2M 2 + t) ; (3.21)
to constrain the �±N transition amplitudes. The quantity I exhibits clearly the crossing-symmetricproperties of the formalism since one has I ↔ −I when s ↔ u. At the same time it simply reduces tothe pion kinetic energy ! at forward direction t=0. We give in Appendix A the detailed derivationof the crucial relation, here we limit ourselves to quote the result, i.e.,
(IB ± I){∓ReB±(I; t)± I
�P∫ ∞
I0
[Im B+(I′; t)
I′ ∓ I+
Im B−(I′; t)I′ ± I
]dI′
I′
}
=g2�±NN
M+ B(0; t)(IB ± I) : (3.22)
The quantities B± represent the invariant amplitudes for elastic �±N scattering, I0 ≡ m� + t=4M ,and IB = (t − 2m2
�)=4M is the location of the nucleon pole, corresponding to s = M 2 or u = M 2,depending upon the momentum transfer t. Eq. (3.22) represents the generalization to t �=0 ofEq. (3.20). Its value at the unphysical point I= IB, corresponding to the single-particle intermediatestate with nucleon propagation, gives the required pion–nucleon coupling constant.
Koch and Pietarinen used Eq. (3.22) in their partial-wave analysis (PWA) of pion–nucleonscattering data. Needless to say, the long-range, isospin-breaking electromagnetic eEects have to be
taken into account before Eq. (3.22) can be applied. How this is achieved is brie?y reviewed inAppendix B.
To use the scattering parameters coming from the PWA in Eq. (3.22), one has to guarantee thatpartial-wave and >xed-t analysis agree with each other and with the experimental data. This has beenachieved by Koch and Pietarinen by requiring the consistency between the transition amplitudes fromthe PWA and those appearing in the >xed-t dispersion relations. As the result of their analysis, theyobtained the value
g2�±NN
4�= 14:28± 0:18 (3.23)
for the pseudoscalar coupling constant, which corresponds to
f2�±NN
4�=(
m�±
2Mp
)2 g2�±NN
4�= 0:079± 0:001 (3.24)
for pseudovector coupling.The �0N coupling has been determined by Kroll [77] by means of an elegant application of
forward dispersion relations to NN scattering. We shall brie?y illustrate how the method works,following essentially Ref. [8]. In the present case it is convenient to choose the center-of-massmomentum squared ||2 as variable to write down the dispersion relation, since one has
s= 4(M 2 + ||2); t = 0; u=−4||2 ≡ −4z : (3.25)
According to the general philosophy of dispersion theory, to establish the dispersion relation let us>rst identify the possible intermediate states dictated by unitarity in the direct (s) and crossed (u)channels. There is obviously the right-hand scattering cut starting at z = 0 (s= 4M 2), plus the cutsstarting at u = n2m2
� (n = 2; 3; : : : ; ), corresponding to multi-pion exchange in the u (N XN → N XN )channel. Because of Eq. (3.25), these cuts map into the left-hand cuts from −n2m2
�=4 in the z-plane.To these contributions one has to add the pole terms associated to one-pion intermediate states. Tobe de>nite, let us consider pp scattering, in which case only �0 propagation is possible in the crossedchannels, both describing pXp → pXp elastic scattering as shown in Fig. 8. There is no single-particleintermediate state in the direct channel, on the other hand the u-channel has the intermediate �0
propagation, which leads to the pole term in the scattering matrix
T (p)u ∼ g2�0NN
u− m2�=− g2�0NN
4z + m2�: (3.26)
A similar contribution is present also in the t-channel,
T (p)t ∼ g2�0NN
t − m2�=−g2�0NN
m2�
: (3.27)
Non-relativistically, these amplitudes can be simply interpreted in terms of the crossed and directone-pion-exchange Born graphs (see Fig. 9). Indeed, for the direct process, in which the two protonsemerge with momenta ′ and −′, one has
Fig. 8. Intermediate �0 propagation in p Xp → p Xp scattering.
Fig. 9. Direct and exchange Born graphs for �0 exchange in pp scattering.
where q = ′ − is the momentum transfer, whereas for the crossed graph, where the role of theemerging protons is exchanged and the momentum transfer is + ′, one gets
T (e) ∼ − g2�0NN
( + ′)2 + m2�: (3.29)
Taking into account that for forward scattering the momentum transfer q vanishes and = ′, oneimmediately gets Eqs. (3.27) and (3.26). Since the two Born terms have to be put together witha minus sign in between, because of the fermionic nature of the nucleons, one >nally gets the polecontribution to the forward pp amplitude
T (pole) =Rz
z + m2�=4
(3.30)
representing a pole at z =−m2�=4, with residue R proportional to the �0pp coupling constant.
All the above analyticity properties can be summarized into the dispersion relation in the z-plane
ReT (z) = ReT (0) +Rz
z + m2�=4
+z�
∫ −m2�
−∞Im T (z′)z′(z′ − z)
dz′ +z�P∫ ∞
0
Im T (z′)z′(z′ − z)
dz′ ; (3.31)
where a subtraction in z = 0 has been performed.One can include all the singularities in the unphysical region (z¡ 0) in a unique integral from
−∞ to 0, and re-write Eq. (3.31) as
ReT (z)− ReT (0)− z�P∫ ∞
0
Im T (z′)z′(z′ − z)
dz′ =z�
∫ 0
−∞Im T (z′)z′(z′ − z)
dz′ ≡ �(z) : (3.32)
The function �(z) clearly collects all the contributions from one- and multi-pion propagation in thepXp system in the unphysical region of the pp scattering process, and is referred to as the discrepancyfunction [78,79]. It is a measure of what is lacking in the information coming from the physicalregion, in order to satisfy the dispersion relation. Stated in another way, the unphysical region canbe regarded as re?ecting the physics of the pXp channel, mostly in a kinematic domain far below thephysical pXp threshold.
In the spin-singlet (S=0) channel, the dispersion relation (3.32) is particularly eScient in isolatingthe pion pole, owing to strong selection rules. In fact, with reference to the N XN system, one has thatparity is given by J=(−1)L+1, L being the orbital angular momentum, whereas for the �� exchangeone has J = (−1)L = (−1)J . It follows that, since J = L in the singlet channel, two-pion exchangeis forbidden by parity conservation. Other selection rules follow from G-parity conservation.
Since G is related to the charge-conjugation operator C and the total isospin by G = C exp(i�Ty),one has, for the corresponding eigenvalue G= (−1)L+S+T for the N XN system, and G= (−1)n for asystem of n pions. As a consequence, an even value of L+ S + T entails the exchange of an evennumber of pions, while an odd value must correspond to the exchange of an odd number of pions.For forward pp scattering (T = 1) in the singlet channel, one >nds again that two-pion exchange isforbidden, while one-pion exchange can occur in the unnatural parity channel 0−. Since, as remarkedbefore, for pp scattering one-pion exchange can occur only through a neutral pion, one has a polelocated at z =−m2
�0=4 according to Eq. (3.30), corresponding to the extrapolated laboratory kineticenergy
T 0Lab =
2M
z =−m2�0
2M� −10 MeV :
Two-pion exchange being totally forbidden, the next singularity, corresponding to three-pion ex-change, is a cut starting at TLab =−9m2
�0=2M � −90 MeV. Thus, the eEects of one-pion exchange,and the associated pole are expected to be well exhibited in the singlet discrepancy function withrespect to other exchange mechanisms in forward pp scattering. These facts have been exploited byKroll to extract the �0pp coupling from pp scattering data. The quantities on the left-hand side ofEq. (3.32) can be all >xed by experiment, the real part of the forward amplitude ReT (z) froma phase-shift analysis or by Coulomb interference, and Im T (z) from the optical theorem. It isworthwhile to note, here, that the singlet discrepancy function �s is a rapidly varying functionin the extrapolation region, since it vanishes by de>nition at TLab = 0, and becomes in>nite at themeson pole near T 0
Lab. For this reason it is more convenient to work with the reduced discrepancyfunction �s
�s ≡ TLab + |T 0Lab|
TLab�s ; (3.33)
which allows a smooth extrapolation to the pion pole. The result of Kroll’s analysis has beeng2�0NN
4�= 14:52± 0:40 (3.34)
for the pseudoscalar coupling, corresponding to
f2�0NN
4�=(
m�±
2Mp
)2 g2�0NN
4�= 0:080± 0:002 : (3.35)
for the pseudovector coupling constant.Eqs. (3.24) and (3.35) have been accepted as the standard values for charged- and neutral-pion
coupling to nucleons for several years. They are consistent with charge independence, or with smallcharge independence breaking (CIB) eEects. This general consensus was shattered in 1983 by theNijmegen group, that found smaller values for the couplings, on the ground of their energy-dependentpartial-wave analysis of the low-energy pp scattering data [80]. The basic philosophy of the Nijmegenapproach is to take into account as closely as possible the energy variation of the phase shifts ina smooth way, while taking advantage of the present knowledge of the tail in the pp interaction.The Nijmegen group has devoted a lot of eEorts to re>ne their analysis; in particular, they extendedthe database to include both pp and np scattering [81], as well as N XN data [82]. The eEects ofvertex structure and the in?uence of the backward np cross section data were studied in particulardetail, and found, according to the authors, to be negligible [83]. The result of all this work can be
for the neutral- and charged-pion coupling constants.Independently from de Swart and collaborators, the VPI group started from their partial-wave
solution of the �±N scattering data [14], using the >xed-t dispersion relation (3.22). They searchedfor an optimal value of g�±NN [22]. To explore the sensitivity of the >tting procedure to the valueof the coupling, the solutions were generated for a grid of �N coupling constants and isoscalarscattering lengths (g2=4�; a(+)), looking for a minimum of C2 in the C2(g2=4�) plane. The result hasbeen
f2�±NN
4�= 0:076± 0:001;
g2�±NN
4�= 13:75± 0:15 : (3.38)
Up to this point, the new determinations of the pion–nucleon coupling constant seemed to point tolower values for these parameters than usually assumed in the past. A con?icting result, however,has been recently obtained by the Uppsala collaboration, in a dedicated experiment on backwardnp scattering [84]. It is expected that backward np cross section ought to be sensitive to the pion–nucleon coupling as remarked by Chew [85] in the 1950s. It is worthwhile here to brie?y recallChew’s physically transparent and elegant argument.
Let us consider two nucleons with CM momenta and ′ before and after the collision. The directBorn amplitude for the exchange of a neutral pion is given by Eq. (3.28)) with q2=2||2(1−cos �),where � represents the scattering angle in the CM system. This amplitude is clearly singular forcos �=1+m2
�=(2||2). In particular, contributes to the pole in the forward direction for pp scattering,as we have already seen when discussing the discrepancy function. If, on the other hand, the analysisrefers to np scattering and �± exchange, so that the incoming proton emerges as a neutron and viceversa, one has the crossed Born graph, with the corresponding Born amplitude
T cB =− g2�±NN
2||2(1 + cos �) + m2�; (3.39)
singular in the backward direction, namely for cos �=−(1+m2�=(2||2)) ≡ cos �P. These considera-
tions prompted Chew to suggest an extrapolation of the backward np cross section to the unphysicalpoint cos �P in order to determine the charged-pion–nucleon coupling constant. Since the cross sec-tion scales as |T c
B|2, it is singular in the backward non-physical region of the scattering angle. Toperform the extrapolation, it is more convenient to look at the smooth function
y(x) ≡ 1g4R
x2d�(x)d
(3.40)
where x ≡ q2 − m2�, and try an expansion in powers of x, so that y(0) gives the >rst term of the
expansion, a0 = (g�±NN =gR)4, with the constant gR >xing a reference scale.
In the Uppsala analysis a modi>cation of the Chew method was employed, to analyze the diEer-ential cross sections measured at 162 MeV, in the angular range 72◦6 �CM6 180◦, >nding
f2�±NN
4�= 0:0803± 0:0014;
g2�±NN
4�= 14:52± 0:26 ; (3.41)
in very good agreement with the Koch–Pietarinen and Kroll determinations, and at variancewith the Nijmegen–VPI conclusions. These results have been recently con>rmed by new analysis at96 MeV [86].
In the light of the above results, it is fair to say that one is here confronted with two diEerentphilosophies. The Nijmegen–VPI approach stresses the smooth, energy-averaged features of the data,and points to achieve, through an energy-dependent analysis an overall view of the �N and NNinteractions. The Uppsala group emphasizes the crucial role played by an accurate analysis of aselected data set, where all details of the procedure are subjected to rigorous scrutiny. Whatever theadopted attitude might be, one obviously expects that diEerent methods, if equally valid, convergesooner or later to consistent results, a goal which presently has not yet been achieved. One mayask, in any case, what consequences that 5% diEerence in the value of g2=4� would have for ourunderstanding of nuclear phenomena.
A >rst constraint for g�NN is given by the Goldberger–Treiman relation [87]. A well-known conse-quence of chiral symmetry, it relates the strong-interaction constant g�NN to quantities characterizingweak interactions, the weak axial coupling gA and the pion decay constant f�,
g�NN = gAMf�
; (3.42)
where both gA and g�NN are evaluated at zero momentum transfer q2 = 0, and not at the pion poleq2 = m2
�. As the experimental determination of f� has improved in the years, Eq. (3.42) has beenveri>ed with increasing accuracy, up to the present 2% level [88], without form-factor eEects. Adecrease of g�NN would improve the agreement with the Goldberger–Treiman relation even more. Insome non-linear �-models this could even overshoot the goal, owing to the presence of additionalquark-mass terms.
Another simple relation between the pion–nucleon coupling and directly observable quantitiesis provided by the Kroll–Ruderman theorem [69]. It relates f�NN to the pion photo-productionamplitude in the long-wavelength limit, and can most easily obtained by considering the static iso-vector Hamiltonian (3.3) in the photon >eld A and making the minimal substitution,∇ → ∇∓ ieA, where ∓ applies for positive or negative pions, respectively. After Fourier analysisof the electromagnetic >eld, one gets when the photon momentum k approaches zero
〈�±(q)|T |+(k)〉 ∼ ±ef�NN
m�� · �B∓ ; (3.43)
where � describes the photon polarization, and the isospin operator B∓ obviously refers to the targetnucleon. In virtue of Eq. (3.43), the pion photoproduction cross sections at threshold are proportionalto f2
�NN . These cross sections are presently known with a ∼ 2% accuracy, so that a 5% change inf2
�NN would imply a change of about 2.5 standard deviations in the cross section values in the samedirection. This means an overall discrepancy of about 3:5 standard deviations, which would not beeasy to accommodate in our present understanding of photo-reaction phenomena.
In principle, a good testing ground of pion physics is represented by the deuteron. In particular,the deuteron quadrupole moment and the asymptotic D=S ratio are known to satisfy a linear relation
Fig. 10. NN scattering processes in the s (NN → NN ) and t (N XN → N XN ) channels.
for any reasonable local NN potential [89]. This allows one to infer that a 2% variation in f2�NN
would lead to a 1% change in the D=S ratio, with the quadrupole moment >xed at its experimentalvalue. The problem here is that diEerent experimental determinations of D=S do not yet agree to therequired precision, in spite of the 2% accuracy claimed for this quantity.
Other consequences of a smaller value for f�NN would emerge in the tensor=vector coupling ratioin �NN coupling, and, obviously, in the value of meson–� couplings, when they are related to f�NN
through quark model considerations. These are the next subjects to be discussed.
3.2.2. Other meson–nucleon couplingsThe determination of the �NN coupling constant is intimately related to the evaluation of the
two-pion-exchange contribution to the NN interaction. The dispersion-theoretic approach to nuclearforces has been vigorously developed in the 1970s, leading to the >rst quantitative model of thenuclear potential at intermediate NN distances, as exhaustively described in Refs. [90,91]. Thisapproach enforces in a consistent way the constraints due to analyticity and unitarity, as well as tothe experimental information about �N scattering and annihilation processes. Let us brie?y recallhow one proceeds. The NN scattering amplitude in the s-channel is related, via crossing symmetry,to the N XN → �� amplitude in the t-channel, as graphically depicted in Fig. 10. The reason forthis indirect way of proceeding is that unitarity in the s-channel would require the knowledge of allpossible processes that could occur in NN scattering, whereas in the crossed channel one naturallyexhibits through unitarity intermediate states with one, two, or any number of pions, which can beregarded as meson exchanges in the other channel. One thus obtains a decomposition of the NNscattering amplitude into terms corresponding to the exchange of an increasing number of pions.In particular, once the 2� unitarity contribution is exhibited in the N XN → �� amplitude, one canhave information about the two-pion-exchange term M2�(s; t) in the NN transition amplitude, throughanalytic continuation of the annihilation amplitude. For two intermediate pions in the relative p-wavestate, the exchanged quantum numbers are J � =1−, T =1, because of Bose statistics, correspondingto the quantum numbers of the � resonance. Since the N XN → �� amplitude can be related, viadispersion theory, to the (p-wave) �� and �N amplitudes, one can link the two-pion component ofthe NN interaction to �� and �N physics.
Assuming Mandelstam analyticity in the variables s and t, M2�(s; t) can be written, ignoring spinand isospin degrees of freedom [8,90,91]
M2�(s; t) =1�
∫ ∞
4m2�
#2�(s; t′)t′ − t
dt′ ; (3.44)
where the spectral function #2�(s; t) can be extracted from the p-wave N XN → �� helicity amplitudes.At low energy (s � 4M 2), in the static approximation t � −|q|2, with q the NN three-momentumtransfer, Eq. (3.44) can be written,
M2�(|q|) = 1�
∫ ∞
4m2�
#2�(t′)t′ + |q|2 dt
′ : (3.45)
After a Fourier transform, the latter can be identi>ed with a superposition of Yukawa potentialsweighted by the spectral function #2� [8].
These considerations can be extended to the spin–isospin-dependent part of the �-exchange NNpotential (3.14), obtained from the eEective �NN Lagrangian (3.13). In analogy to Eq. (3.14) onecan write
V� =−(�1 × q) · (�2 × q)F�(|q|)�1 · �2 (3.46)
with
F�(|q|) ≡ 1�
∫ ∞
4m2�
#2�(t′)t′ + |q|2 dt
′ : (3.47)
If two-pion exchange is approximated with the propagation of a narrow � resonance, so that, con-sistently with dispersion theory, the spectral function #2� is simply given by a pole term
#2�(t) =f2
�NN
m2�
��(t − m2�) ;
Eq. (3.46) reproduces the �-exchange potential (3.14) in the static approximation. This suggests thede>nition
f2�NN
m2�
=1�
∫ ∞
4m2�
#2�(t′) dt′ (3.48)
for the eEective �NN coupling constant which takes into account the dispersive eEects due to the>nite width of the � meson and the 2� continuum. It is related to the vector and tensor couplingsgV and gT by (see Eq. (3.15))
f2�NN
m2�
= g2V(1 + C�)2
4M 2 :
The above considerations show that the overall strength of the �NN coupling in the nucleon–nucleonpotential can be most eSciently measured by the quantity [92]
G2�NN
4�≡ g2V
4�(1 + C�)2 : (3.49)
The best source of information about �NN coupling is represented by the electromagnetic form fac-tors of nucleons. Let us recall that the matrix element between nucleon states of the electromagnetic
current JF as a function of the invariant momentum transfer squared t = (p′ − p)2, is given, on theground of relativistic invariance, by [93]
〈N (p′)|JF(0)|N (p)〉= eu(p′){+FF1(t) +
i2M
�FI(p′ − p)IF2(t)}
u(p) (3.50)
with p and p′ the initial and >nal four momenta of the nucleon, respectively. The Lorentz scalarsF1(t) and F2(t) are the Dirac and Pauli form factors, associated to the charge and anomalous magneticmoments of the nucleon Cp and Cn. For zero momentum transfer they are normalized according to
F (p)1 (0) = 1; F (n)
1 (0) = 0; F (p)2 (0) = Cp; F (n)
2 (0) = Cn : (3.51)
In terms of the exchange of well-de>ned isospin quantum numbers the e.m. current can be moreconveniently decomposed into isoscalar and isovector components, with the corresponding de>nitionof the isoscalar and isovector form factors
F (s)i (t) = 1
2(F(p)i (t) + F (n)
i (t)); F (v)i (t) = 1
2(F(p)i (t)− F (n)
i (t)) ; (3.52)
(i = 1; 2). Because of Eq. (3.51) their normalization for t = 0 is
F (s)1 (0) = F (v)
1 (0) = 12 ; F (s)
2 (0) = 12(Cp + Cn); F (v)
2 (0) = 12(Cp − Cn) : (3.53)
A dynamical theory for the nucleon e.m. form factors can be developed on the assumption thatone-photon exchange dominates electron–nucleon scattering, which is quite reasonable in view ofthe smallness of the >ne-structure constant <, so that the photon–nucleon vertex is the basic buildingblock for the electromagnetic interactions of nucleons. As always in dispersion theory, one thenassumes that the form factors F (l)
i (t) (i=1; 2; l=s; v) are analytic functions of the invariant kinematicvariables so that they satisfy the dispersion relation
F (l)i (t) =
1�
∫ ∞
t0
Im F (l)i (t′)
t′ − t − i0dt′ : (3.54)
The imaginary part of F (l)i (t) is determined by all possible intermediate states between the exchanged
photon and the nucleon. If one assumes that the photon essentially couples to the nucleon throughsingle-vector-meson intermediate states, known as Vector Meson Dominance), Im F (l)
i (t) can beapproximated by a few pole terms,
Im F (l)i (t) = �
∑V
m2V
fVg(l)i �(t − m2
V ) ; (3.55)
so that one gets
F (l)i (t) =
∑V
m2V
fV
g(l)i
m2V − t
: (3.56)
Here, the constants mV represent the masses of the exchanged mesons V , and fV are the +-vector-meson coupling constants, whose empirical values can be inferred from the V → e+e− decay widths2(V → e+e−) through [94,95]
f2V
4�=
<2
3mV
2(V → e+e−): (3.57)
This approximation, though satisfactory for the vector–isoscalar mesons such as the !, is not ade-quate for the � meson, because of its large width, and the small pion mass, which makes the next
unitarity contribution, represented by the 2� continuum, to Im F (v)i non-negligible. Thus, in a disper-
sive approach, Eq. (3.54), the dynamics of the nucleon form factors is intimately related to the pionform factor and to the ��N XN transition amplitude. The latter, however, gets its physical values fort¿ 4M 2, whereas in the isovector form factor it is required in the non-physical region t ∼ 4m2
�. Ananalytic continuation is then required, which cannot be performed directly from experimental data,since this procedure is prone to severe ambiguities when one allows for errors in the data points[96,97]. A dispersion analysis of the 2� contribution to the isovector nucleon form factor has beenperformed by HUohler and Pietarinen (HP) in the mid-1970s [98], starting from their partial-waveanalysis of �N scattering, where t6 0. HP could determine the residue at the � pole of the �NNform factors, to be identi>ed with the eEective vector and tensor coupling constants gV and gT . Theyobtained
g2V4�
= 0:55; C� ≡ gT
gV= 6:6 :
It has to be stressed that the value of the ratio C� of the tensor to vector coupling constant is largecompared with the result emerging from the vector-dominance assumption, C� = 3:7 [99].Clearly, the model-dependent part of the HP analysis is mainly due to the analytic continuation
from the physical into the non-physical region of the partial-wave amplitudes. For t ¡ 0, the t rangeis limited by the use of the partial-wave projection, which gives the helicity amplitudes starting fromthe physical �N amplitudes. If Mandelstam analyticity is assumed, the expansion of these amplitudesinto Legendre polynomials can be shown to converge for t¿− 26m2
� [100]. A truncated Legendreexpansion is still a reasonable approximation up to −45m2
� [98]. At the same time, the integrationin the dispersion integral (3.54) has to be truncated at some >nite value, in order to exclude higherresonances. HP choose tmax=50m2
� ∼ 1 GeV2 in their calculations. Their results have been essentiallycon>rmed by Grein, who combined forward dispersion relations with the information coming fromthe HP partial-wave ��N XN amplitudes to analyze NN and N XN forward scattering data [101]. Heobtained g2V =4�= 0:55; gT =gV = 6:0. Presently, the recommended values are [102]
g2V4�
= 0:55± 0:06; C� =gT
gV= 6:1± 0:6 : (3.58)
The analysis of electromagnetic nucleon form factors has been reconsidered in Ref. [95], imposingnew constraints given by low-energy neutron–atom and electron–proton scattering data, which es-tablish the nucleon mean square radius and proton charge distribution. Constraints from perturbativeQCD where also considered, which >x the behaviour of the form factors at large momentum trans-fers. The form factor parameterization has then three pole terms of the form (3.56), plus the ��contribution as given by HP, and an extra term enforcing the QCD asymptotic behavior, i.e.
ReF (v)i (t) =
1�P∫ 50m2
�
4m2�
Im F�i (t
′)t′ − t
dt′ +∑V
m2V
fV
gi
m2V − t
+ ciFQCDi : (3.59)
The analysis performed with this parameterization turned out again to be consistent with the values(3.58) for the �NN coupling constants.
It is worthwhile to compare G�NN , determined from dispersion analysis, with the values givenin modern NN potentials based upon >eld-theoretic approaches. If the standard values (3.58) are
In the full Bonn potential, on the other hand, one has [103] g2V =4�=0:84, C�=6:1, thereby obtaining
G2�NN
4�� 42 ;
while the Bonn B potential, which parameterizes the full Bonn through one-boson exchange, gives,with g2V =4�= 0:90, C� = 6:1 [67],
G2�NN
4�� 45 :
Clearly, the eEective �-coupling parameter in the Bonn potentials is much closer to the value dictatedby dispersive analysis than to the VDM determination. One arrives at the same conclusion forthe Nijmegen soft-core interaction, once the corresponding coupling constant g2V =4� = 0:795, withC� =4:221, is renormalized by the overall form factor exp(−|q|2=N2), evaluated at the �-meson pole−|q|2 = t = m2
�. With N= 964:52 MeV and m� = 770 MeV one >nds [92]
G2�NN
4�� 41 :
Finally, let us observe that these results allow for a direct comparison between the eEective �-mesoncoupling strength f�NN and the meson–nucleon coupling constant f�NN . Inserting the values (3.58)into Eq. (3.15) one >nds, with f2
�NN =4� � 0:08,
f2�NN
m2�
� 2:0f2
�NN
m2�
: (3.60)
Evidence for a strong �NN coupling C� ∼ 6 from NN scattering observables has been discussedin Ref. [92]. In Fig. 11 the predictions for the 01 mixing parameter by the full Bonn, Bonn B,Paris, and Nijmegen potentials are compared with the experimental data. Predictions by the Reidpotential which has a weak �NN coupling and by a model without �-exchange contributions arealso given. One can see that the “strong �” interactions are in good agreement with the experimentalpoints, whereas the results of “weak �” calculations severely overestimate the data in the energyregion between 200 and 300 MeV, where second-order contributions from the overall tensor forceare negligible [92].
One may conclude that, as far as the �NN coupling is concerned, there is presently a substantialagreement between the results coming from NN studies, and the dispersive analysis of the nucleonelectromagnetic form factors.
Up to now, we have discussed �NN coupling in terms of hadronic degrees of freedom. From thepoint of view of QCD this is an eEective description of the underlying dynamics, one is left whenquark and gluon degrees of freedom have been integrated over. An ab initio determination of the�NN coupling parameters, starting from QCD, would obviously represent a major goal. Similarly
Fig. 11. Predictions for the 01 mixing parameter by the full Bonn, Bonn B, Paris, Nijmegen (solid lines) and Reid (dashedline) potentials compared with a model without �-exchange contributions (dotted line) and the experimental data. Figuretaken from Ref. [92].
to other basic quantities appearing in low- and intermediate-energy Nuclear Physics, this wouldentail the solution of an extremely diScult many-body problem, where quarks and gluons move in ahighly non-trivial vacuum, with non-perturbative eEects playing a dominant role. As a consequence,only a few exploratory calculations of the �NN coupling strengths have appeared up to now, wherenon-perturbative QCD eEects have been taken into account in a more or less approximate way. Thus,the �NN coupling has been considered already several years ago in a two-phase Skyrme model, wherea little quark bag is surrounded by a pionic cloud [104]. If the baryon charge is assumed equallydistributed between the quarks and the pionic cloud, one has that the tensor-to-vector ratio turnsout to be about twice the VDM value, for a quark core radius of about 0:5 fm in the nucleon, asrequired by low-energy phenomenology.
Some QCD-inspired calculations of the �NN coupling parameters have been recently attemptedin the framework of Sum Rules, where the non-perturbative eEects of the vacuum are describedthrough various quark, gluon, or mixed quark–gluon condensates. In [105] the � meson is treated asan external >eld, coupled to the quarks through a vector-type eEective Lagrangian. A value C�=3:6,consistent with the VDM hypothesis, has been obtained. A more recent calculation [106], employingthe operator-product expansion on the light cone to produce a microscopic � wave function, obtainsQCD sum rules for the �-nucleon coupling parameters, which yield values for g2V =4� and C� in goodagreement with Eq. (3.58).
3.2.3. Meson–� couplingsThe coupling constants involving the � resonance are less well-known under the experimental
point of view, given the unstable nature of the � particle. The determination of the parameterscharacterizing the resonance, i.e. its mass M� and width 2�, has been discussed in Section 2.1. Interms of these parameters the J = 3=2+, T = 3=2 partial-wave amplitude is given by the relativistic
where q is the momentum of the emerging pion. Relativistic kinematics dictates its dependence uponenergy and masses of the emerging particles to be [107]
q=
√s2 +M 4 + m4
� − 2sM 2 − 2sm2� − 2M 2m2
�
2√s
:
The width 2�(s) at the resonance peak√s=M� can be related to the �N� coupling constant G�N�
as follows [102,107]:
G2�N�
4�=
M 2�2�(M 2
�)q(M 2
�)3: (3.62)
This is the coupling strength used in SU (3) >ts, where the � is a member of the baryon decuplet.One may look for the relation between this strength and the coupling constant f�N� appearing inthe eEective relativistic Lagrangian L�N� (3.7). This can be accomplished through a straightforward,but somewhat lengthy calculation. To simplify the notation, let us refer to the �++ → �+p case.In the reference frame with the � at rest the two-particle >nal state can be characterized by thevalue |q| and the angular variables (�; /) of the emerging-pion momentum, plus the helicity � ofthe >nal nucleon. On the ground of rotational invariance one expects that the decay matrix elementfor the baryon → � baryon transition T (|q|; �; /; Sz
�) factorizes into purely geometric factors, timesa quantity subsuming the dynamical aspects of the system. As a matter of fact one has [107]
Tfi ≡ T (|q|; �; /; Sz�) = 2 exp[i/(Sz
� − �)]d3=2Sz��(�)V3=2(�) ; (3.63)
where d3=2Sz��(�) are the usual reduced rotation matrices, and Sz
� the third component of the � spin.The vertex V3=2(�) is determined once Eq. (3.63) is compared with the matrix element of theLagrangian (3.7) between the initial � and >nal �N state. In momentum space one gets the quantity(f�N�=m�) Xu(M)qFu
F�, where Xu and uF
� are the Dirac and Rarita–Schwinger spinors for the proton andthe �, respectively. The latter is simply obtained as the coupling of a spin-(1=2) spinor u(M�) and aspin-one four vector 0F, with the proper Clebsch–Gordan coeScients. For the �++ → �+p vertex onesimply has uF
�(+3=2)=u(M�; 1=2)0F(+1). The Dirac invariant Xu(M)u(M�) can be evaluated throughstandard techniques [107], and gives a mass factor ((M� +M)2−m2
�)1=2, while the evaluation of the
scalar product qF0F in the c.m. system allows the identi>cation of V3=2(�). One has
V3=2(�) =1√6
f�N�
m�
√(M� +M)2 − m2
�q : (3.64)
For unpolarized particles, the decay width can be obtained from the decay amplitude Eq. (3.63)integrating over the angular variables, averaging |Tfi|2 over the initial spin states and summing overthe >nal spin states, namely [107]
Fig. 12. Direct and crossed nucleon and � pole contributions to the scattering matrix.
Inserting V3=2(�) given by Eq. (3.64), one obtains the standard decay width
M�2�(M 2�) =
16q(M 2
�)3
M�
f2�N�
4�(M� +M)2 − m2
�
m2�
: (3.66)
Comparing Eq. (3.62) with Eq. (3.66), one >nally gets
f2�N�
4�= 6
m2�
(M� +M)2 − m2�
G2�N�
4�: (3.67)
According to Eq. (3.62), the value of G2�N� is obtained from the determination of the resonance
parameters, >xed through a >t of �N scattering data, giving G2�N�=4� = 14:54 from �+p scattering,
and G2�N�=4� = 14:39 from �−p [102]. Hence, taking for G2
�N�=4� an indicative value of 14.5, onehas from Eq. (3.67)
f2�N�
4�� 0:362 (3.68)
for the �N� coupling constant appearing in the eEective relativistic Lagrangian (3.7).Most applications in Nuclear Physics need a reasonable non-relativistic treatment of the � as
an eEective particle, to be matched with non-relativistic wave functions and eEective operators.As outlined at the beginning of this Section, this can be accomplished through the eEective staticHamiltonian (3.9), which allows a straightforward evaluation of the various N� transition poten-tials. This is indeed possible, as shown many years ago by Sugawara and von Hippel [66] by thenon-relativistic reduction of the Feynmann amplitude corresponding to the Born OPE graph, whichentails a judicious dropping of terms of order
M� −MM� +M
∼ m�
M + m�:
The coupling constant f�N� appearing in (3.9) can be >xed by requiring that the non-relativisticisobar model is able to reproduce the �N scattering data in the same extent as the static Chew–Lowmodel. In this sense f�N� can be expressed in terms of the �N coupling constant. To see how thiscomes out, let us consider p-wave �N scattering, as proceeding through the direct and crossed Borngraphs with intermediate nucleon and � contributions shown in Fig. 12. It is more convenient to
use a formalism based upon a K-matrix approach. In this way unitarity of the S-matrix is fullyguaranteed by the Cayley relation between the S- and K-matrix, see Eq. (2.11). The same doesnot apply for an approximate T -matrix including only the pole contributions and is crucial for ameaningful comparison with the Chew–Low parameterizations of the scattering data. The requiredK-matrix is
K =KN + K� =f2
�NN
4�m2 |q|[(� · q′)(� · q)
−!BbBa +
(� · q)(� · q′)!
BaBb
]
+f2
�N�
4�m2
[(S · q′)(S† · q)
!� − !TbT
+a +
(S · q)(S† · q′)!� + !
TaT+b
]: (3.69)
Here, ! is the pion energy, and !� ≡ M� −M represents the nucleon–� mass shift. The K-matrixgiven by Eq. (3.69) satis>es crossing symmetry, as one can verify through the substitutions ! ↔ −!,q ↔ −q′ and a ↔ b, and possesses the proper pole at the N–� mass diEerence !�.
Invariance with respect to rotations in ordinary and isospin space can be now exploited, for aprojection onto the partial-wave eigenchannels. In particular, for J = T = 3=2 one has
K33(!) = tan �33 =13
|q|34�m2
[4f2
�NN
!+
f2�N�
!� − !+
19
f2�N�
!� + !
]: (3.70)
The last term, comes from the �-crossed graph, and contributes no more than 2% near threshold,so that at low-energy one can write
tan �33 � 13
|q|34�m2
[4f2
�NN
!+
f2�N�
!� − !
]: (3.71)
This result has to be compared with the Chew–Low eEective-range approximation for the J=T=3=2phase shift, which is obtained from the Chew–Low equation keeping only the pole and one-pionintermediate-state contributions. One gets for the �33 phase shift [108]
tan �33 � |q|3m2!
f2�NN
3�!�
!� − !; (3.72)
which fully exhibits the resonant behavior. Clearly, to have Eq. (3.71) coinciding with theChew–Low result (3.72), one needs f2
�N� = 4f2�NN . Taking for f2
�NN =4� the indicative value 0:08one has
f2�N�
4�= 4
f2�NN
4�� 0:32 ; (3.73)
which is not too far from the value (3.68), extracted from the experimental � position and widthby means of the relativistic Breit–Wigner formula.
Many relations among meson–baryon couplings can be obtained by considering the strong-couplinglimit of static >eld-theoretic models of hadrons [109,110], where recoilless sources interact throughthe exchange of diEerent mesons. Although obtained many years ago, these results still comparewell with the experimental data, and represent a beautiful example of how one can exploit thegroup-theoretical symmetries embedded in phenomenological models of strong interactions. As wehave seen, in the simplest case the Chew–Low equation relates the elastic scattering amplitude toall the reaction and production amplitudes allowed by unitarity, in a way that crossing is satis>ed.If several types, <; >; : : : , of mesons are exchanged between diEerent baryonic sources B; B′; : : : , the
Chew–Low equation can be straightforwardly generalized by replacing the transition amplitude T (!)with a set of amplitudes TB′B
>< (!). Disregarding the contributions with two or more mesons in theintermediate states, these transition amplitudes satisfy the generalized Chew–Low equation [109]
TB′B>< (!) = �2
∑B′′
[VB′B′′
> V B′′B<
!+MB −MB′′+
VB′B′′< V B′′B
>
−!+MB′ −MB′′
]
+∑B′′ ;+;p
[TB′′B′+> (!p)∗TB′′B
+< (!p)
!− !p +MB −MB′′+
TB′′B′+< (!p)∗TB′′B
+> (!p)
−!− !p +MB′ +MB′′
]: (3.74)
Here, � is the renormalized strength parameter, and VB′′B< represents the matrix element of the
coupling Hamiltonian between the meson >eld and the baryon states. For a given <, one can re-gard the quantities VB′′B
< as the elements of a matrix V< in the baryon indices. The pole terms inEq. (3.74) clearly satisfy crossing symmetry, and have the proper poles at the physical mass shiftsMB−MB′′ . In the strong-coupling limit � → ∞, the isobar masses are degenerate, and one can writeMB → M + �MB=�2, so that an expansion of the pole terms in Eq. (3.74) in powers of �−2 gives
TB′B>< (!)pole ∼ �2
!([V<; V>])B
′B +1!2 ([V>; [V<; �M ]])B
′B + O(
1�2
); (3.75)
where �M is a diagonal matrix with �MB as diagonal element. Now, unitarity implies that thescattering amplitude must be >nite in the strong-coupling limit. From Eq. (3.75) one then has thecommutation rules
[V<; V>] = 0 (3.76)
for the interaction terms, which can be regarded as dynamical constraints. Other commutation rulescan be obtained under the assumption of invariance under some symmetry group. For the SymmetricPseudoscalar-Meson theory the symmetry group is SU (2)J⊗SU (2)T , implying invariance with respectto spin and isospin rotations. The generators of the associated Lie algebra are, in the sphericalrepresentation, the usual highering and lowering spin and isospin operators J± and T±, and the thirdcomponents Jz and Tz, with standard commutation rules (CR), which must be supplemented withthe ones for the meson sources VFI. They can be written down immediately, by requiring that themeson currents also transform like the regular representation under SU (2)J ⊗ SU (2)T
[J±; VFI] =√
(1∓ F)(2± F)VF±1I; [Jz; VFI] = FVFI : (3.77)
Here, in the meson terms VFI the >rst index refers to spin, and the second one to isospin, so thatthe CR in isospin space can be obtained from Eq. (3.77) simply interchanging the role of F and I.Eqs. (3.76) and (3.77) actually de>ne the Lie algebra associated to the whole symmetry group
[SU (2)J ⊗ SU (2)T ] × T9 of the strong-coupling Symmetric theory, where T9 is the nine-parameterAbelian group. Its presence here can be understood on physical grounds, by noting that the mesonsare described by isovector operators, and interact in p-wave only with the baryons. Relations (3.76)and (3.77) have been solved by Singh [110], with particular reference to the simplest irreduciblerepresentation of the symmetry group, which has the spin–isospin content J = T = 1=2; 3=2; : : : :The explicit realization of the commutation rules implies that the baryon–baryon–meson verticesare essentially given by a product of two Clebsch–Gordan coeScients times a universal coupling
constant N, so that for two baryons with spin and isospin J = T and J ′ = T ′, respectively, one hasin an obvious notation [4]
N�BB′ =
√2J + 12J ′ + 1
N〈J1Mm | J ′M + m〉〈J1MTmT | J ′MT + mT 〉 : (3.78)
Since in the strong-coupling limit the baryon masses belong to a “rotational” band in spin and isospinspace [109,110]
M (J; T ) =M0 +M1J (J + 1) +M2T (T + 1) ; (3.79)
one can identify the lowest baryons of the series with the nucleon and the �, respectively. Thus,the various meson–baryon coupling constants diEer from one another simply by geometric factors,and in particular one >nds
f2�N�
4�=
92f2
�NN
4�; (3.80)
which, for f2�NN =4� � 0:08 gives f2
�N�=4� = 0:36, in surprisingly good agreement with the experi-mental value (3.68).
The simple relation (3.80) between the �NN and �N� couplings is due to the fact that the physicscontained in static >eld-theoretic models can be expressed completely by group-theoretic argumentsin the strong-coupling limit. A similar simpli>cation can be achieved in quark models of hadrons,provided that simple baryon wavefunctions are employed, so as to exploit again the invariance undera suitable symmetry group. This has been clearly shown several years ago by Brown and Weisefor the SU (4) quark model [12], in which the nucleon and the � belong to the same spin–isospinmultiplet, so that here we limit ourselves to recall some of the main points. In this model the �++
isobar, can be described as a bound state of three isospin “up” quarks with their spin “up” u ↑,as detailed in Table 3. The other baryon states can be generated from the �++ wave function byemploying step-down operators in spin and isospin space, expressed in terms of quark creation andannihilation operators. Thus, the spin down operator s(−) is given by
s(−) =3∑
i=1
[a†u↓(i)au↑(i) + a†d↓(i)ad↑(i)] : (3.81)
The analogous operator in isospin space t(−) can be obtained by interchanging the role of spin andisospin in Eq. (3.81). Operating with s(−) and t(−) upon |� 3=2 3=2〉 one can produce all the requirednormalized and symmetric baryon wave functions. Under a mathematical point of view this leads toanalyze the representations of the group SU (4) ⊃ SU (2) ⊗ SU (2). However, one can proceed alsoin a more heuristic way, by introducing the usual non-relativistic coupling Eq. (3.3) at the quarklevel, so as to write
H�QQ =−f�QQ
m�
3∑i=1
(�(i) ·∇)(�(i) · �) ; (3.82)
where the pion >eld � is regarded as an external c-number classical >eld. The �qq coupling constantcan be related to the “macroscopic” �NN coupling strength by requiring the expectation value of
(3.82) in the quark-model proton state |P1=21=2〉 to be equal to the usual meson–proton couplingoperator. One has
f�QQ
m�
⟨P1212
∣∣∣∣∣3∑
i=1
(�z(i)B3(i))
∣∣∣∣∣P1212
⟩∇z/3 =
f�NN
m�∇z/3 : (3.83)
The matrix element with respect to quark degrees of freedom can be evaluated as in simpleshell-model calculations, and one gets [12]
f�QQ = 35f�NN : (3.84)
Similarly, one can >nd out the relation between f�N� and f�NN just equating the matrix element ofthe meson–quark coupling operator between the quark � and proton states |�1=21=2〉 and |P1=21=2〉to the corresponding “macroscopic” matrix element, leading to [12]
f2�N�
4�=
7225
f2�NN
4�; (3.85)
which, for f2�NN =4� � 0:08 gives f2
�N�=4� � 0:23, a value somewhat smaller than the experi-mental one.
The above “simple-minded” quark model has been widely used, in spite of its moderate successin reproducing the experimental �N� coupling, because it can be easily extended to other meson–baryon–baryon couplings. Thus, the ��� strength is related to the �NN coupling constant by [12]
f��� = 45f�NN : (3.86)
Similar considerations allow one to relate the f�N� coupling strength to the already known couplingparameters. With reference to this point we recall that �-exchange plays a crucial role in reducingthe dependence of the N� transition potential upon the regularizing cut-oE parameter [4,111,112]. Tounderstand how the required relation comes out, let us recall the �-exchange N–� interaction (3.18),where the spin and isospin dependence has the same structure as in the meson–nucleon coupling. Asa consequence, in the SU (4) quark model, with eEective coupling to external c-number meson >elds,the only diEerence among the various meson–baryon–baryon vertices amounts to purely geometricfactors, i.e. Clebsh–Gordon coeScients, the reduced matrix elements remaining always the same.One can expect, for instance, that the ratio between f�N� and f�N� is the same as the ratio betweenf�NN and f�NN , namely [4,13,111]
f�N� =f�NN
f�NNf�N� : (3.87)
Similar arguments lead >nally to establish that the analogue of Eq. (3.86) is
f��� = 45f�NN : (3.88)
One may ask what value for f�N� has to be inserted in Eq. (3.87), to >x f�N�. Should one usethe experimental value (3.68), extracted from the � width? Or, to be consistent with the SU (4)quark model, should one use Eq. (3.85), in spite of its moderate agreement with the experimentalinformation? With reference to this point, as already observed by Green several years ago [4], itis worthwhile to recall that the geometric relations (3.87) and (3.88) can be obtained also in theframework of the strong-coupling model, which is able to give the remarkable value 0.36 for the�N� coupling constant. Certainly, from a fundamental point of view, a derivation from more realistic
quark models of hadrons, or, hopefully, from QCD would be much more preferable. To the bestof our knowledge, in spite of the progresses accomplished in this >eld, a satisfactory calculation ofthis type has yet to be performed.
3.2.4. Form factor eDects and correlated exchange of mesonsThe strengths discussed above exhaust the information one needs at the meson–hadron vertices
when the external particles are on their mass or energy shell. In few- and many-body calculations,however, oE-shell situations occur, which require the introduction of vertex form factors. Originallyintroduced to make loop integrations convergent, these form factors take into account the intrinsiccomposite nature of strong-interacting particles, which have to be regarded as collective excitations ofthe underlying QCD degrees of freedom. The actual derivation of form factors from a dynamic theoryrepresents an extremely diScult task. Therefore, they have been generally parameterized throughsimple, phenomenological functions. A consistent treatment of form factors would in principle requiretheir dependence upon all the four momenta of the involved particles, with a parameterization validin all the kinematic domains explored in the various reaction processes where they can be involved[113]. In practice, one limits oneself to the dependence from the four momenta of the exchangedparticle. Thus, for the baryon–baryon interactions considered in the previous section, one simplyreplaces the strengths f<BB′ with multipole form factors, namely
f<BB′(q) = f<BB′
(N2
<BB′ − m2<
N2<BB′ − q2
)n
: (3.89)
The cut-oE parameter N<BB′ >xes, through its inverse N−1<BB′ , the spreading of the vertex function in
con>guration space at the meson–baryon–baryon vertex. The momentum dependence of the cut-oEeEects become smaller and smaller in loop integrations as N increases. When the square of thefour-momentum transfer q2 equals the mass squared of the exchanged particle, the form factorsimply becomes the coupling strength f<BB′ at the considered vertex. The exponent n has to be >xedso as to guarantee convergence in loop integrations; generally, monopole form factor n=1 or dipoleform factor n= 2 is enough to meet this requirement.The cut-oE parameters can be regarded as new phenomenological quantities to be determined in
reproducing, for instance, NN or N� scattering data. Particular attention has been obviously devotedto the �NN cut-oE mass N�NN . Several attempts have been made since the 1970s to infer this basicquantity directly from experimental data, through an analysis of charge-exchange np reactions and pXpscattering [114,115], or from charged pion photoproduction [116]. On the ground of these analysis, avalue for N around 1 GeV was preferred. A major role in >xing the values of the cut-oE parametershas been played by the development of the Bonn NN potential. There, it was found a lower limit of1:3 GeV for N�NN , and an upper limit equal to 1:2 GeV for N�N�, in order to reproduce the asymptoticD-to-S-wave ratio, the quadrupole moment in the deuteron, and low-energy NN scattering phase-shifts[103]. These results had a profound in?uence on subsequent calculations of the oE-shell behavior ofNN amplitudes. In recent years, however, there has been increasing evidence that the above cut-oEmasses are too large, and that softer form factors have to be preferred. This is suggested, for instance,by attempts to resolve the discrepancy of the Goldberger–Treiman relation [117–119], and by thediscrepancy between the pp�0 and pn�+ couplings [120]. In both cases one >nds that a cut-oEaround 0:8 GeV is more adequate. As a matter of fact, one has to keep in mind that the cut-oEparameters, as >xed in reproducing the low-energy NN data, compensate also for meson-exchange
Fig. 13. Uncorrelated �–� and �–� exchange contributions (a)–(h) used in the Bonn NN potential [103] and correlatedtwo-meson exchanges (i) and (j).
Fig. 14. Correlated two-� contributions represented by sharp-mass �′ and � exchange used in the Bonn NN potential[103].
contributions not explicitly taken into account in the considered meson-exchange model. This is mostclearly seen when one compares the value N�NN ∼ 1:75 GeV from one-boson-exchange models withN�NN =1:3 GeV, given by the full Bonn potential, which takes into account uncorrelated �� and ��exchanges with N and � intermediate states, as seen in Fig. 13. It is worthwhile to observe that theexplicit inclusion of the �-isobar propagation allowed also to reduce the unphysical contribution fromthe >ctitious � meson, which characterized early one-boson-exchange models of the NN interaction[121]. In the full Bonn potential, to the uncorrelated two-pion-exchange processes one adds �′ and �one-boson-exchange contributions, which represent in an eEective way the exchange of two correlatedpions in the scalar–isoscalar J=0+; T=0 and vector–isovector J=1−; T=1 states in the t channel.This is graphically depicted in Fig. 14. Clearly, an explicit treatment of �� and �� correlations wouldgive a deeper insight into the dynamics determining the nuclear force at intermediate range, at thesame time paving the way to a further reduction of the �NN cut-oE parameter. It turned out thatexplicit inclusion of the � is necessary for a realistic model of these exchange processes.
It is worthwhile to mention here that a diEerent perspective on the nuclear force is given by arelativistic one-boson-exchange model employing the covariant Gross equation. There, one of thetwo nucleons is kept on the mass-shell, but negative-energy states are allowed for the other fermion.In the framework of this model, the two-pion-exchange contributions to the NN force arising fromthe box graph can be approximated by a large eEective � plus a small � exchange. For details andreferences see Ref. [122].
Correlated-meson exchange can be treated in two ways. Either one resorts to dispersion-relationtechniques, relating the NN potential to semi-empirical information about �N and �� scattering, as
Fig. 15. Model of the �NN vertex function according to Ref. [128].
in the Paris approach [90,123], or one insists on explicit >eld-theoretical models, as advocated bythe Bonn group [67,103]. The latter approach has been considered in Refs. [124,125]. Correlated ��exchange is of particular relevance here, since these contributions in the channel with pionic quantumnumbers can provide a strong tensor-force component in the NN interaction, able to counterbalancethe eEects of a soft �NN form factor. In Ref. [125] correlated �� exchange has been included byrelating the NN → NN scattering amplitude in the s-channel to the N XN → N XN T -matrix in thet-channel via crossing symmetry. These amplitudes can be expressed, through a dispersion relation,in terms of the unitarity contributions due to N XN → �� transitions. The corresponding T -matrixTN XN→�� is given by the direct XNN annihilation, described by the driving term VN XN→��, followed by�, � propagation and rescattering, described by the Green function G�� and the amplitude T��→��,that is [125]
TN XN→�� = VN XN→�� + VN XN→��G��T��→�� : (3.90)
The driving term is given by N and � exchange, plus an ! pole term, whereas T��→�� is obtainedstarting from an explicit meson-exchange model, involving physical �, �, and ! mesons [126].A good >t to the two-nucleon observables can be obtained with a �NN cut-oE N�NN of less than1 GeV. To be consistent with these >ndings, one ought to obtain a comparable reduction of N�NN
in a dynamical model of the �NN vertex, based on a similar meson-exchange description of hadroninteractions. This has indeed been shown by the JUulich group [127,128], by dressing the bare �NNvertex with contributions, where three-pion states are described by �� and �� pairs, as depicted inFig. 15. The corresponding eEective T�<→�> amplitude can be evaluated through a meson-exchangemodel in the crossed, annihilation channel, where t ¿ 0. The �NN form factor in the s channel,where the momentum transfer t is spacelike, is >nally obtained by means of a dispersion integral.The resulting �NN form factor can be >tted by a standard monopole form with a cut-oE N�NN ∼0:8 GeV.
The lesson to be learned from the above example is that a proper microscopic treatment of hadrondynamics can lead to a softer form-factor than the one suggested by a more naive calculation basedon purely phenomenological functions. This is in particular true, when the composite nature ofmeson–baryon–baryon vertices is taken into account. Thus, in Ref. [129], the bare �NN and �N�vertices have been dressed by �N rescattering through the non-pole part of the �N T -matrix, asseen in Fig. 16. The �N transition matrix has been obtained starting from the meson-exchangemodel developed in [130]. Softer dressed form factors are found in respect to the bare ones, withcut-oE masses between 500 and 700 MeV, the result depending upon the treatment of the underlyingreaction dynamics.
Similarly, small N�NN and N�N� are obtained, when composite models are used for the correspond-ing vertices [131,132]. This “softening” of the form factors has been carefully studied in a model
Fig. 16. Dressing of the �NN or �N� vertex due to �N rescattering. The squared box represents the oE-shell �N transitionmatrix T .
calculation by Liu et al. [133]. They considered Feynmann graphs where the vertices have somespeci>c topological structure, and examined the eEects of intermediate particle propagation and >nitemomentum-space cut-oEs for the constituent sub-vertices on the overall vertex function. It turnedout that the non-locality inherent to composite vertices produced smaller cut-oE masses, when theywere parameterized through simple functional forms such as Eq. (3.89).
In the framework of an eEective theory of hadrons, based on meson and baryon degrees of free-dom, the actual shape of <BB′ form factors remains an elusive topic, in spite of the recent progressesmentioned above. Clearly, a direct derivation of these quantities from the underlying QCD would rep-resent a major step forward, since the composite-particle dynamics could be then related to the basicdegrees of freedom. The diSculties in performing realistic calculations in the highly non-perturbativeregime of interest to Nuclear Physics are well-known. Indications from semi-phenomenological ap-proaches, preserving the basic symmetries of QCD, like the Skyrmion model, lead to rather softform factors, with N�NN ∼ 0:6 GeV [134]. Recent lattice calculations of the nuclear axial and elec-tromagnetic form factors lead to a monopole mass N�NN ∼ 0:75 GeV [135].
3.3. The coupled-channel approach to the N� system
As we have seen, we are still unable to derive the meson–baryon–baryon vertices from an underly-ing dynamical theory, and univocally >x the coupling strengths of the � to the various mesons fromthe experimental information. As a matter of fact, coupling constants and cut-oE parameters are stillto be regarded as semi-phenomenological quantities to be >xed in reproducing NN bound-state andscattering data. When the � degrees of freedom are explicitly taken into account, one is naturallylead to face a coupled-channel problem, as graphically depicted in Fig. 7.
The >rst attempt to solve the coupled NN–N� problem was in con>guration space, much in thesame way as one gets coupled SchrUodinger equations in low-energy nuclear and atomic physics [4].In such a case one thinks of the wave function as endowed with both NN and N� components
? = a?NN + b?N� (3.91)
and de>nes a suitable Hamiltonian with nucleon and � kinetic terms plus the N� mass diEerence, andall the relevant baryon–baryon interactions, described in Section 3.1. Projecting onto the availablewave function components one gets
where, FN and F� represent the NN and N� reduced masses and E is the total energy of the system.Only NN → �N and N� → N� transitions were considered, and are represented by the potentialsVN� and V��.Eqs. (3.92) do not pose any new problem with respect to ordinary coupled-channel equations, if
the � is considered as an elementary particle and does not decay. This is not true, however. Owingto the unstable nature of the isobar, its mass M� has to be supplemented with an energy-dependentwidth, namely
M� =M (0) − i2�(E0�)
2; (3.93)
where the internal energy E0� of the isobar is obtained from the total energy by subtracting the
relative-motion term q2=2F�, E0� = E +M − (q2=2F�). In con>guration space q2 has to be regarded
as a diEerential operator, so that Eqs. (3.92) are strictly speaking non-local. This non-localitybecomes harmless in momentum space, where the SchrUodinger Eqs. (3:92) can be replaced by theLippmann–Schwinger set
TNN = VNN + VNNGNTNN + VN�G�T�N ;
T�N = V�N + V�NGNTNN + V��G�T�N (3.94)
with boundary conditions given through the free Green functions GN and G�. In general, if ��states also are taken into account, one will have a larger set of coupled equations, which can bewritten in the compact form
T<> = V<> +∑+
V<+G+T+> ; (3.95)
where <; >; + can be any of the two-particle states with either the nucleon or the �. In such a case,the additional transition potentials V (NN → ��), V (N� → ��), and interaction terms V (�� → ��)of Fig. 7 have obviously to be considered. The Eqs. (3.95) are the basic equations to be solvedin order to >t phenomenological models for the transition potentials to the two-body data. At the sametime they provide on- and oE-shell t-matrices to be employed in few- and many-body calculations.
Eqs. (3:95) use static potentials V<>, and disregard time-retardation eEects characteristic of arelativistic formalism. The question naturally arises to what extent the non-relativistic approach isjusti>ed. The diEerence between the two approaches can be clearly perceived by a comparison ofthe perturbative expansion of Eqs. (3.95) with the corresponding Feynmann graphs. The seconditeration of the basic NN and N� potentials gives the box contributions to NN scattering exhibitedin Fig. 13(a)–(d). Crossed exchange graphs, such as those given in Fig. 13(e)–(h) are thereforedisregarded in the CC approach. Several papers have been devoted since the 1970s to assess therole played by these graphs in the coupled NN–N� problem. The >rst systematic investigationhas been performed by Smith and Pandharipande [136]. They considered all the eight uncorrelatedfourth-order Feynmann graphs one can have for NN scattering in a relativistic >eld-theory with NN ,N� or �� intermediate propagation, as seen in Fig. 13. If intermediate states with positive-energybaryons only are retained, each relativistic graph corresponds to six possible time-ordered diagrams.Thus, graphs (a)–(f) in Fig. 17 represent the time-ordered diagrams one obtains from the boxFeynmann graph with intermediate N� propagation. Clearly, only the four graphs with sequentialmeson exchange can be reproduced by twice-iterated transition potentials, whereas the two “stretched”
Fig. 17. Time-ordered diagrams with positive-energy intermediate N� states.
graphs (e) and (f), with two mesons in ?ight in the intermediate state, as well as the remainingcrossed diagrams cannot have any correspondence in a standard coupled-channel calculation. Smithand Pandharipande studied the diEerence between iterated Born contributions, and the result of thewhole series of time-ordered graphs, for non-relativistic, static nucleons and isobars. They found thatthe twice-iterated OBE potentials can approximate reasonably well the whole series of time-orderedgraphs if scalar–isoscalar mesons are considered. Things are however diEerent for the pion, becausethe crossed graphs give an isospin-dependent contribution with opposite sign with respect to the boxgraphs. Physically this is due to the fact that, after emission of the >rst pion, the baryons can emita second pion while moving in a diEerent total isospin state, a possibility which is not allowed byiterated instantaneous potentials. Therefore they subdivided the crossed contributions into a piecehaving the same isospin dependence as the box graphs, plus a remainder. The overall sum of thebox and crossed contributions with the same isospin factor was overestimated by only about10–15% by the second Born approximation in the coupled-channel approach, while there werestrong cancellations among the crossed terms with the “unpleasant” isospin dependence, therebygiving strong support to standard coupled-channel calculations.
The general validity of these results has been subsequently questioned by the Stony Brook group,on the ground of a comparison of iterated static potentials with calculations based upon a covariant,dispersion-theoretic approach to the NN interaction [137]. By considering the iterative box graphs,they showed that static pion-ranged transition potentials have to be supplemented with shorter-rangeterms. Moreover, it turned out that both the N–� mass diEerence and relativistic eEects play animportant role in the N� system, since they reduce the isobar box contributions by a factor between2 and 3. Needless to say, the evaluation of non-iterative, “stretched” and crossed, contributions ismuch more involved than the iterative ones, because the meson energies occur in all propagators.One can then understand the quest for reasonable approximations, avoiding the direct calculation ofthe non-iterative diagrams.
The whole set of two-pion exchange diagrams has been considered by the Bonn–JUulich group,using time-ordered perturbation theory [138–140]. They showed that the sum of all graphs with N�intermediate states can be written [138–140]
where BN� and CN� represent the box and crossed contributions, respectively. The salient featureof this result is the opposite sign of the terms depending upon the isospin degrees of freedom ofthe external nucleons, i.e. the isovector-exchange pieces. If BN� � CN� the last term cancels out;moreover, it can happen that the twice-iterated transition-potential contribution B(0)
N� overestimates theexact amplitude BN� by a factor of two, namely B(0)
N� � 2BN� � BN� + CN�, so that one can replaceEq. (3.96) with
TN� � 2B(0)N� : (3.97)
The exact contribution is thus replaced by the isoscalar part of the twice-iterated transition potentials,provided that the assumptions about the relative weight of Born, box and crossed contributionsdetailed above are satis>ed. This has to be contrasted with the coupled-channel treatments, wherealso the isovector part is retained and one writes
TN� � (2 + 23�1 · �2)B(0)
N� : (3.98)
Therefore, in the static approximation, retaining a pion-ranged potential for the isovector part of theamplitude also, cannot be considered in general a good approximation, and may lead to a vanishingcontribution in isospin-zero states [137–140].
The approximation (3.97) has been carefully tested in Refs. [138–140]. It works fairly well forthe 1S0 partial wave in NN scattering; in other partial waves, however, things are not so simple. Theexact calculations con>rm many of the conclusions of the Stony–Brook analysis [137]. In higherpartial waves (L¿ 1) the twice-iterated pion-range transition potentials grossly overestimate theexact contributions with N� intermediate states; this can be understood on physical grounds, sincethe realistic contributions are of shorter range and hence much more suppressed for L¿ 1 thanstatic contributions with pion range. As for the isovector term, it turns out to be possibly small onlywhen all intermediate (NN , N�, and ��) states are simultaneously taken into account. These resultsshow that the use of static approximations in coupled-channel calculations require a careful vis-[a-viscomparison with relativistic treatments, to ascertain the possible presence of delicate cancellationeEects.
Even with simpli>ed baryon-baryon transition potentials, the solution of the N� CC problemremains a big task. This can be easily perceived, when rotational and isospin invariance are exploitedto give the transition amplitudes in a partial-wave representation. The lowest NN , N�, and �� T =0; 1; 2 channels are given in Table 4 in the usual spectroscopic notation.
The NN scattering observables are not very sensitive to the N� and �� channels. Moreover, theT=2 and some T=1 states such as the 3S1 are not allowed from >rst principles in the NN sector. Thisimplies that any information on these channels has to come from other indirect processes [141], likeelastic �d scattering or polarization observables, since the unstable � cannot be used as a scatteringprobe. In fact, in Ref. [141] some N� channels were indispensable to reproduce consistently thecross sections and the vector analyzing power in �d elastic scattering.When N� couplings are explicitly taken into account in the NN interaction, bound-state and
scattering data ought to be re->tted starting from a consistent solution of Eq. (3:95). To this end,the diagonal NN potential has to be suitably renormalized by subtracting the pure iterations of theOBE and transition potentials, in order to avoid double-counting when solving the coupled-channelequations [4]. In many calculations, however, this has not been done, but N� transition potentialshave been simply added to some well-established phenomenological or OBE model potential VM
Table 4Partial waves 2S+1LJ for the N� system up to L= 6 in the T = 0; 1 channels and up to L= 2 in the T = 2 channels
NN N� ��
T = 0 3S1–3D13S1 3D1
7D17G1
3D23D2
7D27G2
3D3–3G37S3 3D3
7D33G3
7G37I3
3G47D4
3G47G4
7I43G5–3I5 7D5
3G57G5
3I5 7I51P1
1P15P1
5F11F3
5P31F3
5F35H3
1H55F5
1H55H5
T = 1 1S0 5D01S0 5D0
1D25S2 3D2
5D25G2
5S2 1D25D2
5G21G4
5D43G4
5G45I4 5D4
1G45G4
5I43P0
3P03P0
7F03P1
3P15P1
5F13P1
7F13P2–3F2
3P25P2
3F25F2
3P27P2
3F27F2
7H23F3
5P33F3
5F35H3
7P33F3
7F37H3
3F4–3H43F4
5F43H4
5H47P4
3F47F4
3H47H4
3H55F5
3H55H5
7F53H5
7H53S1 3D1
5D15D1
T = 2 5D03S1 3D1
5D13S1 3D1
7D15S2 3D2
5D23D2
7D23D3
5D37S3 3D3
7D35D4
7D43P03P1
5P11P1
5P13P2
5P25P2
5P35P3
The resulting ansatz is obliged to be phase-equivalent to VMNN at a certain well-de>ned energy E0,
that is
VNN ≡ VMNN − VN�G�(E = E0)V�N : (3.99)
This procedure is clearly reminiscent of what in often done to represent a local interaction through a>nite-rank potential. Its practical applicability strongly depends upon the sensitivity of the approxi-mation when one moves away from the energy E0. It has been used together with the Paris potential,by the Hannover group, to develop their force model for the N� interaction [142–144].
A rather elaborate approach to the CC problem, with few- and many-body applications in mind,has been developed by Haidenbauer et al. [145]. Owing to retardation eEects in the exchange ofmesons, interactions derived from a covariant meson–baryon >eld theory are intrinsically non-localand energy-dependent, unless severe approximations are introduced. Whereas non-locality does notrepresent a big problem, especially in momentum space, energy dependence can make many-body
calculations quite cumbersome. However, the energy dependence can be systematically removedby resorting to folded-diagram techniques. This method, originally developed in the framework ofnuclear shell-model theory [146], has been adapted to the NN problem by Johnson many yearsago [147], to derive instantaneous, energy-independent potentials from energy-dependent exchangeforces. The basic idea consists in requiring the matrix elements of a meson-baryon interaction inthe state space of mesons, nucleons and �’s to be the same as the matrix elements of an eEectivepotential in the restricted space of nucleons and �’s only. This is accomplished order by orderin the time-ordered perturbative expansion of the time-evolution operator. As a consequence, thismethod can be naturally applied to the Bonn interactions, which are obtained through non-covariant,Bloch–Horowitz perturbation theory. Technically, one integrates over all the time variables in theoriginal time-ordered Feynmann graphs, with the exception of the time at which the resulting eEectivepotential acts.
This formalism has been applied in [145] to nucleons and � isobars interacting through theexchange of �; �; �′; ! and � mesons. Both NN ↔ N�, NN ↔ ��, and N� ↔ �� oE-diagonalone-meson exchange transition potentials have been considered. As for the diagonal potentials, thedirect one-meson-exchange interaction V (N� → N�) has been taken into account, whereas theexchange contribution V (N� → �N ) has been ignored. The diagonal �� interaction has been omittedaltogether. Finally, for the nucleon–nucleon potential, uncorrelated two-meson exchange graphs, withbox and crossed contributions, were included. With the above approximations the CC equations(3.95) were solved for the deuteron and NN scattering, thus >tting the meson–nucleon couplingconstants and cut-oE masses, >nding minor deviations with respect to the parameters of the full Bonnpotential [148]. Thanks to a more satisfactory treatment of the N� threshold, the coupled-channelmodel provides an overall improvement in reproducing the low-energy phase shifts, with respect tothe single-channel calculation.
The Argonne group [149] starts from a bare phenomenological NN interaction, the Urbana model[150], and adds extra N� and �� transition operators, with a consistent >tting of low-energytwo-nucleon observables. The transition potentials that involve the isobar have long range OPEPterms, and the short and intermediate regions representing the exchange of the � and ! mesons aredescribed semi-phenomenologically by a combination of generalized spin and tensor operators and aWoods–Saxon shape at short distances. Beside the �N� and ��� coupling constants it has no morefree parameters then the ones already present in the v14 model. There is a central repulsion in theN� and �� channels and some of the box-diagrams generated by two pion exchange process werealso approximately included. In the N� and �� sectors, the intermediate and short range regions arerepresented only by central operators, that produce a repulsive core comparable in size to the onesin the NN channels. The Argonne v28 potential is then a more complete interaction, easy to use incon>guration space, however � degrees of freedom are taken into account in a very phenomenolog-ical way. Possible improvements require theoretical guidance, since NN data show little sensitivityto � excitations. It was used in three-nucleon [151] and in nuclear matter [5,152] calculations, andstrong repulsive eEects were found, as it will be discussed in the following sections.
3.4. Relativistic and unitarity corrections in the N� coupled-channel approach
With increasing energy, it is expected that relativistic eEects may play a more and more importantrole in the description of the coupled NN–N�–�� system. Relativistic requirements can be taken
into account in a minimal way through relativistic propagation. As we shall see later, however, aconsistent relativistic treatment requires the solution of coupled Bethe–Salpeter equations. At thesame time, the opening of inelastic thresholds would require a more satisfactory treatment of uni-tarity requirements than the one oEered by two-body-type formalisms. This is particularly true for� production processes, which proceed mainly through intermediate � formation and dominate inthe energy region between the threshold (� 300 MeV) and 1 GeV in the laboratory system. Greattheoretical activity has been devoted in the 1970s and 1980s to the development of three-body uni-tary models for the NN–NN� system, in which the � degrees of freedom could be accommodatedin a natural way. All these approaches aim at an extension of the non-relativistic Faddeev theoryto the relativistic domain, so as to include eEects due to pion emission or absorption. This is nota trivial task, as can be easily appreciated taking into account that the Faddeev equations can beregarded as a clever resummation of the multiple-scattering series, which requires deep modi>cationswhen the number of particles can change [153]. Theories with pion emission=absorption satisfyingthree-body unitarity have been developed in the past along several diEerent routes. The NN and�NN space can be allowed to communicate through formation and decay of the � particle, which isintroduced in the formalism from the very beginning [144]. Alternatively, one can employ diagram-matic or projection-operator techniques to derive Faddeev-like equations, coupling all the scattering,production and absorption processes one can have in the coupled NN–�NN system [154–156]. The� isobar is then generated dynamically as a consequence of the �N interaction. At the end, onearrives at coupled equations very similar in structure in all approaches. As for the relativistic correc-tions, unitary three-body formalisms have been developed by means of quasi-potential techniques,which yield covariant, three-dimensional equations strictly resembling the Lippmann–Schwinger andFaddeev ones of non-relativistic scattering theory, at the price of some restrictions on the treatmentof the particle propagation in the intermediate states [157,158]. These formalisms, together withphenomenological two-body interactions designed to reproduce two-body observables, have beenemployed to study pion production in NN scattering, NN ↔ �d processes, and �d elastic scatteringwith allowance for the coupling between the various reaction channels. It is really impossible to dofull justice here to this enormous amount of work. It has been reviewed in the book by Garcilazoand Mizutani, where all the relevant references as well as a detailed comparison between modelsand experimental data can be found [159]. Here, we limit ourselves to a brief discussion of theapproach pursued by Tjon and collaborators [160,161], which, being based upon the Bethe–Salpeterequation (BSE), is in our opinion very close in spirit to the papers considered in the previoussections.
As is well-known, the BSE is a four-dimensional, integral equation for the transition amplitude,or the wave function, whose kernel contains the contributions from all two-particle irreduciblediagrams, the ones which cannot be divided into disjoint pieces by drawing a line which cuts noboson line, and each fermion line only once. The exchange Born graph and the crossed fourth-orderdiagram are noticeable examples. As a consequence, the BSE sums up all the two-particle irreduciblediagrams one can have in the considered >eld-theory [162]. In spite of its apparent similarity to thenon-relativistic Lippmann–Schwinger equation, its solution represents a challenging problem. Theconstruction of the driving term is in itself as diScult as solving the full scattering problem. As amatter of fact, one generally limits oneself to the ladder approximation, retaining the Born exchangegraph only in the driving term.
Fig. 18. Diagrammatic representation of the BS equation for coupled-channel NN–N� scattering. The dashed lines representthe exchange of various mesons.
The ladder BS equation has been applied by Tjon to the coupled N� system,
Ti1(p0p; 0k) = Vi1(p0p; 0k)− i4�3
∫d4q
3∑j=1
Vij(p0p; q0q)G0j(q0q)Tj1(q0q; 0k) ; (3.100)
where the incoming and outgoing four momenta are k ≡ (k1− k2)=2= (0; k) and p ≡ (p1−p2)=2=(p0; p), respectively. The former is assumed to be on shell, whereas q ≡ (q1 − q2)=2 = (q0; q)represents the intermediate relative four-momentum as seen in Fig. 18. The indices i; j=1; 2; 3 labelthe NN; N�, and �� channels, respectively, and the two-body propagators G0j are given in termsof factors of the form
G0(q0q) =1
(E + q0)2 − q2 −M 2 + i01
(E − q0)2 − q2 −M 2 + i0; (3.101)
times projection operators for Dirac (N ) or Rarita–Schwinger (�) spinors. The energies and massesappearing in (3.101) correspond to the particles propagating in the speci>c channels. To take intoaccount the onset of pion production processes, in the early calculations the � mass has been givena negative imaginary part, with the requirement that the latter vanishes below the pion productionthreshold [160]. One is thus operating in a two-body framework, three-body unitarity being takeninto account only in an eEective way. It may be expected then, and it has been actually veri>ed,that this may lead to problems near the production threshold.
A rather re>ned treatment of the driving forces characterizes this approach. The NN interaction V11
is described through one-boson-exchange of �, �, #, 0, and � mesons, the �NN and �NN couplingsbeing given by the relativistic Lagrangians (3.1) and (3.13), respectively. The transition interactionV12 ≡ VN� and its inverse are given by one-boson-exchange graphs. The �N� vertex is derived fromthe interaction Lagrangian (3.7), while the eEective Lagrangian
L�N� = if�N�
m�
X I+F+5T
+ · (9IF − 9FI) + h:c : (3.102)
is employed for the �N� vertex.The calculations show that � exchange in the N� coupling reduces the strongly attractive
pionic interaction by contributing a short-range repulsive term, con>rming what comes out fromnon-relativistic coupled-channel calculations [4]. In particular, the resonance-like structures in the 1D2
and 3F3 waves, which in the past have given origin to many speculations about possible dibaryonresonances, turn out weaker than one could obtain without �-exchange contributions. The eEectsof �� intermediate states in the T = 1 NN channels turn out to be small, and the inelasticitiesare dominated by N� coupling. Pion production from �� states is also small. A remarkable pointis the strong sensitivity of the inelastic parameters to the treatment of the � propagator, whichin non-relativistic approaches is always treated in a more or less approximate, phenomenologicalway.
At a >rst stage, the momentum dependence of the � width has been introduced in an eEectiveway. Calculations clearly exhibit the consequences of such an approximation. For a >xed complex� mass one gets a systematic overprediction of inelasticity for the peripheral waves with a rathersteep onset of inelasticity near the pion production threshold. To overcome these limitations, the BSEmodel has been extended to comply with three-body unitarity by dressing the nucleon and isobarpropagators [161] via the Dyson equations
GN = G0 + G0.NGN ;
G� = G�0 + G�
0.�G� ; (3.103)
which are solved in the lowest-order approximation for the self-energy ., identi>ed with the bubblediagram.
For NN scattering, the >eld-theoretic treatment of the � propagator considerably improves thethreshold behavior of the inelasticities, with a much more smooth behavior at threshold. It turns out,however, that propagator dressing Eq. (3.103) in the NN sector introduces considerable attractionin the low partial waves, with a detrimental eEect on the overall >t of the phase shifts. This mightindicate that a more exhaustive treatment of coupling to other channels, like for example the �d one,is required. As is well-known, channel-coupling eEects have been one of the hot topics in unitarythree-body models for the NN–�NN system [159].
3.5. QCD-inspired models
We have seen that the meson-exchange picture can at most provide semi-phenomenological mod-els of the N� interaction, where coupling strengths and cut-oE parameters have to be regarded asquantities to be determined >tting the experimental observables. This task is made even more diScultby the unstable nature of the �, which has up to now hampered the attempts at getting a detailedknowledge of the N� interaction. In particular, the short-range part of the baryon–baryon potentialremains largely elusive, being more strictly related to the internal structure of the strongly interactingparticles, and requires detailed analysis of reaction processes other than NN scattering. A noteworthyexample is provided by Ref. [141], where total and diEerential cross sections, and the tensor ana-lyzing power for �d elastic scattering in the resonance region have been analyzed, looking at eEectsdue to N� rescattering in the intermediate state. The corresponding contributions have been added tothe amplitudes obtained through a Faddeev calculation for the �NN system, and the N� scatteringamplitude has been parameterized so as to respect unitarity bounds. It turned out that calculationscould be brought in good agreement with the experimental data only if short-range interactionswere included in the 5S2 and 5P3 N� channels. Clearly, a detailed mapping of the N� interactionthrough this route would require much more analysis and accurate experimental data than presently
available. These problems would be overcome if the N� interaction could be derived from theunderlying QCD Lagrangian, thereby relating its structure to the fundamental quark and gluondegrees of freedom. This is not yet possible in the highly non-perturbative regime relevant toNuclear Physics, and one has to resort to models, which take into account in an approximate waythe salient non-perturbative features of QCD, i.e. con>nement and spontaneous breaking of chiralsymmetry (CSB).
In QCD gluons, as gauge bosons associated to a non-abelian symmetry, carry a color charge andcan couple to one another; as a consequence, the quark–quark force becomes extremely strong atlow energies and prevents colored objects from propagating far away from the interaction region.In quark models, this long-range phenomenon is generally taken into account introducing a linearor quadratic con>ning quark–quark potential. Spontaneous CSB can explain the large diEerence invalue between current quark masses appearing in the QCD Lagrangian, of the order of a few MeV,and the constituent quark masses of quark models, of the order of 1
3 of the nucleon mass M . Indeed,if the current masses were rigorously zero, the QCD Lagrangian would be chiral invariant. Chiralsymmetry, however, is broken by the QCD vacuum because of non-perturbative, short-range eEects.The consequences of spontaneous CSB are twofold. Quarks acquire a dynamical mass, which can beas large as M=3 at low energies and low momenta; at the same time, Goldstone chiral >elds appear,which couple directly to quarks. In some hybrid quark models these Goldstone bosons are identi>edwith meson >elds, and phenomenological meson-quark–quark couplings are introduced. Needless tosay, these meson >elds are regarded as elementary, and no eEort is made to a description in termsof quark and gluon degrees of freedom.
A hybrid quark model of the type outlined above has been proposed by the TUubingen–Salamancagroup some years ago [163], and subsequently used to derive the N� interaction from a ResonatingGroup Model (RGM) of the two-baryon system [164]. Quarks are coupled to a pseudoscalar chiral>eld, which is identi>ed with the pion in order to be consistent with the well-established picture ofthe long-range part of the NN force. In addition, a scalar Goldstone boson is introduced, reminis-cent of the � boson of phenomenological OBE models. The former leads to a one-meson-exchangeinter-quark potential with the usual spin–isospin and tensor components, the latter to a central ex-change force of the Yukawa type [163]. These interactions are supplemented with a quadratic con->ning potential. Perturbative QCD eEects are, as usual, taken into account through the De R[ujula–Georgi–Glashow OGE interaction [165]. In standard quark models of hadrons OGE is advocated toaccount for the hyper>ne mass splitting within the same multiplet. Here, it concurs, together withthe OPE quark–quark potential, to >x the �–N mass diEerence to its experimental value. In thismodel hadrons are described as colorless clusters of three quarks, and for each state of total spin Sand isospin T the two-baryon wave function is given by the RGM ansatz
where A is the total antisymmetrization operator among the quark degrees of freedom, R is therelative coordinate between the two quarks, and the immaterial total center-of-mass wave function hasbeen omitted. In principle, C(R) ought to be determined solving the Hill–Wheeler RGM equations.In Ref. [163], however, the N� potential has been simply evaluated in the Born–Oppenheimerapproximation, evaluating the total potential energy of the six-quark system between two-clusterstates. One then de>nes the N� interaction as the diEerence between this expectation value for
a given R and its value for R → ∞. In so doing, the � is treated as a stable particle, whose propertiesare modi>ed by a nearby nucleon, because the two baryons can exchange constituent quarks andvirtual bosons between each other. The coupling to the �N continuum has to be taken into accountseparately, through the resonance width, in a way appropriate to any particular problem.
Once the model parameters have been >xed >tting the NN bound-state and scattering observables,and the N� mass diEerence, the N� interaction is univocally determined. It exhibits a short-rangebehavior diEerent from what is predicted by meson-exchange models, with a hard core in severalpartial waves. This can be explained by Pauli correlation eEects between the constituent quarks,much in the same way as one predicts a short-range core in the eEective <–< potential from RGMcalculations. In the region of strong overlap, some relative-motion states are strongly inhibited bythe Pauli principle, and the relative wave function C(R) vanishes near the origin, the position of theinnermost node being almost energy independent. This can be simulated by a strongly repulsive corein the cluster–cluster potential.
An ambitious goal of this model, and as a matter of fact a consistency prescription, is to repro-duce both the NN data and the baryon spectrum with the same set of parameters. In Ref. [166]a reasonable description of the low energy nucleon and � spectrum has been obtained solving thethree-quark SchrUodinger equation through a truncated hyperspherical harmonic expansion. Recently,these conclusions have been questioned on the ground of rigorous Faddeev or accurate variationalcalculations for the three-quark problem [167]. Thus, with the same parameters as in Ref. [166], theN� mass splitting moved from 300 MeV up to 2 GeV. According to Ref. [167] the problem stemsfrom the interplay between the OBE and the OGE quark–quark potential. Once CSB has been takeninto account through eEective chiral bosons, it seems that no much room is left for the usual pertur-bative OGE interaction, in order not to waste the agreement with the experimental baryon spectrum[168]. Whatsoever the right choice may be, weak versus strong OGE interaction in the quark-modelHamiltonian, a detailed comparison between N� model interactions based upon diEerent constituentmodels and experimental data is surely required. As stressed above, this is a diScult task, since itrequires indirect and model dependent analysis of the experimental observables.
Chiral symmetry and its spontaneous breaking represent a basic phenomenon in QCD. It can beexploited to derive eEective Lagrangians which retain this symmetry, and are written in terms ofasymptotically observable hadron >elds. The implementation of this program led to Chiral Pertur-bation Theory, where external momenta low with respect to the chiral scale NC ∼ 1 GeV, and theinverse of the nucleon mass are used as small expansion parameters. These techniques have beenalready used successfully in the low-energy �N sector [169], and extended to deal with the NNproblem [170–172]. When going at higher energies, problems arise because of the appearance ofnucleon resonances. This is particularly true for the �, owing to its strong coupling to the �Nsystem, and the small mass diEerence with respect to the nucleon. Recently, new formulations ofeEective chiral >eld-theory have been developed, which explicitly include the � resonance as a(3=2; 3=2) >eld, and treat the N� mass splitting as a small expansion parameter [173,174]. Presently,these formalism are used to study the chiral dynamics of the �N system in the resonanceregion. It would be outside the scope of this paper to consider in detail these developments;we limit ourselves to observe that they give indications that the experimental �N phase shiftscan be reasonably reproduced with an eEective �N� coupling constant fairly close to thequark-model value. The extension of this ambitious program to the N� system would certainly bea very interesting goal.
4.1. Light nuclei: the percentage of � in the nuclear wave function
A natural question at this point is how one can “see” the eEects of the � in >nite nuclei.Few-nucleon systems (A6 4) are very good candidates to look for these eEects, since in this caseone can resort to microscopic calculations, with a minimum of model approximations. One can de-termine, for instance, the content of � in nuclei by analyzing the weight of the � components in thewave function, with respect to the nucleonic ones. This corresponds to evaluating � percentage P�.The simplest case is represented by the deuteron, even if isospin conservation excludes N� com-
ponents, and one has to look for the possible presence of �� con>gurations. Pioneering calculationshave been performed at the beginning of the 1970s by ArenhUovel and collaborators [175], usingstatic transition potentials with �-exchange only. The isobar con>gurations were generated throughan impulse approximation of the form
?N� � G�V�N?NN : (4.1)
A �-percentage P� ∼ 1% was found. These results, however, can be considered at most qualitative,because of the impulse approximation, and of their strong dependence on the cut-oE required toregularize the �-exchange transition potential at the origin. A major improvement was given by theintroduction of �-exchange [112], and by the advent of CC calculations [176–178]. The overallinteraction due to � and � exchange is much weaker at short distances, leading to a mild cut-oEdependence, whereas CC calculations, summing up the NN–�� couplings to all orders, oEer morereliable results. A � percentage around 0.8% was found. A modern example of CC approach is themodel of Dymarz and Khanna [179–183], where a non-relativistic version of the Bonn potential hasbeen employed for the NN interaction, supplemented with static transition potentials to � channelsof the form (3.10), (3.11). The meson–baryon coupling strengths were chosen according to quarkand=or strong-coupling models, and the full interaction was re>tted to the two-body observables.A probability of �� component P� ∼ 0:4% has been obtained in deuteron wave function.
Three-body systems, 3He and 3H, oEer the possibility of exploring both single- and multiple-�components. The embedding of the coupled-channel NN–N� system in few-body calculations clearlyimplies highly non trivial problems. For A=3 nuclei this was achieved by the Hannover group througha generalization of the Faddeev approach, where baryons can exist in two spin and isospin states,corresponding to the nucleon and the isobar excitation [142,184]. For phenomenological applications,they modi>ed the Paris interaction and described the NN → N� transitions via � and � exchange[142,143,185]. The probability for a con>guration with a single � in the nuclear wave function wasfound about 2.5%, with an underbinding of about 0:8 MeV for the triton binding energy. Thesecalculations, originally restricted to one-� con>gurations only, have been subsequently extendedso as to include �� and ��� components [151]. It was found that the repulsive dispersive ef-fects and the total �-induced three-body attractive contributions to the triton binding energy almostcompletely cancel. Since the Hannover NN -force model was found to be somewhat defective inreproducing the two-body data, similar calculations have been performed with the Argonne v28 inter-action [151]. Again dispersive and � three-body eEects were found to cancel to a large extent. Themain diEerence between the Hannover and v28 potentials is a reduced three-body force contribution.The two calculations predict diEerent �� probabilities; the Hannover model gives P� ∼ 4%, whereas
the v28 produces a more comfortable value P� ∼ 1%. Finally, ��� eEects were always found tobe negligible. It would be interesting to compare these results with N� potentials derived from anunderlying >eld theory, like the ones of Ref. [145]. To the best of our knowledge, this has not beenattempted up to now.
The generalization of the Faddeev formalism to A¿ 3 nuclei is extremely diScult. A muchmore ?exible approach is represented by the variational method, which has met great successesin the last several years in describing the properties of nuclei up to A = 8 starting from realisticphenomenological potentials [186,187]. As is well known, a suitable parameterized trial function isused to calculate an upper bound to the energy which is minimized, and the lowest value taken asthe approximate ground-state energy. The quality of the calculation depends on the starting function,therefore it is useful to consider trial functions with reasonable correlations built in. It is expectedthat the correlations brought by the couplings are re?ected in the wave function, therefore the sameoperators appearing in the NN interaction are used to construct the correlated state. In principle, ifthe NN interaction includes all transitions involving � excitation, like for example the v28 potential[149], this procedure would generate a correlated wave function with � components. This is howeverextremely diScult to achieve in practice, due to the very large number of channels and parameterswhich should be adjusted variationally for each nucleus. An alternative procedure is to use transitioncorrelations which describe two-body bound-state and scattering wave functions, for an interactionwith � degrees of freedom, and assume that these correlations are relatively A independent, sincethey appear to be short-ranged [188]. As a consequence, the NN components of the correlated wavefunction are required to be proportional to the projected NN channels of the full two-body wavefunction, calculated with an interaction with � transitions. The N� and �� channels are obtainedfrom the action of transition correlation operators that simulate the � transitions in the assumedtwo-body interaction, so that one writes [188]
? =
[S∏i¡j
(1 + UTRij )
]?N ; (4.2)
where ?N contains only nucleonic degrees of freedom, S is a symmetrizer taking into account thenon-commutative nature of the intervening operators, and the pair transition correlation operator UTR
ijcontain N�, �N and �� components. In practice, they are constructed with the spin and isospinPauli and transition operators �i, Si, �i, and Ti, times radial functions to be determined by >ttingthe solution of the CC NN–N� problem. This approach is therefore intrinsically non-perturbative.To grasp what a perturbative treatment of the isobar would be in this context, let us go back toEqs. (3.92), and solve it with respect to ?N�. Disregarding the �� interaction, one immediately getsup to lowest-order in V�N Eq. (4.1). The total wave function is therefore in this approximation
? � (1 + G�V�N )?NN : (4.3)
If the kinetic-energy terms are disregarded in the free Green function G�, one sees that the >rst-orderperturbation theory result (4.3) is equivalent to transition correlation operators UTR of the form
UTR � V�N
M −M�; (4.4)
namely, UTR is simply given by the N� transition potentials scaled by the inverse of the N–� massdiEerence.
Correlated wave functions of the type (4.2) have been used to calculate � contributions to elec-troweak observables and radiative and weak capture cross sections at very low energy [188,189].The respective transition matrix elements between initial and >nal states are calculated expanding thewave functions into a pure nucleonic part ?N plus corrections containing the � excitations generatedby Uij, i.e.
? =?N +∑i¡j
Uij?N + · · · : (4.5)
We shall go back to this topic in the next section.
4.2. Mesonic-exchange currents and �
To explore the structure of the nuclear wave function, a privileged source of information is givenby the coupling with an external electromagnetic probe. Below threshold for meson production onecan explain many aspects of nuclear physics in terms of meson-exchange nuclear forces. Sincethe interaction in presence of an external electromagnetic >eld can be determined according tothe minimal electromagnetic coupling ∇ → ∇ − (ie=c)A, one gets speci>c forms for the mesonexchange currents (MEC), in terms of the same operators responsible for the NN interaction [11]These currents add to the electric and magnetic convection currents in the nucleus, and can explainpart of the discrepancy between experiments and calculations for magnetic moments, form factors andradiative capture in few-body systems. In deuteron electrodisintegration by backscattered electrons,in particular, they provide the largest part of the total cross section [8,187]. It is impossible hereto account for the enormous amount of work done in this >eld, and we will refer the reader to theexcellent review papers in the literature [11,187], limiting ourselves to quote few relevant results tothe present topic.
The calculation of magnetic moments, form factors and radiative capture cross sections in few-bodysystems requires the evaluation of matrix elements of the current operator j between the proper initialand >nal states. The current can be expressed as the sum of one-, two- and many-body terms, thatoperate on the nucleon and � degrees of freedom,
j= j(1) + j(2) + · · · : (4.6)
In the impulse approximation (IA) only the one-body contributions are kept.Current conservation implies a relation between j and the nuclear Hamiltonian H = T + VNN
expressed by the continuity equation
∇ · j+ i[H; �] = 0 (4.7)
where � is the charge-density operator. Eq. (4.7) can be immediately separated into continuityequations for the one- and two-body components,
∇ · j(1) + i[T; �] = 0 ; (4.8)
∇ · j(2) + i[VNN ; �] = 0 ; (4.9)
involving the kinetic energy T and the nucleon–nucleon potential VNN , respectively. From symme-try requirements and conservation laws, the currents may be written in terms of longitudinal and
Fig. 19. Feynmann representation of hadronic currents associated to diEerent meson exchanges: one-body currents withoutand with � excitation (a), (b); one-body � current (c); two-body seagull components with � and � exchange (d), (e);mesonic exchange contributions (f), (g); the transverse ��+ and !�+ components (h), (i); pair graphs with � and !exchange (j), (k). The wavy lines represent the photon.
transverse operators [190]. Eqs. (4.8) and (4.9) constrain only the longitudinal part of the currents.In particular, the electric form factors have to be the same in the one- and two-body currents, aswell as in the charge-density operator �. A further constraint is imposed by Eq. (4.9), since thelongitudinal two-body current has to be consistent not only with � but also with the assumed NNinteraction. In the literature this consistency is normally achieved in two ways; one either uses thesame type of meson–baryon vertices in j(2) and VNN [179–183], or one phenomenologically modi>esthe two-body currents so as to be consistent with a phenomenological VNN [11]. This constrainedpart of the two-body current can therefore be considered model-independent, since it contains thesame meson exchanges terms, and no extra parameters than the ones already present in the NNinteraction.
The currents contain contributions due not only to nucleons, but to isobar excitations as well,
The longitudinal and transverse components of the one-body part j(1) arise from direct couplingsof nucleons or �’s, respectively [191], with the probing electromagnetic >eld, as shown inFig. 19(a)–(c), and are expressed in terms of the free baryon electric and magnetic form factors,which take into account the >nite size of the baryon.
For the two-body currents j(2) possible processes are the seagull diagrams shown in Fig. 19(d) and(e) where the probing photon is attached to a vertex of a �- or �-exchange graph, and contributions
Fig. 20. Exchange currents involving excitation of an intermediate �. The dashed lines represent the exchange of � or �mesons.
Table 5Magnetic moment of the deuteron from various calculations: (1) and (2), Ref. [181]; Bonn, Ref. [103]; Paris, Ref. [192];Argonne v14, Ref. [193]; Argonne v18, Ref. [202]
where the photon directly couples to the exchanged meson, as in Fig. 19(f) and (g). In addition,there are purely transverse contributions, which cannot be related to the assumed interaction throughthe continuity equations. Examples of these model-dependent terms are the ��+ and !�+ graphsdepicted in Fig. 19(h) and (i). Other transverse terms are given by the � pair graph of Fig. 19(j),usually combined with the � seagull term, and the ! pair diagram, which is the only ! contribution,since this isoscalar meson cannot directly couple to the electromagnetic >eld.
If the NN interaction does not explicitly contain � transitions, two-body currents involving theisobar will also fall in the model-dependent category. The exchange current operators containing �can be divided into two classes shown in Fig. 20. In graphs 20(a) and (b), the nucleon resonanceis excited by the probing electromagnetic >eld. In the second type 20(c) and (d) the resonance isalready present in the nuclear wave function and couples elastically to the external photon. The lattercontributions are model-independent, once the � excitation and two-meson exchange are explicitlyintroduced in the nuclear force model.
The determination of nuclear magnetic moments amounts essentially to the evaluation of the matrixelement of the current j between initial and >nal states.The >rst column of Table 5 reports the deuteron magnetic moment calculated in impulse approx-
imation for several realistic NN interactions. It is immediately perceived that all these calculationscannot account for the experimental result, and part of the diEerence among them is due to thediEerence in the D-state percentage. The second column shows the non-negligible eEect of the MECquoted above. The explicit inclusion of the � through a coupled-channel calculation gives the >guresreported in the >rst two rows [181], using a Bonn-type NN interaction, a static �� potential re>t-ted to the two-body observables, and the meson–baryon coupling strengths given by the quark and
strong-coupling model, respectively. In the calculation with the Argonne v14 interaction [193] theexchange currents involve the intermediate excitation of �, which was found to give a very smallcontribution, due to cancellations between the �- and �-exchange terms that involve the isobar. Thecoupled-channel calculations of Ref. [181], on the other hand, seems to indicate a non-negligibleimportance of the �� state in the deuteron, which amounts in this model to 0.36%. In fact, thiscomponent in the deuteron form factor accounts to a large extent for the shift of the diEraction min-imum observed in the Rosenbluth structure function for elastic electron–deuteron scattering [183].Similarly, the isobaric degrees of freedom seem to play a signi>cant role at large momentum trans-fers in 2H (e; e′)pn reaction at backward angles [182]. The calculations employing the v14 potentialand based on a conserved current, on the other hand, obtain a fairly good agreement with the ex-perimental data without explicit isobar components in the wave function [193]. One must observe,in any case, that no de>nite conclusion about the � role in electromagnetic currents can be drawn,until more detailed information about the meson–baryon interaction is available.
The considerations made above have all been developed considering non-relativistic current oper-ators. The inclusion of relativistic eEects like retardation eEects, pair N XN contributions, and others,has been the subject of several investigations in the literature [194–199]. A consistent calcula-tion of observables requires relativistic equations, like the Bethe–Salpeter one, to obtain the wavefunction. Calculations along these lines have either included meson-exchange contributions in somequasi-potential approximation [197], or have been limited to the nucleon-propagator singularities inequal-time [198] or ladder approximation [199]. A good description of the experimental data forthe Fd has been obtained [199], with less satisfactory results for the quadrupole moment, whereprobably additional eEects including the isobar have to be taken into account. The introduction ofthese contributions would imply a relativistic coupled-channel framework, such as the one developedby Tjon and co-workers [160,161], plus a covariant description of exchange currents. This seemsstill to be done.
The static properties and the charge and magnetic form factors for 3He and 3H have been calcu-lated with Faddeev techniques by the Hannover group [185]. Both one- and two-body currents areobtained as non-relativistic limits of the standard Feynmann amplitudes, with explicit � contributions,and satisfy the continuity equation only approximately. The good agreement with the experimentalelectromagnetic form factors is obtained mainly from the inclusion of selected relativistic correc-tions, and the use of the Dirac nucleon form factor in place of the Sachs form factor. Subsequentcalculations based upon a slightly diEerent force model have given rather small eEects of the �contributions to the electromagnetic properties of three-nucleon systems [200].
The in?uence of the isobar upon the electroweak properties of very light nuclei has been studiedin detail by means of the variational method. Indeed, within this approach the three-nucleon elec-tromagnetic properties, the Gamow–Teller matrix element in tritium >-decay, and the low-energyneutron radiative capture and proton weak capture on 3He have been calculated both without andwith � degrees of freedom [188,189]. In particular, the electromagnetic structure of trinucleons hasbeen studied [189] with the v18 two-nucleon [202] and Urbana IX [203] three-nucleon interaction;the latter consists of a long-range term due to excitation of an intermediate �, plus a short-rangerepulsive phenomenological contribution, which simulates the dispersive eEects arising when inte-grating out � degrees of freedom. The isobar components in the wave function have been introducedthrough the transition-correlation-operator method described in Section 4.1. As we have seen, theseoperators correspond to transition potentials with a long-range OPE tail and a phenomenological
Fig. 21. Diagrammatic representation of operators included in the one-body (a) and two-body (b) currents in the variationalcalculations of Ref. [189]. The dashed and crossed-dashed lines represent transition-operators insertions.
Table 6Magnetic moments of three-nucleon systems: impulse approximation IA, purely nucleonic current contribution j(3N ),eEect of one-body � currents j(1)(�), full calculation and experimental result. From Ref. [189]
short-range part constrained by two-body observables. A perturbative treatment of � components inthe wave function is simpler, but it may lead to a substantial overprediction of their importancesince they tend to be too large at short distances. Model-independent two-body currents consistentwith the NN interaction have been determined via the continuity equation, supplemented with themodel-dependent contributions with explicit � excitation. The initial and >nal total wave functions?�i and ?�f in 〈?�f|j|?�i〉 determining the three-nucleon electromagnetic form factors were givenin terms of the pure nucleonic wave functions ?N by Eq. (4.2). If ?�f and ?�i are expanded accord-ing to Eqs. (4.5) one gets the contributions to the form factor exhibited in Figs. 21(a) and (b), forthe one- and two-body currents, respectively. Connected three-body terms are neglected. Accordingto this analysis, the contributions associated with � components are small, but help in bringing thecalculated magnetic moments and electromagnetic form factors for 3He and 3H in agreement withthe experimental data. In Table 6 the eEect of � excitation in the currents for trinucleon magneticmoments are presented.
The >rst column represents the result obtained in impulse approximation (IA), whereas the second(j(3N )) gives the outcome of a calculation including purely nucleonic two- and three-body currents.
In j(3N ) + j(1)(�) the contributions of one-body currents with � excitation corresponding toFig. 21(a) have been added. Finally, the result of the full calculation is reported in the last column.In the case of 3He the � currents seem to help the agreement.The low-energy radiative and weak capture cross sections on 3He have been also evaluated in
this model [188], >nding some improvement in the comparison with the data. Similar calculationshave been performed in Ref. [201], where correlated Hyperspherical Harmonic wave functions withisobar admixtures were used to evaluate cross section and polarization observables for the radiativecapture reactions 2H(n; +)3H, and 2H(p; +)3He at low energies, as well as the energy dependenceof the astrophysical S-factor. Again, an overall satisfactory agreement with the observed data wasfound.
With the advances in theoretical approaches to the nuclear few- and many-body problem, onemay expect that increasingly sophisticated microscopic calculations will clarify the intertwined roleof the � and meson-exchange currents in larger and larger nuclei. There are already indicationsthat, including the simplest exchange isobaric contributions, can give important corrections to themagnetic dipole moment in p-shell (A=4–16) nuclei [204]. There, complete 0˝! and (0+2)˝! shellmodel calculations have been performed. Since the considered mass region partially overlaps withthat accessible to modern ab initio Green-Function–Monte-Carlo or variational calculations [187],an analysis able to ascertain to what extent these results are sensitive to the employment of modelwave functions would be welcome.
The results of all these analysis always depend to some extent on the uncertainties in the N� and�� interactions, and on the parameters appearing in the meson–baryon and photon–meson vertices.Thus, any >nal conclusion may be drawn in this cautious perspective. However, a general result is forsure on the ground of modern CC or variational calculations, that the eEects due to � components inthe nuclear wave function are signi>cantly smaller than the ones obtained using perturbation theory.This is particularly true in reactions such as the radiative or weak captures on 3He at very lowenergy, where the small overlap between the main components of the bound-state wave functionstrongly quenches the nucleonic part of the one-body current operator. Another general feature ofthe considered processes is the strong sensitivity of observables to the tensor eEects in the isovectorcomponent of the two-body currents, which can substantially change for diEerent interactions.
4.3. � propagation in nuclei
4.3.1. EDective � interactionThe excitation of � represents the dominant mechanism for pion photo- and electroproduction on
nucleons [8,205], much in the same way as this resonance is the prominent feature for low- andintermediate-energy �N scattering [2,8]. It is strongly excited and propagates as a quasi-particle inthe nuclear medium, as can be clearly seen from the energy dependence of the total �- and +-nucleuscross sections illustrated in the example of Fig. 22. For photo-induced reactions, the cross sectionsre?ect the position and strength of the free � throughout the periodic table, as seen in Fig. 23.In fact, the experimental cross section peaks at the same position as the free nucleon one �(+N ).Similar considerations apply to inclusive electron–nucleus scattering, as exempli>ed in Fig. 24, wherethe (e; e′) cross section for 12C is compared with the sum of the free nucleon cross sections. Herealso one observes a substantial broadening but no shift of the � resonance peak with respect to thefree nucleon case.
Fig. 22. Pion- and photon-12C total cross sections in the resonance region. The full lines represent the free �N and +Ncross sections times the mass number A. From Ref. [8].
Fig. 23. Total photonuclear cross section per nucleon for diEerent targets in comparison with the average single nucleontotal photoabsorption cross section (dashed line). From Ref. [206].
Fig. 24. Inclusive 12C(e; e′) cross section for incident electrons of 620 MeV at � = 60◦. The short-dashed line represents
the sum of the free nucleon cross sections. The dotted and long-dashed lines give theoretical estimates of non-resonantcontributions and kinematical eEects, respectively, whereas the full line is the outcome of a full �-hole model calculation.From Ref. [207].
In �-nucleus scattering, on the other hand, one has a damping of the resonance and a markeddownward shift of its position. These features show that the � survives even in a strongly interactingenvironment, and can therefore be treated as a quasi-particle at the same level as the nucleon. Thisis the basis of the �-hole model [208–210], where the diEerences between pion and photon-induced
Fig. 25. (a) One-pion exchange �-hole interaction; (b) intermediate � → �N decay responsible for the decay width.
reactions are naturally explained in terms of the diEerent coupling of the two probes with the nuclearmedium.
As is well known, in the �-hole model one assumes that nucleons and �’s move in a mean->eld,and excitations of the nuclear medium are described in terms of �-hole states strongly coupled to thepion >eld. The nuclear Hamiltonian is the sum of a single-particle Hamiltonian for non-interactingnucleons and �’s H0, plus the �NN and �N� coupling terms not included in the mean >eld,
H = H0 + H� + H�NN + H�N� ; (4.11)
where H� is the free pion Hamiltonian.The �N� Hamiltonian couples the nucleon and � by absorption or emission of a pion as given
in Eq. (3.9). It naturally introduces an OPE �N →N� interaction (3.11), which provides thelongest-range driving mechanism of the �-hole force of Fig. 25(a), and the possibility of � → �Ndecay illustrated in Fig. 25(b). The decay width is introduced in � propagation. Owing to the inter-action of the � with the surrounding medium, its width 2� acquires a non-trivial energy dependence,and can be written in terms of the free width, plus corrections associated to elastic broadening, Pauliquenching, and coupling to absorption [8],
2�(E) = 2free� + 2el + 2Pauli + 2abs : (4.12)
The absorption cross section can amount up to one third of the total one. The overall eEect of themedium on � can be subsumed into a complex optical potential, whose real part contains bindingeEects in the mean >eld, dispersive shifts associated to absorption, and short-range correlations, whilethe imaginary component takes into account the presence of the absorptive couplings.
The detailed behavior of � in nuclei depends upon the structure of the �-hole interaction. Forthe longest-range part the latter can be determined by the � and � exchange contributions (3.11),(3.18). In the spirit of Landau–Fermi-liquid theory, the eEect of short-range correlations are describedregarding nucleons near the Fermi surface kF as quasi-particles, endowed with an eEective mass M ∗due to the surrounding cloud of N -hole excitations [211]. This leads to the residual contact �-holeinteraction [8]
For the sake of simplicity, we have considered only the �B channel. The overall residual interac-tion can be decomposed into a spin-longitudinal and a spin-transverse part taking into account theanalogue of Eq. (3.16) for S. One gets
V�′h′ ;�h(q) =f2
�N�
m2�{W (LO)(!; q)(S1 · q)(S†
2 · q)
+W (TR)(!; q)(S1 × q) · (S†2 × q)}T1 · T†
2 ; (4.14)
where q ≡ q=|q|. The longitudinal and transverse coeScients W (LO)(!; q) and W (TR)(!; q) dependupon the transferred energy and momentum as well as upon the Landau parameter g′��:
W (LO)(!; q) ≡ g′�� +q2
!2 − |q|2 − m2� + i0
; (4.15)
W (TR)(!; q) ≡ g′�� +f2
�N�
m2�
m2�
f2�N�
q2
!2 − |q|2 − m2� + i0
: (4.16)
The residual interaction (4.14) has by now become the “standard model” for the description of�-hole correlations in nuclei. In practical applications one often uses the quark or strong-couplingrelation (3.87), together with (3.60) to write (f2
�N�=m2�)× (m2
�=f2�N�) � 2.
In a coupled treatment of the N� system Eq. (4.14) has to be supplemented with a NN ↔ �Ntransition interaction. For � exchange this is provided by Eq. (3.10), whereas short-range correlationscan be subsumed into the Fermi–Landau type contact force
V corrNN↔�N =
f�NNf�N�
m2�
g′N��1 · S†2�1 · T
†2 + h:c : (4.17)
A basic question is obviously what can be said about the various Landau parameters g′NN , g′��, and
g′N�, characterizing the short-range behavior of the N - and �-hole excitations in the nuclear medium.In particular, the actual value of g′N� strongly in?uences the role of the isobar in quenching thenuclear response to an isovector spin probe, as we shall see below; larger g′N�, more quenchingeEects are attributable to isobar currents. Big eEorts have been devoted during the 1980s to thisproblem, as extensively discussed in the review paper by Towner [9], to which we refer the readerfor more details. Here, we limit ourselves to touch some essential points of this fascinating but,admittedly, intricate problem.
On the ground of quark-model and=or chiral symmetry arguments it has been proposed that theLandau parameters might satisfy the universality relation [212–214]
g′NN = g′N� = g′�� ≡ g′ : (4.18)
A simple estimate can be inferred for g′ by the minimal requirement that the short-range correlationscancel completely the �-function term in the OPE potential [8,9], which gives the well-known valueg′= 1
3 . Under a phenomenological point of view, the Landau–Migdal interaction in the N -hole sectorreproduces well the collective states in both light and heavy nuclei, where it is probed in the Landaulimit ! → 0, q= k2 − k1 → 0, with the values of the initial and >nal nucleon momenta k1; k2 ∼ pF.One >nds [215,216]
Fig. 26. Direct (a) and exchange (b) NN → �N transition potentials with particle-hole couplings, and higher-order‘induced’ contribution (c). The shaded area represents the sum of RPA particle-hole interactions.
(a) (b) (c) (d) (e) (f)
Fig. 27. Graphical representation of coupled equations to obtain the eEective particle-hole interaction with � transition.Direct (a), exchange (b), and ‘induced’ (c)–(f) contributions.
a result indicating that the short-range spin–isospin interaction is much more repulsive than onecould expect on the ground of a simple picture of non-overlapping nucleons.
On the theoretical side, several attempts have been made to evaluate the Landau parameters micro-scopically, starting from Brueckner–Bethe theory [217–220]. In these approaches one >rst evaluatethe Brueckner G-matrix; the basic contributions to the �-hole interaction are then given by the di-rect and exchange graphs shown in Fig. 26(a) and (b). It turned out that satisfactory results couldbe obtained only by including medium polarization eEects [9], described in Fig. 26(c); omissionof these contributions would otherwise lead to too much attraction in the scalar–isoscalar channeland to instability of nuclear matter with respect to small density ?uctuations. This implies that theoverall particle-hole interaction Fph has to be obtained from the Brueckner G-matrix Gph throughthe self-consistent solution of the non-linear coupled equations
Fph = Gph + FinducedFph (4.20)
as graphically depicted in Fig. 27. Calculations by the JUulich [219] and Tokyo [220] groups gavesomewhat diEerent results for g′N�. Both found, in any case, that core-polarization eEects boost theLandau parameters, with g′ �= g′N�, as shown in Table 7.
Recently, the Landau parameters g′�� and g′N� have been determined through careful analysisof exclusive charge-exchange, quasifree decay, and 2p emission reactions [221,222]. As we shalldiscuss more extensively below, the 12C(3He; t�+)12C(g:s:) process selectively probes the residualinteraction in the longitudinal channel, and in a kinematic domain where the longitudinal responsefunction attains its maximum value. One can expect, therefore, that the cross sections are particularlysensitive to variations �g′�� in the Landau parameter g′��. Indeed, the energy spectra for the above
Table 7Landau parameters g′ and g′N� obtained with diEerent models: I, bare G-matrix; II, bare G-matrix + ‘induced’ interaction;III, bare G-matrix+ induced interaction+ relativistic eEects
Model g′ g′N�
JUulich I 0.49 0.35II 0.58 0.56III 0.56 0.68
Tokyo I 0.52 0.35II 0.61 0.45
reaction exhibit an energy shift, which scales linearly with �g′�� [221]. The experimental data canbe reproduced only by a value of g′�� � 0:33, as required by minimal short-range correlations. Asfor g′N�, the quasifree decay of � and 2p emission, induced by pion absorption and charge-exchange(3He; t) reactions, have been analyzed in the framework of the �-hole model [222]. Since the couplinginteraction is minimally aEected by oE-shell ambiguities in the quasifree decay of the �, one canemploy these processes to study distortion eEects on the wave functions of the outgoing pion andnucleon. The �N → NN transition in 2p emission reactions has been described by the � + � + g′model, and found to be dominated by the Landau–Migdal term. The experimental data for both the(�+; pp) and (3He; tpp) processes could be reproduced with g′N� in the range 0.25–0.35.
4.3.2. Medium eDects in direct and charge-exchange reactionsIn the �-hole model pion- and photon-induced reactions are described in terms of excitation of
the � by the external probe, followed by propagation and decay of the resonance. The diEerencein behavior of the respective cross sections can be simply ascribed to the longitudinal nature of thepion coupling Eq. (3.9) with respect to the transverse +N� coupling,
H+N� ≡ f+N�
m�S† · (�× k)T †
z : (4.21)
The vectors k and � are the momentum and polarization vector of the impinging photon. Indeed,the doorway �-hole state excited by the photon is followed by coherent � propagation via multi-ple scattering, until the >nal photon is created and escapes from the nucleus. The spin-transverse(S† × k) · � coupling of the photon, and the longitudinal S · q coupling of the pion then combine toproduce a (q × k) · � dependence in the transition amplitude. In in>nite nuclear matter, where mo-mentum conservation implies that k must be parallel to q, one has that the coherent � propagationis prohibited. The same is true in a >nite nucleus, apart from minor corrections due to propagationin the non-forward direction. One may conclude that in elastic scattering of photons the probe onlymeasures the broadening of the � resonance due to Fermi motion and binding in the nuclear mean>eld. This situation has to be contrasted with what happens in elastic �-nucleus scattering, wherecoherence eEects are responsible for the observed shift and damping of the resonance. The diEerentbehavior of the cross section in pion- and photon-induced reactions is therefore a direct consequenceof the diEerent spin structure of the �N� and +N� couplings.
� excitations also takes place in charge-exchange reactions, where a downward shift of the res-onance peak position ∼ 70 MeV is observed in nuclear targets, with respect to the single proton
Fig. 28. Experimental diEerential cross section at �=0o for charge-exchange reactions on 12C compared with the p(p; n)�++
reaction at 800 MeV. From Ref. [223].
Fig. 29. Triton spectra seen in (3He; t) reactions at 2 GeV projectile energy for diEerent nuclei. From Ref. [224].
target [10,225]. This is exhibited in Fig. 28, where the experimental zero-degree spectra for the in-clusive p(p; n)�++ and 12C(p; n) reactions are reported. The same trend can be observed in (3He; t)charge-exchange reactions on a variety of nuclei, as shown in Fig. 29, where the triton spectra at2 GeV incident energy are plotted versus the kinetic energy T of the outgoing triton. In terms ofthe excitation energy ! ≡ E3He − T , one has again a downward shift of ∼ 70 MeV with respect tothe free p → � transition. This situation is strongly reminiscent of what is observed in pion–nucleusscattering.
A more careful analysis of the actual displacements of the resonance peak positions, revealsa systematic diEerence between proton and 3He-induced processes. Thus, in the former case theresonance is located at ! � 365 MeV for the proton target, whereas it appears at ! � 295 MeVfor A¿ 12 targets. In the latter, on the other hand, one has resonance energies of � 325 and� 255 MeV for scattering processes on protons and A¿ 12 nuclei, respectively. This apparent shiftof around 40 MeV between proton and 3He induced reactions can be simply explained in terms ofthe composite structure of the 3He projectile. In this case the probability that the triton survives thescattering process is rapidly decreasing with increasing momentum transfer, and the cross sectiongets greater contributions at low excitation energies. This explanation is con>rmed by an analysisof the dependence of 3He–t form factor upon the excitation energy ! and momentum transfer q[226,227]. For a detailed discussion of form-factor eEects for composite projectiles see Ref. [225].
Fig. 30. Experimental neutron (a) and triton (b) zero degree spectra for the reactions 12C(p; n) at E = 800 MeV and12C(3He; t) at 2 GeV, respectively. The theoretical curves for the cross section represent the longitudinal LO, transverseTR components, and the full calculation with and without particle-hole correlations of Ref. [223].
It is by now ascertained that, from the 70 MeV shift observed between proton and nuclear targets,40 MeV can be explained in terms of the Fermi motion of the nucleons and �’s in the nuclearmean >eld [228,229]. The “interesting” part of the eEect is the remaining 30 MeV, which re?ectsthe diEerent kinematic domain explored in charge-exchange reactions and photon- and pion-inducedprocesses. Indeed, in real photon scattering the involved energy and momentum transfers are relatedby !=q, whereas pion scattering implies !=
√q2 + m2
�. In charge-exchange reactions, on the otherhand, the target is probed by the virtual pion- and �-meson >elds of the projectile=ejectile system.From a kinematic point of view, these meson >elds must satisfy the energy-momentum relation!¡q, thereby exploring the response function of the target in a (!; q) region inaccessible to photo-or pion-induced reactions.
The reaction mechanism at the energies of interest can be described in the framework of distortedwave impulse approximation (DWIA), which means that one can think of an eEective projectile–nucleon target–baryon interaction transferring the energy ! and momentum q to the target. Thisexcites N - and �-hole states in the NN and N� sectors. The evolution of the nuclear system dependsthen upon the full residual interaction, which consists of the four couplings Vp′h′ ;ph, where p;p′can be either a nucleon or a �, and contains both spin-longitudinal and spin-transverse components.Consequently, charge-exchange reactions provide a mixed spin-longitudinal (LO) and spin-transverse(TR) probe, whereas pion and photon scattering only probe one of these aspects.
The interplay between the two, LO versus TR, couplings, and the decay properties of � in nucleihave been investigated in a beautiful series of papers by Udagawa, Osterfeld et al. [221–223,230].In particular, it has been shown that the downward shift in the peak position is mainly due tothe strongly attractive �-exchange interaction in the LO channel. This can be seen, for instance,in Figs. 30, where the zero-degree neutron and triton spectra are shown for the inclusive 12C(p; n)and 12C(3He; t) reactions, respectively. In the uncorrelated calculations the residual p-h interactionhas been switched oE. Clearly, p-h correlations improve the agreement with the experimental data.The failure of the model in reproducing the data in the low-energy region of both spectra andin the high-energy tail of the (p; n) cross section can be reasonably attributed to the presence ofbackground contributions. A noteworthy feature of these results is the very good agreement observedin the higher ! part of the 12C(3He; t) spectrum. At variance with the 12C(p; n) reaction, there is nobackground due to projectile excitation on the high-energy tail of the resonance for the 12C(3He; t)
Fig. 31. Angular distributions for coherent pion production in 12C(3He; t�+)12C(g:s:) at 2 GeV (a) and for elastic pionscattering on 12C at 120 MeV (b). The solid line represents the LO component, and the dashed curve the full calculationwith particle-hole excitations. Figure from Ref. [221].
process, since the probability that the excited projectile can decay into a triton plus a free pionis small.
Even more interesting is the comparison between the LO and TR responses exhibited in the >gures.For both reactions one sees that the LO cross section is strongly shifted downwards, whereas theTR one is not. The origin of this diEerent behavior can be clari>ed by looking at a multipoledecomposition of the cross sections [223]. It turns out that the contributions of unnatural-paritystates are lowered in the excitation energy by about 60 MeV with respect to those of the naturalparity excitations. Looking at the behavior of the residual interaction as a function of momentumtransfer, one has a longitudinal component with a singularity at qpole =
√!2 − m2
�, repulsive belowthe pole and attractive above it. In charge-exchange reactions the � in the LO channel is excited by�-exchange with a transferred momentum q¿qpole. The overall eEect is an attractive energy shiftfor all multipoles in the LO channel, with a consequent downward shift of the �-peak position.The above considerations can be extended to exclusive processes, for which data are accumulating
through coincidence experiments [231–233]. Of particular interest here is coherent pion productionvia 12C(3He; t�+)12C(g:s:) scattering [221]. The angular distribution for this reaction at 2 GeV isdisplayed in the upper part of Fig. 31, in comparison with �-12C elastic scattering. Clearly, the LOchannel is selectively populated by the process, which allows an experimental separation of LO andTR responses. The strict proportionality between the production and elastic cross sections shows thatthe same mechanisms are at work in the two coherent processes. Actually, the (3He; t) kinematics
forces the virtual pions to propagate in the nuclear medium along the direction of the momentumtransfer q. The eEective vertices at work in the longitudinal channel are S† · q, acting as an excita-tion operator, and S · q� as de-excitation operator, producing the outgoing pion. As in elastic pionscattering, the angular distribution must therefore scale as (qq� cos ��)2, with a peak at �� = 0◦. Inother words, the coherent pion production in the 12C(3He; t�+)12C(g:s:) reaction can be regarded asa virtual pion scattering oE the target nucleus, where the initial oE-mass-shell pion is converted,through a multiple scattering process, into an on-mass-shell particle, which is >nally emitted fromthe nucleus. As already noticed at the beginning of the section, this is in contrast with pion photo-production reactions. There, due to TR excitation operator S†×q, the diEerential cross section scalesas (qq� sin ��)2, and therefore peaks at �� ∼ 90◦, as con>rmed by the experimental data.
Information on the spin structure of nuclear correlations can be obtained from spin observables[10]. One can measure the forward polarization transfer coeScients Dzz and Dxx, for spin transferalong and perpendicular to the beam axis, respectively. They are related to the longitudinal andtransverse components of the strength function Eq. (4.34) by [234]
Dxx(0◦) =
−|SLO|2|SLO|2 + 2|STR|2 ; (4.22)
Dzz(0◦) =
|SLO|2 − 2|STR|2|SLO|2 + 2|STR|2 : (4.23)
Calculations of these observables [223] show that they are indeed sensitive to the degree of �N−1
correlations in the nuclear system, the uncorrelated results being far more structureless than the cor-related ones. It has to be observed, however, that attempts to reproduce the tensor analyzing powerin the 12C(d; 2p) reaction measured at Saturne [235] have obtained up to now only a moderatesuccess [223].
4.3.3. Quenching phenomenaThe excitation of �-hole states has been advocated as an important ingredient in explaining several
low-energy nuclear phenomena, where the value of observable quantities is quenched with respectto what one can predict on the ground of models involving nucleonic degrees of freedom only. Thebasic physical picture behind this point of view is that a nucleon can polarize the nuclear medium,much in the same way as a magnetic dipole induces spin alignment in a surrounding medium[8]. Actually one can envisage two basic polarization mechanisms. One is due to the tensor forcebetween a valence nucleon and the core, which is very similar to a dipole–dipole interaction, and isintimately related to � exchange. The other mechanism is linked to spin–isospin transitions, due tovirtual �-hole excitations. By their very nature, these quenching phenomena simultaneously involvethe dynamics of � and non-trivial many-body problems.
A classical example of this subject is given by the screening of the �-nucleon coupling in thepresence of virtual �-hole states. In close analogy to what is done in considering �-hole contributionsto the pion self-energy in nuclear matter, one can envisage the excitation and subsequent decayof �-hole states in between the pion and the nucleon, as illustrated in Fig. 32. This produces arenormalization of the eEective �-NN coupling constant by a factor
Fig. 32. Renormalization of the �NN vertex by �-hole excitations. The wiggly line represents the eEective �-hole inter-action minus one-pion exchange.
where g′� ≡ g′�� = g′N� comes from the vertex eEect described in Fig. 32 by the shaded area.The �-hole susceptibility C� is related to the �-hole contribution to the pion self-energy, and isgiven by [8]
C� =89f2
�N�
m2�
�M� −M
; (4.25)
with � the nuclear density. As is well known, the renormalization factor (4.24) has been estimated tobe of the order of 30–35% in nuclear matter [11]. It counteracts the enhancing eEects of nucleon- and�-hole excitations on the pion propagator, thereby preventing medium eEects on the pion exchangeinteraction from being too large at normal nuclear density. Signals of this quenching are ratherclearly seen in the �-nucleus forward dispersion relation, once the pole contributions from low-lyingpion-like states are lumped up into a unique, eEective coupling constant [236]. In the 1990s, mediumeEects on hadron properties, and their dependence upon the density and=or temperature of the nuclearenvironment have attracted increasing attention, in order to identify the >ngerprints of the partialrestoration of chiral symmetry, due to change of the QCD chiral condensate [237].
Other quenching eEects are observed in electromagnetic or weak transitions in nuclei. Two well-known examples are provided by the magnetic moment operator, and by the Gamow–Teller (GT )transitions in >-decay [9,12]. If one considers a valence nucleon around a closed-shell core in theindependent-particle model, the nuclear magnetic moment is the one of the valence particle, namely
� = gll + 12gs� ; (4.26)
with l the orbital angular momentum and gl, gs the orbital and spin g-factors, respectively. Becauseof the interaction of the valence nucleon with the core, however, the magnetic moment operator hasto be interpreted as an eEective one, which implies that Eq. (4.26) must be replaced by [9,238]
�eE = geEl l + 12g
eEs � +
12gp[Y2�][1] ; (4.27)
where the last term represents an induced one-body tensor operator, obviously absent in the baremagnetic moment. The eEective orbital and spin g-factors are renormalized with respect to the bareones, i.e. geEl = gl + �gl, geEs = gs + �gs.
Similarly, the one-body operator for GT transitions in the non-relativistic limit is simply given by
for the (p; n) reaction on various targets, and incoming protons of 200 MeV.From Ref. [10].
where gA is the axial-vector coupling constant, and the ± signs refer to >∓ decays. For a freenucleon one has gA � 1:26. The observed transition strengths in closed-shell-plus-one, or minus one,nuclei, on the other hand, can be reproduced only with an eEective value geEA of the coupling, 20%lower than the free one.
The origin of the above quenching eEects can be traced back to two major contributions. Transi-tion operators are necessarily evaluated in a truncated model space, so that they have to be regardedas eEective quantities, evaluated perturbatively in the eEective interaction expansion. A second con-tribution to the quenching arises from processes in which the external electromagnetic or weak probecouples to the nuclear system in presence of meson exchange. These are the MEC eEects alreadydiscussed in Section 4.2. They can excite a nucleon to a � state, which is subsequently de-excitedby the external probe. In the non-relativistic limit, these contributions can be re-interpreted in thelanguage of the �-hole model. Indeed, in the MEC approach one gets two-body operators O2, whichhave to be evaluated between states of the valence nucleon and core. The leading contributions toO2 can be factorized into the product of a one-body spin–isospin operator, times a meson-exchangetransition potential. One thus recovers the �-hole screening mechanism already considered inFig. 32. This mechanism may be operative both in the quenching of gA and in the renormaliza-tion of the spin g-factor. A simple estimate is again given by Eq. (4.24) [11]. In the light of theprevious discussion it is clear that a careful assessment of the role played by �-hole excitations inquenching phenomena requires the simultaneous evaluation of higher-order perturbative contributionscoming from “ordinary” nuclear-structure eEects. Second-order core polarization processes driven bythe strong tensor interaction between the valence nucleon and core have a major role in renormal-izing both gA and gs. We shall not go into the details of this topic here, referring the reader toprevious reviews [9,239], and prefer to consider in more detail quenching phenomena, as seen froma nuclear-reaction perspective.
Let us >rst consider what is observed in charge-exchange (p; n) reactions at forward angles shownin Fig. 33. A single, prominent peak is seen on top of a large, structureless background, with theexception of the even–even 40Ca nucleus. This peak can be interpreted as a collective spin–isospinoscillation, in which the excess neutrons coherently change the direction of their spins and isospins
without changing their orbital motion [240]. At the same time, shell model and Pauli principlesuggest that a collective spin–isospin excitation ought to be prohibited in a spin saturated nucleussuch as 40Ca. For heavier nuclei, neutron excess increases, a larger neutron fraction can participatein the oscillation, with a corresponding increase in the cross section, in agreement with the trend ofthe experimental data.
To understand better the connection between this phenomenology and quenching eEects, let usrecall that the energies of interest here (100¡E¡ 800 MeV) lead to a one-step, direct mechanism.The inelastic transition is given in DWIA in terms of an eEective projectile-target interaction. Forlarger energies than the Fermi energy in the target (0F ∼ 37 MeV), one can expect that the renormal-ized NN interaction in the medium approaches the free NN transition matrix [242]. This theoreticalexpectation has been implemented by Love and Franey [243,244], to determine the eEective interac-tion directly from the phase-shift analysis of NN scattering data. The eEective interaction has beenwritten as a combination of central, spin–orbit and tensor isoscalar- and isospin-dependent terms,requiring that the corresponding antisymmetrized NN Born T -matrix reproduces the free empiricalone. For high energy and very low momentum transfer, this analysis shows an eEective interactiondominated by a central scalar–isoscalar term, plus a central spin–isospin contribution with a charac-teristic (�1 · �2)(�1 · �2) dependence. On the ground of these results one can expect that the forward(p; n) cross section at high energy can be considerably simpli>ed, and that the nuclear transitioncan be described essentially in terms of spin and isospin operators. As a matter of fact, for q ∼ 0one can write [245]
d�d GT
(�= 0◦) =
M2�
kfki
N�B|T�B(q = 0)|2B+(GT ) ; (4.29)
where T�B(q = 0) is the forward transition amplitude associated to the central spin–isospin term inthe eEective interaction, N�B a distortion factor, and
B±(GT ) ≡∣∣∣∣∣∣〈f|
A∑j=1
�jB±(j)|i〉∣∣∣∣∣∣2
(4.30)
represent the GT B-values for the i → f nuclear transition, well-known from >-decay theory. Itis by now a textbook result in Nuclear Physics that the GT strengths B±, when summed over thepossible >nal states, satisfy the Ikeda sum rule [246]
.GT ≡∑f
B+(GT )−∑f
B−(GT ) = 3(N − Z) : (4.31)
This result is model independent, under the only assumption that the �� operators refer to point-likenucleons. If N�Z , B− referring to >+ decay is strongly suppressed owing to the Pauli principle,and one can simply write
.GT �∑f
B+(GT ) = 3(N − Z) : (4.32)
Eqs. (4.29) and (4.32) state that the forward (p; n) cross section for heavy nuclei, once integratedover the >nal states, directly counts the number of excess neutrons participating in the process, andallow to determine the overall GT strength. When the experimental forward (p; n) cross section is
converted into the GT sum-rule strength, however, one >nds that only about 60% of the expectedstrength is located in the excitation energy region where the major GT peaks occur [247].These results are strongly reminiscent of the quenching of the axial coupling constant mentioned
above, and prompted several physicists to look for an explanation in terms of the same screeningmechanisms. The nuclear response to the exciting probe would then be a mixture of nuclear-structurecon>guration mixings and �-hole excitations in the spin–isospin channels. Indeed, if this is the case,one might expect that
.expGT
.GT=(geEAgA
)2� 0:6 ; (4.33)
which gives for geEA a value about 20% lower than gA, in fair agreement with the experimental indi-cations. The exact role played by the various, possible mechanisms, “conventional” nuclear-structureeEects versus �-hole excitations, in quenching the GT strength has been the subject of many inves-tigations, often leading to con?icting results. On one hand, one may argue that a strong repulsiveeEective force between �- and N -hole states can remove strength from the low-lying excitationspectrum, enhancing the transitions to the high-lying �-hole states. In spite of the large energy gapbetween N - and �-hole states, ∼ 300 MeV, the N–� mass diEerence, this might be possible, due tothe large number of �-hole con>gurations not inhibited by the Pauli principle [248]. On the otherhand, ordinary con>guration mixing, where 1p–1h states couple to high-lying 2p–2h states, couldalso shift the GT strength into the energy region far beyond the GT resonance. Clearly, the relativeweight of �-hole and conventional nuclear-structure mechanisms depends in a crucial way upon theactual values of the corresponding coupling strengths.
For doubly-closed shell nuclei, the excitation energies of the low-lying GT states, and the frag-mentation of the strength between them and the GT resonance can be calculated in the frameworkof the Random Phase Approximation (RPA). The solution of the RPA equations is equivalent tosumming up to all orders processes where particle-hole states are created, annihilated, and propagatethrough the action of the residual interaction. Once the excitation energies EI and the correspondingstates I have been determined, the response of the nuclear many-body system to a probing one-body>eld F can be determined from the strength function [249]
SF(!) ≡∑I
�(!− I)|〈I|F|0〉|2 ; (4.34)
where |0〉 is the ground state. The method is not adequate to describe the width of the GT resonance,an aspect of primary importance for its connections with the quenching of the GT strength in (p; n)reactions. At high excitation energies of relevance, 2p–2h excitations can participate in damping theresonance, and may be taken into account in the framework of the second RPA (SRPA) [250], orvia two-phonon states [251]. These approaches lead to some fragmentation of the strength, with tailsin the high-energy region containing up to 30–40% of the total integrated strength, relative to thesum-rule value 3(N − Z).The results and considerations outlined above show that only a delicate balance of data analysis
and theoretical calculations can provide reliable information on the actual extent and nature of thedamping mechanism of the GT strength. From the experimental point of view, a naive procedurefor background subtraction leads to a large underestimation of the GT strength [10]. This explainswhy detailed theoretical models, employing large-basis RPA calculations and a careful treatment of
Fig. 34. Diagrammatic representation a BBG equation with � transitions. The indices <; >; + specify the channels NN , N�and ��, respectively.
the continuum, are necessary to shed some light on the role of the various quenching mechanismsin GT transitions. Several microscopic or semi-microscopic analysis of (p; n) reactions on doublyclosed-shell targets suggested that part of the strength is indeed shifted from the low-energy partof the spectrum to the high-energy tail because of the mixing of higher-order con>gurations with1p–1h states, whereas the quenching of the strength due to � isobars cannot amount to more than10–20%, in order not to destroy the agreement with the experimental data [10]. These conclusionscan be contrasted with a recent RPA calculation of � contributions for the GT strength, whichcon>rms the role of �-hole con>gurations in quenching [252].
Presently, large-scale shell model calculations can provide a reliable description of the nuclearexcitation spectrum. Great progresses in this >eld were achieved in the nineties, with full-scalecalculations for pf-shell nuclei, through the development of more eScient computational schemesfor direct disorganization [253–256], or by Monte-Carlo techniques [257–259]. These approachesmade possible to compare the outcome of 0˝! computations in the full major shell with the resultsof truncated calculations. It turned out that in general the full calculation recovers much morequenching than the truncated ones, with a reduction factor close to 2. However, to obtain agreementwith the Ikeda sum rule, even the exact results need a renormalization of the axial coupling constantgA by a factor ∼(0:77)2, quite consistent with Eq. (4.33). On the ground of these analysis, inline with previous results for the sd [260] and p shells [261], the quenching of gA appears as agenuine in-medium eEect, once shell correlations have been properly taken into account. How much� excitations are responsible for this is still to be checked.
4.4. The � in nuclear matter
The dynamics of the � resonance has been investigated in the Nuclear Matter regime, where trans-lational invariance allows single-particle plane-wave states, thereby simplifying the treatment of themany-body problem. In spite of being an “ideal” system only realized in the interior region of heavynuclei it is a natural lab to test microscopic interactions. Calculations have been generally performedwithin the non-relativistic Bethe–Brueckner–Goldstone (BBG), or relativistic Dirac–Brueckner (DB)approaches. The usual BBG theory, at the two-hole-level approximation, or DB are extended toaccommodate the coupled-channel nature of the NN–N�–�� system, and the ladder series for thereaction G-matrix is summed up with � excitations in intermediate states, propagating as “virtual”particles. The corresponding coupled-channel BBG equations are illustrated in Fig. 34, and can be
where the index < runs over the three possible baryon channels NN , N�, and ��, while ki and k ′icollectively denote the spin, isospin and momentum of the initial and >nal particles. To simplifythe notation, the summation runs over discrete variables and intermediate momentum integrations. Thetotal energy E< is simply the sum of the single-particle energies ep(k) in channel <, that include thesingle-particle potential experienced by the nucleon or the �, i.e. eN (k) = k2=2M +UN (k); e�(k) =(M� −M)+ k2=2M� +U�(k). Finally, Q<′′(k3; k4) is the Pauli operator, which prevents the nucleonsfrom propagating in occupied states of the Fermi sea, but allows the � to have any momentum.It is expressed in terms of the occupation number of the quasi-particle states.
The single-particle average potentials for the nucleon and � are de>ned in terms of the G-matrixas the Hartree–Fock contribution,
UN (k1) =∑k26kF
〈k1k2NN |G(!)|k1k2NN 〉 ; (4.36)
U�(k1) =∑k26kF
〈k1k2N�|G(!)|k1k2N�〉 ; (4.37)
where kF is the Fermi momentum, and ! = eN (k1) + eN (k2). Two possible choices can be usedconcerning the behavior of the nucleon single-particle potential above the Fermi surface. If the“gap choice” is adopted, Eqs. (4.36) and (4.37) hold only for momenta below kF, while above kFthe nucleon potential is assumed to vanish. If instead the “continuous choice”, is used, the potentialwill be continuous over the Fermi surface and the previous equations are valid for all values of k1.It has been proved that the “continuous choice” is able to include higher-order clusters in the BBGexpansion, beyond the two-body cluster contribution. For U� a similar procedure can be used, andchannels with T = 2 can also contribute.
Equations (4.35)–(4.37) have to be solved self-consistently after partial-wave expansion, andfor simplicity standard approximations, like angle-averaged Pauli operators, are used. The explicitinclusion of � has two main consequences on the BBG G-matrix. First of all, due to Pauli principlefor nucleons, some N� states will be excluded. The second modi>cation occurs because of the massincrease of the energy denominators in N� and �� channels, and can be regarded as a dispersiveeEect. Both eEects have been already considered when dealing with the �-hole model. They reducethe eEective coupling to N� and �� states, and are expected to exhibit some density dependence.Brueckner calculations with explicit � degrees of freedom have been done since the early seventies
and were quite recently reviewed [5]. We will limit ourselves to the discussion of general featuresand comments on the most up-to-date calculations. The >rst problem to face in such microscopiccalculations are the interactions. A BBG calculation needs, in principle, the interaction in all channelswith isotopic spin T =0; 1; 2 shown in Table 4. The two-body observables should be reproduced bythese forces. As discussed in Section 3 many of these channels are not tested by NN scattering, or ifthey are observed, phase shift analysis is not very sensitive to the presence of �’s. This has importantconsequences on the reliability of the conclusions one arrives at through Brueckner calculations.
Fig. 35. Equation of state obtained with the Argonne v28 potential, assuming U� = >UN , with >=0; 1; 1:2. From Ref. [5].
For example, the calculations performed in Refs. [262,263] with the most complete interaction whichhas explicit � degrees of freedom, the Argonne v28, proved that this potential does not account for� transitions in a reliable way. It is simply too phenomenological, with little theoretical guidance,as we have seen in Section 3:1:3 and gives unreasonable repulsion in some of these channels. Asa matter of fact, the repulsive core used in the N� and �� sectors aEect the interaction with thenuclear medium, while having little in?uence on the NN phase shifts [5]. The potential does notseem to have the correct form to be used in channels that do not couple to N–N scattering [262,263].This is presumably due to the simple central form of the corresponding operators. Other, spin- andisospin-dependent terms might provide additional attractive contributions and lead to more realistic� potentials. This subject is still largely to be assessed.
Since the >rst estimation by Day and Coester [264] of � contributions to nuclear binding, thereis a general agreement of the repulsive eEect produced on the saturation properties. The saturationcurve representing the equation of state (EOS) is shifted upwards a few MeV, and the correspondingsaturation density lowered in calculations with diEerent interactions, without the eEect of � mean>eld or simply considering it constant. Unfortunately, U� is far less known than UN , both under theexperimental and the theoretical point of view, but it can have an important role in changing thetrend of EOS. It was shown [262,263] that the repulsive eEects on saturation due to � excitations,can be largely counteracted by the action of a realistic attractive � potential. This is illustrated inFig. 35 where the saturation curve of nuclear matter is determined with diEerent choices for U�,ranging from a constant attractive potential, to a density and momentum-dependent form with thesame structure as UN . Features of � propagation in >nite nuclei are described within the �-holemodel with a >eld similar to the one felt by the nucleon. The calculation of U� from the v28 showeda strongly repulsive mean >eld [262,263], mainly due to the projection of the interaction in statesnot seen in NN scattering. The magnitude of U� is than strongly dependent on the model used forinteraction between nucleons and �’s.
A realistic description of � propagation in nuclei should account also for decay. Therefore, inaddition to the average binding eEects described by U�, coupling to the �N state and modi>cationsof the width due to the presence of the surrounding medium should be considered. To lowest orderthis can be accomplished through the “bubble” graph of Fig. 25, as already outlined with referenceto the �-hole model. Self-energy eEects have been studied in [265] under the assumption that theself-energy contribution .0(k�; E) corresponding to Fig. 25 has a weak momentum dependence, andessentially depends upon the � single-particle energy only. Binding eEects are included through theself-consistent nucleon potential UN (k) in the non-relativistic nucleon propagator. This approximationamounts to assuming an additional constant potential due to the modi>cation of the decay channelin the nuclear medium, and is consistent with the original description of the � as an elementaryparticle with a modi>ed mass. At low density this additional contribution vanishes. The self-energymass shift is determined self-consistently, together with Eqs. (4.35), (4.36) and (3.47), as solutionof the Dyson equation
0�(k�) =M� −M +k2�2M�
+ Re[.(k�; 0�)] ; (4.38)
where the total isobar self-energy is given by
.(k�; 0�) = .0(0; 00�) + U�(k�): (4.39)
Here, 00� is the solution of Eq. (4.38) with k� = 0. It has been veri>ed that U� is largely unaEectedby the inclusion of the decay channel self-energy in the self-consistent procedure [265], the decaychannel contribution giving an additional repulsion of about 20MeV, without modifying howeverthe general trend of the � potential, especially around the saturation point.
It should be stressed, at this point, that many aspects of � behavior in nuclear matter deservefurther investigation. First, one might ask what is the role of three-body diagrams with � intermediatestates, which ought to give the main contribution of three-body correlations to the binding energy.It has been pointed out already several years ago that contributions of n-body ring diagrams do notconverge with increasing n when � degrees of freedom are explicitly included, presumably becauseof the large coupling of the pion to particle-hole and isobar-hole states, so that suitable regrouping ofterms in the perturbation expansion is required [266]. Coupled-channel two-hole-line calculations withthe continuous choice for the average potential implicitly include some three-body contributions withintermediate isobar excitation through the self-consistent procedure [5]. These contributions turn outto be repulsive. The attractive part of the three-body force needs certainly to be included in order tobring the saturation curve in agreement with the experimental trend. This will require three-hole-linecalculations with � degrees of freedom explicitly built in.
The percentage of �’s in the nuclear medium can be obtained from the defect parameter whichcan be regarded as the expansion parameter of the BBG theory. When the � contributions are added,one has three defect parameters, the usual one UNN , expressing the content of two-particle-two-holecomponents in the nuclear ground state, and two parameters UN� and U��, giving an indication ofthe percentage of virtual � particles in the ground state C� = (UN� + U��)=(1 + U), where U is thesum of all the three parameters. The defect parameters are determined from U< = �I XI < where � andI are the nuclear density and the statistical weight of the channel, respectively, and the bar indicates
an average over initial two nucleon and � states in the Fermi sea of the quantities
I< =∫
d3q(2�)3
G<L;NNL′(q; q′; P;E<(q′; P))2
(ENN (q′; P)− E<(q; P))2Q<(q; P) : (4.40)
The values obtained for U< are increased by � contributions, and according to Refs. [262,263], thepercentage of � is 10–12% near the saturation point.
Finally, one may discuss the role of relativistic eEects in nuclear matter with explicit � degreesof freedom. This subject has been investigated by Mal?iet and co-workers in the framework of acoupled-channel Dirac–Brueckner (DB) approach [7,65,267–269]. This formalism can be regardedas a low-order approximation to the Martin–Schwinger hierarchy of many-body Green functions[270], from which it follows when correlation contributions to the self-energy and hole–hole scat-tering are disregarded, and the energy spectrum is restricted to quasi-particle states only [267]. Inpractice, one tries to mimic the non-relativistic Brueckner approach as closely as possible, withsingle-particle states described by eEective spinors, satisfying the Dirac equation with a self-energyterm. Self-consistent equations are obtained, which require the matrix elements of the NN interactionand the single-particle Green functions determined from the solutions of the Dirac equation itself.In [7,267] the relativistic G matrix is constructed starting from a OBE interaction with � transi-tions. The bare transition operator is obtained solving a covariant reduction of the Bethe–Salpeterequation, closely following the approach of van Faassen and Tjon described in Section 3:1:5. Theisobar was taken as an unstable particle, and the resonance parameters >tted reproducing the P33
�N phase shift behavior, by demanding that the full dressed � propagator has a pole at the physical� mass M� − i2�=2. Only positive-energy contributions have been considered for the single-particlestates, and both the Bethe–Salpeter and the BG equations have been solved after a three-dimensionalThompson reduction. At a >rst stage, the � self-energy was taken as a spin-averaged quantity (whichis essentially equivalent to a non-relativistic approximation), whereas the nucleon was described in afully relativistic way, with the large scalar and vector components characterizing relativistic nuclearmodels [7]. Subsequently, a full Lorentz representation has been employed for both the nucleonand the isobar self-energies, thereby treating both particles in essentially the same way [65]. Theresults for nuclear matter near saturation also show a strong repulsion. The � eEective mass is al-most constant as a function of density, and the self-energy is not very much momentum dependent,an indication that the properties of the isobar seem hardly aEected by the medium in this type ofrelativistic models.
5. Conclusions and outlook
We have seen that >ngerprints of the � appear in many nuclear phenomena. It contributes atthe 1–2% level to the total wave function of two- and three-nucleon systems, and has to be takeninto account in the transverse, model-dependent part of two-body electromagnetic currents in order toimprove the agreement between calculated and experimental electromagnetic form factors. To a goodextent, this baryon behaves as a quasi-particle in the nuclear medium, as testi>ed by the success of the�-hole model in explaining photon- and pion-induced reaction data, the diEerence between positionand width of the resonance with respect to the free case being attributable to medium eEects. The �comes into play in renormalizing the eEective pion–nucleon coupling, as well as the weak axial-vectorcoupling constant gA. With reference to this point, modern large-scale shell model calculations give
clear evidence that the quenching of gA can be regarded as a universal phenomenon in nuclei,although the relative weight of �-hole excitations and “standard” con>guration mixing is still to becompletely clari>ed. The mechanism of � excitation and propagation in charge-exchange reactions,and the role played by longitudinal and transverse couplings is now better understood, thanks tomore and more re>ned nuclear-structure and reaction calculations. In spite of these successes, muchremain to be done to better understand the physics of the �.
From a fundamental point of view, its parameters, the isospin mass splitting within the associatedmultiplet, and the long-standing issue of the � quadrupole moment have to be assessed in the broaderframework of QCD or QCD-motivated quark models. Under this point of view, some basic issues,such as the role played by the residual OGE qq interaction, seem still to be clari>ed. Usually, thesecolor-magnetic forces are invoked to remove the degeneracy within the same SU (6) multiplet, inparticular between nucleon and �, and, through their dependence upon the quark masses, concur todetermine the >ne structure of the isospin multiplets, as discussed in Section 2.2. Moreover, theirtensor component is responsible for the D-state admixture in the hadron wave functions, leadingto non-vanishing quadrupole moments. Simple quark models, however, meet some diSculties inreproducing the level orderings of positive and negative-parity excitations in the baryon spectra,and several modi>cations have been proposed in the literature, which give diEerent emphasis toOGE contributions. Thus, we have seen that in hybrid quark models an eEective meson–quarkinteraction is simply added to OGE [271,272], whereas in the Goldstone–Boson-Exchange quarkmodel chiral symmetry arguments are pursued to their extreme consequences, and no room is leftfor the standard OGE potential [168,273]. These diEerent perspectives on the hadron structure havedirect implications on the interpretation of isospin mass splittings and quadrupole moments, where� plays a distinguished role in virtue of its prominent position in the baryon spectrum and its largecoupling to the �N system. From a more technical point of view, the necessity of high-quality,few-body calculations of hadron spectra is emerging as more and more compelling, in order toassess the reliability of diEerent quark models. This is exempli>ed by predictions of the N–� masssplitting for diEerent qq interactions; a good agreement with the experimental value, obtained throughan approximate evaluation of the three-quark baryon wave function, may be completely lost in exactcalculations [167].
Another topic which will certainly require further investigations is the derivation of the meson–baryon–baryon coupling strengths and oE-shell form factors from the underlying QCD, or fromsome sensible approximation to the full QCD dynamics. As a matter of fact, the relations among thevarious strengths based upon symmetry arguments reviewed in Section 3:1:2 can be at most of guidingvalue; similarly, one needs some hint from microscopic models about the proper values to be givento the various cut-oE parameters appearing in the strong (�N�; �N�; : : :) form factors. In the lastseveral years there has been a ?ourishing activity in this direction, ranging from lattice calculationsand Nambu–Jona–Lasinio-type models, to non-perturbative approaches based on Schwinger–Dysonformalism [274]. The relative merits and limitations of all these attempts have still to be assessed onthe ground of a detailed comparison with the experimental data. A more profound knowledge of thecouplings and a deeper information on the short-range behavior from experiments or non-perturbativeQCD calculations would open the way to a complete treatment of the interactions between nucleonsand �’s.A >eld which is presently enjoying an uprise of experimental and theoretical activity is the study
of the electromagnetic structure of the nucleon and its excitations. As a matter of fact, the baryon
spectrum alone is not enough to distinguish between the many quark models proposed in the lit-erature. Other observables, such as the elastic and transition electromagnetic form factors, have tobe considered in order to constrain theoretical calculations. Here also new quark models of baryonscompete with the traditional ones in trying to get a better description of data. An example is givenby the Hypercentral Constituent Quark Model, where baryons are regarded as true three-body boundstates, to be described through hyperspherical coordinates, and the driving interaction is representedby a three-body potential depending upon the hyperradius only [277,278]. From the experimentalpoint of view, thanks to new, high-performance machines like the Thomas JeEerson AcceleratorFacility, the baryon structure and electromagnetic transitions are explored in new kinematic domainswith high-luminosity beams. Theoreticians are therefore confronted with the demanding task of pro-viding models able to bridge the gap between the low-energy region where quark models of hadronsmay perform well, and high-energy, deep-inelastic processes where a perturbative approach to QCDis possible [279]. A direct solution of QCD equations being beyond our present capabilities, onewill have to resort to ingenious approximation schemes. A promising approach is represented bylight-cone >eld theory, which allows for a particularly simple Fock representation of the vacuumstate, so that more and more complex con>gurations can be introduced into the baryon wave func-tions, when needed, in the framework of a Hamiltonian description of the dynamics. Long-rangephenomena associated to spontaneous symmetry breaking, on the other hand, can be associated tothe vacuum zero modes [280]. The � isobar will again play a paramount role in testing new the-oretical models, and might give new hints on the energy domain where constituent quark modelsbreak down, and perturbative QCD applies. In particular, the behavior of the N� transition formfactor at momentum transfers of several GeV=c2, may still have signatures of the non-perturbativeQCD regime.
As we have seen, the understanding of the physical processes involving the � is not completed.Many open problems still remain to be solved. The new experimental facilities might shed new lightand give new hints to theoreticians on this fascinating issue.
Appendix A. Dispersion relation constraints
A.1. Preliminaries
In the dispersion approach to scattering it is intended to explore the analytic properties of theS-matrix regarded as a function of the kinematic variables (s; t; u) to give a complete description ofthe process. The basic requirements are [282–284](a) analyticity,(b) unitarity, and(c) crossing symmetry.Analyticity means that the scattering amplitude can be safely continued from the physical region,where all singularities corresponding to the physical processes lie, into the complex energy (s) or=andmomentum-transfer (t) or u plane. Actually, one requires what is called “minimal analyticity, i.e.the only singularities that S(s; t; u) has in the complex plane of the Mandelstam variables are thosedictated by unitarity.
Due to unitarity, expressing ?ux conservation, the S-matrix has only poles associated to the prop-agation of a single on-mass-shell particle, if allowed by conservation laws, plus branch points atthe scattering cuts associated to multiparticle propagation in intermediate states. The discontinuitiesacross the cuts are >xed by the unitarity relation.
Crossing symmetry allows one to look at the transition amplitudes for diEerent processes as bound-ary values of the same analytic function in diEerent regions of the kinematic variables. However, theT -matrix can contain kinematic factors with diEerent singularities in the various crossed channels.This would make the analytic continuation from one channel to another impossible. To avoid thisproblem, invariant amplitudes have to depend in a symmetric way upon the Mandelstam variables,so as to exhibit in all channels only the singularities due to unitarity. This is simple for spin-lessparticles, but requires non-trivial algebraic relations when some of the particles have spin [107,282].
According to the Mandelstam hypothesis, the various scattering amplitudes for the processes in thes-, t-, and u-channel are related to each other, as boundary values of the same analytical function,determined once the leading intermediate states have been speci>ed.
A.2. Fixed-variable dispersion relations
The analytic properties of the scattering amplitude can be summarized by a dispersion relation. Bythis terminology one means a Hilbert integral transform that expresses the real part of the transitionamplitude through a contour integral in the complex energy plane. There, one assumes analytic-ity for the scattering amplitude, apart from singularities dictated by the unitarity condition, whichare located on the real axis, so that the integration contour is (A.1). Applying the Cauchy theoremone gets
T (s; t) =12�i
∫Cds′
T (s′; t)s′ − s
; (A.1)
where T (s; t) is the scattering amplitude with all kinematic factors removed. For spinless particles thisfactor is just a phase-space quantity, whereas for particles with spin non-trivial transformations arerequired to get invariant amplitudes whose singularities only depend upon the dynamics. The contourC in the complex-energy plane is chosen to avoid all the singularities of T dictated by unitarityrequirements in the various channels that divide the physical region into pieces. The scatteringmatrix is analytical in them, and can be analytically continued from one piece to another, as thesame regular function of the variable s. In order to avoid the singularities one is obliged to gothrough paths connecting the various pieces of analyticity, like the integration contour in Fig. A.1.
Alternatively, one can ascribe an in>nitesimal imaginary part i0 to the various singularities,thereby shifting them out of the real physical region, no longer divided into disconnected pieces.One can thus integrate over the real axis in the Cauchy integral (A.1), and write
T (s; t) =1�
∫ ∞
s0
Im T (s′; t)s′ − s− i0
ds′ +1�
∫ s′0
−∞Im T (s′; t)s′ − s− i0
ds′ +12�i
∫2
T (s′; t)s′ − s
ds′ : (A.2)
Here s0 and s′0 are the lower and upper limits of the singular regions on the real axis, and 2represents the curved part of the contour C. The Schwartz re?ection principle T (z∗) = [T (z)]∗ hasbeen used to write
Fig. A.1. The Cauchy contour for the dispersion integral in the energy plane of the �N scattering amplitude.
The actual nature of the singularities can be identi>ed through unitarity,
Im T<>(s; t) =12
∑k
∫dLips(s; k)T<k(s; t′)T>k(s; t′′) ; (A.3)
where < and > denote the initial and >nal channels. The sum is over all channels which are physicallyallowed, and dLips(s; k) represents the appropriate Lorentz-invariant phase-space diEerential.
The simplest singularities are associated to the propagation of single-particle states in the s-channelallowed by conservation laws. In particular, for the case of �N → �N scattering in the s-channel,a nucleon can propagate as an intermediate on-shell particle. It can be shown that the single-particlecontribution to Eq. (A.3) is [107,282]
Im T<>(s; t)(1) =12
∑m
�(M 2 − s)〈>|T †|jm;X 〉〈jm;X |T |<〉 : (A.4)
A partial-wave decomposition has been performed to exhibit the intrinsic spin j of the intermediateparticle of mass M . The matrix element 〈jm;X |T |<〉, evaluated at s=M 2, measures the strength ofthe coupling between the incoming particles and the intermediate one, and is de>ned as a couplingconstant G<X . In these terms Eq. (A.4) can be rewritten as
Im T<>(s; t)(1) = G<XGX>�(M 2 − s) : (A.5)
When Eq. (A.5) is inserted into the dispersion relation (A.2), and the transition amplitude is assumedto vanish for |s| → ∞ one gets for example in �−N scattering
T (s; t) =G2
M 2 − s+
1�
∫ ∞
s0
Im T (s′; t)s′ − s− i0
ds′ +1�
∫ s′0
−∞Im T (s′; t)s′ − s− i0
ds′ : (A.6)
where G denotes the �−p → n coupling constant. The pole term coming from the u-channel isabsent in this example.
In addition to the single-particle states, the unitarity relation (A.3) contributes with cuts in thes- and u-channels, due to the propagation of two or more on-shell particles. For �N scattering one
has a branch point at s0 = u0 = (m� +M)2. The singularities in the u variable can be transferred intothe complex s-plane in virtue of the Mandelstam identity
s+ t + u=∑
i
m2i = 2m2
� + 2M 2 ; (A.7)
so that, one has in the s-plane a right-hand cut starting at s0=(m�+M)2, and a left-hand cut startingat s′0 =−t + (M − m�)2, coming from unitarity in the u-channel. Using the well-known identity
1s′ − s− i0
= P1
s′ − s+ i��(s′ − s) (A.8)
one >nally gets the >xed-t dispersion relation
ReT (s; t) =G2
M 2 − s+
1�P∫ ∞
s0
Im T (s′; t)s′ − s
ds′ +1�P∫ s′0
−∞Im T (s′; t)
s′ − sds′: (A.9)
In practice, one actually has only one principal-value integral according to the energy region con-sidered. Thus, for s0 ¡s¡+∞ only the >rst one gets a singularity, whereas the second gets it for−∞¡s¡s′0. For real s lying in between s′0 and s0; T (s; t) is real.
Similar considerations in general apply for the crossed u-channel amplitude Tu. For elastic �Nscattering, however, one can choose the s-channel so as to have no single-particle contributions in theu-channel. Thus, in our example, if the elastic �−p process is in the s-channel, the crossed u-channelis �+p → �+p, and no single-particle propagation is allowed. From the Mandelstam hypothesis theamplitudes are boundary values of the same function and one can write [107],
T (s; t; u) = Tu(u; t; s) (A.10)
for any given t.Using Eqs. (A.10) and (A.7), the dispersion relation (A.9) can be >nally written in the form
ReT =G2
M 2 − s+
1�P∫ ∞
s0
Im T (s′; t)s′ − s
ds′ +1�P∫ ∞
s0
Im Tu(s′; t)s+ t + s′ −∑i m
2ids′ (A.11)
with s0 = (m� +M)2.The above derivation can be repeated for the crossed amplitude Tu.Eqs. (A.11) and the equivalent for Tu represent the required >xed-t dispersion relations for the
amplitudes. Crossing symmetric and antisymmetric combinations of these amplitudes can also bede>ned,
T (±) ≡ 12 [T (s; t; u)± Tu(s; t; u)] = 1
2[T (s; t; u)± T (u; t; s)] ; (A.12)
which obviously satisfy
T (±)(u; t; s) =±T (±)(s; t; u) : (A.13)
The whole formalism can be made more transparent by introducing the variable [107]
For forward scattering (t = 0); I gives the incoming pion energy in the laboratory system ! =(s−M 2 −m2
�)=2M . It is straightforward to show that Eqs. (A.11), its equivalent for Tu, and (A.12)lead to
ReT (±)(I; t) = T (±)pole +
1�P∫ ∞
I0
Im T (±)(I′; t)[
1I′ − I
± 1I′ + I
]dI′ ; (A.15)
where I0 = m� + t=4M depends upon the squared momentum transfer t, and the pole terms
T (+)pole =
G2
2MIB
(IB − I)(IB + I); T (−)
pole =G2
2MI
(IB − I)(IB + I)(A.16)
are singular at the nucleon pole IB = (t − 2m2�)=4M , corresponding to s=M 2 or u=M 2.
When spin degrees of freedom are included, suitable invariant transition amplitudes that exhibit theisospin degrees of freedom have to be extracted from the S-matrix, in order to avoid the singularitiesassociated to purely kinematic factors. For �N scattering, let
Tba(q′; q) ≡ 〈�b(q′)|T |�a(q)〉 (A.17)
be the invariant amplitude for a transition from a pion of isospin a and four momenta q, to a pionwith isospin b and four momenta q′. It can be decomposed into isospin even and odd componentsas follows [8]:
Tba(q′; q) = T (+)�ba + 12[Bb; Ba]T
(−) ; (A.18)
where the matrices T (±) are related to the scattering amplitudes in the total-isospin representationby,
T (+) = 13(T
1=2 + 2T 3=2); T (−) = 13(T
1=2 − T 3=2) : (A.19)
Taking into account the eEect of the incoming and outgoing nucleon, it can be proved that the mostgeneral parity-conserving form of the on-shell T -matrix is
where u and u′ denote the incoming and outgoing nucleon spinors, and the Feynmann notationq= ≡ q · + has been employed. The scalar functions A(±)(s; t; u) and B(±)(s; t; u) are the so-calledinvariant amplitudes, free of kinematic singularities. Obviously, they can be expressed in terms ofinvariant amplitudes for the s- and u-channel much in the same way as the transition amplitudesT (±) for scalar particles are related to T and Tu, namely
The singularity structure of the invariant amplitudes is only determined by unitarity, therefore, onecan write down immediately >xed-t dispersion relations similar to Eq. (A.15),
ReA(±)(I; t) =1�P∫ ∞
I0
Im A(±)(I′; t)[
1I′ − I
± 1I′ + I
]dI′ ; (A.24)
ReB(±)(I; t) = B(±)pole +
1�P∫ ∞
I0
Im B(±)(I′; t)[
1I′ − I
∓ 1I′ + I
]dI′ ; (A.25)
where the pole terms
B(+)pole =
g2
MI
(IB − I)(IB + I); B(−)
pole =g2
MIB
(IB − I)(IB + I); (A.26)
are now contained only in the isospin odd amplitudes B(±). To make the present notation consistentwith the Born contribution given by the pseudoscalar >eld-theoretic Lagrangian (3.1). The couplingconstant g has been de>ned for the p → p�0 vertex, with g= G=
√2.
The pole term appears only in the B± amplitudes as Eqs. (A.24) and (A.25) show. In orderto extract the coupling constant from the experimental data, it is more convenient to separate outexplicitly the I= 0 contribution in Eq. (A.25) giving,
ReB(−)(I; t) = B(−)pole +
I�P∫ ∞
I0
Im B(−)(I′; t)[
1I′ − I
− 1I′ + I
]dI′
I′+ B(0; t) (A.27)
with
B(0; t) ≡ 2�P∫ ∞
I0
Im B(−)(I′; t)I′
dI′ : (A.28)
The dispersion relation for B(+) can be identically re-written in the form
ReB(+)(I; t) = B(+)pole +
I�P∫ ∞
I0
Im B(+)(I′; t)[
1I′ − I
+1
I′ + I
]dI′
I′: (A.29)
Combining both Eqs. (A.27) and (A.29) one immediately gets,
(IB ± I){∓ReB±(I; t)± I
�P∫ ∞
I0
[Im B+(I′; t)
I′ ∓ I+
Im B−(I′; t)I′ ± I
]dI′
I′
}
=g2
M+ B(0; t)(IB ± I) ; (A.30)
with B± = B(+) ∓ B(−). Eq. (A.30) can be used as a linear relation in the symmetric variable I toextrapolate the experimental data for elastic �±N scattering processes up to the pion pole I = IB[18,22]. The comparison of the pseudoscalar meson–baryon vertex functions and the T -matrix in theBorn approximation gives then the strength of the interaction g.
The above procedure simpli>es somewhat for forward scattering (t=0). In this case it is customaryto write the dispersion relations in the laboratory system, where the nucleon remains at rest. SinceI equals the incoming pion energy !, and the nucleon pole IB is located at !N ≡ −m2
Owing to the normalization of the nucleon spinors, and the property
u(pN ; �)+Fu(pN ; �′) = ���′pF
M;
the transition amplitude (A.20) is diagonal in the spin quantum numbers as expected for forwardscattering, and simply given by
T (±)(!) = A(±)(!) + !B(±)(!) ; (A.34)
since the nucleon momentum pN is zero. Inserting (A.31) and (A.32) into this result one gets forthe isospin odd amplitude
ReT (−)(!) =12(ReT− − ReT+) =
g2
M2!N!
!2N − !2
+2!�
P∫ ∞
m�
Im T (−)(!′)!′2 − !2 d!′ ; (A.35)
where the shorthand notation T± ≡ T�±N ≡ T (�±N → �±N ) has been employed.This equation can be related to the experimental data by resorting to the optical theorem,
Im T±(!) =k(!)�±(!)
4�(A.36)
with k(!) =√
!2 − m2�. Using the trivial identity
1!′2 − !2 =
1!′2 +
!2
!′2(!′2 − !2);
one arrives at the remarkable result[ReT (−)(!)
!− !2
4�2P∫ ∞
m�
k(!′)!′2
�−(!′)− �+(!′)!′2 − !2 d!′
](!2
N − !2)
=−(m�
M
)2g2 +
!2N − !2
4�2
∫ ∞
m�
k(!′)!′2 (�−(!′)− �+(!′)) d!′ ; (A.37)
which states that the complicated quantity on the left is a linear function of !2, and can be determinedfrom the measured total cross sections, and a phase shift analysis for the forward amplitude T (−)(!).The coupling strength g2 can then be determined again by a linear extrapolation to the nucleonpole !2 = !2
N . These extrapolations, based upon the >xed-t relations (A.30) or upon the forwarddispersion relation (A.37), clearly represent the physicist’s way of determining the residue of thescattering amplitude in correspondence to its essential singularity at I= IB (or != !N for forwardscattering in the laboratory system). The crucial physics coming into play through unitarity is that
this residue represents, apart from mass factors, the coupling between the nucleon and the exchangedmeson. Eq. (A.37) has been used, in conjunction with phase-shift analysis, to obtain the >rst reliabledetermination of the �N coupling constant based upon dispersion theory, yielding the well-knownresult f2
�±=4�= 0:08 for the charged meson–nucleon pseudovector coupling [75].Similar considerations apply for the isospin even amplitude T (+)(!). However, whereas in
Eq. (A.37) the diEerence between the total cross sections appears, the integrals can be assumedwell-behaved with increasing !, since ��+N (!) � ��−N (!), for T (+)(!) the sum of the cross sec-tions comes into play. As a consequence, a subtraction is required to make the corresponding integralconvergent. One generally takes != m� as the subtraction point.
Finally, it is worthwhile to note that one of the >rst applications of forward dispersion relationswas to the unraveling of the energy dependence of the pion–nucleon phase shifts. As a matter offact, the angular distribution data in the 1950s could be reproduced within the experimental errors bysix diEerent partial-wave solutions. By insisting that any set obtained should satisfy the dispersionrelations, one could discard four of the six possible solutions [285]. Of the two remaining sets, onewas characterized by a large phase shift in the J = T = 3=2 state (the Fermi solution), whereas theother had a large phase shift in the J=3=2; T=1=2 state (the Yang solution). By a clever dispersionanalysis Davidon and Goldberger were able to show that the Fermi set could be univocally selected[76]. Thus, dispersion relations played a crucial role since the very beginning of � physics!
Appendix B. The �N scattering matrix and electromagnetic corrections
Let us consider a pion of four momenta q impinging on a nucleon of four momentum p. If q′ andp′ denote the four momenta of the two particles emerging from the scattering process, the non-trivialpart of the S-matrix is related to the invariant transition matrix T by [8]
where u(p) and Xu(p′) are the spinors (de>ned in both spin and isospin space) describing the initialand >nal nucleon, respectively, and
Tba ≡ 〈�b(q′)|T |�a(q)〉 : (B.2)
The pion isospin is here given in cartesian representation. The invariance with respect to rotationsin isospin space can be exhibited by decomposing T into components of well-de>ned total isospin(1=2 or 3=2) through the projectors
P 32= 1
3(2 + t · B); P1=2 = 1− P1=2 = 13(1− t · B) ; (B.3)
to write
T = T (1=2)P1=2 + T (3=2)P3=2 : (B.4)
According to Eq. (B.2), this operator has to be evaluated between cartesian pion states. The identities
with T (±) de>ned according to Eq. (A.19). This is the decomposition of the T -matrix into isospineven and odd components we already considered in Appendix A. For the extraction of the singularity-free amplitudes A(±) and B(±) is convenient to discuss crossing symmetry, and write dispersionrelations for the �N system. However, to exploit rotational invariance, so as to introduce partial-waveamplitudes, it is more convenient to write the transition amplitude in terms of the spin non-?ip andspin-?ip components. To achieve this end the invariant T -matrix is written as the representativekernel of a suitable operator T between two-component nucleon Pauli spinors in spin and isospinspace, namely
M4�
√su(p′)〈�(q′)|T |�(q)〉u(p) = C†TC : (B.8)
Here T is a function of the >nal and initial pion three momenta q′ and q for an on-shell pion,whereas it is still an operator in the space of the (spin and isospin) nucleon degrees of freedom.The normalization is such that the diEerential cross section is given by
d�d
= |C†TC|2 : (B.9)
The kernel operator T can be expressed in terms of the invariant amplitudes A(s; t; u) and B(s; t; u)through a straightforward but lengthy calculation. One gets [8,107]
and n ≡ q ∧ q′=|q ∧ q′| is the usual unit vector perpendicular to the scattering plane. Rotationalinvariance >nally leads to the well-known partial-wave expansions
f(s; �) =∞∑l=0
[(l+ 1)fl+(s) + lfl−(s)]Pl(cos �) ; (B.15)
g(s; �) = sin �
{ ∞∑l=1
[fl+(s)− fl−(s)]P′l(cos �)
}; (B.16)
for the non-?ip and spin-?ip amplitudes, with coeScients only depending upon the total CM scat-tering energy s. As usual, P′
l(x) represents the derivative of the Legendre polynomial Pl(x) withrespect to x, and l± are associated to the total angular momentum eigenvalues J ±1=2, respectively.The unitarity of the S-matrix >nally implies that the partial-wave amplitudes fl±(s) can be writtenin terms of phase and inelasticity parameters,
fl± =12iq
(#l± exp 2i�l± − 1) : (B.17)
Needless to say, the isospin decomposition (B.4) and the partial-wave expansion (B.15) and (B.16)could be combined to introduce inelasticities and phase shifts for each total isospin and angularmomentum state (I; J ).
The above developments apply in presence of short-range, strong interactions only. The eEects ofthe electromagnetic interactions modify this scenario for at least two reasons. First, the partial-waveexpansion does not converge any more, because of the long-range tail of the Coulomb potential;second, isospin symmetry is explicitly broken by the electromagnetic force. Moreover, problemsassociated with the zero mass of the photons arise, so that the diEerential cross section has to bemultiplied by a factor which takes into account soft photon emission [102], namely
where � is a >ctitious photon mass acting as a regularizing parameter, VE the energy resolution, andh(s; t;VE; �) a factor which takes into account the undetected photons of total energy less than VE.For non-forward scattering one >nds that the spin-free and spin-?ip amplitudes f and g vanish in the� → 0 limit, infrared catastrophe. It is not possible here to consider these questions in detail, and werefer the reader to the relevant literature [102,286,287]. Even apart from this problem, the additivityof the interactions does not obviously re?ect itself in the additivity of the amplitudes, so that thetreatment of electromagnetic eEects always imply some model dependence. What is generally doneis to deFne the nuclear partial-wave amplitudes through
f′N (s; �) ≡ f(s; �)− fC(s; �) =
∞∑l=0
[(l+ 1)f′l+(s) + lf′
l−(s)]Pl(cos �) ; (B.19)
g′N (s; �) ≡ g(s; �)− gC(s; �) =∞∑l=1
[f′l+(s)− f′
l−(s)]P′l(cos �) ; (B.20)
where the Coulomb amplitudes fC and gC are obtained from Feynmann graphs with only photonexchange, and eventually meson and proton form factors. Nuclear phase shifts and inelasticities can
be de>ned by factoring out the appropriate Coulomb phases. This is particularly simple for elastic�+N scattering, which involves a pure total isospin state (T = 3=2). One has
f′l± = e2i�l
#′l±e2i�′l± − 12iq
(B.21)
with the Coulomb phase �l derived from fC and gC (the divergence of the partial-wave expansionfor the Coulomb amplitudes being in this context irrelevant). Thus, for scattering of a point chargedmeson oE a point-like nucleon one has the text-book expression
�l = arg2(l+ 1 + i#) (B.22)
in terms of the 2 function, with # the usual Sommerfeld parameter. If the >nite extension of thestrong-interacting hadrons is taken into account, a dynamical treatment of the photon–hadron vertexis required [107], and the Coulomb amplitudes and phase-shifts are expressed in terms of dispersionintegrals [18]. This is not the whole story, however, since the pure hadronic phases and inelasticitiesare not yet those appearing in Eq. (B.21). We will limit ourselves to outline the procedure followedby Tromborg et al. [288–290], which has been employed in the classical determination of the �±Ncoupling constant by Koch and Pietarinen. For elastic �+N scattering one de>nes
(�H )l± = �′l± − �l± (B.23)
and
(#H )l± = #′l± + #l± (B.24)
as hadronic parameters, with corrections �l± and #l± given in tabulated form [288,289]. The situationis more complicated for elastic �−N scattering and the charge-exchange reaction �−p → �0n, becauseof the mixing of the T = 1=2 and T = 3=2 states. In these cases one has >rst to de>ne isospinamplitudes f1
l± and f3l± for the T = 1=2 and 3=2 channel, respectively, plus a mixing amplitude
f13l±, arising from electromagnetic eEects which break isospin symmetry. These amplitudes can be
then parameterized in terms of inelasticities and phase parameters. For elastic �−N scattering onehas
f′l± = 1
3(2f1l± + f3
l± − 2√2f13
l±)e−2i�l± (B.25)
with fIl± given by an expression similar to Eq. (B.21), namely
fIl± =
12iq
(#Il±e
2i�Il± − 1) ; (B.26)
whereas a more complex expression is needed for the mixing amplitude f13l±,
f13l± =
23
√2(#13l± + i#13l±)e
i(�1l±+�3l±)
2iq: (B.27)
Only at this stage one can de>ne the hadronic parameters. For the diagonal (T=1=2; 3=2) amplitudesone has in analogy to the Eqs. (B.23) and (B.24),
For the charge-exchange case extra kinematic factors enter into play, owing to the diEerent massesin the initial and >nal channel. It is worthwhile to observe that in low-energy �N scattering theinelasticities #I
l± and #13l± are mainly due to the coupling to the �−p → n+ reaction.
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