Hypernuclear photoproduction

ANNALS OF PHYSICS 184, 33-61 (1988)

Hypernuclear Photoproduction*

A. S. ROSENTHAL, DEAN HALDERSON, AND KIMBERLY HODGKINSON

Department of Physics, Western Michigan University, Kalamazoo, Michigan 49008

AND

FRANK TABAKIN

Deparrment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Received August, 20, 1987

The nuclear (y. K+) reaction is studied using a distorted wave impulse approximation including the full Coulomb plus kaon-nucleus interaction. The need to include kaon-hypernucleus interaction effects, especially at higher energies, is demonstrated. The sensitivity of the hypernuclear cross sections to two alternate, viable descriptions of the basic photoproduction process is examined in detail. The basic photoproduction operators, including nonstatic terms, are examined. These nonstatic operators come into play to excite a richer spectrum of hypernuclear states as one goes beyond zero angle and threshold production. Hypernuclear wavefunctions, based on a phenomenological A nucleus interaction and including A continuum effects, are used to study the role of the associated form factors in determining the production cross sections for a variety of nuclear targets, and a range of angles and energies. Hypernuclear excitations other than spin-flip modes are then found to be signiticantly excited and kaon distortion effects are found to strongly alter the magnitude of the cross sections. Our conclusions are compared to those found by others in earlier investigations of this reaction. @? 1986 Academic Press, Inc.

I. INTRODUCTION

The construction of a high intensity 4-GeV continuous electron beam facility (CEBAF), and of an associated high resolution K+ spectrometer, will make possible the electromagnetic production of hypernuclei [ 1,2]. The Bates electron accelerator, when extended to 1 GeV, could also be used to explore the (y, K+) process [2] near its threshold of k,,, = 909.6 MeV/c.

The basic idea is to use the process y + p + n + K+ to transform a proton into a /1 particle. The photoproduction of a K+ (mass = 493.67 MeV, strangeness = + 1) along with the n (mass = 1115.6 MeV, strangeness = - 1) is an example of associated production, wherein strange particles are produced in conjugate pairs

* Research supported in part by the U.S. National Science Foundation July, 1987.

33 0003-4916/88 $7.50

Copyright 0 1988 by Academic Press, Inc. All rights ol reproduction in any form reserved.

34 ROSENTHAL ET AL.

S

FIG. 1. The production of an SS pair by a &on, which also occurs in the electroproduction process of Fig. 2.

from an initial system of zero strangeness. At the quark level, this associated production involves the basic process of creating strange (s) and antistrange (S) quarks (Fig. 1). The initial photon could be a real one or the virtual photon created by the electrons, as illustrated in Fig. 2. If the proton is within a nucleus, this reaction turns an ordinary nucleus, which consists of neutrons and protons, into a nucleus with negative strangeness or hypercharge. The produced hypernucleus consists of one n plus neutrons and the remaining protons. Thus the initial nucleus experiences a change in its strangeness AS = - 1 = AY (the hypercharge, Y, is related to the strangeness, S, and the baryon number, B, by Y = S + B).

The study of such hypernuclei is not a new subject [3-lo]. It has been pursued extensively using the reaction KP + n -+ n- + /i, both experimentally and theoretically. The production of Z hypernuclei, using the reaction K- + N + 7c + C, has also been explored [S]. The great advantage of the (K-, rc - ) reaction is that the final n (or C) can be produced with low momentum and therefore can be placed readily within a nucleus. That well-known case is discussed extensively in Ref. [7]. In the K- + n + c + n process, strangeness is transferred from the K- meson (strangeness = - 1) to the n baryon (strangeness = - l), and not created, as

uud

P

FIG. 2. The electroproduction process is illustrated including the constituent quarks and the gluonic production of the sS pair, i.e., the associated production idea.

HYPERNUCLEAR PHOTOPRODUCTION 35

in the y+p+A+K+ reaction. At the quark level, for the K- + n + rc + A process, the s quark in K- is transferred and replaces the d quark in the initial baryon. Therefore the Kp + n + n- + /i reaction is a strangeness exchange reaction, whereas y + p --) ,4 + K+ is an associated production reaction.

Other hadronic induced strangeness exchange reactions, in addition to the (K-, n-) case, have been proposed [3-lo]. In fact the (rr+, K+) reaction has been successfully used in this context. More speculative is the idea of using antiproton beams with the two-step reaction p+ p -+R+K followed by K+N+z+A or direct use of p + p + ;1+ /1 has been suggested as a means of producing heavy hypernuclei [6]. Finally, the production of double hypernuclei, with AS = - 2 and two final ,4 particles, is also possible, using the double strangeness exchange reactions A(K*, K’) B,,,, [7]. The (rc+, K+) reaction is of particular interest to us since it shares some features with the (y, K+) case, as will be discussed.

These various possibilities for producing hypernuclei can be classified not only according to their being strangeness exchange or associated production type reactions but also according to their spin and momentum transfer capabilities. From the work of many authors [3-lo], we learn that the (K-, xc-) reaction, with its low momentum transfer to the baryon, has the characteristic of exciting low total angular momentum’ As=0 states in hypernuclei. For the reaction (z+, K+) the As = 0 states are also preferentially excited. However, as a consequence of the lighter mass of the incident hadron in the (n +, K+ ) reaction, the momentum transfer to the baryon LI is sizable and hence the reaction depends on the high momentum components of the nuclear wavefunctions or, more precisely, on the transition form factors, p(Q), at higher momentum transfer (Q) values. Such form factors tend to be larger for larger J values; hence one finds preferential excitation of high spin states. The (Z + , K+ ) reaction is similar to the y + p + K+ + n reaction in two essential aspects: (a) they both are associated production reactions, wherein s and S quarks are created, and (b) they have similar momentum transfer characteristics. The operators that determine the basic dynamics of the above (rr+, K+) and (y, K+) reactions are however quite distinct. While the (rc+, K+) reaction leads to predominately no spin flip, As = 0, states, the (y, K+ ) reaction appears to preferentially excite As = 1 states. Therein lies the major appeal of the y + p + Kf + A reaction; i.e., it provides a means of exploring the As = 1 states over a range of momentum transfers using the electromagnetic probe. Because of the large momentum transfers involved in both the (n+, K+ ) and (y, K+ ) reactions (Q above 0.2 GeV/c) the cross sections are expected to be small, of the order of a few tenths of a microbarn, but observable with the new accelerator and detector developments. In addition, the hope is to learn about the full X43) baryon-baryon dynamics as in the A-N and C-N interactions. If sufficiently reliable experiments and theories can be developed, it might also be possible to extract information about strange baryon ( Y*) and strange meson (K*, K**) resonances. Finally, if quark dynamics is shown

’ We denote spin by s and strangeness by S.

36 ROSENTHAL ET AL.

to be necessary, it might be possible to extract detailed spin effects and relate that to the basic gluonic production of sS pairs (see Fig. 1).2

In earlier pioneering studies [ 1 l-151, the basic ideas and motivations for studying the (y, K+) process have been nicely delineated. At first, the simplest static photoproduction operator (r. E ̂was used as an effective interaction to lit the extant data on the nucleon and used in a plane wave impulse approximation (PWIA) approach to obtain the zero degree cross sections for several light nuclear targets [ll]. Since the Kf has a relatively weak interaction with nuclei, plane wave calculations were invoked in the first studies. Subsequent work by Cotanch and Hsiao [ 131; Cohen [ 143; and Adelseck, Bennhold, and Wright [ 15) increased the sophistication of the theories to include kaon distortions (DWIA), improved photoproduction operators, more extensive hypernuclear structure, heavier, nuclei, and relativistic effects. In view of the extensive experimental plans to study the electromagnetic production of hypernuclei, we have embarked on independent DWIA studies that are reported here.

In several aspects of our work, we have confirmed prior conclusions concerning the excitation of hypernuclear states at zero degrees; however, significant improvements and extensions have also been introduced. In particular, we explore the influence of different two-body (y, K+) inputs on the predicted cross sections, the detailed effects of distortion, and the use of realistic configuration-mixed wavefunctions as opposed to simple particle-hole models. We also consider the effects of quasi-freee production on the estimated excitation spectra. The con- tributions of the full photoproduction operator, beyond the simple (r. E ̂static case, are examined, including the effect of having the nonstatic or nonlocal photoproduction operators act on the distorted kaon waves and the hypernuclear wavefunctions. As a consequence of such nonstatic photoproduction operators, gradients of the kaon waves and an extended set of form factors appear in our formulation, along the lines previously studied for the (y, rr) case [ 161.

In Section II, the kinematics of the p(y, K+) ,4 reaction along with our notation is presented. The photoproduction dynamics and the associated operators for use in the nuclear case are also presented in Section II. The characteristics of the photoproduction operator having been described, the basic ideas of the DWIA approach are presented in Section III. That is followed by a presentation of results for light (Section IV) and heavy (Section V) nuclei including an analysis of the general features of the reaction dynamics and the effects of using realistic hypernuclear wavefunctions, including the /1 continuum in the shell model. Finally, our conclusions are presented, including a critique of the DWIA method and the need for future developments in this subject.

Z The possibility of observing the SS content of the nucleon, i.e., of the admixture of SS pairs in the nucleon, has been discussed recently by Jaffe in his talk at the PANIC conference, Kyoto, 1987.

HYPERNUCLEARPHOTOPRODUCTION 31

II. THE KAON PHOTOPRODUCTION ON A PROTON

Essential input for any calculation of photokaon nuclear processes is a set of reliable amplitudes for the reaction

lJ+p+K++A. (1)

Once the amplitude is understood, the problem of determining the proper dynamical description of the meson photoproduction in the nuclear context can be coped with using a variety of reasonable schemes. The procedure for generating an operator to be used in a DWIA calculation, starting from a good description of the amplitude, is an issue that we discuss here and in Section III.

Ideally, we would like to understand the y + p + K+ +/i reaction with at least the degree of reliability which has been obtained after years of experimental and theoretical effort for the photopion reaction

y+N-+7c+N. (2)

In view of the limited empirical base, that level of understanding is not yet possible for the photokaon process. Fortunately, with CEBAF more information will become available and the basic amplitude will no doubt be relined. In this section, we describe what is currently known about process (1) and suggest procedures for enhancing our understanding of the photokaon reaction.

Any photomeson reaction of the general type

y + (l/2+ target) -+ O- + (l/2+ product),

when subject to the requirements of Lorentz covariance, parity conservation, and gauge invariance, can be expressed in the CGLN [17] form for on mass shell particles:

Ml = Y& M,=2y,(E.p,k.p,-&.p,k.p,)

M,=IJ,(~~.P,-PE.P,) M, = &,W ‘~2 -6~ .pz).

(3)

The M:s act in the Dirac spinor space and are functions of the photon (k), baryon (pl and p2), and pseudoscalar meson (q= k+ p1 - p2) four vectors. The CGLN amplitudes, Ai, are functions of the s, t, and u Mandelstam variables, and are subject to crossing symmetry and dispersion relation (unitarity) constraints.

38 ROSENTHAL ET AL.

Equivalently, one may give the amplitude for the meson photoproduction reaction using the Pauli spinor amplitude

where the four Q spin-space operators are

Note that all of the operators except Q2 are purely spin operators. The above Fi are functions of the photon (k,), meson (qc) 2CM momenta and of the K+-n scattering angle.

The meson photoproduction cross section and recoil polarization are bilinear combinations of the Ai, or, equivalently, of the Fi. The covariant CGLN amplitudes Ai and the Pauli spinor basis amplitudes Fi are related to each other simply [17, 183, with the Fs offering the advantage of being readily incorporated into nonrelativistic descriptions of nuclei. Of course, it would be preferable to have a consistent relativistic many-body nuclear description plus a relativistic meson photoproduction theory. Momentum space methods offer a natural step toward that admirable goal and significant advances have been made [IS]. At this stage, however, a fully consistent theory which can also incorporate sophisticated shell model continuum effects is not available and hence we need a suitable kaon production operator.

All dynamical models of the photokaon process yield amplitudes of the above form. Several representations of the dynamics, as contained in either A, or Fi, exist in the literature (and another will be proposed here). Unfortunately, all available models suffer from significant empirical and dynamical uncertainties and none seem capable of explaining the kLAB = l-2 GeV data.

To gain insight into the design of a suitable description of photokaon dynamics, which can be applied to hypernuclear studies, let us briefly recall the pion case. An essential part of the physics of the (y, rr) process (2) is the formation and decay of the d( 1236) resonance (Fig. 3). The Born terms of Fig. 4 also contribute significantly for charged pion photoproduction. The r-channel R exchange term is, for certain nuclei (i.e., A = 14, 15), also important, whereas exchanges of heavier

FIG. 3. The A( 1236) resonance excitation process for y + N + K + N.


FIG. 4. Born terms for y + N--f z + N.

charged mesons in the t-channel, such as those of Fig. 5, are important only at energies 2 1 GeV.

Based on the above general characteristics, photopion amplitudes of good quality have been constructed starting from these simple Feynman diagrams. In the CGLN approach the driving terms are based on the form suggested by the Feynman diagrams, but the requirements of crossing and unitarity are also imposed.

For photopion nuclear physics, we need to describe the photoproduction process from the constituent nucleons, which are moving within the nucleus as described by the many-nucleon wavefunction. In that case, it is dificult to carry out the CGLN program. Instead, the Born, A, and pion t-channel exchange Feynman diagrams were used by Blomqvist and Laget [ 191 to generate a dynamical theory, which is at least consistent with the general form Eqs. (3)-(4), symmetries, the Watson theorem [20], etc. In the nuclear case, the graphs must be evaluated in the y-nucleus center- of-momentum frame (y ACM), where the amplitude UMu is rather complicated because of the baryon motion and recoil effects. Blomqvist and Laget [19] have shown how to produce a relatively simple set of photopion operators for use in a nuclear context by neglecting terms of the form (P,JM~)~ or higher. The resulting set of operators, used in DWIA calculations [16], furnished a viable description of the (y, 71) reaction region for most nuclei. Once this procedure is adopted, one can then make the requisite transformation from the photoproduction amplitude to the dynamically off shell photoproduction operator needed in the nuclear context. In a

A P’ fP

FIG. 5. The t-channel exchange of a charged vector meson for y + N + I[ + N.

40 ROSENTHAL ET AL.

coordinate space description that procedure takes the form of replacing the meson and baryon momenta by associated momenta operators, i.e., p + (A/i) V.

These developments in describing the photopion case have inspired several researchers to treat the photokaon reaction similarly. The earliest work was that of Thorn [lS] who lit a set of Breit-Wigner resonances to the photokaon amplitudes and found that, contrary to the pion situation, photokaon production receives only a weak contribution from s-channel baryon resonances. Indeed, Thorn found that the Born terms alone (Fig. 6) seem to give a reasonable description of the data, if supplemented merely by a t-channel K*(892) exchange (the K* is the strange partner of the p-meson). Thorn’s successful lit to data depends strongly on the strength of the KN/1 coupling constant, which he gave as 1.1 < g,,,,/,/& < 2.6. This KNA coupling is quite a bit smaller than the coupling deduced from hadron scattering. For example, the Nijmegen group’s OBE tits to hyperon-nucleon scattering [21] (which includes a hard core) provide an adequate description of the (admittedly sparse) hyperon-nucleon data with 4.16 < gKN,,/$< 4.71. Also, Dover and Walker [7] have deduced g,,, /& = 4.71 from the dispersion relation analysis of the K’p forward amplitudes of Baillon et al. [22]. There has been considerable debate over hadronic determinations of the value of g,,, ifi, with recent analyses mostly favoring values > 3.7. In models of the basic photokaon amplitude of Thorn’s type, it is impossible to lit reaction (1) even qualitatively using values consistent with those found from hadron scattering.

Recently, Adelseck, Bennhold, and Wright [is] reanalyzed the data available to Thorn; their purpose was similar to ours in desiring a description of the dynamics suitable for photohypernuclear studies. They performed several least-squares adjustments of coupling constants for various S- and u-channel intermediate states. We will discuss in detail only one of their characteristic fits, the one they called (Nl, N4). This lit includes

(a) the Born terms;

(b) K* exchange; and (c) two s-channel resonances N*(1470) and N*(1650).

We refer to this particular (Nl, N4) parametrization as the Ohio University (OU) amplitude (see Ref. [ 151 for other sets). The extracted value of gKN,,,/,/‘& for this amplitude is a rather low value of 1.03.

FIG. 6. The Born terms for y + p + K+ + A.


To study the range of permissible variation in existing data fits, we will compare the OU amplitude with an amplitude originally derived by one of us for application to the (K-, y) reaction [23]. This amplitude involves

(a) the s-channel resonances N*(1470), N*(1650), N*( 1710) (our least- squares adjustment required no (1( 1405) contribution);

(b) the t-channel tensor meson K**(1434) as well as the K*(1892); and

(c) the Born terms (see Fig. 6).

Our reason for including an additional t-channel exchange, the K**, was to mock up the background in a theoretically well-motivated fashion. At very high energies (k LAB z 10 GeVjc) the photokaon process is dominated by r-channel exchange and the reaction can be well described in a Regge exchange picture by interference between K* and K** trajectories. It is not unreasonable to expect some manifestation of the K** at lower energies where the K* is also known to be important. Use of this additional background term improves the fit but does not alter the discrepancy between electromagnetic and hadronic determinations of g,,, . Our best lit, which we dub the WMU-PGH set, also requires a rather low value of g,,,;& = 0.9.

In Figs. 7 and 8, the y + p -+ K+ + ,4 differential cross sections for the OU and WMU-PGH amplitude sets are compared at k LAB < 1.2 and 1.4 GeV/c. Both sets are seen to be in broad agreement with the data. A general feature of all such fits is that agreement with the forward angle data at 1.2 GeV yields too large a forward cross section at 1.4 GeV/c. This is a direct result of the Born terms; no value of g,,, permits a good fit to the forward data at all energies.

In view of such difficulties, it is probable that both models are ignoring some relevant physics. Vertex renormalizations and associated form factor effects are the

- WMU-PGH -

0 20 40 60 60 100 120 ocm

FIG. 7. Comparison of fitted y + p -+ K+ + A cross sections at k, = 1.2 GeV/c.

42 ROSENTHAL ET AL.

3.8

- WMU-PGH

FIG. 8. Same as Fig. 7, for kL = 1.4 GeV/c.

most obvious ones. The coupling constant is not strictly a constant but rather a vertex function evaluated at some value of its variables and these kinematic variables differ in the electromagnetic and hadronic experiments. For photokaon production at threshold one has (s, t, u) = (2.9,0.02,0.07) Mk, whereas for kaon-nucleon scattering at low momenta (s, t, u) = (2.3, O., 0.2) Mi; the associated vertex functions could differ appreciably at these two different S, t, u values and possibly account for the discrepancy between the electromagnetic versus hadronic determinations of the coupling constant. Another contributing effect might be a

TABLE I

Coupling Constants for the OU and WMU-PGH Amplitude Sets

ou This work

g,/& 1.03 0.915

%I& -0.807 - 1.22

G v/4x 0.220 0.212

G,(f$Z -0.048 -0.181

1.47 0.903

G6SOl& 0.111 0.080

Gl710/~ - 0.022

G,l4s 0.05

G,/4x 7.46

Note. Here g, is the K/IN coupling; G, denotes the KEN coupling, including the electromagnetic transition magnetic moment; G, and Gr are the two couplings involved in the K* exchange, the electric and magnetic couplings. The G 147,,,65,,dr i,,,, refer to the associated s-channel N*‘s and include the production vertex yNN* and the subsequent decay N*AK+ couplings. The G, and G, are defined as for G,, r, except for the K** meson case. For a detailed definition of the parameters see Ref. [ 15) and Appendix II.


1.0 1.2 1.4 1.6 I.6 2.0 2.2 2.4 2.6 KLAB(GeV)

FIG. 9. Predicted amplitudes for the two different model sets. Solid curves are for WMU-PGH; dashed for OU.

strong final state x+-A interaction. Not much is known about this interaction and at the momenta in question even the overall sign is in doubt. Additional processes that might remedy the presently unsatisfactory situation are two-step reactions such as P(y, n) N(rr, K+) and P(y, ti) C’(k?, K+). A more complete description of the amplitude including such coupled channels effects will become necessary as the data base improves. At the present time, either of the two model amplitude sets is expected to be adequate for the purpose of describing general features of hypernuclear photoproduction near threshold.

The full set of coupling constants for the two models (OU and WMU-PGH) is given in Table I. The procedure for including the K** meson is discussed in Appendix II; for the other interactions described by the Feynman diagrams of Fig. 6, we refer the reader to Ref. [ 151, which we have confirmed in detail. Most of the differences between the two parametrizations can be traced to r-channel effects, especially the different magnetic couplings of the K*(892). The K**( 1434) has a relatively small effect below k,,, = 2 GeV/c. The Fi are shown at 0 = 0” for the two models in Fig. 9. Only F, is nqnvanishing near threshold but the other amplitudes, especially F2, become significant at higher energies. Note that the two models are similar, but have distinctly different F, amplitudes.

Above 1.4 GeV/c, the p(y, K+ ) ,4 data are practically nonexistent and our amplitudes, necessarily obtained by extrapolation, must be regarded as speculative.

III. THE DISTORTED WAVE IMPULSE APPROXIMATION

The cross section for the photoproduction of hypernuclei via the reaction y + p + K+ + /1 can be evaluated using the distorted wave impulse approximation, which was developed earlier for the case of charged pion photoproduction [ 161.

44 ROSENTHAL ET AL.

The DWIA method, albeit not the most sophisticated approach, does offer the advantage of including the basic features of such reactions in a simple, readily accessible, and comprehensible manner. For selected nuclei, such as 12C 160, and “‘Pb, it has yielded reliable predictions for pion photoproduction, although for nuclei that are particularly sensitive to relined aspects of the photoproduction dynamics, such as 14N and i3C, only qualitative results are obtained using the simpler DWIA versions. Clearly, as the (y, K+) subject develops it will be necessary to include the full nonlocalities, relativistic, nuclear medium, and Watson theorem3 effects which have been invoked ‘as the pion nuclear photoproduction data unfolded.

Relativistic corrections to the standard (Schriidinger-based) DWIA have been investigated in a Dirac-based formalism by the Ohio University group [ 151 and by Cohen et al. [ 141 at Indiana University. These authors have used a relativistic mean field theory to model both the initial nuclear and the final hypernuclear wavefunctions and generally find substantial corrections to the standard DWIA. We have three reasons for not choosing this approach. The first is that although the reported corrections are large, they are also model dependent. For example, significant differences are found depending on whether the pAK+ vertex is taken to be of pseudoscalar or pseudovector form. In the simple version of the DWIA which we use here there is no such model dependence; all reasonable y +p + K+ + A r-matrices give essentially the same hypernucleus production rates. Second, we feel that the very weak binding of hypernuclei requires that realistic A wavefunctions be computed in a way which takes into account the effects of coupling to the continuum; in fact, we expect that such effects are likely to be more important than those associated with the lower component of a Dirac-spinor bound state wavefunction. However, the usual R-matrix methods of computing wavefunction in the continuum shell model cannot be readily extended to the Dirac equation [24] and rather than deal with the complications of a discretized continuum, we prefer to work with the Schrodinger equation. Finally, we wish to establish the predictions of an accurate non-Dirac model so that confrontation with experiment can help to decide the necessity of more sophisticated models.

At this stage, we feel that a highly sophisticated DWIA approach is not warranted for the kaon case; indeed, it would be better to see the need for such improvements develop with the future kaon photoproduction data.

The cross section for the process A(y, K+ ) B can be obtained in the y nucleus ACM system from the expression

3 The Watson theorem, using unitarity, states that the phase of the amplitude for photomeson production on a nucleon is specified by the elastic meson-nucleon phase shifts. For the nuclear case the use of the distorted wave xi-) includes that requirement to some extent, but consistency with the nucleon case needs to be incorporated. Blomqvist-Laget included the Watson theorem in their studies, and the nuclear case has been recently examined by Wittman, Davidson, and Mukopadhyay [20].


where the basic DWIA amplitude involves taking matrix elements of the photoproduction operator

HYK= J5 .A&’ V.. , (6)

between the initial and final states. The SU(3) ladder operator V_ changes a proton into a ,4 (or an up to a strange quark). Here the photon field is denoted by A,(r) = 2(k, ,I.) eA.r. The initial state consists of the photon, of momentum k and polarization I, plus the nuclear target described by a nuclear shell model wavefunction with angular momentum JiMi. The final state consists of the final hypernucleus, and its associated shell model wavefunction (with angular momentum JrM,), plus the final kaon wave which is described by the distorted wave x:-b.

The kaon distorted wave incorporates the effect of the final state kaon-hypernucleus interaction, which is unknown. As a first step beyond- using a plane wave kaon, we use a K+ nucleus optical potential, based on the Martin K+N amplitude and a Klein-Gordon equation, to stipulate the kaon-hypernucleus interaction. The K’ optical potential is taken to be of the simple form

uopt= -4nA(fic& 1 +z bp(r), K [ 1 (7)

where p(r) is the nuclear density and b is obtained from the KfN amplitudes using

Heref(0”) is the 2CM forward K+N amplitude and the brackets denote an average over the nucleon fermi sea motion. Typical values of b are shown in Fig. 10; these yield good (but not excellent) fits to the available K+-nucleus elastic scattering data.

k L(GeVlc)

1 1.2 1.4 1.6 1.8 2

k L (G&/c)

FIG. 10. (Left) Isovector kaon optical parameter b, for n ‘*B. (Right) Isoscalar kaon optical parameter b, for !,jB. Solid lines are real parts; dashed lines are imaginary parts.

46 ROSENTHAL ET AL.

The photokaon transition current density is related to the elementary amplitude by

(9)

where F, is the (y, KC) amplitude expressed in the yN center-of-momentum frame (2CM). The DWIA calculation is carried out in the y-nucleus center-of-momentum fram (ACM), so that F, must be calculated from the given dynamical variables k, q, 1. This is accomplished by making the following approximations:

1. The struck nucleon is taken to be at rest in the target nucleus, so that a well-defined 2CM can be constructed. The effects of averaging the Fi amplitudes over the nucleon motion, which should partially correct the frozen nucleon approximation, is found to alter Fi by less than 2%.

2. The Q dependence of the F, is neglected and the F;s are evaluated at @=O”, since the dominant Q dependence of the cross section is contained in the nuclear form factors p, p’, and b’; see later text.

3. The 2CM and ACM angles are assumed to be the-same so that we use cos 9, N cos 8, which neglects the “angle transformation” effect on the Sz, and 52, operators of Section II.

Hence a small angle approximation is used; this approximation improves with decreasing energy and leads to an error of some 5 % in F at 0 = 10” at the highest energy studied. See Appendix I for details of our method for constructing a transition current density.

These approximations, together with the assumption that the kaon’s transverse (2CM) and (ACM) momenta are equal (qcl=ql), allow us to map the 2CM amplitude into the ACM system,

-,F,cr.E^-~~.~q.k+iiFZ.(kxq~ qk kc

+ (E;+Fd F4

kc a.kq.t+qga.qq.E^.

c (10)

In the operator Sz, = IS. 4 @. E ,̂ which involves two meson operators, we use q = pi + k - pf to express the transition current density in terms of the ACM momenta, using the notation of Ref. [16], in the form

(11)


Now the transition current depends only linearly on the momenta pi, pr, and q. We can express the strength of each term in Js in terms of the CGLN amplitudes

=471 s J EA, EA,

F, - iFA - Fzlq F&w,

(F2 + r;, )/qc + kF,lqq,

- F&m

(12)

As discussed earlier, the a;s and the associated Fis are evaluated in a zero angle, no angle-transformation approximation and consequently the momentum and angle dependence of Js is contained in the operators above and not in the ais.

Following Ref. [16], the nuclear matrix elements of J, cause the baryon and meson momenta to be replaced by associated gradient operators. That is, pi -+ (h/i) Vi, pr+ (h/i) V,, and q + (A/i) V are used when the nuclear and meson matrix elements of J, are evaluated. Consequently, from the nuclear gradients, nuclear transition densities of the following types [16],

(13)

appear in our calculation of the (y, K+) amplitudes. Here b refers to the nlj initial single nucleon and a to the final nlj single /i quantum numbers. To construct the many-body transition densities, the above single-particle reduced matrix elements are combined with the many-body matrix elements, as for example in

Here P;Ib are the many-body nuclear reduced matrix elements Pib = (Jr11 [a: x ab]“ll J,), between the initial nucleus and the final hypernucleus. These matrix elements are taken from the continuum hypernucleus shell model calculations of Halderson et al. [lo]. The above matrix elements lead to the following angular momentum triangle addition selection rules,

d(Ji, Jf, JL d(joy jb, J), (15)

W’, S, J), A(& 1, L’),

where S = 0 or 1 and the last condition applies to p’ and p’. From the meson momentum dependence and its associated gradient operator

arises a dependence not only on the meson distorted wave but also on its gradient

595/184/l-4

48 ROSENTHAL ET AL.

V,&‘(r). The coefficients a, determine the role of each of these transition densities and hence we call ai the “transition strengths.”

The moduli of the coefficients [ail as calculated from the two model amplitude sets are shown in Fig. 11. This plot shows how the strength of each operator in J, (Eq. (11)) varies with incident photon momentum k, for the OU and WMU-PGH models. In these figures, a,, u8, and a,, have been multiplied by q, while a, and a13 have been multiplied by pi4 and prq, respectively, so that they are all dimensionally consistent. Clearly, the dominant components of J, . E ̂are a, (due to F,) and u, and a, (due to F,).

Note that the a, coefficient multiplies the operator cos 80. E ̂so that this component interferes with the a, contribution. The phases are such that there is destructive interference for both the OU and the WMU-PGH amplitude sets. The u, component is of a completely different type. It excites As = 0 states and therefore contributes to the richness of the spectrum. Note that the two amplitude sets predict very similar values of a,, a,, and u8.

Of less importance are the a,, a,*, and a,3 components, which arise largely from F3 and F4. The small value of these components is unfortunate since the models disagree substantially here, and they might otherwise have provided a means of discriminating between amplitude sets. Qualitatively, these components also contribute to As = 1 excitations.

These transition strengths combine with the associated transition densities and the photon and meson waves to generate the DWIA amplitude for the nuclear photokaon process. The amplitude can now be expressed as a sum of radial integrals

=jom r’dr f uipyi,(r) ~~~,(~)(JIM~IJ~JM,M). (16) ILL’

1.4 1.6 1.6

hL (GeVlc)

0.06 F a,3 _---- ----7

. , / , , , , I , .

1.4 1.6 1.6

h L (GeVlc)

FIG. 11. (Left) Predicted transition strengths for the two amplitude sets. Solid curves are for WMU- PGH; dashed for OU. (Right) Same as that for (left).


The sum over the index i incorporates the transition strengths associated with each type of operator in J5. The transition densities p, p’, and 0’ are now labelled p’, p*, and p3, respectively. The photon multipoles and the kaon partial waves are combined to form the photon-pion wavefunction @; see Ref. [16] for the detailed expressions. The important point about the photokaon amplitude is that the process is clearly determined not only by the transition strengths ai but also by the overlap of the transition densities pi with @. The kaon distorted wave x(r) is, along with the photon multipole, the major effect in the radial dependence of @. A large cross section results from a sizable ai combined with a good overlap of the above functions, all subject to the aforementioned triangle selection rules (Eq. (15)).

IV. LIGHT NUCLEI

Excitation of the (,p$ n~3,2)J=0+*‘f*2+,3+ “substitutional” multiplet of fi*C, whose angular distributions at k, = 1.47 and 1.84 GeV/c are shown in Fig. 12, conveniently summarizes many properties of this reaction. The distorting potential U,,, has been set to zero in the PWIA cases shown in Fig. 12; DWIA results are in Fig. 13. The most strongly excited state is the stretched J= 3 +. This strong

0.4

ti 10

(desees)

J

kL= 1 84 GeV/c

1

I+ k. 01

.’ 5 10 15 ti (degrees)

20

FIG. 12. Predicted photoproduction differential cross sections for a pure (,p;i ,,pjla) multiplet, calculated with only Coulomb distortions in a harmonic oscillator basis using b, = b, = 1.7 fm.

50 ROSENTHAL ET AL.

FIG. 13. Same as Fig. 12 with effects of full kaon distortion included.

excitation of the stretched state is a common feature of the reaction; in any multiplet the highest J state is the most strongly excited. The reason for the highest J state being excited is the following:

1. The large momentum transfers Q imply that the orbital momentum transfer AL - QR will also be large. We find that the radial transition densities pJL1L. have maximum overlap with the photokaon wavefunction when p/L>, carries the maximum allowable multipolarity AL = Li + L, (see Fig. 12).

2. The reaction proceeds largely via spin flip As = 1.

3. The two conditions above (maximum AL, maximum AS) together imply maximal J transfer: AJ= Ji + J,. The relevant reduced matrix element is (L&; Jr11 [ Y, x rrllJll Lii; Ji) and when L = Li + L, this is maximized when J= Ji + Jf.

The 2+ hypernucleus state, which is excited via the F2 amplitude, is next in importance while the smaller J states are even more difficult to excite (Fig. 12). This pattern persists at all energies although the relative magnitudes change. At higher energies the 2 + is enhanced both absolutely and relative to the 3 + (Fig. 12). At all energies, the angular signatures of these two states are sufficiently different to


permit unambiguous experimental identification, provided that detectors have the requisite energy resolution (that is a major qualification which will be discussed further below).

The effects of kaon distortion at the higher energies is shown in Fig. 13. There is roughly a 30% decrease in the predicted cross sections for 3 + and 2 + with no significant change in angular distribution. We have found this pattern to persist for many other transitions: the effect of distortion on the dominantly excited states can be generally represented by an overall change in normalization. Curiously, this change shows little energy dependence. That is, if one defines a distortion factor as the ratio of cross section maxima D = max(o,,)/max(o,,), then D depends only weakly on energy over the 600-MeV interval studied, although the cross sections themselves vary considerably.

We note that the distortion factors are only weakly dependent on J within a given multiplet, because these depend on the overlap of radial transition densities with kaon and photon wavefunctions and this overlap in turn tends to be dominated by a single transition density p. It is dominated by a single p because only the operators a,, a,, and a8 are important and these all have the same p.

The (,p$ ,,JQ) multiplet has a radial transition density which is relatively surface peaked. For a transition which peaks more in the nuclear interior, such as the (,s$ ,,s~,~)-‘=‘+, I+ multiplet, the distortion effects are much more significant, as shown in Fig. 14.

It is also of some interest to inquire into the kaon partial wave dependence of the differential cross section. We have found that at 1.84 GeV/c the cross sections do not saturate for A = 12 until 24 kaon partial waves are included, as shown in Fig. 15. The kaon momentum here is N 2.5 fin-’ in the ACM so that arguments from elastic scattering which relate kR N 8 to the number of significant partial waves would seem to be violated. These arguments, however, are not applicable to reactions; indeed, for reactions many more partial waves are needed than indicated

BCdegrees)

FIG. 14. Comparison of differential cross section using Coulomb waves and full distorted waves for a pure (,s;: ,,s,,~) J= 1 + state at k, = 1.84 GeV/c.

52 ROSENTHAL ET AL.

0 5 10 15 20

0 (degrees)

FIG. 15. Calculated 3 + cross section including only kaon partial waves L < L,,, for LMAX = 15, 20, and 25.

by the kR value, essentially because of having significant centrifugal barrier effects in the final state only. For photokaon production, 24 kaon partial waves for A = 12 and about 45 for A = 208 are required.

The representation of hypernuclear states as good particle-hole states is useful only as a first orientation. Realistic shell model calculations for A = 12 require realistic ‘*C and ji2B wavefunctions. We have calculated the expected kaon spectrum using the wavefunctions of the recoil-corrected continuum shell model (RCCSM) [25]. Given nonspurious shell model wavefunctions for the A-l system, the RCCSM provides a coupled channels wavefunction for the relative motion of the /1 in an oscillator basis. Such wavefunctions are in excellent agreement with (K-, zn-) and (n+, K+) data [ 10,251. The advantage of the RCCSM formalism is that bound states, resonances, and quasi-free scattering are all included in one consistent formalism.

The essential features are shown in Figs. 16 and 17, where the number of kaons is plotted against energy for 8=0” kaons. Although there is substantial quasi-free scattering, as previously anticipated by Hsiao and Cotanch [l 1 J, this scattering is spread over a wide energy range and we predict that the states of interest will not be obscured by a quasi-free peak.

In Figs. 16 and 17 we show the predicted spectra for ‘*C(y, K+) fi2B at k, = 1.8 GeV/c for 0” and IO”. One sees that the quasi-free region is composed of many spins and parities. The splitting among states in any given multiplet is very small; for example, the chances of resolving the (,p$ n~1,2) states are nil. Some hope exists for determining the 2+-3 + splittings by peak fitting, provided angular distributions are available. The clear need for angular distributions is shown in these figures. Note, for example, that the 2+ spectra change considerably from 0” to 10”.

The lowest 2+ state is primarily an As= 0 excitation and should show up strongly at forward angles in (71 +, K+). In (y, K+), however, the lowest 2+ shows


___ Total - -.- 2- ---- ,-

;:

t&=00

B, (MeV)

~ Total

B,(MeV)

FIG. 16. (Top) The total of all excitation functions and the individual negative parity excitation functions for W(y, K+) ,\*B at 8,, = 0” at E, = 1.8 GeV. The total, 4-, 3-, 2-, and l- excitation functions are shown. (Bottom) Same as that for (top) showing the positive parity excitation functions for 3+, 2+, and I+.

up only away from 0” and appears as a shoulder on the 3 + peak. Clearly, from the discussion above, the stretched states will dominate the spectra of light nuclei. The addition of these unnatural parity states to the known spectra of hypernuclei will provide not only tighter constraints on the effective A-N interaction but also information on the quenching of A spin excitations in the nuclear environment. Possible sources of quenching in conventional nuclei are configuration mixing, core defor- mation, and meson exchange currents. The weak A-N interaction should eliminate configuration mixing as a source of quenching and any observed reduction of spin excitation strength must be attributed to other mechanisms. The lack of configuration mixing in ,4 hypernuclei is already evident from strangeness exchange reactions which show no collectivity in the natural parity states which they excite.

54 ROSENTHAL ET AL.

n I I I ‘2C(y,K+)1$ 1

B, (MeV)

IZ-

2

I I 1 I I I

f IO - 12C(y,K+)‘;B

k a

1

cd* e - Total

‘0 ---3+

3 --------2+ -.-.-0+

BA (MeV)

FIG. 17. (Top) Same as that for Fig. 16 (top) for 0= 10”. (Bottom) Same as that for Fig. 16 (bottom) for 0 = 10”.

This paper presents calculations which include distortions and full elementary amplitudes. Therefore, when combined with realistic hypernucleus wavefunctions, these calculations will predict cross sections which can be compared directly to experiment to determine quenching effects. The most appropriate states for quenching studies are, in fact, the stretched configurations. In addition to being the strongest states in the spectra due to their high spin, they have a minimum of contributing transition densities. A survey of stretched /i-hypernuclear states was made in Ref. [lo] with the RCCSM. Shown in Fig. 18 are the ‘*B(3+), fPN(3+), iE Al( 5 + ), and TK(5 + ) cross sections for (y, K+ ) at kLAB = 1.8 GeV/c calculated with RCCSM wavefunctions. The fpN, ,, 28Al, and 4,OK states are above LI separation threshold and Fig. 23 shows the strength of these states integrated over an energy range of six-half-widths. The :*A1 cross section has been multiplied by 0.78, which is


FIG. 18.

,,,t / \

0 (degrees)

Energy integrated excitation functions for stretched contigurations.

the ratio of %i proton pickup strength in the ground state to total observed pickup strength. Although shown as one state, the ,, 40K(5 + ) will be split into three states according to the z , 5 + I = 2, proton pickup strength to the 39K ground state. One would then be comparing Fig. 18 to the sum of observed 5+ strength.

V. HEAVY NUCLEI

It has been suggested that the relatively weak absorption of K+ mesons by nuclei might imply reasonable counting rates for the (y, K+) reaction when the final hypernuclear state consists of a /l-particle deeply bound in a heavy nucleus. The ‘O’Pb(y, K+) ;08Tl reaction to the (ph,;f2n~,,2)J=6- “state” has been frequently cited as a possible candidate. This suggestion has excited some interest since it

FIG. 19. “sPb(y, K+) ~‘*Tl cross section to an assumed pure (+JI~~,~,,s~$)J=~- state at k LAB = 1.84 GeV/c.

56 ROSENTHAL ET AL.

Hdegrees)

FIG. 20. Same as Fig. 19 for k,,, = 1.2 &V/c.

holds out the possibility of studying a /i-hyperon in nuclear matter, a possibility which is unobtainable by any other means.

We have shown in earlier work that this (y, K+) process is strongly suppressed by kaon distortions at kLAB = 1.8 GeV/c, as shown in Fig. 19. Figure 20 shows that the relative independence of distortion factors on energy implies a similar conclusion at k,,, = 1.2 GeV/c. The distorted wave result is suppressed below that of the plane wave by about a factor of 5 at all energies studied. Configuration mixing, which is not negligible for such a large A, will suppress the cross section even further, so that we are not optimistic that proposed facilities can be used to study this process. Transitions which are more surface peaked should have substantially greater cross sections and may well be observable.

VI. CONCLUSIONS

The primary reason for producing hypernuclei electromagnetically is to study the As #O transitions which can be excited only weakly, if at all, in strangeness exchange reactions. Such states will complete our knowledge of hypernuclear spectra and transition rates and thereby impose restrictions on models of the /1N force. Ultimately, one hopes to relate this force to the basic interactions among quarks. In this section we summarize the major advantages and difficulties of (y, K+ ) studies for this purpose.

An important conclusion of this work is that although our understanding of the basic y + p + K+ + /i interaction in the N* resonance region (between & z 1 GeV and AZ 3 GeV) is sketchy and incomplete, the predicted DWIA hypernuclear excitations tend to be insensitive to the detailed form of the photokaon operator.

HYPERNUCLEARPHOTOPRODUCTION 51

Models which adequately describe the photokaon cross sections and polarizations predict essentially the same rates for production of hypernuclear states, despite their differences in underlying dynamics. This conclusion, however, holds only within the framework of the usual DWIA. Propagator renormalizations within the medium are sensitive to dynamics and have been shown to be significant in a Dirac-based extension of the DWIA [ 13, 151. Such formalisms are not free of theoretical uncertainties and we feel that a breakdown of the standard DWIA should be revealed directly in the photoproduction data before invoking more sophisticated approaches.

It has been shown in earlier studies that each hypernuclear multiplet excited in (y, K+) tends to be dominated by a single state, usually the state of highest J. We find that this domination is modified when nonstatic terms are included in the photoproduction operator and attention is paid to nonzero angles. Thus, for example, the 2 + is roughly half as large as the 3 + in the B, = 0 peak of Fig. 22. These nonstatic terms (mainly those due to F,) have a characteristic angular dependence which will assist in the identification of the broad peaks. Although there is little hope for learning about the energy splittings between the states of interest, good angular distributions will therefore allow us to extract spectroscopic factors for these states.

We note that both kinematic and dynamic considerations lead to the preferential excitation of high J states. These stretched states are expected to be relatively pure so that uncertainties introduced into the analysis by the requirement of estimating a transition form factor at very high Q are at least partially compensated by the relative simplicity of the nuclear structure. This is encouraging in view of our ability to successfully predict (y, rc) cross sections to such simple states in many ordinary nuclei.

Better measurements of the two-body cross sections y + p -+ K+ + ,4 would decrease the uncertainties of these calculations considerably. One should also address the issue of the K+-/i interaction and its possible effects on the optical potential. Neither of these difficulties present insuperable problems and we look forward to the next generation of experiments to decide whether the conventional DWIA (with attention paid to the particularities of hypernuclear structure) can account for the unique nuclear processes that will be explored at the new facilities.

APPENDIX I: CONSTRUCTION OF THE TRANSITION OPERATOR

We use Feynman graphs to construct a Lorentz invariant amplitude M. Evaluated in the 2CM, this leads to a unique set of Pauli spinor amplitudes F,. These amplitudes are evaluated at 0 = 0” and all momentum dependences of the Fi are subsequently ignored; the off shell properties of the transition operator are solely those of the Gi; see Eq. (4). We give an example of this procedure below.

Consider the contribution of the Zhyperon, which contributes to the (y, K+)

58 ROSENTHALETAL.

process via a crossed s-channel diagram as illustrated in Fig. 6 (right). The CGLN amplitudes, Aj, are given in the 2CM by

where gKNZ is the pseudoscalar coupling constant, ~1 is the C” + n + y transition magnetic moment = - 1.37 NM, and u is the usual Mandelstam variable u = (p, -lz)*. Expressed in terms of Pauli spinors, the amplitude tiMu has the following coefficients (see Eq. (lo)),

Fl F2 [I k e%KNz 1

F3 =G 2&f, (u-M$)

F4

Here W= & is the total invariant 2CM energy. The quantities k, P,,, pZ, and q are magnitudes of ACM vectors determined from kLAB by Lorentz transforming, assuming that the kaon and photon are colinear 4 = f. The transformation from 2CM to ACM yields the F;s needed in Eq. (10). The Fi coefficients are simply numbers, as expressed here in momentum space. They are real above; they become complex numbers when complex masses are used to represent unstable intermediate state particles in the relevant Feynman diagrams. Since the baryon momentum to mass ratio is not small, we cannot carry out the Blomqvist-Laget expansion used in the photopion case. Instead, we have used the zero angle approximation to evaluate the above F, quantities and do not replace the momenta vectors therein by gradient operators; only the Q,‘s are made into such gradient operators.

APPENDIX II: CONTRIBUTION OF A TENSOR MESON TO PHOTOPRODUCTION AMPLITUDES

Although only a small contribution, we present our treatment of the tensor meson (K**) contribution to photokaon production here, since it has not previously appeared in print. It also illustrates the procedures used to extract first a 2CM amplitude and then a ACM photomeson operator, starting from Feynman


diagrams. For the other diagrams of Fig. 6, see Ref. [15]. We consider the electromagnetic coupling of a tensor meson, the K**, and the strong coupling of the K** to the N and /1. These couplings contribute to (y, K+) as the t-channel exchange illustrated in Fig. 6.

Denoting the 2 + meson by T, the most general y TK gauge invariant and covariant interaction is

L YTK = -ig~pvor Fp” a” T”, a= K + h.c.,

where Fe’ is the electromagnetic field tensor, K is the pseudoscalar kaon (K+) field, and T denotes the K** tensor field. The above form leads to a (y, K+, K**) vertex of the form

(K(q), T”,(Q) I -iLyTKi Y(&, k)) = - &,,,,(kpEa - kQEp) Q”q’.

Here the k, q, Q, etc. denote four-vectors. Similarly, the most general TN + A vertex is

= FAQ2)Cr& + YJ,I + Fz(Q2)CQpQv - g,,Q21 + Nq2) P,,P,,

where P = p,, + p,,, and Q = P, - P,. Only the first and third terms contribute to the photoproduction amplitude and we obtain the following expression for the invariant amplitude due to the r-channel tensor meson exchange,

M = - id~pv,,W %OL -kY‘) Q’q’} DaPoS { CG(P,Y, + P,Y,) + c4P,P,l>?

where the tensor meson propagator is

D, P”fi = -i[dp, d”fi + da, &‘“- $d”, @]/(@- Q’)

using d,,” = g,, - Q, QY/M$ and with C, = F,(Q2) and C, = F3(Q2) both evaluated at zero degrees. The structure of the above equation simply reflects the associated Feynman diagram; namely a strong interaction part, a K** propagator, and an electromagnetic yK+ K** coupling.

After algebraic reduction, we can cast the tensor meson exchange contribution into the CGLN form of Eq. (3) and express the amplitudes A, as

60 ROSENTHAL ET AL.

where

Y, =4C’,[M,M, A~- +M,s-A4,u]

+C,(M~-Q~~A~[2A~‘-Q~]/M~+C,x[(A~~-Q~]/M~

Y2=C,(MZ,-Q2)A&4-C,4x/M’,

Y, = 4C,(M; - u) - C,( t - M;) A _ (M, + 3M,)/M; + 2C, xM,/MZ,

Y, = 4C,(s - M;) - C,(t - M;) A- (3M, + M,,)/@+ 2C,xM,/M~,

where s, t, and u are the usual Mandelstam variables and A k = M, f M,,

x=MZ,[t-M:+2(s-M;)] +A- A+(MZ,-t).

The coupling constants given in Table I are G, = egC,M* and G, = egC, M3, where M= 1 GeV. Note that we do not expand in powers of (p/M,), which was done by Blomqvist-Laget [19] for pion photoproduction, since the kaon case involves larger kaon and baryon momenta. Instead, the above expression is evaluated in a zero angle, fixed baryon approximation as discussed in the text and in Appendix I. Thus from the above Ai amplitudes, one can extract the Fj and the ACM versions of the ai amplitudes, which can be used in DWIA calculations.

ACKNOWLEDGMENTS

We thank G. E. Walker very much for suggesting this collaboration. We also thank S. R. Cotanch, .I. Cohen, L. E. Wright, and A. M. Bernstein for several helpful discussions.

REFERENCES

1. F. GROSS, in “Proceedings of the CEBAFjSURA 1986 Summer Workshop” (Franz Gross and Ralph Minehart, Eds.), Continuous Electron Beam Accelerator Facility, Newport News, VA, and articles therein.

2. A. M. BERNSTEIN, private communication; Bates proposal, 1987. 3. R. S. HAYANO et al., in “Proceedings of the 1986 INS International Symposium on Hypernuclear

Physics,” (H. Bando, 0. Hashimoto, and K. Ogawa, Eds.), Institute for Nuclear Study, University of Tokyo, Japan.

4. B. POVH, Annu. Rev. Nucl. Part. Sci. (1978), 28. 5. Much of the 1986 INS Symposium, op.cit., was devoted to this subject. 6. M. EPHERRE-REY CAMPAGNOLLE, INS Symposium, op.cit. 7. CARL B. DOVER AND GEORGE E. WALKER, Phys. Rep. 89 (1982), 1. 8. R. H. DALI~Z AND A. GAL, Ann. Phys. 116 (1978), 167. 9. C. MILNER, et al., Phys. Rev. Lett. 54 (1985). 1237.

10. DEAN HALDERSON, et al., Phys. Rev. Lett. 57 (1986), 1117. 11. T. W. DONNELLY, A. M. BERNSTEIN, AND G. N. EPSTEIN, Nucl. Phys. A 358 (1981), 195. 12. S. S. HSIAO AND S. R. COTANCH, Phys. Rev. C 28 (1983), 1668.


13. STEPHEN R. COTANCH AND SHIAN S. HSIAO, Nucl. Whys. A 450 (1986), 419. 14. JOSEPH COHEN, Whys. Lelt. B 153 (1985). 367; Phys. Rev. C 32 (1985), 543; CEBAF Summer

Workshop, op.cit. 15. R. A. ADELSECK, C. BENNHOLD, AND L. E. WRIGHT, Phys. Rev. C 32 (1985), 1681. 16. M. K. SINGHAM AND F. TABAKIN, Ann. Phys. 135 (1981), 71. 17. G. F. CHEW, M. L. GOLDBERGER, F. E. Low, AND Y. NAMBU, Phys. Rev. 106 (1957), 1345. 18. H. THOM, Phys. Rev. 151 (1966), 1322. 19. I. BLOMQVIST AND J. M. LAGET, Nucl. Phys. A 280 (1977), 405. 20. R. WIITMAN, R. DAVIDSON, AND N. C. MUKOPADHYAY, Phys. Left. B 142 (1984), 336. 21. M. M. NAGELS, et al., Phys. Rev. D 12 (1975), 744; 15 (1977). 2547; 17 (1978), 768; 20 (1979), 1633. 22. P. BAILLON, et al., Phys. Lett. E 50 (1974), 383. 23. H. BURKHARDT, J. LOWE, AND A. S. ROSENTHAL, Nucl. Phys. A 440 (1985), 653. 24. A. S. ROSENTHAL, J. Phys. G 13 (1987), 491. 25. D. HALDERSON, P. NING, AND R. J. PHILPOTT, Nucl. Phys. A 458 (1985), 605.

Date post:	14-Nov-2023
Category:	Documents
Upload:	independent
View:	0 times
Download:	0 times

Hypernuclear photoproduction

Documents