PHYSICAL REVIEW D VOLUME 39, NUMBER 7 1 APRIL 1989

Spectrum of QCD and chiral Lagrangians of the strong and weak interactions

John F. Donoghue, Carlos Ramirez, and German ValenciaDepartment of Physics and Astronomy, Uniuersity of Massachusetts, Amherst, Massachusetts 01003

(Received 14 November 1988)

Chiral Lagrangians contain in their coe%cients information concerning the underlying fundamen-tal theory. We show that to a large measure this structure is determined by a duality with the low-mass spectrum of the theory. This is tested in some model theories and then applied to QCD. Inthe real world, only the vector mesons are light enough to pass our criteria for being low mass, andin practice they do seem to dominate the structure of the phenomenological Lagrangians. Thisraises the possibility of a fusion between vector-dominance ideas and rigorous chiral Lagrangianmethods.

I. INTRODUCTION

The theory of effective chiral Lagrangians' providesa compact and elegant method for dealing with the in-teractions of the Goldstone modes of a theory. Theseeffective theories can describe all of the couplings interms of a relatively small number of parameters. How-ever, in the case of a complex theory such as QCD theseparameters are not yet capable of being predicted directlyfrom the theory, and instead are obtained phenomenolog-ically. This is unfortunate, as the parameters contain agood deal of information on the structure of the theory.In this paper, we will explore the idea that the form ofthe effective Lagrangians is constrained by the low-lyingresonance (non-Goldstone) spectrum. The result will bethat in most cases the parameters of the chiral Lagrang-ian, which in principle can be determined from very pre-cise, very-low-energy data, will at somewhat higher ener-gy contain information relating to the spectrum. Forlack of a better name, we will refer to this idea as chiralduality, and will apply it to QCD in order to obtain con-straints on the predictions of the chiral Lagrangian.

The effective Lagrangians are organized in an expan-sion in terms of the energy. The lowest-order Lagrangianin chiral SU(2) is

F2 2 2

Tr(t)„Mt)"M )+ Tr(M+Mt),

(called order F. throughout the paper) that the theoriesmay be distinguished. The generic structure at order Ewill be reviewed in Sec. II, and will contain two hadronicparameters (in the massless limit) undetermined by anysymmetry consideration. These will be the subjects ofthis paper.

The idea in its simplest form follows from the dual roleof scattering processes as being a probe of the particlespectrum while also being governed by the underlyingchiral Lagrangian. Consider for definiteness m-~ scatter-ing in the I= 1,J= 1 channel. We know that if this exper-iment is performed we will see the p resonance at E=770MeV, with the result displayed as idealized data in Fig. 1.The lowest-order chiral Lagrangian "knows" nothing ofthis physics. It predicts scattering as shown by the solidline in Fig. 1, given entirely in terms of F and mHowever, the chiral Lagrangians at order E govern thedeviations from the lowest-order prediction and the infor-mation is contained in the two new parameters. These

where

.7'7T'M =exp i

with m', i=1,2,3, being the pion fields. The parametershere are only F and m, where F is the pion decay con-stant normalized so that the experimental value isF =94 MeV. At this order, there is not much informa-tion in the form of the Lagrangian. This is because alltheories with a slightly broken chiral SU(2) symmetrymust have this same form. For example, the linear o.model, the nonlinear o model, and QCD would all reduceto this form at lowest order, despite being very differenttheories. It is at the next order in the energy expansion

FIG. 1. Scattering of m-~ in a channel where a resonanceoccurs. The figure is similar to the I =1,J=1 channel contain-ing the p(770). The low-energy region I is not very sensitive toorder E corrections because E is small. In the high-energy re-gion III, all powers in E are required and the perturbative ex-pansion in E has broken down. It is region II which is mostsensitive to the E corrections in a chiral Lagrangian. The solidline is the lowest-order prediction of chiral symmetry, while thedashed curve is one fit to include the order E corrections.

39 1947 1989 The American Physical Society

1948 JOHN F. DONOGHUE, CARLOS RAMIREZ, AND GERMAN VALENCIA

could be fit at low energies and, in our idealized situation,would give a total scattering amplitude of the form of thedashed line. Clearly these could never by themselvesreproduce the full p resonance shape. However, theirsign and magnitude are constrained by the need to matchthe cross section rise for the resonance. When this idea isexpanded to include other channels, it seems promisingthat the low-lying spectrum of the theory ean provideconstraints and even a determination of the parameters ofthe chiral Lagrangian.

The main thrust of our interest is theoretical. The in-formation about the spectrum of a theory is often easierto obtain than the corresponding knowledge about thechiral Lagrangians. For example, in QCD, both rigorous(in principle) lattice Monte Carlo techniques and thequark model can address the resonance spectrum, but nocomparable method exists for the chiral Lagrangians.However, if this chiral duality holds, then the spectruminformation can be used to predict or constrain the chiralparameters of that theory In . the case of QCD, the in-terest is primarily to learn what physics input is requiredto solve the low-energy limit of the theory. In other con-tests ideas similar to chiral duality could be more usefulin other ways.

There are many similarities to older work on pole mod-els and vector dominance and to aspects of investigationinto Skyrmions. Closest in intent is the chiral Lagrang-ian work of Gasser and Leutwyler. Since this work hasbeen completed we have also learned that related ideasare under investigation by de Rafael, Ecker, Gasser, andPich.

Our prime focus is on general ehiral theories. In Sec.II we describe in more detail the proposed connection ofchiral Lagrangians with the spectrum. This is then ex-plored in a pair of solvable but nontrivial theories in Sec.III. Section IV is a rather technical interlude where themagnitude of various resonant intermediate states areea1culated in order to decide if an effect is "large. " Thisis then applied to the real world and QCD in Sec. V, lead-ing to a result reminiscent of vector dominance. A sum-mary, Sec. VI, recapitulates the conclusion and givessome suggestions for applications within QCD and in ex-tension of the standard model (technicolor, etc.). In anappendix we comment on the information known aboutE terms in the weak interaction.

II. CHIRAL DUALITY

Consider a chiral SU(2) theory in the massless limit.The effective action can be expressed in an expansion inthe number of derivatives acting on a nonlinear chiralmatrix M:

L —L, +4+F2

L = Tr(B„MB"M ),

L~= [Tr(B„MB"M )]

+ Tr(B MB~ )Tr(B"MB'Mt),4 P

where M contains the pion field

. ~'7T'M =exp i

with i =1,2,3. If one algebraically expands the lowest or-der effective action I 2, and takes matrix elements, oneobtains relations among amplitudes which are the sameas obtained by PCAC (partial conservation of axial-vector-current) current-algebra soft-pion theorems, asmust be the case since the predictions follow only fromthe chiral symmetry. The effective Lagrangian allowsone to address systematically corrections to these lowest-order relations. Such corrections are given by I.4 and byloop diagrams. The reader is referred to Ref. 1 for an ex-position of this method and to Refs. 2 and 7 for its im-plementations.

The parameters a& and o'.2 will govern the deviation of~-m. scattering from the lowest-order predictions of Wein-berg. Elsewhere we have shown that, for the scatteringin a channel with isospin I and angular momentum J, theamplitude rJ has the form (at the tree level)

To =2

s +( l la, +7a2) +O(m ),1 s16m F 3F

1 sT', = s+(az —2a, )96mF F

—1 S 2

To = s —3(ai+2a2)32+F F2

2Q2+ Q )T—4 s

240~F

2' 1 +CL2

T2.— S480vrF

where s is the square of the center-of-repass energy. Inparticular, for m.-~ scattering to reveal the presence of alow-lying resonance, the parameters a; must be such thatthis deviation reproduces the low-energy effect, or "tail, "of the resonance. The notion of chiral duality is that theparameters in the chiral Lagrangians should be such thatthey are compatible with the low-lying spectrum of thetheory. The lower in mass the resonance is, the strongerits effect will be on a;.

When is the resonance the dominant effec'? An appre-ciation of the scales of chiral symmetry is useful inanswering this question. We adopt an argument original-ly given by Cxeorgi. ' Loop diagrams will naturally gen-erate a scale dependence to the renormalized parameters,of the form

a;(p) =a; (po)+b, lnpo/p

where p and po are two different scales which can be usedto define the theory. If one tries to choose a, (po) to bemuch smaller than b, , this choice is not natural, as adifferent choice of scale would yield a,.(p)-b;. However,cz,- -b; is natural and is an indication of the level at whichquantum effects contribute to a;. These scale depen-

39 SPECTRUM OF QCD AND CHIRAL LACxRANGIANS OF THE. . . 1949

r,F'.—a, =a2=48~ =0.0084 (7)

using the real p as an example. The factor of I /m iscommon to all resonances, and it is clear that a resonancewith a low enough mass mill be dominant, and that the pmay well satisfy this in practice.

III. TRIAL MODELS

It is clear if one introduces a single resonance whoseonly coupling is to pions and then integrates out the field,as we do in the next section, that the resulting parametersa; will be compatible with the presence of the resonancein ~-~ scattering. It will work because it is constructedto do so. On the other hand, one cannot test the idea ofthis duality in a complex theory such as QCD, as onecannot solve the theory. What we can do is to consider acouple of models with a resonance explicitly in thetheory, but which also have nontrivial couplings and/orextra fields. In such a case, is the resonance still stronglyvisible or do the extra fields or interactions produce a ma-jor modifications? We will address this in (1) a gaugedversion of the chiral model with vector and axial-vectorfields with Yang-Mills-type coupling among themselvesand (2) the linear o. model treated in the perturbative re-gime. The answer appears to be that it is the resonancewhich is the dominant effect on the parameters.

A. The gauged chiral model

In this model, vector and axial-vector mesons are add-ed to a chiral Lagrangian by treating them as if they weregauge particles of a local chiral symmetry. However, amass term must also be added, so that the result is not atrue gauge theory. In particular, one introduces a covari-ant derivative

dences have been calculated and in our normalizationlead to the following estimates of the size:

1b& =

&0 001

96m

1az-b2= z-0 002,

48m

i.e., a few parts in a thousand. A contribution to thecoefficients will be considered "large" if it is much abovethese values. We will see in Sec. IV that the resonancecontribution can be of order

[p(770) and A(1270) in modern notation]. The Lagrang-ian then has the form

FTr(D„MD "M ) —,'Tr(L—„L"+R„,R" )

+MOTr(L„L"+R„R")+BTr(L„MR "Mt),

L„,=a„L,—a~„—ig [L„,L.],R„.=a„R.—a~„—ig [R„,R.] .

The model then contains two sets of spin-1 particles(L„,R„or V„,A„) with triple and quartic self-couplings,plus couplings to the chiral field M. The last term, withcoefficient 8, is not in principle essential, but allows themodel to be reasonably compatible with phenomenology.

By its construction this model must have a pole in theI=1,J=1 ~-m scattering channel. The axial-vector parti-cle is pushed to higher mass by mixing with the pion.Our prime question is whether the O(E ) coef5cients ofthe Lagrangian clearly reAect the effect of the p, orwhether the extra axial field and Yang-Mills-type self-interactions obscure this feature at low energy. In orderto do this we integrate out the spin-1 fields and obtain theresulting low-energy Lagrangian. Formally,

exp i Jd x L,z(M)

= J [dL„][dR„]exp i Jd x L,z(L„,R„,M)

(10)

The quadratic portion may, of course, be done exactly.Because the coupling g need not be small, a usual pertur-bation series is not applicable. However, we are after anexpansion in the number of derivatives, and working toorder E truncates the tree diagram at those of Fig. 2.We do not include the loop effect for J „and R„,partial-ly because we view this only as an effective theory andpartially because of the problems of massive gauge parti-cles without the Higgs mechanism. The result, after asignificant amount of algebra, is

9„M—+D„M =3„M—igL, „M+igMR„,L„=—,'( V„+A„)=—,'r'( V„'+ A„'),

R„=—,'( V„—2„)=—,'r'( V„'+ A „'), (b&

where V' and A„' are SU(2) triplets which would beidentified with the p and A1 mesons in the real world

FIG. 2. Diagrams which generate higher-order corrections tochiral Lagrangians. The X represents vertices containing thechiral matrix M, while the lines are heavy meson propagators.

1950 JOHN F. DONOGHUE, CARLOS RAMIREZ, AND GERMAN VALENCIA 39

r,F'.o; = —a = —48~1 2

PlP

1

4 2 3/2

I—2m

(12)

B. The linear o. model

F2L, (M) = Tr(a„Ma„M )

2F4+ —,'Tr([a MMt, a~Mt]

8m 4 P

X [a"MM, a"MMt]),

F =F /(1+g F /2m )

g 2F2Sp=g

P

Note that F and p (and not Fo and g) are the physicalpion decay constant and pm+ coupling. Expressed interms of our previous notation and the p width theanswer is

Let us define the effective Lagrangian by integratingout the scalar field

r

exp i f d x L,~(M) = f [ds]exp i f d4x L(s,M)

(18)

This can be organized in a perturbation expansion. Thequartic coupling A, must be taken small enough for per-turbation theory to apply. The diagrams which contrib-ute are also given in Fig. 2. The x in the graphs is of or-der E, so that in order to obtain the Lagrangian at orderE only two x factors need to be considered. At the treelevel this gives

2L(u)= Tr(a Ma"M )

4 p

2

+ f d y Tr(a„Ma&M )„D(x —y)

XTr(aPfa M ) + .

2

Tr(a„Ma"Mt)

Another trial model which can be used to test the effectof interactions is the linear o. model. This is a fully re-normalizable theory which has been worked out to oneloop. Here the extra field is a I=O scalar meson, withpolynomial coupling to pions. We treat the theory in theperturbative regime (A, not too large).

The linear 0. model can be written in terms of the ma-trix field

2

+ [Tr(a MP'M )] + .4m'

S

where in the last step we have Taylor expanded

Tr(a~a'M ) =Tr(a~a"M')

+(y —x) a [Tr(aPfa'M )]„+

(19)

X=—o. +i ~.~',2

L =—'Tr( a Xa"Xt ) + Tr( X"X )4 p 4

(13)(20)

dropped higher-order terms, and integrated over the spropagator. Written in terms of the width of the s,

The ground state of this theory may be chosen at

& X)=(a & =u =&p'/~ .

A convenient choice for the fields after symmetry break-ing is

this yields

16~I, „43 m'

S

3mI (samer)=

64~v

1CZ2=0,

(21)

(22)

X=(u +s)M,

In this case, the Lagrangian expanded about theminimum of the potential is

L = —,'(u+s) Tr(a„Ma"M )

+—'(a sa"s —2p s ) —A.us ——s2 p 7

m, =&2@ . (17)

which contains, as expected, massless pions and a massiveself-interacting scalar s with

exactly the same result as our non-self-interacting scalarof the next sections.

It is interesting that the self-interactions of the s modi-fy the tree-level Lagrangian only starting at order E .Moreover it is clear that pion loops will not dominate fork small enough. This is because the only scale for pioninteractions is u = t/p~/A, , and pion loops cannot involve

p or A, separately. Therefore the dimensionless constantsa& and a2 can only be a pure number (estimated of order1/48vr in Sec. II) and cannot have a factor of I/A, asgiven in Eq. (22). In perturbation theory the I/A, factor islarger than the pure number. Loops of s particles appearin a way as to renormalize A, and v. The full one-loop re-normalization is only modified by a pure number, result-ing in

39 SPECTRUM OF QCD AND CHIRAL LAGRANCsIANS OF THE. . . 1951

ma)(p) = + ln

8g 96~2 p2

1 ms 11a2(p) =

2ln z-

48n p

356

(23)

For a scale p=m„ the resonance contribution will bedominant for k &40, i.e., throughout the perturbative re-gime.

These trial models illustrate how different theories leadto distinctive predictions for the E chiral Lagrangian.In each case, the lowest-energy resonances are the primedeterminate in the chiral coefficients, with results essen-tially the same as found for simple fields with no self-interaction.

IV. RESONANCES AND THEIR CHIRAL COUPLINGS

In this section we consider particles whose quantumnumbers match those of m-m scattering, so that theirspectrums could in principle be revealed in these reac-tions. We couple these to chiral fields and obtain theirlow-energy effect on chiral Lagrangians. This will allow

. us to estimate how low in mass a resonance must lie inorder to strongly inAuence the spectrum.

The five channels of relevant for m-a scattering below1 GeV are (I=0,J=0), (I =2,J=0), (I=1,J=1),(I=O,J=2), and (I=2,J=2). We will use particleswith these quantum numbers, named, respectively, cr, P'~,

p„', f„, and h'~, where i,j=1,2,3 are isospin indices.The chiral couplings of o and f„are easy to write down,and we will use

L =g cr Tr(a„Md"M'),

Lf gff„,Tr( B" MB M )(24)

where g and gf are some coupling strengths which willlater be eliminated in favor of the particle widths. Forisotensor fields, the interaction is a bit more complicat-ed. 9 The field M transforms under SU(2)l X SU(2)i, rota-tions as

M —+M' =I.MR (25)

However the matter fields P'~ and h'~ should not havechiral transformation properties, but should transformvectorially. The Inethod for constructing this involvesthe fields g defined as

(26)

L~=g~g'~Tr(r'g d"Mg rjgd„M g),Li, =gi h„'~ Tr( r'g 8"Mgrj(B M g) .

(29)

The case of the p has already been treated in this frame-work by Gasser and Leutwyler.

At low energies the effect of the heavy mesons on thelow-energy physics can be found by "integrating out" theheavy fields. The result is equivalent to one-meson ex-change with the propagators evaluated at q =0.Specifically we obtain a Lagrangian of the form of Eq. (2),with the five cases contributing:

m

I ~Fa, = ——'a2= —4m-

1 4 2my

I Fp: cz1= cz2 = —48~

mp

160m' fCX = —0!1 3 2

mf

I Fh: a = —7a =280~

1 2 5mp,

(30)

g: m «6F„;m ~7F

p: m «9F

f: m ~12F„;h: m «13F

(31)

Here we have eliminated the coupling constants in favorof the particle's width. Note that all are proportional tothe same combination of widths and masses, up to overallnumerical factors.

For what masses will these contributions be most im-portant? If one is dealing with a general chiral theory,rather than QCD, which we feel is the real theory ofpions, this answer will depend on the scale of F . How-ever, let us make the estimate using a width which hasthe same relation to F as does the width of the p, i.e.,I /F =1.6. If we require that the resonance contribu-tion is a few times our estimate of the "natural" size ofa;, the resonance must generate n, ~0.004; this says thatthe resonance will dominate for masses

These may have the transformation properties of

g~g=LgU = Ug'R (27)

(The scalar s field in Sec. III had a higher bound becauseit's width was larger. ) These make more definite what wemean by "low-lying" resonance.

where U is nonlinear matrix, U = U(L, R, vr) in vectorialSU(2). The isotensor fields would then transform as

V. PION LOOPS

PJ Q4 D (U)DD(U)Q J

D'J(U)= —'Tr(v'Ur~U ) .

This allows the chirally invariant couplings

(28)

When pion loops are considered, the coefficients a;need to be renormalized and become functions of thescale used to define the theory. In what sense then canthe renormalized values be predicted in terms of the spec-trum? In order to answer this, we turn to the operational

1952 JOHN F. DONOGHUE, CARLOS RAMIREZ, AND GERMAN VALENCIA 39

procedure for phenomenologically determining the pa-rameters. The data and the chiral model are comparedover range of energy and the deviations from the lowest-order result tell us the best phenomenological parame-ters. This is depicted schematically, but not unrealistical-ly, in Fig. 1. At low energies, the effect of s terms aretoo small to be visible, while at Vs equal to the resonancemass m& the expansion in the energy has become non-linear and yet higher powers of s are required. Thegreatest strength in determining the parameters willoccur at &s =m~/2. Therefore, physically we will wantto adjust the parameters in order to match the tail of theresonance at this energy.

Let us demonstrate this by using the dimensionally reg-ularized form of the amplitude in the chiral limit. Here

M(~ H +~ra —)= A (s, t, u)5 p~s+ A (t, s, u)5 5&&

1 re g2

C,fr= 44s' a,(p)+,—,ln

144m 96m 4p

+[s +(t —u) ]

7X a~(p)+288m

Ply2

ln48m 4p

a, (p)+ 1

144~1 m,2

q =~']'"',96~ 4p

This method is designed to match the amplitude in theregion most sensitive to the chiral coe%cients. Opera-tionally then, the parameters will be adjusted to matchthe effect of the tail of the resonance in this region. Thistells us that there is shift of the parameters such that

+ A (u, t, s)5 s5p

where s, t, and u are the Mandelstan variables

(32)7a2(p)+

288m.

mg2

~(&ree)

48' 4p

s =(p +p&)',

t =(p —p )',u =(p —ps)' .

(33)

The amplitude A (s, t, u) can be decomposed into alowest-order result, loop corrections, and terms due tothe chiral parameters in L4.

(38)

Note that this procedure correctly gives the p depen-dence of the chiral parameters.

It would be these parameters which would be identifiedwith the effect of integrating out the resonances, such aswas given in Eq. (31), i.e., for the case of the p:

m I Fa, (p)+ — ln = —48

144m 96m 4p m

A (s, t, u)=2 +B(s, t, u)+C(s, t, u),p2

B(s,t, u) = 3s I(s)+t(t —u)I(t)+u (u t)I(u)—1

[21s +5(t —u) ]96~

C(s, t, u)= [4s'a, +a,[s'+(t —u)']],(34)

I(a) = lnz

—216m p

If it were not for the logarithmic factors in I, the efFect ofloops could all be absorbed into a new set of coeScientsa;. However, in the moderate energy region &s =mz/2,the logarithmic factors do not vary too rapidly, and wecan approximate the loop function by setting &s equal tomz/2 inside the logarithm. If we do this then the result-ing amplitude will again be a polynomial and the efFect ofthe loops can be absorbed into a shift of the chiral param-eters. In particular, in this approximation

—a& =+2=48m r,r'.

-3~2 =0.0084 .4m

2foal

m 1—5P

(39)

VI. THE CHIRAL LAGRANGIAN FOR QCD

The best chiral Lagrangian compatible with data is ofcourse known phenomenologically, and we expect QCDto describe the real world, so that in this sense the chiralLagrangian of QCD is known. However, what we aretrying to do is to see to what extent a Lagrangian can beobtained from purely theoretical considerations. Here weimagine that we know about the QCD spectrum from lat-tice gauge theory and/or the quark model, but have notyet performed m-~ scattering experiments. What wouldwe predict?

The low-energy spectrum has two potentially relevantfeatures. One is the p meson, which appears as the light-est non-Goldstone particle. The other is the possibility ofa scalar glueball. Present lattice calculations put this at1.2 GeV or slightly higher. Other resonance states of ap-propriate quantum numbers are above 1.2 GeV, such asthe J =2++,I =0 F2(1270). Because of the 1/mfeature in the chiral couplings, the p is the only clear res-onance which should dominate the low-energy constants.It would yield

A (s, t, u)= +C,tt(s, t, u),Q2

where

(35) (Here we have reinstated the pion mass into the decaywidth to properly account for phase space. This usage istechnically yet higher order in the chiral expansion, and

39 SPECTRUM OF QCD AND CHIRAL LACxRANGIANS OF THE. . . 1953

a, (m )= —0.010, a2(m )=0.003 . (40)

Theoretically, one might also include an effect from thescalar glueball but this would only change o. by +0.0004for I =300 MeV, nz = 1 GeV.

These estimates are in remarkably good agreementwith the phenomenological fits, where we previouslyfound

could be optional. This gives a minimal estimate of thetheoretical uncertainty in the procedure of 20%%uo. ) If weuse the procedure advocated in Sec. V for predicting thescale-dependent one-loop renormalized coupling, thiswould yield

dependence. The data are in good agreement with thevector-dominance prediction.

A similar situation occurs in the q variation of them —+yy amplitude. The chiral loop results for this haverecently been described in Ref. 10. At one loop, theWess-Zumino-Witten anomaly Lagrangian generates a qvariation for M' —+yy via the Lagrangian. The finiteremainder and the p variation are both very small com-pared to the prediction obtained by integrating out the p.In this case the data are not very precise, but are con-sistent with the vector-dominance prediction.

Overall, the parameters in the QCD Lagrangians seemto refiect primarily the presence of the p meson and itsSU(3) partners.

a, = —0.0092, ~2= +0.0080

as the best tree-level parameters, and

(41)VII. SUMMARY

a, (m )= —0.011, az(m )=+0.0057 (42)

L =. . . + F„,Tr(d„UQB U +r)„U Qd U), (44)2

such that, at the tree level,

CX9

F, (q)=1+ 2 q

In order to match the vector-dominance result onechooses

(45)

F2(x9= =0.014 . (46)

Vlp

This is in fact the result which Gasser and Leutwylerfound for this coefficient when coupling the p to chiralfields and integrating out the p.

The criteria proposed in Sec. II, for determining whena contribution is large, is well satisfied by the p in thiscase. When chiral loops are considered, they introduce ascale dependence of the form

a9(p )=a9(po)+ b 91n 2pPo

(47)b9 2

0 0011

96m

Thus the effect of the p is much larger than the scale

at the one-loop level. The results are better than wewould have expected, and do seem to indicate that the pis the dominant feature of the spectrum.

This result is very similar to the ideas of vector-mesondominance which have been heavily used since being in-troduced by Sakurai in the early 1960s. In fact, a look atthe chiral Lagrangian treatment of the pion electromag-netic form factor will emphasize the similarities. Invector-meson dominance the pion form factor is deter-mined by the coupling to the p meson to have the form

2

F, (q)=2

—1+ +. (43)Pl

p2

Ulp

In a chiral Lagrangian, on the other hand, the q behav-ior is controlled by a coefftcient in the 0 (E ) Lagrangian

We have explored, phenomenologically and throughsolvable trial models, the notion that the low-lying spec-trum of a theory gives the major contribution tocoefficients of the chiral Lagrangian at order E . In allcases studied the idea proved correct. Criteria weredeveloped for deciding whether a given resonance shouldmake a "large" contribution. When applied to QCD the

p contribution appears to be the only significant one,leading to a connection of chira1 Lagrangian with the old-er idea of vector-meson dominance.

Besides the intrinsic interest in the origin of the chiralLagrangian of QCD, there may be applications of thisidea. For example, integrating out the p resonance alsoimplies a set of effects at order E and higher. One cansum these by use of the fu11 p propagator. Perhaps in ap-plications of chiral Lagrangians this separation of the peffect, treated to all orders in the energy expansion, plusthe remaining (small) O(E ) effects can lead to an im-proved phenomenology. Formally no error is made indoing this as the extra terms are of order E and higher.However, numerically this may be an improvement. Forexample, in the pion electromagnetic form factor it wouldlead to the full monopole form rather than simply thelinear term, which in fact is a favorable improvement.

In a different context, these ideas may be profitably ap-plied to extensions of the standard model. All proposedextensions, even without the fundamental Higgs mecha-nism, lead to the same 1owest-order predictions forWL WL scattering as would the standard model with aheavy Higgs boson. " This is a consequence of the under-lying symmetry. However, deviations from this lowest-order result may depend on new physics, or on the trueunderlying theory. In an analogy with the above work,the standard fundamental Higgs mechanism is most likethe linear o. model, while a theory such as technicolor'would more likely be similar to QCD. These could bedistinguished by correction to the lowest-order result. Atheory such as the strongly coupled standard model'would have yet a third pattern. These issues are underinvestigation.

APPENDIX: THE WEAK INTERACTIONS

In the strong interactions one can convert the reso-nance coupling strength involving the chiral field into the

1954 JOHN F. DONOGHUE, CARLOS RAMIREZ, AND GERMAN VALENCIA 39

width of the resonance. In the case of the weak interac-tion, there is no way to normalize the weak coupling ofthe resonance to chiral fields. Here the criterion for a"large" contribution to the Lagrangian coeScient is lost.Nevertheless, one would expect that whatever plays amajor role in generating the AI =

—,' rule would leave its

imprint on the chiral Lagrangians as well. Since there isvery little consensus on the origin of this puzzlingenhancement in nonleptonic transitions, it seemsworthwhile to determine phenomenologically what theknown features are. This appendix is meant as a simpleexploration of the data.

Chiral symmetry relates K —+3~ to %~2m. . Typicallythe K~3~ amplitude is expanded about the center of theDalitz plot:

A (KL ~~+a ~ ) =a+/3Y

+ Y2 +5Y+ Xr 3

'3

"w

(o)

(b)

A (KL~vr n ~ )=3 a+5 Y + X

Y=(sg sp)/sp

X=(s, —s2)/so,

s, =(k —p;)2m~

sp g (si +$2 +s3 )= +m

(A1)

Focusing on the AI =—,' amplitude only, the data are

a=9. 15 X 10

/3=13. 46X10 ',y = —3.40x10-', (A2)

2~ q~, d=1'

q [ 8k 'PoP+ 'P-3so

+4(/' P+Po P-+k P-Po P+ )l

2, l4k PoP+ P-

3so

+4(.k'P+Po P +k'P Po'P+ )i (A—5)—

5= —1.01x10 ',taken from the review of Devlin and Dickey. '

In Ref. 15, it was shown that any chiral Lagrangianwhich does not lead to quadratic terms in the amplitude,y and 5, has the same prediction for a and /3:

a=7. 5 x 10(A3)

P=9.4X 10

up to possible corrections of order m . This includes theunique order E form

I.p =g Tr(A, 6d„Md"M ), (A4)

plus many E Lagrangians. Of the four' remaining ones,only two kinematic invariants are possible, which can bewritten as

FIG. 3. Pole diagrams for %~3m. In these, the order Eeffects of the strong interaction generate E effects in %~3~.

Besides generating the X and Y terms thesesignificantly improve the predictions of a and P. It isclear that our information on the E terms in the weakinteraction is limited, which will hinder our ability todraw firm conclusions.

One general feature which is easy to extract from theseamplitudes is that they are more compatible with astrong p presence rather than a scalar particle such as ao.. This is clear because a p never couples to two ~ 's andhence can only contribute to the parameter y but not 5.An isoscalar in contrast would generate y =5. Since y issignificantly larger than 5, the p contribution could bedominant.

This conclusion will turn out, under further inspection,to be both true and potentially misleading. This is be-cause some of the contributions to y and 5 come from thestrong-interaction E Lagrangian. A statement to thecontrary in Ref. 15 is erroneous. We already know thatthe p is a dominant feature in the strong interaction.However, a relevant question is, once the strong-interaction effect is subtracted off; what is the characterof the purely weak-interaction remainder? We will seethat the remainder is much smaller than the original size,so that we are cautious about drawing any conclusions.

The strong interactions enter through the pole diagramof Fig. 3(a). That of Fig. 3(b) is suppressed by a factor ofm /mk, and will be neglected. If we write

y=y, +y, 5=5, +5the strong contribution is

&w Os2 —7

F6 1 2(2a —a ) = —2. 8 X 10

$25, = 2(a, +a~) = —0.26X 10

SPECTRUM OF QCD AND CHIRAL LAGRANGIANS OF THE. . . 1955

which leaves, for the weak-interaction contribution,

y = —0.6X10 ', 5.= —0.75X10 'el is

L =g Tr(A, 6D„MD"M ), (A7)

This remainder is not much larger than scale dependenceinduced by the strong interactions alone:

with D„ is given before [Eq. (8)]. In terms of the renor-malized weak coefficient Ao, this model predicts

$2 2

h5, —s (0.006)ln -0.64X 10 (A6)

3~0 g p&+m 1y„=+1.4X 10 =g

Thus it is not clear if there is much which an be learnedfrom the weak terms.

It is of interest to ask whether the one model with adefinite prediction for these coefficients —the gaugedchiral model —produces these coefficients. Here the mod-

5„=0, (A8)

using methods such as described in Sec. III. This gives anumerically small contribution (as was observed above)but of the wrong sign.

S. Weinberg, Physica (Utrecht) 96A, 327 (1979); H. Georgi,Peak Interactions and Modern Particle Physics (Benjamin,Palo Alto, 1985).

~J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984).3J. F. Donoghue, in Proceedings of the 3rd International Confer

ence on Intersections Between Particle and Nuclear Physics,Rockport, Maine, 1988, edited by G. Bunce (AIP Conf. Proc.No. 176) (AIP, New York, 1988).

4For a review of vector dominance, see J. J. Sakurai, in Proceed-ings of the Fourth International Symposium on Electron andPhoton Interactions, Daresbury, England, 1987, edited by D.Braben and R. Rand (unpublished); an early reference wouldbe J. J. Sakurai, Ann. Phys. (N.Y.) 11, 1 (1960).

~T. H. R. Skyrme, Proc. R. Soc. London A260, 127 (1961);Nucl.Phys. 31, 556 (1962); G. Adkins, C. Nappi, and E. Witten,ibid. 8228, 552 (1983); T. N. Pham and T. N. Truong, Phys.Rev. D 31, 3027 (1985); M. Lacombe, B. Loiseau, R. VinhMau, and W. N. Cottingham, ibid. 38, 1491 (1988); I. J. R.Aitchison, C. M. Fraser, and P. Miron, ibid. 33, 1994 (1986);M. P. Mattis, Phys. Rev. Lett. 56, 1103 (1986).

E. de Rafael, G. Ecker, J. Gasser, and A. Pich (private com-munication). After submission of our paper, we received a

copy of the de Rafael et al. manuscript [Nucl. Phys. B (to bepublished)]. These authors agree with us on the dominance ofthe vector-meson contribution. They also consider the con-tribution of axial vectors and scalars to others of the low-energy parameters.

7J. F. Donoghue, C. Ramirez, and G. Valencia, Phys. Rev. D38, 2195 (1988).

O. Kaymakcalan, S. Rajeev, and J. Schechter, Phys. Rev. D 30,594 (1984); H. Gomm, O. Kaymakcalan, and J. Schechter,ibid. 30, 2345 (1984);J. Schechter, ibid. 34, 868 (1986). -

S. Coleman, J. Wess, B. Zumino, and C. Callan, Phys. Rev.177, 2239 (1969); 177, 2246 (1969).

oJ. F. Donoghue and D. Wyler, Nucl. Phys. (to be published).'M. Chanowitz, M. Golden, and H. Georgi, Phys. Rev. D 36,

1490 (1987).E. Farhi and L. Susskind, Phys. Rep. 74, 277 (1981).

' M. Claudson, E. Farhi, and R. L. Jaffe, Phys. Rev. D 34, 873(1986).R. Devlin and J. Dickey, Rev. Mod. Phys. 51, 237 (1979).

~5J. F. Donoghue, E. Golowich, and B. R. Holstein, Phys. Rev.D 30, 587 (1984).E. Golowich, Phys. Rev. D 36, 3516 (1988).