Chiral lagrangians for massive spin-1 fields

Volume 223, number 3,4

PHYSICS LETTERS B

15 June 1989

CHIRAL LAGRANGIANS FOR MASSIVE SPIN-1 FIELDS ~"

G. ECKER Institut ~ r Theoretische Physik, Universitdt Wien, A-1090 Vienna, Austria

J. GASSER, H. LEUTWYLER Institut J~r Theoretische Physik, Universitiit Bern, CH-3012 Bern, Switserland

A. PICH Departament de Fisica Tebrica and Institut de Fisica Corpuscular, CS.L C., Universitat de Valbncia, E-46100 Burjassot (Valencia), Spain

and E. DE RAFAEL Centre de Physique ThOorique ~, Section 11, CNRS Luminy, F- 13288 Marseille, France and Departament d'Estructura i Constituents de la Materia, Universitat de Barcelona, Barcelona, Spain Received 30 March 1989

There are different ways to include massive spin-1 particles in the effective low-energy lagrangian. We analyse these ambiguities in the context of chiral perturbation theory to order p4. We show that, provided consistency with QCD asymptotic behaviour is incorporated, the structure of the effective couplings induced by vector and axial-vector exchange is model independent.

1. At low energies, the strong interactions between pseudoscalar mesons can be described by an effective chiral lagrangian [ 1-3 ]. This lagrangian depends on a number of coupling constants which are not determined by symmetry requirements alone. It was recently shown [4] that the coupling constants which occur at O ( p 4 ) in the low-energy expansion are dominated ~ by the low-lying vector, axial-vector, scalar and pseudoscalar resonances. In particular, chiral symmetry was found to confirm the phenomenologically successful notion of vector meson dominance (VMD) [ 6 ]: whenever vector (and axial-vector) mesons contribute at all, they practically saturate the corresponding coupling constants of the O ( p 4 ) effective lagrangian.

In this letter we address the question how much these results depend on the use ofantisymmetric tensor fields to describe spin-1 resonances [ 2,4]. It will be shown that, once the proper conditions guaranteeing the consistency of VMD with the asymptotic behaviour of QCD are satisfied, one may as well employ the more conventional vector field formalism. 2. We denote by Z the generating functional for the Green functions of vector, axial-vector, scalar and pseudoscalar quark currents: eiZ[ v,a,s,p]

~r Work supported in part by Schweizerischer Nationalfonds. Laboratoire propre du Centre National de la Recherche Scientifique. ~ Similar results were obtained in refs. [2,5].

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

= (01Texp ( i I dxOt[~',(vU+,sa u ,

-(s-ip~5) lq)10).

(1)

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The external fields vu, au, s and p are hermitean 3 × 3 matrices in flavour space, with tr vu=tr au=0. The scalar field s contains the quark mass matrix and I0 ) denotes the QCD ground state in the chiral limit mu=ma=ms=O. Chiral symmetry implies that Z admits a low-energy representation of the form [2,3 ]

eiZ[v,a,s,p]

15 June 1989

~(4) also contains a piece LPwz to account for the chiral anomaly [ 7 ]: o~(4) = ~/gWZ"11-~ 4 , 10

LP4= ~ LiP~+contact terms in external fields, i=l

Pl = (Du UtDuU) z, P2 = (Du U*D~ U) ( D " U t D ~ U ) ,

=~[dU]exp(i~dx~ff(U,v,a,s,p)).

(2)

/°3 = (Du UtDUUD~ UtD"U), P4--- (Du U *DuU) ( z t U + z U * ) ,

The right-hand side is a meson field theory, the meson fields being described by a unitary 3 × 3 matrix U (det U= 1 ) which transforms as G

U---+gRUg[,

G=SU(3)R xSU(3)L

(3)

under local chiral transformations. The effective lagrangian LP~rfconsists of a string of terms ~eff = ~ ( 2 ) .9I_ ~,.~(4) .~_ ~ ( 6 ) ..~ . . . .

(4)

We refer to L~~zn) as a term of order p2": the field U counts as a quantity of O (pO), the derivative 0u and the external fields vu, a u are booked as O ( p ) , and the fields s, p count as O(p2). The first term L~~2) is the non-linear a-model lagrangian in the presence of external fields:

~ 2 ) = ¼f 2
1°5 = (Du UtDuU(z *U+ UtZ) ),

P6 = ( x t U + z U t ) 2, P7 = ()~tU-~.U~f ) 2, P8 = (g* UZ* U+zU*zU*), P9 = - i ( F ~ " D u UD~ U * + F ~ D u UtD~ U),

Plo = ( U*F ~~UFLu~ ).

(7)

The quantity F ~ ~ ( F ~ " ) stands for the field strength associated with the non-abelian external field vu +au(vu-au). L~ .... , L~o are ten real coupling constants which absorb the divergences of the one-loop graphs generated by &or2). We denote the corresponding renormalized, scale-dependent couplings by Lf(/t).

D u U= 0u U - i ( v u +au) U+iU(v u - a u ) , z=ZBo(s+ip), f---f~ = 93.3 MeV, (0laUl0)=--f2Bo[l+O(mq~k)].

(5)

The symbol ( A ) stands for the trace of the 3 X 3 matrix A. The low-energy expansion of Z is given by the expansion of the effective meson field theory in the number of loops,

Z = Z(2)-Jv z ( a ) t - Z(6)-t- ....

(6)

The leading term Z ~2) coincides with the classical action associated with ~ z ) . In this letter we are concerned with the next-toleading-order term Z ~4). It contains two types of contributions: one-loop graphs generated by L~tz) and tree graphs involving one vertex from L~~4~. In addition to the general chiral invariant lagrangian Lf4o f O (p4), 426

3. To evaluate the contribution of vector and axialvector exchange to L f, one has to include fundamental spin- 1 fields in LP~fr.Following ref. [ 2 ], these were described in ref. [4 ] in terms of antisymmetric tensor fields Vu~, Au~ transforming under a non-linear realization of spontaneously broken chiral symmetry. Only octet spin-1 resonances are relevant to O ( p 4) [4]. The realization of G on these fields is given by G

Ru~--~h(~o)Ru,h(~o)* , R = V , A ,

(8)

where the non-linear realization [8 ] h(~) defines the action of G on a coset element u(~0) via G u(~o)----+gRu(~o)h(q~ ) t =h(~o)u(~o)g[.

(9)

~0i ( i = 1..... 8) are the Goldstone fields parametrizing coset space and U= u 2 in the chosen gauge. With the covariant derivative

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15 June 1989

VuR = O.R + [F~, R I,

~.~=V.f"~-V~(:..

F u = ½{u*[ 0 . - i ( v . +au) ]u

The relation to the massive Yang-Mills [ 9 ] or hidden-gauge [ 10 ] formulation will be clarified later. For pure kinematical reasons, the couplings of I5". to the pseudoscalar mesons and external fields start at O (p 3). The complete list of such couplings linear in the octet vector field f:. (jec= 1- - ) is

+ u [ G - i ( v . - a u ) ]u*},

(10)

the kinetic spin-1 lagrangian takes the form [4]

~in(R.v) = -- 21< V2R2. Vv R v.__ 2MRRu.R 1 2 . v ).

(11)

For the purpose of presentation, we now limit the discussion to the vectorial case and comment on the axial-vector resonances afterwards. Including the most general interaction lagrangian linear in Vu~ to order p 2, we arrive at model I [4]

( 17"u~fu+~),

(14)

( V..tu", u"] >,

( P . I u . , f U ] ),

E..v.,(I'/*'u"uPu°),

((:'.[uU, x _ ] ) , e..w(f/U{u~,f~}),

(15)

where

where Z +_= u *xu * +-uy.tu. Since the resonance propagator cannot decrease the chiral counting, vector meson exchange with the couplings (15) can only produce terms starting at O(p6). However, as we shall discuss below, some of the terms in ( 15 ) will nevertheless be relevant for£#4. Including only those couplings which will contribute to the Green functions investigated below, we arrive at model II:

f ~"=uF~"ut + u t F ~ u ,

-~,,=&.(¢'.)

~=&.(V..) + ~

1

(Fv ( V,,.fu+~ ) +iGv ( V.~[u", u ~] ) ), (12)

u. = iu*D. Uu* = u'u,

- ~

and Fv, Gv are two real coupling constants not restricted by chiral symmetry. To lowest order p4 in the chiral expansion, vector meson exchange induces a local lagrangian of the type (7) with [ 4 ]

G~ LV=2L v, Lv=-6L v, LV-8MZv,

1

0Cv( (:..fuv)+igv(P..[uU,

uV])). (16)

The effective action SN (N=I, II) for single vector meson exchange can then be written in the compact form aN =1

f dxdy(J~'(x)dN~.o,~(x-y)J~(y)).

(17)

FvGv F~ L V - 2M~ ' L V o = - a M 2 v , L v =L5v ---L6 v =L7v =L8v =0.

(13)

With the available information on the parameters My, Fv and Gv, the contributions (13) (including L~o from axial-vector exchange) were found to saturate the renormalized coupling constants L~(/~) ( i = 1, 2, 3, 9, 10) if ~ is chosen to be in the resonance mass region.

The connection part in 17". is not included in (17) because it will not contribute to the Green functions discussed below. The antisymmetric currents J ~ are defined as

J~v~ = ~

(c~f~Y + c f i t u ", u q ),

c]=Fv/Mv, c]I =fv,

4. We now turn to a description of vector mesons in terms of vector fields I2. transforming again (for octet resonances) as in (8) with

1

c~2=Gv/Mv,

c~'=gv.

(18)

The meson propagators are

~kin(~Z.u) = -- I ( ~r v ~ . r . v _ _ 2 M 2 ~.r,u~ r u ) ,

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Volume 223, number 3,4

PHYSICS LETTERSB (r2)~ 1=0.

u f d4k e-ikx 3u,~oo(x) = (2n) 4 k 2 - M 2 +i~

× [g~k.k,.-g,,~k~kp - (lt~-+v)], n + (g~,,gup--gupgv~)a4(x). zl~v,p~(x)=a,,v,a~(x) (19) We make the following observations: (i) The two models are equivalent up to chiral invariant local terms of O (p4) generated by the contact term in zl~,,p~. In particular, the structure of the spin1 pole contributions is the same in both models with

fv=Fv/Mv,

gv=Gv/Mv.

15 June 1989

(20)

(24)

A dispersive analysis o f F ( t ) shows on a more formal level why the proper description of the vector meson contribution is given by (21), not by (22). Indeed, consider the spectral function Im Hv (t) of the I = 1 vector current 2-point function. In the framework of QCD, one finds I m H v ( t ) ~ c o n s t . as t - - . ~ [11]. Since each intermediate state yields a positive contribution to Im Hv (t), I m H v ( t ) >~ ~

1

( 1 - 4 m 2 / t ) 3/2

× IF(t)120(t-4m2~),

(25)

These pole terms are O (p6). (ii) The models differ in the off-shell behaviour: the local terms of O (p2) in model I lead immediately to the L v given in (13). There is no O ( p 4) contribution in SH because the contact term is absent in the n Pro Pa gatorA -~1,~,p,*"

and we conclude that F ( t ) obeys a dispersion relation with at most one subtraction

5. It thus seems as if vector mesons would have no relevance at O(p 4) according to model II. However, we will now show that the presence of these local terms of O (/7 4) is in fact absolutely necessary to correctly implement VMD in the effective action. To illustrate this, we consider the electromagnetic form factor of the pion. From the effective actions in ( 17 ) one obtains #2, respectively,

Im F ( t ) = K n m z ~ ( t - m 2)

F ' ( t ) = 1+

FvGv t f~M~,-~t

(21)

F(t)=l+~j

t f dt' Im F ( t ' ) t'(t'-t-iO"

In the narrow-width approximation, p-exchange gives

FH(t)=I+

(27)

and therefore xt F(t) = 1 + M 2 -~t"

(28)

While model I produces a form factor of this type, model II does not. As already announced, this disease can be remedied by adding an appropriate local term of O (p4) to ~ . With the improved lagrangian "~1I = "~11 "1-y l I p 9 ,

and

(26)

0

(29)

where P9 was defined in (7), the form factor is now --

t2 M2v_ t.

(22)

Consider now the pion charge radius ( r 2) ~= 6F' (0). In model I we have 6Fv Gv (r2)~ - f 2 M 2 ,

fvgv t 2 + ~2 y~,t. Fii(t) = 1 + f 2 M2v -~t

Requiring consistency with QCD via (28) determines y~' = -~fvgv,

(23)

(30)

(31)

in perfect agreement with experiment and with VMD. On the other hand, in model II one obtains the disastrous result

uniquely, bringing at the same time the form factors in the two models into complete agreement in view of (20). QCD has forced upon model II an explicit local term of O(p 4) which the propagator Au~,vo ii could not produce.

~2 The correct normalization ofF(t) at t=0 is obtained by adding the tree graph contribution from ~(2) in ( 5 ).

6. In order to make this procedure more systematic, let us consider new lagrangians ~r, ~ n which

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differ from ~ , ~1t by local terms of O(p 4) Lf~ = gfN+

~

ffP;,

N = I , II.

Thus, vector mesons contribute with (32)

i= 1,2,3,9,10

From the outset we limit the additional local lagranglans to such terms which are contained in the product of currents J~V"JN;,~ [cf. (18) ]. The lagrangians (32) generate in turn local lagrangians

N= I, II, i= 1,2,3,9,10

with

Li

=Li

+Yi,

Li

-

(33)

" •

We now impose the following five conditions abstracted from QCD (or from general quantum field theory for eq. (37)) to fix the constants yu. Since the L; are independent of quark masses, we shall investigate these conditions in the chiral limit. (i) Two-point functions. Consider the following Green functions for SU (3) vector and axial-vector currents: i J dx

15 June 1989

T,j(v)-T;j(O)lv=

xisv2 M~c_v2.

Although there are many possible channels (ij), only three of the conditions (37) are independent. We exploit (37) for n+n °, n°n ° and n+K - elastic scattering. SU (3) symmetry then guarantees that (37) holds for all channels. The five conditions (28), (35) and (37) (for n+n °, n°n° and n+K - ) determine the ~,~ in (32) uniquely. As expected, one finds ~,~=0

( i = 1 , 2 , 3, 9, 10)

(38)

and ~ll -- 1.2 1 -- g~;V,

=

, , l I - - '~ ,II /'2 --'~fl ,

~)ll =

--6Y~ I ,

= -Jv.

vgv,

× (01T[ V~(x) V~(y) -A;u(x)A~(y ) l [0)

=5~k[ (p;,P, -guvp2)H[~)(p 2) +pup,H[°)(p z) 1. (34) According to QCD the spin-1 invariant function H[~ ) (s) satisfies an unsubtracted dispersion relation [11]. Vector (and axial-vector) mesons thus contribute with

rvM~ H[h)(S) ]v,A - - M2v_S

rAM2A M2_s "

(35)

(ii) Pion form factor. From eq. (28) Kt

F ( t ) = 1 + M2v ------~t" (iii) Elastic meson-meson scattering. The forward amplitudes T;j(v) for the elastic scattering of pseudoscalar mesons M; on Mj (i, j are SU (3) octet indices) are even in the variable v = ( s - u) / 2. It follows from the Froissart bound [ 12] that T;?(v) obeys a once-subtracted dispersion relation

T°(u)=Tij(O)+U---nf dv'ImTu(v')v' / f 2 _ v2

(36)

(39)

Taking into account (13), (20), (33) and (39), we obtain L~ =L~ I ( i = 1, 2, 3, 9, 10)

eip(x-y)

(37)

(40)

and thus models I and II yield identical Green functions to O(p4). The better high-energy behaviour of the antisymmetric tensor versus the vector field exchange has already been traced back to the different structure of their respective propagators (19). Note that the vector meson exchange contribution to the coupling constants LI and L2 obeys the Zweig rule [3]

L2=2LI. 7. Having established the equivalence between the antisymmetric tensor and vector field formulations to O(p4), we now make contact with the most widely used representation for vector mesons. As far as we are aware, all formulations involving massive YangMills [ 9 ] or hidden-gauge [ 10 ] vector mesons boil down, at least after gauge fixing and neglecting additional couplings of the type ( 15 ), to the lagrangian ~ i = - ~ ( 17;,~ff;'~) + ½M2v( [ Vu - (i/g)Fu]2), (41) where the vector field ~, transforms as -

G

-

i

Vu----,h(~o) Vuh((o)* + g h(~)Ouh(~o)*

(42)

0

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and Y~,~=OulT~-O~ff~,- ig [ Vu, V~ ]. Clearly, the field Vu = V/a- ( i / g ) F u then transforms homogeneously as in (8). Using the identity

15 June 1989

we can write (41) in the form

the 1 ÷ mesons are described by an antisymmetric tensor field rather than by a vector field, the problem does not occur because bilinear couplings of the type (47) do not exist - certainly not a disadvantage of the tensor representation. For the proper spin-1 axial-vector field A~, ( j e c = 1 ÷ + ) there is again a whole list of possible couplings (dropping the prime on Au from now on):

1 ~ltl = ~ I 1 "31-6--~g2 < [U/a, U v ] [U/a , U,] >

(flu~fuZ),

i

1

l?u~ = l?u~ - i g [ ITu, I7"~]+ ~g [uu, u~] + ~gf+U~, (43)

i 1 ( UtF~VUFL/a ~ ) - ----5 8g
(44)

if one sets

fv=l/xc~g,

gv=l/2x/~g,

(45)

in ~ l . Comparing ~ n with (32) and (39), we observe that Nn is a special case of model H fulfilling all five QCD conditions (28), (35 ) and (37). Therefore, to O (p4), model III is equivalent to model I with the additional relation Fv = 2 G v .

(46)

We shall come back to this relation below. The fact that model I is consistent with chiral symmetry for any value of F v / G v shows that (46) does not follow from chiral symmetry alone. 8. We still need to discuss the axial-vector mesons. For an axial-vector field A. transforming as in (8) one encounters the complication that chiral symmetry allows for terms which induce mixing between .4~ and the pseudoscalar mesons. The simplest coupling (.4/a u~ )

(47)

arises already at 0 (p ~). Such couplings must and can always be transformed away by a transformation of the form

A,---,A; =A u +cu u +...,

(48)

which removes the spin-0 component of.~ u. (In the literature, this diagonalization is often replaced by a linearized transformation which ruins the chiral transformation properties of the field Au and gives rise to all sorts of apparent symmetry breakings. ) If 430

(.q/a[u~,f~_"]),

(A/au,ul'u~),

(d/a{u", u%}),

(d~u")
(d~,u.) ( uUu ~),

eu~po( d U { u " , f ~ } >

(49)

with "4u. = Vu'4~ - V.A~. However, only one of these O ( p 3) couplings is relevant for 5e4 for the same reasons as in the vector case. Including axial-vector mesons, model I assumes its final form [ 4 ]

~ev+A= Z ~,.(R~.) R=V.A

+ ~

1

(Fv ( Vu.fu+~ ) + i G v ( Vj,.[u u, u"] )

+FA (Au.fu_~) ),

(50)

giving LVo+A =

rZv F2 -__ 4M 2 +4M2A,

(51)

all other coupling constants being unaffected by Aexchange. Correspondingly, model II is now given by ~IEI+A =

E

R=9,£

¢~kin( R a )

1

2x/~ ( f v ( Vu~fu+" ) +igv ( flu,,[uU, u"] )

+fA ( A~u ~ f _Uv > ) +

Z i= 1,2,3,9,10

~iI I ei.

(52)

The only difference to the previous discussion is that condition (35) now requires y IIl o = _ ~ 1f v +2

ZfA, I 2

(53)

bringing once again models I and H into complete agreement withfA=FA/MA. The natural connection F u on coset space is a vector field. A natural axial connection does not exist on this space and there is therefore no natural way to introduce an axial-vector gauge field analogous to (42) (see, however, ref. [101).

Volume 223, number 3,4

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9. In the above we invoked a once-subtracted dispersion relation for the pion form factor. In fact, all empirical evidence and theoretical prejudice suggests that F(t) vanishes sufficiently fast for t-+oo to obey even an unsubtracted dispersion relation

15 June 1989

satisfy an unsubtracted dispersion relation oo

1f

dt' Im Gm(t' ) -tT~_t_--Z~E ,

(61)

Im GA(/) =F2nr(t-M2A) + I m GA(t) rest.

(62)

GA(t)= ~ j

0

with l i d t ' Im F ( t ' )

f ( t ) = -~

t'-t-ie

(54)

0

Restricting the discussion for the rest of this letter to model I, we may write ImF(t)=

(55) where Im F (t) r e s t contains all contributions except the p-pole. (54) then implies

FvGv -~I ~ Im F(/)~e~t" f-----5--+

(56)

Identifying the residual contribution with the twopion continuum, evaluated at leading order in the chiral expansion [ 2 ], and cutting the integral off at t = M 2, one finds &,, 0.04.

(57)

Assuming higher mass contributions to be small, one obtains the approximate relation (58)

A similar analysis can be performed for the axial form factor GA(t) defined by (7(q)

IA~(0)I#(0))

- v/--~e E3ijGA(t ) [qupe(q) -¢u(q)pq] +pionpoleterm,

In this case, there is no contribution from pion loops to O (t9 4) [ 2 ]. Assuming again that the residual part of Im GA(t) is negligible one is led back [cf. (46)] to Fv=2Gv, which is sometimes called KSFR I relation [ 10 ]. Taking (46) and (58) together, one gets

Gv =fJx/~ = 66 MeV,

t = (p_q)2,

(59)

which is relevant for the decay n~ev7 [ 13 ]. A ~ is the axial-vector current and i, j are isospin indices. Model I predicts 2FvGv - F ~ , F~ Gg(t)ME + M2 _----~.

(60)

(64)

in good agreement with Fv = 154 MeV and Gv = 69 MeV obtained from F(p° ~ e + e - ) and F(p-+ 2n), respectively. Using also the Weinberg sum rules [ 14 ] in resonance approximation F2v = F A2+ f = 2,

FvGv =f2.

(63)

Fv = x/~f~ = 132 MeV,

0

FvGv - l - J , f2

2FvGv-F2v 7 dt f2 = a T I m GA(t)rest. 0

FvGvM2v nr(t-M2v) + I m F ( t ) rest, f2

F ( 0 ) = 1=

Setting t = 0, we arrive at the sum rule

~x2V ~~2 _- - z v~R2 ~2 V l A Jr A ,

zra

(65)

we find in addition

FA=f~, MA = x/2Mv,

(66)

leaving a single parameter Mv = Mp to calculate the V-,A-induced coupling constants ( 13 ) and (51 ). In table 1 we compare this prediction for L~, L2, L3, L9 and L~o with the phenomenological values and with the values derived in ref. [4] where L~9(Mp) was taken as input. The agreement is remarkable justifying a posteriori the assumptions that have gone into (58) and (46). The most compact way to exhibit the relations

Fv=xli2f~, Gv=f
FA=f=, MA=x/~Mv (67)

in a lagrangian form employs the vector (axial-vector) field lagrangian

As for the pion form factor, we now assume GA(t) to 431

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Table 1 Comparison between the renormalized coupling constants L~,(Mp) (i= 1,2, 3, 9, 10) in units of 10-3 and the predictions of V-,A-dominance using the relations (67) and with the results of ref. [4] where Fv was taken from/'(p°~e+e- ) and L~9(Mp)was used to fix Gv. The Weinberg rules were applied in both cases to determine FA, MA.

15 June 1989

stages o f this work. A.P. a n d E. de R. are grateful for the hospitality o f the Physics D e p a r t m e n t o f Brookh a v e n N a t i o n a l L a b o r a t o r y where part o f this work was done. T h e work o f A.P. has b e e n partly supported by C A I C Y T , Spain, u n d e r grant No. AE-0021.

L[(Mp) a)

rV+A b)

Lv+A c)

References

L3 L9 Llo

0.7±0.3 1.3±0.7 -4.4±2.5 6.9±0.7 --5.2±0.3

0.9 1.8 --5.5 7.3 --5.5

0.6 1.2 -3.6 6.9 d) --6.0

a) Ref. [3].

b) With (67). c) Ref. [4]. d) input.

[I]S. Weinberg, Phys. Rev. 166 (1968) 1568; Physica A 96 (1979) 327; R. Dashen and M. Weinstein, Phys. Rev. 183 (1969) 1261. [ 2 ] J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142. [ 3 ] J. Gasser and H. Leutwyler, Nucl. Phys. B 250 ( 1985 ) 465, 517, 539. [4 ] G. Ecker, J. Gasser, A. Pich and E. de Rafael, CERN preprint CERN-TH. 5185/88, to appear in Nucl. Phys. B. [5] J.F. Donoghue, C. Ramirez and G. Valencia, University of Massachusetts preprint UMHEP-298 ( 1988 ). [ 6 ] J.J. Sakurai, Currents and mesons (University of Chicago Press, Chicago, IL, 1969). [7] J. Wess and B. Zumino, Phys. Lett. B 37 ( 1971 ) 95. [ 8 ] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 ( 1969 ) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247. [9] U.-G. Meissner, Phys. Rep. 161 (1988) 213, and references therein. [10] M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164 (1988) 217, and references therein. [ 11 ] E.G. Floratos, S. Narison and E. de Rafael, Nucl. Phys. B 155 (1979) 115, and references therein. [ 12] M. Froissart, Phys. Rev. 123 ( 1961 ) 1053; A. Martin, Phys. Rev. 129 (1963) 1432. [ 13] D.A. Bryman, P. Depommier and C. Leroy, Phys. Rep. 88 (1982) 151. [14] S. Weinberg, Phys. Rev. Lett. 18 (1967) 507.

Li

L2

~m= - ¼ < v , , . w ~> + ~Mv ' 2 < [ 17"u- ( 2 i f J M v ) F u ] 2 > - ~ (dumA u~ > +M2v < [Au + (fff2Mv)Uu]2>, where 17, d ~ t r a n s f o r m u n d e r chiral t r a n s f o r m a t i o n s as in ( 4 2 ) a n d ( 8 ) , respectively. T h i s l a g r a n g i a n is e q u i v a l e n t to m o d e l I with the r e l a t i o n s ( 6 7 ) a n d it e x p l a i n s the p h e n o m e n o l o g i c a l success o f the Y a n g Mills i n s p i r e d a p p r o a c h to v e c t o r a n d axial-vector r e s o n a n c e s [ 9,10 ].

Acknowledgement We t h a n k U . - G . M e i s s n e r for discussions. O n e o f us ( G . E . ) wishes to express his g r a t i t u d e to the m e m bers o f the I n s t i t u t e o f T h e o r e t i c a l Physics, U n i v e r sity o f Bern for their k i n d h o s p i t a l i t y d u r i n g the final

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