30 July 1998

Ž .Physics Letters B 432 1998 397–401

A Nambu-Jona-Lasinio like model from QCD at low energies

Jose Luis Cortes a,1, Jorge Gamboa b,2, Luis Velazquez c,3´ ´ ´a Departamento de Fısica Teorica, UniÕersidad de Zaragoza, 50009 Zaragoza, Spain´ ´

b Departamento de Fısica,UniÕersidad de Santiago de Chile, Casilla 307, Santiago, Chile´c Departamento de Matematica Aplicada, UniÕersidad de Zaragoza, 50015 Zaragoza, Spain´

Received 24 March 1998; revised 20 May 1998Editor: R. Gatto

Abstract

A generalization to any dimension of the fermion field transformation which allows to derive the solution of the masslessSchwinger model in the path integral framework is identified. New arguments based on this transformation for a

Ž .Nambu-Jona-Lasinio NJL like model as the low energy limit of a gauge theory in dimension greater than two arepresented. Our result supports the spontaneous chiral symmetry breaking picture conjectured by Nambu many years ago andthe link between QCD, NJL and chiral models. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 12.38-t; 12.38.Aw

Although the theory of strong interactions isknown already for a long time our understanding ofhadronic physics is still very limited. While at highenergies a simple description in terms of quark andgluon degrees of freedom is possible no similarsimple description of the low energy nonperturbativedomain is still available and the basic mechanisms

Žbehind the low energy properties confinement and.chiral symmetry breaking are not well understood.

A candidate for a model incorporating dynamicalsymmetry breaking and goldstone bosons is the

w xNambu-Jona-Lasinio model 1 as a bridge betweenthe high energy description in terms of quark andgluons and the chiral Lagrangian description at low

1 E-mail: cortes@posta.unizar.es.2 E-mail: jgamboa@lauca.usach.cl.3 E-mail: velazquez@posta.unizar.es.

energies. The absence of low mass glueballs can alsobe used as an argument in favour of an effectivefermionic theory at intermediate energies. ANambu-Jona-Lasinio like effective model of QCD at

w xintermediate energies 2,3 provides a new frame-work for studying its non-perturbative behavior. Acomparison of the Nambu-Jona-Lasinio with experi-

w xmental data and extensions are considered in 4 . Thepossibility to use the relation of the Nambu-Jona-Lasinio model with QCD in order to make someprogress in the understanding of the low energyhadronic properties depends on the knowledge of the

Ž .details of this possible relation. The aim of thiswork is to present new arguments, in the path inte-gral formulation, in favor of a fermionic model as anapproximation to a gauge theory.

The main idea is to introduce an effectivefermionic field such that, as a consequence of thechange of variables, an effective mass for the gauge

0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00694-7

( )J.L. Cortes et al.rPhysics Letters B 432 1998 397–401´398

field is generated. If one considers the low energyexpansion of the integration over the gauge field, aLagrangian with contact interactions for the effectivefield is obtained.

Ž .The Schwinger model QED in Ds2 is anexample where a transformation of the fermionic

w xfield 5 , generating a mass term for the gauge field,is used. The transformation is in this case an axialŽ .U 1 transformation with a parameter identified as

the scalar field whose derivatives give the transversepart of the gauge field. A peculiarity of the twodimensional case is that, simultaneously to the gener-ation of an effective mass for the gauge field, theeffective fermionic field decouples and the model

w xreduces to a free theory 6 .Ž .An axial U 1 transformation in D)2 is not the

required change of variables since the Jacobian ofw xthe transformation 5 will not be linear in the gauge

field. An alternative is to consider a transformationc™x where

c x sV x x x ,Ž . Ž . Ž .i

mnV x sexp v x g g 1Ž . Ž . Ž .m n2

with v mn syv nm. In two dimensions ig 0g 1 sg 5

Ž .and the change of variables in 1 reduces to an axialŽ .U 1 transformation.

w x Ž .The Jacobian JJ v, A of the transformation 1is a ratio of fermionic determinants,

w xdet iEuqAuJJs . 2Ž .

det V iEuqAu VŽ .

This result is obtained by comparing the result of thefermionic integration using the original field c asintegration variable with the same integral evaluatedin terms of the effective fermionic field x .

In the evaluation of the fermionic determinants,mŽ . mn Ž .the gauge field A x and the parameters v x

are external fields coupled to the fermionic fieldthrough the fermionic part of the gauge theory La-grangian which, written in terms of the field x , takesthe following form:

w xLL sx V v iEuqAu V v x . 3Ž . Ž . Ž .g

In the case of an infinitesimal transformation, thedominant contribution to the Jacobian can be ob-tained from the one loop diagram with two fermionic

propagators connecting a vertex with a gauge field asexternal line with another vertex where the gaugefield is replaced by v mn.

An ultraviolet regularization is required in orderŽto have a well defined Jacobian and field transfor-

.mation for D)2. A trivial calculation gives

w x Dy 2 D mnJJ v , A s1qL d x E v A q . . . 4Ž .H n m

where L is an energy scale proportional to theultraviolet cut-off with a proportionality factor de-

Žpending on the details of the regularization co-w xvariant momentum cut-off, heat kernel method 7 ,

.. . . . Higher order terms in the Jacobian includecorrections supressed by factors v andror 1rL.

The next step is to make an appropriate choice ofthe effective fermionic field, i.e. a choice for v mn ,allowing to integrate over the gauge field and toobtain an effective fermionic model. Given the gen-eral decomposition of a vector field

A sE aqe E n b m1 m2 . . . mDy 2 , 5Ž .m m mnm m . . . m1 2 Dy2

it is possible to consider a change of fermionicŽ .variables 1 depending on the gauge field where

1 m m . . . m1 2 Dy2v s e e b . 6Ž .mn mnm m . . . m4 1 2 Dy2

As a consequence of the Jacobian of the change ofvariables one has, at leading order in e , an additionalterm in the Lagrangian

E mE n

1 Dy 2 mnLL s e L A g y A 7Ž .J m n4 ž /I

which can be seen as a mass term for the vector fieldA in the Lorentz gauge, E mA s0.m m

It is a difficult dynamical issue to see whetherthere is a physical mass scale corresponding to alimit e™0, L™` with a fixed value for theproduct eL2 in Ds4. If this is the case then, in therange of energies E such that E2

<eL2, the effectof the gauge interaction reduces to a point-like four-fermion coupling and after integration over the gaugefield one has a Lagrangian for the effective fermionicfield

Ž1. 2 mLL sx iEux q1r eL xg x xg xŽ .fer m

E Em nm nqa xg x xg x . 8Ž .I

( )J.L. Cortes et al.rPhysics Letters B 432 1998 397–401´ 399

The last term has a coefficient a depending on thegauge fixing used in the integration over A but them

gauge invariance of the theory guarantees that thea-dependence will cancel at the level of observables.One can take in particular a gauge fixing such thatas0 and the fermionic effective Lagrangian is justa Thirring model in four dimensions.

The fermionic system is a non-renormalizabletheory that has to be understood as an effectivetheory with a limited energy range of validity. As

w xany effective theory 8 it will have a natural scale MŽ .scale of the energy expansion which allows tointroduce a dimensionless parameter for each term inthe effective Lagrangian. One has for example adimensionless effective self-coupling g associated tothe current-current interaction

M 2

gs . 9Ž .2eL

Ž .In the derivation of the fermionic Lagrangian 8 theeffect of the change of variables at the level of theLagrangian has not been taken into account. Whenthe gauge theory action is written in terms of theeffective fermionic field x one has an expansion inpowers of e or equivalently an expansion in M 2rL2.If the scale of the effective fermionic theory can bemade arbitrarily small in units of the ultravioletcut-off of the original gauge theory then the correc-tions due to the expansion in e will be arbitrarily

Ž .small justifying the approximation used to get 8 .Note that as a consequence of the quadratic ultravio-let divergence, the Jacobian generates a term in thefermionic Lagrangian that, although proportional toe , is not suppressed by powers of M 2rL2.

The fluctuations for the fermionic field x in thelow energy effective theory are restricted to theenergy range E-M. This restriction is crucial forthe consistency of the approximation where the massterm from the Jacobian is retained while the effect ofthe change of variables in the Lagrangian is ne-glected. In fact if the fermionic field is integrated outdirectly before eliminating the gauge field then thestandard fermionic determinant of a gauge invariant

Ž .term has to be reobtained so the quadratic term in 7has to be canceled with a similar term from the x

field integration over fluctuations with E-M.The previous arguments suggest a possible ‘‘de-

rivation’’ of a fermionic effective theory in a Wilso-

nian approach as a consequence of the integrationŽ .over energy bands. The result in 8 differs from the

generic renormalization group result which wouldcontain a linear combination of all possible four-fermion terms. This difference is more striking if oneconsiders several fermionic fields andror severalgauge fields.

In fact all the arguments leading to the fermionicrepresentation of the gauge theory can be trivially

Ž .generalized to the non-abelian SU N case with onec

or several multiplets of fermionic fields in the funda-Ž .mental representation. One has to include a flavor I

Ž .and a color i index in the fermionic field. Thevector field A becomes a Lie-algebra valued gaugem

field A sAaT a and, at the same time, one has tom m

Ž .consider a fermionic field transformation 1 withv sv a T a. If the parameter v a is obtained frommn mn mn

a Ž . Ž .the component A of the gauge field by 5 and 6 ,m

the result for the Jacobian of the transformation willŽ .be still given by 7 with a trace over color indices

and no derivative terms if the Lorentz gauge condi-tion E mAa s0 is used. The four-fermion couplingm

generated in this case after integration over the gaugefield is given by

Žn .a .. 2 m a aLL s1r eL x g T x x g T x . 10Ž . Ž .4 f I I J m J

It can be surprising at first look that the stronggluonic self-interactions at low energies do not playany role in the approximation used for the gaugefield integration. Actually this is not the case becausesuch strong non-perturbative effects are crucial in thedynamics responsible of the implicitly assumed dy-namical mass M. The effects of the gluonic self-in-teractions of the non-abelian gauge theory will alsobe incorporated in the effective fermionic theorythrough the determination of the values of the pa-rameters of the effective theory, like the dimension-less four-fermion coupling g, by the matching ofboth theories.

Using the identity

11a aT T s d d y d d 11Ž .i j k l i l k j i j k l2 Nc

for the generators in the fundamental representationand the Fierz identity in Ds4 for the product

( )J.L. Cortes et al.rPhysics Letters B 432 1998 397–401´400

g mmg , one has a Lagrangian for the fermionicm

system

LL Žn .a ..fer

gm 5 5ws x ig E x q x x x x yx g x x g xI m I I J J I I J J I2M

m 5 m 5 xqx g x x g x yx g g x x g g xI J J m I I J J m I

1 M 2

qOO , q‘‘higher dimensional terms’’ ,2ž /N Lc

12Ž .

Ž 2 2 .where the OO M rL are due to terms proportionalto e coming from the change of fermionic variablesin the action, the ‘‘higher dimensional terms’’comefrom corrections to the approximation used in the

Ž .gauge field integration and there are OO 1rN termscŽ . afrom the identity 11 for the generators T in the

fundamental representation.The fermionic model obtained at leading order in

1rN is an extension of the Nambu-Jona-LasiniocŽ .ENJL model. This result had been proposed previ-

w xously 9 as a conjecture on the behavior of a gaugetheory at intermediate energies, compatible with thechiral symmetry of the gauge theory at leading orderin 1rN , which allows to calculate the low energyc

w xparameters 10 . In fact the fermionic model is aparticular case with the constraint G s4G on theS V

w xtwo couplings of the general ENJL model 10 . Thisrelation among couplings, considered previously as a

Ž .perturbative estimate one gluon exchange valid onlyat short distances, has been reobtained in this work atthe nonperturbative level as a consequence of theintroduction of the effective fermionic field x . A

Ž .redefinition of the fermionic transformation 1 wouldlead to a reformulation of the same fermionic systemwith different variables. Then the constraint on theparameters of the ENJL model is a consequence ofthe dynamical content of the derivation of the effec-tive fermionic model and it goes beyond pure sym-metry arguments at low energies. The same con-clusion can be derived from a comparison of thevery particular form of the 1rN termc

2 2 mŽ w x .yg r 2 N M x g x x g x with the generalc I I J m Jw xform 3 for the 1rN terms in an ENJL model. Thec

particular form of the ‘‘higher dimensional terms’’inŽ .12 would only be required if one considers the

effective fermionic theory at energies close to thescale M but in this range of energies it is moreconvenient to consider the original gauge theory.

A strong phenomenological argument in favour ofthe validity of the derivation of the fermionic modelpresented in this work is that values for the cou-plings of the chiral Lagrangian in reasonable agree-ment with the experimental values, are obtained from

Ž .the ENJL model including the case G s4G inS VŽ .the case of QCD N sN s3 . It would be interest-f c

ing to go beyond the dominant term in the 1rNc

expansion in the derivation of the chiral Lagrangianfrom the fermionic model.

Besides the constraint among the two couplings ofw xa general ENJL model 10 , or more precisely the

very particular form for the effective fermionic La-Ž .grangian in 8 as compared with the most general

form for the four-fermion interaction satisfying thew xsymmetry requirements set by QCD 3 , another

result of the derivation of the fermionic model from2 Ž 2 .the gauge theory is the relation grM s1r eL

Ž 2 .among the parameters g, M of the fermionicmodel, the ultraviolet cut-off of the gauge theoryused in the calculation of the Jacobian of thefermionic transformation and the parameter e ap-pearing in the definition of the effective fermionicfield. In order to determine the two parametersŽ 2 .g, M a matching of the results of the effectivetheory at E-M with those of the gauge theoryshould be considered. In the case of QCD an alterna-tive is to determine indirectly these parameters froma fit to the low energy parameters.

Although there are no essential differences at thelevel of the change of fermionic variables betweenthe abelian and nonabelian cases it can happen thatthe generation of a dynamical mass is a consequenceof the growth at low energies of the gluonic self-in-teractions and then that an effective fermionic de-scription is only possible in the non-abelian case.Alternatively one can consider the possibility ofspontaneous symmetry breaking and dynamical massgeneration in the abelian case as a manifestation in afermionic model of a strongly coupled QED phase.

To summarize, a new method to derive fermionicfields and a fermionic effective Lagrangian for gaugetheories has been presented. The main idea is toidentify a change of variables, sensitive to the ultra-violet domain of the theory, which shows the pres-

( )J.L. Cortes et al.rPhysics Letters B 432 1998 397–401´ 401

ence of an intermediate scale M in the theory. Weassume that this scale can be interpreted as a dynam-ically generated mass scale and also that below thisscale a description of the system in terms of onlyfermionic fields is possible. When this idea is ap-plied to QCD an ENJL model is obtained at leadingorder in 1rN whose low energy limit is compatiblec

with the low energy properties as described througha chiral Lagrangian. It could be interesting to exploreother possible applications of the change of variablesintroduced in this work considering different gaugegroups, different representations for the matter fieldsand dimension of space-time. Another possible ex-tension of the present work can be based on different

Žchoices for the transformation i.e. different choices.for v .mn

Acknowledgements

This work was partially supported by CICYTŽ .contract AEN 97-1680 J.L.C., L.V. , Fondecyt-Chile

Ž .1950278, 1960229 and Dicyt-USACH J.G. . One ofŽ .us J.G. is a recipient of a John S. Guggenheim

Fellowship.

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