The Nambu-Jona-Lasinio model and the chiral anomaly

ANNALS

OF PHYSICS

193, 287-325

(1989)

The Nambu-Jona-Lasinio Model and the Chiral Anomaly* M. WAKAMATSU~ Instiiute of Theoretical D-8400 Regensburg, Received

Physics. Federal September

University Republic

of Regenshurg,

qf German?

21, 1988

An approximate bosonization of the Nambu-Jona-Lasinio model is shown to lead to either form of two low energy effective Lagrangians: i.e., the massive Yang-Mills form or the hidden local symmetry form of Bando et al. The specific underlying quark Lagrangian restricts the freedom in obtaining the anomalous effective action on one and the other scheme. This enables us to show the equivalence of the two schemes not only for the physics of the nonanomalous sector but also for the physics of the anomalous sector. Our model in either representation however breaks low energy theorems for some of the anomalous processes. In pursuit of its origin, we investigate the general structure of the hadronic currents in the present model, putting special emphasis upon the underlying Lagrangian at the quark level. As an interesting byproduct of the present analysis. we gain a new insight into the dynamical meaning of the parameter a, which appears as a free parameter in the hidden symmetry model of Bando et al. I(” 1989 Academic Press, Inc.

1. INTRODUCTION Growing attention has recently been paid to the low energy effective Lagrangians of QCD which incorporate vector (and axial-vector) mesons in addition to the Goldstone pion field [l-3]. This renewal of interest in the low energy effective Lagrangians owes greatly to the conjecture that, at low energy and in the limit of large number of colors IV,, QCD reduces to a nonlinear effective theory of weakly interacting mesons [4,5]. Since baryons emerge as topological solitons in such a theory, their properties are intimately connected with the physical parameters of the meson sector [6-91. There are two popular ways of introducing vector (and axial-vector) mesons into the original chiral Larangian, i.e., the non-linear sigma model. The first one is the so-called massive Yang-Mills approach [ 10-121. The guiding principles are chiral symmetry and non-abelian gauge symmetry: the latter is “minimally” broken due to mass terms inserted by hand. The second approach, which has recently received increasing attention is the hidden local symmetry approach due to Bando er al. * Work supported in part by BMFT, Grant MEP 0234 REA. ’ Permanent address: Department of Physics, Osaka University,

Japan.

287 0003-4916189

$7.50

Copyright :K-’ 1989 by Academtc Press. Inc All rghts of reproduclmn in any form reserved

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M. WAKAMATSU

[13, 143. They suggested that the vector mesons p, o, K*, and 4 are dynamical gauge bosons associated with the hidden [U(3),,],,,,, symmetry in the U(3), x U(3),/U(3), nonlinear chiral Lagrangian. Under the assumption that the composite gauge boson acquires a kinetic term, they obtained an effective Lagrangian compatible with the vector meson phenomenology which is manifest, e.g., in the KSFR relation [15, 163, the universality of the p couplings, the p-dominance of the pion electromagnetic form factor etc. [12]. In consideration of the phenomenological success of both approaches, we naturally come to the following question. Is there any fundamental relationship between these two apparently different schemes? An answer to this question was recently given by Meigner and Zahed [ 171 and also by Yamawaki [IS] at a formal level; it was given by us at a dynamical level in a recent paper [19] (hereafter referred to as (I)). To be explicit, it has been revealed through several recent studies [19-221 that either form of two low energy effective meson Lagrangians, i.e., the massive Yang-Mills one or the hidden symmetry one of Bando et al. [13, 143, can be derived from a single quark theory, i.e., the generalized Nambu-Jona-Lasinio (NJL) model, as a result of an approximate bosonization procedure [20,21]. A crucial observation there was that the field variables corresponding to the vector meson (and also the axial-vector meson) in both schemes are related to each other through a chiral transformation which depends on the Goldstone pion field. (It is sometimes called the Stiickelberg transformation in the literature [17, 181.) Although the forms of these two effective Lagrangians are quite different, it was claimed that the predictions of the two schemes are identical as far as the mass relations and the (non-anomalous) on-mass-shell vertices are concerned [3]. Then, a natural question is whether or not the above equivalence between the two schemes can be extended to the physics of the anomalous sector that concerns intrinsicparity violating processes. The physics of the chiral anomaly, especially in the presence of the hadronic vector and axial-vector fields, is rather involved, and there still remain some open questions [23-341. A standard prescription adopted in the massive Yang-Mills scheme [23-321 is to assume that the intrinsic parity violating processes originate in the gauged Wess-Zumino-Witten action [35, 361 (in the Bardeen subtracted form [37]) in which the gauge fields are identified with the phenomenological vector meson (and possibly the axial-vector meson too). The vertices involving the external gauge fields are assumed to be present in the non-anomalous part only, precisely in the form of a direct coupling between the vector meson and the external gauge fields [ 121. This immediately leads to the vector meson dominance of the intrinsic parity violating processes (as well as the intrinsic parity conserving ones). The above procedure is known to give a good phenomenological description but it is generally believed to have two unpleasant features [27, 29, 33, 343. First, in this theoretical framework, we are obliged to introduce the axial-vector meson as a gauge field simultaneously with the vector meson in order to preserve U(n) x U(n), symmetry. But the gauging of the anomalous axial-vector channel has a danger that the equations of motion may be mutually inconsistent, as was pointed out by

NAMBU-JONA-LASINIO

MODEL

289

Kaymakcalan, Rajeev, and Schechter [27--291. Second, the introduction of the axial-vector meson in this way forces us to choose Bardeen’s form of the anomaly because the left-right symmetric form no longer respects the vector current conservation. The Bardeen form of the anomaly, however, explicitly breaks the chiral U(n) x U(n), symmetry. Being aware of the above difficulties, Fujiwara et al. treated the anomaly problem in a different way in the hidden symmetry approach [33]. (See also Ref. 34).) The anomaly is assumed to sit in the part of the effective action that includes the Goldstone bosons and the external gauge fields alone, which is nothing but the gauged Wess-Zumino-Witten action in its original version. The vertices which contain the internal (or hadronic) vector meson are assumed to be included in the chiral gauge invariant (anomaly-free) but intrinsic parity violating part of the effective action, It should be emphasized that this part containing several arbitrary coefficients can be fixed only phenomenologically [33]. In a recent paper [38], Scoccola, Rho, and Min performed a comparative analysis of the anomalous process, K+ Kp -+ 3n, using both the massive Yang-Mills scheme (they call it the external gauging scheme) and the hidden symmetry scheme of Fujiwara et al. [33]. In the hidden symmetry scheme, the above 5-Goldstone vertex is saturated by the contribution of the Wess-Zumino action which depends on the Goldstone fields alone. Any other contributions from the vector meson mediated diagrams exactly cancel among themselves in the low energy limit [33]. On the other hand, in the massive Yang-Mills scheme (it is important to remember that for the purpose of comparison they had to eliminate the axial-vector field in this scheme in a particular way explained later), Scoccola et al. found that the prediction based on the Wess-Zumino term, which is known to reproduce the low energy theorem, receives an intolerably large modification due to the vector meson mediated processes. According to their analysis, there appears to be no natural way in the massive Yang-Mills scheme to suppress additional significant contributions from vector mesons. From this fact, they concluded that the massive YanggMills scheme is not consistent with QCD anomalies, and only the hidden symmetry scheme is acceptable as a consistent framework. In this paper, we wish to reexamine this conclusion. Our conclusion is in fact that neither of the schemes is superior to the other. It is not a problem of scheme (or representation) but rather how to treat the chiral anomaly. Let us now clarify this point in the rest of the paper. The paper is organized as follows. In Section 2, the two forms of effective Lagrangians, i.e., the massive Yang-Mills and hidden symmetry ones, are derived through an approximate bosonization of the generalized Nambu-Jona-Lasinio model. The description of this derivation is divided into three subsections. In the first subsection, we give a precise definition of the effective meson action based on the proper-time regularization scheme of the fermion path integral. Then, the nonanomalous and anomalous parts of the effective action are separately discussed in the following two subsections. Section 3 shows a comparative study of several non-

290

M. WAKAMATSU

anomalous processes using both the massive Yang-Mills and hidden symmetry schemes. The corresponding analysis of the anomalous processes are given in Section 4. In Section 5, we shall analyze the general structure of the hadronic currents in the present model, paying special attention to the underlying Lagrangian at the quark level. Concluding remarks are given in Section 6.

2. BOS~NIZATION 2.1. Definition

OF THE NAMBU-JONA-LASINIO

MODEL

of Effective Meson Actions

The starting point of our analysis is the following Lasinio (NJL) model [ 3942, 20, 211: .2-

YNJL = W‘ drq + 2G, 1

generalized Nambu-Jona-

1

((4T”q)’

+ (qiy’ T”q)2}

Cl=0

rl-

2G2

c 0=0

1 {GWT”d2

+

@“WW2}.

(2.1)

Here q are the quark fields, n is the number of the flavor degrees of freedom, and T, are generators of the flavor U(n) group normalized as tr(TaTb) = IS,,. (The color indices of quarks are not explicitly shown.) Throughout the present study, we shall neglect the bare quark masses, for simplicity. In this limit, the above Lagrangian has exact global U(n), x L(n), symmetry. Naturally, the NJL model cannot be a “true” substitute for the QCD Lagrangian, but there are several good reasons to assume that it might be a good low energy approximation to the QCD Lagrangian. It correctly describes the spontaneous chiral symmetry breaking of the QCD vacuum. Furthermore, owing to the approximate bosonization techniques we can develop a clear understanding about the relationship between the NJL model and some low energy effective meson Lagrangians, as far as the physics of the nonanomalous sector is concerned [19-22, 433. Then, we expect that it should provide useful information also for the physics in the anomalous sector. In our treatment, the introduction of the electroweak interaction is uniquely done at the quark level, by gauging an appropriate subgroup of the flavor [U(n), x U(n),],,,,,, group [ 19,201. (Here we may have in mind, for instance, application to the standard SU(2), x U( 1) y electroweak interaction model.) In general, one may try to gauge an arbitrary subgroup H of U(n), x U(n),, with generators K”, e = 1, .... r. Each K” is a linear combination of generators TF and T> of U(n), and U(n),, Ku= T”,+ Ts. (Either Tz or T: may vanish for some value of a.) Thus, the electroweak interaction is introduced through the following minimal replacement of the quark kinetic part of the Lagrangian

NAMBU-JONA-LASINIO

where PRiL = $ (1-I y’), gauge fields. After introducing the scalar), V,(vector), A, valued) [19-22, 40-433,

and gp = -i%‘;T”,

291

MODEL

and Yp = -iU;1,TO,

are the external

color singlet collective meson fields S (scalar), P (pseudo(axial-vector) in the standard way (they are all matrixthe gauged NJL Lagrangian can be cast into the form:

-&trCS2+ 1

P21 -&

2

tr[Rt+

(2.3)

LfL] + YC,,.

Here and hereafter we interchangeably use the vector and axial-vector notation (V,, A,) and the right- and left-handed notation (R, = V,, + A,,, L,, = V,, - A,,). Let us furthermore introduce the change of the field variables as S+iP=
(2.4)

in terms of a hermitian matrix C and a unitary matrix 5 [19-221. The latter is parametrized as t(x) = einCx)uain terms of the Goldstone pion field n(~). (A more general parametrization S + iP =