Large mass invariant asymptotics of the effective action

Abstract

We study the large mass asymptotics of the Dirac operator with a nondegenerate mass matrix m = diag(m1, m2, m3) in the presence of scalar and pseudoscalar background fields taking values in the Lie algebra of the U (3) group. The corresponding one-loop effective action is regularized by the Schwinger's proper-time technique. Using a well-known operator identity, we obtain a series representation for the heat kernel which differs from the standard proper-time expansion, if m1 = m2 = m3. After integrating over the proper-time we use a new algorithm to resum the series. The invariant coefficients which define the asymptotics of the effective action are calculated up to the fourth order and compared with the related Seeley-DeWitt coefficients for the particular case of a degenerate mass matrix with m1 = m2 = m3.11.10. Ef, 03.65.Db, 12.39.Fe, 11.30.Rd The effective action plays the central role in lagrangian quantum field theory. Dividing fields into a classical background and quantum fluctuations and integrating out the last ones one obtains the effective action which properly accumulates the short distance dynamics of quantum fields [1,2]. In this letter we study the real part of the oneloop effective action with virtual heavy fermions of different masses using the Schwinger's proper-time technique [3]. In QCD the nondegenerate mass matrix of heavy (constituent) quarks results from the spontaneous breakdown of chiral symmetry, as it is the case, for instance, in the Nambu -Jona-Lasinio model [4], or from the specially added invariant term to the QCD Lagrangian that regulates the infrared behaviour of the effective action [5]. We do not consider here the manifest chiral symmetry breaking effect on the effective action [6]. A careful analysis of this problem would lead us too far away from the subject, leaving the present result without changes. It might be well to emphasize, however, that a nondegenerate mass matrix of heavy fermions in theories with spontaneous breakdown of chiral symmetry appears as a consequence of manifest symmetry breaking by the nondegenerate mass matrix of light fermions. The mathematical formalism being presented in this letter is a necessary element of the approach which faithfully mirrors the vacuum structure of such a theory. This approach was formulated for the theory with an explicit and spontaneous breakdown of the global SU (2) × SU (2) chiral symmetry in [6] and generalized to all orders of the asymptotic expansion in [7]. Here we extend this scheme to the SU (3) × SU (3) chiral theory where the mass matrix has a form m = diag(m 1 , m 2 , m 3 ).The explicit forms of dominant effective local vertices induced by virtual heavy fermions in general spontaneously broken gauge theories have been obtained in [8,9]. However both the method of our evaluation of the heat kernel as well as the result, which is cast as an asymptotic series for the effective action with the order by order chiral invariant structure of the derived asymptotic coefficients, are different from the cited papers and ...

π0−η−η′mixing in a generalized multiquark interaction scheme

“…These coefficients are totally defined in terms of h a and the parameters of the model. Now to the fermion determinant related to the integration over the fermion fields: We expand it using a heat-kernel technique that takes appropriately into account the quark mass differences, being chiral covariant at each order of the expansion [55][56][57],…”

Derivative expansion of the heat kernel in curved space

“…Extensions to finite temperature can be found in [38,39]. Expansions around non c-number mass terms are given in [40,41,42]. The extension to non commutative quantum field theory has been presented in [43].…”

Light quark masses in multi-quark interactions

“…Fortunately, the result is known. One can find all necessary details of such calculations, for instance, in [38], where we used the modified heat kernel technique [39][40][41] developed for the case of explicit chiral symmetry breaking. Here we quote the main outcome.…”

“…For that we must exclude quark degrees of freedom in (13), e.g., by integrating them out from the corresponding generating functional. The standard Gaussian path integral leads us to the fermion determinant, which we expand by using a heat-kernel technique [38][39][40][41]. The remaining part of the Lagrangian, L aux , depends on auxiliary fields which do not have kinetic terms.…”