Abstract
The Fock-Schwinger proper-time method is used to derive the effective action in the field theory with the chiral $U(3)\times U(3)$ symmetry explicitly broken by unequal masses of heavy particles. The one-loop effective action is presented as a series in inverse powers of heavy masses. The first two Seeley-DeWitt coefficients of this expansion are explicitly calculated. This powerful technique opens a promising avenue for studying explicit flavor symmetry breaking effects in the effective field theories.
Highlights
The heat kernel technique [1] was introduced to quantum theory in works of Fock [2,3] and later pushed forward by Nambu [4] and Schwinger [5]
The method allows the essentially nonperturbative and nonlocal extensions [8,9,10,11]. It has been widely used in QCD to construct effective meson Lagrangians [12,13], in chiral gauge theories to study chiral anomalies [14], in cosmology to calculate geometric entropy [15], in QED to find Casimir energies and forces [16], etc
The Pauli-Villars regularization we used is usually applied in the NJL model, where the cutoff Λ is a scale of spontaneous chiral symmetry breaking
Summary
The heat kernel technique [1] was introduced to quantum theory in works of Fock [2,3] and later pushed forward by Nambu [4] and Schwinger [5]. The result is an asymptotic expansion for the effective action of the theory in powers of the proper time with Seeley-DeWitt coefficients anðx; yÞ. These coefficients are polynomials in the background fields and describe, in the coincidence limit y → x, the local vertices of the corresponding effective Lagrangian. The asymptotic coefficients an do not change Such long wavelengths (λ ≫ 1=m) expansion allows one to obtain an action that takes into account effectively the leading low-energy effect of virtual heavy states. Many important technical points related to our calculations are collected in six appendixes
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