Large Mass Invariant Asymptotics of the Effective Action

TL;DR

This paper investigates the large mass asymptotics of the Dirac operator with a nondegenerate mass matrix in a U(3) background, deriving new series representations and calculating invariant coefficients up to fourth order.

Contribution

It introduces a novel resummation algorithm for the heat kernel series in the nondegenerate mass case, extending the standard proper-time expansion.

Findings

Derived a series representation for the heat kernel differing from standard expansion when masses are unequal.

Calculated invariant coefficients up to the fourth order for the effective action.

Compared these coefficients with the degenerate mass case to highlight differences.

Abstract

We study the large mass asymptotics of the Dirac operator with a nondegenerate mass matrix m={diag}(m_1,m_2,m_3) 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 m_1\ne m_2\ne m_3. 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 m_1=m_2=m_3.