Computation of the $p^6$ order low-energy constants with tensor sources

TL;DR

This paper calculates the $p^4$ and $p^6$ order low-energy constants for the chiral Lagrangian with tensor sources, extending previous work and clarifying the relations among operators for two and three flavors.

Contribution

It generalizes previous calculations to include tensor sources and identifies the number of independent operators, also showing the non-existence of odd-intrinsic-parity terms with tensor sources.

Findings

Identified 98 independent terms for n-flavor case.

Found 92 terms for three-flavor, 65 for two-flavor cases.

Established relations among $p^6$ operators with tensor sources.

Abstract

We present the results of calculations of the $p^4$ and $p^6$ order low-energy constants for the chiral Lagrangian with tensor sources for both two and three flavors of pseudoscalar mesons. This is a generalization of our previous work on similar calculations without tensor sources in terms of the quark self-energy $\Sigma(p^2)$, based on the first principle derivation of the low-energy effective Lagrangian and computation of the low-energy constants with some rough approximations. With the help of partial integration and some epsilon relations, we find that some $p^6$ order operators with tensor sources appearing in the literature are related to each other. That leaves 98 independent terms for $n$-flavor, 92 terms for three-flavor, and 65 terms for two-flavor cases. We also find that the odd-intrinsic-parity chiral Lagrangian with tensor sources cannot independently exist in any order of low-energy expansion.