Isospin-Breaking quark condensates in Chiral Perturbation Theory

TL;DR

This paper investigates isospin-breaking effects on quark condensates using Chiral Perturbation Theory, including electromagnetic contributions, and explores their phenomenological implications and constraints on low-energy constants.

Contribution

It provides explicit expressions for isospin-breaking corrections in ChPT and extends sum rules to include electromagnetic effects, offering new insights into vacuum asymmetry and low-energy constants.

Findings

Electromagnetic effects influence quark condensates similarly to quark mass differences.

Sum rule extension allows estimation of vacuum asymmetry with EM corrections.

Constraints on electromagnetic low-energy constants are derived from condensate behavior.

Abstract

We analyze the isospin-breaking corrections to quark condensates within one-loop SU(2) and SU(3) Chiral Perturbation Theory including $m_u\neq m_d$ as well as electromagnetic (EM) contributions. The explicit expressions are given and several phenomenological aspects are studied. We analyze the sensitivity of recent condensate determinations to the EM low-energy constants (LEC). If the explicit chiral symmetry breaking induced by EM terms generates a ferromagnetic-like response of the vacuum, as in the case of quark masses, the increasing of the order parameter implies constraints for the EM LEC, which we check with different estimates in the literature. In addition, we extend the sum rule relating quark condensate ratios in SU(3) to include EM corrections, which are of the same order as the $m_u\neq m_d$ ones, and we use that sum rule to estimate the vacuum asymmetry within ChPT. We also discuss the matching conditions between the SU(2) and SU(3) LEC involved in the condensates, when both isospin-breaking sources are taken into account.