Abstract

We present two-loop results for the quark condensate in an external magnetic field within chiral perturbation theory using coordinate space techniques. At finite temperature, we explore the impact of the magnetic field on the pion-pion interaction in the quark condensate for arbitrary pion masses and derive the correct weak magnetic field expansion in the chiral limit. At zero temperature, we provide the complete two-loop representation for the vacuum energy density and the quark condensate.

Highlights

  • The quark condensate—order parameter of spontaneous chiral symmetry breaking—is a crucial quantity in particle physics

  • Our calculation within the framework of chiral perturbation theory (CHPT) goes up to two-loop order, but in contrast to the available CHPT studies—see Refs. [1,2,3,4,5,6,7,8,9,10]—we use a novel representation for the kinematical functions that we established in Ref. [11]

  • As we show in Appendix A 1, all effective low-energy constants become singular in the physical limit d → 4, and the divergence is contained in the parameter λ: λ

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Summary

INTRODUCTION

The quark condensate—order parameter of spontaneous chiral symmetry breaking—is a crucial quantity in particle physics. In a recent article [11], the present author has pointed out that—in the chiral limit—the two published one-loop series for the finite-temperature quark condensate in a weak magnetic field, independently derived by different authors, are erroneous. We clarify the situation by providing the correct weak magnetic field expansion of the finite-temperature quark condensate in the chiral limit. In the chiral limit and in weak magnetic fields, the series at order T2—organized by the expansion parameter ε 1⁄4 jqHj=T2 (q is the electricpcffihffi arge of the pion)—involves a leading square-root term ∝ ε, a term linear in ε, followed by a half-integer power ε3=2 and a logarithmic contribution ε2 ln ε. While Appendix B is devoted to the chiral limit in nonzero magnetic fields at T 1⁄4 0, in Appendix C we consider the same situation at finite temperature which boils down to an analysis of the various kinematical functions required

CHIRAL PERTURBATION THEORY EVALUATION
QUARK CONDENSATE IN A MAGNETIC FIELD
Finite-temperature quark condensate
CONCLUSIONS
Low-energy effective constants at NLO and NNLO
Isolating UV divergences
Findings
S pffiffi zSðzÞ: ðC8Þ z
Full Text
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