Abstract
While the partition function for QCD in a magnetic field $H$ has been calculated before within chiral perturbation theory up to two-loop order, our investigation relies on an alternative representation for the Bose functions which allows for a clear-cut expansion of thermodynamic quantities in the chiral limit. We first focus on the pion-pion interaction in the pressure and show that -- depending on magnetic field strength, temperature and pion mass -- it may be attractive or repulsive. We then analyze the thermodynamic properties in the chiral limit and provide explicit two-loop representations for the pressure in the weak magnetic field limit $|qH| \ll T^2$.
Highlights
The low-energy properties of quantum chromodynamics (QCD) can be understood on the basis of its relevant lowenergy degrees of freedom: the Goldstone bosons. This is the path pursued by chiral perturbation theory (CHPT), and the low-temperature properties of QCD in a magnetic field have been explored within CHPT in many studies up to two-loop order [1,2,3,4,5,6,7,8,9,10,11,12]
While details on the two-loop CHPT evaluation are discussed in Appendix A, the rather technical analysis of the pressure in weak magnetic fields in the chiral limit is presented in Appendix B
Within chiral perturbation theory—based on a coordinate space representation for the thermal propagators—we have analyzed the impact of the magnetic field on the partition function up to the two-loop level
Summary
The low-energy properties of quantum chromodynamics (QCD) can be understood on the basis of its relevant lowenergy degrees of freedom: the Goldstone bosons. Our analysis goes beyond the literature by (i) analyzing how the nature of the pion-pion interaction in the pressure— repulsive or attractive—is affected by the magnetic field, as well as the temperature and pion mass, and (ii) providing the chiral expansion for the pressure in weak magnetic fields (jqHj ≪ T2) in the chiral limit. While details on the two-loop CHPT evaluation are discussed in Appendix A, the rather technical analysis of the pressure in weak magnetic fields in the chiral limit is presented in Appendix B
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