Condensates and pressure of two-flavor chiral perturbation theory at nonzero isospin and temperature
Prabal Adhikari, Jens O. Andersen, Martin A. Mojahed
Abstract
Abstract We consider two-flavor chiral perturbation theory ( $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> PT) at finite isospin chemical potential $$\mu _I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:math> and finite temperature T . We calculate the effective potential and the quark and pion condensates as functions of T and $$\mu _I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:math> to next-to-leading order in the low-energy expansion in the presence of a pionic source. We map out the phase diagram in the $$\mu _I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:math> – T plane. Numerically, we find that the transition to the pion-condensed phase is second order in the region of validity of $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> PT, which is in agreement with model calculations and lattice simulations. Finally, we calculate the pressure to two-loop order in the symmetric phase for nonzero $$\mu _I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:math> and find that $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>χ</mml:mi> </mml:math> PT seems to be converging very well.