Pion and kaon condensation at zero temperature in three-flavor χPT at nonzero isospin and strange chemical potentials at next-to-leading order

Abstract

A bstract We consider three-flavor chiral perturbation theory ( χ PT) at zero temperature and nonzero isospin ( μ I ) and strange ( μ S ) chemical potentials. The effective potential is calculated to next-to-leading order (NLO) in the π ± -condensed phase, the K ± -condensed phase, and the $$ {K}^0/{\overline{K}}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msup> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msup> </mml:math> -condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, $$ \left|{\mu}_I\right|={m}_{\pi },\left|\frac{1}{2}{\mu}_I+{\mu}_S\right|={m}_K $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>π</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mfenced> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> , and $$ \left|-\frac{1}{2}{\mu}_I+{\mu}_S\right|={m}_K $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> , respectively at tree level and remains unchanged at NLO. The transition between the two condensed phases is first order. The effective potential in the pion-condensed phase is independent of μ S and in the kaon-condensed phases, it only depends on the combinations $$ \pm \frac{1}{2}{\mu}_I+{\mu}_S $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>±</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>S</mml:mi> </mml:msub> </mml:math> and not separately on μ I and μ S . We calculate the pressure, isospin density and the equation of state in the pion-condensed phase and compare our results with recent (2 + 1)-flavor lattice QCD data. We find that the three-flavor χ PT results are in good agreement with lattice QCD for μ I &lt; 200 MeV, however for larger values χ PT produces values for observables that are consistently above lattice results. For μ I &gt; 200 MeV, the two-flavor results are in better agreement with lattice data. Finally, we consider the observables in the limit of very heavy s -quark, where they reduce to their two-flavor counterparts with renormalized couplings. The disagreement between the predictions of two and three flavor χ PT can largely be explained by the differences in the experimental values of the low-energy constants.