Speed of sound and trace anomaly in a unified treatment of the two-color diquark superfluid, the pion-condensed high-isospin matter, and the 2SC quark matter

In a unified perturbative treatment from the high-density side, we compute the speed of sound and the trace anomaly as functions of the chemical potential <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>μ</a:mi></a:mrow></a:math> for the two-color diquark superfluid, the pion-condensed high-isospin matter, and the two-flavor color superconducting (2SC) quark matter. We find that the corrections induced by the gap energy <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="normal">Δ</c:mi></c:math> involve nontrivial interplay between the dimensionless magnitude <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mo stretchy="false">|</f:mo><f:mi mathvariant="normal">Δ</f:mi><f:mo>/</f:mo><f:mi>μ</f:mi><f:mo stretchy="false">|</f:mo></f:math> and the derivative <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo stretchy="false">|</k:mo><k:mo>∂</k:mo><k:mi mathvariant="normal">Δ</k:mi><k:mo>/</k:mo><k:mo>∂</k:mo><k:mi>μ</k:mi><k:mo stretchy="false">|</k:mo></k:math>. Even though the gap equation has a common structure for these phases of our interest, different numerical constants cause drastic changes in the speed of sound corrections. As long as <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mo stretchy="false">|</p:mo><p:mi mathvariant="normal">Δ</p:mi><p:mo>/</p:mo><p:mi>μ</p:mi><p:mo stretchy="false">|</p:mo></p:math> is dominant over the derivative, the gap effects increase the speed of sound, which is consistent with the expected behavior of exhibiting a peak at intermediate density. We then discuss the trace anomaly which is pushed down generally by the gap effects and turns out to be negative widely in the high density regime. For demonstration of nonperturbative enhancement, we take account of the instanton-induced interaction. We confirm a further increase in the speed of sound for the two-color diquark superfluid and the pion-condensed high-isospin matter, while the speed of sound for the 2SC quark matter deviates far below the conformal limit due to the derivative contribution.