Pion condensation versus color superconductivity, speed of sound, and charge neutrality effects in the quark-meson diquark model

We employ the two-flavor quark-meson diquark model as a low-energy model for QCD at nonzero quark and isospin chemical potentials <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>μ</a:mi> </a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>μ</c:mi> <c:mi>I</c:mi> </c:msub> </c:math> , and at zero temperature. We map out the phase diagram in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>μ</e:mi> <e:mi>–</e:mi> <e:msub> <e:mi>μ</e:mi> <e:mi>I</e:mi> </e:msub> </e:math> plane, which has four phases: a vacuum phase, a phase with condensed charged pions/Cooper pairs of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>u</g:mi> </g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mover accent="true"> <i:mi>d</i:mi> <i:mo stretchy="false">¯</i:mo> </i:mover> </i:math> quarks, a normal quark matter (NQM) phase, and a color superconducting phase (2SC phase). The global symmetry breaking <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>S</m:mi> <m:mi>U</m:mi> <m:mo stretchy="false">(</m:mo> <m:mn>3</m:mn> <m:msub> <m:mo stretchy="false">)</m:mo> <m:mi>c</m:mi> </m:msub> <m:mo stretchy="false">→</m:mo> <m:mi>S</m:mi> <m:mi>U</m:mi> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:msub> <m:mo stretchy="false">)</m:mo> <m:mi>c</m:mi> </m:msub> </m:math> in the 2SC phase gives rise to a number of Nambu-Goldstone bosons. We classify them and briefly discuss their properties. We calculate the speed of sound <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msub> <t:mi>c</t:mi> <t:mi>s</t:mi> </t:msub> </t:math> in the two special cases, finite <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>μ</v:mi> <v:mi>I</v:mi> </v:msub> </v:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>μ</x:mi> <x:mo>=</x:mo> <x:mn>0</x:mn> </x:math> , and finite <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>μ</z:mi> </z:math> and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi>μ</bb:mi> <bb:mi>I</bb:mi> </bb:msub> <bb:mo>=</bb:mo> <bb:mn>0</bb:mn> </bb:math> . In both cases, the speed of sound exhibits a maximum and approaches the conformal limit from above as the density increases. For nonzero isospin <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:msub> <db:mi>μ</db:mi> <db:mi>I</db:mi> </db:msub> </db:math> , this behavior is in agreement with the speed of sound obtained from lattice simulations. In the 2SC phase, the behavior is qualitatively the same if we impose local charge neutrality. Finally, we discuss the possibility of having a mixed phase of negatively charged NQM and positively charged 2SC matter with global color charge neutrality imposed on the latter.