Topological susceptibility and axion potential in two-flavor superconductive quark matter

We study the potential of the axion, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>a</a:mi></a:math>, of Quantum Chromodynamics, in the two-flavor color superconducting phase of cold and dense quark matter. We adopt a Nambu-Jona-Lasinio-like model. Our interaction contains two terms, one preserving and one breaking the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>U</c:mi><c:mo stretchy="false">(</c:mo><c:mn>1</c:mn><c:msub><c:mo stretchy="false">)</c:mo><c:mi>A</c:mi></c:msub></c:math> symmetry: the latter is responsible of the coupling of axions to quarks. We introduce two quark condensates, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>h</g:mi><g:mi>L</g:mi></g:msub></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>h</i:mi><i:mi>R</i:mi></i:msub></i:math>, describing condensation for left-handed and right-handed quarks respectively; we then study the loci of the minima of the thermodynamic potential, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="normal">Ω</k:mi></k:math>, in the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mo stretchy="false">(</n:mo><n:msub><n:mi>h</n:mi><n:mi>L</n:mi></n:msub><n:mo>,</n:mo><n:msub><n:mi>h</n:mi><n:mi>R</n:mi></n:msub><n:mo stretchy="false">)</n:mo></n:math> plane, noticing how the instanton-induced interaction favors condensation in the scalar channel when the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>θ</r:mi></r:math> angle, <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>θ</t:mi><t:mo>=</t:mo><t:mi>a</t:mi><t:mo>/</t:mo><t:msub><t:mi>f</t:mi><t:mi>a</t:mi></t:msub></t:math>, vanishes. Increasing <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>θ</v:mi></v:math> we find a phase transition where the scalar condensate rotates into a pseudoscalar one. We present an analytical result for the topological susceptibility, <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>χ</x:mi></x:math>, in the superconductive phase, which stands both at zero and at finite temperature. Finally, we compute the axion mass and its self-coupling. In particular, the axion mass <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msub><z:mi>m</z:mi><z:mi>a</z:mi></z:msub></z:math> is related to the full topological susceptibility via <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mi>χ</bb:mi><bb:mo>=</bb:mo><bb:msubsup><bb:mi>m</bb:mi><bb:mi>a</bb:mi><bb:mn>2</bb:mn></bb:msubsup><bb:msubsup><bb:mi>f</bb:mi><bb:mi>a</bb:mi><bb:mn>2</bb:mn></bb:msubsup></bb:math>; hence our result for <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>χ</db:mi></db:math> gives an analytical result for <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msub><fb:mi>m</fb:mi><fb:mi>a</fb:mi></fb:msub></fb:math> in the superconductive phase of high-density Quantum Chromodynamics. Published by the American Physical Society 2024