<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>θ</mml:mi> </mml:math> -term effects on isospin asymmetric hot and dense quark matter

We investigate the impact of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>C</a:mi> <a:mi>P</a:mi> </a:math> -violating <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>θ</c:mi> </c:math> term on isospin symmetry breaking in quark matter and compact star properties using a two-flavor Nambu-Jona-Lasinio model. By incorporating the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>θ</e:mi> </e:math> parameter through the Kobayashi-Maskawa-’t Hooft determinant interaction, we derive the thermodynamic potential and gap equations under finite temperature, baryon chemical potential, and isospin chemical potential. At zero temperature and baryon density, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>θ</g:mi> </g:math> suppresses conventional chiral ( <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>σ</i:mi> </i:math> ) and pion ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>π</k:mi> </k:math> ) condensates while promoting pseudoscalar ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>η</m:mi> </m:math> ) and scalar-isovector ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>δ</o:mi> </o:math> ) condensates, thereby reducing the critical isospin chemical potential <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msubsup> <q:mi>μ</q:mi> <q:mi>I</q:mi> <q:mrow> <q:mi>crit</q:mi> </q:mrow> </q:msubsup> </q:math> for spontaneous symmetry breaking. For <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>θ</s:mi> <s:mo>=</s:mo> <s:mi>π</s:mi> </s:math> , a first-order phase transition emerges at <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msubsup> <u:mi>μ</u:mi> <u:mi>I</u:mi> <u:mrow> <u:mi>crit</u:mi> </u:mrow> </u:msubsup> <u:mo>=</u:mo> <u:mn>0.021</u:mn> <u:mtext> </u:mtext> <u:mtext> </u:mtext> <u:mi>GeV</u:mi> </u:math> , accompanied by <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>C</w:mi> <w:mi>P</w:mi> </w:math> symmetry restoration. Extending the investigation to finite temperature and baryon chemical potential reveals that these <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>θ</y:mi> </y:math> -term-induced effects persist. Axion effects (modeled via <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>θ</ab:mi> <ab:mo>≡</ab:mo> <ab:mi>a</ab:mi> <ab:mo>/</ab:mo> <ab:msub> <ab:mi>f</ab:mi> <ab:mi>a</ab:mi> </ab:msub> </ab:math> ) stiffen the equation of state of nonstrange quark stars, increasing their maximum mass and radii, in agreement with multimessenger constraints from pulsar observations and gravitational wave events. These results establish <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>θ</cb:mi> </cb:math> as a critical parameter modulating both the quantum chromodynamics phase structure and compact star observables.