Pion condensation and pion star from holographic QCD

The properties of quantum chromodynamics (QCD) matter at finite isospin densities are investigated employing holographic hard-wall and soft-wall anti–de Sitter (AdS)/QCD models. It is confirmed that at high enough isospin densities, charged pions start to condense and the pion superfluid phase appears in the system. It is shown that the chiral condensate and the pion condensate can be transformed to each other and form a chiral circle in the superfluid phase. We derived the equation of state (EoS) for pionic matter, calculated the normalized trace anomaly <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="normal">Δ</a:mi></a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mo stretchy="false">(</d:mo><d:mi>ε</d:mi><d:mo>−</d:mo><d:mn>3</d:mn><d:mi>p</d:mi><d:mo stretchy="false">)</d:mo><d:mo>/</d:mo><d:msubsup><d:mi>m</d:mi><d:mi>π</d:mi><d:mn>4</d:mn></d:msubsup></d:math>, and analyzed the sound speed and adiabatic index. Additionally, we provided data on the mass-radius relation and tidal deformability of pion stars. The results indicate that the holographic models align well with lattice QCD concerning isospin density, axial-vector condensation, EoS, and trace anomaly, though discrepancies in sound speed and adiabatic index emerge at higher isospin chemical potentials. The holographic models closely match the leading-order results from chiral perturbation theory (<h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>χ</h:mi><h:mi>PT</h:mi></h:math>) in certain observables, suggesting that they may effectively capture aspects of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>χ</j:mi><j:mi>PT</j:mi></j:math> dynamics within a five-dimensional framework under the probe approximation.