We investigate the impact of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math> term on isospin symmetry breaking in quark matter and compact star properties using a two-flavor Nambu-Jona-Lasinio model. By incorporating the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></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, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math> suppresses conventional chiral (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>) and pion (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi></math>) condensates while promoting pseudoscalar (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math>) and scalar-isovector (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>δ</mi></math>) condensates, thereby reducing the critical isospin chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>μ</mi><mi>I</mi><mrow><mi>crit</mi></mrow></msubsup></math> for spontaneous symmetry breaking. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi><mo>=</mo><mi>π</mi></math>, a first-order phase transition emerges at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>μ</mi><mi>I</mi><mrow><mi>crit</mi></mrow></msubsup><mo>=</mo><mn>0.021</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, accompanied by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> symmetry restoration. Extending the investigation to finite temperature and baryon chemical potential reveals that these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math>-term-induced effects persist. Axion effects (modeled via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi><mo>≡</mo><mi>a</mi><mo>/</mo><msub><mi>f</mi><mi>a</mi></msub></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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math> as a critical parameter modulating both the quantum chromodynamics phase structure and compact star observables.

China

Lei Zhang

Lu-Meng Liu

Mei Huang

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

We study the symmetries and conserved quantities in $$f(R)$$ gravity for the static, spherically symmetric Reissner–Nordström spacetime using two complementary frameworks: Noether symmetries and Mei symmetries. Starting from a canonical Lagrangian for radial metric functions and the curvature scalar $$R$$ , we derive the associated Hamiltonian and show that the Legendre map is regular whenever both the first derivative of $\textit{f}$($\textit{R}$) with respect to $\textit{R}$ and the second derivative with respect to $\textit{R}$ is non-zero. Within Noether’s approach (variational and Lie-derivative forms), we obtain general, canonical, and internal symmetry classes and identify explicit generators; for the quadratic model $$f(R)=R^{2}$$ these include radial translations and scaling symmetries. We then formulate Mei symmetry conditions as invariance of the Euler–Lagrange equations under the first prolongation, which yields an overdetermined partial differential equation (PDE) system for the generator components. Solving this system for $$f(R)=R^{2}$$ , we find eight independent Mei generators and construct the corresponding conserved currents, some with no direct Noether analogue. The analysis demonstrates that Mei symmetries extend the standard Noether framework for higher-order Lagrangians and provide additional conserved quantities relevant to black-hole dynamics in modified gravity. We conclude with a comparison of the two symmetry schemes and outline applications to broader $$f(R)$$ models and to rotating spacetimes.

Egypt

Tahia Dabash

United States

Moataz Emam

Germany

Lukas Schöppner

Symmetry and conserved quantities in $$ f(R) $$ -gravity: Mei vs. Noether approaches

Based on a machine learning holographic QCD model, we construct a systematical framework to investigate the properties of the scalar glueballs continuously from zero temperature to finite temperature. By using both the quasinormal frequencies and the spectral functions, we extract the pole masses, thermal widths, screening masses and dispersion relation of the scalar glueballs in hot medium. It is shown that the pole masses almost remain the vacuum values at temperatures far below the critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math>, and then decrease with the increasing of temperature until a temperature lower than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math>. This result qualitatively agrees with earlier lattice simulations. While the pole masses increase monotonically above the critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math>, which agrees with recent lattice calculation. Meanwhile, it is shown that the thermal widths increase monotonically with temperature, which also agrees with the near <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math> lattice simulations. The screening mass exhibits a similar temperature-dependent behavior to the pole mass, while the dispersion relation increasingly deviates from the relativistic one as the temperature rises. It is interesting to note that we obtain the imaginary corrections in the thermal correlators, which contains both the temperal and spatial information and might be helpful for the four-dimensional calculations. Furthermore, by comparing the quasinormal modes and the spectral functions, we note that it requires more careful analysis when applying the spectral functions in studying thermal hadrons from holography, since there could be other types of quasinormal modes which are not related with bound states while they may contribute to the peaks of the spectral functions.

Ruixiang Chen

Danning Li

Thermal properties of the scalar glueballs from holography

We study dynamical properties of strongly coupled chiral matter by using the holographic method. We demonstrate, at both linear and nonlinear levels, that perturbations on thermodynamically unstable backgrounds within the spinodal region of chiral first-order phase transitions exhibit dynamic instability. The corresponding magnitude of dynamic instability can be characterized by the critical momentum. Furthermore, we found that, within a certain temperature range, the quasinormal-mode spectrum contains purely imaginary diffusive modes. As spatial momentum increases, a transition occurs in the system’s long-time dynamics. The dominant contribution shifts from diffusive mode to propagating mode. When the diffusive mode becomes dominant, the spectral function exhibits a transport peak structure in the low-frequency region. A heuristic argument suggests that this particular transition can be related to the chiral symmetry breaking and restoration.

Pei Zheng

Yidian Chen

Yu-xin Liu

Dynamical instability and transport peak of chiral matter from holography

We present a Bayesian holographic model constructed by integrating the equation of state and baryon number susceptibility at zero chemical potential from lattice quantum chromodynamics (QCD). The model incorporates error estimates derived from lattice data. With this model, we systematically investigate the thermodynamic properties of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>-flavor QCD system. Using Bayesian inference, we perform precise calibration of the model parameters and determined the critical end point (CEP) position under the maximum a posterior (MAP) estimation to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msup><mi>T</mi><mi>c</mi></msup><mo>,</mo><msubsup><mi>μ</mi><mi>B</mi><mi>c</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0.0859</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>,</mo><mn>0.742</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo stretchy="false">)</mo></math>. Additionally, we predict the CEP positions within 68% and 95% confidence levels, yielding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msup><mi>T</mi><mi>c</mi></msup><mo>,</mo><msubsup><mi>μ</mi><mi>B</mi><mi>c</mi></msubsup><msub><mo stretchy="false">)</mo><mrow><mn>68</mn><mo>%</mo></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn>0.0820</mn><mo>−</mo><mn>0.0889</mn><mo>,</mo><mn>0.71</mn><mo>−</mo><mn>0.77</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msup><mi>T</mi><mi>c</mi></msup><mo>,</mo><msubsup><mi>μ</mi><mi>B</mi><mi>c</mi></msubsup><msub><mo stretchy="false">)</mo><mrow><mn>95</mn><mo>%</mo></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn>0.0816</mn><mo>−</mo><mn>0.0898</mn><mo>,</mo><mn>0.71</mn><mo>−</mo><mn>0.79</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, respectively. Moreover, to validate the reliability and predictive power of our approach, we conduct a comprehensive comparison between our predictions and potential CEP locations proposed by other theoretical models. This work not only establishes a novel Bayesian framework for holographic modeling but also provides valuable insights and theoretical support for exploring phase transitions in strongly interacting matter under extreme conditions.

Liqiang Zhu

Italy

Xun Chen

Kai Zhou

Hanzhong Zhang

Bayesian inference of the critical end point in a (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor system from holographic QCD

By using the holographic approach, we investigate the interplay between the order parameter and Goldstone modes in the real-time dynamics of the chiral phase transition. By quenching the system to a different thermal bath and obtaining different kinds of initial states, we solve the real-time evolution of the system and find that an additional prethermalization stage emerges at non-critical temperature after introducing the Goldstone modes. Some basic properties related to this additional prethermalization stage have been discussed in detail. More interestingly, an additional scaling relation is observed, indicating the appearance of a non-thermal fixed point in the dynamical region.

Yuxin Liu

Non-equilibrium dynamics of Goldstone excitation from holography

The jet quenching parameter and energy loss of light and heavy quarks have been investigated in the framework of holographic quantum chromodynamics (QCD) models with a critical end point (CEP) at finite baryon chemical potential in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> systems. The properties of the jet quenching parameter and energy loss around phase boundary have been carefully studied. It is found that the dimensionless jet quenching parameter and the energy loss of light and heavy quarks exhibit evident features around the phase boundary. Specifically, all these quantities increase rapidly near the CEP phase transition temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>CEP</mi></mrow></msub></math> with fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mrow><mi>CEP</mi></mrow></msub></math>. Moreover, the velocity dependent behavior of heavy quark energy loss differs significantly from charged particle energy loss in quantum electrodynamics (QED) matter. For electromagnetic interaction, the energy loss of charged particle can be described by the Bethe-Bloch formula and the Lindhand-Scharff-Schiott theory at low and high velocities, respectively. However, the heavy quark energy loss at CEP is approximately proportional to velocity at low velocities and aligns with Bjorken’s results at high velocities, which indicates that the heavy quark energy loss is predominantly collisional at low velocities and gluon radiation dominant at high velocities. For light quark energy loss, the behavior of the energy loss per unit length and the total energy loss differs significantly. However, the total energy loss and stopping distance exhibit similar behavior. This implies that the stopping distance predominantly determines the total energy loss. Thus, even with increased energy loss per unit length at higher temperatures or chemical potentials, the total energy loss decreases due to the reduced stopping distance.

Zhibin Li

Jet quenching and quark energy loss 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>ε</mi><mo>−</mo><mn>3</mn><mi>p</mi><mo stretchy="false">)</mo><mo>/</mo><msubsup><mi>m</mi><mi>π</mi><mn>4</mn></msubsup></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 (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>χ</mi><mi>PT</mi></math>) in certain observables, suggesting that they may effectively capture aspects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>χ</mi><mi>PT</mi></math> dynamics within a five-dimensional framework under the probe approximation.

Mingshan Ding

Iran

Kazem Bitaghsir Fadafan

Pion condensation and pion star from holographic QCD

Affleck-Dine baryogenesis generated high baryon density in the early Universe. The baryon chemical potential enhanced the potential barrier and significantly reduced the decay rate of false vacuum, which decreased from infinity at the critical end point to zero at the critical nucleation point. When the decay rate reached zero, the false vacuum of high baryon density quark matter was unlikely to decay and could persist over cosmological time scales. Therefore, primordial quark nuggets (PQNs) could form and survive in the early Universe as the seeds of compact stars. This new mechanism for the formation of PQNs is different from Witten's stable droplet of quark matter.

Jingdong Shao

Hong Mao

Nanohertz gravitational waves and primordial quark nuggets from dense QCD matter in the early Universe * Supported in part by the National Natural Science Foundation of China (12235016, 12221005), and the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030000)

We investigate the signal of the chiral magnetic effect (CME) in <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Au</mi><mo>+</mo><mi>Au</mi></mrow></math> collisions and isobar collisions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mmultiscripts><mi>Ru</mi><mprescripts></mprescripts><mn>44</mn><mn>96</mn></mmultiscripts><mo>+</mo><mmultiscripts><mi>Ru</mi><mprescripts></mprescripts><mn>44</mn><mn>96</mn></mmultiscripts></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mmultiscripts><mi>Zr</mi><mprescripts></mprescripts><mn>40</mn><mn>96</mn></mmultiscripts><mo>+</mo><mmultiscripts><mi>Zr</mi><mprescripts></mprescripts><mn>40</mn><mn>96</mn></mmultiscripts></mrow></math> in the newly developed chiral anomaly transport module based on the state-of-the-art model “a multiphase transport.” Our numerical simulation results for the ratio charge correlation <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Δ</mi><mi>γ</mi></mrow></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Ru</mi><mo>+</mo><mi>Ru</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Zr</mi><mo>+</mo><mi>Zr</mi></mrow></math> collisions are close to the latest experimental data. The simulation shows that the CME signal is larger in <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Ru</mi><mo>+</mo><mi>Ru</mi></mrow></math> collisions than that in <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Zr</mi><mo>+</mo><mi>Zr</mi></mrow></math> collisions, while the background is smaller, and the upper limit of the CME signal is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>15</mn><mo>%</mo></mrow></math> in the isobar collisions.

Zilin Yuan

Anping Huang

Guannan Xie

Wen-Hao Zhou

Guo-Liang Ma

Exploring the chiral magnetic effect in isobar collisions through chiral anomaly transport