We construct a 2 + 1-flavor holographic QCD model within the Einstein-dilaton-flavor framework and investigate its phase structure. At zero chemical potential, the model reproduces the equation of state and the chiral transition in quantitative agreement with lattice QCD results. By varying the light and strange quark masses, we map out the quark-mass dependence of the transition order and obtain the corresponding phase diagram, which is consistent with phase structures extracted from lattice simulations and other nonperturbative approaches. In particular, the predicted first-order region is found to be small, in line with the most recent lattice QCD analyses. We further examine the critical behavior along the second-order boundaries and in the tricritical region, finding that the critical exponents exhibit mean-field scaling characteristic of classical holographic constructions. Taken together, these results provide a coherent holographic description of QCD thermodynamics and its phase structure across a wide range of quark-mass regimes.

China

Jin-Yang Shen

Xin-Yi Liu

Jin-Rui Wu

Yue-Liang Wu

Zhen Fang

Phase structure of 2+1-flavor QCD from an Einstein-dilaton-flavor holographic model

We introduce FANG (Focused Angular $\textit{N}$ -body event Generator), a new Monte Carlo tool for efficient event generation in restricted Lorentz-Invariant Phase Space (LIPS). Unlike conventional approaches that sample the full 4$\textit{π}$ solid angle, FANG directly generates events in which selected final-state particles are constrained to fixed directions or finite angular regions in the laboratory frame. Because of the way the generator is constructed, angular constraints can be imposed directly in the laboratory frame while maintaining the correct LIPS structure, enabling differential and total cross sections or decay rates to be computed with high efficiency. The method is validated against analytic results and existing event generators, showing excellent agreement. By reducing the computational cost of full phase-space event generation by several orders of magnitude, FANG provides a robust and versatile framework applicable to particle, nuclear, and detector physics.

Israel

Itay Horin

Arik Kreisel

Or Alon

Focused angular $\textit{N}$ -body event generator (FANG)

<p>We present a fully backreacted Einstein-Maxwell-Dilaton–flavor model with dynamical light and strange sectors, calibrated to lattice QCD using a machine-learning–assisted spectral method. The model reproduces the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>-flavor equation of state and chiral dynamics with quantitative accuracy, and maps the Columbia plot with a tricritical point at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>tri</mi></mrow></msubsup><mo>≃</mo><mn>21</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and a critical mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>c</mi></msub><mo>≃</mo><mn>0.785</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, consistent with lattice results. At finite density, it yields a crossover-to-first-order transition and predicts a critical endpoint at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>C</mi></msub><mo>=</mo><mn>75.4</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>C</mi></msub><mo>=</mo><mn>768</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, within the reach of heavy-ion experiments. These findings establish a unified holographic framework for the QCD phase structure across quark masses and baryon density, providing the first consistent and quantitative description of both deconfinement and chiral transitions within a single holographic model.</p>

Toward a holographic realization 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 phase structure

We investigate the QCD phase transition and its phase structure within Einstein–Maxwell-Dilaton-scalar system and compare the results with those obtained from the Einstein–Maxwell-Dilaton system. It is shown that both models reproduce behavior consistent with lattice QCD. In particular, the Einstein–Maxwell-Dilaton-scalar system exhibits a first-order phase transition in the pure gauge sector, aligning with predictions from Yang–Mills theory. Based on these models, we construct a holographic model for neutron stars, incorporating leptons to satisfy electric charge neutrality, and examine the cold equation of state, the mass-radius relation, and tidal deformability of neutron stars. It is demonstrated that the Einstein–Maxwell-Dilaton-scalar system enables us to describe neutron star properties that meet current astrophysical constraints.

Exploring QCD phase transitions and neutron star properties via holographic models

We investigate the chemical potential effects of the equation of state and the chiral transition in an Einstein-Maxwell-dilaton-scalar system, which is obtained from an improved soft-wall anti–de Sitter/QCD model coupled with an Einstein-Maxwell-dilaton system. The equations of state obtained from the model are in quantitative agreement with the lattice results at both zero and nonzero chemical potentials. The sensible chiral transition behaviors can be realized in the model. The QCD phase diagram with a critical end point has also been obtained from the model.

Xiao-Chang Peng

Holographic study on QCD phase transition and phase diagram with two flavors

In this paper, we study the processes of e + e − annihilation into two body final states, either two pseudoscalar mesons or one meson with a photon. The hadronic vacuum polarization form factors are calculated within the framework of resonance chiral theory in the energy region of E ≲ 2 GeV, with final state interactions taken into account. A joint analysis on the processes of e + e  −  → π + π  − , K + K  − ,   <math> <msubsup> <mi>K</mi> <mi>L</mi> <mn>0</mn> </msubsup> <msubsup> <mi>K</mi> <mi>S</mi> <mn>0</mn> </msubsup> </math>  $$ {K}_L^0{K}_S^0 $$ , π 0 γ, and ηγ has been performed, and the latest experimental data are included. Based on the vacuum polarization form factors of these processes, we estimate their contributions to the lowest order of anomalous magnetic moment of the muon, (g – 2) μ . Combined with other contributions from hadronic vacuum polarization and other interactions from the standard model, the discrepancy between theoretical prediction and experimental measurement is ∆a  μ  = (24.1 ± 5.4) × 10 −10, i.e., 4.5σ.

Shi-Jia Wang

Ling-Yun Dai

Two body final states production in electron-positron annihilation and their contributions to (g − 2) μ

We consider an improved soft-wall AdS/QCD model coupled to an Einstein-dilaton system, which offers a way to study the deconfining and chiral transitions simultaneously. The correlation between these two transitions has been investigated in detail in the Einstein-dilaton-scalar system with the bulk scalar field representing the vacuum of matters in the flavor sector of the model. We find that the effects of the scaling dimension <math><mi>Δ</mi></math> of the dual operator of the dilaton manifest in chiral transitions, although the equations of state can all be matched with the two-flavor lattice results for <math><mrow><mi>Δ</mi><mo>=</mo><mn>2.5</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>3.5</mn></mrow></math> in the decoupling case of <math><mi>β</mi><mo>=</mo><mn>0</mn></math>. In the weak-coupling case with smaller <math><mi>β</mi></math>, both the equation of state and the chiral transition exhibit a crossover behavior and turn into first-order phase transitions with the increase of <math><mi>β</mi></math>.

Ying-Ying Li

Xing-Lin Liu

Correlations between the deconfining and chiral transitions in holographic QCD

We studied the magnetic effects on the chiral transition and the melting properties of vector and axial-vector mesons in the improved soft-wall AdS/QCD model under a charged magnetic background, which is solved perturbatively from an Einstein–Maxwell system with a negative cosmological constant. The phase diagrams for both chiral transition and meson melting have been obtained. We show that the inverse magnetic catalysis emerged naturally in the improved soft-wall model. We also find that the magnetic field can induce meson melting, at least for the vector and axial-vector mesons, in our holographic setup.

QCD phase diagram with a background magnetic field in an improved soft-wall AdS/QCD model

We construct an improved soft-wall AdS/QCD model with a cubic coupling term of the dilaton and the bulk scalar field. The background fields in this model are solved by the Einstein-dilaton system with a nontrivial dilaton potential, which has been shown to reproduce the equation of state from the lattice QCD with two flavors. The chiral transition behaviors are investigated in the improved soft-wall AdS/QCD model with the solved gravitational background, and the crossover transition can be realized. Our study provides the possibility to address the deconfining and chiral phase transitions simultaneously in the bottom-up holographic framework.

Equation of state and chiral transition in soft-wall AdS/QCD with a more realistic gravitational background Supported in part by the National Natural Science Foundation of China (NSFC) (11851302, 11851303, 11690022, 11747601), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB23030100) as well as the CAS Center for Excellence in Particle Physics (CCEPP). Z. F. is also Supported by the NSFC (11905055) and the Fundamental Research Funds for the Central Universities (531118010198)

We give a further study on chiral phase diagram in the improved soft-wall AdS/QCD model with <math><mn>2</mn><mo>+</mo><mn>1</mn></math> flavors. The equations of motion for the octet pseudoscalar, vector, and axial-vector mesons are derived to compute the octet meson spectra and relevant decay constants, by which the model parameters are determined. The chemical potential effects on thermal transition of chiral condensate are investigated, which enables us to obtain the chiral phase diagram in the <math><mi>μ</mi><mo>−</mo><mi>T</mi></math> plane. We find that the critical endpoint linking the crossover transition with the first-order phase transition still exists and locates at <math><mo>(</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>,</mo><msub><mi>T</mi><mi>c</mi></msub><mo>)</mo><mo>≃</mo><mo>(</mo><mn>390</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>,</mo><mn>145</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>)</mo></math>, which along with the crossover line are consistent with lattice result and experimental analysis from relativistic heavy ion collisions.