The QCD phase diagram and critical end point at finite magnetic field from holography with neural networks

We study the magnetic-field-driven critical end point (CEP) of the QCD thermal transition at zero baryon chemical potential using a five-dimensional Einstein–Maxwell–dilaton (EMD) model. We calibrate the model to lattice entropy at zero and finite magnetic fields using a neural-network formulation of holographic radial evolution. As the magnetic field increases, the thermal transition changes from a crossover to a first-order transition. The merger of temperature turning points, a divergent specific heat, and a vanishing squared sound speed identify $(e{B}_c,T_c)\simeq (7.22{\mathrm{G}\mathrm{e}\mathrm{V}}^2,95.9\mathrm{M}\mathrm{e}\mathrm{V})$. The transition-region inflection of $s\mathrm{/}{T}^3$ defines the crossover temperature, while an equal-pressure Maxwell construction determines the first-order coexistence line. The two lines meet at the CEP. The change of transition order agrees qualitatively with lattice results. The phase boundary is a prediction of the calibrated model.