Temperature derivative divergence of the electric conductivity and thermal photon emission rate at the critical end point from holography

TL;DR

The study targets a CEP signature in the QCD phase diagram via holography, using a machine‑learning–calibrated Einstein–Maxwell–dilaton model with $N_f=2+1$ to compute the thermal photon emission rate $dΓ/dk$ and the DC electric conductivity $σ_Q$. Observables are derived from the transverse current correlator $Π^T$ in AdS/CFT, with a probe Maxwell action featuring a dilaton‑dependent coupling $f_Q(φ)$. The authors find that the temperature derivatives of both $dΓ/dk$ and $σ_Q$ diverge at the CEP, revealing a robust, equilibrium signature of criticality that complements lattice results at $μ_B=0$ and aligns with the $\mathcal{N}=4$ SYM limit at high temperature. Finite‑density analysis locates the CEP at $(T_{CEP}, μ_{CEP}) ≈ (0.094 \, GeV, 0.74 \, GeV)$ in this model, and shows inflection‑point behavior and enhanced photon production near the CEP, supporting the usefulness of electromagnetic probes for CEP studies. The work highlights how holographic transport calculations, anchored to lattice EOS via machine learning, can illuminate nonperturbative QCD phase structure with potential phenomenological relevance for heavy‑ion collision observables.

Abstract

The thermal photon emission rate $\frac{dΓ}{dk}$ and DC eletric conductivity $σ_{Q}$ of the strongly coupled quark-gluon plasma (sQGP) are investigated around the critical end point in a $N_f=2+1$ holographic QCD model with parameters obtained from machine-learning. It is found that both thermal photon emission rate and eletric conductivity grow most obviously around $T_c$, which agrees with the previous studies, and the result of eletric conductivity at zero chemical potential resembles the lattice results. Moreover, it is found that both the temperature derivative of the eletric conductivity and thermal photon emission rate diverge at the critical end point.

Figures (5)

• Figure 1: Holographic results of DC electric conductivity scaled by $T C_{em}$ as a function of $T/T_c$ for three different coupling functions $f_Q(\phi)$, comparing with lattice results. The lattice results are marked by brown 2019ELEconductivityLattice and blue GertAartsAmato2013PRLGertAartsAmato2014JHEP with error bars, while the horizontal lightblue line corresponds to the $\mathcal{N}=4$ SYM result Caron-Huot. The other three curves represent holographic results of electric conductivity using three different sets of parameters: 1) black dashed line for $f_Q=\alpha_1=1.19$; Orange solid line for $\alpha_1=0.81,\alpha_2=0.41,\beta=0.4$; Green dashed line for $\alpha_1=0,\alpha_2=1.8,\beta=0.4$. We also normalize the temperature using (pseudo)transition temperature $T_c$ at zero chemical potential.
• Figure 2: Left: Electric conductivitis calculated using parameters in Set 2 as a function of the temperature for $\mu_B=0$ (red), $0.5$ (orange) and $0.74$ (blue) GeV. Right: First order derivatives of electric conductivity at $\mu_B=0$ (red), $\mu_B=0.5$ GeV (orange) and $\mu_B=\mu_{\mathrm{CEP}}=0.74$ GeV. Each curve exhibits a peak which corresponds to the inflection point in the left pannel.
• Figure 3: Temperature dependence on $z_h$ at $\mu_B=0$ (red), $0.5$ GeV (orange) and $0.74$ GeV (blue). At both high and low temperature, all three curves converge into a single one. We only consider the region of midium temperature where the temperature changes more significantly with $\mu_B$.
• Figure 4: Thermal photon emission rate plotted agaginst the momentum $k/T$, divied by fine structure constant $\alpha_\mathrm{EM}$ and temperature cubed. Left: $\mu_B=0$. The dashed lines represent thermal photon emission rate calculated using parameter Set 2 at different temperatures. From red to purple, the temperature increases. The SYM and pQCD (upto NLO) result are also shown for comparison, they are plotted using black and light blue solid line respectively. Right: $\mu_B=\mu_\mathrm{CEP}=0.74$ GeV case. The coupling function is also parameterized by Set 2. The emission result computed at CEP is plotted by the green solid curve.
• Figure 5: Photon emission rate at $\mu_B=\mu_{\mathrm{CEP}}=0.74$GeV. Each curve corresponds to one particular temperature. From red to purple, the corresponding temperatures rise from $0.5T_c$ to $4T_c$. In particular, the green solid line represents the photon emission rate at CEP. The overall amplitude of photon emission rate increases with temperature. Similar to the conductivity before, the most rapid growth occurs at the critical temperature as shown by the obvious slope in this 3D figure. In order to show the slope tendency more clearly, a black line connecting the peaks of all curves is depicted.