Phase structure and critical phenomena in two-flavor QCD by holography

We explore the phase structure of quantum chromodynamics (QCD) with two dynamical quark flavors at finite temperature and baryon chemical potential, employing the nonperturbative gauge/gravity duality approach. Our gravitational model is tailored to align with state-of-the-art lattice data regarding the thermal properties of multiflavor QCD. Following a rigorous parameter calibration to match equations of state and the QCD trace anomaly at zero chemical potential derived from cutting-edge lattice QCD simulations, we investigate thermodynamic quantities and order parameters. We predict the location of the critical endpoint (CEP) at $({\mu }_{\mathrm{CEP}},T_{\mathrm{CEP}})=(219,182)\mathrm{MeV}$ at which a line of first-order phase transitions terminate. We compute critical exponents associated with the CEP and find that they almost coincide with the critical exponents of the quantum 3D Ising model.

Figure 1

Comparison of thermodynamics at ${\mu }_B=0$ between our hQCD model (solid curves) and the lattice QCD (data with error bars). Left panel: the energy density $\epsilon$, pressure $P$ and trace anomaly (also called interaction measure) $\epsilon -3P$, as a function of temperature, where the lattice data comes from [20]. Right panel: the temperature dependence of baryon number susceptibility ${\chi }_2^B$, where the lattice result is from [61].

Figure 2

Left panel: the ratio of pressure and energy density $P/\epsilon$ versus energy density $\epsilon$ at ${\mu }_B=0$ with the lattice data being from [20]. Right panel: the baryon number density $n_B$ as a function of temperature $T$ at fixed ${\mu }_B/T$, where the lattice data is from [61]. Our holographic results are all denoted by solid lines.

Figure 3

The entropy density $s$, pressure $P$, energy density $\epsilon$, and trace anomaly $\epsilon -3P$ as a function of $T$ at different values of ${\mu }_B$. These quantities are all enhanced by increasing the chemical potential.

Figure 4

The squared speed of sound $c_s^2$ and baryon number susceptibility ${\chi }_2^B$ as a function of temperature at different chemical potentials. At small ${\mu }_B$, there is only a crossover. For sufficiently large ${\mu }_B$, a first-order phase transition is triggered.

Figure 5

Higher-order baryon number susceptibilities ${\chi }_4^B$ (left) and ${\chi }_6^B$ (right) as a function of temperature at ${\mu }_B=0$. The holographic results of susceptibilities are qualitatively consistent with the lattice data [61].

Figure 6

The Polyakov loop $⟨\mathcal{P}⟩$ (left) and free-energy density $\mathrm{\Omega }$ at different ${\mu }_B$. The phase transition becomes first order when ${\mu }_B>219\mathrm{MeV}$.

Figure 7

The phase diagram of QCD matter in our two-flavor holographic model. The green curve shows the phase boundary for the first-order phase transition, and the blue dash line denotes the first-order axis. The green first-order line terminates at the CEP $({\mu }_{\mathrm{CEP}},T_{\mathrm{CEP}})=(219\mathrm{MeV},182\mathrm{MeV})$ (red point). The locations of CEP predicted by other approaches are presented as well, including FRG, DSE, the combination of FRG and DSE, NJL effective chiral model coupled to the PNJL, quasiparticle model (QPM), lattice taylor expansion (LTE), hadronic bootstrap (HB), and the coalescence model for light nuclei production. FRG-1 is from [14]. DSE and FRG is from [19]. Coalescence model is from [18]. DSE is from [2]. PNJL-1 is from [11] and PNJL-2 is from [12]. QPM is from [70]. LTE is from [71]. HB is from [72]. The magenta star represents the CEP of ($2+1$)-flavor QCD obtained by our previous model in [49]. We also indicate the directions of approach of the various critical exponents.

Figure 8

The entropy density $s$ as a function of $T$ for several values of ${\mu }_B$ near the CEP. For ${\mu }_B<{\mu }_{\mathrm{CEP}}$, the curve $s(T)$ is single-valued (left), while for ${\mu }_B>{\mu }_{\mathrm{CEP}}$ it becomes multivalued (right). At ${\mu }_B={\mu }_{\mathrm{CEP}}$ and $T=T_{\mathrm{CEP}}$, the slope is infinite (middle).

Figure 9

The dimensionless specific heat ${\widehat{C}}_n=C_n/T^3$ in a log-log plot with $\widehat{T}=\frac{T-T_{\mathrm{CEP}}}{T_{\mathrm{CEP}}}$ near the critical endpoint along the first-order axis. The slope of the best-fit line to our data (blue line) yields $\alpha =0.113_{-0.010}^{+0.010}$ at a 95% confidence level with the coefficient of determination $R^2=0.98705$, indicating a very high level of fit between the linear model and the data.

Figure 10

The discontinuity in the dimensionless entropy density $\widehat{s}=s/T^3$ as one approaches the CEP on a log-log plot. The value of $\beta$ obtained from the slope is $\beta =0.322_{-0.000}^{+0.000}$ with a 95% confidence interval and the coefficient of determination $R^2=0.99785$.

Figure 11

The baryon number susceptibility ${\chi }_2^B$ as a function of temperature $T$ as the CEP is approached on a log-log plot. We obtain the value of $\gamma$ from the slope at a 95% confidence level, i.e., $\gamma =1.243_{-0.008}^{+0.008}$. The coefficient of determination $R^2$ is 0.99945, illustrating a very tight fit.

Figure 12

The trajectory of $s$ nearing $s_{\mathrm{CEP}}$ is plotted as ${\mu }_B$ approaches ${\mu }_{\mathrm{CEP}}$ on the critical isotherm. The derived slope from this plot gives us $\delta =4.854_{-0.383}^{+0.455}$ with a 95% confidence interval and a coefficient of determination $R^2$ at 0.98502, denoting a very high degree of fit to the data.