Locating the QCD critical endpoint through finite-size scaling

Considering the 3D Ising universality class of the QCD critical endpoint, we use a universal effective action for the description of the baryon-number density fluctuations around the critical region. Calculating the baryon-number multiplicity moments and determining their scaling with system’s size, we show that the critical region is very narrow in the direction of the baryon chemical potential $\mu$ and wider in the temperature ($T$) direction. In this context, published experimental results on local proton density-fluctuation measurements, obtained by intermittency analysis in transverse momentum space in NA49 central $A+A$ collisions at $\sqrt{s_{NN}}=17.2\mathrm{GeV}$ ($A=\mathrm{C},\mathrm{Si},\mathrm{Pb}$), restrict significantly the location $({\mu }_c,T_c)$ of the QCD critical endpoint. The main constraint is provided by the freeze-out chemical potential of the $\mathrm{Si}+\mathrm{Si}$ system, which shows nonconventional baryon density fluctuations. An indicative solution, ignoring experimental uncertainties, is $119\mathrm{MeV}\le T_c\le 162\mathrm{MeV}$, $252\mathrm{MeV}\le {\mu }_c\le 258\mathrm{MeV}$.

Figure 1

The baryon-number susceptibility scaled by $T{M}^{2/3}/T_c^2$ versus $(T-T_c)/T_c$ plotted in the double logarithmic scale for three different values of the size $M$: $10^3$ (red circles), $10^4$ (green circles), and $10^5$ (blue circles). The slope in the linear fit determines the critical exponent $\gamma$. All sets converge to the same value for the scaled susceptibility for $T\to T_c$, verifying the finite-size scaling relation (6).

Figure 2

(a) The critical region (red shaded area) in the plane $(\mu -{\mu }_c)/T_c$, $(T-T_c)/T_c$ where the scaling law $⟨N⟩\sim M^{\tilde{q}}$ with $\tilde{q}\in (0.75,1)$ holds. The blue line is the line for which $\tilde{q}=0.96$. The freeze-out states for central $A+A$ collisions in the NA49 experiment at maximum SPS energy according to Ref. [17] are given by the stars: $A=\mathrm{Pb}$ (black), $A=\mathrm{Si}$ (blue), and $A=\mathrm{C}$ (orange). (b) The line describes the possible pairs $(T_c,{\mu }_c)$ compatible with the finite-size analysis in the current work. The red patterned region is determined employing the different lattice QCD results for the critical temperature $T_c$ [9, 10].

Figure 3

(a) The first moment of the baryon-number density $⟨N⟩$ as a function of $M$ for $t=0$, 0.02, 0.06 (corresponding to $\widehat{m}=0$, 0.07, 0.15) and $\zeta ={\zeta }_c=1$. (b) The mean value $⟨N⟩$ as a function of $M$ for $T=T_c$ ($\widehat{m}=0$) and $\zeta =1$, 1.01, 1.02.