Strongly interacting quark-gluon plasma, and the critical behavior of QCD at imaginary $\mu$

We explore the highly nonperturbative hot region of the QCD phase diagram close to $T_c$ by use of an imaginary chemical potential $\mu$ which avoids the sign problem. The simulations use four flavors of staggered fermions and are carried out on a ${16}^3\times 4$ lattice. The number density and the quark number susceptibility are consistent with a critical behavior associated with the transition line in the negative ${\mu }^2$ half-plane. We compare the analytic continuation of these results with various phenomenological models, none of which provides a satisfactory description of data, a failure on which we make some comments. These results complement and extend the information obtained via the analysis of the susceptibilities evaluated at zero $\mu$, yielding a simple description of the candidate strongly interacting quark-gluon plasma phase. As a byproduct of our analysis we investigate the Polyakov loop and its hermitian conjugate. Our data offer vivid evidence of the importance of the complex nature of the functional integral measure, which results in $L(\mu )\ne \overline{L}(\mu )$ for a real chemical potential $\mu$.

Figure 1

Schematic phase diagram for four-flavor QCD in the $T$, ${\mu }^2$ plane. The candidate sQGP phase is bound by the chiral (pseudo)critical line in the negative ${\mu }^2$ half-plane.

Figure 2

$R_F({\mu }_I)=n({\mu }_I)/n({\mu }_I{)}_{\mathrm{free}}$ as a function of ${\mu }_I$, showing very clear evidence of a deviation from a free field behavior.

Figure 3

$R_x({\mu }_I)$ (defined in Eqs. (7, 8)) as a function of ${\mu }_I$, showing the limitations of a linear approximations, as well as the deviations from the simplest trigonometric parametrizations motivated by a hadron resonance gas model.

Figure 4

$n(\mu )$ fitted to the form predicted by the simple critical behavior at imaginary $\mu$ (Eq. (9)).

Figure 5

The quark number susceptibility obtained by numerical differentiation of the results for the quark number.

Figure 6

Numerical results for the chiral condensate, and the results of the fit to a functional form inferred from the Maxwell equation (Eq. (13)).

Figure 7

The Polyakov loop fitted to the form predicted by a simple critical behavior at imaginary $\mu$ (Eq. (17)).

Figure 8

The imaginary part of the Polyakov loop, divided by the coefficient of the linear term of a fit at small $\mu$, as a function of the imaginary chemical potential, demonstrating the relevance of the phase of the determinant for a real chemical potential; in the same plot we show the number density, again normalized by the coefficient of the linear term.

Figure 9

Analytic continuation to real chemical potential of $n(\mu )$ based on the fits at imaginary $\mu$ described in Sec. 2, together with the associated error bands.

Figure 10

$n(\mu )$ from analytic continuation, together with a free field behavior is shown for comparison. The fits suggest that the slower increase observed in the interacting case with respect to the free case can be described by an overall exponent smaller than 1.

Figure 11

Results of the fits of $n({\mu }_I)$ to the trigonometric functions (Eq. (24)), see text for details.