Higher order quark number fluctuations via imaginary chemical potentials in $N_f=2+1$ QCD

We discuss analytic continuation as a tool to extract the cumulants of the quark number fluctuations in the strongly interacting medium from lattice QCD simulations at imaginary chemical potentials. The method is applied to $N_f=2+1$ QCD, discretized with stout improved staggered fermions, physical quark masses and the tree level Symanzik gauge action, exploring temperatures ranging from 135 up to 350 MeV and adopting mostly lattices with $N_t=8$ sites in the temporal direction. The method is based on a global fit of various cumulants as a function of the imaginary chemical potentials. We show that it is particularly convenient to consider cumulants up to order two, and that below $T_c$ the method can be advantageous, with respect to a direct Montecarlo sampling at $\mu =0$, for the determination of generalized susceptibilities of order four or higher, and especially for mixed susceptibilities, for which the gain is well above one order of magnitude. We provide cumulants up to order eight, which are then used to discuss the radius of convergence of the Taylor expansion and the possible location of the second-order critical point at real $\mu$: no evidence for such a point is found in the explored range of $T$ and for chemical potentials within present determinations of the pseudocritical line.

Figure 1

Ranks of the $A_N$ matrices for some values of the order $N$, for simulation points chosen along the lines described in Eq. (13) and measuring as an input both quark number densities and second-order susceptibilities. Red circles correspond to the number of independent ${\chi }_{ijk}(T,\mu =0)$ at a given order.

Figure 2

Same as in Fig. 1, but with just quark number densities taken as an input.

Figure 3

Same as in Fig. 1, apart from the fact that also quark number susceptibilities of order three are measured and added as an input.

Figure 4

Same as in Fig. 1, but considering less or more lines of simulation points. Triangles correspond to the case in which two additional lines, $(i{\mu }_I,0,i{\mu }_I)$ and $(i{\mu }_I,i{\mu }_I,-i{\mu }_I)$, have been added, while diamonds to the case in which two lines, $(i{\mu }_I,-i{\mu }_I,0)$ and $(i{\mu }_I,i{\mu }_I,i{\mu }_I)$, are not considered.

Figure 5

Sketch of the phase diagram in the $T-{\mu }_I$ plane. In this case (${\mu }_{u,d,s}=i{\mu }_I$) RW lines are exactly vertical and located at ${\mu }_I/T=(2n+1)\pi /3$. Solid lines represent first-order phase transitions separating sectors with different orientation of the Polyakov loop while dashed lines correspond to the analytic continuation of the pseudocritical line.

Figure 6

Example of the global fit for $T=149\mathrm{MeV}$. We show only a subsample of a total of 54 polynomial fits which are performed at the same time (three densities plus six second-order susceptibilities fitted along six different trajectories). The best-fit functions are taken according to Eq. (12) with a truncation to order eight. The reduced ${\tilde{\chi }}^2$ is 1.3. Notice that in the global fit we did not take into account cross-correlations between susceptibilities measured at the same chemical potential, hence the covariance matrix has a simple diagonal form.

Figure 7

An example of the procedure followed to determine systematic errors. The errorbars represent the statistical error obtained in the global best fit. The polynomial degree is increased every time that the global best fit yields nonacceptable values of the reduced chi-squared ${\tilde{\chi }}^2$. Circles, triangles and diamonds refer to a global fit performed with a polynomial of order 6, 8, 10, respectively, while the grey bands represent the final estimate. Data refer to simulations on the $32^3\times 8$ lattice at $T=135\mathrm{MeV}$.

Figure 8

Variation of some quark susceptibilities with the volume size at $T=170\mathrm{MeV}$ on the $N_t=6$ lattice.

Figure 9

Same as in Fig. 8 for $T=350\mathrm{MeV}$ on the $N_t=8$ lattice.

Figure 10

Temperature dependence of the ratio ${\chi }_4^B/{\chi }_2^B$ of baryonic cumulants. Blue points correspond to our determinations while red points, corresponding to data obtained on a $N_t=8$ lattice using our own discretization, are taken from [7].

Figure 11

We show how the precision attained for some susceptibilities changes when adding more and more cumulants to the global fit. The first three graphs correspond to a global fit performed using ${\mu }_{\max }/T=0.4\pi$ and a polynomial of degree 8, whereas for the last three ${\mu }_{\max }/T=0.2\pi$ and a polynomial of degree 4 has been used. Data refer to simulations at $T=143\mathrm{MeV}$.

Figure 12

Comparison of results on sixth-order cumulant of the electric charge fluctuations between this work and Ref. [46]. Diamonds refer to the determinations of Ref. [46] obtained on a $N_t=8$ lattice with the highly improved staggered quark (HISQ) action and almost physical quark masses.

Figure 13

Same as in Fig. 12 for the fourth-order cumulant of the net baryon number fluctuation.

Figure 14

Our determination of the sixth-order cumulant of the net baryon number fluctuation.

Figure 15

Comparison between results on second-order susceptibilities obtained in this work and in [45] on the $N_t=8$ lattice and with our own discretization.

Figure 16

Our lattice determination of the baryon number density as a function of the imaginary (baryon-)chemical potential is shown. Bands correspond to polynomial truncations at various orders of the series expansion around ${\mu }_B=0$. Data refer to $T=135\mathrm{MeV}$.

Figure 17

Radius of convergence estimates for various temperatures below $T_c$. Circles/diamonds correspond to our estimate for ${\rho }_{n,m}^{f/\chi }$ while black lines are values predicted from the HRG model.

Figure 18

The values of ${\rho }_{n,m}^{\chi }$ are shown along with the $\mathcal{O}({\mu }_B^2)$ determination for the pseudocritical chiral line.

Figure 19

Radius of convergence estimates for our statistical toy model. Filled points represent the radius of convergence estimates for the Taylor expansion of the susceptibility while the unfilled ones the estimates for the Taylor expansion of the free energy [see Eq. (16)].