Apparent convergence of Padé approximants for the crossover line in finite density QCD

We propose a novel Bayesian method to analytically continue observables to real baryochemical potential ${\mu }_B$ in finite density QCD. Taylor coefficients at ${\mu }_B=0$ and data at imaginary chemical potential ${\mu }_B^I$ are treated on equal footing. We consider two different constructions for the Padé approximants, the classical multipoint Padé approximation and a mixed approximation that is a slight generalization of a recent idea in Padé approximation theory. Approximants with spurious poles are excluded from the analysis. As an application, we perform a joint analysis of the available continuum extrapolated lattice data for both pseudocritical temperature $T_c$ at ${\mu }_B^I$ from the Wuppertal-Budapest Collaboration and Taylor coefficients ${\kappa }_2$ and ${\kappa }_4$ from the HotQCD Collaboration. An apparent convergence of $[p/p]$ and $[p/p+1]$ sequences of rational functions is observed with increasing $p$. We present our extrapolation up to ${\mu }_B\approx 600\mathrm{MeV}$.

Figure 1

Apparent convergence of the multipoint Padé approximants $C_N$ (solid lines) determined from $T_c$ values at ${\mu }_B^I$ in the CQM model in comparison to the Taylor expansion of order $N-1$ around ${\mu }_B=0$ (dashed lines). In the main plot the parametric curves for the Padé approximants are obtained as $(\sqrt{{\widehat{\mu }}_B^2}{C}_N({\widehat{\mu }}_B^2),C_N({\widehat{\mu }}_B^2))$. The bivaluedness of the subdiagonal approximants (and the curves of the odd-order Taylor expansion) reflects that ${\mu }_B≔\sqrt{{\widehat{\mu }}_B^2}{C}_N({\widehat{\mu }}_B^2)$ has a maximum. The inset shows the percentage difference between the approximated and exact values of $T_c$, with the vertical dotted line indicating the radius of converges of the Taylor expansion.

Figure 2

Result of a mock analysis showing the effect of the error of the data and of the sampling range on the quality of the analytic continuation obtained via multipoint Padé approximants of various order and by including or omitting information (the latter is denoted by “no $c_i$” in the key) on the error of the Taylor coefficients in the evaluation of ${\chi }^2$. For additional information see the main text.

Figure 3

Choice of the interpolating (node) points, whose position is indicated by the vertical dotted line, in comparison to the actual lattice data. The labels indicate the use of the interpolating point in the multipoint Padé approximant (bottom) and the Padé-type approximant (top) of a given order, characterized by the number of independent parameters $N$. At the value of ${\widehat{\mu }}_B^2$ corresponding to the lattice data points we show the mean and the error of the $T_c$ computed from the multipoint Padé approximants generated with importance sampling (for the sake of the presentation the abscissa is shifted). The band indicates the standard deviation of the normal distribution used in Method 1 to generate $\{T_c^i\}$ instances, i.e., it indicates the prior distribution, excluding the factor that removes the spurious poles. The values ${\overline{T}}_c({\widehat{\mu }}_{B,\mathrm{i}}^2)$ at the node points ${\widehat{\mu }}_{B,\mathrm{i}}^2$ used in the expression (5) of the prior are from the dotted curve in the band, which interpolates the mean values of the lattice data.

Figure 4

Result of the analytic continuation via the diagonal and subdiagonal sequences of Padé-type approximants constructed based on $\{T_c^i\}$ instances generated using importance sampling. The inset shows the percentage difference with respect to the $T_c(0)(1-{\kappa }_2{\mu }_B^2/T_c^2(0))$ curve plotted in the main figure, for which we used $T_c(0)=158.01\mathrm{MeV}$ and ${\kappa }_2=0.012$.