Asymptotic behavior of the Fourier coefficients and the analytic structure of the QCD equation of state

In this paper we study the universal properties of the baryon chemical potential Fourier coefficients in quantum chromodynamics. We show that by following a well-defined strategy, the Fourier coefficients can be used to locate Yang-Lee edge singularities associated with chiral phase transition (and by extension with the Roberge-Weiss) in the complex chemical potential plane. We comment on the viability of performing this analysis using lattice QCD data.

Figure 1

Complex chemical potential plane with the chiral and RW YLE. The integration path along the imaginary chemical potential axis can be deformed to the integration around the branch point singularities and the associated cuts (Stokes lines).

Figure 2

Computation of the Fourier coefficient for the function with one singularity in the right half-plane of complex chemical potential.

Figure 3

Illustration of the positions of the singularities under the scaling assumptions in QCD. The stars (dots) indicate the positions of RW (chiral) YLE. Numerical labels indicate the corresponding temperatures in MeV. The horizontal dashed lines indicate $\mathrm{Im}\widehat{\mu }=\pi /2$ and $\pi$.

Figure 4

Illustration of different qualitative behavior of the Fourier coefficients. Note that although we plotted the coefficients at small values of $k$, Eq. (13) is only formally valid for asymptotically large $k$. (a) $T$ = 120 MeV, (b) $T$ = 140 MeV, (c) $T$ = 160 MeV, (d) $T$ = 170 MeV, (e) $T$ = 180 MeV, (f) $T$ = 200 MeV.

Figure 5

Mean-field Fourier coefficients ${\tilde{b}}_k=k^{1+\sigma }{b}_k$ (${\sigma }_{\mathrm{MF}}=1/2$) for $T=150\mathrm{MeV}$ (left) and $T=180\mathrm{MeV}$ (right) and the corresponding fits.

Figure 6

LPA FRG Fourier coefficients ${\tilde{b}}_k=k^{1+\sigma }{b}_k$ (${\sigma }_{\mathrm{LPA}}=1/5$) and the corresponding fits for $T=150\mathrm{MeV}$ (left) and $T=180\mathrm{MeV}$ (right).

Figure 7

Comparison of the Fourier coefficients from the plain piecewise Filon quadrature on $N$ subintervals with the exact results for the case of the mean-field quark-meson model at $T=150\mathrm{MeV}$.