Flow equations for spectral functions at finite external momenta

In this work we study the spatial-momentum dependence of mesonic spectral functions obtained from the quark-meson model using a recently proposed method to calculate real-time observables at finite temperature and density from the functional renormalization group. This nonperturbative method is thermodynamically consistent, symmetry preserving and based on an analytic continuation from imaginary to real time on the level of the flow equations for two-point functions. Results on the spatial-momentum dependence of the pion and sigma spectral function are presented at different temperatures and densities, in particular near the critical endpoint in the phase diagram of the quark-meson model.

Figure 1

Diagrammatic representation of the flow equation for the effective average action. Dashed (solid) lines represent bosonic (fermionic) propagators and circles represent bosonic (fermionic) regulator insertions ${\partial }_k{R}_k^{B,F}$.

Figure 2

Diagrammatic representation of the flow equation for the mesonic two-point functions. Mesonic and quark-meson three-point vertices are represented by triangles, respectively, and mesonic four-point vertices are represented by squares. The bosonic and fermionic regulator insertions are depicted by circles.

Figure 3

Graphical representation of the possible time-like, $p^2={\omega }^2-{\overrightarrow{p}}^2>0$, and space-like, $p^2<0$, processes. Asterisks denote off-shell particles while others represent on-shell particles from a heat bath. First column: Time-like decay channels for an off-shell sigma, ${\sigma }^{*}$. Second column: Time-like decay channels for an off-shell pion, ${\pi }^{*}$. Third column: Absorption or emission of a space-like sigma excitation. Fourth column: Absorption or emission of a space-like pion excitation.

Figure 4

The sigma (left) and pion (right) spectral function, ${\rho }_{\sigma }(\omega ,\overrightarrow{p})$ and ${\rho }_{\pi }(\omega ,\overrightarrow{p})$, are shown versus external energy $\omega$ at $T=100\mathrm{MeV}$ and $\mu =0\mathrm{MeV}$ for different external momenta $|\overrightarrow{p}|$ ranging from 0 MeV to 700 MeV, as indicated by the inset labels.

Figure 5

The sigma and pion spectral functions, ${\rho }_{\sigma }(\omega ,\overrightarrow{p})$ and ${\rho }_{\pi }(\omega ,\overrightarrow{p})$, are shown versus external energy $\omega$ at $T=10\mathrm{MeV}$ and $\mu =292.97\mathrm{MeV}$ for different external momenta $|\overrightarrow{p}|$ ranging from 0 MeV to 500 MeV, as indicated by the inset labels.

Figure 6

The sigma spectral function, ${\rho }_{\sigma }(\omega ,\overrightarrow{p})$, is shown as a function of external energy $\omega$ (left) and as a function of the invariant $\sqrt{{\omega }^2-{\overrightarrow{p}}^2}$ (right) at $T=0\mathrm{MeV}$ and $\mu =0\mathrm{MeV}$ for different external momenta $|\overrightarrow{p}|$ ranging from 0 MeV to 700 MeV, as indicated by the inset labels.

Figure 7

The sigma (left) and pion (right) spectral functions, ${\rho }_{\sigma }(\omega ,\overrightarrow{p})$ and ${\rho }_{\pi }(\omega ,\overrightarrow{p})$, are shown versus external energy $\omega$ at $T=0\mathrm{MeV}$ and $\mu =0\mathrm{MeV}$ for different values of the parameter $\epsilon$ ranging from 0.01 MeV to 10 MeV, as indicated by the inset labels.