In-medium spectral functions of vector- and axial-vector mesons from the functional renormalization group

In this work, we present the first results on vector- and axial-vector meson spectral functions as obtained by applying the nonperturbative functional renormalization group approach to an effective low-energy theory motivated by the gauged linear sigma model. By using a recently proposed analytic continuation method, we study the in-medium behavior of the spectral functions of the $\rho$ and $a_1$ mesons in different regimes of the phase diagram. In particular, we demonstrate explicitly how these spectral functions degenerate at high temperatures as well as at large chemical potentials, as a consequence of the restoration of chiral symmetry. In addition, we also compute the momentum dependence of the $\rho$ and $a_1$ spectral functions and discuss the various timelike and spacelike processes that can occur.

Figure 1

Flow equation for the effective action of the model defined by Eq. (4) in diagrammatic form. The dashed blue line is associated with propagators of the $\sigma$ and $\pi$ mesons, the solid black line is associated with the quark propagator, and the crossed circles represent the regulator insertions ${\partial }_k{R}_k$.

Figure 2

Flow equations for the $\rho$ and $a_1$ two-point functions in diagrammatic form. Vertices are indicated by black filled dots, regulator insertions by crossed circles. The color of the lines and regulators represents the type of field: blue for scalar and pseudoscalar mesons, black for fermions, and purple for vector mesons.

Figure 3

Flow of the scale-dependent Euclidean curvature masses of mesons and the constituent quark mass with the RG scale $k$ at $T=0\mathrm{MeV}$ and $\mu =0\mathrm{MeV}$.

Figure 4

Euclidean curvature masses of mesons and constituent quark mass at the IR scale vs temperature at $\mu =0\mathrm{MeV}$.

Figure 5

Phase diagram of the quark-meson model as a contour plot of the order parameter for chiral symmetry ${\sigma }_0(T,\mu )$. The value of ${\sigma }_0(T,\mu )$ decreases with increasing temperature and chemical potential as indicated with a darker color. The CEP is indicated as a red dot, whereas the first-order phase boundary is indicated by a black line.

Figure 6

Euclidean curvature masses of mesons and constituent quark mass vs chemical potential at $T=10\mathrm{MeV}$.

Figure 7

Pole masses of $\rho$ and $a_1$ mesons vs temperature at $\mu =0\mathrm{MeV}$.

Figure 8

Spectral functions of $\rho$ (solid blue) and $a_1$ (dashed purple) meson vs external energy $\omega$ at $|\overrightarrow{p}|=0$ shown for increasing temperature at $\mu =0\mathrm{MeV}$ (left column) and for increasing chemical potential at $T=10\mathrm{MeV}$, towards the CEP (right column). For the last plot on the right-hand side we chose $T=50\mathrm{MeV}$ and $\mu =600\mathrm{MeV}$ in order to avoid a thermodynamically problematic regime in the phase diagram (see Sec. 3b for details).

Figure 9

The transverse $\rho$ (left) and $a_1$ (right) spectral functions, ${\rho }_{\rho }(\omega ,\overrightarrow{p})$ and ${\rho }_{a_1}(\omega ,\overrightarrow{p})$, are shown versus external energy $\omega$ and external spatial momentum $\overrightarrow{p}$ at $T=100\mathrm{MeV}$ and $\mu =0\mathrm{MeV}$. The timelike regime, $\omega >|\overrightarrow{p}|$, is Lorentz-boosted to higher energies as the momentum increases, while the spacelike regime, $\omega <|\overrightarrow{p}|$, is homogeneously filled up by the spacelike processes.

Figure 10

Real (left) and imaginary (right) parts of the retarded two-point function of the $\rho$ (blue) and $a_1$ (dashed red) meson vs external energy $\omega$ at $T=0\mathrm{MeV}$ and $\mu =0\mathrm{MeV}$. The different contributions to the imaginary parts are shown by light dashed lines (right): ${\rho }^{*}/a_1^{*}\to \psi +\overline{\psi }$ (upper dashed lines) and ${\rho }^{*}\to \pi +\pi$ for the $\rho$ meson and $a_1^{*}\to \pi +\sigma$ for the $a_1$ meson (lower dashed lines).

Figure 11

Spectral functions of the $\rho$ (blue) and $a_1$ (dashed red) meson at $T=0\mathrm{MeV}$ (normal color) and $T=150\mathrm{MeV}$ (light color), both for vanishing chemical potential in linear scales with (left) and without (right) a quark-antiquark decay channel.

Figure 12

Vacuum spectral functions of the $\rho$ (blue) and $a_1$ (dashed red) meson without a quark-antiquark decay channel.

Figure 13

Collection of the possible timelike, $p^2={\omega }^2-{\overrightarrow{p}}^2>0$, and spacelike, $p^2<0$, processes. Asterisks denote off-shell particles with total energy $\omega$ and spatial momentum $\overrightarrow{p}$, while others represent on-shell particles from the heat bath. First column: Timelike decay channels for the $\rho$ meson. Second column: Timelike decay channels for the $a_1$ meson. Third column: Absorption of a spacelike $\rho$ excitation. Fourth column: Absorption of a spacelike $a_1$ excitation. In addition, particle-hole excitations on the Fermi surface of (anti)quarks are possible (see text for details).