Vector mesons at finite temperature and QCD sum rules

Finite energy sum rules for vector and axial-vector currents are derived in a thermal medium to provide constraints for the spectral behavior of $\rho$ and $a_1$ mesons at nonvanishing temperature and hence to study the tendency toward chiral symmetry restoration. The parity-mixing ansatz for the $\rho$ and $a_1$ spectra, including finite widths, is investigated as a function of temperature. Characteristic differences between vector and axial-vector channels are discussed with regard to the implementation of the chiral-symmetry-breaking scale, $4\pi$ times the pion decay constant, in the sum-rule approach.

Figure 1

Continuum threshold $\sqrt{s_V}$ in the vector meson channel as a function of $T$ obtained from the sum rules for the $0\text{th}$ (black solid line) and $1\text{st}$ (red dashed line) moment of the spectral function. The $T$ dependence of chiral scale $4\pi f_{\pi }(T)$ (blue dotted line) is also displayed.

Figure 2

Axial-vector continuum threshold $\sqrt{s_A}$ as a function of $T$ obtained from the sum rules for the $0\text{th}$ (black solid line) and $1\text{st}$ (red dashed line) moment of the $a_1$ spectral function. In contrast to the $\rho$ meson, the $a_1$-meson mass (black dotted line) decreases with rising temperature.

Figure 3

Vector (black curve) and axial-vector (red curve) spectral distributions in vacuum, compared to $e^{+}{e}^{-}\to n\pi$ data with $n$ even [18, 19] and data from hadronic $\tau$ decays [20, 21]. Here, $s_V$ and $s_A$ stand for the continuum thresholds in vector and axial-vector channels, respectively. The ramping function (blue dashed line) shows an example of smooth threshold modeling (see text and appendix).

Figure 4

The $\rho$ and $a_1$ meson spectra at finite temperature.

Figure 5

The $\rho$ meson resonance at finite temperature calculated in the GHLS model.

Figure 6

Dependence of $\sqrt{s_V}$ [determined from Eqs. (A3, A4, A5)] on the slope $(s_2-s_1){}^{-1}$ of the ramp function $W(s)$ describing the onset of the continuum for the example of $T=100\hspace{0.5em}\mathrm{MeV}$. The grey band indicates the uncertainty range of the result obtained with step-function parametrization of the continuum.