Exact vector channel sum rules at finite temperature and their applications to lattice QCD data analysis

We derive three exact sum rules for the spectral function of the electromagnetic current with zero spatial momentum at finite temperature. Two of them are derived in this paper for the first time. We explicitly check that these sum rules are satisfied in the weak coupling regime and examine which sum rule is sensitive to the transport peak in the spectral function at low energy or the continuum at high energy. Possible applications of the three sum rules to lattice computations of the spectral function and transport coefficients are also discussed: we propose an Ansatz for the spectral function that can be applied to all three sum rules and fit it to available lattice data of the Euclidean vector correlator above the critical temperature. As a result, we obtain estimates for both the electrical conductivity $\sigma$ and the second-order transport coefficient ${\tau }_J$.

Figure 1

The contour $C$ used in the integral of Eq. (1).

Figure 2

The right-hand sides of Eq. (15) (upper panel) and Eq. (17) (lower panel), divided by $T^4$ and shown as a function of temperature $T$. To extract the temperature dependence of the condensates, lattice QCD data provided in Ref. [28] were used. $T_f^{00}$ in Eq. (17) was estimated within a free pion gas model, which is reliable at low $T$, and leading-order perturbative QCD, which should give the correct behavior at high $T$. We used the value $e^2=0.092$ for the plots. For the free pion gas model, we employ the free pion mass averaged over the three isospin states, $m_{\pi }=138\mathrm{MeV}$. The cross at the lower right side of the lower panel marks the of our fitted spectral function given in Eq. (37). For details, see Sec. 3.

Figure 3

Ansatz A (red solid line), Ansatz B (blue dotted line), and the Ansatz used in Ref. [13] (green dashed line) as functions of $\omega$. Note that Ansätze A and B are identical for $\omega <{\omega }_{\min }=4.0T$, where they overlap. The unit of the vertical axis is $C_{em}T\omega$ while that of the horizontal axis is $T$.

Figure 4

Lattice data for the Euclidean vector correlator, adapted from Ref. [13] (black points), the fit result using Eqs. (26)–(28) (red solid line), and the fit result using the improved functional form of Eq. (30) (blue dashed line).