Spectral sum rules for the quark-gluon plasma

We derive sum rules involving the spectral density of the stress-energy tensor in $\mathcal{N}=4$ super-Yang-Mills theory and pure Yang-Mills theory. The sum rules come from the hydrodynamic behavior at small momenta and the conformal (in the case of $\mathcal{N}=4$ super-Yang-Mills theory) or asymptotically free (as for the pure Yang-Mills theory) behavior at large momenta. These sum rules may help constrain quark-gluon plasma transport coefficients obtained from lattice QCD.

Figure 1

Integration contour for the integral (6).

Figure 2

Numerical results for the spectral function $\delta \rho (\omega ,0)$ for strongly coupled $\mathcal{N}=4$ SYM. The results shown correspond to the integrand of the sum rules (1, 2), respectively. The inset in (a) demonstrates the near-exponential drop in the amplitude of $\delta \rho (\omega ,0)$. Horizontal dashed lines are visual aids.

Figure 3

The LHS of the bulk sum rule [Eqs. (3, 62)] (solid line), as evaluated from $SU(3)$ lattice data [36]. The result is negative for all temperatures shown, and should not be associated with the value of the bulk viscosity in $SU(3)$. For comparison, the result from evaluating Eq. (c7) (corresponding to the sum rule by Kharzeev and Tuchin [11]) is shown (dashed lines). The horizontal dotted line is a visual aid.