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Spectral sum rules for the quark-gluon plasma

Paul Romatschke, Dam Thanh Son

Abstract

We derive sum rules involving the spectral density of the stress-energy tensor in 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 N=4 SYM theory) or asymptotically free (as for the pure Yang-Mills theory) behavior at large momenta. These sum rules may help constrain QGP transport coefficients obtained from lattice QCD.

Spectral sum rules for the quark-gluon plasma

Abstract

We derive sum rules involving the spectral density of the stress-energy tensor in 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 N=4 SYM theory) or asymptotically free (as for the pure Yang-Mills theory) behavior at large momenta. These sum rules may help constrain QGP transport coefficients obtained from lattice QCD.

Paper Structure

This paper contains 12 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Kramers-Kronig relation
  3. Shear sum rules in ${\cal N}=4$ SYM and pure Yang-Mills theory
  4. AdS/CFT Calculation of $f_q(0)$
  5. Rederivation of $f_q(0)$ in ${\cal N}=4$ SYM from OPE
  6. Calculation of $f_q(0)$ in pure Yang-Mills theory
  7. Numerical verification of sum rules in AdS/CFT
  8. The bulk sum rule in QCD
  9. Conclusion
  10. The coefficient $\kappa$
  11. The coefficient $f_q(0)$ in weakly coupled SU($N$)
  12. Non-commutativity of the $\omega\to0$ and $q\to0$ limits of the $\langle\theta\theta\rangle$ correlator

Figures (3)

  • Figure 1: Integration contour for the integral (\ref{['KK1']}).
  • Figure 2: Numerical results for the spectral function $\delta \rho(\omega,0)$ for strongly coupled ${\cal N}=4$ SYM. The results shown correspond to the integrand of the sum rules (\ref{['shearSYM-sr']}),(\ref{['shearSYM-sr2']}), respectively. The inset in (a) demonstrates the near-exponential drop in the amplitude of $\delta \rho(\omega,0)$. Horizontal lines are visual aids to the eye.
  • Figure 3: The LHS of the bulk sum rule [Eqs. (\ref{['bulkQCD-sr']}) and (\ref{['prebulk']})] (full line), as evaluated from SU(3) lattice data Boyd:1996bx. 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. (\ref{['sumrule-q']}) (corresponding to the sum rule by Kharzeev and Tuchin Kharzeev:2007wb) is shown (dashed lines). The horizontal line is a visual aid to the eye.