Lattice constraints on the thermal photon rate

J. Ghiglieri, O. Kaczmarek, M. Laine, and F. Meyer

Phys. Rev. D 94, 016005 – Published 28 July 2016

Abstract

We estimate the photon production rate from an SU(3) plasma at temperatures of about $1.1 T_{c}$ and $1.3 T_{c}$ . Lattice results for the vector current correlator at spatial momenta $k \sim (2 - 6) T$ are extrapolated to the continuum limit and analyzed with the help of a polynomial interpolation for the corresponding spectral function, which vanishes at zero frequency and matches to high-precision perturbative results at large invariant masses. For small invariant masses the interpolation is compared with the next-to-leading-order (NLO) weak-coupling result, hydrodynamics, and a holographic model. At vanishing invariant mass we extract the photon rate which for $k ≳ 3 T$ is found to be close to the NLO weak-coupling prediction. For $k ≲ 2 T$ uncertainties remain large but the photon rate is likely to fall below the NLO prediction, in accordance with the onset of a strongly interacting behavior characteristic of the hydrodynamic regime.

Received 6 May 2016

DOI:https://doi.org/10.1103/PhysRevD.94.016005

Physics Subject Headings (PhySH)

Finite temperature field theory Perturbative QCD Quark-gluon plasma

Lattice gauge theory

Particles & Fields

Authors & Affiliations

J. Ghiglieri¹, O. Kaczmarek², M. Laine¹, and F. Meyer²

¹AEC, ITP, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
²Faculty of Physics, University of Bielefeld, 33501 Bielefeld, Germany

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Vol. 94, Iss. 1 — 1 July 2016

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Figure 1
Fitted imaginary-time correlators at nonzero momenta. The “best estimate from pQCD” (perturbative QCD) is based on Refs. [17, 18, 20], and has been constructed as explained in footnote . “Polynomial interpolations” correspond to $n_{\max} = 0$ , but similarly good fits are obtained for $n_{\max} = 1$ .
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Figure 2
We show $χ^{2} / d . o . f .$ (top) and $D_{eff} T$ [bottom; cf. Eq. (5.2)] as functions of the matching point $ω_{0}$ for $n_{\max} = 0$ . In the right panel, the upper curves are for $T = 1.2 T_{c}$ and the lower curves are for $T = 1.3 T_{c}$ on the perturbative side (the lattice data is fixed but it is not known precisely to which temperature it corresponds, cf. Table 1). A local minimum of $χ^{2} / d . o . f .$ is generally found close to the point where $ω_{0} = \sqrt{k^{2} + (π T)^{2}}$ ; it is very shallow for the smallest $k$ .
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Figure 3
The spectral functions corresponding to Fig. 1 ( $n_{\max} = 0$ ). The vertical bars locate the light cone. The “best estimate from pQCD” is based on Refs. [17, 18, 20], and has been constructed as explained in footnote . The AdS/CFT result comes from Ref. [28], and has been rescaled to agree with the noninteracting QCD result at large $ω / T$ . (This rescaling choice is rather arbitrary.)
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Figure 4
Lattice results for $D_{eff}$ defined in Eq. (5.2) (data points), compared with the NLO perturbative prediction from Ref. [17] (continuous curves). The lattice errors have been obtained by carrying out fits with $n_{\max} = 1$ to the bootstrap ensemble. The data points at $k = 0$ (cf. the Appendix) have been slightly displaced for better visibility. For comparison note that the heavy-quark diffusion coefficient, determined with different methods, has been estimated as $D T \sim 0.6 \dots 1.1$ at $T \sim 1.5 T_{c}$ [40], and the light-quark value as $D T \sim 0.2 \dots 0.8$ at $T = 1.1 T_{c}$ and $D T \sim 0.2 \dots 0.5$ at $T = 1.3 T_{c}$ [37]. The predictions of Ref. [17] are only reliable for $k ≫ g T$ , but LO perturbative values at $k = 0$ can be obtained by dividing the results of Ref. [39] through the lattice susceptibility according to Eq. (2.9), yielding $D T \approx 2.9$ at $T = 1.1 T_{c}$ and $D T \approx 3.1$ at $T = 1.3 T_{c}$ . The AdS/CFT value is $D T = 1 / (2 π)$ [27].
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Figure 5
Like in Fig. 1 but at zero momentum. Only the spatial components of the vector current have been included here. The “best estimate from pQCD” is based on Refs. [24, 41, 42], and has been constructed as explained in footnote .
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Figure 6
The spectral functions corresponding to Fig. 5 from fits with $n_{\max} = 0$ . The “best estimate from pQCD” is based on Refs. [24, 41, 42], and has been constructed as explained in footnote . The AdS/CFT result comes from Ref. [28], and has been rescaled to agree with the noninteracting QCD result at large $ω / T$ (cf. caption of Fig. 3). As discussed in the Appendix and illustrated with the arrows, the intercepts at $ω = 0$ are lower bounds, and the widths of the transport peaks are upper bounds.
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