Photon emission from quark-gluon plasma out of equilibrium

The photon emission from a nonequilibrium quark-gluon plasma is analyzed. We derive an integral equation that describes photon production through quark-antiquark annihilation and quark bremsstrahlung. It includes coherence between different scattering sites, also known as the Landau-Pomeranchuk-Migdal effect. These leading-order processes are studied for the first time together in an out-of-equilibrium field theoretical treatment that enables the inclusion of viscous corrections to the calculation of electromagnetic emission rates. In the special case of an isotropic, viscous, plasma the integral equation only depends on three constants, which capture the nonequilibrium nature of the medium.

Figure 1

Photon production through quark bremsstrahlung and the annihilation of a quark and an antiquark. For the first process, the diagram with the photon emitted left of the vertex is included but not shown, as is that with the gluon connecting to the antiquark in the second process.

Figure 2

Quark-gluon vertices in the $r/a$ basis in the real-time formalism.

Figure 3

Definition of momenta in the argument for bremsstrahlung and pair annihilation contribution at leading order.

Figure 4

The diagrams for the LPM effect.

Figure 5

One of the diagrams contributing to ${\mathrm{\Pi }}_{\mathrm{ret}}={\mathrm{\Pi }}_{ar}$.

Figure 6

Diagrams contributing to ${\mathrm{\Sigma }}_{\mathrm{ret}}$ at leading order in $g$.

Figure 7

Definition of the four-point function $S_{nmkl}(x_1,x_2;y_1,y_2)$ in position space. $n$, $m$, $k$, $l$ are either 1 or 2. Also shown is the diagram we need to evaluate, i.e., $S_{1122}$ in momentum space.

Figure 8

Analysis of quark rails in the $r/a$ basis.

Figure 9

Vertices that contribute at leading order.

Figure 10

The pairs of propagators that give pinching poles.

Figure 11

$S_{rara}$ and $S_{arar}$. These diagrams do not contribute at leading order because there are no pinching poles.

Figure 12

The only way of placing $r/a$ indices in $S_{rarr}$ at leading order. In a general diagram with arbitrarily many gluon rungs $S_{rr}$ must still be on the far left.

Figure 13

The only way of placing $r/a$ indices in $S_{aarr}$ at leading order.

Figure 14

An example of a diagram that does not contribute to $S_{rrrr}$ at leading order because the propagators in the middle do not give pinching poles.

Figure 15

Leading-order contributions to $S_{rrrr}$ where the $S_{rr}$ propagators are on top of each other.

Figure 16

Leading-order contributions to $S_{rrrr}$ where the $S_{rr}$ propagators are immediately diagonal to each other.

Figure 17

All leading-order contributions to $S_{rrrr}$ after cancellation.

Figure 18

Leading-order diagrams contributing to $S_{aarr}$.

Figure 19

The LPM diagrams that contribute at leading order to the photon polarization tensor ${\mathrm{\Pi }}_{12}^{\gamma }$. Red propagators are $S_{ar}$ and blue propagators are $S_{ra}$.

Figure 20

Procedure for summing up the diagrams in Fig. 19.

Figure 21

Definition of the quantities in Eq. (68).