Photon Emission from Ultrarelativistic Plasmas

TL;DR

This work resolves the leading-order photon emission rate from a hot, weakly coupled ultrarelativistic plasma by summing collinearly enhanced inelastic processes with the Landau-Pomeranchuk-Migdal (LPM) effect. Using real-time thermal field theory, it identifies ladder diagrams with soft exchanges $Q\sim gT$ as the dominant class, while showing crossed ladders and ultrasoft exchanges cancel or are subleading. It derives a linear integral equation for the transverse emission amplitude ${\bf f}({\bf p}_{\perp};{p_{\parallel}},k)$ that resums the multiple scatterings and yields a Migdal-type formulation for the photon emission rate, expressed as $\frac{d\Gamma_\gamma^{\rm LPM}}{d^3\mathbf{k}} = \text{prefactor} \int dp_{\parallel} \int d^2\mathbf{p}_{\perp} \, A(p_{\parallel},k) \, \mathrm{Re}\{2 \mathbf{p}_{\perp} \cdot {\bf f}\}$. The results show the leading rate is insensitive to nonperturbative $g^2T$ dynamics, with extensions to fermions and off-shell/softer-photon regimes discussed and a companion paper AMY2 to supply explicit solutions for specific theories. This framework provides a controlled, first-principles handle on hard photon production in quark-gluon plasmas relevant to heavy-ion phenomenology and early-universe cosmology.

Abstract

The emission rate of photons from a hot, weakly coupled ultrarelativistic plasma is analyzed. Leading-log results, reflecting the sensitivity of the emission rate to scattering events with momentum transfers from $gT$ to $T$, have previously been obtained. But a complete leading-order treatment requires including collinearly enhanced, inelastic processes such as bremsstrahlung. These inelastic processes receive O(1) modifications from multiple scattering during the photon emission process, which limits the coherence length of the emitted radiation (the Landau-Pomeranchuk-Migdal effect). We perform a diagrammatic analysis to identify, and sum, all leading-order contributions. We find that the leading-order photon emission rate is not sensitive to non-perturbative $g^2 T$ scale dynamics. We derive an integral equation for the photon emission rate which is very similar to the result of Migdal in his original discussion of the LPM effect. The accurate solution of this integral equation for specific theories of interest will be reported in a companion paper.

Figures (20)

• Figure 1: The two processes of interest, bremsstrahlung and inelastic pair annihilation. In each case, the emerging photon is hard, with energy $\sim T$, but is nearly collinear with the quark from which it is radiated. One or more interactions which exchange momentum with other excitations in the plasma are required for these processes to occur; the exchange momenta are all soft, with energy and momentum $\sim gT$. Time should be viewed as running from left to right.
• Figure 2: $2\leftrightarrow 2$ particle processes which contribute to the photo-emission rate at $O(e^2 g^2 \, T^4)$.
• Figure 3: A cartoon of the real-space appearance of scattering and photon emission. The photon emission, which is sensitive to the interference of unscattered and scattered waves, occurs over a region of spatial extent $1/g^2 T$, which is the same as the mean free path for additional scatterings of the quark.
• Figure 4: The generic diagram which will contribute to bremsstrahlung and inelastic pair annihilation at leading order. The solid outer lines represent charged particles whose momenta are hard but collinear with the (hard, lightlike) incoming photon; These propagators can all be approximately on-shell simultaneously. Therefore, self-energy resummation, including the imaginary part of the self-energy, is required. The cross-rungs represent soft gauge boson exchange with momenta $Q^2 \sim g^2 T^2$ which are restricted to satisfy $Q \cdot K \sim g^2 T^2$ in order to maintain the collinearity condition.
• Figure 5: Diagrams contributing to the current-current correlation function. The shaded blob in (A) represents the amputated four-point correlation function of the electrically charged fields which generate the current. Tadpole diagrams (B) make no contribution to the correlator of interest and may be ignored.