In-medium loop corrections and longitudinally polarized gauge bosons in high-energy showers

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD), avoiding soft-emission approximations. In this particular paper, we show (i) how the "instantaneous" interactions of Light-Cone Perturbation Theory must be included in the calculation to account for effects of longitudinally-polarized gauge bosons in intermediate states, and (ii) how to compute virtual corrections to LPM emission rates, which will be necessary in order to make infrared-safe calculations of the characteristics of in-medium QCD showering of high-energy partons. In order to develop these topics in as simple a context as possible, we will focus in the current paper not on QCD but on large-$N_f$ QED, where $N_f$ is the number of electron flavors.Comment: 43 pages + appendices for 89 pages total; 43 figures. Difference from v2: Overall sign of eqs. (F30,F33,F39,F42) fixed; correction to eq. (H14); final results of paper are unchange

The LPM effect in sequential bremsstrahlung

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We analyze the case when the coherence lengths of two consecutive splitting processes overlap, which is important for understanding corrections to standard treatments of the LPM effect in QCD. Previous authors have analyzed this problem in the case of overlapping double bremsstrahlung where at least one of the bremsstrahlung gluons is soft. Here we show how to generalize to include the case where both splittings are hard. A number of techniques must be developed, and so in this paper we simplify by (i) restricting attention to a subset of the interference effects, which we call the "crossed" diagrams, and (ii) working in the large-$N_c$ limit. We first develop some general formulas that could in principle be implemented numerically (with substantial difficulty). To make more analytic progress, we then focus on the case of a thick, homogeneous medium and make the multiple scattering approximation (also known as the $\hat q$ or harmonic approximation) appropriate at high energy. We show that the differential rate $d\Gamma/dx\,dy$ for overlapping double bremsstrahlung of gluons with momentum fractions $x$ and $y$ can then be reduced to the calculation of a 1-dimensional integral, which we perform numerically. [Though this paper is unfortunately long, our introduction is enough for getting the gist of the method.]Comment: 85 pages, 30 figures [only change from v5: fixed trivial typo of a missing bar in eq. (2.20a). The authors are obsessive.

Strong- vs. weak-coupling pictures of jet quenching: a dry run using QED

High-energy partons ($E \gg T$) traveling through a quark-gluon plasma lose energy by splitting via bremsstrahlung and pair production. Regardless of whether or not the quark-gluon plasma itself is strongly coupled, an important question lying at the heart of philosophically different approaches to energy loss is whether the high-energy partons of an in-medium shower can be thought of as a collection of individual particles, or whether their coupling to each other is also so strong that a description as high-energy `particles' is inappropriate. We discuss some possible theorists' tests of this question for simple situations (e.g. an infinite, non-expanding plasma) using thought experiments and first-principles quantum field theory calculations (with some simplifying approximations). The physics of in-medium showers is substantially affected by the Landau-Pomeranchuk-Midgal (LPM) effect, and our proposed tests require use of what might be called `next-to-leading order' LPM results, which account for quantum interference between consecutive splittings. The complete set of such results is not yet available for QCD but is already available for the theory of large-$N_f$ QED. We therefore use large-$N_f$ QED as an example, presenting numerical results as a function of $N_f\alpha$, where $\alpha$ is the strength of the coupling at the relevant high-energy scale characterizing splittings of the high-energy particles.Comment: 31 pages + appendices for 48 pages total, 21 figures. [Difference from version 2: Main change was to eliminate some summary formulas of NLO rates in section III.B, made unnecessary by a clear summary of formulas having been added to ref. [13].

The LPM effect in sequential bremsstrahlung 2: factorization

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. In this paper, we continue analysis of the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD), avoiding soft-gluon approximations. In particular, this paper analyzes the subtle problem of how to precisely separate overlapping double splitting (e.g.\ overlapping double bremsstrahlung) from the case of consecutive, independent bremsstrahlung (which is the case that would be implemented in a Monte Carlo simulation based solely on single splitting rates). As an example of the method, we consider the rate of real double gluon bremsstrahlung from an initial gluon with various simplifying assumptions (thick media; $\hat q$ approximation; large $N_c$; and neglect for the moment of processes involving 4-gluon vertices) and explicitly compute the correction $\Delta\,d\Gamma/dx\,dy$ due to overlapping formation times.Comment: 59 pages, 37 figures. The major changes from v1: new section I.A.4 added to give kinetic theory analogy to better explain the importance of the subtraction defining Delta[d(Gamma)/dx dy]; new appendix F added to compare/contrast with issues raised by Blaizot, Dominguez, Iancu, and Mehtar-Tani [22

The LPM effect in sequential bremsstrahlung: dimensional regularization

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. Of recent interest is the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD). In previous papers, we have developed methods for computing such corrections without making soft-gluon approximations. However, our methods require consistent treatment of canceling ultraviolet (UV) divergences associated with coincident emission times, even for processes with tree-level amplitudes. In this paper, we show how to use dimensional regularization to properly handle the UV contributions. We also present a simple diagnostic test that any consistent UV regularization method for this problem needs to pass.Comment: 59 pages, 8 figures [main change from v1: addition of the new appendix B summarizing more about use of the i*epsilon prescription in earlier work

Are gluon showers inside a quark-gluon plasma strongly coupled? a theorist's test

We study whether in-medium showers of high-energy gluons can be treated as a sequence of individual splitting processes $g{\to}gg$, or whether there is significant quantum overlap between where one splitting ends and the next begins. Accounting for the Landau-Pomeranchuk-Migdal (LPM) effect, we calculate such overlap effects to leading order in high-energy $\alpha_{\rm s}(\mu)$ for the simplest theoretical situation. We investigate a measure of overlap effects that is independent of physics that can be absorbed into an effective value $\hat q_{\rm eff}$ of the jet-quenching parameter $\hat q$.Comment: 6 pages, 3 figures. Main change for v3: minor clarifications adde

The LPM effect in sequential bremsstrahlung: analytic results for sub-leading (single) logarithms

Consider the in-medium splitting $g \to gg$ of a very high-energy gluon traversing a QCD medium, accounting for the Landau-Pomeranchuk-Migdal (LPM) effect. It has been known for some time that soft radiative corrections to that splitting generate a double-log correction to the splitting rate, whose effects can be absorbed into running of the medium parameter $\hat q$ describing the rate of transverse momentum kicks to high-energy particles due to small-angle scattering from the medium. Less has been known about sub-leading, *single* logarithms in this context. In this paper, we find analytic formulas for those single logs (with various caveats and clarifications).Comment: 54 pages, 20 figure

Thermalization of weakly coupled non-Abelian plasmas at next-to-leading order

We employ the QCD kinetic theory, including next-to-leading(NLO) order corrections in coupling constant, to study the evolution of weakly coupled non-Abelian plasmas towards thermal equilibrium. For two characteristic far-from-equilibrium systems with either under- or over-occupied initial conditions, the NLO corrections remain well under control for a wide range of couplings, and the overall effect of NLO corrections is a reduction in the time required for thermalization