Landau-Pomeranchuk-Migdal effect in sequential bremsstrahlung: From large- N QCD to N=3 via the SU(

Abstract

Consider a high-energy parton showering as it traverses a QCD medium such as a quark-gluon plasma. Interference effects between successive splittings in the shower are potentially very important but have so far been calculated (even in idealized theoretical situations) only in soft emission or large-N c limits, where N c is the number of quark colors. In this paper, we show how one may remove the assumption of large N c and so begin investigation of N c =3 without soft-emission approximations. Treating finite N c requires (i) classifying different ways that four gluons can form a color singlet and (ii) calculating medium-induced transitions between those singlets, for which we find application of results for the generalization of Wigner 6-j symbols from angular momentum to SU(N c ). Throughout, we make use of the multiple scattering (q) approximation for high-energy partons crossing quark-gluon plasmas, and we find that this approximation is self-consistent only if the transverse-momentum diffusion parameterq for different color representations satisfies Casimir scaling (even for strongly-coupled, and not just weakly-coupled, quark-gluon plasmas). We also find that results for N c =3 depend, mathematically, on being able to calculate the propagator for a coupled non-relativistic quantum harmonic oscillator problem in which the spring constants are operators acting on a 5-dimensional Hilbert space of internal color states. Those spring constants are represented by constant 5×5 matrices, which we explicitly construct. We are unaware of any closed form solution for this type of harmonic oscillator problem, and we discuss prospects for using numerical evaluation.

The LPM effect in sequential bremsstrahlung: nearly complete results for QCD

“…When combined with the leading-order rate Γ LO given by (3.7), we have 28) which indeed involves the same correction to [dΓ/dx] LO , and so toq, as (3.18).…”

“…So there is no divergence of S NLO associated with x → 0 or x → 1. 28 In summary, the only IR divergences coming from S NLO are the uncanceled double-log divergences associated with y → 0.…”

Useful relations among the generators in the defining and adjoint representations of SU(N)

“…The quadratic Casimir operator, C 2 ≡ R a R a , commutes with all the su(N ) generators. 1 Hence in light of Schur's lemma, C 2 is proportional to the d × d identity matrix. In particular, the quadratic Casimir operator in the defining representation of su(N ) is given by…”

“…Recently, two authors contacted me concerning a first draft of these notes, which they apparently found to be quite useful [1,2]. They urged me to make these notes more widely available, and I am pleased to accommodate their request.…”

Wilson line correlators beyond the large-Nc

“…Our calculation is also very pertinent for improving our understanding of color dynamics in the medium, for instance in the context of multi-gluon emissions with overlapping formation times [28] and to understand hadronization after exiting the QGP [29]. In the process of evaluation of the multi-Wilson line correlators, the only assumption made is the exact form of the medium average, see eq.…”