Perturbative and non-perturbative aspects non-Abelian Boltzmann–Langevin equation

Abstract

We study the Boltzmann-Langevin equation which describes the dynamics of hot YangMills fields with typical momenta of order of the magnetic screening scale g 2 T . It is transformed into a path integral and Feynman rules are obtained. We find that the leading log Langevin equation can be systematically improved in a well behaved expansion in log(1/g) −1 . The result by Arnold and Yaffe that the leading log Langevin equation is still valid at next-to-leading-log order is confirmed. We also confirm their result for the next-to-leading-log damping coefficient, or color conductivity, which is shown to be gauge fixing independent for a certain class of gauges. The frequency scale g 2 T does not contribute to this result, but it does contribute, by power counting, to the transverse gauge field propagator. Going beyond a perturbative expansion we find 1-loop ultraviolet divergences which cannot be removed by renormalizing the parameters in the Boltzmann-Langevin equation.

Diagrammatic and Kinetic Equation Analysis of Ultrasoft Fermionic Sector in Quark-Gluon Plasma

“…In fact, the Boltzmann equation should be used instead of the Vlasov equation in the calculation of the gluon n-point function in this energy region [111][112][113][114][115][116][117]. We will show in Chapter 3 that the other interaction effects such as the asymptotic thermal mass should be taken into account as well as the collision, in the analysis of the ultrasoft fermion propagator.…”

“…(1.32), respectively. The Boltzmann equation in the analysis of the gluon n-point function can be translated into a diagrammatic language, and the resultant diagrammatic method is not a simple one-loop approximation (HTL approximation), but the resummed perturbation which resums the damping rate of the hard particles and sums up the ladder diagrams [111][112][113][114][115][116][117].…”

“…Though there are some suggestion on existence of the ultrasoft mode as is written in the previous chapter, It is not a simple task to establish that fermionic mode exist in the ultrasoft region on the complete leading order calculation because of the infrared divergence called pinch singularity [64-79, 111-117, 137, 138] that breaks a naive perturbation theory, as will be briefly reviewed in the next section. We remark that the same difficulty arises in the calculation of transport coefficients [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79] and the gluon self-energy [111][112][113][114][115][116][117] in the ultrasoft energy region. Therefore, in this chapter, we analyze the fermion propagator in the ultrasoft energy region in Yukawa model and QED/QCD using a similar diagrammatic technique in Refs.…”

“…As is described in Chapter 1, there is a correspondence between some perturbation schemes and the kinetic equations: the HTL approximation is equivalent to the Vlasov equation [108][109][110], and the resummed perturbation theory which enables the analysis of the ultrasoft gluon is equivalent to the Boltzmann equation [111][112][113][114][115][116][117]. Therefore it is natural to expect that the resummed perturbation theory which is used in the analysis of the ultrasoft fermion in Chapter 2, is equivalent to some kinetic equation.…”

Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations

“…We base our analysis on Bödeker's effective theory, despite the fact that Bödeker has also derived a generalised Boltzmann-Langevin equation which is valid to all orders in [log(1/g)] −1 [9], and of which Bödeker's effective theory is merely the leading logarithmic approximation. We choose this approximation because the more general Boltzmann-Langevin equation not only is far more complicated, but is also not renormalisable by power counting [10]. The effective theory on the other hand is ultraviolet finite, and is known to still be valid at next-to-leading logarithmic order provided one uses the next-toleading logarithmic order colour conductivity σ [11].…”