# NLO Dispersion Laws for Slow-Moving Quarks in HTL QCD

###### Abstract

We determine the next-to-leading order dispersion laws for slow-moving quarks in hard-thermal-loop perturbation of high-temperature QCD where weak coupling is assumed. Real-time formalism is used. The next-to-leading order quark self-energy is written in terms of three and four HTL-dressed vertex functions. The hard thermal loops contributing to these vertex functions are calculated ab initio and expressed using the Feynman parametrization which allows the calculation of the solid-angle integrals involved. We use a prototype of the resulting integrals to indicate how finite results are obtained in the limit of vanishing regularizer.

###### pacs:

11.10.Wx 12.38.Bx 12.38.Cy 12.38.Mh## I Introduction

The past ten years or so have witnessed an abundant activity that tries to understand the properties of the quark gluon plasma, the mechanism(s) of deconfinement and the characteristics of the transition from hadronic matter to quarks and gluons. Experimentally, collaborations at RHIC rhic-collabos () and now ALICE at the LHC alice () are among the main efforts dedicated to this aim. On the other hand, lattice simulations lqcd () as well as hydrodynamic modeling hydro () help investigate the thermodynamic and transport properties of the plasma.

From a perturbative QCD standpoint, calculations at high temperature use the so-called hard-thermal-loop (HTL) summation of Feynman diagrams HTL-perturbation (). For example, one determines the pressure and the quark number susceptibilities from the thermodynamic potential calculated to two and three-loop order two-three-loop-htl (), or the electric and magnetic properties of the plasma liu-luo-wang-xu ().

HTL summation came about in order to overcome early problems encountered in
the standard loop-expansion of high-temperature QCD early-htl ().
However, it makes the next-to-leading order (NLO) dispersion relations for
slow-moving quasiparticles, quarks and gluons, difficult to calculate as
they involve the use of the fully HTL-dressed propagators and vertices. The
first quantity calculated in NLO fully-HTL-dressed perturbation is the
non-moving gluon damping rate gamt0 (). That was followed by the
calculation of the non-moving quark damping rate^{1}^{1}1
This quantity was later calculated in carrington–PRD75-2007-045019 ()
using the real-time formalism. gamq0 (). These calculations have been
performed in the imaginary-time formalism of finite-temperature quantum
field theory imaginary-time-formalism (); they extract the damping rates
from the imaginary part of the fully-HTL-dressed one-loop order
self-energies after analytic continuation to real energies is taken. In a
line of works, we have used this formalism and looked into the infrared
behavior of fully-HTL-dressed one-loop-order damping rates of slow-moving
longitudinal gaml () and transverse gluons gamt (), quarks gamq (), fermions^{2}^{2}2
Note that the fermion damping rate at zero momentum in finite-temperature
QED is independently calculated in carrington–PRD75-2007-045019 ()
using the real-time formalism. The same result is found. fermions ()
and photons photons () in QED, and quasiparticles sqed () in scalar
QED^{3}^{3}3
In scalar QED, we have also calculated the NLO energy of the quasiparticle..

In this logic, the natural step forward is to try to calculate the NLO energies of the quasiparticles. That would come from the real part of the fully-HTL-dressed one-loop order self-energies. This is notoriously much harder than extracting the imaginary part. The first contribution in this direction is the determination of the pure-gluon plasma frequency at next-to-leading order in the long wavelength limit schulz (). Imaginary-time formalism is used and the number of quark flavors is set to zero from the outset. A gauge-invariant result is found:

(1) |

In this result, is the strong coupling constant, the temperature and the number of colors. The next contribution came some time later carrington-et-al–EPJC50-2007-711 (), carrington–PRD75-2007-045019 () and carrington-et-al–PRD78-2008-045018 (), namely the determination of the NLO fermion mass in high- QCD (and QED). For quarks, the result found is carrington-et-al–PRD78-2008-045018 ():

(2) |

This calculation was performed in the real-time formalism of quantum field theory (for reviews on this formalism, see real-time-formalism ()).

One should note in this respect that the NLO contributions to such quantities come for soft one-loop diagrams. Indeed, a general power-counting analysis performed in mirza-carrington–PRD87-2013-065008 () using the real-time formalism shows that, except for the photon self-energy where two-loop diagrams with hard internal momenta do contribute, next-to-leading order contributions come from soft one-loop diagrams with HTL-dressed vertices and propagators. The work carrington–PRD75-2007-045019 () shows that the usual power-counting in imaginary-time formalism overestimates a number of terms that are in effect subleading.

The present work aims at determining the NLO dispersion relations, real part (energy) and imaginary part (damping rate), for slow-moving quasi-quarks in a quark-gluon plasma at high temperature with bare masses taken to zero. We use the closed-time-path formulation of the real-time formalism of finite-temperature quantum field theory martin-schwinger (); keldysh (). The advantage is that we avoid the analytic continuation from discrete Matsubara frequencies to continuous real energies and all that comes with it which, in a sophisticated calculation like the determination of the dispersion relations, can make it difficult to extract the analytic behavior of the physical quantities. But as everything comes with a price, one disadvantage is that, as a result of the so-called doubling of degrees of freedom, each -point function acquires a tensor structure with components to start with, which means a significant increase in the number of say one-loop diagrams involving three and four-point 1PI vertex functions. In addition, this calculation will not benefit from nice simplifications that arise when we set the quark momentum to zero, like the replacement of momentum contractions of HTL vertices with appropriate HTL self-energy differences via Ward identities carrington-et-al–PRD78-2008-045018 ().

This article is organized as follows. After this introduction, we define in section two the HTL-dressed quark and gluon propagators, as well as the quark energies and damping rates at next-to-leading order . These quantities are directly related to the NLO fully-HTL-dressed quark self-energy . This quantity is calculated in section three. We give there an explicit expression of in terms of the three and four HTL-dressed vertex functions. These functions are derived ab initio as discrepancies between different results in the literature are found defu-et-al–PRD61-2000-085013 (); fueki-nakkagawa-yokota-yoshida (). Then, in section four, we introduce a Feynman parametrization to help perform the solid-angle integrals present in the vertex hard thermal loops.

Still, the subsequent integration task remains formidable. In section five, we take a prototype and show how one can carry out with such integrals. The work is mainly numerical. We choose to avoid using the spectral decompositions of the HTL-dressed propagators and aim at getting a finite result with the multi-integral as defined. We indicate in this section how it is possible to obtain a stable behavior down to in unit of the quark thermal mass .

Brief concluding remarks populate section six. An appendix is dedicated to the derivation of the three and four-vertex hard thermal loops.

## Ii The NLO dispersion relations

We consider QCD with colors and flavors. The quark dispersion relations can been cast as:

(3) |

Here, is the quark external soft four-momentum and is the quark self-energy, which can be decomposed into two components in the following manner:

(4) |

In this expression, , with and the four Dirac matrices. Relation (3) is equivalent to the following two dispersion relations:

(5) |

On shell, the (complex) quark energy can be decomposed in powers of the coupling constant :

(6) |

This follows a similar decomposition of , namely:

(7) |

where is the lowest-order contribution, formed by the hard thermal loops of order , and the NLO contribution, of order . The contribution is thus of lowest order , and is the NLO contribution of order . The dispersion relations (5) can therefore be decomposed as:

(8) |

Remembering that , We have:

(9) |

Here stands for . The real parts of are the NLO corrections to the plasma quark energies, and the negatives of the imaginary parts are their damping rates .

The quantities are the solutions to the lowest-order dispersion relations in which only are retained in (5). These latter are known:

(10) |

where is the quark thermal mass to lowest order with . The lowest-order quark energies are real; they are displayed in Fig. 1. Note how quickly the ultra-relativistic behavior sets in, at already for . This indicates that the soft region is effectively narrow. For soft , they can be obtained in power series:

(11) |

Also, from the definition of the HTL self-energies , one can rewrite:

(12) |

The HTL self-energies define also the HTL-dressed quark propagator, which can also be decomposed into two components:

(13) |

The HTL-dressed gluon propagator is also a quantity we need. In the Landau
gauge^{4}^{4}4
The Landau gauge is part of a class of covariant gauges for which the soft
one-loop order corrections to the lowest-order dispersion relations are
independent of the gauge schulz ()., it is given by the following
relation:

(14) |

in which are the usual transverse and longitudinal projectors respectively:

(15) |

where, in the plasma rest-frame, . The quantities are the transverse and longitudinal gluon HTL-dressed propagators respectively, given by:

(16) |

In this expression, is the gluon thermal mass to lowest order.

## Iii The NLO quark self-energy

There are two one-loop HTL-dressed diagrams that contribute to the NLO quark self-energy , displayed axodraw () in Fig. 2 and Fig. 3.

The diagram in Fig. 2 writes as follows:

(17) |

with , and the diagram in Fig. 3 writes as:

(18) |

There are three summation structures: Lorentz (explicit), Dirac, and RTF. We introduce now the Keldysh indices (“r/a” basis) of the closed-time-path (CTP) formulation of the finite-temperature real-time formalism keldysh (); real-time-formalism (). The retarted (R), advanced (A), and symmetric (S) propagators are given by the following definitions:

(19) |

where stands for bosons and for fermions, and are related to the Bose-Einstein Fermi-Dirac distributions via the relations:

(20) |

We then have for the two components of the following explicit expressions:

(21) | |||||

and for the two components of the following expressions:

(22) | |||||

Note that these expressions of in Eq. (21) and in Eq. (22) have been written in carrington–PRD75-2007-045019 () using a different notation.

The HTL-dressed vertex functions are derived in the literature defu-heinz–EPJC7-1999-101 (); defu-et-al–PRD61-2000-085013 (); fueki-nakkagawa-yokota-yoshida () . However, there are discrepancies in these results defu-et-al–PRD61-2000-085013 (); fueki-nakkagawa-yokota-yoshida (), which led us to rederive all three and four-point HTL-dressed vertex functions ab initio in the CTP formalism. We recover the results of fueki-nakkagawa-yokota-yoshida (); these are presented in appendix A. For our needs, we have:

(23) |

for the two-quarks-one-gluon vertices, and:

(24) |

for the two-quarks-two-gluons vertices. In these relations, the quantities ’s are hard thermal loops given by these solid-angle integrals:

(25) | |||||

where and the indices and take the values or . Using these results, we can rewrite and above as simply:

(26) | |||||

(27) |

We now work out the contractions over the Dirac and Lorentz indices. Starting with , the contributions that do not include a hard thermal loop write generically as:

(28) | |||||

In this expression, . Also, with and similarly for , with summation understood over . The superscripts X and Y indicate the RTF indices (R, S, and A). The contributions that involve one hard thermal loop vertex function writes:

Here and take the values . Due to the symmetry in , The other contribution with one hard-thermal-loop vertex function is equal to the one above when changing into , namely:

(30) |

The contribution involving two hard-thermal-loop vertex functions is longer. It generically writes:

The integrand in is faster to write:

From these expressions, we can write the NLO HTL-dresses quark self-energy in a compact form:

(33) | |||||

## Iv HTL vertex functions and Feynman parametrization

The next step is to find a way to evaluate the solid-angle integrals involved in the hard-thermal-loop vertex functions. The way we do this is to rewrite these integrals using the Feynman parametrization. From Eqs. (III), (III), and (III) above, we see that we have two kinds of solid-angle integrals to deal with, namely:

(34) | |||||

The simplest of all these integrals is the integral:

(35) |

Remember that is the 4-vector ( and the integration is over the solid angle of . Let us put aside the prescription for the moment. Using the Feynman parametrization:

(36) |

we write:

(37) |

where . In this case, the integral over can be done formally to get:

(38) |

with the notation and . Noticing that , the reintroduction of the ’s in the final result (38) is a matter of shifting and . This also applies to the two next integrals.

The next HTL vertex function to consider is the solid-angle integral:

(39) |

Using a Feynman parametrization and the notation in which and are redefined with the corresponding ’s, we obtain the result:

(40) | |||||

Little useful comes from pushing further the integration over as the final result will not have a reasonably simple formal expression; it is better to leave this expression as it is and let the integration over be done numerically. However, the Feynman parametrization is useful as it reduces the number of integrations to be performed from two (solid angle) to one (over ) or zero when this latter is explicitly feasible.

The third HTL-vertex solid-angle integral is: