A non-perturbative contribution to jet quenching

It has been argued by Caron-Huot that infrared contributions to the jet quenching parameter in hot QCD, denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{q}$\end{document}, can be extracted from an analysis of a certain static-potential related observable within the dimensionally reduced effective field theory. Following this philosophy, the order of magnitude of a non-perturbative contribution to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{q}$\end{document} from the colour-magnetic scale, g2T/π, is estimated. The result is small; it is probably below the parametrically perturbative but in practice slowly convergent contributions from the colour-electric scale, whose all-orders resummation therefore remains an important challenge.


Introduction
One of the classic qualitative indications for the generation of a thermal medium in heavy ion collision experiments is the fact that high-p T jets get quenched [1,2]. In fact, not only do jets get quenched but they appear to do so more effectively than naive estimates suggest [3,4]. This motivates not only a complete leading-order weak-coupling analysis [5], but also developing methods to go beyond the leading order [6], taking into account large corrections from the infrared (IR) scales that are characteristic of thermal field theory [7,8]. (The weak-coupling regime has recently also been discussed in Ref. [9].) In a weakly coupled non-Abelian plasma, there are two momentum scales in the IR: the colour-magnetic scale, g 2 T /π, and the colour-electric scale, gT (g 2 = 4πα s is the QCD gauge coupling). We refer to these through the gauge coupling, g 2 E /π , and the mass parameter, m E , of the dimensionally reduced effective theory [10,11]. The contributions of the two scales have been disentangled for several thermodynamic observables, and a particularly rich source of information is the "spectrum" measured through screening masses [12]. For instance, the smallest screening mass (at temperatures below ∼10 GeV), which is parametrically of a e-mail: laine@itp.unibe.ch the form 2m E + O(g 2 E /π), gets a numerically insignificant contribution from the colour-magnetic scale [13], whereas the Debye screening mass, which is parametrically of the form m E + O(g 2 E /π), is in practice all but dominated by the colour-magnetic scale [14]: The purpose of the current study is to estimate whether one of these extreme scenarios could be relevant forq.

Colour-electric contribution
The rate of jet broadening (transverse momentum diffusion) is often parametrized by a phenomenological coefficient, called the jet quenching parameter and denoted byq [15]. Although, strictly speaking, the definition ofq is ambiguous even at leading order [16], it can be argued that the ambiguities can be hidden by considering a quantity called the transverse collision kernel, C(q ⊥ ); then It is now the choice of the upper bound q * which reflects the ambiguities. The form of C(q ⊥ ) for q ⊥ ∼ πT was determined in Ref. [5] and for q ⊥ ∼ gT in Ref. [6]; although the NLO contribution from q ⊥ ∼ gT toq is parametrically suppressed by O(g), it is numerically large. More precisely, denoting by m E = gT √ N c /3 + N f /6 + O(g 3 T ) the Debye mass parameter and by g 2 E = g 2 T + O(g 4 T ) the coupling constant of the dimensionally reduced effective field theory, the next-to-leading order (NLO) expression for the IR part (q ⊥ ∼ m E ) of the collision kernel can be written as [6] Here The terms involving q * cancel against contributions from hard momenta, q ⊥ πT [5], and the q * -independent middle term inside the curly brackets then represents the physical NLO contribution toq [6].

Colour-magnetic contribution
For small momenta, q ⊥ m E , the NLO part of Eq. (3) behaves as We note from Eqs. (3), (5) that, as also suggested by the operator definition [6], the small-q ⊥ limit (q ⊥ m E ) of C(q ⊥ ) goes over into minus the momentum-space static potential of three-dimensional pure Yang-Mills theory: where [17,18] (In principle higher-dimensional "hybrid" potentials could also appear, but their contributions toq are suppressed by (g 2 E /πm E ) n , with some positive n, with respect to those fromṼ (q ⊥ ).) However, at the next order the perturbative expansion breaks down: a direct momentum-space computation ofṼ (q ⊥ ) at 2-loop order produces in d = 3 − 2 dimensions the (gauge-independent) expression [17,18] A comparison with Eq. (2) suggests the presence of a nonperturbative contribution toq at O(g 6 E ), both because the integral is logarithmically divergent at the lower end, and because the coefficient in Eq. (8) is IR divergent.
For future reference we note that the Fourier transform of Eq. (7), V (r) = q ⊥ (e iq ⊥ ·r − 1)Ṽ (q ⊥ ), yields the coordinate space potential where r * is a regularization-dependent constant.

Transformation to configuration space
In view the delicate nature of the IR contribution toq, it is useful to re-express the small-q ⊥ part of Eq. (2) in another way. The idea is to make use of Eq. (6), in combination with a non-perturbative understanding of V (r), in order to get a handle on IR physics. To be concrete, letθ Λ (q ⊥ ) be some cutoff function, where Λ is chosen to be formally in the range g 2 E /π Λ m E . Then we can rephrase the IR part of Eq. (2), to be denoted byq Λ , aŝ where θ Λ (r) ≡ d 2 q ⊥ (2π) 2 e −iq ⊥ ·rθ Λ (q ⊥ ) is the configuration space version of the cutoff function. For instance, if θ Λ (q ⊥ ) = θ(Λ − q ⊥ ), then θ Λ (r) = ΛJ 1 (Λr)/2πr. This very choice is not particularly convenient, however, since the Bessel function J 1 is oscillatory. We find it more transparent to choose a Gaussian, because both functions are elementary and positive. Now, because of rotational symmetry and the form of Eq. (9), the behaviour of ∇ 2 V (r) appearing in Eq. (10) is where we have adopted the notation F (r) ≡ V (r) and c(r) ≡ r 3 V (r)/2 from Ref. [19]. We subsequently need to subtract the known perturbative terms, Eq. (9), from the IR contribution toq Λ , given that they were already included in Eqs. (3), (4). The corresponding potential, i.e. the term of O(g 6 E r 2 ) in Eq. (9), is denoted by δV . Thereby we obtain an expression for the remaining IR contribution, let us call it δq Λ : where we rescaled the integration variable asr = rΛ and defined As Eq. (13) shows, only the short-distance part of φ(r), terms linear in r (modulo logarithms), contributes to δq Λ ; this corresponds to terms quadratic in r in δV (r).
The nature of the short-distance behaviour of δV (r) can be discussed within a framework similar to the Operator Product Expansion [20,21]. The lowest-dimensional condensate in three-dimensional pure SU(3) evaluates to where the errors come from Monte Carlo simulations (MC) [22] and a scheme conversion (NSPT) [23], and the numerical values apply for N c = 3. Just cancelling the scale dependence from Eq. (16) one might expect a short-distance behaviour of the form However, according to Ref. [24] the shape at g 2 E r 1 is really more complicated, ∼g 6 E r 2 ln[ln(1/g 2 E r)]/ ln(1/g 2 E r), where g 2 E N c ln(1/g 2 E r) represents the difference of octet and singlet potentials. Note that such a tail does not contribute in Eq. (13) for Λ g 2 E /π . Unfortunately the analysis is not valid for the range g 2 E r ∼ 1 that is relevant for us here, so we resort to modelling in the following.

Modelling of lattice data
In Fig. 1, numerical data for r 2 0 φ(r) from Ref. [19] is shown. The parameter r 0 is defined from r 2 0 F (r 0 ) = 1.65 (18) for any given lattice spacing [25]. In the continuum limit [19], and we have inserted this estimate in order to combine the last term of Eq. (14) with lattice data. (Obviously, it would be nice to add data at shorter distances, but this task is nontrivial because strong cutoff effects appear and a careful extrapolation to the continuum limit is needed.) Now, motivated by the naive Eq. (17) and the definition in Eq. (14), we describe the data through the function The data (at β = 20) are well modelled ( (1), c = 0.18 (1). We remark that around r/r 0 ≈ 2.0 this function is close to the asymptotic value given by the non-perturbative string tension minus the perturbative subtraction, {[0.553(1)] 2 − 7/16π}g 4 E r 2 0 ≈ 0.81 [26]. We have checked that adding a cubic term to the ansatz of Eq. (20) does not change the results in any significant way, for instance a = 0.74 (14), b = 0.52 (17) then. In the following it is therefore assumed that Eq. (20) reflects the magnitude of the true function for g 2 E r ∼ 1. Carrying out the integral in Eq. (13), we get Fig. 1 The function φ defined by Eq. (14), in units of the parameter r 0 , defined by Eq. (18). The lattice data is from tables 5 and 7 of Ref. [19] The coefficient a/r 3 0 , which comes together with a logarithmic dependence on the cutoff Λ, should be an analytically computable function, since the cutoff should cancel against an NNLO contribution from the colour-electric scale; in this sense it is not a "genuine" colour-magnetic contribution. In contrast, the coefficient b/r 3 0 represents a genuine colourmagnetic contribution within the model (although, because of the freedom in choosing the argument of the logarithm, this notion is somewhat ambiguous).

Phenomenological interpretation
Omitting from Eq. (21) terms vanishing for Λr 0 1; replacing the cutoff inside the logarithm with the scale ∼ m E at which other physics sets in; inserting r 0 from Eq. (19); and estimating m E /g 2 E ∼ 1 (cf. Fig. 1  This can be compared with the middle term from the curly brackets in Eq. (4), The colour-magnetic contribution, Eq. (22), is clearly below the NLO perturbative contribution from the colour-electric scale, Eq. (23).
In conclusion, the contribution toq from the colourmagnetic scale, q ⊥ ∼ g 2 E /π , may well be of modest magnitude. Thus the phenomenological motivation for its theoretically consistent determination may be feeble; rather, it should probably be measured as a part of the total IR contribution from both the colour-electric and colour-magnetic scales, e.g. along the lines suggested in Ref. [6].