The jet quenching parameter describes the momentum broadening of a
high-energy parton moving through the quark-gluon plasma. Following an approach
originally proposed by Caron-Huot, we discuss how one can extract information
on the collision kernel associated with the parton momentum broadening, from
the analysis of certain gauge-invariant operators in dimensionally reduced
effective theories, and present numerical results from a lattice study.Comment: 7 pages, 3 pdf figures, talk presented at the 31st International
  Symposium on Lattice Field Theory "Lattice 2013" (29 July - 3 August 2013,
  Mainz, Germany

Panero, Marco

Rummukainen, Kari

Schäfer, Andreas

English

arXiv

Institutional Research Information System University of Turin

PoS(LATTICE 2013)173Momentum broadening of partons on the light conefrom the latticeMarco Panero∗Department of Physics and Helsinki Institute of Physics, University of Helsinki,P.O. Box 64, FI-00014 Helsinki, FinlandE-mail: marco.panero@helsinki.fiKari RummukainenDepartment of Physics and Helsinki Institute of Physics, University of Helsinki,P.O. Box 64, FI-00014 Helsinki, FinlandE-mail: kari.rummukainen@helsinki.fiAndreas SchäferInstitute for Theoretical Physics, University of Regensburg,D-93040 Regensburg, GermanyE-mail: andreas.schaefer@physik.uni-regensburg.deThe jet quenching parameter describes the momentum broadening of a high-energy parton movingthrough the quark-gluon plasma. Following an approach originally proposed by Caron-Huot,we discuss how one can extract information on the collision kernel associated with the partonmomentum broadening, from the analysis of certain gauge-invariant operators in dimensionallyreduced effective theories, and present numerical results from a lattice study.HIP-2013-16/TH31st International Symposium on Lattice Field Theory LATTICE 2013July 29 – August 3, 2013Mainz, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LATTICE 2013)173Momentum broadening of partons on the light cone from the lattice Marco Panero1. IntroductionIn this contribution, we present our recent study of the jet quenching phenomenon in thedeconfined QCD plasma [1]. More precisely, following an idea originally proposed in ref. [2],we discuss the strategy for a lattice computation of the jet quenching parameter qˆ, which describesthe average increase in squared transverse momentum per unit length experience by a high-energyquark propagating in the quark-gluon plasma (QGP). Recent related works include refs. [3 – 5],while a different lattice approach to the same physical problem was put forward in ref. [6].Jet quenching is defined as the suppression of particles with high transverse momentum andof back-to-back correlations in heavy nuclei collisions [7]. It is considered as a “gold-plated”observable providing evidence for a strongly coupled quark-gluon plasma: the interpretation of thephenomenon is that a highly energetic parton moving through the deconfined medium undergoesenergy loss and momentum broadening, due to interactions with the QGP constituents [8].2. Theoretical approachJet quenching belongs to the class of hard probes of the QGP: the hard scale involved is themomentum of the parton Q, which can be of the order of many GeV. For a certain collision of heavynuclei, a complete theoretical description of the probability σ(M+N→hadron) of observing final-statehadrons with given momenta necessarily involves both perturbative and non-perturbative aspects.QCD factorization theorems, however, allow one to “separate” these different contributions in awell-defined way [9],σ(M+N→hadron) = fM(x1,Q2)⊗ fN(x2,Q2)⊗σ(x1,x2,Q2)⊗Dparton→hadron(z,Q2), (2.1)where fM(x1,Q2) and fN(x2,Q2) are parton distribution functions, Dparton→hadron(z,Q2) is a fragmen-tation function, while σ(x1,x2,Q2) describes the short-distance cross-section. In the following weconcentrate our attention on the latter, focusing on the propagation of a highly energetic quark inthe QGP. Following the standard formalism [10], one can describe the process in terms of multiplesoft scatterings, in the eikonal approximation, assuming that the hard, ultrarelativistic parton moveson the light cone: then, the leading effect is the broadening of the parton momentum. The averageincrease of the squared transverse momentum component per unit length defines the jet quench-ing parameter qˆ, which can be evaluated if the collision kernel C(p⊥), describing the differentialcollisional rate between the parton and plasma constituents, is known:qˆ=〈p2⊥〉L=∫ d2p⊥(2pi)2p2⊥C(p⊥). (2.2)The collision kernel is related to the two-point correlation function of light-cone Wilson lines.Therefore, as it is always the case for phenomena involving real-time dynamics, a direct computa-tion of C(p⊥) via numerical simulations on a Euclidean lattice is not straightforward.3. Soft physics contribution from a Euclidean setupA full lattice computation of the collision kernel would necessarily require analytical con-tinuation. However, we will now show that it is possible to extract the large, non-perturbativecontributions to C(p⊥) directly from Euclidean lattice simulations.2PoS(LATTICE 2013)173Momentum broadening of partons on the light cone from the lattice Marco PaneroThe idea can be understood in the limit of very high temperatures T , in which the physical cou-pling g becomes small, and a hierarchy of parametrically separated momentum scales emerges: inaddition to the typical scale of thermal excitations (piT ), one can identify “soft” (gT ) and “ultrasoft”(g2T/pi) scales, which are relevant for the long-wavelength modes of the QGP. The ultrasoft scale,in particular, is intrinsically non-perturbative in nature: the long-wavelength modes of the plasmaare strongly coupled at any temperature, and large spatial Wilson loops are always confining. Awell-defined, systematic way to treat the physics at these different scales is by means of dimension-ally reduced effective theories [11]. More precisely, the dynamics of the soft scale can be describedby a static, purely Euclidean, effective theory, which corresponds to three-dimensional Yang-Millstheory coupled to an adjoint scalar field (“electrostatic QCD” or EQCD). In turn, the low-energylimit of EQCD is captured by an effective theory which is just three-dimensional Yang-Mills the-ory, encoding the physics at the ultrasoft scale (“magnetostatic QCD” or MQCD). The typicallylarge next-to-leading order (NLO) corrections affecting perturbative expansions in thermal QCDare related to soft, essentially classical, fields.As observed in ref. [2], for physics on the light cone, it turns out that the contributions from softmodes are insensitive to the precise value of the parton velocity: in particular, they would remainunchanged even for superluminal partons. This idealized case is obviously unphysical, but it wouldmake the parton world-line space-like: then, it ought to be possible to extract the soft contributionto the correlation function of light cone Wilson lines from a Euclidean setup. Indeed, this canbe proven rigorously [2, 3]: in momentum space, the two-point correlation function of spatiallyseparated (|t| < |z|) light-like Wilson lines G<(t,x⊥,z) can be written in terms of the differencebetween a retarded (G˜R) and an advanced (G˜A) correlator,G<(t,x⊥,z) =∫dωd2p⊥dpz[12+nB(ω)][G˜R(ω, p⊥, pz)− G˜A(ω, p⊥, pz)]e−i(ωt−x⊥·p⊥−zpz),(3.1)where nB denotes the Bose-Einstein distribution. Shifting the momentum component along the di-rection of motion as p′z = pz−ωt/z, the integration over frequencies can be carried out by analyti-cal continuation into the upper (lower) complex half-plane for the retarded (advanced) contribution.This leads to a sum over Matsubara frequencies involving the Euclidean correlator G˜E,G<(t,x⊥,z) = T ∑n∈Z∫d2p⊥dp′zG˜E(2pinT, p⊥, p′z+2piinTt/z)ei(x⊥·p⊥+zp′z). (3.2)All non-zero modes are suppressed at large separations. The soft contribution, however, is entirelyencoded in the n= 0 mode, which is time-independent and therefore can be evaluated in EQCD.4. Lattice implementationThe lattice computation of the two-point correlation function of light-cone Wilson loops isthen straightforward. We regularize the super-renormalizable continuum EQCD LagrangianL =14Fai jFai j+Tr((DiA0)2)+m2ETr(A20)+λ3(Tr(A20))2(4.1)on the lattice using the Wilson formulation, and fix its parameters (the 3D gauge coupling and themass- and quartic-term coefficients) by matching to high-temperature QCD [11]. We chose a setup3PoS(LATTICE 2013)173Momentum broadening of partons on the light cone from the lattice Marco PaneroL13LL1L3 HrFigure 1: The gauge-invariant operator describing the two-point correlation function of light-cone Wilsonlines appearing in eq. (4.2) involves a set of Hermitian parallel transporters along the real time directionH(x), obtained from exponentiation of the scalar field A0(x).corresponding to QCD with n f = 2 light dynamical quarks, at two temperatures T ' 398 MeV and2 GeV, respectively equal to about twice and ten times the deconfinement temperature [12].The two-point correlator of light-cone Wilson lines can be made gauge-invariant by “closingit” (with the inclusion of transverse parallel transporters) to a loop and taking its trace, W (`,r).Then, its lattice implementation becomes straightforward. An important point to remark, however,is the following: although the effective theory that we consider is purely spatial, the operator mustdescribe real time evolution. For this reason, the lattice discretization of the two light-cone Wilsonlines involves insertions of the local Hermitian operator H(x) = exp[−ag2EA0(x)], representing aparallel transporter along the real-time direction:〈W (`,r)〉=〈Tr(L3L1L†3L†1)〉, (4.2)with L3 =∏(U3H) and L1 =∏U1, see fig. 1. This lattice operator has well-defined renormaliza-tion properties [13]. From the expectation value of W (`,r) (that we compute with the multilevelalgorithm [14]) one can extract a “potential” V (r),〈W (`,r)〉= exp [−`V (r)] , (4.3)which is nothing but the transverse Fourier transform of the collision kernel C(p⊥).5. Results and discussionOur results for V (r) are shown in the left-hand-side panel of fig. 2: they exhibit very goodscaling properties and are compatible with the perturbative predictions that V (r) should vanish forr = 0, and that at distances r ' 1/g2E it should include a linear contribution, with the slope of thedashed line displayed in the figure. Note that this short-distance behavior is dramatically differentfrom the one of the usual potential extracted from ordinary Wilson loops in 3D Yang-Mills theory(which we also computed, for comparison with the MQCD study discussed in ref. [4]), shown inthe right-hand-side panel of fig. 2.4PoS(LATTICE 2013)173Momentum broadening of partons on the light cone from the lattice Marco Panero0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6r g2E-0.500.511.522.533.54V  /  g2 Eβ = 12β = 14β = 16β = 18β = 24β = 32β = 40β = 54β = 67β = 80perturbative slope0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6r g2M-2-1.5-1-0.500.511.52V  /  g2 Mβ = 12β = 14β = 16β = 18β = 24β = 32β = 40β = 54β = 67β = 80Laine modelFigure 2: Left-hand-side panel: Our results for V (r) extracted from eq. (4.3), for our lower temperatureT ' 398 MeV. The slope predicted perturbatively for rg2E ' 1 is also shown. Right-hand-side panel: Thestatic-quark potential extracted in three-dimensional Yang-Mills theory. The dashed curve is a short-distanceparametrization based on the analysis carried out in ref. [4].Since qˆ is given by the second moment associated with the C(p⊥) kernel, it can be relatedto the curvature of V (r). Following the approach discussed in ref. [4], we arrive at the resultthat the soft NLO contribution to qˆ is quite large: 0.55(5)g6E for T ' 398 MeV, and 0.45(5)g6E forT ' 2 GeV. Taking also the other known contributions into account, our analysis leads to a roughestimate for the final value of qˆ in the ballpark of 6 GeV2/fm for RHIC temperatures. Interestingly,this result is comparable with strong-coupling predictions from holography [15], as well as withphenomenological computations [16].In summary, we have shown that a lattice approach is possible for certain real-time problemsinvolving physics on the light cone. In particular, here we have focused on soft physics in thermalQCD (but related ideas have also been discussed in the context of QCD at zero temperature [17]).We emphasize that the outlined approach is systematic: this is not a model. The effective theorystudy that we carried out in this work is based on the modern approach to thermal QCD, providinga consistent framework to describe the physics at different momentum scales relevant for the QGP.Although, by construction, the effective theory that we simulated does not capture phenomena atthe hard thermal scale, we stress that this is not a limit, but a virtue, of the formalism, allowing oneto disentangle effects of different physical origin.As for possible extensions of this work, we plan to improve our extrapolation to the continuumlimit at short distances by simulations on finer lattices, and/or using improved actions (as originallysuggested in ref. [18]). We also plan to study in detail the temperature dependence of qˆ and com-pare it with holographic expectations [15]. In the same spirit, it would also be interesting to repeatthis study in theories with more than N = 3 colors, in which the dynamics simplifies consider-ably [19]. Recent lattice studies show that the static equilibrium properties of the QGP have verylittle dependence on N, both in four [20] and in three spacetime dimensions [21], and, given thatall holographic computations are carried out in the large-N limit, it would be interesting to see ifthis is also the case for quantities related to real-time dynamics.5PoS(LATTICE 2013)173Momentum broadening of partons on the light cone from the lattice Marco PaneroAcknowledgementsThis work is supported by the Academy of Finland (project 1134018) and by the German DFG(SFB/TR 55). Part of the numerical simulations was carried out at the Finnish IT Center for Science(CSC) in Espoo, Finland.References[1] M. Panero, K. 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Momentum broadening of partons on the light cone from the lattice

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Momentum broadening of partons on the light cone from the lattice

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