Jet quenching from the lattice

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ScienceDirect

Nuclear Physics A 931 (2014) 393–398

www.elsevier.com/locate/nuclphysa

Jet

quenching

from

the

lattice

Marco Panero

a,∗

_,

_{Kari Rummukainen}

b

_,

_{Andreas Schäfer}

c

a_Instituto_de_Física_Téorica,_Universidad_Autónoma_de_Madrid_&_CSIC,_E-28049_Cantoblanco,_Madrid,_Spain b_Department_of_Physics_&_Helsinki_Institute_of_Physics,_P.O._Box_64,_FI-00014_University_of_Helsinki,_Finland

c_Institute_for_Theoretical_Physics,_University_of_Regensburg,_D-93040_Regensburg,_Germany

Received 8July2014;receivedinrevisedform 24July2014;accepted 25July2014 Availableonline 4August2014

Abstract

Wepresentalatticestudy ofthemomentumbroadening experiencedbyahardpartonin thequark– gluonplasma.Inparticular,thecontributionstothisreal-timephenomenonfromsoftmodesareextracted fromasetofgauge-invariantoperatorsinadimensionally-reduced effectivetheory(electrostaticQCD), whichcanbesimulatedonaEuclideanlattice.Atthetemperaturesaccessibletopresentexperiments,the softcontributionstothejetquenchingparameterarefoundtobequitelarge.Wecompareourresultsto phenomenologicalmodelsandtoholographiccomputations.

Keywords: Jetquenching;LatticeQCDcalculations;Quark–gluonplasma

1. Introduction

Jet quenching, namely the suppression of particles with large transverse momenta and of correlations between back-to-back hadrons detected after a heavy-ion collision, is an effect di-rectly related to the energy loss and momentum broadening experienced by a hard parton moving in the deconfined medium, due to its interactions with the quark–gluon plasma (QGP) con-stituents[1].

Under the assumption that the parton is much harder than the typical momenta of thermal excitations in the QGP, the standard formalism to describe jet quenching theoretically relies on a multiple soft-scattering picture, in the eikonal approximation[2–6]. The average increase in the

* _{Corresponding}_author.

http://dx.doi.org/10.1016/j.nuclphysa.2014.07.037

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(squared) transverse momentum component of the hard parton per unit length is constant, and defines the phenomenological jet quenching parameter ˆq,

ˆq =p⊥2 L = d2p_⊥ (2π )2p 2 ⊥C(p⊥), (1)

expressed as the second moment of the differential collision rate between the parton and the QGP constituents, C(p_⊥). In turn, the latter quantity is directly related to the two-point correlation function of Wilson lines on the light cone.

What tools can be used to calculate this two-point correlator of null Wilson lines? Analyti-cal weak-coupling expansions are a well-defined first-principles approach; however, the infrared divergences characteristic of thermal QCD pose limitations on the order to which they can be pushed[7,8]—and the quantitative accuracy of perturbative computations truncated at the leading (LO) or next-to-leading order (NLO) is generally observable-dependent, and may be question-able at RHIC and LHC temperatures T , at which the QCD coupling g is not very small[9]. On the other hand, holographic computations based on the gauge/string correspondence are an ideal tool to investigate the strong-coupling limit of the plasma; however, they are not derived from the microscopic formulation of QCD, but rather from some models, like the N = 4 super-symmetric Yang–Mills theory[10]. Finally, numerical lattice calculations (which do not rely on either strong- or weak-coupling assumptions) are based on a Euclidean formulation, hence they are generally unsuited for the whole class of phenomena involving real-time dynamics in the QGP[11].

2. Soft contributions from lattice EQCD

As pointed out in Ref.[12](see also Ref.[13]), however, it is possible to show that the contri-bution to C(p_⊥)from soft QGP modes (i.e., those at momentum scales up to gT ) can be exactly evaluated in a dimensionally reduced, low-energy effective theory, namely electrostatic QCD (EQCD)[14–21], which is nothing but Yang–Mills theory in three spatial dimensions, coupled to an adjoint scalar field. The EQCD Lagrangian is

L =1 4F a ijF a ij+ Tr (DiA0)2 + m2 ETr A2₀+ λ3 TrA2₀2; (2)

its parameters can be fixed by matching to high-temperature QCD. For example, at LO the gauge coupling, the squared mass and the quartic coupling of the scalar are related to the QCD param-eters via g_E2= g2T + . . . , m2_E= 1+nf 6 g2T2+ . . . , λ3= 9− nf 24π2 g 4_T _{+ . . . ,} (3) where nf denotes the number of dynamical light quark flavors. This effective theory can be regularized on a lattice[22]and studied non-perturbatively by means of Monte Carlo simulation. The parameters of our study correspond to QCD with nf = 2 light quarks at T 398 MeV and at T 2 GeV (roughly equal to twice and ten times the deconfinement temperature). To get sufficient accuracy at these “low” temperatures, we included subleading corrections in the EQCD parameter definitions.

Although this effective theory is purely spatial, the operator of interest for our computation of ˆq must describe dynamical evolution in real time[23]. This operator can be interpreted as the dimensionally-reduced counterpart of (a gauge-invariant version of) the light-cone Wilson line correlator, and can be written as the trace of a “decorated Wilson loop”:

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W (, r)= TrL3(x, )L1(x+ ˆ3, r)L−1₃ (x+ r ˆ1, )L†₁(x, r)

, (4)

having denoted the point at which the loop starts (and ends) as x, the direction of the spatial component of the light-cone Wilson lines as ˆ3, and the direction of the spatial separation between the lines as ˆ1, with

L3(x, )= /a−1

n=0

U3(x+ anˆ3)H

x+ a(n + 1)ˆ3, H (x)= exp−ag2_EA0(x) , L1(x, r)= r/a₋₁ n=0 U1(x+ anˆ1). (5)

Note that H (x) represents a parallel transporter along a real-time interval of length equal to the lattice spacing a, and is a Hermitian (rather than unitary) matrix. The W operator enjoys well-defined renormalization properties[24].

3. Numerical results

The exponential decay of W(, r) exp[−V (r)] at large can be studied accurately using a multilevel algorithm [25]and defines the quantity V (r), which equals minus the transverse Fourier transform of the collision kernel C(p_⊥)(up to a constant). Eq.(1)implies that (the soft contribution to) the jet quenching parameter ˆq is related to the curvature of V (r) near the origin. Fitting our lattice results for V (r) to a functional form which includes linear, quadratic, and logarithmic-times-quadratic terms (and including the contribution from hard modes, which can be reliably computed perturbatively and is numerically subdominant) we get a final estimate for

ˆq around 6 GeV2_/

fm for T 398 MeV (i.e. at a temperature comparable to those realized at RHIC), with total uncertainty around 15–20%.

This result indicates that the non-perturbative contribution to ˆq from soft modes is non-negligible, and significantly larger than expected from a naïve parametric analysis in perturbation theory. It is interesting to note that the mismatch between our non-perturbative results and the perturbative NLO predictions[12,13]can be related to the existence of large non-perturbative contributions to the Debye mass mD: as shown in Fig. 1, plotting our results for V (r) in units of the non-perturbatively estimated Debye mass[26]brings our results in agreement with the curve predicted perturbatively at NLO, and makes the curves obtained at the two different temperatures compatible with each other (within uncertainties). Plugging the value of the non-perturbative De-bye mass into the analytical expression for ˆq

g4T2mDCfCa

3π2+ 10 − 4 ln 2

32π2 (6)

(where Cf= 4/3 and Ca= 3 denote the eigenvalues of the quadratic Casimir operators for the fundamental and for the adjoint representation of SU(3)) results, again, in a final value of ˆq around 6 GeV2/fm at RHIC temperatures.

4. Discussion and conclusions

In this contribution, we reported on our recent lattice study of the momentum broadening experienced by a light quark in the QGP[23]. Our computation is based on the idea of sepa-rating the contribution from hard thermal excitations (which can be evaluated analytically in a

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Fig. 1.Thecoordinate-spacecollisionkernelV (r)computednon-perturbativelyinEQCDsimulations,atT 398 MeV (left-hand-sidepanel)andatT 2 GeV (right-hand-sidepanel),inunitsofthenon-perturbativeDebyescreeningmass mD[26].Symbolsofdifferentcolorscorrespondtosimulationsatdifferentlatticespacingsa,withβ= 6/(ag2E).The

dashedblackline(andthegrayband)showthecontinuumextrapolation(andthecorrespondinguncertainty),whilethe solidblackcurveistheperturbativepredictionatNLO[12,13].

weak-coupling calculation) from those due to modes up to the soft scale, which we extracted non-perturbatively from Monte Carlo simulations of a dimensionally reduced, low-energy effective theory, EQCD. Related studies have also been carried out in magnetostatic QCD (which describes the physics of “ultrasoft”, O(g2T /π ), modes of thermal QCD)[27,28], where it was found that the contribution to ˆq from the ultrasoft scale was essentially negligible. By contrast, our results indicate that, at least at experimentally accessible temperatures, non-perturbative contributions in the soft sector are non-negligible. In particular, our final result for ˆq at RHIC temperatures is around 6 GeV2/fm. This value is close to estimates obtained from holographic studies[29–31], and also from certain phenomenological model computations [32,33]. Although more recent studies of this type tend to favor smaller values[34], one should note that a quantitative compari-son is difficult, because the precise numerical value of ˆq depends on details of the kinematics that is assumed. Interestingly, we also found that, by expressing our results for V (r) (the collision kernel in transverse coordinate space) in units of the non-perturbatively evaluated Debye mass

mDbrings our results to agree with the perturbative calculation.

The approach underlying our computation allows one to bypass the intrinsic challenges of

ab initio studies of real-time phenomena on a lattice with Euclidean signature, following the seminal observation[12]that soft contributions to light-cone physics can be exactly computed in the purely spatial (and bosonic) effective theory describing the thermal excitations of the QGP up to momenta O(gT ). A closely related observation is that the screening masses of the QGP can be related to light-cone real-time rates[35]. An explicit check of the fact that the soft contribution to

C(p_⊥)can be extracted “crossing the light cone” was carried out in classical lattice gauge theory in Ref.[36].

The approach followed in the present work could be used to investigate various other real-time phenomena on the lattice. For quantities requiring a delicate control of lattice discretization effects, it might be suitable to resort to improved lattice actions, for which sophisticated error-reduction algorithms already exist[37].

Other interesting extensions of this study include a more detailed investigation of the depen-dence of ˆq on the temperature (beyond the purely dimensional expectation ˆq ∝ T3) and on the

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number of color charges N . The latter plays an important rôle in the context of holographic computations (see Ref.[38, subsect. 2.6]and references therein), hence it would be important to check if quantities related to real-time dynamics in thermal QCD also exhibit a mild dependence on N , as equilibrium quantities do[39–42].

Acknowledgements

This work is supported by the Spanish MINECO (grant FPA2012-31686 and “Centro de Ex-celencia Severo Ochoa” programme grant SEV-2012-0249), by the Academy of Finland (project 1134018), by the German DFG (SFB/TR 55), and partly by the European Community (FP7 pro-gramme HadronPhysics3, 283286).

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