Calculating the jet quenching parameter q in lattice gauge theory
Abhijit MajumderDepartment of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA, and
Department of Physics, Ohio State University, Columbus, Ohio 43210, USA(Received 8 May 2012; revised manuscript received 30 January 2013; published 18 March 2013)
We present a framework where first-principles calculations of jet modification may be carried out in a nonper-turbative thermal environment. As an example of this approach, we compute the leading-order contribution to thetransverse momentum broadening of a high-energy (near on-shell) quark in a thermal medium. This involves a fac-torization of a nonperturbative operator product from the perturbative process of scattering of the quark. An oper-ator product expansion of the nonperturbative operator product is carried out and related via dispersion relations tothe expectation of local operators. These local operators are then evaluated in quenched SU (2) lattice gauge theory.
As of this time, the Large Hadron Collider (LHC) hascompleted three successful runs with heavy ions. There isnow a wealth of data on the modification of hard jets fromthe Relativistic Heavy-Ion Collider (RHIC) [1,2] and the LHC[3–5]. With the similarity between the various soft observablesbetween the RHIC and the LHC the study of jets has moved tothe forefront of heavy-ion programs at both of these colliders.
In the last several years, the science of jet quenchinghas undergone considerable evolution. There are nowfour different successful jet quenching formalisms basedon perturbative QCD (pQCD) [6–19] and a collection offormalisms based on Anti-DeSitter space/Conformal FieldTheory [20–23]. While one would have expected a largedisparity between the physical pictures underlying thestrong and weak coupling approaches, there are actuallyconsiderable differences between the various pQCD-basedapproaches [24,25]. Besides the differences in the descriptionof the perturbative gluon emission process, the description ofthe medium is quite different in the various approaches: In boththe Armesto-Salgado-Weidemann (ASW) and the higher-twist(HT) approach, one assumes that the transverse momentumexchanged in numerous interactions with the medium issoft enough that one may approximate the distribution as aGaussian and retain only the leading two moments (mean andvariance). The variance of this Gaussian transverse momentumdistribution is often referred to as q. In the Gyulassy-Levai-Vitev (GLV) formalism, the exchanged momentum isassumed to have a considerable hard tail, such that it cannot beapproximated as a Gaussian broadening. In the Arnold-Moore-Yaffe (AMY) formalism one describes the medium usinghard-thermal-loop-improved perturbation theory [26,27].
With the exception of the AMY formalism, none of the otherpQCD-based formalisms can be said to be a first-principlescalculation. In all cases the transport parameter q (eitheraveraged or a normalized function of space-time in a fluiddynamical simulation) is a fit parameter in the calculation, setby comparison to one data point. Even in the AMY formalismthe strong coupling constant αs is varied to fit one data point.Thus, even the AMY formalism is not, strictly speaking, a first-principles calculation. The strong coupling approaches, thoughfirst-principles calculations, are not sufficiently sophisticated
to address the great variety of jet modification data. Thepredictions from such calculations also seem to be inconsistentwith the rising RAA observed at the LHC [28].
The goal of the present paper is to suggest a setup wherea first-principles calculation of jet modification can be carriedout using a combination of perturbative and nonperturbativemethods. The perturbative sector will be similar in form withthe higher-twist approach in that it will involve a factorizationof the perturbative sector describing the propagation of hardpartons from operator products that will be used to describethe medium. The computation of these operator products inthe nonperturbative sector will be carried out using finitetemperature lattice gauge theory. We point out that alreadyat this stage a completely first-principles calculation can neverbe directly compared with data. It will, however, provideconstraints on the number, structure, and normalization ofthe various transport coefficients that one routinely uses toconstruct a phenomenological analysis of the data.
The paper is organized as follows: In Sec. II, we describe thesetup where calculations can be carried out and in particularwe attempt to justify why the current method to identify andestimate jet transport coefficients is the better alternative. InSec. III we focus on the particular process of a hard quarkpropagating through a medium and set up the formalism forthis process. In Sec. IV the various regions of phase spaceare explored. In Sec. V, dispersion relations that are used toevaluate the operator products are set up. In Sec. VI we discussthe details of the lattice gauge theory calculation. We concludein Sec. VII with an outlook for future work.
II. pQCD PROCESSES IN A QGP BRICK
The notion that jet transport coefficients represent proper-ties of the medium and thus should be calculable in latticeQCD has definitely been informally considered for some timenow. The most naive approach would be to simply take theexpression for a given transport coefficient, say q, as derivedin an appropriate effective theory in Ref [29], where
ABHIJIT MAJUMDER PHYSICAL REVIEW C 87, 034905 (2013)
and attempt to compute this on the lattice. (In the equationabove F⊥μν is a gauge field strength operator, one of whoseindices are either 1 or 2.) This particular form of the transportcoefficient is obtained in either covariant gauge or light-conegauge.
The equation above is not manifestly gauge invariant andrequires the introduction of Wilson lines. At first sight, thepath taken by the Wilson lines seems arbitrary. However,following the arguments in Ref. [30], one obtains four differentWilson lines that need to be included, two along the light-conedirection and two along the transverse direction. The fullygauge invariant expression for q is now given as
q = 4π2αs
Nc
∫dy−d2y⊥d2k⊥
(2π )3ei
k2⊥y−2q− −ik⊥y⊥
× 〈P |Tr[Fa⊥
+μ(y−, y⊥)U †(∞−, y⊥; 0−, y⊥)
× T †(∞−, �∞⊥; ∞−, y⊥)T (∞−,∞⊥; ∞−, 0⊥)
×U (∞−, 0⊥; 0−, 0⊥)Fb⊥
+,μ]|P 〉. (2)
In the equation above, U represents a Wilson line along the (−)light-cone direction and T represents a Wilson line along thetransverse light-cone direction. If the calculation were beingcarried out in covariant gauge, only the light-cone Wilsonlines would contribute, while for the calculation in light-conegauge, only the transverse Wilson lines would contribute. Thuswhile the exact expressions are rather different in the twogauges, both may be derived from Eq. (2). Given the extentof the Wilson lines (and the issues related with analyticallycontinuing a Euclidean operator product to one that is almostlightlike separated), it appears almost impossible to evaluatethese on a finite size lattice.
However, there exists an alternative, based on the similaritybetween q and the gluon distribution function and the methodby which parton distribution functions (PDFs) are evaluatedon the lattice [31–34], i.e., using the method of operatorproduct expansions. Imagine a high-energy process, e.g., thedeep inelastic scattering (DIS) of an electron with momentumk off a single quark prepared with momentum p, at oneedge of a finite volume V which is maintained at a fixedtemperature T ∼ �QCD. At this temperature the volume willbe filed with strongly interacting matter, which at temperaturessomewhat below �QCD will be a hadronic gas and at veryhigh temperatures will be quark gluon plasma. We maintainthe chemical potential μ = 0 so that the contents have theconserved charges of the vacuum. On scattering off theelectron, the quark will produce a hard virtual quark whichwill then propagate through the medium. In vacuum such aparton would undergo a perturbative shower, spraying partonswith ever lower virtuality until the scale becomes comparableto �QCD and hadronization begins to set in. In the presenceof a strongly interacting medium the produced shower willscatter off the constituents in the medium, diffuse in transverseand longitudinal momentum, and be induced to radiate morepartons leading to a further degradation in the energy of thepart of the jet that escapes the medium.
If the medium is not larger than E/μ20, where E is the energy
of the jet and μ0 is the minimum scale below which pQCD isno longer applicable, a portion of the jet will hadronize outside
the medium. The differential cross section for any particularoutcome from such a hard scattering process can be expressedusing the standard factorized formula,
dσh = α2
kpQ4LμνdWμν, (3)
where Lμν is the usual leptonic tensor and dWμν is thedifferential hadronic tensor for the particular process ofinterest; all interactions that involve the QCD coupling g arecontained within the hadronic tensor.
Say further that in the hadronic tensor we could factorizethe initial distribution of the hard quark, the hard scattering offthe photon, and the final propagation through the medium as
dWμν =∫
dxf (x)dσ μνD({pf }). (4)
In the equation above, f (x) represents the distribution of theinitial quark; in the case of a quark inside a proton this wouldsimply be the parton distribution function. In the case of asingle quark it is simply δ(1 − x). The term σ μν represents thehard cross section for the scattering of a quark off a virtualphoton. The function D, which is a function of the set ofmeasured final state momenta {pf }, includes all final stateeffects after the hard collision of the quark with the photon.The general structure of D may be written as
D({pf }) =∑j,k
〈M|Oj |M〉〈0|Q†j |{pf }X〉〈{pf }X|Qj |0〉, (5)
where |M〉 represents the medium where the jet interacts and|{pf }X〉 represents an inclusive hadronic state containing thedetected hard momenta {pf } and other states that are not part ofthe medium. The operators Q and Q† represent the part of theprocess that occurs outside the medium and fragments to yieldthe detected “nonmedium” final state. The remaining operatorOj represents the part of the process that occurs within themedium. In a real heavy-ion collision such a distinction maybe impossible to even formulate. However, in the theoreticalscenario of a hard jet propagating through a finite medium,such a separation can be carried out order by order.
In the case of a single inclusive measured hadronic momen-tum, D would become the standard fragmentation function (ifOi = 1 there would be no medium effect; otherwise one wouldobtain the medium modified fragmentation function). Formore exclusive observables (with more specified momenta),D would represent a more complicated object [35]. We shouldmake it clear that the momenta which specify D do not needto be hadronic and may be completely partonic; in fact theparticular D that we consider is completely partonic.
In the remainder of this paper, we consider evaluating Oi
by perturbing in the weak coupling of the hard producedquark with the medium. Note that this does not assume thatthe coupling within the medium is perturbatively weak. Weencode the effect of the medium on the hard quark in termsof an infinite series of local, power-suppressed operators(suppressed by powers of the hard scale Q2). Thus Oi is
obtained as a series of local operators Oi
n and ever more
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suppressed perturbative coefficients cin/[Q2]n,
Oi =∑
n
cin
[Q2]nOi
n. (6)
While perturbation theory is valid for the interactions ofthe hard quark, it is not valid for the local operator products.Any evaluation in perturbation theory necessarily requires thespecification of a gauge and the calculations in this paper are nodifferent. Each choice of gauge results in a slightly different setof perturbative terms along with a slightly different set of localoperator products. For gauge-invariant observables such as Dthe total sum is gauge invariant. To demonstrate this however,one needs to be able to evaluate the operator products (for atleast the first couple of terms).
In all prior attempts to evaluate D, the nonperturbative sec-tor has never been evaluated exactly. In the HT scheme, whichis closest in spirit to the present discussion, the operator prod-ucts (or some combination of them) are treated as parametersof the theory. A model is assumed for how they would dependon intrinsic properties of the medium such as the temperatureT . The overall normalization is set by comparing with one datapoint. In this paper we present the first effort to estimate theseoperator products nonperturbatively on the lattice.
The primary motivation for this effort is to test if suchan approach is at all feasible. There is no attempt to beexhaustive and only the simplest process of jet broadeningis considered: the broadening of a single quark by a singlescattering with momentum exchange k⊥ in a hot medium.Dividing the mean k2
⊥ by the length of the medium yieldsthe transport coefficient q. The question that is addressedin this paper is if such an approach is at all possible. Tothis end, we calculate the perturbative part only in A− =0 gauge and the nonperturbative part in quenched SU (2)lattice gauge theory. In this sense, this paper should beviewed as a “proof of principle” of such a methodology.Issues related to renormalization on both the perturbative sideand the nonperturbative side are ignored. The evaluation ofthe perturbative coefficient functions in an alternate gauge,the computation of the modification of the shower patternof the jet, and the evaluation of the nonperturbative operatorproducts in SU (3) are left for future efforts.
We note in passing that while, in this paper, we assumedthe factorization of the hard scattering from the final statescattering, this (assumption) is not strictly necessary in such aframework. Indeed one may consider e+e−annihilation withinsuch an enclosure and calculate the modification of the back-to-back pair of jets. Depending on the choice of observable andgauge this will lead to a unique expansion in the form of Eq. (6).
III. LEADING-ORDER DERIVATION
In this section, the operator expectation D is factorizedinto a perturbative and nonperturbative part. As pointed outabove, we consider the simplest process of jet broadening atleading order in the medium. To this end, we consider thepropagation of a hard virtual quark through a hot mediumwith the quantum numbers of the vacuum. The large scaleassociated with this parton allow for the use of perturbation
theory and we compute the first perturbative contribution thatoccurs only in the presence of a medium.
Imagine a quark in a well-defined momentum state, |q〉 ≡|q+, q−, 0⊥〉, impinging on a medium, |M〉, and then exitingin the state
|q + k〉 ≡∣∣∣∣ (k2
⊥ + Q2)
[2(q− + k−)], q− + k−, �k⊥
⟩,
with the medium state absorbing this change in momentumand becoming |X〉. The quark is assumed to be spacelikeoff-shell with virtuality Q2 = 2q+q− � 0 with the negativez axis defined as the direction of the propagating quark. Ina physical situation, one would have a gluon radiated off aquark, with either the gluon or the quark spacelike off-shell(or both). The spacelike parton would be placed closer to itsmass shell by scattering in the medium. The rate of scattering iscontrolled by the transport coefficient q. To mimic this processwe have considered the very simple process of a spacelikequark scattering off the glue field in an extended medium. Thecase of an on-shell quark is included in the limit of Q2 → 0.
Consider the reaction in the rest frame of the medium. Inthis frame q0 > 0, and we have defined the z axis such thatqz < 0. In this choice of frame, for a spacelike quark we haveq+ = (q0 + qz)/
√2 � 0 and q− = (q0 − qz)/
√2 > 0. If the
z axis were chosen such that qz > 0, q+ and q− would simplyswitch roles. For a spacelike quark we have q0 � |qz|, andthis implies that q− > q+. For a jet one requires q− � q+.Alternatively stated
√|q2
0 − q2z | � q0 ∼ −qz.
The spin-color-averaged transition probability (or matrixelement) for this process, in the interaction picture, is given as
W (k) = 1
2Nc
〈q−; M|T ∗ei∫ t
0 dtHI (t)|q− + k⊥, X〉
× 〈q− + k⊥, X|T e−i∫ t
0 dtHI (t)|q−,M〉, (7)
where we have averaged over the initial color and spin ofthe quark, assuming that the medium is in a fixed state.In the case of a thermal medium one may use the densitymatrix to average out the initial state. We assume that allthis is implicitly included in |M〉. In the equation above,HI = ∫
d3xψ(x)igtaγ μAaμ(x)ψ(x) and T (T ∗) represents
time(anti-time) ordering. Expanding the exponential toleading order yields
where Aμ = taAaμ. To deal with the factors of time t and
volume V , we introduce box normalization for the quarkwave functions and later take the limit of t, V → ∞. In boxnormalization, ψ(x)|q−〉 = e−iqxu(q)/
√V , we get
W (k) = g2
2NcV
∫d4xd4yTr
{〈M| /q
2Eq
/A(y)
× Disc
[(/q + /k)
(q + k)2 + iε
]/A(x)|M〉
}e−ik(y−x). (9)
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If we shift the x and y integrations, the four volume maybe extracted (
∫d4x = tV ) and divided out by the factors in
the denominator. The mean k2⊥ that yields q has the obvious
definition
q =∑
k
k2⊥
W (k)
t, (10)
where we have summed over all values of the four-vector kwith the restriction that the final outgoing quark remain onshelland t represents the time spent by the hard quark in thethermal volume V . With the overall factor of four-volumeremoved we can take t, V → ∞.
We now demonstrate that in the limit that q goes near on-shell, i.e., q− � q+, the expression above reduces to the well-known expression for the transport coefficient q. Taking thelimit that Q2 = 2q+q− → 0 while q− → ∞, we can simplifythe Dirac trace as
〈M|Tr[/q /A(/q + /k) /A]|M〉= 8(q−)2Tr[tatb]〈M|A+
a (y)A+b (x)|M〉. (11)
The imaginary part of the propagator yields the on-shell δfunction, which may also be simplified as
δ[(q + k)2] � 1
2q− δ
(k+ − k2
⊥2q−
). (12)
Because k− has been ignored, compared to q− it may beintegrated over to yield 2πδ(y+). The k2
⊥ may be combinedwith the vector potentials to yield ∇⊥A+ � F+
⊥ . Absorbingboth factors of k⊥, we obtain an expression containing onlyfield strength tensors.
Substituting the above simplifications, one obtains
q = 4π2αs
Nc
∫dy−d2y⊥
(2π )3d2k⊥e
−ik2⊥
2q− ·y−+i�k⊥· �y⊥
× 〈M|F+,⊥(y−, y⊥)F+⊥ (0)|M〉. (13)
This is the standard definition of q. Note that nothing isspecified about |M〉, it may indeed be an arbitrary medium. If|M〉 is a thermal medium, then it must be averaged over in thesum over all initial states. Averaging with a Boltzmann weightyields
q = 4π2αs
Nc
∫dy−d2y⊥
(2π )3d2k⊥e
−ik2⊥
2q− ·y−+i�k⊥· �y⊥
× 〈n|e−βEn
ZF+,⊥(y−)F+
⊥ (0)|n〉. (14)
Note that in the above derivation, no ordering is introducedbetween the two field strength operators. The expression aboveis not gauge invariant, but is gauge covariant. This impliesthat, if one were to carry out an operator product expansion interms of local operators, one could reorganize the expansionto only contain gauge-invariant local operators. Any gaugedependence would then only be contained in the coefficientfunctions.
IV. THE OFF-SHELL REGIME ANDTHE NONPHYSICAL REGIME
In the preceding section, we considered the process of a nearon-shell quark propagating through a hot medium, at leadingorder in the scattering off the medium. In this section, thecase of a slightly off-shell quark is considered. The quarkvirtuality or off-shellness is still small compared to the energy.Once the operator products are isolated, we consider theprocess in the region of very high virtuality, of the order ofthe energy, and consider an expansion in a power series withincreasing negative powers of the virtuality.
Consider the imaginary part of the propagator in Eq. (9).In the limit where q− is very large, and q+ is vanishinglysmall, there is a pole at the point where k+ = (k2
⊥)/(2q−).In the regime where q+ � q− but q+ is not vanishinglysmall (i.e., the parton has a nonnegligible virtuality) weobtain small additive contributions to the gauge-covariantstructure derived above. In this section we consider the morephysical limit where q+q− ∼ k2
⊥ ∼ λ2(q−)2, where λ is asmall dimensionless constant. In this case, the Dirac matrixstructure is simplified by taking the trace as
Tr[/q /A(0)(/q + /k) /A(y)] = 4Aμ(0)GμνAν(y),
with Gμν = [qμ(q + k)ν + qν(q + k)μ − (q + k)qgμν] Ex-panding this out, we obtain
A(0)GA(y)
= 2q−A+(0)q−A+(y) + q−A+(0)(q+ + k+)A−(y)
+ q+A−(0)q−A+ + (q+ + k+)A−(0)q−A+(y)
+ q−A+(0)q+A−(y) + 2q+(q+ + k+)A−(0)A−(y)
− q−A+(0)k⊥A⊥(y) − k⊥A⊥(0)q−A+(y)
− q+A−(0)k⊥A⊥(y) − k⊥A⊥(0)q+A−(y)
− [q−(q+ + k+) + q+q−]
× [A+(0)A−(y) + A−(0)A+(y) − A⊥(0)A⊥(y)]. (15)
We now consider this expression in A− = 0 gauge, wherewe may drop terms that scale as Q2/q− ∼ λ2q−. This leads toa considerable simplification of the final expression:
AGA = 2q−A+(0)q−A+(y)
+ q−A+(0)k⊥,μAμ⊥(y) + k⊥,μA
μ⊥(0)q−A+(y)
− [q−(k+ + q+) + q−q+][A⊥,μ(0)Aμ⊥(y)]. (16)
The exponential phase factor is
eiφ = exp
[i
{(k2⊥
2q− − q+)
y− + k⊥,μyμ⊥
}], (17)
where the general (⊥)-four-vector implies A⊥ ≡ [0, 0, �A⊥].Using these relations, we may simplify,
2(q−)2(−kμ⊥k⊥,μ)A+(0)A+(y)eiφ(y)
= −2(q−)2∇μ⊥A+(0)∇⊥,μA+(y)eiφ(y). (18)
The next set of terms simplify as
eiφq−A+(0)k⊥,μAμ⊥(y)k2
⊥= 2(q−)2i∇⊥,μA+[q+ − i∂+]Aμ
⊥(x)
= 2(q−)2[∇⊥,μA+∂+Aμ⊥(x) + i∇⊥,μA+q+A
μ⊥(y)]. (19)
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The first term in the bracket above can be combined withEq. (18) to produce the field strength tensor at location x.There is another term similar to the one above that can becombined to form the field strength tensor at the origin. Thelast line in Eq. (16) may be reexpressed as
−2(q−)2[∂+A⊥,μ(0)∂+Aμ⊥(y) + 2iq+A⊥,μ∂+A
μ⊥(y)
−iq+∂+A⊥,μ(0)Aμ⊥ + 2(q+)2A⊥,μ(0)Aμ
⊥(y)]. (20)
The first set of terms in the equations above [Eqs. (18)–(20)]can be combined to obtain the known form that appears inthe definition of the on-shell q, i.e., 2(q−)2F+
⊥,μFμ,+⊥ . Note
that all terms in Eq. (20) are rather small [they scale asλ22(q−)2∇⊥,μA+(0)∇μ
⊥A+(y)] and thus the remaining termsmay be ignored.
We now have an expression for the transport coefficientq over a range of values of q+ where q+ � q− but is stilllarge enough that Q2 = 2q+q− � �2
QCD. We can now takethis particular operator product and consider its behavior overthe entire complex plane of q+.
We now analytically continue to the region where q+ <0 and |q+| ∼ q− � k. Consider the analytically continued,unphysical expression
Q = 4π2αs
Nc
∫d4yd4k
(2π )4eiky 2(q−)2
√2q−
× 〈M|F+⊥(0)F+⊥,(y)|M〉
(q + k)2 + iε. (21)
We introduce a new object Q to indicate that the expressionabove is not the jet transport coefficient q. The discontinuityof the above expression in the region −q− � q+ � q−corresponds to q.
In the regime where q+ ∼ q− � k, one can expand out thedenominator as
1
(Q2 − k2⊥ + 2qk)
� 1
Q2
∞∑n=0
(−2qk + k2⊥
Q2
)n
. (22)
The instances of the gluon momentum k may be replaced withderivatives. Adding, gluon scattering terms, we can convert theregular derivatives into covariant derivatives. Thus we obtaina series of gauge covariant expressions for the jet transportcoefficient:
Q = 4π2αs
Nc
∫d4yd4k
(2π )4eiky
√2q−
Q2
×〈M|F+μ⊥ (0)
∞∑n=0
(−qiD − D2⊥
Q2
)n
F+⊥,μ(y)|M〉. (23)
With all instances of k removed from the integrand (except forthe phase factor), the integrals over all components of k canbe carried out to yield four δ functions over the position y.This yields a very simple expression for q in A− = 0 gauge,in terms of local gauge-invariant operators,
Q = 4√
2π2αsq−
NcQ2〈M|F+μ
⊥
∞∑n=0
(−qiD − D2⊥
Q2
)n
F+⊥,μ|M〉.
(24)
The above expression requires some discussion. The dis-continuity in the expression above across the real axis ofq+ corresponds to the transport coefficient q when −q− �q+ � q−. For q+ ∼ q− and positive, there is another sourceof a discontinuity, from real hard gluon emission. This partis perturbatively calculable as long as Q2 = 2q+q− � �2
QCDand does not depend on any properties of the medium. In theregion where Q2 is spacelike or q+ � −�QCD there is nodiscontinuity across the real axis. Alternatively speaking, inthe deep spacelike region the internal quark line cannot goon-shell. For virtualities that are not in the deep spacelikeregion, the quark can still absorb a gluon from the mediumand go on-shell and there will be a discontinuity.
V. DISPERSION RELATIONS
In the preceding section, the expression for q was gen-eralized to the region of (a physically realizable) nonzerovirtuality and then considered in the region of (unphysical)very high virtuality. In the current section the two expressionsare related via dispersion relations in the complex q+ plain.The expansion in the unphysical region [Eq. (24)] is used toestimate the value of q in the physical region.
In order to evaluate q = Disc[Q] for q+ ∼ λ2q−, we usethe method of dispersion relations: We evaluate a similarintegral in a region of the q+ complex plain where there isno discontinuity and use methods of contour integration torelate the evaluated integral to q.
Consider the integral
Im =∮
dq+
2πi
Q(q+)
(q+ + Q0)m, (25)
where Q0 is large and positive. The contour is taken as asmall counterclockwise circle around the point q+ = −Q0.The residue of this integral is given as
Im = dm−1
dm−1q+ Q(q+)
∣∣∣∣q+=−Q0
. (26)
While this analysis can be carried out for arbitrary m, weconsider, for definiteness, the case of m = 1. In the limit where|q+| � λQ, we obtain Eq. (24) with Q2 replaced by −2q−Q0,i.e.,
I1 =4√
2π2αs〈M|F+μ⊥
∑∞n=0
(−qiD−D2⊥
2q−Q0
)nF+
⊥,μ|M〉Nc2Q0
. (27)
Since q+, q− � k2⊥, the above operator relation in simplified
as
I1 = 4√
2π2αs
Nc2Q0〈M|F+μ
⊥
∞∑n=0
(−iD+
2Q0+ −iD−
2q−
)n
F+⊥,μ|M〉,
= 4√
2π2αs
Nc2Q0〈M|F+μ
⊥
∞∑n=0
n∑m=0
(nm
)(−iD+
2Q0
)m
×(−iD−
2q−
)n−m
F+⊥,μ|M〉,
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= 4√
2π2αs
Nc2Q0〈M|F+μ
⊥
∞∑m=0
1
m!
(−iD+
2Q0
)m
×∞∑
k=0
(m + k)!
k!
(−iD−
2q−
)k
F+⊥,μ|M〉. (28)
We can now deform the contour and evaluate it over thebranch cut from q+ > −λ2Q to q+ → ∞. This yields
I1 = 4π2αS
Nc
∫dq+ d4yd4k
(2π )4eiky
δ(k+ + q+ − k2
⊥2q−
)2q−
× 〈M|F+μ(0)F+μ,(y)|M〉
(q+ + Q0)
=∫ λ2Q
−λ2Q
dq+ q(q+)
q+ + Q0+
∫ ∞
0dq+V (q+). (29)
The second term in the equation above refers to the contri-bution to the operator above from vacuum gluon radiation,i.e., the Bremsstrahlung radiation of gluons from an off-shellquark. As such, it contributes only in the region where thevirtuality of the incoming quark is timelike and is independentof the temperature of the medium. Thus for a fixed T thesecond term above is a constant, while the first depends on thetemperature of the medium.
The limits on the first integral in the equation above allowfor a simple expansion of the denominator. The factor Q0 ∼ Qis much larger than the q+ ∼ λ2Q in this region and thus weobtain the following much simplified relation,
∫dq+ q(q+)
Q0
∞∑n=0
[−q+
Q0
]n
� I1 −∫ ∞
0dq+V (q+). (30)
To obtain q, a general functional form in the vicinity of−λ2Q � q+ � λ2Q must be used. We start with the assump-tion that q at a fixed q− is a slowly varying function of q+. Thisallows us to use a truncated Taylor expansion for q [We shouldpoint out that using the first few terms of the Taylor expansionis, in itself, an assumption regarding the functional form ofq(q+)]. To provide a simple illustration of the procedure, wetake only three terms; in the final numerical results we onlyuse those results where the first term greatly dominates overall subsequent terms (note that an arbitrary number of termsin the Taylor expansion may be retained for a more accuratedetermination of q),
q(q+) = q + q ′q+ + q ′′(q+)2
2. (31)
In the above equation q ′ = ∂q/∂q+|q+=0.Using the above truncated Taylor expansion we obtain
I1 =∫ Q+
−Q+ dq+[q + q
(q+Q0
)2 − q ′ (q+)2
Q0 + q ′′ (q+)2
2
]Q0
+∫ ∞
0dq+V (q+) = 2qQ+
Q0+ q ′′(Q+)3
3Q0
− q ′2(Q+)3
3Q20
+ q2(Q+)3
3Q30
+ q ′′ (Q+)5
5Q30
. (32)
In the equation above, Q+ represents the limit of integrationover q+ for the jet. For a jet with maximum virtuality μ2 and(−) momentum q−, Q+ = μ2/(2q−). One may now simplycompare with the expression for I1 from Eq. (28) and equatethe vacuum subtracted coefficients of Qn
0.The methodology outlined above can be made even more
precise and straightforward by setting a definite value forQ0 = q−. While this will readjust the relative importance ofthe various terms in the series it allows for a simpler set ofoperators that need to be evaluated numerically. This simplifiesI1 in Eq. (28) to
I1 = 2√
2π2αs
Ncq− 〈M|F+μ⊥
∞∑n=0
(−iD0
q−
)n
F+⊥,μ|M〉 (33)
and similarly simplifies Eq. (32) with Q0 replaced by q−. Fora virtuality μ2 such that �2
QCD � μ2 � (q−)2, we can definea q+ or virtuality averaged q as
ˆq(Q+)2Q+ =∫ Q+
−Q+dq+q(q+) � 2qQ+ + q ′′(Q+)3
3, (34)
where the second line is only valid in the limit that q is aslow function of q+ (or alternatively stated Q+ � q−). Wecan obtain an estimate of this by studying the second term inEq. (33). If this term is comparable to the first term then theabove approximation is no longer valid. If this term is small,then one may obtain a good estimate of q from just the firstterm in the series in Eq. (33). In the subsequent section theforms of the operators and their evaluation on the lattice arediscussed.
VI. LATTICE CALCULATIONS
In the preceding sections, the jet transport parameter q,as obtained in the physical regime of jet momenta q+ ∼λ2q− � q−, was related via dispersion relations to a seriesof local operators in an unphysical regime where q+ = −q−.The availability of a series of local operators suppressed bypowers of the hard scale q− allow for the calculation of suchnonperturbative operator products on the lattice. In essence,our task is to compute the finite temperature Minkowski spacecorrelator,
D>(t) =∑
n
〈n|e−βHO1(t)O2(0)|n〉, (35)
in the limit where t → 0. In the equation above, β isthe inverse temperature (β = 1/T ), H is the Hamiltonianoperator, and |n〉 represents an eigenstate of the Hamiltonian.Using the standard relations of the imaginary time formalismof finite temperature field theory, we can relate the Minkowskicorrelator with the Matsubara correlator in Euclidean space,
D>(−iτ ) = �(τ ) = Tr[e− ∫ β
0 dτH (τ )O1(τ )O2(0)]. (36)
For the case where there are no time derivatives in O1 and O2,the above equation yields D>(−iτ ) = iNt �(τ ) for a total ofNt time derivatives in D>(t). As a result, we obtain the simplerelation that
D>(t = 0) = iNt �(τ = 0). (37)
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Using the above relation, the local operator products inMinkowski space may be obtained from the local operatorsin Euclidean space.
In the following, we list out the operators that must beevaluated and reexpress them in a form where they may beeasily calculated on the lattice. In this first exploratory attempt,the calculation is carried out for an SU (2) gauge theory on aspace-temperature lattice in the simplified quenched approx-imation. Quarkless SU (2) possesses a negative β functionas in full QCD. Because issues of higher-order contributionsand renormalization were ignored in the perturbative sector,renormalization is dealt with in a very simplified fashion, in thenonperturbative sector. The extension to more sophisticatedsimulations in quenched (or unquenched) SU (3) is left forfuture efforts. In defense of the current effort, we point outthat, in the context of jet transport coefficients in heavy-ioncollisions, quenched calculations may provide a very realisticestimate, as the early dense plasma is believed to be gluondominated.
In the language of links, the field strength tensor taF aμν may
be expressed as
Fμν ≡ taF aμν = Uμν − U †
μν
2iga2L
, (38)
where Uμν represents a plaquette in the μν plane and aL isthe lattice spacing. Similarly, terms with a covariant derivativemay be expressed as
D4Fμν(x)= Fμν(x4+aL, �x) − U4(x4, �x)Fμν(x4, �x)
aL
, (39)
where U4 represents a gauge link in the four-direction. In thispaper, we have only used the right derivative as we seek onlyan order of magnitude estimate of terms with a time derivative,as argued below.
The first operator to be evaluated is
〈M|F+μ⊥ F⊥+
μ |M〉 =∑
〈n|e−βH F+μ⊥ F⊥,+
μ |n〉≡
∑e−βEn〈n|F+μ
⊥ F⊥,+μ |n〉, (40)
where |n〉 represents an eigenstate of the full Hamiltonian. Wedo not indicate the location of the two F field strength tensorinsertions because both are at the same location.
We now discuss the rotation of the operator products toEuclidean space. This involves the following two rotations:
x0 → −ix4 and A0 → iA4 ⇒ F 0i → iF 4i . (41)
As a result,
〈[F 01 + F 31][F 01 + F 31]〉 → 〈F 31F 31〉 − 〈F 41F 41〉. (42)
In the above equation, we have ignored terms such as (F 31F 41),because their vacuum subtracted contributions turn out to berather small in the region where we attempt to estimate q [thisis plotted in Fig. (4) and is discussed below].
In the following, we first discuss the lattice calculation ofthe operator
∑i=1,2(F 3iF 3i − F 4iF 4i). The reader will note
that, with the addition of the extra term (F 21F 21 − F 43F 43),this will become the operator for the entropy density (up tonormalization constants). For an isotropic lattice, one could
q
k
q
k
FIG. 1. A quark scattering off a gluon in medium |M〉.
even estimate the value of∑
i=1,2(F 3iF 3i − F 4iF 4i) as 2/3times the entropy density. In our calculation, the jet travels inthe z direction. As a result, the entire problem (perturbativeand nonperturbative sectors) is not isotropic, even though thelattice part of the calculation is isotropic. Also the remainderof the operators required for the calculation of q have nosimple relation with well-known operators. Hence, we directlyevaluate the operator mentioned above and do not try toestimate its value from the known results of the entropy density.The notion that q may be proportional to the entropy densityhas been prevalent in jet modification phenomenology and hasbeen used in various calculations of jet modification, e.g., seeRef. [36].
We report results on a (4 × nt )3 × nt lattice where nt isvaried from 3 to 6. Note that finite temperature calculations aremeant to be carried out in the limit where nt � ns . For nt = 2we also carried out a calculation with ns = 12 and ns = 8,these are not presented because they show very little variationwith ns for a fixed nt . We did not repeat the calculation withsmaller values of nt and the largest value of ns = 24 becausethe results for nt = 2 and 3 do not seem to be have anydependence on ns for ns > 12.
For this first attempt we use the Wilson gauge action forSU (2) [37,38]. The scale (or lattice spacing) is set on the latticeusing two different renormalization group formulas. The firstis based on the two-loop perturbative Renormalization Group(RG) equation for the string tension [37,38], which yields thefollowing formula for the lattice spacing,
aL = 1
�L
(11g2
24π2
)− 51121
exp
(−12π2
11g2
), (43)
where g represents the bare lattice coupling and �L representsthe one dimensionfull parameter on the lattice. Comparingwith the vacuum string tension, we have used �L = 5.3 MeV.For a lattice at finite temperature or one with nt � ns , thetemperature is obtained as
T = 1
ntaL
. (44)
The results for the field-strength-field-strength correlation∑i=1,2(F 3iF 3i − F 4iF 4i)/2 with this choice of formula for
the lattice spacing are presented in Fig. 2. The resultingcorrelation is scaled by T 4 as obtained from the formula above.
The formula above does not provide the best means to setthe scale on the lattice at finite temperature [39–41]. However,
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FIG. 2. (Color online) The temperature dependence of the localoperator 〈F +iF +i〉, scaled by T 4 to make it dimensionless. The latticespacing is set using Eq. (43). The expectation of the operator productshows a transition in the vicinity of T ∼ 250–350 MeV. See text fordetails.
it constitutes a simple formula that is very easy to use. Wealso set the scale using a nonperturbative approach as outlinedin Ref. [42] where the formula for the lattice spacing isexpressed as the product of that obtained from Eq. (43) and anonperturbative function λ(g2) that has been dialed to ensurethat Tc/�L is independent of g2 (comparing with a vacuumstring tension of
√K = 400 MeV, this procedure yields �L =
10.3 MeV). The results for the field-strength-field-strengthcorrelation
∑i=1,2(F 3iF 3i − F 4iF 4i)/2 with this next choice
of formula for the lattice spacing are presented in Fig. 3.Again, the correlation results are scaled by T 4. In both plots,gauge configurations are generated using a simple heat bathalgorithm [38]. Calculations consist of 5000 heat bath sweeps
FIG. 3. (Color online) Same as Fig. 2, except for the use ofnonperturbative RG factors (from Ref. [42]) to evaluate the latticespacing. See text for details.
for each data point. The error represents the standard error asdefined in Ref. [43].
Figures 2 and 3 represent our results for the calculation ofthe uncrossed operator product
∑i=1,2(F 31F 31 − F 41F 41)/2
as a function of the temperature, as measured on the lattice.We find that while the calculations with nt = 3 and 4 do notshow scaling with lattice size, the calculations with nt = 4, 5,and 6 show good scaling especially in Fig. 3 where we clearlynote the independence of the transition temperature on thelattice size. The transition is around Tc ∼ 250–350 MeV forthe curves with perturbative renormalization, while it is aroundTc ∼ 150 MeV for the curves with the nonperturbative factor,with the same choice of �L. Thus, while the behavior aroundthe transition is sensitive to the choice of how the scale is set onthe lattice, the behavior of the correlation at a temperature T �1.25 − 2Tc, or in more definite terms T > 400 MeV, seems tobe unchanged, i.e., the correlator yields the value of ∼0.5T 4.
Given the behavior around the transition temperature alongwith the larger fluctuations in this region, we focus on dis-cussing the value of the field-strength-field-strength correlatorat temperatures above 1.25–2Tc where the expectation for thecorrelator has begun to scale with T 4. The goal is to evaluatethe series of terms outlined in Eq. (33) in this region. The plotsin Figs. 2 and 3 represent the evaluation of a part of the firstcorrelator in this series, as discussed in Eq. (42). The remainingterms are the cross terms 〈F 3xF 4x + F 3yF 4y〉, which we haveso far neglected. We now present a computation of theseterms in Fig. 4 for the case of nt = 3. The plot represents thedifference of the finite temperature and vacuum calculationsof the same operator product scaled by T 4, with the latticespacing set by Eq. (43). This plot should be compared withFig. 2. While the crossed correlator is of the same size as theuncrossed correlator, measured in Fig. 2, in the phase transitionregion, above a temperature of T = 400 MeV, the crossedcorrelator is rather small compared to the uncrossed correlator∑
i=1,2(F 31F 31 − F 41F 41)/2.
FIG. 4. The temperature dependence of the local operator〈F 3xF 4x + F 3yF 4y〉 (thermal contribution minus vacuum contribu-tion) scaled by T 4 to make it dimensionless. The lattice spacing is setusing Eq. (43).
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FIG. 5. (Color online) Unscaled expectation of the lattice-size-independent correlator
∑i=1,2 a4
L(F 31F 31 − F 41F 41)/2 at finite tem-perature (red squares) versus expectation in vacuum (black circles)as a function of β = 4/g2 (g is the bare lattice coupling). The plot isfor nt = 3 and ns = 12.
In the calculation of the cross term, we have considered thedifference of the thermal and vacuum expectation values ofthe operator product. For the case of nt = 3, as presented inFig. 4, this is not a very time intensive calculation. However,the calculation of the vacuum expectation values becomesincreasingly numerically intensive with growing nt for boththe crossed and uncrossed operator product. For the caseof nt = 6, the calculation of the vacuum expectation valuehas become prohibitively difficult. As a result, in Figs. 2and 3, only the thermal expectation value of the uncrossedcorrelator is plotted. This engenders a small systematic errorbecause the uncrossed correlators are a difference of twooperator products, both of which have vacuum expectationvalues of similar size. As a result, the vacuum expectationvalues of
∑i=1,2(F 3iF 3i − F 4iF 4i)/2 are small compared to
the thermal expectation, particularly in a region far abovethe transition. To illustrate the small size of the vacuumexpectation values, we plot the thermal expectation of theuncrossed operator product as a function of the bare couplingon the lattice, i.e., without any scaling relation for the latticespacing. This is plotted for the case of nt = 3 in Fig. 5, for thecase of nt = 4 in Fig. 6, and for the case of nt = 5 in Fig. 7.As mentioned above, the calculation of the expectation of theoperator in the vacuum for the case of nt = 6 has turned outto be prohibitively difficult with current resources. As a result,this is not presented. To plot consistent results, the plots inFigs. 2 and 3 do not contain any vacuum subtraction.
Careful observation of all these curves indicates that, atT > 1.25Tc, the vacuum expectation of the operator product∑
i=1,2(F 3iF 3i − F 4iF 4i)/2 is considerably smaller than thethermal expectation, and so is ignored in the remainder ofthe discussion. We reiterate that, had the focus been on thelow-temperature region at and below Tc, one would not beable to ignore the vacuum expectation.
FIG. 6. (Color online) Same as Fig. 5, but with nt = 4 and ns =16.
In the preceding paragraphs, we outlined the neglect of avariety of corrections to the leading operator product that hasto be evaluated to calculate the jet quenching coefficient q.These corrections tend to be large at lower temperatures, atand below Tc. We now consider the behavior of the series athigher temperatures. In our view, the most important correctionin this region is brought on by the higher derivative terms inEqs. (32) and (33). To estimate the value of q solely from thefirst term in the expansion in Eq. (33) requires that the higherderivative terms be small. As an estimate of the size of theseterms, we compare the modulus of the expectation of the firsttwo operators in Eq. (33), for the case of nt = 6 in Fig. 8, wherethe lattice spacing is set using Eq. (43), and in Fig. 9, wherethe lattice spacing is set using the nonperturbative approachof Ref. [42]. The next operator in the series is of the form∑
i=1,2 F 3i D4
q− F 3i − F 4i D4
q− F 4i . We plot it both with (greendiamonds) and without (red diamonds) the large factor of q−in the denominator. We reiterate again that for this expansion
FIG. 7. (Color online) Same as Fig. 5, but with nt = 5 andns = 20.
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FIG. 8. (Color online) The temperature dependence of absolutevalues of the the local operator 〈F +iF +i〉 and the second operatorproduct 〈[F +i i∂4F +i]〉, both with (green diamonds) and without (reddiamonds) the large factor of q− in the denominator. The latticespacing is set using Eq. (43). See text for details.
to be a useful estimate of q, there must be a large jet scale inthe problem. The results in Fig. 8 are for q− = 20 GeV.
The plots in Figs. 8 and 9 demonstrate that for temper-atures below T = 600 MeV, the expectation of the operator[F+i i∂4F+i]/q− for q− ∼ 20 GeV is less than 25% of thefirst operator product (in either method of determinationof the lattice spacing). It is remarkable that in the moreaccurate method of determining the lattice spacing, using thenonperturbative method of Ref. [42], the leading operator isabout a factor of 10 larger than the first correction in thevicinity of T ∼ 600 MeV. Based on the plots in Figs. 8 and 9,for temperatures below 600 MeV, for q− ∼ 20 GeV, we may
FIG. 9. (Color online) Same as Fig. 8, except for the use ofnonperturbative RG factors (from Ref. [42]) to evaluate the latticespacing. See text for details.
obtain an estimate of the transport coefficient q from only theleading term in this lattice calculation.
As pointed out in the prior discussion of Figs. 2–7, thecorrections from the vacuum expectation of the operatorproduct, the uncertainty from scale setting, and the largerfluctuation around the transition are small enough only for T >400 MeV. Thus one can extract q from such a calculation onlyin the range 400 MeV < T < 600 MeV. This range coincideswith the highest temperatures reached at the RHIC and theLHC and thus will allow future, more sophisticated, effortsto compare meaningfully with the values of q obtained fromphenomenological analysis of the RHIC and LHC data. Thisconstitutes the primary result of the current manuscript: thedemonstration that the framework developed in Secs. III, IV,and V can be used to obtain reliable estimates of jet transportcoefficients in a hot medium. Of course, comparisons withexperiment will require both a more sophisticated perturbativeanalysis and a much more developed lattice calculation.
VII. ESTIMATING q AND CONCLUDING DISCUSSIONS
In this concluding section, we attempt a simple-mindedextraction of the jet transport coefficient q from the latticecalculation outlined above. We would like to clearly pointout that what follows is, for the most part, a hand wavingestimate. Later calculations, which will involve the partonbeing produced far off its mass shell and radiating gluons asit propagates, will involve many more issues in the extractionof q. Recall that our calculation required that the hard quarkmoves through the medium without undergoing any radiation.This constrains the highest virtuality that the quark maypossess for such an approach to make sense. In a future effort,partons with a higher initial virtuality will be considered. Thesewill undergo radiative splitting in the medium and may showsensitivity to a somewhat different series of operator products.
We choose the region around the third last point in the〈F+iF+i〉 curve in Figs. 2 and 8. This corresponds to atemperature of T � 400 MeV, which is in between the toptemperature reached at the RHIC and LHC collisions. AtT � 400 MeV, 〈F+iF+i〉 = 0.01 GeV4. Also we considera lattice with a length given by 4 × ntaL = 4/0.4 GeV−1 =10 GeV−1. This states that the maximum virtuality of a jet (withq− = 20 GeV) that traverses such a length without undergoingradiation is given as μ2 = E/L = 20/10/
√2 � 1.4 GeV2.
Thus Q+ = 1.4/40 GeV. With these estimates, we obtain
ˆq = 2√
2π2αs(μ2)
Nc2Q+q− 〈M|F+iF+i |M〉. (45)
Using αs(1.4 GeV2) = 0.375 [44], we obtain ˆq =0.186 GeV2/fm for an SU (2) quark traversing a quenchedSU (2) plasma. In most phenomenological estimates onequotes the q of the gluon. If the above calculation were donefor an SU (2) gluon, the q would differ only by the overallCasimir factor of CA/CF = 2N2
c /(N2c − 1) = 8/3, yielding
qG = 0.5 GeV2/fm, at T = 400 MeV.In future efforts, the calculation will be extended to higher
statistics runs, along with a more careful treatment includingthe crossed correlators, to evaluate the q across the phase
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CALCULATING THE JET QUENCHING PARAMETER q . . . PHYSICAL REVIEW C 87, 034905 (2013)
transition. The next step is to evaluate the required operatorproducts for a realistic jet which starts at a higher virtualityand undergoes radiative splitting in the medium. In such acalculation, the range of operators that will need to be evaluated[i.e., the number of terms in the series of Eq. (33) that need tobe retained] will depend on the particular parton in the shower,in particular on that parton’s energy and virtuality.
Finally, to be of use to jet modification at the RHIC and theLHC, the calculation will have to be extended to unquenchedSU (3). This will involve a nontrivial extension, not only dueto the increase in the level of computational difficulty butalso due to the issues arising from the larger gauge group.Beyond these extensions, more sophisticated renormalizationfactors will have to be introduced, and better means to setthe lattice spacing will have to be used. At this stage we mayonly set suggestive limits on such a future estimation: thequenched SU (2) calculation has three colors of gluons as thefundamental fields in its Lagrangian, whereas there are eightcolors of gluons in quenched SU (3), along with three colorsof quarks and antiquarks in the unquenched SU (3) calculation(note that even if the plasma were completely perturbative,quarks would contribute differently to the calculation of qthan gluons [45], or rather the lattice calculation could changeconsiderably with the introduction of dynamical fermions).Ignoring such subtleties, assuming two flavors of light quarks,and assuming a simple scaling law with number of fields inthe Lagrangian, we estimate that the full q at the RHIC would
lie in the range
q(T = 400 MeV) = 1.3 GeV2/fm to 3.3 GeV2/fm. (46)
(If we had instead used Figs. 3 and 9 to estimate q wewould have obtained a range from 0.9 to 2.3 GeV2/fm.) Weshould point out that while the above estimate is very specificto a particular range of q+, q− of the propagating parton,the estimate obtained from phenomenological analysis of theRHIC collisions is an average over a wide range of partonenergies and virtualities. Despite the many shortcomings ofthe above calculation, we find the very encouraging result thatour estimate for q at T = 400 MeV is comparable with thatextracted from phenomenological analysis of the RHIC andLHC data [36,46].
ACKNOWLEDGMENTS
The author thanks J. Drut, C. Gale, S. Gavin, U. Heinz,Y. Kovchegov, B. Muller, A. Schafer, and J. Shigemitsu forhelpful discussions. This work was supported in part by theNational Science Foundation under Grant No. PHY-1207918.Part of this work was carried out while the author wasemployed at Ohio State University, where it was supportedin part by the US Department of Energy under GrantsNo. DE-SC0004286 and (within the framework of the JETCollaboration) No. DE-SC0004104.
[1] K. Adcox et al., (PHENIX Collaboration), Nucl. Phys. A 757,184 (2005).
[2] J. Adams et al., (STAR Collaboration), Nucl. Phys. A 757, 102(2005), arXiv:nucl-ex/0501009.
[3] J. Schukraft, J. Phys. Conf. Ser. 381, 012011 (2012).[4] P. Steinberg (ATLAS Collaboration), arXiv:1110.3352.[5] B. Wyslouch (CMS Collaboration), J. Phys. G 38, 124005
(2011).[6] N. Armesto, C. A. Salgado, and U. A. Wiedemann, Phys. Rev.
D 69, 114003 (2004).[7] C. A. Salgado and U. A. Wiedemann, Phys. Rev. D 68, 014008
(2003),.[8] U. A. Wiedemann, Nucl. Phys. B 588, 303 (2000).[9] P. Arnold, G. D. Moore, and L. G. Yaffe, J. High Energy Phys.
06 (2002) 030.[10] S. Jeon and G. D. Moore, Phys. Rev. C 71, 034901 (2005).[11] G.-Y. Qin, J. Ruppert, C. Gale, S. Jeon, and G. D. Moore, Phys.
Rev. C 80, 054909 (2000).[12] M. Gyulassy, P. Levai, and I. Vitev, Nucl. Phys. B 594, 371
(2001).[13] M. Gyulassy, P. Levai, and I. Vitev, Phys. Rev. Lett. 85, 5535
(2000).[14] M. Gyulassy, P. Levai, and I. Vitev, Phys. Lett. B 538, 282
(2002).[15] X.-F. Guo and X.-N. Wang, Phys. Rev. Lett. 85, 3591 (2000).[16] X.-N. Wang and X.-F. Guo, Nucl. Phys. A 696, 788 (2001).[17] A. Majumder and B. Muller, Phys. Rev. C 77, 054903
(2008).
[18] A. Majumder, R. J. Fries, and B. Muller, Phys. Rev. C 77, 065209(2008).
[19] A. Majumder, Phys. Rev. D 85, 014023 (2012).[20] S. S. Gubser, Phys. Rev. D 74, 126005 (2006).[21] S. S. Gubser, S. S. Pufu, F. D. Rocha, and A. Yarom,
arXiv:0902.4041.[22] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe,
J. High Energy Phys. 07 (2006) 013.[23] J. Casalderrey-Solana and D. Teaney, J. High Energy Phys. 04
(2007) 039.[24] A. Majumder and M. Van Leeuwen, Prog. Part. Nucl. Phys. A
66, 41 (2011).[25] N. Armesto, B. Cole, C. Gale, W. A. Horowitz, P. Jacobs et al.,
Phys. Rev. C 86, 064904 (2012).[26] E. Braaten and R. D. Pisarski, Phys. Rev. Lett. 64, 1338 (1990).[27] E. Braaten and R. D. Pisarski, Nucl. Phys. B 337, 569 (1990).[28] W. Horowitz, AIP Conf. Proc. 1441, 889 (2012).[29] A. Idilbi and A. Majumder, Phys. Rev. D 80, 054022 (2009).[30] M. Garcia-Echevarria, A. Idilbi, and I. Scimemi, Phys. Rev. D
84, 011502 (2011).[31] A. S. Kronfeld and D. M. Photiadis, Phys. Rev. D 31, 2939
(1985).[32] G. Martinelli and C. T. Sachrajda, Phys. Lett. B 196, 184 (1987).[33] G. Martinelli and C. T. Sachrajda, Nucl. Phys. B 316, 355 (1989).[34] G. Martinelli and C. T. Sachrajda, Phys. Lett. B 217, 319 (1989).[35] A. Majumder and X.-N. Wang, Phys. Rev. D 70, 014007 (2004).[36] A. Majumder and C. Shen, Phys. Rev. Lett. 109, 202301
