The nonperturbative contribution to asymptotic masses
PPrepared for submission to JHEP
The nonperturbative contribution to asymptoticmasses
Guy D. Moore, a Niels Schlusser a,b a Institut f¨ur Kernphysik, Technische Universit¨at DarmstadtSchlossgartenstraße 2, D-64289 Darmstadt, Germany b Department of Physics & Helsinki Institute of PhysicsP.O. Box 64, FI-00014 University of Helsinki
E-mail: [email protected],[email protected]
Abstract:
In gauge theories, charged particles obey modified dispersion due to mediuminteractions (forward scattering), leading at high energies E ≥ T to an asymptotic mass-squared m ∞ . We calculate the infrared part of this mass nonperturbatively for the theoryof the strong interactions, QCD, through a lattice treatment of its low-energy effective de-scription, Electrostatic QCD (EQCD). Incorporation of these results into a nonperturbativedetermination of the effective thermal mass will require a still-incomplete next-to-leadingorder perturbative matching of this quantity to full QCD. Keywords: quark-gluon plasma, dimensional reduction, effective theories, kinetic theory,lattice gauge theory a r X i v : . [ h e p - l a t ] S e p ontents The Quark-Gluon-Plasma (QGP) is an exotic state of matter created at collider facilities[1–4] and delivering important insights into the nature of strongly interacting matter. Theexperimental effort to measure the features of the QGP with the highest possible preci-sion goes hand in hand with the theoretical goal of giving precise predictions for these verymeasurements. The properties of the QGP can in essence be divided into two classes: ther-modynamics and transport properties. Typical examples for the former are the pressure,perturbatively determined up to O ( g ln g ) [5–10] and investigated nonperturbatively onthe lattice [11, 12], and charge susceptibilities and other quark-number correlations, whichare well investigated on the lattice [13]. Since thermal equilibrium is time-translation in-variant, these results can be derived in Euclidean time, which simplifies the calculationsconsiderably. In contrast, the calculation of transport properties usually requires an ex-plicit treatment of Minkowski time, which is conceptually more demanding. Nevertheless,there has been significant progress in recent years, for instance for the shear viscosity ηs [14, 15] or the thermal photon production rate [16, 17].Anther interesting transport phenomenon to investigate is how the medium modifiesa jet , whose theoretical treatment is for example summarized in [18]. For our purpose, jets– 1 –re strongly interacting particles that carry a large momentum and move at (almost) thespeed of light. Because of the large momentum scale, asymptotic freedom of QCD suggeststhat the strong coupling constant g is small [19, 20], and that they can be well describedby an expansion in powers of g . In traversing through the medium, however, they receivehighly nontrivial modifications [21, 22], pictured by a large number of soft bumps betweenjet and medium constituents. This interaction eventually leads to jet broadening , i.e. abroadening of the jet’s transversal momentum distribution. Related to that, jet quenching refers to a jet losing energy to the medium [23, 24], predominantly via stimulated emissiondue to multiple soft scatterings with medium constituents.A quantity that influences both phenomena is the asymptotic mass . It was alreadyfound by Klimov and by Weldon that massless particles, be it gluons [25, 26] or fermions[27, 28], tend to follow the dispersion relation of massive particles ω p = p + m p (1.1)with their mass m p depending on the momentum p and generated by the interaction ofthe particle with the medium (essentially, forward scattering). At large momenta, theparticle’s velocity in the plasma frame approaches the speed of light v → (1 , ˆ p ), and thisthermal mass approaches a constant which we will call the asymptotic mass m ∞ . At theleading perturbative order, m ∞ can be computed from the imaginary part of the transverseHard-Thermal-Loop-resummed self-energy Π T near the light cone [29, 30]. The gluon’sasymptotic mass, for instance, reads [31] m ∞ ≡ Π T ( k = k ) = m m = (2 N c + N f ) g T / Z g and afermionic contribution Z f : m ∞ = g C R ( Z g + Z f ) , (1.3)with the (running) gauge coupling g , and the Casimir operator of the representation C R ofthe particle that makes up the jet. The two condensates are [32, 33] Z f ≡ d R (cid:28) ψ v µ γ µ v · D ψ (cid:29) (1.4) Z g ≡ − g d A (cid:28) v σ F σµ v · D ) v ρ F ρµ (cid:29) , where d A = N c − d R is the dimension of the representation R of the fermion, for instancethe fundamental representation of SU (3) for quarks in QCD, and the angular bracketsrepresent an average over all thermal fluctuations. Note that our definition of the QCDfield strength F σµ is the geometrical one, F µν = i [ D µ , D ν ] with D µ = ∂ µ − iA µ ; g is– 2 –bsorbed into the coefficient on the field strength in the action, S = (cid:82) g Tr F + . . . . Thesecondensates represent the contribution of forward scattering in the medium, with 1 / ( v · D )or ( v · D ) accounting for the lightlike (eikonalized) propagation through the medium and¯ ψ, ψ or v σ F σµ representing an interaction with a medium excitation. The leading-order,long-distance approximation to these condensates reproduce the forward-scattering part ofthe hard thermal loop effective action along the light cone.The gauge part of (1.3) can be related to an integral over a correlator in position space[15] Z g = − g d A (cid:90) ∞ d x + x + (cid:104) v k µ F µνa ( x + , , ⊥ ) U ab A ( x + , , ⊥ ; 0 , , ⊥ ) v k ρ F ρb ν (0 , , ⊥ ) (cid:105) , (1.5)where the x + integration and the modified adjoint Wilson line U ab A arise from 1 / ( v · D ) .Locations in space-time are described by three-vectors ( x + , x − , x ⊥ ) in light-cone coordi-nates. Because the jet is highly relativistic, we have x − = t − z = 0 in the plasma frame.The problem therefore effectively reduces to three dimensions, two in the transverse plane x ⊥ , where rotational invariance is present, and a coordinate that encompasses the elapsedtime t for the jet as well as its covered distance z in the medium x + = ( t + z ) / ≡ L .We use the convention that the z-component of k is the jet’s direction of propagation, so v = (1 , , , Figure 1 . Pictorial motivation for the form of the correlator (1.5).
At this point, we would like to briefly build up some intuition for the specific formof (1.5). We can distinguish two sorts of jet-medium interactions. First, there is truescattering, where the transverse momentum of the jet particle changes; this is describedby the transverse momentum broadening kernel C ( b ⊥ ) [22]. But there is also forwardscattering, where the jet particle temporarily changes direction before scattering back intoits original state (or more technically, a scattering creates an amplitude in a differentmomentum state, which subsequently scatters again to return to the original momentum,introducing a phase shift). Such forward scattering is what causes a dispersion correction.It can be understood qualitatively as the jet particle making a scattering-induced detour, aspictured in Fig. 1. The sum of all such forward scattering possibilities induces an effectivemass term [23]. – 3 –elativistic kinematics tells us the length of the detour (cid:96) = L + F T m (cid:48) γ ( v ) ≡ L + ∆ (cid:96) , (1.6)where m (cid:48) is a dummy jet mass that will eventually drop out of the calculation, γ ( v ) is thegamma-factor of the jet at velocity v , and T is the time it takes for the jet to cover thedistance L . If we consider this problem from a frame of reference in which we do not takeinto account the additional detour, the kinematic invariant reads p µ p µ = m (cid:48) γ ( v ) (cid:18) − (cid:96) − ∆ (cid:96) T (cid:19) T → (cid:96), m (cid:48) → −−−−−−−→ − F (cid:96) ≈ − F L (1.7)in the ultra-relativistic limit, in which also the contribution of longitudinal forces to theeffective mass vanish. Consequently, the (transversal) gauge force must appear twice in thecorrelator resulting in the mass. Furthermore, a full quantum treatment requires a sumover all possible paths instead of just considering one as in Fig. 1. Hence, our estimate forthe effective mass yields (cid:88) L F L → (cid:90) d L LF ∼ m ∞ , (1.8)reflecting the qualitative features of (1.5).The state-of-the-art perturbative result for Z g at next-to-leading-order (NLO) [33] is Z g = Z LOg + δZ g = T − T m D π . (1.9)The Debye mass [26], in turn, depends on the coupling g : in QCD ( N c = 3) it reads m = g T (cid:18) N f (cid:19) + O ( g ) . (1.10)Formally the NLO correction is therefore suppressed by a single power of the coupling g .Numerically, it is not suppressed even at T = 100 GeV temperatures, indicating a verypoor convergence for the perturbative series. This is a familiar situation for perturbativecalculations in hot QCD, and as usual, it arises because there are large corrections arisingfrom the large-occupancy, infrared sector of QCD. Specifically, the convergence of theperturbative series is spoiled by the high occupancy of the gluon 0-Matsubara-frequencymode. When this mode circulates in a loop, it introduces an additional factor of n B ∼ /g [34–36]. A possible solution to this problem is provided by treating the badly-convergentcontributions separately in a framework called Electrostatic Quantum Chromodynamics (EQCD) [37].In contrast, the situation for Z f appears to be better under control. Fermions do notfeature a 0-mode so the above reasoning does not apply here. Indeed, it was found that[33] Z f = Z LOf + δZ f = T
12 + 0 , (1.11) We use the Minkowski-metric with the negative sign in the time-component. – 4 –o the assumption of a convergent series for the fermionic part of m ∞ may be justified.According to [38], correlators like (1.4) can be computed in a setup that quantizesthe plasma on the light-cone. Their dynamics is dominated by multiple soft scatteringsthat have to be resummed. But because they are infrared, they can be computed in threedimensional EQCD instead of four dimensional full QCD in real-time, with modified Wilsonlines that account for the parallel transport of the gauge configurations. In the past, thistechnique has been predominantly used for the computation of the transverse collisionkernel C ( b ⊥ ) [38–40]. Similarly to the calculation of C ( b ⊥ ), the EQCD operator agreeswith its full-QCD-equivalent only in the infrared (IR) regime where EQCD is supposed tobe a valid description of full QCD. In the ultraviolet regime, however, a difference betweenthe two operators will emerge. This difference should be under perturbative control andcan be included as part of a matching calculation between thermal QCD and EQCD, whenthe matching is extended to include the operators of Eq. (1.5).In the course of the present work, we will evaluate the soft operator in EQCD andleave the purely perturbative matching calculation for later investigation. As for C ( b ⊥ ),perturbation theory, even in EQCD, is not always sufficient to deal with the nonpertur-bative nature of some high-temperature phenomena. Sometimes, it is necessary to solveEQCD directly on the lattice, which has been done recently for C ( b ⊥ ) [39, 40]. This willalso be our method of choice for the determination of the soft part of asymptotic jet massesin the following.We address the problem as follows. In Sec. 2, we outline how we would compute Z g if there were no interactions. To this end, we briefly introduce how our setup translatesinto EQCD, show how the observables appear in this theory and finally present the freesolutions. Solving the full interacting theory requires discretizing the theory and the cor-responding observables on the lattice, done in Section 3. This part of the present workalso outlines how to deal with the contributions of Eq. (1.5) at the shortest and longestdistances, where the lattice determination becomes problematic. Section 4 involves a de-tailed presentation of our results together with a thorough discussion and, where possible,quantification of the errors that our results are fraught with. The consequences of ourresults and how to improve on them is discussed in Sec. 5. Appendix A provides a detailedoverview of our lattice configurations. Electrostatic Quantum Chromodynamics, or EQCD for short, is an effective field theory forhot QCD. At high temperatures, the scaling behavior changes, for lengths longer than 1 /T or energies lower than T , to the behavior of a 3D theory, which is more strongly coupled inthe IR. Within the Matsubara formalism for describing thermal QCD, this arises becausethe usual frequency integral (cid:82) dω/ π is replaced by a discrete sum T (cid:80) ω =2 nπT , with the n = 0 (zero Matsubara frequency) mode massless (at tree level), leading to an O (2 πT /p )enhancement of small momentum p parts of diagrams. For the electric sector this is cut offby the Debye mass, but for the magnetic sector it is not cut off until magnetic degrees of– 5 –reedom become strongly coupled and confine. For an overview see Refs [34–36]. The ideaof EQCD is to integrate out – in a Wilsonian renormalization-group-sense – all Matsubaramodes in QCD except for these 0-modes, which can then be treated nonperturbatively.Since there is only one Matsubara-mode left, one has effectively eliminated the Euclideantime direction. Therefore the transition to EQCD is often called dimensional reduction . Atime direction that only consists of one time layer does not allow for derivatives in timedirection any more. Consequently, this direction no longer features gauge symmetry, whichis what allows the temporal gauge field A , now responsible for all electrical phenomena,to acquire a mass m . Loop level effects induce a self-coupling for this field, denoted as λ .A scale is set by the now dimensionful gauge coupling g = g T . The full action of EQCDin the continuum reads S EQCD , c = (cid:90) d x (cid:18) g Tr F ij F ij + Tr D i Φ D i Φ + m Tr Φ + λ (cid:0) Tr Φ (cid:1) (cid:19) . (2.1)The parameters in (2.1) can be related to the full theory via a perturbative matchingprocedure [37]. We employ a O ( g )-accurate result [41]. Input parameters are the temper-ature T and the number of dynamical massless fermion flavors N f translating into the pairof (dimensionless) EQCD parameters x ≡ λ/g and y ≡ m /g | µ = g . Note that m varies logarithmically with scale due to 2-loop effects; therefore in defining y we must spec-ify the renormalization point. Following convention we choose µ = g . We will considerthe values given in Tab. 1. EQCD also has a nontrivial phase structure [42, 43], with atransition between a Z -symmetric and broken phase that is of second order at large x andturns into a first order one for smaller x , including all physically relevant x values. Note the( x, y )-pairs which describe full QCD prove to lie in the region where the 3D theory prefersthe Z -broken phase. Therefore we are to consider metastable Z -symmetric states in aparameter range where the broken phase is actually thermodynamically preferred. Thuschoosing sufficiently large volumes is crucial; they keep the system from accidentally fallinginto the broken phase. It is believed that dimensional reduction to EQCD is valid for all T (cid:38) T c and possibly even further down [44], where T c is the transition temperature of fullQCD. T n f x y
250 MeV 3 0 . . . . . . . . Table 1 . EQCD parameters at four different temperatures and number of fermion flavors.
As already mentioned, one promotes (1.5) to EQCD by replacing Wilson lines along thelight cone with their modified EQCD-counterparts [38]˜ U R ( x ⊥ , L ; x ⊥ ,
0) = P exp (cid:18)(cid:90) L d z ( iA az ( x ⊥ , z ) + g Φ a ( x ⊥ , z )) T a R (cid:19) , (2.2)– 6 –nd F i with D i Φ. Applying rotational invariance in the transversal plane, we find Z g , = − d A (cid:90) ∞ d L L (cid:16) (cid:104) ( D x Φ( L )) a ˜ U ab A ( L,
0) ( D x Φ(0)) b (cid:105) + (cid:104) F xza ( L ) ˜ U ab A ( L, F xzb (cid:105) (cid:17) . (2.3)This correlation function is UV safe within EQCD; here it is essential that the modifiedWilson loop of Eq. (2.2) is used, and not the standard Wilson loop without the Φ a T a termin Eq. (2.2); the standard Wilson loop contains a logarithmically UV divergent perimeterlaw suppression which is canceled by the inclusion of the scalar.As a matter of choice, we work in the fundamental representation of the correlators,related to the adjoint ones via (cid:104) ( D x Φ( L )) a ˜ U ab A ( L ; 0) ( D x Φ(0)) b (cid:105) = 2 Tr (cid:104) D x Φ( L ) ˜ U F ( L ; 0) D x Φ(0) ˜ U − ( L ; 0) (cid:105) (2.4) (cid:104) F xza ( L ) ˜ U ab A ( L ; 0) F xzb (cid:105) = 2 Tr (cid:104) F xz ( L ) ˜ U F ( L ; 0) F xz (0) ˜ U − ( L ; 0) (cid:105) . (2.5)In the following, we will drop the tilde on top of the modified Wilson lines ˜ U for the sakeof readability, and write half the right-hand side of (2.4) as Tr (cid:104) E ( L ) U E (0) U − (cid:105) and halfthe right-hand side of (2.5) as Tr (cid:104) B ( L ) U B (0) U − (cid:105) .Let us stress again that the argument of Eq. (2.3) agrees with that of Eq. (1.5) onlyup to IR-safe terms which can and must be determined in a matching calculation betweenfull thermal QCD and EQCD. This calculation has not been carried out for these specificoperators, though it should proceed along similar lines to the matching already conductedfor C ( b ⊥ ) in [15, 38].Through insertions of the scale g , everything can be recombined into dimensionlessquantities and we end up with the master formula for our observable Z g g = − (cid:90) ∞ g L Emin d (cid:0) g L (cid:1) g L Tr (cid:104) D x Φ( g L ) U F ( g L ; 0) D x Φ( g L ) U − ( g L ; 0) (cid:105) g (2.6) − (cid:90) ∞ g L Bmin d (cid:0) g L (cid:1) g L Tr (cid:104) F xz ( g L ) U F ( g L ; 0) F xz (0) U − ( g L ; 0) (cid:105) g , where g L Emin and g L Bmin are two minimal separations at which we can measure ourobservable in the EQCD effective picture on the lattice. Beyond that, we have to rely onanalytical solutions, as we will elaborate in Sec. 3.
The short distance behavior of the full, continuum-extrapolated correlators in (2.4) and(2.5) is expected to match the perturbative expectations and eventually, at sufficientlysmall separations, the free solutions. Since they not only provide an important crosscheckfor our data, but also will be of practical use later, we will briefly present their instructivebut simple derivation in the following. – 7 –e assume the transverse direction to be oriented along the x -axis for simplicity. Attree-level, i.e. at O ( g ), (2.4) reduces toTr (cid:104) D x Φ( L ) U F ( L, D x Φ( L ) U − ( L, (cid:105) = ∂ x Tr (cid:104) Φ( x, L )Φ(0 , (cid:105) (cid:12)(cid:12) x → = C A C F ∂ x (cid:32) e − m D √ x + L π √ x + L (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → = − e − m D L πL (1 + m D L ) , (2.7)specializing to SU (3) in the last step, and in a similar fashion for (2.5),Tr (cid:104) F xz ( L ) U F ( L, F xz U − ( L, (cid:105) = ∂ x Tr (cid:104) A z ( x, L ) A z (0 , (cid:105) (cid:12)(cid:12) x → + ∂ z Tr (cid:104) A x ( x, z ) A x (0 , (cid:105) (cid:12)(cid:12) x → , z = L = − C A C F πL + 2 C A C F πL = 1 πL . (2.8)Plugging these results into (1.5), we see that the O ( L − ) and O ( L − ) parts cancel. Thedominant short-distance behavior is of O (1 /L ), which when integrated over L d L leads toa finite short-distance behavior in Z g . The superrenormalizable nature of EQCD meansthat, at the next loop order, the small- L behavior can be at worst O ( L − ). However, ifthere is no cancellation between the electric and magnetic contributions at this order, therecould be complications in the short-distance behavior, leading potentially to logarithmicshort-distance contributions in Eq. (2.3). If this is the case, their role must be clarified bythe matching to the 4D theory. Our lattice implementation agrees with the one in [40], based on a modification of MartinL¨uscher’s openQCD-1.6 [45]. Due to [46] and [43], we are able to discretize the EQCDaction on a three-dimensional spatial lattice without any errors at O ( a ). As an updatealgorithm, we used a composite of 2 heatbath sweeps followed by 4 overrelaxation sweepsthrough the volume in a checkerboard-order. To improve the signal-to-noise ratio, weapplied the multi-level algorithm [47, 48], dividing the volume into four subvolumes alongthe z -axis and updating each subvolume 80 times before a sweep through the entire volumetook place. For the F xz -insertions in (2.5), we use the standard clover discretization of the field-strengthtensor [49, 50], with four 1 × D x Φ in (2.4) arediscretized as D x Φ( x ) = 12 a (cid:16) U ( x )Φ( x + a e x ) U † ( x ) − U † ( x − a e x )Φ( x − a e x ) U ( x − a e x ) (cid:17) , (3.1)– 8 –eeping in mind that the rescaling of the Φ-fields to their lattice equivalents yields anotherfactor of 1 /a . Note that the clover magnetic field operator is spatially larger than theelectric field operator we use; therefore its O ( a ) corrections are larger, and we need morelattice spacings of separation between the B operators before continuum behavior setsin than for the E operators. Consequently we cannot pursue as short distances in theTr (cid:104) B ( L ) U B (0) U − (cid:105) correlators as in the Tr (cid:104) E ( L ) U E (0) U − (cid:105) correlators.All of our observables are tree-level O ( a ) accurate. Nevertheless, there are still O ( a )discretization effects present generated by loop effects in lattice perturbation theory. Thesegenerate multiplicative O ( a )-corrections, which we will denote by Z E and Z B in the fol-lowing. For details about the discretization of the modified Wilson line, we again refer to[40] and [39]. The O ( a )-renormalization of an isolated modified Wilson line, or two well-separated modified Wilson lines, was calculated analytically in [51]. We use the resultingprescription, which is to scale A in Eq. (2.2) by a Z factor which is determined there.However for the current case this is insufficient, since there are new UV effects which canoccur because the two modified Wilson lines in Eq. (2.4), Eq. (2.5) are coincident. Thiswill lead to new, so far uncalculated, O ( a ) perimeter-law renormalizations of the Wilsonline, which must be accounted for. The overall lattice expressions eventually readTr (cid:104) E ( L ) U E (0) U − (cid:105) latt g a = (cid:18) Tr (cid:104) E ( L ) U E (0) U − (cid:105) cont g + O ( a ) (cid:19) e g aL Z P (cid:0) g a Z E (cid:1) , (3.2)Tr (cid:104) B ( L ) U B (0) U − (cid:105) latt g a = (cid:18) Tr (cid:104) B ( L ) U B (0) U − (cid:105) cont g + O ( a ) (cid:19) e g aL Z P (cid:0) g a Z B (cid:1) . (3.3)An analytical calculation of Z E , Z B , and Z P is probably possible but appears to betechnically somewhat complex, since the Wilson line operator is an extended object. How-ever, by analyzing what diagrams could give rise to such linear in a effects, we find thatthe rescalings Z B and Z E do not depend on the length L (provided it is long in latticeunits), and the perimeter contribution Z P is strictly linear in L . Furthermore, the EQCDparameters x and y would first enter at a higher loop order than the NLO effect givingrise to O ( a ) corrections. Therefore the Z E , Z B , and Z P factors are common to all ( x, y )values and all separations. They represent only 3 total fitting parameters over all of ourdata at multiple ( x, y ) and L values. Such global parameters make the fit more compli-cated, introducing correlations between the fits to multiple lattice spacings, separations,and ( x, y ) pairs. This made the fitting procedure somewhat unstable, which we cured byusing a finite-difference solver to minimize χ . The correlations between all datapointsmust also be considered when performing the final numerical integrations. But because weonly lose fitting 3 degrees of freedom over all data, the impact on the overall error budgetis relatively modest.Fig. 2 shows a generic example for the continuum extrapolation of the two correlatorsat a given length g L = 1 .
0. We clearly see that O ( a )-contributions are still present, butthe consideration of parabolae with joint linear coefficients according to (3.2) and (3.3)– 9 – < E ( L ) E ( ) > / g g ax = 0.0178626x = 0.0463597x = 0.0677528x = 0.0889600 (a) Tr (cid:104) E ( L ) UE (0) U − (cid:105) g < B ( L ) B ( ) > / g g ax = 0.0178626x = 0.0463597x = 0.0677528x = 0.0889600 (b) Tr (cid:104) B ( L ) UB (0) U − (cid:105) g Figure 2 . Two continuum limits for g L = 1 . results in curves that fit our data reasonably well. The obtained continuum values andtheir respective quadratic coefficients are close to each other, but can still be distinguished.We considered four ( x, y ) pairs, each at 8 separations L for the Tr (cid:104) E ( L ) U E (0) U − (cid:105) correlator and 7 separations for the Tr (cid:104) B ( L ) U B (0) U − (cid:105) correlator, and at 7 differentlattice spacings. (Note that not every lattice spacing is useful for each length L ; and wecannot evaluate the Tr (cid:104) B ( L ) U B (0) U − (cid:105) correlator at the smallest L value because as notedbefore, the clover B operator is rather large, requiring larger L/a values before one entersthe continuum regime.) For each ( x, y ) pair and separation, the Tr (cid:104) E ( L ) U E (0) U − (cid:105) andTr (cid:104) B ( L ) U B (0) U − (cid:105) correlator are each continuum extrapolated, fitting to a constant andquadratic term, plus linear terms determined by the three global parameters Z E , ZB , and Z P . The likelihood of the grand continuum extrapolation was reasonable with a χ ≈ −
123 = 153 degrees of freedom.
Even without the contributions from the 4D to 3D matching calculation, we can stillattempt to perform the integral shown in Eq. (2.6), at least for L above some small- L cutoff. Our lattice data is available only within a finite range of separations g L . Weintegrate this data using the trapezoidal rule. Ideally we would want data at a denselyspaced set of distances L , but this is impractical because the L spacing is limited by thelattice spacings of the coarsest lattices used. Also, using denser-spaced L values wouldnot actually help very much, because the different L -value data are from the same sim-ulations, and data at nearby separations L are highly correlated. Nevertheless, the fi-nite spacing in g L introduces another error in Z g which we have to bear in mind. Wechoose to measure Tr (cid:104) E ( L ) U E (0) U − (cid:105) at g L = 0 . , . , . , . , . , . , . , . (cid:104) B ( L ) U B (0) U − (cid:105) at g L = 0 . , . , . , . , . , . , .
0, where we elaborate on thischoice in App. A. We will eventually find that using the trapezoid rule only introduces asubdominant discrete-integration-error. – 10 – .4 Integration of the tails
The correlator (1.5) requires an integration of the condensate up to infinite separations.Since the signal-to-noise ratio drops dramatically with increasing L , correlators at separa-tions beyond g L ≥ . L , we can fit the tail to the largest data wehave available. Similar things have been done, for instance, in the context of the hadroniccontribution to the muon g − (cid:10) F xz ( L ) U F ( L, F xz U − ( L, (cid:11) = Ag L e − B g L . (3.4)Even though they consider a conventional Wilson line connecting the two B -fields, thisform is still applicable. However in our case the coefficient B has a valid continuum limit,rather than having logarithmically UV divergent perimeter contributions. Each coefficientis nonperturbative; we cannot predict them.Based on similar physical reasoning, and consistent with the free solution (2.7), weexpect the functional form of the large-separation limit of Tr (cid:104) E ( L ) U E (0) U − (cid:105) also tobe (3.4), with different coefficients to be determined from the fit. We choose our fittingrange for the large- g L -tail to be [1 . , . ξ ∼ /m ∞ . However, we will eventually find that 1 . (cid:29) g /m ∞ .At the other end, we do not have data at separations below certain minimal separations g L Emin and g L Bmin . Below these minimal separations, one would have to connect to anEQCD perturbative expression. Since the (yet unknown) difference between the EQCDeffective description and the “true” full-QCD version of (1.5) is expected to contributesignificantly there, we leave the consideration of this region entirely for future study.
Before we give our numerical results, we present a thorough discussion of the multiplepossible sources of both statistical and systematic errors throughout our calculation. Asnumerical results, we provide continuum-extrapolated data for Tr (cid:104) E ( L ) U E (0) U − (cid:105) andTr (cid:104) B ( L ) U B (0) U − (cid:105) in Figs. 3 and 4, the corresponding values can be found in Tab. 2,and details about the simulations appear in Tab. 3 of App. A. The estimation of the error has to take several sources into account. They are:1. The statistical fluctuations of our data induced by the Monte-Carlo simulation of thepath-integral2. The uncertainty in choosing the model for the long- L -tail and in the determinationof its parameters – 11 –. The discretized evaluation of the L d L -integration4. The matching to the perturbative solutions at small g L
5. The reliability of reducing full QCD to EQCD6. The perturbative determination of Z f We will now discuss these aspects in detail, and give and explain quantitative estimateswhere possible, see Tab. 2.The Monte-Carlo error of Tr (cid:104) E ( L ) U E (0) U − (cid:105) and Tr (cid:104) B ( L ) U B (0) U − (cid:105) can be deter-mined by a standard binning and Jackknife analysis. Since the continuum extrapolationwas performed in a grand fit, one has to account for correlations of all lattices at all sep-arations g L . Even if their g a naively would not contribute, it does so via its influenceon the values of Z P , Z B , and Z E . This error, ∆ MC , is given in the first brackets in Tab. 2.Another dominantly statistical error is created when we match the tails of our modelsfor Tr (cid:104) E ( L ) U E (0) U − (cid:105) and Tr (cid:104) B ( L ) U B (0) U − (cid:105) , see (3.4). To be precise, the origin ofthis error is twofold; the assumption of the model itself introduces a systematic error,and the determination of the model parameters from our data at large g L generates astatistical error. A valid estimate for this is provided by determining the coefficients in(3.4) by a fit, performing the integral, repeating this procedure for each Jackknife bin, andrunning the standard Jackknife-analysis over the different values for the tail-integral. Thisprocedure includes the correlations between the fitting parameters in Eq. (3.4). In practice,we found that the statistical error amounts to about 100% of the correction induced bythe large- g L -tail, which seems to be a reasonable estimate of the overall error of thelarge- g L -integration. This error, ∆ LT , is given in the second brackets of Table 2.Our data in the available window was integrated using the trapezoidal rule. Theremainder ∆ TR of such a discretized integration bounded by g L min and g L max reads∆ TR = g L max − g L min
12 max g L | I (cid:48)(cid:48) ( g L ) | , (4.1)where the maximum of the second derivative of the integrand I in (2.3) within the interval (cid:2) g L min , g L max (cid:3) has to be found, involving the additional factor of g L from the L d L -integration. This is tricky since the overall functional form of our data is precisely whatwe do not know. However, we know the functional form of the tails which we take asa basis. This might not be a rigorous choice but we will see that this error is in anycase subdominant compared to other error sources. For this consideration, we extend therange of validity of the free solutions up to g L = 1 . g L -tails down to the same value. Consequently, we have three trapezoids of length δ ( g L ) = 0 .
25 with free-solution estimates of the second derivative and four trapezoidsof length δ ( g L ) = 0 . O ( g ) relative to theleading behavior. Second, there is the accuracy of the matching for the specific operator weare considering. So far this has only been performed at tree level, which as we have alreadynoted is insufficient. A calculation at the next order would represent parametrically O ( g )contributions to Z g , which is formally O ( g ) relative to the leading behavior. Withoutthis matching there are formally O ( g ) relative errors from the matching calculation; withan NLO calculation this would be reduced to formally O ( g ) errors. The reason an NLOmatching is needed is that nonperturbative EQCD effects are also formally of relativeorder O ( g ); therefore without an NLO matching calculation for the operator, we have notimproved the formal order of the uncertainty. It is difficult to estimate the size of these(systematic) matching errors. An NLO calculation would shed considerable light on thispoint.Last but not least, the current perturbative determination of Z f neglects terms sup-pressed by a factor of g . These could be determined in an NLO calculation of this con-densate within the full 4D theory.All in all, the values of Z g in Tab. 2 are of the formbest estimate(∆ MC )(∆ LT )(∆ TR ) . (4.2)Here ∆ MC , ∆ LT , and ∆ TR are respectively the errors from Monte-Carlo statistics includingthe continuum extrapolation, the long-distance tail, and the use of the trapezoid rule.The overall quantified error of our results is dominated by integration of the long-distance tail. This introduces a relative error of ∼ Z f would not allow for higher precision anyway, so an improvementis not mandatory. The plots Figs. 3 and 4 show agreement of our continuum data with the free solutions from(2.7) and (2.8) at small g L . Take note that, in contrast to Tr (cid:104) E ( L ) U E (0) U − (cid:105) , we donot have data for Tr (cid:104) B ( L ) U B (0) U − (cid:105) at g L = 0 .
25 since the discretization effects aremore severe for this operator and we did not compute fine enough lattices to perform avalid continuum extrapolation at this distance. At the other end of the range, we fit thefunctional form (3.4) to the correlators through the last four points. The fit is dominatedby the two innermost of these four points since the signal-to-noise-ratio drops dramaticallyat increasing g L . In turn, this phenomenon leads to some of the points in Fig. 4 at g L = 3 . x and y fromTab.1) diminishes in going to small g L . For Tr (cid:104) B ( L ) U B (0) U − (cid:105) in Fig. 4, this becomesevident by looking at the free solution (2.8) which has neither a dependence on x nor on y ,and is jointly approximated by all four scenarios. For Fig. 3 displaying Tr (cid:104) E ( L ) U E (0) U − (cid:105) ,– 13 – - < E ( L ) E ( ) > / g g Lx = 0.0178626x = 0.0463597x = 0.0677528x = 0.0889600
Figure 3 . Continuum-extrapolated version of − Tr (cid:104) E ( L ) UE (0) U − (cid:105) g , connecting to the free solution(2.7) at small g L and to an exponential tail of the form (3.4) with the coefficients determined fromour data at g L ≤ .
5. The short-dashed lines indicate the respective free solutions correspondingto a pair of x and y . The long-dashed lines refer to the respective fitted tail function of the form(3.4). the situation is less obvious. Expanding the exponential in (2.7) for small m D L , we seethat the dependence on m D , or √ y respectively, comes with a suppression of O ( L ) andtherefore becomes negligible at small distances.In both cases, we observe a strict hierarchy: the correlators − Tr (cid:104) E ( L ) U E (0) U − (cid:105) andTr (cid:104) B ( L ) U B (0) U − (cid:105) grow with increasing x (corresponding to decreasing the temperature).We also find that − Tr (cid:104) E ( L ) U E (0) U − (cid:105) and Tr (cid:104) B ( L ) U B (0) U − (cid:105) become closer to eachother as one increases the temperature, which is expected due to the screening mass m D increasing with temperature and reducing scalar correlations over large distances.Adding Tr (cid:104) E ( L ) U E (0) U − (cid:105) and Tr (cid:104) B ( L ) U B (0) U − (cid:105) and integrating according to(2.6), as discussed in the previous section, one observes that Tr (cid:104) E ( L ) U E (0) U − (cid:105) out-weighs Tr (cid:104) B ( L ) U B (0) U − (cid:105) and therefore drives the integral negative in all four scenarios.Multiplying by − · d A = − , one indeed obtains a positive Z g .Our results for Z g /g , shown in Table 2, are significantly larger than the leading-ordercontribution from Eq. (1.9). However note that we perform the Tr (cid:104) E ( L ) U E (0) U − (cid:105) inte-gration, which contributes positively, down to a smaller cutoff than the Tr (cid:104) B ( L ) U B (0) U − (cid:105) – 14 – < B ( L ) B ( ) > / g g Lx = 0.0178626x = 0.0463597x = 0.0677528x = 0.0889600
Figure 4 . Continuum-extrapolated version of Tr (cid:104) B ( L ) UB (0) U − (cid:105) g , connecting to the free solution(2.8) at small g L and to an exponential tail of the form (3.4) with the coefficients determinedfrom our data at g L ≤ .
5. The black dashed line indicates the free solution that is independentof x and y . The dashed lines in color refer to the respective fitted tail function of the form (3.4). integration, which contributes negatively. This mismatch dominates the presented result.However we still present the result in this way, because it incorporates all results for whichour numerical treatment gives statistically useful information. This mismatch can be fixedtogether with the incorporation of the NLO matching, which we expect to give a procedurefor incorporating the missing short-distance information. In the present work, we have shown how to compute the asymptotic masses m ∞ in QCD athigh temperature nonperturbatively. We showed that the real-time continuum expression,Eq. (1.4) and Eq. (1.5), can be matched to EQCD operators, Eq. (2.3), to compute thecrucial nonperturbative infrared contributions. We then evaluated these operators numer-ically within EQCD on a 3D lattice, performing a precise extrapolation to the continuumlimit. Our data are presented in Table 2, and we discussed a procedure for numericallyintegrating these data to determine the IR contribution to the asymptotic masses, and forassessing all sources of statistical and systematic error in using the EQCD data.– 15 – = 0 . x = 0 . y = 0 . y = 0 . N f = 3 ,
250 MeV N f = 3 ,
500 MeV g b ⊥ (cid:104) EE (cid:105) (cid:104) BB (cid:105) (cid:104) EE (cid:105) (cid:104) BB (cid:105) − . − . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . − . Z g g . . x = 0 . x = 0 . y = 0 . y = 1 . N f = 4 , N f = 5 ,
100 GeV g b ⊥ (cid:104) EE (cid:105) (cid:104) BB (cid:105) (cid:104) EE (cid:105) (cid:104) BB (cid:105) − . − . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . − . Z g g . . Table 2 . Results for the correlators Tr (cid:104) E ( L ) U E (0) U − (cid:105) /g and Tr (cid:104) B ( L ) U B (0) U − (cid:105) /g at fourtemperatures over a range of separations. The sum of the integrated correlators according to (2.6)yields our estimate of Z g . We would like to point out once more that the integration still lacksUV-contributions below g L Emin and g L Bmin , respectively.
Unfortunately, we find that to use the numerical results properly, we need to completethe matching calculation for the operators of Eq. (1.5) between full QCD and EQCD a next-to-leading order in the perturbative expansion. Therefore we are in the unusual situationthat the numerical study of EQCD is currently ahead of the analytical understanding of thematching calculation. Such a matching calculation would improve the short-distance partof the EQCD calculation by providing an NLO result for the small-separation correlationfunctions, and it would ensure that all effects up to the desired m ∞ ∼ g T level of accuracyare accounted for. In terms of future improvements, this is clearly the most pressing, sincewithout this matching calculation our results cannot yet be used. Further improvementsto our numerical calculation are possible, in particular by sharpening the error bars andcomputing down to shorter distances, but generally the data appears to be in good shapeand the matching calculation is the most urgent need.– 16 –ogether with the transverse collision kernel C ( b ⊥ ) from [43], the then-complete- m ∞ will hopefully contribute significantly to understanding how a jet is modified by the mediumthat it traverses. We are eager to see the impact of both results on phenomenologicaltransport calculations and how they match experimental observations. Acknowledgments
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) – project number 315477589 – CRC TRR 211. Calculations for thisresearch were conducted on the Lichtenberg high performance computer of the TU Darm-stadt. N. S. acknowledges support from Academy of Finland grants 267286 and 320123.We thank S¨oren Schlichting and Ismail Soudi for fruitful discussions on related topics inthe course of which the idea for this project was born.
A Simulation parameters
The parameters of our simulations can be found in Tab. 3. The volumes were chosen largeenough such that the suppression of finite-volume effects according to [54] was maintained.All separations of both Tr (cid:104) E ( L ) U E (0) U − (cid:105) and Tr (cid:104) B ( L ) U B (0) U − (cid:105) were measured onthe same lattice configurations, introducing correlations between the different g L thathad to be accounted for in the error analysis. The measurement of the very smallestseparation, g L = 0 .
125 on the g a = 1 / T = 250 MeV-lattices was mostly higher sincewe used this case to investigate how large separations g L could be taken into account. Oneach of the other temperatures, about 12000 CPU · hrs on Intel-Xeon E5-2670-processorswere spent. – 17 – a x cont y cont N x N y N z L/a statistics1 / . . ×
48 4 , , , ,
12 945601 / . . ×
72 6 , ,
18 887401 / . . ×
96 4 , , , , , ,
24 614201 /
12 0 . . ×
144 6 , , , , ,
36 88201 /
16 0 . . ×
192 4 , , , , , ,
40 24001 /
24 0 . . ×
288 6 , , ,
24 4801 /
32 0 . . ×
384 4 , , , ,
32 2001 / . . ×
48 4 , , , ,
12 502001 / . . ×
72 6 , ,
18 294801 / . . ×
96 4 , , , , , ,
24 239401 /
12 0 . . ×
144 6 , , , , ,
36 86801 /
16 0 . . ×
192 4 , , , , , ,
40 36001 /
24 0 . . ×
288 6 , , ,
24 4801 /
32 0 . . ×
384 4 , , , ,
32 3601 / . . ×
48 4 , , , ,
12 623501 / . . ×
72 6 , ,
18 364401 / . . ×
96 4 , , , , , ,
24 293801 /
12 0 . . ×
144 6 , , , , ,
36 87801 /
16 0 . . ×
192 4 , , , , , ,
40 37601 /
24 0 . . ×
288 6 , , ,
24 4601 /
32 0 . . ×
384 4 , , , ,
32 3001 / . . ×
48 4 , , , ,
12 623401 / . . ×
72 6 , ,
18 365001 / . . ×
96 4 , , , , , ,
24 295801 /
12 0 . . ×
144 6 , , , , ,
36 87401 /
16 0 . . ×
192 4 , , , , , ,
40 37001 /
24 0 . . ×
288 6 , , ,
24 4801 /
32 0 . . ×
384 4 , , , ,
32 300
Table 3 . Parameters for all EQCD multi-level simulations.
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