Hard partons moving through a dense QCD medium lose energy by radiative
emissions and elastic scatterings. Deposition of the radiative contribution
into the medium requires rescattering of the radiated gluons. We compute the
total energy loss and its deposition into the medium self-consistently within
the same formalism, assuming perturbative interaction between probe and medium.
The same transport coefficients that control energy loss of the hard parton
determine how the energy is deposited into the medium; this allows a parameter
free calculation of the latter once the former have been computed or extracted
from experimental energy loss data. We compute them for a perturbative medium
in hard thermal loop (HTL) approximation. Assuming that the deposited
energy-momentum is equilibrated after a short relaxation time, we compute the
medium's hydrodynamical response and obtain a conical pattern that is strongly
enhanced by showering.Comment: 4 pages, 3 figures, revtex4, intro modified, typos correcte

A. Majumder

G.-Y. Qin

H. Song

U. Heinz

V. N. Gribov

Yu. L. Dokshitzer

English

arXiv

For a hard parton moving through a dense QCD medium, we compute self-consistently the energy loss and the fraction deposited into the medium due to showering and rescattering of the shower, assuming weak coupling between probe and medium. The same transport coefficients thus determine both the energy loss and its deposition into the medium. This allows a parameter free calculation of the latter once the former are computed or measured. We compute them for a weakly interacting medium. Assuming a short thermalization time for the deposited energy, we determine the medium’s hydrodynamical response and obtain a conical pattern that is strongly enhanced by showering

Qin, G.-Y.

Majumder, A.

Song, H.

Heinz, U.

KnowledgeBank at OSU

Energy and Momentum Deposited into a QCDMedium by a Jet ShowerG.-Y. Qin, A. Majumder, H. Song, and U. HeinzDepartment of Physics, The Ohio State University, Columbus, Ohio 43210, USA(Received 18 March 2009; published 8 October 2009)For a hard parton moving through a dense QCD medium, we compute self-consistently the energy lossand the fraction deposited into the medium due to showering and rescattering of the shower, assumingweak coupling between probe and medium. The same transport coefficients thus determine both theenergy loss and its deposition into the medium. This allows a parameter free calculation of the latter oncethe former are computed or measured. We compute them for a weakly interacting medium. Assuming ashort thermalization time for the deposited energy, we determine the medium’s hydrodynamical responseand obtain a conical pattern that is strongly enhanced by showering.DOI: 10.1103/PhysRevLett.103.152303 PACS numbers: 12.38.Mh, 11.10.Wx, 25.75.DwJet quenching (the modification of hard jets in densemedia) is one of the most studied discoveries at theRelativistic Heavy-Ion Collider (RHIC) [1]. It is expectedto play a key role in the study of the quark-gluon plasma(QGP) produced in heavy-ion collisions at the LargeHadron Collider (LHC). Numerous experiments [2] haveestablished the suppression of hadrons with high transversemomenta; others indicate that the lost energy manifestsitself as conical flow in the soft sector [3].Calculations of jet modification tend to focus on one oftwo separate questions: the modification of the final hadrondistribution from the hard parton due to its energy loss, orthe response of the medium to the energy deposited.Numerous studies of the former, based on perturbativeQCD (pQCD), have yielded near-rigorous measures ofthe two nonperturbative transport coefficients q^ ¼ dp2?dLand e^ ¼ dEdL which codify the transverse (to the jet axis)momentum diffusion and longitudinal drag experienced bya fast parton [4]. Computations of the medium responseconsist of two parts: an ansatz for the space-time profile ofthe energy-momentum deposition, and a calculation of thedynamical response to this ‘‘source’’ of excess energy andmomentum. Based on its success at RHIC, ideal fluiddynamics has been used to compute this medium response[5], assuming that the energy lost by the jet is entirelydeposited into the medium at a constant rate and thermal-izes instantaneously.So far there exists no first principles calculation of themagnitude and space-time profile of the energy-momentum deposited in a medium by a hard parton thatcan be considered on par with the pQCD energy losscalculations [6]. A noteworthy attempt to calculate thedeposition profile in pQCD is the semiphenomenologicalapproach of Neufeld and Mu¨ller [7] who use the differen-tial single gluon emission spectrum of Ref. [8] and inter-pret this as the rate of gluon emission in the medium. Anondiffusive Fokker-Planck equation is then motivated tocompute how this distribution changes due to elastic en-ergy loss of the emitted gluons. As anticipated in [9], theyfind that not all of the energy lost to gluon radiation isdeposited in the medium. However, since the underlyingformalism [8] lacks information about virtuality evolution,this calculation does not include gluon multiplication byshowering, i.e., the splitting of a radiated gluon into twolower virtuality gluons. The transverse momentum depo-sition thus cannot be computed and, due to the stricteikonal limit used in [8], the parent parton does not loseenergy after radiation. We present a new formalism inwhich the radiative and elastic energy loss of the fastparton, its virtuality evolution by radiation, the showeringand multiplication of the radiated gluons, and the energydeposited by them in the medium are all calculated con-sistently in the same approach.Hard jets in vacuum or in heavy-ion collisions are pro-duced with considerable virtuality. As the jets proceedthrough vacuum or medium, this virtuality is lost by se-quential radiative emissions. The effect of this perturbativeshower on the nonperturbative hadronization process iscomputed using Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations [10] for the fragmentation func-tion. These equations express the radiation of multiplepartons, which hadronize independently, via an evolutionin virtuality of the parent parton. In a medium, one canderive analogous equations where the gluon radiationprobability is modified by the scattering of hard partonand emitted gluons off medium constituents. These arereferred to as ‘‘medium modified evolution equations.’’In addition to stimulating gluon emission, the scatteringof the hard parton causes it to lose forward light-conemomentum by elastic exchanges with the medium[11,12]. At the same time the parton gains transversemomentum from the medium [13] and imparts to it anequal amount in return. In an arbitrary medium theseeffects are encoded in two nonperturbative transport coef-ficients, q^ and e^, defined in terms of in-medium gluon fieldcorrelation functions [12,13]. The medium modification ofthe standard vacuum evolution depends on these transportcoefficients.PRL 103, 152303 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending9 OCTOBER 20090031-9007=09=103(15)=152303(4) 152303-1  2009 The American Physical SocietyIn-medium evolution equations where the mediummodified fragmentation function (MMFF) is affectedonly by q^ were derived in [14]. We point out that thesame processes can be used to compute the amplificationof the energy deposited through multiple radiations stimu-lated by transverse broadening. Formally, this can be com-puted by replacing the operator expression for thefragmentation function with that for the energy deposited;this is identical to e^. Using this calculation of the modifiede^ as the energy deposited through all elastic scatterings ofthe shower places it on the same footing as energy loss. Thediagrams involved and the resulting expressions for the in-medium splitting functions (IMSF) are identical. The so-lution of the in-medium evolution equation for e^ no longerrepresents the elastic energy loss by one parton, but ratherthe energy deposited by the jet shower.Imagine a hard quark or gluon with large light-conemomentum q (and thus energy E ¼ q= ﬃﬃﬃ2p ) and virtual-ity  entering a medium of fixed length L held at aconstant temperature T. Let us assume that the rate ofenergy deposition by this jet in the medium as a functionof length  , denoted as dEd ðE; ;2Þ, is known (i.e., can becalculated or measured). Note that both the depositedenergy E and  are actually the light-cone quantitiesq and . For brevity we refer to these simply asdeposited energy and distance traveled. Given the abovefunction, the total energy deposited by a jet originating atlocation i and propagating to f is given asEðE;2Þfi ¼Z fiddEdðE; ;2Þ ’1 partonðf  iÞe^;(1)where the last approximate equality is solely for the case ofa single parton propagating without radiation.If the scale is much larger thanQCD, the change withvirtuality in the partonic shower pattern may be calculatedperturbatively: a leading quark at the higher virtuality maysplit into a quark and a gluon with lower virtuality, andsimilarly for a gluon. As a result, there is change in theenergy deposited in the medium due to the increase of thenumber of partons depositing energy. Using the IMSF from[14], the change in the energy deposition by a quark withenergy E from i to f due to the increase in virtuality can be expressed as [15]dEqðE;2Þfid lnð2Þ ¼sð2Þ2Z 10dyZ fidPq!qgðy; ;2; EÞ fEqðE;2Þi þEqðyE;2ÞfþEg½ð1 yÞE;2f g: (2)Here the first term in square brackets represents the energydeposited by a quark with energy E and virtuality2, fromthe initial location i to the intermediate location ; thesecond and third terms represent the energy deposited bythe quark and the emitted gluon with reduced energies yEand ð1 yÞE, respectively, from the intermediate location to the final location f. In Eq. (2) the quark IMSF Pq!qgis given as [14,16]Pq!qg ¼ q^CF221þ y21 y2 2 cos22Eyð1 yÞ: (3)The increase in the energy deposited due to the splitting ofthe parton is reduced by the virtual correction which re-stores unitarity to the evolution equations. The effect ofsuch corrections on Eq. (2) is incorporated by subtractingfrom it the virtual termV ¼ sð2Þ2EqðE;2ÞfiZ 10dyZ fidPq!qgðy; ; 2; EÞ: (4)Along with the energy deposition from a quark jet one hasto evolve the one from a gluon jet of virtuality, using asimilar evolution equation that includes the splitting of agluon into two gluons or a q q pair. Similar to the MMFFs,one solves a coupled set of evolution equations forEqðE;2Þfi and EgðE;2Þfiboth of which are func-tions of three variables E, i, f at the scale 2.The evolution Eqs. (2)–(4) for a quark jet and thecoupled equations for gluon jets are motivated by existingrigorous derivations of the medium modification of thefragmentation functions due to gluon emission [16], theaccumulation of transverse momentum [13] and longitudi-nal drag [12] by propagating hard partons in a QCDmedium, and the effect of such accumulated momentumon radiative processes [17]. The IMSF (3) accounts forinterference between diagrams where the gluon is emittedat the origin or at the location  . In propagating up to  thequark loses a fraction of its energy; while this is included inthe total energy deposited, its effect on the interferencepattern in Eq. (3) is ignored; this is justified in the eikonallimit for the propagating parton. Yet another approxima-tion is the neglect of the energy lost by the radiated (re-absorbed) gluon in the virtual correction. Since the radiatedgluon in the virtual correction exists in only one amplitude,with a single parton in the complex conjugate, its energyloss is balanced by the quark propagating in the loop.In the eikonal approximation, the hard jet loses light-cone momentum and remains close to on shell; thus, thez component of the deposited light-cone momentum isapproximately equal to the energy deposited (pz ’E). Note that the negative light-cone momentum (q)is not conjugate to  and thus it is not inconsistent tocompute the  dependence of the q deposited. Theremaining two components that may be computed are thetransverse momentum deposited by the jet as a function of. This can again be directly estimated from a pQCDcalculation: A parton traversing a medium gains transversemomentum squared with length asPRL 103, 152303 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending9 OCTOBER 2009152303-2hp2?iðE;2Þfi¼Z fiddhp2?iðE;2Þd’1 partonðf  iÞq^:(5)By momentum conservation this equals the p2? depositedin the medium by the same parton.For a hard virtual quark the total transverse momentumdeposited increases due to parton splitting. This can becalculated using an equation similar to that for light-conemomentum deposition. For a quark with energy E andvirtuality 2, traversing a medium from i to f, thechange of the transverse momentum deposited with vir-tuality is obtained asdhp2?iqðE;2Þfidlnð2Þ ¼sð2Þ2Z 10dyZ fidPq!qgðy;;2;EÞfhp2?iqðE;2Þiþhp2?iqðyE;2Þfþhp2?ig½ð1yÞE;2f g: (6)The splitting function here is identical to that in Eq. (3),and the meaning of the three terms in the bracket isanalogous to Eq. (2). Further, one must include a virtualcorrection and couple Eq. (6) to a similar equation for thep2? deposited by a virtual gluon.Using Eqs. (2) and (6) (along with the coupled ones forgluon jets), we can compute the 3-momentum q, ~pTdeposited by a hard virtual parton, disintegrating into ashower of partons, in a dense medium as a function of thelength  traversed. Similar to the case of in-mediumevolution equations for the MMFF [14], these equationsrequire an initial condition. For the case of the MMFF, theonly possible choice was to insist that the part of the jetwith virtuality below a minimum 20 exited the mediumand use the known vacuum FF at that scale as an input. Forthe deposited part of the energy-momentum we here as-sume that the medium is weakly coupled; thus, when thevirtuality of the parton is 0 ’ 4T the deposited energyand p2? can be obtained from the expressions for e^ and q^ inan HTL plasma [11]:dEð0; EÞ=d ¼ CRsð20Þm2D log½ð4ET=m2DÞ1=4;dhp2?ið0; EÞ=d ¼ CRsð20ÞTm2D log½4ET=m2D:(7)Here mD is the Debye screening length and CR is therepresentation-specific Casimir factor. The integrated en-ergy and p2? deposited from Eq. (7), as a function of thelength traveled, is plotted for gluons (circled) and quarks assolid lines in Figs. 1 and 2. For a consistent description, weimpose that partons with an energy E< 4T become part ofthe thermal medium. This condition is maintained throughthe evolution equations.Using Eq. (7) as input, we may calculate the increase inthe energy and p2? deposition in the medium as a functionof  for initially highly virtual hard partons that evolve intoa radiative shower. Starting from the scale of 0 ¼ 4T (inall calculations we pick T ¼ 300 MeV and a partonicplasma with 3 quark flavors) we evolve up to an initialscale ¼ E=2. These are plotted as dashed lines in Figs. 1and 2. One notes immediately that both quantities increaseas we evolve up in virtuality. For comparison, we alsoestimate the total energy lost by the hard parton due toelastic, radiative inelastic and flavor changing interactions(dash-dotted lines in Fig. 1). The last type of energy lossrefers to the case where a quark splits with the gluoncarrying a larger fraction of the momentum, or a gluonsplits into a quark-antiquark; in this case we assume thatthe entire energy of that parent parton has been lost. Thisleads to a somewhat artificial enhancement of the totalenergy loss.As an illustration of the effect of this energy-momentumdeposition in the medium, we compute its hydrodynamicresponse to the following source term:J dEð;EÞd; 0; 0;dpzð;EÞd2ð~r?Þðt zÞ: (8)In this first attempt we ignore the transverse momentumcontribution to the source current. Following Refs. [18], weassume that the energy deposited is a small perturbationand solve for the linear response of the medium:T’T0 þT; @T0 ¼0; @T¼J: (9)T0 is the unperturbed energy-momentum tensor of ahomogeneous and static partonic medium in equilibrium.The small excess T is decomposed asT00  ; T0i  gi;Tij ¼ ijc2s sð@igj þ @jgi  23ijr  ~gÞ:(10) is the excess energy density, ~g is the momentum current0 1 2 3 4 5L (fm)05101520∆E (GeV)collisional depositcoll.+rad. depositloss, coll.+rad.+flavorgluon: with circlesFIG. 1 (color online). Dash-dotted: Total energy lost by a hardgluon (circled) or quark by radiative, elastic, and flavor changingprocesses. Solid: Energy deposited in the medium by a hardparton which does not radiate. Dashed: The same for a virtualparton devolving into a partonic shower.PRL 103, 152303 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending9 OCTOBER 2009152303-3density, and s ¼ sT is the sound attenuation length. Forthe specific shear viscosity we took s ¼ 12 . We delay theresponse to the source J by a time 	rel ¼ 1mD to account forthermalization of the deposited energy.In Fig. 3 we show the azimuthal projection of the energydensity jxj at t ¼ 5 fm=c after the parton is created, fora single nonradiating parton (left) and a parton-initiated jetshower (right). A gluon (bottom row) deposits more energythan a quark (top row), due to its larger color factor thatenters both in the elastic energy loss and shower productionrate. One immediately notes that, while the basic Machcone structure is not changed, showering leads to an en-hancement by a factor of 3 in the overall magnitude of theresponse. For quark jets our results are qualitatively similarto Ref. [7].In this Letter, we have presented a consistent pQCDbased calculation of the light-cone and transverse momen-tum (q, p2T) deposited by a jet in a medium, as afunction of distance traversed. Assuming a short thermal-ization time for the deposited energy we also computed thehydrodynamic response. The pQCD shower has the effectof a large part of the energy being deposited later in thehistory of the jet [7] which tends to enhance the Machcone-like structure formed.We thank B. Mu¨ller and R. B. Neufeld for helpful dis-cussions. This work was supported by the U.S. Departmentof Energy under Grant No. DE-FG02-01ER41190.[1] I. Arsene et al., Nucl. Phys. A757, 1 (2005); B. B.Back et al., ibid. A757, 28 (2005); J. Adams et al., ibid.A757, 102 (2005); K. Adcox et al., ibid. A757, 184(2005).[2] K. Adcox et al., Phys. Rev. Lett. 88, 022301 (2001); C.Adler et al., Phys. Rev. Lett. 89, 202301 (2002).[3] A. Adare et al., Phys. Rev. Lett. 98, 232302 (2007); B. I.Abelev et al., Phys. Rev. Lett. 102, 052302 (2009).[4] A. Majumder, J. Phys. G 34, S377 (2007).[5] A. K. Chaudhuri and U. Heinz, Phys. Rev. Lett. 97,062301 (2006); T. Renk and J. Ruppert, Phys. Lett. B646, 19 (2007); B. Betz et al., Phys. Rev. C 79, 034902(2009).[6] For an alternate approach that assumes that the hard partoncouples strongly to the medium see P.M. Chesler andL.G. Yaffe, Phys. Rev. Lett. 99, 152001 (2007); S. S.Gubser, S. S. Pufu, and A. Yarom, ibid. 100, 012301(2008).[7] R. B. Neufeld and B. Mu¨ller, Phys. Rev. Lett. 103, 042301(2009).[8] U. A. Wiedemann, Nucl. Phys. B588, 303 (2000); C. A.Salgado and U.A. Wiedemann, Phys. Rev. D 68, 014008(2003).[9] A. Majumder, E. Wang, and X.N. Wang, Phys. Rev. Lett.99, 152301 (2007).[10] V. N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15, 438(1972); Yu. L. Dokshitzer, Sov. Phys. JETP 46, 641(1977); G. Altarelli and G. Parisi, Nucl. Phys. B126,298 (1977).[11] E. Braaten and M.H. Thoma, Phys. Rev. D 44, R2625(1991); S. Wicks et al., Nucl. Phys. A784, 426 (2007);G. Y. Qin et al., Phys. Rev. Lett. 100, 072301 (2008).[12] A. Majumder, arXiv:0810.4967.[13] A. Majumder and B. Mu¨ller, Phys. Rev. C 77, 054903(2008).[14] A. Majumder, arXiv:0901.4516.[15] A. Majumder (to be published).[16] X. f. Guo and X.N. Wang, Phys. Rev. Lett. 85, 3591(2000); X.N. Wang and X. f. Guo, Nucl. Phys. A696,788 (2001).[17] A. Majumder, R. J. Fries, and B. Mu¨ller, Phys. Rev. C 77,065209 (2008).[18] J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney,J. Phys. Conf. Ser. 27, 22 (2005); R. B. Neufeld, Phys.Rev. C 79, 054909 (2009).202x fm024z fm0.2500.250.5x δ202x fm024z fm0.7500.751.5x δ202x fm024z fm0.500.51x δ202x fm024z fm1.501.53x δFIG. 3 (color online). The linear fluid dynamical response tothe energy deposited by a single parton (left) or by a parton-initiated shower (right), when the parton is a quark (top) or agluon (bottom). Note the different vertical scales.0 1 2 3 4 5L (fm)05101520〈p _|_2 〉 (GeV2 )collisional depositcoll.+rad. depositgluon: with circlesFIG. 2 (color online). The hp2?i deposited by a hard gluon(circled) or quark without radiative emission (solid) and with afull radiative shower (dashed).PRL 103, 152303 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending9 OCTOBER 2009152303-4

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Energy and Momentum Deposited into a QCD Medium by a Jet Shower

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Energy and momentum deposited into a QCD medium by a jet shower

Energy and momentum deposited into a QCD medium by a jet shower

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