Radiative energy loss and radiative p_T-broadening of high-energy partons in QCD matter
NNuclear Physics B Proceedings Supplement 00 (2015) 1–4
Nuclear Physics BProceedingsSupplement
Radiative energy loss and radiative p ⊥ -broadening of high-energy partons inQCD matter Bin Wu
Institut de Physique Th´eorique, CEA Saclay, 91191, Gif-sur-Yvette Cedex, France
Abstract
I give a self-contained review on radiative p ⊥ -broadening and radiative energy loss of high-energy partons in QCDmatter. The typical p ⊥ of high-energy partons receives a double logarithmic correction due to the recoiling e ff ect ofmedium-induced gluon radiation. Such a double logarithmic term, averaged over the path length of the partons, canbe taken as the radiative correction to the jet quenching parameter ˆ q and hence contributes to radiative energy loss.This has also been confirmed by detailed calculations of energy loss by radiating two gluons. Keywords:
QCD matter, jet quenching, double logarithmic corrections
1. Introduction
High energy partons serve as hard probes to bulkQCD matter created in ultra-relativistic heavy-ion col-lisions at RHIC and the LHC. The motion of such hardprobes, determined by the properties of bulk matter, canbe theoretically studied by first-principle (pQCD) cal-culations. The nature of such a medium can be de-ciphered only by confronting theoretical calculationswith experimental data. Calculations beyond leadingorder (LO) in pQCD are of great importance to testingthe reliability of theoretical studies and hence help im-proves our understanding of the properties of bulk mat-ter. Significant progress has recently been made on thistopic. And I shall only limit myself to a self-containedsummary on the leading-logarithmic correction propor-tional to α s ln ( L / l ) obtained in a slightly generalizedBDMPS-Z formalism[1, 2]. Here, L is the length ofthe QCD medium and l is the size of its constituents.Such a correction is universally present in radiative p ⊥ -broadening[3, 4], the transport coe ffi cient ˆ q [5, 6] andradiative energy loss[5, 7, 8]. Let us start with the dis-cussion about the typical and averaged transverse mo-mentum broadening at LO in [9, 10]. Email address:
[email protected] (Bin Wu)
2. Radiative p ⊥ -broadening ⊥ -broadening at LO ˆ q R is defined as the typical transverse momentumsquared per unit length transferred from the medium toa high energy parton in color representation R [9]. Interms of the gluon distribution xG and the number den-sity ρ of the medium constituents, it can be written inthe following form[9]ˆ q R ≡ ρ (cid:90) dq ⊥ q ⊥ d σ R dq ⊥ = π α s C R N c − ρ xG ( x , ˆ q R L ) (1)with d σ R / dq ⊥ the di ff erential cross section for singlescattering. In a hot quark-gluon plasma (QGP) the gluondistribution in the wave function of each plasma parti-cle is given by xG i ( x , ˆ q R L ) = α s C i π ln (cid:18) ˆ q R Lm D (cid:19) with i = F for (anti-)quarks and i = A for gluons and the (total)number density of each species is ρ i = d i n i with d i thedimension of color representation i . Here, m D ( (cid:28) / L )is the Debye mass and, in terms of the phase-space dis-tribution f i , n i ≡ (cid:90) d p (2 π ) f i (1 + (cid:15) i f i ) with (cid:15) A / F = ± . (2) a r X i v : . [ h e p - ph ] S e p Nuclear Physics B Proceedings Supplement 00 (2015) 1–4 The following discussions shall be applicable to bothcold and hot matter in terms of the corresponding xG .A high-energy parton traversing a medium of length L picks up a typical p ⊥ equal to ˆ q R L , in whichonly the contribution from multiple soft scattering isincluded[9]. This can be easily understood in terms ofthe mean free path of the high-energy parton (with q ⊥ atypical momentum transfer to the medium)1 λ R ≡ σ R ρ xG ∼ α s C R q ⊥ ρ xG ∼ ˆ q R q ⊥ . (3)By comparing λ R with L , one can distinguish typicalmultiple soft scattering with q ⊥ (cid:46) ˆ q R L in each indi-vidual scattering from rare single hard scattering with q ⊥ (cid:29) ˆ q R L [10]. The distribution of transverse momentaof the parton takes the form dNd p ⊥ = (cid:90) d x ⊥ (2 π ) e ρ L (cid:82) d q ⊥ d σ Rd q ⊥ ( e i p ⊥· x ⊥ − ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) S ( x ) . (4)Including only typical multiple scattering gives, in thelogarithmic approximation, dNd p ⊥ ≈ π ˆ q R L e − p ⊥ ˆ qRL . (5)And it is modified by rare hard single scattering only at p ⊥ (cid:29) ˆ q R L in which case one has dNd p ⊥ ≈ ρ L d σ R d p ⊥ ∝ p − ⊥ . (6)Quantities such as the average p ⊥ do receive largecorrections from such rare scattering at high energies.However, the distribution dN / d p ⊥ carries more infor-mation and its shape is mainly characterized by the typ-ical p ⊥ . This motivates us to focus only on multiplesoft scattering and typical p ⊥ -broadening shall be sim-ply denoted by (cid:104) p ⊥ (cid:105) in all the following discussions. ⊥ -broadening In QCD matter the high-energy parton undergoesmultiple scattering and radiates gluons. It hence gen-erates a recoiling transverse momentum. Such a radia-tive correction to (cid:104) p ⊥ (cid:105) has been studied in [3, 4] us-ing a formalism as an extension of that by BDMPS-Z[1, 2]. The complete analytic evaluation of the contri-bution from one-gluon emission is complicated by thepresence of multiple scattering but double logarithmicterms, ln ( L / l ), and single logarithmic terms, ln( L / l ),can be evaluated analytically.The radiated gluon and the high-energy parton arein a coherent state within the formation time t ≡ ω k ⊥ with ω the gluon’s energy and k ⊥ its transverse momen-tum. The evolution of the coherent pair is governed by aSchr¨odinger-type evolution equation with a potential[3] V ≈ − iN c ˆ q R B ⊥ C R for B ⊥ (cid:39) k ⊥ (cid:38) q R L . (7)The inverse of V is the typical time scale for the pair ofpartons to undergo one individual collision and hencereferred to as the mean free path of the coherent state λ c ≡ C R k ⊥ N c ˆ q R ⇒ (cid:40) λ c (cid:38) t : single scattering ,λ c (cid:46) t : multiple scattering . (8)By taking ˆ q R (cid:39) m D λ R one can see that λ c becomes largerthan that of a single parton λ R when k ⊥ (cid:38) N c m D / (4 C R ).The double logarithmic contribution only comes froma single scattering. As discussed above, the singlescattering phase space can be singled out by requiringthe formation time of the gluon, radiated inside of themedium, is not larger than the mean free path of the co-herent pair, that is, L (cid:38) λ c (cid:39) k ⊥ ˆ q (cid:38) t and L (cid:38) t (cid:38) l . (9)Within this region, the double logarithmic correctioncan be easily reproduced from the di ff erential cross sec-tion for the emission process in single scattering d σ R → Rg d ω dk ⊥ d p ⊥ = α s N c π q ⊥ ω k ⊥ p ⊥ d σ R d q ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) q ⊥ = (cid:126) k ⊥ + (cid:126) p R ⊥ (10)with the limits of the integral given by (9). And onehas[4] ∆ p ⊥ ( L ) ≡ α s N c π ˆ q R L (cid:90) Ll dtt (cid:90) ˆ q R L ˆ q R t dk ⊥ k ⊥ = ˆ q R L ∆ ( L ) with ∆ ( L ) ≡ α s N c π ln (cid:16) Ll (cid:17) . (11)Subleading single logarithmic terms have also beenevaluated by crossing di ff erent boundaries between dif-ferent regions in the phase space of the radiated gluon.A resummation of leading double logarithmic termsfrom radiating an arbitrary number of gluons induced bya single scattering can be easily carried out by repeatingthe calculation in (11)[4]. Let us start with two-gluonemission. As illustrated in Fig. 1, the term proportionalto ln ( L / l ) comes only from the phase space in whichthe second gluon is radiated within the formation timeof the first one and the transverse momentum of the sec-ond gluon is smaller than that of the first one. Accord-ingly, the next term in the leading double logarithmic Nuclear Physics B Proceedings Supplement 00 (2015) 1–4 k l lk Figure 1: Schematic illustration of diagrams for leading double log-arithmic terms. The blob here denotes a constituent of the mediumand l denotes the momentum transfer between the constituent and thehigh-energy parton (dipole) with n radiated gluons. series is given by (cid:104) p ⊥ (cid:105) = (cid:18) α s N c π ˆ q R L (cid:19) (cid:90) Ll dt t (cid:90) ˆ q R L ˆ q R t dk ⊥ k ⊥ × (cid:90) t l dt t (cid:90) k ⊥ ˆ q R t dk ⊥ k ⊥ = ( ˆ q R L ) ∆ ( L )] . (12)By induction, one can show that the leading double log-arithmic contribution from n -gluon emission is given by (cid:104) p ⊥ (cid:105) n = ( ˆ q R L ) n ( n + n ! [2 ∆ ( L )] n (13)and hence obtain [4]. (cid:104) p ⊥ (cid:105) = ˆ q R L [2 ∆ ( L )] − I (cid:16) [8 ∆ ( L )] (cid:17) . (14)The p ⊥ -broadening in (11) is the typical p ⊥ and, afterbeing averaged over the path length, can be taken as ane ff ective (renormalized) ˆ q R [5, 6]ˆ q renR ( L ) ≈ ˆ q R [1 + ∆ ( L )] . (15)Even though the correction includes contributions fromradiating gluons with a life time ∼ L , it can be stilltaken as a correction to the (local) transport coe ffi -cient as a consequence of the property of logarithmicintegrations[5]. Let us only focus on leading doublelogarithmic terms from radiating n gluons without over-lapping formation times. They can be evaluated by re-peating the calculation in (11) for each radiated gluon.Specifically, S ( x ) receives a correction of the form ∆ S n ( x ⊥ , L ) = − α s N c π n ˆ q R Lx ⊥ (cid:90) Ll d ln t ln (cid:18) Lt (cid:19) × ∆ S n − ( x ⊥ , L − t ) (16)with ∆ S ( x ⊥ , L ) ≈ −
14 ˆ q R L ∆ ( L ) x ⊥ e − ˆ q R Lx ⊥ . (17) Each gluon has a formation time larger than l and hence ∆ S n ( x ⊥ , L ) vanishes if L ≤ nl . This simply indicatesthat the double logarithmic approximation breaks downwhen n ∼ L / l . Therefore, we have to restrict ourselvesto the case L (cid:46) nl . In this case leading logarithmicapproximation applies and one has ∆ S n ( x ⊥ , L ) ≈ e − ˆ q R Lx ⊥ n ! (cid:32) − ∆ p ⊥ x ⊥ (cid:33) n . (18)From this, one can justify the exponentiation of the dou-ble logarithmic integral in (11) S ( x ⊥ ) = e − ˆ q R Lx ⊥ (cid:88) n (cid:46) L / l ∆ S n ≈ e − (ˆ q R L +∆ p ⊥ ) x ⊥ , (19)if x ⊥ satisfies( ∆ p ⊥ x ⊥ ) L / l ( L / l )! < e − ∆ p ⊥ x ⊥ . (20)At L (cid:29) l it asymptotically gives x ⊥ (cid:46) Ll ∆ p ⊥ (cid:29) ∆ p ⊥ and, in this case, plugging (19) into (4) gives dNd p ⊥ ≈ e − p ⊥ ˆ qRL (1 +∆ ( L )) ˆ q R L (1 + ∆ ( L )) . (21)This justifies that the leading logarithmic result in (11)is the typical value of p ⊥ .
3. Radiative parton energy loss
At high energies, the parton loses its energy mainlydue to medium-induced gluon radiation (the LPM ef-fect). Since the formation time of a radiated gluongrows with its energy, the average energy loss withinany period of time t (cid:38) λ R is dominated by radiatingone gluon with the formation time comparable with t .Within t (cid:39) ω/ k ⊥ , the gluon picks up a transverse mo-mentum broadening k ⊥ = ˆ qt . Accordingly, one has t (cid:39) (cid:112) ω/ ˆ q and the maximum energy that the gluonmay carry away is given by ω c ( t ) = ˆ qt . From this,one can get parametrically two well-known results: theLPM spectrum[1, 2] ω dId ω ∼ α s N c Lt ( ω ) ∼ α s N c (cid:114) ˆ qL ω (22)and the average energy loss per unit length [9] − dEdz ∼ α s N c ω c ( L ) L = α s N c ˆ qL . (23)Will these LO results be modified by the radiative cor-rection? The answer is: in double logarithmic accuracy Nuclear Physics B Proceedings Supplement 00 (2015) 1–4 the only thing one needs to do is replace ˆ q by ˆ q ren in(15), as justified in [5, 6]. This has been further con-firmed by a detailed calculation of average energy lossfrom radiating two and more gluons[7] and a more elab-orated and careful analysis of two-gluon emission be-yond the double logarithmic approximation[8]. In thefollowing I shall only take average energy loss for ex-ample to illustrate the underlying physical picture.The energy loss due to two-gluon emission ∆ E isgreatly simplified in the double logarithmic region dis-cussed in the previous section. The first gluon with en-ergy ω , similar to that in the case with one-gluon emis-sion, typically has a formation time t (cid:39) L , ω (cid:39) ˆ qL and k ⊥ (cid:39) ˆ qL . In the double logarithmic region, thesecond gluon with energy ω typically has k ⊥ (cid:46) k ⊥ , ω (cid:46) ω , t (cid:46) (cid:114) ω ˆ q A (cid:46) (cid:114) ω ˆ q A (cid:39) t . (24)Therefore, the second gluon plays the same role as thatin the calculation of the radiative p ⊥ -broadening in (11).After integrating out the contribution from the secondgluon, we have[7] ∆ E ≈ α s N c
12 ˆ qL ∆ ( L ) , (25)which has exactly the same prefactor as the LO result(for the parton approaching the medium from outside)in[1], i.e., ∆ E = α s N c ˆ qL .The resummation of the double logarithmic correc-tion in eq. (25) can be carried out in exactly the sameway as that in the calculation of (cid:104) p ⊥ (cid:105) . For ( n + n gluons undergoes one single scattering. Using the sameorderings in the formation time and transverse momentaof these n gluons that lead to (13) gives a contributionof the form ∆ E n + = α s N c L ˆ qLn !( n + (cid:34) α s N c π ln Ll (cid:35) n . (26)Therefore, the total energy loss is given by[7] ∆ E = ∞ (cid:88) n = ∆ E n + = α s N c L (cid:104) p ⊥ (cid:105) (27)with (cid:104) p ⊥ (cid:105) given by (14), including all the leading doublelogarithmic contributions from radiating arbitrary num-ber of gluons.
4. Discussions
Our discussions so far have been focused on themedium-induced radiative corrections. Rather, in the above results the limit ˆ q → ff ect neglected from such a subtractionis the running of the QCD coupling constant. We havealso discussed this e ff ect in the radiative correction to p ⊥ from one-gluon emission and find a factor 2 largerthan in the fixed coupling case[4]. How the inclusion ofthe running coupling modifies the evolution of ˆ q ren hasalso been discussed in more detail in [11].At the end, I briefly comment on the di ff erences be-tween our double logarithmic radiative correction to p ⊥ and some previous results beyond LO in [12, 13].In [12], the NLO ( O ( g )) correction to transverse mo-mentum broadening has been obtained analytically in ahot ( g (cid:28)
1) quark-gluon plasma using a perturbative-kinetic approach. This result was calculated fromthe soft contribution to the two-body collisional ker-nel from soft collisions with q ⊥ ∼ gT . In contrastour result comes from radiation, which is of O ( g ) andwould be present at next-next-to-leading order in thatapproach[14].The authors of Ref. [13] (see [15] for a recentprogress) have also calculated the radiative correctionto p ⊥ -broadening in a QGP. The e ff ect of multiple (soft)scattering was not considered in this formalism. As aresult, the regime of double logarithmic approximation(Sec. V) has been extended into Q > k ⊥ > µ and E / ( ˆ qL ) > t > l . This is di ff erent from ours in (9) inwhich the phase space includes only single scattering in-side the medium (multiple scattering sets the boundaryat k ⊥ (cid:39) ˆ qt ). It will be intriguing to see how to formulatemultiple scattering in this formalism. References [1] R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schi ff , Nucl.Phys. B , 403 (1998).[2] B. G. Zakharov, JETP Lett. (1996) 952; JETP Lett. (1997)615.[3] B. Wu, JHEP , 029 (2011).[4] T. Liou, A. H. Mueller and B. Wu, Nucl. Phys. A , 102(2013).[5] J. P. Blaizot and Y. Mehtar-Tani, Nucl. Phys. A , 202 (2014).[6] E. Iancu, JHEP , 95 (2014).[7] B. Wu, JHEP , 081 (2014).[8] P. Arnold and S. Iqbal, JHEP , 070 (2015).[9] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne andD. Schi ff , Nucl. Phys. B , 265 (1997).[10] P. B. Arnold, Phys. Rev. D , 025004 (2009).[11] E. Iancu and D. N. Triantafyllopoulos, Phys. Rev. D , no. 7,074002 (2014).[12] S. Caron-Huot, Phys. Rev. D , 065039 (2009).[13] J. Casalderrey-Solana and X. N. Wang, Phys. Rev. C77