Drag induced radiative energy loss of semi-hard heavy quarks
NNuclear Physics B Proceedings Supplement 00 (2017) 1–4
Nuclear Physics BProceedingsSupplement
Drag induced radiative energy loss of semi-hard heavy quarks
Raktim Abir a , Shanshan Cao b , Abhijit Majumder a , Guang-You Qin c a Department of Physics and Astronomy, Wayne State University, 666 W. Hancock St., Detroit, MI 48201, USA, b Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA, c Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, 430079, China.
Abstract
We revisited gluon bremsstrahlung o ff a heavy quark in nuclear matter within higher twist formalism. In this work,we demonstrate that, in addition to transverse momentum di ff usion parameter ( ˆ q ), the gluon emission spectrum for aheavy quark is quite sensitive to ˆ e , which quantify the amount of light-cone drag experienced by a parton. This e ff ectleads to an additional energy loss term for heavy-quarks. From heavy flavor suppression data in ultra-relativisticheavy-ion collisions one can now estimate the value of this sub-leading non-perturbative jet transport parameter (ˆ e )from our results. Keywords: energy loss, drag, heavy-quark, bremsstrahlung radiation.
1. Introduction
By now a substantial amount of work has alreadybeen done to understand unexpectedly large suppres-sion of single electrons or open heavy mesons com-ing from the fragmentation of a heavy-quark in nuclearmedium. There are two main categories associated withthese developments [1]: ( a ) Calculations that extendedradiated energy loss formalism for light flavors to in-clude mass dependent terms, as well as a drag term[2, 3]. ( b ) Calculations that have totally ignored therole of radiative loss and only focussed on drag loss.In both sets of calculations, radiative loss is the resultsof transverse momentum di ff usion experienced by theheavy quark, which is denoted by the jet transport coef-ficient ˆ q . Till now no calculation of heavy flavor energyloss has investigated the possibility that the drag coe ffi -cient ˆ e (or the longitudinal di ff usion coe ffi cient ˆ e ) mayalso lead to an additional prominent source of radiativeenergy loss, beyond that provided by ˆ q . In the highertwist framework, the drag (and longitudinal di ff usion)coe ffi cient ˆ e (ˆ e ) defined as the loss of light-cone mo-mentum (fluctuation in light-cone momentum) per unit light-cone length as,ˆ e = d (cid:104) ∆ p − (cid:105) dL − , ˆ e = d (cid:104) ∆ p − (cid:105) dL − . (1)Here we assume that the parton is moving in the nega-tive light-cone direction. Though this sub-leading trans-port coe ffi cients have little e ff ect to the o ff -shellnessof a near on-shell massless quark, it has a consider-able impact on the o ff -shellness of a near on-shell mas-sive quark. In particular this sub-leading transport co-e ffi cients will only have an e ff ect on the radiative en-ergy loss of a patron where the momentum p is com-parable to the mass M . We also point out that thismass dependent e ff ect is by no means limited only tothe higher-twist scheme, but intrinsic to all other for-malisms that have considered the radiative loss froma heavy-quark in a nuclear medium. To delineatethe relative importance of these additional new drageterms, we have used power-counting techniques bor-rowed from Soft-Collinear-E ff ective-Theory (SCET).This helps one to identify the regime where these massdependent terms really cause detectable e ff ects on theactual gluon bremsstrahlung spectrum. In this work, be-ing a first attempt, we have considered only the case ofsingle scattering and single emission o ff the propagating a r X i v : . [ nu c l - t h ] M a y Nuclear Physics B Proceedings Supplement 00 (2017) 1–4 heavy quark.
2. Deep inelastic scattering and the semi-hard heavyquark
We consider the case of deep-inelastic scattering of ahard virtual photon o ff a hard heavy quark within a largenucleus with mass number A [4]. We also factorized thepropagation of the heavy-quark from the hard scatter-ing vertex which produces the outgoing slow movingheavy-quark. The exchanged virtual photon possessesno transverse momentum, in the Breit frame, q ≡ [ q + , q − , q ⊥ ] = (cid:2) q + , q − , (cid:3) . (2)The scattering process under consideration is the fol-lowing: e ( L ) + A ( P ) → e ( L ) + J Q ( L Q ) + X . (3)As there is no valence heavy-quark within the nucleon,the photon will have to strike a heavy quark fromrare Q ¯ Q fluctuations within the sea of partons. In therest frame of the nucleus we assume that the quarkand antiquark are almost at rest i . e . (cid:16) p + Q , p − Q , p Q ⊥ (cid:17) ∼ (cid:16) M / √ , M / √ , (cid:17) . Now if the nucleus is boosted bya large boost factor γ in the “ + ” direction, momen-tum components of the incoming heavy quark will thenscales as, p Q = (cid:104) p + Q , p − Q , p Q ⊥ (cid:105) ≡ (cid:34) γ M √ , γ M √ , (cid:35) . (4)Momentum components of the incoming photon havebeen assumed as, q = (cid:34) − γ M √ + M q − , q − − γ M √ , (cid:35) . (5)As there is a large boost factor ( γ ), one can reasonablyassume that γ M (cid:29) M ∼ q − (cid:29) M /γ . This would let usto define the hard scale Q as Q = − q (cid:39) γ Mq − / √ p f = p Q + q = (cid:34) M q − , q − , (cid:35) . (6) λ parameter In this study, we have introduced the dimensionlesssmall parameter λ in order to set up the power counting.We borrowed the concept from soft collinear e ff ectivetheory (SCET) to represent semi-hard scales as λ Q andsofter scales as λ Q . Here we have retained leading and next-to-leading terms in λ power counting and neglect-ing all terms that scale with λ or a higher power of λ .We have chosen the scaling variable λ so that perturba-tion theory may be applied down to momentum transferscales even at λ / Q ∼ Λ QCD . In this study,1 (cid:29) √ λ (cid:29) λ (cid:29) λ . (7)The virtuality of the hard photon defines the hardestscale in the problem, Q . The incoming or initial heavyquark has momentum components p i ∼ ( λ − , λ , Q ,the outgoing heavy quark has momentum components p ∼ ( √ λ, √ λ, Q . The mass of the semi-hard heavyquark scales as M ∼ √ λ Q . Leading contribution togluon emission arises from the region where real emit-ted gluons have momenta which scale as l ∼ ( λ, λ, λ ) Q .The fraction of light cone momenta carried out by thegluon is y = l − / p − ∼ √ λ . Scaling of the virtualGlauber gluons is as follows, k ∼ ( λ , λ , λ ) Q with k = k + k − − k ⊥ (cid:39) − k ⊥ .
3. Induced gluon radiation o ff the heavy quark p p ′ q qP Pl Q k k ′ ly y ′ y y ′ Figure 1: One of the 11 single gluon emission diagram, where gluonemission is induced by single scattering. Dashed lines indicate threeseparate cuts, denoted as central, left and right.
In total, there are 11 separate topologically distinctdiagrams similar to that in Fig. 1 [5]. We have definedthe following momentum fractions for convenience, y = l − q − , η = k − l − , ζ = − y − y + η y , x = p + i P + , x = k + P + , χ = y M l ⊥ , x L = l ⊥ P + q − y (1 − y ) , x D = k ⊥ − l ⊥ k ⊥ P + q − , x K = k ⊥ P + q − , κ = + (1 − y ) , x M = M P + q − . (8) Nuclear Physics B Proceedings Supplement 00 (2017) 1–4 One obtain the full contribution to the hadronic tensorby adding all the contributions from all the diagrams,We decompose the hadronic tensor as, W µν = g π ( − g µν ⊥ ) H c , l , r . (9)The entire contribution from all real diagrams is con-tained in the factor H c , l , r . In virtual contributions thereare no cuts in the final state radiated gluon. We havenot considered virtual contributions in this e ff ort. Thisentire factor H c , l , r is obtained as, H c , l , r = (cid:88) m , n = C c , l , rm , n + C , + C , = πα s N c (cid:90) dl ⊥ H c , l , r × exp (cid:34) i (cid:32) x B + x L + x M − y (cid:33) P + (cid:16) y (cid:48)− − y − (cid:17) + i (cid:32) ζ x D + ( ζ − x M − y − ζ η y − y x L (cid:33) × P + (cid:16) y (cid:48)− − y − (cid:17)(cid:105) × (cid:104) A | ¯ ψ ( y (cid:48) ) γ + A + ( y (cid:48) ) A + ( y ) ψ ( y ) | A (cid:105) (10)Now one needs to expand the cross-section in k ⊥ and in k − in order to calculate the next-to-leading power contri-bution to the semi-inclusive hard partonic cross-section.We then extract the corresponding transport coe ffi cientsinside the gluon emission spectrum kernel for the heavyquark. All factors of k ⊥ and k − are absorbed as deriva-tives within the definition of the transport coe ffi cientsusing integration by parts [e.g. k ⊥ A + ( (cid:126) y ⊥ ) exp( i (cid:126) k ⊥ · (cid:126) y ⊥ ) = − i ∇ ⊥ A + ( (cid:126) y ⊥ ) exp( i (cid:126) k ⊥ · (cid:126) y ⊥ ) (cid:39) iF + ⊥ ( (cid:126) y ⊥ ) exp( i (cid:126) k ⊥ · (cid:126) y ⊥ )]. Thefour point non-perturbative operator have also been fac-tored using the usual phenomenological factorization,which in case of transverse scattering may be expressedas, (cid:104) A | ¯ ψ ( y (cid:48) ) γ + F + ⊥ ( y (cid:48) ) F + ⊥ ( y ) ψ ( y ) | A (cid:105)(cid:39) C Ap (cid:104) p | ¯ ψ ( y (cid:48) ) γ + ψ ( y ) | p (cid:105)× ρ p + (cid:104) p | F + ⊥ ( y (cid:48) ) F + ⊥ ( y ) | p (cid:105) . The first operator product yield the incoming quark dis-tribution function within one nucleon. The second op-erator product yield the transport coe ffi cient due to thescattering of the final state, o ff a gluon. We have as-sumed the average condition that both nucleons have amomentum p = P / A . The nucleon density within thenucleus is denoted by the factor ρ , and C Ap is an overallnormalization constant that contain the nucleon density.The factor of ρ/ (2 p + ) is written separately as that would part R AA CMS J/ Ψ , p T = 6.5-30GeVResult, B, p T = 10-50GeV Figure 2: Momentum integrated nuclear modification factor, R AA , asfunction of N part . be absorbed within the definition of the transport coef-ficient. Expressions for the transverse di ff usion ˆ q , thedrag (and longitudinal di ff usion) coe ffi cient ˆ e (ˆ e ) canbe obtained through derivatives of the kernel with re-spect to the transverse and ( − )-light-cone component ofthe exchange momentum respectively, (cid:104) ∇ k ⊥ , ∇ k − , ∇ k − (cid:105) H c , l , r (cid:12)(cid:12)(cid:12) k ⊥ , k − = = C A (cid:32) + (1 − y ) y (cid:33) l ⊥ [ l ⊥ + y M ] ˜ H ˆ q , ˆ e , ˆ e c , l , r . (11)In the equation above, the factor ˜ H ˆ q , ˆ e , ˆ e c , l , r represents sev-eral terms, depending on the cut taken i . e . , central c ,left l , or right r , and the momentum component withrespect to which the Taylor expansion is carried out, i . e . , ˆ q for the second derivative in terms of k ⊥ , ˆ e forthe first derivative with respect to k − and ˆ e for the sec-ond derivative with respect to k − . Once the derivativeshave been taken, both factors of the momentum k ⊥ , k − must be set to zero. Finally all the terms can then becombined to obtain the real single gluon emission spec-trum. In this work we have retained terms only upto O ( √ λ ), the approximation that has been justified inthis study. All terms which scale as O ( λ ) or greater,have been neglected. Preliminary estimation of inte-grated nuclear modification factor both for B and J /ψ with number of participants using our result is ratherencouraging, see Fig. [2]. This shows that the gluonbremsstrahlung spectrum from a semi-hard heavy quarkis strongly modified by drag induced radiation. And thegluon bremsstrahlung spectrum of heavy quark is para-metrically sensitive to ˆ e which quantifies the amount ofdrag the moving quark experiences. We finally expressthe gluon spectrum per unit light-cone length as, Nuclear Physics B Proceedings Supplement 00 (2017) 1–4 dN g dydl ⊥ d τ = απ P ( y ) 1 l ⊥ (cid:32) + χ (cid:33) sin (cid:32) l ⊥ l − (1 − y ) (1 + χ ) τ (cid:33) × (cid:34)(cid:26)(cid:18) − y (cid:19) − χ + (cid:18) − y (cid:19) χ (cid:27) ˆ q + l ⊥ l − χ (1 + χ ) ˆ e + l ⊥ ( l − ) χ (cid:32) − χ (cid:33) ˆ e (cid:35) . (12)Three transport coe ffi cients contain the non-perturbativeexpection of the gluon field strength operators as fol-lows, ˆ q = ς (cid:90) dy − π ρ p + (cid:104) A | F + ⊥ ( y − ) F ⊥ + (0) | A (cid:105) e − i ¯ ∆ P + y − , ˆ e = ς (cid:90) dy − π ρ p + (cid:104) A | i ∂ − A + ( y − ) A + (0) | A (cid:105) e − i ¯ ∆ P + y − , ˆ e = ς (cid:90) dy − π ρ p + (cid:104) A | F − + ( y − ) F − + (0) | A (cid:105) e − i ¯ ∆ P + y − , with, ς = π C R α s N C − . (13)In the equations above, one can observe the appearanceof a new modified momentum fraction:¯ ∆ = ζ x D + ( ζ − x M − y − ζ η y − y x L . (14)The presence of such a momentum fraction, clearly in-dicates that the range of momentum fractions in the def-inition of ˆ q , ˆ e and ˆ e for heavy quark scattering is dif-ferent from that for light flavor energy loss. This alsoindicates that the actual value of ˆ q , ˆ e or ˆ e for heavyquarks may be di ff erent from that for light quarks. Care-ful analysis of heavy-quark energy loss may then lead toan understanding of the x -dependence of the in-mediumgluon distribution function that sources transport coef-ficients. This is essential to understand the degrees offreedom within dense media, where heavy quark losesits energy most.
4. Conclusion
In this work [6] we have considered a hard virtualphoton scattering o ff a heavy quark (within a proton),that converts it to a slow moving heavy quark whichthen moves back through the remainder of the nucleus before escaping and fragmenting into a jetwith a lead-ing heavy meson. Here both transverse broadening aswell as the longitudinal drag and longitudinal di ff usion,have been studied on an equal footing. We have cat-egorically focussed our study on “semi-hard” quarkswhere the mass and momentum scale as M , p ∼ √ λ Q ,as these are the quarks for which mass modifications ismost prominent. Our result can be used to estimate thevalue of this sub-leading non-perturbative jet transportparameter (ˆ e ) from heavy flavor data of heavy-ion col-lider experiments. These extra additive contributions tothe gluon bremsstrahlung spectrum may lead to an even-tual solution of the heavy quark puzzle.
5. Acknowledgements
This work was supported in part by the US NationalScience Foundation under grant number PHY-1207918and by the US Department of Energy under grant num-ber de-sc00013460. This work is also supported in partby the Director, O ffi ce of Energy Research, O ffi ce ofHigh Energy and Nuclear Physics, Division of NuclearPhysics, of the U.S. Department of Energy, through theJET topical collaboration. References [1] A. Majumder and M. Van Leeuwen, Prog. Part. Nucl. Phys. A , 41 (2011).[2] G. Y. Qin and A. Majumder, Phys. Rev. Lett. , 262301(2010).[3] R. Abir, U. Jamil, M. G. Mustafa and D. K. Srivastava, Phys.Lett. B , 183 (2012).[4] R. Abir, G. D. Kaur and A. Majumder, arXiv:1407.1864 [nucl-th].[5] B. W. Zhang, E. k. Wang and X. N. Wang, Nucl. Phys. A ,493 (2005) [hep-ph //