Towards tomography of quark-gluon plasma using double inclusive forward-central jets in Pb-Pb collision
aa r X i v : . [ h e p - ph ] N ov Towards tomography of quark-gluon plasma using doubleinclusive forward-central jets in Pb-Pb collision
Michal Deak and Krzysztof Kutak
Instytut Fizyki Jadrowej, Radzikowskiego 152, 31-342 Kraków, Poland
Konrad Tywoniuk
Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
September 18, 2018
Abstract
We propose a new framework, merging High Energy Factorization with final-state jetquenching effects due to interactions in a quark-gluon plasma, to compute di-jet rates atmid- and forward-rapidity. It allows to consistently study the interplay of initial-state effectswith medium interactions, opening the possibility for understanding the dynamics of hardprobes in heavy-ion collisions and the QGP evolution in rapidity.
The Large Hadron Collider (LHC) offers unprecedented possibilities to study properties of nu-clear matter in extreme conditions. One of the paramount results obtained at LHC, and earlierat the Relativistic Heavy Ion Collider (RHIC), is the strong evidence of a state of matter calledquark-gluon plasma (QGP) in ultrarelativistic heavy-ion collisions through its strong quenchingof perturbative probes, such as heavy-quark and QCD jet production. In particular, jets areexcellent tools in the study of the properties of the QGP due to their coupling to the deconfinedplasma degrees of freedom, and they allow one to study new aspects of QCD dynamics, forreviews see [1, 2]. In particular, recent measurements of single-inclusive jet modifications [3–5]and di-jet asymmetry [6–8] have also attracted interest from theory [9, 10]; see also [11] andthe references therein. On theoretical grounds, the latter phenomenon was understood to be aconsequence of fluctuations of final-state jet energy loss, leading to an on average dispersion ofthe energy difference. Because the typical medium scales, related through multiple scatteringto the jet quenching coefficient ˆ q , are modest compared to the jet energies, the di-jet angularcorrelation remains practically unmodified compared to its vacuum baseline. The studies per-formed so far were limited to a strictly back-to back configuration of the di-jets at mid-rapiditywell suited for calculations within collinear factorization.It is worth pointing out that the di-jet azimuthal distribution away from the back-to-backconfiguration is dominated by vacuum effects related to initial-state (space-like) emissions. Itwould therefore be interesting to study in greater detail the interplay of these contributions andthe novel, final-state modifications arising in heavy-ion collisions. This was first addressed in thecontext of di-jet azimuthal decorrelation due to in-medium transverse momentum broadening [12,13]. In the current study, we focus instead on effects related to medium-induced radiative energyloss. Additionally, we allow for jet production at more forward rapidities than considered in1eavy ion collisions so far. This opens for an interesting exploration of the nuclear wave-functionin tandem with medium effects. Such a framework would be of general interest for providing aconsistent cross-referencing of observables calculated across various hadronic colliding systems,especially proton-nucleus and nucleus-nucleus collisions. Finally, it makes new use of jets astomographic probes of the rapidity profile of the QGP.The framework which allows us to study from the first principles the full angular dependenceof decorrelations of forward-central jet configuration in vacuum is the hybrid High Energy Fac-torization [14, 30, 32]. In this approach, the kinematics is treated exactly from the outset, thematrix elements are calculated with one the incoming parton’s momenta (carrying low longitu-dinal momentum fraction of parent hadron) off-shell and one on-shell (carrying large fractionof parent hadron). The incoming off-shell parton carries transversal momenta which allow forthe decorrelation of final-state jets. In this approach, the kinematics is treated exactly fromthe outset, the matrix elements are calculated with one of the incoming parton’s momenta(carrying small longitudinal momentum fraction of the parent hadron momentum) off-shell andone on-shell (carrying large fraction of the parent hadron momentum). The incoming off-shellparton carries transversal momenta which allow for decorrelation of final state jets. In thisapproach, in order to calculate cross sections, the matrix element needs to be convoluted withthe transverse momentum dependent (TMD) parton density function as well as with standardPDF parametrizing partons carrying large longitudinal momentum fraction of parent hadron (inour case a typical x on the ’projectile’ side is − ). In particular the transversal momentumdependent PDF could be provided by the BFKL [15–17] equation, when the longitudinal mo-menta are small but the system is sufficiently dilute to obey linear dynamics, or by the KMRWframework, when the longitudinal momenta are moderate as considered here sue up (in our casea typical on the ’target’ side is x ≃ − ). The latter framework allows for a transformation ofthe collinear gluon density to the TMD PDF by the so called Sudakov resummation.In the more extreme situations, i.e. when the parton densities are probed at low- x , oneneeds to account for eventual saturation effects [18]. This complicates the factorization formulasince, besides taking into account the dipole gluon density which is a solution of the Balitsky-Kovchegov equation [19–21], one also needs to take into account the Weizsäcker-Williams gluondensity [22].By combining the hybrid HEF with final-state rescatterings in a hot and dense medium cre-ated during nucleus-nucleus collisions, we propose a framework which allows for well controlledstudy of the full azimuthal dependence of the cross section and for investigations of the longi-tudinal structure of QGP at the same time. Encouraged by the success of HEF in describingvarious data [23, 24] we shall apply it to central-forward di-jet production in heavy-ion collisionsby including effects relevant for jets passing through a hot and dense QCD medium into theHEF Monte Carlo generator KaTie [25]. We argue that di-jet observables in HEF are moresuitable to study rapidity/rapidity-azimuthal structure of the quark-gluon plasma formed in aheavy-ion collisions. Owing to the factorization of soft, medium-induced radiation from the hardvertex, the final-state modifications are then implemented as energy-loss probabilities affectingfinal-state particles [26, 27].Recently, the importance of jet substructure fluctuations on the di-jet asymmetry and thegeneric energy-loss mechanism was pointed out [28, 29]. In this exploratory study, we will, how-ever, not consider further details of jet fragmentation. We structure the paper in the followingway. We present the details of the framework and implementation of medium effects in Sect. 2.Numerical results for the production of central-forward di-jets in heavy-ion collisions at the LHCare presented in Sect. 3, and finally we discuss our results and provide a brief outlook in Sect. 4.2 General framework and implementation of medium effects
In order to calculate the cross section for the double inclusive jet production with medium effectsincluded one needs to generalize the vacuum framework, which in our case refers to proton-protoncollisions. The generalization is two-fold: • replacement of collinear PDF by nPDF and by replacement of TMD by nTMD; • acconting for energy loss.The formula for hybrid HEF in dilute-dilute scattering reads [30, 31]: dσ acd dy dy dp t dp t d ∆ φ = p t p t π ( x x S ) |M ag ∗ → cd | x f a/A ( x , µ ) F g/B ( x , k t , µ ) 11 + δ cd , (1)with k t = p t + p t + 2 p t p t cos ∆ φ , and x = 1 √ S ( p t e y + p t e y ) , x = 1 √ S (cid:0) p t e − y + p t e − y (cid:1) , where |M ag ∗ → cd | is the hard matrix element for scattering of on-shell parton a off a space-likegluon to partons c and d . The matrix elements can be found in [32] or evaluated using helicitymethods [33]. The distribution F ( x, k t , µ ) is an unintegrated gluon density parametrizing thepartonic content of a hadron carrying a small longitudinal momentum fraction x of the parenthadron and some transverse momentum k t . This PDF depends, in general, on some factorizationscale µ . It is obtained via the application of the KMRW framework, i.e., by performing aresummation of soft gluons using the Sudakov form factor [34,35]. The formulation is such that,upon integration over the transversal momentum up to hard scale µ , one recovers the collineargluon density. The function x f a/A ( x , µ ) is a collinear PDF characterizing partons carryinglarge longitudinal momentum fractions and probed at the hard scale µ .In order to calculate the cross section for propagation of di-jets through medium producedin heavy-ion collision we need to extend the HEF framework to account for the energy loss ofjets traversing the medium. For high- p t jets, one can safely assume the dominance of radiativeprocesses from medium-induced bremsstrahlung. The emission spectrum of medium-inducedgluons can be factorised from the hard process and is given by ω dI R ( χ ) dω = α s C R ω Re Z χω d q (2 π ) Z ∞ dt ′ Z t ′ dt Z d z exp (cid:20) − i q · z − Z ∞ t ′ d s n ( s ) σ ( z ) (cid:21) × ∂ z · ∂ y (cid:2) K ( z , t ′ ; y , t | ω ) − K ( z , t ′ ; y , t | ω ) (cid:3) y =0 , (2)in terms of the gluon energy ω and transverse momentum q with respect to the jet axis [36–40].The spectrum Eq. (2) is, in fact, independent of the jet direction. The function K ( z , t ′ ; y , t | ω ) = Z r ( t ′ )= zr ( t )= y D r exp (Z t ′ t d s (cid:20) i ω r − n ( s ) σ ( r ) (cid:21)) , (3)is the solution to a 2D Schrödinger equation describing rescattering in the medium governed by amedium gluon density n ( s ) along the path of propagation. Finally, σ ( r ) is related to the mediuminteraction potential. In Eq. (2), we have explicitly subtracted the vacuum contribution K ≡ lim n ( s ) → K , which corresponds to the free gluon Green function. Further vacuum showering isnot considered, in accordance with Eq. (7). The spectrum is proportional to the colour factor The contribution from off-shell quarks for the studied jet configuration is negligible.
3f the projectile, for a fast quark (gluon) C R = C F ( C R = N c ) and is a function of the factor χ , which parameterizes the angular range of the emitted gluons. In Eq. 2, χ = sin Θ where Θ is the angle between jet axis and radiated emission. We will assume χ = 1 , corresponding togluons’ emitted angles ≤ π/ .Due to the steeply falling spectrum of hard particles, energy loss will be dominated bymultiple emissions of soft gluons [26]. Due to the typical short formation time, this warrants adescription in terms of multiple independent emissions; for recent improvements see [41]. Sincewe are interested in computing the energy emitted off a high-energy projectile, we will onlyresum primary emissions and neglect, for the moment, further cascading. This can be furtherjustified by the lack of cone definition in our setup. Hence, the probability of emitting a totalenergy ǫ can be written as P R ( ǫ ) = ∆( L ) ∞ X n =0 n ! n Y i =1 Z L d t Z d ω i d I R ( χ ) d ω i d t δ ǫ − n X i =1 ω i ! , (4)with ∆( L ) ≡ exp (cid:18) − Z L d t Z ∞ d ω d I R ( χ ) d ω d t (cid:19) (5)being the Sudakov form factor that represents the probability of not radiating between 0 and L.Concretely, we will use the numerical implementation utilized in [27].We employ standard parametric estimates to argue that the timescale for the hard processis much smaller than the timescales related to soft, medium-induced radiation in the final state.Furthermore, momentum broadening effects are neglected due to the smallness of the mediumparameters compared to the typical jet energies which result only in very small deflection angles.This allows us to generalize the HEF formula asd σdy dy dp t dp t d ∆ φ = X a,c,d Z ∞ dǫ Z ∞ dǫ P a ( ǫ ) P g ( ǫ ) dσ acd dy d y dp ′ t dp ′ t d ∆ φ (cid:12)(cid:12)(cid:12)(cid:12) p ′ t = p t + ǫ p ′ t = p t + ǫ , (6)where the Pb-Pb vacuum cross section is given by dσ acd dy dy dp t dp t d ∆ φ = p t p t π ( x x S ) |M ag ∗ → cd | x f P ba/A ( x , µ ) F P bg/B ( x , k t , µ ) 11 + δ cd , (7)Equation (6) accounts for nuclear effects in the partonic content of nuclei as well as energy lossof the final-state jet particles (Fig. 1). We stress that the formula above is a conjecture andassumes factorization of vacuum emissions and medium rescatterings. As explained there areindications that the formula can be justified when the jets are hard, i.e. the medium modifiestheir properties slightly. Furthermore the particular choice of thge factorization scale allows oneto separate initial-state emissions from final-state ones.We will apply the harmonic approximation, consisting in writing n ( s ) σ ( r ) ≈ ˆ q ( s ) r / , asa simple model for interactions in the QGP. One of the crucial elements of the formula forincorporating the medium effects comes from the transport coefficient ˆ q . Assuming a thermalizedQGP, it is associated to the local temperature and parametrically ˆ q ∼ g T , where g is the in-medium coupling. In our studies we use a model linking it with the energy density describedin [42]; see also [43]. It reads ˆ q = 2 K ε / , (8)where K is a constant quantifying the deviation from expectations in a weakly coupled QGP.The energy density ε is parameterized according to the data of bulk particle production, and itreads ε = ε tot W ( x , y ; b ) H ( η ) (9)4 (a) (b) Figure 1:
Illustration of jets passing through the nuclear medium. Length the jets pass through themedium L . (a) The azimuthal cross section of the nuclear medium. (b) Longitudinal cross section of thenuclear medium. q - - - - ) q H ( Figure 2: Profile of the function H ( η ) .where ǫ tot is a free parameter [43, 44]. We have updated the model in order to describe therapidity distribution of particle production at LHC, as H ( η ) = 1 √ π ( a b − a b ) h a e −| η | / ( b ) − a e −| η | / ( b ) i , (10)with fitting parameters a , b , a and b [45]. Since the distribution (10) is normalized, ǫ tot corresponds to the total energy density distributed in the whole rapidity range. Finally, weassume a simplified geometry of the QGP, having all particles traversing the same length L inthe medium so that ˆ q is only a function of the rapidity; see Fig. 1a. This amounts to putting W ( x , y ; b ) → and neglect the sampling over production points and impact parameters. Wechoose a realistic value of L = 5 fm.In the numerical calculations we have used the following values of the parameters. First,we fix K = 1 , demanding that the value of ˆ q at mid-rapidity corresponds to GeV /fm. Thismeans that ǫ tot ≈ GeV/fm . Varying the parameter K allows us to scan a range of realisticvalues for ˆ q . The remaining parameters we fit to the data on charged particles in − centralcollisions [45], giving a = 2108 . , b = 3 . , a = 486 . and b = 1 . . The resultingshape of the H ( η ) function is plotted in Fig. 2.A realistic energy-loss probability distribution P ( ξ, r ) [40], where ξ = ǫ/ω c with ω c = ˆ qL / and r = ˆ qL / , contains two components: a discrete and a continuous component, P ( ξ, r ) = C δ ( ξ ) + C D ( ξ, r ) . (11)5 [GeV] tc p
100 110 120 130 140 150 160 170 180 190 200 R medium(K=1)/vacuummedium(K=2)/vacuum
100 110 120 130 140 150 160 170 180 190 200 ] - [ nb G e V t c d N / dp - - vacuummedium K=1medium K=2 <3 f h <1 2< c h -1< >100 GeV tf >30 GeV p tc p =5 TeV NN S (a) [GeV] tf p
40 60 80 100 120 140 160 180 200 R medium(K=1)/vacuummedium(K=2)/vacuum
40 60 80 100 120 140 160 180 200 ] - [ nb G e V tf d N / dp - - vacuummedium K=1medium K=2 <3 f h <1 2< c h -1< >100 GeV tf >30 GeV p tc p =5 TeV NN S (b) Figure 3: (a) On top: Central jet p tc before and after jets pass through the medium. (b) On top:Forward jet p tf before and after the jets pass through the medium. The red histogram represents the p tc/f of the jet without passing the medium. The light green histogram represents the p tc/f spectrumof a jet quenched by the medium with constant K = 1 (8). The blue histogram represents the p tc/f spectrum of a jet quenched by the medium with K = 2 . On bottom: ratios of the histograms. The coefficient C gives the probability that no suppression occurs. The function D ( ξ, r ) de-scribes the continuous component of the probability distribution. With the parameters chosenabove we find that ω c ≈ . GeV and r ≈ . In the implementation of the probabilitydistribution, with α s = 1 / as default, for a given event, we first generate a random number C r from to . If C r < C , then no suppression in the medium occurred, ξ = 0 and the weightcoming from the medium correction is w M = 1 ; therefore, the total weight is w T = w M w = w ,where w is the original weight for the event in the vacuum. If C r > C , one can employ theMetropolis algorithm to generate ξ according to the distribution D ( ξ, r ) , then w M = 1 , or gen-erate ξ according to a simple distribution with a corresponding weight w S , then w M = D ( ξ, r ) and w T = w S w M w . To outline the workings of the implementation of the model we have chosen five observables:transversal momentum of the central jet p tc (Fig. 3a), transversal momentum of the forward jet p tf (Fig. 3b), rapidity distance between the jets ∆ η (Fig. 4a), azimuthal angle between the jets ∆ φ (Fig. 4b) and a relative transversal momentum difference of the jets A j = (cid:0) p tc − p tf (cid:1) / (cid:0) p tc + p tf (cid:1) (Fig. 5). In each figure we plot the unsuppressed cross section of jets passing through vacuumand two suppressed cross sections of jets passing through nuclear medium with two differentnuclear medium parameters ˆ q modified by choosing the value of the constant K = 1 and K = 2 .Each of the plots is accompanied by a plot of medium suppression calculated as a ratio ofsuppressed over unsuppressed cross sections.We have chosen the transversal momentum of the central jet is p tc > GeV. The rapidityof the central jet was − < η c < . The transversal momentum of the forward jet p tf > GeV.The rapidity of the forward jet is moderate, < η f < , in order to be within the reachof current experimental capabilities at LHC. To evaluate the cross sections, we have used thenCTEQ15FullNuc_208_82 [49] nuclear (lead) PDF for the collinear parton. For the off-shell6luon density, we have used the novel nuclear TMD PDF constructed applying the KMRWprocedure to the nCTEQ15FullNuc_208_82 collinear set.We can see in Fig. 3a that the suppression by the medium is stronger for lower transversalmomenta and gets weaker by increasing p tc , which is consistent with other available results inthe literature. The same behaviour with suppression decreasing with increasing momentum ispresent in Fig. 3b. The peak in the p tf spectrum at GeV corresponds to the back-to-back di-jet configuration. The medium suppression ratio grows with the momentum for p tf > GeV,but for p tf < GeV the behaviour is the opposite with the ratio growing with decreasing p tf .The latter indicates that smaller p tf values are associated with bigger p tc values.In the rapidity difference spectrum in Fig. 4a we can see that the suppression grows slightlyfor increasing ∆ η as a consequence of the rapidity dependence model of the ˆ q parameter (10).The behaviour seen in the ∆ φ distribution in Fig. 4b is due to the effect of Sudakov resum-mation. Similar structures have already been been observed in [47], albeit for at higher rapiditiesand transverse momenta. This effect arises because of the reshuffling of events from the strictlyback-to-back limit to lower momenta, conserving the total number of events. The resultingstructure is partly suppressed by medium effects. In the region where ∆ φ < , for di-jets notpassing through the medium, the distribution is completely flat. On the other hand, for di-jetswhich have passed through the medium, the dependence on ∆ φ emerges with a distributionslowly falling with decreasing ∆ φ .The peak, in the Fig. 5, at A j ≈ (100 − / (100+30) ≈ . corresponds to the back-to-backpeak in the Fig. 3b. Furthemore, as one can see in the ratio plot, the medium suppression isreshuffling di-jets from configurations with p tc ≈ p tf to configurations with unequal momenta.This effect becomes stronger after increassing the medium transport coefficient ˆ q by increasingthe constant K (8) from K = 1 to K = 2 . We have proposed a new framework merging HEF with final-state processes in a deconfinedmedium. The framework allows to investigate the longitudinal structure of QGP and the patternof decorrelations in QGP that ultimately is the result an interplay of medium effects and vacuumeffects. In order to carry out such a task, we have also introduced a new TMD nuclear PDF fora realistic modeling of initial-state nuclear effects.The proposed framework could serve to disentangle effects related to energy loss from thoserelated to angular decorrelation whether due to initial-state/saturation or final-state broadeningin the quark-gluon plasma. It also allows to calculate observables that are potentially sensitiveto physics at forward rapidity; in particular, the longitudinal structure of the plasma. We havecalculated distributions involving cuts and parameters realistic for heavy-ion experiments at theLHC.Our study confirms that the bulk component of the decorrelations is due to a vacuum initial-state shower. This is mainly because medium-induced energy loss mainly shifts the p t -spectraof the outgoing jets. The presence of medium interactions changes the normalization and tosome extend shape of distributions of the studied observables. This is a clear prediction fromthis particular model, and would be interesting to compare with experimental data.We are currently limiting ourselves to high-energy processes, where medium-modificationsfactorise from the hard cross section and affect mainly the resulting p t -distributions of the out-going jets. In the future we plan to study more forward processes and therefore to generalize theframework to account for saturation effects. We also plan to study the impact of jet substructurefluctuations, in the spirit of [29] that are crucial for describing high- p t data at mid-rapidity [28].Furthermore, in order to shed more light on the role of final-state broadening, we plan to imple-7 D R medium(K=1)/vacuummedium(K=2)/vacuum [ nb ] hD d N / d vacuummedium K=1medium K=2<3 f h <1 2< c h -1< >100 GeV tf >30 GeV p tc p =5 TeV NN S (a) fD R medium(K=1)/vacuummedium(K=2)/vacuum [ nb ] fD d N / d vacuummedium K=1medium K=2<3 f h <1 2< c h -1< >100 GeV tf >30 GeV p tc p =5 TeV NN S (b) Figure 4: (a) On top: Rapidity difference between the jets ∆ η before and after the jets pass throughthe medium. The red histogram represents the ∆ η of the jet before passing the medium. The lightgreen histogram represents the ∆ η spectrum of a jet quenched by the medium with transport coefficient K = 1 (8). The blue histogram represents the ∆ η spectrum of a jet quenched by the medium withtransport coefficient K = 2 . On bottom: ratios of the histograms. (b) On top: Azimuthal angle betweenthe jets ∆ φ before and after the jets pass through the medium. The red histogram represents the ∆ φ of the jet without passing the medium. The dark green histogram represents the ∆ φ spectrum of ajet quenched by the medium with K = 1 . The green histogram represents the ∆ φ spectrum of a jetquenched by the medium with K = 2 . On bottom: ratios of the histograms. j A R medium(K=1)/vacuummedium(K=2)/vacuum [ nb ] j d N / d A - vacuummedium K=1medium K=2 <3 f h <1 2< c h -1< >100 GeV tf >30 GeV p tc p =5 TeV NN S Figure 5:
On top: Relative transversal momentum difference between the jets A j before and after thejets passe through the medium. The red histogram represents the A j of the jet without passing themedium. The green transparent histogram represents the A j spectrum of a jet quenched by the mediumwith K = 1 (8). The blue transparent histogram represents the A j spectrum of a jet quenched by themedium with K = 2 . On bottom: ratios of the histograms. Acknowledgments
The work of M.D. and K.K. was supported by Narodowe Centrum Nauki with Sonata Bis grantDEC-2013/10/E/ST2/00656. K.T. has been supported by a Marie Sklodowska-Curie IndividualFellowship of the European Commission’s Horizon 2020 Programme under contract number655279 ResolvedJetsHIC. We acknowledge discussions with Andreas van Hameren on the detailsof implementation of medium effects into the KaTie event generator and discussions with DogaCan Gulhan on the experimental aspects of production of forward jets in Pb-Pb collision. Figure1 was made with the JaxoDraw package [50].
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