Parton energy loss in the mini quark-gluon plasma and jet quenching in proton-proton collisions
aa r X i v : . [ h e p - ph ] A p r Parton energy loss in the mini quark-gluon plasmaand jet quenching in proton-proton collisions
B.G. Zakharov
L.D. Landau Institute for Theoretical Physics, GSP-1, 117940,Kosygina Str. 2, 117334 Moscow, RussiaE-mail: [email protected]
Abstract.
We evaluate the medium suppression of light hadron spectra in pp collisions at RHIC and LHC energies in the scenario with formation of a mini quark-gluon plasma. We find a significant suppression effect. For p T ∼
10 GeV we obtainedthe reduction of the spectra by ∼ [20 − , − , − √ s = [0 . , . , AA and pA collisions.
1. Introduction
The experiments at RHIC and LHC have provided clear evidence that in AA collisionsthe hadroproduction goes through the formation of a fireball of hot and dense quark-gluon plasma (QGP). This follows from the observation of strong suppression of high- p T particle spectra (the so-called jet quenching phenomenon) and from the results of thehydrodynamic simulations of AA collisions. In the pQCD paradigm the jet quenchingis due to radiative [1, 2, 3, 4, 5, 6, 7] and collisional [8] energy loss in the QGP whichsoften the parton → hadron fragmentation in AA collisions (for recent comprehensivereviews, see [9, 10]). The suppression of the high- p T particle spectra in AA collisions ischaracterized by the nuclear modification factor R AA defined as the ratio of the particlespectrum in AA collisions to the binary-scaled spectrum in pp collisions [11] R AA = dσ ( AA → hX ) /d p T dyN bin dσ ( pp → hX ) /d p T dy . (1)Presently, in theoretical calculations of the R AA for the inclusive cross section dσ ( pp → hX ) /d p T dy in the denominator in (1) predictions of the pQCD are used. However, ifthe QGP is produced in pp collisions as well, the real inclusive cross section differs fromthat calculated in pQCD by its own medium modification factor R pp , i.e., dσ ( pp → hX ) /d p T dy = R pp dσ pert ( pp → hX ) /d p T dy . (2)In this scenario the theoretical quantity which should be compared with the experimental R AA given by (1) can be written as R AA = R stAA /R pp , (3)where R stAA is the standard nuclear modification factor calculated using the pQCDpredictions for the particle spectrum in pp collisions. Of course, the R pp is unobservabledirectly because experimentally we do not have the baseline spectrum with the finalstate interactions in the QGP switched off. Nevertheless, the presence of the R pp in(3) may be important for theoretical predictions for jet quenching in AA collisions. Forexample, for the jet flavor tomography of the QGP [12, 13, 14, 15] due to differentsuppression of light and heavy flavors in pp collisions.Presently, it is widely believed that in pp collisions in the studied energy rangea hot QCD matter is not produced in the typical inelastic minimum bias events dueto small energy density. But in high multiplicity (HM) pp events the energy densitymay be comparable to that in AA collisions at RHIC and LHC energies. And if thethermalization time, τ , is small enough, say τ ∼ < . ∼ − AA collisions.In recent years the possibility that the mini-QGP may be created in HM pp collisionshas attracted increasing interest (see, for instance, Refs. [16, 17, 18, 19, 20, 21, 22]).Actually, we already have some experimental indications in favor for the formation ofthe mini-QGP in HM pp collisions. It is possible that the ridge correlation structurein HM pp events at √ s = 7 TeV observed by the CMS collaboration [23] is due tothe transverse flow of the QGP. In [19], employing Van Hove’s idea [24] that phasetransition should lead to anomalous behavior of the mean transverse momentum h p T i as a function of multiplicity, it has been argued that the data on h p T i signal possibleplasma formation in the domain dN ch /dη ∼ −
24. Some intriguing similarities betweenthe results of the femtoscopic analyses of pp and AA collisions at RHIC [25] and LHC [26]also may signal the formation of the collective QCD matter in the HM pp events. Thepreliminary data from ALICE [27], indicating that for the HM pp events jets undergo asofter fragmentation, also support this idea.From the point of view of jet quenching it is important that the conditions for theQGP production in pp collisions are better in events with jets, because the multiplicityof soft off-jet particles (the so-called underlying events (UE), see [28] for a review) isenhanced by a factor of 2 − K ue ) as comparedto the minimum bias multiplicity. And even at RHIC energies √ s ∼ . γ -triggered and inclusive jets in HM pp collisions, and have presented preliminary resultsfor medium suppression of hadron spectra. We have found that the medium effects aresurprisingly strong. In the present work we perform a detailed analysis of the mediummodification of the hadron spectra in pp collisions due to parton energy loss in themini-QGP. We evaluate R pp of charged hadrons in the central rapidity region ( y = 0) atRHIC ( √ s = 0 . √ s = 2 .
76 and 7 TeV) energies. We also address theeffect of R pp on R AA at RHIC and LHC energies and on R pA in the context of the recentdata from ALICE [31] on R pP b at √ s = 5 .
02 TeV. The analysis is based on the light-conepath integral (LCPI) approach [3, 4] to induced gluon emission. It treats accurately thefinite-size and Coulomb effects (which are very important for the mini-QGP), the masseffects, and is valid beyond the soft gluon approximation. We evaluate the mediummodified FFs within the scheme developed previously for AA collisions [32]. It takesinto account both radiative and collisional energy loss. Previously in [33, 14, 15] theapproach has been successfully used for description of jet quenching in AA collisions.The paper is organized as follows. In the next section we discuss the parameters ofthe mini-QGP for the UE pp events at RHIC and LHC. In section 3 we discuss the basicaspects of the theoretical framework. In section 4 we present the numerical results onparton energy loss in the mini-QGP and the medium modification factors for pp , AA and pA collisions, section 5 summarizes our work.
2. Mini-QGP in proton-proton collisions
We neglect the transverse expansion of the mini-QGP and use 1+1D Bjorken’s model[34], which gives T τ = T τ . For τ < τ we take medium density ∝ τ . In thebasic variant we take τ = 0 . τ is used in most studiesof jet quenching in AA collisions. For the QGP produced in AA collisions with thelifetime/size L ≫ τ the medium modification of hadron spectra is not very sensitiveto variation of τ . But this may be untrue for the mini-QGP in pp collisions when theplasma size is not very large as compared to τ . To understand the sensitivity of R pp to τ , which is not well constrained by the hydrodynamic modeling of AA collisions [35], wealso perform calculations for τ = 0 . AA collisions [32, 33, 14],we neglect variation of the initial temperature T with the transverse coordinates. Tofix T we use the entropy/multiplicity ratio C = dS/dy . dN ch /dη ≈ .
67 obtained in[36]. The initial entropy density can be written as s = Cτ πR f dN ch dη , (4)where R f is the radius of the created mini-QGP fireball. We ignore the azimuthalanisotropy, and regard the R f as an effective plasma radius, which includes pp collisionsat all impact parameters. This approximation seems to be plausible since anyway thejet production should be dominated by the almost head-on collisions for which theazimuthal effects should be small. This is supported by calculation of the distributionof jet production cross section in the impact parameter plane using the MIT bag modelwhich says that only 25% of jets come from pp collisions with the impact parameterlarger than the bag radius. It says that typically the fireball has a relatively smalleccentricity. Anyway, we are interested in R pp , which is averaged over the azimuthalangle, and it is practically insensitive to the fireball eccentricity.One can expect that for pp collisions the typical radius of the fireball should be aboutthe proton radius R p ∼ R f obtained for pp collisionsat √ s = 7 TeV in numerical simulations performed in [21] within the IP-Glasma model[37]. The R f from [21] grows approximately as linear function of ( dN g /dy ) / and thenflatten. The flat region corresponds to almost head-on collisions. In this regime thefluctuations of multiplicity are dominated by the fluctuations of the glasma color fields[21]. We use the R f from [21] parametrized in [38] via dN g /dy in the form R f = 1 fm × f pp (cid:18) q dN g /dy (cid:19) (5)with f pp ( x ) = ( .
387 + 0 . x + 0 . x − . x if x < . .
538 if x ≥ .
4. (6)We evaluate R f taking dN g /dy = κdN ch /dη with κ = C / π ξ (3) ≈ .
13. Possibleincrease of the R f from RHIC to LHC should not be important since our results are notvery sensitive to variation of R f .The multiplicity density of the UEs grows with momentum of the leading chargedjet hadron at p T ∼ < − E jet ∼ > −
20 GeV). To fix the dN ch /dη in (4) at √ s = 0 . K ue from PHENIX [39] obtained by dihadron correlation method. Taking for the minimumbias non-diffractive events dN mbch /dη = 2 . ± .
34 from STAR data [43], we obtainedfor the UEs in the plateau region dN ch /dη ≈ .
5. To evaluated the UE multiplicity at √ s = 2 .
76 and 5 .
02 TeV we use the data from ATLAS [40] at √ s = 0 . dN ch /dη ≈ . .
9. Assuming that dN ch /dη ∝ s δ by interpolating between √ s = 0 . dN ch /dη ≈ . . √ s = 2 .
76 and 5 .
02 TeV,respectively. With the above values of the UE multiplicity densities in the plateauregions we obtain the following values for the fireball radii R f [ √ s = 0 . , . , . , ≈ [1 . , . , . , .
51] fm . (7)With these radii, using (4) and the ideal gas formula s = (32 /
45 + 7 N f / T (with N f = 2 . T [ √ s = 0 . , . , . , ≈ [199 , , , . (8)One can see that the values of T lie well above the deconfinement temperature T c ≈ −
170 MeV [44, 45] ‡ . For such initial temperatures the purely plasma phasemay exist up to τ QGP ∼ − . τ QGP the hot QCD matter will evolvein the mixed phase up to τ max ∼ R f where the transverse expansion should lead to afast cooling of the system. Since in the interval τ QGP < τ < τ max the QGP fraction inthe mixed phase is approximately ∝ /τ [34] we can use in calculating jet quenchingthe 1 /τ dependence of the number density of the scattering centers in the whole rangeof τ (but with the Debye mass defined for T ≈ T c at τ > τ QGP ).Although we neglect the transverse expansion of the QCD matter, it should notlead to large errors in our predictions. As was demonstrated in [46] the transverse ‡ In fact, if one uses the entropy from the lattice calculations [44, 45] the fireball temperatures in (8)will be higher by ∼ − motion does not affect strongly jet quenching in AA collisions. Physically it is due toan almost complete compensation between the enhancement of the energy loss causedby increase of the medium size and its suppression caused by reduction of the mediumdensity. In pp collisions the effect should be even smaller since the typical formationlength for induced gluon emission is of the order of R f or larger. In such a regime theparton energy loss is mostly controlled by the mean amount of the matter traversedby fast partons, and the details of the density profile along the jet trajectory are notvery important. Also, in pp collisions the QCD matter spends much time in the mixedphase, where the sound velocity becomes small and the transverse expansion should beless intensive than in AA collisions.We conclude this section with two additional remarks. First, naively one couldthink that for evaluation of the medium suppression of the minimum bias pp spectrumone should use the minimum bias multiplicity density in evaluating the mini-QGPparameters. But it would be wrong. Indeed, the minimum bias events include eventswith and without jet production, and the minimum bias high- p T spectrum is related toevents with jet (at least one) production. The corresponding multiplicity density forsuch events is exactly the UE dN ch /dη .The second remark concerns the formula (4). It implicitly assumes that the UEmultiplicity distribution dN ch /dη , likewise the minimum bias multiplicity density, hasa central plateau in rapidity (we assume that jet is produced at y = 0). For typicalinelastic events the existence of the central plateau is a consequence of the approximatelongitudinal boost invariance. In the glasma picture [21] it naturally appears due toboost invariance of the initial glasma color fields. However, for the UE events thisinvariance is broken by the jet production at y = 0. And in principle there may be abump in the UE multiplicity distribution near the jet rapidity, say due to the initialstate radiation. For (4) to be applicable the half-width of the bump should satisfy theinequality ∆ η ∼ > c s ln( τ f /τ ) , (9)where τ f is the freezeout time and c s is the sound velocity of the matter. This inequalityensures that in the whole interval from τ to τ f the edges of the bump do not affectthe τ -dependence of the entropy s ∝ /τ . For the relevant temperature range c s ∼ < . τ f ∼ τ max and τ max /τ ∼ η ∼ >
1. The factthat data on the UE dN ch /dη from ATLAS [40] obtained for | η | < . | η | < .
3. Sketch of the Calculations
We now turn to the jet quenching in the mini-QGP produced in pp collisions. Ourtreatment is similar to that for AA collisions in our previous analysis [32] to which werefer the reader for details. Here we give only a brief sketch of the calculations, focusingon aspects relevant for jet quenching in the mini-QGP, and present parameters of themodel. As usual we write the perturbative inclusive cross section in (2) in terms of the vacuumparton → hadron FF D h/i dσ pert ( pp → hX ) d p T dy = X i Z dzz D h/i ( z, Q ) dσ ( pp → iX ) d p iT dy , (10)where dσ ( pp → iX ) /d p iT dy is the ordinary hard cross section, p iT = p T /z is the partontransverse momentum. We write the real inclusive cross section, which accounts forthe final state interactions in the QCD matter, in a similar form but with the mediummodified FF D mh/i dσ ( pp → hX ) d p T dy = X i Z dzz D mh/i ( z, Q ) dσ ( pp → iX ) d p iT dy . (11)Here it is implicit that D mh/i is averaged over the geometrical variables of the hard partonprocess and the impact parameter of pp collision.The formula (11) can be viewed as an analogue of the formula for the minimum bias R AA defined in the whole centrality (impact parameter) range. However, there is oneimportant difference between pp and AA collisions. In AA collisions at a given impactparameter the fluctuations of the multiplicity (and of the parameters of the fireball)are small. And this allows to relate the centrality (defined through the multiplicity)to the impact parameter. In pp collisions one cannot relate the multiplicity density tothe impact parameter, since for each impact parameter and jet production point in thetransverse plane (which can be localized with the accuracy ∼ z/p T ) the fluctuations ofthe multiplicity density are large. These fluctuations, together with the event-by-eventfluctuations of the impact parameter and the jet production point, give the observablefluctuating UE dN ch /dη , which can be translated into the fluctuating fireball parameters.However, the detailed dynamics of the UEs and of the multiplicity fluctuations in suchevents is far from being clear. In particular, we do not know whether the enhancementof the UE multiplicity is only due to the fact that the jet production is biased to morecentral collisions and to which extent it may be related to the increase of the softgluon density in jet production due to the initial state radiation. Therefore an accurateaccounting for the fluctuations of the parameters of the mini-QGP fireball is impossible.In the present study in evaluating D mh/i we take into account (approximately) only theevent-by-event variations of the geometrical parameters (see below), but ignore thefluctuations of the UE dN ch /dη . And evaluate the parameters of the fireball simply usingthe typical UE multiplicity density, although technically the inclusion of the fluctuationsof the UE dN ch /dη in our formalism is quite simple, and we do it to estimated theaccuracy of our approximation (see below).As in [32], we calculated the hard cross sections in the LO pQCD with the CTEQ6[47] parton distribution functions (PDFs). To simulate the higher order effects incalculating the partonic cross sections we take for the virtuality scale in α s the value cQ with c = 0 .
265 as in the PYTHIA event generator [48]. This gives a fairly gooddescription of the p T -dependence of the spectra in pp collisions. Of course, in principle,in the scenario with the QGP formation for a fully consistent treatment of R pp (and R AA ) one should use a bootstrap procedure and compare with the experimental datanot the perturbative cross section (10) but the real one given by (11), and namely thelatter should be adjusted (say, by varying PDFs, FFs, and α s ) to describe experimentaldata. However, since the hadron spectra have very steep p T -dependence (as comparedto a relatively weak p T -dependence of R pp ) this inconsistency may be safely ignored incalculating R pp (the same is true for R AA and R pA ).For the hard scale Q in the FFs in (10), (11) we use p T /z . We calculate thevacuum FFs D h/j as a convolution of the KKP [49] parton → hadron FFs at soft scale Q = 2 GeV with the DGLAP parton → parton FFs D DGLAPj/i describing the evolutionfrom Q to Q . The latter have been computed with the help of PYTHIA [48]. Thisprocedure reproduces well the whole Q -dependence of the KKP [49] parametrizationof the vacuum FFs. For a given fast parton path length in the QGP the mediummodified FFs D mj/i have been calculated in a similar way but inserting between theDGLAP parton → parton FFs and the KKP parton → hadron FFs the parton → partonFFs D indj/i which correspond to the induced radiation stage in the QGP. The inducedradiation FFs D indj/i have been calculated from the medium induced gluon spectrumusing Landau’s method [50] imposing the flavor and momentum conservation (again, werefer the interested reader to [32] for details).Note that, since in both the vacuum and the medium modified FFs the DGLAPevolution is accounted for in the same way, the medium effects vanish strictly at zeromatter density, as it must be. The above approximation with the time ordered andindependent DGLAP and induced radiation stages, suggested for the large-size plasmaproduced in AA collisions [32], seems to be reasonable for the mini-QGP as well (atleast in the jet energy region ∼ < −
50 GeV where the suppression effect appears tobe strongest) since the typical formation time for the most energetic DGLAP gluons isof the order of (or smaller) than the thermalization time τ . It is worth noting that,although the time ordering of the DGLAP and induced radiation stages seems to bephysically reasonable, the permutation of these stages in the above convolution gives avery small effect [32].Since we do not consider the azimuthal effects, the averaging of the mediummodified FFs over the geometrical variables of the hard parton process and pp collisionsin the impact parameter plane is simply reduced to averaging over the parton path length L in the QGP. It cannot be performed accurately since the distribution of hard processesin the impact parameter plane is not known yet. But one can expect that the effectof L fluctuations should be relatively small for any more or less centered distributionof energetic partons in the proton wave function. We have performed averaging over L using the distribution of hard processes in the impact parameter plane obtained with thequark distribution from the MIT bag model (we assume that the valence quarks and thehard gluons radiated by the valence quarks follow approximately the same distribution inthe transverse spacial coordinates). Calculations within this model show that practicallyin the full range of the impact parameter of pp collisions the distribution in L is sharplypeaked around L ≈ p S ov /π , where S ov is the overlap area for two colliding bags. Itmeans that our fireball radius R f (which includes all centralities) at the same time givesthe typical path length for fast partons. Our calculations show that the effect of the L -fluctuations on R pp is relatively small. As compared to L = R f they reduce the mediummodification by ∼ − AA collisions we treat the collisionalenergy loss, which is relatively small [51], as a small perturbation to the radiativemechanism. We incorporate it in the above procedure simply by renormalizing theQGP temperature in calculating the medium modified FFs (see [32] for details). Weassume that the collisional energy loss vanishes at τ < τ in the pre-equilibrium stagewhich probably is populated by strong collective glasma color fields, and the conceptof the collisional energy loss is hardly applicable in this region. On the contrary, it isclear that the coherent glasma fields can give some contribution to the radiative energyloss (probably rather small [52]). For this reason the use of the linearly growing plasmadensity at τ < τ seems to be a plausible parametrization to model the transition fromthe glasma phase to the hydrodynamically evolving QGP, which of course cannot beabrupt. As in [32] we evaluate the medium induced gluon spectrum dP/dx ( x = ω/E is thegluon fractional momentum) for the QGP modeled by a system of the static Debyescreened color centers [1]. We use the Debye mass obtained in the lattice calculations[53] giving µ D /T slowly decreasing with T ( µ D /T ≈ . T ∼ T c , µ D /T ≈ . T ∼ T c ). For the quasiparticle masses of light quarks and gluon in the QGP we take m q = 300 and m g = 400 MeV supported by the analysis of the lattice data [54]. Butthe results are not very sensitive to the m g , and practically insensitive to the value of m q . We evaluated the induced gluon spectrum using the representation suggested in[55]. It expresses the x -spectrum for gluon emission from a quark (or gluon) through thelight-cone wave function of the gq ¯ q (or ggg ) system in the coordinate ρ -representation.The z -dependence of the wave function is governed by a two-dimensional Schr¨odingerequation with the “mass” µ = x (1 − x ) E ( E is the initial parton energy) in which thelongitudinal coordinate z plays the role of time and the potential v ( ρ ) is proportional tothe local plasma density/entropy times a linear combination of the dipole cross sections σ ( ρ ), σ ((1 − x ) ρ ) and σ ( xρ ). Note that the physical pattern of induced gluon emissionin the mini-QGP differs from that for the large-size QGP. For the mini-QGP whenthe typical path length in the medium L ∼ − . L f ∼ > L , where L f ∼ ω/m g is the gluon formation length in the low densitylimit. It is the diffusion regime in the terminology of [56], in which the finite-size effectsplay a crucial role. In this regime the dominating contribution comes from the N = 1rescattering and the Coulomb effects are very important [56]. On the contrary, for theQGP in AA collisions a considerable part of the induced energy loss comes from gluonswith L f ∼ < L . Indeed, in the bulk of the large-size QGP L f ∼ ωS LP M /m g , where S LP M is the LPM suppression factor. For RHIC and LHC typically S LP M ∼ . − . ω ∼ L f ∼ . − . L for the QGP in AA collisions. In this regime the finite-size effects are much less important and inducedgluon radiation is (locally) approximately similar to that in an infinite extent matter.From the point of view of jet quenching in pp collisions it is important that inducedradiation in the mini-QGP is more perturbative than in the QGP in AA collisions.Indeed, let us consider induced radiation for the mini-QGP. From the Schr¨odingerdiffusion relation one can obtain for the typical transverse size of the three partonsystem ρ ∼ ξ/ω , where ξ is the path length after gluon emission. Then, using thefact that σ ( ρ ) is dominated by the t -channel gluon exchanges with virtualities up to Q ∼ /ρ [57] we obtain Q ∼ ω/ξ . For ω ∼ ξ ∼ . − Q ∼ − . The virtuality scale for α s in the gluon emissionvertex has a similar form but smaller by a factor of ∼ . /ξ dependenceof Q persists up to ξ ∼ L f . For the large-size QGP in the above formulas one shouldreplace ξ by the real in-medium L f (which contains S LP M ) which is by a factor of ∼ ξ for the mini-QGP. It results in a factor of ∼ AA collisions. In this sense the calculations for the mini-QGPare more robust than for the large-size QGP.As in [32, 33, 14, 15] we perform calculations of radiative and collisional energyloss with running α s frozen at some value α frs at low momenta. For gluon emission invacuum a reasonable choice is α frs ∼ . − . α frs . However, in principle, the extrapolation from the vacuum gluonemission to the induced radiation is unreliable due to large theoretical uncertainties ofjet quenching calculations. For this reason α frs should be treated as a free parameterof the model. To evaluate the medium suppression in pp collisions it is reasonableto use the information on the values of α frs necessary for description of jet quenchingin AA collisions. Previously we have observed [15] that data on R AA are consistentwith α frs ∼ . α frs ∼ . α frs from RHICto LHC may be related to stronger thermal effects in the QGP due to higher initialtemperature at LHC. But the analysis [15] is performed under assumption that there isno medium suppression in pp collisions. The inclusion of R pp should increase the valuesof α frs . However, in [15] we used the plasma density vanishing at τ < τ , whereas in thepresent work we use the QGP density ∝ τ in this region which leads to stronger mediumsuppression. As a result, as we will see below, the values of α frs , which are preferablefrom the standpoint of the description of the data on R AA , remain approximately thesame, or a bit larger, as obtained in [15]. If the difference between the preferable values0of α frs for AA collisions at RHIC and LHC is really due to the thermal effects, thenfor the mini-QGP with T as given in (8) a reasonable window is α frs ∼ . − .
7. Inprinciple for the mini-QGP the thermal reduction of α s may be smaller than that forthe large-size plasma (at the same temperature). Since at L f ∼ < L , which typically holdsfor the mini-QGP, the dominating contribution to the induced gluon spectrum comesfrom configurations with interference of the emission amplitude and complex conjugateone when one of them has the gluon emission vertex outside the medium and is notaffected by the medium effects at all. We perform the calculations for α frs = 0 .
5, 0 . .
7. Note that, in principle R pp should be less sensitive to α frs than R AA since, aswe said above, the typical virtualities for induced gluon emission in the mini-QGP arelarger than that in the large-size QGP. As will be seen from our numerical results, thesensitivity to α frs is really quite weak.
4. Numerical Results
Before presenting the results for the medium modification factors it is worthwhile firstto show the results for radiative and collisional energy loss that may give some insightinto the magnitude of the medium effects generated by the mini-QGP in pp collisions.In Fig. 1 we show the energy dependence of the total (radiative plus collisional) andcollisional energy loss for partons produced in the center of the mini-QGP fireball forRHIC and LHC conditions for α frs = 0 .
6. Both the radiative and collisional contributionsare calculated for the lost energy smaller than half of the initial parton energy. Thefireball radius R f and the initial temperature T have been calculated with the UEmultiplicity density dependent on the jet energy E using the data [39, 40]. In [39, 40]the UE activity has been measured vs the transverse momentum of the leading chargedjet hadron (we denote it as p lT ). To obtain the UE dN ch /dη as a function of the jetenergy E we neglect the fluctuations of p lT for a given E and use the rigid relation p lT = h z l i E , where h z l i is the average fractional momentum of the leading jet hadron.For the h z l i we take the PYTHIA predictions, which gives in the relevant energy region( E ∼ <
10 GeV) h z l i ∼ .
26 for gluon jets and h z l i ∼ .
36 for quark jets. The jet energydependence of the parameters of the fireball becomes important only for partons with E ∼ < −
15 GeV. At higher energies the UE dN ch /dη is flatten and R f and T arevery close to that given by (7) and (8). And radiative and collisional energy loss maybe calculated using (7), (8). To illustrate it in Fig. 1 we presented the results for thetotal energy loss obtained for the fireball parameters for the UE dN ch /dη in the plateauregion ( p lT ∼ > E ∼ >
10 GeV. This says that the decrease of the UE multiplicity density at p lT ∼ < R pp ( p T ) already at p T ∼ > −
10 GeV.Fig. 1 shows that the parton energy loss in the mini-QGP turns out to be quite1
E [GeV] E ( E ) [ G e V ] ∆ g, s =0.2 TeV g, s =2.76 TeVq, s =0.2 TeV q, s =2.76 TeV radiative+collisionalcollisional Figure 1.
Energy dependence of the energy loss of gluons (upper panels) and lightquarks (lower panels) produced in the center of the mini-QGP fireball at √ s = 0 . √ s = 2 .
76 TeV (right). Solid line: total (radiative plus collisional) energyloss calculated with the fireball radius R f and the initial temperature T obtained withthe UE dN ch /dη dependent on the initial parton energy E ; dashed line: same as solidline but for collisional energy loss; long-dashed line: same as solid line but for R f and T obtained with the UE dN ch /dη in the plateau region as given by (7) and (8). Allthe curves are for α frs = 0 . L [fm] E ( L ) [ G e V ] ∆ radiative T =199 MeVradiative T =320 MeV (x(199/320) )collisional T =199 MeV L [fm] E ( L ) [ G e V ] ∆ radiative T =217 MeVradiative T =420 MeV (x(217/420) )collisional T =217 MeV Figure 2.
Left: Radiative (solid) and collisional (dashed) gluon energy loss vs the path length L in the QGP with T = 199 MeV for (bottom to top) E = 20 and 50 GeV. The dotted linesshow radiative energy loss for T = 320 MeV rescaled by the factor (199 / . All curves arecalculated for α frs = 0 .
6. Right: same as in the left figure but for T = 217 and 420 MeV andthe rescaling factor (217 / for dotted lines. E ∼ −
20 GeV for gluons the total energy loss is ∼ −
15% of the initialenergy. The contribution of the collisional mechanism is relatively small. The energyloss for the mini-QGP shown in Fig. 1 is smaller than that obtained in [15] for thelarge-size QGP in AA collisions by only a factor of ∼ L -dependence of the parton energy loss in our model in Fig. 2 weshow the results for the radiative and collisional gluon energy loss vs the path length L for E = 20 and 50 GeV for T = 199 and 217 MeV, corresponding to √ s = 0 . .
76 TeV. To show the difference between the QGP produced in pp and AA collisionswe present also predictions for radiative energy loss for T = 320 MeV corresponding tocentral Au + Au collisions at √ s = 0 . T = 420 MeV corresponding tocentral P b + P b collisions at √ s = 2 .
76 TeV ( the procedure that leads to these valuesof T is described in [15]). To illustrate the temperature dependence better we rescaledthe predictions for AA collisions by the factor ( T ( pp ) /T ( AA )) . One can see that at L ≥ τ the radiative energy loss is approximately a linear function of L . At L < τ the radiative energy loss is approximately ∝ L (since the leading N = 1 rescatteringcontribution to the effective Bethe-Heitler cross section is ∝ L [56, 60] and integrationover the longitudinal coordinate of the scattering center gives additional two powers of L ). The comparison of the radiative energy loss for T = 199 and 217 MeV to that for T = 320 and 420 MeV shows deviation from the T scaling by factors of ∼ . ∼ L ∼ T scaling comes mostly from the increase of the LPM suppression (and partlyfrom the increase of the Debye mass) for the QGP produced in AA collisions. pp collisions As we have seen from Fig. 1 the dependence of the UE multiplicity density from[39, 40] on the momentum of the leading jet hadron p lT practically does not affectthe parton energy loss at E ∼ > −
15 GeV, which from the standpoint of the particlespectra corresponds approximately to p T ∼ > −
10 GeV. To account for the effect ofthe p lT -dependence of the UE dN ch /dη on R pp ( p T ) at p T ∼ <
10 GeV we use the rigidapproximation p lT = h z l i E as in the above calculations of the energy loss. And inaddition ignore the fluctuations of the variable z in (10) (since the integrand of (10) isquite sharply peaked about h z i ). In this approximation we can write p lT = p T /η , where p T is the momentum of the observed particle in (10) and η = h z i / h z l i . Jet simulationwith PYTHIA [48] shows that for jets with energy E ∼ < −
15 GeV, that can feel theenergy dependence of the UE multiplicity, one can take η ∼ . √ s = 0 . η ∼ . √ s ∼ > .
76 TeV. The uncertainties from this prescription arerestricted to the region p T ∼ < −
10 GeV. However, even in this region it should workon average (in the sense that the fluctuations will just smear the p T -dependence of themedium suppression in this region). This problem becomes completely irrelevant for R pp at p T ∼ >
10 GeV.In Fig. 3 we present the results for R pp of charged hadrons at √ s = 0 .
2, 2 .
76 and 73 (a) s =0.2 TeV (b) s =2.76 TeV (c) s =7 TeV p T [GeV] R pp ( p T ) Figure 3. R pp of charged hadrons at √ s = 0 . .
76 (b), 7 (c) TeV for (top tobottom) α frs = 0 .
5, 0 . . τ = 0 . . TeV for α frs = 0 .
5, 0 . .
7. To illustrate the sensitivity of the results to τ we showthe curves for τ = 0 . . τ = 0 . p T ∼ <
20 GeV both for RHICand LHC. Fig. 3 shows that for τ = 0 . R pp does not exhibits a strong dependenceon α frs . Although the plasma density is smaller at √ s = 0 . √ s = 2 .
76 and 7 TeV. It is due to a steeper slope ofthe hard cross sections at √ s = 0 . √ s = 2 . √ s = 7 TeV is relatively small.To understand the sensitivity of R pp to the fireball radius we also performedthe calculations for the fireball radii calculated with (5), (6) times 0 . .
3. Weobtained in these two cases the reduction of the medium suppression by ∼
3% and10%, respectively. The weak dependence on the value of R f is due to a compensationbetween the enhancement of the energy loss caused by increase of the fireball size and itssuppression caused by reduction of the fireball density. § . The stability of R pp againstvariations of R f gives a strong argument that the errors due to the neglect of thevariation of the plasma density in the transverse coordinates should be small. Indeed,the dominating N = 1 rescattering contribution to the radiative energy loss is a linearfunctional of the plasma density profile along the fast parton trajectory. It means thatthe results for a more realistic distribution of the initial plasma density in the impactparameter plane with a higher density in the central region can be roughly approximatedby a linear superposition of the results obtained for the step density distributions (withdifferent R f ) that should lead to approximately the same R pp as our calculations. Notealso that since the variation of the plasma density in our test is very large (by a factor of ∼ .
5) this stability at the same time indicates indirectly that the effect of the neglected § The fact that for 0 . R f and 1 . R f the variations of R pp have the same sign is not very surprisingsince we use a wide window in the the fireball size. In this situation the second order term in the Taylorexpansion of R pp around R f may be bigger than the linear term dN ch /dη is impossible since it should be done on the event-by-even basis (in the sense of theimpact parameter and the jet production point), and requires detailed information aboutdynamics of the UEs. To understand how large the theoretical uncertainties, related tothe event-by-event fluctuations of the UE dN ch /dη , might be we evaluated R pp assumingthat the distribution in the UE dN ch /dη is the same at each impact parameter and jetproduction point. We performed the calculations using the distribution in dN ch /dη fromCMS [41] measured at √ s = 0 . dN ch /dη the magnitude of (1 − R pp )is reduced by only ∼ −
6% both for RHIC and LHC energies. This says that ourapproximation without the event-by-event fluctuations of the fireball parameters is quitegood, since it is very unlikely that an event-by-event analysis may change significantlythe results obtained using the total fluctuations of the UE multiplicity density. AA collisions Although R pp is unobservable quantity it can alter the results of the jet tomography of AA collisions. To illustrate the possible effect of the mini-QGP in pp collisions on R AA we show in Fig. 4 the comparison of our results for R AA with the data for π -mesons incentral Au + Au collisions at √ s = 0 . P b + P b collisions at √ s = 2 .
76 TeV (b,c) from ALICE [63]and CMS [64]. We show the results obtained with (solid) the 1 /R pp factor, i.e. for R AA defined by (3), and the results without (dashed) this factor, i.e. for R stAA . We use the R pp obtained with α frs = 0 .
6. We present the curves for R stAA obtained with α frs = 0 . . √ s = 0 . α frs = 0 . . √ s = 2 .
76 TeV. Since thesevalues give better agreement with the data of the R AA given by (3). In calculatingthe hard cross sections for AA collisions we account for the nuclear modification of thePDFs with the EKS98 correction [65]. As in [15] we take T = 320 MeV for central Au + Au collisions at √ s = 0 . T = 420 MeV for central P b + P b collisionsat √ s = 2 .
76 TeV obtained from hadron multiplicity pseudorapidity density dN ch /dη from RHIC [66] and LHC [67, 68]. One can see that at p T ∼
10 GeV for RHIC theagreement of the theoretical R AA (with the 1 /R pp factor) with the data is somewhatbetter for α frs = 0 .
6, and for LHC the value α frs = 0 . p T -dependence of R AA is evidently not perfect (especially forLHC). One sees that the theory somewhat underestimates the slope of the data. Andthe regions of large p T support α frs = 0 . . R pp even reduces a little the slope of R AA (since R pp in thedenominator on the right hand side of (3) grows with p T ). However, this discrepancy5 (a) Au+Au, s =0.2 TeV (b) Pb+Pb, s =2.76 TeV (c) Pb+Pb, s =2.76 TeV π ο h ch ALICE 0-5% h ch CMS 0-5% p T [GeV] R AA ( p T ) PHENIX 0-5%
Figure 4. (a) R AA of π for 0-5% central Au + Au collisions at √ s = 0 . α frs = 0 . . /R pp factor in (3). (b,c) R AA for charged hadrons for 0-5% central P b + P b collisionsat √ s = 2 .
76 TeV from our calculations for (top to bottom) α frs = 0 . . /R pp factor in (3). The solid curves are obtained withthe factor 1 /R pp calculated with α frs = 0 .
6. Data points are from PHENIX [62] (a),ALICE [63] (b) and CMS [64] (c). Systematic experimental errors are shown as shadedareas. does not seem to be very dramatic since the theoretical uncertainties may be significant.From Fig. 4 one can see that the effect of R pp on R AA for the central AA collisionscan approximately be imitated by simple reduction of the α frs . However, of course, itdoes not mean that all the theoretical predictions for jet quenching in AA collisions areinsensitive to the medium modification of high- p T spectra in pp collisions. It is clearthat the effect of the R pp should be important for v and centrality dependence of R AA (simply because in the scenario with the mini-QGP formation in pp collisions the valuesof α frs become bigger). It should also be important for the flavor dependence of the thetheoretical R AA since the suppression effect for heavy quarks in pp collisions is smaller(by a factor of ∼ . − pA collisions The medium suppression factor R pp should also be taken into account in calculatingthe nuclear modification factor for pA collisions. Similarly to (3) the correct formulareads R pA = R stpA /R pp . Comparison with data on R pA may be even more crucial forthe scenario with the formation of the mini-QGP in pp collisions since the sizes andthe initial temperatures of the plasma fireballs in pp and pA collisions should not differstrongly. And for this reason the predictions for R pA should not have much uncertaintiesrelated to variation of α s or the temperature dependence of the plasma density andthe Debye mass. The ALICE data [31] on R pP b at √ s = 5 .
02 TeV exhibit a smalldeviation from unity at p T ∼ >
10 GeV, where the Cronin effect should be weak. In6 (b) (c) h ch ALICE p T [GeV] R p A ( p T ) (a) p+Pb, s =5.02 TeV Figure 5. (a) R pP b for charged hadrons at √ s = 5 .
02 TeV from our calculationsfor α frs = 0 . /R pp factor for (top to bottom) K ue = 1, 1 .
25 and 1 . R f ( pP b ) from (12). (b,c) same as (a) but for R f ( pP b )times 1 . .
4. The dotted line shows R pp . The dot-dashed line shows R pP b due tothe EKS98 correction [65] to the nucleus PDFs. Data points are from ALICE [31]. the scenario with the formation of the QGP in pp and pA collisions this is possibleonly if the magnitudes of the medium suppression in both the processes are very closeto each other. Unfortunately, presently we have not data on the UE multiplicity in pP b collisions. However, it is clear that it cannot be smaller than the minimum biasmultiplicity density dN mbch /dη = 16 . ± .
71 [69]. In principle, it is possible that in thetypical minimum bias events the energy deposited in the central rapidity region is alreadysaturated due to a large number of the nucleons which interact with the proton in each pP b collision, and the enhancement of the multiplicity due to jet production is relativelysmall. The preliminary PHENIX data [39] on the UE in dAu collisions at √ s = 0 . pP b collisions in the scenario with themini-QGP formation we simply calculate R pP b for dN ch /dη = K ue dN mbch /dη for K ue = 1,1 .
25, and 1 . R pP b we also need the fireball radius R f ( pP b ) which may be bigger thanthat in pp collisions. In our calculations as a basic choice we use the parametrizationof the R f ( pP b ) as a function of the multiplicity density given in [38] obtained fromthe results of simulation of the pP b collisions performed in [21] within the IP-Glasmamodel [37]. The R f ( pP b ) from [37] is close to the R f ( pp ) in the region where R f ( pp ) ∝ ( dN g /dy ) / , but flattens at higher values of the gluon density. Using theparametrization for R f ( pP b ) of Ref. [38] and formula (4), we obtained for our set of theenhancement factors for the UE multiplicity K ue = [1 , . , . R f ( pP b ) ≈ [1 . , . , .
98] fm , (12) T ( pP b ) ≈ [222 , , . (13)7Comparison of our results with the data on R pP b at √ s = 5 .
02 TeV from ALICE[31] is shown in Fig. 5. To illustrate the sensitivity to the fireball size in Fig. 5 we alsopresent the results for the R f ( pP b ) 1 . . AA collisions weshow the curves with (solid) and without (dashed) the 1 /R pp factor. Similarly to R AA weaccount for the nuclear modification of the PDFs with the EKS98 correction [65] (whichgives a small deviation of R pP b from unity even without parton energy loss). The resultsfor R pp are also shown (dotted). All the curves are obtained with α fr = 0 .
6. However,our predictions for R pP b (with the 1 /R pp factor) are quite stable against variation of α frs since the medium suppression is very similar for pp and pP b collisions.From Fig. 5 one can see that at p T ∼ >
10 GeV, where the Cronin effect should besmall, our predictions (with 1 /R pp factor) obtained with K ue = 1 agree qualitativelywith the data. The agreement becomes better with increase of the R f ( P b ). However,similarly to R pp the variation of R pP b with the fireball size is relatively weak. Thecurves for the higher UE multiplicities ( K ue = 1 .
25 and 1 .
5) lie below the data. Thus,Fig. 5 shows that the data from ALICE [31] may be consistent with the formation of theQGP in pp and pP b collisions if the UE multiplicity is close to the minimum bias one.This condition may be somewhat weakened if the size of the fireball in pP b collisions isconsiderably bigger than predicted in [21]. But it seems to be rather unrealistic sincethe required increase of the fireball size is too large. Say, for a good agreement withthe data for the UE multiplicity enhancement factor K ue = 1 . R f ( pP b ) by a factor of ∼ . One remark is in order about the description of the QGP in the ideal gas modelin our study. Of course, the QGP temperature formally defined in this model fromexperimental multiplicity densities is somewhat incorrect. Say, at T ∼
200 MeV theideal gas formula for the entropy underestimates the plasma temperature by 10 − v ( ρ ) in the two-dimensional Schr¨odinger equation, which is used for evaluation of the induced gluon x -spectrum in the LCPI approach [3], is proportional to the entropy. Since in ourcalculations in each case we use the entropy extracted directly from the experimentalmultiplicity densities, the temperature enters our calculations only through the Debyemass in the dipole cross section. But the latter depends weakly on the Debye mass.For this reason 10 −
15% errors in the temperature can be safely ignored. Anaccurate definition of the temperature does not make much sense since anyway thenonperturbative effects should modify the form of the potential. Also, one should bearin mind that presently there are many other theoretical uncertainties in the jet quenchingcalculations, and practically the theory cannot give absolute predictions for the mediumsuppression. However, one can expect that it can be used to describe the variation ofjet quenching from one experimental situation to another (when the parameters of the8model are already fitted to some experimental data). And we do follow this strategyin the present work. We calculate the medium suppression for the small-size plasmain pp collisions using the information about the values of α frs which are necessary fordescription of the data on R AA . Without this information one could obtain only avery crude estimate of the effect. Of course, the extrapolation from AA to pp collisionsassumes that for the real QGP the potential v is approximately proportional to theentropy, as it is for the ideal QGP. But this assumption seems to be physically veryreasonable. Anyway the extrapolation from AA to pp collisions should not give largeerrors since the difference of the plasma temperatures in these two cases is not very big.It is worth to emphasize that for a reliable extrapolation of the theoreticalpredictions from AA to pp collisions the calculations should be performed with accuratetreatment of the LPM suppression (which is very important for AA collisions) and finite-size and Coulomb effects (which are very important for the mini-QGP produced in pp collisions). Also, the calculations should be made with running α s since the typicalvirtualities for induced gluon emission in the mini-QGP are higher than that in thelarge-size QGP in AA collisions. The LCPI [3] approach used in the present analysissatisfies all these requirements.Note that, in principle, the assumption that the produced QCD matter exists inthe form of an equilibrated QGP is not crucial for our main result that there must be arather strong jet quenching in pp collisions. Since even if the created matter is some kindof a hadron resonance gas the parton energy loss will be approximately the same as forthe QGP because for a given entropy density the intensities of multiple scattering for thehadron matter and the QGP are very similar [70]. Note that, since the most importantquantity, which controls induced gluon emission, is the number density of the colorconstituents in the medium, from the standpoint of jet quenching, it is even not veryimportant whether the created QCD matter is equilibrated or not. Therefore one can saythat in the pQCD picture of jet quenching the significant medium suppression of hadronspectra in pp collisions is an inevitable consequence of the observed UE multiplicitiesin pp collisions and the medium suppression of hadron spectra in AA collisions (whichallows to fix free parameters).From the point of view of the pQCD the medium suppression of the high- p T spectrain pp collisions may be regarded as a higher twist effect. And of course it would beinteresting to observe it through a deviation of the experimental spectra from predictionsof the standard pQCD formulas. But it is difficult since the medium suppression shouldhave a very smooth onset in the energy region where the regime of free streaming hadronstransforms to a relatively slow collective expansion of the fireball. Probably it could stilloccur at √ s ∼ −
40 GeV, where the UE pseudorapidity multiplicity density may be ∼ − T ∼ T c . For this reason direct observation of this effect by comparingthe pQCD predictions with experimental spectra is hardly possible since it is fairlyhard to differentiate it from the variations of the theoretical predictions related to smallmodifications of the PDFs and of the FFs or other higher twist effects not related tothe mini-QGP. Also, presently the uncertainties of the pQCD predictions remain large9[71, 72, 73, 74, 75], and the deviation of the ratio data/theory from unity at energies √ s ∼ <
50 GeV [71, 72, 74] is often considerably bigger than the found medium effects.In this situation it is difficult to identify a relatively small effect from the mini-QGP.Nevertheless, it worth noting that the results of the most recent NLO pQCD analysisof the inclusive charged particle spectra in pp and ¯ pp collisions at √ s = 0 . − √ s data. But quantitatively the observed effect is considerablylarger than what can be associated with the difference between R pp at RHIC and LHCenergies found in the present analysis.It is worth noting that in principle the preliminary data from ALICE [27] for √ s = 7 TeV support the existence of jet quenching in pp collisions. These data clearlyindicate that the jet fragmentation becomes softer with increase of the UE multiplicity.It is important that the effect is well seen already for the UE dN ch /dη smaller thanthe average one by a factor of ∼ dN ch /dη at √ s = 0 .
5. Summary
Assuming that a mini-QGP may be created in pp collisions, we have evaluated themedium modification factor R pp for light hadrons at RHIC ( √ s = 0 . √ s = 2 .
76 and 7 TeV) energies. We have found an unexpectedly large suppressioneffect. For p T ∼
10 GeV we obtained R pp ∼ [0 . − . , . − . , . − .
7] at √ s = [0 . , . ,
7] TeV. We analyzed the role of the R pp in the theoretical predictionsfor the nuclear modification factor R AA in central AA collisions at RHIC and LHCenergies. We found that the presence of R pp does not change dramatically the descriptionof the data on R AA for light hadrons in central AA collisions, and its effect may beimitated by some renormalization of α s . Nevertheless, the effect of the QGP formationin pp collisions may be potentially important in calculating other observables in AA collisions. For example, it should affect v and the centrality dependence of R AA , and,due to the flavor dependence of R pp , its effect may be important for description of theflavor dependence of R AA . We leave analysis of these effects for future work. We alsocalculated the nuclear modification factor R pP b at √ s = 5 .
02 TeV. Comparison withthe data from ALICE [31] shows that the scenario with the formation of the QGP in pp and pP b collisions may be consistent with the data only if the UE multiplicity densityin pP b collisions (which is unknown yet) is close to the minimum bias one. Acknowledgements
I am grateful to I.P. Lokhtin for discussion of the results. I also thank D.V. Perepelitsa0for useful information. I am indebted to the referee, who pointed out on the recentanalysis [76]. This work is supported in part by the grant RFBR 12-02-00063-a.
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