Partonic energy loss in ultrarelativistic heavy ion collisions: jet suppression versus jet fragmentation softening

We discuss the modification of a jet fragmentation function due to medium-induced partonic energy loss in context of leading particle observables in ultrarelativistic nucleus-nucleus interactions. We also analyze the relation between in-medium softening jet fragmentation function and suppression of the jet rates due to energy loss outside the jet cone. The predicted anti-correlation between two effects allows to probe a fraction of partonic energy loss carried out of the jet cone and truly lost to the jet.


Introduction
Jets play one of the central roles as a promising tool to study properties of quark-gluon plasma (QGP) expected to be created in heavy ion collisions at RHIC and LHC.Medium-induced energy loss of energetic partons, the so-called jet quenching, has been proposed to be very different in cold nuclear matter and in QGP, resulting in many challenging observable phenomena (see for the review [1] and references therein).In particular, softening jet fragmentation function and, as a consequence, suppression of high-p T hadron spectrum in nucleus-nucleus collisions relative to their production in independent nucleon-nucleon interactions, are considered [2,3,4,5,6,7,8].Recent RHIC data on inclusive high-p T charge and neutral hadron production from STAR [9], PHENIX [10], PHOBOS [11] and BRAHMS [12] experiments show such kind of suppression and are in agreement with the jet quenching hypothesis.However, since at the moment direct event-by-event reconstruction of jets and their characteristics is not available in RHIC experiments, the assumption that integrated yield of all high-p T particles originates only from jet fragmentation is not fully clear (see, for example, [13]).At the LHC, a new regime is reached where hard and semi-hard QCD multi-particle production can certainly dominate over underlying soft events [14].CMS experiment at LHC [15] will be able to provide adequate jet reconstruction using calorimetric measurements [16,14].Thus identification of leading particle in a jet (i.e.particle carrying the maximal fraction of jet transverse momentum) allows the measurement of jet fragmentation function (JFF) to be done.Comparison of JFF in AA and pp collisions (or in central and peripheral AA interactions) may give information about in-medium modification of JFF.
The crucial related question here is: how much energy loss falls outside the typical jet cone and is truly lost to the jet?There are some discussions on this subject in the literature [17,18,19,20,21].In fact, since coherent Landau-Pomeranchuk-Migdal radiation induces a strong dependence of the radiative energy loss of a jet on the angular cone size, it will soften particle energy distributions inside the jet, increase the multiplicity of secondary particles, and to a lesser degree, affect the total jet energy.On the other hand, collisional energy loss turns out to be practically independent of jet cone size and causes the loss of total jet energy, because the bulk of "thermal" particles knocked out of the dense matter by elastic scatterings fly away in almost transverse direction relative to the jet axis [17].Moreover, the total energy loss of a jet will be sensitive to the experimental capabilities for low-p T particles, products of soft gluon fragmentation.For example, in the CMS case, most of these low-p T hadrons may be cleared out of the central calorimeters by the strong magnetic field [15,16].
In this Letter we analyze relation between in-medium softening JFF and suppression of jet spectrum due to energy loss outside the jet cone.

Medium-modified jet fragmentation
Let us recall that in the leading order of perturbative QCD the jet production cross section with transverse momentum p T and rapidity y in a nucleon-nucleon collision is given by where x i = p i /p a , x j = p j /p b are the initial momentum fractions of nucleons a, b carried by the interacting partons of types i, j; f i a (x i , Q 2 ), f j b (x j , Q 2 ) are the parton distribution functions for the colliding nucleons a, b; s, t and u are the Mandelstam variables of hard parton subprocess; d σ/d t(ij → kl) is the Born cross section for the hard ij → kl scattering subprocess; Q 2 = p 2 T = ( t u)/ s; t/ u = (x i /x j ) exp (−2y); ξ = p T / √ s(exp (y)/x i + exp (−y)/x j ) is the momentum fraction of parton k carried by the jet; √ s = s/(x i x j ) is the center of mass energy of the colliding nucleons.To calculate the inclusive cross section for "jet-induced" hadron production with transverse momentum p h T and rapidity y h in a nucleon-nucleon collision, one convolutes the jet production cross section with the fragmentation function D h k (z ′ , p 2 T ) for the parton of type k into hadron h: where y h = y, and z ′ = p h /p k = p h T /p T is the momentum fraction of a parton k carried by the final observable hadron.The parton distribution functions f i a (x i , Q 2 ) are measured in deep inelastic scattering experiments such as those at HERA [22], while fragmentation functions D h k (z ′ , Q 2 ) are extracted from e + e − annihilation from PETRA, PEP and LEP [23] and from hadronic collisions from UA1 [24] (gluon fragmentation function).
In nuclear interactions medium-induced energy loss of fast partons can modify the cross sections for high-p T hadrons and jets together with other potentially important nuclear effects like, for instance, parton shadowing playing significant role at small values of the momentum fraction carried by the interacting partons.However for sufficiently hard jets and hadrons under consideration (x i,j > ∼ 0.2) this effect is negligible as well as next-to-leading order (NLO) corrections (K factor ∼ 1) [25].As a result the rate of k-type jets of finite angular cone size θ 0 in mid-rapidity with transverse momentum p jet T in AA collisions at given impact parameter b can be estimated as where r 1,2 (b, r, ψ) are the distances between the nucleus centers and the jet production vertex V (r cos ψ, r sin ψ); r max (b, ψ) ≤ R A is the maximum possible transverse distance r from the nuclear collision axis to the V ; R A is the radius of the nucleus A; T A (r 1,2 ) is the nuclear thickness function (see Ref. [25] for detailed nuclear geometry explanations).The effective shift ∆p jet T (r, ψ, θ 0 ) of jet momentum spectrum depends on the jet cone size θ 0 .The partons will not hadronize inside QGP.In a hadronic medium, we assume that the fragmentation functions can be approximated by their forms in vacuum (see, however, Refs.[26,27] where the corrections to the parton fragmentation function in a thermal medium were discussed).We take into consideration the fragmentation of leading partons only and omit the fragmentation of emitted gluons, because we are interested here in the leading particle in a jet.
Then the rate of high-p T jet-induced hadrons can be estimated as where the shift ∆p T of hadron momentum distribution generally is not equal to the mean in-medium partonic energy loss due to the steep fall-off of the p T -spectrum [3].
The integral jet suppression factor Q can be introduced by the natural way as the ratio of jet rate with energy loss to jet rate without one, JFF is defined here as and coincides approximately with D h k (z, (p jet T min ) 2 ) for the case without energy loss and with the type of jet specified.Note that z ≡ p h T /p jet T (= z ′ p T /p jet T ) is experimentally observable quantity depending on jet cone size θ 0 .Further we are interested also in the ratio where the integral hadron suppression factor is distinguished from the differential hadron quenching factor usually defined as [3,6,7,8] Qh ( 3 Model In order to generate the initial jet distributions in nucleon-nucleon sub-collisions at √ s = 5.5 TeV, we have used PYTHIA 5.7 [28].After that we perform event-by-event Monte-Carlo simulation of rescattering and energy loss of partons in QGP (for model details one can refer to our previous papers [17,25]).The approach relies on an accumulative energy losses, when gluon radiation is associated with each scattering in expanding medium together including the interference effect by the modified radiation spectrum as a function of decreasing temperature dE/dl(T ).The basic kinetic integral equation for the energy loss ∆E as a function of initial energy E and path length L has the form where l is the current transverse coordinate of a parton, dP/dl is the scattering probability density, dE/dl is the energy loss per unit length, λ = 1/(σρ) is in-medium mean free path, ρ ∝ T 3 is medium density at temperature T , σ is the integral cross section of parton interaction in the medium.Such numerical simulation of free path of a hard jet in QGP allows us to obtain any kinematical characteristic distributions of jets in final state.Besides the different scenarios of medium evolution can be considered.
For the calculations we have used collisional part of loss [25], and the dominant contribution to the differential cross section for scattering of a parton with energy E off the "thermal" partons with energy (or effective mass) m 0 ∼ 3T ≪ E.Here C = 9/4, 1, 4/9 for gg, gq and qq scatterings respectively, α s is the QCD running coupling constant for N f active quark flavours, and Λ QCD is the QCD scale parameter which is of the order of the critical temperature, Λ QCD ≃ T c ≃ 200 MeV.The integrated cross section σ is regularized by the Debye screening mass squared µ 2 D (T ).The energy spectrum of coherent medium-induced gluon radiation and the corresponding dominated part of radiative energy loss was estimated using BDMS formalism [18]: ) where τ 1 = L/(2λ g ), y = ω/E is the fraction of the hard parton energy carried by the radiated gluon, and C R = 4/3 is the quark colour factor.A similar expression for the gluon jet can be obtained by substituting C R = 3 and a proper change of the factor in the square bracket in (13), see Ref. [18].The integral ( 13) is carried out over all energies from ω min = E LP M = µ 2 D λ g (λ g is the gluon mean free path), the minimal radiated gluon energy in the coherent LPM regime, up to initial jet energy E. Note that although the radiative energy loss of an energetic parton dominates over the collisional loss by up to an order of magnitude, the relative contribution of collisional loss of a jet growths with increasing jet cone size due to essentially different angular structure of loss for two mechanisms [17].
The medium was treated as a boost-invariant longitudinally expanding quark-gluon fluid, and partons as being produced on a hyper-surface of equal proper times τ [29].For certainty we used the initial conditions for the gluon-dominated plasma formation expected for central Pb−Pb collisions at LHC [30]: τ 0 ≃ 0.1 fm/c, T 0 ≃ 1 GeV, ρ g ≈ 1.95T 3 .For non-central collisions we suggest the proportionality of the initial energy density to the ratio of nuclear overlap function and effective transverse area of nuclear overlapping [25].
Thus we consider that in each i-th scattering off the comoving particle (with the same longitudinal rapidity) a fast parton loses energy collisionally and radiatively, ∆e i = t i /(2m 0 ) + ω i , where t i and ω i are simulated according to Eqs. ( 11) and ( 13) respectively, and the distribution of jet production vertex -according to Eq. ( 3).Finally we suppose that in every event the energy of an initial parton decreases by value ∆p T (r, ψ) = i ∆e i and the jet energy loss, ∆p jet T , is the product of ε times energy loss of an initial parton, ∆p jet T (r, ψ, θ 0 ) = ε • ∆p T (r, ψ).We examine here the fraction ε of partonic energy loss carried out of the jet cone as a phenomenological parameter, since the treatment of angular spectrum of emitted gluons is rather sophisticated and model-dependent [17,18,19,20,21] 1 .We was interested in relatively realistic values of ε in the range from 0 to 1 (although ε can be even larger than 1 at small θ 0 , see [18,20]).
In our simulations we have considered channel with neutral pions only as leading particles in jets.One can expect the similar results for leading charged hadrons.From methodical point of view tracking in heavy ion environment at LHC is rather complicated (although solvable) task, while the reconstruction of electromagnetic clusters in calorimeters being at the moment more understandable [14,16].At high enough transverse momentum of π 0 ( > ∼ 15 GeV at CMS case [14]), two photons from pion decay fall into one crystal of electromagnetic calorimeter, and traditional technique for reconstructing π 0 's using two-photon invariant mass spectrum does not work.However, such electromagnetic cluster can be identified as a leading π 0 , if it belongs a hard jet and carries the significant part of jet transverse energy.

Numerical results and conclusions
Figure 1 shows JFF (6) (without the jet-type specification and therefore experimentally observable) for leading π 0 's for the cases without and with medium-induced energy loss obtained in the frameworks of our model [17,25] in central and minimum-bias Pb−Pb collisions.The threshold for jet reconstruction p jet T min = 100 GeV was used [16] 2 .If ε close to 0 ("small-angular radiation" dominates), then the factor of jet suppression Q jet (5) is close to 1 (there is almost no jet rate suppression), and effect on JFF softening is maximal.Increasing ε (the contribution from "wide-angular radiation" and collisional loss grows) results in stronger jet suppression (Q jet value decreases), but effect on JFF softening becomes smaller, especially for highest z. Figure 2 presents the ε-dependences for jet suppression factor Q jet (without the jet-type specification) and ratio (7) of JFF with energy loss to JFF without loss, D(z > z 0 )/D(z > z 0 , ∆p T = 0), for z 0 = 0.5 and 0.7.Note that in the case without jet quenching, fraction of events when leading π 0 carries larger 50% (70%) of jet transverse momentum is 6.3 • 10 −3 (9.3 • 10 −4 ).One can see the distinctive anti-correlation between strengthening jet suppression and JFF softening (ratio (7) can be even greater than 1 at large enough ε and z values) determined mainly by fraction of partonic energy loss carried outside the jet cone.The physical reason for the effect to be 1 Of course, this parameter is regulated by the jet cone size θ 0 and its dependence on θ 0 was numerically investigated [18,20].We can formally recalculate our result as a function of θ 0 using such numerical estimation or taking into consideration the angular structure of radiative and collisional energy loss in each scattering complicating our simulation.However this analysis demands also the jet finding procedure specification and other technical details.We believe that our simplified treatment here is enough to demonstrate the anti-correlation between jet suppression and JFF softening, moreover the integral hadron suppression factor Q h (p jet T min , z 0 ) in Eq. ( 7) depends on ε implicitly (only via z-definition). 2 The estimated statistics for > 100 GeV jets in CMS acceptance is high enough, at the level ∼ 10 7 jet pairs per 1 month of LHC Pb−Pb run [16,14] opposite in the jet suppression factor and the fragmentation function is it follows.Increasing ε results in decreasing final jet transverse momentum, p jet T = p jet T (∆p jet T = 0) − ε • ∆p jet T (which is the denominator in definition of z ≡ p h T /p jet T in JFF ( 6)) without any influence on the numerator of z and, as a consequence, in reducing effect on JFF softening, while the integral jet suppression factor (5) becomes larger.The remarkable prediction here is that the effect on jet rate suppression becomes comparable with the effect on JFF softening at quite reasonable value ε ∼ 0.3.
In summary, we have analyzed the relation between in-medium softening jet fragmentation function (in leading π 0 channel) and suppression of the jet spectrum due to energy loss outside the jet cone.We believe that this kind of analysis can be performed for the heavy ion collisions at LHC experiments.The observation of significant JFF softening without substantial jet rate suppression would be an indication of the fact that small-angular gluon radiation is dominating mechanism for medium-induced partonic energy loss.Increasing contribution of wide-angular gluon radiation and collisional loss can result in jet rate suppression, while the effect on JFF softening becomes less in this case.If the contribution of the "out-of-cone" jet energy loss is large enough, the jet rate suppression may be even more significant than JFF softening.