Medium-modified average multiplicity and multiplicity fluctuations in jets

The energy evolution of average multiplicities and multiplicity fluctuations in jets produced in heavy-ion collisions is investigated from a toy QCD-inspired model. In this model, we use modified splitting functions accounting for medium-enhanced radiation of gluons by a fast parton which propagates through the quark gluon plasma. The leading contribution of the standard production of soft hadrons is enhanced by a factor √ Ns while next-to-leading order (NLO) corrections are suppressed by 1/ √ Ns, where the parameter Ns > 1 accounts for the induced-soft gluons in the medium. Our results for such global observables are cross-checked and compared with their limits in the vacuum. E-mail: [email protected] ha l-0 03 38 94 7, v er si on 2 19 M ay 2 00 9 Recent experiments at the Relativistic Heavy Ion Collider (RHIC) have established a phenomenon of strong high-transverse momentum hadron suppression [1], which supports the picture that hard partons going through dense matter suffer a significant energy loss prior to hadronization in the vacuum (for recent review see [2]). Predictions concerning multi-particle production in nucleus-nucleus collisions can be carried out by using a toy QCD-inspired model introduced by Borghini and Wiedemann in [3]; it allows for analytical computations and may capture some important features of a more complete QCD description. In this model, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions q → gq̄ and g → gg [4] of the QCD evolution equations were distorted so that the role of soft emissions was enhanced by multiplying the infra-red singular terms by the medium factor Ns. The model [3] was further discussed and used on the description of final states hadrons produced in heavy-ion collisions [5]. Within the model, we make predictions for the medium-modified average multiplicity NA in quark and gluon jets (A = q, g) produced in such reactions, for the ratio r = Ng/Nq and finally for the second multiplicity correlators 〈NA(NA − 1)〉/N2 A, which determines the width of the multiplicity distribution. The starting point of our analysis is the NLO or Modified-Leading-Logarithmic-Approximation (MLLA) master evolution equation for the generating functional [4] which determine the jet properties at all energies together with the initial conditions at threshold at small x, where x is the fraction of the outgoing jet energy carried away by a single gluon. Their solutions with medium-modified splitting functions can be resummed in powers of √ αs/Ns and the leading contribution can be represented as an exponential of the medium-modified anomalous dimension which takes into account the Ns-dependence: NA ≃ exp { ∫ Y γmed (αs(Y )) dY } , (1) where γmed(αs) can be expressed as a power series of √ αs/Ns in the symbolic form: γmed (αs) ≃ √ Ns × √ αs ( 1 + √ αs Ns +O ( αs Ns )) . Within this logic, the leading double logarithmic approximation (DLA, O( √ Nsαs)), which resums both soft and collinear gluons, and NLO (MLLA, O(αs)), which resums hard collinear partons and accounts for the running of the coupling constant αs, are complete. The choice dY = dΘ/Θ, where Θ ≪ 1 is the angle between outgoing couples of partons in independent partonic emissions, follows from Angular Ordering (AO) in intra-jet cascades [4]. In order to obtain the hadronic spectra, we advocate for the Local Parton Hadron Duality (LPHD) hypothesis [6]: global and differential partonic observables can be normalized to the corresponding hadronic observables via a certain constant K that can be fitted to the data, i.e. N g,q = K ×Ng,q. The evolution of a jet of energy E and half-opening angle Θ involves the DLA anomalous dimension γ0 related to the coupling constant αs through γ2 0 = 2Ncαs/π, with αs = 2π/4Ncβ0(Y + λ), where Y = ln(Q/Q0) (Q = EΘ is the hardness or maximum transverse momentum of the jet), λ = ln(Q0/Λ QCD ) is a parameter associated with hadronization (Q0 is the collinear cut-off parameter, kT > Q0, and Λ QCD is the intrinsic QCD scale) and β0 = 1 4Nc ( 11 3 Nc − 4 3TR ) , where TR = nf/2, nf being the number of active flavors. At MLLA, as a consequence of angular ordering in parton cascading, the average multiplicity inside a gluon and a quark jet, Ng,q, obey the system of two-coupled evolution equations [7] 1 ha l-0 03 38 94 7, v er si on 2 19 M ay 2 00 9 (the subscript Y denotes d/dY ) NgY = ∫ 1 0 dx γ 0 [ Φg (Ng(x) +Ng(1− x)−Ng) + nfΦg (Nq(x) +Nq(1− x)−Ng) ] , (2) NqY = ∫ 1 0 dx γ 0 [ Φq (Ng(x) +Nq(1− x)−Ng) ] , (3) which follow from the MLLA master evolution equation for the generating functional; Ng,q ≡ Ng,q(Y ), Ng,q(x) ≡ Ng,q(Y + lnx), Ng,q(1 − x) ≡ Ng,q(Y + ln(1 − x)), ΦA denotes the medium-modified DGLAP splitting functions: Φg(x) = Ns x − (1− x)[2− x(1− x)], Φg(x) = 1 4Nc [x + (1− x)], Φq(x) = CF Nc ( Ns x − 1 + x 2 ) , (4) which accounts for parton energy loss in the medium by enhancing the singular terms like Φ ≈ Ns/x as x ≪ 1 as proposed in the Borghini-Wiedemann model [3]. Thus, when Ns increases the DLA becomes dominant and energy-momentum conservation plays a less important role. For Y ≫ lnx ∼ ln(1− x), Ng,q(x) (Ng,q(1− x)) can be replaced by Ng,q in the hard partonic splitting region x ∼ 1 − x ∼ 1 (non-singular or regular parts of the splitting functions), while the dependence at small x ≪ 1 is kept in the singular term Φ(x) ≈ Ns/x as done in the vacuum. Furthermore, the integration over x can be replaced by the integration over Y (x) = ln (