Pis'ma v ZhETF, vol.88, iss. 12, pp.899-904
© 2008 December 25
Jet quenching with running coupling including radiative and collisional
energy losses
B.G. Zakharov
L.D. Landau Institute for Theoretical Physics RAS, 117334 Moscow, Russia Submitted 5 November 2008
We calculate the nuclear modification factor for RHIC and LHC conditions accounting for the radiative and collisional parton energy loss with the running coupling constant. We find that the RHIC data can be explained both in the scenario with the chemically equilibrium quark-gluon plasma and purely gluonic plasma with slightly different thermal suppression of the coupling constant. The role of the parton energy gain due to gluon absorption is also investigated. Our results show that the energy gain gives negligible effect.
PACS: 24.85.+p
1. It is widely believed that suppression of the high-Pt hadrons in AA-collisions (jet quenching (JQ)) observed at RHIC (for a review, see [1]) is dominated by the induced gluon emission [2-7] in the hot quark-gluon plasma (QGP) produced at the initial stage of AA-collisions. There are currently considerable theoretical efforts in the development of the quantitative methods for computation of JQ [8-13] which can be used for the tomographic analysis of the QGP. In the present paper we study JQ using the light-cone path integral (LCPI) approach to the radiative energy loss [3, 4]. In this formalism the probability of gluon emission is expressed through the Green's function of a two-dimensional Schrodinger equation with an imaginary potential. This approach has not the restrictions on the applicability of both the BDMPS approach [2] (valid only for massless partons in the limit of strong Landau-Pomeranchuk-Migdal effect) and the GLV formalism [6] (applicable only to a thin plasma in the regime of small Landau-Pomeranchuk-Migdal suppression). We perform the calculations with accurate treatment of the Coulomb effects. If one neglects these effects the gluon spectrum can be expressed in terms of the oscillator Green's function and the medium may be characterized by the well-known transport coefficient q [2, 4]. However, the oscillator approximation can lead to uncontrolled errors since it gives a physically absurd prediction that for massless partons the dominating N = 1 rescattering contribution vanishes [14, 15]. Besides the radiation energy loss we include the collisional energy loss. Both the contributions are calculated with the running coupling constant. Also, we investigate the impact of the parton energy gain due to gluon absorption from the QGP on JQ.
We calculate the nuclear modification factor 11aa, which characterizes JQ, accounting for the fluctuations
of the parton path lengths in the QGP. In the treatment of multiple gluon emission we use a new method which takes into account time ordering of the DGLAP and the induced radiation stages. We compare the theoretical results with the data obtained at RHIC by the PHENIX Collaboration[16] and give prediction for LHC. Our principle purpose in comparing with the RHIC data is to understand whether the observed JQ is consistent with the entropy of the QGP required by the hydrody-namical simulations of the AA-collisions for reproducing the observed particle multiplicities. Our results show that JQ and particle multiplicities can be naturally reconciled. Contrary to the conclusion of Ref. [17] that the observed at RHIC JQ is consistent only with purely gluonic plasma, we find that the scenario with the chemically equilibrium QGP is also possible. A good description of the JQ RHIC data can be obtained in this scenario with the thermal suppression of the coupling constant qualitatively consistent with the lattice results.
2. As usual we define the nuclear modification factor for AA-collisions as
= dN(A + A^h + X)/dpTdy AA( ) TAA(b)dcr(N + N h + X)/dpxdy' 1 ' where p/ is the hadron transverse momentum, y is rapidity (we consider the central region y = 0), b is the impact parameter, Taa(&) = fdpTA(p)TA(p — b), Ta is the nucleus profile function. The differential yield for high-px hadron production in AA-collision can be written in the form
dN(A + A^h + X) _ dpTdy
where dam(N + N h + X)/dpxdy is the medium-modified cross section for the N + N h + X process.
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899
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In analogy to the ordinary pQCD formula, we write it in the form
dam(N + N h + X)
dpjdy
=w
dz
Z—i I 72
, H z
ЩгМ)
dcr(N + N ^i + X) dplTdy
(3)
Here pj- = pt/z is the parton transverse momentum, D™n is the medium-modified fragmentation function (FF) for transition of the parton i to the observed hadron h, and da(N + N —t i + X)/dplTdy is the ordinary hard cross section. For the parton virtuality scale Q we take the parton transverse momentum p%T. We assume that hadronization of the fast partons occurs after escaping from the QGP. This hadronization process should be described by the FFs at relatively small fragmentation scale, fih- Indeed, from the uncertainty relation AEAt > 1 one can obtain for the L dependence of the parton virtuality Q2(L) ~ max(Q/L,Qg), where we have introduced some minimal nonperturbative scale Qo ~ 1 - 2 GeV. For RHIC and LHC conditions the size of the QGP is quite large (> Ra, where Ra is the nucleus radius), and from the above formula one sees that for partons with energy E < 100 GeV the hadronization of the final partons may be described by the FFs at the scale Hh ~ Qo- Then we can write
Г dz1
i(z,Q)« / —Dwiz/z'MDfyiz'^Q),
J z z
(4)
where Dh/j(z,Qo) is the FF in vacuum, and DJj^z', Qo, Q) is the medium-modified FF for transition of the initial parton i with virtuality Q to the parton j with the virtuality Qo- Presently there is no a systematic method for calculation of the medium-modified FFs which treats on an even footing the DGLAP and induced radiation processes. In the present paper we use the picture based on the time ordering of the DGLAP and the induced radiation stages which should be a reasonable approximation for not very high parton energies, say, E < 100 GeV. It uses the fact that at such energies the typical length/time scale of the DGLAP stage is smaller than the longitudinal scale of the induced radiation stage. The gluon emission scale for the DGLAP stage can be estimated using the gluon
formation length lp(x,k;
2Ex(l - x)/(k
where x is the gluon fractional longitudinal momentum, and c in terms of the effective parton masses reads
e2 = то2®1
to2( 1 — a;). Using the vacuum spectrum of
the gluon emission from a quark
dN _ СРав(кт)
dJbrpdx
7ГХ
(1
V/2)
^T
{Щ, + e2)2
(5)
one can obtain for the typical formation length lp ~ ~ 0.3-1 fin for E < 100 GeV (if one takes mg ~ 0.3 GeV and nig ~ 0.75GeV [18]). This estimate is obtained in the one gluon approximation. However, it should be qualitatively correct since in the energy interval of interest the number emitted gluons is small Ng < 2, and the first hardest gluon dominates the DGLAP energy loss. Thus we see that the DGLAP time scale is about the formation time for the QGP, To ~ 0.5 — lfm. Since the induced radiation is dominated by the distances from L ~ To up to L ~ Ra one can neglect the interference between the DGLAP and the induced radiation stages. In this approximation we can write
D™fi(z,Qo,Q) =
Г1 rlr1
= / a-^riy^l(zlzl,El)Df/^(zl,Qo,Q)
J z z
(6)
where Ei = Qz', D™^ is the induced radiation FF (it depends on the parton energy E, but not the virtuality), and D®fLAF is the DGLAP partonic FF. In numerical calculations the DGLAP FFs have been evaluated with the help of the PYTHIA event generator [19].
The induced radiation FFs have been calculated making use the probability distribution of the 1^2 partonic processes obtained in the LCPI approach. We have taken into account only the processes with gluon emission, and the process g qq which gives a small contribution has been neglected. For calculation the one gluon emission distribution we use the method elaborated in [20]. To calculate the D™^ one needs to take into account the multiple gluon emission. Unfortunately, up to now, there is no an accurate method of incorporating the multiple gluon emission. We follow the analysis [8] and employ the Landau method developed originally for the soft photon emission. In this approximation the quark energy loss distribution has the form
oo , Г n „
dujiX
, dP(wi
dw
Wi exp
i=1
(7)
where dP/dw is the probability distribution for one gluon emission. This approximation leads to the leakage of the probability to the unphysical region of AE > E
х
Jet quenching with running coupling including .
901
[9]. To avoid the quark charge non-conservation we define the renormalized distribution P(AE) = KqP(AE) with Kq = /0°° dAEP(AE)dAEP(AE). We use the renormalized distribution to define the in-medium FF D™^q(z) = P(AE = E(l — z)). To ensure the momentum conservation we take into account the q g transition as well. At the one gluon level the corresponding FF can be written as DtJl^(z) = dP(u} = zE)/dw. This automatically leads to the FFs which satisfy the momentum sum rule. We use the same form of the q g FF for the case with the multiple gluon emission. To satisfy the momentum sum rule (which are not valid after the renormalization of the q q distribution) we multiply it by a renormalization coefficient Kg defined from the total momentum conservation. This procedure seems to be reasonable since the nuclear modification factor is only sensitive to the behavior of the FFs at z close to unity [8] where the form of the q g distribution should not be very sensitive to the multiple gluon emission. In the case of the g g transition we use the following prescription. In the first step we define D at z > 0.5 through the Landau distribution P(AE), and in the soft region z < 0.5 (where the multiple gluon emission and the Sudakov suppression strongly compensate each other) we use the one gluon distribution. Then we multiply this FF by a renormalization c
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