Pis'ma v ZhETF, vol. 101, iss. 9, pp. 659-664
© 2015 May 10
Radiative parton energy loss in expanding quark-gluon plasma with
magnetic monopoles
B. G. Zakharov^
Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia Submitted 30 March 2015
We study radiative parton energy loss in an expanding quark-gluon plasma with magnetic monopoles. We find that for realistic number density of thermal monopoles obtained in lattice simulations parton rescatter-ings on monopoles can considerably enhance energy loss for plasma produced in AA collisions at RHIC and LHC energies. However, contrary to previous expectations, monopoles do not lead to the surface dominance of energy loss.
DOI: 10.7868/S0370274X15090015
I. Introduction. It is widely accepted that the jet quenching phenomenon in AA collisions observed at RHIC and LHC is a manifestation of parton energy loss in the hot quark-gluon plasma (QGP) produced in the initial stage of AA collisions. The dominating contribution to parton energy loss comes from induced gluon radiation due to parton multiple scattering in the QGP [1-7]. The effect of collisional energy loss is relatively small [8]. The RHIC and LHC data on suppression of the high-p^ hadrons in AA collisions can be reasonably well described within the light-cone path integral (LCPI) approach to induced gluon emission [3] in a scenario of purely perturbative QGP (pQGP) [912] with the quasiparticle parton masses borrowed from the quasiparticle fit [13] to lattice results (which are close to that predicted by the HTL scheme [14]). Although, in the relevant range of the plasma temperatures T < Tc(2—3), the non-perturbative effects may be important, one could hope that the pQGP model is reasonable since radiative energy loss is mostly sensitive to the number density of the color constituents of the QGP. And the internal dynamics of the matter is practically unimportant from the standpoint of the energy loss calculations. This assumption may be wrong, however, if the non-perturbative effects lead to formation of new effective scattering objects that are absent in the pQCD picture. Evidently, thermal magnetic monopoles in the so called "magnetic scenario" of the QGP [15-17] are such objects that can be potentially very important for parton energy loss.
The thermal magnetic monopoles are now under active investigation [18-21] (and references therein). Lattice calculations show that monopoles in the QGP are
-^e-mail: bgz@itp.ac.ru
compact and heavy objects [19]. For this reason from the point of view of parton rescatterings they can act as practically point-like static scattering centers. Similarly to QED (for a review on monopoles in QED see, for instance, [22]) the differential cross section for par-ton scattering off thermal monopoles has the Rutherford form. It is important that, contrary to the ordinary pQCD parton cross sections, for monopoles, due to the Dirac charge quantization condition constraint, there are no the running and thermal effects. Lattice results show that the monopole number density, nm, may be quite large nm/T3 ~ 0.4 - 0.9 at T ~ Tc( 1-3) [20, 19]. Although, it is smaller by a factor of ~ 5—10 than the number density of ordinary thermal partons in pQGP, the scattering cross section for monopoles is considerably higher than that for thermal quarks and gluons. As a result, the monopoles can give a considerable contribution to induced gluon emission (and to photon emission from quarks). In [23] within the classical non-relativistic approach it was shown that interaction of quarks with monopoles may be important for photon emission from the QGP. The effect of monopoles on jet quenching in AA collisions has been addressed in recent analysis [24] within the GLV approach [6] in the approximation of N = 1 rescatter-ing.
In the present paper we address within the LCPI scheme [3] the question to which extent monopoles can be important for parton energy loss in the expanding QGP for RHIC and LHC conditions. The advantage of the LCPI formalism is that it includes any number of parton rescatterings (that is very important for the QGP with monopoles (below we denote it as mQGP) due to large cross section of parton interaction with monopoles). The LCPI approach treats accurately the
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mass and finite-size effects, and is valid beyond the soft gluon approximation.
II. Theoretical framework. In the LCPI approach [3] the induced gluon x-spectrum for a fast quark (or gluon) may be written through the in-medium light-cone wave function of the gqq (or ggg) system in the coordinate yo-representation. The z-dependence of this light-cone wave function is governed by a two-dimensional Schrodinger equation in which the longitudinal coordinate 2 (z-axis is chosen along the fast parton momentum) plays the role of time. We use the representation for the gluon spectrum obtained in [25] which is convenient for numerical calculations. For a fast quark (produced at 2 = 0) the gluon spectrum reads
dP
dx
= dz n(z
dx
(1)
where n(z) is the medium number density, duf^/dx is an effective Bethe-Heitler cross section accounting for both the LPM and finite-size effects. The da™/dx reads
da™ (a
PQa(x)
dx
TTM
(2)
p=0
Here Pq(x) = Cp[ 1 + (1 — x)2]/x is the usual splitting function for q —>• gq process, M = Ex( 1 — x) is the reduced "Schrodinger mass", E is the initial parton energy, Q2(£) = aM/e, with a « 1.85 [8], ^ is the solution to the radial Schrodinger equation for the azimuthal quantum number m = 1
-v(p,x,z -£) +
1
~2M 4to2 - 1 8Mfß
d_
dp
1
Lf]
(3)
with the boundary condition \I>(£ = 0, p) = = ^paqqg(p,x,z)eKl(ep) (K\ is the Bessel function), Lf = 2M/e2 with e2 = m2x2 + to2( 1 — x)2, aqqg(p, x, z) is the cross section of interaction of the qqg system (in the yo-plane q is located at the center of mass of qg) with a medium constituent located at 2. The potential v in (3) reads
v(p,x,z)
■ n(z)aqqg(p, x,z)
(4)
(summing over the species of the medium constituents is implicit here). The aqqg may be written through the dipole cross section aqq [26]
9 1
aqqg{p,x,z) = -{aqq(p, z)+aqq[(l-x)p, z]}--aqq(xp, z)
(5)
The dipole cross section for scattering of the qq pair on a medium constituent c may be written as
°qq(P, z) = \ J dq[! - exp(¿qp)]
d<7q, dq2
(6)
where daqc/dq2 is the qc —> qc differential cross section. For scattering on thermal quarks and gluons the differential cross section (in the approximation of static Debye screened color centers [1]) reads
dxjqc CtCf ira2(q2
[q2
(7)
where Cf,t are the color Casimir for the quark and thermal parton (quark or gluon), mo is the local De-bye mass. For the QGP with monopoles we should account for in the potential (4) the contribution from rescatterings on monopoles. The formula (6) is valid for monopoles as well. In QCD there are two different species of monopoles related to the Cartan generators T3 and Xg of the SU(3) group. Lattice calculations [20] show that both the species of monopoles have the same number density. For fast quarks thermal monopoles M^ act as Abelian scattering centers. For gluons in the color basis of definite color isospin and hyper charge (see below) it is true as well. In vacuum the differential cross section for scattering of a charged particle with electric charge qe off a monopole with magnetic charge qm has the Rutherford form [27]
da dtf
attd2
(8)
where D = qeqm/A-K. The Dirac charge quantization condition says that \D\ = nj2 where n is an arbitrary integer. We will assume that in the QGP for both the color species of monopoles \D\ = 1/2 (here we mean the minimal value of \D\ for parton-monopole interactions, for some parton color states it can be bigger). This value is supported by extraction of the magnetic coupling am = = q^-J47r from the monopole-(anti)monopole correlations in lattice simulations [19, 20] which give am ^2—4 at T/Tc ~ 1 — 2. Making use this am by inspecting the qM-scattering one can easily obtain that the condition \D\ = 1/2 gives as = 1 /am ~ 0.25—0.5 which is quite reasonable for as in the QGP at T/Tc ~ 1—2. The value \D\ = 1 leads to a four times bigger as which seems to
be unrealistic. For the scattering the coupling to
the vector potential of the monopole color field is given by and the possible values of \D\ are 1/2 and
1. We write the qM(M) differential cross section in the form
daqM _ CFTrF2(q2) dq2 ~{q2+m'2)2' {>
Here we introduced a phenomenological form-factor F accounting for the finite size of the monopole, m'D is the magnetic Debye mass, Cp is the color factor arising from averaging over the quark and monopole color states which for the SU(3) group gives {[l2 + ( —1)2]/3 + [l2 + l2 + (—2)2]/3}/2 = 4/3. From (7) and (9) one can see that in comparison to qq scattering the qM(M) cross section is enhanced by a factor of 3/2a2 (if we ignore F and possible difference in the electric and magnetic Debye screening masses).
For energetic partons energy loss is dominated by the small x-region. In the limit x —> 0 the three-body cross section (5) reduces to the cross section for the gg-color dipole. In pQCD for scattering on thermal partons (jgg = ^aqq. One can show that this relation is valid for scattering off monopoles as well. Indeed, similarly to our analysis of the synchrotron-like gluon emission [28], the scattering amplitude for interaction of glu-ons with monopoles may be diagonalized by introducing the gluon fields having definite color isospin, Qa, and color hypercharge, Qb • In terms of the usual gluon vector potential, G, the diagonal color gluon states read (we denote Q = (Qa,Qb)) X = (G1+iG2)/V2 (Q = = (-1, 0)), Y = (G4 + iGz)/V2 (Q = (-1/2, -V3/2)), Z = (G6 + iGr)/V2 (Q = (1/2, —a/3/2)). The neutral gluons A = Gs and B = with Q = (0,0), do not interact with monopoles at all. Then using this basis one can easily show that the averaged over the color states of the gluon and of the monopole differential cro
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