May 1996
[0.1cm] BITP 96/21
CUTP–759
LPTHEOrsay 9634
RADIATIVE ENERGY LOSS OF HIGH ENERGY QUARKS AND
GLUONS IN A FINITE VOLUME QUARKGLUON PLASMA
R. Baier , Yu. L. Dokshitzer , A. H. Mueller^{1}^{1}1Supported in part by the U.S. Department of Energy under grant DEFG0294ER40819 , S. Peigné and D. Schiff
Fakultät für Physik, Universität Bielefeld, D33501 Bielefeld, Germany
Theory Division, CERN, 1211 Geneva 23, Switzerland^{2}^{2}2Permanent address: Petersburg Nuclear Physics Institute, Gatchina, 188350 St. Petersburg, Russia
LPTHE^{3}^{3}3 Laboratoire associé au Centre National de la Recherche Scientifique  URA D0063 , Université ParisSud, Bâtiment 211, F91405 Orsay, France
Physics Department, Columbia University, New York, NY 10027, USA^{4}^{4}4Permanent address
Abstract
The medium induced energy loss spectrum of a high energy quark or gluon traversing a hot QCD medium of finite volume is studied. We model the interaction by a simple picture of static scattering centres. The total induced energy loss is found to grow as , where is the extent of the medium. The solution of the energy loss problem is reduced to the solution of a Schrödingerlike equation whose “potential” is given by the singlescattering cross section of the high energy parton in the medium. These results should be directly applicable to a quarkgluon plasma.
1 Introduction
The determination of the radiative energy loss of a high energy charged particle as it passes through matter is a problem studied some time ago, in QED, by Landau, Pomeranchuk and Migdal [1–3]. There is recent data from SLAC [4] on radiative energy loss in QED [5]. New interest in this problem [6–9] has arisen because the corresponding problem in QCD, that of the energy loss of a high energy quark or gluon due to medium stimulated gluon radiation, may be important as a signal for quarkgluon plasma formation in high energy heavy ion collisions.
Recently this problem was considered for infinite matter in [8], hereinafter referred to as BDPS. Using the GyulassyWang (GW) model [6] for hot matter they observed that the QED and QCD problems are mathematically equivalent if one identifies the emission angle of radiated photons in the QED case with the transverse momentum of radiated gluons in the QCD case. The energy loss per unit length, , in hot QCD matter was found to be proportional to for an incident parton of energy . The growth of was unexpected and suggested that energy losses of high energy jets in a quarkgluon plasma might be large. The procedure used by BDPS was not adequate for determining the exact logarithmic prefactor in . The problem was revisited in [9], hereinafter referred to as BDMPS, where a simple differential equation was given to determine the spectrum for radiated photons (in QED) and gluons (in QCD) which corrects the prefactor of the result in BDPS.
The present paper generalizes the BDMPS approach to finite length hot matter. Although our discussion is carried out in the context of QCD it is a simple matter to change variables in the QCD results to get the corresponding QED results. Our main interest is in the situation where the incident quark or gluon is sufficiently energetic so that the length of the matter, , satisfies , with the gluon mean free path and the Debye screening mass of the medium. Our principal results are for , given in (5.16), and for given in (6.6) and (6.7). The total energy loss in hot QCD matter of length is
The outline of the paper is as follows:
In section 2 the emission probability of a soft gluon from a high energy quark traversing hot QCD matter is given in the GW model. The basic emission vertex is calculated in detail as are the subsequent rescatterings of the quarkgluon system passing through the matter as the gluon is becoming free. While, for simplicity, much of the discussion is given in the large limit, final formulas are given with exact colour factors. After observing that the QED and QCD emission formulas are identical, with the identification of corresponding variables, the formula for the radiation intensity spectrum for infinite volume hot QCD matter is given as a direct consequence of the QED spectrum derived by BDMPS.
In section 3 a heuristic discussion of energy loss is given both for and for . By requiring that these results match at a direct connection between for and for is made.
Section 4 is concerned with deriving the general equations governing radiative energy loss in a hot QCD plasma of extent . The basic equation determining energy loss is a Schrödingerlike equation whose “potential” is given in terms of a single scattering cross section, in impact parameter space, of a high energy parton. We expect this same formalism, but with a different “potential”, to apply to the energy loss problem in cold nuclear matter [10]. We presume, however, that the magnitude of the energy loss in hot and cold matter may be quite different.
In section 5 we give an approximate solution for the radiation spectrum valid, at large , for those gluon energies dominating the energy loss of the primary parton. Our main result is given in (5.16). We note that as the spectrum agrees with the result previously given in BDMPS.
Formulas for the total energy loss due to medium induced radiation are given in section 6. We expect the most likely place that these results may have direct phenomenological application is in high energy jet production when a quarkgluon plasma is formed in heavy ion collisions. We also expect results much like (6.6) to hold in jet production in cold nuclear matter, but that is the subject for a subsequent paper [10].
Gluon emission with double scattering is given in Appendix A while rules for dealing with colour factors for multiple scattering are given in Appendix B. The calculation of the planar diagrams necessary to obtain (2.31) is given in Appendix C. A curious integral which arises in evaluating the energy loss is calculated in Appendix D.
2 General expression of the medium induced radiation spectrum in QCD
Here we derive the general form of the gluon energy spectrum induced by the propagation of a high energy quark in a finite nonabelian medium. Results are given for the case of an incident parton of arbitrary colour representation . We also establish a formal analogy between QED and QCD.
2.1 Model for multiple scattering
In order to describe the successive interactions of a high energy incident parton with a hot QCD medium, we use the model introduced by Gyulassy and Wang and recently used in BDMPS to study the photon energy spectrum induced by multiple QED scattering of a fast charged particle. The main feature of the model consists in assuming that scattering centres are static. This allows one to focus on purely radiative processes, since the collisional energy loss then vanishes. The centre located at creates a screened Coulomb potential
(2.1) 
with Fourier transform
(2.2) 
where is the QCD coupling constant. We suppose that the range of the potentials is small compared to the mean free path of the incident parton,
(2.3) 
where is the Debye mass induced by the medium. This means that successive scatterings are independent, since the incident parton cannot scatter simultaneously off two distinct centres. As a consequence, its propagation is “timeordered”, and we may number the scattering centres according to the interaction time (or equivalently the longitudinal coordinate) of the radiating parton. (See [9] for a justification in terms of Feynman diagrams). Moreover, in the context of one gluon emission, this assumption allows us to neglect amplitudes involving fourgluon vertices of the type shown in Fig. 1.
Finally, we work in the limit of very high energy for the incident parton and in the soft gluon approximation,
(2.4) 
Let us exhibit the implications of these two assumptions by giving the basic emission amplitude in a single scattering.
2.2 Gluon emission induced by a single scattering
The emission amplitude is depicted in Fig. 2. It includes the emission off the projectile (from now on chosen to be a quark) given by and the emission off the exchanged gluon given by .
The colour indices of the static centre and of the incident quark are denoted by , and , respectively. The indices of the exchanged and radiated gluons are and . Neglecting screening for the moment, we write the amplitude for elastic scattering off a static source as
(2.5a) 
Here
(2.5b) 
where we neglected spin effects in the high energy limit. The static source can be viewed as if it were a heavy quark.
In Feynman gauge, the amplitude (Fig. 2) for soft gluon emission may be expressed as the elastic scattering amplitude times a radiation factor as
(2.6) 
where denotes the gluon polarization state. The generators of the fundamental representation of are , satisfying . In the same way we get
(2.7) 
In addition to and , there is a term coming from gluon radiation off the static source. The sum of the three terms is gauge invariant. In a physical gauge such as lightcone gauge, is down by a factor of compared to and . In the calculation given below we use lightcone gauge and assume .
In a hot plasma the source is screened as indicated by (2.1) and (2.2) in the GW model. The reader may have doubts as to the general gauge invariance of that model. These doubts may be put to rest by the following arguments. It is straightforward to show that remains gauge invariant when the emitted and exchanged gluons are given the same mass . As we shall see later, the emitted gluon has a small impact parameter for the physical problem we consider. As a consequence of the small impact parameter, one may neglect the mass for the emitted gluon; keeping the mass only for the exchanged gluon leads to the GyulassyWang model.
In lightcone gauge
(2.8) 
In the high energy limit
(2.9) 
where is the transverse momentum of the gluon with respect to the direction of the incident particle. Thus,
(2.10) 
In QED [9], the photon radiation amplitude vanishes in the limit . In QCD, in the high energy limit only the purely nonabelian contribution to the gluon radiation spectrum survives. This is underlined by the presence of the commutator in (2.10). As a result we can use the eikonal approximation where the trajectory of the projectile is taken to be a straight line. Also,
(2.11) 
Finally, the radiation amplitude induced by one scattering of momentum transfer reads
(2.12) 
where the emission current is defined as
(2.13) 
We are interested in the gluon energy spectrum, which is given by the ratio between the radiation and elastic cross sections. Up to a common flux factor
(2.14a)  
(2.14b) 
where is defined as . Thus we obtain, for ,
(2.15) 
As the amplitude has been evaluated for a fixed momentum transfer , an average over has to be performed. For this we use the probability density deduced from the elastic scattering cross section which is easily obtained from (2.2). Thus in (2.15) we define
(2.16) 
where the normalized cross section for elastic quark scattering reads
(2.17) 
with . We have used the fact that the longitudinal transfer is negligible with respect to when . As we aim to derive the radiation density induced by multiple scattering, it is convenient to keep the colour structure together with the current and introduce
(2.18) 
The fact that the colour structure is the same as the threegluon vertex allows one to give a compact diagrammatic representation of the effective current as shown in Fig. 3.
Then the differential energy spectrum is simply written as
(2.19) 
A comparison between (2.12) and (2.18) allows one to set the proper colour factor in order to normalize to the elastic scattering cross section. The square includes the sum over all colour indices and the in the denominator cancels the sum over initial quark colours while corresponds to the colour factor of the normalizing elastic cross section. We see that the spectrum (2.19) has exactly the same form as in QED [9], up to the replacement of the photon angle by the gluon transverse momentum (and up to colour factors).
The introduction of the effective current given in (2.18) or in Fig. 3 will provide an important simplification in the case of multiple scattering. Let us indicate how this simplification appears in the case of two scatterings.
2.3 Effective radiation amplitudes for double and multiple scattering
Double scattering.
For two scatterings, the radiation amplitude is given by a collection of seven diagrams. These are simply calculated in the framework of timeordered perturbation theory. We show in Appendix A that all amplitudes may be grouped in effective radiation amplitudes induced by momentum transfers or at times and ; each is associated with a corresponding phase factor
(2.20)  
(see Appendix A for the notation concerning colour factors). This expression multiplies the elastic double scattering amplitude and may be represented diagrammatically as an effective emission current as in Fig. 3. This is shown in Fig. 4.
We thus have
(2.21) 
where we use an obvious notation for the phases. The first term on the righthand side of (2.21) and Fig. 4 corresponds to gluon emission at followed by rescattering of the quark at . The second is gluon production at while the third is gluon production at followed by rescattering of the gluon at . As seen from (2.21), quark rescattering does not affect the phase.
Multiple scattering.
The generalization of this simple result to scatterings is straightforward. After integrating over the time of emission it is always possible to collect three pieces in order to construct the effective radiation amplitude induced by at time . Consider the three lightcone perturbation theory graphs of Fig. 5.
For each diagram, integrating over yields the difference of two exponential phase factors (see Appendix A). Keeping only the one depending on , we collect three terms having the same phase,
(2.22) 
The sum of these three terms gives the effective current
(2.23) 
as in (2.18).
Similarly to the case considered in (2.20), the radiated gluon can rescatter on centres , so that the momentum and the colour factor have to be changed accordingly in (2.23). For example, if the gluon emitted at centre rescatters on centre () the sum of the corresponding three terms results in an expression analogous to (2.22), with replaced by , since labels the final real emitted gluon. In this case, one obtains
(2.24a) 
This is diagrammatically shown below
(2.24b) 
The associated phase is shifted according to
(2.25) 
In the total radiation amplitude, we should include, for centre , the possibilities (labelled by ) for the quarkgluon system to rescatter on the remaining centres. For to , the associated phase gets modified each time the gluon rescatters (the phase is unchanged by quark rescattering). Thus we write
(2.26) 
where the colour structure is included in .
2.4 Expression for the radiation spectrum induced by scatterings
As for a single scattering, we square the radiation amplitude given in (2.26) and normalize by the multiple elastic scattering cross section to get the radiation spectrum induced by scatterings
(2.27) 
This expression deserves some comments.

In , the index refers to the centre which induces the effective emission current. For a simple calculation of colour factors, it is convenient to represent interference terms in the form of connected diagrams, where the “conjugate amplitude” appears in the lower part of the diagram (Fig. 6).

The colour factor in the denominator of (2.27) corresponds to the normalization to the elastic scattering cross section, depicted in Fig. 7.
This is easily calculated from the rules given in Appendix B.

The sum over all possible gluon rescatterings is implicit in (2.27) (the sum over in (2.26)). We should take into account all possible ways the quarkgluon system has to rescatter on centres , in particular, for . However this part of the diagram describes the multiple scattering of the produced quarkgluon system after centre , which has no influence on the energy radiation spectrum we are interested in.

Between centres and , it matters whether it is the quark or the gluon which absorbs the transverse momentum , because it changes the relative phase . To simplify our derivation, we will first consider the large limit, where all nonplanar diagrams may be dropped, which corresponds to neglecting quark scatterings on centres for .
The products in (2.27) include a colour sum as indicated in (2.4). We may rewrite (2.27) as
(2.29a) 
As in the case of QED [9], we have the equivalent expression
(2.29b) 
In the large limit is given by two sets of planar diagrams denoted by and and shown in (2.4).
(2.30a)  
(2.30b) 
The second term of (2.29b) is the socalled factorization term, which corresponds to the limit of vanishing phases. In this limit, all emission amplitudes from the internal lines vanish (see Appendix A). Two cases have to be distinguished.

If the incident quark is produced at a time , we see from Appendix A that only emission amplitudes from initial and final lines remain. The factorization term contribution is then equivalent to the contribution induced by a single scattering of momentum transfer , and thus has a weak logarithmic medium dependence, as in the QED case [9].

In a realistic situation where the incident quark is produced, through a hard scattering, at a time , only emission from the final line remains (see the table of Appendix A). In this case the factorization term has no medium dependence at all, so that the medium induced spectrum is exactly given by the first term of (2.29b). It should be directly accessible experimentally by comparing hard scattering on a nucleus with that on a proton.
We show in Appendix C that after dropping the mediumindependent factorization term, (2.29b) leads to the following mediuminduced radiation spectrum in the large limit
(2.31) 
where the ’s are the transverse momenta of the gluon expressed in units of ,
(2.32) 
and is the rescaled emission current
(2.33) 
The dimensionless parameter is
(2.34) 
The radiation spectrum for the infinite medium QED case was given in [9] for . We observe that the expression (2.31) has the same form as in QED [9], with the replacements
(2.35a) 
(2.35b) 
This analogy allows us to give directly the result for the infinite QCD medium. Thus,
(2.36) 
Note that the radiation density is obtained by normalizing (2.31) to the distance , where is the mean free path of the incident quark [8].
The changes necessary to include all corrections as well as the case of an arbitrary incident parton of colour represention have been worked out in [8]. These changes are
(2.37a)  
(2.37b) 
This leads to the general formula
(2.38) 
where is the gluon mean free path.
We note that apart from the overall normalization, proportional to the squared colour charge , there is no dependence of the induced radiation spectrum on the nature of the initial parton.
3 Heuristic discussion of the energy loss in finite length media
When a very energetic parton of energy is propagating through a medium of finite length the gluon radiation spectrum shows characteristic features depending on the gluon energy . For discussing the radiation density three different regimes may be distinguished [8,9] : the BetheHeitler (BH) regime with small gluon energies, the coherent regime (LPM) for intermediate , and the highest energy regime corresponding to the factorization limit. The coherent regime corresponds to the condition (cf. (4.19a) in [9])
(3.1) 
with given in (2.34) for the QCD case. Thus, for the finite media under consideration a reasonably large number of scatterings will be assumed, .
For the following qualitative derivations we neglect logarithmic factors. Thus we ignore numerical factors of order 1, and do not distinguish between propagating quarks and gluons. However we explicitly keep the parameters representing the medium.
In terms of the gluon energy the condition (3.1) is
(3.2) 
Obviously, only holds when is less than the critical length,
(3.3) 
We note that the case is consistent with the soft gluon approximation for the induced spectrum.
The radiation spectrum per unit length behaves in the limit as
(3.4) 
for a finite length . These main features are illustrated schematically in Fig. 8. In the BH regime the radiation is due to incoherent scatterings, whereas in the factorization regime the medium behaves as one single scattering centre. In the LPM regime elementary centres act as a single scattering centre.
In order to obtain the total energy loss we integrate the spectrum (3.4) over and , with and . In addition to mediumindependent contribution to the energy loss proportional to (a factorization contribution), we find the induced loss
(3.5) 
It has been already pointed out in [11], that the total energy loss increases quadratically with the length , and is independent of the parton energy in the high energy limit. This interesting case is investigated in more detail in what follows.
Here we conclude this heuristic discussion by considering the case of finite , but with . This situation occurs for parton energies (see (3.3)). By extending the coherent soft spectrum (3.4) up to , the total induced loss is
(3.6) 
which is dependent and linear in . The results given in (3.5) and (3.6) are schematically summarized in Fig. 9, where the dependence of is plotted for fixed , and in Fig. 10, where is fixed and the dependence is shown.
The loss given by (3.6) is also relevant for an infinite medium, . An dependent loss per unit length of propagating partons is found [8],
(3.7) 
4 General equation for the induced radiation spectrum
In this section we shall derive the general equations which govern the induced radiation spectrum for finite length materials. These equations generalize (4.34) and (4.40) of BDMPS. Our starting point, for the sake of simplicity, is the large formula (2.31). We divide by , the length of the material, and we allow the sum over scatterings to be arbitrary in number giving
(4.1) 
where is the distance between the first and last scatterings. Expressing the sum over as an integral of the position in the medium of the gluon emission vertex and neglecting initial and final state scattering one arrives at
(4.2) 
where now is the distance between the emission vertices and . Although the sum in (4.2) formally goes from to we expect the typical number of scatterings to be . It is convenient to scale all distances by the mean free path . To that end let
Then
(4.3) 
where we have done the integral over . In (4.3) the dependence on and is contained only in the product of currents
(4.4) 
and in and . As in BDMPS we integrate first over and holding , fixed. Defining an averaged elementary current
(4.5) 
we note that
(4.6) 
The spectrum can be written as
(4.7) 
where
(4.8) 
The first term on the righthand side of (4.8) is the (nearest neighbours, ) term of (2.31). The rest of the terms in (4.8) correspond to arbitrary numbers of gluon rescatterings between the currents 1 and .
Equation (4.8) has a BetheSalpeter structure so it is straightforward to check that satisfies the natural integral equation
(4.9) 
where
(4.10) 
and where we have used
(4.11) 
It is convenient to eliminate the integration on the righthand side of (4.9). This is achieved by taking a derivative which gives
(4.12) 
Although (4.12) has been derived for an incident quark in the large limit, essentially the same equation holds for partons of colour representation even without taking the large limit. In the general case