# The LPM effect in QCD and Radiative energy loss in a quark-gluon plasma

###### Abstract

The non-abelian analog of the Landau-Pomeranchuk-Migdal effect is investigated in perturbative QCD. Extending our previous studies, the suppression of induced soft bremsstrahlung due to multiple scatterings of quarks in the spinor representation is considered. The effective formation time of gluon radiation due to the color interference is shown to depend on the color representation of the emitting parton, and an improved formula for the radiative energy loss is derived that interpolates between the factorization and Bethe-Heitler limits.

###### pacs:

24.85.+p, 12.38.Mh, 25.75.+r, 13.87.Ce^{†}

^{†}preprint: LBL-35980

## I Introduction

Ultrarelativistic heavy ion collisions at future collider energies of BNL Relativistic Heavy Ion Collider (RHIC) and CERN Large Hadron Collider (LHC) are expected to reveal novel QCD dynamics not accessible at present fixed target energies. When the transverse momentum transfer involved in each nucleon-nucleon collision is small, , effective models based on meson-exchange and resonance formation are sufficient to describe multiple interaction between hadrons. Those interactions lead to collective behavior in low energy heavy ion collisions as first observed in Bevalac experiments[1] and recently at AGS energies[2]. When becomes large enough to resolve individual partons inside a nucleon, the dynamics is best described on the parton level via perturbative QCD. Though hard parton interactions occur at CERN-SPS energies ( AGeV), they play a negligible role in the global features of heavy ion collisions. However, at collider energies ( AGeV) the importance of hard or semihard parton scatterings is clearly seen in high-energy and collisions [3]. They are therefore also expected to be dominant in heavy ion collisions at RHIC and LHC energies [4, 5].

Since hard parton scatterings occur very early ( fm/c) and their rates are calculable via perturbative QCD (pQCD), we proposed in [6, 7] that high parton jets could serve as a unique probe of the quark-gluon plasma formed due to copious mini-jet production over longer time scales ( fm/c). The systematics of jet quenching provides information on the stopping power, , of high energy partons in dense matter[8]. The stopping power is in turn controlled by the color screening scale, , in the medium. Thus, jets provide information on that interesting dynamical scale in deconfined matter. At very high energy densities, that scale is expected to be large compared to confinement scale[9]. At temperatures, , for example, we expect . In that case, most partonic interactions have high momentum transfers and perturbative QCD methods may apply to the calculation of multiple collision amplitudes.

In Ref.[10] a systematic study of QCD multiple collision theory was initiated with the aim of deriving the non-abelian analog of the Landau-Pomeranchuk-Migdal (LPM) effect. That effect, first derived in the case of QED[11, 12] long ago, involves the destructive interference between Bremsstrahlung radiation amplitudes. It suppresses radiation relative to the Bethe-Heitler formula in kinematic regions where the radiation formation time is long compared to the mean free path. In QCD a similar effect is also expected because it mainly follows from general relativistic uncertainty principle arguments. However, we showed in [10] that specific non-abelian effects influence the detailed interference pattern in the case of QCD.

The LPM effect in QCD is especially important for estimating the energy loss, , of an energetic parton traversing a dense QCD medium. We note that there exists considerable controversy in the literature on the magnitude and energy dependence of the energy loss in QCD [13, 14, 15, 16, 17]. In this paper we extend our derivation in [10] to clarify further this effect and improve our previous estimates of .

In the next section, we consider in detail the problem of induced gluon radiation from a spin 1/2 quark suffering two elastic scatterings. This provides an insight into the applicability of the effective potential model used in Ref. [10] to calculate multiple collision amplitudes. We show why radiation from the target partons is negligible although such amplitudes are absolutely necessary to ensure gauge invariance. On the other hand, gauge invariance constrains all gluon propagators to be regulated by the same screening mass in the potential model. In section 3, the QCD radiation interference pattern is shown to be expressible as a function of an effective formation time that depends on the color representation of the jet parton. In section 4, that formation time is used to calculate the average radiative energy loss and derive a simple formula that interpolates between the factorization and Bethe-Heitler limits as a function of a dimensionless ratio depending on the incident parton energy , the screening scale and the mean free path. Finally, section 5 contains a summary and closing remarks.

## Ii The Potential Model and Gauge Invariance

To analyze multiple parton scatterings and the induced gluon radiation, certain simplifications have to be made for the interaction. Consider the scattering of a high energy jet parton in a color neutral quark-gluon plasma. If the average distance, , between two successive scatterings is large compared to the color screening length, , the effective average random color field produced by the target partons can be modeled by a static Debye screened potential:

(1) |

where is the color screening mass, are the generators of corresponding to the representation of the target parton at . The initial and final color indices, , , of the target parton are averaged and summed over when calculating the ensemble averaged cross sections. This model potential was used in Ref. [10] to calculate cross sections of multiple scattering and induced radiation. However, to obtain a gauge invariant amplitude of gluon radiation in QCD, every diagram with the fixed number of final gluon lines has to be taken into account, including gluon radiation from the target parton line. Radiation from target partons cannot be described by the above static model potential. The relative importance of different diagrams to the net energy loss depends on the choice of the gauge. In the light-cone gauge one expects that radiation from target legs to be negligible as compared to that from the high energy beam parton.

To estimate the importance of target radiation to the energy loss of a fast parton and to see how the gauge invariance constrains the potential model, consider the simplest case of induced radiation from quark-quark scatterings. The Born amplitude for through one gluon exchange is,

(2) |

where , , , and are the initial and final color indices of the beam and target partons, respectively. The corresponding elastic cross section is,

(3) |

where is the color factor for a single elastic quark-quark scattering and ,, and are the Mandelstam variables.

For induced radiation, there are all together three groups of diagrams as shown in Fig. 1. If we rewrite the three amplitudes as

(4) |

then gauge invariance implies that

(5) |

where and are, respectively, the polarization and momentum of the radiated gluon. In the soft radiation limit, we can neglect all the terms proportional to in which does not contribute to (). The complete and gauge invariant amplitude for induced gluon radiation is,

(6) | |||||

where is the color index of the radiated gluon, and

(7) |

are the color matrices associated with gluon radiation from the internal gluon line (Fig. 1c). The three terms in Eq. (6) correspond to the radiation from the projectile (Fig. 1a), target parton (Fig. 1b) and the internal gluon line (Fig. 1c). As one can see, no term alone is separately invariant under a gauge transformation . Only the total amplitude is gauge invariant.

In the potential model, one simply neglects the radiation from the target lines and replace the gluon-exchange amplitude associated with the target parton, , by an effective potential . The potential given by Eq. (1) is regularized by the color screening mass squared, . However in Eq. (6), if any of the gluon propagators is regulated, gauge invariance is preserved only if all propagators are regulated with the same mass squared, . In particular, the two propagators in the internal radiation diagram must both be regulated by the same scale.

While gauge invariance is manifest only if all diagrams are added, we now show that, in the gauge, only the projectile diagrams contribute significantly to the radiation of soft gluons in the dominant kinematic range for the net energy loss. We consider the case that a gluon with light-cone momentum and polarization,

(8) |

is radiated off a high energy quark with initial momentum

(9) |

During the interaction, the beam quark exchanges a momentum

(10) |

with a target quark which has a typical thermal momentum,

(11) |

with in the plasma rest frame. We focus on the limit defined by , however, with the condition, . By requiring both of the final quarks to emerge on mass shell, one finds that,

(12) |

The final momenta of the beam and target partons are, respectively,

(13) |

With the above kinematics, one can obtain the momentum elements of the radiation amplitudes:

(14) |

(15) |

In the large limit, we see that the magnitude of the matrix elements involving target parton radiation in Eq. (15) are much smaller than those involving projectile radiation in Eq. (14). As we will see below, the LPM effect limits the radiation to and . Therefore, as long as , the contribution to the energy loss in the main kinematic range is dominated by the projectile radiation in this gauge. However, as we have demonstrated, the small target contributions to the radiation amplitude are amplified if is replaced by . As a result of this amplification, those amplitudes cannot be neglected when considering gauge invariance even though they can be neglected for calculating the energy loss.

Taking into account only the dominant contributions to the radiation amplitude, we have the factorized amplitude,

(16) |

where is the elastic amplitude as given in Eq. (2), and is defined as the radiation amplitude induced by a single scattering. For later convenience, all the color matrix elements are included in the definition of the radiation amplitude . With the above approximations, we recover the differential cross section for induced gluon bremsstrahlung by a single collision as originally derived by Gunion and Bertsch [18],

(17) |

where the spectrum for the radiated gluon is,

(18) |

In the square modulus of the radiation amplitude, an average and a sum over initial and final color indices and polarization are understood. We see that the spectrum has a uniform distribution in central rapidity region (small ) which is a well known feature of QCD soft radiation [19]. This feature is consistent with the hadron distributions predicted by Lund string models and the “string effects” in three jets events [20, 21], all being the results of interference effects of pQCD radiation. To demonstrate this a little in detail, let us consider only the radiation amplitude from the beam quark in Eq. (6),

(19) |

The corresponding gluon spectrum is (beside a factor ),

(20) |

where and are the second order Casimir for quarks in the fundamental and for gluons in the adjoint representation, respectively. Note that the first term is identical to gluon radiation induced by an abelian gauge interaction like a photon exchange and does not contribute to the gluon spectrum in central rapidity region due to the destructive interference of the initial and final state radiations. The second term arises from the non-abelian interactions with the target partons and is the main contribution to the central region.

Another nonabelian feature in the induced gluon radiation amplitude, Eq. (16), is the singularity at due to induced radiation along the direction of the exchanged gluon. For , we note that the induced radiation from a three gluon vertex can be neglected as compared to the leading contribution . However, at large , this three gluon amplitude is important to change the gluon spectrum to a behavior, leading to a finite average transverse momentum. Therefore, may serve as a cut-off for when one neglects the amplitude with the three gluon vertices as we will do when we consider induced radiation by multiple scatterings in the next section. If one wishes to include the three gluon amplitude, then the singularity at has to be regularized. As we have discussed above, the regularization scheme has to be the same as for the model potential or the gluon propagator in as required by gauge invariance. In our case, a color screening mass will be used.

## Iii Effective Formation Time

The radiation amplitude induced by multiple scatterings has been discussed in Ref. [10]. We discuss here the special case of double scatterings to gain further insight into the problem. Consider two static potentials separated by a distance , which is assumed to be much larger than the interaction length, . In the abelian case, the radiation amplitude associated with double scatterings is (see Appendix),

(21) |

where is the four-momentum of the intermediate parton line which is put on mass shell by the pole in one of the parton propagators, , and are the four-coordinates of the two potentials with . This formula has been recently used to discuss the interference effects on photon and dilepton production in a quark gluon plasma [22, 23]. We notice that the amplitude has two distinct contributions from each scattering. Especially, the diagram (Fig. 2b) with a gluon radiated from the intermediate line between the two scatterings contributes both as the final state radiation for the first scattering and the initial state radiation for the second scattering. The relative phase factor

determines the interference between radiations from the two scatterings and is simply the ratio of the path length to the formation time defined as

(22) |

The Bethe-Heitler limit is reached when . In this limit, the intensity of induced radiation is additive in the number of scatterings. However, when , the final state radiation amplitude from the first scattering is mostly cancelled by the initial state radiation amplitude from the second scattering. The radiation pattern then looks as if the parton has only suffered a single scattering from to . This destructive interference is often referred to as the Landau-Pomeranchuk-Migdal (LPM) effect. The corresponding limit is usually called the factorization limit.

The radiation amplitude in QCD is similar to Eq. (21), except that one has to include different color factors for each diagram in Fig. 2. In the high energy limit, . The momentum dependence of each contribution can be factorized out and the radiation amplitude for diagrams in Fig. 2 is,

(23) |

where we have included two color matrices from the potentials, and again represents the color index of the radiated gluon. The radiation amplitude from diagrams with three gluon vertices has the same phase and color structures as in Eq. (23), but the momentum dependence cannot be factorized, since each term depends on the transverse momentum transfer which differs from one scattering to another. However, since we are interested in the soft radiation limit, , the contributions from internal gluon line emissions can be neglected as shown in [10]. As we discussed above, however, those amplitudes serve to provide an effective cutoff for .

The extrapolation of Eq. (23) to the general case of number of scatterings is straightforward with the result

(24) |

The above amplitude contains terms each having a common momentum dependence in the high energy limit, but with different color and phase factors. The above expression is also valid for a gluon beam jet, with the corresponding color matrices replaced by those of an adjoint representation. In Eq. (24), we also assumed that all the potentials have a color structure of a fundmental representation. One can generalize to the case in which each individual potential could have any arbitrary color representation. However, our following results on the gluon spectrum and interference pattern will remain the same. With this in mind, we have the spectrum of soft bremsstrahlung associated with multiple scatterings in a color neutral ensemble, similar to Eq. (18),

(25) |

where is the color factor for the elastic scattering cross section without radiation. , defined as the “radiation formation factor” to characterize the interference pattern due to multiple scatterings, can be expressed as

(26) |

where the color coefficients, as computed in Ref. [10], are

(27) |

For a gluon beam jet, one can simply change the dimension to that of an adjoint representation and replaces by the corresponding second order Casimir . We then obtain a general form for the radiation formation factor for a high energy parton jet,

(28) |

where

(29) |

Similar to the special case of double scatterings, there are a few interesting limits for the above general form of radiation formation factor and the induced gluon spectrum. When , . We recover the gluon spectrum induced by a single scattering in Eq. (18) in the small limit. For multiple scatterings in the Bethe-Heitler limit when for all , the phase factors average to zero and the intensity of the radiation is additive in the number of scatterings, i.e., . In the factorization limit, one has for all . In this case, the phase factors can be set to unity and the summations in Eq. (28) can be carried out to give

(30) |

In contrast to the Bethe-Heitler limit, the factorization limit is independent of the number of collisions as , and the radiation formation factor approaches . It is interesting to note that the destructive interference for quarks in the fundamental representation is so effective that the radiation spectrum induced by many scatterings is even slightly less, , than by a single scattering. For gluon jets, however, the interference is not as complete as for quarks. The induced radiation approaches 2 times that from a single scattering. Using these values of in Eqs. (25) and (18), the radiation intensity induced by multiple scatterings is proportional to as compared to in the single scattering case. The gluon intensity radiated by a gluon jet is therefore higher than that by a quark due to the interference in multiple scatterings. This dependence of LPM effect in QCD on the color representation of the beam parton is a unique non-abelian effect. As we will discuss in the following, the effective formation time of the radiation in a QCD medium should also take this non-abelian effect into account.

To see analytically how interpolates between the Bethe-Heitler and factorization limits, let us average over the interaction points according to a linear kinetic theory. We take an eikonal approximation [10] for the multiple scatterings so that the transverse phase factors can be neglected in the soft radiation limit, . In a linear kinetic theory, the longitudinal separation between successive scatterings, , has a distribution,

(31) |

which is controlled by the mean free path, , of the scatterings. The averaging of the phase factors,

(32) |

enables us to complete the summation in Eq. (28). Neglecting terms proportional to for relatively large , we have,

(33) |

The non-abelian LPM effect in QCD is therefore controlled by the dimensionless ratio of the mean free path to an effective formation time,

(34) |

where the effective formation time in QCD depends on the color representation of the jet parton and is related to the usual abelian formation time in Eq.(22) as,

(35) |

This formula for the radiation formation factor is illustrated in Fig. 3 as a function of for the case of five collisions and shows how interpolates between the Bethe-Heitler limit for small value of and the factorization limit for large value of . For radiation with average transverse momentum the additive Bethe-Heitler region is limited to rapidities . In general, the radiation formation factor is proportional to in the limit of large . We can therefore regard the radiation as being additive to the number of scatterings with the radiation from each scattering simply suppressed by the factor due to nonabelian LPM effect. Since this effective formation time is a result of the unique color interference effect in QCD, we should use it in the following to estimate the radiative energy loss by a high energy parton traversing a color neutral quark-gluon plasma.

## Iv Radiative Energy Loss

We now apply the effective formation time to derive a simple approximate formula for the induced radiative energy loss extending our previous result in [10]. As shown in the previous section the radiation spectrum is given by Eq. (18) multiplied by the radiation formation factor Eq. (33). That factor simply restricts the additive kinematic region to and the the incremental change in that factor for each successive collision can be approximated by a pocket formula (see Fig.3). The additive radiative energy loss for each collision beyond the first one is then

(36) |

where is given by Eq. (35), the second -function is for energy conservation, and the regularized gluon density distribution induced by a single scattering is,

(37) |

As discussed earlier, gauge invariance requires that the singularity at in Eq. (18) must be regularized by the same color screening mass as in the elastic cross section in the potential model. Since the transverse momentum transfer is the result of elastic scatterings, we have to average any function of by the elastic cross section,

(38) |

where is the average energy squared for the scattering of a jet parton with energy off the thermal partons at temperature . For the dominant small angle scattering, the elastic cross sections are,

(39) |

where respectively for , and scatterings. This average can be approximated by replacing in the numerator of Eq. (37) with its average value,

(40) |

In the denominator, we simply replace by after the angular integration. The remaining integration in Eq. (36) over the restricted phase space approximately leads to the simple analytic formula

(41) |

(42) | |||||

(43) |

In small regime, the phase space is mainly restricted by a small effective formation time, , which gives the first term proportional to . For large , the radiation becomes additive in a restricted phase space constrained by energy conservation. That region contributes to the second term which appears to be proportional to the incident energy . However, in the high energy limit the function and hence the radiated energy loss grows only as .

The above derivation assumed that the mean free path is much larger than the interaction range specified by . As we shall discuss below, this is satisfied in a quark gluon plasma at least in the weak coupling limit. Therefore, we can neglect the second term in . For a high energy jet parton, , we can also neglect the second term in . The resulting radiative energy loss reduces in that case to the simple form,

(44) |

which depends on a dimensionless variable,

(45) |

Since we have used gluon spectrum from a single scattering in Eq. (18) which is valid for all values of , the full integration over results in the logarithmic energy dependence of . This logarithmic dependence was absent in our derivation in Ref.[10] since there only the contribution from the very soft region was considered.

We see that the radiative energy loss thus obtained interpolates between the factorization and Bethe-Heitler limits as a function of the dimensionless ratio . In the factorization limit, we fix and let , so that . In this case, we can neglect the second term in Eq. (44) and have,

(46) |

Thus, the radiative energy loss in the factorization limit has only a logarithmic energy dependence (in addition to the energy dependence of ). Due to the non-abelian nature of the color interference, the resultant energy loss for a gluon ( is 9/4 times larger than that for a quark (). In the other extreme limit, we fix and let , so that . In this case, the mean free path exceeds the effective formation time. The radiation from each scattering adds up. We then recover the linear dependence of the energy loss on the incident energy (modulo logarithms),

(47) |

as in the Bethe-Heitler formula. In both cases, the radiative energy loss is proportional to the average of the transverse momentum transfer, , which is controlled by the color screening mass as in Eq. (40).

To see more clearly how the factorization limit is approached, we now estimate for a parton propagating inside a high temperature quark-gluon plasma. From Eq. (39) and the perturbative QCD expressions for the quark and gluon densities at temperature, , the mean free path for 3 quark flavors is

(48) |

(49) |

where . We emphasize that the above mean free path corresponds approximately to the color relaxation mean free path, , and not the momentum relaxation mean free path, . As shown in Ref.[24], is generally the shorter of the two in QCD. The reason why the color relaxation mean free path controls the radiation pattern is that the color current responsible for emitting the gluons is coherent only over a distance scale . It take a much longer path length to stop a parton. However, unlike in QED, this longer momentum relaxation mean free path is irrelevant for nonabelian radiation.

Using Eqs. (48) and (49) and the perturbative color electric screening mass, , we see that appearing in the logarithms has a common energy and temperature dependence for both quarks and gluons,

(50) |

With the above expression for , we plot the radiative energy loss in Fig. 4 as a function of the beam energy inside a plasma at temperature MeV with . The solid line is the full expression in Eq. (44) while the dashed line is the factorization limit corresponding to the first term in Eq. (44). We see that Eq. (46) approximates Eq. (44) quite well in this parameter range. The energy dependence of the radiative energy loss is due to the double logarithmic function in the formula one of which comes from the energy dependence of the average transverse momentum in Eq. (40).

The energy loss of a quark in dense matter due to elastic scattering was first estimated by Bjorken [25] and later was studied in detail [26] in terms of finite temperature QCD. For our purpose, a simple estimate taking into account both the thermal average and color screening will suffice. In terms of elastic cross sections and the density distributions for quarks and gluons in a plasma, we have,

(51) |

where is the energy transfer of the jet parton to a thermal parton with energy during an elastic scattering, is the average transverse momentum transfer given by Eq. (40). Similar to Eqs. (48) and (49), we have,

(52) |

(53) |

The elastic energy loss of a fast parton inside a quark gluon plasma at temperature is then given by,

(54) |

For comparison, we plot this elastic energy loss in Fig. 4. In general, it is much smaller than the radiative energy loss and has a weaker energy dependence (single logarithmic).

Using Eqs. (40), (46) and (50), the total energy loss can be expressed as,

(55) |

It is interesting to note that both the elastic and radiative energy loss have the same color coefficient . For high energy partons, the radiative energy loss is dominate over the elastic one. For GeV, MeV, and , the total energy loss for a propagating quark is GeV/fm. Only about 25% of this amount comes from elastic energy loss.

## V Summary and Discussions

We extended our previous derivation by considering the role of gauge invariance and target radiation in the case of spin 1/2 quarks to improve our estimate of radiative energy loss of a fast parton inside a quark gluon plasma. Our main result [Eq. 44)] interpolates between the factorization and Bethe-Heitler limits, and has unique nonabelian properties. The factorization limit is of course consistent with the general bound [16] imposed by the uncertainty principle, but reveals peculiar energy and temperature dependence of the mean square radiation transverse momentum controlling that energy loss. The total energy loss is very sensitive to the color screening scale in the plasma. The double logarithmic energy dependence of is the result of non-abelian aspects of the LPM effect in QCD. The same effect should be responsible for the limited gluon equilibration rate as discussed in Ref. [27].

Our derivation improves that in [8, 10] in a number of ways. First, an effective formation time in QCD radiation was used to account for the color interference due to multiple scatterings. The dependence of this effective formation time on the color representation of the jet parton gave rise to the different color factors, proportional to , for the radiative energy loss of a quark and gluon. In contrast both are proportional to in the case of a single scattering. Secondly, the gluon spectrum including radiation from both the jet line and the internal gluon line and regulated consistently with the requirement of gauge invariance was used. However, Eq. (44) is still an idealization to the physically realizable situation in nuclear collisions because a number of strong assumptions were made in its derivation. The strongest is the extrapolation of pQCD in a regime and the assumption that the interaction range is small compared to the color relaxation mean free path. We have therefore left with an explicit dependence on in since strong nonperturbative variations of occur in the vicinity of . The basic result that is proportional to is however very general and consistent with the uncertainty bounds in[16]. Therefore, in a separate paper [28], we will investigate the phenomenological consequences of interesting temperature dependence of suggested by lattice calculations.

###### Acknowledgements.

We would like to thank J. Cleymans, J. Knoll, and B. Müller for stimulating discussions. This work was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098 and DE-FG02-93ER40764.## References

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##
The radiation amplitude within multiple scattering theory has been
derived in Ref. [10] in a general from. In this appendix we use
the radiation induced by double scattering as an example to demonstrate
how the general formula arises from the multiple scattering theory.
We consider the scattering of a high energy particle off potentials
as given by Eq. (1). For simplicity, let us first neglect the
color indices as in the case of QED. The amplitude for a single
scattering is then

(56)

(57)

where and the amplitude is defined such that the
differential cross section is given by

(58)

and , are the initial and final polarizations which
should be averaged and summed over in the calculation of cross sections.
One can check that with the definition of in Eq. (1),
the above formula leads to .

One can similarly write down the amplitude for double scatterings,

(59)

where the energy conservation at each potential vertex sets the energy
of the internal line to . Amplitudes involving backscattering
are suppressed at high energies because of the limited momentum transfer
that each potential can impart. When the mean free path (or distance
between two sequential scatterings) is much larger than the interaction
range () of the potential in Eq. (1), the singularity
in can be neglected and the integration over (with
respect to the direction of
) gives us

(60)

where is the
four-momentum of the internal line. One can derive the classical
Glauber multiple collision cross section from this amplitude by
averaging and summing over initial and final state ensemble of the
target[10]. In the limit of high energy and small angle scattering,
one can neglect the phase factor in the above equation and obtain
the amplitude [as defined in Eq. (56)],

(61)

where comes from the different ordering of the target potentials
and is the total
transverse momentum transfer due to the multiple scatterings. The
eikonal function is defined as the
Fourier transform of the single scattering amplitude (besides a
factor ),

(62)

In the definition of the product of eikonal functions, summation over
the polarizations of the intermediate lines is implied,

The radiation amplitude within multiple scattering theory has been derived in Ref. [10] in a general from. In this appendix we use the radiation induced by double scattering as an example to demonstrate how the general formula arises from the multiple scattering theory. We consider the scattering of a high energy particle off potentials as given by Eq. (1). For simplicity, let us first neglect the color indices as in the case of QED. The amplitude for a single scattering is then

(56) |

(57) |

where and the amplitude is defined such that the differential cross section is given by

(58) |

and , are the initial and final polarizations which should be averaged and summed over in the calculation of cross sections. One can check that with the definition of in Eq. (1), the above formula leads to .

One can similarly write down the amplitude for double scatterings,

(59) | |||||

where the energy conservation at each potential vertex sets the energy of the internal line to . Amplitudes involving backscattering are suppressed at high energies because of the limited momentum transfer that each potential can impart. When the mean free path (or distance between two sequential scatterings) is much larger than the interaction range () of the potential in Eq. (1), the singularity in can be neglected and the integration over (with respect to the direction of ) gives us

(60) |

where is the four-momentum of the internal line. One can derive the classical Glauber multiple collision cross section from this amplitude by averaging and summing over initial and final state ensemble of the target[10]. In the limit of high energy and small angle scattering, one can neglect the phase factor in the above equation and obtain the amplitude [as defined in Eq. (56)],

(61) |

where comes from the different ordering of the target potentials and is the total transverse momentum transfer due to the multiple scatterings. The eikonal function is defined as the Fourier transform of the single scattering amplitude (besides a factor ),

(62) |

In the definition of the product of eikonal functions, summation over the polarizations of the intermediate lines is implied,