Jet quenching in thin quark–gluon plasmas I: formalism

Nuclear Physics B 571 Ž2000. 197–233 www.elsevier.nlrlocaternpe

Jet quenching in thin quark–gluon plasmas I: formalism M. Gyulassy a , P. Levai ´ a,b, I. Vitev a a

b

Department of Physics, Columbia UniÕersity, New York, NY, 10027, USA KFKI Research Institute for Particle and Nuclear Physics, PO Box 49, Budapest, 1525, Hungary Received 26 July 1999; accepted 4 November 1999

Abstract The modification and amplification of the gluon angular distribution produced along with hard jets in nuclear collisions is computed. We consider the limit of a thin quark–gluon plasma, where the number of rescatterings of the jet and gluons is small. The focus is on jet quenching associated with the formation of highly off-shell partons in hard scattering events involving nuclei. The interference between the initial hard radiation amplitude, the multiple induced Gunion–Bertsch radiation amplitudes, and gluon rescattering amplitudes leads to an angular distribution that differs considerably from both the standard DGLAP evolution and from the classical limit parton cascading. The cases of a single and double rescattering are considered in detail, and a systematic method to compute all matrix elements for the general case is developed. A simple power law scaling of the angular distribution with increasing number of rescatterings is found and used for estimates of the fractional energy loss as a function of the plasma thickness. q 2000 Elsevier Science B.V. All rights reserved. PACS: 12.38.Bx; 12.38.Mh; 24.85.qp Keywords: Radiative energy loss; Non-abelian plasma; Parton cascade

1. Introduction One of the new predicted observables in nuclear collisions at RHIC energies Ž 6s ; 200 A GeV . is jet quenching w1–5x. While collisional energy loss of a hard jet propagating through quark–gluon plasma has been estimated to be modest w1,2x, dEcollrdx < 1 GeVrfm, for accessible plasma temperatures, the induced radiative energy loss, as estimated w3,4x using the Gunion–Bertsch w6x non-abelian generalization of the Bethe–Heitler formula for QED is expected to be significantly larger with dEradrdx ) few GeVrfm. An uncertainty principle analysis in w7x leads to a simple estimate 2 : ²kH dE ; , Ž 1. dx 2 2 : where ² k H is the mean transverse momentum of the radiated gluons. In Ref. w8x a 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 7 1 3 - 0

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model for non-abelian energy loss was developed and a class of multiple scattering diagrams lead to dErdx ; 1 GeVrfm independent of the path length. However, in BDMPS w9x gluon rescattering diagrams in the medium were shown to induce a 2 : non-linear enhancement of the energy loss because ² k H s m2 Lrl. The result for an incident very high energy jet penetrating a plasma of thickness, L, in which the mean free path, l s 1rŽ sr ., and the color electric fields are screened on a scale m , is w9x dE dx

f a s² p 2H : A a s m2

L

l

.

Ž 2.

Here ² p 2H : is the jet broadening due to multiple elastic scatterings of the jet in matter, and a factor of order unity is suppressed. This estimate is derived for the situation when the radiation formation length is large compared to the target thickness and the target is thick compared to the mean free path:

D L f s Erm2 4 L 4 l . In the other theoretical limit corresponding to Landau–Pomeranchuk–Migdal ŽLPM. w10–13x regime in QED, L 4 D L f 4 l, and the energy loss was shown to saturate as dErdxA a s r E . The remarkable prediction that the non-abelian dErdx increases linearly with the plasma thickness in nuclear collisions, suggests that the total jet energy loss could increase quadratically with the nuclear size. However, several practical problems limit the application of the above results directly to observable jet quenching in nuclear collisions. First, jets are produced in such reactions with high initial virtualities in hard pQCD processes, and thus induced radiation from secondary interactions must compete with the usual broad gluon shower accompanying a hard pQCD event. Second, the observable energy loss depends on the detailed angular distribution of the gluons. Unlike in electrodynamics where the energy of the final electron and photon can be directly measured, the energy of a single parton is only measurable when it is kinematically well separated from other jets. This is because the observable hadron showers have wide angular distributions. Nearly collinear partons in particular are not resolvable because they are absorbed into the coherent hadronic wavefunctions. Therefore, in order to test experimentally the non-abelian energy loss of jets, dNgrdy d 2 k H of the induced gluon radiation must be known. The angular integrated energy loss, dErdx, is insufficient to predict its experimental consequences. Third, for finite nuclei Lrl - 10 is not very large, and a large fraction of the jets are produced in the nuclear ‘‘corona’’ with Lrl ( 3. Thus the form of the angular distribution for only a few collisions is needed. Fourth, a problem specific to nuclear collisions is that the very large transverse energy background due to minijets Žthe partons from which the dense quark–gluon plasma is made. prevents the application of traditional transverse energy jet cone measurements of jets w14x. Therefore, jet quenching must be inferred from the systematics of leading hadrons at moderate p H ; 3–10 GeV relative to well-known spectra in elementary nucleon–nucleon reactions. This step requires the interface of the parton level shower calculations with hadronization phenomenology. This paper is part I of a study of jet quenching concentrating on the formal problem of computing the angular distribution gluon radiation from off-shell jets produced in a

'

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dense but thin medium. In the subsequent paper II w15,16x, we encode the results of this paper into a nuclear collision event generator ŽHIJING1.36 w17,18x using the Lund JETSET w19x string fragmentation scheme for hadronization. and compute the hadron inclusive jet quenching pattern. Note that in a separate study w20x, the transverse momentum dependence of the LPM effect in QED was discussed in another formalism and compared to the analytic results for QCD derived in this paper. Multiple interactions in QCD are complicated by the color charge of the radiated gluons. This leads to interference between the production amplitudes and the rescattering amplitudes that makes it impossible to factor the cross section into a product of the elastic cross section and the gluon distribution. In BDMPS, a great technical simplification was achieved by considering only the angular integrated dNgrd v , energy distribution. The freedom to shift the transverse momentum, k H , integrations was used often to reduce the multiple collision problem into tractable form. In addition, a large n s s Lrl 4 1 and Nc 4 1 approximations were employed to convert the problem into an elegant equivalent Schrodinger equation problem valid under the conditions that ¨ 1rm < l < L < L f s Erm2 . An alternative path integral approach proposed in w21,22x obtained similar results Žsee also Ref. w20x.. Unfortunately, the continuum approximations developed in w23–25x are not applicable to thin plasmas and more importantly they do not apply at all to the problem of computing the angular distributions. The main complication is that the angular distributions require a complete treatment of all gluon and jet final state interactions. In this paper, we simply follow the direct albeit tedious route of computing the sum of squares of all diagrams with finite number of rescatterings. In the thin plasma limit, the main simplicity arises from the fact that for n s ( 3 the number of amplitudes 2 n sq1 y 1 is still small enough for practical calculations. Historically, the on-shell single rescattering case, n s s 1, was first considered by Gunion and Bertsch w6x ŽGB.. They estimated the hadron rapidity distribution due to gluon bremsstrahlung associated with a single elastic valence quark scattering in the Low–Nussinov model. The important difference compared to the problem addressed in this paper is that GB computed the non-abelian analog of the Bethe–Heitler formula while we are interested in the analog of radiative energy loss in sudden processes similar to the beta decay. The GB problem is to compute the soft radiation associated with a single scattering of an incident on-shell quark prepared in the remote past, i.e. t 0 s y`, relative to the collision time, t 1. In our case, the radiation associated with a hard jet processes in nuclear reactions must take into account the fact that the jet parton rescatters within a short time, t 1 y t 0 ; R A , after it is produced. The sudden appearance of the jet color dipole moment within a time interval ; 1rE results in a broad angular distribution of gluons even with no final state scattering. The induced radiation cased by rescattering necessarily interferes strongly with this hard radiation as indicated in w26x. To model a thin plasma, we employ the static color-screened potential model as considered in Gyulassy–Wang ŽGW. w8x. The scattering potential has Fourier components 4pa s Vi s V Ž qi . eyi q i P x i Ta Ž R . m Ta Ž i . , V Ž qi . s 2 . Ž 3. qi q m2

200

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Here TrŽTaŽ R .Tb Ž R .. s d a b CR DRrDA is proportional to the second-order Casimir of the DR-dimensional representation R of the jet parton, and DA s Nc2 y 1. In principle, the i t h target could be in the Di-dimensional representation with Casimir C2 Ž i . so that TrŽTaŽ i .Tb Ž j .. s d a b d i j C2 Ž i . DirDA . Note that in Eq. Ž3. Vi corresponds to A i Ž q .rŽy2 iP 0 . in the notation of Ref. w8x. For a thermally equilibrated medium at temperature T, the color screening mass in pQCD is given by m s 4pa s T 2 . In addition, as in ordinary plasmas, no gluon modes propagate below the plasma frequency, v pl ; mr63. In practice, lattice QCD calculations of m indicate sizable non-perturbative corrections to the pQCD estimates for achievable temperatures. Therefore, we simply take here m ; v pl ; 0.5 GeV as a characteristic infrared scale of the medium. In perturbation theory, there is a relation between the screening scale m and the mean free path l,

m2rl f 4pa s2 r ,

Ž 4.

where r is the density of plasma partons weighed by appropriate color factors. Another important scale for our problem is the Bethe–Heitler frequency,

v BH ' 12 m2l 4 m

Ž 5.

in the dilute plasma approximation assumed here. This paper is organized as follows. In Section 2 we derive the spectrum of the gluons, radiated as a result of the sudden production of a high energy parton in a highly localized wave-packet J Ž p . in free space. The large radiative energy loss, which we call ‘‘self-quenching’’, grows as D E ; E0 logŽ E0rm .. The strong destructive interference of this initial amplitude with the subsequent induced radiative and cascade amplitudes is considered in Section 3. The main formal results of this paper are derived in that section. In Subsection 3.1 jet acoplanarity is reviewed for the case of multiple elastic scattering without radiative energy loss. The formal structure of joint jet plus gluon probability density including multiple rescatterings is derived in Subsection 3.2. A systematic procedure for extracting all the relevant matrix elements in the eikonal approach is developed using a simple binary encoding scheme in Subsection 3.3 to help automate the computation of phase factors and the color algebra. In Subsection 3.4 the interference form factors and their dependence on the detailed plasma geometry is derived. These form factors control how close the exact result is to the extreme factorization or the classical parton cascade limits. In Subsection 3.5, the physical interpretation of the relative conditional probabilities computed from the square of the sum of amplitudes and the wavefunction renormalization necessary to compute the gluon angular distribution as a function of n s is presented. Section 4 illustrates in detail the single rescattering case. We rearrange the amplitudes into Gunion–Bertsch and gluon rescattering terms that reduce in special limits to the known cases. An additional form factor associated with the transverse profile of the target is derived that helps to identify the physical interpretation of the rearranged terms. In the simpler ‘‘broad’’ target case, those form factors drop out. The induced gluon bremsstrahlung Žbeyond self-quenching. is shown to be strongly suppressed relative to the classical limit due to the destructive LPM effect in QCD at small angles. However, the angular enhancement at large angles due to gluon rescattering remains a key feature. The more involved case of two scatterings is treated in Section 5. The analytical result shows a power law enhanced angular broadening of

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the gluon distribution relative to the case of n s s 1. This approximate power law scaling, A Ž1 q CArCR . n s , is used in Section 6 to extract an approximate n s independent reduced quantum modification factor of the gluon angular distribution. We apply this scaling to compute the angular integrated gluon energy spectrum, dI Ž n s .rd v and the jet fractional energy loss D E Ž n s .rE. The numerical results show surprisingly an approximate linear dependence of those quantities on the plasma thickness at least up to L s n s l ; 3 fm. The jet energy loss remains small because the wavefunction renormalization factor also grows rapidly with n s . As emphasized in the Summary these estimates are only lower bounds because multi-gluon emission is not treated in this paper. The generalization to multi-gluon effects is deferred to Refs. w15,16x. Details of the n s s 2,3 matrix elements and their rearrangement into the physical hard, Gunion– Bertsch, and gluon cascading components as well as the relevant color factors are recorded in Appendix A and C to ease the flow of the presentation. In Appendix B we compare the n s s 2 case to the approximation of BDMPS where some of the terms in the amplitude were neglected. In Appendix D we derive the curious color ‘‘wheel diagram’’ that arises in the n s s 3 case. 2. Hard radiation and self-quenching Consider first the amplitude for the production of a hard jet with momentum P0 localized initially near x 0m s Ž t 0 ,0.. We denote this color independent amplitude by MJ s J Ž P . e i P x 0 .

Ž 6. Ž . The hard vertex is localized within a distance ; 1rP0 , and the amplitude, J P , is assumed to vary slowly with P on the infrared screening scale m of the medium. This spectrum of hard jets is of course softened due to gluon radiation to first order in g s as shown in Fig. 1. The final state in that case is characterized by a gluon and a jet with momenta m

k s Ž v ,k z , k H . s

zPq 0 ,

2 kH

zPq 0

,kH , 2

m

p s Ž p 0 , pz ,Q H yk H . s Ž 1 y z .

Pq 0 ,

Ž Q H yk H . ,Q H yk H , Ž 1 y z . Pq 0

Ž 7.

where Ž . . . . denote their 4-momentum and w . . . x denote their light-cone momentum coordinates.

Fig. 1. The MJ hard jet production amplitude and the MJ m M0 soft gluon radiation amplitude.

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We emphasize that the ‘‘ z’’ direction in this paper refers to the unperturbed hard jet direction. In nuclear reaction, the jets are directed perpendicular to the beam axis in the frame in which its longitudinal rapidity vanishes. Therefore k x ,k y ,k z actually correspond to a 90 degree rotation about the y axis of the conventional frame in which the z axis is parallel to the beam. The initial ‘‘transverse’’ momentum Q H of the jet thus accounts for both its longitudinal rapidity as well as its azimuthal Žsymmetric. dependence about the beam. The energy loss, v , leads to a suppression of the high p H distribution of the fragments emerging from the jet. . x The gluon polarization in the Aqs 0 gauge is e Ž k . s w0,Ž2 e H Pk H .rŽ zPq 0 , e H . We consider here eikonal kinematics where

(

2 v s k z2 q k H f zE0 q

2 kH

2 zE0

, 2

0

p s

(

pz2 q p 2H

Ž Q H yk H . f Ž 1 y z . E0 q . 2 Ž 1 y z . E0

Ž 8.

The amplitude to emit a soft gluon and the jet neglecting spin effects can be expressed as MJ m M0 s i d 4 P J Ž P . e i P x 0

H

Ž yig s Tc . e P Ž 2 p q k . Ž pqk.

2

d 4 Ž P yk yp.

f e i E 0 t 0 J Ž k q p . M0 Ž k , p . ,

Ž 9.

where M0 is the radiation amplitude that in the eikonal limit is given by M0 s y2 ig s

e H Pk H 2 kH

2

e i t 0 k H r2 v c.

Ž 10 .

Here, c ' Tc Ž R . is the SUŽ Nc . color generator Tc in the DR-dimensional representation of the jet parton with TrŽ cc . s CR DR . 2 A simple direct way to recover the phase factor and the 1rk H singularity of M0 is through the use of time ordered perturbation theory as emphasized in w9x. This follows from integrating over the emission time of the gluon `

y´ < t
Ht dt e

yi

2v s

0

2 kH

2

e i t 0 k H r2 v ,

Ž 11 .

where 2 E f y Ei s v q p 0 y E 0 f k H r2 v .

This method is particularly useful when final state interactions are taken into account in the next sections. The probability distribution for the jet production is given by d 3 D Ž0. s DR < MJ < 2

d3p 3 Ž 2p . 2 p 0

' r Ž0. Ž p .

d3p 3 Ž 2p . 2 < p <

,

Ž 12 .

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where r Ž0. Ž p . is proportional to the invariant hard production probability density with no elastic rescatterings of the jet nor radiative energy loss. The pQCD jet distribution is approximately a power law 1rp 2Hq relative to the beam axis and uniformly distributed in longitudinal rapidity. The power q ; 3 depending on the beam energy and the p H range under consideration. However, in terms of our rotated variables with pz p x

r Ž0. Ž p . ; cr Ž pz2 q p y2 .

l

q

independent of our p x that is parallel to the beam direction. We emphasize this point because unlike the usual energy loss problem of a beam passing through matter, the peculiar longitudinally expanding cylindrical geometry of the matter in nucleus collisions makes applications of simple formulas in the rotated frame somewhat unintuitive. The double inclusive jet plus gluon probability distribution is given by Ž0. Ž0. d 6 Drad s rrad Ž k , p.

d3k

d3p

3 3 Ž 2p . 2 v Ž 2p . 2 p 0

,

Ž 13 .

where Ž0. rrad Ž k , p . ' Tr < MJ m M0 < 2

Ž 14 .

is the double inclusive probability density for producing a jet and a gluon in the case of no rescatterings. As seen from Eqs. Ž9., Ž10. the double inclusive density factorizes for soft gluons into a production probability of an off-shell jet with k q p and a distribution of gluons.

v

d 3 NgŽ0. d3k

f

CR a s 1

' R Ž0. g Ž k. ,

2 p 2 kH

Ž 15 .

where the dependence on p is only implicit through the kinematic boundaries. As in QED the radiation pattern associated with a hard jet corresponds to a uniform rapidity distribution along the jet axis with a broad transverse transverse momentum distribution 2 that is uniform in log k H . If the off-shell dependence of the source amplitude is weak and the gluon is soft in the sense 1
< JŽ < p
1 < pqk<

< J Ž < p q k <, p q k . < 2 ,

then we can write 3

Ž0. rrad Ž k , p . f 2 Ž 2p . r Ž0. Ž p q k . R Ž0. Ž k . .

Ž 16 .

The gluon radiation associated with any hard pQCD process leads to what one can think of as ‘‘self-quenching’’ of the jet in the sense that the initially produced high energy jet parton loses a substantial fraction of its energy

D E0 s C R

as

1

dz

H HE p 0 z m 2

2

2 dk H 2 kH

zE0 ; CR

2 as

p

E0 log Ž E0rm . .

Ž 17 .

This occurs even in free space where there are no final state interactions. In practice, the large logarithm implies that multiple gluon emission must also be considered. This leads

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to a Sudakov form factor for the jet and a probabilistic Alterelli–Parisi parton shower w27x. The final state multigluon shower can be calculated most readily by one of the many Monte Carlo event generators such as PYTHIA w28–30x. Those generators encode empirical ‘‘parton to hadron’’ fragmentation functions and thus allow the detailed study of the effect of parton showering on the final hadron distributions. The magnitude of the self-quenching effect, i.e. initial and final state radiation, is found to be significant and is essential to describe accurately the high p H data as shown in w31x. Our main interest here is to derive the extra quenching associated with gluon radiation induced by final state interactions of the jet in a thin plasma.

3. Gluon radiation with rescatterings In this section, the self-quenching case is generalized to take into account possible multiple scattering of the jet and gluon following the hard jet production. 3.1. Multiple elastic scattering The amplitude that jet undergoes n s elastic scatterings with momentum transfers qi s Ž0,qi z ,qi H . is given by w8x ns

MJŽ n s . s d 4 P J Ž P . e i P x 0

H

H is1 Ł

d 3 qi

ž

Ž 2p .

Ž 2 P 0 Ta Ž i . D Ž P q Qiy1 . V Ž qi . eyi q x . i

3

i

=d 4 Ž P y p q Ý qi . Ta1Ž R . . . . Ta n Ž R . ,

ž

/

Ž 18 .

/

s

i

where DŽ p . s Ž p 2 y m 2 q i e .y1 is the jet propagator neglecting spin effects, and the cumulative momentum transfer is Q i ' Ýijs1 q j . The invariant jet probability distribution together with n s recoil partons in the target and no gluon radiation is to lowest order Žwithout Sudakov factor. given by d 3 DelŽ n s . s Tr < M ŽJn s . < 2

¦

d3p

; Ž 2p . 2 p x

3

0

s relŽ n s . Ž p .

d3p 3 Ž 2p . 2 p 0

.

Ž 19 .

The ensemble average over the location of the scattering centers is taken as in w8x ns

² . . . :x s

H is1 Ł

ž

`

dx i H T Ž x i H .

H0

d Ž z i y z iy1 .

li

/

ey Ž z iyz iy 1 . r l i u Ž z i y z iy1 . Ž . . . . .

Ž 20 . In a static medium, the average longitudinal separation between the centers is specified by the mean free path, l s l i . An expanding medium, on the other hand, has l i ) l iy1 and can only be approximately treated with a constant mean free path. The transverse coordinates are distributed according to a normalized overlap function, T Ž x H . ; 1rŽp R 2H .. We assume that the transverse size R H of the target Žrelative to the jet axis. is large compared to the range of the interactions, 1rm. In the high energy Ž p 0 s E 4 v 4 m . eikonal limit, going back to the picture including the momenta of the jet, the integrations over pi z are fixed by residues

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approximately to pi z f E y pi2H rŽ2 E .. The contributions from the poles of the static potential are suppressed by a factor A expŽyml.. In this model we neglect the small collisional energy loss associated with target: dEel rdx f 4a s2 T 2 log Erm ; 0.3 GeVrfm w1,2x. Elastic scattering modifies the jet distribution in Eq. Ž19. in this eikonal limit as ns

relŽ n s . Ž p . f

H is1 Ł

ž )

d 2 qi

d 2 qiX 2

Ž 2p . Ž 2p .

2

V Ž qi . V ) Ž qiX . d2 QH

X

Ž DR J Ž p y Q . J Ž p y Q . . f H

Ž 2p .

=

ž

2

d 2 qiX

Ž 2p .

CR C2 Ž i . DA

T Ž qi H yqiX H .

ns

r Ž p yQ. Ž0.

/

H is1 Ł 2

ž ž

2

d qi

d 2sel d 2 qi H

T qX 2p . d 2 Q H y 2 Ž i. Ž

/ /

ns

Ý qi H js1

/

,

Ž 21 . which includes the expected jet acoplanarity w32,33x resulting from random walk but no recoil energy loss. The major simplification in the above derivation arises from the assumption that < qi y qiX < ; 1rR H < m < l so that the main source of non-diagonal dependence arises through the T Ž qi H yqiX H . factors. This is what allows us to recover the above classical elastic parton cascade form in terms of the effective differential cross sections d 2sel d 2 qi H

s

CR C2 Ž i .

4a s2

DA

Ž qi2H q m2 .

2

.

Ž 22 .

The elastic cross section is then sel s Ž CR C2 Ž i .rDA .Ž4pa s2rm2 . which for a gluon jet propagating in a gluon plasma with m s 0.5 GeV is typically sel ; 2 mb. ŽSee Ref. w34x for a new Monte–Carlo implementation of such an elastic parton cascade.. We will neglect the influence of elastic multiple scattering on the inclusive distribution of jets because r Ž0. Ž p . is broadly distributed and the elastic energy loss is relatively small. In this case, r Ž0. Ž p y Q . f r Ž0. Ž p . can be factored out of Eq. Ž21. leading to ns

relŽ n s . Ž p . f r Ž0. Ž p . Ž sel T Ž 0 . . ,

Ž 23 .

which is simply proportional to the free space spectrum times a geometrical factor Žthe Glauber multiple collision probability.. 3.2. Gluon radiation with quantum cascading The amplitude to emit a gluon together with the jet including final state interactions with n s scattering centers is given by ns

MJŽ n s . m Mn s s d 4 P J Ž P . e i P x 0

H

H is1 Ł

ž

d 3 qi

Ž 2p .

3

Ž y2 iP 0 . V Ž qi . eyi q i x i T Ž i .

= Mn sŽ k , p;q1 , . . . ,qn s . d 4 Ž P y k y p q Ý qi . .

/ Ž 24 .

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This leads to the conditional double inclusive jet and gluon probability distribution Ž n s. Ž n s. d 6 Drad s rrad Ž k , p.

d3k

d3p

3 3 Ž 2p . 2 v Ž 2p . 2 p 0

ns Ž n s. rrad Ž k , p . s < M ŽJn s . m Mn s < 2 '

¦

;

ž

1

Ł

is1

Di

/

,

Tr < M ŽJn s . m Mn s < 2 x ,

¦

;

Ž 25 .

which involves averaging over all initial target colors and summing over all final colors and the locations x i of the target centers. After integrating over the energy-conserving delta functions we arrive at the momentum space distribution density ns Ž n s. rrad

ž

Ž k , p . sH Ł

is1

¦

=

d 3 qi

d 3qiX 3

2

Ž 2p . Ž 2p .

ns

X i

;

Ł eyi Ž q yq . x i

is1

3

Ž 2 p 0 . V Ž qi . V ) Ž qi . Ž C2 Ž i . rDA . X

/

J Ž k q p y Ý qi . J ) Ž k q p y Ý qiX .

i

x

=Tr Ž Mn sŽ k , p;q1 , . . . ,qn s . Mn†s Ž k , p;q1X , . . . ,qnX s . . .

Ž 26 .

The amplitude Mn S is a sum over all time-ordered graphs involving n s scatterings in which one gluon is radiated. For a slowly varying source, J Ž p q k y Q . f J Ž p q k . can be factored out of the integral in Eq. Ž26. leading to ns Ž n s. rrad

Ž k, p. f< J Ž p qk . < 2 0 2

=

Ž2 p .

H is1 Ł

C2 Ž i .

DA

ž

d 3 qi

d 3 qiX 3

Ž 2p . Ž 2p .

3

V Ž qi . V ) Ž qiX . ns

T Ž qi H yqiX H .

/¦Ł

;

X

eyi Ž q i zyq i z . z i

is1

=Tr Ž Mn sŽ k , p;q1 , . . . ,qn s .

Mn†s

z

X

X

Ž k , p;q1 , . . . ,qn . . . s

Ž 27 .

The integration over the qi z can again be performed by contour integration leading to a sum over many terms with longitudinal phases set by energy differences as in w8,9x. In the eikonal limit, a simple shortcut to that calculation is possible when the scattering of potentials localized in the longitudinal direction at z i can be approximated by impulse interactions at times, t i f z irc w9x. In that case, time ordered perturbation theory can be used to express Eq. Ž27. as ns Ž n s. rrad Ž k, p. f< J Ž p qk . < 2

=

C2 Ž i . DA

H is1 Ł

ž

d 2 qi

Ž 2p .

d 2 qiX 2

Ž 2p .

2

ns

T Ž qi H yqiX H .

V Ž qi . V ) Ž qiX . 2m

/¦ ž Ý Ý Tr

Mn s , m ,l Ž k , p;q1 H , . . . ,qn s H .

ms0 ls1

/

X

ns

=

2m

žÝ Ý X

X

m s0 l s1

Mn†s , mX ,lX Ž k , p;q1X H , . . . ,qnX s H .

/;

, t

Ž 28 .

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Fig. 2. Example of the reduced M5,1,10 radiative amplitude for n s s 5.

where the sums go over the 2 n sq1 y 1 different time ordered diagrams that we list in the next sections. The reduced radiation amplitudes, Mn s , m,l , for a given n s are labeled by two integers, 0 ( m ( n s and 0 ( l. Here m specifies the time interval t m - t - t mq1 in which the gluon is radiated and l specifies which of the 2 n sym final state interaction patterns with centers m q 1, . . . ,n s occurs. 3.3. Reduced radiatiÕe amplitudes It is convenient to associate with each amplitude diagram, Mn s , m,l , an n s-dimensional binary array, s , that decodes the label l for a given m as follows:

s s Ž s 1 s 0, . . . , sm s 0, smq1 , . . . , sn s . m

mls 2 1

mq1

ns

ž

Ý sj 2 j js1

/

,

Ž 29 .

where si s 0 for 1 ( i ( m while si s 0 or 1 for m q 1 ( i ( n s depending on the integer part of lr2 iymy1. This is the base 2 representation of l = 2 mq 1 s sn s . . . s 1 so that 0 ( l ( 2 n sym y 1. For a fixed m, l s 0 corresponds then to no final state gluon rescattering, while l s 2 n sym y 1 corresponds to the gluon rescattering with all n s y m remaining scattering centers after t m . For example, Fig. 2 shows M5,1,10 as the time ordered diagram with n s s 5 scattering centers, where the gluon was emitted from the jet between t 1 and t 2 and l s 10 corresponding to s s Ž0,0,1,0,1. encodes the fact that the gluon received impulses q H 3 ,q H 5 at times, t 3 and t5 . The binary representation is useful as a mask to extract the correct kinematic variables automatically. The transverse momentum of the gluon between interactions i and i q 1 is simply ns

K iŽ l . s k H y

Ý

Žl. sj q j H s K iy1 q si q i H ,

Ž 30 .

jsiq1

with K nŽ ls. ' k H , the final gluon transverse momentum. For the example in Fig. 2, K 0Ž10. s K 1Ž10. s K 2Ž10. s k H yq3 H yq5 H and K 3Ž10. s K 4Ž10. s k H yq5 H . The complete amplitude matrix can be constructed in a few steps. First, the time integration over the emission times between t m and t mq1 and the n s impulses gives rise to the factor

v vm l

Ž e i v m l t mq 1 y e i v m l t m . e iF l m ,

Ž 31 .

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where the energy shift at the emission point is denoted 2

ns

vm l s

Ž K mŽ l . .

2

ž

kHy

Ý sj q j H js1

/

. Ž 32 . 2v 2v 2 The apparent singularity at v m l s 0 s Ž K mŽ l . . due to the energy denominator is cancelled for finite t mq 1 y t m - `. The singular t n sq1 s ` phase is always suppressed by an adiabatic damping factor Žnot shown.. The only singular amplitude is in fact the one corresponding to the factorization limit in which final state interactions factor out. Including the eikonal gluon emission vertex 2 ig s emŽ K mŽ l . . p m f g s e H PK mŽ l ., the special case for m s n s , l s 0 reduces to e H PK nŽ0.s i t K Ž 0 . 2r2 v Mn s , n s ,0 s y2 ig s e nsŽ n . c Ž a n s . . . a1 . 2 Ž 0. K Ž ns . s

s

2

s e iŽ t n syt 0 . k H r2 v M0 Ž a n s . . . a1 . , where M0 is given by Eq. Ž10.. All the other amplitudes remain bounded < Mn s , m,l < C Ž t mq 1 y t m .. On the other hand, in the formal t 0 y` limit, all the 2 n s y 1 singularities at v m l s 0 become exposed. The overall eikonal phase, Fm l , acquired by the gluon through final state interactions is ns ns tj 2 Žl . 2 Fm l s Ý V m l j t j s Ý K jŽ l . . y Ž K jy1 Ž . js1 js1 2 v

™

ž

ns

s

Ý jsmq1

sj

tj 2v

žŽK

Žl . 2 j

.

/

2

y Ž K jŽ l . y q j H . ,

Ž 33 .

/

where V m l j s EFm lrE t j . The amplitude for color exchange leaving the final gluon with color c s 1, . . . , DA is specified by a particular color matrix, Tm l c . The scattering of the jet with center i brings in a color matrix a i ' Ta iŽ R . in the R representation. Gluons interact with center i, on the other hand with Ta iŽ A. in the adjoint representation. It is numerically convenient, however, to express the complete color structure involving gluon final state interactions in terms of commutators in the jet Ž R dimensional. representation. Label sequentially in increasing order all n g s Ý k s k color matrices, a k where s k s 1, by c1 , . . . ,c n g. Label all remaining n s y m y n g matrices with k ) m and s k s 0 by b 1 , . . . ,bn symyn g. Then Tm l c s Ž bn symyn g . . . b 1 . . . . w c,c n g x , . . . ,c1 Ž a m . . . a1 . .

Ž 34 .

In the example of Fig. 2, T1,10,c s a 4 a 2 ww c,a5 x,a 3 x a1. Eq. Ž34. defines a straightforward algorithm for numerical computation of color factors, Cm l mX lX s Ý Tr Ž Tm l c Tm†X lX c . , c, a1 , . . . , a n

needed as weights of the different interference terms in Eq. Ž28.. Finally, the reduced radiation amplitudes with final state interactions are given by 2 e H PK mŽ l . i t Žl. 2 Ž l . r2 v mq 1 Ž K m . Mn s , m ,l s 2 ig s e y e i t m Ž K m . r2 v e iF l m Tm l c . Ž 35 . Žl . 2 Ž Km .

ž

/

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3.4. Interference form factors If the ‘‘width’’, R H 4 l 4 1rm, of the target transverse to the jet direction is large compared to its thickness, L, Ži.e. R H 4 L ) l 4 1rm ., then the off-diagonal dependence in < qi H yqiX H < ; 1rR H in Eq. Ž28. can be ignored in all factors except T Ž qi H yqiX H . Žsee Subsection 4.1 for more detailed discussion on this.. In this case, the interference terms arising from the distribution of scattering times, t i , involve the following form factors: X X

Fm l mX lX '² ei ŽF m ly F m l . Ž e i t mq 1 v m l y e i t m v m l . Ž eyi t mXq1 v mX l y eyi t mX v mX l . :. X

X

X

Ž 36 .

For m,m - n s , the form factors insure that Fm l m l rŽ v m l v m l . have no singularities except for m,mX s n s , and control the magnitude of the destructive LPM interference effects. The diagonal form factors are given by X X

X X

Fm l m l s 2 y 2 ²cos Ž Ž t mq1 y t m . v m l . : ,

Ž 37 .

except that Fn s 0 n s 0 s 1 since t n sq1 s ` q i e . The extreme limits of quantum cascading correspond to Ž1. the incoherent Classical Parton Cascade limit and Ž2. the coherent Factorization limit. In the first case, all non-diagonal form factors vanish, while in the second case all but the m s n s diagonal form factor vanish: Fm l mX lX s 2 dm , mX d l ,lX

Classical Parton Cascade,

Fm l mX lX s dm , n sdmX , n s

Coherent Factorization.

Ž 38 .

Simple analytic expressions can be obtained for the form factors if the ensemble average over the interaction times is taken as in Eq. Ž20.. The form factors involve a sum of four terms of the form ns

ns

ns

e i Ý Ž t jyt 0 . v j s e i Ý t j v˜ j s Ł

¦

;¦

js 1

js1

;

`

H js1 0

ns

dt j

lj

tj

ey l j Ž1yi l j v˜ j .

1

sŁ

js1

1 y i l j Ž v j q . . . qv n s .

,

Ž 39 .

where t j ' t j y t jy1 , v˜ j s v j q . . . qv n s. Here l j s ²t j : is the average distance between the j y 1 and j th scattering centers. This ensemble is characterized by the following distribution of t s t k y t 0 : Pk Ž t . s ² d Ž t k y t 0 y t . : dv

s

ns

s

k yi v Ž t k yt 0 .

H 2p e Ý js1

eyt r l j

lj

Ł

js1

k

ž

1 1 y i vl j 1

Ł Ž1 y l rl . l/j l j

/

™

l ks l

eyt r l Ž trl .

l

ky 1

Ž k y 1. !

The geometry of the plasma is particularly simple if l k s l, with ² Ž tk y t0 . : s k l ,

² Ž t j y t 0 . Ž t k y t 0 . : s k Ž j q 1 . l2 ,

j 0 k.

.

Ž 40 .

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For more realistic ensembles corresponding to an expanding target with centers distributed according to r Ž z,t .. The form factors must be evaluated numerically via ns

1

e i Ý Ž t jyt 0 . v j s

¦

;

js 1

Pn

`

Ht dt

n

ey x Ž` ,t n .

0

tn

Ht

dt ny1 . . .

0

t2

Ht

dt 1

žŁs

el

r Ž z k s t k ,t k .

k

0

=e i Ž t jyt 0 . v j ey x Ž t k ,t ky 1 . ,

/

x Ž t ,t X . s

t

Ht d tˆs

el

X

r Ž z s tˆ,tˆ . ,

x n s Ž `,t 0 .

Pn s s ey x Ž` ,t 0 .

ns!

.

Ž 41 .

Here x Ž t,t X . is the average number of interactions a jet moving with the speed of light through the expanding target suffers between t X and t with sel given by Eq. Ž22.. Note that Pn s is the probability that only n s out of a large possible N 4 1 target interactions occur in the medium. In this more general case, the interactions are distributed as Pk Ž t k . s Pn s

dg Ž t k . dt k

Ž1 y g Ž tk . .

n syk

g ky 1 Ž t k . ,

g Ž t . s x Ž t ,t 0 . rx Ž `,t 0 . .

Ž 42 .

For a static uniform slab of thickness L with density r 0 u Ž L y z . ns

e i Ý tjv j s

¦

js 1

;

ns! ns

L

L

H0 dt

ns

e i t n sv n s . . .

t2

H0 dt e 1

i t1 v 1

,

Ž 43 .

It is instructive to compare Eq. Ž43. with Eq. Ž39.. In the slab geometry setting t 0 s 0 ² tk : s

kL

Ž n s q 1.

,

² t j tk : s

k Ž j q 1 . L2

Ž n s q 2. Ž n s q 1.

,

j 0 k.

Thus, if we set l ' LrŽ n s q 1. ² t j t k : s k Ž j q 1 . l2

² tk : s k l ,

Ž n s q 1. , Ž n s q 2.

j 0 k.

For n s 4 1 the geometry of both ensembles is quite similar if set l appropriately. Near s the factorization limit, where Ý njs1 lv j < 1 we can expand ns

i

¦e Ý

js 1

ns Ž t jyt 0 . v j

ns

j

2

;f 1 q i l Ý jv y rl Ý Ý k Ž j q 1. v v , j

js1

j

k

Ž 44 .

jsi ks1

where the only difference between Eq. Ž39. and Eq. Ž43. is that r s 1 in the first case and r s Ž n q 1.rŽ n q 2. in the second case. The deviations from factorization can therefore be reasonably well approximated by our simpler ensemble. On the other hand, for large lv j , the slab form factors oscillate due to the sharp boundary while the form factors in Eq. Ž39. ensemble decrease monotonically. The deviations from the classical parton cascade limit is therefore more sensitive to the detailed form of the geometry of the medium. We will continue to use below Eq. Ž39. to reduce the complexity of the calculation.

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Since the final pattern of jet quenching depends on the actual distributions and correlations among scattering centers in general, one should evaluate Eq. Ž36. numerically for any actual application to nucleus–nucleus collisions w15,16x. 3.5. RelatiÕe conditional probabilities It is tempting to identify as in Eqs. Ž15., Ž16. R Žgn s . Ž k . s

Ž n s. rrad Ž k, p.

1

Ž 45 .

16p 3 relŽ n s . Ž p q k .

as the invariant soft gluon number distribution associated with n s interactions. However, because the radiated gluon can rescatter in addition to the incident jet, the conditional Ž n s. , cannot be factored into a product of the elastic double inclusive density, rrad distribution and the gluon number distribution as in the case of QED. The problem is Ž n s. includes the that the production and gluon rescattering amplitudes interfere, and rrad process where the gluon is radiated before the first collision and the n s recoil target partons are produced by elastic scattering with the produced gluon rather than the incident jet. Therefore, we interpret R Žgn s . as a relative conditional probability density to find a gluon in the final state with momentum k given a final observed jet with momentum p and given that n s target recoil partons are observed Žwith any momentum.. Let Pn s be the a priori probability that there are n s scattering centers or target recoil partons. The final jet quenched spectrum is then given by

rJ Ž p . s Ý ns

Pn s Zn s

ž

rŽ 0 . Ž p . q

d3k

H

v

/

r Ž0. Ž p q k . R Žgn s . Ž k . ,

Ž 46 .

where r Ž0. Ž p . is the initial hard jet distribution in Eq. Ž12., and where the Žwavefunction. renormalization factor d3k

RŽ n s. Ž k . Ž 47 . v g insures that the integrated total number of jets is conserved. It in fact corresponds to the first-order in a s term in the expansion of the inverse Sudakov factor for the probability that no radiation accompanies the jet. In this paper we only compute, R Žgn s . Ž k ., which limits the discussion to single gluon emission: R Žgn s . Ž k . ' R Žgn s , n gs1 . Ž k .. Unfortunately, even in the n s s 0 case the single gluon approximation is not accurate because CR a s Z0 s 1 q log 2 Ž E0rm . Ž 48 . p can be significantly greater than one, indicating that multigluon emission from the hard vertex is necessary. Summing the double logarithms leads to a modified Bessel function multiplicity distributions as discussed in w27x. The DLA leads to w27x Zn s s 1 q

H

Z0 s 1rI0 Ž r Ž E . . ,

(

r Ž E . s 8CR arp log Ž Erm . , ² ng : s

r I1 Ž r . 2 I0 Ž r .

f

r Ž E. 2

.

Ž 49 .

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In the general case of multiple gluon emission, as required in practical applications, we must generalize Eqs. Ž46., Ž47. as Pn s

rJ Ž p . s Ý

Zn s

ns

ng

`

ž

rŽ 0 . Ž p . q

Ý n g s1

=R Žgn s , n g . Ž k 1 , . . . , k n g

.

/

H is1 Ł

d3ki

vi

ž

r Ž0. p q Ý k j

,

j

/ Ž 50 .

where the Žwavefunction. renormalization factor becomes `

Zn s s 1 q

Ý n gs1

ng

H is1 Ł

d3ki

vi

R Žgn s , n g . Ž k 1 , . . . , k n g . .

Ž 51 .

The implementation of this approach requires Monte–Carlo methods that will be discussed in the subsequent paper w15,16x. From the above formulas we can infer that the exclusive n g gluon number distributions associated with n s scattering is

v 1 . . . vng

dNgŽ n s , n g . 3

3

1 s

d k1 . . . d k ng

Zn s

R Žgn s , n g . Ž k 1 , . . . , k n g . .

Ž 52 .

4. Case of one-scattering center We consider in this section the simplest case of gluon radiation in hard processes followed by one rescattering. 4.1. Reduced amplitudes and transÕerse profile dependence The three graphs that generalize the Gunion–Bertsch analysis w6x to this case are shown in Fig. 3.

Fig. 3. Three contributions to the soft gluon radiation amplitude MJ m M1 in case of one scattering.

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Following BDMPS w9x, the three radiative amplitudes are given by M1,0,0 s 2 ig s

M1,0,1 s 2 ig s

e H Pk H 2 kH

2 1 H

Ž eit k

r2 v

e H P Ž k y q1 . H

Ž

2 k y q1 H

2

y e i t 0 k H r2 v . a1 c, 2

Ž 53 .

2

e i t 1w k H yŽ kyq1 . H xr2 v

.

2

2

= Ž e i t 1 w Ž kyq1 . H x r2 v y e i t 0 w Ž kyq1 . H x r2 v . w c,a1 x , M1,1,0 s 2 ig s

e H Pk H 2 kH

2 1 H

Ž ye i t k

r2 v

Ž 54 .

. ca1 .

Ž 55 .

Here t 1 is the impulse time when q1 H transverse momentum is imparted to the jet relative to its axis p. The color charge flow is modified by an additional matrix a1 , that appears in different orderings relative to the radiation vertex c in these amplitudes. A new infrared singularity k H yq1 H s 0 emerges due to the dipole radiation associated with the exchanged gluon. However, the phase factors cancel at that point and the amplitude remains finite proportional to t 1 y t 0 . To keep the somewhat cumbersome notation compact we introduce the following reduced amplitudes: Hs

kH 2 kH

,

CŽ i 1 i 2 . . . i m . s

Bi ' H y C i ,

Ž k y qi y qi y . . . yqi . H , 2 Ž k y qi y qi y . . . yqi . H 1

2

m

1

2

m

BŽ i 1 i 2 . . . i m .Ž j1 j 2 . . . i n . ' CŽ i 1 i 2 . . . j m . y CŽ j1 j 2 . . . j n . ,

which we call ‘‘hard’’, ‘‘cascade’’, ‘‘Gunion–Bertsch’’, and ‘‘Gunion–Bertsch cascade’’. The time integrals involve energy differences 2 v0 skH r2 v s 1rt f ,

v 0 i ' v 0 y vi ,

2

vŽ i1 i 2 . . . i m . s Ž k y qi 1 y qi 2 y . . . yqi m . H r2 v ,

vŽ i 1 i 2 . . . i m .Ž j1 j 2 . . . i n . ' vŽ i 1 i 2 . . . i m . y vŽ j1 j 2 . . . i n . ,

that control the formation physics. A quantity with two indices is defined to be the difference of two quantities with a single index Žnote that an array in parentheses counts as a single index.. Thus, for example t 10 ' t 1 y t 0 , v 10 ' v 1 y v 0 . To make contact with the Gunion–Bertsch bremsstrahlung radiation distribution w6x and facilitate the physics interpretation we rearrange the sum of the three amplitudes as follows: M1 s y2 ig s e i t 0 v 0 e H P  Ha1 c q B1 e i t 10 v 0 w c,a1 x q C1 eyi t 10 Ž v 1yv 0 . . w c,a1 x 4 . Ž 56 . Including an adiabatic damping factor eye t 1 , we see that for t 1 radiation formula for the n s s 0 case is recovered,

™limq` M s a

t1

1

1

M0 .

™ ` the pure hard Ž 57 .

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On the other hand, in the t 0 M1

™ y2 ig e s

H PB1 e

™ y` limit,

i t1 v 0

w c,a1 x s MGB ,

Ž 58 .

corresponding to the isolated bremsstrahlung amplitude. This leads to the characteristic GB radiation spectrum dNgŽGB. dy d 2 k H

s CA

q12

as

,

2 2 p 2 kH Ž k y q1 . H

Ž 59 .

that has to be aÕeraged over the transferred momentum. Here only the adjoint Casimir CA s Nc enters and the result is independent of CR . Note that in this limit, the k H s q1 H is exposed and in w6x is regulated with an appropriate hadronic form factor. For the theoretical application to a parton penetrating through infinite matter, this singularity is regulated by the in-medium self-energy of the gluon, P mn Ž k .. The transverse plasmon modes satisfy v 2 s k 2 q P T Ž v ,k . in an infinite plasma medium, with P T Ž v plŽ k .,k . f v pl2 f m2r3 and m2 s g 2 T 2 Ž2 Nc q Nf .r6 is the Debye screening 2 mass in thermal pQCD. For k z 4 T on the other hand, v plŽ k . f k z q Ž k H q 2 2 m r2.rŽ2 k z .. Thus the infrared singularities are regulated by a scale ; m . Near the light cone, the one loop approximation for the self-energy is not likely to remain accurate though. In fact, gauge invariance requires that the same m2 screening scale regulate both the potential singularity and the above infrared divergences w8,9x. In contrast to the above, t 0 y` limit, the k H s q1 H singularity is also regulated by canceling phases for t 10 finite. In this case of interest here, it is the maximum of m2 and vrt10 that regulates that singularity. For large v 4 v BH s m 2lr2, most important for energy loss, the latter scale in fact sets the characteristic transverse momentum of the gluon in the k H ) m domain. Summing over final polarizations and colors and averaging over initial colors, the n s s 1 gluon distribution from Eqs. Ž28., Ž53. – Ž55. is

™

Ž1. rrad

Ž k , p. f < J Ž p qk . <

¦

= Tr

ž

2

d 2 q1

H Ž 2p .

Ý

d 2 q1X 2

Ž 2p .

2

V Ž q1 . V ) Ž q1X .

C2 Ž i . DA

T Ž q1 H yq1X H .

M1, m ,l Ž k , p;q1 H . M1,† mX ,lX Ž k , p;q1X H . X

1,m ,l , m ,l

X

/; .

Ž 60 .

t

The main complexity in the above expression arises from the mixing of different q1 and q1X components. However, if the transverse profile T Ž x H . of the target is approximately a constant over dimensions R 4 1rm, then the difference < q1 y q1X < ; 1rR can be neglected in some of the terms. The only sensitive dependence of that difference arises through the phase factors in Eq. Ž53. – Ž55.. A systematic expansion in that difference is possible changing variables to Q s Ž q1 q q1X .r2 and q s q1 y q1X and expanding in q. To leading order in this expansion Ž1. rrad

Ž k , p.

s 4 g s2

< JŽ pqk. <2

H Ž 2p .

¦ ž Ž Ha c q B

= Tr

1

d 2 Q C2 Ž i .

Qe

2

i t 10 v 0

DA


d 2q

H Ž 2p .

2

TŽq .

w c,a1 x . P Ž Hca1 q BQ eyi t10 v 0 w a1 ,c x . /

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215

X

qTr < CQ < 2 eyi t 10 Ž v 1y v 1 . w c,a1 xw a1 ,c x

ž qTr ž Ž Ha c q B qTr ž Ž Ha c q B

i t 10 v 0

/

w c,a1 x . P CQ eqi t10 Ž v 1yv 0 . w a1 ,c x /

1

Qe

1

yi t 10 v 0 Qe

X

w c,a1 x . P CQ eyi t10 Ž v 1yv 0 . w a1 ,c x /;t . 1 Ž 61 .

.2

X

.2

Note that v 1 s Ž k y Q y qr2 r2 v , v 1 s Ž k y Q q qr2 r2 v , and therefore, the extra phase factor multiplying the diagonal cascade contribution has a phase b 1 P q s t 10 Ž k y Q . P qrv , where we drop O Ž t 10 m2rv . terms. The integration over d 2 q transforms T Ž q . therefore into transverse spatial profile T Ž b 1 . evaluated at the impact parameter b 1 s t 10 Ž k y Q . rv , at which a gluon produced at the origin with transverse momentum k y Q would pass by the scattering center at z 1 s ct 1. This tends to suppress the cascade contribution as intuitively expected for kinematic configurations where the gluon ‘‘misses’’ the target. On the other hand, the diagonal hard and Gunion–Bertsch contributions remain unsuppressed since the rescattering of the jet occurs at zero impact parameter in the high energy limit. For a broad target, the transverse profile is approximately a constant T Ž b 1 . f T Ž0. ; 1rp R 2 . However, in general the cascade term is modulated by a relative target transverse profile factor T Ž b 1 . ' T Ž b 1 . rT Ž 0 . .

™

The relative conditional probability for gluon radiation in this n s s 1 case for a fixed momentum transfer Q H q1 H therefore reduces to as R Ž1.  H 2 q R Ž B12 q T Ž b1 . C12 . y R Ž H P B1cos Ž t10 v 0 . . g Ž k ,q1 H . s C R p2 yR T Ž b 1r2 . Ž H P C 1cos Ž t 10 v 10 . y 2 C 1 P B1cos Ž t 10 v 1 . . 4 . Ž 62 . Note that the transverse profile factor modulates the interference term between C and H q B at the average impact parameter, b 1r2 of the jet and gluon at the scattering center. Note that for t 10 0 there is a perfect cancellation of all but the hard radiation term since b 1 A t 10 0 and T Ž b s 0. s 1. In fact, this factorization limit is recovered more generally when t 10 rv 0. The first three terms corresponds to the diagonal terms which are independent of the time between the hard production point and the rescattering point. The next three terms are oscillating interference terms. The color factor R s CArCR is 1 for a gluon jet and 9r4 for a quark jet. It is interesting to note that if the momentum transfer q1 in the scattering vanishes, then B1 s 0, C s H, and v 1 s v 0 and the above distribution reduces to the zero-scattering case in the previous section. This is a manifestation of gauge invariance requiring a finite q1 to generate a color dipole that can modify the hard radiation pattern. It is clear that the classical incoherent parton cascade limit corresponds to assuming that all the oscillating interference term average to zero. In that case, the final gluon distribution is a sum of the initial hard radiation plus an isolated Gunion–Bertsch bremsstrahlung plus a rescattering ‘‘cascade’’ contribution reflecting the modification of

™ ™™

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the final transverse momentum spectrum due to rescattering of a gluon produced at t 0 . That latter term broadens the transverse distribution of the emitted gluons. 4.2. RelatiÕe conditional probability for n s s 1 The magnitude of the interference effects in Eq. Ž62. is controlled by the dimensionless variables

k'

2 kH

m2

lm2 j'

,

v BH

s 2v

v

2

k1 '

,

Ž k y q1 . H

u1 '

,

m2

q12H

m2

.

Ž 63 .

Note that this averaged interference cosine is small when jk i ) 1. This is the kinematic region where the relevant formation time, t i s 1rv i s 2 vrŽ k H yqi H . 2 , is smaller than the mean free path. For a fixed energy v this restricts the angle of induced emission radiation to exceed umin s 1r6Ž vl.. Below umin complete destructive interference cancels all but the hard production amplitude. The time averaged relative conditional distribution for fixed q1 H can be written as Ž0. Ž1. R Ž1. Ž k H ,q1 H , j . , g Ž k ,q1 H . s R g Ž k . F

Ž 64 .

Ž . where R Ž0. g is given by Eq. 15 and the modification factor due to n s s 1 rescattering is given by

k2 F Ž1. s

k˜

2

qR

ž

k2 k˜

2

q2

kk 1 k 12

˜

kDk

yR

k2 s

k˜

2

y

ž

ž / kk ˜ ˜1

½ /ž

qRj 2

kDk kk ˜ ˜1

kDk y2

kk ˜ ˜1

1 1 q j 2 Ž k˜ y k˜ 1 .

k2 2 2

1 q j k˜

q2

2

/ ž yR

q2 R

ž

k2 k˜

2

kDk y

kk ˜ ˜1

kDk y

kk ˜ ˜1

kk 1 k 12

˜

/ /

1 1 q j 2k˜ 2 1 1 q j 2k˜ 12

kk 1 1 q j 2k˜ 12 2

2 k˜ 12 1 q j 2k˜ 12

y

k˜ 2 Ž k˜ y k˜ 1 . q 2 1 q j 2k˜ 2 1 q j 2 Ž k˜ y k˜ 1 .

/5

,

Ž 65 .

where Dk s k y k H Pq1 H rm2 s Ž k 2 q k 12 y u 12 .r2. The tilde denotes k˜ i ' k i q e 2 , that results when a plasmon mass shift, v pl2 s e 2m2 , is introduced in all propagators. Eq. Ž65. shows that for finite j , the k H s q1 H singularity of free space perturbation theory Eq. Ž59. is automatically regulated by the interference terms. For finite j therefore we can safely take the e s 0 limit and write

½ ž /ž

F Ž1. s 1 q R j 2

Dk

y

k1

k2 2 2

1qj k 2 k 12

1 q j 2k 12

q2

y

kk 1 1 q j 2k 12

Ž k y k1 .

2

1 q j 2 Ž k y k1 .

k2 2

q

1 q j 2k 2

/5

.

Ž 66 .

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

However, in the formal classical parton cascade limit with j apparent singularity is exposed unless e / 0:

k2 F Ž1. Ž k H ,q1 H , j s ` . s

k˜

2

k

Ž1 q R. q2 R

k˜ 1

ž

k1 k˜ 1

Dk y

™™ Ž H q RŽ B q C . .rH .

e

0

2

2 1

2 1

k˜

™` Žv4v

217

BH

. the

/

2

Ž 67 .

To compare the full quantum parton cascade modification factor Ž66. to the incoherent classical parton cascade result, we regulated the singular classical limit with e s 1 FclŽ 1 . Ž k H ,q1 H ,` . FclŽ1. Ž k H ,q1 H . 'F

™

™

e s1 .

Ž 68 .

In the opposite j 0 limit, on the other hand, we see that the factorization theorem, F Ž1. 1, is recovered. Another interesting limit is k H 4 q1 H ; m. In that kinematic range, the induced GB bremsstrahlung can be ignored, Ž B12 f 0. but the cascade term survives Ž C 12 H 2 .. Therefore, in the m < k H - v kinematic regime

™

™

2 kH 4m 2

F Ž1. Ž k H ,q1 H .

1 q R.

Ž 69 .

This enhancement of the high k H tail arises because rescattering of the gluon is enhanced by a color factor R s CArCR relative to the jet. As we shall see, this is a 2 special case of the general asymptotic k H 4 n s m 2 form in the case of n s collisions: F Ž n s . Ž k H ,q1H , . . . qn s H .

™

2 kH 4n s m 2

n

Ž1 qR. s .

Ž 70 .

Although it is possible to integrate over the momentum transfers analytically, the result even in the case of one-scattering center is not instructive. Therefore, we average F Ž1. over q1 H numerically for different values of j s v BH rv : F Ž1. Ž k , j . s ² F Ž1. : s

H0sr4m

2

du 1

Ž u 1 q 1.

2p

2

H0

df 1 2p

F Ž1. Ž k , u 1 , f 1 , j . .

Ž 71 .

Note that we keep the large sr4m2 f 6 Ejet Tr4m2 upper bound in the integration finite 2 because the kinematic boundaries include regimes where E 2 0 v 2 0 k H 0 s. In Fig. 4a the three components of the classical parton cascade envelop Ž j s `. for an initial quark jet are shown for the case m s 0.5 GeV as a function of the logarithmic ‘‘angle’’ log 10 Ž k .. The hard factorization term corresponds to the unit curve. The GB induced radiation component dominates at small k and the cascade contribution is only important at large k . The bump at k ; 5 is the remnant of the logarithmic singularity exposed in the j s ` limit, but regulated here with a plasmon mass v pl s m s 0.5 GeV. Fig. 4b shows the ensemble averaged quantum modification factor, F Ž1., with m s 0.5 GeV, l s 1 fmrc Ž v BH f 0.63 GeV. for seven different gluon energies v . These correspond to j s 0.013, 0.031, 0.063, 0.13, 0.31, 0.63, and 1.3 that cover an energy range 0.5 GeV - v - 50 GeV. In each case, the curves are cut off at the kinematic bound

v2 k max s

m2

s

Ž lm . 4j 2

2

,

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

218

Fig. 4. Ža. The normalized time-independent envelopes and separate contributions to the conditional probability distribution of gluons associated with a single rescattering n s s1 of quark jet Ž Ejet s 50 GeV. is shown as a 2 function of the logarithmic ‘‘angle’’, log 10 k H r m2 with m s 0.5 GeV. These curves correspond to the Ž . incoherent parton cascade limit. b For finite t10 s1 fmrc, destructive interference limits the corrections to the hard self-quenched distribution to higher angles as shown for different gluon energies.

which corresponds to right angle radiation relative to the jet axis. Here Ž lm . 2r4 f 1.5 . The curves illustrate well how the total destructive interference ŽLPM effect. for kj - 1 reduces F Ž1. to unity. In particular, the induced Gunion–Bertsch radiation component in Fig. 4a is almost completely suppressed due to interference with the hard production amplitude. Only the cascade term modifies the spectrum considerably at large k H . The onset of significant medium modification can be seen to follow qualitatively the point k c s 1rj . It is convenient to define in general the reduced modification factor f Ž n s . Ž k , j . through F

Ž n s.

ns

Ž k , j . ' 1 q Ž Ž 1 q R . y 1. f

Žns .

Ž k ,j . 4u

ž

Ž lm . 4j 2

2

/

yk ,

Ž 72 .

which isolates the kinematic boundary and the asymptotic color scaling factor. Note that the incoherent classical parton cascade envelope is not reached for any reasonable values of lm due the finite kinematic boundaries. 5. Case of two-scattering centers Ž . The relative conditional probability R Ž2. g k can be obtained through the averaging of the square of the matrix element M2 on the scattering potentials in the q1 and q2 space. In Appendix A we listed the matrix elements connected to the seven planar diagrams, contributing to the soft gluon radiation. The sum of the matrix elements can be written as M2 s y2 ig s e i t 0 v 0 = e H P  Ha2 a1 c q B1 e i t 10 v 0 a 2 w c,a1 x q B2 e i t 20 v 0 w c,a2 x a1 qB2Ž12. e iŽ t 20 v 0yt 21 v 2 . w c,a 2 x ,a1 q C 1 eyi t 10 v 10 a 2 w c,a1 x qC 2 eyi t 20 v 20 a1 w c,a 2 x q CŽ12. e iŽ t 20 v 0yt 21 v 2yt 10 v Ž12. . w c,a2 x ,a1

4.

Ž 73 .

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219

In the limit where all t i y t j s 0, t i ) t j , i, j s 1,2 one can easily see that all the terms but the ‘‘hard’’ scattering cancel M2 s y2 ig s e i t 0 v 0 e H PHa2 a1 c.

Ž 74 .

™

The GB t 0 y` limit and the connection to the BDMPS w9x result we discuss in Appendix B. After calculating the appropriate color factors in Eqs. ŽA.10., ŽA.11. we obtain Tr Ž M2 M2† . s CR3 DR =  H 2 q R Ž B12 q B22 q C12 q C 22 . q R 2 Ž B22 Ž 12 . q C 2Ž 12 . . 4 q interference terms.

Ž 75 .

ŽSee Eq. ŽA.13. for the full result.. We can isolate the color prefactor together with the 2 leading parabolic dependence 1rk H ; R Ž0. g in the form Ž0. Ž2. R Ž2. Ž k H ,q1 H ,q2 H , j 1 , j 2 . . g Ž k ,q1 H . s R g Ž k . F

Ž 76 .

2 We note that when the interference terms drop at asymptotic k H 4 2 m2 the enhancement factor scales as in Eq. Ž70.

™

2 kH 42 m 2

F Ž2. Ž k H ,q1 H ,q2 H .

2 Ž1 qR. .

Ž 77 .

The longitudinal ensemble averaging can be performed in Eq. Ž75. over the times t 10 s t 1 y t 0 , t 21 s t 2 y t 1 as indicated in Eqs. Ž20., Ž39.. For the more general case l1 / l2 we find

²cos Ž va t10 q vb t21 . :s

1 y l1 l2 va vb

Ž 1 q l12 va2 . Ž 1 q l22 vb2 .

.

Ž 78 .

The integration over the momentum transfer due to the rescatterings from the static centers is most readily performed in terms of the dimensionless variables

ks

2 kH

m2

,

ui s

qi2H

m2

,

ji s

Ž t i y t iy1 . m2

,

i s 1,2,

2p

df 1 df 2

2v

Ž 79 .

leading to an averaged quantum correction factor F Ž2. Ž k , j 1 , j 2 . s² F Ž 2 . Ž k ,q1 H ,q2 H , j 1 , j 2 . : s

H0sr4m

2

du 1

du 2 2

Ž u 1 q 1. Ž u 2 q 1.

2

H0

2p 2p

F Ž2. .

Ž 80 .

In Fig. 5a we plot the contributions to FclŽ2. – the incoherent parton cascade limit where all the exposed singularities in the propagators have been regulated by the Debye screening mass m. At small transverse momenta of the radiated gluon the quantum 2 modification is dominated by the ‘‘Gunion–Bertsch terms’’. In the limit of large k H rm2 the only terms that survive apart from the hard scattering are the ‘‘cascade terms’’. This Ž . represents the expected behavior of R Ž2. g k in the Bethe–Heitler limit. It is important to notice that simple picture of the process is only possible in the incoherent case where one easily decouples production from rescattering and the separate terms acquire physical interpretation.

220

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

™

Fig. 5. Ža. The structure of the regulated Ž m s 0.5 GeV. contributions to the j ` limit Žquark jet of Ejet s 50 GeV, n s s 2.. Žb. Characteristic behavior of F Ž2. for j s 0.013, 0.031, 0.063, 0.13, 0.31, 0.63 and 1.3.

Fig. 5b shows F Ž2. plotted for the same set of gluon energies v as in Fig. 4b Žand the same values of j i , i s 1,2.. The picture of classical parton cascading is modified by the presence of interference terms that control behavior of the probability distribution in the intermediate LPM region. Due to the destructive interference one observes a smooth 2 Ži.e. the factorization limit. to the transition from the ‘‘hard term’’ at small k H Bethe–Heitler limit. It is remarkable that the detailed angular profile of the probability distribution is very similar Žapart from the explicit rescaling factor. to the one in Fig. 4b. 6. Power law scaling in n s and estimated energy loss The amplitudes in the case of n s s 3 are tabulated in Appendix C. The main novelty at this order is the appearance of an interesting ‘‘color wheel’’ diagram derived in Appendix D. The amplitude, M3 is rearranged in Eq. ŽC.16. in terms of the hard, GB, and cascade amplitudes. The diagonal Žclassical parton cascade. terms in < M3 < 2 with 2 color factors ŽC.18. involve Ž1. the usual H 2 s 1rk H with a relative color weight 1, Ž2. three simple GB induced radiation terms and three one-scattering gluon cascade terms each with color weight of R s CArCR , Ž4. three double gluon cascade terms and four GB-cascade terms with a weight R 2 , and Ž5. one triple gluon cascade term with a weight R 3. At small k H , only the three simple GB Ž B 2 H 2 . terms contribute since the finite cascade terms are negligible relative to the divergent hard and GB terms. In this small k H region therefore the H 2 term is simply amplified by a factor 1 q 3 R as illustrated in Fig. 6a. For large angles, on the other hand, all the GB and GB-cascade terms vanish while the cascade terms C 2 H 2 . Therefore, in that region the hard radiation is amplified by a factor Ž1 q R . 3. In between, near k H ; m all terms contribute and a bump that is sensitive to the screening scale m. Fig. 6a summarizes therefore the general scaling with n s of the classical parton cascade angular distribution. The power law enhancement Ž1 q R . n s at large angles is the most important non-linear feature since the bump near k ; 1 is killed by destructive interference shown in Fig. 6b.

™

™

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221

™

Fig. 6. Ža. The classical parton cascade limit Ž j `. angular enhancement factor from Eq. Ž72. for a quark jet 2 Ž Ejet s 50 GeV, m s 0.5 GeV. versus log 10 Ž k H r m2 .. Žb. The reduced modification factor f Ž n s . including kinematic cut-offs for n s s1,2 shows an approximate n s independence.

In the general case, the small k H enhancement, 1 q n s R, expected in the classical cascade limit also is cancelled by destructive interference as seen in Figs. 4 and 5. Here we note a remarkable simple scaling between the n s s 1,2 cases. In Fig. 6b the reduced modification factors f Ž n s . Ž k , j . extracted from F Ž n s . Ž k , j . in Eq. Ž72. are shown. The reduced factors are nearly identical in spite of the much more complex functional form Ž1. of the R Ž2. g than R g . The approximate scaling from n s s 1 to n s s 2 can be summarized as follows: n

s Ž1. R Žgn s . Ž v , k H . f R Ž0. Ž k ,j . 4 . g Ž k H . 1 q Ž Ž 1 q R . y 1. f

Ž 81 .

The generic analytic structure of the reduced modification factor can be roughly approximated by f Ž1. Ž k , j . ; k 2j 2rŽ1 q k 2j 2 ., which shows that the classical parton cascade result is approached only for k ) 1rj , i.e. k H ) vrl . For higher n s the validity of this scaling will be tested in w15,16x. Here we extrapolate Eq. Ž81. to n s s 3 to make a rough estimate of the energy loss spectrum dIrd v and the fractional energy loss D ErE to L ; 3 fm. We emphasize that these estimates are included in this paper only to illustrate qualitative features. For quantitative estimates, multi-gluon showers that require a Monte Carlo analysis must be considered as deferred to Refs. w15,16x. The normalization factor Zn s 4 1 for even modest n s ; 3 in Eq. Ž47. signals a high probability of multi-gluon emission. This is well known in the n s s 0 case as emphasized in Eqs. Ž48., Ž49.. For example, Z0 ) 2 from Eq. Ž48. for E ) 10 GeV while DLA gives, Z0 f 3. Eq. Ž49. then leads to ² n g : f 1.2 for this energy, implying that two gluon final states must be taken into account. With the above caveats, Fig. 7 illustrates the qualitative features of the angular integrated dIrd v and D ErE for a quark jet of energy 5 GeV penetrating a thin Ž n s s 1,2,3. plasma with m s 0.5 GeV, l s 1 fm, a s s 0.3. As seen from Eq. Ž15., the intensity distribution for the self-quenching Ž n s s 0. case increases linearly as a function of logŽ vrm . and the fractional energy loss due to self-quenching is proportional to logŽ Erm ..

'

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

222

Fig. 7. Ža. The gluon radiation intensity dIr dw for a 5 GeV quark jet Ž n s s 0,1,2,3. based on Ž81. is illustrated. Žb. The fractional energy loss is shown for the ‘‘self-quenching’’ and medium modified cases.

With increasing n s , the approximate scaling in Eq. Ž81. leads to

v

dNgŽ n s , n gs1 . 3

f

Zn s p

d k

Zn s f 1 q

1 CR a s 1

CR a s

p

E

Hm

dv

v

2

2 kH

v2

Hm

2

1 q Ž Ž 1 q R . n y 1. f Ž1. Ž k , j . 4 ,

2 dk H 2 kH

s

½

n

1 q Ž Ž 1 q R . s y 1. f Ž1.

ž

2 kH v BH , 2 v m

/5

.

Ž 82 .

Without the Z factor, the gluon distribution would scale approximately exponentially with plasma thickness at high k H ) mr6j due to the multiple collisions of the gluon in the target. However, the normalization factor compensates this increase to a very large extent as seen in Fig. 7. In fact, at this single gluon level of approximation, the energy loss is surprisingly small, ; 300 MeVrfm, and only grows approximately linearly with increasing thickness L s n s l in spite of the large modification of the gluon angular distribution.

7. Summary We investigated the angular distribution of radiated gluons and the energy loss of hard jet in thin plasmas, where the size of the plasma L is comparable to the mean free path, l. The small number of rescattering terms made it possible to calculate analytically the full matrix element within the eikonal approximation. This paper established a systematic method for computation of all relevant partonic matrix elements for higher n s which are needed as input to a numerical Monte Carlo study of the actual hadronic observables associated with jet quenching to be presented in a subsequent paper w15,16x. We developed an efficient algorithm to automate the cumbersome color algebra and keep track of the many canceling phases involved in the non-abelian Landau– Pomeranchuk–Migdal effect. Detailed discussion for the n s s 1,2,3 cases were presented to clarify the how the competing physical processes interfere in different kinematic regions. An approximate power law scaling of the angular broadening

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

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emerged with the increasing number of the rescattering centers. However, the rapidly growing wavefunction renormalization was shown to limit the growth of final energy loss. The shape and normalization of the induced gluon radiation calculated in this paper assumed implicitly an exclusive measurement of jet, gluon, and recoil target partons. For inclusive gluon distributions, on the other hand, unitarity corrections must be included w36x. This will be treated in a forthcoming paper. Acknowledgements We thank Urs Wiedemann for extensive discussions related to this problem. This work was supported by the DOE Research Grant under Contract No. De-FG-02-93ER40764, partly by the US-Hungarian Joint Fund No.652 and OTKA No. T025579. Appendix A. Case of two-scattering centers In the approximation of a high energy incident parton and soft gluon bremsstrahlung the seven planar diagrams in Fig. 8 can be treated in the eikonal approximation spinless limit. We denote by t 0 the time at which the particle is produced in the medium and by t i , i s 1,2 the times of the successive rescatterings. In the approximation when l 4 my1 ‘‘backward’’ scattering can be neglected and the process is approximately time ordered w9x. We denote by k the momentum of the radiated gluon and by qi , i s 1,2 are the

Fig. 8. The contributions to MJ m M2 soft gluon radiation amplitude in case of two scatterings.

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

224

momentum transfers. In QCD with each vertex we associate a generator of the SUŽ3. group. We denote them by c for the radiated gluon and a i , i s 1,2 for the momentum transfer vertices. The polarization of the radiated gluon is determined by e H . The corresponding amplitudes can be obtained with the procedure described in Eqs. Ž30., Ž33. – Ž35.: M2,0,0 s 2 ig s

M2,0,1 s 2 ig s

e H Pk H 2 kH

2 1 H

Ž eit k

e H P Ž k y q1 . H

Ž

2 k y q1 H

.

r2 v

2

y e i t 0 k H r2 v . a 2 a1 c, 2

Ž A.1 .

2

e i t 1w k H yŽ kyq1 . H xr2 v Ž e i t 1Ž kyq1 . 2H r2 v y e i t 0 Ž kyq1 . 2H r2 v .

=a2 w c,a1 x , M2,0,2 s 2 ig s

Ž A.2 .

e H P Ž k y q2 . H

Ž

2 k y q2 H

.

2

2

e i t 2 w k H yŽ kyq 2 . H xr2 v Ž e i t 1Ž kyq 2 . 2H r2 v y e i t 0 Ž kyq 2 . 2H r2 v .

=a1 w c,a 2 x , M2,0,3 s 2 ig s

Ž A.3 .

e H P Ž k y q1 y q 2 . H 2

Ž k y q1 y q 2 . H

2

2

2

= Ž e i t 1Ž kyq1yq 2 . 2H r2 v y e i t 0 Ž kyq1yq 2 . 2H r2 v . w c,a2 x ,a1 , M2,1,0 s 2 ig s

M2,1,1 s 2 ig s

e H Pk H 2 kH

Ž eit

2 2 kH

e H P Ž k y q2 . H

Ž

2 k y q2 H

.

r2 v

2

y e i t 1 k H r2 v . a2 ca1 , 2

e H Pk H 2 kH

Ž A.4 . Ž A.5 .

2

e i t 2 w k H yŽ kyq 2 . H xr2 v Ž e i t 2 Ž kyq 2 . 2H r2 v y e i t1Ž kyq 2 . 2H r2 v .

= w c,a 2 x a1 , M2,2,0 s 2 ig s

2

e i t 2 w k H yŽ kyq 2 . H xr2 v e i t 1wŽ kyq 2 . H yŽ kyq1yq 2 . H xr2 v

Ž A.6 .

Ž ye i t

2 2kH

r2 v

. ca2 a1 .

Ž A.7 .

After regrouping the sum of the above amplitudes five distinct color factors remain in Eq. Ž73.. We label them as follows: C2 s a2 w c,a1 x ,

C1 s a2 a1 c, C4 s a1 w c,a 2 x ,

C3 s w c,a 2 x a1 ,

C5 s w c,a 2 x ,a1 .

Ž A.8 .

In the case of QCD the ‘‘color’’ and ‘‘kinematic’’ parts of a diagram factor. Diagrammatic techniques have been developed to treat the color part alone w35x. We denote by CR the Casimir of the representation of the incident parton. For SUŽ N . following the standard normalization for the generators we have CF s

N 2y1 2N

,

CA s N.

Ž A.9 .

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225

The trace of the color factors can be taken separately in the square of the matrix element, yielding Diagonal part: Tr C1C1† s CR3 DR , Tr C2 C2† s Tr C3 C3† s Tr C4 C4† s CA CR2 DR , Tr C5 C5† s CA2 CR DR .

Ž A.10 .

Off-diagonal part: Tr C1C2† s y 12 CA CR2 DR , Tr C1C3† s Tr C1C4† s y 21 Ž CR y 21 CA . CA CR DR , Tr C2 C3† s Tr C2 C5† s y 41 CA2 CR DR , Tr C3 C4† s yTr C4 C5† s Ž CR y 12 CA . CA CR DR , Tr C3 C5† s 12 CA2 CR .

Ž A.11 .

Diagrams with 0: Tr C1C5† s Tr C2 C4† s 0.

Ž A.12 .

With R s CArCR , the relative conditional probability is given by R Ž2. g Žk .s

as p2

½

CR H 2 q R Ž B12 q B22 q C12 q C 22 . q R 2 Ž B22 Ž 12 . q C 2Ž 12 . . R

yR

ž / 2

B1 P B2 cos Ž t 21 v 0 . y R

R

ž / 2

C1 P B2 cos Ž t 21 v 0 q t 10 v 1 .

q2 R 2 CŽ12. P B2Ž12. cos Ž t 10 vŽ12. . y R Ž H P B1cos Ž t 10 v 0 . qH P C 1cos Ž t 10 v 10 . y 2 C1 P B1cos Ž t 10 v 1 . .

ž

yR 1 y

R 2

/Ž

H P B2 cos Ž t 20 v 0 . q H P C 2 cos Ž t 20 v 20 .

y2C 2 P B2 cos Ž t 20 v 2 . . y R

R

ž /Ž 2

C1 P B2 Ž 12 . cos Ž t 10 v 1 y t 21 v 20 .

qC1 P C Ž 12 . cos Ž t 10 v 1 Ž 12 . y t 21 v 20 . q B1 P B2 Ž 12 . cos Ž t 21 v 20 . qB1 P C Ž 12 . cos Ž t 10 v Ž 12 . q t 21 v 20 . . y R 2 Ž C 2 P B2 Ž 12 . cos Ž t 10 v 2 . qC 2 P C Ž 12 . cos Ž t 10 v 2 Ž 12 . . y B2 P B2 Ž 12 . cos Ž t 21 v 2 .

5

yB2 P C Ž 12 . cos Ž t 10 v Ž 12 . q t 21 v 2 . . , where we have used the concise notation introduced in Section 4.

Ž A.13 .

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226

™ I` limit in the case of n s 2 Going back to Eq. Ž73. it is easy to check the t ™ y` limit and make contact with BDMPS w9x. In the phases we will assume small adiabatic switch-off factors "i ´ when taking the corresponding limits. The limit of t ™ y` corresponds to removing the interference with the jet source. In this limit only the two Gunion–Bertsch terms and the Gunion–Bertsch cascade term Appendix B. The t 0

s

0

0

survive, M2 s y2 ig s e H P  B1 e i t 1 v 0 a 2 w c,a1 x q B2 e i t 2 v 0 w c,a 2 x a1 qB2Ž12. e iŽ t 2 v 0yt 21 v 2 . w c,a 2 x ,a1

4.

Ž B.1 .

This is the two-scattering generalization of the Gunion–Bertsch one-scattering center formula w6x and corresponds exactly to Eq. Ž2.22. of BDMPS in w9x. However, when squared there are several extra terms compared to the one effectiÕe term retained in the approximations employed there. In particular, from Eq. ŽA.13.

™ y`

lim v

t0

dNgŽ2. dv

s d2 kH

H

s

as p2

dy d 2 k H

¦H ½

d 2 k H B12 q B22 q R B22 Ž 12 .

CA R

y

dNgŽ2.

ž /Ž 2

B1 P B2 cos Ž t 21 v 0 . q B1 P B2 Ž 12 . cos Ž t 21 v 20 .

y2 B2 P B2 Ž 12 . cos Ž t 21 v 2 . .

5;

.

Ž B.2 .

After taking the kinematic limits to infinity and performing the substitution k H s k XH yq2 H in the B1 P B2 contribution one arrives at a result that in the large Nc limit, i.e. R s 2, the weighted energy distribution of the gluons has the form

™ y`

lim v

t0

dNgŽ2. dv

sv

dNgŽ2 .fact . dv

y

sv

as p2

¦H

CA

dNgŽ2 .fact .

y

dv

as p2

q3

p2

¦H

CA

;

d 2 k H Ž B2 P B2 Ž 12 . Ž cos Ž t 21 v 2 . y 1 . .

;

d 2 k H Ž B1 P B2 Ž 12 . Ž cos Ž t 21 v 20 . y 1 . .

qv

¦H

CA

as

dNgŽ2 .BDMPS dv

;

d 2 k H Ž B1 P B2 Ž 12 . Ž cos Ž t 21 v 20 . y 1 . . ,

Ž B.3 .

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

227

where direct comparison to Eq. Ž2.31. in Ref. w9x shows that

v

Ž2. dNBD MPS

dv

s3

P

s3

CA a s

p

ž

2

kH 2 kH

CA a s

p2

Ž k y q2 . H

¦H ž d2 k

y

Ž

2 k y q2 H

y

2

Ž k y q2 . H

/

2

Ž k y q1 y q 2 . H

.

Ž k y q2 . H

Ž k y q1 y q 2 . H

/

;

Ž cos Ž t 21 v 2 . y 1 .

¦H ¦

;

d 2 k B2 P B2Ž12. Ž cos Ž t 21 v 2 . y 1 . .

™

Ž B.4 .

If we also require that t 2 y t 1 ` then the Gunion–Bertsch cascade term also vanishes and we reproduce two independent Gunion–Bertsch scatterings with gluon radiation.

Appendix C. Case of three scattering centers The interaction of the incident high energy parton with three scattering centers accompanied by emission of one gluon is described Žin the eikonal approach. by the fifteen diagrams in Fig. 9. The corresponding matrix elements are easy to obtain through the procedure described in Subsection 3.3: M3,0,0 s 2 ig s

M3,0,1 s 2 ig s

e H Pk H 2 kH

2 1 H

Ž eit k

e H P Ž k y q1 . H

Ž

2 k y q1 H

.

r2 v

2

y e i t 0 k H r2 v . a 3 a 2 a1 c, 2

Ž C.1 .

2

e i t 1w k H yŽ kyq1 . H xr2 v Ž e i t 1Ž kyq1 . 2H r2 v y e i t 0 Ž kyq1 . 2H r2 v .

=a3 a 2 w c,a1 x , M3,0,2 s 2 ig s

Ž C.2 .

e H P Ž k y q2 . H

Ž

2 k y q2 H

.

2

2

e i t 2 w k H yŽ kyq 2 . H xr2 v Ž e i t 1Ž kyq 2 . 2H r2 v y e i t 0 Ž kyq 2 . 2H r2 v .

=a3 a1 w c,a2 x , M3,0,3 s 2 ig s

Ž C.3 .

e H P Ž k y q1 y q 2 . H 2

Ž k y q1 y q 2 . H

2

2

2

= Ž e i t 1Ž kyq1yq 2 . 2H r2 v y e i t 0 Ž kyq1yq 2 . 2H r2 v . a 3 w c,a 2 x ,a1 , M3,0,4 s 2 ig s

e H P Ž k y q3 . H

Ž

2 k y q3 H

=a2 a1 w c,a3 x ,

.

2

e i t 2 w k H yŽ kyq 2 . H xr2 v e i t 1wŽ kyq 2 . H yŽ kyq1yq 2 . H xr2 v

2

Ž C.4 .

2

e i t 3w k H yŽ kyq 3 . H xr2 v Ž e i t 1Ž kyq 3 . 2H r2 v y e i t 0 Ž kyq 3 . 2H r2 v .

Ž C.5 .

228

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

Fig. 9. The contributions to MJ m M3 soft gluon radiation amplitude in case of three scatterings.

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

M3,0,5 s 2 ig s

e H P Ž k y q1 y q 3 . H

Ž

2 k y q1 y q 3 H

2

2

229

2

2

e i t 3w k H yŽ kyq 3 . H xr2 v e i t 1wŽ kyq 3 . H yŽ kyq1yq 3 . H xr2 v

.

= Ž e i t 1Ž kyq1yq 3 . 2H r2 v y e i t 0 Ž kyq1yq 3 . 2H r2 v . a 2 w c,a 3 x ,a1 , M3,0,6 s 2 ig s

e H P Ž k y q2 y q3 . H

Ž

2

2

2

Ž C.6 . 2

e i t 3w k H yŽ kyq 3 . H xr2 v e i t 2 wŽ kyq 3 . H yŽ kyq 2yq 3 . H xr2 v

2 k y q2 y q3 H

.

= Ž e i t 1Ž kyq 2yq 3 . H2 r2 v y e i t 0 Ž kyq 2yq 3 . 2H r2 v . a1 w c,a 3 x ,a 2 , M3,0,7 s 2 ig s

Ž C.7 .

e H P Ž k y q1 y q 2 y q 3 . H 2

Ž k y q1 y q 2 y q 3 . H 2

2

2

2

=e i t 3w k H yŽ kyq 3 . H xr2 v e i t 2 wŽ kyq 3 . H yŽ kyq 2yq 3 . H xr2 v 2

2

=e i t 1wŽ kyq 2yq 3 . H yŽ kyq1yq 2yq 3 . H xr2 v = Ž e i t 1Ž kyq1yq 2yq 3 . H2 r2 v y e i t 0 Ž kyq1yq 2yq 3 . 2H r2 v .

w c,a3 x ,a2 ,a1 , Ž C.8 .

M3,1,0 s 2 ig s

M3,1,1 s 2 ig s

e H Pk H 2 kH

Ž eit

2 2 kH

r2 v

e H P Ž k y q2 . H

Ž

2

y e i t 1 k H r2 v . a3 a 2 ca1 , 2

Ž C.9 .

2

e i t 2 w k H yŽ kyq 2 . H xr2 v Ž e i t 2 Ž kyq 2 . 2H r2 v y e i t1Ž kyq 2 . 2H r2 v .

2 k y q2 H

.

=a3 w c,a 2 x a1 , M3,1,2 s 2 ig s

Ž C.10 .

e H P Ž k y q3 . H

Ž

2 k y q3 H

.

2

2

e i t 2 w k H yŽ kyq 3 . H xr2 v Ž e i t 2 Ž kyq 3 . 2H r2 v y e i t 1Ž kyq 3 . 2H r2 v .

=a2 w c,a 3 x a1 , M3,1,3 s 2 ig s

Ž C.11 .

e H P Ž k y q2 y q3 . H

Ž

2

2

2

.

= Ž e i t 2 Ž kyq 2yq 3 . H2 r2 v y e i t 1Ž kyq 2yq 3 . H2 r2 v . w c,a 3 x ,a 2 a1 , M3,2,0 s 2 ig s

M3,2,1 s 2 ig s

e H Pk H 2 kH

Ž eit

2 3kH

e H P Ž k y q3 . H

Ž

2 k y q3 H

.

r2 v

2

y e i t 2 k H r2 v . a 3 ca2 a1 , 2

e H Pk H 2 kH

Ž ye i t

Ž C.12 . Ž C.13 .

2

e i t 3w k H yŽ kyq 3 . H xr2 v Ž e i t 3 Ž kyq 3 . 2H r2 v y e i t 2 Ž kyq 3 . 2H r2 v .

= w c,a 3 x a 2 a1 , M3,3,0 s 2 ig s

2

e i t 3w k H yŽ kyq 3 . H xr2 v e i t 2 wŽ kyq 3 . H yŽ kyq 2yq 3 . H xr2 v

2 k y q2 y q3 H

Ž C.14 . 2 3kH

r2 v

. ca3 a2 a1 .

Ž C.15 .

230

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

Organizing the sum of the matrix elements in groups with common color and phase factors we obtain M3 s y2 ig s e i t 0 v 0 = e H P  Ha3 a 2 a1 c qB1 e i t 10 v 0 a 3 a 2 w c,a1 x q B2 e i t 20 v 0 a 3 w c,a 2 x a1 qB3 e i t 30 v 0 w c,a 3 x a2 a1 q B2Ž12. e iŽ t 20 v 0yt 21 v 2 . a 3 w c,a2 x ,a1 qB3Ž13. e iŽ t 30 v 0yt 31 v 3 . a2 w c,a 3 x ,a1 q B3Ž23. e iŽ t 30 v 0yt 32 v 3 . w c,a 3 x ,a 2 a1 qBŽ23.Ž123. eyi Ž t 32 v 30qt 21 v Ž23.0 yt 10 v 0 . w c,a 3 x ,a 2 ,a1 qC1 eyi t 10 v 10 a3 a 2 w c,a1 x q C 2 eyi t 20 v 20 a 3 a1 w c,a 2 x qC 3 eyi t 30 v 30 a 2 a1 w c,a3 x q CŽ12. eyiŽ t 10 v Ž12.0 qt 21 v 20 . a 3 w c,a 2 x ,a1 qCŽ13. eyi Ž t 10 v Ž13.0 qt 31 v 30 . a 2 w c,a 3 x ,a1 qCŽ23. eyi Ž t 20 v Ž23.0 qt 32 v 30 . a1 w c,a 3 x ,a 2 qCŽ123. eyi Ž t 32 v 30qt 21 v Ž23.0 qt 10 v Ž123.0 . w c,a 3 x ,a 2 ,a1

4.

Ž C.16 .

Eleven distinct color factors are present for n s s 3. We label them as follows: C1 s a3 a 2 a1 c,

C2 s a 3 a 2 w c,a1 x ,

C4 s w c,a3 x a 2 a1 ,

C5 s a 3 a1 w c,a 2 x ,

C7 s a3 w c,a 2 x ,a1 , C10 s a1 w c,a 3 x ,a 2 ,

C3 s a 3 w c,a 2 x a1 , C6 s a2 a1 w c,a3 x ,

C8 s a 2 w c,a 3 x ,a1 , C11 s

C9 s w c,a 3 x a 2 a1 ,

w c,a3 x a2 ,a1 .

Ž C.17 .

Computing the quantum amplitude requires knowing the color traces. These traces can be organized into a symmetric matrix: Diagonal terms: Tr C1 C1† s CR4 DR , Tr C2 C2† s Tr C3 C3† s Tr C4 C4† s Tr C5 C5† s Tr C6 C6† s CA CR3 DR , † Tr C7 C7† s Tr C8 C8† s Tr C9 C9† s Tr C10 C10 s CA2 CR2 DR , † Tr C11 C11 s CA3 CR DR .

Ž C.18 .

Diagrams with 0: Tr C1C7† s Tr C1C8† s Tr C1C9† s Tr C2 C5† s Tr C2 C6† s Tr C2 C9† s 0, † † † † Tr C3 C6† s Tr C3 C8† s Tr C5 C11 s Tr C6 C11 s Tr C7 C10 s Tr C8 C10 s 0.

Ž C.19 .

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

231

Traces inÕolÕing the ‘‘Wheel’’ diagram Žsee Appendix D.: † † Tr C1C10 s Tr C2 C11 s Tr C5 C6† s Tr C7 C8† s W y 18 CA3 CR DR , † † Tr C1C11 s Tr C2 C10 s Tr C5 C8† s Tr C6 C7† s yW q 18 CA3 CR DR .

Ž C.20 .

Remaining off-diagonal terms: Tr C1C2† s y 12 CA CR3 DR , Tr C1C3† s Tr C1C5† s y 21 Ž CR y 21 CA . CA CR2 DR , 2

Tr C1C4† s Tr C1C6† s y 12 Ž CR y 12 CA . CA CR DR , Tr C2 C3† s Tr C2 C7† s Tr C3 C4† s Tr C3 C9† s y 41 CA2 CR2 DR , † † Tr C2 C4† s Tr C2 C8†sTr C3 C10 sTr C4 C5†sTr C5 C9†sTr C5 C10

s y 14 Ž CR y 12 CA . CA2 CR DR , Tr C3 C5† s Ž CR y 12 CA . CA CR2 DR , Tr C3 C7† s Tr C4 C9† s yTr C5 C7† s yTr C6 C8† s 12 CA2 CR2 DR , † † Tr C3 C11 s Tr C4 C7† s Tr C7 C9† s Tr C7 C11 s y 81 CA3 CR DR , 2

Tr C4 C6† s Ž CR y 12 CA . CA CR DR , † † Tr C4 C8† s Tr C4 C10 s yTr C6 C9† s yTr C6 C10 s 21 Ž CR y 21 CA . CA2 CR DR , † † Tr C4 C11 s yTr C8 C9† s yTr C8 C11 s 41 CA3 CR DR , † Tr C9 C10 s Ž CR y 12 CA . CA2 CR DR , † † Tr C9 C11 s yTr C10 C11 s 12 CA3 CR DR .

Ž C.21 .

Appendix D. The color ‘‘ wheel’’ diagrams In the case of three-scattering centers some non-trivial color factors appear for the first time in Eq. ŽC.20.. They correspond diagrammatically to the ‘‘wheel’’ diagrams shown in Fig. 10.

Fig. 10. The ‘‘wheel’’ diagram in the fundamental and the adjoint representations.

M. Gyulassy et al.r Nuclear Physics B 571 (2000) 197–233

232

Table 1 The Nc dependence of the wheel diagram to be found numerically Nc

2

3

4

5

WF WA

9r4 24

33r2 189

135r2 840

405r2 2775

For the fundamental representation

 Ta ,Tb 4 s

1 N

d a b q d a b c Tc ,

Ž D.1 .

It is useful to introduce the complex tensor h a b c s d a b c q if ab c ,

h a b c s h b c a s h c ab ,

h a a b s 0.

Ž D.2 .

If the physical case of SUŽ3., we find WF s Tr Ž TaTb Tc Td . f k al f l b m f m cn f n d k s Ž 121 d a b dc d q 81 h a b r h r c d . 41 Ž 5d a b dc d q d ac d b d q 5d a d d b c y 6 d acr d r b d . s 332

Ž D.3 .

for the fundamental representation and WA s Ž f k al f l b m f m cn f n d k . Ž f r au f u d t f t c s f sb r . ss 14 Ž 5d a b dc d q d ac d b d q 5d a d d b c y 6 d acr d r b d . 14 Ž 5d a d dc b q d ac d b d q5d a b d d c y 6 d acr d r d b . s189

Ž D.4 .

for the adjoint representation – see Table 1.

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