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Reaction Operator Approach to Multiple Elastic Scatterings

We apply the GLV Reaction Operator formalism to compute the effects of multiple elastic scatterings of jets propagating through dense matter. We derive the elastic Reaction Operator and demonstrate that the recursion relations have a closed form solution that reduces to the familiar Glauber form. We also investigate the accuracy of the Gaussian dipole approximation for jet transverse momentum broadening.


I. INTRODUCTION
New experiments at the Relativistic Heavy Ion Collider (RHIC) will provide high statistics high p T measurements of photon, lepton and hadron production in A + A reactions [ 1,2]. These observables provide novel tests of pQCD multiple scattering theory. Quantitative comparisons [ 3,4,5] to data require understanding of nuclear effects such as parton shadowing [ 6], the Cronin effect [ 7,8,9,10] and non-abelian jet energy loss [ 11,12,13]. In this paper we apply the Reaction Operator method [ 13], developed to solve the radiative energy loss problem, to derive the well known multiple elastic Glauber distribution [ 14] in order to illustrate the power of this recursion technique. The summation to all orders in the opacity, χ = σ(z)ρ(z)dz = L/λ, of the matter is given in closed form. In addition we evaluate the accuracy of the dipole approximation, leading to the Gaussian approximation to Moliere scattering, for the case of moderate opacities χ < 10 relevant for applications in nuclear physics.
We describe multiple jet scattering as a series expansion in χ, the mean number of scatterings that a hard parton undergoes along its trajectory [ 13,15]. Jet production is modeled here by an initial wave packet j(p) for a spinless parton in color representation R prepared at finite time t 0 and centered at x 0 = (z 0 , x 0 ). The momentum space amplitude in the absence of interactions is then given by where the d R dimensional unit matrix accounts for the jet colors. Multiplying |M 0 | 2 by the invariant one particle phase space element d 3 p/((2π) 3 2| p|) and taking the color trace one arrives at the unperturbed inclusive distribution of jets in the wave packet In the presence of nuclear matter the multiple interactions of a fast projectile propagating through the medium are modeled by a scattering potential assumed to be of the form where T an (R) is the generator of SU (N c ) in the d R dimensional representation of the jet, and T an (n) is the corresponding generator in the d n dimensional representation of the target. The scattering center is localized at position x n , and v( q n ) is the Fourier transform of the spatial part of the potential. We here use p to denote the 3D spatial part of a vector p and p represents its 2D transverse component. While there are no a priori limitations on the functional form of the scattering potential and the corresponding differential elastic scattering cross section we here consider potentials with a range 1/µ ≪ λ, the mean free path of the jet in the medium. This condition together with the hard jet approximation ensures the path ordering of the successive scatterings along the jet trajectory. Diagrams with crossed momentum transfers (at points typically separated by z n − z n−1 ≃ λ) are suppressed by factors ∼ exp(−λµ). One particular example used in Refs. [ 11,13] is the Yukawa color-screened potential where µ 2 n = µ 2 n⊥ = µ 2 + q 2 n and λµ ≫ 1. This paper is organized as follows: in Sec. II we present a systematic recursive way of keeping track of the multiple collision Feynman diagrams in terms of "direct" and "virtual" interactions. We discuss the color and kinematic structure of the single Born (direct) operatorD and the double Born (virtual) operatorV . In Sec. III the corresponding elastic Reaction OperatorR =D †D +V † +V is computed. We find a recursion relation for the observed inclusive jet distribution at any fixed order in opacity and use impact parameter space resummation to recover a classical Glauber-like formula. In Sec. IV we discuss the implications of our results in relation to the frequently employed Gaussian approximation of transverse momentum distribution of jets. We show, however, that for χ < 10 the Gaussian approximation fails to reproduce the power law tails of the jet p T distribution as well as the ∼ log Q 2 max enhancement of p 2 T .

II. TENSORIAL BOOK-KEEPING AND DIAGRAMMATIC CALCULUS
The interaction Hamiltonian in a medium with N scattering centers in the high jet energy spinless approximation is given by Each interaction of the colored parton propagating through the nuclear matter results in one power of the scattering potential Eq. (3) in the diagrammatic expansion. There is no limit on the number of scatterings the jet can have at a particular scattering center at position z n . Using the terminology of Ref. [ 13] we call such interactions single Born, double Born, e.t.c. It can be seen from Eq. (4) that two momentum transfers resulting from two interactions with the scattering potential are needed to produce one power of the elastic scattering cross section σ el . In performing the average over the transverse (relative to the jet trajectory) position of the scattering center · · · A ⊥ = d 2 b/A ⊥ ( · · · ) one order in the optical thickness (opacity) N σ el /A ⊥ = L/λ is thus generated in three ways: a.) single Born or "direct" interaction in the amplitude and the complex conjugate amplitude; b.) double Born or "virtual" interaction in the amplitude and no interaction in the complex conjugate amplitude; c.) no interaction in the amplitude and double Born interaction in the complementary complex conjugate amplitude. More momentum transfers at a fixed longitudinal position z n , e.g. a triple Born interaction, will produce O(α 2 s ) corrections to the opacity expansion and are here neglected.
It is therefore sufficient to consider single and double Born terms in the opacity expansion approach, i.e. when the jet propagates by a scattering center at position z n it can either miss the center, interact once, or interact twice as illustrated in Fig. 1.
..,i n-1 p,c z n q n ,a n A i 1 ,...,i n-1 p,c z n q n ′,b n q n ,a n The kinematic and color structure of the scatterings is contained in the unit (Î ), direct (D ) and virtual (V ) operators. The path ordering of the projectile interactions in the medium in the high energy eikonal regime facilitates the introduction of a convenient tensorial book-keeping notation that classifies the diagrams according to the type of scatterings that the parton has experienced along its trajectory. At order n in the opacity expansion there are 3 n possible amplitudes classified by a set of n indices i 1 · · · i n . Here i m = 0 indicates that the jet misses the m th target, i m = 1 means that it scatters once, and i m = 2 indicates that it scatters twice. The diagrams in Fig. 1 can be algebraically represented as follows: The set of amplitude indices in Eq. (7) thus encodes the complete history of the jet interactions. Repeating the basic operator steps in Eq. (7) any amplitude that includes parton scatterings inside the medium can be iteratively derived from the unperturbed jet production amplitude Time (or longitudinal coordinate) ordering is implicit in the above formula for the high energy eikonal limit under consideration here.
To proceed we need to know the color an kinematic structure of the direct and virtual interactions. We use the notation ∆(p) ≡ (p 2 + iǫ) −1 and Γ(p) = p 0 for the momentum space representation of the propagator and the vertex factor. The vertex factor is treated here in the high energy spinless limit. The direct iteration step (with z n > z n−1 > ...) in Eq. (7) reads where we have taken into account that the amplitude of order n − 1 has to be evaluated at momentum p − q n if the jets emerges on shell with momentum p. The double Born amplitude at the same external potential is similarly given by is expanded in the opacity series, where the contribution to order χ n is given by The amplitudesĀ i1···in (p, c) in Eq. (11) are the complementary amplitudes given bȳ The trace is taken over the color matrices and the binomial coefficient C N n = N !/n!(N − n)! ≈ N n /n! is introduced to take into account the number of combinations of n scattering sites out of the total N .
Performing the sum over the first n−1 interaction points and using Eqs. (8,12) we obtain a simple recursion identity which relates dN (n) to dN (n−1) through the Reaction OperatorR Consider first the direct part dN (n) (Dir.) ∝Ā i1···in−1D †D A i1···in−1 of the Reaction Operator. Performing the color traces we take into account the identity Tr T a (i)T b (j) = δ ij δ ab C 2 (T )d T /d A , which enforces that the positions of the ordered scattering centers in the amplitude and its complementary are identical. We use the form of the scattering potential specified by Eq. (3) as well as the the high energy eikonal approximation where the deviations due to the in-medium interactions from the initial jet trajectory are small and E + ≃ 2E.
The q z n , q ′ z n integral can be performed by closing the contour in the lower/upper half plane. We note that due to the short range of the scattering potential relative to the mean free path the residues from the q z n = −i q 2 n + µ 2 and q ′ z n = +i q ′ 2 n + µ 2 poles are exponentially suppressed and only the residues at q z n ≈ −iǫ + q 2 n /E + and q ′ z n ≈ +iǫ + q ′ 2 n /E + contribute. In the high energy limit E + ≫ µ ≫ q 2 n /E + the quadratic terms from the residues in the exponent can be neglected. This removes the correlations between the scattering centers and there are no coherence effects at the jet level. We note that this is an important difference from the case of induced gluon radiation discussed in Ref. [ 13]. Emitted gluons are much softer than the parent parton and non-negligible φ ∼ (x j − x 0 ) · (k − Q) 2 /k + phases control the non-abelian analog of the Landau-Pomeranchuk-Migdal effect [ 16]. Taking into account that the Glauber thickness function (at a fixed impact parameter b 0 ) T (b 0 ) = dz ρ(b 0 , z) = N/A ⊥ is expected to vary slowly with impact parameter: Eq. (15) is a key simplifying relation valid in the case √ A ⊥ ≫ 1/µ because it diagonalizes Eq. (14) in the transverse momentum variables.
This leads to a simple recursion relation for the Direct contribution to the jet spectrum to the momentum shifted distribution of order n − 1.
One notes that the momentum space integrations in this case could have been performed already at the amplitude level, Eq. (10). Using the same set of approximations as for the Direct part we first perform the q z n integral as well as the impact parameter average and note that the latter constrains q n + q ′ n = 0. Picking the residue at q z n ≈ −q ′ z n − iǫ: It was shown in Ref. [ 13] that for a broad class of real spherically symmetric potentials of short range ≪ λ the remaining q ′ z n integral can be performed. The screened Yukawa potential, Eq. (3), belongs in this category when µλ ≫ 1. We note that the contour integration produces a real result and a factor of 1 2 that cancels the factor of 2 arising from the Virtual terms in both the amplitude and its complementary. The recursion relation for the virtual part of the Reaction Operator can then be written as Combining Eqs. (16,19) one recovers the full structure of the elastic Reaction Operator where we used the C N n /N n ≈ 1/n! (Poisson) approximation. Eq. (20) is the main result of this paper and has transparent physical interpretation illustrated in Fig. 2. The first term in the kinematic structure of the Reaction Operator corresponds to the direct elastic scattering part. The next two terms have delta function strength in the forward direction and correspond to the virtual corrections (in the amplitude and its complementary complex conjugate). These terms enforce unitarity in the GLV Reaction Operator formalism. Given any initial parton distribution one can recursively solve from Eq. (20) for the final inclusive distribution of jet that have penetrated the medium characterized here by its Glauber thickness function T (b 0 ). The method used here is quite general and has several important advantages. Within the eikonal approximation it provides exact answers to problems related to multiple elastic and inelastic processes in QCD media. In particular it was applied to the case of gluon radiation * from jets produced in heavy ion reactions [ 13]. For more details on its uses in jet tomography see Refs. [ 4,5]. Since the solution is known to all orders in opacity, cases ranging from thin media with few scatterings to thick media with formally very large number of scatterings can be studied. In the case of multiple elastic scattering, the reaction operator is so simple that it is possible to sum over all orders in closed form. As usual it is most convenient to perform the summation in impact parameter space. The distribution in impact parameter space conjugate to the transverse momentum p is where the Fourier transform of the differential cross section was denoted by (2π) 2 dσ el /d 2 q(b) ≡σ el (b) andσ el (0) = σ tot el . The impact parameter space representation Eq. (21) depends only on the differencẽ σ el (b) −σ el (0). Transforming back to momentum space, we recover the parton version of the Glauber multiple collision series [ 14] dN where χ = σ el T (b 0 ) = L/λ. Eq. (22) has the familiar physical interpretation of a Poisson random walk in transverse momentum space distributed according to dσ el /σ el . * In the case of medium induced gluon bremsstrahlung the Reaction OperatorR retains its general operator form Eq. (13) but its color and kinematic structures are different since the on-shell t = ∞ cuts (see Fig. 2) go through both the jet and gluon lines that interact with the double Born term.

IV. DISCUSSION
Impact parameter space resummation is often used in conjunction with small impact parameter approximation of the differential cross section, Eq. (4). We show below that although such an approach is analytically appealing, it gives jet distributions that may differ substantially from the exact formula.
The Fourier transform of the normalized cross section is given by: where b = |b| and in the quadratic term in Eq. (23) the log 2/(1.08 µ b) multiplicative factor has been absorbed into a b-independent constant ξ due to its small logarithmic variation. We consider the transverse momentum broadening [ 17,18] of a jet propagating in the "ẑ" direction, i.e. dN (0) /d 2 p = δ 2 (p). In the approximation that dominantly small impact parameter scatterings are important, the momentum space distributions from Eqs. (21,22) reduce to the classic Moliere form [ 15] The resulting distribution is of Gaussian form and has a width of χ µ 2 ξ, i.e. p 2 = χ µ 2 ξ.  (Fig. 3a) and χ = 10 (Fig. 3b). We compare the full result (without the delta function contribution at pT ∼ 0) to the Moliere Gaussian approximation with ξ = 1 and ξ = log χ. In this example we use µ 2 = 0.25 GeV 2 .
To asses the accuracy of the Gaussian ansatz for finite χ relevant in nuclear physics, we compare the Fourier transform of Eq. (21) to Eq. (24). Note that in going from Eq. (21) back to momentum space, a e −χ δ 2 (p) component arises. This component is of course not displayed, and for finite p numerical integration converges much more rapidly if unity is subtracted from exp[σ(b)T (b 0 )] in the integrand (22).
In Figs. 3a, 3b numerical results are shown for opacities χ = 3 and χ = 10. For large opacities χ ≥ 10 (see Fig. 3b) the Gaussian approximation gives better results than for small ones χ ≤ 5. However, as is well known, at larger transverse momenta it fails to account for the power law p T tail of the inclusive distribution. Unlike familiar Moliere scattering in atomic matter, there are no form factors to limit the growth of the high p T Rutherford tail out to the kinematic limit Q 2 max ∼ E 0 µ in the case of QCD. Our results for the rms transverse momentum kick shown in Fig. 4 demonstrate that in semi-hard and hard pQCD processes there is a logarithmic in Q enhancement of the mean squared transverse momentum, p 2 ≃ χ µ 2 log(1 + Q 2 /χµ 2 ) that is missed in the dipole approximation. Of course for large Q radiative energy loss also contributes substantially to the p T broadening and should be taken into account in applications. In summary, we showed that the Reaction Operator formalism developed in [ 13] easily recovers the well known Glauber limit for elastic multiple collisions in pQCD matter and sheds light on the accuracy of the dipole approximation. In the future it would be interesting to combine both elastic and the LPM suppressed inelastic processes in this method and apply the results to jet acoplanarity observables suggested in Refs. [ 17,18].