Gluon Bremsstrahlung in Relativistic Heavy Ion Collisions

We study the process $qq\rightarrow qqg$ at lowest order in QCD perturbation theory to understand gluon radiation in the fragmentation region of relativistic heavy-ion collisions. We arrive at a formula for gluon multiplicity that interpolates between $\sim 1/k_{\bot}^2$ behavior at low $k_{\bot}$, to $\sim 1/k_{\bot}^4$ at large $k_{\bot}$.


I. INTRODUCTION
Understanding gluon distributions in the fragmentation region [1,2] is interesting for a couple of reasons: first, it will improve our knowledge of the initial conditions that feed into hydrodynamics; secondly, we might also gain insight into the behavior of matter at high baryon density. In any such collision, one nucleus may be designated the target and the other the projectile; we will be concerned with gluon radiation in the fragmentation region of the target. Supposing that the nuclei are of the same size, one is immediately presented an asymmetry between the saturation scales of the target and projectile (Q targ s Q proj s ), an asymmetry which is enhanced if the projectile nucleus is larger. This is because the saturation scale of a nucleus (or hadron) is proportional to the gluon rapidity density [3,4], dN/dy, which grows like an exponential in the rapidity-difference τ ≡ ln 1/x = y nucl − y [5,6]. In the fragmentation region of the target nucleus, this rapidity-difference is by definition very small; but for the projectile, it is large, and hence the respective gluon densities are very different. In the ultrarelativistic case in which we are insterested, not only may the gluon fields be treated classically-as is typically the case in saturation physics [7]-but the asymmetry of this problem allows one to solve classical Yang-Mills equations [8,9] by treating projectile as a strong background field A µ , while the target field, δA µ , is taken to first order since it is much weaker. This asymmetry of saturation scales is not unique to the case of rapidities far from the central region, in fact it has been exploited to calculate gluon radiation in the central region of collisions involoving particles with different sizes (p-A for example) [10]. Many features of the results in that scenario should be shared by a calculation in the fragmentation region.
In a previous work, Kajantie, McLerran, and Paatelainen [11,12], proceeding in this spirit of classical Yang-Mills, took steps towards calculating small-k ⊥ gluon radiation in the fragmentation region of nucleus-nucleus collisions. We will now give a very brief overview of their key results, highlighting the issues this paper aims to address. The problem considered therein is that of gluon radiation produced when a sheet of colored glass interacts with a classical particle that has an associated color-charge vector T. One then finds two sources of radiation, the first is ED-like bremsstrahlung of a charged particle getting a momentum kick from p to p ; the second, which they term the Gunion-Bertsch contribution [13][14][15], is from the acceleration of the coulomb field as it passes through a sheet of colored glass. The former is calculated from the following radiation current, with the resulting gluon distribution [11] 16π 3 |k| dN where k is the gluon's momentum and p T is the transverse-momentum kick of the charged particle. The above result falls off like ∼ 1/k 2 ⊥ no matter the value of k ⊥ . The Gunion-Bertsch contribution has so far proven intractable in general, but it is calculable in the large-k ⊥ limit and found to fall off like ∼ 1/k 4 ⊥ . To understand why this result is troublesome let's recount some findings from Ref. [10], which, as discussed above, is analogous to the study of the fragmentation region. In the region 1 Q s , the gluon distribution has a ∼ 1/k 2 ⊥ behavior and for k ⊥ > Q s , the fields of both particles are weak enough to permit a perturbative calculation. Hence as far as one can calculate, the Gunion-Bertsch contribution is correct and probably interpolates between the ∼ 1/k 2 ⊥ behavior at low k ⊥ to the 1/k 4 ⊥ at large transverse momentum. This result is and Q (1) s are respectively the saturation momenta of the large nucleus and the proton. In the case of the fragmentation region, these correspond to the saturation momenta of the projectile and target nucleus respectively.  Figure 1: The lowest order tree-level diagrams for gluon bremsstrahlung, and the Born term.
remarkable, considering that this calculation is non perturbative and classical. However, the ED-like radiation shown above is not correct, at least for large transverse momentum k ⊥ > Q proj s . Indeed the calculation as done in Ref. [11] was only meant to be valid for small k ⊥ , i.e when the gluon does not carry away a significant fraction of the quark's momentum.
In the present work, our aim is to understand how this formula may be remedied and produce one that has the correct behavior for all values of k ⊥ . Since we want to deal with large transverse momentum we can turn to a simpler but related perturbative problem. We calculate gluon radiation from quark-quark scattering to lowest order in QCD perturbation theory, where we consider one quark to be at rest and the other ultrarelativistic.
This paper is organized as follows. In Section II we derive the multiplicity distribution of gluons perturbatively using the qq → qqg process. Section III is a discussion of the results, we study the gluon multiplicty distribution in various kinematic limits with a special focus on the fragmentation region. Section IV is the conclusion.

II. GLUON BREMSSTRAHLUNG
We calculate gluon bremsstrahlung perturbatively using the process qq → qqg (FIG. 1). Both beam and target could move on the light cone, the beam fragmentation region can be studied in the forward limit [16,17], but in our calculation we just consider the target to be at rest. This treatment is similar to that of Gunion and Bertsch [13], and it is also just the Lipatov vertex [18] at high energy. We use the following kinematics (in Light-Cone where p µ 1 is the initial momentum of the quark at rest with M = m/ √ 2 (where m is the quark mass), p µ 2 is initial momentum of the incident quark with energy P , k µ is the momentum of the radiated gluon, and x is the fractional light-cone momentum carried by the radiated gluon. The momentum p µ 1 and p µ 2 are the final momentum of the two quarks, respectively. The momentum transfer can be written as where we have dropped terms of order ∼ 1/P . We calculate in the Light-Cone gauge, and hence the gluon propagator takes the following form where n = (0, 1, 0). With this choice, diagrams with gluon emissions from the bottom quark line do not contribute to the amplitude-squared, and hence it suffices to consider only the diagrams in FIG. 1a, FIG. 1b, and  FIG. 1c. The amplitude-squared is the sum of six terms which are respectively displayed in diagramatic form in FIG. 2. We get the following results for each of these terms where We have only kept terms that are proportional to P 2 , which turns out to be the leading power in the large momentum P .
If we ignore Eq.(10), then the terms above separate into two sets according to their color factors: either a term is proportional to C F or it is proportional to C 2 F /N c . Taking alook at Eq.(10), we notice that it comes with the following color factor and so we see that it contributes to both sets. In fact it contributes precisely so as to cancel out all the terms that would go as ∼ 1/k 2 ⊥ at large k ⊥ , leaving only those that go as ∼ 1/k 4 ⊥ . The full amplitude-squared can then be expressed as To get the multiplicity distribution of gluons we need to divide this result by the Born diagram shown in FIG. 1d, which is given by resulting in the following multiplicity distribution

III. DISCUSSION
To start analyzing the Brehmstrahlung probability in Eq. (14), we will assume that M 2 k 2 ⊥ , q 2 ⊥ , which allows one to neglect M 2 in the numerators and in D A , but not in D B,C where M 2 regularizes singularities when A few remarks are in order here: the part proportional to C F can be viewed as Brehmstrahlung from a fermion line (including the interference), whereas terms proportional to C A involve a 3-gluon vertex and a C A part of fermion interference (x 2 / (D B D A )). Importantly, a term from the 3-gluon vertex squared that should naively be proportional to 1/D 2 C cancels out (except for a term proportional to M 2 that we neglected). This is why the full result is proportional to (x 2 − 2x + 2), which is connected to the DGLAP probability, Note also that in the large N c limit 2C F → N c = C A , and both terms have the same color factor. Let us first check limits coming from x = 1 or x = 0 (note that in these limits the mass terms that we neglected vanish): This is the Bertsch-Gunion formula. Note that it comes entirely from the interference term.
• For x = 1 we have This looks like the BG formula, but with a different color factor. The numerical coefficient in front is 2 rather than 4 due to a different value of (x 2 −2x+2) at x = 1 and 0. Of course physically there is no radiation in this limit as the Born term goes to zero as 1 − x.
Now we shall investiagate three limits in k ⊥ : This formula does have a 1/k 4 ⊥ part, which is however suppressed in the large N c limit.
(23) This formula is "x-safe" so that we can take both x = 0 and x = 1 limits and it agrees with the large k ⊥ limit of (19) and (20).
Unpacking these results a bit further, the condition k ⊥ q ⊥ means we are looking at soft gluons in the sense that there is virtually no recoil of the emitting quark. Staying with the soft-gluon case, the condition k ⊥ xq ⊥ is equivalent to requiring that the emitted gluons have rapidity between zero and the final rapidity of the kicked quark-we're essentially looking at the fragmentation region. Eq. (21) then says: for recoil-less quarks, in the fragmentation region, the contribution from QED-like bremstralhung falls off like 1/k 2 ⊥ while the BG contribution is constant in k ⊥ . The next case, xq ⊥ k ⊥ q ⊥ , still considers recoilless qaurks but this time in the central region. Here the QED-like bremstralhung falls off rapidly as 1/k 4 ⊥ while the BG contribution is dominated by a 1/k 2 ⊥ fall-off as seen in Eq. (22). In the final case, k ⊥ q ⊥ , we are looking at high recoil and Eq. (23) shows a ∼ 1/k 4 ⊥ fall-off in all regions. Here the fragmentation region corresponds to x ≥ 1 2 .

IV. CONCLUSION
We have computed the contribution to gluon radiation of a particle scattering from the strong field of a nucleus. As noted in Ref. [12], the classical treatment of the particle computation breaks down in this region. This result should allow a proper matching onto the high transverse momentum region of the emitted gluon, as in this region one can compute the radiation perturbatively, and the contribution we present should be of leading order. This paper therefore completes the determination of the ingredients necessary to properly determine the initial conditions for matter produced in the fragmentation region of high energy heavy ion collisions.