Double inclusive small-x gluon production and their azimuthal correlations in a biased ensemble

We consider double gg → g production in the presence of a bias on the unintegrated gluon distribution of the colliding hadrons or nuclei. Such bias could be due to the selection of configurations with a greater number of gluons or higher mean transverse momentum squared or, more generally, due to a modified spectral shape of the gluon distribution in the hadrons. Hence, we consider reweighted functional averages over the stochastic ensemble of small-x gluons. We evaluate explicitly the double inclusive gluon transverse momentum spectrum in high-energy collisions, and their azimuthal correlations, for a few simple examples of biases.


I. INTRODUCTION
Observables in high-energy scattering in QCD are computed by expressing them in terms of expectation values of various Wilson line operators O; see, for example, ref. [1]. The expectation value O corresponds to a statistical average [2] over the distribution of small-x gluon fields. Hence, the Wilson lines from which O is constructed are computed in the soft gluon field sourced by the valence color charge density ρ which is the large component of the light-cone color current due to the partons with large light-cone momenta [3]: Here we have assumed that the fast hadron propagates in the positive z-direction and that ρ( x ⊥ ) is the source in covariant gauge. We may also compute the average leading twist (covariant gauge) gluon distribution itself via The weight functional is assumed to be normalized to Dρ W [ρ] = 1. The constraint effective potential for is given by [4] e −V eff [X] = Dρ δ X( k) − g 4 k 4 tr ρ( k)ρ(− k) W [ρ] .
This integrates out fluctations of ρ which do not affect the covariant gauge gluon distribution. The most likely gluon distribution from eq. (2) can then be obtained (at leading power in N c ) as the stationary point of the effective potential, Given an observable which is a functional of X( q) the ensemble average now reads For a Gaussian color charge density weight functional W [ρ] one has, explicitly [4], and where A ⊥ denotes the transverse area over which the gluon distribution has been integrated over. The function µ 2 (k) parameterizes the Gaussian ensemble for the color charge density, W [ρ] ∼ exp[− d 2 k/(2π) 2 tr ρ( k)ρ(− k)/2µ 2 ( k)]. However, the corresponding effective potential for X( k) is not quadratic but of "linear minus log" form 1 .
To probe configurations away from the peak of the distribution it is standard in statistical physics to compute biased (or reweighted) expectation values, Just like W [ρ], the bias b[ρ] in general is supposed to be a gauge invariant functional of the color charge density. Here, we impose the bias directly on the gluon distribution X( k), In particular, we may choose b[X] so that the most likely gluon distribution in the reweighted ensemble is shifted to where η( k) ≥ 0 is some prescribed function of transverse momentum 2 . Defining this is achieved via In fact, b[X] is nothing but the generating functional for the moments of X( k), while log b[X] is the cumulant generating functional. The gluon distribution function in principle depends not only on transverse momentum but also on rapidity y. It is straightforward to generalize the above to rapidity dependent biases by writing X( q, y) = η( q, y) X s (q, y) so that then t( q, y) also depends on rapidity via eq. (14). One could then reweight towards rare evolution trajectories, for example. However, in this paper we only consider the MV model [3] effective theory of color charge density fluctuations which does not exhibit a dependence on y.
For the Gaussian action for ρ, from eqs. (7,8) and (14) we have explicitly A particularly simple example for a gluon distribution in a biased ensemble would be This simply boosts the number of gluons with transverse momenta from Λ 2 to Q 2 by a constant factor of 1 + η 0 .
[One may also interpret this as a boost of the transverse momentum of the gluons by a factor of (1 + η 0 ) 1/4 .] Other examples will be considered below. To any given "distortion" η( k) one can associate a potential V [η( k) X s (k 2 )]. The greater this potential, the smaller the weight of the function X( k) = η( k) X s (k) in the ensemble average (6). Hence, a stronger bias is required to make this the dominant gluon distribution in the reweighted ensemble. Explicitly, the "penalty action" for any given η( k) is Thus, the gluon distribution X( k) = η( k) X s (k) occurs in the unbiased ensemble with a probability density (in the space of functions) relative to the saddle point of p is finite or else the gluon distribution X( k) = η( k) X s (k) is not part of the ensemble. A given η( k) corresponds to an excess gluon multiplicity, in the hadron or nucleus, of [4] Likewise, to any η( k) one may associate an increased mean transverse momentum (see definition of k 2 T in ref. [5], for example) and so on. We note, however, that our approach allows us to compute expectation values in an ensemble defined by a functional bias on the gluon distribution X( k) rather than to bias merely by gluon number, mean transverse momentum etc.
One may sample the gluon distributions in the biased ensemble eq. (11) via a Metropolis algorithm. While these gluon distributions are part of the original ensemble, the standard approach of generating configurations without bias and then rejecting those that do not meet given criteria would be prohibitive. Importance sampling with the action V eff [X] − log b[X] strongly increases the overlap with the desired target ensemble. We consider the following three biases for illustration: We take Λ = 2 and Q = 6; the units may be taken as GeV although the energy scale is arbitrary since b[X] is dimensionless. Also, we choose A ⊥ = 10π and g 4 µ 2 = 2 in eq. (7). This bias does not impose a specific transverse momentum dependence of X(k) b . Rather, we let the Monte-Carlo determine the optimal spectral shape.
2. E T bias corresponding to Once again, here we do not impose a specific transverse momentum distribution of the gluons but let the Monte-Carlo determine the optimal spectral shape.
with t( k) = 1 − η −1 (k) k 4 g 4 µ 2 and the prescribed function η(k) = k/Λ.    1 shows the results. Not surprisingly, the N g bias adds gluons mostly near Λ since high-k gluons come with greater penalty action. The E T -bias produces a harder spectrum of excess gluons. Lastly, the t[η] bias multiplies the gluon distribution between Λ and Q by the prescribed function η(k) = k/Λ.
In a collision of two hadrons or nuclei, one is required to average over the color charge distributions of both projectile and target, One may then bias either one or both of the ensembles as described above.
The single-inclusive gluon production cross section in a biased ensemble has been computed previously in ref. [6]. The main purpose of the current paper is to illustrate the effect of a bias on azimuthal angular correlations of two small-x, high-p T gluons produced in a high-energy collision. We recompute the so-called "glasma graphs" for a biased gluon distribution different from its expectation value X s (k) in the unbiased small-x ensemble. These diagrams for high-p T double gluon production have originally been introduced in refs. [7,8]. Their applicability, and corrections to this approximation, have been studied in refs. [9][10][11].
The literature on azimuthal correlations of small-x gluons is rather extensive and we do not attempt to summarize it here. Instead, we refer the reader to the review articles in refs [12,13]. Our main focus here is on effects due to a bias on the gluon distributions of the colliding hadrons or nuclei, an issue which has rarely been addressed. Notable exceptions are refs. [14] where the authors assumed that high multiplicity p+p and p+Pb events correspond to an enhanced saturation scale Q s (x 0 ) of the proton at the initial rapidity for small-x evolution. Ref. [15] considered a constant multiplicative rescaling of the color charge density in the proton to discuss the multiplicity dependence of azimuthal moments [as defined in eq. (37) below] in p+Pb collisions. Ref. [16] analyzed angular correlations in a combinatoric model for multi-particle production with color interference effects, and their dependence on multiplicity. Ref. [17], finally, applied a hydrodynamic model to look into the effect of final state interactions on angular correlations, as a function of the particle multiplicity in the event. Here, we perform a first analysis of the "glasma graphs" in the presence of a functional bias on the gluon distribution.
The remainder of this paper is organized as follows. In section II we write the two gluon inclusive distribution at high transverse momentum for general η( k). In sec. III, we shall analyze several specific momentum dependences to see how they affect the double gluon spectrum and their angular correlations.

II. TWO GLUON INCLUSIVE DISTRIBUTION IN A BIASED ENSEMBLE
The cross section for inclusive production of two small-x gluons with transverse momenta p, q much greater than the saturation scales of the projectile and target is given by the so-called "glasma graphs". These graphs correspond to a k T -factorization approximation in terms of unintegrated gluon distributions [18], From now on we consider a constant, k-independent µ 2 for simplicity. This amounts to the classical MV model [3] approximation where one neglects the anomalous dimension of the gluon distribution. While it is possible, in principle, to generalize our analysis to account for the anomalous dimension due to small-x evolution, our current focus is on better understanding the effect of a bias on the glasma graphs.
In a biased ensemble then, Beyond the dilute limit one needs to evaluate a correlator of two eikonal Wilson lines in the reweighted ensemble, see ref. [6]. Here we restrict to high transverse momentum where the approximation of dilute projectile and target should be applicable. We start from the expression for the two gluon transverse momentum distribution from the glasma graphs given in ref. [19] Here, Λ denotes an infrared cutoff for applicability of the leading twist, weak field approximation. A ⊥ Λ 2 will be taken to be ∼ 1 or greater. Furthermore, A corresponds to the disconnected diagram for inclusive double gluon production shown in fig. 2. C are the HBT-like [20] parts proportional to δ 2 ( p ± q), shown in fig. 3. Finally, the rest is given by diagrams B (fig. 4) and has been interpreted as Bose enhancement [21]. Note that B and C correspond to connected two gluon production diagrams. Explicitly, (28) Here, The final contribution is Note that the contributions B and C from the connected diagrams do not come with a second power of the transverse area A ⊥ nor with a second factor of N 2 c − 1 as there is a single connected color flow loop. It is sufficient for our present purposes to consider a bias on the target's ensemble of gluon distributions, only. We first compute the contributions denoted as A and B. In the computation of B we may assume that p = ± q, i.e. that p ± q are hard momenta themselves, much greater than the saturation scales of the colliding protons or nuclei. In fact, when the magnitudes of p and q are close then one needs to also account for the back-to-back dijet contribution (see e.g. ref. [14]) when computing angular correlations of high-p T gluons. Therefore, while we do give the expression for C for general η( k) later in this section for completeness, we do not consider these contributions further in sec. III.
In this paper, we will only consider reflection symmetric gluon distributions, η(− p) = η( p). To perform the integrations over the transverse momenta of the gluons in projectile and target we expand the integrands in eqs. (27, 28, 30) around the singularities of the Coulomb propagators, and keep the leading terms. For example, The first term is the DGLAP logarithm [22]. To compute the integral in the second term we first write η( k) − 1 = η(k) Θ(Q 2 − k 2 ) to display explicitly the finite support of the modification to the gluon distribution. Now, if Q 2 is on the order of p 2 then the contribution from small k 2 p 2 to the integral is This contribution is absent 4 if Q 2 p 2 . For any Q 2 ∼ p 2 or less the integral on the r.h.s. of eq. (31) also receives a contribution from the region ( p − k) 2 p 2 , provided that η( )/ 2 has a pole at → 0: Then eq. (27) is approximated as In an unbiased ensemble where η( k) − 1 = η(k) = 0 this contribution for independent production of two gluons does not depend on the angle between p and q, of course. The same is true if η( k) is isotropic.
For the contribution from connected two-gluon production diagrams we obtain Here, is one fourth the leading term of f ( k, p, q) in the limit k 2 p 2 , q 2 . The expansion in eq. (35) includes terms that explicitly depend on the azimuthal angle, φ, between p and q, even though they may be subleading at large p 2 , q 2 . In contrast, we have dropped a term in eq. (35) that does not depend on φ, and that would be subleading when A and B are added. However, we have not dropped the last term in eq. (35) which exhibits power-sensitivity to low transverse momenta but is independent of φ when η( k) is isotropic. In sec. III we will compute the angular moments Reflection symmetry under the simultaneous p → − p, q → − q implies invariance under φ → −φ, and so e inφ is real. When η( k) is isotropic then eq. (35) gives the leading φ-dependent terms. However, in sec. III D we shall see that when η( k) is anisotropic then eq. (34) will also contribute to the angular moments; here, the angular correlations in the "disconnected diagrams" actually arise due to the bias.
Using the same approximation for the integrations over the 2d Coulomb propagators, the "HBT diagrams" for general η( k) evaluate to (38)

III. SPECIFIC ENSEMBLES
In this section we evaluate explicitly the contributions from diagrams A and B for a few choices of η( k). As already mentioned above we will focus on the case where the transverse momenta p and q do not have very similar magnitudes, so we ignore diagrams of type C.
We study a biased ensemble where the number of gluons (defined from the covariant gauge gluon distribution) with squared transverse momenta between Λ 2 and Q 2 is boosted. In this section, we consider ensembles of the form where η 0 is a dimensionless constant, a controls the transverse momentum dependence, and b ≥ 0 the anisotropy (in the directionÊ). We consider isotropic ensembles with a = 0 (section III B) and a = 1 (section III C), as well as anisotropic ensembles with a = 0 and b = 1 (sec. III D). We will also briefly discuss the case a = −1, b = 0 at the end of section III C.
Only the φ-dependent terms enter into the numerator of eq. (37), from which we can see that the odd moments vanish.
To compute the even moments, we will need the integrals which can be derived with contour integration. Here q 2 < = min(p 2 , q 2 ) and q 2 > = max(p 2 , q 2 ). To use eqs. (41 , 42) in (40) we need to neglect the dependence of ( p ± q) 2 on φ when the former appears inside a logarithm. This is justified in leading logarithmic approximation. The even angular moments for n ≥ 1 then read where 2 = p 2 + q 2 − 2pq. This formula predicts that e 2niφ decreases with n like [min(p 2 , q 2 )/max(p 2 , q 2 )] n . When 5 p 2 ≈ q 2 , eq. (43) simplifies to B. Constant boost of the gluon density between k 2 = Λ 2 and k 2 = Q 2 In this section, we consider a boost of the gluon density between k 2 = Λ 2 and k 2 = Q 2 by a constant factor 1 + η 0 : This corresponds to the "penalty" action and to a gluon number excess Hence, for η(k) like in eq. (45) a substantial gluon number excess is much more likely to occur due to many additional gluons with small transverse momenta not too far above Λ, so that η 0 is large but Q 2 /Λ 2 is moderate. However, such configurations do not increase k 2 T much. [Biasing towards gluon distributions like eq. (45) with large k 2 T would rather favor smaller η 0 and larger Q 2 .] We first calculate the two-gluon spectrum for Q 2 ∼ p 2 , q 2 , ( p ± q) 2 . Factoring (1 + η 0 ) 2 from eqs. (27) and (28) it simply becomes (48) dN dy p d 2 p dy q d 2 q bias = (1 + η 0 ) 2 dN dy p d 2 p dy q d 2 q unb. .
We conclude that, in the limit where Q 2 is on the order of the momenta of the produced gluons (or greater), the angular moments in this ensemble are the same as in the unbiased ensemble, c.f. eq. (43). This is analogous to the k-independent rescaling of the color charge density of the proton considered in ref. [15]. Note, however, that such gluon distributions have very small probability in the original ensemble since V eff ∝ Q 2 .
On the other hand, when Q 2 p 2 , q 2 , ( p ± q) 2 , we can drop contributions like in eq. (32). Then It may be surprising at first glance that a "distortion" of the gluon distribution up to Q 2 much less than the momenta of the produced gluons would affect the cross section. This is due to the fact that the production occurs via Lipatov fusion [23] of one gluon from the projectile with one gluon from the target, where typically one of the fusing gluons carries much smaller transverse momentum than the produced gluon. The even angular moments for n ≥ 1 resulting from eq. (49) are We recall that 2 = p 2 + q 2 − 2pq.
We have yet to make any assumptions about the magnitude of η 0 . Consider η 0 = O(1) and Q 2 less than min(p 2 , q 2 , 2 ). This corresponds to a class of high gluon multiplicity configurations with k 2 T moderately higher than in absence of the bias. Here, the corrections to the numerator and denominator of eq. (50) due to the bias are suppressed only logarithmically (relative to the unbiased ensemble) ! On the other hand, if η 0 1, we can simplify the previous expression by ignoring the terms ∼ η 2 0 to compute the ratio of the moments in the biased and unbiased ensembles 6 . For 2 < min(p 2 , q 2 ), for example, this is e 2niφ bias e 2niφ unb.
Thus, when p and q have comparable magnitudes (→ 2 p 2 , q 2 ) then the correction is positive, and so the angular moments in the biased ensemble increase with increasing η 0 . In the limit p 2 q 2 (or vice versa), on the other hand, the correction in eq. (50) is negative, and so the angular moments decrease with increasing η 0 . Thus, we see that even a very simple "distortion" of the gluon distribution may give rise to a fairly intricate behavior of the angular correlations.
C. Transverse momentum dependent boost of the gluon density We now consider a transverse momentum dependent increase of the gluon density between k 2 = Λ 2 and k 2 = Q 2 . We will mostly confine ourselves to an η(k) that decreases with increasing k (but will briefly take up the case of η(k) increasing with k at the end of this section). Specifically, we take with Q 2 Λ 2 . Such a gluon distribution comes with a penalty action of Also, the number of high-k T gluons in the hadron increases by Hence, here both V eff and ∆N g approach a constant as Q 2 /Λ 2 1 at fixed η 0 .
The resulting gluon production distribution is No logarithms of Q 2 appear here since the gluon distribution in this biased ensemble (53) drops more rapidly than 1/k 4 . The angular moments are .
To simplify this expression we now consider various parametric magnitudes for η 0 . The case η 0 ∼ < 1 is not very interesting here as it does not lead to rare configurations with a substantial increase in the number of gluons; we have It is more interesting to let η 0 ∼ log Q 2 Λ 2 so that the amplitude of the shift of the gluon distribution and its transverse momentum cutoff are related. ∆N g then increases in proportion to log Q 2 Λ 2 while V eff ∝ log Q 2 Λ 2 log log Q 2 Λ 2 . This means that high gluon multiplicities can be reached with much higher probability than for the bias considered in the previous section. Furthermore, the angular moments will increase with η 2 0 for any choice of p 2 or q 2 . For example, when p 2 ≈ q 2 , e 2niφ bias e 2niφ unb. ≈ 1 + 1 40 and the correction increases like the square of ∆N g , i.e. proportional to log Q 2 .
We briefly comment on the case where the gluon distribution at the saddle point of the reweighted ensemble is shifted to again with Q 2 Λ 2 . This leads to a very strong increase of k 2 T as compared to the unbiased ensemble. On the other hand, to have V eff ∝ Q 2 like in sec. III B rather than V eff ∝ Q 4 we must choose very small amplitude, η 0 ∝ 1/Q 2 . And then, ∆N g asymptotes to a constant at large Q 2 . In other words, these are rare configurations of the hadron where the excess mean squared transverse momentum of the gluons is large but their number excess is not. In this paper we restrict to cases corresponding to large ∆N g .

D. Anisotropic η( k)
In this section, we explore anisotropic gluon distributions such that the average gluon distribution X s (k) in the unbiased ensemble is multiplied by The vectorÊ specifies an arbitrary direction in the transverse plane. Studying such anisotropic configurations has been suggested by Kovner and Lublinsky [24] (also see ref. [25]). In their work the anisotropy is due to fluctuations from configuration to configuration of the hadron or nucleus. In this scenario, after computing the angular correlator e inφ , we would have to perform an average over the directions ofÊ. However, we will also consider the possibility that 2D rotational symmetry is explicitly broken due to the external bias (like a spin model in an external magnetic field) so that eq. (60) represents the gluon distribution averaged over the reweighted ensemble. It is beyond the scope of the present paper to discuss specific phenomenological models for how such an anisotropic bias may arise in p+p or p+A collisions. Nevertheless, it is an interesting exercise to compute two-particle correlations in an anisotropic ensemble as this leads to new contributions to the angular moments. Specifically, the disconnected diagram, A, will now contribute φ-dependent terms. From eqs. (32,34), we see that these terms only occur when Q 2 ∼ p 2 , q 2 , and that they will be proportional to η 2 0 (p ·Ê) 2 (q ·Ê) 2 b . Therefore, only moments less than or equal to 2b will receive such contributions.
When a = 1, the leading angular contribution to A is proportional to Λ 4 p 4 log 2 p 2 Λ 2 , which is smaller than the angular contributions we saw in sec III B. Hence, we consider the case a = 0 and b = 1. We then have for η 0 1, Q 2 Λ 2 : and The contribution from the disconnected diagram now becomes (for Q 2 ∼ p 2 , q 2 ) Because this is the contribution from the "disconnected diagram", and there is a constant k-independent boost of the gluon distribution relative to the unbiased ensemble, we recover the two DGLAP logarithms even in the anisotropic contribution.
exp(2iφ) increases with the gluon number excess ∆N g when p and q have comparable magnitude, but decreases with ∆N g when p q. Another example of at least theoretical interest is that of a bias that introduces a preferred direction in the transverse impact parameter plane, inducing an anisotropy of the gluon distribution. Such a bias gives rise to an angular dependence even of the disconnected (in terms of color flow) diagram for double gluon production. Hence, the leading dependence of the cross section on the azimuthal angle φ between p and q emerges at leading power in N c , and comes with the same DGLAP logarithms log p 2 log q 2 as the diagrams for uncorrelated production in an unbiased ensemble.
Our analysis could be continued by looking at more complicated biases and how they would reweight the functional integral over gluon distributions. In particular, a very important step would be to develop a more direct connection between the reweighting functional b[X] used here and specific event ensembles that one might be able to construct in practice, either from experiment or from an event generator.