Azimuthal asymmetries and the emergence of"collectivity"from multi-particle correlations in high-energy pA collisions

We show how angular asymmetries $\sim\cos 2\phi$ can arise in dipole scattering at high energies. We illustrate the effects due to anisotropic fluctuations of the saturation momentum of the target with a finite correlation length in the transverse impact parameter plane, i.e.\ from a domain-like structure. We compute the two-particle azimuthal cumulant in this model including both one-particle factorizable as well as genuine two-particle non-factorizable contributions to the two-particle cross section. We also compute the full BBGKY hierarchy for the four-particle azimuthal cumulant and find that only the fully factorizable contribution to $c_2\{4\}$ is negative while all contributions from genuine two, three and four-particle correlations are positive. Our results may provide some qualitative insight into the origin of azimuthal asymmetries in p+Pb collisions at the LHC which reveal a change of sign of $c_2\{4\}$ in high-multiplicity events.


I. INTRODUCTION
Large azimuthal asymmetries have been observed in p+Pb collisions at the LHC [1][2][3][4] and in d+Au collisions at RHIC [5]. These asymmetries are usually measured via multi-particle angular correlations (see below) and were found to extend over a long range in rapidity. Causality then requires that the correlations originate from the earliest times of the collision [6]. Furthermore, the data shows that the asymmetries persist up to rather high transverse momenta, well beyond p ⊥ ∼ 1 GeV. Recent data by the ATLAS collaboration, for example, shows that large "elliptic" (v 2 ) asymmetries in p+Pb collisions at √ s = 5 TeV persist up to p ⊥ = 10 GeV [7]. Therefore, it is important to develop an understanding of their origin in terms of semi-hard (short distance) QCD dynamics [8][9][10][11][12][13][14][15].
The ALICE collaboration has measured the two-and four-particle v 2 cumulants in p+Pb collisions at 5 TeV as a function of multiplicity, see Figs. 1 and 4 in Ref. [2]. These cumulants are defined as [16] Here, · denotes an average over the corresponding azimuthal angles weighted by the two-or four-particle distribution, respectively. The two-particle cumulant with a rapidity gap suppresses contributions from resonance decays and jet fragmentation; it depends weakly on multiplicity and is positive over the entire range of multiplicity. On the other hand the four-particle cumulant, c 2 {4}, decreases monotonically and changes sign to become negative in high multiplicity events, an effect also seen by the CMS collaboration (see second paper in [4]). As shown below, this requires an anisotropy of the single-particle angular distribution. In the soft, long wavelength regime, c 2 {4} is negative when hydrodynamic flow dominates over "non-flow" correlations [17]. In this paper we perform a first computation of all connected and disconnected contributions to the cumulants in the short distance regime using a model that allows for anisotropic "domains" of the color-electric fields E of the target [10,18].

II. CALCULATION
Our discussion is based on the dipole model of high-energy interactions [19]. We consider scattering of a dipole of size r ∼ 1/p ⊥ from the target described by a particular configuration of the (color) electric field E i ∼ F +i . For a small dipole r ≡ x − y the leading C-even interaction with the target is given by with a C-odd correction at order (igr) 3 which is not considered here because it does not contribute to ∼ cos 2φ asymmetries [18]. Equation (3) arises from an expansion of the S-matrix, tr V ( x)V † ( y)/N c , in powers of r, where is the path-ordered Wilson line describing the propagation of a charge in the field of the (right-moving) target. We focus on the S-matrix for a fundamental charge though the calculation could be repeated for a charge in the adjoint representation yielding the same results for c 2 {2} and c 2 {4}. To obtain the cross section the scattering matrix is averaged over the configurations of the E field of the target. Averaging over all such configurations leads to in the leading log approximation, log 1/rΛ ≫ 1. Here, Q s ( b) denotes the saturation scale below which non-linear effects become significant. In what follows we shall assume a very large nucleus and drop the dependence of the average saturation momentum on b. Equation (5) corresponds to the single-particle cross section averaged over all configurations of E( b) in the target and is, of course, isotropic. On the other hand, for any particular configuration the S-matrix does exhibit an angular dependence, c.f. for example Fig. 7 in Ref. [20]. The idea that anisotropic fluctuations of the saturation momentum would induce v n = 0 has been presented previously in Refs. [10,18,21]. Hence, to evaluate the amplitude of the angular modulation of the S-matrix we perform the average subject to the constraint That is, we divide the target ensemble into classes such that for a given class the anisotropic part of the electric field correlator in the vicinity of b (within a given "domain") points in a specific direction. The summation over all classes, which corresponds to an integration over the directionsâ, is performed only after the m-particle angular cumulant has been evaluated. The quantity A in eq. (6) is the amplitude of anisotropy of the electric field correlator. For simplicity, as we mentioned above, in our current analysis we singled out only fluctuations ofâ while possible fluctuations of Q s and A are averaged out in Eq. (6). The results could be extended to account for fluctuations of Q s and A in the future.
The domain structure of the field is described by the two-point correlation function where ξ denotes the correlation length. We assume a Gaussian correlation function, other options do not change our results qualitatively. To simplify the notation we introduce which is the area of a domain divided by the area of the collision zone, in other words, the inverse number of domains. Equation (7) essentially describes the correlations of the saturation momentum Q s in the transverse plane.
We can now compute the angular distribution for scattering of a single dipole, for a fixedâ. Using Eqs. (6) and performing a Fourier transform to momentum space, as well as an average over the impact parameter, we arrive at Hence, the one- To avoid confusion let us stress that here · refers to a different average than the average over E-field configurations from above; it is simply an average over the azimuthal angle φ k weighted by the distribution (9). We now proceed to two-particle distributions. The averages over E-field configurations shall be performed assuming a Gaussian action [22] and a color diagonal four-point function although in general additional contributions could appear [10,23]. Then the two-particle S-matrix for fixedâ is given by The factorizable (disconnected) contribution involves the correlations of the directions of E( b) in the impact parameter plane; we employ C(â, . Averaging over impact parameters gives In this expression the prefactor 1/N D arises due to the fact that the orientation of the electric field is approximately constant only over distance scales of order the correlation length ξ. Multiplying the Fourier transform of this expression by exp(2i(φ 1 − φ 2 )) and averaging over the azimuthal angles leads to the disconnected (single-particle factorizable) contribution to (v 2 {2}) 2 : Note that this is independent of the global directionâ relative to which we define φ 1 and φ 2 and so the final average over a is trivial. The additional term in the denominator originates from the connected contribution to the normalization. The connected contribution from Eq. (13) is Averaging over impact parameters produces a factor so that the connected contribution to the two-particle cumulant becomes As before, here the average · on the l.h.s. is an average over φ 1 and φ 2 but does not involve averaging over E-field configurations since the one-and two-particle distributions have already been averaged over all such configurations corresponding to a givenâ. However, the r.h.s. is independent ofâ so that the final average over its direction is trivial. Also, for A = O(1/N c ) the first factor on the r.h.s. of Eqs. (17,22) can be approximated by 1/N D so that in all, v 2 {2} is then given by .

(23)
The first term is the square of the single-particle v 2 {1}; it is scaled by 1/N D since both particles have to scatter from the same domain. The second contribution corresponds to genuine non-factorizable two-particle correlations. Both contributions are positive; nonetheless Eq. (23) reveals the existence of two distinct regimes. For the ellipticity is mainly due to the asymmetry of the single-particle distribution induced by the E-field domains. In the opposite limit v 2 {2} is mainly due to genuine two-particle correlations. Expression (23) applies when both particles have sufficiently high transverse momenta as we have approximated both of their S-matrices by their leading small-r behavior ∼ tr ( r i · E) 2 . On the other hand, experimentally one typically considers angular correlations of a hard with a softer particle. Recent numerical computations [24] of c 2 {2} which do not expand the S-matrices show that hard-soft correlations exhibit a fall-off with the transverse momentum of the hard particle. This is due to a decorrelation of the anisotropy axis in a high-p T bin with that of the bulk.
The four particle cumulant exhibits qualitatively different behavior in the regimes of "small" vs. "large" A. For general A, c 2 {4} is given by which determines the azimuthal anisotropy from four particle correlations: v 2 {4} = (−c 2 {4}) 1/4 . Before addressing the corrections written in Eqs. (28,29) we compute the fully connected contribution and show that it is positive. The fully connected contribution to the S-matrix is given by where i + 1 is defined modulo 4. Averaging over impact parameters generates a factor of 1/(4N 3 D ). We may now perform the Fourier transform and sum the 48 contractions of the amplitudes / conjugate amplitudes of dipoles 1 to 4. This leads to Here, corrections of order ∼ 1/(N 2 c − 1) to the normalization have been neglected, see related discussion for v 2 {2} above. As promised, the fully connected contribution to c 2 {4} is positive; thus if the anisotropy A is zero, the elliptic harmonic v 2 {4} would be complex. Furthermore, the magnitude of the fully connected contribution relative to v 2 {1} 4 is ∼ 1/(A 4 N 6 c ). Hence, parametrically c 2 {4} crosses zero when A ∼ 1/N The terms from Eqs. (28,29), to leading order in N c , are given by They provide manifestly positive contributions to c 2 {4}. When A is of order of N −3/2 c , which is the regime where c 2 {4} changes sign, we can write our final result in the form Here the additional terms listed in Eqs. (28,29) are suppressed by additional powers of N −2 c .

III. DISCUSSION
An anisotropic single-particle distribution, v 2 {1} = 0, requires an angular dependence of the dipole S-matrix ∼ tr ( r · E) 2 for individual configurations of E. We describe this by the term ∼ A(r ·â) 2 in Eq. (6).
The first term corresponds to the square of the asymmetry of the one-particle distribution while the second term is due to non-factorizable, genuine two-particle correlations. The transition between the two regimes occurs at A ∼ 1/N c . In practice, using N c = 3 and the estimate A ≃ 0.2 from Ref. [18] we conclude that the magnitudes of both terms are comparable. The elliptic asymmetry from four-particle correlations, This expression applies when , where c 2 {4} changes sign. The first term on the r.h.s. corresponds to the fully factorized distribution and is the only negative contribution to c 2 {4}. Thus, parametrically this transition to c 2 {4} < 0 occurs before the one-particle factorizable contribution dominates c 2 {2}. That is, in the vicinity of c 2 {4} = 0 the two-particle cumulant c 2 {2} is dominated at leading order in 1/N 2 c by connected diagrams. We repeat, also, that all contributions in eqs. (36,37) computed within small-x QCD are long range in rapidity.
Our analysis naturally raises a question about the magnitude of the E-field polarization amplitude A and its dependence on multiplicity. Averaging over all target configurations without a multiplicity bias gives A ∼ 0.1 − 0.15 at small x [25]. In fact, A(r) exhibits a (weak) dependence on r at small r and this function has been found [25] to coincide with the distribution of linearly polarized gluons (for the MV model) obtained in refs. [26]. The effect of a multiplicity bias remains to be investigated. In order for the disconnected contribution to dominate in high multiplicity events, A would have to grow with multiplicity.
Although our present discussion is restricted to high-p ⊥ particles, i.e. small dipoles, it suggests that the measurement by the ALICE and CMS collaborations of a sign change of c 2 {4} corresponds to the fully factorizable contribution becoming dominant. The emergence of "collectivity" in pA collisions could be viewed as multi-particle correlation functions becoming dominated by fully disconnected diagrams, analogous to the BBGKY hierarchy. It will be important to understand specifically how this emerges from small-x QCD dynamics.