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Anisotropy of the semi-classical gluon field of a large nucleus at high energy

The McLerran-Venugopalan model describes a highly boosted hadron/nucleus as a sheet of random color charges which source soft classical Weizs\"acker-Williams gluon fields. We show that due to fluctuations, individual configurations are azimuthally anisotropic. We present initial evidence that impact parameter dependent small-x JIMWLK resummation preserves such anisotropies over several units of rapidity. Finally, we compute the first four azimuthal Fourier amplitudes of the S-matrix of a fundamental dipole in such background fields.


I. INTRODUCTION
To explain azimuthal asymmetries observed in high-energy pA collisions [1][2][3][4][5] Refs. [6][7][8] argued that individual configurations of the light-cone electric fields of the target should be anisotropic, leading to a non-trivial azimuthal distribution of a projectile parton scattered off such a target. That is, configuration by configuration, two-dimensional rotational symmetry is broken by E-field "domains" of finite size in the impact parameter plane. These, in contrast to Weiss magnetic domains separated by domain walls, arise purely due to fluctuations of the valence (large-x) random color charge sources for the soft, small-x E field.
Assuming said azimuthal anisotropy of the light-cone electric fields several features of the data could be described, at least qualitatively [7,8]. On the other hand, a direct calculation of the anisotropic distributions, in particular for a large nucleus and small x (i.e. high energy), has so far been lacking. It is our goal here to compute scattering of a dipole off a large nucleus, and specifically, to determine its angular dependence. That is, we compute the (first four) Fourier amplitudes of the dipole S-matrix with respect to the azimuthal orientation of the dipole. We should stress that we do not address the fluctuations of S(r, b) in impact parameter space b (see Ref. [9] for a recent study) but rather its dependence on the size and orientation of the dipole vector r which is the variable conjugate to the transverse momentum of the parton in the final state.

II. THE MODEL
In the McLerran-Venugopalan model [10] the large-x valence partons are viewed as random, recoilless color charges ρ a (x) which source the semi-classical small-x gluon fields. We first provide a brief description of how these color charge configurations are generated on a lattice; more detailed discussions can be found in the literature [11,12].
The effective action describing color charge fluctuations is taken to be quadratic, with µ 2 ∼ g 2 A 1/3 proportional to the thickness of a nucleus [10]; here A denotes the number of nucleons in the nucleus. The variance of color charge fluctuations determines the saturation scale Q 2 s ∼ g 4 µ 2 [13]. The coarse-grained effective action (1) applies to (transverse) area elements containing a large number of large-x "valence" charges, Hence, in the first step we construct a random configuration of color charges on a 2D lattice according to the distribution exp(−S[ρ]). Their (non-Abelian) Weizsäcker-Williams fields are pure gauges; in covariant gauge, This also satisfies A − = 0 and thus the only non-vanishing field strength is F +i = −∂ i A + . The (light-cone) electric field is The propagation of a fast charge in this field is described by an eikonal phase given by a light-like SU(3) Wilson line V (x): where P denotes path-ordering in x − . The absolute value squared of this amplitude gives the S-matrix for scattering of this charge off the given target field configuration, Thus, following the ideas leading to the MV model we assume that every particular scattering event probes one particular configuration in the target, i.e. that the S-matrix is computed with a frozen ρ a (x). The main purpose of this paper is to analyze the dependence of the S-matrix on the angular orientation of the dipole vector r, conjugate to the transverse momentum, at fixed transverse impact parameter (coordinate) b. The S-matrix for a fundamental charge is complex (for three or more colors). Its real (imaginary) part corresponds to C-even (C-odd) exchanges [14]: C conjugation transforms ρ a (x) → −ρ a (x) and V (x) → V † (x). The dipole scattering amplitude D ρ (r) = D ρ (−r) is even under r → −r and generates even azimuthal v 2n harmonics while the odderon O ρ (r) = −O ρ (−r) generates the odd v 2n+1 [7]. It is useful to consider the limit of small dipoles, rQ s 1. Then the real part of the S-matrix from Eq. (6) is To compute the elliptic (dipole) asymmetry, Refs. [6][7][8] considered the following angular dependence of the two-point function whereâ corresponds to the "event plane" orientation, and ∆(b 1 − b 2 ) describes the E-field correlations in the transverse impact parameter plane. It is implicit that for each configuration E(b) is rotated to point in a particular, fixed directionâ before performing the ensemble average. In fact, Eq. (9) is the MV model analogue of the gluon TMD for an unpolarized target [15,16], Thus, the amplitude A from Eq. (9), which we shall denote A 2 (r) below, is basically h ⊥g 1 at small x. However, beyond the MV model the relation between these functions may be more involved.
The action (1) is C-even and so O ρ (r) = 0 while D ρ (r) ∼ r 2 Q 2 s (at small r) is proportional to the thickness of the nucleus, A 1/3 . A C-odd operator with κ 3 ∼ g 3 A 2/3 could be added to the action 1 which would then induce an expectation value ∼ A 1/3 for the odderon [17]. This is beyond the scope of the present paper, we focus here on azimuthal anisotropies due to fluctuations of the charge densities ρ a (x) and their associated electric fields E a (x).

III. IMPLEMENTATION
To generate the random configurations ρ a (x − , x) via Monte-Carlo techniques we discretize the longitudinal and transverse coordinates. The number of sites in the longitudinal direction is taken to be N − = 100 while the number of sites in either transverse direction is N ⊥ = 1024. All of our results presented here have been obtained with g 2 µa = 0.05, hence g 2 µL = 51.2, where a ≡ L/N ⊥ denotes the transverse lattice spacing. We have determined numerically that Q s ≈ 0.7125g 2 µ as defined from S ρ (r = √ 2/Q s ) = exp(−1/2). The physical value for the lattice spacing could be determined by assigning a physical value to Q s ; instead, we choose to measure distance scales in units of 1/Q s or 1/g 2 µ and so this step is not required.
We use periodic boundary conditions in the transverse directions and solve the Poisson equation (2) by Fast Fourier Transform. The amplitude of the zero mode of ρ a (k) is set to zero before inversion which ensures color neutrality of each configuration.
We have generated about 10 3 configurations; for each of them we measured D ρ (r) and O(r) at b = 0. Both functions were decomposed into their Fourier series to extract the amplitudes of azimuthal anisotropy: The function N (r) is the isotropic part of the dipole S-matrix, see for example Ref. [12]. Each amplitude A n contains a random phase exp(inψ) which fluctuates from configuration to configuration. This corresponds to a random global rotation of the charge distribution ρ a (x) from configuration to configuration. Azimuthal harmonics v n are defined from multi-particle correlation functions in such a way that they are invariant under a global shift of the azimuthal angles of all particles by the same amount. Consequently, we discard this random phase by defining A n = |A n |.
Averaging over configurations we finally obtain A 1 , · · · , A 4 as well as the variances of A 1 and A 2 .

IV. RESULTS
Before presenting our results for the azimuthal amplitudes we show two examples for S ρ (r) in Figs. 1 and 2. Either of these corresponds to one particular (random) configuration of color charges. The real parts display predominantly a ∼ cos(2φ) angular dependence, with φ the angle between r and E(b = 0). On the other hand, the imaginary part for the configuration shown in Fig. 1 is predominantly ∼ cos(φ) while that from Fig. 2 is mainly ∼ cos(3φ), modulo a random phase shift as mentioned above. The figures show, also, that the angular structures appear at a resolution on the order of r g 2 µ ∼ 1; this is consistent with the requirement µ 2 ∆A ⊥ 1 mentioned above (which sets the regime of applicability of the effective theory) at weak coupling: 1/g 2 1. Figure 3 shows our results for the averaged amplitudes of the first four azimuthal harmonics. As expected, the biggest one is the quadrupole amplitude A 2 which reaches > ∼ 12% at r < ∼ 1/Q s . Such values are in the range of the asymmetries extracted phenomenologically [7] for high-multiplicity p+Pb collisions at LHC energies. However, here we have not made any attempts to bias the configurations towards "high multiplicities". The fact that the variance √ (δA 2 ) 2 is not much smaller than A 2 indicates that some configurations generate much larger elliptic asymmetries than others. Also, we observe that A 2 is approximately constant for r < 1/Q s since up to quadratic order the real part of the S-matrix is at small r. To derive this expression one performs a gradient expansion of Re tr V (x)V † (y), assuming that the electric field is smoothly varying over scales of order r. The leading term on the r.h.s., if scaled by 1/r 2 , is independent of r.
For not too large dipoles our numerical result agrees well with the behavior derived in Ref. [16]: Equation (15) provides a perfect fit of A 2 for r Q s < ∼ 3, as shown in Fig. 3. The second term in Eq. (14) generates a hexadecupole asymmetry at the next to leading order in r 2 . However, the numerical result for A 4 (r) shown in Fig. 3 is essentially constant at small r. We interpret this as due to corrections to the gradient expansion which leads to Eq. (14); a ∼ cos(4φ) angular component appears already at O(r 2 ) albeit with a much smaller amplitude than the ∼ cos(2φ) harmonic.
We now turn to the odd amplitudes A 1 and A 3 . As already mentioned above, the expectation value of the odderon over a C-even ensemble such as that generated by the action (1) is of course zero. Nevertheless, each particular configuration of semi-classical small-x fields (2) does contain a C-odd component and iO(r) as defined in Eq. (7) is non-zero. This is due to fluctuations of the saturation momentum Q s in impact parameter space [19], The expression on the r.h.s. corresponds to an expansion in powers of r; Q c is a cutoff for the spectrum of fluctuations of Q s (b) which was otherwise assumed to be scale invariant, and B is their amplitude [7]. Eq. (16) shows that for small dipoles, after we divide by the isotropic normalization factor N (r) ∼ r 2 , that we should expect A 1 ∼ r as well as a smaller A 3 ∼ r 3 . The lattice results appear consistent with A 1 ∼ r at r 1/Q s but so is A 3 , albeit with a smaller slope. Future simulations on larger lattices may be able to push to smaller r, and the analytical derivation of Eq. (16) based on a simple fluctuation spectrum could perhaps be refined as well.
Just as for the elliptic asymmetry we have also computed the standard deviation of the amplitude A 1 . Again, we find that √ (δA 1 ) 2 is not much smaller than A 1 , i.e. that some configurations generate much larger dipole asymmetries than others.
We have also analyzed the effect of "smearing" the impact parameter of the projectile over a region corresponding to its size [20]. If the E-field anisotropy exhibits a non-zero correlation length in the impact parameter plane [6][7][8], specifically a correlation length that exceeds the size of the dipole, then the azimuthal moments should remain approximately the same.
Hence, we have also computed the azimuthal amplitudes A n from "smeared" configurations: and similarly for iO ρ (r, b). On the r.h.s. the points x = b + r/2 and y = b − r/2 are now determined by r and b . Equation (17) averages the S-matrix over an area πr 2 . The result is shown in Fig. 4 which can be compared to Fig. 3 from above. Except for a slight suppression of their magnitudes, we do not observe any substantial modification of the amplitudes A n . The behavior for large dipoles is different, c.f. Fig. 5. For a fixed impact parameter the harmonic amplitudes approach a common non-zero function at large r 1/Q s . This is consistent with universal (angular) scale invariant fluctuations of the azimuthal dependence of the S-matrix. Indeed, if D(r, b) and O(r, b) are first averaged over an area πr 2 , see Eq. (17), then the resulting A n are strongly suppressed. This shows that the direction of E is not correlated over distances much beyond ∼ 1/Q s . Also, we note that the resummed analytical result [16] written in Eq. (15) does not provide a good fit for r Q s > ∼ 3. This is not unexpected since the derivation involves adhoc infrared cutoffs which need to be introduced by hand (c.f. related discussion in Ref. [12]). On the other hand, the non-perturbative lattice computation does not require IR cutoffs beyond imposing global color neutrality.

V. SUMMARY
Following the conjecture by Kovner and Lublinsky [6], we have analyzed azimuthal anisotropies of the S-matrix S(r) for scattering of a dipole off a large nucleus. They arise due to fluctuations of the configuration of color charges ρ a (x), described here within the Gaussian McLerran-Venugopalan model [10]; an alternative picture in terms of classical fluctuations of the energy-momentum tensor of a holographic shock wave has been discussed in Ref. [21].
For a projectile in the fundamental representation of color SU(3) these fluctuations generate both C-even as well as C-odd target field configurations which correspond to cos(nφ) moments. For small dipoles, r < ∼ 1/Q s we find that A 2 and A 4 are approximately constant and that the amplitude of the elliptic harmonic is much larger than that of the quadrangular harmonic, A 2 A 4 . Odd harmonics appear at higher order in r [7,19] and so their amplitudes decrease with decreasing r. The fluctuations of both A 1 and A 2 are comparable to their mean values, indicating that some configurations exhibit much larger anisotropies than others.
For large dipoles, r > ∼ 1/Q s , we find that all amplitudes A 1 (r), · · · , A 4 (r) asymptotically approach a universal function if the S-matrix is evaluated at fixed impact parameter. This points at angular scale invariant fluctuations of the direction of E over large distances. Accordingly, if the S-matrix is averaged over an area πr 2 the resulting cos(nφ) amplitudes are strongly suppressed.
Our calculations confirm that individual small-x target field configurations do exhibit angular dependence which would play an important role in understanding azimuthal v n harmonics in pp and pA collisions [6][7][8]. In particular, the amplitude of elliptic anisotropies A 2 ∼ 10 − 15% is on the order of the v 2 harmonic observed in p+Pb collisions at the LHC. In other words, our result supports the conjecture that the h ⊥g 1 (x, k 2 ) gluon distribution of a nucleus at small x is significant [15,16].
We have here considered the classical fields of a large nucleus at moderately small x within the McLerran-Venugopalan model [10]. In the future it will be important to incorporate QCD evolution effects. Mean-field evolution of the dipole has been shown to wash out initial elliptic anisotropies after several units of rapidity [6]. The effects of fluctuations in small-x evolution on the anisotropies deserves further study.