Subleading corrections in nonlinear small evolution
Yuri V. Kovchegov, Janne Kuokkanen,
Kari Rummukainen, and Heribert Weigert
Department of Physics, The Ohio State University, Columbus,
OH 43210, USA
Department of Physical Sciences, University of Oulu, P.O. Box 3000,
FI90014 Oulu, Finland
1 Introduction
Little is known about the features of small evolution in the Color Glass Condensate (CGC) picture [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] beyond the Balitsky–Kovchegov (BK) truncation [23, 24, 25, 21, 22] of the Balitsky hierarchy of evolution equations [23, 24, 25]. Besides the theoretical work deriving the JalilianMarian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) equations that summarize the Balitsky hierarchies in a compact form, only a single numerical study of generic properties of the full evolution equations is available, carried out by Rummukainen and Weigert [29]. All other studies employ some additional approximation, typically in form of the BK truncation or even more schematic approximation. The BK truncation, as the Mueller dipole model [3, 4, 5] it is based on, explicitly neglects corrections to the full evolution of QCD observables at high energy. Nevertheless, both JIMWLK evolution and its BK truncation correctly reproduce the dependence of the linear Balitsky–Fadin–Kuraev–Lipatov (BFKL) [30, 31] evolution equation in their respective low density limits. This implies that in the linear, low density (BFKL) domain subleading corrections are manifestly absent from JIMWLK evolution. The influence of corrections on the nonlinear part of the full, untruncated evolution equations is much harder to estimate.
The only study of the full leading JIMWLK equation available [29] has established, albeit only summarily, that the corrections appear to be much smaller than the naively expected for the gluondominated evolution. Instead of expected corrections, the JIMWLK solution for the scattering amplitude of a quark dipole on a target nucleus found in [29] differs from the solution of the BK equation for the same quantity by only . This has established the BK equation as a reasonable tool to predict the energy dependence of CGC cross sections, at least after running coupling and some DGLAP corrections are included [32, 33, 34, 35].^{1}^{1}1Kuokkanen, Rummukainen, Weigert, in preparation. However, the question remains whether the unexpectedly small difference found in [29] is accidental, being perhaps due to either some intrinsic properties of the calculated dipole amplitude or to some features of the numerical setup used in [29]. In this paper we argue that the smallness of the corrections found in [29] is not accidental. In fact it is imposed by an interplay of group theoretical properties and saturation effects of the CGC. As a result nonlinear small evolution turns out to be an example of a system in which the corrections are much smaller than naively expected.
We should emphasize that our discussion remains strictly within the context of JIMWLK evolution and within that only explores the contextual neighborhood of the BK truncation. The JIMWLK evolution equation is valid for scattering on a large target that provides a strong gluon field, e.g. for a nucleus with a large atomic number . It does not include contributions of diagrams which are not enhanced by the strong target field or, for a large nucleus, are subleading in powers of . This excludes, right from the start any discussion of pomeron loop contributions, as they are not included in the JIMWLK framework. Indeed for small targets with a weaker gluon field like a proton, which has , pomeron loops are not parametrically suppressed anymore. While pomeron loop induced fluctuations have also recently been identified in [36] as a source of possible large factorization violations for such small targets with some parametric uncertainty, [37] had found that running coupling corrections tend to strongly numerically suppress such fluctuations, so that we feel that our exclusion of pomeron loops from the analysis should not lead to a very severe restriction for the applicability of our results.
In addition to the large gluon field in the target required for JIMWLK evolution, the BK evolution equation induces a correlator factorization assumption that is valid only in the large limit. [See Eq. (6) below.] Hence the BK factorization (6) has two types of corrections: those suppressed by the powers of and those suppressed by powers of . In this paper we are interested in the second kind of corrections only, in corrections, which are resummed to all orders in the JIMWLK equation but are excluded in the BK equation.
Even within the purview of JIMWLK evolution we restrict ourselves to a subset of phenomena: We only discuss how suppressed contributions affect dipole evolution. Quantities that have no good approximation in terms of (multi) dipole projectiles scattering on dense targets are beyond the scope of our discussion. An example not addressed here would be scattering at high energies: any realistic description of a proton projectile lies far outside the standard dipole large approximation, despite the fact that JIMWLK evolution does cover this example faithfully. Obviously, in situations like this, where even a leading order large dipole description is unavailable any discussion of the size of corrections is moot.
One should also note that JIMWLK and BK equations were both first derived at leading order , but have a whole tower of suppressed corrections, of which only the next to leading order (NLO) terms are partially available. Running coupling corrections [32, 33, 34] have been calculated and partial results for the remaining contributions (new physics channels) are available [34, 38]. These corrections will change quantitative features to some extent, but should not completely distort the qualitative structures found at leading order. Our discussion and simulations will therefore focus on the leading order situation and only comment on NLO corrections where possible.
In Sect. 2 we prepare the ground for our arguments, reminding the reader about the differences between the BK equation and the JIMWLK equation for a dipole scattering amplitude. The removal of subleading corrections in the BK equation is operationally achieved by factorizing the expectation value of the product of a pair of dipole operators into a product of their expectation values. For this reason we refer to the BK equation as a factorized truncation of the JIMWLK equation. The difference of the unfactorized and the factorized expectation values measures the size of the factorization violations. The factorization violation is defined in Sect. 2 [see (8)], where we also present its main features. At one loop accuracy, a vanishing would imply a complete decoupling of all corrections from dipole correlators and is thus the crucial quantity to explore.^{2}^{2}2At NLO, running coupling corrections primarily modify the evolution kernel and thus mainly modify how strongly a nonvanishing affects evolution speed [see also Sect. 5]. Other NLO corrections generically introduce new suppressed contributions but are accompanied by an additional power of .
In Sect. 3 we will clarify the reason for the smallness of the factorization violations observed in [29], for “typical” factorization violations . A more in depth discussion of this issue than that offered in [29] must first note that the correlator measuring the factorization violation itself has in fact contributions that do reach all the way up to their natural size of in certain regions of configuration space. (Note that depends on three transverse coordinates: the positions of the original quark and antiquark, and of the emitted gluon. Varying those coordinates gives different values of .) However, as observed in [29], the typical contributions to in the majority of configuration space are in fact tiny compared to . In Sect. 3 we will systematically map out configuration space to identify all regions with factorization violations. We will argue on general grounds that the factorization violation is indeed much smaller than in the majority of its configuration space, in agreement with the result of numerical simulations presented in [29]. We will also demonstrate analytically that the evolution kernel wipes out all contributions from the only region where the factorization violation is of the naively expected order . We thus will complete the proof of the statement that corrections to BK evolution, which are consistently included into JIMWLK evolution, are indeed much smaller than . This constitutes our first main result.
The basis of our mapping out of configuration space in a systematic way is the insight that the origin of the factorization violations is to be found in a set of group theoretical identities that apply to coincidence points of (schannel) point functions involved in the Balitsky hierarchies (i.e., the limits in which any pair of the transverse coordinates overlaps). The identities are shown in (12). They are, by construction, respected in JIMWLK evolution, but automatically broken at the level by the correlator factorization assumption underlying the BK truncation.
In Sect. 4 we note that it is possible to extend the BK equations in a minimal manner that reinstates these group theoretical constraints for all eikonal correlators in high energy scattering. The inspiration comes from calculating all involved Wilson line correlators in the quasiclassical approximation known as the McLerranVenugopalan (MV) model [6, 7, 8]. One can sum up all GlauberMueller (GM) multiple rescatterings [39] to calculate various 2 and 3point functions (see e.g. [12, 40, 41, 42, 43]). Using the resulting correlation functions one can construct the factorization violation and study its properties. This allows us to revisit our earlier general observation on the structure of in configuration space and amend it with explicit expressions for the correlators, albeit within a model. However, as was noted in [43] and as we will explain in Sect. 4.1, one can also insert the 2 and 3point correlators obtained in the GM/MV approximation into the JIMWLK evolution equation for the 2point correlator (the lowest order equation in the corresponding Balitsky hierarchy). One can then suggest treating the resulting equation as an evolution equation in its own right [43], though no parametric justification/proof of this statement exists. This equation (see (22) below) is thus only a guess for the evolution equation beyond the leading BK equation. The result will be referred to as a Gaussian truncation (GT) of the Balitsky hierarchies or equivalently the JIMWLK equation. This Gaussian truncation had been introduced originally in [43] as an “exponential parametrization” for dipoles and a certain set of other correlators, with an evolution equation derived explicitly for the dipole operator. On this level it was also explicitly used in [27] to unify a diversity of ‘‘McLerranVenugopalan models.’’^{3}^{3}3This is different in content as well as in spirit from the Gaussian approximation discussed in [44].
The relationship of the Gaussian truncation to the BK equation turns out to be unexpectedly subtle: On the one hand it extends the BK truncation in the sense that it includes a set of subleading corrections, those “minimally” required to reinstate the coincidence limits violated in the BK factorization. Consistently, the Gaussian truncation reduces to BK in the large limit. On the other hand, Eq. (22), the evolution equation in the Gaussian truncation turns out to be equivalent to the BK evolution equation with respect to dynamical content. The only changes occur in the way this content is mapped onto the expressions for correlators.
In Sect. 4.2 we compare the factorization violation given by GT and by JIMWLK, and find them similar qualitatively, but still quite different quantitatively. Since we view GT as a truncation of JIMWLK evolution and hence the Balitsky hierarchies, we will also clarify where GT breaks consistency with JIMWLK: GT remains only an approximation to the full JIMWLK evolution.
Our analytical arguments are complemented in Sect. 5 by a new numerical study of the JIMWLK evolution equation that goes beyond that of [29] with simulations on much larger lattices in the transverse space, extending the range covered earlier with simulations on lattices. We emphasize that the simulations presented here are done for fixed coupling only: at present the numerical simulation of the exact JIMWLK kernel with the running coupling corrections found in [32, 33, 34] would render the numerical cost prohibitive. To efficiently include them would require us to find an alternative representation that allows a factorized form of the JIMWLK Hamiltonian akin to that used at leading order. This remains beyond the scope of this paper. Nevertheless, the additional numerical effort allows us to reduce extrapolation errors considerably (they arise mostly from the infinite volume limit as it turns out) and establish reliably that JIMWLK evolution is in fact slightly slower than factorized BK evolution: Subleading corrections indeed slow down evolution, just as was observed earlier with running coupling corrections. Evolution speed turns out to be particularly sensitive to factorization violations, which is in keeping with the integral expressions of Eq. (31) below. At one loop order, we observe numerically a 35% slowdown induced by factorization violations where our simulations approach the scaling region. We argue that running coupling corrections should suppress the UV part of phase space leading to a strong reduction of this difference of evolution speeds between JIMWLK and BK. We cannot estimate the influence of other NLO corrections which may well have their own offsetting effects, but they should not completely distort the leading order picture. We conclude that, while the net effect remains small, corrections pull our predictions towards evolution speeds compatible with experiment, not in the opposite direction. This qualitative slowdown effect is our second main conclusion.
In Sect. 6 we concentrate on the physical origin of corrections to the BK evolution equation. As noted above, the BK truncation reproduces the dependence of the linear BFKL evolution equation, corresponding to a tworeggeon state in the channel. However, among other corrections, the BK truncation neglects contributions of multiple reggeon exchanges [45, 46, 47, 48]. Some of those omitted higher reggeon exchanges, like the odderon contribution corresponding to a odd threereggeon exchange [49, 50, 51, 52, 53], have been included in the BKtruncated CGC formalism by a minimal modification of the truncated evolution equations [54, 55, 56]. Higherorder reggeon exchanges usually require substantial modification of Mueller’s dipole model to a more generic channel picture as one generically is required to include suppressed multipole correlators on top of simple dipoles: see [57] for an analogue of the Bartels–Jaroszewicz–Kwiecinski–Praszalowicz (BJKP) evolution equation [45, 46, 47, 48] in the channel formalism. Generically not much has been done to identify the contributions of higher reggeon exchange contributions [58] to nonlinear JIMWLK evolution in any systematic way.
Nevertheless, the fact that the odderon [54, 55, 56] and 4reggeon [57] exchanges are included in the channel evolution picture allows us to conjecture that all multi–reggeon exchanges are included in the JIMWLK evolution equation. The JIMWLK equation also probably includes some multi–reggeon vertices containing more legs on the target side of the evolution than on the projectile side. If this conjecture is true, one concludes that the difference between the dipole amplitude given by BK and by JIMWLK is at least partially due to an aggregate of multiple–reggeon effects. The smallness of this difference then, in turn, would indicate the smallness of multiplereggeon exchange effects.
The link of multi–reggeon exchanges with subleading corrections gives a natural explanation for the slowdown of JIMWLK evolution compared to BK observed in Sect. 5. Generically one would argue that nonlinear effects will work to temper any influence of multi–reggeon contributions, which would complement the power suppression of contributions via the kernel observed earlier in our line of argument. If true, this is testable numerically, but it is not a priori clear how to test this. Identifying the Gaussian truncation with iterated two reggeon exchange gives a handle on this as well: we may filter out the multi–reggeon exchanges by comparing the Gaussian truncation with full JIMWLK evolution. It turns out that the Gaussian truncation has a distinctive feature that is naturally violated by multi–reggeon exchanges: the Gaussian truncation would predict strict Casimir scaling of dipole correlators in different representations. (Casimir scaling is defined in (23).) In Sect. 6 we illustrate this statement by extending GT to include the simplest multi–reggeon contribution in the form of an odderon exchange: we then show that it indeed violates the Casimir scaling. Therefore we argue that the size of Casimir scaling violations can quantify the net contribution of all multi–reggeon exchanges. We thus can numerically explore the effect of multi–reggeon exchanges by measuring the violations of Casimir scaling of the dipole correlators. Casimir scaling violation in the numerical solution of JIMWLK that we performed is studied in Sect. 6. It turns out that the Casimir scaling violations (which summarize the collective effect of all multi–reggeon exchanges included in JIMWLK evolution) are generically small and do not grow with energy (see e.g. Fig. 10). This is our third main result.
We review our results and methods in Sect. 7.
2 Dipole evolution in JIMWLK and BK frameworks
JIMWLK evolution is equivalent to sets of coupled infinite hierarchies of evolution equations, the simplest of which is based on the equation for the dipole correlators for the scattering on a target at high energies in which the scattering of the and is expressed via lightlike Wilson lines in the fundamental representation or respectively (at fixed transverse positions ),
(1) 
This operator is gauge invariant in the sense that the contributions that close the trace at are unity to leading order in .^{4}^{4}4This is true strictly speaking only if one does not force the factors shown to be one by gauge choice. In this case one would need to display the contributions that connect the trace at to a closed Wilson loop. They would carry the full contribution, but would remain independent of the paths used to connect to . Averaging the operator in Eq. (1) over all states in the target wave function yields the dependent Smatrix for the scattering of a dipole on that specific target. The evolution equation for this average, , involves a gluon Wilson line operator in the adjoint representation on its righthand side. At fixed coupling can be written either as [24, 23, 18]
(2) 
or, using (1) and the Fierz identity
(3) 
as
(4) 
The integral kernel in both (2) and (4) is given by [3, 21]
(5) 
Eqs. (2) and (4) are completely equivalent versions of the first equation in the Balitsky hierarchy of the quark dipole operator (1). Eqs. (2) and (4) obviously do not represent closed equations since the evolution of depends on an operator with an additional gluon operator insertion. The evolution equation of that new operator, , in turn will involve yet one more insertion of a gluon operator , iteratively creating an infinite coupled hierarchy of evolution equations, the Balitsky hierarchy of the quark dipole operator (1) [24, 23]. JIMWLK evolution summarizes the totality of all such hierarchies, based on any (gauge invariant) combination of multipole operators but can only be solved numerically [29] at considerable numerical cost. The situation can be simplified for the price of introducing an additional approximation that truncates the hierarchy. The most widely used truncation is known as the BK approximation. It assumes the factorization
(6) 
which turns Eq. (4) into a closed equation in terms of only and thus decouples the rest of the Balitsky hierarchy. The BK truncation is valid and is parametrically justified in the large limit for scattering on a large dilute nuclear target. Using (6) in (4) we obtain the BK evolution equation
(7) 
Provided that the dipole correlator shapes are not too different (this notion with be refined in Sec. 5) in JIMWLK (without factorization (6)) and BK (with factorization, as shown in (7)) the factorization violation that creates the difference between the two
(8a)  
can be simply interpreted as the difference of the correlators on the righthand side of Eq. (2) and its BK counterpart (7), i.e.  
(8b)  
(the term is the same under these conditions and cancels trivially) or as a fluctuation away from a mean field value  
(8c) 
The first interpretation directly leads us to consider factorization violations as a source for a difference in evolution speed in JIMWLK and BK, the second interpretation will lead us to the question of what kind of degrees of freedom (which are absent in BK but included in JIMWLK) would be associated with these fluctuations. We meet the latter question repeatedly in all remaining sections, here we will first look at the individual terms in Eq. (8b) to get a generic idea of the structure of configuration space and how it affects evolution and then give a first glimpse at how JIMWLK evolution via (4) might differ from BK evolution (7).
In both cases in an otherwise translationally invariant system with a given parent dipole the integrands (correlators and kernel separately – that is why we will leave the latter aside) have a twofold mirror symmetry in the plane: one with respect to the axis, the other with respect to an axis perpendicular to , through the midpoint . The latter only holds if , i.e., it is real (and thus symmetric in ) as is the case if we study its contribution to the total DIS cross section at high energy [23, 21]. In this context it is useful to introduce a coordinate with respect to as the origin
(9) 
There are strong zeroes in the correlators on the righthand side of the evolution equations as well as in when or . They are needed to cancel the kernel singularities at these points and have their origin in real virtual cancellations. Generically, these zeroes are not isolated but lie on lines that separate the positive from the negative contributions to the integrand of the evolution equations. Picking out the positive sign regions in the integrand, the mirror symmetries allow two situations: one in which there are two separate such regions adjacent to the and respectively, and another where the regions are joined together [generically in situations with dipole correlators not too dissimilar from the GolecBiernat–Wüsthoff (GBW) case [59] (also known as GlauberMueller multiple rescatterings) which serves as our initial conditions; the initial conditions are in fact radially symmetric]. The generic patterns are shown in Fig. 1, which presents contour plots of the righthand side of the BK equation (7) (divided by ) obtained by performing a numerical solution of that equation. The horizontal and vertical axis on each panel show and , which are the two components of the twodimensional vector . The coordinates are plotted in the units of the initial correlation length of the system (formally defined as the distance at which the dipole correlator falls to , i.e. via ). can be thought of as the inverse of the saturation scale : . is the rapidity of the initial conditions for (7). The dots in Fig. 1 denote the positions and of the quark and the antiquark, with taken in Fig. 1 to be equal to . They lie on the contour lines that separate positive from negative regions. The left panel of Fig. 1 corresponds to the initial conditions for the BK evolution (), while the right panel corresponds to a higher rapidity where , i.e., after running the evolution for some time.
For fixed parent dipole size the most extreme situations arise when we vary along the axis and the axis perpendicular to it, all other directions in the plane interpolate smoothly. The two axes, along with one intermediate axis, are also shown in both panels of Fig. 1.
Evolution speed, in either JIMWLK or BK, is then a consequence of a numerically delicate balance of the negative and positive regions of the quantity plotted in Fig. 1. Since we are talking of evolution for in which generically is driven to smaller values at fixed dipole sizes as rapidity increases, the negative regions in Fig. 1 push evolution forward (these contributions are generically those at large ), while the positive regions in Fig. 1 (generically near and ) slow it down. At fixed coupling, any change of evolution speed can be mapped onto a change of relative weight of these two contributions. Starting from a nonscaling initial condition like the GBW model, evolution typically speeds up until scaling is reached and evolution speed is maximal. (Scaling here is defined as the situation in which all rapidity dependence is carried by the saturation scale so that observables like become functions of scaling ratios like only [60].) This is mirrored perfectly in a shrinking of the positive regions from the radially symmetric situation of the GBW initial condition (Fig. 1, left) to a situation in which there are two separate positive regions near the and positions (Fig. 1, right).
The plane symmetries of the dipole evolution equations translate directly into the factorization violations from Eqs. (8), and also the zeroes at the and positions carry over. In [29] two of us (Rummukainen and Weigert) had observed numerically that all factorization violations tested were positive (i.e., qualitatively acted to slow down evolution compared to BK), and unexpectedly small, at least in the regions that contribute to evolution: instead of at one found contributions roughly another magnitude smaller.
Fig. 2 reillustrates the observation of[29], namely that the factorization violation is about an order of magnitude smaller than the naively expected in regions relevant for evolution. Fig. 2 plots (henceforth referred to as simply without the arguments) from the numerical solution of JIMWLK evolution equation which we will describe below. is plotted in Fig. 2 as a function of the parent dipole size
(10) 
for fixed with the angle between and being (left panel), (middle panel) and (right panel). These directions were also shown in Fig. 1. In each panel of Fig. 2 the factorization violation is plotted for three different rapidities: , , and , with the exact numerical value of being irrelevant here (along with the value of the fixed coupling constant ), as our goal in this Section is only to demonstrate the size of the typical factorization violations.
While the correlator geometries in Fig. 2 differ slightly from those shown in [29], the magnitudes are comparable. We see again that instead of naively expected one gets , which is an order of magnitude smaller. As rapidity increases beyond the values shown in Fig. 2, the factorization violation does not grow significantly beyond the values achieved in the figure. In [29] no attempt was made to clarify in which regions of configuration space the factorization violation is small, and no generic discussion of relative importance of configuration space regions was given. To fully understand the statement of the smallness of corrections one must expand on the discussion given there and first gain a better understanding of where to expect sizable contributions, since mapping out all of the configuration space in , and is otherwise not feasible. This will also provide the underlying reason for the observed smallness.
3 Origin and smallness of the factorization violation: an interplay of saturation and coincidence limits
Smallness of the specific factorization violations (8) are only one facet of a more generic question: what kind of deviations from full JIMWLK evolution are caused by the factorization assumption (6) with its associated truncation of the Balitsky hierarchy of the quark dipole operator?
Full JIMWLK evolution does not only couple in a full hierarchy of evolution equations for the quark dipole operator, it has an even wider scope: It consistently incorporates hierarchies based on any point correlator. Examples for such distinct hierarchies are obtained by considering the infinite set of dipole correlators labeled by all finite dimensional unitary representations . Each of them has its own distinct evolution equation, that can be summarily written as
(11) 
Here refers to the group element in the representation , with analogous notations for the trace, generators and conjugate representation. denotes the second Casimir of the representation of the dipole, i.e., for the correlator of the BK case it equals or for a dipole it would be . The gluon produced in the evolution step is denoted by and is of course always in the adjoint representation.
The hierarchies based on the dipole equations (11) are by no means all independent (group constraints and coincidence limits may reveal that the dependence of the same multi– correlator does appear in several hierarchies), nor do they exhaust all the information contained in JIMWLK evolution (for instance operators with nonvanishing triality are absent from the family of dipole hierarchies). What is important here is that JIMWLK evolution treats this multitude of correlator equations consistently – as long as no truncation assumptions are made.
The BK approximation greatly simplifies this intricately interlinked set of hierarchies and may not capture all of its features in the process: since there is no simple generalization of the Fierz identity (3) for in an arbitrary representation , it may become impossible to consistently generalize the BK approximation to even this class of equations, despite the fact that one can write expressions for the BK (large ) limit of generic dipole operators. (For the cases of being fundamental or adjoint representations the BK approximation is indeed possible and is done routinely.) To consistently include all equations (11) is to go at least one step beyond the BK approximation and below, in Sect. 4, we shall see that this can indeed be achieved quite elegantly.
The key feature satisfied by JIMWLK evolution that is violated by BK factorization beyond the leading limit is a set of group identities for the three point correlators on the righthand side of the evolution equations (11). In what follows we will generically use the term coincidence limits to refer to the limits were any pair of points ( and , and , or and ) or all three of them (, , and ) coincide with each other. At the coincidence limits the correlator should inherit relationships that JIMWLK evolution respects on the operator level (see Appendix A for their derivation):
(12a)  
(12b)  
(12c) 
where stands for the dimension of the representation ( for the fundamental representation, for adjoint, etc.) and denotes the trace in the adjoint representation. While the third statement is merely a normalization statement, the first two are remarkable: we read off that in the limit of small parent dipole the three point operator on the lefthand side reduces to a gluon dipole, no matter what representation refers to, while in the limit or it reduces to an dipole. The latter, (12b), is crucial to ensure the real virtual cancellations in (11).
For correlators, the implications of (12) go far beyond the isolated points featuring in the limits shown. Since the correlation (saturation) length is the only dimensionful parameter, the only scale in the problem, (12) determines the generic behavior of in all of configuration space.
Configuration space is first divided into two classes in which is either smaller or larger than . For each of these classes one has to distinguish two cases according to whether the distance between the gluon and the nearest quark is larger or smaller than . The configurations are shown in Fig. 3 and exhaust all physically different situations (labels “a” through “d” in the figure are in correspondence to the equation labels in (13) below).
One infers from (12) that in regions “a” and “b” the falloff is dipolelike
(13a)  
(13b)  
i.e., vanishes like a gluon or dipole where or (in the latter region the gluon is near either the or , implying either respectively. This only occurs when the angle between and is near ). It also vanishes trivially in region “c”, where , and the gluon is far from both and (). In this region all three interparticle distances are large and force exponential suppression, although (12) gives no additional information about the falloff, leaving us with  
(13c)  
This leaves only one region, labeled “d” in Fig. 3, in which the contributions are not suppressed. In this remaining region, all scales are small simultaneously, as one would naively expect in a system with a finite correlation length . In region “d” we have  
(13d) 
Fig. 4 illustrates this theoretical discussion with contour plots of the three point function as obtained from actual JIMWLK simulations. The plots show dependence on separation and the distance of the gluon location with respect to the midpoint , with perpendicular to and parallel to , i.e., along two of the lines indicated in Fig. 1. One may notice that the contributions on the axes, and have no angular dependence: the first corresponds to zero size parent dipoles in which case does not single out any direction to refer to, the second keeps firmly in the middle of the pair while varying its size so that again the angle does not play a role.
Eqs. (12) and (13) represent but one example of a much larger set of group constraints for more complicated correlators that are all inherently true in the full JIMWLK setting, but broken by the BK factorization. The BK truncation is geared towards quark dipoles (where is the fundamental representation), where it approximates the Fierz identity (3) by dropping the term
(14) 
This distorts the coincidence limit of the quark dipole version of Eqs. (12) into their large approximations
(15a)  
(15b)  
(15c) 
i.e., it approximates the gluon dipole operator on the righthand side of Eq. (12a) by the square of the quark dipole operator, and replaces the constants in the remaining equations by their large counterparts.
For correlators, the implications of (15) mirror the conclusions drawn in Eq. (13) up to corrections of order , hence one naively expects the factorization violations to be of precisely that order, unless there is a stronger cancellation at work in the coefficient of that term. Eq. (13) contains all that is needed to assess this issue if one uses (3) to recast in terms of two and three point correlators only (the first two terms represent the unfactored correlator):
(16) 
The four distinct regions of Fig. 3 and Eq. (13) can then be addressed in turn (all individual correlators are real and positive):

Region “a”, : Both the first and the last term inside the brackets of (16) are exponentially small, but the second term approaches . In the extreme case , one finds . Contrary to , contains a independent additive term that survives this limit. If region “a” were to contribute to evolution at all, this would destroy infrared safety of JIMWLK evolution (see below).

Region “b”, (gluon near or ): Since there are always two large distances involved, all three of the terms in (16) are exponentially suppressed and the contribution is naturally much smaller than .

Region “c”, : With all three interparticle distances large, all terms are exponentially suppressed individually, rendering their sum much smaller than .

Region “d”, : The terms inside the brackets are order , , and respectively. Moreover, in the strict coincidence limit , they cancel exactly! This guarantees a very strong (albeit not exponential) reduction of the coefficient in front of . The cancellation is slightly less pronounced farther from exact coincidence, for scales of order , before large distance damping at the boundary to the previous regions sets in.
One concludes that is strictly bounded from above by , but there is only one region left in which this bound is actually reached – region “a”. In all other regions strong cancellations reduce the contributions to values significantly below this bound.
So far we have used general arguments based on coincidence limits (12) and on the effects of saturation on the dipole scattering amplitude (twoWilson line correlators) to argue that is in fact much smaller than for much of its phase space. To understand the impact of on the evolution let us rewrite (4) using (8a)
(17) 
Eq. (17) shows how enters the full, untruncated evolution equations.
As we saw above, somewhat surprisingly, region “a” where the maximal possible factorization violation occurs is characterized by small parent dipole size but with the gluon produced far away, with . This region, however, has no impact on evolution at all: it is completely power–suppressed by the evolution kernel (5) in (17) which goes to zero as the sixth power of distances involved:
(18) 
Were it not for this kernel suppression, JIMWLK evolution (represented via the Balitsky hierarchy) would receive large distance contributions from this region – infrared safety would be lost. Notably, region “a” is the only large distance region that requires suppression from the kernel. The remaining large distance regions (“b” and “c”) are exponentially suppressed on the correlator level and automatically decouple from evolution. That leaves the last region (region “d”) with its strong cancellation of contributions as dictated by the properties of the coincidence limits: it remains as the sole channel through which the nonfactorized contributions affect the energy dependence of the dipole. It is this region which was quoted in [29] to contribute the “typical” factorization violations without connecting this to the coincidence limits (12).
While the argument given in this section does not give a parametric estimate for the size of the factorization violations, it does explain why the contributions are naturally much smaller than . We see that factorization violation is bounded by from above. However this value is reached only in a small subset of configuration space (in region “a”), which is suppressed by the evolution kernel. The relative suppression of the integral of the factorization violation over all ’s in (17) compared to the first term on the righthand side of the equation is therefore much stronger than the one would naively expect. Note that the generic arguments given here also do not allow to determine the sign of the contribution and thus do not allow to infer if one should expect JIMWLK evolution to be slower or faster than the factorized BK truncation.
We might now just push ahead and map out configuration space of the JIMWLK 3point correlators using numerical results from our simulations, to systematically supplement the numerical results of[29] and Fig. 2 with contributions from the regions not shown there (numerical results will be shown in Figs. 5 and 6). Let us instead first give a simple generalization of BK factorization that treats the set of equations (11) consistently and respects the coincidence limits (12). This generalization will restore at least part of the true factorization violation and respect the configuration space pattern deduced from (13) and (16). This will likely improve agreement with JIMWLK evolution and give some insight into the question in which direction evolution speed is changed by the factorization violations.
4 Gaussian truncation of JIMWLK
4.1 A step beyond BK
Our argument for the suppression of corrections in the previous section were based on saturation effects and coincidence limits. We observed that the BK equation, while incorporating the saturation effects, violates the coincidence limits at the subleading level.
Approaches that both incorporate saturation physics and respect the coincidence limits of the general argument given in Sec. 3 are well established in the literature. They take the form of variants of the McLerranVenugopalan model [7, 8, 9] and the closely related GlauberMueller approximation to high energy scattering [2, 3, 4], which can be rigorously established by summing QCD diagrams without taking into account small evolution. All these descriptions fall into a class of approximations of the JIMWLK average over Wilson lines , that is characterized by a longitudinally local Gaussian averaging procedure that can be cast as
(19) 
We refer the reader to [27] for a discussion of how various well known models can be recovered from the generic form shown in Eq. (19) by choosing specific expressions for . In the quasiclassical limit encodes a two gluon exchange with the target in the channel. That this same generic approach automatically satisfies the coincidence limits has also been demonstrated in [27] for the case of being the fundamental representation.
Stepping beyond the quasiclassical limit in [43], Kovner and Wiedemann have suggested an all evolution equation that merges BK principles with the Gaussian treatment of correlators incorporated in Eq. (19) by what amounts to applying the averaging prescription to quark and gluon dipoles.
In fact, Eq. (19) allows us to extend the treatment of [43] beyond the specific case of quark (fundamental) and gluon dipole evolution. This results in a selfconsistent treatment of the evolution of all generic dipole operators in which one replaces the and by colored objects in arbitrary representations and . Doing so, one finds completely generic expressions for the previously discussed correlators (see Appendix B for calculational details and also [12, 40, 41, 42, 43] for similar calculations):
(20a)  
(20b) 
For convenience we have introduced
(21) 
to denote the combination in which the tchannel gluons enter these expressions. Note that as required by consistency in (20a). Quick inspection reveals that (20b) indeed complies with (12) as advertised. In fact, this property is not specific to this particular set of correlators. Any correlator calculated using (19) (or any generalization thereof) will automatically satisfy all necessary group constraints by construction.
This procedure then is a candidate to generalize the BK factorization in which one simply trades an evolution equation for for an evolution equation for . The procedure at least qualitatively repairs the flaw that is the source of factorization violation in the BK equation. We will find below that it provides quite good qualitative insights on factorization violation but is not sufficient to obtain quantitatively correct results.
The equation for has already been derived in [27] (and in a somewhat different form earlier in [43]^{5}^{5}5To connect with the form given in [43], Eq. (5.3), one should reconstruct the evolution equation for the dipole operator by multiplying (22) with and note that our corresponds to in [43]. ), starting from the dipole evolution equation (2). We reiterate that this average treats all dipole equations consistently: Inserting (20) into the generic dipole evolution equation (11) yields one and the same equation for ,
(22) 
irrespective of the representation . Note that (22) is similar to (though not exactly the same as) the AyalaGay DucatiLevin (AGL) evolution equation [61, 62, 63].
Contrary to BK evolution which systematically discards all suppressed terms contained in JIMWLK, the Gaussian truncation has no expansion parameter justifying the approximation. Nevertheless we expect it to lead to a good approximation of JIMWLK evolution since

the equation incorporates a subset of these corrections that is sufficient to restore the coincidence limits;

the low density limit of Eq. (22) (viewed as its small limit) reduces to the BFKL equation;

it has a large limit that is compatible with the BK equation as will be seen below.
Despite (22) being surprisingly more generic than the BK equation, in the sense that the procedure addresses arbitrary dipoles irrespective of representation, one remains with an approximation to the true JIMWLK evolution, and does not obtain an exact solution of the JIMWLK equation: the evolution equation for resulting from JIMWLK would impose additional conflicting conditions on , and can only be satisfied by introducing degrees of freedom beyond . The Gaussian truncation still deviates from JIMWLK evolution at the level of evolution equations for three point functions.
It is worth noting two particular features that the average (19) entails. First, Eq. (20a) implies Casimir scaling for dipole correlators:^{6}^{6}6Approximate Casimir scaling has been observed for Wilson line correlators in the context of heavy quark potentials in [64, 65]. Given two representations the normalized dipole correlators are related by a simple power law
(23) 
Second, somewhat surprisingly, Eq. (22) can be mapped back onto the BK equation. This implies that the dynamical content of the Gaussian truncation is the same as that of the BK equation. The main improvement is how this information is mapped onto the correlators. As we shall see, this leads to a slightly better approximation of JIMWLK results. On the practical side, this turns into a time saver: one can recycle the numerical tools written to solve the BK equation, provided one relates correlators and initial conditions accordingly.
One way to see that the dynamical content is the same is based on a simple reparametrization of the BK matrix in as close an analogy to (20a) as possible. We write
(24) 
should be thought of as simply being defined by the solutions of the BK equation through (24). The dependent constant is a convention chosen in keeping with the lore of the Mueller dipole model and the BK equation. Next one inserts this into the BK equation for ,
(25) 
and obtains
(26) 
which is identical to (22), thus establishing our claim of identical dynamical content, despite the different treatment of correlators.
The procedure to obtain solutions for the Gaussian truncation that would serve to determine, say, the –proton cross section for DIS at HERA would then be to choose an initial condition for , read off via (20a), insert it in place of in (24) to determine the initial condition on to be used in the BK equation (25). After solving Eq. (25) to obtain at all rapidities one reverses the procedure to recover via (24) and (20a) at each rapidity . This then determines all correlators of the Gaussian truncation through (19) and the special cases shown in Eq. (20).
As an immediate consequence we conclude that the asymptotic scaling shape for the dipole correlators in the Gaussian truncation can be obtained from those of the BK equation using a simple power law relationship (23). As an immediate corollary also evolution speeds of the Gaussian truncation and BK evolution must coincide in that region. This link does not extend to the preasymptotic regime and we will see that GT tends to be slower that BK in that range in Sec. 5.
Two further points are worth noting: (ı) one may recover BK factorization (wherever it can be meaningfully applied) as the leading contribution of the new procedure. (This can be verified for the contributions entering the BK equation by taking the large contributions in the exponents in (20).) (ıı) In the small density limit, i.e., the limit of weak target fields where is small its evolution equation, Eq. (22), consistently reduces to the BFKL equation for , in keeping with the underlying interpretation.
In summary, one might think of the Gaussian truncation as a minimal extension of the BK factorization to consistently incorporate group constraints with an associated set of “minima