On perturbative limits of quadrupole evolution in QCD at high energy
aa r X i v : . [ h e p - ph ] N ov On perturbative limits of quadrupole evolution in QCD at high energy
Jamal Jalilian-Marian , Department of Natural Sciences, Baruch College, CUNY17 Lexington Avenue, New York, NY 10010, USA The Graduate School and University Center, City University of New York,365 Fifth Avenue, New York, NY 10016, USA
We consider the perturbative (weak field) limit of the small x QCD evolution equation forquadrupole, the normalized trace of four Wilson lines in the fundamental representation, whichappears in di-hadron angular correlation in high energy collisions. We linearize the quadrupoleevolution equation and then expand the Wilson lines in powers of g A µ where A µ is the gauge field.The quadratic terms in the expansion ( ∼ g A ) satisfy the BFKL equation as has been recentlyshown. We then consider the quartic terms ( ∼ g A ) in the expansion and show that the linearizedquadrupole evolution equation, written in terms of color charge density ρ , reduces to the well-knownBJKP equation for the imaginary part of four-reggeized gluon exchange amplitude. We commenton the possibility that the BJKP equation for the evolution of a n -reggeized gluon state can beobtained from the JIMWLK evolution equation for the normalized trace of n fundamental Wilsonlines when non-linear (recombination) terms are neglected. I. INTRODUCTION
The recent experimental observation of disappearance of the away side peak in di-hadron angular correlation inthe forward rapidity region in deuteron-gold collisions at RHIC [1] has generated a lot of interest in multi-partoncorrelations at high energy (small x ). Unlike structure functions in DIS and single inclusive particle production inhadronic collisions which are sensitive to dipoles (normalized trace of two Wilson lines), di-hadron correlations probecorrelators of higher number of Wilson lines [2, 3]. Therefore one has the opportunity, for the first time, to investigatethese higher correlators experimentally through studies of angular and rapidity correlations in di-hadron productioncross section in high energy hadronic collisions. Such studies can teach us much about the intrinsic correlations inthe hadronic or nuclear wave functions which are not accessible in single inclusive particle production or in studies ofstructure function in DIS.Higher correlators of Wilson lines appear in two-hadron production cross section in any dilute-dense collision at highenergy where analytic calculations are possible. Classic examples of such asymmetric collisions are proton-nucleuscollisions (see [4] for a review) in the fragmentation region of the proton, and in DIS close to the virtual photonremnants . Two-gluon production cross section in DIS has been considered in [2] while two-parton production crosssection in proton-nucleus collisions has been investigated in [2, 5–8]. In all cases, the cross section involves correlatorsof higher (more than two) number of Wilson lines, the most important being the quadrupole operator. Evolutionequations for these higher point correlators have been derived [2, 9, 10] and approximate analytic expressions for themhave been developed using a Gaussian model [11] and approximate analytic solutions have been proposed [12]. Veryrecently, powerful lattice gauge theory techniques have been applied to solve the JIMWLK evolution equation whichthen allows a systematic and detailed numerical study of the properties of these higher point correlators [13].Here we study the evolution equation for the quadrupole operator in the weak field limit. A first study of thishas already been performed in [10] where it is shown that the quadrupole evolution equation reduces to a sum ofBFKL equations for the dipole operator in the limit where the dipole is expanded in powers of the gluon field andquadratic terms in gluon field are kept. Here, we go beyond the quadratic expansion and show that the quartic termsin the expansion of the linearized quadrupole evolution equation satisfy an equation which is identical to the BJKPequation [14, 15] for the imaginary part of the four-reggeized gluon exchange amplitude. This should be very usefulsince there is an extensive literature on the properties of the BJKP equation which may give us further insight on theproperties of the JIMWLK equation in the limit where one may ignore non-linear terms. Particle production in the very forward rapidity region in proton-proton collisions at very high energy falls into this category also.
II. QUADRUPOLE EVOLUTION EQUATION
We start by defining the quadrupole operator Q as Q ( r, ¯ r, ¯ s, s ) ≡ N c tr V r V † ¯ r V ¯ s V † s (1)where V r ≡ V ( r t ) is a Wilson line in the fundamental representation in the covariant gauge V ( r t ) ≡ ˆ P e − ig R dx − A + (2)and A µ ( x − , r t ) = δ µ + δ ( x − ) α ( r t ). The gauge field α ( r t ) is related to the color charge density via ∂ t α a ( r t ) ∼ g ρ a ( r t ) and r, ¯ r, ¯ s, s etc. denote two-dimensional coordinates on the transverse plane. The evolution equation forthe quadrupole was derived in [2] in the large N c limit and using Feynman diagram techniques. It has been recentlyre-derived [10] using the JIMWLK equation where it was shown that there are no N c suppressed corrections. Here weoutline the derivation using the JIMWLK formalism [16] where the evolution ( y = log /x ) of any operator is givenby ddy h O i = 12 (cid:28)Z d x d y δδα bx η bdxy δδα dy O (cid:29) , (3)with η bdxy = 1 π Z d z (2 π ) ( x − z ) · ( y − z )( x − z ) ( y − z ) (cid:2) U † x U y − U † x U z − U † z U y (cid:3) bd . (4)and U is a Wilson line in the adjoint representation. The derivation of the quadrupole evolution equation is straight-forward but tedious. It involves functional differentiation of the Wilson lines and repeated use of the identity[ U ( r )] ab t b = V † ( r ) t a V ( r ). The result is ddy h Q ( r, ¯ r, ¯ s, s ) i = N c α s (2 π ) Z d z (*(cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( r − s ) ( r − z ) ( s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) (cid:21) Q ( z, ¯ r, ¯ s, s ) S ( r, z )+ (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21) Q ( r, z, ¯ s, s ) S ( z, ¯ r )+ (cid:20) (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − (¯ r − s ) ( s − z ) (¯ r − z ) (cid:21) Q ( r, ¯ r, z, s ) S (¯ s, z )+ (cid:20) ( r − s ) ( r − z ) ( s − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21) Q ( r, ¯ r, ¯ s, z ) S ( z, s ) − (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) + ( r − s ) ( r − z ) ( s − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) (cid:21) Q ( r, ¯ r, ¯ s, s ) − (cid:20) ( r − s ) ( r − z ) ( s − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21) S ( r, s ) S (¯ r, ¯ s ) − (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) (cid:21) S ( r, ¯ r ) S (¯ s, s ) +) (5)where the S matrix is defined as S ( r, ¯ r ) ≡ N c tr V r V † ¯ r (6)We will refer to the first four lines in this equation as ”real” and the last three terms as ”virtual” terms in coordinatespace. This is to distinguish them from the real and virtual terms in momentum space after we Fourier transform theequation since there is no one to one correspondence between the real and virtual terms in coordinate and momentumspaces. We have also verified that this equation is exact in the sense that there are no N c suppressed terms in theequation itself (note that models used to evaluate the color averaging denoted by < · · · > may introduce sub-leading N c terms). It also agrees with the previous results for the quadrupole evolution equation [2, 10]. The S matrixsatisfies the BK evolution equation [17] given by ddy h S ( r − s ) i = N c α s π Z d z ( r − s ) ( r − z ) ( s − z ) (cid:20) h S ( r − z ) i h S ( z − s ) i − h S ( r − s ) i (cid:21) (7)Unlike the dipole kernel in the BK equation which allows a probabilistic interpretation in coordinate space, the sameis not true in the quadrupole evolution equation due to terms with negative signs. Even though the individual kernelsin eq. (5) are just the standard dipole kernels [18], it is still perhaps useful to explain in a more intuitive way, thevarious terms that appear in eq. (5). The first four lines in eq. (5) are the ”real” corrections and come from thethird and fourth terms in eq. (4). One can rewrite any kernel in eq. (5) in a way which may look more familiar andfacilitates the comparison with the standard dipole emission kernel. For example, the kernel in the first line on theright hand side of eq. (5) can be written as as ∼ (cid:20) r − z ) − ( r − z ) · (¯ r − z )( r − z ) (¯ r − z ) − ( r − z ) · ( s − z )( r − z ) ( s − z ) + (¯ r − z ) · ( s − z )(¯ r − z ) ( s − z ) (cid:21) (8)with a similar form for all the other kernels. Here the first term corresponds to a gluon being radiated by a quark linerepresented by V ( r ). If it is absorbed by the same quark line in the amplitude, it leaves the quadrupole unchangedand will correspond to a ”virtual” correction. On the other hand if it is absorbed by the same quark line but in thecomplex conjugate amplitude (so the gluon line crosses the cut), it will multiply the quadrupole with the coordinate r replaced by z and a dipole with coordinates r, z . This will be part of the ”real” corrections. The second term abovecorresponds to the case when the quark line, represented by V ( r ), in the quark anti-quark system represented by V ( r ) and V (¯ r ) radiates a gluon with transverse coordinate z . If the radiated gluon does not cross the cut line and isabsorbed by the anti-quark line at ¯ r it becomes part of the ”virtual” corrections. On the other hand if the radiatedgluon at z crosses the cut and is then absorbed by an anti-quark line in the complex conjugate amplitude, it breaksthe original quadrupole into a quadrupole with coordinate r replaced by z and a dipole at r, z . This is part of the”real corrections. All other terms have a similar interpretation.To investigate the weak field limit of this evolution and to make our approximations more transparent, it is moreuseful to work with the T matrices, defined as T Q ≡ − Q and T ≡ − S . It is easy to see that all kernels multiplying1 (when we switch from Q, S to T Q , T ) add up to zero. Therefore, eq. (5) is re-written as ddy h T Q ( r, ¯ r, ¯ s, s ) i = N c α s (2 π ) Z d z (*(cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( r − s ) ( r − z ) ( s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) (cid:21)(cid:20) T Q ( z, ¯ r, ¯ s, s ) + T ( r, z ) − T Q ( z, ¯ r, ¯ s, s ) T ( r, z ) (cid:21) + (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21)(cid:20) T Q ( r, z, ¯ s, s ) + T ( z, ¯ r ) − T Q ( r, z, ¯ s, s ) T ( z, ¯ r ) (cid:21) + (cid:20) (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( s − ¯ r ) ( s − z ) (¯ r − z ) (cid:21)(cid:20) T Q ( r, ¯ r, z, s ) + T (¯ s, z ) − T Q ( r, ¯ r, z, s ) T (¯ s, z ) (cid:21) + (cid:20) ( r − s ) ( r − z ) ( s − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21)(cid:20) T Q ( r, ¯ r, ¯ s, z ) + T ( z, s ) − T Q ( r, ¯ r, ¯ s, z ) T ( z, s ) (cid:21) − (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) + ( r − s ) ( r − z ) ( s − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) (cid:21) T Q ( r, ¯ r, ¯ s, s ) − (cid:20) ( r − s ) ( r − z ) ( s − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21)(cid:20) T ( r, s ) + T (¯ r, ¯ s ) − T ( r, s ) T (¯ r, ¯ s ) (cid:21) − (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) (cid:21)(cid:20) T ( r, ¯ r ) + T (¯ s, s ) − T ( r, ¯ r ) T (¯ s, s ) (cid:21)+) (9) A. The weak field limits
It is useful to consider the above equation for T Q in the weak field (dilute) limit where all sizes are much smallerthan the inverse saturation scale, i.e., | a − b | << Q s for any external coordinates a, b . In this limit the non-linearterms ( T Q T and T T ) in eq. (9) may be dropped and we get ddy h T Q ( r, ¯ r, ¯ s, s ) i = N c α s (2 π ) Z d z (*(cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( r − s ) ( r − z ) ( s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) (cid:21)(cid:20) T Q ( z, ¯ r, ¯ s, s ) + T ( r, z ) (cid:21) + (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21)(cid:20) T Q ( r, z, ¯ s, s ) + T ( z, ¯ r ) (cid:21) + (cid:20) (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( s − ¯ r ) ( s − z ) (¯ r − z ) (cid:21)(cid:20) T Q ( r, ¯ r, z, s ) + T (¯ s, z ) (cid:21) + (cid:20) ( r − s ) ( r − z ) ( s − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21)(cid:20) T Q ( r, ¯ r, ¯ s, z ) + T ( z, s ) (cid:21) − (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) + ( r − s ) ( r − z ) ( s − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) (cid:21) T Q ( r, ¯ r, ¯ s, s ) − (cid:20) ( r − s ) ( r − z ) ( s − z ) + (¯ r − ¯ s ) (¯ r − z ) (¯ s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) (cid:21)(cid:20) T ( r, s ) + T (¯ r, ¯ s ) (cid:21) − (cid:20) ( r − ¯ r ) ( r − z ) (¯ r − z ) + ( s − ¯ s ) ( s − z ) (¯ s − z ) − ( r − ¯ s ) ( r − z ) (¯ s − z ) − (¯ r − s ) (¯ r − z ) ( s − z ) (cid:21)(cid:20) T ( r, ¯ r ) + T (¯ s, s ) (cid:21)+) (10)To proceed further, we first consider the two-gluon exchange limit, i.e., the BFKL equation [19]. Since T Q and T include multiple gluon exchanges, we need to linearize them, i.e., take the single (reggeized) gluon exchange limit.This corresponds to expanding each of the Wilson lines in the definition of T Q and T to first order in the gauge field α and then keeping terms of the order α . In this limit (note the relative sign which appears when taking both α ’sfrom either V ’s or V † ’s rather than taking one α from a V and another α from a V † ) T Q ( r, ¯ r, ¯ s, s ) → T ( r, ¯ r ) + T (¯ s, s ) − T ( r, ¯ s ) − T (¯ r, s ) + T ( r, s ) + T (¯ r, ¯ s ) (11)Using eq. (11) in both sides of eq. (10) we get the BFKL equation for each T of a given argument. For example, ddy h T ( r, s ) i = N c α s π Z d z ( r − s ) ( r − z ) ( s − z ) (cid:20) h T ( r, z ) i + h T ( z, s ) i − h T ( r, s ) i (cid:21) (12)where T in eq. (12) and right hand side of (11) stands for T ( r, ¯ r ) → Γ( r − ¯ r ) ∼ g α a ( r ) α a (¯ r ) (13)This limit was already considered in [10] and the correspondence with BFKL was shown. We also note that thisrelation still holds when the evolution equation is written in terms of the color charge density ρ rather than the gaugefield α .The next interesting case is to consider O ( α ) and see whether our evolution equation reduces to the well-knownBJKP equation governing the evolution of four reggeized-gluon state in the dilute limit. To do this, again we firstignore the non-linear terms in the evolution equation, then we expand the Wilson lines and keep terms of the order α in eq. (10). Since the BJKP equation is written in momentum space, we will start by Fourier transforming T Q (ignoring T at the moment) to momentum space and disregard any contribution which leads to a vanishing externalmomentum. We define T ( l , l , l , l ) ≡ Z d r d ¯ r d ¯ s d s e i ( l · r + l · ¯ r + l · ¯ s + l · s ) T ( r, ¯ r, ¯ s, s ) (14)where l , l , l , l are two-dimensional external transverse momenta satisfying overall transverse momentum conserva-tion so that there are only three independent momenta. This corresponds to having a choice in picking the origin ofthe coordinate space on the transverse plane. One can then right away see that the last term in (8) convoluted with We will use the notation T here to denote the ∼ O ( α ) terms in the expansion of T Q so that T ≡ N c T r [ α α α α ]. T ( z, ¯ r, ¯ s, s ) will give a δ ( l ) since it does not depend on coordinate r . A similar argument shows that the last termin each kernel in the first 4 lines in eq. (10) (the ”real” terms) will lead to a delta function which sets one of theexternal momenta to zero. Since the external momenta of the reggeized-gluons are assumed to be finite (non-zero),all these terms can be safely ignored. We now consider the contribution of the ”virtual” terms, line 5 in eq. (10).Upon Fourier transforming, we get − N c α s (2 π ) Z d p t p t T ( l , l , l , l ) + 4 N c α s (2 π ) Z d p t p t T ( p t + l , l − p t , l , l ) + · · · (15)with a cyclic permutation of the external momenta in the second term understood. The first term is part of thevirtual corrections while the second term is part of the real corrections in momentum space. Let us consider now thecontribution of ”real” terms. Fourier transforming the non-zero terms in the first line of eq. (10) gives2 N c α s (2 π ) Z d p t (cid:20) p t · ( p t − l ) p t ( p t − l ) T ( l , l , l , l ) + 2 p t · l p t l T ( p t + l , l − p t , l , l ) (cid:21) (16)The first term in eq. (16) is part of the virtual corrections (in momentum space) while the second term is part of thereal corrections. With a slight rearrangement of the first term one can rewrite the contribution of the first line in eq.(10) as 2 N c α s (2 π ) Z d p t (cid:26)(cid:20) p t − l p t ( p t − l ) (cid:21) T ( l , l , l , l ) + 2 p t · l p t l T ( p t + l , l − p t , l , l ) (cid:27) (17)It is clear that the first term in the square bracket in eq. (17) partially cancels the first term in eq. (15). Thiscancellation becomes complete when we include the similar contributions from the lines 2 − T with the same argument, at the moment) gives ddy T ( l , l , l , l ) = N c α s π Z d p t (cid:20) p t + p t · l p t l − p t · l p t l − l · l l l (cid:21) T ( p t + l , l − p t , l , l ) + · · ·− N c α s (2 π ) Z d p t (cid:20) l p t ( l − p t ) + { l → l , l , l } (cid:21) T ( l , l , l , l ) (18)where · · · stands for real contributions obtained by appropriate permutation of the external momenta. Finally wenote that the term proportional to l · l comes from keeping O ( ∼ α ) in the expansion of V z and setting one of theother V ’s to unity, for example, taking V ¯ r = 1 and α ( z ) in the first line of eq. (10). It is clear that the virtualterms in eq. (18) are already in exact agreement with one gets from BJKP equation [14, 15] but the real terms lookdifferent. To show agreement of the real terms with the BJKP equation, we rewrite this equation for color chargedensity ρ rather than the gauge field α (this does not affect the virtual corrections). To this end, we note that thesquare bracket in the real term in eq. (18) can be rewritten as (cid:20) p t + p t · l p t l − p t · l p t l − l · l l l (cid:21) = 12 (cid:20) ( p t + l ) p t l + ( p t − l ) p t l − ( l + l ) l l (cid:21) Recalling the relation between gauge field α and color charge density ρ , α ( p t ) ∼ ρ ( p t ) p t (19)and defining ˆ T ( l , l , l , l ) = N c T r ρ ( l ) ρ ( l ) ρ ( l ) ρ ( l ), we multiply both sides of eq. (18) with l l l l whicheffectively removes the external legs. Eq. (18) can then be written as ddy ˆ T ( l , l , l , l ) = N c α s π Z d p t (cid:20) p i p t − ( p i − l i )( p t + l ) (cid:21) · (cid:20) p i p t − ( p i − l i )( p t + l ) (cid:21) ˆ T ( p t + l , l − p t , l , l ) + · · ·− N c α s (2 π ) Z d p t (cid:20) l p t ( l − p t ) + { l → l , l , l } (cid:21) ˆ T ( l , l , l , l ) (20)This is our final result and corresponds to the evolution of ˆ T after one step in rapidity y as depicted (the real part)in Fig. (1). We have checked that it agrees with the expressions given in [14, 15]. p + l l l l l pp l − p FIG. 1: Evolution of the four-point function ˆ T after one step in rapidity as given by eq. (20). Shown is one of the real diagramsonly and the dashed line represents a cut. There are several points that need to be clarified; first, we have completely disregarded the dipole terms ( ∼ T ) hereeven though they also contain O ( α ) terms. Since T ( r, s ) depends only on two external transverse coordinates r, s , O ( α ) terms will necessarily involve two pairs of gauge fields at the same point. Assuming rotational invariance onthe transverse coordinate plane, this leads to setting two of the external momenta equal to each other which takesone back to the BFKL ladders. Therefore, these terms are not relevant for our purpose. A second point is the coloraveraging denoted by < · · · > . We have not made any assumptions about the color averaging [20] and the evolutionequation derived is independent of how one performs this averaging. Furthermore, the overall color structure of theequation seems to be more general than the BJKP equation since here one has a trace of four color matrices in thefundamental representation on both sides of the equation. This trace could be written in terms of products of thegroup structure constants δ ab , f abc , d abc whereas the BJKP equation is for the exchange of four reggeized-gluon statein a symmetric color singlet state. One expects that δ δ terms would lead to a topology which is equivalent to exchangeof two independent BFKL pomerons which would then be disregarded. Therefore, one would only consider the colorsymmetric structures involving d ’s.In summary, we have shown in this preliminary study that the JIMWLK evolution equation for the quadrupoleoperator can be reduced to the BJKP equation for the real part of the four reggeized-gluon exchange amplitude.To do this, we first ignore the non-linear (recombination) terms in quadrupole evolution equation, and then expandthe Wilson lines in terms of the gauge field (or equivalently, the color charge density). This approximation shouldbe valid when the external momenta are larger that the saturation scale, i.e., in the dilute region. The quadrupoleevolution equation reduces to a sum of independent BFKL equations in O ( ρ ) and to the BJKP equation whenone looks at the terms of order ∼ ρ . This suggests that the JIMWLK evolution equation for the n -pole operator N c < T r V ( x ) V † ( x ) · · · V † ( x n ) > in the linear limit (dilute region) may be equivalent to the BJKP heirarchy forthe imaginary part of the n reggeized-gluon exchange amplitude. This would be very useful since there is much thatis known about the BJKP equation and its properties but not much is known about the properties of the JIMWLKequation in analytic form. Proving the equivalence between linearized JIMWLK and BJKP equations may not be sodifficult since the JIMWLK evolution equation for N c < T r V ( x ) V † ( x ) · · · V † ( x n ) > can almost be written downby inspection in analogy with the pattern seen in eq. (5). The problem reduces to keeping track of which quark lineradiates a gluon and counting all the possibilities since all emission kernels are just the standard dipole kernel. Itwould also be interesting to investigate the connection between the non-linear terms in the JIMWLK equation andmulti-pomeron vertices employed in reggeized-gluon approach to high energy scattering. These issues are beyond thescope of this preliminary work and will be reported elsewhere. Acknowledgments
We thank F. Dominguez, A. Dumitru, Y. Kovchegov, A. Mueller and B. Xiao for useful discussions. This work issupported by the DOE Office of Nuclear Physics through Grant No. DE-FG02-09ER41620 and by The City Universityof New York through the PSC-CUNY Research Program, grant 62625-41. Figures are made using JaxoDraw [21]. [1] E. Braidot [ STAR Collaboration ], Nucl. Phys.
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