Remarks on Diffractive Dissociation within JIMWLK Evolution at NLO

We discuss the high energy diffractive dissociation in DIS at the Next to Leading Order. In the large $N_c$ dipole limit we derive the NLO version of the Kovchegov-Levin equation. We argue that the original structure of the equation is preserved, that is it coincides with the Balitsky-Kovchegov equation at NLO.


II. HIGH ENERGY SCATTERING
In this section we recap some basic formalism and introduce notations. The total S-matrix of the high energy scattering process at a given rapidity Y in the CGC formalism is computable via the following factorization formula Here for a composite projectile which has some distribution of gluons in its wave function the eikonal S-matrix is ρ(x) is the color charge density in the projectile wave function at a given transverse position; W P Y −Y0 [ρ] is its probability distribution in the projectile, while α is a target color field. In eq.(2.1), the projectile-averaged S-matrix Σ P P is further averaged over the distribution of the color fields α with the weight W T Y0 [S(x)]. The fields α are parametrized by the eikonal S-matrix S(x) for a single parton at transverse position x to scatter on a given configuration of α.
In Eq. (2.1) we have chosen the frame where the target has rapidity Y 0 while the projectile carries the rest of the total rapidity Y − Y 0 . Lorentz invariance requires S to be independent of Y 0 . The high energy evolution of both W P.T is driven by an effective high energy Hamiltonian, which in this note will be assumed to be the NLO JIMWLK The NLO JIMWLK Hamiltonian [10] is Here S A is a unitary matrix in the adjoint representation -the gluon scattering amplitude. The left and right SU (N c ) rotation generators, when acting on functions of S have the representation Here T a are SU (N c ) generators in the fundamental representation. We use the notations of ref.
All Js in (2.4) are assumed not to act on S in the Hamiltonian.
Here µ is the normalization point in the M S scheme and b = 11 Our goal will be to study diffractive dissociation at NLO within the dipole model approximation. The S-matrix for a quark-antiquark dipole is where F denotes fundamental representation. The S matrix of a projectile made of several but not too many dipoles is therefore some function of the variable s only In the large N c limit, the NLO JIMWLK Hamiltonian acting on a function of dipoles only, reduces to the action of the NLO dipole Hamiltonian. To derive this Hamiltonian, we have to act with the NLO JIMWLK Hamiltonian on one, two and three dipoles. Four and more dipoles cannot be coupled by the NLO evolution because there are only three Js in the Hamiltonian. The action on a single dipole is by construction [10] reproduces the result of [8] and, in the large N c limit is equivalent to the action of the dipole Hamiltonian The connected parts of two-and three-dipole evolutions are subleading in N c and it is a matter of a straightforward color algebra to see that. When acting with the NLO JIMWLK on two dipoles, the connected part arises when both dipoles are rotated by at least one J. The N c counting of such a "dipole merging" is most easily done when S A (z) and S A (z ′ ) in the Hamiltonian are set to one. All the "dipole merging" terms generated by the operators in the NLO JIMWLK Hamiltonian are found to be subleading in N c compared to uncorrelated terms generated by H N LO dipole . As a result, there are no leading N c δ 2 /δs 2 and (δ 3 /δs 3 with the initial condition The target average in eq.(2.1) still allows to accommodate nontrivial, non-factorized multi-s correlators s(x 1 , y 1 ) · · · s(x n , y n ) T , which have been recently argued [18,19] to be of relevance to various two-particle correlations, such as the "ridge". Further simplification is achieved if one assumes that the dipoles scatter on the target independently. This amounts to factorization of the target averages of the dipole s-matrices With this assumption, one replaces the ensemble average over target fields with a fixed initial function s Y0 (x, y). We refer to this factorization property as the target mean field approximation. Within the target mean field approximation (2.20) As has been stressed in the past, this mean field approximation does not follow from the dipole model approximation for the evolution kernel eq.(2.15), but is an additional assumption about the properties of the target.

III. DIFFRACTIVE DISSOCIATION
We will be interested in identifying the evolution governing the diffractive observables both with respect to the total rapidity of the process Y and the rapidity gap. We now consider processes where the projectile diffracts within rapidity interval Y P . This interval is not necessarily small, so this type of observable can be evolved independently over the total rapidity Y and the width of the diffractive interval Y P . The target can either scatter elastically or can in principle also diffract. We consider here the process where the scattering on the target side is elastic with the In ref. [16] we have developed a formalism that makes it possible to generalize (2.1) for semi-inclusive processes. Particularly, for elastic/diffractive processes one has to introduce two independent S-matrix variables, S andS for the amplitude and its conjugate. For the observable in question the cross section reads What we have here is the target evolved through the gap Y gap both in the amplitude and its conjugate, while the projectile is evolved inclusively though the diffractive interval Y P . The derivation of this result did not rely on any explicit form of the evolution Hamiltonian and is thus valid for the case of the NLO JIMWLK. We would like now to project this result into the dipole picture at large N c . At leading order, this observable in the dipole limit has been discussed by Kovchegov and Levin [13]. In the dipole limit the evolution of Σ P P with respect to the diffractive interval at fixed Y gap is given by eq. (2.16). Within the gap, due to independent target averaging over S andS in eq. (3.1), the "composite" dipole made ofS † S factorizes into the product of two As has been already noticed in [16], the derivation in [13] was originally done for a single dipole projectile, Σ P P [s] = s. Thanks to the fact that at both LO and NLO, all the dipoles evolve independently, Eq. (3.3) provides generalization to a more complex projectile wave function.