Abstract
In Ref. [1] we presented the JIMWLK Hamiltonian for high energy evolution of QCD amplitudes at the next-to-leading order accuracy in $\alpha_s$. In the present paper we provide details of our original derivation, which was not reported in [1], and provide the Hamiltonian in the form appropriate for action on color singlet as well as color nonsinglet states. The rapidity evolution of the quark dipole generated by this Hamiltonian is computed and compared with the corresponding result of Balitsky and Chirilli [2]. We then establish the equivalence between the NLO JIMWLK Hamiltonian and the NLO version of the Balitsky's hierarchy [3], which includes action on nonsinglet combinations of Wilson lines. Finally, we present complete evolution equation for three-quark Wilson loop operator, thus extending the results of Grabovsky [4].
Highlights
The Wilson line S(x), in the high energy eikonal approximation represents the scattering amplitude of a quark at the transverse coordinate x
In ref. [1] we presented the JIMWLK Hamiltonian for high energy evolution of QCD amplitudes at the next-to-leading order accuracy in αs
Thanks to the above mentioned major progress in the NLO computations, in [1] we have presented the NLO JIMWLK Hamiltonian which reproduces these results by simple algebraic application to the relevant amplitudes
Summary
The JIMWLK Hamiltonian defines a two-dimensional non-local field theory of a unitary matrix (Wilson line) S(x). The last virtual term vanishes when acting on both, the dipole and the baryon operator, and one needs additional information to determine the kernel KJJJ. The kernels KJJSJ and KJJSSJ are fixed by acting with the Hamiltonian on the operator B and comparing the result to that of ref. These additional terms can be inferred by considering the action of the Hamiltonian on nonsinglet products of Wilson line and comparing the results to [3] This is the subject of the section. The modifications of the kernels involve only extra terms that are independent of either x or y These additional terms do not contribute to evolution of gauge invariant operators. In order to compare with [3], we only need to consider connected terms, that is the terms in which each factor of S in the operator is acted upon at least one charge density operator J in the Hamiltonian
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