Abstract
We consider the Bethe–Salpeter approach to the BFKL evolution in order to naturally incorporate the property of the Hermitian separability in the BFKL approach. We combine the resulting all order ansatz for the BFKL eigenvalue together with reflection identities for harmonic sums and derive the most complicated term of the next-to-next-to-leading order BFKL eigenvalue in SUSY N=4. We also suggest a numerical technique for reconstructing the unknown functions in our ansatz from the known results for specific values of confomal spin.
Highlights
For the BFKL Hamiltonian H and the BFKL eigenvalue E which is related to the pomeron intercept
In the present paper we focus on one major feature of the leading order (LO) and the next-to-leading order (NLO) functions, namely the so-called Hermitian separability first discussed by Kotikov and Lipatov [8,9]
The LO eigenvalue is manifestly Hermitian separable, whereas the NLO eigenvalue [6] is not. It was demonstrated by Kotikov and Lipatov [8,9] that color singlet NLO eigenvalue in N = 4 SYM can be written as a combination of a product of two hermitian separable functions and a hermitian separable function f N L O (z, z) = f L O (z, z)g(z, z) + ρ(z, z), (4)
Summary
The Balitsky–Fadin–Kuraev–Lipatov (BFKL) [1,2,3,4,5] equation is traditionally schematically written in the form of the linear Schrödinger equation. The function f L O , f N L O , g and ρ all have Hermitian separable form of Eq (3) This sort of non-linearity is difficult to explain by the Schrödinger equation approach to the BFKL dynamics, where the two degrees of freedom corresponding to the complex variables z and zare mixed at the Hamiltonian level. All currently available results show that the BFKL eigenvalue is built of polygamma functions and its generalizations Those functions are either logarithmically divergent at infinite value of the argument or give transcendent constants.. This concept despite not being rigorously proven is very useful and widely used in building functional bases for different ansätze Another observation, which is widely used is that the maximal transcendentality is increased by two units for each order of the perturbations theory. This significantly increases a space of functions defining a functional basis for any ansatz
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