Exact result in $$ \mathcal{N} $$ = 4 SYM theory: generalised double-logarithmic equation

Abstract

We present the new results for the generalised double-logarithmic equation, obtained from the analytical continuation of the seven-loop anomalous dimension of twist-2 operators in the planar $$ \mathcal{N} $$ N = 4 SYM theory. The double-logarithmic equation is related to the special asymptotic of the scattering amplitudes, when the large logarithms of the energy of scattering particles are appeared and should be summed in all order of perturbative theory. These large logarithms correspond to the poles of the analytically continued anomalous dimension. The generalised double-logarithmic equation includes the subleading logarithms. We have found, that the expansion of the generalised double-logarithmic equation can be ressumed in the form of rational functions with simple denominator. The solution of the generalised double-logarithmic equation provides a lot of information about the poles of the analytically continued anomalous dimension in all orders of perturbative theory. We have found also the generalised double-logarithmic equation for the analytically continued anomalous dimension near the value, which is related with BFKL-equation.

Analytic continuation of harmonic sums: dispersion representation

“…The dispersion representation for the nested harmonic sums provides exactly the information that is needed for such analysis. Such study can help to extended the generalized double-logarithmic equation [25], known at this moment for the case N = −2 + ω, to other values of N. * * We used Eqs. (4.15) and (4.16) from arXiv-version of Ref.…”

“…(15) † See [21]; similar representation for the analytically continued harmonic sums was used in Refs. [22,23,24,25] and in unpublished work of L. Lipatov and A. Onishchenko (2004).…”

Analytic continuation of harmonic sums near the integer values

“…These poles are related to the evaluation equations for parton distributions, mostly to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [4][5][6] and the double-logarithmic equation [7,8]. Knowledge of the analytic continuation of nested harmonic sums made it possible to find higher order corrections to the BFKL equation [9][10][11] and to the double-logarithmic equation [12][13][14]. Analytic continuation of nested harmonic sums can be performed directly by following Refs.…”

“…In our previous work [17], we proposed a more effective way to find the expansion of the nested harmonic sums near the negative integers by extractinf ln x-terms in the inverse Mellin transform for the corresponding harmonic sums using summer [1] and harmpol [18] packages for FORM [19][20][21] along with database [22]. The obtained database for analytic continuation of nested harmonic sums near negative and positive integers allowed us to generalised the double-logarithmic equation [14]. Knowing the pole expressions for the nested harmonic sums, one can use the dispersion representation [23] to obtain the value for any complex argument.…”