L.N. Lipatov

BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes

After a brief review of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach to Regge processes in QCD and in supersymmetric gauge theories we propose a strategy for calculating the next-to-next-toleading order corrections to the BFKL kernel. They can be obtained in terms of various cross sections for Reggeized gluon interactions. The corresponding amplitudes can be calculated in the framework of the effective action for high-energy scattering. In the case of N ¼ 4 supersymmetry it is also possible to use the Bern-Dixon-Smirnov ansatz. For this purpose the analytic properties of the Bern-Dixon-Smirnov amplitudes at high energies are investigated, in order to verify their self-consistency. It is found that, for the number of external particles being larger than 5, these amplitudes, beyond one loop, are not in agreement with the BFKL approach which predicts the existence of Regge cuts in some physical channels.

On the highest transcendentality in N = 4 SUSY ∗

Abstract We investigate the Eden–Staudacher equation for the anomalous dimension of the twist-2 operators at the large spin s in the N = 4 super-symmetric gauge theory. This equation is reduced to a set of linear algebraic equations with the kernel calculated analytically. We prove that in perturbation theory the anomalous dimension is a sum of products of the Euler functions ζ ( k ) having the property of the maximal transcendentality with the coefficients being integer numbers. The radius of convergency of the perturbation theory is found. It is shown, that at g = ∞ the kernel has an essential singularity. The analytic properties of the solution of the Eden–Staudacher equation are investigated. In particular for the case of the strong coupling constant the solution has an essential singularity on the second sheet of the variable j appearing in its Laplace transformation. Similar results are derived also for the Beisert–Eden–Staudacher equation which includes the contribution from the phase related to the crossing symmetry of the underlying S -matrix. We show, that its singular solution at large coupling constants reproduces the anomalous dimension predicted from the string side of the AdS/CFT correspondence.

N=4 supersymmetric Yang–Mills scattering amplitudes at high energies: the Regge cut contribution

We further investigate, in the planar limit of N=4 supersymmetric Yang–Mills theories, the high energy Regge behavior of six-point MHV scattering amplitudes. In particular, for the new Regge cut contribution found in our previous paper, we compute in the leading logarithmic approximation (LLA) the energy spectrum of the BFKL equation in the color octet channel, and we calculate explicitly the two-loop corrections to the discontinuities of the amplitudes for the transitions 2→4 and 3→3. We find an explicit solution of the BFKL equation for the octet channel for arbitrary momentum transfers and investigate the intercepts of the Regge singularities in this channel. As an important result we find that the universal collinear and infrared singularities of the BDS formula are not affected by this Regge-cut contribution. Any improvement of the BDS formula should reproduce this cut to all orders in the coupling.

BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N=4 SUSY☆

Abstract We calculate the eigenvalues of the next-to-leading kernel for the BFKL equation in the adjoint representation of the gauge group SU ( N c ) in the N = 4 supersymmetric Yang–Mills model. These eigenvalues are used to obtain the high energy behavior of the remainder function for the 6-point scattering amplitude with the maximal helicity violation in the kinematical regions containing the Mandelstam cut contribution. The leading and next-to-leading singularities of the corresponding collinear anomalous dimension are calculated in all orders of perturbation theory. We compare our result with the known collinear limit and with the recently suggested ansatz for the remainder function in three loops and obtain the full agreement providing that the numerical parameters in this ansatz are chosen in an appropriate way.

MHV amplitude for 3→3 gluon scattering in Regge limit

Abstract We calculate corrections to the BDS formula for the six-particle planar MHV amplitude for the gluon transition 3 → 3 in the multi-Regge kinematics for the physical region, in which the Regge pole ansatz is not valid. The remainder function at two loops is obtained by an analytic continuation of the expression derived by Goncharov, Spradlin, Vergu and Volovich to the kinematic region described by the Mandelstam singularity exchange in the crossing channel. It contains both the imaginary and real contributions being in agreement with the BFKL predictions. The real part of the three loop expression is found from a dispersion-like all-loop formula for the remainder function in the multi-Regge kinematics derived by one of the authors. We also make a prediction for the all-loop real part of the remainder function multiplied by the BDS phase, which can be accessible through calculations in the regime of the strong coupling constant.

Pomeron in the N=4 supersymmetric gauge model at strong couplings

Abstract We find the BFKL Pomeron intercept at N = 4 supersymmetric gauge theory in the form of the inverse coupling expansion j 0 = 2 − 2 λ − 1 / 2 − λ − 1 + 1 / 4 λ − 3 / 2 + 2 ( 1 + 3 ζ 3 ) λ − 2 + O ( λ − 5 / 2 ) with the use of the AdS/CFT correspondence in terms of string energies calculated recently. The corresponding slope γ ′ ( 2 ) of the anomalous dimension calculated directly up to the fifth order of perturbation theory turns out to be in an agreement with the closed expression obtained from the recent Basso results.

NLO Corrections to the kernel of the BKP-equations

Abstract We present results for the NLO kernel of the BKP-equations for composite states of three reggeized gluons in the Odderon channel, both in QCD and in N = 4 SYM. The NLO kernel consists of the NLO BFKL kernel in the color octet representation and the connected 3 → 3 kernel, computed in the tree approximation.

Integrable spin chains and scattering amplitudes

In this review, we show that the multi-particle scattering amplitudes in SYM at large Nc and in the multi-Regge kinematics for some physical regions have the high energy behavior appearing from the contribution of the Mandelstam cuts in the complex angular momentum plane of the corresponding t-channel partial waves. These Mandelstam cuts or Regge cuts result from gluon composite states in the adjoint representation of the gauge group SU(Nc). In the leading logarithmic approximation (LLA), their contribution to the six-point amplitude is in full agreement with the known two-loop result. The Hamiltonian for the Mandelstam states constructed from n gluons in LLA coincides with the local Hamiltonian of an integrable open spin chain. We construct the corresponding wavefunctions using the integrals of motion and the Baxter?Sklyanin approach.In this review we show that the multi-particle scattering amplitudes in N = 4 SYM at large Nc and in the multi-Regge kinematics for some physical regions have the high energy behavior appearing from the contribution of the Mandelstam cuts in the complex angular momentum plane of the corresponding t-channel partial waves. These Mandelstam cuts or Regge cuts are resulting from gluon composite states in the adjoint representation of the gauge group SU(Nc). In the leading logarithmic approximation (LLA) their contribution to the six point amplitude is in full agreement with the known two-loop result. The Hamiltonian for the Mandelstam states constructed from n gluons in LLA coincides with the local Hamiltonian of an integrable open spin chain. We construct the corresponding wave functions using the integrals of motion and the Baxter-Sklyanin approach. This article is an invited review for a special issue of Journal of Physics A devoted to Scattering Amplitudes in Gauge Theories.

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory

A bstractWe consider the multi-Regge limit for N=4 SYM NMHV leading color amplitudes in two different formulations: the BFKL formalism for multi-Regge amplitudes in leading logarithm approximation, and superconformal N=4 SYM amplitudes. It is shown that the two approaches agree to two-loops for the 2 → 4 and 3 → 3 six-point amplitudes. Predictions are made for the multi-Regge limit of three loop 2 → 4 and 3 → 3 NMHV amplitudes, as well as a particular sub-set of two loop 2 → 2 + n NkMHV amplitudes in the multi-Regge limit in the leading logarithm approximation from the BFKL point of view.

Double-logarithms in Einstein-Hilbert gravity and supergravity

A bstractWe study the interplay between graviton reggeization and double-logarithmic in energy contributions to four-graviton scattering in theories with and without supersymmetry. Predictions to all orders in the gravitational coupling are given for these double-logarithms. As the number of supersymmetries grows these terms generate a convergent behaviour for the amplitudes at very high energies. At two-loop level, we find agreement with previous exact results for N