High-energy evolution to three loops

TL;DR

The paper derives the three-loop Balitsky-Kovchegov evolution equation in planar N=4 SYM by leveraging a deep link to non-global logarithms and implementing a Lorentz-invariant subtraction scheme to control infrared and collinear divergences. It builds the three-loop kernel from triple-soft and double-soft currents, plus renormalization counter-terms, and expresses the result in terms of convergent angular integrals tied to the Bern-Dixon-Smirnov remainder. The linearized analysis yields the BFKL Pomeron trajectory, including explicit eigenvalues for low angular momentum, which agree with integrability predictions and twist-two anomalous dimensions; the work also provides a framework to study resummations in high-energy QCD-like theories. Overall, the results offer a stringent perturbative test of high-energy evolution at three loops and establish techniques potentially extendable to QCD and finite coupling.

Abstract

The Balitsky-Kovchegov equation describes the high-energy growth of gauge theory scattering amplitudes as well as nonlinear saturation effects which stop it. We obtain the three-loop corrections to this equation in planar $\mathcal{N}=4$ super Yang-Mills theory. Our method exploits a recently established equivalence with the physics of soft wide-angle radiation, so-called non-global logarithms, and thus yields at the same time the three-loop evolution equation for non-global logarithms. As a by-product of our analysis, we develop a Lorentz-covariant method to subtract infrared and collinear divergences in cross-section calculations in the planar limit. We compare our result in the linear regime with a recent prediction for the so-called Pomeron trajectory, and compare its collinear limit with predictions from the spectrum of twist-two operators.

Figures (7)

• Figure 1: Soft wide-angle radiation: radiation is allowed in some region but excluded in another. To keep track of the allowed radiation we use a color density matrix, defined by applying an angle-dependent color rotation $U(\theta_i)$ between the matrix element and its conjugate for each final state particle. In the planar limit this configuration reduces to a product of two color dipoles.
• Figure 2: Extracting squared soft currents from the four-particle integrand: cuts which give the squares of single (a) and double (b) emissions by taking the cut internal propagators to be soft.
• Figure 3: One-loop virtual correction to double soft current contributing to the cross-section at three loops.
• Figure 4: The linear kernel $K^{(3){\rm lin}}$ in coordinate space in the physical region $\bar{x} = x^*$.
• Figure 5: The BFKL eigenvalue for $m=0$ along the real $\nu$ axis at various orders for $\lambda = g_{\rm YM}^2 N_c = 6$. Convergence near the maximum is visibly slower than away from it. The "resummation of leading-order" is defined below eq. (\ref{['level_crossing']}).