Leading and subleading UV divergences in scattering amplitudes for $D=8$ $\mathcal{N}=1$ SYM theory in all loops

We consider the leading and subleading UV divergences for the four-point on-shell scattering amplitudes in $D=8$ $\mathcal{N}=1$ supersymmetric Yang-Mills theory in the planar limit. This theory belongs to the class of maximally supersymmetric gauge theories and presumably possesses distinguished properties beyond perturbation theory. We obtain the recursive relations that allow one to get the leading and subleading divergences in all loops in a pure algebraic way staring from the one loop (for the leading poles) and two loop (for the subleading ones) diagrams. As a particular example where the recursive relations have a simple form we consider the ladder type diagrams. The all-loop summation of the leading and subleading divergences is performed with the help of the differential equations which are the generalization of the RG equations for nonrenormalizable theories. They have explicit solutions for the ladder type diagrams. We discuss the properties of the obtained solutions and interpretation of the results.

Figure 1

The universal expansion for the four-point scattering amplitude in SYM theories in terms of master integrals. The connected strokes on the lines mean the square of the flowing momentum.

Figure 2

The application of the ${\mathcal{R}}^{\prime }$ operation to the ladder type diagrams. At the right it is shown to which term of expansion (4) it corresponds to.

Figure 3

The behavior of $u(z)$ evaluated numerically.

Figure 4

Comparison of the exact solution with the perturbation series including 20 terms for ${\mathrm{\Sigma }}_A$ (left) and for ${\mathrm{\Sigma }}_{sB}$ (right). The solid line is the exact (numerical) solution and the dotted line is the PT series summation result.

Figure 5

The ratio of the coefficients $bs[n]$: $R=bs[n+1]/bs[n]$.