Two-loop N = 1 SYM Amplitudes via SUSY Decomposition and Massive Spinor-Helicity

Authors: Henrik Johansson, Gregor Kälin, Gustav Mogull and Bram Verbeek

Preprint number: UUITP–42/23

We obtain a color-kinematics-dual representation of the two-loop four-vector amplitude in a renormalizable massless N=1 SYM theory, including internal matter as chiral supermultiplets. The integrand is constructed to be compatible with dimensional regularization and supersymmetry by employing two strategies (implicitly defining our regularization scheme): supersymmetric decomposition and matching to massive spinor-helicity amplitudes.  All internal vector components inherit their D-dimensional properties by relating them to the previously constructed D ≤ 6, N=2 SQCD amplitude using supersymmetric decomposition identities of individual diagrams. This leaves only diagrams with internal matter lines as unknown masters, which are in turn constrained on D-dimensional unitarity cuts by reinterpreting the extra-dimensional momentum components as masses for the chiral supermultiplets. We rely on the massive spinor-helicity formalism and massive on-shell N=1 superspace, generalized here to complex masses. Finally, we extend the kinematic numerator algebra to include three-term identities that are dual to color identities linear in the matter Clebsch-Gordan coefficients, as well as two new optional identities satisfied by mass-deformed N=4 and N=2 SYM theories that preserve N=1 supersymmetry. Altogether, these identities makes it possible to completely reduce the two-loop integrand to only two master numerators.