Two-loop $$ \mathcal{N} $$ = 2 SQCD amplitudes with external matter from iterated cuts

Abstract

We develop an iterative method for constructing four-dimensional generalized unitarity cuts in N = 2 supersymmetric Yang-Mills (SYM) theory coupled to fundamental matter hypermultiplets (N = 2 SQCD). For iterated two-particle cuts, specifically those involving only four-point amplitudes, this implies simple diagrammatic rules for assembling the cuts to any loop order, reminiscent of the rung rule in N = 4 SYM. By identifying physical poles, the construction simplifies the task of extracting complete integrands. In combination with the duality between color and kinematics we construct all four-point massless MHV-sector scattering amplitudes up to two loops in N = 2 SQCD, including those with matter on external legs. Our results reveal chiral infrared-finite integrands closely related to those found using loop-level BCFW recursion. The integrands are valid in D ≤ 6 dimensions with external states in a four-dimensional subspace; the upper bound is dictated by our use of six-dimensional chiral N = (1, 0) SYM as a means of dimensionally regulating loop integrals.

Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes

“…To organize the particle content of four-point MHV amplitudes in a QCD-like theory, we use κ (ij)(kl) introduced in refs. [45,48] where i < j and k < l. A particle label appearing in both parentheses of the subscript corresponds to a negative-helicity gluon g − ; a label not appearing at all corresponds to a positive-helicity gluon g + ; a label appearing only in the first parenthesis is a quark q; a label appearing in the second is an antiquarkq. When all external states are gluons, we also abbreviate…”

“…In ref. [45] this amplitude was presented in a form that makes manifest the colorkinematics duality [46][47][48][49][50]. Its color-dual numerators are nonzero for boxes and triangles, but only the former give rise to non-vanishing integrals: 8…”

“…We are aided by the N = 4 matching conditions, which the kinematic numerators satisfy by construction [44,45]. where to the right of each line we have displayed its divergent behavior.…”

“…Achieving these results has been facilitated by remarkable properties of the N = 2 SQCD integrands obtained previously in refs. [44,45]. In particular, diagrams containing internal matter lines are observed to diverge in fewer IR regions, as soft and collinear divergences arise only from virtual gluon exchanges.…”

“…Purely gluonic diagrams (or those related to them by supersymmetry), which have the strongest IR divergences, are naturally eliminated from the remainder function W Finally, we also take this opportunity to complete the work begun in ref. [45] by presenting fully integrated two-loop four-point amplitudes with matter on external legs. The results provide us with expressions for the cusp, gluonic, and quark anomalous dimensions in N = 2 SQCD up to two-loop order, which are conveniently cross-checked by comparison between the different amplitudes.…”