Full-Color Two-Loop Four-Gluon Amplitude in N=2 Supersymmetric QCD.

Claude Duhr,Gustav Mogull,Bram Verbeek,Henrik Johansson,Gregor Kälin

doi:10.1103/physrevlett.123.241601

Abstract

We present the fully integrated form of the two-loop four-gluon amplitude in N=2 supersymmetric quantum chromodynamics with gauge group SU(N_{c}) and with N_{f} massless supersymmetric quarks (hypermultiplets) in the fundamental representation. Our result maintains full dependence on N_{c} and N_{f}, and relies on the existence of a compact integrand representation that exhibits the duality between color and kinematics. Specializing to the N=2 superconformal theory, where N_{f}=2N_{c}, we obtain remarkably simple amplitudes that have an analytic structure close to that of N=4 super-Yang-Mills theory, except that now certain lower-weight terms appear. We comment on the corresponding results for other gauge groups.

Highlights

The formidable goal of one day solving a four-dimensional gauge theory such as quantum chromodynamics (QCD) has inspired spectacular progress related to analytic computations of scattering amplitudes
In this Letter we study a two-loop amplitude in SUðNcÞ N 1⁄4 2 supersymmetric QCD (SQCD)—a theory which has tuneable matter content like QCD, namely Nf supersymmetric quarks, as well as a weakly coupled superconformal phase, like the N 1⁄4 4 SYM theory, at the critical point Nf 1⁄4 2Nc
Equations (11) and (13)–(15) give the full analytic result for the remainder Rð42Þ. We find it striking that the complete result for four-gluon scattering at two loops in N 1⁄4 2 SCQCD can be cast in a compact form which fits into a few lines

Summary

Published by the American Physical Society

Each containing a quark and two complex scalars. We only consider the limit where the quarks and superpartners are massless. We present a closed analytic form of the two-loop, four-gluon amplitude in N 1⁄4 2 SQCD, for arbitrary Nc and Nf. Our starting point is the integrand of Ref. The color-dual integrand.—N 1⁄4 2 SQCD has a running coupling constant αSðμ2RÞ, and loop amplitudes need to be renormalized to remove ultraviolet (UV) divergences. [34] was constructed to make manifest separations at the diagrammatic level between distinct gauge-invariant contributions It manifests the difference between the N 1⁄4 4 SYM theory and the N 1⁄4 2 superconformal theory (SCQCD), with Nf 1⁄4 2Nc, as a combination of simple diagrams that are manifestly UV finite. We note that any gluon amplitude in N 1⁄4 2 SQCD can be decomposed into three independent blocks that have different characteristics after integration, MðnLÞ 1⁄4 MðnLÞ1⁄2N 1⁄44 þ RðnLÞ þ ðCA − NfÞSðnLÞ: ð4Þ. Þ, which we denote by Rðð1−Þ−1⁄20ÞðþþÞ (and analogously for other helicity configurations), giving [49]

TðT þ

The coefficient