Full-Color Two-Loop Four-Gluon Amplitude in $\mathcal{N}=2$ Supersymmetric QCD

We present the fully integrated form of the two-loop four-gluon amplitude in $\mathcal{N}=2$ supersymmetric quantum chromodynamics with gauge group $\mathrm{SU}(N_c)$ 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 $\mathcal{N}=2$ superconformal theory, where $N_f=2N_c$, we obtain remarkably simple amplitudes that have an analytic structure close to that of $\mathcal{N}=4$ super-Yang-Mills theory, except that now certain lower-weight terms appear. We comment on the corresponding results for other gauge groups.

Figure 1

Ten cubic diagrams that describe the four-gluon two-loop amplitude of $\mathcal{N}=2$ SQCD. At the conformal point, $N_f=2N_c$, diagrams (d), (e), (i), and (j) manifestly cancel out from the integrand.