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We obtain a color-kinematics-dual representation of the two-loop four-vectoramplitude a general renormalizable massless $\mathcal{N}=1$ SYM theory,including internal matter as chiral supermultiplets. The integrand isconstructed to be compatible with dimensional regularization and supersymmetryby employing two strategies (implicitly defining our regularization scheme):supersymmetric decomposition and matching to massive spinor-helicityamplitudes. All internal vector components inherit their $D$-dimensionalproperties by relating them to the previously constructed $D\leq6$,$\mathcal{N}=2$ SQCD amplitude using supersymmetric decomposition identities ofindividual diagrams. This leaves only diagrams with internal matter lines asunknown masters, which are in turn constrained on $D$-dimensional unitaritycuts by reinterpreting the extra-dimensional momentum components as masses forthe chiral supermultiplets. We rely on the massive spinor-helicity formalismand massive on-shell $\mathcal{N}=1$ superspace, generalized here to complexmasses. Finally, we extend the kinematic numerator algebra to includethree-term identities that are dual to color identities linear in the matterClebsch-Gordan coefficients, as well as two new optional identities satisfiedby mass-deformed $\mathcal{N}=4$ and $\mathcal{N}=2$ SYM theories that preserve$\mathcal{N}=1$ supersymmetry. Altogether, these identities makes it possibleto completely reduce the two-loop integrand to only two master numerators.

Two-loop $\mathcal{N}=1$ SYM Amplitudes via SUSY Decomposition and Massive Spinor-Helicity