Connecting Supersymmetry to Non-Supersymmetric theories: the Gross-Neveu-Yukawa example

We construct a generalized Lagrangian that unifies the Gross-Neveu-Yukawa, Nambu-Jona-Lasinio-Yukawa, and Wess-Zumino models, allowing for arbitrary scalar and fermion flavors in $D$-dimensional regularization. This framework clarifies how emergent supersymmetry arises at critical points and reveals structural connections between these theories. The unified formulation provides additional supersymmetry Ward identities that simplify loop calculations, even for non-supersymmetric models. As an application, we show how this technique can reduce the computational cost of determining anomalous dimensions of twist-two operators.