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N=(0,2) Supersymmetry and a Nonrenormalization Theorem

In this paper we continue the study of perturbative renormalizations in an $\mathcal{N}=(0,2)$ supersymmetric model. Previously we analyzed one-loop graphs in the heterotically deformed CP$(N-1)$ models. Now we extend the analysis of the $β$ function and appropriate $Z$ factors to two, and, in some instances, all loops in the limiting case $g^2\to 0$. The field contents of the model, as well as the heterotic coupling, remain the same, but the target space becomes flat. In this toy $\mathcal{N}=(0,2)$ model we construct supergraph formalism. We show, by explicit calculations up to two-loop order, that the $β$ function is one-loop-exact. We derive a nonrenormalization theorem valid to all orders. This nonrenormalization theorem is rather unusual since it refers to (formally) $D$ terms. It is based on the fact that supersymmetry combined with target space symmetries and "flavor"? symmetries is sufficient to guarantee the absence of loop corrections. We analyze the supercurrent supermultiplet (i.e., the hypercurrent) providing further evidence in favor of the absence of higher loops in the $β$ function.