 One-loop exactness of the holomorphic gauge kinetic function

In a supersymmetry-preserving local Wilsonian effective action, the gauge kinetic function receives only a one-loop perturbative correction. For one simple gauge factor, $\mu df/d\mu =b_0/(8{\pi }^2)$ with $b_0=3C_2(G)-{\sum }_{i}T(R_i)$. A perturbative shift of the topological angle leaves the running coefficient invariant; the Cauchy-Riemann equations then make its holomorphic beta function independent of the gauge coupling. Holomorphic regularity and spurionic R-charges exclude higher-loop Yukawa combinations, which would require antichiral couplings. The constant is fixed by one-loop matching. This does not assert one-loop exactness of a canonically normalized physical coupling, or forbid nonperturbative terms.

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