Evanescent Operators, Scheme Dependences and Double Insertions

TL;DR

The paper analyzes how evanescent operators—vanishing in $D=4$ but contributing $1/\varepsilon$-poles in dimensional regularization—affect the renormalization of four-quark operators. It derives exact next-to-leading order scheme-transformation rules for Wilson coefficients and the anomalous-dimension matrix, demonstrating that different definitions of evanescent operators correspond to renormalization schemes and that physical observables remain scheme-independent when matched consistently. The authors prove that evanescent operators do not mix into physical operators (block-triangular γ) under a finite renormalization, extend the treatment to Green functions with double insertions, and address inclusive decays, providing a coherent NLO framework. The practical upshot is a flexible yet consistent methodology to handle evanescent-scheme dependences, enabling simpler operator bases and reliable RG-improved calculations for rare processes and meson mixing. All scheme-dependent artifacts cancel in physical predictions, provided the evanescent sector is treated with the described finite renormalizations and scheme-transformations.

Abstract

The anomalous dimension matrix of dimensionally regularized four-quark operators is known to be affected by evanescent operators, which vanish in $D=4$ dimensions. Their definition, however, is not unique, as one can always redefine them by adding a term proportional to $(D-4)$ times a physical operator. In the present paper we compare different definitions used in the literature and find that they correspond to different renormalization schemes in the physical operator basis. The scheme transformation formulae for the Wilson coefficients and the anomalous dimension matrix are derived in the next-to-leading order. We further investigate the proper treatment of evanescent operators in processes appearing at second order in the effective four-fermion interaction such as particle-antiparticle mixing, rare hadron decays or inclusive decays.

Figures (3)

• Figure 1: Diagrams contributing to $\langle \hat{Q}_k ^{\hbox{\tiny bare} } \rangle ^{(1)}$.
• Figure 2: The lowest order diagram contributing to the Green function in (\ref{['GFdouble']}).
• Figure 3: The lowest order self-energy diagram needed for the calculation of inclusive decays via the optical theorem (method 2).