On the geometry of operator mixing in massless QCD-like theories
Physics Department, INFN Roma1, Piazzale A. Moro 2, 00185, Rome, Italy
Accepted: 9 August 2021
Published online: 20 August 2021
We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation, , is diagonalizable to all perturbative orders. As a key step, we provide a differential-geometric interpretation of renormalization that allows us to apply the Poincaré-Dulac theorem to the problem above: We interpret a change of renormalization scheme as a (formal) holomorphic gauge transformation, as a (formal) meromorphic connection with a Fuchsian singularity at , and as a Wilson line, with the matrix of the anomalous dimensions and the beta function. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues of the matrix , in nonincreasing order , satisfy the nonresonant condition for and k a positive integer, then a renormalization scheme exists where is one-loop exact to all perturbative orders. If in addition is diagonalizable, is diagonalizable as well, and the mixing reduces essentially to the multiplicatively renormalizable case. We also classify the remaining cases of operator mixing by the Poincaré–Dulac theorem.
© The Author(s) 2021
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