**81**: 749

https://doi.org/10.1140/epjc/s10052-021-09543-5

Letter

## On the geometry of operator mixing in massless QCD-like theories

Physics Department, INFN Roma1, Piazzale A. Moro 2, 00185, Rome, Italy

^{a}
marco.bochicchio@roma1.infn.it

Received:
21
July
2021

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, $Z(x,\mu )$, 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, $-\frac{\gamma (g)}{\beta (g)}$ as a (formal) meromorphic connection with a Fuchsian singularity at $g=0$, and $Z(x,\mu )$ as a Wilson line, with $\gamma (g)={\gamma}_{0}{g}^{2}+\cdots $ the matrix of the anomalous dimensions and $\beta (g)=-{\beta}_{0}{g}^{3}+\cdots $ the beta function. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues ${\lambda}_{1},{\lambda}_{2},\dots $ of the matrix $\frac{{\gamma}_{0}}{{\beta}_{0}}$, in nonincreasing order ${\lambda}_{1}\ge {\lambda}_{2}\ge \cdots $, satisfy the nonresonant condition ${\lambda}_{i}-{\lambda}_{j}-2k\ne 0$ for $i\le j$ and *k* a positive integer, then a renormalization scheme exists where $-\frac{\gamma (g)}{\beta (g)}=\frac{{\gamma}_{0}}{{\beta}_{0}}\frac{1}{g}$ is one-loop exact to all perturbative orders. If in addition $\frac{{\gamma}_{0}}{{\beta}_{0}}$ is diagonalizable, $Z(x,\mu )$ 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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Funded by SCOAP^{3}