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, 
, 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
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3
