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
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