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

TL;DR

This paper applies differential geometry and the Poincaré-Dulac theorem to analyze operator mixing in massless QCD-like theories, identifying conditions for diagonalizability of the mixing matrix across all perturbative orders.

Contribution

It introduces a geometric framework and uses the Poincaré-Dulac theorem to determine when the renormalized mixing matrix can be diagonalized in massless QCD-like theories.

Findings

A scheme exists where the anomalous dimension matrix is one-loop exact under certain eigenvalue conditions.

Diagonalizability of the mixing matrix is linked to the eigenvalues of the ratio of anomalous dimensions to the beta function.

Remaining cases of operator mixing are classified using the Poincaré-Dulac theorem.

Abstract

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 Poincar\'e-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 Poincar\'e-Dulac theorem, if the eigenvalues $\lambda_1, \lambda_2, \cdots $ of the matrix $\frac{\gamma_0}{\beta_0}$, in nonincreasing order $\lambda_1 \geq \lambda_2 \geq \cdots$, satisfy the nonresonant condition $\lambda_i -\lambda_j -2k \neq 0$ for $i\leq j$ 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 Poincar\'e-Dulac theorem.