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

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 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 ${\lambda }_1,{\lambda }_2,\cdots$ 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.

10 pages, paper shortened, minor changes, typos corrected

References in corpus (1)

• The large-N Yang-Mills S-matrix is ultraviolet finite, but the large-N QCD S-matrix is only renormalizable

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