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

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, $$Z(x, \mu )$$ Z ( x , μ ) , 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, $$-\frac{\gamma (g)}{\beta (g)}$$ - γ ( g ) β ( g ) as a (formal) meromorphic connection with a Fuchsian singularity at $$g=0$$ g = 0 , and $$Z(x,\mu )$$ Z ( x , μ ) as a Wilson line, with $$\gamma (g)=\gamma _0 g^2 + \cdots $$ γ ( g ) = γ 0 g 2 + ⋯ the matrix of the anomalous dimensions and $$\beta (g)=-\beta _0 g^3 +\cdots $$ β ( g ) = - β 0 g 3 + ⋯ the beta function. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues $$\lambda _1, \lambda _2, \ldots $$ λ 1 , λ 2 , … of the matrix $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 , in nonincreasing order $$\lambda _1 \ge \lambda _2 \ge \cdots $$ λ 1 ≥ λ 2 ≥ ⋯ , satisfy the nonresonant condition $$\lambda _i -\lambda _j -2k \ne 0$$ λ i - λ j - 2 k ≠ 0 for $$i\le j$$ i ≤ j and k a positive integer, then a renormalization scheme exists where $$-\frac{\gamma (g)}{\beta (g)} = \frac{\gamma _0}{\beta _0} \frac{1}{g}$$ - γ ( g ) β ( g ) = γ 0 β 0 1 g is one-loop exact to all perturbative orders. If in addition $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 is diagonalizable, $$Z(x, \mu )$$ Z ( x , μ ) 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.