Publication Date:
2021-08-01
Description:
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, $$-frac{gamma (g)}{ eta (g)}$$ - 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 $$ eta (g)=- eta _0 g^3 +cdots $$ 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 $$frac{gamma _0}{ eta _0}$$ γ 0 β 0 , in nonincreasing order $$lambda _1 ge lambda _2 ge cdots $$ λ 1 ≥ λ 2 ≥ ⋯ , satisfy the nonresonant condition $$lambda _i -lambda _j -2k
e 0$$ λ i - λ j - 2 k ≠ 0 for $$ile j$$ i ≤ j and k a positive integer, then a renormalization scheme exists where $$-frac{gamma (g)}{ eta (g)} = frac{gamma _0}{ eta _0} frac{1}{g}$$ - γ ( g ) β ( g ) = γ 0 β 0 1 g is one-loop exact to all perturbative orders. If in addition $$frac{gamma _0}{ eta _0}$$ 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.
Print ISSN:
1434-6044
Electronic ISSN:
1434-6052
Topics:
Physics
Permalink