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