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Abstract Recently, the short-distance asymptotics of the generating functional of n-point correlators of twist-2 operators in SU(N) Yang–Mills (YM) theory has been worked out in Bochicchio et al. (Phys Rev D 108:054023, 2023). The above computation relies on a basis change of renormalized twist-2 operators, where $$-\gamma (g)/ \beta (g)$$ - γ ( g ) / β ( g ) reduces to $$\gamma _0/ (\beta _0\,g)$$ γ 0 / ( β 0 g ) to all orders of perturbation theory, with $$\gamma _0$$ γ 0 diagonal, $$\gamma (g) = \gamma _0 g^2+\cdots $$ γ ( g ) = γ 0 g 2 + ⋯ the anomalous-dimension matrix and $$\beta (g) = -\beta _0 g^3+\cdots $$ β ( g ) = - β 0 g 3 + ⋯ the beta function. The construction is based on a novel geometric interpretation of operator mixing (Bochicchio in Eur Phys J C 81:749, 2021), under the assumption that the eigenvalues of the matrix $$\gamma _0/ \beta _0$$ γ 0 / β 0 satisfy the nonresonant condition $$\lambda _i-\lambda _j\ne 2k$$ λ i - λ j ≠ 2 k , with $$\lambda _i$$ λ i in nonincreasing order and $$k\in {\mathbb {N}}^+$$ k ∈ N + . The nonresonant condition has been numerically verified up to $$i,j=10^4$$ i , j = 10 4 in Bochicchio et al. (Phys Rev D 108:054023, 2023). In the present paper we provide a number theoretic proof of the nonresonant condition for twist-2 operators essentially based on the classic result that Harmonic numbers are not integers. Our proof in YM theory can be extended with minor modifications to twist-2 operators in $$\mathcal {N}=1$$ N = 1 SUSY YM theory, large-N QCD with massless quarks and massless QCD-like theories. |
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