Determine non-logarithmic NLO condensate coefficients

Determine the non-logarithmic next-to-leading-order Wilson-coefficient contributions b_{10}, c_{10}, and e_{10} for the gluon condensate \langle\alpha_s G^2\rangle, the dimension-six gluon condensate \langle g_s^3G^3\rangle, and the factorized four-quark condensate \alpha_s\langle\bar uu\rangle\langle\bar dd\rangle, respectively, since the renormalization-group methodology used determines only their leading-logarithmic coefficients.

Background

The paper extends the light tensor (J^P=2^+) hybrid correlation-function calculation beyond leading order in the strong coupling. For the non-perturbative operator-product-expansion terms, the authors use renormalization-group equations to derive the leading-logarithmic next-to-leading-order coefficients b_{11}, c_{11}, and e_{11} associated with the gluon condensate, dimension-six gluon condensate, and factorized four-quark condensate.

The resulting parametrizations also contain non-logarithmic NLO coefficients b_{10}, c_{10}, and e_{10}. These constants are not fixed by the leading-order renormalization-group analysis, so a complete NLO treatment of the corresponding condensate contributions would require an explicit calculation beyond the leading-logarithmic approximation.