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 (JP=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.

References

Note that the RGE methodology only determines the leading-log terms, and hence cannot determine b_{10}, c_{10}, and e_{10}.

— Next-to-Leading-Order Calculation of the Light Tensor $(J^P=2^+)$ Hybrid Correlation Function  (2609.09669 - Kleiv et al., 9 Sep 2026) in Section 6, “Leading-Log NLO Contributions to Condensates,” immediately following Eq. (\ref{eq:b_c_d_terms})

For the dimension-six four-quark Wilson coefficient considered here, however, higher-order corrections remain to be calculated and could modify the magnitude of the mass shift.

— One-nucleon intermediate-state contribution to the pion-coupled four-quark condensate in the $N$--$J/ψ$ interaction  (2610.08260 - Yeo et al., 6 Oct 2026) in Section 3, “Results”