NLO corrections to dimension-seven and heavy-heavy spectator contributions

Derive the next-to-leading-order QCD corrections for the dimension-seven heavy-light spectator operators and for the heavy-heavy dimension-six operator contribution involving the bottom–charm pair in inclusive Xi_{bc} and Omega_{bc} lifetime calculations.

Background

The heavy-quark expansion used to calculate inclusive bcq-baryon lifetimes contains dimension-seven heavy-light spectator operators and a heavy-heavy contribution involving both the bottom and charm quarks. In the numerical analysis, the partonic coefficients are available through NNLO and several lower-dimensional spectator terms through NLO, but the dimension-seven and heavy-heavy sectors are treated only at leading order. Improving these perturbative ingredients is relevant because spectator effects substantially control the predicted lifetime hierarchy and the dependence on the bc-diquark mixing angle.

References

For the dimension-seven operators and O{bc}, for which NLO corrections are not yet known, we use the LO values of C_{1,2}.

— Mixing-angle sensitivity of inclusive $Ξ_{bc} $ and $Ω_{bc}$ lifetimes  (2609.00660 - Cheng et al., 1 Sep 2026) in Section 2, Theoretical framework