Four-body contributions to B ¯ → X s γ $$ \overline{B}\to {X}_s\gamma $$ at NLO

Abstract

Ongoing efforts to reduce the perturbative uncertainty in theB → X s γ decay rate have resulted in a theory estimate to NNLO in QCD. However, a few contributions from multi-parton final states which are formally NLO are still unknown. These are parametrically small and included in the estimated error from higher order corrections, but must be computed if one is to claim complete knowledge of theB → X s γ rate to NLO. A major part of these unknown pieces are four-body contributions corresponding to the partonic process b → sqqγ. We compute these NLO four-body contributions toB → X s γ, and confirm the corresponding tree-level leading-order results. While the NLO contributions arise from tree-level and one-loop Feynman diagrams, the four-body phase-space integrations make the computation non-trivial. The decay rate contains collinear logarithms arising from the mass regularization of collinear divergences. We perform an exhaustive numerical analysis, and find that these contributions are positive and amount to no more than ∼ 1% of the total rate in the Standard Model, thus confirming previous estimates of the perturbative uncertainty.

Weak radiative decays of the B meson and bounds on $$M_{H^\pm }$$ M H ± in the Two-Higgs-Doublet Model

“…[29,30]. Such effects are small in B sγ (unless one goes well below E 0 = 1.6 GeV) but become much more relevant in B dγ where the tree-level b → duūγ transitions are not CKM-suppressed with respect to the leading b → dγ one.…”

The (Q7, Q1,2) contribution to B ¯ → X s γ $$ \overline{B}\to {X}_s\gamma $$ at O α s 2 $$ \mathcal{O}\left({\alpha}_{\mathrm{s}}^2\right) $$

“…where T (0) ij describe small tree-level contributions to b → sqqγ from Q u 1,2 and Q 3,...,6 [23,24]. At the NLO and NNLO, numerically dominant effects come from G (n) 77 , G (n) 17 and G (n) 27 .…”

“…[29,32,36]. Last but not least, the current analysis includes the previously unknown m c -independent part of G (2) 78 [37], all the relevant BLM corrections to G (2) ij with i, j = 7 [31,38,39], tree-level contributions T (0) ij [23,24], four-body NLO corrections [24], as well as the updated non-perturbative corrections [14,40,41]. The only contributions to G (2) ij with i, j ∈ {1, 2, 7, 8} that remain neglected are the unknown (n ≥ 3)-body final state contributions to the non-BLM parts of G (2) ij with i, j = 7.…”

Rare B decays as tests of the Standard Model

“…2. The calculation of the matrix elements has seen significant progress in recent years [42,102,43,[103][104][105][106][107][108][109][110][111], allowing an improved NNLO prediction of thē B → X s γ branching fraction for a photon energy E γ > 1.6 GeV [112]:…”