The authors' preliminary semileptonic calculation points to a higher value of Vub inferred from inclusive B decays. They say the N3LO corrections further decrease the partial rate, Γ_u, and appear to widen the gap with the exclusive determination. The result is not final: the perturbative tail of the shape function and a full uncertainty analysis have not yet been included.
The paper calculates heavy-to-light form factors at third order in QCD and applies those ingredients to inclusive B-decay calculations in the radiative channel B̄ → X_sγ and the semileptonic channel B̄ → X_uℓν̄_ℓ, where corrections through three loops are used to determine partial rates.
Building the three-loop ingredients
The form-factor setup uses a massless quark and one on-shell massive quark of mass m. All possible current insertions are calculated in full QCD through three loops, with the fermion mass neglected. The amplitudes are then converted into finite matching coefficients for the decay calculations.
To manage the calculation, the authors report 429 master integrals, the building blocks used to assemble the larger result. An independent tensor calculation at zero momentum transfer, written as , uses 246 master integrals, with s denoting the momentum-transfer invariant. Remaining master integrals are evaluated semi-analytically by matching series expansions around regular and singular points.
Infrared subtraction uses a universal factor from SCET. The finite hard matching coefficient is defined by , where C is the coefficient, Z is the universal infrared subtraction factor and F is the ultraviolet-renormalized form factor. Those coefficients are then used as inputs for the decay calculations.
A new hard function for radiative decay
For B̄ → X_sγ, the paper extracts the N3LO hard function as the square of a particular linear combination of tensor-current matching coefficients. N3LO denotes the third order included here, making this a higher-order short-distance input for the radiative decay calculation.
At the m_b scale, the reported hard-function expansion has coefficients -4.5483, -19.2861 and -181.1617. The paper says the three-loop hard function is completely analytic in the leading-colour approximation, a restricted colour structure rather than the full colour result.
The semileptonic result remains provisional
Within the semileptonic factorization calculation, QCD corrections through three loops are implemented to determine partial rates in B̄ → X_uℓν̄_ℓ. The initial shape-function treatment uses an exponential model and applies shape-function-scheme parameter constraints at three-loop level.
On that basis, the preliminary direction is a further decrease in Γ_u and a possible increase in the inclusive Vub determination. The authors say this appears to widen the gap with the exclusive determination. Because the calculation is still being developed, this should not be read as a final numerical shift.
Resummation in the calculation uses the QCD beta function and cusp anomalous dimension through five loops. The cusp anomalous dimension is known only through four loops, so its five-loop term is approximated. The four-loop jet anomalous dimension is supplied through the relation , where γ_J is the jet anomalous dimension, γ_q is the light-quark anomalous dimension and γ_virt is the virtual anomalous dimension.
The authors also report that the heavy-quark parameter relations in the shape-function scheme can be calculated through four loops, with the results slated for publication. The semileptonic prediction depends on a nonperturbative shape function, and the authors stress that the perturbative tail and full uncertainty analysis still need to be added.
A useful calculation with work still ahead
Taken together, the findings are new perturbative inputs and an early phenomenological indication, not a definitive revision of inclusive Vub. The semileptonic result remains preliminary, and the complete analytic three-loop hard function is available only in the leading-colour approximation.
The document is an arXiv version 1 preprint dated 28 Aug 2026. Funding is acknowledged from DFG grant 396021762 (TRR 257), the DFG Excellence Strategy Cluster of Excellence “Color meets Flavor” (EXC 3107, Project-ID 533766364), SNSF contract TMSGI2_211209, and EU MSCA Grant 101204018.
Paper data and sources
Original title: Three-loop QCD corrections to heavy-to-light form factors and applications to inclusive $B$ decays
Authors: Matteo Fael, Tobias Huber, Fabian Lange et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text