Four-loop hard anomalous dimension

Determine the four-loop hard anomalous dimension required for the N$^3$LO resummation of partial decay rates in inclusive $\bar B \to X_u l \bar\nu_l$ decays within the BLNP factorization framework.

Background

The BLNP factorization formula requires renormalization-group exponents evaluated to sufficiently high perturbative order. At N3^3LO, the hard anomalous dimension γ′\gamma' must be known through four loops. The paper explains that this quantity is unavailable and that the lower-loop results give a poor convergence for a direct Padé approximation, so the authors instead relate it to the collinear light- and heavy-quark anomalous dimensions and approximate the heavy-quark contribution.

A direct determination of the four-loop hard anomalous dimension would replace this approximation and improve the theoretical control of the N3^3LO predictions for partial semileptonic decay rates and the inclusive extraction of ∣Vub∣|V_{ub}|.

References

The hard anomalous dimension $\gamma\prime$ has to be provided to four loops, which is also unknown.

— Three-loop QCD corrections to heavy-to-light form factors and applications to inclusive $B$ decays  (2608.28172 - Fael et al., 28 Aug 2026) in Section “Semileptonic $\bar B \to X_u l \bar \nu_l$ decays to three loops,” subsection “Resummation ingredients”

The only caveat is that we need the soft anomalous dimension $\gamma{\mathrm{soft}$ to four loops, which is unknown.

— Three-loop QCD corrections to heavy-to-light form factors and applications to inclusive $B$ decays  (2608.28172 - Fael et al., 28 Aug 2026) in Section “Semileptonic $\bar B \to X_u l \bar \nu_l$ decays to three loops,” subsection “Shape-function scheme”