The photon energy spectrum in B-> X_s + γin perturbative QCD through O(α_s^2)

TL;DR

This paper computes the dominant part of the O(α_s^2) perturbative correction to the photon energy spectrum in B→X_s γ, focusing on the O_7 operator.Using an optical-theorem approach with a photon-energy constraint and advanced multiloop techniques, it derives the normalized spectrum including non-BLM contributions and decomposes them into color structures. The study shows non-BLM corrections are at the percent level and that the usual z→1 singular terms are not reliable for moderate z, informing improved predictions for cut-dependent observables like R(E_cut) and the first moment ⟨E_γ⟩. The results, together with phenomenological estimates for E_cut≈1.8 GeV, tighten the theoretical control over heavy-quark parameters and the extraction of the b-quark mass from radiative B decays.

Abstract

We derive the dominant part of the O(α_s^2) correction to the photon energy spectrum in the inclusive decay B-> X_s+gamma. The detailed knowledge of the spectrum is important for relating the theoretical calculations of the B-> X_s + γdecay rate and the experimental measurements where a cut on the photon energy is applied. In addition, moments of the photon energy spectrum are used for the determination of the b-quark mass and other fundamental parameters of heavy quark physics. Our calculation reduces the theoretical uncertainty associated with uncalculated higher orders effects and shows that, for B-> X_s+γ, QCD radiative corrections to the photon energy spectrum are under theoretical control.

Figures (6)

• Figure 1: The normalized perturbative spectrum Eq.(\ref{['result']}) in $b\to s+\gamma$ retaining ${\cal O}(\alpha_s^2)$ (solid), BLM (dots) and ${\cal O}(\alpha_s)$ terms.
• Figure 2: The non-BLM ${\cal O}(\alpha_s^2)$ correction to the photon energy spectrum relative to the ${\cal O}(\alpha_s)$ correction.
• Figure 3: The non-BLM ${\cal O}(\alpha_s^2)$ correction (solid), its $z \to 1$ approximation (dots-dashes), the $z \to 1$ approximation minus its value for $z=0$ (dots) and the $z \to 1$ asymptotics times $z^3$ (dashes).
• Figure 4: Same as Fig.\ref{['q4']}. Each curve is normalized to the non-BLM ${\cal O}(\alpha_s^2)$ correction.
• Figure 5: The function $\Delta_R(z)$ Eq.(\ref{['R']}) for an experimentally relevant range of $z_{cut}$. Only the non-BLM corrections are used in the calculation.