Rare $W \to B_c + \gamma$ decay up to the NNLO and NLL accuracy in QCD

TL;DR

This paper provides a comprehensive theoretical analysis of the rare $W o B_c + ext{gamma}$ decay, calculating form factors up to NNLO in QCD and resumming large logarithms to NLL accuracy, revealing significant corrections and scale dependence effects.

Contribution

It introduces a combined NNLO and NLL resummation approach for the decay, improving the precision of theoretical predictions for this rare process.

Findings

Decay width reduced by 19% at NLO and 31% at NNLO.

NLL resummation significantly alters fixed-order predictions.

Radiative corrections increase scale dependence, mitigated by NLL resummation.

Abstract

We perform a detailed theoretical study of the rare radiative decay of the $W$ boson into a $B_c$ meson and an on-shell photon. The decay amplitude is described by two independent form factors, which are calculated up to the next-to-next-to-leading order (NNLO) in QCD within the nonrelativistic QCD (NRQCD) factorization formalism. Since the two typical energy scales, the $W$-boson mass $m_W$ and the $B_c$-meson mass $m_{B_c}$, involved in the process are widely separated, large logarithms of $m_W^2/m_{B_c}^2$ present in the NRQCD short-distance coefficients are also resummed to all orders in $\alpha_s$ up to the next-to-leading logarithmic (NLL) accuracy, by employing the light-cone factorization approach. Taking into account all these corrections, we then perform a phenomenological exploration of this rare decay. It is found that, relative to the leading-order result, the decay width of the process is reduced by the next-to-leading-order and NNLO corrections, with a net effect of $\sim19\%$ and of $\sim31\%$, respectively. Furthermore, the NLL resummation can considerably alter the fixed-order NRQCD predictions, especially for the $\mathcal{O}(\alpha_s)$ correction. We also find that the radiative corrections increase the renormalization scale dependence of the branching fraction, which is however significantly reduced by the NLL resummation. The dependence of the branching fraction on the heavy-quark masses $m_{b,c}$ is also investigated, which shows a monotonic decrease (increase) with $m_c$ ($m_b$).