$B$-meson decay width up to $1/m_b^3$ corrections within and beyond the Standard Model

TL;DR

This paper develops a model independent HQE framework to quantify beyond Standard Model effects in non-leptonic, tree-level $b$-quark decays relevant for $B$-meson lifetimes. It derives analytic LO matching coefficients for all two-quark HQE operators up to mass-dimension six, including the previously missing weak-annihilation contributions and the Darwin and chromomagnetic operators, while regularising infrared divergences with both a light-quark mass regulator and dimensional regularisation. The results span the CKM favored channels $b\to c\bar{u}d$ and $b\to c\bar{c}s$ and also provide SM QCD-penguin induced coefficients up to dim-six, clarifying interference and quadratic penguin effects. These calculations enable robust constraints on generic BSM Wilson coefficients from lifetime observables such as $\tau(B^0_s)/\tau(B^0_d)$ and pave the way for precision tests combining lifetimes with other flavour data, including potential mappings to $W'$ models, diquarks, or extended Higgs sectors.

Abstract

Starting from the most general effective Hamiltonian describing non-leptonic $b$-quark decays $b\to q_1 \bar q_2 q_3$, we compute analytic expressions for all matching coefficients of the two-quark operator contributions in the heavy quark expansion~(HQE) of a $B$ meson, up to mass-dimension-six. In addition, we calculate the weak-annihilation contributions, which enter the matching of four-quark operators in the HQE at dimension-six and were previously missing. Our results complete the calculation of beyond Standard Model effects in non-leptonic, tree-level, $b$-quark decays relevant for $B$ meson lifetimes and lifetime ratios such as $τ(B^0_s)/τ(B^0_d)$. Such effects naturally arise in extensions of the Standard Model (SM) that aim to address the observed tensions between experimental measurements and theoretical predictions based on QCD factorisation in several colour-allowed non-leptonic $B$-meson decays. As a by-product of our calculation, we also determine the matching coefficients in the HQE induced by the QCD-penguin operators within the SM, including both the interference between current-current and penguin operators and the contributions quadratic in the penguin operators. Owing to the suppression of the QCD-penguin Wilson coefficients within the SM, these effects are typically regarded as corrections of order $α_s$ and $α_s^2$ in the strong coupling, respectively. Our results reproduce the known expressions at dimension-three and provide new results for the coefficients of the chromomagnetic operator at dimension-five and of the Darwin operator at dimension-six.

Figures (3)

• Figure 1: Schematic representation of the HQE for $B$-meson decays due to non-leptonic $b\to q_1 \bar{q}_2 q_3$ decays, including two-quark contributions (two-loop) and four-quark contributions (one-loop). The crossed red vertices indicate the insertion of the $\abs{\Delta B}=1$ effective Hamiltonian. The imaginary part is taken across the dotted line.
• Figure 2: One soft-gluon correction to the matrix element of the dimension-six $\Delta B = 0$ four-quark operators with light $q$ quarks, leading to operator mixing with the Darwin operator. The inclusion of these contributions ensures the cancellation of the IR divergences appearing in the matching coefficient of the Darwin operator.
• Figure 3: Diagrammatic representation of the $\Delta B = 0$ four-quark operator contributions in the HQE from the $b\to q_1 \bar{q}_2 q_3$ transition: WE (left), PI (middle), and WA (right).