Photonic penguins at two loops and m_t-dependence of BR[ B -> X_s l^+ l^-]

TL;DR

This work performs a complete two-loop matching of the Standard Model onto an effective theory for the inclusive decay $B \to X_s \ell^+\ell^-$, reducing the dominant renormalization-scale uncertainty tied to the top-quark mass and achieving a NNLO-like refinement in the low-$\hat{s}$ region. By deriving explicit two-loop matching conditions and evolving the Wilson coefficients through the renormalization group, the authors obtain refined expressions for the effective coefficients $\tilde{C}_7^{Q\,eff}$, $\tilde{C}_9^{Q\,eff}(\hat{s})$, and $\tilde{C}_{10}^{Q\,eff}(\hat{s})$, including ${\cal O}(\alpha_s)$ corrections. The phenomenological analysis shows a substantial reduction in $\mu_0$-dependence (to about $2.5\%$ at $\hat{s}\approx0.2$) and predicts $BR[ B \to X_s \ell^+\ell^-]_{\hat{s}\in[0.05,0.25]} = (1.46 \pm 0.19) \times 10^{-6}$, with perturbative uncertainties around 13% and smaller non-perturbative effects. Remaining theoretical uncertainties arise mainly from uncalculated two-loop matrix elements of four-quark operators, motivating future calculations of these matrix elements to further sharpen the prediction and probe potential new physics through the sign of $\tilde{C}_7^{eff}(\mu_b)$.

Abstract

We calculate two-loop matching conditions for all the operators that are relevant to B -> X_s l^+ l^- decay in the Standard Model. In effect, we are able to remove the +_16% uncertainty in the decay spectrum, which was mainly due to the renormalization-scale dependence of the top-quark mass. We find 1.46 * 10^-6 for the branching ratio integrated in the domain 0.05 < m_{l^+l^-}^2/m_b^2 < 0.25, for l= e or mu. There remains around 13% perturbative uncertainty in this quantity, while the non-perturbative effects are expected to be smaller.

Figures (8)

• Figure 1: Perturbative and non-perturbative versions of Re$[h(m_c^2/m_b^2,\hat{s})-h(m_c^2/m_b^2,0)]$ and Im$[h(m_c^2/m_b^2,\hat{s})-h(m_c^2/m_b^2,0)]$ as functions of $\hat{s}$ (see the text).
• Figure 2: Reduction of $\mu_0$-dependence of $R^{l^+l^-}_{quark}(\hat{s})$.
• Figure 3: Remaining $\mu_b$-dependence of $R^{l^+l^-}_{quark}(\hat{s})$.
• Figure 4: One-loop 1PI diagrams for $b \to s \gamma$ in the SM. The charged would-be Goldstone boson is denoted by $\pi^{\pm}$. There is no $W^{\pm}\pi^{\mp}\gamma$ coupling in the background-field gauge.
• Figure 5: Two-loop 1PI diagrams for $b \to s \gamma$ in the SM. The wavy lines denote either the $W$-boson or the charged would-be Goldstone boson. The external photon can couple at any of the places marked by small circles.