$\mathrm{SU}(2)\times \mathrm{U}(1)$ Gauge Invariance and the Shape of New Physics in Rare $B$ Decays

New physics effects in $B$ decays are routinely modeled through operators invariant under the strong and electromagnetic gauge symmetries. Assuming the scale for new physics is well above the electroweak scale, we further require invariance under the full standard model gauge symmetry group. Retaining up to dimension-six operators, we unveil new constraints between different new physics operators that are assumed to be independent in the standard phenomenological analyses. We illustrate this approach by analyzing the constraints on new physics from rare $B_q$ (semi-)leptonic decays.

Figure 1

In the upper panel we show the limits at 68% C.L. and 95% C.L. on the scalar Wilson coefficients that are induced by the experimental ${\overline{\mathcal{B}}}_{q\mu }$ in Eq. (10), where the corrections by mixing have been taken into account. For the electronic modes in the lower panel, we only show the 95% C.L. allowed regions (11). In both cases the Wilson coefficients are understood to be renormalized at $\mu =m_b$.