We compute the isospin asymmetries in $B\to (K^{*},\rho )\gamma$ and $B\to (K,K^{*},\rho )l^{+}{l}^{-}$ for low lepton pair invariant mass $q^2$, within the Standard Model (SM) and beyond the SM in a generic dimension six operator basis. Within the SM the $CP$-averaged isospin asymmetries for $B\to (K,K^{*},\rho )ll$, between $1{\mathrm{GeV}}^2\le q^2\le 4m_c^2$, are predicted to be small (below 1.5%) though with significant cancellation. In the SM the non-$CP$-averaged asymmetries for $B\to \rho ll$ deviate by $\approx \pm 5\%$ from the $CP$-averaged ones. We provide physical arguments, based on resonances, of why isospin asymmetries have to decrease for large $q^2$ (towards the endpoint). Two types of isospin violating effects are computed: ultraviolet isospin violation due to differences between operators coupling to up and down quarks, and infrared isospin violation where a photon is emitted from the spectator quark and is hence proportional to the difference between the up- and down-quark charges. These isospin violating processes may be subdivided into weak annihilation (WA), quark loop spectator scattering, and a chromomagnetic contribution. Furthermore we discuss generic selection rules based on parity and angular momentum for the $B\to Kll$ transition as well as specific selection rules valid for WA at leading order in the strong coupling constant. We clarify that the relation between the $K$ and the longitudinal part of the $K^{*}$ only holds for leading twist and for left-handed currents. In general the $B\to \rho ll$ and $B\to K^{*}ll$ isospin asymmetries are structurally different yet the closeness of ${\alpha }_{\mathrm{CKM}}$ to $90°$ allows us to construct a (quasi)null test for the SM out of the respective isospin symmetries. We provide and discuss an update on $\mathcal{B}(B^0\to K^{*0}\gamma )/\mathcal{B}(B_s\to \varphi \gamma )$ which is sensitive to WA.

• Received 5 June 2013

DOI:https://doi.org/10.1103/PhysRevD.88.094004