Isospin asymmetries inB→(K*,ρ)γ/l+l−andB→Kl+l−in and beyond the standard model

Abstract

We compute the isospin asymmetries in $B\ensuremath{\rightarrow}({K}^{*},\ensuremath{\rho})\ensuremath{\gamma}$ and $B\ensuremath{\rightarrow}(K,{K}^{*},\ensuremath{\rho}){l}^{+}{l}^{\ensuremath{-}}$ 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 $B\ensuremath{\rightarrow}(K,{K}^{*},\ensuremath{\rho})ll$, between $1\text{ }\text{ }{\mathrm{GeV}}^{2}\ensuremath{\le}{q}^{2}\ensuremath{\le}4{m}_{c}^{2}$, are predicted to be small (below 1.5%) though with significant cancellation. In the SM the non-$CP$-averaged asymmetries for $B\ensuremath{\rightarrow}\ensuremath{\rho}ll$ deviate by $\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}5%$ 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 $B\ensuremath{\rightarrow}Kll$ 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 $B\ensuremath{\rightarrow}\ensuremath{\rho}ll$ and $B\ensuremath{\rightarrow}{K}^{*}ll$ isospin asymmetries are structurally different yet the closeness of ${\ensuremath{\alpha}}_{\mathrm{CKM}}$ to $90\ifmmode^\circ\else\textdegree\fi{}$ allows us to construct a (quasi)null test for the SM out of the respective isospin symmetries. We provide and discuss an update on $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{K}^{*0}\ensuremath{\gamma})/\mathcal{B}({B}_{s}\ensuremath{\rightarrow}\ensuremath{\phi}\ensuremath{\gamma})$ which is sensitive to WA.