NRQCD matrix elements for S-wave bottomonia and Γ[ηb(nS)→γγ] with relativistic corrections

We determine the leading-order nonrelativistic quantum chromodynamics (NRQCD) matrix element 〈O1〉ϒ and the ratio 〈q2〉ϒ, for ϒ=ϒ(nS) with n=1, 2, and 3 by comparing the measured values for Γ[ϒ→e+e−] with the NRQCD factorization formula in which relativistic corrections are resummed to all orders in the heavy-quark velocity v. The values for 〈q2〉ϒ, which is the ratio of order-v2 matrix element to 〈O1〉ϒ, are new. They can be used for NRQCD predictions involving ϒ(nS) and ηb(nS) with relativistic corrections. As an application, we predict the two-photon decay rates for the spin-singlet states: Γ[ηb(1S)→γγ]=0.512−0.094+0.096 keV, Γ[ηb(2S)→γγ]=0.235−0.043+0.043 keV, and Γ[ηb(3S)→γγ]=0.170−0.031+0.031 keV.