NRQCD matrix elements for S-wave bottomonia and Γ[η_b(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 ηb(3S)→γγ=0.170_-0.031 ^+0.031 keV.