NRQCD matrix elements for S-wave bottomonia and Gamma[eta_b(nS) -> gamma gamma] with relativistic corrections

Abstract

We determine the leading-order nonrelativistic quantum chromodynamics (NRQCD) matrix element < O_1 >_Upsilon and the ratio < q^2>_Upsilon, for Upsilon=Upsilon(nS) with n=1, 2, and 3 by comparing the measured values for Gamma[Upsilon -> 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 < q^2 >_Upsilon, which is the ratio of order-v^2 matrix element to < O_1 >_Upsilon, are new. They can be used for NRQCD predictions involving Upsilon(nS) and eta_b(nS) with relativistic corrections. As an application, we predict the two-photon decay rates for the spin-singlet states: Gamma[eta_b(1S) -> gamma gamma] = 0.512^{+0.096}_{-0.094} keV, Gamma[eta_b(2S) -> gamma gamma] = 0.235^{+0.043}_{-0.043} keV, and Gamma[eta_b(3S) -> gamma gamma] = 0.170^{+0.031}_{-0.031} keV.