Soft, collinear and non-relativistic modes in radiative decays of very heavy quarkonium

We analyze the end-point region of the photon spectrum in semi-inclusive radiative decays of very heavy quarkonium (mαs ≫ ΛQCD). We discuss the interplay of the scales arising in the Soft-Collinear Effective Theory, m, m(1 − z) and m(1 − z) for z close to 1, with the scales of heavy quarkonium systems in the weak coupling regime, m, mαs and mα 2 s . For 1 − z ∼ αs only collinear and (ultra)soft modes are seen to be relevant, but the recently discovered soft-collinear modes show up for 1 − z ≪ αs . The Sand P -wave octet shape functions are calculated. When they are included in the analysis of the photon spectrum of the Υ(1S) system, the agreement with data in the end-point region becomes excellent. The NRQCD matrix elements 〈 1S1|O8(S0)|1S1 〉