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Soft, collinear and nonrelativistic modes in radiative decays of very heavy quarkonium
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We analyze the end-point region of the photon spectrum in semi-inclusive radiative decays of very heavy quarkonium (m alpha_s^2 >> Lambda_QCD). We discuss the interplay of the scales arising in the Soft-Collinear Effective Theory, m, m(1-z)^{1/2} and m(1-z) for z close to 1, with the scales of heavy quarkonium systems in the weak coupling regime, m, m alpha_s and m alpha_s^2. For 1-z \sim alpha_s^2 only collinear and (ultra)soft modes are seen to be relevant, but the recently discovered soft-collinear modes show up for 1-z << alpha_s^2. The S- and P-wave octet shape functions are calculated. When they are included in the analysis of the photon spectrum of the Upsilon (1S) system, the agreement with data in the end-point region becomes excellent. The NRQCD matrix elements <1^3S_1| O_8(^1S_0) | 1^3S_1> and <1^3S_1| O_8(^3P_J) | 1^3S_1> are also obtained.
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GARCIA I TORMO, Xavier and SOTO RIERA, Joan. Soft, collinear and nonrelativistic modes in radiative decays of very heavy quarkonium. Physical Review D. 2004. Vol. 69, num. 11, pags. 114006-1-114006-9. ISSN 0556-2821. [consulted: 11 of August of 2026]. Available at: https://hdl.handle.net/2445/12481