Resumming the color-octet contribution to radiative Upsilon decay

TL;DR

This work tackles the endpoint of radiative Upsilon decay, where NRQCD's OPE breaks and color-octet contributions with large Sudakov logs become relevant. It introduces a collinear-soft effective field theory coupled to NRQCD to sum Sudakov logarithms and to describe endpoint physics via nonperturbative lightcone structure functions, enabling a resummed, convolution-form expression for the color-octet contribution. A phenomenological study with a simple shape function demonstrates how resummation broadens and shifts the endpoint spectrum and suggests that endpoint data could constrain color-octet matrix elements, albeit with the caveat that color-singlet resummation is also necessary for a full comparison. The findings highlight the importance of incorporating appropriate low-energy degrees of freedom and nonperturbative inputs to accurately describe the endpoint region and extract NRQCD matrix elements.

Abstract

At the upper endpoint of the photon energy spectrum in Upsilon -> X gamma, the standard NRQCD power counting breaks down and the OPE gives rise to color-octet structure functions. Furthermore, in this kinematic regime large Sudakov logarithms appear in the octet Wilson coefficients. The endpoint spectrum can be treated consistently within the framework of a recently developed effective field theory of collinear and soft particles. Here we show that within this approach the octet structure functions arise naturally and that Sudakov logarithms can be summed using the renormalization group equations. We derive an expression for the resummed energy spectrum and, using a model lightcone structure function, investigate the phenomenological importance of the resummation.

Figures (5)

• Figure 1: Matching onto operators in the effective field theory with one and two gluons in the final state.
• Figure 2: Collinear and soft diagrams needed to calculate the renormalization of the vertices in the effective theory.
• Figure 3: The leading OPE: tree level matching of the time ordered product in the collinear-soft theory to a nonlocal operator in the soft theory.
• Figure 4: The differential decay spectra near the endpoint region $0.7<z$ in arbitrary units. The solid curve is the perturbative resummation convoluted with the structure function and the dashed curve is the perturbative resummation $C_i(1-z)/\tilde{C}^{(0)}_i$. The dotted curve is the plus distribution terms in the one-loop result (\ref{['WCoct1']}), and the dot-dashed curve is these terms convoluted with the structure function.
• Figure 5: The differential decay spectra in the region $0.5<z$. The dashed curves are the fully resummed result convoluted with the shape function for two choices of the octet matrix elements. In addition we have interpolated the fully resummed result with the NLO result in the region away from the endpoint. The dotted curves are the NLO result convoluted with the structure function for two choices of the matrix elements. The solid curve is the color-singlet contribution.