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The short-distance coefficients for the color-octet 3 S 1 term in the fragmentation function for a gluon to split into heavy quarkonium states is calculated to order α 2 s . The gauge-invariant definition of the fragmentation function by Collins and Soper is employed. Ultraviolet divergences are removed using the MS renormalization procedure. The longitudinal term in the fragmentation function agrees with a previous calculation by Beneke and Rothstein. The next-to-leading order correction todoi:10.1142/s0217751x01006565 fatcat:e2kzwtm54zgvvhukl2f2q3yly4