Gluon Fragmentation into ${}^3{P}_J$ Quarkonium

Abstract

The functions of the gluon fragmentation into ${}^3{P}_J$ quarkonium are calculated to order ${\alpha }_s^2$. With the recent progress in analysing quarkonium systems in QCD we show explicitly how the socalled divergence in the limit of the zero-binding energy, which is related to $P$-wave quarkonia, is treated correctly in the case of fragmentation functions. The obtained fragmentation functions satisfy explicitly at the order of ${\alpha }_s^2$ the Altarelli-Parisi equation and when $z\to 0$ they behave as $z^{-1}$ as expected. Some comments on the previous results are made.

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