Next-to-leading order calculation of a fragmentation function in a light-cone gauge

The short-distance coefficients for the color-octet ${}^{3}S_1$ term in the fragmentation function for a gluon to split into polarized heavy quarkonium states are recalculated to order ${\alpha }_s^2$. The light-cone gauge remarkably simplifies the calculation by eliminating many Feynman diagrams at the expense of introducing spurious poles in loop integrals. We do not use any conventional prescriptions for spurious pole. Instead, we only use gauge invariance with the aid of Collins-Soper definition of the fragmentation function. Our result agrees with a previous calculation of Braaten and Lee in the Feynman gauge, but disagrees with another previous calculation.

Figure 1

Leading-order Feynman diagram for $g\to Q\overline{Q}$.

Figure 2

The Feynman diagrams of order ${\alpha }_s^2$ for $g\to Q\overline{Q}$ with $Q\overline{Q}$ final states. There are additional contributions from the complex-conjugate diagrams.

Figure 3

One-loop correction diagrams for $g^{*}\to Q\overline{Q}$.

Figure 4

The Feynman diagrams of order ${\alpha }_s^2$ for $g\to Q\overline{Q}$ with $Q\overline{Q}g$ final states. There are a total of 25 diagrams, but only the left halves of the diagrams are shown.