Two-loop form factors for $P$-wave quarkonium production and decay

Abstract

We present the analytical results for the two-loop form factors needed for ${\chi }_{Q,J}$ production and decay. We consider the two-loop corrections to the process $\gamma \gamma \leftrightarrow {}^3{P}_J^{[1]}$, that has been known only numerically before, and the processes $gg\leftrightarrow {}^3{P}_J^{[1]}$, $\gamma g\leftrightarrow {}^3{P}_J^{[8]}$ and $gg\leftrightarrow {}^3{P}_J^{[8]}$, which have not been computed before. We observe that the NRQCD pole structure of the two-loop amplitude in the $gg$ channel is more involved for the spin-triplet $P$-wave case than for the pseudo-scalar $S$-wave case. It involves in addition to the standard Coulomb singularity also a new singularity whose cancellation requires the inclusion of the $gg\leftrightarrow {}^3{S}_1^{[8]}$ form factor. We give the high precision numerical results for the hard functions that can be used to compute ${\chi }_{Q,J}$ production and decay up to NNLO accuracy.

68 pages, 2 figures, v2: version published in JHEP

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