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

A bstract We present the analytical results for the two-loop form factors needed for χ Q , J production and decay. We consider the two-loop corrections to the process $$ \gamma \gamma \leftrightarrow {}^3{P}_J^{\left[1\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γγ</mml:mi> <mml:mo>↔</mml:mo> <mml:mmultiscripts> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mi>J</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>3</mml:mn> </mml:mmultiscripts> </mml:math> , that has been known only numerically before, and the processes $$ gg\leftrightarrow {}^3{P}_J^{\left[1\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>gg</mml:mi> <mml:mo>↔</mml:mo> <mml:mmultiscripts> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mi>J</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>3</mml:mn> </mml:mmultiscripts> </mml:math> , $$ \gamma g\leftrightarrow {}^3{P}_J^{\left[8\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γg</mml:mi> <mml:mo>↔</mml:mo> <mml:mmultiscripts> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mi>J</mml:mi> <mml:mfenced> <mml:mn>8</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>3</mml:mn> </mml:mmultiscripts> </mml:math> and $$ gg\leftrightarrow {}^3{P}_J^{\left[8\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>gg</mml:mi> <mml:mo>↔</mml:mo> <mml:mmultiscripts> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mi>J</mml:mi> <mml:mfenced> <mml:mn>8</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>3</mml:mn> </mml:mmultiscripts> </mml:math> , 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 $$ gg\leftrightarrow {}^3{S}_1^{\left[8\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>gg</mml:mi> <mml:mo>↔</mml:mo> <mml:mmultiscripts> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> <mml:mfenced> <mml:mn>8</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>3</mml:mn> </mml:mmultiscripts> </mml:math> form factor. We give the high precision numerical results for the hard functions that can be used to compute χ Q , J production and decay up to NNLO accuracy.