Order-v4 corrections to heavy quark fragmentation to S-wave heavy quarkonium

A bstract Within the framework of nonrelativistic quantum chromodynamics (NRQCD) factorization, we compute the $$ \mathcal{O}\left({v}^4\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mfenced> </mml:math> relativistic corrections to the fragmentation of a heavy quark into the color-singlet $$ {}^1{S}_0^{\left[1\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>1</mml:mn> </mml:mmultiscripts> </mml:math> and $$ {}^3{S}_1^{\left[1\right]} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> <mml:mprescripts/> <mml:none/> <mml:mn>3</mml:mn> </mml:mmultiscripts> </mml:math> quarkonium states. Using the Collins-Soper definition of the fragmentation function, we reproduce the known $$ \mathcal{O}\left({v}^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:math> results. We find that the $$ \mathcal{O}\left({v}^4\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mfenced> </mml:math> correction gives a positive contribution relative to the leading order result over a wide range of the light-cone momentum fraction z , while its magnitude remains much smaller than that of the $$ \mathcal{O}\left({v}^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:math> correction. This behavior indicates a good convergence of the NRQCD relativistic expansion in this process. We further extend the calculation to the fragmentation functions in the unequal-mass case at $$ \mathcal{O}\left({v}^4\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mfenced> </mml:math> and obtain the corresponding analytical expressions.