Pion fragmentation functions from a quark-jet model in a functional approach

The elementary fragmentation function that describes the process <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>q</mi> <mo>→</mo> <mi>π</mi> </mrow> </math> is predicted by applying crossing and charge symmetry to the cut diagram of the pion valence quark distribution function. This elementary probability distribution defines the ladder-kernel of a quark jet fragmentation equation, which is solved self-consistently to obtain the full pion fragmentation function. The hadronization into a pion employs the complete Poincaré invariant Bethe-Salpeter wave function, though the overwhelming contribution to the fragmentation function is due to the leading Bethe-Salpeter amplitude. Compared to a Nambu–Jona–Lasinio model prediction, the fragmentation function we obtain is enhanced in the range <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>z</mi> <mo>≲</mo> <mn>0.8</mn> </mrow> </math> but otherwise in good qualitative agreement. The full pion fragmentation function is overall greater than the elementary fragmentation function below <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>z</mi> <mo>≲</mo> <mn>0.6</mn> </mrow> </math> .