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

The elementary fragmentation function that describes the process <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>q</a:mi><a:mo>→</a:mo><a:mi>π</a:mi></a:mrow></a: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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>z</b:mi><b:mo>≲</b:mo><b:mn>0.8</b:mn></b:mrow></b:math> but otherwise in good qualitative agreement. The full pion fragmentation function is overall greater than the elementary fragmentation function below <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>z</c:mi><c:mo>≲</c:mo><c:mn>0.6</c:mn></c:mrow></c:math>.