Comment on “QCD factorization with multihadron fragmentation functions”

We make several comments on the recent work in Rogers [] while also reaffirming and adding to the work in Pitonyak []. We show that the factorization formula for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>e</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>e</a:mi> <a:mo>−</a:mo> </a:msup> <a:mo stretchy="false">→</a:mo> <a:mo stretchy="false">(</a:mo> <a:msub> <a:mi>h</a:mi> <a:mn>1</a:mn> </a:msub> <a:mo>⋯</a:mo> <a:msub> <a:mi>h</a:mi> <a:mi>n</a:mi> </a:msub> <a:mo stretchy="false">)</a:mo> <a:mi>X</a:mi> </a:math> in Rogers is equivalent to a version one can derive using the definition of a <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>n</f:mi> </f:math> -hadron fragmentation function (FF) introduced in Pitonyak . In addition, we scrutinize how to generalize the number density definition of a single-hadron FF to a <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>n</h:mi> </h:math> -hadron FF, arguing that the definition given in Pitonyak should be considered the standard one while the definition in Rogers has no clear interpretation. We also emphasize that the evolution equations for dihadron FFs (DiFFs) in Pitonyak have the same splitting functions as those for single-hadron FFs. Therefore, the DiFF (and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>n</j:mi> </j:math> -hadron FF) definitions in Pitonyak have a natural number density interpretation and, contrary to what is stated in Rogers , are consistent with collinear factorization using the standard hard factors and evolution kernels.