Numerical solution of image evolution equations for fragmentation functions Original Research Article

Semi-inclusive hadron-production processes are becoming important in high-energy hadron reactions. They are used for investigating properties of quark–hadron matters in heavy-ion collisions, for finding the origin of nucleon spin in polarized lepton–nucleon and nucleon–nucleon reactions, and possibly for finding exotic hadrons. In describing the hadron-production cross sections in high-energy reactions, fragmentation functions are essential quantities. A fragmentation function indicates the probability of producing a hadron from a parton in the leading order of the running coupling constant image. Its image dependence is described by the standard DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi) evolution equations, which are often used in theoretical and experimental analyses of the fragmentation functions and in calculating semi-inclusive cross sections. The DGLAP equations are complicated integro-differential equations, which cannot be solved in an analytical method. In this work, a simple method is employed for solving the evolution equations by using Gauss–Legendre quadrature for evaluating integrals, and a useful code is provided for calculating the image evolution of the fragmentation functions in the leading order (LO) and next-to-leading order (NLO) of image. The renormalization scheme is image in the NLO evolution. Our evolution code is explained for using it in oneʼs studies on the fragmentation functions.