Resummation of soft gluon logarithms in the DGLAP evolution of fragmentation functions

We define a general scheme for the evolution of fragmentation functions which resums both soft gluon logarithms and mass singularities in a consistent manner and to any order, and requires no additional theoretical assumptions. Using the double logarithmic approximation and the known perturbative results for the splitting functions, we present our scheme with the complete contribution from the double logarithms, being the largest soft gluon logarithms. We show that the resulting approximation is more complete than the modified leading logarithm approximation even with the fixed order contribution calculated to leading order only, and find, after using it to fit quark and gluon fragmentation functions to experimental data, that this approximation in our scheme gives a good description of the data from the largest $x_p$ values to the peak region in $\xi =\ln (1/x_p)$, in contrast to other approximations. In addition, we develop a treatment of hadron mass effects which gives additional improvements at large $\xi$.