On the quark distribution in an on-shell heavy quark and its all–order relations with the perturbative fragmentation function

I present new results on the quark distribution in an on-shell heavy quark in perturbative QCD and explore its all–order relations with heavy–quark fragmentation. I first compute the momentum distribution function to all orders in the large–β0 limit and show that it is identical to the perturbative heavy–quark fragmentation function in the same approximation. I then analyze the Sudakov limit of the distribution and the fragmentation functions using Wilson lines and prove that the corresponding Sudakov exponents in the non-Abelian theory are the same to any logarithmic accuracy. The anomalous dimension is then determined to two–loop order, corresponding to next–to–next–to–leading logarithmic accuracy in the exponent, in two ways: the first by extracting the singular terms from a recent calculation of the fragmentation function and the second by performing the two–loop Wilson–line calculation in configuration space. I find perfect agreement between the two.