Matching of the shape function

Abstract

The shape function f( k +) describes Fermi motion effects in inclusive semi-leptonic decays such as B→ X u + e+ ν near the end-point of the lepton spectrum. We compute the leading one-loop corrections to the shape function f( k +) in the effective theory with a hard cut-off regularization. The matching constant onto full QCD is infrared safe, i.e. the leading infrared singularity represented by the term log 2 k + cancels in the difference of integrals. We compare the hard cut-off result with the result in dimensional regularization, the latter containing an additional factor of two in the coefficient of the log 2 k + term, and consequently requiring an oversubtraction.