Matching of the Shape Function

TL;DR

This paper calculates one-loop corrections to the shape function in semi-leptonic B decays using a hard cut-off, ensuring infrared safety and comparing results with dimensional regularization.

Contribution

It provides the first computation of the shape function corrections with a hard cut-off and analyzes the differences with dimensional regularization methods.

Findings

The matching constant onto full QCD is infrared safe.

The hard cut-off result differs from dimensional regularization by a factor of two in the log^2(k_+) coefficient.

Dimensional regularization requires oversubtraction due to additional factors.

Abstract

The shape function f(k_+) describes Fermi motion effects in inclusive semi-leptonic decays such as B -> X_u+e+nu 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.