By Alexander Mitov
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In what follows, we will describe the method of Catani and Trentadue [53] originally applied to Drell-Yan and DIS. We will apply their method in the next two chapters for the calculation of the resummed expressions for the coefficient functions for top-quark decay and inclusive CC DIS. The equivalence of their result with the Sterman’s [54] was shown in [55]. The coefficient function represents the normalized cross-section for production of a massless quark after the corresponding collinear singularities are subtracted.
The most convenient way to regularize those divergences is to work with dimensional regularization since it preserves both Poincare and gauge CHAPTER 2. PRELIMINARIES: PERTURBATIVE QCD 31 invariance (unlike gluon-mass regularization). It was understood long ago [37] that the problem of IR divergences is rooted in the way the physical observables are defined. It is intuitively clear that a state containing a hard parton cannot be distinguished from a state containing in addition arbitrary number of soft (or collinear) gluons.
Thus, the effective perturbation parameter is not αS anymore but αS × (a power of ln m2 /Q2 ). The latter can be quite large and can even invalidate the perturbation series. In effect, small but non-zero quark mass leaves the result finite but unphysical; one should sum up to all orders terms of this type in order to be able to make definite perturbative predictions. Such large logs are called quasi-collinear logs and are classified in the following way: a term at order αSn has the form n αSn k=0 ck lnk m2 .