# DGLAP: Kollinear divergence in splitting functions?

1. Nov 6, 2008

### blue2script

Hello all!

I am just preparing for an oral exam in QCD and try to figure out the interplay of UV and IR divergences, regularisation, renormalization and virtual diagrams.

As to now, my idea of the whole thing is this:

1) In lowest order no divergences occur
2) In second order we get UV divergences from loops and IR divergences from emissions of particles (in DIS this is one gluon in NLO)
3) To calculate with divergent subjects we may introduce some regulation scheme
4) UV divergences, if not canceled against other diagrams, are subject to renormalization
5) IR divergences always have to cancel (well, somewhat, at least for DIS and DY, there are also non-infrared safe quantities)

Especially for DIS we have:

5) No loops in real contributions -> no UV divergences
6) However, we encounter IR divergences as collinear and soft-gluon divergences
7) These divergences can be split into two parts:
a) collinear and soft-gluon divergences of the incoming quark
b) collinear and soft-gluon divergences of the outgoing quark
8) Soft-gluon divergences (the "classical" IR divergence) has to cancel against the UV divergences of the (virtual) vertex corrections
9) The collinear divergence of the outgoing quark can not be part of the quark density since this density can not be dependent on the outgoing state. Neither can it be part of the hard scattering amplitude since this would spoil the factorization theorem (why?) -> thus this divergence will also be canceled by the virtual diagrams
10) The collinear divergence emerging from the gluon emitted from the incoming quark however is part of the quark density and as such contributes to the evolution equation.

Up to now: Am I right? Is this scheme correct? Now comes my question: Where do I find the collinear divergence of the incoming quark in the DGLAP equations? One DGLAP equation reads:

$$\frac{\partial f_q\left(x_{Bj},Q^2\right)}{\partial\ln Q^2} = \frac{\alpha_s}{\pi}\int\limits_{x_{Bj}}^{1}{\frac{d y}{y} P_{qq}\left(\frac{x_{Bj}}{y}\right)f_q\left(y,Q^2\right)}$$

with

$$P_{qq}\left(z\right) = \frac{4}{3}\left(\cfrac{1+z^2}{1-z}\right)_+$$

The plus-prescription in the splitting function will cancel the soft-gluon divergence (for a collinear gluon with zero momentum). However, where in this equation do I find the collinear divergence? To me this integral looks divergent... or am I wrong?

Another question: If the parton density in first order would be divergent, wouldn't we have to renormalize it? In a paper I found some thing of renormalized parton density, however in Greiner or in my lecture script this is not mentioned at all.

Thanks a lot for all comments, either on the scheme or on the collinear divergence! Guess understanding this part by heart will be crucial for my oral exam...

Blue2script