DGLAP: Kollinear divergence in splitting functions?

[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

 

[eljose79]

Dear Blue2script,

Thank you for sharing your thoughts and questions on the interplay of UV and IR divergences, regularisation, renormalization, and virtual diagrams. It seems like you have a good understanding of the general concepts, but allow me to clarify and expand on some points.

Firstly, your understanding of the occurrence of divergences in different orders is correct. In lowest order, no divergences occur as there are no loops. In higher orders, both UV and IR divergences can arise. The UV divergences come from virtual loops, while the IR divergences come from real emissions of particles.

Secondly, you are correct in your understanding that we can introduce a regularization scheme to deal with divergences. This is necessary to be able to perform calculations and make predictions in quantum field theories. However, it is important to note that different regularization schemes can give different results, so it is important to choose a scheme that is consistent and appropriate for the problem at hand.

Thirdly, when it comes to renormalization, it is important to understand that this is a process of removing the divergences in a systematic way. This allows us to make meaningful predictions and calculations in quantum field theories. The UV divergences, if not canceled against other diagrams, are indeed subject to renormalization. However, IR divergences do not need to be renormalized as they are already finite and physical.

Fourthly, your understanding of the cancellation of IR divergences in DIS and DY processes is correct. The collinear and soft-gluon divergences can be split into two parts, and the soft-gluon divergence must cancel against the UV divergences of the virtual diagrams. The collinear divergence of the outgoing quark also cancels, as it cannot be part of the quark density or the hard scattering amplitude.

To answer your question about the DGLAP equations, the collinear divergence of the incoming quark can be found in the splitting function P_{qq}(z). This function is divergent at z=1, which corresponds to a collinear gluon with zero momentum. This divergence is canceled by the plus-prescription, which is why the integral does not appear divergent.

Finally, to address your question about the renormalized parton density, it is true that the parton density in first order would be divergent. However, the renormalization process takes care of

 

[Entropia]

Hello Blue2script,

Your understanding of the interplay between UV and IR divergences in QCD is correct. In particular, your description of the DGLAP equations for parton evolution is also correct. However, you are missing the crucial point that the collinear divergence of the incoming quark is already taken care of in the definition of the parton density function. Let me explain in more detail.

The DGLAP equations describe the evolution of the parton density function, which is defined as the probability of finding a parton (quark or gluon) with a certain momentum fraction x inside a hadron. This definition already takes into account the collinear divergence of the incoming quark. In other words, the parton density function is already renormalized and there is no need to renormalize it separately.

To see this, let's look at the definition of the parton density function in terms of the hadronic tensor W_{\mu\nu} in DIS:

f_q\left(x_{Bj},Q^2\right) = \frac{1}{4\pi x_{Bj}} W_{\mu\nu} \frac{d^2\sigma}{d\Omega dE} dx_{Bj} dQ^2

Here, x_{Bj} is the Bjorken scaling variable, Q^2 is the virtuality of the exchanged photon, and \frac{d^2\sigma}{d\Omega dE} is the differential cross section for the scattering process. The hadronic tensor W_{\mu\nu} contains the collinear divergence of the incoming quark, which is already canceled by the factor of \frac{1}{4\pi x_{Bj}}. This factor is often called the "flux factor" and it takes into account the fact that in DIS, the incoming quark can emit a collinear gluon with a momentum fraction x_{Bj} before interacting with the photon. Therefore, the collinear divergence of the incoming quark is already accounted for in the definition of the parton density function.

In summary, the collinear divergence of the incoming quark is not explicitly present in the DGLAP equations because it is already included in the definition of the parton density function. I hope this clarifies your doubts. Good luck with your oral exam!