Kln theorem and initial-state singularities

In summary, the KLN theorem rescues from IR singularities in final states by summing over all degenerate states. However, this does not work for initial-state singularities, such as in the emission of a soft gluon before hitting a target. This is because the IR divergence is "renormalized" into the parton distribution function of the initial state. There is a question about whether an inclusive summation over all initial degenerate states would also eliminate IR divergences, but this has not been observed in current treatments.
  • #1
eoghan
207
7
Hi!
If I have a pair [itex]q\bar q g[/itex] in a final state, I know that the gluon has a IR singularity. But KLN theorem rescues me: if I sum over all degenerate states the IR singularity cancels away.
Otherwise, if the emission of the soft gluon is in an initial state, then the IR divergence cannot be canceled. Why KLN doesn't work for initial-state singularities?
I'm thinking of the hard scattering of a parton from an hadron with a target. If the parton radiates a gluon before it hits the target, then I can't cancel the IR singularity, unless I "renormalize" the parton distribution function. Why can't I sum over all initial degenerate states to cancel the singularity?
 
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  • #2
This is a post from 2012 but I was wondering the same question ^^ - the remnant IR divergences coming from collinear/soft gluons in the initial state are 'put under the rug' into the PDF associated with the mother hadron from which the parton came from. This is the treatment I have seen everywhere but if I did an inclusive summation over all initial degenerate states would I get no IR divergences as the KLN theorem perhaps tells me and, if so, what would such initial states look like?
 

1. What is the Kln theorem?

The Kln theorem, also known as the Kln-Schnirelmann theorem, is a fundamental result in topology that states that any compact, connected, orientable 3-manifold can be decomposed into a finite number of 3-dimensional cells.

2. What is an initial-state singularity?

An initial-state singularity is a point in space-time where the curvature and density of matter are infinite, making it impossible to predict the behavior of the universe at that point using the laws of physics.

3. How does the Kln theorem relate to initial-state singularities?

The Kln theorem provides a framework for understanding the structure of 3-dimensional space, which is crucial for investigating the existence of initial-state singularities. It allows for the decomposition of a 3-manifold into smaller, simpler cells, which can then be studied individually.

4. Are initial-state singularities a proven concept?

No, the existence of initial-state singularities is still a topic of debate among scientists. While some theories, such as the Big Bang theory, suggest the presence of an initial-state singularity, there is currently no conclusive evidence to support its existence.

5. How do scientists study initial-state singularities?

Scientists use mathematical models and simulations to study the behavior of the universe at or near the time of the initial-state singularity. They also use data from observations and experiments, such as the cosmic microwave background radiation, to gather evidence and refine their understanding of these singularities.

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