# The Factorization Theorem in Particle Physics

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• Elmo
In summary, the conversation discusses the task of calculating amplitudes of a B meson decaying to a photon and lepton/lepton anti-neutrino pair, the factorization theorem for this process, and the role of LCDA and Hard Kernel in the calculation. The paper attached provides examples of using QCD Feynman rules to calculate amplitudes, but there is confusion about neglecting the internal dynamics of a hadron. The conversation also mentions questions about the intuitive and mathematical basis of the factorization theorem, where else it is applied, and the necessary use of light cone coordinates. The purpose of the exercise is to obtain the Hard Kernel after evaluating all terms till one loop order, and the LCDA is defined in
Elmo
TL;DR Summary
Came across this thing as part of my graduate degree and I have been told to use it in calculating a meson decay. I am not sure about the basis of this formula and the extent of its usage.
I have been tasked with calculating amplitudes of a B meson decaying to a photon and lepton/lepton anti-neutrino pair ,upto one loop and have pretty much never seen this thing before. I will ask my questions along the way as I describe what I am doing.
This factorization theorem seems to go thus :
Amplitude = LCDA ⊗ Hard Kernel and this has an expansion in terms of orders ,eqtn(23)

The paper (attached below) has solved a couple of examples where they calculate the amplitude by QCD Feynman rules but I am not sure how can one neglect the internal dynamics of a hadron in calculating the Amplitude ?
The tree and loop amplitudes for some of these are given in eqtns (13,26)
The LCDA seems to be the amplitude of a hadron decaying to vacuum ( defined eqtn(14) ) and has its own Feynman rules for calculating it at tree and loop level but I don't know why they use light cone coordinates for this. Is it absolutely necessary ? I am asking this because the insistence on using only these types of coordinates really restricts the types of answers you can get.

The Hard Kernel is obtained algebraically after doing the convolution integral (⊗ ).
Now I am not really sure what's the purpose of all this exercise, in general. My supervisor has told me to just obtain the Hard Kernel after evaluating all terms till one loop order.
I also have some questions about the intuitive and mathematical basis of this theorem. As in where did it even come from. I have seen it in very few places and its only ever stated without any background. In what other places is it applied ?
Is it something that you always have to do when dealing with processes involving hadrons ?

I vaguely get that factorization has to do with separating long and short distance physics, in that Hard Kernel ultimately comes out to have large momenta and LCDA with soft momenta but some more detailed explanation will be very helpful.
Also would be nice to have explanation of why the LCDA has to be defined in terms of operator involving a a finite Wilson line eqtn(14).Also,is it necessary that the length scale of the Wilson line ([0,z]) should be of order Rhadron ,as the interaction binding the quarks together is non-perturbative.

#### Attachments

• Nuclear Physics B 650 (2003) 356–390.pdf
390.5 KB · Views: 153
ohwilleke

There didn't have to be a factorization theorem. Things could have been a coupled mess. We are lucky QCD, to a good approximation, works this way.

vanhees71 and ohwilleke

There didn't have to be a factorization theorem. Things could have been a coupled mess. We are lucky QCD, to a good approximation, works this way.
FWIW, 194 pages. Great resource.

ohwilleke said:
FWIW, 194 pages. Great resource.
FWIW ?

Elmo said:
FWIW ?
"For What It's Worth" In other words, I was pointing out that it wasn't a quick read, even though it is a great resource.

## 1. What is the Factorization Theorem in Particle Physics?

The Factorization Theorem in Particle Physics is a mathematical framework used to calculate the interactions between particles in high energy collisions. It allows us to separate the calculation of the hard scattering process from the soft interactions between the colliding particles.

## 2. Why is the Factorization Theorem important in Particle Physics?

The Factorization Theorem is important because it allows us to make precise predictions for the outcomes of high energy collisions. It also helps us understand the underlying structure of particles and their interactions.

## 3. How is the Factorization Theorem applied in experiments?

In experiments, the Factorization Theorem is used to calculate the cross section, or probability, of a specific particle interaction occurring. This allows us to compare the theoretical predictions with experimental data and test the validity of the Standard Model of Particle Physics.

## 4. Are there any limitations to the Factorization Theorem?

Yes, there are limitations to the Factorization Theorem. It is only applicable to high energy collisions where the particles involved are moving close to the speed of light. It also assumes that the particles involved are point-like and do not have internal structure.

## 5. How does the Factorization Theorem relate to other theories in Particle Physics?

The Factorization Theorem is closely related to other theories in Particle Physics, such as Quantum Chromodynamics (QCD) and the Renormalization Group. It provides a framework for calculating the effects of QCD in high energy collisions and helps us understand the behavior of particles at different energy scales.

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