- #1

Elmo

- 38

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- 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 R

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 R

_{hadron},as the interaction binding the quarks together is non-perturbative.