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Scalar QCD renormalization

  • #1
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Hello all, I hope you can give me a hand with a QFT homework I'm working on. We are to compute the beta equation of a Non-abelian SU(N) theory with: Complex scalars (massless), bosons, ghosts. My question is referring to the Boson self-energy scalar loop correction.

1. Homework Statement

We have the boson self energy correction involving a scalar loop.
This loop is formed of 2 3-vertex of Boson-scalar-scalar:
scalar_loop.jpg


Homework Equations


The Feynman rules I derived for this diagrams are:
feynman_rules.jpg

Where solid lines are scalars and dashed lines are ghosts.

The Attempt at a Solution


This is what I get:
boson_selfe.jpg

(We are setting the scalars to be massless). I know the boson loop and the ghost loop are corrrect as I checked them on a book (I'm using Bailin & Love).
The reason of my confusion is, when I add them all I get, besides some factors:
zeta_boson.jpg

But on the lecture our teacher told us we should get:
image.png


I have a sign and a factor on 2 wrong, and It's coming from the scalar loop diagram.

As we can see from the feynman rules, the 3-vertex for boson-scalar is the same as the one for boson-ghost, except for a factor. Doesn't these mean that both self-energy corrections should give me the same answer, except for such factor? And, therefore, if I know the ghost loop is correct, then I also know the scalar loop should have the same form (given the scalars are massless, as mentioned previously). But then, I am missing the (-) sign and the factor of two.

Can you help me with this please? Can someone confirm the Feynman rule I got for boson-scalar-scalar is correct? If so, where might the problem be?

Thank you very much
Regards
 

Answers and Replies

  • #2
nrqed
Science Advisor
Homework Helper
Gold Member
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Hello all, I hope you can give me a hand with a QFT homework I'm working on. We are to compute the beta equation of a Non-abelian SU(N) theory with: Complex scalars (massless), bosons, ghosts. My question is referring to the Boson self-energy scalar loop correction.


Hi, I am trying to make sure I understand the question correctly. Where exactly do you think you are off by a factor of 2 and a minus sign? What do you get for your ##Z_3##? (I can see what I would get using your expression but I want to make sure we are on the same wavelength in terms of notation)
 
  • #3
14
0
Hi, I am trying to make sure I understand the question correctly. Where exactly do you think you are off by a factor of 2 and a minus sign? What do you get for your ##Z_3##? (I can see what I would get using your expression but I want to make sure we are on the same wavelength in terms of notation)
Adding the contributions from the diagrams shown, I get what is shown in the first line of the last pic:
boson_terms.png


The counter term in my lagrangean should look like:
(propagator) * (something)

So, at this point I would like to have something like:
hope.png


Alas, I cannot do this becuase there is a 1/6 S that does not let me factorize my result like this. That term comes from the scalar loop, that is why I think that is the prpblem.

By the way, I should've mentioned before that I am working in Feynman gauge, so I should only have the propagator counter-term, not a gauge fixing counter-term.
 

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