# Homework Help: Scalar QCD renormalization

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1. May 7, 2016

### idmena

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. The problem statement, all variables and given/known data

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

2. Relevant equations
The Feynman rules I derived for this diagrams are:
Where solid lines are scalars and dashed lines are ghosts.

3. The attempt at a solution
This is what I get:

(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:

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

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

2. May 7, 2016

### nrqed

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. May 7, 2016

### idmena

Adding the contributions from the diagrams shown, I get what is shown in the first line of the last pic:

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

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

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.