MSbar Renormalisation + Gamma Functions

In summary, the author is trying to calculate the NLO vertex function, and is having difficulty understanding how to do so. The author has found a paper that explains the method for calculating Zlambda, but is struggling to apply it to the problem at hand.
  • #1
Hi all,

I am currently reading and calculating stuff on Next-To-Leading-Order calculations and I came across a paper that I want to calculate by myself and try to get the same results.

There is one part, though, where I have absolutely no idea how this works. It comes in the context of renormalisation which is a topic I know something about but in which there remain still a lot of mysteries for me:

In this paper, the author calculated the NLO Vertex function, which I evaluated also:

[itex]\tilde{\Gamma} = \frac{\alpha}{4\pi} C_F \left(\frac{-q^2}{4\pi \mu^2}\right)^{\frac{\epsilon}{2}} \frac{\Gamma^2(1+\frac{\epsilon}{2})\Gamma(1-\frac{\epsilon}{2})}{\Gamma(2+\epsilon)} \left[-\frac{2}{\epsilon^2}(2+\epsilon)^2\right][/itex]

Then comes the text
"The renormalisation constant Zlambda is defined through the relation
[itex]Z_\lambda^{-1} =1 + \tilde{\Gamma_{UV}}[/itex]
and of course depends on the way the ultraviolet divergent part GammaUV is isolated. Within the MSbar scheme, it can easily be ascertained to be

[itex]Z_\lambda = 1+\frac{\alpha}{4 \pi \Gamma(1+\frac{\epsilon}{2})}C_F \left(\frac{1}{4\pi}\right)^{\frac{\epsilon}{2}} \frac{8}{\epsilon}[/itex]

I have no clue how this solution can be derived. I know that you can do Zlambda = 1/(1+Gamma) = 1 - Gamma to order alpha but that's as far as I get. I know how to expand the Gamma function in terms of epsilon and gamma,but the easy-peasy way I learned it in Peskin SChroeder only has one Gamma function with one epsilon and one gamma and one 4pi part, where the MSbar scheme explicitely says how to absorb this. But now I have four Gamma functions, which are all basically convergent, and I end up with only one Gamma function left (I would have assumed that I expand all of them) and this denominator Gamma function somehow is different from the denominator Gamma function you start with.

Can anyone give me a hint how you would actually start rewriting the gamma functions and how you in fact would then apply the msbar scheme formally to get the UV part?
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  • #2
The denominator gamma function (and the power of 4pi) are only there to convert from MS to MS-bar. To leading order in e=epsilon, Gamma(1+e/2)=exp(-g e/2), where g is Euler's constant.
  • #3
Ah I see. I never heard about this method of shifting between these two schemes.

I think then I understand how the rest works- Thank you!
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1. What is MSbar Renormalisation?

MSbar renormalisation is a mathematical technique used in quantum field theory to remove divergences in physical quantities and make them finite. It involves subtracting an infinite term from the original quantity and replacing it with a finite renormalised value.

2. How does MSbar Renormalisation work?

MSbar renormalisation works by subtracting the divergent part of a quantity, which is typically expressed as a power series in the coupling constant, and replacing it with a finite value that depends on a chosen renormalisation scale. This process ensures the physical quantity does not depend on the arbitrary choice of scale.

3. What are the advantages of using MSbar Renormalisation?

One of the main advantages of MSbar renormalisation is that it allows for a systematic and consistent way to remove divergences in physical quantities. It also preserves the symmetries of the underlying theory and ensures that the renormalised quantities are finite and well-defined.

4. What are Gamma Functions in the context of MSbar Renormalisation?

In MSbar renormalisation, the Gamma function is used to regularize divergent integrals that arise in the renormalisation process. It is a generalisation of the factorial function and is often used to calculate the coefficients of the power series that represents the divergent part of a physical quantity.

5. How is MSbar Renormalisation related to the strong coupling constant?

MSbar renormalisation is closely related to the strong coupling constant, also known as the QCD coupling constant. In fact, the value of the strong coupling constant is typically determined through renormalisation schemes such as MSbar. This allows for a consistent and precise calculation of the strong coupling constant and its evolution with energy scale.

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