QED: Vertex renormalisation; Please have Peskin and Schroeder book handy

[theoryiscool]

Hi guys,

I have a question regarding a point in the QFT book of peskin and schroeder. Iv been working through renormalisation in QED, The 2-point corrections are simple enough, however in this book the 3-point function is a little more involved and I have some issues. Essentially id to produce the result of page 334, equation 10.45. I am happily standing at page 192, equation 6.47. Now however; I don't want to regularize with Pauli-Villas, but use the dimensional regularization. The parts that i really don't get are buts of page 334 equation that have epsilonxy, for example, i don't reallly understand where this is coming from. I know that anything with an index can be altered to involve informatino about dimension, but i don't see this part... equivalently I don't see why we need the -epsilon(1-z)^2 part...same sort of issue crops in equation 10.41 page 333; where 4--> 4-epsilon, In desperation i reason this via 4 being a the original dimension; then making this expansion, but its unsettling and probably wrong reasoning, can any please eludicate what going on here?

Cheers!

p.s. real sorry for the lack of latex use, last time I used this program seriously was long ago, Iv forgotten how it works, and tried to use it to no avail...sorry:(

 

[mark726]

Hello,

Thank you for bringing up your question regarding the QFT book of Peskin and Schroeder. I understand that the 3-point function and dimensional regularization can be a bit challenging to grasp at first. Let me try to explain the parts you are having trouble with.

Firstly, the reason we use dimensional regularization instead of Pauli-Villars is because it is a more elegant and powerful method of regularization. It allows us to perform calculations in a higher dimensional space (usually d = 4-epsilon) and then take the limit as epsilon approaches 0. This allows us to avoid the divergences that arise in QFT calculations.

Regarding the epsilonxy terms, they come from the dimensional regularization process. In dimensional regularization, we introduce a new parameter epsilon, which represents the deviation from the original dimension (in this case, d = 4). This is why you see terms like 4-epsilon or epsilonxy in the equations. These terms are necessary to account for the deviation from the original dimension.

As for the -epsilon(1-z)^2 term in equation 10.45, this comes from the integration over the Feynman parameter z. When we use dimensional regularization, the integration over z can be extended to infinity, which introduces the epsilon term. The (1-z)^2 term comes from the Feynman parameterization of the propagator.

I hope this helps clarify your doubts. If you have any further questions, please don't hesitate to ask. Good luck with your studies!