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4

Your way of thinking is, essentially, correct. When it comes to this $\tilde\lambda\Lambda^2$, there is this famous quote (citing from memory, don't remember which book it's from, but it's famous), "Even though it is infinitely large, we will assume that it is finite, and that is furthermore infinitely small." In most QFTs the perturbative series diverges ...


0

What you describe is usually called regularization, as distinct from renormalization, although the terms are related. It could help to cite the paper that you are reading, but in any case, it often happens that long wavelength physics do not depend exactly on the details of short distance physics. For example, if you are scattering a particle off of a ...


2

When you introduce an auxiliary variable, such as a regularization parameter, at the end of the calculation you have to take the limit that sets the expression back to the original one. If you introduce multiple auxiliary variables, you have to do this for all of them. Otherwise you're just doing a different integral. In this case specifically, ...



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