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In non-perturbative approaches to QFT the $\Lambda \rightarrow \infty$ limit shows that the QFT is well-defined as a continuous theory. However perturbative approaches sometimes have the different goal of computing $S$-matrix elements. In the latter we demand that the parameters of the theory (such as the mass, coupling constant...etc) take certain values at a reference energy scale. In other words, we treat the QFT as an effective field theory. From a Wilsonian renormalization group point of view, we can say that we are looking for a Lagrangian where we have integrated out all momenta above our reference energy scale when there is a cutoff.

What has been confusing me is that in particle physics books they still are interested in either the $\Lambda \rightarrow \infty$ limit or results independent of $\Lambda$ even in these perturbative scattering computations. Given what I wrote in the first paragraph this seems counter-intuitive. We already have a reference energy scale that gives us our physical parameters, so why do we care about the $\Lambda \rightarrow \infty$ limit? Why is having a cutoff-dependent result a bad thing if, to my understand, that is essentially what it means to demand that a Lagrangian parameter takes a certain value at a certain energy scale?

A related question is then: What role does the cutoff play in both perturbative QFT and Wilson's renormalization group? But perhaps this better suited for a separate post.

Qmechanic
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CBBAM
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1 Answers1

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In perturbation theory for the Standard Model, we can find a suitable cutoff (for example, at the GUT scale or the Planck scale), but such high-energy physics does not manifest at the energy scales typical of the Standard Model. Therefore, in most cases, one is interested in the theory obtained by tending this cutoff to infinity.

To keep physical parameters finite in a theory with the cutoff scale tends to infinity, the theory must be on a critical surface (since, critical theories have a finite correlation length). In other words, from the perspective of Wilsonian renormalization group, the problem of tending the cutoff to infinity corresponds to the question of how to take the continuum limit from a lattice theory (or a theory with a cutoff) and obtain a theory on the critical surface. The location of this critical surface (or how to set the parameters to achieve the critical theory) can only be determined by varying the cutoff scale and observing the response by suitable differential equations. This is the essence of Wilsonian renormalization group.

To reiterate, the renormalization procedure in a perturbation theory with a cutoff is equivalent to the question of how to fix the theoretical parameters to bring the theory to the critical surface in the perspective of Wilsonian renormalization group. I am not aware of any good textbooks or notes on this specific point, but perhaps someone on this site might know of one. For reference, I recommend Wilson-Kogut, and Polchinski, though both are somewhat dated, and there may well be better textbooks that I am unaware of. Or perhaps a kind person might explain more details on this site.

Siam
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