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I have a question about the "QFT in a nutshell"-book by A. Zee, chapter 3.1 (page 148-149). It's about renormalization and regularization, and I still don't get the exact point.

Zee looks at meson-meson scattering in [itex]\lambda^{4}[/itex] theory. The [itex]\lambda^{2}[/itex]-term is a diverging integral, as can easily be seen. Now, the introduction of a cutt-off [itex]\Lambda[/itex] is clear to me; you don't expect theories to be valid for all energies, so you regularize your integral. After some rewriting, the scattering amplitude M up to second order in [itex]\lambda[/itex] becomes

[tex]

M = -i\lambda + iC\lambda^{2}[\log{\Lambda^{2}/s}]+ O(\lambda^{3})

[/tex]

Here the [itex]\log{\Lambda^{2}/s}[/itex]-term is actually the sum of 3 terms with kinematic variables in them, but their exact form doesn't concern us; we focus on one kinematic variable s.

Now, the question in this chapter is: what does [itex]\lambda[/itex] exactly mean? Zee introduces [itex]\lambda_{P}[/itex], a physical coupling constant as measured by an actual experiment. Then, after formula (3) he states that "according to our theory",

[tex]

-i\lambda_{P} = -i\lambda + iC\lambda^{2}[\log{\Lambda^{2}/s_{0}}]+ O(\lambda^{3})

[/tex]

where [itex]s_{0}[/itex] is the value found of the kinematic variable s of the experiment. Why is this the case? Why is [itex]M = -i\lambda_{P} [/itex]? Does this physical coupling include ALL orders of [itex]\lambda[/itex] and so gives directly the physical scattering amplitude M because an experiment concears all of these lambda-orders?

He also states that [itex]\lambda[/itex] is a function of [itex]\Lambda[/itex] in order that the actual scattering amplitude M doesn't depend on [itex]\Lambda[/itex]. But for my feeling, [itex]\lambda[/itex] and [itex]\Lambda[/itex] are 2 different things, and I don't see intuitively why they should be related besides the invariance-argument.

Can anyone clarify things up to me? I've read quite some QFT-stuff and I'm also quite familiar with the idea of renormalization and regularization, but these pages keep troubling me. Thanks!

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# Cutting off our Ignorance:renormalization

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