# Clarification of a passage from the Schwartz QFT text

## Main Question or Discussion Point

Hi, would like some clarification on this passage from Matthew D. Schwartz's QFT text regarding renormalization on page 297, 15.4.1 Renormalization of $\lambda$:

"First of all, notice that, while $M(s)$ (the $\phi\phi\rightarrow\phi\phi$ scattering matrix for $\phi^4$ theory) is infinite, the difference between $M(s_1)$ and $M(s_2)$ at two different scales is finite:$$M(s_1)-M(s_2)=\frac{\lambda^2}{32\pi^2}\ln(\frac{s_2}{s_1}).$$ Should we also expect that $M(s)$ itself be finite? After all, $M^2$ is supposed to be a physical cross section.
To answer this, let us think more about $\lambda$. It should be characterizing the strength of the $phi^4$ interaction. So to measure $\lambda$ we would simply measure the cross section for $\phi\phi\rightarrow\phi\phi$ scattering, or equivalently, $M$. But this matrix element is not just proportional to $\lambda$ but also has the $\lambda^2$ correction above. Thus, it is impossssible to simply extract $\lambda$ from this scattering process."

I'm a bit confused by the part in bold, can't we determine $\lambda$ from the difference in scattering cross-sections that he just wrote down?

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vanhees71
I'm not sure what Schwartz is after. I've the book not here, but I guess what he means is the following: The $\lambda$ is the bare coupling and is infinite and unobservable. However, you can use the elastic scattering cross section to define a finite renormalized value for $\lambda$ at some given scale $s_R$ by measuring the cross section. Then you express everything in terms of renormalized quantities and everything is finite. However, the renormalized coupling constants (and wave-function normalization as well as masses) have to be changed, when the renormalization scale $s_R$ is changed, such that the cross section stays unchanged. That demand leads to the running of the renormalized coupling constant, wavefunction normalization constants, and masses governed by the socalled renormalization-group equation. For my treatment of renormalization, see