- #1

- 635

- 103

## 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##.

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?

"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?