Clarification of a passage from the Schwartz QFT text

  • #1
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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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  • #2
vanhees71
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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

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf
 

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