In Zee's QFT book he writes an amplitude as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]M(p)=\lambda_0+\Gamma(\Lambda,p,\lambda_0) [/tex]

He then states that you make a measurement:

[tex]M(\mu)=\lambda_0+\Gamma(\Lambda,\mu,\lambda_0) \equiv \lambda_R [/tex]

and substitute that into M(p) to get:

[tex]M(p)=\lambda_R+\left[\Gamma(\Lambda,p,\lambda_0)-\Gamma(\Lambda,\mu,\lambda_0) \right][/tex]

which is independent of [itex]\Lambda[/itex]. But isn't this only true if [itex]\Gamma[/itex] is logarithmically divergent in a ratio [itex]\Lambda^2/p^2[/itex]? What if this is not the case?

But generally speaking, doesn't the RG equation say that if:

[tex]M(p)=\lambda_0+\Gamma(\Lambda,p,\lambda_0) [/tex]

then it must be true that:

[tex]M(p)=\lambda_R+\Gamma(\mu,p,\lambda_0) [/tex]

Doesn't this force a log dependence, because:

[tex]M(p)=\lambda_R+\Gamma(\mu,p,\lambda_0)=

\lambda_R+\left[\Gamma(\Lambda,p,\lambda_0)-\Gamma(\Lambda,\mu,\lambda_0) \right] [/tex]

which gives an equation involving [itex]\Gamma[/itex], and doesn't that equation force a log dependence of [itex]\Gamma[/itex]?

But surely you can have divergences that aren't log!

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# Logarithmic divergences, RG

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