Are all divergences in quantum field theory logarithmic?

[geoduck]

In Zee's QFT book he writes an amplitude as:

$$M(p)=\lambda_0+\Gamma(\Lambda,p,\lambda_0)$$

He then states that you make a measurement:

$$M(\mu)=\lambda_0+\Gamma(\Lambda,\mu,\lambda_0) \equiv \lambda_R$$

and substitute that into M(p) to get:

$$M(p)=\lambda_R+\left[\Gamma(\Lambda,p,\lambda_0)-\Gamma(\Lambda,\mu,\lambda_0) \right]$$
which is independent of $\Lambda$. But isn't this only true if $\Gamma$ is logarithmically divergent in a ratio $\Lambda^2/p^2$? What if this is not the case?

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

$$M(p)=\lambda_0+\Gamma(\Lambda,p,\lambda_0)$$

then it must be true that:

$$M(p)=\lambda_R+\Gamma(\mu,p,\lambda_0)$$

Doesn't this force a log dependence, because:

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

which gives an equation involving $\Gamma$, and doesn't that equation force a log dependence of $\Gamma$?

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

 

[andrien]

Logarithmic divergence arises when the coupling has mass dimension zero.All other coupling having mass dimension positive are known to be renormalizable. we can prove that zero mass dimension theory(like $\phi^4$,quantum electrodynamics) is renormalizable however it is not very easy to tell if the theory is renormalizable or not if it's mass dimension is zero.If the dimension is negative, then it is nonrenormalizable.So in order to have a renormalizable theory you should have at least the logarithmic divergence.If the theory is linearly or quadratically divergent,mass dimension goes as -1,-2,they are nonrenormalizable.