Understanding Renormalization in Quantum Field Theory

[Sombrero]

When was reading about renormalization I did no understand the main Idea of the last :(
It has been considered photon propagator with virtual pair of electron/pozitron. Takeing that loop integral the M^2 cuttoff is introduced, which tends then to ininifity, M^2 is "sopped up" in renormalized coupling constant, but even then when M^2 tends to infinity the coupling constant runs to infinity? Yes? Am I clear?

Pls help

 

[Sombrero]

Nobody answers? :(

and also could anyone answer thread: https://www.physicsforums.com/showthread.php?t=106533

Thank you

 

[Sombrero]

One help needed please

Gents,
I am not familiar with physics and that's why my questions may seem stupid for you, but could you pls help me and explain the Idea of formula 7.27 in attached file from Halzen F., Martin A.D. Quarks and leptons.
Does not the e_R tends to infinity when M^2 goes to infinity?
Thank you very much.

 

[vanesch]

No,
\lim_{x\rightarrow\infty} Log[x^2]=0
So,
\lim_{x\rightarrow\infty} 1-Log[x^2]=1
So it does not go to infinity as M goes to infinity.

?

 

[George Jones]

?

My feelings exactly!

Regards,
George

 

[Norman]

Sorry, in my sickly, feverish state I didn't elaborate. He is asking about how Halzen and Martin go about doing renormalization of the electric charge. They introduce the cut-off, M, and show that the renormalized charge goes like:
e_r \approx e(1-Log[M^2])^{\frac{1}{2}}
With a bunch of stuff I cannot remember off hand. So my understanding of the question was basically a simple math question. What does the limit as M-> infty do to e_r.
Did I misinterperet or just not state enough in my original post? Maybe my math is completely wrong, that tends to happen with a fever...

 

[Perturbation]

The limit of log(x²) as x->infin isn't zero.

 

[gulsen]

isn't it 2 \log x and \log x approaches to infinity at x->\infty?

 

[Norman]

OMG...
Well, the good news is I am feeling better and now see where I went wrong... I even checked it with mathematica, funny thing is if you switch the variable and the base in mathematica, you get a very different answer...

time to get some rest... I suppose this means all the work I did today needs to be rechecked.

Edit: I deleted the post- my shame is too deep...