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latentcorpse
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Hi,
(1) I'm trying Q3 of this paper
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2009/Paper48.pdf
and have got to the 2nd last bit where it asks me to define an appropriate running coupling i should solve for the massless case.
Using a procedure outlined in my notes which is roughly the same as that on p60 of these notes
http://www.damtp.cam.ac.uk/user/ho/Notes.pdf
I have established
[itex]G_n( \{ x_i \} ; g(\mu) , \mu) = G_n( \{ x_i \} ; g(\mu_0) , \mu_0) e^{- \int_{\mu_0}^\mu \frac{ds}{s} \gamma(g(s))}[/itex]
I believe this is correct however, I think he wants me to solve it explicitly i.e. substitute in a value for [itex]\gamma(g(s))[/itex]. I can't find any such value? Or how it was related to the coefficients in the original Callan-Symanzik equation?
Even if my solution is fine as it is for this part of the question, in the next bit I have to substitute in for [itex]\gamma(g(s))[/itex] so I clearly am expected to know its relationship with the original coefficients in the equation!
(2) Also in the same paper on 1,b) I don't really understand what's going on. I think the Taylor expansion in question is
[itex]I=\int_{-\infty}^{+\infty} e^{-\frac{m^2}{2}x^2}e^{-\frac{x^4}{4!}}(1-\lambda+\frac{\lambda^2}{2!} - \dots)[/itex]
But I have no idea how to get any Feynman rules out of this? Or indeed how we need to use Feynman rules to determine the coefficients - from that expansion, they look to me as if they are already fixed, aren't they?
Thanks for any help!
(1) I'm trying Q3 of this paper
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2009/Paper48.pdf
and have got to the 2nd last bit where it asks me to define an appropriate running coupling i should solve for the massless case.
Using a procedure outlined in my notes which is roughly the same as that on p60 of these notes
http://www.damtp.cam.ac.uk/user/ho/Notes.pdf
I have established
[itex]G_n( \{ x_i \} ; g(\mu) , \mu) = G_n( \{ x_i \} ; g(\mu_0) , \mu_0) e^{- \int_{\mu_0}^\mu \frac{ds}{s} \gamma(g(s))}[/itex]
I believe this is correct however, I think he wants me to solve it explicitly i.e. substitute in a value for [itex]\gamma(g(s))[/itex]. I can't find any such value? Or how it was related to the coefficients in the original Callan-Symanzik equation?
Even if my solution is fine as it is for this part of the question, in the next bit I have to substitute in for [itex]\gamma(g(s))[/itex] so I clearly am expected to know its relationship with the original coefficients in the equation!
(2) Also in the same paper on 1,b) I don't really understand what's going on. I think the Taylor expansion in question is
[itex]I=\int_{-\infty}^{+\infty} e^{-\frac{m^2}{2}x^2}e^{-\frac{x^4}{4!}}(1-\lambda+\frac{\lambda^2}{2!} - \dots)[/itex]
But I have no idea how to get any Feynman rules out of this? Or indeed how we need to use Feynman rules to determine the coefficients - from that expansion, they look to me as if they are already fixed, aren't they?
Thanks for any help!
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