I have a very basic question about exactly how to match the experimentally measured masses and coupling constants to the parameters in the lagrangian density in a given QFT. Let me specialize to a particular theory and perhaps people here can help me out.(adsbygoogle = window.adsbygoogle || []).push({});

I've just computed the tadpole and self-energy diagram up to 1 loop in [tex]\phi^{3}[/tex] theory using dimensional regularization. If you do power counting you'll find these are the only divergent diagrams up to 1 loop. (I know [tex]\phi^{3}[/tex] is probably not a good QFT because the Hamiltonian is not bounded from below, but this is for practice.)

[tex] \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2} m^{2} \phi^{2} - \frac{\lambda \mu^{\epsilon}}{3!} \phi^{3} [/tex]

I got, up to factors of 2 and minus signs (I can never get this right; if anyone could go thru the computation and tell me what they got that'll be great, because they don't match Cheng and Li's gauge theory problem book's solutions)

1-point function:

[tex] \tau \equiv \frac{i \lambda m^{2}}{32 \pi^{2}} \left( \frac{1}{\epsilon} - \gamma +1 + Log[ \frac{\mu}{m^{2}} ] \right) [/tex]

(I'm not sure why I have [tex]Log[{\mu}/{m^{2}}][/tex] - the dimensions aren't right.)

2-point function

[tex] \Sigma[\lambda, m, p^{2}, \mu] \equiv \frac{i \lambda^{2} \mu^{2 \epsilon}}{16 \pi^{2}} \left( \frac{1}{\epsilon} - \gamma + Log[4 \pi] - \int_{0}^{1} Log[m^{2}-p^{2}x(1-x)] dx \right) [/tex]

When I (actually Mathematica) did the remaining integral in the 2-point function I found

[tex] -2 + 2 \kappa[m^{2}, p^{2}] ArcTan[\frac{1}{\kappa[m^{2}, p^{2}]}] + Log[m^{2}] [/tex]

where

[tex] \kappa[m^{2}, p^{2}] \equiv \sqrt{\frac{4 m^{2}}{p^{2}}-1} [/tex]

I know I need to introduce counterterms. Say I do the MS scheme and so the re-normalized 1- and 2-point functions are as above but without the [tex]1/{\epsilon}[/tex] term. Then what? How do I get the physical mass? Physical coupling constant?

I think I should do

[tex]

\mathcal{L} = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2} m^{2} \phi^{2} - \frac{\lambda \mu^{\epsilon}}{3!} \phi^{3} + \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi (Z_{\phi}-1) - \frac{1}{2} m^{2} \phi^{2}(Z_{m} Z_{\phi} - 1)- \frac{\lambda \mu^{\epsilon}}{3!} \phi^{3} (Z_{\lambda}Z_{\phi}^{3/2}-1)

[/tex]

but I'm actually not sure if I need to introduce additional operators. For example I'm confused about how to deal with the [tex]p^{2}[/tex] dependence. I am probably doing something wrong? If I get a term linear in [tex]p^{2}[/tex] it would have allowed me to determine the wavefunction renormalization factor [tex]\phi_{0} = \phi_{R} \sqrt{Z_{\phi}}[/tex].

As one can see I have lots of problems. Any help is very much appreciated.

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# What is the physical mass and coupling?

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