I've been given a small project to ease me into life as a PhD student and part of it involves working out Feynman rules for scalar bosons coupling to gauge bosons.(adsbygoogle = window.adsbygoogle || []).push({});

I've been reading through Peskin and Schroder Chapter 9 and want to clarify if I've got my understanding right of how to do it.

The QED Lagrangian is [tex]L = \int d^{4}x \,\mathcal{L} = \int d^{4}x \, \bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - e\psi\gamma^{\mu}\psi A_{\mu}[/tex]

The electron propogator's Feynman diagram rule is obtained by finding the Green's function of the Dirac operator [tex]i\gamma^{\mu}\partial_{\mu}-m[/tex] because it just describes an electron moving without interaction.

Similarly, the photon propogator is found by rearranging [tex]-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/tex] into the form [tex]A_{\mu}\Delta^{\mu\nu}A_{\nu}[/tex] and finding the Green's function of [tex]\Delta^{\mu\nu}[/tex].

One thing I'm a bit hazy on (it seems to work but I want to be sure) is getting the vertex Feynman rules.

Suppose [tex]\mathcal{L} = \mathcal{L}_{0}+\mathcal{L}_{I}[/tex]. Suppose the interaction term involves 3 fields (like [tex]\psi[/tex], [tex]\bar{\psi}[/tex] and [tex]A_{\mu}[/tex]. Is the vertex rule, in momentum space, then simply

[tex]\frac{\delta}{\delta \psi}\frac{\delta}{\delta \bar{\psi}}\frac{\delta}{\delta A_{\mu}}\int d^{4}x \mathcal{L}_{I}[/tex]

That would give the QED one as [tex]-ie\gamma^{\mu}\int d^{4}x[/tex] which is right. Is this true for all interaction Lagrangians? I follow most of the "Computing the correlation function" bits and the [tex]Z = e^{iS...}[/tex] bits involving functional derivatives, but when it comes to then computing the Feynman rules I just kind of noticed that relation between the Lagrangian and the rules rather than find any step by step guide in P&S.

Am I going about it the right way or have I just hit on the right answer by (dumb) luck? Thanks :)

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# Deriving Feynman rules from Lagrangian

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