# Derivation of Feynman rules from generating functional

I have to factorise the Sudakov form factor in six dimensional $$\phi^3$$-theory, but first I want to determine the Feynman rules using path integrals. The Lagrangian of the theory reads

$$\mathcal{L} = \dfrac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi + \partial_{\mu}\varphi^* \partial^{\mu} \varphi - \lambda\varphi^* \varphi \phi - \dfrac{g}{3!}\phi^3,$$ where $$\phi$$ is a real scalar and $$\varphi$$ a complex scalar field.

Schwartz mentioned in his book that the generating functional is the holy grail of any particular field theory, but how do I find an exact closed-form expression for the generating functional $$Z[J]$$ of this theory?

• The only reason anyone cares about Feynman rules is because we can't find an exact closed form for the holy grail of interacting field theories. Commented Dec 21, 2021 at 17:29

The corresponding action is given by $$\int d^dx\left[-\phi\partial^2\phi-2\varphi^*\partial^2\varphi-g\frac{g\phi^3}{3!}\right],$$ where I have used the integration by parts. Next step is to write down $$S[\phi,\varphi,j,J,J^*]=S_0[\phi,\varphi]+S_{\text{int}}[\phi]+\int d^dx\,[j\phi+J\varphi^*+J^*\varphi],$$ where $$S_0[\phi,\varphi]=\int d^dx\left[-\phi\partial^2\phi-2\varphi^*\partial^2\varphi\right].$$ Finally, you can write down the generating functional, $$Z[j,J,J^*]=\int\mathcal{D}[\phi,\varphi,\varphi^*]\,e^{iS[\phi,\varphi,j,J,J^*]}.$$ In order to obtain Feynman rules from the generating functional, we can start simply from the interacting part, which is $$\text{scalar \phi self-interaction}\sim -ig\int d^dx\,\phi(x)^3.$$ It tells us that in theory there is a vertex where three fields interacts to each other, $$\text{vertex}_1\sim -ig\int d^dx.$$ The second vertex comes from Yukawa-type interaction, $$\text{vertex}_2\sim -i\lambda\int d^dx,$$ where two fields $$\varphi$$ interacts with $$\phi$$. In order to obtain Feynman rule for vertex in momentum space, just perform Fourier transform for each field and require momentum-conservation. Next, in order to find bare propagators, you should consider the generating functional without $$S_{\text{int}}$$ term, $$Z_0[j,J,J^*]=\int\mathcal{D}[\phi,\varphi,\varphi^*]\exp\left\lbrace iS_0[\phi,\varphi,\varphi^*]+i\int d^dx\,[j\phi+J\varphi^*+J^*\varphi]\right\rbrace.$$ The functional integration can be performed "exactly" because this integrals are Gaussian in fields. From the obtained functional you can find bare propagators of theory by simply acting functional derivatives, $$\delta/\delta j$$ and $$\delta/\delta J$$ twice.
• Thank you for your thorough reply, Artem. But where did the scalar Yukawa interaction term $-\lambda \varphi^* \varphi \phi$ go? Commented Dec 21, 2021 at 19:20