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?
Best Answer
If you can do this, you'll be nothing more than god.
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.
Considering all the above, we obtain the set of rules and computation tools wich allow us to compute the holy grail. Nowadays noone can find an exact and explicit closed form for this holy grail. We can simply draw some pictures and say something.