# Do we take care of the external momenta when finding the scalar 3-point function for $\phi^3$ theory?

The scalar 3-point function (1-loop correction) for $$\phi^3$$ theory could be found using the following diagram:

$$iV_3 = (-ig)^3\int\frac{d^4k}{(2\pi)^4}\frac{i^3}{[k^2-m^2][(k+p_1)^2-m^2][(k+p_1+p_2)^2-m^2]}.$$

Does this look right? On my solution, it was written as

$$iV_3 = (-ig)^3\int\frac{d^4k}{(2\pi)^4}\frac{i^3}{[k^2-m^2]^3}.$$

Therefore I'm confused about whether we should include the external momenta when finding the scalar 3-point function for $$\phi^3$$ theory. These momenta are included in the 2-point function, so maybe the reason for dropping external momenta here is for simplicity and approximation purpose?

• Your loop integral looks fine. No idea why the external momenta are missing in your "solution". Dec 27, 2022 at 14:58
• @Hyperon Thank you!!
– IGY
Dec 27, 2022 at 15:20

It really depends on what you are interested in. If you really want a three-point function $$\langle \phi(x_1)\phi(x_2)\phi(x_3)\rangle$$, be it in position or momentum space, then the external lines do matter.