Typical Momentum Invariants of a 3-Point Function

[gobbles]

According to Peskin, p.414, at the bottom, as part of calculating the $\beta$ functions of a theory, we need to fix the counter terms by setting the "typical invariants" built from the external leg momenta to be of order $−M^2$. For a 4-point function, these invariants are s, t and u obviously. What are the typical invariants of a three-point function? Are they just any combination of the incoming momenta, like $p^2_1$ or $p_1\cdot p_2$ or $\not{\!p}_1\cdot\not{\!p}_2$, etc., where $p_1, p_2$ are two momenta on the external legs?

 

[fzero]

If the external momenta are on shell, then $p_i^2 = -m_i^2$, so our other invariants are the 3 combinations of $p_i\cdot p_j$. However momentum conservation also let's us write $p_1\cdot p_2 + p_1\cdot p_3 = m_1^2$ and $p_2\cdot p_1 + p_2\cdot p_3 = m_2^2$, so we can choose just one of them, say $p_1\cdot p_2$ as our parameter for the amplitude.

For the 4pt function, you can use similar relationships to reduce to, say, $p_1\cdot p_2$ and $p_1\cdot p_4$, which would be related to $s$ and $t$. $u$ is not independent because $s+t+u = \sum m_i^2$ (up to a sign that I am too lazy to check).

 

[gobbles]

Thank you fzero!
If I understand correctly, any Lorentz invariant expressions made of external momentum 4-vectors can be used as the invariant momenta and when setting the renormalization conditions at a certain scale, we just say that all those invariant momenta are of the order of magnitude of that scale, which, in Peskin is $-M^2$.

 

[fzero]

Yes, generally any amplitude will itself be a Lorentz invariant, so it will automatically be expressed in terms of Lorentz-invariant quantities. For a scalar field theory, these will all be products of the momenta, though with spinors we have $\gamma^\mu$ to dot into momenta and with vectors we will have polarizations $\epsilon^\mu$ as well. As I described above, there will inevitably be relations among the invariant quantities, so one will typically exploit that if it helps to simplify the mathematical form of an expression.

Peskin indeed determines the renormalization condition at an arbitrary scale $M$, so we would set all momentum invariants to be $-M^2$ in order to determine the counterterm coefficients.