 Scalar four-fermion scattering

For ${\mathcal{L}}_{\mathrm{int}}=g^2(\bar{\psi }\psi {)}^2/2$ with one species of Dirac field, the tree-level Feynman diagrams for $f(p_1)\bar{f}(p_2)\to f(p_3)\bar{f}(p_4)$ giveThe fermionic sign is the antisymmetry of the two external contractions. The initial spin average is one quarter of the spin sum. The two diagonal contributions are $g^4(t-4m^2{)}^2$ and $g^4(s-4m^2{)}^2$; the interference Dirac trace is $R=(s-4m^2{)}^2+(t-4m^2{)}^2-(u-4m^2{)}^2$, giving the displayed result after subtracting $g^{4}R/2$. Here the Mandelstam variables satisfy $s+t+u=4m^2$. At threshold this gives $16g^4{m}^4$; for zero mass it gives $g^4(s^2+t^2+u^2)/2$. The formula describes the leading scattering amplitude of an effective interaction and is not a statement of perturbative renormalizability.