Relevent and irrelevant operators
In a general EFT $$\mathcal{L} = \sum_n c_n \frac{\mathcal{O}_n}{\Lambda^{d_n - 4}}$$ with operators $\mathcal{O}_n$ with dimension $d_n$ at energies below $\Lambda$ we define
irrelevant operators: mass dimension $d_n > 4$ relevant operators: mass dimension $d_n < 4$ marginal operators: mass dimension $d_n = 4$
- contributions of irrelevant operators vanish if the scale of the EFT goes to zero
- In a 4 dimensional field theory there are only three relevant operators
- $d_n = 0$: unit operator $1$
- $d_n = 2$: boson mass term $\phi^dagger \phi$
- $d_n = 3$: fermion mass term $\overline{\psi}\psi$, cubic scalar interaction $\phi^3$
- marginal and relevant operators are renormalizable due to their mass dimension
Example: Toy theory with scalar fields: $$\mathcal{L} = \frac{1}{2}(\partial \phi)^2 + \frac{1}{2} (\partial \varphi)^2 - \frac{1}{2}m^2\varphi^2 - \frac{1}{2}M^2\phi^2 - \frac{\lambda}{2}\varphi^2\phi$$
- coupling constants have mass dimension $1$
- First two operators are marginal, last three are relevent
- assume $m \ll M$
Amplitude for $\varphi\varphi \to \varphi\varphi$ $$\mathcal{A}(\varphi\varphi \to \varphi\varphi)\propto \frac{\labmda^2}{q^2 - M^2}\frac{1}{E}$$
- For $E \gg M$: $$\sigma \propto |A|^2 \propto \frac{1}{E^2}(\frac{\lambda}{E})^4$$
- For $E \ll M$: $$\sigma \propto |A|^2 \propto \frac{1}{E^2}(\frac{\lambda}{M})^4$$
$\Rightarrow$ At low energies the cross section increases because $\varphi^2\phi$ is relevent operator