Euler-Heisenberg Lagrangian
The Euler-Heisenberg Lagrangian is the most general Lagrangian for $\gamma$-$\gamma$ scattering. We want to describe the process for $E_\gamma \ll m_e$.
Derived from the fundamental QED Lagrangian according to recepie EFT recipe:
$$\mathcal{L} = \bar{\psi}(i D\negthinspace\negthinspace\negthickspace/ - m_e)\psi - \frac{1}{4} F_{\mu\nu}F^{\mu\nu}$$ $$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$$
- Energy scales: $\Lambda_H = m_e$, $\Lambda_L = E_\gamma \ll m_e$
- Degrees of freedom: Since photon energy so low, electrons cannot be resolved. Only effective d.o.f. is photon. (electrons integrated out)
- Symmetries: QED symmetry: $U(1)$, lorentz-inv, charge conjugation and parity invariance
- parity: $A_\mu(t, \vec{\alpha})\to A^\mu(t, -\vec{\alpha})$
- charge conj.: $A_\mu(t, \vec{\alpha})\to -A_\mu(t, \vec{\alpha})$ (prevents terms with three field strength tensors)
- Moste general Lagrangian:
Euler-Heisenberg Lagrangian $$\mathcal{L}\text{EH} = -\frac{1}{4}F{\mu\nu}F^{\mu\nu} + \frac{a}{m^4_e}\left(F_{\mu\nu}F^{\mu\nu}\right)^2 + \frac{b}{m^4_e}F^{\mu\nu}F_{\nu\sigma}F^{\sigma\rho}F_{\rho\mu} + \mathcal{O}\left(\frac{\partial^2 F^4}{m^6_e}\right)$$ Dimension six operators vanish due to C-parity, or for terms with deviations, equation of motion $\partial_\nu F^{\lambda\nu} = j^\lambda = 0$. Therefore higher order contributions only at dimension eight.
power counting: Counting mass dimension of terms:
$-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} \propto F_{\mu\nu}F^{\mu\nu}$
$\frac{a}{m^4_e}\left(F_{\mu\nu}F^{\mu\nu}\right)^2 \propto \frac{E^4_\gamma}{m^4_e} F_{\mu\nu}F^{\mu\nu}$
$\frac{b}{m^4_e}F^{\mu\nu}F_{\nu\sigma}F^{\sigma\rho}F_{\rho\mu} \propto \frac{E^4_\gamma}{m^4_e} F_{\mu\nu}F^{\mu\nu}$
$\frac{\partial^2 F^4}{m^6_e} \propto \frac{E^6_\gamma}{m^6_e}F^{\mu\nu}F_{\mu\nu}$
Choose accuracy: We only consider terms up to order $\frac{E^4_\gamma}{m^4_e}$
Matching: Equating QED box diagram with the EFT 4 photon interaction diagrams, which gives $a,b \propto \alpha^2$.