Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part IV, Chapter 5 | Page 3 of 4

Photon-Photon Scattering

The fermion box diagram, Euler-Heisenberg Lagrangian, and nonlinear vacuum effects

5.9 Light-by-Light Scattering

In classical electrodynamics, Maxwell's equations are perfectly linear: electromagnetic waves pass through each other without interaction. In QED, however, photons can scatter off each other through virtual electron-positron loops. This is a purely quantum effect with no classical analog.

Since photons couple only to charged particles (not directly to each other), the leading contribution to $\gamma\gamma \to \gamma\gamma$ requires at least one fermion loop. The lowest-order diagram is the box diagram with four photon vertices connected by four electron propagators:

$$i\mathcal{M} = (-ie)^4 \int \frac{d^4k}{(2\pi)^4} \frac{\text{Tr}\!\left[\gamma^{\mu_1}(\not{k}+m)\gamma^{\mu_2}(\not{k}+\not{q}_2+m)\gamma^{\mu_3}(\not{k}+\not{q}_2+\not{q}_3+m)\gamma^{\mu_4}(\not{k}-\not{q}_1+m)\right]}{[k^2-m^2][(k+q_2)^2-m^2][(k+q_2+q_3)^2-m^2][(k-q_1)^2-m^2]}$$

multiplied by the polarization vectors $\epsilon_{\mu_i}$ of the four external photons. There are six distinct box diagrams corresponding to the different orderings of the four photons around the loop (three independent permutations, each with a clockwise and counterclockwise orientation).

The amplitude is finite (no UV divergence) due to gauge invariance — the Ward identity ensures that all potential divergences cancel among the six diagrams. This is an $\mathcal{O}(\alpha^2)$ process since each vertex contributes a factor of $e$, giving $e^4 = (4\pi\alpha)^2$.

💡Why Is It Finite?

Naive power counting suggests the box diagram could diverge logarithmically. However, gauge invariance (Ward identities) requires that the amplitude vanishes when any external photon polarization is replaced by its momentum:$q_i^{\mu_i}\mathcal{M}_{\mu_1\mu_2\mu_3\mu_4} = 0$. This constraint reduces the effective degree of divergence and renders the integral finite.

5.10 The Euler-Heisenberg Effective Lagrangian

When the photon energies are much smaller than the electron mass ($\omega \ll m_e$), the box diagram simplifies dramatically. We can "integrate out" the heavy electron and describe the low-energy photon-photon interaction by an effective Lagrangian.

The result, first derived by Euler and Heisenberg in 1936, adds nonlinear corrections to the free Maxwell Lagrangian:

$$\mathcal{L}_{\text{EH}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\alpha^2}{90\,m_e^4}\left[\left(F_{\mu\nu}F^{\mu\nu}\right)^2 + \frac{7}{4}\left(F_{\mu\nu}\tilde{F}^{\mu\nu}\right)^2\right]$$

where $\tilde{F}^{\mu\nu} = \tfrac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$is the dual field-strength tensor. In terms of the electric and magnetic fields:

$$F_{\mu\nu}F^{\mu\nu} = 2(\vec{B}^2 - \vec{E}^2), \qquad F_{\mu\nu}\tilde{F}^{\mu\nu} = -4\,\vec{E}\cdot\vec{B}$$

The effective Lagrangian can also be written as:

$$\mathcal{L}_{\text{EH}} = \frac{1}{2}(\vec{E}^2 - \vec{B}^2) + \frac{2\alpha^2}{45\,m_e^4}\left[(\vec{E}^2 - \vec{B}^2)^2 + 7(\vec{E}\cdot\vec{B})^2\right]$$

The nonlinear terms become significant when the field strength approaches the Schwinger critical field:

$$E_{\text{cr}} = \frac{m_e^2 c^3}{e\hbar} \approx 1.3 \times 10^{18} \;\text{V/m}$$

💡Effective Field Theory Logic

The Euler-Heisenberg Lagrangian is a textbook example of effective field theory. At energies $\omega \ll m_e$, we cannot resolve the virtual electron loop and instead see an effective four-photon vertex. The coupling scales as$\alpha^2/m_e^4$: two powers of $\alpha$ for the loop, and$1/m_e^4$ from dimensional analysis (the lowest-dimension operator involving four field strengths).

5.11 Cross Section and Experimental Status

From the effective Lagrangian, the four-photon vertex contributes an amplitude that scales as $\mathcal{M} \sim \alpha^2 \omega^2/m_e^4$ (two powers of$\omega$ from the derivative coupling of photons). The cross section, proportional to $|\mathcal{M}|^2/s$, gives:

$$\boxed{\sigma(\gamma\gamma \to \gamma\gamma) \sim \frac{\alpha^4 \omega^6}{m_e^8}} \qquad (\omega \ll m_e)$$

The precise numerical result for unpolarized photons is:

$$\sigma = \frac{973}{10125\pi}\frac{\alpha^4\omega^6}{m_e^8} \approx 0.031\,\frac{\alpha^4\omega^6}{m_e^8}$$

For visible light ($\omega \sim 2\;\text{eV}$), this gives$\sigma \sim 10^{-65}\;\text{cm}^2$ — far too small to observe directly with conventional light sources. However, high-energy photon-photon scattering has been observed:

  • ATLAS (2017): First evidence of light-by-light scattering in ultraperipheral Pb-Pb collisions at the LHC, where the strong electromagnetic fields of the lead nuclei act as sources of quasi-real photons
  • ATLAS (2019): Observation with 8.2$\sigma$ significance, measuring $\sigma = 120 \pm 17\;\text{(stat)} \pm 13\;\text{(syst)} \pm 4\;\text{(lumi)}\;\text{nb}$
  • CMS (2019): Independent confirmation of light-by-light scattering

💡Why the LHC Can See Photon-Photon Scattering

The $\omega^6$ dependence means the cross section grows rapidly with photon energy. In ultraperipheral heavy-ion collisions, the photon energies reach$\omega \sim$ GeV, where $(\omega/m_e)^6 \sim 10^{18}$, dramatically enhancing the rate compared to optical photons. The coherent electromagnetic fields of the entire nucleus ($Z = 82$ for lead) further boost the effective luminosity by $Z^4$.

5.12 Delbruck Scattering and Vacuum Birefringence

Delbruck scattering

Delbruck scattering is the elastic scattering of photons by the Coulomb field of a nucleus: $\gamma + Z \to \gamma + Z$. One can view it as photon-photon scattering where two of the photons are virtual (from the nuclear Coulomb field). The amplitude is proportional to $Z^2\alpha^3$:

$$\mathcal{M}_{\text{Delbrück}} \sim Z^2 \alpha^3 \cdot f(\omega/m_e)$$

Delbruck scattering was first observed in the 1970s by scattering high-energy photons (1–10 MeV) off heavy nuclei. It competes with Compton scattering off nuclear electrons and nuclear Thomson scattering. The Delbruck contribution was isolated by its characteristic angular distribution and energy dependence.

Vacuum birefringence

The Euler-Heisenberg Lagrangian predicts that the vacuum in the presence of a strong external magnetic field acts as a birefringent medium: photons polarized parallel and perpendicular to the field propagate with different velocities.

From the effective Lagrangian, the indices of refraction for the two polarizations are:

$$n_\parallel = 1 + \frac{7\alpha}{90\pi}\left(\frac{B}{B_{\text{cr}}}\right)^2, \qquad n_\perp = 1 + \frac{4\alpha}{90\pi}\left(\frac{B}{B_{\text{cr}}}\right)^2$$

where $B_{\text{cr}} = m_e^2/(e) \approx 4.4 \times 10^9\;\text{T}$. The difference in indices is:

$$\Delta n = n_\parallel - n_\perp = \frac{3\alpha}{90\pi}\left(\frac{B}{B_{\text{cr}}}\right)^2 = \frac{\alpha}{30\pi}\left(\frac{B}{B_{\text{cr}}}\right)^2$$

Experimental efforts to detect vacuum birefringence include:

  • PVLAS experiment: Laboratory measurement using high-finesse optical cavities in strong magnetic fields. Current sensitivity is approaching but has not yet reached the QED prediction
  • Magnetar observations: The IXPE X-ray polarimetry satellite has reported evidence consistent with vacuum birefringence in the extreme magnetic fields ($B \sim 10^{10}\;\text{T}$) near magnetars

💡The Vacuum as a Nonlinear Medium

In quantum electrodynamics, the vacuum is not truly empty — it seethes with virtual electron-positron pairs. An external electromagnetic field polarizes these virtual pairs, causing the vacuum to respond like a dielectric medium. The Euler-Heisenberg Lagrangian captures this response at leading order. When $E \sim E_{\text{cr}}$, the field can actually rip real pairs from the vacuum (Schwinger pair production), and the effective Lagrangian breaks down.

Key Concepts (Page 3)

  • • Photon-photon scattering proceeds via the fermion box diagram at one loop — purely quantum
  • • The Euler-Heisenberg Lagrangian captures the low-energy limit as an effective four-photon vertex
  • • Cross section scales as $\sigma \sim \alpha^4\omega^6/m_e^8$ — $\omega^6$ dependence from dimensional analysis
  • • ATLAS observed light-by-light scattering at 8.2$\sigma$ in ultraperipheral Pb-Pb collisions
  • • Delbruck scattering: photon scattering off a nuclear Coulomb field
  • • Vacuum birefringence: strong B-field makes vacuum act as a birefringent medium
← Page 2
Page 3 of 4
Chapter 5: Elementary QED Processes II
Next Page →