SQUARE45

Let

\mathcal{L}_{QED} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \bar{\psi} (i \gamma^{\mu} D_{\mu} - m) \psi

be the classical Lagrangian density for Quantum Electrodynamics. The effective Lagrangian

\mathcal{L}_{eff}

incorporating vacuum polarization is given by the one-loop correction to the photon propagator. Define the vacuum polarization tensor

\Pi_{\mu\nu}(k^2)

via the electron loop diagram:

\Pi_{\mu\nu}(k^2) = -e^2 \int \frac{d^4 k}{(2\pi)^4} \text{Tr} \left[ \gamma_{\mu} S_F(k) \gamma_{\nu} S_F(k-k') \right]

, where

S_F(k) = \frac{i \gamma^{\alpha} (k_{\alpha} + m)}{k^2 - m^2 + i\epsilon}

is the fermionic propagator. The modified photon propagator

D'_{\mu\nu}(k^2)

in momentum space is then determined by the Dyson-Schwinger equation: \begin{equation} D'^{-1}(k^2) = D_0^{-1}(k^2) + \Pi(k^2) \end{equation}, where

D_0^{-1}(k^2) = k^2 g_{\mu\nu}

and

\Pi(k^2) = \Pi_{\mu\nu}(k^2) - k^2 g_{\mu\nu} \frac{\Pi(k^2)}{k^2}

. The resulting effective coupling constant

\alpha_{eff}(k^2)

is related to the running coupling constant

\alpha(k^2) = \frac{e^2}{4\pi} \left( 1 + \frac{\alpha}{3\pi} \Pi(k^2) \right)

, demonstrating the momentum-dependent renormalization of the electromagnetic interaction.

Vacuum Polarization