A virtual electron-positron pair appears in the photon propagator, forming a bubble. The pair screens the bare charge, reducing the effective coupling at long distances. This is the mechanism behind the running of the fine structure constant.
Computed by Schwinger, Tomonaga, and Feynman in 1947-1949. The Uehling potential it generates was computed by Uehling in 1935.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
A virtual electron-positron pair forms a closed loop (bubble) inside the photon propagator. The $(-1)$ from the closed fermion loop and the trace over spinor indices are characteristic of this topology.
Feynman rules expression
$$\Pi^{\mu\nu}(q) = (-1)\int\frac{d^4k}{(2\pi)^4}\,\mathrm{Tr}\left[(-ie\gamma^\mu)\frac{i(\not{k}+m)}{k^2-m^2}(-ie\gamma^\nu)\frac{i(\not{k}-\not{q}+m)}{(k-q)^2-m^2}\right]$$
Two electron propagators and two vertices form the loop. The overall $(-1)$ is the fermion loop sign. The trace is over spinor indices.
Ward identity constrains the tensor structure
$$\Pi^{\mu\nu}(q) = (q^2 g^{\mu\nu} - q^\mu q^\nu)\,\Pi(q^2)$$
Gauge invariance (the Ward identity $q_\mu \Pi^{\mu\nu} = 0$) forces the vacuum polarization to be transverse. This means only one scalar function $\Pi(q^2)$ needs to be computed.
Regularized result
$$\Pi(q^2) = -\frac{\alpha}{3\pi}\left[\frac{1}{\epsilon} - \int_0^1 dx\,6x(1-x)\ln\frac{m^2 - x(1-x)q^2}{\mu^2}\right]$$
After Feynman parametrization, loop integration in $d=4-\epsilon$ dimensions, and evaluation of the trace. The $1/\epsilon$ pole is the UV divergence, absorbed into charge renormalization.
Running of $\alpha$
$$\alpha(Q^2) = \frac{\alpha(\mu^2)}{1 - \frac{\alpha(\mu^2)}{3\pi}\ln\frac{Q^2}{\mu^2}}$$
The finite part of $\Pi(q^2)$ gives the running of the fine structure constant from its low-energy value $\alpha(m_e) \approx 1/137$. In electron-only QED the running is slow — the electron loop alone produces a small logarithmic increase with energy. (The dramatic running to $\alpha(m_Z) \approx 1/128$ measured at LEP includes contributions from all charged fermions.)
Physical contribution
Drives the running of alpha from its low-energy value alpha(m_e) = 1/137.036. In electron-only QED, the electron loop is the sole source of running. (The full Standard Model running to alpha(m_Z) = 1/128 includes all charged fermions.) Gives the Uehling correction to the Coulomb potential in hydrogen.