The electron emits and reabsorbs a virtual photon. This dresses the bare electron propagator, shifting the electron mass and field normalization. Both UV and IR divergent: the UV divergence is absorbed into mass and field renormalization; the IR divergence cancels against soft real photon emission.
The Lamb shift, first measured by Lamb and Retherford in 1947, was the experimental impetus for computing this diagram.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
The electron emits a virtual photon at vertex 1 and reabsorbs it at vertex 2. Between the two vertices, the electron propagates with modified momentum $p - k$, where $k$ is the photon loop momentum.
Feynman rules expression
$$-i\Sigma(p) = \int\frac{d^4k}{(2\pi)^4}\,(-ie\gamma^\mu)\frac{i(\not{p}-\not{k}+m)}{(p-k)^2-m^2+i\epsilon}(-ie\gamma_\mu)\frac{-i}{k^2+i\epsilon}$$
Two vertices contribute $(-ie\gamma^\mu)$ each. The internal electron propagator carries momentum $p-k$. The internal photon propagator carries momentum $k$. In Feynman gauge the photon propagator is $-ig_{\mu\nu}/k^2$, and contracting $\gamma^\mu \cdots \gamma_\mu$ will simplify the numerator.
Simplify the Dirac algebra
$$-i\Sigma(p) = -e^2\int\frac{d^4k}{(2\pi)^4}\frac{\gamma^\mu(\not{p}-\not{k}+m)\gamma_\mu}{[(p-k)^2-m^2]\,k^2}$$
Pull out coupling constants. Use the contraction identity $\gamma^\mu \gamma^\alpha \gamma_\mu = -2\gamma^\alpha$ (in 4 dimensions) and $\gamma^\mu m \gamma_\mu = 4m$ to simplify the numerator: $\gamma^\mu(\not{p}-\not{k}+m)\gamma_\mu = -2(\not{p}-\not{k}) + 4m$.
Feynman parametrization
$$\frac{1}{[(p-k)^2-m^2]\,k^2} = \int_0^1 dx\,\frac{1}{[\ell^2 - \Delta]^2}$$
Combine the two propagator denominators using one Feynman parameter $x$. Shift the loop momentum $k \to \ell + xp$ to complete the square. The combined denominator becomes $[\ell^2 - \Delta]^2$ with $\Delta = -x(1-x)p^2 + (1-x)m^2$.
Loop integration in $d = 4-\epsilon$ dimensions
$$-i\Sigma(p) = \frac{-e^2}{(4\pi)^{d/2}}\int_0^1 dx\left[\frac{-2(1-x)\not{p}+4m}{\Delta^{2-d/2}}\right]\Gamma\!\left(2-\frac{d}{2}\right)$$
Wick rotate $\ell^0 \to i\ell_E^0$ and evaluate the standard Euclidean loop integral. The result involves the Euler gamma function $\Gamma(2-d/2)$, which diverges as $d \to 4$: $\Gamma(2-d/2) = \Gamma(\epsilon/2) = 2/\epsilon - \gamma_E + O(\epsilon)$.
Extract the divergent and finite parts
$$\Sigma(p) = \frac{\alpha}{4\pi}\left[\left(\frac{1}{\epsilon}+\text{finite}\right)(2m - \not{p}) + 4m\left(\frac{1}{\epsilon}+\text{finite}\right)\right]$$
Expanding around $d=4$, the self-energy has the structure $\Sigma(p) = A(p^2) + B(p^2)\not{p}$, where both $A$ and $B$ contain $1/\epsilon$ poles. The coefficient of $m$ gives the mass renormalization; the coefficient of $\not{p}$ gives the field strength renormalization.
Renormalization: counterterm cancellation
$$\Sigma(p) + \Sigma_{\text{ct}}(p) = \Sigma^{\text{ren}}(p) \quad\text{(finite)}$$
The counterterm diagram contributes $\Sigma_{\text{ct}} = -(\delta Z_2 \not{p} - \delta m - \delta Z_2 m)$. The renormalization conditions fix $\delta m$ and $\delta Z_2$ so that the sum $\Sigma + \Sigma_{\text{ct}}$ is finite. In the on-shell scheme: $\delta m$ is chosen so the pole of the dressed propagator is at $p^2 = m^2$ (the physical mass), and $\delta Z_2$ is chosen so the residue at the pole is 1.
Physical contribution: the Lamb shift
The finite renormalized self-energy modifies the electron's interaction with external fields. In the hydrogen atom, this shifts the energy levels — the Lamb shift. The dominant contribution comes from the logarithmic dependence $\Sigma^{\text{ren}}(p) \sim (\alpha/3\pi) \ln(m/\langle E \rangle)$, where $\langle E \rangle$ is a mean excitation energy. Bethe's 1947 calculation (famously done on a train) gave 1040 MHz for the $2s$-$2p$ splitting, compared to the measured 1057 MHz.
Physical contribution
Generates mass renormalization (delta_m divergent) and contributes to the Lamb shift.