The one-loop electron self-energy (rainbow) with a vacuum polarization bubble inserted into the virtual photon line. Two vertices on the main fermion line, two on the internal fermion loop. Factorizes as the product of the rainbow topology and the bubble topology, but is 1PI and contributes independently at two loops.
This diagram is conceptually the simplest two-loop self-energy because it factorizes into one-loop subdiagrams. It was among those computed by Karplus and Kroll (1950) in the first two-loop QED calculations.
The computational chain
From diagram to number—every step of the amplitude calculation.
Diagram topology: rainbow with VP bubble
The electron emits a virtual photon at vertex $v_1$ and reabsorbs it at vertex $v_2$, just as in the one-loop self-energy. But now the photon line does not go directly from $v_1$ to $v_2$ — it passes through a vacuum polarization bubble. Vertices $v_3$ and $v_4$ sit on the bubble, where the photon splits into a virtual $e^+e^-$ pair that annihilates back into the photon. The overall topology is still 1PI (one-particle irreducible): cutting any single internal line does not disconnect the diagram.
Two-loop Feynman rules expression
$$-i\Sigma_{\text{VP}}(p) = \int\frac{d^dk}{(2\pi)^d}\frac{d^d\ell}{(2\pi)^d}\,(-ie\gamma^\mu)\frac{i(\not{p}-\not{k}+m)}{(p-k)^2-m^2}\,\frac{-ig_{\mu\alpha}}{k^2}\left[i\Pi^{\alpha\beta}(k)\right]\frac{-ig_{\beta\nu}}{k^2}\,(-ie\gamma^\nu)$$
There are two independent loop momenta: $k$ circulates in the outer self-energy loop (the photon and the internal electron line), while $\ell$ circulates in the VP bubble. The bubble subdiagram $\Pi^{\alpha\beta}(k)$ is itself a one-loop integral over $\ell$, involving a trace over the internal fermion loop with the $(-1)$ fermion loop sign. Two photon propagators ($1/k^2$ each) flank the bubble, giving $1/k^4$ before the bubble is contracted in.
Factorization: the bubble is a known one-loop object
$$-i\Sigma_{\text{VP}}(p) = \int\frac{d^dk}{(2\pi)^d}\,(-ie\gamma^\mu)\frac{i(\not{p}-\not{k}+m)}{(p-k)^2-m^2}(-ie\gamma^\nu)\,\frac{-i}{k^2}\left[g_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}\right]\Pi(k^2)$$
The Ward identity forces $\Pi^{\alpha\beta}(k) = (k^2 g^{\alpha\beta} - k^\alpha k^\beta)\Pi(k^2)$, so one factor of $k^2$ cancels against the extra photon propagator. The scalar function $\Pi(k^2)$ is exactly the vacuum polarization computed at one loop — the same function that appears in the VP diagram page. This factorization is the key simplification: the two-loop integral separates into a known inner piece ($\Pi$) inserted into a known outer topology (the rainbow self-energy).
The subdivergence: renormalizing the VP bubble first
$$\Pi(k^2) = \Pi^{\text{ren}}(k^2) + \delta Z_3 + O(\alpha^2)$$
The VP bubble is UV divergent — it contains the same $1/\epsilon$ pole as the standalone vacuum polarization diagram, corresponding to charge renormalization. Before the outer $k$-integral can be evaluated, this subdivergence must be removed. The charge renormalization counterterm $\delta Z_3$ (which lives on the photon propagator) cancels the $1/\epsilon$ pole inside $\Pi(k^2)$, leaving the finite renormalized function $\Pi^{\text{ren}}(k^2)$. This is a concrete instance of the BPHZ procedure: renormalize subdivergences before addressing the overall divergence.
After VP renormalization: a one-loop integral with a dressed photon
$$-i\Sigma_{\text{VP}}^{\text{sub-ren}}(p) = \int\frac{d^dk}{(2\pi)^d}\,(-ie\gamma^\mu)\frac{i(\not{p}-\not{k}+m)}{(p-k)^2-m^2}(-ie\gamma_\mu)\,\frac{-i}{k^2}\,\Pi^{\text{ren}}(k^2)$$
Once the bubble is renormalized, what remains is structurally a one-loop self-energy integral — identical to the rainbow diagram, except the bare photon propagator $-i g_{\mu\nu}/k^2$ has been replaced by a modified propagator carrying the extra factor $\Pi^{\text{ren}}(k^2)$. This integral still has its own overall UV divergence (from the large-$k$ region), which is absorbed by the usual mass and field counterterms $\delta m$ and $\delta Z_2$ at order $\alpha^2$. The nested renormalization — first the bubble, then the outer loop — is what makes this diagram the most transparent example of the two-loop renormalization procedure.
Physical meaning: the self-energy with a running coupling
$$\Sigma_{\text{VP}}^{\text{ren}}(p) \sim \frac{\alpha}{4\pi}\int_0^1 dx\,\left[-2(1-x)\not{p}+4m\right]\,\frac{\alpha}{3\pi}\ln\frac{\Delta(x)}{\mu^2}$$
At large virtual photon momentum $k^2$, the renormalized VP function behaves as $\Pi^{\text{ren}}(k^2) \approx (\alpha/3\pi)\ln(-k^2/m^2)$. Inserting this into the self-energy integral effectively replaces the fixed coupling $\alpha$ with the running coupling $\alpha(k^2)$ inside the loop. The result: this diagram computes the electron self-energy not with the bare or low-energy coupling, but with a coupling that runs with the virtual photon's momentum. It is the leading correction from charge screening to the electron's mass shift and field renormalization.
Connection to the running of $\alpha$
$$\frac{-ig_{\mu\nu}}{k^2} \;\longrightarrow\; \frac{-ig_{\mu\nu}}{k^2}\,\frac{1}{1-\Pi(k^2)} \approx \frac{-ig_{\mu\nu}}{k^2}\left[1 + \Pi^{\text{ren}}(k^2) + \cdots\right]$$
The VP insertion is the first term in the geometric series that dresses the photon propagator. Every QED process that exchanges a virtual photon receives corrections from vacuum polarization through this dressed propagator. In the self-energy, this diagram is precisely where those corrections enter: it ties the running of $\alpha$ to the running of the electron mass and wave function. Among the three two-loop self-energy diagrams (VP insertion, nested rainbow, and the crossed/overlapping rainbow), this one is unique in its clean factorization and its direct physical interpretation as the effect of charge screening on the electron's self-interaction.
Physical contribution
Contributes to the electron mass and field renormalization at order alpha^2. Encodes the effect of running alpha on the self-energy: replaces the bare photon propagator with the dressed (vacuum-polarization-corrected) propagator.