Two virtual photons whose arcs cross: the first photon is emitted at $v_1$ and absorbed at $v_3$, while the second is emitted at $v_2$ and absorbed at $v_4$, with ordering $v_1, v_2, v_3, v_4$ along the fermion line. This overlapping topology cannot be decomposed into a product of one-loop subdiagrams — it is irreducibly two-loop.
The computational chain
From diagram to number—every step of the amplitude calculation.
The overlapping (crossed rainbow) topology
$$v_1 \xrightarrow{\quad} v_2 \xrightarrow{\quad} v_3 \xrightarrow{\quad} v_4 \qquad \text{photon}_1: v_1 \leftrightarrow v_3, \quad \text{photon}_2: v_2 \leftrightarrow v_4$$
Four vertices sit on the fermion line in order $v_1, v_2, v_3, v_4$. One photon connects $v_1$ to $v_3$; the other connects $v_2$ to $v_4$. Drawn as arcs above the line, these arcs *cross* — neither is contained inside the other. Compare with the nested topology $(v_1 v_4)(v_2 v_3)$, where the inner photon arc is entirely enclosed by the outer one. The crossing is what makes this diagram irreducibly two-loop: there is no way to cut a single internal line and separate it into two independent one-loop diagrams.
Feynman rules expression
$$-i\Sigma_{\text{ov}}(p) = (-ie)^4 \int\frac{d^dk}{(2\pi)^d}\frac{d^d\ell}{(2\pi)^d}\;\gamma^\mu\,S(p\!-\!k)\,\gamma^\nu\,S(p\!-\!k\!-\!\ell)\,\gamma_\mu\,S(p\!-\!\ell)\,\gamma_\nu\;\frac{1}{k^2}\frac{1}{\ell^2}$$
Two independent loop momenta $k$ and $\ell$ flow through the diagram. Route $k$ through the first photon ($v_1 \to v_3$) and $\ell$ through the second ($v_2 \to v_4$). The three internal fermion propagators then carry momenta $p-k$ (between $v_1$ and $v_2$), $p-k-\ell$ (between $v_2$ and $v_3$), and $p-\ell$ (between $v_3$ and $v_4$), where $S(q) = i(\not{q}+m)/(q^2 - m^2)$. Each vertex contributes $(-ie\gamma^\mu)$ or $(-ie\gamma^\nu)$, and Lorentz indices are contracted between the two ends of each photon. The crucial feature is that $k$ and $\ell$ are entangled in the middle propagator $S(p-k-\ell)$ — the two loop momenta cannot be factored apart.
Why overlapping divergences are qualitatively harder
$$\underbrace{\int d^dk\;\text{(div)}}_{\gamma_1:\{v_1,v_2,v_3\}} \quad \cap \quad \underbrace{\int d^d\ell\;\text{(div)}}_{\gamma_2:\{v_2,v_3,v_4\}} \qquad \gamma_1 \cap \gamma_2 \neq \varnothing,\;\; \gamma_1 \not\subset \gamma_2,\;\; \gamma_2 \not\subset \gamma_1$$
The diagram has two one-loop subdivergences: $\gamma_1$ (holding $\ell$ fixed, integrating over $k$) and $\gamma_2$ (holding $k$ fixed, integrating over $\ell$). These subdiagrams *overlap* — they share the vertices $v_2, v_3$ and the propagator $S(p-k-\ell)$, but neither contains the other. This is the defining property of overlapping divergences. For the nested topology, one subdivergence is entirely inside the other, so you can renormalize from the inside out. Here, you cannot: subtracting the $\gamma_1$ divergence changes the $\gamma_2$ integrand and vice versa. The subdivergences are coupled, and naive sequential subtraction fails.
The Zimmermann forest formula for overlapping divergences
$$R\,\Gamma = \sum_{F \in \mathcal{F}(\Gamma)} \prod_{\gamma \in F} (-t_\gamma)\;\Gamma$$
The BPHZ approach resolves overlapping divergences through the *forest formula*. A forest $F$ is a set of divergent subdiagrams of $\Gamma$ that are either nested or disjoint — no overlapping pairs allowed. The operator $t_\gamma$ extracts the divergent part (Taylor expansion to the degree of divergence) of subdiagram $\gamma$. The sum runs over *all* forests, including the empty forest (the unsubtracted diagram) and the full forest (overall counterterm). For our overlapping diagram, the forests are: $\varnothing$, $\{\gamma_1\}$, $\{\gamma_2\}$, $\{\Gamma\}$, $\{\gamma_1, \Gamma\}$, and $\{\gamma_2, \Gamma\}$. There is no forest $\{\gamma_1, \gamma_2\}$ because $\gamma_1$ and $\gamma_2$ overlap — but $\{\gamma_i, \Gamma\}$ is allowed since $\gamma_i \subset \Gamma$ (nested, not overlapping). The formula automatically handles the interplay: the counterterms for $\gamma_1$ and $\gamma_2$ are applied independently, and the overall subtraction $t_\Gamma$ removes whatever divergence remains. Zimmermann proved that $R\,\Gamma$ is finite to all orders — this is the content of Bogoliubov's $R$-operation.
Divergence structure
$$\Sigma_{\text{ov}}(p) \sim \frac{A}{\epsilon^2} + \frac{B}{\epsilon} + \text{finite} \qquad (\text{with } d = 4 - \epsilon)$$
By power counting, the overall diagram has degree of divergence $D = 1$ (as for any electron self-energy in QED), so it is linearly divergent overall. Each one-loop subdivergence $\gamma_1, \gamma_2$ is individually logarithmically divergent. In dimensional regularization, the overlapping subdivergences produce a $1/\epsilon^2$ double pole — one power of $1/\epsilon$ from each subdivergence. The single $1/\epsilon$ pole contains both the overall divergence and the leftover from incomplete cancellation between the overlapping counterterms. The double-pole coefficient is fixed by the one-loop anomalous dimension (it is not independent new information), but the single-pole coefficient contains genuinely new two-loop data that enters the two-loop beta function and anomalous dimensions.
Physical contribution at order $\alpha^2$
$$\delta m^{(2)}_{\text{ov}} \sim \frac{\alpha^2}{\pi^2}\,m\left(\frac{c_1}{\epsilon^2} + \frac{c_2}{\epsilon} + c_3 + \cdots\right), \qquad \delta Z_2^{(2)}\big|_{\text{ov}} \sim \frac{\alpha^2}{\pi^2}\left(\frac{d_1}{\epsilon^2} + \frac{d_2}{\epsilon}\right)$$
After renormalization, the overlapping diagram contributes finite corrections to the electron mass shift and the field strength renormalization constant $Z_2$ at order $\alpha^2$. Together with the nested and vacuum-polarization-insertion topologies, it completes the two-loop electron self-energy. The overlapping diagram's contribution cannot be inferred from one-loop results — it is genuinely new at two loops. These corrections enter physical observables: the two-loop anomalous magnetic moment, the $\alpha^2$ Lamb shift, and the running of the electron mass.
The appearance of $\zeta(3)$: a signature of irreducible two-loop structure
$$\Sigma_{\text{ov}}^{\text{ren}}(p)\big|_{\text{finite}} \;\ni\; \frac{\alpha^2}{\pi^2}\,\zeta(3)\,(\text{Dirac structures})$$
After all subtractions, the finite part of the overlapping diagram contains the Riemann zeta value $\zeta(3) = \sum_{n=1}^\infty n^{-3} \approx 1.202$. This is characteristic of irreducible two-loop integrals and does not appear at one loop or in factorizable two-loop diagrams. Technically, it arises because the Feynman parameter integrals, after combining three fermion propagators and two photon propagators, produce iterated integrals of the form $\int_0^1 dx\int_0^1 dy\;\text{Li}_2(\cdots)/(1-\cdots)$, where Li$_2$ is the dilogarithm. These evaluate to weight-3 multiple polylogarithms, whose rational-argument values reduce to $\zeta(3)$. The nested rainbow diagram, by contrast, factorizes at the level of Feynman parameters and produces only $\zeta(2) = \pi^2/6$ (weight 2). The appearance of $\zeta(3)$ is thus a direct fingerprint of the overlapping topology — a concrete consequence of the two loop momenta being irreducibly entangled in the middle propagator.