A counterterm insertion on the electron propagator, drawn as an X on the fermion line. Absorbs both the mass divergence ($\delta m$) and the field strength renormalization divergence ($\delta Z_2$) from the one-loop self-energy. The values are fixed by renormalization conditions: the physical mass equals the measured mass, and the propagator residue at the pole is correctly normalized.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
An X on the electron propagator — a counterterm insertion representing the combined mass and field strength renormalization. This is not a loop diagram but a tree-level vertex with a divergent coefficient.
Feynman rule
$$i\Sigma_{\text{ct}}(p) = i(\delta Z_2\not{p} - \delta m - \delta Z_2 m) = i\delta Z_2(\not{p}-m) - i\delta m$$
The counterterm contributes at the same order in $\alpha$ as the one-loop self-energy. It has the same Lorentz structure — a scalar piece (mass) and a $\not{p}$ piece (field normalization).
Renormalization conditions determine the coefficients
$$\delta m = \Sigma(m)\big|_{\text{div}}, \qquad \delta Z_2 = \left.\frac{\partial\Sigma}{\partial\not{p}}\right|_{\not{p}=m,\,\text{div}}$$
In the on-shell renormalization scheme: $\delta m$ is fixed by requiring the propagator pole to be at the physical mass ($p^2 = m^2$), and $\delta Z_2$ is fixed by requiring the residue at the pole to be 1. Each is determined by evaluating the one-loop self-energy and extracting the divergent part. The counterterm is not a free parameter — it is completely determined by the loop calculation.
Divergence cancellation
$$\Sigma^{\text{ren}}(p) = \Sigma(p) + \Sigma_{\text{ct}}(p) = \text{finite}$$
The $1/\epsilon$ poles in $\Sigma(p)$ and $\Sigma_{\text{ct}}(p)$ cancel exactly, leaving a finite renormalized self-energy. This is guaranteed by the renormalizability of QED — the divergence structure of the loop diagram matches the counterterm structure of the Lagrangian.