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The fermion self-energy of an electron in quantum electrodynamics. After renormalization, its contribution in a bound chemical atom is one source of the Lamb shift; other radiative corrections and nuclear structure also affect the observed energy levels.

Fix the sign convention by taking the proper two-point insertion to be $i\Sigma (/p)$. This convention matches the loop expression and mass conversion printed later. Dyson resummation of successive fermion self-energy insertions givesThe subscript denotes the renormalized self-energy. If the insertion is instead named $-i{\Sigma }_{\mathrm{usual}}$, then ${\Sigma }_{\mathrm{usual}}=-{\Sigma }_R$ and the denominator is written $/p-m-{\Sigma }_{\mathrm{usual}}$. These are the same physical convention.

Lorentz covariance allows ${\Sigma }_R=\mathcal{A}(p^2)/p+\mathcal{B}(p^2)m$. The physical mass-shell condition isIt locates the mass-shell singularity of the Dirac propagator. In infrared-regulated perturbation theory this is the pole mass; near a simple pole the quantum field theory propagator has the form $i{Z}_{\mathrm{pole}}(/p+m_{\mathrm{phys}})/(p^2-m_{\mathrm{phys}}^2+i0)$. The mass and pole residue are different quantities: the former fixes the singularity's location, the latter the field normalization. The calculation below uses this standard perturbative pole-mass definition.

After modified minimal subtraction, the surviving logarithm in the given fermion self-energy is $\log [{\mu }^2/((1-x)(m^2-p^{2}x))]$. For a one-loop mass shift, set $p^2=m^2$ and let $/p$ act as $m$ on an on-shell spinor inside that correction; changing these arguments by the mass shift contributes only at order $e^4$. ThusThe needed integrals areFor the second, put $u=1-x$ and use ${\int }_0^1\log udu=-1$ and ${\int }_0^{1}u\log udu=-1/4$. ConsequentlyThe pole mass condition $m_{\mathrm{phys}}-m+{\Sigma }_R=0$ gives $m_{\mathrm{phys}}=m[1+e^2(5+3\log ({\mu }^2/m^2))/(16{\pi }^2)]+O(e^4)$. Invert this relation and replace $m$ by $m_{\mathrm{phys}}$ inside the already one-loop term to obtainThe negative sign in the running-mass conversion follows from the explicitly chosen self-energy convention. Subtracting poles alone, instead of the overbarred combination, would leave additional $\log 4\pi -{\gamma }_E$ terms.

Repeated insertions of the fermion self-energy form a geometric Dyson resummation:The physical fermion mass is the pole mass: after analytic continuation, $m_{\mathrm{phys}}$ is determined by the zero of the exact inverse propagator at $p^2=-m_{\mathrm{phys}}^2$. Ifthen to all orders the pole obeyswith the signs fixed by the displayed Dyson convention.

In quantum electrodynamics, the modified minimal subtraction scheme running mass differs from the perturbative pole mass by a finite correction. The displayed relation follows by evaluating the renormalized fermion self-energy at the mass shell and imposing a zero of the inverse Dirac propagator. The sign assigned to the self-energy insertion must be matched to its denominator convention; changing that naming convention does not change the conversion. A wave-function coefficient counterterm and multiplicative mass renormalization are also distinct: the coefficient of the mass term includes both at first order.