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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.

The fermion self-energy is the amputated one-particle-irreducible two-point function $\Sigma (/p)$. It corrects both the coefficient of $/p$ and the fermion mass and thereby shifts the pole and residue of the exact propagator.

The first panel is the one-particle-irreducible Feynman diagram for photon vacuum polarization. The two internal Dirac propagators carry $k$ and $k+q$, and the external photons carry $q$. The other panels fix the routings used in parts c and d.

The closed fermion loop sign contributes a minus sign; there is no extra symmetry factor. Applying the given Feynman rules, with the external propagators amputated,Equivalently, the required amplitude isThe Feynman slash notation is $/k={\gamma }^{\rho }{k}_{\rho }$ and $\mathrm{tr}1=4$ in the chosen dimensional regularization convention.

The middle panel in part a has an internal photon of momentum $k$ joining two points on a fermion line; the internal fermion has momentum $p-k$. It is a one-particle-irreducible Feynman diagram with no closed fermion loop sign. In Feynman gauge,orThe decomposition in the source gives $\mathrm{tr}(/p\Sigma )=4p^{2}S(p^2)$. In particular,For a direct ultraviolet divergence extraction, use a Feynman parameter $x$ multiplying the massive denominator, shift $r=k-xp$, and define $\Delta =x{m}^2-x(1-x)p^2$. Then $/p-/k=(1-x)/p-/r$ andOdd $r$ terms integrate to zero. Taking the trace explicitly supplies the weight $(2-d)(1-x)$, whose integral tends to $-1$. HenceAs a check on the same fermion self-energy calculation,This is the ultraviolet pole of QED fermion self-energy. It is safe to extract it at generic external momentum; an on shell or massless limit must distinguish any infrared divergence from this ultraviolet divergence.

In $d=4-2ϵ$, the modified minimal subtraction scheme subtracts ${\Delta }_{\overline{\mathrm{MS}}}=1/ϵ-{\gamma }_E+\log (4\pi )$ for each one-loop pole, with no further finite, kinematics-dependent subtraction. Equivalently, use a rescaled renormalization scale ${\overline{\mu }}^2=4\pi e^{-{\gamma }_E}{\mu }_R^2$ and subtract only $1/ϵ$ in that convention. This differs from the minimal subtraction scheme by a fixed scale convention.

Let $C=e^2{\Delta }_{\overline{\mathrm{MS}}}/(16{\pi }^2)$. With the source's counterterm insertion signs, cancellation proceeds as follows. The vertex has $-ie{\gamma }^{\mu }C$ from the loop and $-ie{\gamma }^{\mu }{\delta }_1$ from its counterterm, giving ${\delta }_1=-C$. The kinetic part of the fermion self-energy insertion is $+i/pC$ and its counterterm is $+i/p{\delta }_2$, giving ${\delta }_2=-C$. Finally the photon vacuum polarization insertion is $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu })(4C/3)$ and its counterterm is $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu }){\delta }_3$, givingWriting $Z_i=1+{\delta }_i$ therefore verifies the Ward identity for QED renormalization constants, $Z_1=Z_2$, at one loop. The proper-vertex Ward-Takahashi identity in QED independently gives ${\Lambda }^{\mu }(p,p)=-\partial \Sigma (p)/\partial p_{\mu }$, whose divergent part reproduces the opposite signs of $F_1^{(1)}$ and $S$ found above. The equality holds in a common gauge fixing and subtraction prescription; the individual constants $Z_1,Z_2$ are gauge dependent.

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.