Renormalizing the Electron and Photon

QED is renormalized by relating bare fields and parameters to finite, scheme-dependent ones and adding local counterterms with the same operators as the original action. Electron and photon propagator poles, the electric charge, and a complete inclusive observable can be fixed physically; the individual field-renormalization constants and finite counterterms are not observables and generally depend on gauge and scheme.

Required background. The QED action fixes the model and signs. Renormalized perturbation theory supplies the general counterterm construction.

Helpful background. Renormalization conditions, schemes, and finite parts explains why different finite subtractions describe the same physics.

Bare and renormalized QED

Use dimensional regularization with $d=4-2\epsilon$ and a covariant gauge. The bare Lagrangian is

$$\begin{aligned} \mathcal L_0={}&-\frac14F_{0\mu\nu}F_0^{\mu\nu} -\frac{1}{2\xi_0}(\partial\cdot A_0)^2 +\bar\psi_0(i\partial\!\!\!/-m_0)\psi_0 \\ &-e_0\bar\psi_0\gamma^\mu\psi_0A_{0\mu}. \end{aligned}$$

Define

$$\psi_0=Z_2^{1/2}\psi, \qquad A_0^\mu=Z_3^{1/2}A^\mu, \qquad m_0=Z_m m, \qquad e_0=\mu^\epsilon Z_e e, \qquad \xi_0=Z_3\xi.$$

It is convenient to package the interaction renormalization as

$$Z_1\equiv Z_eZ_2Z_3^{1/2}.$$

Writing $Z_i=1+\delta Z_i$, the renormalized Lagrangian plus counterterms becomes

$$\begin{aligned} \mathcal L={}&-\frac14F_{\mu\nu}F^{\mu\nu} -\frac{1}{2\xi}(\partial\cdot A)^2 +\bar\psi(i\partial\!\!\!/-m)\psi -\mu^\epsilon e\bar\psi\gamma^\mu\psi A_\mu \\ &-\frac14\delta Z_3F_{\mu\nu}F^{\mu\nu} +\delta Z_2\bar\psi i\partial\!\!\!/\psi -\delta Z_{m2}\,m\bar\psi\psi -\mu^\epsilon e\,\delta Z_1\bar\psi\gamma^\mu\psi A_\mu, \end{aligned}$$

where $\delta Z_{m2}=Z_2Z_m-1$. Keeping $Z_m$ distinct from the mass-term counterterm prevents a common bookkeeping error.

The three primitively divergent functions

Lorentz invariance and parity decompose the one-particle-irreducible electron self-energy as

$$\Sigma(p)=p\!\!\!/\,\Sigma_V(p^2)+m\Sigma_S(p^2),$$

so the renormalized inverse propagator is

$$S_R^{-1}(p)=p\!\!\!/-m-\Sigma_R(p).$$

Gauge invariance makes the photon self-energy transverse,

$$\Pi^{\mu\nu}(q) =(q^2\eta^{\mu\nu}-q^\mu q^\nu)\Pi(q^2), \qquad q_\mu\Pi^{\mu\nu}=0.$$

The proper vertex is written

$$\Gamma^\mu(p',p)=\gamma^\mu+\Lambda^\mu(p',p).$$

The counterterms $\delta Z_2$ and $\delta Z_{m2}$ cancel local ultraviolet pieces of $\Sigma$; $\delta Z_3$ cancels the ultraviolet part of $\Pi$; and $\delta Z_1$ cancels the vertex divergence. These are separate operations even though the Ward–Takahashi identity later relates $Z_1$ and $Z_2$. Detailed one-loop constructions appear in Schwartz 2014, Chapters 16, 18, and 19, pp. 302–354 and Weinberg 1995, §§ 10.3–11.4, pp. 436–498.

Two useful renormalization schemes

The same regulated functions support different, internally consistent finite definitions.

Quantity	On-shell conditions	$\overline{\mathrm{MS}}$ conditions
electron mass	pole at $p^2=m_{\rm pole}^2$	subtract only $1/\bar\epsilon$; $m(\mu)$ runs
electron field	perturbative pole residue set to one	subtract only the ultraviolet pole
photon field and charge	$\Pi_R(0)=0$; charge fixed in the zero-momentum limit	subtract ultraviolet pole at scale $\mu$
finite parts	fixed by physical low-momentum conditions	retain $\ln(\mu)$ dependence

Here

$$\frac{1}{\bar\epsilon} \equiv\frac1\epsilon-\gamma_E+\ln4\pi.$$

For the electron, the on-shell conditions can be written

$$S_R^{-1}(p)u(p)\big|_{p^2=m^2}=0, \qquad \lim_{p\!\!\!/\to m} \frac{S_R(p)}{i/(p\!\!\!/-m)}=1$$

order by order. The second equation is a perturbative normalization prescription. Because a charged particle is accompanied by arbitrarily soft photons, the exact charged propagator does not have an isolated one-particle pole with a nonzero residue. Infrared physics must therefore be kept distinct from the ultraviolet definition.

For the photon, $\Pi_R(0)=0$ makes the long-distance Coulomb charge the renormalized input. A mass-independent scheme instead defines $e(\mu)$ by ultraviolet subtraction. Directly comparing these two numerical couplings without a finite conversion is meaningless.

A one-loop ultraviolet checkpoint

An especially clean benchmark uses massless QED with all external invariants nonexceptional and Euclidean. Off-shell momenta remove infrared poles, so every $1/\bar\epsilon$ below is ultraviolet. With the covariant propagator

$$D_{\mu\nu}(k)=\frac{-i}{k^2+i0} \left[\eta_{\mu\nu}-(1-\xi)\frac{k_\mu k_\nu}{k^2+i0}\right],$$

one unit-charge Dirac fermion gives

$$\delta Z_2^{\rm UV} =-\frac{\alpha\xi}{4\pi\bar\epsilon}, \qquad \delta Z_1^{\rm UV} =-\frac{\alpha\xi}{4\pi\bar\epsilon},$$ $$\delta Z_3^{\rm UV} =-\frac{\alpha}{3\pi\bar\epsilon}, \qquad \delta Z_m^{\rm UV} =-\frac{3\alpha}{4\pi\bar\epsilon}.$$

The equality of the first two expressions is required for every $\xi$, including $\xi=0,1,3$; it is not an accident of Feynman gauge. The mass and photon coefficients are gauge independent at this order, while $Z_2$ is not. From $Z_1=Z_2$ one obtains

$$Z_e=Z_3^{-1/2} =1+\frac{\alpha}{6\pi\bar\epsilon}+O(\alpha^2).$$

This checkpoint isolates ultraviolet structure. If the external electron is instead put on shell, dimensional regularization can use the same symbol $1/\epsilon$ for infrared and ultraviolet poles. They must be labelled before cancellation; an ultraviolet counterterm must never be chosen to remove an infrared singularity.

Pole cancellation and scheme translation

After counterterms are included, each renormalized two-point function is finite at fixed nonexceptional momentum. A robust check is to retain the loop and counterterm pieces separately and verify

$$\operatorname*{Res}_{\epsilon=0} \bigl[\Sigma_{\rm loop}+\Sigma_{\rm ct}\bigr]=0, \qquad \operatorname*{Res}_{\epsilon=0} \bigl[\Pi_{\rm loop}+\Pi_{\rm ct}\bigr]=0.$$

Finite scheme changes are coordinate changes on parameter space. If

$$g'=g+c g^3+O(g^5)$$

and an observable has expansion

$$\mathcal O=g^2\mathcal O_0+g^4\mathcal O_1+O(g^6),$$

then the same prediction expressed in $g'$ is

$$\mathcal O=g'^2\mathcal O_0 +g'^4(\mathcal O_1-2c\mathcal O_0)+O(g'^6).$$

Changing the input coupling without transforming the coefficient creates a spurious scheme dependence at the order being claimed. After consistent conversion, two schemes differ only by terms beyond the retained order.

Common pitfalls

Bare, on-shell, and running masses are not interchangeable. Each belongs to a different parameter definition; conversion terms must accompany a change of scheme.

A field-renormalization constant is not an observable probability. $Z_2$ is gauge and scheme dependent, and its exact charged-particle interpretation is obstructed by infrared physics.

One symbol for two pole types is dangerous. Label $1/\epsilon_{\rm UV}$ and $1/\epsilon_{\rm IR}$ whenever on-shell integrals contain both.

Counterterms do not authorize arbitrary finite changes. A finite redefinition must be applied to parameters, amplitudes, and matching conditions consistently.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), Chapters 16 and 18–19, doi:10.1017/9781139540940.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press (1995), Chapters 10–12, doi:10.1017/CBO9781139644167.