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. In perturbation theory, mass and charge can be fixed by physical renormalization conditions; infrared-finite predictions require a complete inclusive or dressed observable. The pole and residue conditions below are perturbative prescriptions with an explicit infrared treatment. 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. Here $e>0$ is the electron charge magnitude: $q=-e$, so the global interaction $+q\bar\psi\gamma^\mu\psi A_\mu$ is $-e\bar\psi\gamma^\mu\psi A_\mu$. 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 reduced inverse propagator is

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

The Feynman propagator includes an extra factor $i$: $S_{F,R}(p)=iS_R(p)$. Keeping that factor separate makes the inverse-propagator and Ward identities unambiguous.

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

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

Its photon-vertex rule is $-i\mu^\epsilon e\Gamma^\mu=+i\mu^\epsilon q\Gamma^\mu$; $\Gamma^\mu$ itself has the coupling and factor $i$ removed.

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 relates $Z_1$ and $Z_2$. The counterterm construction and the electron, photon, and vertex conditions are developed in Schwartz 2014, §§ 19.1–19.4, pp. 340–349.

Two useful renormalization schemes

The same regulated functions support different, internally consistent finite definitions. For this on-shell comparison, keep the electron mass nonzero and retain an infrared prescription until infrared-safe quantities have been formed.

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 logarithms of $\mu$ divided by a mass or momentum scale

Here

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

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

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

order by order. The second equation fixes the residue of the regulated perturbative propagator, not an observable probability. Its counterterm generally contains infrared singularities; dimensional regularization may regulate both ultraviolet and infrared regions, provided their poles are distinguished. The on-shell derivative and its infrared divergence are explicit in Schwartz 2014, § 18.3.1, pp. 332–333. These conditions do not establish an isolated charged one-particle pole in the exact theory: the soft-photon limit requires an inclusive or dressed prescription.

For the photon with massive charged matter, $\Pi_R(0)=0$ makes the long-distance Coulomb charge the renormalized input. This zero-momentum subtraction is not an infrared-finite prescription for strictly massless charged matter; use a nonzero subtraction scale in that case. A mass-independent scheme defines $e(\mu)$ by ultraviolet subtraction. Directly comparing these two numerical couplings without a finite conversion is meaningless.

A one-loop ultraviolet checkpoint

A clean benchmark takes all external invariants nonexceptional and Euclidean. The electron and photon kinetic terms and vertex can be treated at zero electron mass. To determine $Z_m$, however, retain the coefficient linear in a nonzero mass, or insert the scalar mass operator before taking the massless limit: the strictly massless two-point function cannot determine that coefficient. Off-shell momenta keep this extraction infrared finite, so every $1/\bar\epsilon$ below is ultraviolet. Write $\alpha=e^2/(4\pi)$. 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

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

Extracting the electron counterterms

The mass qualification can be seen directly in Feynman gauge, $\xi=1$. Keep $m\ne0$ and $p^2<0$. Combining the electron and photon denominators with a parameter $x$, then shifting the loop momentum, gives $\Delta=(1-x)(m^2-xp^2)$. In the convention $S_R^{-1}=p\!\!\!/-m-\Sigma_R$, the amputated self-energy graph is $-i\Sigma_{\rm loop}$. The resulting one-loop expression is

$$\Sigma_{\rm loop}(p) =\frac{e^2\mu^{2\epsilon}\Gamma(\epsilon)}{(4\pi)^{2-\epsilon}} \int_0^1\!\mathrm dx\, \frac{(2-d)x\,p\!\!\!/+dm}{\Delta^\epsilon}.$$

The numerator follows from $\gamma^\mu(k\!\!\!/+m)\gamma_\mu=(2-d)k\!\!\!/+dm$; the term odd in shifted loop momentum integrates to zero. Only the four-dimensional numerator is needed for the pole coefficient: its $O(\epsilon)$ corrections affect finite terms. Expanding $\Gamma(\epsilon)(4\pi\mu^2/\Delta)^\epsilon$ therefore gives

$$\begin{aligned} \Sigma_{\rm loop}^{\rm UV}(p) &=\frac{\alpha}{4\pi\bar\epsilon} \int_0^1\!\mathrm dx\,[-2x\,p\!\!\!/+4m]\\ &=\frac{\alpha}{4\pi\bar\epsilon}[-p\!\!\!/+4m]. \end{aligned}$$

The Lagrangian counterterm changes the inverse propagator by $\delta Z_2 p\!\!\!/-\delta Z_{m2}m$. Consequently its contribution in the self-energy convention used here is

$$\Sigma_{\rm ct}(p) =-\delta Z_2 p\!\!\!/+\delta Z_{m2}m.$$

Matching the two independent structures in $\Sigma_{\rm loop}^{\rm UV}+\Sigma_{\rm ct}^{\rm UV}=0$ gives $\delta Z_2^{\rm UV}=-\alpha/(4\pi\bar\epsilon)$ and $\delta Z_{m2}^{\rm UV}=-4\alpha/(4\pi\bar\epsilon)$. At one loop $\delta Z_{m2}=\delta Z_2+\delta Z_m$, so

$$\delta Z_m^{\rm UV} =\delta Z_{m2}^{\rm UV}-\delta Z_2^{\rm UV} =-\frac{3\alpha}{4\pi\bar\epsilon}.$$

Setting $m=0$ before matching would erase the second equation, while leaving the field counterterm determined by the off-shell two-point function. The loop and subtraction are developed in Schwartz 2014, §§ 18.2.1–18.2.2, pp. 326–329, Eqs. (18.8)–(18.28), with modified minimal subtraction in Schwartz 2014, § 18.4, p. 335, Eqs. (18.59)–(18.62). His dimensional regulator obeys $4-d=\epsilon_{\rm S}=2\epsilon$ here, and his self-energy enters the inverse propagator with the opposite sign; these translations reproduce the coefficients above.

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

For the infrared-finite off-shell checkpoint with ultraviolet minimal subtraction just described, each renormalized two-point function is finite at fixed nonexceptional momentum. Retain the loop and counterterm pieces separately and verify

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

These are ultraviolet cancellation tests. On-shell field counterterms can retain infrared singularities even when evaluated in an off-shell Green function; they must not be included in an undifferentiated pole-cancellation test. Weinberg exhibits precisely this separation after electron self-energy subtraction in Weinberg 1995, § 11.4, pp. 493–496, especially Eq. (11.4.14).

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.

Exercises

Which mass coefficient belongs in the counterterm? In the Feynman-gauge ultraviolet calculation, put $a=\alpha/(4\pi\bar\epsilon)$. A proposed subtraction uses $\delta Z_2=-a$ and $\delta Z_m=-3a$ correctly, but writes the mass counterterm in the Lagrangian as $-\delta Z_m m\bar\psi\psi$. Compute the remaining ultraviolet self-energy and identify the missing term. Why would checking only $m=0$ miss the error?

Solution

The proposed counterterm contributes $\Sigma_{\rm ct}^{\rm wrong}=a p\!\!\!/-3am$. Therefore

$$\Sigma_{\rm loop}^{\rm UV}+\Sigma_{\rm ct}^{\rm wrong} =a(-p\!\!\!/+4m)+a p\!\!\!/-3am =am.$$

The mass coefficient in the Lagrangian must be $\delta Z_{m2}=\delta Z_2+\delta Z_m=-4a$ at this order. Its missing $\delta Z_2m=-am$ contribution to $\Sigma_{\rm ct}$ cancels the residual. The error is invisible at $m=0$, which is why the mass coefficient must be extracted before that limit.

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.