Renormalization and the Running Coupling

Tree-level QED is finite and already accurate to a percent. Pushing further means including loop diagrams, and the first loop integral one writes down is infinite. The resolution — regularization to make the infinity finite, then renormalization to absorb it into measured quantities — is not a trick to hide a defect. It is the statement that the parameters in the Lagrangian are not what a experiment measures, and once the substitution is made the theory predicts finite corrections that agree with experiment to twelve digits. The physical residue of the whole procedure is that the coupling constant depends on the energy scale of the process. This lesson traces that logic from the divergent integral to the running of $α$ .¹

Natural units throughout; $α = e^{2} /4 π \approx 1/137$ at low energy.

The three primitive divergences

At one loop, three diagrams correct the tree amplitudes of QED, and each contains a closed loop whose momentum $q$ is not fixed by the external kinematics but integrated over all values $\int d^{4} q / (2 π)^{4}$ .

Vacuum polarization (photon self-energy): a photon momentarily converts into a virtual electron-positron pair that reannihilates. It corrects the photon propagator and, physically, the effective charge.
Electron self-energy: an electron emits and reabsorbs a virtual photon. It corrects the electron propagator and shifts the mass.
Vertex correction: a virtual photon spans the two fermion legs at a vertex. It corrects the coupling and produces the anomalous magnetic moment of the following lesson.

Each loop integral grows at large loop momentum. Counting powers, the vacuum polarization integrand behaves as $\int d^{4} q / q^{2}$ at large $q$ , which is quadratically divergent by naive counting and logarithmically divergent once gauge invariance is enforced; the self-energy and vertex integrals are logarithmically divergent. These are ultraviolet divergences — they come from the high-momentum, short-distance end of the integral, not from any large-distance behavior.

The three primitive one-loop diagrams of QED. Vacuum polarization inserts a virtual fermion loop into a photon line; the self-energy inserts a virtual photon loop into a fermion line; the vertex correction bridges the two fermion legs at a vertex with a virtual photon. Each carries an integral over the unfixed loop momentum that diverges in the ultraviolet.

Regularization

A divergent integral has no value until it is made finite by a regulator, a deformation of the theory controlled by a parameter that is removed at the end. The regulator makes the divergence explicit as a dependence on the parameter.

Momentum cutoff: forbid loop momenta above a scale $Λ$ . The divergence reappears as $ln Λ$ , and the physics must not depend on $Λ$ .
Pauli-Villars: subtract a copy of the loop with a heavy fictitious mass $M$ ; the difference is finite and the divergence lives in $ln M$ .
Dimensional regularization: compute in $d = 4 - ϵ$ dimensions, where the loop integral converges, and the divergence appears as a pole $1/ ϵ$ as $ϵ \to 0$ . This regulator preserves gauge invariance and is the modern standard.

Whatever the regulator, the one-loop vacuum polarization produces a correction to the photon propagator of the schematic form

Π (q^{2}) \sim \frac{α}{3 π} ln \frac{Λ ^{2}}{- q ^{2}} + (finite),

a logarithm of the cutoff times the finite, momentum-dependent piece that carries the real physics. The task is to give the logarithm a home.

Renormalization

The parameters in the QED Lagrangian — the bare charge $e_{0}$ , mass $m_{0}$ , and field normalizations — are not measured directly. What a experiment measures is the charge seen at some reference scale and the physical mass of the electron, both of which already include the loop corrections. Renormalization is the statement that the bare parameters are functions of the cutoff, chosen so that the measured quantities come out finite and correct.

Concretely, write each bare quantity as a renormalized quantity times a divergent renormalization constant $Z$ ,

e_{0} = Z_{e} e, m_{0} = Z_{m} m, ψ_{0} = Z_{2}^{1/2} ψ, A_{0} = Z_{3}^{1/2} A,

and let the $Z$ absorb the loop divergences. Every one-loop divergence in QED is soaked up by these constants; no divergence is left over that needs a new parameter. A theory with this property — finitely many renormalization constants handling all orders — is renormalizable, and QED is the cleanest example. Renormalizability is what makes the perturbation series predictive rather than a sequence of new infinities.

The finite remainder $Π (q^{2}) - Π (0)$ , with the divergent $Π (0)$ absorbed into the charge, is unambiguous and measurable. Its momentum dependence is the running coupling.

The running of the coupling

Once the divergent constant is absorbed at a reference scale, the leftover momentum dependence of the vacuum polarization makes the effective coupling depend on the scale $Q$ at which it is probed. Summing the chain of one-loop vacuum polarization insertions (a geometric series) gives the running coupling

α (Q^{2}) = \frac{α ( μ ^{2} )}{1 - \frac{α ( μ ^{2} )}{3 π} ln \frac{Q ^{2}}{μ ^{2}}},

for a single fermion species, with the coefficient generalizing to $\frac{α}{3 π} \sum_{f} Q_{f}^{2}$ when several charged fermions circulate in the loop. The closed form is the sum of a chain: the bare propagator, plus one vacuum polarization insertion, plus two, and so on, a geometric series whose resummation turns the single logarithm into a coupling valid to high energy.

The running coupling as a resummed chain. Adding the bare photon propagator, one vacuum-polarization bubble, two bubbles, and so on gives a geometric series; summing it converts the leading logarithm into the effective coupling that grows with energy.

The denominator decreases as $Q$ grows, so $α (Q^{2})$ increases with energy. Equivalently, the beta function

β (α) = \frac{d α}{d ln Q} = \frac{2 α ^{2}}{3 π} f \sum Q_{f}^{2} > 0

is positive: QED is not asymptotically free. Numerically, starting from $α (0) = 1/137.036$ measured at large distance and running up through all the charged fermions, the coupling reaches

α (M_{Z}) \approx \frac{1}{128}

at the $Z$ mass $M_{Z} \approx 91 GeV$ — a shift of roughly seven percent that is confirmed by precision electroweak data.² The coupling that undergraduate textbooks quote as a constant is the low-energy limit of a scale-dependent quantity.

The effective fine-structure constant versus the energy scale on a logarithmic axis. It sits at one over one hundred thirty-seven at low energy and climbs to about one over one hundred twenty-eight at the Z mass, because virtual fermion loops screen the charge and probing at higher energy penetrates closer to the bare charge.

Charge screening: the physical picture

The running of $α$ has a direct physical reading. A bare charge polarizes the vacuum around it: virtual electron-positron pairs orient with the positrons drawn inward and the electrons pushed outward, exactly as a dielectric medium screens a charge. A test particle at large distance sees the bare charge partly cancelled by the surrounding cloud, hence the smaller low-energy coupling. Probing at higher energy means approaching closer, inside part of the screening cloud, so more of the bare charge is exposed and the effective coupling grows. This is why $α (Q^{2})$ rises with $Q$ : short distance and high energy are the same thing, and both see a less-screened charge.³

The sign of the effect is a feature of QED specifically. The vacuum of QED screens charge because only fermion loops contribute to the photon polarization, and they always screen. In quantum chromodynamics the gluon self-coupling adds loops of the opposite sign, the vacuum anti-screens, and the coupling instead falls at high energy — asymptotic freedom, taken up in the QCD module. The contrast between the two is the sign of one beta function.

Charge screening by the vacuum. Virtual pairs polarize around a bare charge with opposite charges drawn inward, so a probe at large distance sees a reduced effective charge; a probe at short distance penetrates the cloud and sees more of the bare charge, which is why the coupling grows toward high energy.

Summary

Loop diagrams in QED diverge in the ultraviolet: the vacuum polarization, self-energy, and vertex corrections each integrate over an unfixed loop momentum and grow logarithmically at large $q$ . A regulator — cutoff, Pauli-Villars, or dimensional — exposes the divergence as $ln Λ$ or $1/ ϵ$ . Renormalization absorbs it into the bare charge, mass, and field normalizations via the constants $Z_{e}, Z_{m}, Z_{2}, Z_{3}$ ; because a finite set suffices to all orders, QED is renormalizable. The finite remainder makes the coupling run: $α (Q^{2}) = α / [1 - (α /3 π) ln (Q^{2} / μ^{2})]$ rises from $1/137$ at low energy to $\approx 1/128$ at $M_{Z}$ , because the vacuum screens charge and higher energy probes penetrate the screening cloud. The vertex correction that appears here also produces the electron's anomalous magnetic moment, the most precise test of the whole framework, treated next.

The one-loop structure of QED — vacuum polarization, self-energy, vertex correction — and the running of the coupling are treated in Griffiths, §7.9 and Ch. 6; Halzen & Martin, Ch. 7 (regularization and renormalization); and Thomson, Ch. 6. Tong, The Standard Model (Cambridge Part III), §3.1, gives the beta-function derivation and the screening/anti-screening picture, damtp.cam.ac.uk/user/tong/standardmodel.html. ↩
The low-energy value $α^{- 1} = 137.035999$ and the effective coupling at the $Z$ mass, $α^{- 1} (M_{Z}) \approx 128$ , together with $M_{Z} = 91.1876 GeV$ , are from the Particle Data Group, pdg.lbl.gov. Tong quotes $α (μ) \approx 1/127$ at $E \approx 100 GeV$ , §3.1. ↩
The dielectric-screening interpretation of vacuum polarization and the contrast with QCD anti-screening are given in Tong, The Standard Model (Cambridge Part III), §3.1.2 (Anti-Screening and Paramagnetism), damtp.cam.ac.uk/user/tong/standardmodel.html. ↩