Vacuum Polarization, Running Charge, and Screening

Vacuum polarization inserts a charged-fermion loop into the photon propagator. Current conservation makes the insertion transverse; renormalization turns its scalar coefficient into either a momentum-dependent effective charge or a running scheme parameter. Both descriptions encode screening, but they are not numerically identical until their subtraction conventions are matched.

Required background. Electron and photon renormalization supplies $Z_3$ and the subtraction conditions. Beta functions and anomalous dimensions supplies the general renormalization-group definition.

Helpful background. Subtracted dispersion relations explains how the timelike spectral cut determines the spacelike function.

The transverse photon self-energy

For one Dirac fermion of mass $m$ and charge magnitude $e$, the one-loop proper photon two-point function is

$$i\Pi^{\mu\nu}(q) =-(-ie)^2\mu^{2\epsilon} \int\frac{\mathrm d^d\ell}{(2\pi)^d} \operatorname{tr}\left[ \gamma^\mu\frac{i(\ell\!\!\!/+m)}{\ell^2-m^2+i0} \gamma^\nu\frac{i(\ell\!\!\!/+q\!\!\!/+m)}{(\ell+q)^2-m^2+i0} \right].$$

The initial minus sign is the closed-fermion-loop sign. A symmetry-preserving regulator and consistent momentum routing give

$$q_\mu\Pi^{\mu\nu}(q)=0,$$

so Lorentz invariance fixes a single scalar function. Choose the positive-spectral current-correlator convention

$$\Pi^{\mu\nu}(q) =(q^\mu q^\nu-q^2\eta^{\mu\nu})\Pi_M(q^2).$$

Any residual $\eta^{\mu\nu}$ term not accompanied by the required $q^2$ is a photon-mass artifact and violates the Ward identity. The loop calculation and its Coulomb-potential interpretation are developed in Schwartz 2014, § 16.2, pp. 304–314 and Weinberg 1995, § 11.2, pp. 473–482.

A subtracted Euclidean effective charge

Let $Q^2=-q^2>0$ be spacelike. To make the sign convention explicit, define the positive Euclidean screening function

$$\Pi_E(Q^2) \equiv-\bigl[\Pi_M(-Q^2)-\Pi_M(0)\bigr].$$

Zero-momentum subtraction then gives the finite one-loop result

$$\boxed{ \Pi_E(Q^2)=\frac{2\alpha}{\pi} \int_0^1\mathrm dx\,x(1-x) \ln\left[1+\frac{Q^2}{m^2}x(1-x)\right]}$$

with $\alpha=e^2/(4\pi)$. Dyson summation of transverse insertions motivates the zero-momentum-subtracted Euclidean effective charge

$$\alpha_{\rm eff}(Q^2) =\frac{\alpha}{1-\Pi_E(Q^2)}.$$

This definition is tied to the chosen subtraction and spacelike kinematics. It should not be silently identified with an $\overline{\mathrm{MS}}$ coupling.

At low momentum, expand the logarithm:

$$\begin{aligned} \Pi_E(Q^2) &=\frac{2\alpha}{\pi}\frac{Q^2}{m^2} \int_0^1\mathrm dx\,x^2(1-x)^2 +O(Q^4/m^4)\\ &=\frac{\alpha}{15\pi}\frac{Q^2}{m^2} +O(Q^4/m^4). \end{aligned}$$

The $Q^2/m^2$ suppression is decoupling: a massive charged species cannot produce logarithmic running far below its pair threshold. At the independent checkpoint $Q^2=m^2$,

$$\Pi_E(m^2) =\frac{\alpha}{9\pi} \left[7+\sqrt5\ln(9-4\sqrt5)\right].$$

At $Q^2\gg m^2$,

$$\Pi_E(Q^2) =\frac{\alpha}{3\pi} \left[\ln\frac{Q^2}{m^2}-\frac53\right] +O\!\left(\frac{m^2}{Q^2}\ln\frac{Q^2}{m^2}\right).$$

The positive logarithm makes $\alpha_{\rm eff}$ increase as shorter distances are resolved.

Timelike threshold and analyticity

The Euclidean integral has no imaginary part. Continuing the underlying Minkowski scalar $\Pi_M$ to timelike $s=q^2>0$ requires the $+i0$ prescription and a declared logarithm branch. The first physical cut begins at

$$s=4m^2,$$

where the photon can create a real fermion–antifermion pair. With the scalar convention above, the spectral density is

$$\rho(s)=\frac{1}{\pi}\operatorname{Im}\Pi_M(s+i0) =\frac{\alpha}{3\pi} \left(1+\frac{2m^2}{s}\right) \sqrt{1-\frac{4m^2}{s}}\, \theta(s-4m^2).$$

A once-subtracted dispersion relation reconstructs the spacelike function:

$$\Pi_E(Q^2) =Q^2\int_{4m^2}^{\infty} \frac{\mathrm ds}{s(s+Q^2)}\rho(s).$$

This formula supplies an independent sign check: $\rho(s)\ge0$ implies $\Pi_R(Q^2)\ge0$ for $Q^2>0$. Using the Euclidean logarithm directly above timelike threshold would miss both the imaginary part and the correct sheet.

The renormalization-group coupling

In $\overline{\mathrm{MS}}$, the one-loop photon counterterm for $N_f$ active unit-charge Dirac fermions is

$$Z_3=1-\frac{N_f\alpha}{3\pi\bar\epsilon}+O(\alpha^2).$$

Together with $e_0=\mu^\epsilon Z_3^{-1/2}e$, holding the bare coupling fixed gives

$$\boxed{ \frac{\mathrm d\alpha}{\mathrm d\ln\mu} =\frac{2N_f}{3\pi}\alpha^2+O(\alpha^3)}.$$

For charges $Q_f$ in units of $e$, replace $N_f$ by $\sum_f Q_f^2$. At fixed $N_f$, integration gives the one-loop invariant

$$\alpha^{-1}(\mu) +\frac{2N_f}{3\pi}\ln\frac{\mu}{\mu_{\rm ref}} =\text{constant},$$

or

$$\alpha^{-1}(\mu) =\alpha^{-1}(\mu_0) -\frac{2N_f}{3\pi}\ln\frac{\mu}{\mu_0}.$$

The factor is written for $\ln\mu$. Differentiating with respect to $\ln\mu^2$ halves it; mixing the two conventions is a common error. The general counterterm derivation appears in Schwartz 2014, §§ 23.1–23.2, pp. 417–426.

Threshold matching

A mass-independent scheme does not automatically decouple a heavy field. Below a mass $M$, match the full theory onto an EFT without that active species and run with the lower $N_f$. At leading order the coupling is continuous at $\mu=M$; higher orders add a finite decoupling coefficient.

For the exact benchmark

$$N_f=1\quad(\mu<M), \qquad N_f=2\quad(\mu>M), \qquad \alpha(M)=\frac{1}{10},$$

one-loop running with leading matching gives

$$\alpha^{-1}(M/e)=10+\frac{2}{3\pi}, \qquad \alpha^{-1}(eM)=10-\frac{4}{3\pi}.$$

Running through $M$ with one unchanged beta coefficient fails the threshold check.

To compare $\alpha_{\overline{\rm MS}}(\mu)$ with $\alpha_{\rm eff}(Q^2)$, choose a matching point, compute the finite conversion there, and only then compare slopes. Their raw values belong to different definitions.

Screening and the Landau-pole caution

Virtual charged pairs polarize the vacuum so that a distant probe sees a more screened charge than a short-distance probe. Equivalently, the beta function is positive and the renormalized coupling grows toward the ultraviolet. This interpretation is gauge invariant when formulated through the current–current amplitude or a physical effective charge, not through a gauge-dependent photon field normalization alone.

Formally extending the one-loop solution produces a scale where its denominator vanishes. That “Landau pole” lies outside the regime in which the one-loop approximation can justify itself. It signals that perturbative QED does not supply a controlled ultraviolet completion; it is not a prediction of a physical pole at a trustworthy energy.

Common pitfalls

An $\overline{\mathrm{MS}}$ coupling is not a measured potential by definition. Match it to a physical effective charge before comparing numbers.

Euclidean and timelike polarization are not the same real function. Above $4m^2$, the $+i0$ prescription and branch cut are essential.

Mass-independent running does not implement decoupling automatically. Change the active theory and match at each threshold.

The one-loop Landau pole is not controlled evidence of a state. It occurs where the approximation used to derive it has already failed.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), §§ 16.2 and 23.1–23.2, doi:10.1017/9781139540940.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press (1995), § 11.2, doi:10.1017/CBO9781139644167.