Chapter 6: Renormalization

Loop diagrams in QED produce ultraviolet (UV) divergences — integrals that blow up at high momenta. Renormalization is the systematic procedure for absorbing these infinities into redefinitions of the physical parameters (mass, charge, field normalization), yielding finite, predictive results. Far from being a defect, renormalization reveals that physical quantities depend on the energy scale at which they are measured.

UV Divergences in Loop Diagrams

Consider the one-loop correction to any QED process. Each loop involves an integral over an undetermined internal momentum $\ell$. The superficial degree of divergence for a diagram with $E_f$ external fermion lines and $E_\gamma$ external photon lines is:

$D = 4 - \frac{3}{2}E_f - E_\gamma$

This power-counting formula tells us which diagrams are divergent. The three primitive divergences in QED are:

Electron Self-Energy ($E_f=2, E_\gamma=0, D=1$)

The electron propagator receives a correction from a virtual photon loop. Superficially linearly divergent, but gauge invariance (specifically, chiral symmetry in the massless limit) reduces this to a logarithmic divergence.

Photon Vacuum Polarization ($E_f=0, E_\gamma=2, D=2$)

The photon propagator is modified by a virtual electron-positron loop (vacuum polarization). Superficially quadratically divergent, but the Ward identity ensures that$\Pi^{\mu\nu}(q) = (q^2 g^{\mu\nu} - q^\mu q^\nu)\Pi(q^2)$, reducing the divergence to logarithmic.

Vertex Correction ($E_f=2, E_\gamma=1, D=0$)

The QED vertex receives a one-loop correction that is logarithmically divergent. This correction modifies both the charge (Dirac form factor $F_1$) and generates an anomalous magnetic moment (Pauli form factor $F_2$).

A remarkable property of QED is that it is renormalizable: all divergences can be absorbed into a finite number of counterterms. The key insight is that only diagrams with $D \geq 0$ diverge, and there are only three such structures. Higher-point functions (e.g., light-by-light scattering with $E_\gamma = 4$) have$D < 0$ and are finite.

Dimensional Regularization

Dimensional regularization is the most elegant and widely used regularization scheme. The idea, due to 't Hooft and Veltman, is to analytically continue the spacetime dimension from $d = 4$ to $d = 4 - \varepsilon$, where $\varepsilon$ is a small positive parameter. In $d < 4$ dimensions, the previously divergent integrals become finite, with the divergences reappearing as poles in $1/\varepsilon$.

Master Formula

The fundamental integral in dimensional regularization is:

$\int \frac{d^d\ell_E}{(2\pi)^d} \frac{1}{(\ell_E^2 + \Delta)^n} = \frac{1}{(4\pi)^{d/2}} \frac{\Gamma(n - d/2)}{\Gamma(n)} \frac{1}{\Delta^{n-d/2}}$

where the integral is in Euclidean space (after Wick rotation $\ell^0 \to i\ell_E^0$) and $\Delta$ is a function of external momenta and masses determined by Feynman parameterization. As $d \to 4$, the gamma function $\Gamma(n-d/2)$develops poles:

$\Gamma(\varepsilon/2) = \frac{2}{\varepsilon} - \gamma_E + O(\varepsilon)$

where $\gamma_E \approx 0.5772$ is the Euler-Mascheroni constant. The$1/\varepsilon$ pole is the dimensional regularization analog of the$\ln\Lambda$ divergence in cutoff regularization.

Mass Scale $\mu$

In $d \neq 4$ dimensions, the coupling constant acquires a mass dimension. To keep$e$ dimensionless, we introduce a mass scale $\mu$:

$e \to e\mu^{\varepsilon/2}$

This scale $\mu$ is the renormalization scale. Physical predictions must be independent of $\mu$, and this requirement leads to the renormalization group equations.

Electron Self-Energy

The one-loop electron self-energy is the correction to the electron propagator from a virtual photon loop. The self-energy function $\Sigma(\not{p})$ modifies the propagator:

$S_F(p) = \frac{i}{\not{p} - m - \Sigma(\not{p})}$

At one loop in Feynman gauge, the self-energy integral is:

$-i\Sigma(\not{p}) = (-ie)^2 \int \frac{d^d\ell}{(2\pi)^d} \gamma^\mu \frac{i(\not{p}-\not{\ell}+m)}{(p-\ell)^2-m^2} \gamma_\mu \frac{-i}{\ell^2}$

Using Feynman parameterization and the master integral formula, the result in $d = 4 - \varepsilon$ dimensions decomposes as:

$\Sigma(\not{p}) = \frac{\alpha}{4\pi}\left[\not{p}\left(-\frac{1}{\varepsilon} + \text{finite}\right) + m\left(\frac{4}{\varepsilon} + \text{finite}\right)\right]$

The self-energy has two Lorentz structures: one proportional to $\not{p}$(wavefunction renormalization) and one proportional to $m$ (mass renormalization). Both contain $1/\varepsilon$ poles that must be cancelled by counterterms.

Photon Vacuum Polarization

The vacuum polarization tensor $\Pi^{\mu\nu}(q)$ describes the correction to the photon propagator from a virtual electron-positron loop. By the Ward identity, it takes the transverse form:

$\Pi^{\mu\nu}(q) = (q^2 g^{\mu\nu} - q^\mu q^\nu)\Pi(q^2)$

The one-loop calculation gives:

$\Pi(q^2) = -\frac{\alpha}{3\pi}\left[\frac{1}{\varepsilon} - \gamma_E + \ln\frac{4\pi\mu^2}{m^2} + \frac{5}{3} - 4\int_0^1 dx\, x(1-x)\ln\left(1 - \frac{q^2 x(1-x)}{m^2}\right)\right]$

The vacuum polarization has a physical interpretation: the virtual electron-positron pairs screen the bare charge of the electron. This screening effect makes the effective charge$\alpha_\text{eff}(q^2)$ increase at shorter distances (higher $|q^2|$), a phenomenon known as charge screening. For spacelike momenta $q^2 < 0$:

$\alpha_\text{eff}(q^2) = \frac{\alpha}{1 - \Pi(q^2)} \approx \alpha\left(1 + \frac{\alpha}{3\pi}\ln\frac{|q^2|}{m^2} + \cdots\right)$

For timelike momenta above the pair-production threshold $q^2 > 4m^2$, the logarithm develops an imaginary part, corresponding to real $e^+e^-$ pair production.

Optical Theorem and Unitarity

Renormalized loop amplitudes must satisfy unitarity — the optical theorem relates the imaginary part of forward scattering amplitudes to total cross sections:

$2\,\text{Im}\,\mathcal{M}(k \to k) = \sum_f \int d\Pi_f |\mathcal{M}(k \to f)|^2$

For the vacuum polarization, the imaginary part above the pair-production threshold$q^2 > 4m^2$ is directly related to the $e^+e^- \to \text{pairs}$ cross section. The Cutkosky cutting rules provide a systematic method: to compute$\text{Im}\,\mathcal{M}$, replace each cut propagator by its on-shell delta function:

$\frac{i}{p^2 - m^2 + i\varepsilon} \to 2\pi\delta(p^2 - m^2)\theta(p^0)$

For the vacuum polarization function, the imaginary part in the spacelike region is:

$\text{Im}\,\Pi(q^2) = \frac{\alpha}{3}\left(1 + \frac{2m^2}{q^2}\right)\sqrt{1 - \frac{4m^2}{q^2}}\,\theta(q^2 - 4m^2)$

This discontinuity across the branch cut $q^2 > 4m^2$ encodes the physical process of pair creation. The full vacuum polarization can be reconstructed from its imaginary part using a dispersion relation, providing a powerful consistency check on the calculation.

Counterterms and Renormalization Schemes

The divergences are absorbed by adding counterterms to the Lagrangian. We write the bare quantities in terms of renormalized quantities:

$\psi_0 = \sqrt{Z_2}\,\psi, \quad A_0^\mu = \sqrt{Z_3}\,A^\mu, \quad m_0 = m + \delta m, \quad e_0 = \frac{Z_1}{Z_2\sqrt{Z_3}}\,e\mu^{\varepsilon/2}$

The counterterm Lagrangian is:

$\mathcal{L}_\text{ct} = (Z_2 - 1)\bar{\psi}i\not{\partial}\psi - (Z_2 m + \delta m)\bar{\psi}\psi - \frac{1}{4}(Z_3 - 1)F^2 - (Z_1 - 1)e\bar{\psi}\gamma^\mu\psi A_\mu$

On-Shell Renormalization

In the on-shell scheme, renormalization conditions are imposed at physical values:

• $\Sigma(\not{p} = m) = 0$ — the pole of the propagator is at the physical mass

• $\frac{d\Sigma}{d\not{p}}\big|_{\not{p}=m} = 0$ — the residue at the pole is unity

• $\Pi(q^2 = 0) = 0$ — the photon remains massless, charge defined at $q^2=0$

• $\Gamma^\mu(p, p)|_\text{on-shell} = \gamma^\mu$ — vertex normalization

$\overline{\text{MS}}$ Scheme

The modified minimal subtraction ($\overline{\text{MS}}$) scheme is the most commonly used scheme in modern calculations. In this scheme, the counterterms are chosen to cancel only the $1/\varepsilon$ poles together with the universal constants$\gamma_E - \ln(4\pi)$ that always accompany them:

$\delta Z_i^{\overline{\text{MS}}} = \text{pole terms} \times \left(\frac{1}{\varepsilon} - \gamma_E + \ln 4\pi\right) \text{ only}$

The $\overline{\text{MS}}$ scheme is technically simpler than the on-shell scheme because the counterterms are pure poles (plus the universal constants), independent of masses and external momenta. The tradeoff is that the renormalized parameters (mass, coupling) depend on the renormalization scale $\mu$ and do not directly correspond to physical observables.

Vertex Correction and the Anomalous Magnetic Moment

The one-loop vertex correction modifies the electron-photon coupling. By Lorentz covariance and the Ward identity, the corrected vertex takes the form:

$\Gamma^\mu(p', p) = \gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu}q_\nu}{2m}F_2(q^2)$

where $q = p' - p$ is the momentum transfer. The Dirac form factor $F_1(0) = 1$(charge normalization) and the Pauli form factor $F_2(0)$ gives the anomalous magnetic moment. Schwinger's celebrated one-loop calculation yields:

$a_e = \frac{g - 2}{2} = \frac{\alpha}{2\pi} \approx 0.00116$

This is one of the most precisely verified predictions in all of physics. The current theoretical value, including terms up to fifth order in $\alpha$ (five-loop diagrams involving over 12,000 Feynman diagrams), agrees with the experimental measurement to better than one part in $10^{10}$. This extraordinary agreement is a triumph of quantum field theory and the renormalization program.

Feynman Parameterization and Wick Rotation

The standard technique for evaluating loop integrals involves two key steps: combining denominators with Feynman parameters and rotating to Euclidean space.

Feynman Parameters

The Feynman parameter identity combines products of propagators into a single denominator:

$\frac{1}{A_1 A_2 \cdots A_n} = (n-1)!\int_0^1 dx_1 \cdots dx_n \frac{\delta(1-\sum x_i)}{[x_1 A_1 + x_2 A_2 + \cdots + x_n A_n]^n}$

For the simplest case of two propagators:

$\frac{1}{AB} = \int_0^1 dx \frac{1}{[xA + (1-x)B]^2}$

After introducing Feynman parameters, the loop momentum integral takes the standard form with a single denominator. The combined denominator defines the quantity $\Delta$ that appears in the master integral formula:

$\Delta = m^2 - x(1-x)p^2$

for the simplest one-loop self-energy diagram.

Wick Rotation

Loop integrals in Minkowski space have poles from the propagator denominators. The Wick rotation $\ell^0 \to i\ell_E^0$ rotates the integration contour from Minkowski to Euclidean space, where the integral is better behaved:

$\int \frac{d^d\ell}{(2\pi)^d}\frac{1}{(\ell^2 - \Delta)^n} = \frac{i(-1)^n}{(2\pi)^d}\int \frac{d^d\ell_E}{(\ell_E^2 + \Delta)^n}$

The Euclidean integral is then evaluated using the $d$-dimensional solid angle formula $\Omega_d = 2\pi^{d/2}/\Gamma(d/2)$, reducing to the master formula involving gamma functions. The Wick rotation is justified whenever the integrand has no poles in the first or third quadrants of the complex $\ell^0$ plane, which is guaranteed by the $i\varepsilon$ prescription in the propagators.

The Predictive Power of Renormalization

A common misconception is that renormalization is merely "sweeping infinities under the rug." In reality, renormalization is what makes quantum field theory predictive. The key insight is that a renormalizable theory requires only a finite number of measured inputs to predict infinitely many observables.

Renormalizable vs Non-Renormalizable

In QED, three measurements fix the theory completely:

• The electron mass $m_e = 0.511$ MeV (fixes the mass counterterm $\delta m$)

• The fine structure constant $\alpha = 1/137.036$ (fixes the charge counterterm)

• The electron field normalization (fixes $Z_2$, absorbed into the LSZ formula)

Given these three inputs, QED predicts all electromagnetic processes at any energy to any order in perturbation theory. Every scattering cross section, every bound-state energy level, every form factor is a parameter-free prediction.

In contrast, a non-renormalizable theory (like Fermi's four-fermion interaction with coupling $G_F$) generates new divergent structures at each loop order, requiring infinitely many parameters. Such theories are still useful as effective field theories at low energies but lose predictive power above their cutoff scale.

The Wilsonian Perspective

Kenneth Wilson revolutionized our understanding of renormalization by connecting it to the renormalization group. In the Wilsonian picture, we integrate out high-energy modes above a cutoff $\Lambda$ to obtain an effective action at scale $\Lambda$:

$e^{-S_\text{eff}[\phi_<]} = \int \mathcal{D}\phi_> \, e^{-S[\phi_< + \phi_>]}$

This procedure generates all possible operators consistent with the symmetries. The renormalizable couplings (relevant and marginal operators) dominate at low energies, while non-renormalizable terms (irrelevant operators) are suppressed by powers of $E/\Lambda$. Renormalizability is thus not a fundamental requirement but a consequence of the low-energy limit of any well-defined UV theory.

Naturalness and Hierarchy

In the Wilsonian framework, the parameters at scale $\Lambda$ are set by the UV physics. If a parameter is much smaller than its natural value$\sim \Lambda^{[\text{dim}]}$, this requires fine-tuning — a "naturalness problem." The electron mass in QED is technically natural because its smallness is protected by chiral symmetry: setting $m_e = 0$ enhances the symmetry from $U(1)_V$to $U(1)_V \times U(1)_A$, ensuring that $\delta m_e \propto m_e$ at every order.

Computational Analysis: Loop Integral Regularization

We visualize the UV divergences in loop integrals, compare cutoff and dimensional regularization, compute the vacuum polarization function, and examine the QED vertex form factors including the anomalous magnetic moment.

Loop Integral Regularization Visualization

Python

script.py256 lines

#!/usr/bin/env python3
"""
renormalization.py - Loop Integral Regularization Visualization
Demonstrates UV divergences, dimensional regularization, and counterterms
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

print("=" * 72)
print("  RENORMALIZATION: LOOP INTEGRALS & REGULARIZATION")
print("=" * 72)
print()

# ============================================================
# 1. UV Divergences: One-Loop Self-Energy (Cutoff Regularization)
# Sigma(p) ~ integral d^4k / [(k^2-m^2)((p-k)^2-m^2)]
# In Euclidean space after Wick rotation
# ============================================================
print("1. ELECTRON SELF-ENERGY: Cutoff Regularization")
print("-" * 50)

# Simulate the 1D radial part of the loop integral
# After angular integration in d dimensions:
# I(p, Lambda) = integral_0^Lambda dk k^(d-1) / [(k^2+m^2)((p-k)^2+m^2)]
# For d=4, this is logarithmically divergent

m_e = 1.0  # electron mass (units)
p_ext = 2.0  # external momentum magnitude

# Cutoff values
Lambda_values = np.array([10, 50, 100, 500, 1000, 5000])

# Numerical integration of the radial part (Euclidean, simplified 1D model)
def self_energy_cutoff(p, m, Lambda, N=5000):
    """Compute simplified self-energy integral with cutoff Lambda."""
    k = np.linspace(0.001, Lambda, N)
    dk = k[1] - k[0]
    # Simplified integrand representing the k^3 dk / [(k^2+m^2)((p-k)^2+m^2)]
    # This captures the log divergence
    integrand = k**3 / ((k**2 + m**2) * ((p - k)**2 + m**2 + 0.01))
    return np.sum(integrand) * dk / (16 * np.pi**2)

print(f"  External momentum p = {p_ext}, mass m = {m_e}")
print(f"  {'Lambda':>10s}  {'Sigma(p)':>15s}  {'log(Lambda/m)':>15s}")
for L in Lambda_values:
    sigma = self_energy_cutoff(p_ext, m_e, L)
    print(f"  {L:>10.0f}  {sigma:>15.6f}  {np.log(L/m_e):>15.4f}")

print()
print("  -> Sigma(p) grows as log(Lambda): LOGARITHMIC DIVERGENCE")
print()

# ============================================================
# 2. Dimensional Regularization: d = 4 - epsilon
# I(p) = mu^epsilon * Gamma(epsilon/2) / (4*pi)^(d/2)
# 1/epsilon pole replaces log(Lambda)
# ============================================================
print("2. DIMENSIONAL REGULARIZATION")
print("-" * 50)

# The key integral in dim reg:
# integral d^d k / (k^2 + Delta)^n = pi^(d/2) Gamma(n-d/2) / [Gamma(n) Delta^(n-d/2)]

def dim_reg_integral(Delta, n, d):
    """Compute standard dim-reg integral I_n(Delta) in d dimensions."""
    from math import gamma as gamma_func
    try:
        prefactor = np.pi**(d/2)
        num = gamma_func(n - d/2)
        den = gamma_func(n) * Delta**(n - d/2)
        return prefactor * num / den
    except (ValueError, ZeroDivisionError):
        return float('inf')

# Show how 1/epsilon pole emerges
epsilon_values = np.logspace(-3, 0, 50)
Delta = 1.0  # Feynman parameter combination

print("  Standard integral: I_n(Delta) = pi^(d/2) Gamma(n-d/2) / [Gamma(n) Delta^(n-d/2)]")
print()
print(f"  For n=2 (self-energy), Delta = {Delta}:")
print(f"  {'epsilon':>10s}  {'d = 4-eps':>10s}  {'I_2(Delta)':>15s}  {'1/epsilon':>12s}")
for eps in [1.0, 0.5, 0.1, 0.01, 0.001]:
    d = 4 - eps
    I_val = dim_reg_integral(Delta, 2, d)
    print(f"  {eps:>10.3f}  {d:>10.3f}  {I_val:>15.6f}  {1/eps:>12.3f}")

print()
print("  As epsilon -> 0, the integral develops a 1/epsilon pole.")
print("  This pole is the dim-reg analog of log(Lambda).")
print()

# ============================================================
# 3. Vacuum Polarization Pi(q^2)
# Pi(q^2) = -(alpha/3*pi) * [1/epsilon - gamma_E + log(4*pi*mu^2)
#           - integral_0^1 dx 2x(1-x) log(Delta/mu^2)]
# where Delta = m^2 - x(1-x)*q^2
# ============================================================
print("3. VACUUM POLARIZATION Pi(q^2)")
print("-" * 50)

alpha = 1.0 / 137.036
m = 1.0  # mass scale
mu = 1.0  # renormalization scale

q2_values = np.linspace(-20, 20, 500)

def vacuum_pol_finite(q2, m, mu):
    """Finite part of vacuum polarization (after MS-bar subtraction)."""
    x = np.linspace(0.001, 0.999, 200)
    Delta = m**2 - x * (1 - x) * q2
    # Avoid log of negative numbers for timelike q2
    Delta_safe = np.maximum(Delta, 1e-10)
    integrand = 2 * x * (1 - x) * np.log(Delta_safe / mu**2)
    return -(alpha / (3 * np.pi)) * np.trapz(integrand, x)

Pi_values = np.array([vacuum_pol_finite(q2, m, mu) for q2 in q2_values])

print(f"  Renormalization scale mu = {mu}")
print(f"  {'q^2':>10s}  {'Pi_finite(q^2)':>15s}")
for q2 in [-10, -5, -1, 0, 1, 5, 10]:
    Pi = vacuum_pol_finite(q2, m, mu)
    print(f"  {q2:>10.1f}  {Pi:>15.8f}")

print()
print("  Pi(0) = 0 after on-shell renormalization (charge defined at q^2=0).")
print()

# ============================================================
# 4. Vertex Correction (Form Factors)
# Gamma^mu(p',p) = gamma^mu F_1(q^2) + (i*sigma^{mu nu}*q_nu)/(2m) * F_2(q^2)
# F_1(0) = 1 (charge), F_2(0) = alpha/(2*pi) (anomalous magnetic moment)
# ============================================================
print("4. VERTEX CORRECTION & ANOMALOUS MAGNETIC MOMENT")
print("-" * 50)

# Schwinger result: a_e = (g-2)/2 = alpha/(2*pi)
a_e_schwinger = alpha / (2 * np.pi)
a_e_experimental = 0.00115965218091
print(f"  Schwinger correction: a_e = alpha/(2*pi) = {a_e_schwinger:.10f}")
print(f"  Experimental value:   a_e = {a_e_experimental:.10f}")
print(f"  Ratio (shows higher-order corrections needed): {a_e_experimental/a_e_schwinger:.6f}")
print()

# Form factors as functions of q^2 (simplified model)
q2_ff = np.linspace(-10, 0, 300)  # spacelike momentum transfer

# F_1(q^2) at one loop (simplified, captures log behavior)
F1_values = 1 + (alpha/(2*np.pi)) * np.log(np.abs(q2_ff)/m**2 + 1) * 0.5
# F_2(q^2) at one loop
F2_values = (alpha/(2*np.pi)) / (1 - q2_ff/(4*m**2))

print(f"  {'q^2':>10s}  {'F_1(q^2)':>12s}  {'F_2(q^2)':>12s}")
for q2 in [0, -1, -2, -5, -10]:
    f1 = 1 + (alpha/(2*np.pi)) * np.log(np.abs(q2)/m**2 + 1) * 0.5
    f2 = (alpha/(2*np.pi)) / (1 - q2/(4*m**2))
    print(f"  {q2:>10.1f}  {f1:>12.8f}  {f2:>12.8f}")

print()

# ============================================================
# Plotting
# ============================================================
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0f0a1a')

# --- Plot 1: Self-energy vs cutoff (log divergence) ---
ax1 = axes[0, 0]
ax1.set_facecolor('#1a1025')
for spine in ax1.spines.values():
    spine.set_color('#4c1d95')
ax1.tick_params(colors='#c4b5fd')
ax1.grid(True, color='#4c1d95', alpha=0.3)

Lambda_scan = np.logspace(0.5, 4, 100)
sigma_vals = [self_energy_cutoff(p_ext, m_e, L) for L in Lambda_scan]
log_fit = [0.04 * np.log(L/m_e) + 0.02 for L in Lambda_scan]

ax1.plot(Lambda_scan, sigma_vals, color='#a78bfa', linewidth=2.5, label='Sigma(p)')
ax1.plot(Lambda_scan, log_fit, color='#f472b6', linewidth=2, linestyle='--', label='~ log(Lambda/m) fit')
ax1.set_xlabel('Cutoff Lambda', color='#a78bfa', fontsize=11)
ax1.set_ylabel('Self-Energy Sigma(p)', color='#a78bfa', fontsize=11)
ax1.set_title('Electron Self-Energy vs Cutoff', color='#e9d5ff', fontsize=13, fontweight='bold')
ax1.set_xscale('log')
ax1.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)

# --- Plot 2: Dim reg integral vs epsilon ---
ax2 = axes[0, 1]
ax2.set_facecolor('#1a1025')
for spine in ax2.spines.values():
    spine.set_color('#4c1d95')
ax2.tick_params(colors='#c4b5fd')
ax2.grid(True, color='#4c1d95', alpha=0.3)

eps_plot = np.linspace(0.01, 2.0, 200)
I2_vals = [dim_reg_integral(1.0, 2, 4-e) for e in eps_plot]
one_over_eps = [1.0/e for e in eps_plot]

ax2.plot(eps_plot, I2_vals, color='#a78bfa', linewidth=2.5, label='I_2(Delta=1, d=4-eps)')
ax2.plot(eps_plot, one_over_eps, color='#22d3ee', linewidth=2, linestyle='--', label='1/epsilon')
ax2.set_xlabel('epsilon (d = 4 - epsilon)', color='#a78bfa', fontsize=11)
ax2.set_ylabel('Integral value', color='#a78bfa', fontsize=11)
ax2.set_title('Dimensional Regularization: 1/epsilon Pole', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax2.set_ylim(0, 50)

# --- Plot 3: Vacuum polarization Pi(q^2) ---
ax3 = axes[1, 0]
ax3.set_facecolor('#1a1025')
for spine in ax3.spines.values():
    spine.set_color('#4c1d95')
ax3.tick_params(colors='#c4b5fd')
ax3.grid(True, color='#4c1d95', alpha=0.3)

ax3.plot(q2_values, Pi_values * 1e4, color='#a78bfa', linewidth=2.5)
ax3.axhline(y=0, color='#4c1d95', linewidth=1, linestyle='-')
ax3.axvline(x=4*m**2, color='#ef4444', linewidth=1.5, linestyle='--', label=f'Threshold q^2 = 4m^2 = {4*m**2}')
ax3.set_xlabel('q^2 [mass^2 units]', color='#a78bfa', fontsize=11)
ax3.set_ylabel('Pi_finite(q^2) x 10^4', color='#a78bfa', fontsize=11)
ax3.set_title('Vacuum Polarization (finite part)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)

# --- Plot 4: Form factors F_1 and F_2 ---
ax4 = axes[1, 1]
ax4.set_facecolor('#1a1025')
for spine in ax4.spines.values():
    spine.set_color('#4c1d95')
ax4.tick_params(colors='#c4b5fd')
ax4.grid(True, color='#4c1d95', alpha=0.3)

ax4.plot(q2_ff, F1_values, color='#a78bfa', linewidth=2.5, label='F_1(q^2) [Dirac]')
ax4.plot(q2_ff, F2_values * 1000, color='#22d3ee', linewidth=2.5, label='F_2(q^2) x 1000 [Pauli]')
ax4.axhline(y=1.0, color='#4c1d95', linewidth=1, linestyle='--', alpha=0.5)
ax4.set_xlabel('q^2 [mass^2 units]', color='#a78bfa', fontsize=11)
ax4.set_ylabel('Form Factor', color='#a78bfa', fontsize=11)
ax4.set_title('QED Vertex Form Factors', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)

plt.tight_layout(pad=2.0)

buf = BytesIO()
plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor='#0f0a1a')
buf.seek(0)
img_b64 = base64.b64encode(buf.read()).decode()
print(f'<img src="data:image/png;base64,{img_b64}" alt="Renormalization Loop Integrals" />')
buf.close()

print()
print("Key results: UV divergences appear as log(Lambda) with cutoff or 1/epsilon")
print("in dimensional regularization. Counterterms absorb these divergences,")
print("leaving finite, physically measurable quantities.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary: Renormalization

UV Divergences

Loop integrals diverge at high momenta. In QED, only three structures diverge: the electron self-energy, photon vacuum polarization, and vertex correction. This finite number of divergent structures makes QED renormalizable.

Dimensional Regularization

Working in $d = 4 - \varepsilon$ dimensions preserves gauge invariance and Lorentz covariance. Divergences appear as $1/\varepsilon$ poles, which are absorbed by counterterms in the Lagrangian.

Renormalization Schemes

The on-shell scheme defines parameters at their physical values; the $\overline{\text{MS}}$ scheme minimally subtracts poles. Physical observables are scheme-independent, but intermediate expressions depend on the choice.

Physical Consequences

Vacuum polarization screens the electron charge (running coupling). The vertex correction gives the anomalous magnetic moment $a_e = \alpha/(2\pi)$, verified to extraordinary precision. These finite results emerge naturally from the renormalization program.

← Previous: QED Basics Next: Running Couplings & RG →