Chapter 5: Quantum Electrodynamics

Quantum Electrodynamics (QED) is the quantum field theory of the electromagnetic interaction. It describes how charged fermions (electrons, muons, quarks) interact via the exchange of photons. QED was the first successful relativistic quantum field theory and remains the most precisely tested theory in all of physics, with predictions verified to better than 10 significant figures. We derive the QED Lagrangian, extract the Feynman rules, and compute tree-level cross sections for the fundamental scattering processes.

The QED Lagrangian

QED is constructed from the principle of local $U(1)$ gauge invariance. We start with the free Dirac Lagrangian for a spin-1/2 fermion and demand invariance under local phase transformations $\psi(x) \to e^{i\alpha(x)}\psi(x)$. This requirement forces us to introduce a gauge field $A_\mu$ — the photon field — and replace ordinary derivatives with covariant derivatives.

Covariant Derivative

The covariant derivative is defined as:

$D_\mu = \partial_\mu + ieA_\mu$

Under a gauge transformation $\psi \to e^{i\alpha(x)}\psi$, the gauge field transforms as$A_\mu \to A_\mu - \frac{1}{e}\partial_\mu\alpha$, ensuring that $D_\mu\psi$ transforms the same way as $\psi$ itself. The field strength tensor is:

$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$

The Full QED Lagrangian

Combining the gauge-covariant Dirac term with the kinetic energy of the gauge field:

$\mathcal{L}_\text{QED} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}$

Expanding the covariant derivative explicitly:

$\mathcal{L}_\text{QED} = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu} - e\bar{\psi}\gamma^\mu\psi A_\mu$

The three terms are: (1) the free Dirac Lagrangian for the electron, (2) the free Maxwell Lagrangian for the photon, and (3) the interaction term $-e\bar{\psi}\gamma^\mu\psi A_\mu$ that couples the electron current $j^\mu = \bar{\psi}\gamma^\mu\psi$ to the photon field. The coupling constant $e$ is the electric charge, related to the fine structure constant by $\alpha = e^2/(4\pi) \approx 1/137$.

Gauge Fixing

To quantize the photon field, we must fix the gauge redundancy. Adding a gauge-fixing term:

$\mathcal{L}_\text{gf} = -\frac{1}{2\xi}(\partial_\mu A^\mu)^2$

Common choices are Feynman gauge ($\xi = 1$), Landau gauge ($\xi = 0$), and unitary gauge ($\xi \to \infty$). Physical observables are independent of $\xi$, but Feynman gauge simplifies calculations considerably.

Feynman Rules for QED

The Feynman rules for QED are derived from the Lagrangian by expanding the path integral perturbatively in the coupling $e$. Each element of a Feynman diagram has a precise mathematical factor.

Electron Propagator

An internal electron line carrying momentum $p$ contributes:

$\frac{i(\not{p} + m)}{p^2 - m^2 + i\varepsilon} = \frac{i}{\not{p} - m + i\varepsilon}$

where $\not{p} = \gamma^\mu p_\mu$ is the Feynman slash notation. The numerator$\not{p} + m$ is the spin sum projector.

Photon Propagator

An internal photon line carrying momentum $k$ contributes (Feynman gauge):

$\frac{-ig_{\mu\nu}}{k^2 + i\varepsilon}$

In general $R_\xi$ gauge: $\frac{-i}{k^2+i\varepsilon}\left(g_{\mu\nu} - (1-\xi)\frac{k_\mu k_\nu}{k^2}\right)$. The photon is massless, so there is no mass term in the denominator.

QED Vertex

Each electron-photon vertex contributes:

$-ie\gamma^\mu$

This is the fundamental interaction vertex of QED. Momentum is conserved at each vertex. The vertex factor comes directly from the interaction term $-e\bar{\psi}\gamma^\mu\psi A_\mu$ in the Lagrangian.

External Lines

Incoming electron: $u(p, s)$

Outgoing electron: $\bar{u}(p, s)$

Incoming positron: $\bar{v}(p, s)$

Outgoing positron: $v(p, s)$

Incoming photon: $\varepsilon_\mu(k, \lambda)$

Outgoing photon: $\varepsilon_\mu^*(k, \lambda)$

Additional Rules

• A factor of $(-1)$ for each closed fermion loop (Fermi statistics)

• Integrate over each undetermined loop momentum: $\int \frac{d^4\ell}{(2\pi)^4}$

• Impose momentum conservation $(2\pi)^4\delta^4(\sum p_i)$ at each vertex

Compton Scattering

Compton scattering $e^-\gamma \to e^-\gamma$ proceeds through two tree-level diagrams: the $s$-channel (electron absorbs photon then emits) and the $u$-channel (crossed diagram). The total amplitude is:

$i\mathcal{M} = (-ie)^2 \bar{u}(p')\left[\gamma^\nu \frac{\not{p}+\not{k}+m}{(p+k)^2-m^2}\gamma^\mu + \gamma^\mu \frac{\not{p}-\not{k'}+m}{(p-k')^2-m^2}\gamma^\nu\right]u(p)\,\varepsilon_\mu(k)\varepsilon_\nu^*(k')$

After averaging over initial spins and summing over final spins and polarizations, we obtain the Klein-Nishina formula. In the lab frame where the electron is initially at rest:

$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{2m_e^2}\left(\frac{\omega'}{\omega}\right)^2\left(\frac{\omega'}{\omega} + \frac{\omega}{\omega'} - \sin^2\theta\right)$

where $\omega' = \omega/[1 + (\omega/m_e)(1-\cos\theta)]$ is the scattered photon energy. In the low-energy limit $\omega \ll m_e$, we recover the Thomson cross section:

$\sigma_\text{Thomson} = \frac{8\pi\alpha^2}{3m_e^2} \approx 0.665 \text{ barns}$

At high energies $\omega \gg m_e$, the cross section falls as $\sigma \sim (\alpha^2/\omega m_e)\ln(2\omega/m_e)$, reflecting the point-like nature of the electron at short distances.

Møller Scattering

Møller scattering $e^-e^- \to e^-e^-$ involves the exchange of a virtual photon between two electrons. There are two contributing diagrams at tree level: $t$-channel (direct exchange) and $u$-channel (exchange with crossed final-state electrons). The relative minus sign between diagrams is required by Fermi-Dirac statistics.

$i\mathcal{M} = (-ie)^2\left[\frac{\bar{u}(p_3)\gamma^\mu u(p_1)\,\bar{u}(p_4)\gamma_\mu u(p_2)}{t} - \frac{\bar{u}(p_4)\gamma^\mu u(p_1)\,\bar{u}(p_3)\gamma_\mu u(p_2)}{u}\right]$

After spin-averaging, the unpolarized squared amplitude is:

$\overline{|\mathcal{M}|^2} = 2e^4\left[\frac{s^2+u^2}{t^2} + \frac{s^2+t^2}{u^2} + \frac{2s^2}{tu}\right]$

The $1/t^2$ and $1/u^2$ terms produce strong forward and backward peaks in the angular distribution. The interference term $2s^2/(tu)$ reflects the quantum mechanical identity of the two electrons. The differential cross section in the center-of-mass frame is:

$\frac{d\sigma}{d\Omega} = \frac{\overline{|\mathcal{M}|^2}}{64\pi^2 s}$

Bhabha Scattering

Bhabha scattering $e^+e^- \to e^+e^-$ is the electron-positron elastic scattering process. It proceeds through two channels: $t$-channel photon exchange (similar to Møller) and $s$-channel annihilation (the electron and positron annihilate into a virtual photon which then produces a new pair).

$i\mathcal{M} = (-ie)^2\left[\frac{\bar{u}(p_3)\gamma^\mu u(p_1)\,\bar{v}(p_2)\gamma_\mu v(p_4)}{t} - \frac{\bar{v}(p_2)\gamma^\mu u(p_1)\,\bar{u}(p_3)\gamma_\mu v(p_4)}{s}\right]$

The unpolarized squared amplitude after spin averaging:

$\overline{|\mathcal{M}|^2} = 2e^4\left[\frac{t^2+u^2}{s^2} + \frac{s^2+u^2}{t^2} + \frac{2u^2}{st}\right]$

The $s$-channel term $(t^2+u^2)/s^2$ represents annihilation and is isotropic at high energies. The $t$-channel term $(s^2+u^2)/t^2$ produces the characteristic forward peak from Coulomb scattering. The interference term $2u^2/(st)$ is negative at forward angles, partially cancelling the forward peak. Bhabha scattering serves as the primary luminosity monitor at $e^+e^-$ colliders because its cross section is large and theoretically well-understood.

Crossing Symmetry

The three processes — Compton, Møller, and Bhabha scattering — are related by crossing symmetry. By analytically continuing momenta from incoming to outgoing particles (i.e., replacing $p \to -p$ for a crossed particle), the amplitude for one process can be obtained from another. For example, the Bhabha amplitude is obtained from the Møller amplitude by the crossing $s \leftrightarrow u$ with appropriate sign changes for fermion statistics.

Ward Identity and Gauge Invariance

The Ward identity is the quantum manifestation of gauge invariance. It states that replacing a photon polarization vector with its momentum in any QED amplitude yields zero:

$k_\mu \mathcal{M}^\mu = 0$

This identity has profound consequences. It ensures that the unphysical longitudinal and timelike photon polarizations decouple from physical processes. It also guarantees that the photon remains massless to all orders in perturbation theory, and it constrains the form of radiative corrections through the Ward-Takahashi identity:

$q_\mu \Gamma^\mu(p+q, p) = S_F^{-1}(p+q) - S_F^{-1}(p)$

where $\Gamma^\mu$ is the exact vertex function and $S_F$ is the exact electron propagator. This identity relates the vertex correction to the electron self-energy and is crucial for the consistency of renormalization: it ensures that the charge renormalization constant $Z_1$ equals the wavefunction renormalization $Z_2$ (i.e., $Z_1 = Z_2$), so the physical charge is renormalized only by the photon wavefunction factor $Z_3$.

Spin Sums and Trace Technology

Computing QED cross sections requires evaluating the squared amplitude$|\mathcal{M}|^2$, averaged over initial spins and summed over final spins. The key tools are the spin completeness relations and gamma matrix trace identities.

Spin Completeness Relations

For electrons and positrons, the spin sums produce projection operators:

$\sum_{s} u(p,s)\bar{u}(p,s) = \not{p} + m$

$\sum_{s} v(p,s)\bar{v}(p,s) = \not{p} - m$

For photon polarizations, in Feynman gauge the polarization sum is:

$\sum_{\lambda} \varepsilon_\mu(k,\lambda)\varepsilon_\nu^*(k,\lambda) = -g_{\mu\nu}$

Using these, the spin-averaged squared amplitude becomes a trace over gamma matrices:

$\frac{1}{4}\sum_\text{spins}|\mathcal{M}|^2 = \frac{e^4}{4}\text{Tr}[\gamma^\mu(\not{p}_3+m)\gamma^\nu(\not{p}_1+m)]\text{Tr}[\gamma_\mu(\not{p}_4+m)\gamma_\nu(\not{p}_2+m)] / t^2 + \cdots$

Fundamental Trace Identities

$\text{Tr}[\gamma^\mu\gamma^\nu] = 4g^{\mu\nu}$

$\text{Tr}[\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma] = 4(g^{\mu\nu}g^{\rho\sigma} - g^{\mu\rho}g^{\nu\sigma} + g^{\mu\sigma}g^{\nu\rho})$

$\text{Tr}[\text{odd number of } \gamma\text{'s}] = 0$

$\text{Tr}[\gamma^5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma] = -4i\varepsilon^{\mu\nu\rho\sigma}$

Together with contraction identities like $\gamma^\mu\gamma_\mu = 4$ in 4D and $\gamma^\mu\not{a}\gamma_\mu = -2\not{a}$, these allow systematic evaluation of any QED amplitude. In practice, the traces are often computed using symbolic algebra packages (such as FeynCalc or FORM), especially for multi-loop calculations where the number of terms grows rapidly.

Pair Annihilation and Production

The process $e^+e^- \to \mu^+\mu^-$ is one of the cleanest QED predictions and was a cornerstone measurement at early electron-positron colliders. At tree level, it proceeds through a single $s$-channel photon exchange:

$i\mathcal{M} = (-ie)^2 \frac{-ig_{\mu\nu}}{s}\bar{v}(p_2)\gamma^\mu u(p_1)\,\bar{u}(p_3)\gamma^\nu v(p_4)$

In the high-energy limit $s \gg m_\mu^2$, the unpolarized differential cross section is:

$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4s}(1 + \cos^2\theta)$

The $1 + \cos^2\theta$ angular distribution is characteristic of spin-1/2 pair production from a spin-1 intermediate state. Integrating over angles gives the total cross section:

$\sigma(e^+e^- \to \mu^+\mu^-) = \frac{4\pi\alpha^2}{3s}$

This cross section serves as a benchmark: the ratio $R = \sigma(e^+e^- \to \text{hadrons})/\sigma(e^+e^- \to \mu^+\mu^-)$is a direct measure of the number of quark colors and charges. At energies below charm threshold, $R = N_c \sum_q Q_q^2 = 3(4/9 + 1/9 + 1/9) = 2$, providing early evidence for three colors.

The Pair Annihilation Process

The time-reversed process $e^+e^- \to \gamma\gamma$ is related to Compton scattering by crossing symmetry. At tree level, two diagrams contribute ($t$ and $u$ channel electron exchange). In the center-of-mass frame at high energies:

$\frac{d\sigma}{d\Omega}(e^+e^- \to \gamma\gamma) = \frac{\alpha^2}{2s}\frac{1+\cos^2\theta}{\sin^2\theta}$

The forward and backward singularities ($\theta \to 0, \pi$) reflect the$t$-channel and $u$-channel poles. These singularities are regulated by the electron mass, which we have neglected in the high-energy limit.

QED as the Prototype Gauge Theory

QED established the paradigm for modern particle physics. Several features of QED generalize directly to the non-abelian gauge theories that describe the weak and strong interactions:

Gauge Principle

The entire structure of QED is dictated by local $U(1)$ symmetry. The interaction Lagrangian, the form of the coupling, and the masslessness of the photon all follow from this single principle. In the Standard Model, the gauge group is$SU(3)_C \times SU(2)_L \times U(1)_Y$, and the same logic determines all interactions.

Renormalizability

QED was the first example of a renormalizable quantum field theory. The key requirements — gauge invariance, power-counting renormalizability, and the Ward identities — carry over to non-abelian theories, where they were proven by 't Hooft and Veltman (Nobel Prize 1999).

Precision Tests

The anomalous magnetic moment $a_e = (g-2)/2$ has been measured to$0.24$ parts per billion, and the theoretical prediction (including five-loop QED, hadronic, and electroweak corrections) matches to extraordinary precision. This success validates the entire framework of perturbative quantum field theory.

Computational Analysis: QED Tree-Level Cross Sections

We compute and compare the differential and total cross sections for Compton, Møller, and Bhabha scattering at tree level. The Klein-Nishina formula gives the exact Compton cross section, while the Møller and Bhabha formulas follow from the squared amplitudes derived above.

QED Tree-Level Cross Sections

Python

script.py289 lines

#!/usr/bin/env python3
"""
qed_basics.py - Tree-Level QED Cross Sections
Computes Compton, Moller, and Bhabha scattering at tree level
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

print("=" * 72)
print("  QED TREE-LEVEL CROSS SECTIONS")
print("=" * 72)
print()

# ============================================================
# Physical constants (natural units, hbar = c = 1)
# ============================================================
alpha_em = 1.0 / 137.036  # fine structure constant
e_charge = np.sqrt(4 * np.pi * alpha_em)
m_e = 0.511e-3  # electron mass in GeV

print(f"Fine structure constant: alpha = {alpha_em:.6f}")
print(f"Elementary charge: e = {e_charge:.4f}")
print(f"Electron mass: m_e = {m_e*1e3:.3f} MeV")
print()

# ============================================================
# 1. Compton Scattering: e + gamma -> e + gamma
# Klein-Nishina formula (unpolarized, averaged)
# d(sigma)/d(cos theta) = pi*alpha^2/(m_e^2) * (omega'/omega)^2
#   * [omega'/omega + omega/omega' - sin^2(theta)]
# where omega' = omega / (1 + (omega/m_e)(1-cos(theta)))
# ============================================================
print("1. COMPTON SCATTERING (Klein-Nishina formula)")
print("-" * 50)

theta = np.linspace(0.01, np.pi - 0.01, 500)
cos_theta = np.cos(theta)

omega_values = [0.01, 0.1, 0.5, 1.0, 5.0]  # photon energy in units of m_e

compton_dsigma = {}
compton_total = {}

for omega_ratio in omega_values:
    omega = omega_ratio * m_e
    omega_prime = omega / (1 + (omega / m_e) * (1 - cos_theta))
    ratio = omega_prime / omega

# Klein-Nishina differential cross section
    dsigma = (alpha_em**2 / (2 * m_e**2)) * ratio**2 * (ratio + 1.0/ratio - np.sin(theta)**2)
    compton_dsigma[omega_ratio] = dsigma

# Numerical integration for total cross section
    sigma_total = np.trapz(dsigma * 2 * np.pi * np.sin(theta), theta)
    compton_total[omega_ratio] = sigma_total
    print(f"  omega/m_e = {omega_ratio:.2f}: sigma_total = {sigma_total:.6e} GeV^-2")

# Thomson limit
sigma_thomson = 8 * np.pi * alpha_em**2 / (3 * m_e**2)
print(f"  Thomson cross section (low-energy limit): {sigma_thomson:.6e} GeV^-2")
print()

# ============================================================
# 2. Moller Scattering: e- + e- -> e- + e-
# t-channel and u-channel exchange
# |M|^2 = 2e^4 [ (s^2+u^2)/t^2 + (s^2+t^2)/u^2 + 2s^2/(t*u) ]
# ============================================================
print("2. MOLLER SCATTERING (e- e- -> e- e-)")
print("-" * 50)

sqrt_s_values = [0.01, 0.1, 1.0, 10.0, 100.0]  # GeV

moller_dsigma = {}
for sqrt_s in [1.0, 10.0, 100.0]:
    s = sqrt_s**2
    E_cm = sqrt_s / 2
    if E_cm <= m_e:
        continue
    p_cm = np.sqrt(E_cm**2 - m_e**2)

t_vals = -2 * p_cm**2 * (1 - cos_theta)
    u_vals = -2 * p_cm**2 * (1 + cos_theta)

# Avoid singularities
    t_reg = np.where(np.abs(t_vals) < 1e-10, -1e-10, t_vals)
    u_reg = np.where(np.abs(u_vals) < 1e-10, -1e-10, u_vals)

M_sq = 2 * e_charge**4 * (
        (s**2 + u_reg**2) / t_reg**2 +
        (s**2 + t_reg**2) / u_reg**2 +
        2 * s**2 / (t_reg * u_reg)
    )

dsigma = np.abs(M_sq) / (64 * np.pi**2 * s)
    moller_dsigma[sqrt_s] = dsigma

sigma_t = np.trapz(dsigma * 2 * np.pi * np.sin(theta), theta)
    print(f"  sqrt(s) = {sqrt_s:.1f} GeV: sigma = {sigma_t:.6e} GeV^-2")

print()

# ============================================================
# 3. Bhabha Scattering: e+ e- -> e+ e-
# s-channel and t-channel
# |M|^2 = 2e^4 [ (t^2+u^2)/s^2 + (s^2+u^2)/t^2 + 2u^2/(s*t) ]
# ============================================================
print("3. BHABHA SCATTERING (e+ e- -> e+ e-)")
print("-" * 50)

bhabha_dsigma = {}
for sqrt_s in [1.0, 10.0, 100.0]:
    s = sqrt_s**2
    E_cm = sqrt_s / 2
    if E_cm <= m_e:
        continue
    p_cm = np.sqrt(E_cm**2 - m_e**2)

t_vals = -2 * p_cm**2 * (1 - cos_theta)
    u_vals = -2 * p_cm**2 * (1 + cos_theta)

t_reg = np.where(np.abs(t_vals) < 1e-10, -1e-10, t_vals)

M_sq = 2 * e_charge**4 * (
        (t_reg**2 + u_vals**2) / s**2 +
        (s**2 + u_vals**2) / t_reg**2 +
        2 * u_vals**2 / (s * t_reg)
    )

dsigma = np.abs(M_sq) / (64 * np.pi**2 * s)
    bhabha_dsigma[sqrt_s] = dsigma

sigma_t = np.trapz(dsigma * 2 * np.pi * np.sin(theta), theta)
    print(f"  sqrt(s) = {sqrt_s:.1f} GeV: sigma = {sigma_t:.6e} GeV^-2")

print()

# ============================================================
# 4. Comparison: total cross sections vs energy
# ============================================================
print("4. CROSS SECTION ENERGY DEPENDENCE")
print("-" * 50)

sqrt_s_scan = np.linspace(0.005, 50.0, 500)
sigma_compton_scan = []
sigma_moller_scan = []
sigma_bhabha_scan = []

for sqrt_s in sqrt_s_scan:
    s = sqrt_s**2
    E_cm = sqrt_s / 2
    if E_cm <= m_e:
        sigma_compton_scan.append(0)
        sigma_moller_scan.append(0)
        sigma_bhabha_scan.append(0)
        continue

p_cm = np.sqrt(E_cm**2 - m_e**2)
    omega = E_cm  # photon energy ~ E_cm for Compton

# Compton
    omega_p = omega / (1 + (omega / m_e) * (1 - cos_theta))
    ratio = omega_p / omega
    ds_c = (alpha_em**2 / (2 * m_e**2)) * ratio**2 * (ratio + 1.0/ratio - np.sin(theta)**2)
    sigma_compton_scan.append(np.trapz(ds_c * 2 * np.pi * np.sin(theta), theta))

# Moller
    t_v = -2 * p_cm**2 * (1 - cos_theta)
    u_v = -2 * p_cm**2 * (1 + cos_theta)
    t_r = np.where(np.abs(t_v) < 1e-10, -1e-10, t_v)
    u_r = np.where(np.abs(u_v) < 1e-10, -1e-10, u_v)
    M2_m = 2 * e_charge**4 * ((s**2+u_r**2)/t_r**2 + (s**2+t_r**2)/u_r**2 + 2*s**2/(t_r*u_r))
    ds_m = np.abs(M2_m) / (64 * np.pi**2 * s)
    sigma_moller_scan.append(np.trapz(ds_m * 2 * np.pi * np.sin(theta), theta))

# Bhabha
    M2_b = 2 * e_charge**4 * ((t_r**2+u_v**2)/s**2 + (s**2+u_v**2)/t_r**2 + 2*u_v**2/(s*t_r))
    ds_b = np.abs(M2_b) / (64 * np.pi**2 * s)
    sigma_bhabha_scan.append(np.trapz(ds_b * 2 * np.pi * np.sin(theta), theta))

sigma_compton_scan = np.array(sigma_compton_scan)
sigma_moller_scan = np.array(sigma_moller_scan)
sigma_bhabha_scan = np.array(sigma_bhabha_scan)

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

# --- Plot 1: Compton differential cross section ---
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)

colors = ['#22d3ee', '#a78bfa', '#f472b6', '#fbbf24', '#34d399']
theta_deg = np.degrees(theta)
for i, omega_ratio in enumerate([0.01, 0.1, 1.0, 5.0]):
    ds = compton_dsigma[omega_ratio]
    ds_norm = ds / ds.max()
    ax1.plot(theta_deg, ds_norm, color=colors[i], linewidth=2, label=f'omega/m_e = {omega_ratio}')

ax1.set_xlabel('Scattering angle (degrees)', color='#a78bfa', fontsize=11)
ax1.set_ylabel('d sigma / d Omega (normalized)', color='#a78bfa', fontsize=11)
ax1.set_title('Compton Scattering (Klein-Nishina)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax1.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=9)
ax1.set_yscale('log')
ax1.set_ylim(1e-3, 2)

# --- Plot 2: Moller differential cross section ---
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)

for i, sqrt_s in enumerate([1.0, 10.0, 100.0]):
    ds = moller_dsigma[sqrt_s]
    ds_norm = ds / ds.max()
    ax2.plot(theta_deg, ds_norm, color=colors[i], linewidth=2, label=f'sqrt(s) = {sqrt_s} GeV')

ax2.set_xlabel('Scattering angle (degrees)', color='#a78bfa', fontsize=11)
ax2.set_ylabel('d sigma / d Omega (normalized)', color='#a78bfa', fontsize=11)
ax2.set_title('Moller Scattering (e- e- -> e- e-)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=9)
ax2.set_yscale('log')
ax2.set_ylim(1e-4, 2)

# --- Plot 3: Bhabha differential cross section ---
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)

for i, sqrt_s in enumerate([1.0, 10.0, 100.0]):
    ds = bhabha_dsigma[sqrt_s]
    ds_norm = ds / ds.max()
    ax3.plot(theta_deg, ds_norm, color=colors[i], linewidth=2, label=f'sqrt(s) = {sqrt_s} GeV')

ax3.set_xlabel('Scattering angle (degrees)', color='#a78bfa', fontsize=11)
ax3.set_ylabel('d sigma / d Omega (normalized)', color='#a78bfa', fontsize=11)
ax3.set_title('Bhabha Scattering (e+ e- -> e+ e-)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=9)
ax3.set_yscale('log')
ax3.set_ylim(1e-4, 2)

# --- Plot 4: Total cross sections vs energy ---
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)

mask = sqrt_s_scan > 0.01
ax4.plot(sqrt_s_scan[mask], sigma_compton_scan[mask], color='#22d3ee', linewidth=2.5, label='Compton')
ax4.plot(sqrt_s_scan[mask], sigma_moller_scan[mask], color='#a78bfa', linewidth=2.5, label='Moller')
ax4.plot(sqrt_s_scan[mask], sigma_bhabha_scan[mask], color='#f472b6', linewidth=2.5, label='Bhabha')

ax4.set_xlabel('sqrt(s) [GeV]', color='#a78bfa', fontsize=11)
ax4.set_ylabel('sigma_total [GeV^-2]', color='#a78bfa', fontsize=11)
ax4.set_title('QED Total Cross Sections vs Energy', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax4.set_xscale('log')
ax4.set_yscale('log')

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="QED Tree-Level Cross Sections" />')
buf.close()

print()
print("Key results: Compton scattering approaches the Thomson limit at low energies.")
print("Moller and Bhabha scattering are dominated by t-channel exchange (forward peak).")
print("At high energies, all QED cross sections fall as 1/s (point-like behavior).")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary: Quantum Electrodynamics

QED Lagrangian

$\mathcal{L}_\text{QED} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ is fully determined by local $U(1)$ gauge invariance. The single coupling constant $e$ (or equivalently $\alpha = e^2/4\pi \approx 1/137$) governs all electromagnetic interactions.

Feynman Rules

The vertex factor $-ie\gamma^\mu$, electron propagator $i(\not{p}+m)/(p^2-m^2+i\varepsilon)$, and photon propagator $-ig_{\mu\nu}/(k^2+i\varepsilon)$ are the building blocks for computing any QED process order by order in $\alpha$.

Tree-Level Processes

Compton, Møller, and Bhabha scattering illustrate the interplay of $s$, $t$, and $u$ channel diagrams. Crossing symmetry relates these processes, and all cross sections scale as $\alpha^2/s$ at high energies.

Ward Identity

Gauge invariance ensures $k_\mu\mathcal{M}^\mu = 0$, keeping the photon massless and constraining the structure of radiative corrections through $Z_1 = Z_2$.

← Previous: Feynman Diagrams Next: Renormalization →