Chapter 4: Feynman Diagrams & Scattering

Feynman diagrams are the visual language of quantum field theory. Each diagram represents a term in the perturbative expansion of scattering amplitudes, with precise mathematical rules translating pictures into numbers. We derive the Feynman rules from the path integral, compute tree-level scattering in $\phi^4$ and Yukawa theories, and connect amplitudes to measurable cross sections via the LSZ reduction formula.

Derivation 1: Feynman Rules from the Path Integral

Feynman diagrams arise systematically from the perturbative expansion of the path integral. Each element of a diagram — propagator, vertex, external line — has a precise mathematical correspondence.

Perturbative Expansion

Consider a scalar field with interaction $\mathcal{L}_\text{int} = -\frac{\lambda}{4!}\phi^4$. The generating functional is:

$Z[J] = \int \mathcal{D}\phi \, e^{i\int d^4x (\mathcal{L}_0 + \mathcal{L}_\text{int} + J\phi)}$

Expanding the interaction exponential in powers of $\lambda$:

$Z[J] = \sum_{n=0}^{\infty} \frac{1}{n!} \left(\frac{-i\lambda}{4!}\right)^n \int d^4z_1 \cdots d^4z_n \int \mathcal{D}\phi \, \phi(z_1)^4 \cdots \phi(z_n)^4 \, e^{i\int(\mathcal{L}_0 + J\phi)}$

Each factor of $\phi^4(z_i)$ can be replaced by functional derivatives acting on$Z_0[J]$, and then Wick's theorem pairs up the fields into propagators.

The Feynman Rules (Momentum Space)

1. Propagator (Internal Line)

Each internal line carrying momentum $k$ contributes:

$\frac{i}{k^2 - m^2 + i\varepsilon}$

2. Vertex

Each $\phi^4$ vertex with four lines meeting contributes:

$-i\lambda$

Plus a momentum-conserving delta function $(2\pi)^4 \delta^4(\sum k_i)$.

3. External Lines

Each external line contributes a factor of 1 for scalars (or $u(p), \bar{u}(p)$ for fermions, $\epsilon_\mu(k)$ for photons).

4. Loop Integration

For each independent loop momentum, integrate:

$\int \frac{d^4\ell}{(2\pi)^4}$

5. Symmetry Factor

Divide by the symmetry factor $S$ of the diagram — the number of permutations of internal lines and vertices that leave the diagram unchanged.

Power counting: A diagram with $V$ vertices,$I$ internal lines, and $L$ loops contributes at order $\lambda^V$. The number of loops is $L = I - V + 1$. Tree diagrams ($L = 0$) are the leading (classical) approximation; loops give quantum corrections proportional to $\hbar^L$.

Feynman Rules for Dirac Fermions

For fermion fields, the Feynman rules include additional structure from the spinor algebra:

Fermion propagator: $\frac{i(\not{p} + m)}{p^2 - m^2 + i\varepsilon}$ — a $4 \times 4$ matrix in spinor space

External fermions: incoming: $u(p)$ or $\bar{v}(p)$; outgoing: $\bar{u}(p)$ or $v(p)$

Fermion loop: each closed fermion loop contributes an extra factor of $(-1)$ from anticommutation

Derivation 2: $\phi^4$ Scattering at Tree Level

The simplest interacting QFT is $\phi^4$ theory. We compute the $2 \to 2$scattering amplitude at tree level.

The Tree-Level Amplitude

For $\phi(p_1) + \phi(p_2) \to \phi(p_3) + \phi(p_4)$, there is only one tree-level diagram: a single vertex connecting all four external lines. The amplitude is:

$i\mathcal{M} = -i\lambda$

The amplitude is a constant — independent of scattering angle!

Mandelstam Variables

The kinematics of $2 \to 2$ scattering are described by three Lorentz-invariant variables:

$s = (p_1 + p_2)^2, \quad t = (p_1 - p_3)^2, \quad u = (p_1 - p_4)^2$

These satisfy the constraint:

$s + t + u = \sum_i m_i^2 = 4m^2$

Physical interpretation:

• $s$: center-of-mass energy squared ($\sqrt{s}$ is the total CM energy)
• $t$: momentum transfer squared (related to scattering angle)
• $u$: crossed momentum transfer

In the center-of-mass frame with equal masses:

$t = -2|\mathbf{p}|^2(1 - \cos\theta), \quad u = -2|\mathbf{p}|^2(1 + \cos\theta)$

where $|\mathbf{p}|^2 = s/4 - m^2$ is the CM momentum squared

Derivation 3: From Amplitudes to Cross Sections

The amplitude $\mathcal{M}$ is not directly observable. What we measure are cross sections — rates of scattering events per unit flux.

The Differential Cross Section

For $2 \to 2$ scattering of equal-mass particles in the CM frame:

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

Deriving the Phase Space Factor

The differential cross section involves the Lorentz-invariant phase space (LIPS):

Step 1: The S-matrix element squared gives the transition rate:

$d\sigma = \frac{1}{2E_1 2E_2 |v_1 - v_2|} |\mathcal{M}|^2 \, d\Pi_\text{LIPS}$

Step 2: The two-body LIPS in the CM frame:

$d\Pi_\text{LIPS} = \frac{|\mathbf{p}_f|}{16\pi^2 \sqrt{s}} d\Omega$

Step 3: The flux factor in the CM frame:

$4E_1 E_2 |v_1 - v_2| = 4|\mathbf{p}_i|\sqrt{s}$

Combining (with $|\mathbf{p}_f| = |\mathbf{p}_i|$ for equal masses):

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

The Total Cross Section

Integrating over solid angle (with a factor of $1/2$ for identical particles):

$\sigma_\text{total} = \frac{1}{2} \int \frac{d\sigma}{d\Omega} d\Omega = \frac{\lambda^2}{32\pi s}$

(for $\phi^4$ theory, where $|\mathcal{M}|^2 = \lambda^2$ is angle-independent)

The Optical Theorem

Unitarity of the S-matrix ($S^\dagger S = 1$) leads to the optical theorem, relating the total cross section to the forward scattering amplitude:

$\boxed{\sigma_\text{total} = \frac{1}{s} \, \text{Im} \, \mathcal{M}(s, t=0)}$

At tree level, $\mathcal{M}$ is real, so the optical theorem is trivially satisfied ($0 = 0$). At one loop, $\mathcal{M}$ develops an imaginary part from intermediate states going on-shell — the total cross section equals the probability of all possible scattering processes.

Derivation 4: Yukawa Theory and the Yukawa Potential

Yukawa theory describes the interaction of fermions with a scalar field via the coupling$g\phi\bar{\psi}\psi$. It was originally proposed to explain the nuclear force, with the pion as the exchange particle.

The Yukawa Interaction

$\mathcal{L}_\text{int} = -g\phi\bar{\psi}\psi$

This describes a vertex where a fermion line emits or absorbs a scalar particle. The Feynman rules give a vertex factor of $-ig$.

t-Channel Exchange: Deriving the Yukawa Potential

Consider fermion-fermion scattering via scalar exchange in the $t$-channel. The amplitude is:

$i\mathcal{M} = (-ig)^2 \frac{i}{t - m_\phi^2} = \frac{-ig^2}{t - m_\phi^2}$

In the non-relativistic limit, $t = -(p_1 - p_3)^2 \approx -|\mathbf{q}|^2$ where$\mathbf{q} = \mathbf{p}_1 - \mathbf{p}_3$ is the momentum transfer. The Born approximation relates the amplitude to the potential:

$\mathcal{M}_\text{NR} = -\tilde{V}(\mathbf{q}) = \frac{g^2}{|\mathbf{q}|^2 + m_\phi^2}$

Fourier transforming to position space:

$V(r) = -\int \frac{d^3q}{(2\pi)^3} \frac{g^2}{|\mathbf{q}|^2 + m_\phi^2} e^{i\mathbf{q}\cdot\mathbf{r}}$

Converting to spherical coordinates and performing the angular integral:

$V(r) = -\frac{g^2}{2\pi^2 r} \int_0^\infty \frac{q \sin(qr)}{q^2 + m_\phi^2} \, dq$

The integral is evaluated by contour integration (closing in the upper half-plane, picking up the pole at $q = im_\phi$):

$\boxed{V(r) = -\frac{g^2}{4\pi} \frac{e^{-m_\phi r}}{r}}$

The Yukawa potential — an attractive, screened interaction

Key features:

• Range: $r_0 \sim 1/m_\phi$. Massive exchange particles produce short-range forces.
• Coulomb limit: As $m_\phi \to 0$, $V(r) \to -g^2/(4\pi r)$ — recovering the Coulomb potential. This is how QED gives electrostatics!
• Nuclear force: With $m_\phi \approx m_\pi \approx 140$ MeV, the range is$\sim 1.4$ fm, matching the observed nuclear force range.

Derivation 5: The LSZ Reduction Formula

The Lehmann-Symanzik-Zimmermann (LSZ) reduction formula is the rigorous bridge between the time-ordered correlation functions (computable from the path integral) and the S-matrix elements (which give scattering amplitudes).

The Problem

We know how to compute correlation functions like $\langle \Omega | T\{\phi(x_1)\cdots\phi(x_n)\} | \Omega \rangle$from the path integral. But what we need for scattering predictions is the S-matrix:

$\langle p_3, p_4, \ldots | S | p_1, p_2 \rangle = ?$

The LSZ Formula

For the scattering of $n$ incoming particles to $m$ outgoing particles, the LSZ formula states (for scalar fields):

$\langle p_1', \ldots, p_m' | S | p_1, \ldots, p_n \rangle =$

$\prod_{i=1}^{n} \left[\frac{i}{\sqrt{Z}} \int d^4x_i \, e^{ip_i \cdot x_i} (\Box_{x_i} + m^2)\right] \prod_{j=1}^{m} \left[\frac{i}{\sqrt{Z}} \int d^4y_j \, e^{-ip_j' \cdot y_j} (\Box_{y_j} + m^2)\right]$

$\times \langle \Omega | T\{\phi(x_1) \cdots \phi(x_n) \phi(y_1) \cdots \phi(y_m)\} | \Omega \rangle$

Key Elements

Field Strength Renormalization Z

The factor $\sqrt{Z}$ accounts for the fact that the interacting field is not canonically normalized: $\langle \Omega|\phi(0)|p\rangle = \sqrt{Z}$, where$Z < 1$ due to virtual particle dressing.

The Klein-Gordon Operator

The operator $(\Box + m^2)$ acting on the correlation function "amputates" the external propagators. In momentum space, this replaces the full propagator near the mass shell$p^2 = m^2$ with the residue $iZ/(p^2 - m^2)$, and the $(\Box + m^2)$ cancels the pole, leaving just the amputated amplitude.

The Physical Mass

The mass $m$ in the LSZ formula is the physical (pole) mass, which may differ from the bare mass in the Lagrangian due to quantum corrections. It is defined as the pole of the exact propagator: $G(p)^{-1}|_{p^2=m^2} = 0$.

The Practical Consequence

In practice, the LSZ formula tells us:

$i\mathcal{M} = Z^{-(n+m)/2} \times (\text{amputated Feynman diagrams, on-shell})$

At tree level, $Z = 1$, and the S-matrix element is simply the amputated diagram evaluated with on-shell external momenta. The LSZ formula guarantees that this recipe works to all orders in perturbation theory, provided we properly account for $Z$ and the physical mass.

Simulation: Cross Sections & Potentials

This simulation computes tree-level cross sections in $\phi^4$ theory (showing the$1/s$ energy dependence), visualizes the Mandelstam variables as functions of scattering angle, compares the Yukawa potential for different exchange masses against the Coulomb limit, and shows how light exchange particles produce strongly forward-peaked angular distributions.

Feynman Diagrams: Cross Sections & Yukawa Potential

Python

script.py218 lines

#!/usr/bin/env python3
"""
feynman_diagrams.py - Tree-Level Cross Sections & Yukawa Potential
Computes phi^4 scattering cross sections and Yukawa vs Coulomb potentials
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# ============================================================
# Part 1: phi^4 Theory - 2->2 Scattering at Tree Level
# M = -i*lambda (constant at tree level!)
# d(sigma)/d(Omega) = lambda^2 / (64 * pi^2 * s) in CM frame
# ============================================================

print("=" * 72)
print("  FEYNMAN DIAGRAMS & SCATTERING - NUMERICAL ANALYSIS")
print("=" * 72)
print()

# phi^4 theory: coupling constant
lam = 1.0  # lambda

# Center-of-mass energy scan
sqrt_s_vals = np.linspace(0.5, 10, 500)
s_vals = sqrt_s_vals**2

# Total cross section: sigma_total = lambda^2 / (32 * pi * s) (in natural units)
# Phase space factor for 2->2 identical scalars
m_phi = 0.5  # scalar mass

sigma_phi4 = np.zeros_like(s_vals)
for i, s in enumerate(s_vals):
    if s > (2 * m_phi)**2:  # above threshold
        p_cm = np.sqrt(s / 4 - m_phi**2)
        # |M|^2 = lambda^2 (tree level, no angular dependence)
        sigma_phi4[i] = lam**2 / (32 * np.pi * s) * (1.0 / (2.0))  # 1/2 for identical particles
    else:
        sigma_phi4[i] = 0

print("phi^4 Theory (tree level):")
print("-" * 50)
print(f"  lambda = {lam}, m_phi = {m_phi}")
print(f"  Threshold: sqrt(s) = 2*m = {2*m_phi}")
print(f"  |M|^2 = lambda^2 = {lam**2} (momentum-independent!)")
print()
for sv in [1.0, 2.0, 5.0, 10.0]:
    s = sv**2
    if s > (2*m_phi)**2:
        sig = lam**2 / (32 * np.pi * s) * 0.5
        print(f"  sqrt(s) = {sv:.1f}: sigma = {sig:.6f}")
print()

# ============================================================
# Part 2: Mandelstam Variables
# s = (p1 + p2)^2, t = (p1 - p3)^2, u = (p1 - p4)^2
# s + t + u = 4m^2
# ============================================================

print("Mandelstam Variables for 2 -> 2 scattering:")
print("-" * 50)
sqrt_s = 5.0
s = sqrt_s**2
p_cm = np.sqrt(s/4 - m_phi**2)
theta_cm = np.linspace(0, np.pi, 100)

t_vals = -2 * p_cm**2 * (1 - np.cos(theta_cm))
u_vals = -2 * p_cm**2 * (1 + np.cos(theta_cm))

print(f"  At sqrt(s) = {sqrt_s}, p_cm = {p_cm:.4f}")
print(f"  s = {s:.2f}")
print(f"  t range: [{t_vals.min():.2f}, {t_vals.max():.2f}]")
print(f"  u range: [{u_vals.min():.2f}, {u_vals.max():.2f}]")
print(f"  s + t + u = {s + t_vals[50] + u_vals[50]:.4f} (should be {4*m_phi**2:.4f})")
print()

# ============================================================
# Part 3: Yukawa Theory - t-channel Exchange
# M = -g^2 / (t - m_exchange^2)
# d(sigma)/d(Omega) = g^4 / (64 * pi^2 * s) * 1/(t - m_ex^2)^2
# ============================================================

g = 1.0           # Yukawa coupling
m_fermion = 1.0   # fermion mass
m_exchange = [0.001, 0.5, 1.0, 2.0]  # exchange particle masses

print("Yukawa Theory (t-channel exchange):")
print("-" * 50)

# Yukawa potential: V(r) = -g^2/(4*pi) * exp(-m*r) / r
r = np.linspace(0.01, 8, 1000)

print("Yukawa Potential V(r) = -g^2/(4*pi) * exp(-m*r)/r")
for m_ex in m_exchange:
    V = -g**2 / (4 * np.pi) * np.exp(-m_ex * r) / r
    print(f"  m_exchange = {m_ex}: V(r=1) = {-g**2/(4*np.pi)*np.exp(-m_ex)/1:.6f}, "
          f"range ~ 1/m = {1/m_ex if m_ex > 0 else 'inf':.2f}")
print()

# ============================================================
# Part 4: Differential cross section angular dependence
# For Yukawa: M ~ 1/(t - m_ex^2), t = -2p^2(1-cos(theta))
# ============================================================

sqrt_s_yukawa = 4.0
s_y = sqrt_s_yukawa**2
E_cm = sqrt_s_yukawa / 2
p_y = np.sqrt(E_cm**2 - m_fermion**2) if E_cm > m_fermion else 0
theta = np.linspace(0.01, np.pi - 0.01, 500)
t_y = -2 * p_y**2 * (1 - np.cos(theta))

print(f"Differential Cross Section at sqrt(s) = {sqrt_s_yukawa}:")
print("-" * 50)

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

# --- Plot 1: phi^4 total cross section vs sqrt(s) ---
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)

ax1.plot(sqrt_s_vals, sigma_phi4, color='#a78bfa', linewidth=2.5)
ax1.axvline(x=2*m_phi, color='#ef4444', linewidth=1.5, linestyle='--', label=f'Threshold 2m = {2*m_phi}')
ax1.set_xlabel('sqrt(s) [GeV]', color='#a78bfa', fontsize=11)
ax1.set_ylabel('sigma [natural units]', color='#a78bfa', fontsize=11)
ax1.set_title('phi^4 Total Cross Section (tree level)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax1.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax1.set_yscale('log')
ax1.set_ylim(1e-5, 1)

# --- Plot 2: Mandelstam variables ---
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)

theta_deg = np.degrees(theta_cm)
ax2.plot(theta_deg, np.full_like(theta_deg, s), color='#22d3ee', linewidth=2.5, label=f's = {s:.1f}')
ax2.plot(theta_deg, -t_vals, color='#a78bfa', linewidth=2.5, label='-t')
ax2.plot(theta_deg, -u_vals, color='#f472b6', linewidth=2.5, label='-u')
ax2.set_xlabel('Scattering angle theta (degrees)', color='#a78bfa', fontsize=11)
ax2.set_ylabel('Mandelstam variable', color='#a78bfa', fontsize=11)
ax2.set_title(f'Mandelstam Variables (sqrt(s) = {sqrt_s})', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)

# --- Plot 3: Yukawa potential vs Coulomb ---
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)

# Coulomb (m -> 0 limit)
V_coulomb = -g**2 / (4 * np.pi * r)
ax3.plot(r, V_coulomb, color='#22d3ee', linewidth=2.5, linestyle='--', label='Coulomb (m -> 0)')

colors_y = ['#a78bfa', '#f472b6', '#fbbf24', '#34d399']
for i, m_ex in enumerate([0.1, 0.5, 1.0, 2.0]):
    V_yukawa = -g**2 / (4 * np.pi) * np.exp(-m_ex * r) / r
    ax3.plot(r, V_yukawa, color=colors_y[i], linewidth=2, label=f'Yukawa m = {m_ex}')

ax3.set_xlabel('Distance r', color='#a78bfa', fontsize=11)
ax3.set_ylabel('V(r)', color='#a78bfa', fontsize=11)
ax3.set_title('Yukawa vs Coulomb Potential', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=9)
ax3.set_ylim(-5, 0.5)
ax3.set_xlim(0, 6)

# --- Plot 4: Angular distribution of Yukawa scattering ---
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)

theta_deg2 = np.degrees(theta)
for i, m_ex in enumerate([0.1, 0.5, 1.0, 2.0]):
    t_yuk = -2 * p_y**2 * (1 - np.cos(theta))
    M_sq = g**4 / (t_yuk - m_ex**2)**2
    # Normalize for comparison
    M_sq_norm = M_sq / M_sq.max()
    ax4.plot(theta_deg2, M_sq_norm, color=colors_y[i], linewidth=2, label=f'm_ex = {m_ex}')

ax4.set_xlabel('Scattering angle theta (degrees)', color='#a78bfa', fontsize=11)
ax4.set_ylabel('|M|^2 (normalized)', color='#a78bfa', fontsize=11)
ax4.set_title(f'Yukawa: Angular Distribution (sqrt(s) = {sqrt_s_yukawa})', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax4.set_yscale('log')
ax4.set_ylim(1e-4, 2)

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="Feynman Diagram Calculations" />')
buf.close()

print()
print("Key insight: light exchange particles produce forward-peaked (Rutherford-like)")
print("scattering, while heavy exchange particles give more isotropic scattering.")
print("In the massless limit, the Yukawa potential reduces to the Coulomb potential.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Crossing Symmetry and Beyond Tree Level

Crossing Symmetry

A remarkable property of relativistic amplitudes is crossing symmetry: the same analytic function $\mathcal{M}(s,t,u)$ describes different physical processes depending on which variables are in the physical region.

• $s$-channel: $A + B \to C + D$ (scattering), physical for $s > (m_A + m_B)^2$
• $t$-channel: $A + \bar{C} \to \bar{B} + D$ (related by crossing), physical for $t > 0$
• $u$-channel: $A + \bar{D} \to \bar{B} + C$ (related by crossing), physical for $u > 0$

Crossing symmetry is a deep consequence of CPT invariance and the analyticity of scattering amplitudes. It means that particle-antiparticle scattering is related to particle-particle scattering by analytic continuation.

Loop Corrections: A Preview

At one loop, diagrams contain a closed internal line requiring integration over an unconstrained loop momentum:

$\int \frac{d^4\ell}{(2\pi)^4} \frac{i}{\ell^2 - m^2 + i\varepsilon} \frac{i}{(\ell - q)^2 - m^2 + i\varepsilon}$

These integrals are often ultraviolet divergent — they blow up as$|\ell| \to \infty$. This signals that the theory requires renormalization: the divergences can be absorbed into redefinitions of the coupling constant, mass, and field normalization, leaving finite, predictive results. This is the subject of Part II.

Power counting: In $\phi^4$ theory in 4 dimensions, the superficial degree of divergence of a diagram with $E$ external lines is$D = 4 - E$. Diagrams with $E = 2$ ($D = 2$: quadratically divergent) renormalize the mass; $E = 4$ ($D = 0$: logarithmically divergent) renormalize the coupling. Only finitely many types of divergences occur — the theory isrenormalizable.

Decay Rates

Feynman diagrams also compute particle decay rates. For a particle of mass $M$ decaying to $n$ particles, the partial width is:

$d\Gamma = \frac{1}{2M} |\mathcal{M}|^2 \, d\Pi_\text{LIPS}$

The total decay rate $\Gamma$ gives the lifetime $\tau = 1/\Gamma$. For example, the muon lifetime is computed from the Fermi theory diagram $\mu^- \to e^- \bar{\nu}_e \nu_\mu$, giving excellent agreement with the measured value of $\tau_\mu \approx 2.2 \times 10^{-6}$ s.

Summary: The Feynman Diagram Toolkit

Feynman Rules

Propagators, vertices, and symmetry factors follow directly from the path integral and Wick's theorem. Each diagram is a term in the perturbative expansion in the coupling constant.

Scattering Amplitudes

$\phi^4$ theory gives angle-independent scattering; Yukawa theory gives forward-peaked scattering through $t$-channel exchange. The Mandelstam variables$s, t, u$ parameterize all $2 \to 2$ kinematics.

Cross Sections and Observables

$d\sigma/d\Omega = |\mathcal{M}|^2/(64\pi^2 s)$ connects theory to experiment. The optical theorem enforces unitarity at every order.

LSZ Reduction

The LSZ formula rigorously connects correlation functions to S-matrix elements, justifying the Feynman diagram approach. External propagators are amputated and put on shell; field renormalization $Z$ accounts for wavefunction dressing.

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