Cross sections σ: Probability for scattering (measured in barns, 1b = 10^-28m²)
Decay rates Γ: Probability per unit time for a particle to decay (related to lifetime τ = 1/Γ)

💡What is a Cross Section?

Imagine shooting particles at a target. The cross section σ is the effective target area per scattering center.

If you fire N_beam particles at a target with n scattering centers, the number of scattering events is:

N_scatter = N_beam × n × σ

Larger σ → more likely to scatter!

2.2 Differential Cross Section

For 2 → 2 scattering (a + b → c + d), the differential cross section in the center-of-mass frame is:

$$\boxed{\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 s}\frac{|\vec{p}_f|}{|\vec{p}_i|}|\mathcal{M}|^2}$$

where:

s = (p_a + p_b)² is the Mandelstam variable (total energy squared)
|p⃗_i| is the initial momentum magnitude in CM frame
|p⃗_f| is the final momentum magnitude in CM frame
M is the invariant amplitude (Lorentz scalar)
dΩ is the solid angle element

For equal mass particles with |p⃗_i| = |p⃗_f| = |p⃗|, this simplifies to:

$$\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 s}|\mathcal{M}|^2$$

2.3 Total Cross Section

Integrate over all angles:

$$\boxed{\sigma_{\text{tot}} = \int d\Omega \frac{d\sigma}{d\Omega}}$$

For spherically symmetric scattering (|M|² independent of angle):

$$\sigma_{\text{tot}} = \frac{4\pi}{64\pi^2 s}|\mathcal{M}|^2 = \frac{1}{16\pi s}|\mathcal{M}|^2$$

Example: φ⁴ Scattering

For φ⁴ theory with M = -λ (tree level), we get:

$$\sigma_{\text{tot}} = \frac{\lambda^2}{16\pi s}$$

Cross section decreases as energy increases! This is typical for point-like interactions.

2.4 Phase Space Integrals

For n final particles, we must integrate over phase space:

$$d\Phi_n = \prod_{i=1}^n \frac{d^3p_i}{(2\pi)^3 2E_i}(2\pi)^4\delta^4\left(p_{\text{in}} - \sum_{j=1}^n p_j\right)$$

This is Lorentz-invariant phase space (LIPS). The factors ensure:

Lorentz invariance
Energy-momentum conservation (delta function)
On-shell particles: E_i = √(p⃗_i² + m_i²)

2.5 Decay Rates

For a particle A decaying to n particles, the decay rate (or decay width) is:

$$\boxed{\Gamma = \frac{1}{2m_A}\int d\Phi_n |\mathcal{M}|^2}$$

The lifetime is τ = 1/Γ. Particles with larger Γ decay faster!

Two-Body Decay

For A → B + C in the rest frame of A:

$$\Gamma = \frac{|\vec{p}|}{8\pi m_A^2}|\mathcal{M}|^2$$

where |p⃗| is the momentum of B (or C) in the rest frame:

$$|\vec{p}| = \frac{1}{2m_A}\sqrt{[m_A^2 - (m_B + m_C)^2][m_A^2 - (m_B - m_C)^2]}$$

2.6 Mandelstam Variables

For 2 → 2 scattering, define three Lorentz-invariant variables:

\begin{align*} s &= (p_1 + p_2)^2 = (p_3 + p_4)^2 \quad \text{(total energy squared)} \\ t &= (p_1 - p_3)^2 = (p_2 - p_4)^2 \quad \text{(momentum transfer squared)} \\ u &= (p_1 - p_4)^2 = (p_2 - p_3)^2 \quad \text{(crossed channel)} \end{align*}

These satisfy:

$$s + t + u = \sum_{i=1}^4 m_i^2$$

Only two are independent! Different physical regions correspond to different scattering channels.

Key Takeaways (Page 1)

Differential cross section: $\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2 s}|\mathcal{M}|^2 \frac{|\vec{p}_f|}{|\vec{p}_i|}$
Total cross section: Integrate $\frac{d\sigma}{d\Omega}$ over all angles
Phase space: Lorentz-invariant measure for final states
Decay rate: $\Gamma = \frac{1}{2m_A}\int d\Phi_n |\mathcal{M}|^2$
Lifetime: τ = 1/Γ (shorter lifetime = larger decay rate)
Mandelstam variables: s, t, u describe kinematics

← Interaction Picture

Page 1 of 3

Chapter 2: Cross Sections & Decay Rates

Differential Cross Section for e+e- to mu+mu-

Python

script.py45 lines

import numpy as np

# e+ e- -> mu+ mu- via virtual photon (QED tree level)
# dsigma/dOmega = alpha^2 / (4*s) * (1 + cos^2(theta))
# sigma_total = 4*pi*alpha^2 / (3*s)

alpha = 1.0 / 137.036  # fine structure constant
m_e = 0.511e-3   # electron mass in GeV
m_mu = 0.10566   # muon mass in GeV
hbar_c_sq = 0.3894e6  # (hbar*c)^2 in GeV^2 * pb

print("QED Cross Section: e+e- -> mu+mu- (tree level)")
print("dsigma/dOmega = alpha^2/(4s) * (1 + cos^2 theta)")
print("=" * 55)

# Total cross section vs center-of-mass energy
energies = [0.5, 1.0, 2.0, 5.0, 10.0, 30.0, 91.2, 200.0]
print(f"\n{'sqrt(s) [GeV]':>14s}  {'sigma [pb]':>12s}  {'sigma [nb]':>12s}")
print("-" * 42)

for sqrt_s in energies:
    s = sqrt_s**2
    if sqrt_s < 2 * m_mu:
        print(f"{sqrt_s:14.1f}  {'below threshold':>12s}  {'---':>12s}")
        continue
    beta_mu = np.sqrt(1 - 4*m_mu**2/s)
    sigma = 4 * np.pi * alpha**2 / (3 * s) * beta_mu * hbar_c_sq
    print(f"{sqrt_s:14.1f}  {sigma:12.4f}  {sigma/1000:12.6f}")

# Angular distribution at sqrt(s) = 10 GeV
sqrt_s = 10.0
s = sqrt_s**2
print(f"\nAngular distribution at sqrt(s) = {sqrt_s} GeV:")
print(f"{'cos(theta)':>12s}  {'dsigma/dOmega [pb/sr]':>22s}")
print("-" * 38)

cos_theta = np.linspace(-0.9, 0.9, 10)
for ct in cos_theta:
    dsigma = alpha**2 / (4 * s) * (1 + ct**2) * hbar_c_sq
    print(f"{ct:12.3f}  {dsigma:22.6f}")

print("\nNote: sigma ~ 1/s (dimensional analysis)")
print("At Z pole (91.2 GeV), resonance enhancement occurs")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Mandelstam Variables and Crossing Symmetry

Fortran

program.f9055 lines

program mandelstam
  implicit none
  double precision, parameter :: pi = 3.14159265358979d0
  double precision :: m1, m2, m3, m4
  double precision :: s, t, u, sqrt_s, E_cm, p_cm, cos_th
  double precision :: sum_m2
  integer :: i

! e+ e- -> mu+ mu-
  m1 = 0.000511d0  ! electron mass (GeV)
  m2 = 0.000511d0  ! positron mass
  m3 = 0.10566d0   ! mu- mass
  m4 = 0.10566d0   ! mu+ mass

sum_m2 = m1**2 + m2**2 + m3**2 + m4**2

print *, "Mandelstam Variables for e+e- -> mu+mu-"
  print *, "s + t + u = m1^2 + m2^2 + m3^2 + m4^2"
  print *, "========================================="
  print '(A, F12.8, A)', " Sum of masses^2 = ", sum_m2, " GeV^2"
  print *, ""

print '(A8, A12, A14, A14, A14, A8)', &
    "sqrt_s", "cos_th", "s", "t", "u", "check"
  print *, "-------------------------------------------------------------------"

do i = 1, 10
    sqrt_s = 1.0d0 + 2.0d0 * dble(i)
    s = sqrt_s**2

! CM frame kinematics
    E_cm = sqrt_s / 2.0d0
    if (E_cm .lt. m3) cycle
    p_cm = sqrt(E_cm**2 - m3**2)

cos_th = 0.5d0  ! 60 degrees

! t = (p1 - p3)^2 for massless limit
    t = -0.5d0 * (s - 4.0d0*m3**2) * (1.0d0 - cos_th)
    u = sum_m2 - s - t

print '(F8.2, F12.4, F14.6, F14.6, F14.6, L8)', &
      sqrt_s, cos_th, s, t, u, &
      abs(s + t + u - sum_m2) < 1.0d-8
  end do

print *, ""
  print *, "Crossing symmetry channels:"
  print *, "  s-channel: e+e- -> mu+mu-  (annihilation)"
  print *, "  t-channel: e-mu- -> e-mu-  (scattering)"
  print *, "  u-channel: e-mu+ -> e-mu+  (exchange)"

end program mandelstam

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Cross Sections & Decay Rates

🔗Course Connections

📚Prerequisites

Video Lecture

2.1 From Theory to Experiment