π⁰ → γ γ in the laboratory

A compact end-to-end example that combines construction, algebra, and transformations. We model $\pi^0 \to \gamma\gamma$ in the lab: the parent pion has mass $m_{\pi^0}$ and a finite lab momentum; each photon is massless ($m_\gamma = 0$).

Strategy:

1. Build photon 1 in a frame where its direction is simple, with a light-like proxy scaled to energy $m_{\pi^0}/2$ in the π⁰ rest frame.
2. Boost to the lab with Bz(boost_gamma(p_pi0)).
3. Rotate with Ry and Rz so the kinematics align with the parent’s polar axis (spherical_coordinates).
4. Obtain photon 2 by four-momentum subtraction — momentum conservation in the lab.

using FourVectors
using Test

Parent pion in the lab

Mass in GeV (PDG value is $\approx 135\,\mathrm{MeV}$); spatial components are illustrative lab momentum.

m_pi0 = 0.135
p_pi0 = FourVector(1.0, 1.0, 30.0; M = m_pi0)

4-element FourVector{Float64} with indices SOneTo(4):
  1.0
  1.0
 30.0
 30.033618246891265

Photon 1: decay frame → lab → parent axis

In the π⁰ rest frame the two photons are back-to-back with energy $E_\gamma = m_{\pi^0}/2$. We start from a unit direction $(\sin\theta_\gamma, 0, \cos\theta_\gamma)$ with $E=1$, then multiply by $m_{\pi^0}/2$ so that $m^2 = E^2 - |\vec{p}|^2 \approx 0$ for this proxy four-vector.

Bz(boost_gamma(p_pi0)) takes the configuration to the lab frame along the pion’s boost. Ry(acos(Ω.cosθ)) and Rz(Ω.ϕ) with Ω = spherical_coordinates(p_pi0) align the decay plane with the parent direction.

theta_gamma = 0.3
Omega = spherical_coordinates(p_pi0)

p_gamma1 =
    let
        p = (m_pi0 / 2) * FourVector(sin(theta_gamma), 0.0, cos(theta_gamma); E = 1.0)
        p |> Bz(boost_gamma(p_pi0)) |> Ry(acos(Omega.cosθ)) |> Rz(Omega.ϕ)
    end

4-element FourVector{Float64} with indices SOneTo(4):
  0.9917625170514055
  0.9917625170514054
 29.329252809918138
 29.36276989945852

Photon 2 from momentum conservation

In the lab, $\;p_{\pi^0}^\mu = p_{\gamma_1}^\mu + p_{\gamma_2}^\mu\;$, so $p_{\gamma_2} = p_{\pi^0} - p_{\gamma_1}$. Both photons should remain light-like ($m^2 \approx 0$), and the sum should reproduce the parent.

p_gamma2 = p_pi0 - p_gamma1

@test mass2(p_gamma1) < 1e-10
@test mass2(p_gamma2) < 1e-10
@test mass(p_gamma1 + p_gamma2) ≈ mass(p_pi0)
@test p_gamma1 + p_gamma2 ≈ p_pi0

Test Passed

Inspect components

Components are $(p_x, p_y, p_z, E)$; the slight mismatch in $m^2$ is at the level of floating-point error after boosts.

(collect(p_pi0), collect(p_gamma1), collect(p_gamma2))

([1.0, 1.0, 30.0, 30.033618246891265], [0.9917625170514055, 0.9917625170514054, 29.329252809918138, 29.36276989945852], [0.008237482948594499, 0.00823748294859461, 0.6707471900818618, 0.6708483474327451])