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The tree amplitude is, up to an overall convention-dependent sign,

M = - λ \overset{u}{ˉ} (q_{2}) \neq p v (q_{1}), p = q_{1} + q_{2} .

\neq q_{1} v (q_{1}) = - m v (q_{1}), \overset{u}{ˉ} (q_{2}) \neq q_{2} = m \overset{u}{ˉ} (q_{2}) .

Therefore

M = - λ \overset{u}{ˉ} (q_{2}) (\neq q_{1} + \neq q_{2}) v (q_{1}) = 0.

Although

μ > 2 m

permits the relativistic two-body decay kinematically, the matrix element vanishes. Thus

spins \sum ∣ M ∣^{2} = 0, Γ (ϕ \to \overset{ˉ}{ψ} ψ) = 0.

In the parent's rest frame, momentum conservation makes the massive daughter and photon momenta equal and opposite, of magnitude

p

p = \frac{M ^{2} - m ^{2}}{2 M} c, E_{R} = \frac{M ^{2} + m ^{2}}{2 M} c^{2} .

Since

v = p c^{2} / E_{R}

and

r = m / M

v = c f (r), f (r) = \frac{1 - r ^{2}}{1 + r ^{2}} .

γ = \frac{E _{R}}{m c ^{2}} = \frac{1 + r ^{2}}{2 r} .

If it decays after proper time

τ

t = γ τ

, while uniform motion gives

x = v γ τ

. Therefore the event coordinates in the parent frame are

(c t, x) = (\frac{1 + r ^{2}}{2 r} c τ, \frac{1 - r ^{2}}{2 r} c τ) .

P^{μ} = (E / c, p) = mγ (c, v), P^{2} = m^{2} c^{2},

while for a massless particle

P^{μ} = (E / c, p), E = c ∣ p ∣, P^{2} = 0.

For

P_{1} = P_{2} + P_{3}

, invariance gives

(P_{1} - P_{2})^{2} = P_{3}^{2} .

In the rest frame of particle 1,

P_{1} = (m_{1} c, 0)

, so

E_{2} = \frac{m _{1}^{2} + m _{2}^{2} - m _{3}^{2}}{2 m _{1}} c^{2}, E_{3} = \frac{m _{1}^{2} + m _{3}^{2} - m _{2}^{2}}{2 m _{1}} c^{2},

with

E_{1} = m_{1} c^{2}

. Equal and opposite final momenta are real exactly when

m_{1} \geq m_{2} + m_{3},

 Incoming links: Relativistic two-body decay