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Massless formal annihilation angular contrast

Facts & Assumptions

Given: Use the conditional assumptions and conventions of Full-mass unpolarized formal pair-production coefficient, including its formal tree coefficient, positive-shell inputs, center-of-momentum angle, and Countable Choice premise. Set $m = M = 0$ and $s > 0$ : write $p_{i} = (E_{i}, p_{i})$ and $k_{i} = (ε_{i}, k_{i})$ with $E_{i} = ∣ p_{i} ∣ > 0$ and $ε_{i} = ∣ k_{i} ∣ > 0$ , and retain $p_{1} + p_{2} = k_{1} + k_{2} = P$ with $h (P, P) = s$ .

[F1]

Full-mass unpolarized formal pair-production coefficient: the exact differential and total formal coefficients as functions of $β_{e}, β_{μ}, θ, s$ .

Example

For this massless specialization, compare the formal differential coefficient at $θ = 0$ and at $θ = π / 2$ . This is a coefficient calculation, not an observed event ratio or an exact continuum-QED cross section.

Verification

1.1givenF1algebra

Setting $β_{e} = β_{μ} = 1$ in [F1] gives $d σ_{c o e f f} / d Ω = α^{2} (ℏ c)^{2} (1 + \cos^{2} θ) / (4 s)$ . At $θ = 0$ this is $α^{2} (ℏ c)^{2} / (2 s)$ ; at $θ = π / 2$ it is $α^{2} (ℏ c)^{2} / (4 s)$ , so the ratio is $2$ .

2.1step 1.1F1algebra∎

Integrating the same density over the sphere uses $\int d Ω = 4 π$ and $\int \cos^{2} θ d Ω = 4 π / 3$ , giving $σ_{c o e f f} = 4 π α^{2} (ℏ c)^{2} / (3 s)$ , in agreement with [F1]. No count data or continuum scattering operator is introduced.

Depends on

Full-mass unpolarized formal pair-production coefficient

Massless formal annihilation angular contrast

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources