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Massless formal annihilation angular contrast

Physical scope

A massless specialization of the adopted formal one-photon tree coefficient. It compares two angular coefficients and integrates their formal coefficient over the sphere; it is not a measured event ratio or exact continuum QED cross section.

Statement

Use the massless $m = M = 0$ case of the formal tree theorem Formal tree scattering and scalar-decay coefficients, including its stated ${A C}_{ω}$ hypothesis, with $s > 0$ . Explicitly take incoming $p_{1}, p_{2}$ and outgoing $k_{1}, k_{2}$ on the positive massless shell, so each $(E, q)$ has $E = ∣ q ∣ > 0$ , with $p_{1} + p_{2} = k_{1} + k_{2} = P$ and $h (P, P) = s > 0$ . Use the theorem's proper orthochronous center-of-momentum chart $Λ P = (\sqrt{s}, 0)$ ; $θ$ is the Euclidean angle between the nonzero spatial parts of $Λ p_{1}$ and $Λ k_{1}$ . The differential coefficient at $θ = 0$ is twice its value at $θ = π / 2$ ; integrating over the full sphere gives $σ_{c o e f f} = \frac{4 π α^{2} (ℏ c)^{2}}{3 s} .$ All three quantities are formal coefficients in the adopted map, not event counts or exact measured cross sections.

Facts & Assumptions

Given: $s > 0$ , $m = M = 0$ , the theorem's Countable Choice condition, the positive-shell/common-total and center-of-momentum angle data in the Statement, and the coefficient conventions of Formal tree scattering and scalar-decay coefficients and Formal diagram coefficient-to-observable adoption.

[F1]

Formal tree scattering and scalar-decay coefficients: the differential coefficient is $α^{2} (ℏ c)^{2} β_{μ} [3 - β_{e}^{2} - β_{μ}^{2} + β_{e}^{2} β_{μ}^{2} \cos^{2} θ] / (4 s β_{e})$ and the total coefficient is its full-sphere integral.

[F2]

QFT example units, metric, gauge, and pairing conventions: $α$ is dimensionless, $[ℏ c] = J m$ , and $[s] = J^{2}$ .

[A1]

Formal diagram coefficient-to-observable adoption: the selected tree result is a leading formal coefficient with no exact continuum or empirical identification.

Example

The angular contrast is a simple consequence of the massless tree angular polynomial.

Verification

1.1givenF1F2algebra

Setting $β_{e} = β_{μ} = 1$ in [F1] gives $\frac{d σ_{c o e f f}}{d Ω} = \frac{α^{2} (ℏ c)^{2}}{4 s} (1 + \cos^{2} θ) .$ Thus its values at $0$ and $π / 2$ are respectively $α^{2} (ℏ c)^{2} / (2 s)$ and $α^{2} (ℏ c)^{2} / (4 s)$ , and their ratio is $2$ . The numerator has unit $J^{2} m^{2}$ and $s$ has unit $J^{2}$ , so each coefficient has area unit $m^{2}$ .

2.1step 1.1F1A1algebra∎

Integrating the same density uses $\int_{S^{2}} d Ω = 4 π$ and $\int_{S^{2}} \cos^{2} θ d Ω = 4 π / 3$ , so $σ_{c o e f f} = \frac{α^{2} (ℏ c)^{2}}{4 s} (4 π + \frac{4 π}{3}) = \frac{4 π α^{2} (ℏ c)^{2}}{3 s} .$ The adopted reading [A1] introduces no count or measurement record.

Massless formal annihilation angular contrast

Statement

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources