Two-body phase-space and formal coefficient data

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and use the energy-vector, metric, and unit conventions of QFT example units, metric, gauge, and pairing conventions. A rest-energy parameter $\mathsf{M}\ge 0$ and every energy-vector component are measured in joules. Set $D_{\mathsf{M}}={\mathbb{R}}^3$ when $\mathsf{M}>0$ and $D_0={\mathbb{R}}^3\setminus \{0\}$. Define ${\iota }_{\mathsf{M}}:D_{\mathsf{M}}\to {\mathbb{R}}^4$ by ${\iota }_{\mathsf{M}}(\mathbf{p})=(\sqrt{\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+{\mathsf{M}}^2},\mathbf{p}),{\mathcal{H}}_{\mathsf{M}}^{+}:={\iota }_{\mathsf{M}}(D_{\mathsf{M}})\mathrm{.}$ Thus ${\mathcal{H}}_{\mathsf{M}}^{+}$ is the positive-energy shell, with its subspace Borel structure. Define the Borel shell measure ${\nu }_{\mathsf{M}}$ as the pushforward under ${\iota }_{\mathsf{M}}$ of the measure absolutely continuous with respect to three-dimensional Lebesgue measure on $D_{\mathsf{M}}$, with density $((2\pi {)}^{3}2\sqrt{\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+{\mathsf{M}}^2}{)}^{-1}$. In particular, the massless apex is excluded.

Let ${\mathsf{M}}_1,{\mathsf{M}}_2\ge 0$ and let $s>({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2$. Define $k_{\ast }=\frac{\sqrt{[s-({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2][s-({\mathsf{M}}_1-{\mathsf{M}}_2{)}^2]}}{2\sqrt{s}},E_i^{\ast }=\sqrt{k_{\ast }^2+{\mathsf{M}}_i^2}\mathrm{.}$ Both factors under the square root are positive, so $k_{\ast }>0$. On the Euclidean unit sphere $S^2$, let $\mathrm{\Omega }$ be the chart surface Borel measure of Agreement with the existing polar sphere measure. Define the finite Borel measure ${\mathrm{\Phi }}_2$ on $S^2$ and the shell-pair map ${\mathrm{\Psi }}_2:S^2\to {\mathcal{H}}_{{\mathsf{M}}_1}^{+}\times {\mathcal{H}}_{{\mathsf{M}}_2}^{+}$ by $d{\mathrm{\Phi }}_2=\frac{k_{\ast }}{16{\pi }^2\sqrt{s}}d\mathrm{\Omega },{\mathrm{\Psi }}_2(\mathbf{n})=((E_1^{\ast },k_{\ast }\mathbf{n}),(E_2^{\ast },-k_{\ast }\mathbf{n}))\mathrm{.}$ The center-of-momentum measure on the shell-pair space is the pushforward $({\mathrm{\Psi }}_2{)}_{\mathrm{\#}}{\mathrm{\Phi }}_2$. The angular measure ${\mathrm{\Phi }}_2$ and its shell-pair pushforward are distinct typed measures.

For incoming energy four-vectors $p_i\in {\mathcal{H}}_{{\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}}^{+}$, where ${\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}\ge 0$ for $i=1,2$, define the invariant flux when its radicand is nonnegative by $F(p_1,p_2)=4\sqrt{(p_1\cdot p_2{)}^2-{\mathsf{M}}_{1,\mathrm{i}\mathrm{n}}^2{\mathsf{M}}_{2,\mathrm{i}\mathrm{n}}^2},p\cdot q=h(p,q)\mathrm{.}$ Let ${\mathcal{S}}_{\mathrm{i}\mathrm{n}}$ and ${\mathcal{S}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ be finite nonempty sets of initial and final spin labels, respectively, and let $w_i\ge 0$ for $i\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}}$ satisfy ${\sum }_i{w}_i=1$. A spin-resolved formal amplitude is a family of dimensionless Borel functions ${\mathcal{A}}_{if}:S^2\to \mathbb{C}$, indexed by $i\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}}$ and $f\in {\mathcal{S}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$. Define the Borel function $\overline{\mathrm{\mid }\mathcal{A}(\mathbf{n}){\mathrm{\mid }}^2}:={\sum }_{i\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}}}{w}_i{\sum }_{f\in {\mathcal{S}}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\mathrm{\mid }{\mathcal{A}}_{if}(\mathbf{n}){\mathrm{\mid }}^2\mathrm{.}$ For a specified final two-particle channel, set $S_f=2$ when its two final particles are identical and $S_f=1$ when they are distinguishable. When the incoming flux is positive, $F>0$, define the angular cross-section coefficient measure ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}$ on $S^2$ by $d{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{(\mathrm{\hslash }c{)}^2}{F{S}_f}\overline{\mathrm{\mid }\mathcal{A}(\mathbf{n}){\mathrm{\mid }}^2}d{\mathrm{\Phi }}_2\mathrm{.}$ For a parent at rest with energy ${\mathsf{M}}_0>0$ and a specified single-channel Borel formal decay amplitude $\mathcal{G}:S^2\to \mathbb{C}$ measured in joules, define the extended nonnegative coefficient ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{1}{2\mathrm{\hslash }{\mathsf{M}}_0{S}_f}{\int }_{S^2}\mathrm{\mid }\mathcal{G}(\mathbf{n}){\mathrm{\mid }}^{2}d{\mathrm{\Phi }}_2\mathrm{.}$ It is a finite rate coefficient when this integral is finite; otherwise the displayed integral has value $+\mathrm{\infty }$. These are mathematical coefficient maps for the explicitly specified formal amplitudes; their definitions assert no exact interacting scattering operator or decay law.