Scalar symmetry factor and conditional lifetime model

Assume Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as required by the cited phase-space map and the Borel restriction of Lebesgue measure.

Let $s>0$, dimensionless $\lambda \in \mathbb{R}$, parent energy ${\mathsf{M}}_0>0$, and $G_E\in \mathbb{C}\setminus \{0\}$ in joules. For scattering, take the incoming $p_1,p_2$ and outgoing $k_1,k_2$ on the positive massless scalar shell, each $(E,\mathbf{q})$ having $E=\mathrm{\mid }\mathbf{q}\mathrm{\mid }>0$, with common total $p_1+p_2=k_1+k_2=P$, $h(P,P)=s>0$, and evaluate the coefficient in a proper orthochronous center-of-momentum chart $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$ as in the phase-space supplier. For decay, take massless positive-shell daughters summing to the rest parent $P_0=({\mathsf{M}}_0,\mathbf{0})$. With these data the identical final-state quartic total coefficient is ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{{\lambda }^2(\mathrm{\hslash }c{)}^2}{32\pi s},$ and the distinguishable-final coefficient is twice this value. The identical-daughter decay-rate coefficient is ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\mathrm{\mid }{G}_E{\mathrm{\mid }}^2}{32\pi \mathrm{\hslash }{\mathsf{M}}_0}[{\mathrm{s}}^{-1}]\mathrm{.}$ Set $\mathrm{\Gamma }:={\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}>0$. For the lifetime calculation only, let $(\mathrm{\Xi },\mathcal{F},\mathbb{P})$ be a probability space in the sense of Probability measures and probability spaces: $\mathcal{F}$ is a sigma-algebra on $\mathrm{\Xi }$, and $\mathbb{P}$ is a countably additive measure with $\mathbb{P}(\mathrm{\Xi })=1$. Let $\mathcal{B}([0,\mathrm{\infty }))$ be the sigma-algebra generated by sets $U\cap [0,\mathrm{\infty })$ for open $U\subseteq \mathbb{R}$, as in The Borel sigma-algebra of a topological space. Let $T:\mathrm{\Xi }\to [0,\mathrm{\infty })$ be a parent lifetime measured in seconds and measurable in the explicit sense that $T^{-1}(B)\in \mathcal{F}$ for every $B\in \mathcal{B}([0,\mathrm{\infty }))$. Define its law by ${\mu }_T(B):=\mathbb{P}(T^{-1}(B))$ on those Borel sets. Let $\mathrm{\ell }$ denote one-dimensional Lebesgue measure from Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, restricted to $\mathcal{B}([0,\mathrm{\infty }))$; this restriction is a measure by Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable under the stated Countable Choice assumption. Separately assume the exponential lifetime model, defined by ${\mu }_T(B)={\int }_B\mathrm{\Gamma }{e}^{-\mathrm{\Gamma }t}d\mathrm{\ell }(t)\text{for every }B\in \mathcal{B}([0,\mathrm{\infty }))\mathrm{.}$ It gives $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}(t):=\mathbb{P}(T^{-1}((t,\mathrm{\infty })))=e^{-\mathrm{\Gamma }t}$ for $t\ge 0$. Define the mean lifetime as $\mathbb{E}[T]:={\int }_{[0,\mathrm{\infty })}td{\mu }_T(t)$, and define the half-life $t_{1\mathrm{/}2}$ by $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}(t_{1\mathrm{/}2})=1\mathrm{/}2$. Then the mean lifetime is $1\mathrm{/}\mathrm{\Gamma }$ and the half-life is $\ln 2\mathrm{/}\mathrm{\Gamma }$. The exponential law is not a consequence of the formal coefficient calculation.

The massless special case exposes both the full-sphere identical-particle factor and the distinction between a rate coefficient and a decay law.