Scalar symmetry factors and decay coefficient units

Assume Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as required by the phase-space and coefficient-map suppliers used below.

Let $m\ge 0$ be a scalar rest energy in joules, and let $p_1,p_2,k_1,k_2$ be the incoming and outgoing scattering vectors on the positive $m$-shell of Two-body phase-space and formal coefficient data, so each $(E,\mathbf{q})$ satisfies $E=\sqrt{\mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^2+m^2}>0$. Require $p_1+p_2=k_1+k_2=P$ and $s=h(P,P)>4m^2$, with $h$ as in QFT example units, metric, gauge, and pairing conventions. Evaluate the coefficient map in a center-of-momentum frame $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$ as defined in Complete two-body phase space and coefficient flux map. For one scalar initial label of weight $1$ and one specified final channel, take the dimensionless real quartic tree amplitude $\mathcal{A}=-\lambda$. The coefficient map of Complete two-body phase space and coefficient flux map gives $\frac{d{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}}{d\mathrm{\Omega }}=\frac{{\lambda }^2(\mathrm{\hslash }c{)}^2}{128{\pi }^{2}s},{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{{\lambda }^2(\mathrm{\hslash }c{)}^2}{32\pi s}$ for identical final scalars integrated over the full labeled sphere. For distinguishable final scalars the two coefficients are twice these values.

For a parent at rest, $P_0=({\mathsf{M}}_0,\mathbf{0})$ with energy ${\mathsf{M}}_0>2m$, decaying through a specified single channel to two identical scalars on the positive $m$-shell whose total four-vector is $P_0$, under a constant amplitude $G_E$ in joules, the rate coefficient is ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\mathrm{\mid }{G}_E{\mathrm{\mid }}^2}{32\pi \mathrm{\hslash }{\mathsf{M}}_0}\sqrt{1-\frac{4m^2}{{\mathsf{M}}_0^2}}\mathrm{.}$ This result concerns the defined formal coefficients. It does not assert an exact exponential survival law.