Clifford traces and normalized spin completeness — Examples

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 $p_i=(E_i,{\mathbf{p}}_i)$ and $k_i=({\epsilon }_i,{\mathbf{k}}_i)$ with $E_i=\mathrm{\mid }{\mathbf{p}}_i\mathrm{\mid }>0$ and ${\epsilon }_i=\mathrm{\mid }{\mathbf{k}}_i\mathrm{\mid }>0$, and retain $p_1+p_2=k_1+k_2=P$ with $h(P,P)=s$.

Full-mass unpolarized formal pair-production coefficient: the exact differential and total formal coefficients as functions of ${\beta }_e,{\beta }_{\mu },\theta ,s$.

Given: Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Use the formal scalar coefficient conventions of Scalar symmetry factors and decay coefficient units and Complete two-body phase space and coefficient flux map. Set the equal scalar rest energy to $m=0$ and take $s>0$. For scattering, write $p_i=(E_i,{\mathbf{p}}_i)$ and $k_i=({\epsilon }_i,{\mathbf{k}}_i)$ with $E_i=\mathrm{\mid }{\mathbf{p}}_i\mathrm{\mid }>0$ and ${\epsilon }_i=\mathrm{\mid }{\mathbf{k}}_i\mathrm{\mid }>0$, and require $P=p_1+p_2=k_1+k_2$ and $h(P,P)=s$, evaluating the scattering coefficients in the center-of-momentum frame of Complete two-body phase space and coefficient flux map. For decay, take both daughters on that same positive shell with total parent $P_0=({\mathsf{M}}_0,\mathbf{0})$, ${\mathsf{M}}_0>0$. Let the constant quartic amplitude $\lambda \in \mathbb{R}$ and cubic parent amplitude $G_E\in \mathbb{C}$ have the units specified by those suppliers.

Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is assumed; no stronger choice axiom is used.

Scalar symmetry factors and decay coefficient units: identical and distinguishable scalar scattering coefficients and the equal-daughter decay-rate coefficient.

Complete two-body phase space and coefficient flux map: the phase-space integral and the units and definitions of the formal coefficient maps.

Given: Use the formal spin term and magnetic-moment convention of Minimal Dirac magnetic square and formal Pauli benchmark. Let $m>0$ be its mass in kilograms, $q=-e$ with $e>0$ in coulombs, and define $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ by $A(\mathbf{x})=\frac{1}{2}{B}_0{\mathbf{e}}_3\times \mathbf{x}$, where $B_0>0$ is in tesla; this is smooth with tesla-metre components and its curl is the constant field $\mathbf{B}=\mathrm{\nabla }\times A:{\mathbb{R}}^3\to {\mathbb{R}}^3$, $\mathbf{B}(\mathbf{x})=B_0{\mathbf{e}}_3$, where ${\mathbf{e}}_3=(0,0,1)$. Use ${\chi }_{+}=(1,0{)}^{\mathsf{T}}$ and ${\chi }_{-}=(0,1{)}^{\mathsf{T}}$ in ${\mathbb{C}}^2$, which satisfy ${\sigma }_3{\chi }_{\pm }=\pm {\chi }_{\pm }$ for the matrix in the supplier.

Minimal Dirac magnetic square and formal Pauli benchmark: the formal magnetic term is $-(q\mathrm{/}m)\mathbf{S}\cdot \mathbf{B}$ with $\mathbf{S}=\mathrm{\hslash }\mathbf{\Sigma }\mathrm{/}2$ and the minimal coefficient convention has $g_s=2$.