Formal diagram coefficient-to-observable adoption

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

For a specifically named finite tree calculation, adopt its selected formal diagram coefficient as a leading perturbative coefficient with a stated phenomenological reading. For QED, use the vertex and free-line conventions of the current draft supplier Explicit formal QED vertex/free-line convention. A Dirac species of natural charge $q_n$ has vertex $i{q}_n{\gamma }^a$; the electron has $q_n=-e_n$ and vertex $-i{e}_n{\gamma }^a$. The rational fermion line is $i\frac{\mathrm{p̸}+\mu }{h(p,p)-{\mu }^2},$ for $\mu >0$ and off-shell $p$ with $h(p,p)-{\mu }^2\mathrm{\ne }0$. In Feynman gauge the auxiliary photon line is $-i\frac{h_{ab}}{h(k,k)+i0}=i\frac{{\eta }_{ab}}{h(k,k)+i0}\mathrm{.}$ For a finite matrix kernel the coefficientwise inverse in Explicit formal QED vertex/free-line convention is part of the formal convention and asserts no convergence. If a time-ordered distribution is used, use the actual free carrier specified there; that identification is conditional on The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). The displayed pole notation does not define loop products. In the pair-production channel below the selected muon also has model charge $-e$, and only the photon line is internal.

For declared incoming spin probabilities, sum over declared final spin labels, and a specified formal amplitude $\mathcal{A}$, adopt the cross-section coefficient map of Complete two-body phase space and coefficient flux map. Its scattering inputs are positive-shell energy vectors $p_i=(E_i,{\mathbf{p}}_i)$ with $E_i=\sqrt{\mathrm{\mid }{\mathbf{p}}_i{\mathrm{\mid }}^2+{\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}^2}>0$, ${\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}\ge 0$, and common future-timelike total $P=p_1+p_2=k_1+k_2$. The outgoing $k_j$ are on their specified positive shells. Require $s=h(P,P)>({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2$ for the outgoing rest energies and positive incoming flux $F>0$, precisely the supplier's admissible scattering domain. In the center-of-momentum frame it uses the invariant incoming flux $F$, the two-body measure $d{\mathrm{\Phi }}_2$, and the final-state factor $S_f=2$ for identical particles or $S_f=1$ for distinguishable particles, integrated over the full labeled sphere. When this coefficient is read phenomenologically, the initial preparation, channel, threshold, and approximation are part of the model statement. No universal error estimate for omitted terms is implied.

For the scalar examples, separately adopt the constant formal amplitudes $\mathcal{A}=-\lambda$ for the quartic contact channel and $\mathcal{G}=G_E$ for the cubic parent decay, with the units and external normalization stated at each use. The scalar scattering inputs lie on their stated positive shells with a common total and the same strict threshold and positive-flux conditions. For decay, the parent is $P_0=({\mathsf{M}}_0,\mathbf{0})$, ${\mathsf{M}}_0>{\mathsf{M}}_1+{\mathsf{M}}_2$, and its two positive-shell daughters sum to $P_0$. Their coefficient map is the one in Complete two-body phase space and coefficient flux map. If a rate $\mathrm{\Gamma }$ is reported as an energy width, the convention is explicitly ${\mathrm{\Gamma }}_E:=\mathrm{\hslash }\mathrm{\Gamma }$; it does not supply a survival law.

The readings above do not identify a formal coefficient with an exact continuum cross section, probability, decay rate, or measured estimate. In particular they do not construct an interacting continuum $S$-matrix or LSZ limit, establish convergence of a perturbation series, bound loop or soft-radiation effects, specify beam or detector cuts, resolve threshold resummation, or reproduce the statistical analysis of an experiment. A comparison with observations would require the actual apparatus, sampling, calibration, uncertainty, and radiative-inclusiveness conditions. The reported electron magnetic-moment measurements, including the Fan et al. report listed in the source references, remain source-reported empirical results and are not premises of this adoption.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Use the fixed Cartesian Minkowski chart $x=(ct,\mathbf{x})$, the geometric metric $\eta =(-,+,+,+)$, and the distinct algebraic contraction $h=-\eta =(+,-,-,-)$ of QFT example units, metric, gauge, and pairing conventions. Let $m,M\ge 0$ be the electron and muon rest energies in joules. In this chart, a future energy four-vector is $q=(E,\mathbf{q})\in {\mathbb{R}}^4$ with $E>0$; it has rest energy $\mathsf{M}\ge 0$ when $h(q,q)={\mathsf{M}}^2$. All four-vector components are in joules, and $\mathbf{q}$ is $c$ times physical spatial momentum. Let $p_1,p_2$ be future electron and positron energy four-vectors with rest energy $m$, and $k_1,k_2$ future muon and antimuon energy four-vectors with rest energy $M$, satisfying $p_1+p_2=k_1+k_2=P$ and $s=h(P,P)>4\max (m^2,M^2)\mathrm{.}$ By Complete two-body phase space and coefficient flux map, choose a proper orthochronous $h$-Lorentz matrix $\mathrm{\Lambda }\in GL(4,\mathbb{R})$, meaning ${\mathrm{\Lambda }}^{T}h\mathrm{\Lambda }=h$, $\det \mathrm{\Lambda }=1$, and preservation of the future causal cone $C_{+}:=\{(E,\mathbf{q}):E>0,h((E,\mathbf{q}),(E,\mathbf{q}))\ge 0\}$, with $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$. Its boost construction supplies such a map because $P$ is future timelike. The center-of-momentum inertial chart has coordinates $x^{\mathrm{C}\mathrm{M}}=\mathrm{\Lambda }x$ on spacetime and corresponding four-vector components $q^{\mathrm{C}\mathrm{M}}=\mathrm{\Lambda }q$; from here on, four-vector components are in that chart, so $P=(\sqrt{s},\mathbf{0})$. The strict threshold makes ${\mathbf{p}}_1$ and ${\mathbf{k}}_1$ nonzero. Define the scattering angle $\theta \in [0,\pi ]$ by $\theta :=\arccos \text{\hspace{0.17em}⁣}\left(\frac{{\mathbf{p}}_1\cdot {\mathbf{k}}_1}{\mathrm{\mid }{\mathbf{p}}_1\mathrm{\mid }\mathrm{\mid }{\mathbf{k}}_1\mathrm{\mid }}\right),$ using the Euclidean inner product on ${\mathbb{R}}^3$. Write $g=\sqrt{4\pi \alpha }>0$, as in QFT example units, metric, gauge, and pairing conventions, and ${\beta }_e=\sqrt{1-\frac{4m^2}{s}},{\beta }_{\mu }=\sqrt{1-\frac{4M^2}{s}}\mathrm{.}$ For the selected formal one-photon tree coefficient, take equal incoherent weight $1\mathrm{/}4$ for the four initial spin-label pairs and sum over final spin labels. Take the selected muon model charge to be $-e$ as well. Up to an irrelevant common phase, its amplitude is ${\mathcal{M}}_{\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{e}}=\frac{g^2}{s}[\bar{v}(p_2){\gamma }^{a}u(p_1)][\bar{u}(k_1){\gamma }_{a}v(k_2)],$ where the brackets denote the product of the two currents. This selected formal amplitude adopts the current draft QFT-G02 line/vertex convention; it is not an exact amplitude of a constructed continuum theory.

Conditional on this amplitude adoption and Formal diagram coefficient-to-observable adoption, the spin-averaged squared coefficient is $\overline{\mathrm{\mid }{\mathcal{M}}_{\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{e}}{\mathrm{\mid }}^2}=g^4[3-{\beta }_e^2-{\beta }_{\mu }^2+{\beta }_e^2{\beta }_{\mu }^2{\cos }^2\theta ],$ The distinguishable-particle differential and total cross-section coefficients are $\frac{d{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}}{d\mathrm{\Omega }}=\frac{{\alpha }^2(\mathrm{\hslash }c{)}^2{\beta }_{\mu }}{4s{\beta }_e}[3-{\beta }_e^2-{\beta }_{\mu }^2+{\beta }_e^2{\beta }_{\mu }^2{\cos }^2\theta ],$ ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\pi {\alpha }^2(\mathrm{\hslash }c{)}^2}{s}\frac{{\beta }_{\mu }}{{\beta }_e}[3-{\beta }_e^2-{\beta }_{\mu }^2+\frac{1}{3}{\beta }_e^2{\beta }_{\mu }^2]\mathrm{.}$ For $\delta =4m^2\mathrm{/}s\le 1\mathrm{/}2$, the ratio of this differential coefficient to the coefficient at $m=0$ with the same $M,s,\theta$ belongs to $[1,1+4\delta ]$. This bound is only the incoming-electron-mass effect within the selected tree coefficient.

The on-shell equations also give $P_a\bar{v}(p_2){\gamma }^{a}u(p_1)=0,P_a\bar{u}(k_1){\gamma }^{a}v(k_2)=0.$ These are the two current contractions for this finite external-spinor calculation; they do not assert an all-orders or loop Ward theorem.

For the separate scalar conventions of Formal diagram coefficient-to-observable adoption, let $m_s\ge 0$ be a scalar rest energy, let the incoming $r_1,r_2$ and outgoing $l_1,l_2$ lie on the positive $m_s$-shell, meaning each $(E,\mathbf{q})$ has $E=\sqrt{\mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^2+m_s^2}>0$, and require $r_1+r_2=l_1+l_2=P_s$ with $s=h(P_s,P_s)>4m_s^2$. Evaluate in a proper orthochronous center-of-momentum chart taking $P_s$ to $(\sqrt{s},\mathbf{0})$ as in [F4]. Let $\lambda \in \mathbb{R}$ be the dimensionless constant quartic amplitude parameter. The identical-final full-sphere and distinguishable total coefficients are respectively ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{{\lambda }^2(\mathrm{\hslash }c{)}^2}{32\pi s},{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{i}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}}=\frac{{\lambda }^2(\mathrm{\hslash }c{)}^2}{16\pi s}\mathrm{.}$ For a parent at rest of energy ${\mathsf{M}}_0>2m_s$ and a constant identical daughter amplitude $G_E$ in joules, take the daughters on the positive $m_s$-shell with total $P_0=({\mathsf{M}}_0,\mathbf{0})$. The rate coefficient is ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\mathrm{\mid }{G}_E{\mathrm{\mid }}^2}{32\pi \mathrm{\hslash }{\mathsf{M}}_0}\sqrt{1-\frac{4m_s^2}{{\mathsf{M}}_0^2}}[{\mathrm{s}}^{-1}]\mathrm{.}$ If an energy width is named, its convention is ${\mathrm{\Gamma }}_E:=\mathrm{\hslash }{\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}$ in joules. None of these coefficient statements implies an exact survival law. The QFT-G02 postulate is a current draft carrier; its use here retains draft status and does not imply independent acceptance.

Assume Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as in the mathematical Dirac-matrix supplier chain. Also assume the adopted local Maxwell–Dirac action of Explicit charged Dirac/Maxwell local coupling model with electron charge $q=-e$, $e>0$, and rest mass $m>0$, with $q,e$ in coulombs and $m$ in kilograms. Use the oriented, time-oriented affine Minkowski spacetime and inertial chart $x=(ct,\mathbf{x})$ with $\eta =\mathrm{diag}(-1,1,1,1)$ of Quantum field quantities and compatible SI normalizations. On ${\mathbb{R}}^4$, the static lower potential is $A_{\mathrm{S}\mathrm{I},a}(x^0,\mathbf{x})=(0,\mathbf{A}(\mathbf{x}))$: it is independent of $x^0$, with $\varphi =0$ and $\mathbf{A}\in C^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{R}}^3)$ in SI units $\mathrm{T}\mathrm{m}$. Define $\mathbf{B}=\mathrm{\nabla }\times \mathbf{A}$ in tesla, ${\pi }_i=-i\mathrm{\hslash }{\mathrm{\partial }}_i-q{A}_i,H_D=c{\sum }_{i=1}^3{\alpha }_i{\pi }_i+\beta m{c}^2,$ as spatial differential expressions on $C^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{C}}^4)$. Let $\mathcal{D}=C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{C}}^4)$ be their common test domain; smooth coefficients make ${\pi }_i$ and $H_D$ map $\mathcal{D}$ into itself. Use the Dirac matrices and adjoint of Dirac Clifford matrices and energy projections and ${\mathrm{\Sigma }}_i=\mathrm{diag}({\sigma }_i,{\sigma }_i),S_i=\frac{\mathrm{\hslash }}{2}{\mathrm{\Sigma }}_i\mathrm{.}$ The matrices ${\alpha }_i,\beta ,{\mathrm{\Sigma }}_i$ are dimensionless, $S_i$ has units $\mathrm{J}\mathrm{s}$, ${\pi }_i$ has units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and $H_D$ has units joules. Let $\psi :{\mathbb{R}}^4\to {\mathbb{C}}^4$ be a smooth spacetime spinor field. For a compactly supported direction $\xi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^4;{\mathbb{C}}^4)$, holding $A$ fixed and using real $\epsilon$, define ${\delta }_{\xi }{S}_{\mathrm{c}\mathrm{l}}:={\frac{d}{d\epsilon }\mid }_{\epsilon =0}\mathrm{\hslash }{\int }_{{\mathbb{R}}^4}({\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}[\psi +\epsilon \xi ,A]-{\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}[\psi ,A])d^{4}x,$ where ${\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}$ is the adopted density in Explicit charged Dirac/Maxwell local coupling model. The integrand has compact support, so this variation is finite. Assume ${\delta }_{\xi }{S}_{\mathrm{c}\mathrm{l}}=0$ for every such $\xi$. Then the compactly supported spinor-variation Euler–Lagrange equation of the adopted action holds pointwise as a differential equation. In the natural action normalization of Quantum field quantities and compatible SI normalizations, $\psi$ has engineering length degree $-3\mathrm{/}2$, and this equation is $i\mathrm{\hslash }{\mathrm{\partial }}_t\psi =H_D\psi$ for this pure magnetic specialization. Independently of this condition on $\psi$, on $\mathcal{D}$ the spatial differential expression satisfies $H_D^2=m^2{c}^4{I}_4+c^2({\mathit{\pi }}^2-q\mathrm{\hslash }\mathbf{\Sigma }\cdot \mathbf{B})\mathrm{.}$ Writing the formal first-order square-root coefficient as $m{c}^2+K\mathrm{/}(2m)$, where $K={\mathit{\pi }}^2-q\mathrm{\hslash }\mathbf{\Sigma }\cdot \mathbf{B}$, the spin term is $-(q\mathrm{/}m)\mathbf{S}\cdot \mathbf{B}$. Under the explicitly chosen magnetic-energy convention $H_{\mathrm{m}\mathrm{a}\mathrm{g}}=-{\mathit{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{g}}\cdot \mathbf{B},{\mathit{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{g}}=g_s\frac{q}{2m}\mathbf{S},$ where $H_{\mathrm{m}\mathrm{a}\mathrm{g}}$ is an energy-valued matrix coefficient in joules, ${\mathit{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{g}}$ is a $4\times 4$ matrix-valued magnetic moment coefficient with units $\mathrm{A}{\mathrm{m}}^2$, and $g_s$ is dimensionless. Comparing the coefficient of $\mathbf{S}\cdot \mathbf{B}$ as a linear-form coefficient gives $g_s=2$ for this minimal tree coefficient. The Dirac equation is asserted only for fields satisfying the stated stationarity condition. The square and coefficient are formal on the stated test domain; no magnetic self-adjoint closure, full spectrum, radiative correction, or empirical measurement is asserted.