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000001 | \subsection{Renormalized, physical, and scheme readout layers}
000002 | 
000003 | \begin{definition}[Admissible effective average action]
000004 | \label{def:admissible-effective-average-action}
000005 | Let $R_k$ be an admissible infrared regulator commuting with the structural sector selection.
000006 | Define
000007 | \[
000008 | e^{W_k[J]}
000009 | :=
000010 | \int_{\mathcal H_{\mathrm{adm}}}\mathcal D\varphi\,
000011 | \exp\!\Bigl(
000012 | -S_{\mathrm{adm}}[\varphi]
000013 | -\frac12\langle \varphi,R_k\varphi\rangle
000014 | +\langle J,\varphi\rangle
000015 | \Bigr),
000016 | \]
000017 | and set
000018 | \[
000019 | \Gamma_k[\phi]
000020 | :=
000021 | \sup_J\{\langle J,\phi\rangle-W_k[J]\}
000022 | -\frac12\langle \phi,R_k\phi\rangle.
000023 | \]
000024 | The renormalized admissible effective action is
000025 | \[
000026 | \Gamma_{\mathrm{TFPT}}^{\mathrm{ren}}:=\lim_{k\to0}\Gamma_k.
000027 | \]
000028 | \end{definition}
000029 | 
000030 | \begin{theorem}[Exact admissible renormalization-group flow]
000031 | \label{thm:exact-admissible-rg-flow}
000032 | The admissible effective average action satisfies the exact functional flow equation
000033 | \[
000034 | \partial_t \Gamma_k[\phi]
000035 | =
000036 | \frac12\,
000037 | \operatorname{STr}_{\mathcal H_{\mathrm{adm}}}
000038 | \Bigl[
000039 | \bigl(\Gamma_k^{(2)}[\phi]+R_k\bigr)^{-1}
000040 | \partial_t R_k
000041 | \Bigr],
000042 | \qquad
000043 | t:=\log k.
000044 | \]
000045 | Thus the running couplings of the admissible theory are generated dynamically on $\Gamma_k$, not by
000046 | the static projector $\Padm$.
000047 | \end{theorem}
000048 | 
000049 | \begin{proof}
000050 | Differentiate the Legendre transform defining $\Gamma_k$ at fixed mean field $\phi$. Using
000051 | \[
000052 | \frac{\delta^2 W_k}{\delta J\,\delta J}
000053 | =
000054 | \bigl(\Gamma_k^{(2)}+R_k\bigr)^{-1},
000055 | \]
000056 | one obtains
000057 | \[
000058 | \partial_t \Gamma_k
000059 | =
000060 | \frac12 \operatorname{STr}
000061 | \bigl[
000062 | \bigl(\Gamma_k^{(2)}+R_k\bigr)^{-1}\partial_t R_k
000063 | \bigr].
000064 | \]
000065 | The supertrace is taken on the admissible physical sector because the regulated functional integral
000066 | itself is defined there.
000067 | \end{proof}
000068 | 
000069 | \begin{corollary}[One-loop shadow of the exact admissible flow]
000070 | \label{cor:one-loop-shadow-exact-flow}
000071 | On the canonical branch with three families and one Higgs doublet, the exact admissible flow has
000072 | the perturbative shadow
000073 | \[
000074 | \partial_t g_i^{-2}(k)=-\frac{b_i}{8\pi^2}+O(g_i^0),
000075 | \qquad
000076 | (b_1,b_2,b_3)=\left(\frac{41}{10},-\frac{19}{6},-7\right),
000077 | \]
000078 | with $g_1$ in the standard hypercharge normalization. Equivalently,
000079 | \[
000080 | \partial_t \alpha_1^{-1}(k)=-\frac{41}{20\pi}+O(\alpha_1).
000081 | \]
000082 | \end{corollary}
000083 | 
000084 | \begin{proof}
000085 | Expanding \Cref{thm:exact-admissible-rg-flow} around the Gaussian region yields the standard
000086 | one-loop coefficients determined by the field content of the canonical branch. The carrier and
000087 | family theorems fix exactly three families and one light Higgs doublet, so the resulting shadow
000088 | coefficients are the displayed Standard Model values.
000089 | \end{proof}
000090 | 
000091 | \begin{corollary}[Infrared boundary condition for the electromagnetic coupling]
000092 | \label{cor:em-ir-boundary-condition}
000093 | Let $\alpha_\ast$ be the exact positive solution of the admissible electromagnetic closure equation.
000094 | Then
000095 | \[
000096 | \alpha_{\mathrm{em}}(0)=\alpha_\ast
000097 | \]
000098 | is the infrared boundary value of the exact admissible flow. For $\mu>0$ one has
000099 | \[
000100 | \alpha_{\mathrm{em}}^{-1}(\mu)
000101 | =
000102 | \alpha_\ast^{-1}
000103 | -\int_0^{\log\mu}\beta_{\alpha^{-1}}(t)\,dt
000104 | -\Delta_{\mathrm{thr}}(\mu),
000105 | \]
000106 | where $\Delta_{\mathrm{thr}}$ records electroweak mixing and threshold decoupling.
000107 | \end{corollary}
000108 | 
000109 | \begin{proof}
000110 | \Cref{thm:alpha-canonical} fixes the unique branch value $\alpha_\ast$ at the infrared endpoint.
000111 | The exact flow of \Cref{thm:exact-admissible-rg-flow} then propagates that value to finite scales,
000112 | while threshold effects are collected in the declared correction term $\Delta_{\mathrm{thr}}$.
000113 | \end{proof}
000114 | 
000115 | \begin{theorem}[Admissible 1PI graph expansion]
000116 | \label{thm:admissible-1pi-graph-expansion}
000117 | Around any admissible background field $\phi$, the effective action admits the loop expansion
000118 | \[
000119 | \Gamma_k[\phi]
000120 | =
000121 | S_{\mathrm{adm}}[\phi]
000122 | +\frac{\hbar}{2}\operatorname{STr}\log\!\bigl(S_{\mathrm{adm}}^{(2)}[\phi]+R_k\bigr)
000123 | +\sum_{\ell\ge 2}\hbar^\ell\Gamma_{\ell,k}^{\mathrm{1PI}}[\phi].
000124 | \]
000125 | Each coefficient $\Gamma_{\ell,k}^{\mathrm{1PI}}$ is the sum of admissible one-particle-irreducible
000126 | Feynman graphs with propagator
000127 | \[
000128 | G_{k,\phi}:=\bigl(S_{\mathrm{adm}}^{(2)}[\phi]+R_k\bigr)^{-1}
000129 | \]
000130 | and vertices given by the higher functional derivatives $S_{\mathrm{adm}}^{(n)}[\phi]$.
000131 | \end{theorem}
000132 | 
000133 | \begin{proof}
000134 | Set $\varphi=\phi+\sqrt{\hbar}\eta$ and expand
000135 | \[
000136 | S_{\mathrm{adm}}[\phi+\sqrt{\hbar}\eta]
000137 | =
000138 | S_{\mathrm{adm}}[\phi]
000139 | +\frac{\hbar}{2}\langle \eta,S_{\mathrm{adm}}^{(2)}[\phi]\eta\rangle
000140 | +\sum_{n\ge 3}\frac{\hbar^{n/2}}{n!}S_{\mathrm{adm}}^{(n)}[\phi](\eta^n).
000141 | \]
000142 | The regulated functional integral therefore becomes a Gaussian measure with covariance
000143 | $G_{k,\phi}$ dressed by higher interaction vertices. Taking the logarithm produces the connected
000144 | diagram expansion, and the Legendre transform removes one-particle-reducible pieces. The resulting
000145 | coefficients are exactly the admissible 1PI graph sums recorded above.
000146 | \end{proof}
000147 | 
000148 | \begin{definition}[Canonical renormalized 1PI package]
000149 | \label{def:renormalized-low-energy-package}
000150 | Let $\GrenTFPT$ denote the category whose objects are renormalized low-energy packages
000151 | \[
000152 | \mathcal G=
000153 | \bigl(
000154 | \{\Sigma_f^{\mathrm{TFPT}}(p)\}_f,\,