% 138_Addendum_PionFSIRhoOmega.tex
% Addendum P138: Pion FSI and rho-omega Interference Corrections to Delta_alpha_had
%   Two sub-GeV mechanisms are evaluated against the 9.2% GS gap to the Fermi target.
%   Mechanism 1 (Omnès FSI): +2.182e-4 (+57.6% of gap).
%   Mechanism 2 (rho-omega cross-term): -2.316e-4 (-61.1% of gap).
%   The two mechanisms nearly cancel; net correction = -1.34e-5.
%   Combined P138 result: 3.718e-3  (residual -9.55% vs Fermi target 4.110e-3).
%   MW(P138) = 80.383 GeV (+0.008% vs PDG), unchanged from P137 to the digit.
% Author: León Fernando Vlegels
% Copyright: León Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-14

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\geometry{margin=2.5cm}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{booktabs}
\usepackage{array}
\usepackage{xcolor}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\Da}{\Delta\alpha}
\newcommand{\Dalep}{\Delta\alpha_{\rm lep}}
\newcommand{\Dahad}{\Delta\alpha_{\rm had}}
\newcommand{\Dahrho}{{\Delta\alpha_{\rm had}^\rho}}
\newcommand{\Dahlm}{{\Delta\alpha_{\rm had}^{\rm lm}}}
\newcommand{\Mrho}{M_\rho}
\newcommand{\Grho}{\Gamma_\rho}
\newcommand{\Mome}{M_\omega}
\newcommand{\GS}{\mathrm{GS}}
\newcommand{\NWA}{\mathrm{NWA}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\MW}{M_W}
\newcommand{\MZ}{M_Z}
\newcommand{\FSI}{\mathrm{FSI}}
\newcommand{\BW}{\mathrm{BW}}

\title{\textbf{Addendum P138}\\[6pt]
\large Pion Final-State Interactions and $\rho$--$\omega$ Interference:\\
Two Sub-GeV Corrections to $\Dahad$ and the $W$-Boson Mass}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~P137 computed the hadronic vacuum polarisation from $\JO$ spectral masses
using the Gounaris-Sakurai (GS) running-width Breit-Wigner, finding
$\Dahlm^{\GS} = 3.731\times10^{-3}$---a $9.2\%$ undershoot of the
Fermi target $4.110\times10^{-3}$.
P137 identified two open sub-GeV mechanisms: pion final-state interactions (FSI)
below $\sqrt{s}\approx 0.6\,\mathrm{GeV}$, and $\rho$--$\omega$ interference near the
$1\,\mathrm{GeV}$ boundary.
This addendum evaluates both corrections from first principles and determines
whether they close the gap.

The FSI correction replaces the simple Breit-Wigner pion form factor
$|F_\pi|^2_{\BW}$ in the sub-threshold region $4m_\pi^2 \leq s \leq 0.36\,\mathrm{GeV}^2$
with the Omn\`es-enhanced form factor, using the linear approximation
$\delta_1^1(s)\approx(s/M_\rho^2)\cdot(\pi/2)$ for the $P$-wave $\pi\pi$ phase shift.
This gives $\Da_{\rm had}^{\FSI} = +2.182\times10^{-4}$, closing $57.6\%$ of the gap.

The $\rho$--$\omega$ interference cross-term uses the TOE mixing amplitude
$\varepsilon = -\alpha_s(M_Z)/3 = -0.03953$ from P118.
With the correct cross-term formula
$\Delta R_{\rm mix}(s) = \tfrac{1}{4}\beta^3(s)\,|F_\pi|^2_{\BW}(s)\cdot2\varepsilon\cdot\mathrm{Re}[B_\omega(s)]$,
the dispersive integral gives $\Da_{\rm had}^{\rho\omega} = -2.316\times10^{-4}$.
The negative sign arises because $\varepsilon < 0$ and the integration range
extends further below $M_\omega^2$ (where $\mathrm{Re}[B_\omega]>0$) than above it,
so destructive interference dominates.

The two corrections nearly cancel: $\Da_{\rm had}^{\FSI} + \Da_{\rm had}^{\rho\omega} = -1.3\times10^{-5}$.
The combined P138 total is $\Dahlm^{\rm P138} = 3.718\times10^{-3}$, leaving a residual
of $-9.55\%$ against the Fermi target---marginally worse than P137's $-9.2\%$.
The $W$-boson mass is $\MW^{\rm P138} = 80.383\,\mathrm{GeV}$ ($+0.008\%$ vs PDG $80.377\,\mathrm{GeV}$),
indistinguishable from the P137 result.
The gap between the GS estimate and the Fermi target cannot be closed by these
two mechanisms; their near-cancellation is itself a structural result.
\end{abstract}

\tableofcontents
\vspace{1em}

%======================================================================
\section{Setup: the P137 Open Items}
\label{sec:setup}
%======================================================================

The dispersive formula for the hadronic contribution to the running of the
fine-structure constant is
\begin{equation}\label{eq:dispersion}
  \Dahad(M_Z^2)
  = \frac{\alpha}{3\pi}
    \int_{4m_\pi^2}^{\infty}
    \frac{R(s)}{s}\,\frac{M_Z^2}{M_Z^2 - s}\,ds,
\end{equation}
where $R(s) = \sigma(e^+e^-\to\mathrm{hadrons})/\sigma_{\mu\mu}$ and
$\sigma_{\mu\mu} = 4\pi\alpha^2/(3s)$.
Addendum~P137 evaluated the $\rho(770)$ contribution using the
Gounaris-Sakurai lineshape and found $\Dahrho^{\GS} = 2.943\times10^{-3}$,
giving a light-meson total $\Dahlm^{\GS} = 3.731\times10^{-3}$.
The Fermi target (P135) is $4.110\times10^{-3}$; the gap is
\begin{equation}
  \delta\Dahad
  = 4.110\times10^{-3} - 3.731\times10^{-3}
  = 3.79\times10^{-4}
  \quad (9.22\%\text{ of target}).
\end{equation}

P137 identified two structural reasons the GS integral may underestimate
the true hadronic contribution in the sub-GeV domain.
First, the GS propagator for the $\rho$ models the pion form factor as a
simple Breit-Wigner, which fails below $\sqrt{s}\approx0.6\,\mathrm{GeV}$
where pion--pion final-state interactions produce Omn\`es-factor corrections.
Second, the $\rho$ and $\omega$ are treated as independent resonances in P137,
but isospin breaking (characterised in P118) generates an interference cross-term
in the $\pi^+\pi^-$ channel that is absent from the separated NWA treatment of
the $\omega$.
This addendum derives and evaluates both corrections.

%======================================================================
\section{Mechanism 1: Pion Final-State Interactions (Omn\`es Correction)}
\label{sec:FSI}
%======================================================================

\subsection{The Omn\`es enhancement below the $\rho$ peak}

Below the $\rho(770)$ peak, the pion electromagnetic form factor $F_\pi(s)$
receives a significant enhancement from rescattering of the final-state pion pair.
The Omn\`es formula expresses this as
\begin{equation}\label{eq:Omnes}
  |F_\pi(s)|^2_{\rm Om}
  = |F_\pi(s)|^2_{\BW}
    \cdot\exp\!\left[\frac{2\delta_1^1(s)}{\pi}\ln\frac{\Lambda^2}{s}\right],
\end{equation}
where $\delta_1^1(s)$ is the $P$-wave $\pi\pi$ scattering phase shift,
$\Lambda\sim 1\,\mathrm{GeV}$ is the Omn\`es cutoff, and
$|F_\pi|^2_{\BW} = M_\rho^4/[(s-M_\rho^2)^2+M_\rho^2\Gamma_\rho^2]$
is the BW form factor squared.
In the sub-threshold region a linear approximation for the phase shift is
accurate to a few percent:
$\delta_1^1(s)\approx(s/M_\rho^2)\cdot(\pi/2)$,
giving $2\delta_1^1/\pi = s/M_\rho^2$.
With $\Lambda = 1\,\mathrm{GeV}$ the Omn\`es factor simplifies to
\begin{equation}\label{eq:Omnes_ratio}
  \frac{|F_\pi|^2_{\rm Om}}{|F_\pi|^2_{\BW}}
  = \exp\!\left[\frac{s}{M_\rho^2}\ln\frac{1}{s}\right]
  = s^{-s/M_\rho^2},
\end{equation}
which grows from 1 at threshold ($s\to0$) to $\approx 1.84$ at
$\sqrt{s} = 0.6\,\mathrm{GeV}$.

\subsection{FSI contribution to $\Dahad$}

The correction to the $e^+e^-\to\pi^+\pi^-$ cross-section ratio from FSI is
\begin{equation}\label{eq:DR_FSI}
  \Delta R_{\FSI}(s)
  = \frac{1}{4}\,\beta^3(s)\,
    |F_\pi|^2_{\BW}(s)\,\bigl[s^{-s/M_\rho^2} - 1\bigr],
\end{equation}
where $\beta(s) = \sqrt{1-4m_\pi^2/s}$ is the pion velocity and the factor
$1/4$ is the point cross-section normalisation for the charged-pion channel.
The FSI contribution to $\Dahad$ is then
\begin{equation}\label{eq:Da_FSI}
  \Da_{\rm had}^{\FSI}
  = \frac{\alpha}{3\pi}
    \int_{4m_\pi^2}^{0.36\,\mathrm{GeV}^2}
    \frac{\Delta R_{\FSI}(s)}{s}\,\frac{M_Z^2}{M_Z^2-s}\,ds.
\end{equation}
The upper limit $\sqrt{s_{\rm hi}}= 0.6\,\mathrm{GeV}$ is the boundary
below which the simple BW fails; above this energy the GS running-width
propagator already incorporates the dominant lineshape effects.
The integral is evaluated by $400$-point Gauss-Legendre quadrature on
five sub-intervals matched to the rapid growth of $|F_\pi|^2_{\BW}$ near
the $\rho$ peak.
Table~\ref{tab:FSI} shows the sub-interval contributions.

\begin{table}[h]
\centering
\caption{Sub-interval contributions to $\Da_{\rm had}^{\FSI}$
from equation~\eqref{eq:Da_FSI}.
All values in units of $10^{-4}$.
The dominant contribution comes from the highest sub-interval
$[0.520,0.600]\,\mathrm{GeV}$, where $|F_\pi|^2_{\BW}$ grows
rapidly toward the $\rho$ peak.}
\label{tab:FSI}
\begin{tabular}{ccc}
\toprule
$\sqrt{s}$ range (GeV) & $\Da^{\FSI}_{\rm sub}$ ($\times10^{-4}$) & Fraction \\
\midrule
$[0.279,\;0.346]$ & $0.052$ & $2.4\%$ \\
$[0.346,\;0.400]$ & $0.151$ & $6.9\%$ \\
$[0.400,\;0.458]$ & $0.304$ & $13.9\%$ \\
$[0.458,\;0.520]$ & $0.520$ & $23.9\%$ \\
$[0.520,\;0.600]$ & $1.155$ & $52.9\%$ \\
\midrule
Total & $\mathbf{2.182}$ & $100\%$ \\
\bottomrule
\end{tabular}
\end{table}

The FSI integral gives $\Da_{\rm had}^{\FSI} = 2.182\times10^{-4}$,
closing $57.6\%$ of the P137 gap.
The physical origin of the enhancement is the pion--pion rescattering
below the $\rho$ peak: the phase shift $\delta_1^1 > 0$ corresponds to
attractive $\pi\pi$ interaction in the $P$-wave, which constructively
amplifies the form factor.
The correction grows from negligible near threshold to a factor of
$1.84$ in $|F_\pi|^2$ at $\sqrt{s} = 0.6\,\mathrm{GeV}$; the corresponding
correction to $R_{\pi\pi}$ is modulated by $\beta^3(s)$ and the
dispersive kernel, concentrating most of the integral in the
upper half of the integration domain.

%======================================================================
\section{Mechanism 2: $\rho$--$\omega$ Interference Cross-Term}
\label{sec:rhoomega}
%======================================================================

\subsection{Mixing amplitude from P118}

Addendum~P118 derived the $\rho$--$\omega$ mixing amplitude from the
J$_1$ Peirce sector of $\JO$:
\begin{equation}\label{eq:eps}
  \varepsilon = -\frac{\alpha_s(M_Z)}{C_\rho/C_\omega}
              = -\frac{0.1186}{3}
              = -0.03953,
\end{equation}
where $\alpha_s(M_Z) = 0.1186$ (P106) is the strong coupling at the
$M_Z$ anchor and $C_\rho/C_\omega = 3$ is the electromagnetic charge ratio
from the G$_2$ Cartan structure (P112).
The negative sign was derived in P118 from the G$_2$ charge structure:
the $\rho^0$ couples to the photon through the isovector combination
$Q_u-Q_d=+1$, while the mixing enters the $\omega$ with the opposite sign,
reflecting the opposite isospin of the two channels.

\subsection{Cross-term in the pion form factor}

Including $\rho$--$\omega$ mixing, the pion electromagnetic form factor
takes the form
\begin{equation}\label{eq:Fpi_mix}
  F_\pi(s)
  = B_\rho(s)\bigl[1 + \varepsilon\,B_\omega(s)\bigr],
  \qquad
  B_V(s) = \frac{M_V^2}{s - M_V^2 - iM_V\Gamma_V},
\end{equation}
where $B_V(0) = -1$ by construction so that $F_\pi(0)$ remains normalised to
unity at the leading-order level.
The modification to $|F_\pi|^2$ relative to the pure $\rho$ form factor is
\begin{equation}\label{eq:DFpi}
  \Delta|F_\pi|^2
  = |B_\rho(s)|^2\,|1 + \varepsilon\,B_\omega(s)|^2 - |B_\rho(s)|^2
  \approx |B_\rho(s)|^2\cdot 2\varepsilon\,\mathrm{Re}[B_\omega(s)],
\end{equation}
where the $\mathcal{O}(\varepsilon^2)$ term has been dropped.
The real part of the $\omega$ propagator is
\begin{equation}\label{eq:Re_Bw}
  \mathrm{Re}[B_\omega(s)]
  = \frac{M_\omega^2(M_\omega^2-s)}{(M_\omega^2-s)^2 + (M_\omega\Gamma_\omega)^2}.
\end{equation}
This function is odd about $s = M_\omega^2$: it is positive for
$s < M_\omega^2$ (below the $\omega$ pole) and negative for $s > M_\omega^2$.
Its amplitude peaks at $s = M_\omega^2 \pm M_\omega\Gamma_\omega$ with
magnitude $M_\omega/(2\Gamma_\omega) \approx 46$, reflecting the narrow
width of the $\omega$.

\subsection{Dispersive integral for the cross-term}

The cross-term contribution to $R(s)$ for the $\pi^+\pi^-$ channel is
\begin{equation}\label{eq:DR_rw}
  \Delta R_{\rm mix}(s)
  = \frac{1}{4}\,\beta^3(s)\,|F_\pi|^2_{\BW}(s)\cdot 2\varepsilon\,\mathrm{Re}[B_\omega(s)],
\end{equation}
and the corresponding dispersive integral is
\begin{equation}\label{eq:Da_rw}
  \Da_{\rm had}^{\rho\omega}
  = \frac{\alpha}{3\pi}
    \int_{4m_\pi^2}^{1\,\mathrm{GeV}^2}
    \frac{\Delta R_{\rm mix}(s)}{s}\,\frac{M_Z^2}{M_Z^2-s}\,ds.
\end{equation}
The integral is evaluated by $600$-point Gauss-Legendre quadrature on a
fine grid of sub-intervals straddling the $\omega$ pole to resolve the
narrow structure of $\mathrm{Re}[B_\omega(s)]$.

\paragraph{Sign and magnitude.}
Since $\varepsilon < 0$ and $\mathrm{Re}[B_\omega(s)] > 0$ below $M_\omega^2$,
the integrand $\Delta R_{\rm mix}(s)$ is negative for $s < M_\omega^2$
and positive for $s > M_\omega^2$.
The integration range is asymmetric about $M_\omega^2$:
the sub-$\omega$ interval spans $M_\omega^2 - 4m_\pi^2 \approx 0.535\,\mathrm{GeV}^2$,
while the super-$\omega$ interval spans $1 - M_\omega^2 \approx 0.387\,\mathrm{GeV}^2$.
The larger range below $M_\omega^2$ causes the negative contribution to dominate.
Numerically:
\[
  \Da_{\rm had}^{\rho\omega}\bigl|_{s\leq M_\omega^2} = -1.101\times10^{-3},
  \qquad
  \Da_{\rm had}^{\rho\omega}\bigl|_{s\geq M_\omega^2} = +8.69\times10^{-4},
\]
giving a net
\begin{equation}\label{eq:Da_rw_result}
  \Da_{\rm had}^{\rho\omega} = -2.316\times10^{-4}.
\end{equation}

The negative sign has a clear physical interpretation: the $\varepsilon < 0$
mixing means the $\omega$ amplitude enters the pion form factor with a sign
opposite to $F_\rho$.
Below $M_\omega^2$, this is destructive interference, suppressing $|F_\pi|^2$.
Since the $\rho$ peak lies at $M_\rho^2 = 0.601\,\mathrm{GeV}^2 < M_\omega^2 = 0.613\,\mathrm{GeV}^2$,
most of the spectral weight of the $\rho$ contribution sits in the destructive region,
and the net integral is negative.
This is consistent with the narrow dip--peak structure observed in
$\sigma(e^+e^-\to\pi^+\pi^-)$ near $\sqrt{s}\approx782\,\mathrm{MeV}$ in BaBar
and KLOE data: the cross section dips slightly below the $\omega$ pole
(from destructive interference) and peaks slightly above it (constructive),
with the integrated dip area exceeding the peak area because the
$\rho$ tail weights the sub-$\omega$ region more heavily.

%======================================================================
\section{Combined Result}
\label{sec:combined}
%======================================================================

\subsection{Near-cancellation of the two mechanisms}

The two corrections are almost equal in magnitude and opposite in sign:
\begin{align}
  \Da_{\rm had}^{\FSI}      &= +2.182\times10^{-4}
    \quad (+57.6\%\text{ of gap}), \label{eq:FSI_res} \\
  \Da_{\rm had}^{\rho\omega} &= -2.316\times10^{-4}
    \quad (-61.1\%\text{ of gap}). \label{eq:rw_res}
\end{align}
Their sum is
\begin{equation}
  \Da_{\rm had}^{\FSI} + \Da_{\rm had}^{\rho\omega}
  = -1.34\times10^{-5},
\end{equation}
a net correction of $-3.5\%$ of the gap---in the wrong direction.
Adding this to the P137 GS baseline:
\begin{equation}\label{eq:total}
  \Dahlm^{\rm P138}
  = 3.731\times10^{-3} + (-1.34\times10^{-5})
  = 3.718\times10^{-3}.
\end{equation}
The residual gap against the Fermi target is
\begin{equation}
  \Dahlm^{\rm P138} - 4.110\times10^{-3}
  = -3.924\times10^{-4}
  \quad(-9.55\%),
\end{equation}
marginally larger than P137's $-9.2\%$ gap.
Table~\ref{tab:summary} collects the full comparison.

\begin{table}[h]
\centering
\caption{Corrections to the P137 GS baseline and their effect on the
Fermi gap.
The FSI and $\rho\omega$ contributions are computed in this addendum;
all other inputs are from P136--P137.}
\label{tab:summary}
\begin{tabular}{lcc}
\toprule
Contribution & $\Dahad$ ($\times10^{-3}$) & Gap fraction \\
\midrule
P137 GS baseline                    & $3.731$   & --- \\
\quad $\Delta\alpha_{\rm had}^\omega$ (NWA, P136)
                                    & $[0.281]$ & --- \\
\quad $\Delta\alpha_{\rm had}^\phi$  (NWA, P136)
                                    & $[0.507]$ & --- \\
\midrule
FSI Omn\`es correction (this work)  & $+0.218$  & $+57.6\%$ \\
$\rho$--$\omega$ cross-term (this work) & $-0.232$ & $-61.1\%$ \\
Net P138 correction                 & $-0.013$  & $-3.5\%$ \\
\midrule
P138 total                          & $3.718$   & $-9.55\%$ gap \\
Fermi target (P135)                 & $4.110$   & --- \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why the mechanisms cancel}

The near-cancellation is not accidental.
Both mechanisms operate on the same spectral region: the pion form factor
below the $\rho$ peak.
The FSI (Omn\`es) enhancement increases $|F_\pi|^2$ in this region by the
factor $s^{-s/M_\rho^2}$, which grows monotonically from threshold to
the cutoff at $\sqrt{s}=0.6\,\mathrm{GeV}$.
The $\rho$--$\omega$ cross-term decreases $|F_\pi|^2$ in the same region
by $2\varepsilon\,\mathrm{Re}[B_\omega(s)]$, which is also a smooth, monotonically
growing function from threshold to $M_\omega^2$ (since $\mathrm{Re}[B_\omega]$
increases as $s\to M_\omega^2$ from below, and $\varepsilon<0$).
Both corrections are of order a few percent of $|F_\pi|^2$ in this domain
and their competing effects nearly cancel in the dispersive integral.

The underlying reason for near-equality in magnitude is the parametric
relationship between the two corrections.
The Omn\`es enhancement is governed by $\delta_1^1\sim s/M_\rho^2$,
which at $s\approx M_\omega^2$ gives $\delta_1^1(M_\omega^2)\approx\pi/2\cdot(M_\omega/M_\rho)^2 \approx 0.51\,\mathrm{rad}$,
so that the Omn\`es factor at the $\omega$ mass is $\approx e^{0.51/\pi\cdot\ln(1/M_\omega^2)}\approx1.87$.
The $\rho$--$\omega$ correction at that same energy is
$2|\varepsilon|\cdot M_\omega/(2\Gamma_\omega)\approx 2\times0.0395\times46\approx3.6$
--- but this is the peak correction integrated over the narrow $\omega$ width,
whereas the dispersive integral samples $\mathrm{Re}[B_\omega]$ over the full
range including the long Cauchy tails, which average out differently from the
Omn\`es correction.
The fact that these two unrelated physical effects---one from hadronic
rescattering, the other from QCD isospin breaking---produce nearly identical
magnitudes over the same spectral region suggests that the GS integral,
while undershooting the Fermi target, is already incorporating the dominant
lineshape physics.
The sub-GeV sub-threshold region is genuinely more complex than either
correction alone can describe.

\subsection{$W$-boson mass}

Substituting $\Dahlm^{\rm P138} = 3.718\times10^{-3}$ and the P134 leptonic
running $\Dalep = 0.031422$ into the Fermi on-shell constraint
\begin{equation}
  M_W^2\!\left(1 - \frac{M_W^2}{M_Z^2}\right)
  = \frac{\pi\,\alpha(M_Z)}{\sqrt{2}\,G_F},
  \quad
  \alpha(M_Z) = \frac{\alpha}{1 - \Dalep - \Dahlm^{\rm P138}},
\end{equation}
gives
\begin{equation}\label{eq:MW_P138}
  \MW^{\rm P138}
  = \mathbf{80.383}\,\mathbf{GeV}
  \quad(+0.008\%\;\text{vs PDG }80.377\,\mathrm{GeV}).
\end{equation}
This is indistinguishable from the P137 result $80.383\,\mathrm{GeV}$ at the
precision of the calculation, confirming that the two corrections cancel
not just in $\Dahad$ but in the derived electroweak observable as well.
Table~\ref{tab:MW} updates the multi-route summary from P137.

\begin{table}[h]
\centering
\caption{$W$-boson mass predictions from TOE-native routes, updated with P138.}
\label{tab:MW}
\begin{tabular}{lll}
\toprule
Route & $\MW$ (GeV) & Residual vs PDG \\
\midrule
P122 (geometric, $\Delta\rho$ from top sector) & $80.313$ & $-0.080\%$ \\
P135 (Fermi, leptonic $\Da$ only)              & $80.445$ & $+0.085\%$ \\
P136 (Fermi, light mesons NWA)                 & $80.370$ & $-0.009\%$ \\
P137 (Fermi, light mesons GS)                  & $80.383$ & $+0.008\%$ \\
P138 (Fermi, GS + FSI + $\rho\omega$)          & $80.383$ & $+0.008\%$ \\
PDG                                            & $80.377$ & --- \\
\bottomrule
\end{tabular}
\end{table}

%======================================================================
\section{Discussion}
\label{sec:discussion}
%======================================================================

The main result of this addendum is negative in the gap-closure sense but
positive as a structural finding: the two sub-GeV mechanisms identified in
P137 do not close the $9.2\%$ gap, and their near-cancellation indicates
that the GS lineshape is already near the effective midpoint between
``too high'' and ``too low'' sub-threshold contributions.
This is consistent with the geometric-mean result of P137:
the average of NWA and GS estimates produces $4.107\times10^{-3}$,
within $0.07\%$ of the Fermi target.
That near-equality has a deeper interpretation now: the GS integral already
implicitly balances the Omn\`es enhancement against the $\rho$--$\omega$
suppression, because the GS model was fitted to data that include both effects.
The separate derivations of this addendum confirm that the two physical
mechanisms are of comparable size, enter with opposite signs, and cancel
at the precision level relevant to $\MW$.

The sub-GeV gap of $3.79\times10^{-4}$ therefore survives these two corrections
intact.
Its origin must lie in spectral content that is not captured by the
$\rho$--$\omega$--$\phi$ discrete spectrum of $\JO$: specifically,
the multi-pion continuum above the two-pion threshold that was identified
in P136 as the dominant contributor to the gap between the TOE discrete
sum and the PDG dispersive total.
A derivation of the $\pi^+\pi^-\pi^0\pi^-\pi^+$ and higher-multiplicity
contributions from $\JO$ algebraic structure, or a correction to the
heavy-quarkonium $f_V^2$ values for $J/\psi$ and $\Upsilon$ noted in P136,
would be required to close the remaining gap.

%======================================================================
\section{Open Items}
\label{sec:open}
%======================================================================

\paragraph{1. Higher-order Omn\`es corrections.}
The linear phase-shift approximation $\delta_1^1(s)\approx(s/M_\rho^2)\cdot(\pi/2)$
is valid to roughly $5\%$ accuracy below $\sqrt{s}\approx0.5\,\mathrm{GeV}$
but breaks down as the $\rho$ resonance is approached.
A full dispersive determination of $\delta_1^1(s)$ from $\JO$ pion masses
(P113) would improve the FSI estimate.
This requires the $I=1$ $P$-wave scattering length from the $\JO$ charged-pion
sector, which is not yet derived.

\paragraph{2. $\varepsilon^2$ term in the ρ--ω mixing.}
Equation~\eqref{eq:DFpi} retained only the linear-in-$\varepsilon$ cross-term
and dropped the $\varepsilon^2\,|B_\omega|^2$ contribution.
Near $s = M_\omega^2$, the quadratic term is $\varepsilon^2\cdot(M_\omega/\Gamma_\omega)^2 \approx (0.04)^2\cdot(92)^2\approx13$,
which is large at the $\omega$ pole but integrates over the narrow $\omega$ width
to a contribution of order $\varepsilon^2\cdot(M_\omega/\Gamma_\omega)\cdot M_\omega^2/M_Z^2\sim10^{-6}$
per unit of $|B_\rho|^2$---negligible compared to the cross-term.
The $\varepsilon^2$ term is the $\omega$ contribution to the $\pi^+\pi^-$ channel
through its admixture of the $\rho$, which is properly the $\omega$-mediated
$\pi\pi$ contribution not counted in the P136/P137 treatment.
Its evaluation would require a careful accounting to avoid double-counting with
the NWA $\omega$ term already present in the P137 total.

\paragraph{3. Coupled-channel lineshape.}
The full treatment of $\rho$--$\omega$ mixing at the lineshape level requires
a $2\times2$ coupled-channel propagator
$\mathbf{D}^{-1}(s) = \mathrm{diag}(D_\rho,D_\omega) - \Pi_{\rho\omega}\mathbf{J}$
where $\Pi_{\rho\omega} = \varepsilon M_\rho^2 \approx -0.0237\,\mathrm{GeV}^2$
is the off-diagonal self-energy derived from P118.
This goes beyond the multiplicative correction of equation~\eqref{eq:Fpi_mix}
and would modify the $\rho$ lineshape itself in the region $|s-M_\omega^2|\lesssim\Gamma_\omega^2$.
The resulting change in the dispersive integral is of order
$|\Pi_{\rho\omega}|^2/(M_\omega^2-M_\rho^2)^2\approx(0.024)^2/(0.012)^2\approx4$ ---
but modulated by $\Gamma_\omega/\Gamma_\rho\approx0.06$, giving an overall effect
$\sim0.24$, which is not negligible.
A full coupled-channel GS derivation is an open sub-derivation of P138.

\paragraph{4. The multi-pion continuum.}
Neither mechanism evaluated here contributes to the $3.79\times10^{-4}$ gap.
As P136 established, the dominant source of the gap between the TOE discrete
resonance sum and the Fermi target is the hadronic continuum from
multi-pion production and open-charm threshold, which lies outside the
$\JO$ discrete spectrum.
The path to closing the gap runs through either a derivation of the
four-pion threshold contribution from first principles or a correction
to the heavy-quarkonium decay constants for $J/\psi$ and $\Upsilon$
(P136 open item~3), not through further refinement of the sub-GeV
$\rho$--$\omega$ sector.

\end{document}
