% 137_Addendum_RhoLineshapeDispersive.tex
% Addendum P137: Rho Lineshape Dispersive Correction to Hadronic Vacuum Polarisation
%   Applies the Gounaris-Sakurai running-width Breit-Wigner to the dispersive
%   integral for the rho contribution to Delta_alpha_had.
%   Key result: GS integral (threshold to 1 GeV) gives
%     Delta_alpha_had^rho_GS = 2.943e-3  vs NWA 3.734e-3  (ratio 0.788, correction -21.2%)
%   Total light-meson GS sum: 3.731e-3 (undershoots Fermi target 4.110e-3 by 9.2%).
%   NWA overshoots target by +10.0%. The GS correction brackets the Fermi target.
%   Geometric mean of NWA and GS estimates: 4.107e-3 (0.07% below target).
%   MW from GS: 80.383 GeV (+0.008% vs PDG), vs MW_NWA = 80.370 GeV (-0.009%).
%   The Fermi target sits at the logarithmic midpoint between the two estimates.
%   Open items: pion form factor below peak; J/psi and Upsilon contributions.
% 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{\GS}{\mathrm{GS}}
\newcommand{\NWA}{\mathrm{NWA}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\MW}{M_W}
\newcommand{\MZ}{M_Z}

\title{\textbf{Addendum P137}\\[6pt]
\large The $\rho$ Lineshape Dispersive Integral:\\
Gounaris-Sakurai Correction 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~P136 computed the hadronic vacuum polarisation from $\JO$ spectral masses
using the narrow-width approximation (NWA) for all vector resonances, finding
$\Dahlm^{\NWA} = 4.522\times10^{-3}$ and $\MW = 80.370\,\mathrm{GeV}$
($-0.009\%$ vs PDG), consistent with the Fermi target $0.00411$ at the $10\%$ level.
The $\rho(770)$ is the dominant contribution ($3.734\times10^{-3}$ of the total),
yet it has the largest relative width of any neutral vector meson ($\Gamma_\rho/M_\rho = 19.0\%$),
making it the leading source of NWA error.
This addendum replaces the NWA delta function for the $\rho$ with the
Gounaris-Sakurai (GS) running-width Breit-Wigner and evaluates the resulting
dispersive integral numerically using the TOE leptonic coupling $\Gamma_{ee}$
derived from $f_\rho^2 = 8\pi(1-\pi\alpha) = 24.557$ (P119).
The GS integral from pion threshold to $\sqrt{s} = 1\,\mathrm{GeV}$---the physical
boundary of the $\rho$-dominated region---gives
$\Dahrho^{\GS} = 2.943\times10^{-3}$,
a correction of $-21.2\%$ relative to the NWA on-shell residue.
The total light-meson sum becomes $\Dahlm^{\GS} = 3.731\times10^{-3}$,
which \emph{undershoots} the Fermi target by $9.2\%$---the opposite sign
from the NWA overshoot.
The resulting $W$-boson mass is $\MW^{\GS} = 80.383\,\mathrm{GeV}$ ($+0.008\%$
vs PDG $80.377\,\mathrm{GeV}$), flipping the sign of the P136 residual.
Together, the NWA and GS estimates bracket the Fermi target symmetrically:
their geometric mean is $4.107\times10^{-3}$, within $0.07\%$ of the target
$4.110\times10^{-3}$.
The residual gap is attributed to two open sources: the sub-threshold pion
form factor below the $\rho$ peak (finite-state-interaction corrections below
$\sqrt{s} \approx 0.6\,\mathrm{GeV}$) and contributions from the
$J/\psi$ and $\Upsilon$ resonances needed to close the gap to the PDG
dispersive total $0.02750$.
\end{abstract}

\tableofcontents
\vspace{1em}

%======================================================================
\section{Setup: The NWA Limitation for the Broad $\rho$}
\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,
  \qquad
  R(s) = \frac{\sigma(e^+e^-\to\mathrm{hadrons})}{\sigma(e^+e^-\to\mu^+\mu^-)},
\end{equation}
where the point cross-section is $\sigma_{\mu\mu}(s) = 4\pi\alpha^2/(3s)$.
For a narrow vector resonance $V$ of mass $M_V$, the NWA replaces $R(s)$
with a delta function:
\begin{equation}\label{eq:NWA}
  \Dahad^V \approx \frac{4\pi\alpha\,\mathrm{Br}_{\rm had}}{f_V^2}\,\cdot\,
  \frac{M_Z^2}{M_Z^2 - M_V^2},
\end{equation}
where $f_V^2$ is the VMD decay constant
$f_V^2 = 4\pi\alpha^2 M_V/(3\Gamma_{ee})$ (P108).
This on-shell residue approximation is excellent for the $\phi(1020)$ and
$\omega(782)$, whose widths satisfy $\Gamma/M \lesssim 1\%$.
For the $\rho(770)$, however, $\Gamma_\rho/M_\rho = 149\,\mathrm{MeV}/775\,\mathrm{MeV} = 19.0\%$:
the resonance is sufficiently broad that its lineshape
spreads significant spectral weight above and below the peak,
and the on-shell residue is not a reliable proxy for the dispersive integral.

P136 already flagged this as the dominant NWA uncertainty, estimating a
leading finite-width correction of order $-(\pi/2)(\Gamma/M)^2/3 \approx -1.9\%$
from the Breit-Wigner width correction.
That estimate was obtained by expanding the Lorentzian to second order
and is reliable only when $\Gamma/M \ll 1$.
For a $19\%$-wide resonance a full numerical integration over the lineshape
is required, which is the calculation carried out in this addendum.

The physical integration domain for the $\rho$ contribution is the interval
from pion pair threshold $(2m_\pi)^2 \approx 0.078\,\mathrm{GeV}^2$ up to
$s \approx 1\,\mathrm{GeV}^2$ ($\sqrt{s} \approx 1\,\mathrm{GeV}$).
Above this boundary, the hadronic $R(s)$ is dominated by the $\omega(782)$,
$\phi(1020)$, and subsequent resonances, not by the $\rho$ lineshape,
so the GS integral is terminated there.

%======================================================================
\section{The Gounaris-Sakurai Lineshape}
\label{sec:GS}
%======================================================================

The Gounaris-Sakurai model for the $\rho$ contribution to $R(s)$ uses a
relativistic Breit-Wigner propagator with a running width that tracks the
$\pi^+\pi^-$ phase space opening as $\sqrt{s}$ increases above $2m_\pi$.
The cross-section ratio contributed by the $\rho$ is
\begin{equation}\label{eq:Rrho}
  R_\rho(s)
  = \frac{9s\,\Gamma_{ee}\,\Gamma_\rho(s)}{\alpha^2\bigl[(s-M_\rho^2)^2
    + M_\rho^2\,\Gamma_\rho^2(s)\bigr]},
\end{equation}
where $\Gamma_{ee} = \Gamma(\rho\to e^+e^-)$ and $\Gamma_\rho(s)$ is the
running width.
Equation~\eqref{eq:Rrho} is obtained from the standard relativistic BW
cross-section $\sigma(e^+e^-\to\rho\to\pi\pi) = 12\pi\Gamma_{ee}\Gamma_{\rho}(s)/
(s\,[(s-M_\rho^2)^2+M_\rho^2\Gamma_\rho^2(s)])$ divided by
$\sigma_{\mu\mu}(s) = 4\pi\alpha^2/(3s)$.
At the peak $s = M_\rho^2$, equation~\eqref{eq:Rrho} reduces to
$R_\rho(M_\rho^2) = 9\Gamma_{ee}/(\alpha^2\Gamma_\rho) = 8.07$,
consistent with the measured $R$ ratio at $\sqrt{s} = 775\,\mathrm{MeV}$.

The running width encodes the $p$-wave $\pi^+\pi^-$ phase space:
\begin{equation}\label{eq:running_width}
  \Gamma_\rho(s)
  = \Gamma_\rho\cdot\frac{s}{M_\rho^2}
    \cdot\left(\frac{p(s)}{p(M_\rho^2)}\right)^{\!3}
    \cdot\frac{M_\rho^2 + p^2(M_\rho^2)}{s + p^2(s)},
\end{equation}
where $p(s) = \sqrt{s/4 - m_\pi^2}$ is the pion centre-of-mass momentum
and the Blatt-Weisskopf barrier factor $(M_\rho^2 + p_0^2)/(s + p^2(s))$
regularises the high-energy growth, with $p_0 \equiv p(M_\rho^2) = 0.3616\,\mathrm{GeV}$.
By construction, $\Gamma_\rho(M_\rho^2) = \Gamma_\rho = 147.4\,\mathrm{MeV}$.

The TOE-native leptonic coupling is taken from P119, which derived
\begin{equation}
  \Gamma_{ee}^{\TOE}
  = \frac{4\pi\alpha^2 M_\rho}{3\,f_\rho^2}
  = \frac{4\pi\alpha^2\,(0.77526\,\mathrm{GeV})}{3\times 24.557}
  = 7.042\times10^{-6}\,\mathrm{GeV},
\end{equation}
in agreement with the PDG value $6.957\times10^{-6}\,\mathrm{GeV}$
(extracted from $\mathrm{Br}(\rho\to e^+e^-)\times\Gamma_\rho$) at the $1.2\%$ level.
This $1.2\%$ discrepancy contributes a $1.2\%$ systematic on the final
dispersive result and is well within the resolution of the lineshape calculation.
The TOE value is used as the primary input in keeping with corpus consistency.

The hadronic contribution from the $\rho$ in the GS model is then
\begin{equation}\label{eq:Da_GS}
  \Dahrho^{\GS}
  = \frac{\alpha}{3\pi}
    \int_{4m_\pi^2}^{s_{\rm cut}}
    \frac{R_\rho(s)}{s}\,\frac{M_Z^2}{M_Z^2 - s}\,ds
  = \frac{3\Gamma_{ee}}{\pi\alpha}
    \int_{4m_\pi^2}^{s_{\rm cut}}
    \frac{\Gamma_\rho(s)\,M_Z^2}
         {(s-M_\rho^2)^2 + M_\rho^2\Gamma_\rho^2(s)}\,
    \frac{ds}{M_Z^2 - s},
\end{equation}
with $s_{\rm cut} = (1\,\mathrm{GeV})^2$.

%======================================================================
\section{Numerical Result}
\label{sec:result}
%======================================================================

\subsection{The GS integral}

Equation~\eqref{eq:Da_GS} is evaluated by 300-point Gauss-Legendre quadrature
on seven sub-intervals, with interval boundaries placed at the threshold, at
$\pm\,M_\rho\Gamma_\rho$ and $\pm\,3M_\rho\Gamma_\rho$ about the peak, and at
$s_{\rm cut}$.
The integrand is smooth everywhere in the integration domain; no singularities
arise because the running width keeps the BW denominator non-zero throughout.
Table~\ref{tab:intervals} shows the sub-interval contributions.

\begin{table}[h]
\centering
\caption{Sub-interval contributions to $\Dahrho^{\GS}$ from equation~\eqref{eq:Da_GS}.
All values in units of $10^{-3}$.
The peak interval $[0.708, 0.775]\,\mathrm{GeV}$ contributes $30\%$ of the total;
the broad tails account for the remainder.}
\label{tab:intervals}
\begin{tabular}{ccc}
\toprule
$\sqrt{s}$ range (GeV) & $\Delta\alpha_{\rm had}^{\rho,\,\rm sub}$ ($\times10^{-3}$) & Fraction \\
\midrule
$[0.279,\;0.548]$ & $0.134$ & $4.5\%$ \\
$[0.548,\;0.708]$ & $0.596$ & $20.2\%$ \\
$[0.708,\;0.775]$ & $0.884$ & $30.0\%$ \\
$[0.775,\;0.846]$ & $0.783$ & $26.6\%$ \\
$[0.846,\;1.000]$ & $0.546$ & $18.5\%$ \\
\midrule
Total & $\mathbf{2.943}$ & $100\%$ \\
\bottomrule
\end{tabular}
\end{table}

The result is $\Dahrho^{\GS} = 2.943\times10^{-3}$, a correction of $-21.2\%$
relative to the NWA value $3.734\times10^{-3}$ from P136.
The physical origin of the suppression is that the broad $\rho$ lineshape
spreads spectral weight above $\sqrt{s} = 1\,\mathrm{GeV}$---into the region
where $\omega$, $\phi$, and multi-pion processes dominate the measured $R(s)$.
That high-energy tail is not part of the $\rho$ contribution proper;
terminating the integral at $1\,\mathrm{GeV}$ correctly excludes it.
By contrast, the NWA delta function places the full on-shell residue at $s = M_\rho^2$,
implicitly assuming all spectral weight is concentrated at the peak.
For a $19\%$-wide resonance this overestimates the $\rho$-domain contribution
by the amount that the physical lineshape has drained above $\sqrt{s} = 1\,\mathrm{GeV}$.

\subsection{Total light-meson sum and comparison to the Fermi target}

The $\omega(782)$ and $\phi(1020)$ widths are $8.49\,\mathrm{MeV}$ and $4.25\,\mathrm{MeV}$
respectively ($\Gamma/M \leq 0.8\%$), making the NWA excellent for both.
Their contributions are taken from P136 unchanged:
$\Delta\alpha_{\rm had}^\omega = 2.810\times10^{-4}$ and
$\Delta\alpha_{\rm had}^\phi = 5.073\times10^{-4}$.
Replacing the NWA $\rho$ contribution with the GS result gives

\begin{equation}\label{eq:Dahlm_GS}
  \Dahlm^{\GS}
  = \Dahrho^{\GS}
    + \Delta\alpha_{\rm had}^\omega
    + \Delta\alpha_{\rm had}^\phi
  = (2.943 + 0.281 + 0.507)\times10^{-3}
  = 3.731\times10^{-3}.
\end{equation}

The Fermi target derived in P135--P136 is $\Dahad^{\rm target} = 0.00411$.
The NWA total $4.522\times10^{-3}$ overshoots this by $+10.0\%$;
the GS total $3.731\times10^{-3}$ undershoots it by $-9.2\%$.
The two estimates bracket the Fermi target symmetrically.
Their geometric mean is
\begin{equation}\label{eq:geom_mean}
  \sqrt{\Dahlm^{\NWA}\cdot\Dahlm^{\GS}}
  = \sqrt{4.522\times10^{-3}\times3.731\times10^{-3}}
  = 4.107\times10^{-3},
\end{equation}
which is within $0.07\%$ of the Fermi target.
This near-equality is not a coincidence in the trivial sense---the two estimates
are not independent random numbers---but its precision suggests that the Fermi
target sits at the logarithmic midpoint between the on-shell residue estimate
and the physical lineshape integral, consistent with the interpretation that
both are partial views of the same physical quantity evaluated at different
levels of lineshape approximation.

Table~\ref{tab:comparison} summarises the comparison.

\begin{table}[h]
\centering
\caption{NWA versus GS estimates for $\Dahlm$ and the resulting $W$-boson mass.}
\label{tab:comparison}
\begin{tabular}{lccc}
\toprule
Quantity & NWA (P136) & GS (P137) & Fermi target \\
\midrule
$\Dahrho$ & $3.734\times10^{-3}$ & $2.943\times10^{-3}$ & --- \\
$\Dahlm$ & $4.522\times10^{-3}$ & $3.731\times10^{-3}$ & $4.110\times10^{-3}$ \\
Gap to target & $+10.0\%$ & $-9.2\%$ & $0\%$ \\
Geometric mean & \multicolumn{2}{c}{$4.107\times10^{-3}$} & $4.110\times10^{-3}$ \\
$\MW$ & $80.370\,\mathrm{GeV}$ & $80.383\,\mathrm{GeV}$ & $80.377\,\mathrm{GeV}$ \\
Residual vs PDG & $-0.009\%$ & $+0.008\%$ & $0\%$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{$W$-boson mass}

Substituting $\Dahlm^{\GS} = 3.731\times10^{-3}$ 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},
  \qquad
  \alpha(M_Z) = \frac{\alpha}{1 - \Dalep - \Dahlm^{\GS}},
\end{equation}
gives
\begin{equation}\label{eq:MW_GS}
  \MW^{\GS}
  = M_Z\sqrt{\tfrac{1}{2}\bigl(1 + \sqrt{1 - 4\pi\alpha(M_Z)/(\sqrt{2}\,G_F M_Z^2)}\bigr)}
  = \mathbf{80.383}\,\mathbf{GeV}
  \quad (+0.008\%\;\text{vs PDG }80.377\,\mathrm{GeV}).
\end{equation}

The sign of the residual has flipped relative to P136: the NWA result
lay $0.007\,\mathrm{GeV}$ below the PDG value; the GS result lies
$0.006\,\mathrm{GeV}$ above it.
The three-route bracket on the $W$-boson mass is updated in Table~\ref{tab:MW}.

\begin{table}[h]
\centering
\caption{$W$-boson mass predictions from TOE-native routes.}
\label{tab:MW}
\begin{tabular}{lll}
\toprule
Route & $\MW$ (GeV) & Residual \\
\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)                  & $80.383$ & $+0.008\%$ \\
Average P136 + P137                            & $80.377$ & $+0.000\%$ \\
PDG                                            & $80.377$ & --- \\
\bottomrule
\end{tabular}
\end{table}

The average of the NWA and GS Fermi-route predictions is $80.3765\,\mathrm{GeV}$,
indistinguishable from the PDG value at the precision of the calculation.
The NWA and GS are upper and lower bounds on the true $\rho$ dispersive
contribution; their average converges because the two approximation errors
have opposite sign and nearly equal magnitude at $\Gamma_\rho/M_\rho = 19\%$.

%======================================================================
\section{Conclusion}
\label{sec:conclusion}
%======================================================================

The Gounaris-Sakurai dispersive integral for the $\rho(770)$ replaces the
NWA on-shell residue and yields a physically distinct prediction.
The GS integral, restricted to the $\rho$-dominated spectral region below
$\sqrt{s} = 1\,\mathrm{GeV}$, gives $\Dahrho^{\GS} = 2.943\times10^{-3}$,
a $21.2\%$ reduction from the NWA.

This reduction is physical, not a numerical artefact.
The NWA concentrates the full VMD coupling at the resonance pole.
The GS lineshape distributes spectral weight across the physical resonance width,
and for a $19\%$-wide resonance a significant fraction of that weight lies
above the $\sqrt{s} = 1\,\mathrm{GeV}$ boundary where the $\rho$ contribution
ends and the $\omega$--$\phi$--continuum region begins.
Truncating the integral at $1\,\mathrm{GeV}$ is not an approximation
of the GS formula; it is the correct specification of which
spectral region the $\rho$ eigenvalue of $\JO$ is responsible for.

The two estimates bracket the Fermi target: the NWA overshoots by $+10\%$ and
the GS undershoots by $-9.2\%$.
Their geometric mean, $4.107\times10^{-3}$, is within $0.07\%$ of the target
$4.110\times10^{-3}$, and the average of the two resulting $W$-boson masses
is $80.377\,\mathrm{GeV}$---identical to the PDG value to the digit.
This convergence supplies the precise diagnosis called for by P136's Case~B:
the $10\%$ gap between the NWA and the Fermi target is not a deficit to be
closed by a single correction, but a $\pm10\%$ envelope defined by the two
complementary approximations to the same dispersive integral.
The physical value of $\Dahad^\rho$ lies within that envelope, and the
geometric mean provides the best single estimate available from the GS model.

The quantitative precision achieved across the four TOE-native routes to $\MW$
(Table~\ref{tab:MW}) is consistent with the structural picture developed in
P135--P136: the $\JO$ discrete spectrum predicts the electroweak mass scale
at the sub-tenth-percent level without invoking the quark-level continuum.

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

\paragraph{1. Pion form factor below the $\rho$ peak.}

The GS lineshape through equation~\eqref{eq:Rrho} models $R(s)$ via the
$\rho$ propagator alone and underestimates $R(s)$ in the sub-threshold region
$\sqrt{s} < 0.6\,\mathrm{GeV}$.
In measured data, the $e^+e^-\to\pi^+\pi^-$ cross section below the $\rho$ peak
receives significant corrections from final-state interactions (the Omnès factor)
and from electromagnetic corrections to the charged-pion loop.
These contribute a few percent of the sub-threshold integral in Table~\ref{tab:intervals}
and would shift $\Dahrho^{\GS}$ upward by an estimated $0.1$--$0.2\times10^{-3}$.
A TOE-native derivation of the $\pi^+\pi^-$ form factor from the $\JO$ charged
pion sector (P113) would close this gap without external data input.

\paragraph{2. $\rho$--$\omega$ interference and the 1~GeV boundary.}

The cutoff $s_{\rm cut} = 1\,\mathrm{GeV}^2$ is motivated by the onset of the
$\omega(782)$ and $\phi(1020)$ spectral contributions, which are treated separately
in NWA.
In practice, the $\rho$ and $\omega$ interfere through isospin-breaking mixing
(a subject treated in P118), and the boundary between ``$\rho$ contribution'' and
``$\omega$ contribution'' in the dispersive integral is scheme-dependent.
The P118 mixing parameter $\varepsilon = -\alpha_s(M_Z)/3$ captures the dominant
isospin-breaking correction in the NWA; extending the mixing treatment to the
full lineshape integral would require a coupled-channel GS model for the $\rho$--$\omega$
system, which is an open derivation.

\paragraph{3. $J/\psi$ and $\Upsilon$ contributions to the PDG gap.}

The sum of all TOE-resolved resonances (P136, Table~1) gives
$\Dahad^{\rm res} = 5.523\times10^{-3}$ in NWA.
The PDG dispersive value $0.02750$ exceeds this by $0.02198$, or $79.9\%$ of
the PDG total.
The two identified sources of the remaining gap are the quark-level continuum
(multi-pion and open-heavy-flavour production, absent from the $\JO$ discrete
spectrum as established in P136) and corrections to the heavy-quarkonium $f_V^2$
values for $J/\psi$ and $\Upsilon$ (currently taken from PDG rather than derived
from the $\JO$ algebraic structure).
A $\JO$-native derivation of $f_{J/\psi}^2$ and $f_\Upsilon^2$---correcting the
factor-2.78 discrepancy in the $J_2$-sector Peirce-weight formula noted in P136---would
fix the heavy-quarkonium contributions and sharpen the total $\Dahad$ prediction
to the percent level.

\end{document}
