\documentclass[ aps,% 12pt,% final,% notitlepage,% oneside,% onecolumn,% nobibnotes,% nofootinbib,% superscriptaddress,% noshowpacs,% centertags]% {revtex4} \begin{document} \selectlanguage{english} \title{Dispersive estimation of the LO hadronic contribution to the muon ${g-2}$: what can we get from the ${e^+e^-}$ data? } \author{\firstname{V.~V.}~\surname{Bryzgalov}} %\\ \email{Valery.Bryzgalov@ihep.ru} \affiliation{% NRC ``Kurchatov Institute'' -- IHEP, Protvino, Russia }% \author{\firstname{O.~V.}~\surname{Zenin}} %\\ \email{Oleg.Zenin@ihep.ru} \affiliation{% NRC ``Kurchatov Institute'' -- IHEP, Protvino, Russia }% % %\date{\today} \begin{abstract} Using an up-to-date compilation of $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ data we estimated the LO hadronic contribution to the muon anomalous magnetic moment, $\ahad$. Incompatibilities between $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ measurements by independent experiments are mitigated by extra systematic uncertainties estimated using a requirement of uniformity of $\chi^2$ distribution over degrees of freedom in the joint fit of $\sigma_{\mathrm{tot}}$. Tensions in the $e^+e^-$ input data translate into an expanded uncertainty of the % $\ahad = (697.7 \pm {9.8}_{e^+e^-} \pm 3.6_{sys}) \times 10^{-10}$. % Given this, we obtain the SM prediction for the muon anomaly $a_\mu^{\mathrm{SM}} = 11659185.5(10.6) \times 10^{-10}$, below the experimental world average $a_\mu^{\mathrm{exp}}$ at {$2\sigma$} level. \end{abstract} \maketitle \section{Introduction} \label{sec:intro} The anomalous magnetic moment of the muon $a_\mu$ is the most precisely measured quantity in particle physics \cite{Muong-2:2026qnz} sensitive to physics beyond the SM \cite{Aliberti:2025beg}. The accuracy of $a_\mu$ calculation in the SM is limited by the leading order hadronic contribution to the photon vacuum polarization (VP) operator in the triangle diagram (Fig.~\ref{fig:hadlo}). As the amplitude is dominated by momenta running through the photon line in the loop $|Q^2| \sim m_\mu^2 < 1$~GeV$^2$, it cannot be calculated in the perturbative QCD. Despite a remarkable progress in precision of the lattice QCD calculations \cite{Aliberti:2025beg}, it is desirable to have an independent method of the amplitude evaluation. Currently, the dispersive calculation of the photon VP based on measurements of the total hadronic cross section with $e^+e^-$ beams \cite{Petermann:1957ir} \footnote{Optionally supplemented by hadronic form-factors from $\tau$ lepton decays.} is the only viable alternative.% The total cross section $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ is measured in exclusive final states ($\pi^+\pi^-$, $\pi^+\pi^-\pi^0$, $\pi^0\gamma$, $K\bar{K}$, {\it etc.}) at $\sqrt{s} < 2$~GeV and inclusively ($\ge$ 2 hadrons) at $\sqrt{s} > 2$~GeV. The leading contribution ($\simeq 70\%$) to the dispersion integral \cite{Petermann:1957ir}, % \begin{eqnarray} a_\mu({\mathrm{had, LO}}) &=& 4\alpha_0^2 \int^{\infty}_{m_\pi^2} \frac{ds}{s} K(s)\, \frac{1}{\pi}\, \mathrm{Im}\, \Pi^{\mathrm{had}}(s) = \frac{\alpha_0^2}{3\pi^2} \int^{\infty}_{m_\pi^2} \frac{ds}{s} K(s) R^{\mathrm{had}} (s) \label{eq:dispersion} \\ % R^{\mathrm{had}}(s) &=& \sigma_{\mathrm{tot}} (e^+e^- \to \gamma^* \to hadrons,\, \mathrm{bare}\,\, e^+e^-\,\, \mathrm{vertex\,\, and}\,\, \gamma^*) \, \left/ \, \frac{4\pi\alpha_0^2}{3s} \right. \label{eq:R} \\ % K(s) &=& \int^1_0 dx \frac{x^2 (1-x)}{x^2 + (1-x) (s/m_\mu^2)} \, , \nonumber % \end{eqnarray} % comes from $\sigma_{\mathrm{tot}}(e^+e^- \to \pi^+\pi^-(\gamma))$ at $2m_\pi < \sqrt{s} < 2$~GeV, where the cross section was measured with $\sim 1\%$ precision by BaBar \cite{BaBar:2012bdw}, KLOE-2 \cite{KLOE-2:2017fda} and CMD-3 \cite{CMD-3:2023alj} experiments. However, despite the claimed precision, these measurements are mutually incompatible at $\sim 3-5\sigma$ level (Fig.~\ref{fig:2pi}a). Possible sources of tensions are widely discussed in the literature, to no definite conclusion so far~\cite{Aliberti:2025beg}. Being agnostic about instrumental origin of the tensions, we assume that incompatible measurements are affected by systematic effects that might not be controlled by the experiments {\it in situ}, but still can be estimated by comparing their published results. The estimate of the uncontrolled systematic uncertainty facilitates more consistent fits of $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ in all final states and, consequently, the dispersive evaluation of $\ahad$ with a realistic uncertainty. The numerical procedures are described elsewhere \cite{Bryzgalov:2024ebj,Bryzgalov:2026yaz}. The program code and an up-to-date index of the $e^+e^-$ input data are available online \cite{the-code,ihep-cs}. \section{Fitting (incompatible) $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ data} Evaluation of the dispersion integral (\ref{eq:dispersion}) involves rescaling of published $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ measurements to the $R$-ratio (\ref{eq:R}), a sufficiently smooth parameterization of the latter for each hadronic final state (channel, in what follows), and fitting the parameterized $R^{\mathrm{had}}(s)$ by minimization of the $\chi^2$. % A low probability of the fit may indicate either an inadequacy of the $R^{\mathrm{had}}(s)$ parameterization or significant tensions between its measurements in different experiments, the latter being the case in our analysis. Resolution of the tensions via commonly recommended scaling of {\it all} experimental uncertainties, both statistical and systematic, by overall Birge factor $\sqrt{\chi^2/n_{\mathrm{dof}}}$ \cite{ParticleDataGroup:2026aaa} is inadequate for (at least) the following reason. The statistical uncertainty is determined by an exactly known number of reconstructed signal events and a normalization factor depending on luminosity, detector acceptance, reconstruction efficiency, etc. Birge scaling of statistical uncertainties in all experiments implies simultaneous scaling of their normalizations by the same factor, which was not intended. Thus, it is feasible to modify only (possibly underestimated) systematic uncertainties. The extra systematic uncertainty $\epsilon$ accounting for the tensions is estimated as the root mean square of relative integral pulls of experiments selected by an anomalously large $\chi^2$ contribution from the projection of the pull onto ``systematic'' unit eigenvector of the covariance matrix, $\Delta\chi^2_{\mathrm{sys}} > \chi^2_{\mathrm{thr}}$ \cite{Bryzgalov:2026yaz}. The $\epsilon$ is added in quadrature as an extra normalization uncertainty to {\it all} experiments in the channel, assuming no correlation of pulls between them. The fit is then reiterated with the modified covariance matrix until all $\Delta\chi^2_{\mathrm{sys}}$ drop below the modification threshold $\chi^2_{\mathrm{thr}} = 10$. In the $\pi^+\pi^-(\gamma)$ channel featuring most prominent tensions, the process converges after one iteration with $\epsilon = 5.7\%$. The resulting uncertainty of the fit is of an order of discrepancy between BaBar, CMD-3 and KLOE-2 measurements (Fig.~\ref{fig:2pi}b), far exceeding the underestimated Birge scaled uncertainty of the fit with the unmodified covariance matrices (Fig.~\ref{fig:2pi}a). % The extra systematic uncertainties are identified only in $\pi^+\pi^-(\gamma)$, $\pi^+\pi^-\pi^0$, $K^+K^-$, $\eta\gamma$ and $\omega\pi$ channels (Table~\ref{tab:channels}).% % \footnote{Tensions below our $\Delta\chi^2_{\mathrm{sys}} > \chi^2_{\mathrm{thr}}$ criterion are also present in the inclusive measurements at $\sqrt{s} > 2$~GeV. See, e.g., the discussion of tensions under open charm threshold in Ref.~\cite{Kataev:2026gea} and references therein. } % Residual tensions are accounted for by scaling the uncertainty of the fit by Birge factor $\sqrt{\chi^2/n_{\mathrm{dof}}}$, in case $P(\chi^2, n_{\mathrm{dof}}) < 0.05$. \section{Results} The contributions to $\ahad$ from individual channels are shown in Table~\ref{tab:channels}.% % \footnote{In regions uncovered by the $e^+e^-$ data, analytic parameterizations of $R^{\mathrm{had}}(s)$ \cite{Bryzgalov:2024ebj} are used for the dispersion integral (\ref{eq:dispersion}). At $\sqrt{s} > 11.2$~GeV the 3-loop pQCD expression for $R^{\mathrm{had}}(s)$ is used. } % The total LO hadronic contribution to $a_\mu$ is % \begin{equation}\label{eq:ahad} \ahad = \left( 697.7 \pm 9.8_{e^+e^-} \pm 1.1_{\chi^2_{\mathrm{thr}}} \pm 2.3_{\mathrm{par}} \pm 2.5_{\mathrm{rad}} \right) \times 10^{-10}\, , \end{equation} % where the first uncertainty is due to the $e^+e^-$ input data (including the extra systematic uncertainty accounting for tensions between them), the second is related to variation of the modification threshold within $6 < \chi^2_{\mathrm{thr}} < 25$ interval, the third and the last one are, respectively, due to $R^{\mathrm{had}}(s)$ parameterization and radiative corrections. \section {Conclusion} Adding (\ref{eq:ahad}) to the known electromagnetic, electroweak and higher order hadronic $a_\mu$ terms \cite{Aliberti:2025beg}, we obtain the SM value of the muon anomaly % \begin{equation} a_\mu^{\mathrm{SM}} = (11659185.5 \pm 10.6) \times 10^{-10}\, , \nonumber \end{equation} % below the experimental world average $a_\mu^{\mathrm{exp}}$ \cite{Muong-2:2026qnz} at $2\sigma$ level. The uncertainty is dominated by tensions between $e^+ e^- \to \pi^+\pi^-(\gamma)$ total cross sections measured in BaBar, KLOE-2 and CMD-3 experiments with 1\% level precision. Summary of theoretical $a_\mu$ estimates compared to its experimental world average is shown in Fig.~\ref{fig:status}. % The current precision of the $e^+e^-$ based dispersive $\ahad$ estimate is insufficient to claim a significant discrepancy between $a_\mu^{\mathrm{SM}}$ and $a_\mu^{\mathrm{exp}}$. % New precise measurements of $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$ are anticipated, via direct energy scan or radiative return at operating BEPCII, SuperKEKB, VEPP-2000, VEPP-4M, and future STCF \cite{STCF}, VEPP-6 \cite{VEPP-6} $e^+e^-$ colliders. %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgments} The authors are grateful to V.~B.~Anikeev, A.~L.~Kataev, A.~G.~Myagkov %, A.~A.~Solodkov and K.~Yu.~Todyshev for useful discussions. \end{acknowledgments} \section*{BIBLIOGRAPHY} %\bibliographystyle{JHEP} \bibliographystyle{maik} %\biboptions{numbers,sort&compress} \bibliography{biblio.bib} %\newpage \selectlanguage{russian} \begin{center} \large \bfseries \MakeTextUppercase{% Дисперсионная оценка адронного вклада ведущего порядка в $g-2$ мюона: что можно извлечь из данных $e^+e^-$? } \end{center} % \begin{center} \bfseries В.~В.~Брызгалов${}^{1)}$, О.~В.~Зенин${}^{1)}$ \end{center} % \begin{center} \begin{minipage}{\textwidth - 2cm} \small \hspace{4mm} {\it 1) НИЦ ``Курчатовский институт'' - ИФВЭ, Протвино, Россия} \\[2mm] Адронный вклад ведущего порядка $\ahad$ в аномальный магнитный момент мюона $a_\mu$ оценен дисперсионным методом на актуальном наборе измерений $\sigma_{\mathrm{tot}}(e^+e^- \to hadrons)$. Рассогласования между измерениями в отдельных независимых экспериментах при их совместной подгонке компенсируются введением дополнительных систематических неопределенностей, оцененных с использованием критерия равномерности распределения вкладов в $\chi^2$ по степеням свободы, что приводит к увеличению неопределенности оценки % $\ahad = (697.7 \pm {9.8}_{e^+e^-} \pm 3.6_{sys}) \times 10^{-10}$. % C учетом данного вклада получено значение $a_\mu^{\mathrm{SM}} = 11659185.5(10.6) \times 10^{-10}$, лежащее ниже экспериментально измеренного $a_\mu^{\mathrm{exp}}$ на уровне двух стандартных отклонений. \end{minipage} \end{center} %\newpage %%%%%%%%%%%%% figure & tables \selectlanguage{english} \begin{figure}[h!] %\setcaptionmargin{5mm} \onelinecaptionstrue \includegraphics[width=0.13\textwidth,clip]{had_LO-1.eps} \captionstyle{normal} \caption{ The leading order hadronic contribution to the $a_\mu$. The blob represents 1-particle irreducible VP operator induced by hadronic EM currents. } \label{fig:hadlo} \end{figure} \begin{figure}[h!] %\setcaptionmargin{5mm} \onelinecaptionstrue \includegraphics[width=0.41\textwidth,clip]{rho-omega-1.eps}% \includegraphics[width=0.41\textwidth,clip]{rho-omega-x.eps}\\[-3ex] \hspace*{0.35\textwidth} (a) \hfill (b) \hspace*{0.30\textwidth} \vspace*{-2ex} % \captionstyle{normal} \caption{ $R^{\pi^+\pi^-(\gamma)}$ in the $\rho$--$\omega$ interference region, most representative to demonstrate tensions between BaBar, CMD-3 and KLOE-2. Total experimental uncertainties are shown by vertical error bars with ticks indicating the statistical uncertainty. (a) The fit with unmodified experimental uncertainties. (b) The fit with the additional 5.7\% systematic uncertainty accounting for tensions between the experiments (the expanded total uncertainties are shown by shaded rectangles). $\chi^2/n_{\mathrm{dof}}$ values for the full $0.3 < \sqrt{s} < 2.0$~GeV range are shown. The fit uncertainty (green band) is scaled by $\sqrt{\chi^2/n_{\mathrm{dof}}}$. } \label{fig:2pi} \end{figure} \begin{figure}[H] %\setcaptionmargin{5mm} \onelinecaptionstrue \includegraphics[width=0.38\textwidth,clip]{sm-vs-exp.eps} \vspace*{-2ex} \captionstyle{normal} \caption{ $a_\mu^{\mathrm{exp}}$ {vs} theoretical $a_\mu^{\mathrm{SM}}$ using the $\ahad$ values (top to bottom): average of dispersive estimates \cite{Aoyama:2020ynm} before publication of CMD-3 $\pi^+\pi^-$ data; lattice QCD estimate~\cite{Muong-2:2026qnz}; our dispersive estimates with $\pi^+\pi^-(\gamma)$ part based on KLOE-2 \cite{KLOE-2:2017fda}, BaBar \cite{BaBar:2012bdw}, CMD-3 \cite{CMD-3:2023alj} (with OLYA and BCF \cite{ihep-cs} to cover $1 < \sqrt{s} < 2$ GeV, without tensions); their weighed average with Birge scaled uncertainty; our value (\ref{eq:ahad}). } \label{fig:status} \end{figure} \begin{table}[h!] \caption{% Contributions to $\ahad$ from individual final states. $\chi^2/n_{\mathrm{dof}}$ values are shown for the final fit with the extra systematic uncertainty $\epsilon$ accounting for tensions between the experiments. The sources of uncertainties are: (exp.) -- experimental uncertainty of the $e^+e^-$ input data (including the extra uncertainty $\epsilon$ and scaled by $\sqrt{\chi^2/n_{\mathrm{dof}}}$ if $P(\chi^2, n_{\mathrm{dof}}) < 0.05$); ($\chi^2_{\mathrm{thr}}$) -- variation of the $\chi^2_{\mathrm{thr}}$ threshold; (par.) -- $R^{\mathrm{had}}(s)$ parameterization; (rad.) -- radiative corrections. } \label{tab:channels} \begin{center} \scriptsize \renewcommand{\arraystretch}{0.8} \input{channels.tex} \end{center} \end{table} \renewcommand{\arraystretch}{1.} \end{document}