# Data-driven determination of the light-quark connected component of the
intermediate-window contribution to the muon $g-2$

Genessa Benton  Department of Physics and Astronomy, San Francisco State
University,
San Francisco, CA 94132, USA  Diogo Boito  Instituto de Física de São Carlos,
Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP, Brazil  Maarten
Golterman  Department of Physics and Astronomy, San Francisco State
University,
San Francisco, CA 94132, USA  Alexander Keshavarzi  Department of Physics and
Astronomy, The University of Manchester,
Manchester M13 9PL, United Kingdom  Kim Maltman  Department of Mathematics and
Statistics, York University,
Toronto, ON Canada M3J 1P3  CSSM, University of Adelaide, Adelaide, SA 5005
Australia  Santiago Peris  Department of Physics and Astronomy, San Francisco
State University,
San Francisco, CA 94132, USA  Department of Physics and IFAE-BIST, Universitat
Autònoma de Barcelona,
E-08193 Bellaterra, Barcelona, Spain

###### Abstract

We present the first data-driven result for $a_{\mu}^{\rm win,lqc}$, the
isospin-limit light-quark connected component of the intermediate-window
Hadronic-Vacuum-Polarization contribution to the muon anomalous magnetic
moment. Our result, $(198.8\pm 1.1)\times 10^{-10}$, is in significant tension
with eight recent mutually compatible high-precision lattice-QCD
determinations, and provides enhanced evidence for a puzzling discrepancy
between lattice and data-driven determinations of the intermediate window
quantity, one driven largely by a difference in the light-quark connected
component.

Since the pioneering work of Schwinger [1], and the subsequent experimental
confirmation of his result [2], the study of the lepton magnetic moments,
$\vec{\mu}$, has played a central role in the development of Quantum
Electrodynamics (QED) and, later, in that of the Standard Model (SM) of
particle physics. The anomalous magnetic moment of the muon,
$a_{\mu}=(g_{\mu}-2)/2$, defined in terms of the muon charge, $q$, mass,
$m_{\mu}$, and spin, $\vec{s}$, as

$\vec{\mu}=g_{\mu}\left(\frac{q}{2m_{\mu}}\right)\vec{s}$ (1)

is, at present, one of the most accurately known quantities in physics. The
new 2021 and 2023 Fermilab E989 measurements of $a_{\mu}$, performed by the
Muon $g-2$ collaboration, reached a precision of 0.20 ppm [3, 4, 5], and are
fully compatible with the previous BNL E821 experiment results [6]. As a
consequence, the experimental average of $a_{\mu}$ is known nowadays to 0.19
ppm. In order to stringently test the SM, the theory prediction for this
quantity must reach a similar level of accuracy.

In 2020, the $g-2$ Theory Initiative released a white paper (WP) [7] where,
based on the works of Refs. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], the SM expectation for $a_{\mu}$
was determined to 0.37 ppm. The SM result is a sum of pure QED, electroweak
and Higgs physics, hadronic vacuum polarization (HVP), and hadronic light-by-
light scattering (HLbL) contributions. The latter two, which involve Quantum
Chromodynamics (QCD), are particularly difficult to assess. The final
uncertainty in the WP SM result for $a_{\mu}$ is strongly dominated by the HVP
contribution, $a_{\mu}^{\rm HVP}$. It is thus a fundamental task to control
this contribution as well as possible.

The $a_{\mu}$ assessment of the WP is based on data-driven evaluations of the
HVP contribution [12, 14, 13, 15]. As is well known, this result showed a
$4.2\sigma$ tension with the then-current experimental average, motivating
many works aimed at finding potential beyond-the-SM (BSM) explanations for
this disagreement. Many developments, however, have taken place since the
publication of the the WP assessment. In particular, the Budapest-Marseille-
Wupertal (BMW) collaboration has published a complete, sub-percent lattice-QCD
evaluation of $a_{\mu}^{\rm HVP}$ [32]. If this result is used in the SM
assessment of $a_{\mu}$, the outcome is compatible with the experimental
average at the 1.5$\sigma$ level. In this situation, in order to conclude
whether the discrepancy between the SM assessment(s) of $a_{\mu}$ and
experimental results is due to BSM effects, it is, first of all, crucial to
understand the discrepancy between data-driven and lattice-QCD based HVP
results.

The complete computation of the HVP in lattice QCD is a very challenging task
and, as of today, only the BMW result is sufficiently precise to allow for a
detailed comparison with data-driven HVP determinations. This has motivated
the introduction of the window quantities by the RBC/UKQCD collaboration [33].
The so-called intermediate window, which cuts out short- and long-distance
regions of the $a_{\mu}^{\rm HVP}$ integral on the lattice-QCD side, is of
particular importance because it significantly suppresses the associated
systematic lattice uncertainties related to continuum limit extrapolation and
finite volume effects. The intermediate-window contribution to $a_{\mu}^{\rm
HVP}$, henceforth $a_{\mu}^{\rm win}$, is now known, with excellent precision
and very good agreement between results from different groups, from four
different lattice-QCD collaborations [32, 34, 35, 36]. A recent data-driven
assessment of $a_{\mu}^{\rm win}$ using electroproduction input [37], however,
shows an important tension with these lattice results (see e.g. Ref. [38]).
(This tension is reduced if, instead, one uses $\tau$ data for the two-pion
contribution [39], estimating the required model-dependent isospin-breaking
corrections with the models discussed in Ref. [40].)

The lattice evaluation of $a_{\mu}^{\rm HVP}$ is split into several building
blocks with the dominant contributions arising from the isospin-limit (defined
by taking $m_{\pi}=m_{\pi^{0}}$) light- and strange-quark connected and
disconnected parts, with additional, smaller, contributions from charm and
bottom quarks. Isospin breaking (IB) effects, both of electromagnetic (EM) and
strong (SIB) origin, are accounted for perturbatively, keeping terms to first
order in an expansion in the fine-structure constant $\alpha$ and the up-down
quark-mass difference $m_{u}-m_{d}$. The light-quark connected (lqc)
contribution to $a_{\mu}^{\rm win}$ in the isospin symmetric limit, denoted
$a_{\mu}^{{\rm win,lqc}}$, is known now with very good precision from eight
different lattice determinations (see the blue data points in Fig. 1). These
eight determinations are all in excellent agreement and have small relative
errors, ranging from 0.3 to 1.1%. Since this contribution gives about 87% of
$a_{\mu}^{\rm win}$, and appears to be under good control, given the agreement
among the eight different lattice determinations, it is highly desirable to
obtain a precise data-driven estimate of $a_{\mu}^{{\rm win,lqc}}$ in order to
further scrutinize the discrepancy between lattice-QCD and data-driven
determinations of $a_{\mu}^{\rm HVP}$. It is the aim of this letter to present
this estimate.

We turn now to a short review of the theoretical framework for our data-driven
determination of $a_{\mu}^{{\rm win,lqc}}$. To be able to compute
$a_{\mu}^{\rm HVP}$ on the lattice-QCD side, one determines the Euclidean-time
zero-momentum two-point correlation function given by

$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\sum_{i=1}^{3}\int d^{3}x\langle j_{i}^{\rm
EM}(\vec{x},t)j_{i}^{\rm EM}(0)\rangle$ (2) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{m_{\pi}^{2}}^{\infty}ds\,\sqrt{s}\,e^{-\sqrt{s}t}\,\rho_{\rm
EM}(s)\quad(t>0),$

where $m_{\pi}$ is the neutral pion mass, $j_{\mu}^{\rm EM}(x)$ is the EM
current, and $\rho_{\rm EM}$ is the associated inclusive hadronic spectral
function. In terms of $C(t)$, the intermediate-window contribution to
$a_{\mu}^{\rm HVP}$ is given by

$a_{\mu}^{\rm win}=2\int_{0}^{\infty}dt\,w(t)W_{\rm win}(t)C(t),$ (3)

where the function $w(t)$ can be obtained from its counterpart in $s$-space
[41] and $W_{\rm win}(t)$ is the weight function associated with the RBC/UKQCD
intermediate-window [33], defined as

$W_{\rm
win}(t)=\frac{1}{2}\left(\tanh\frac{t-t_{0}}{\Delta}-\tanh\frac{t-t_{1}}{\Delta}\right),$
(4)

with $t_{0}=0.4$ fm, $t_{1}=1.0$ fm, and $\Delta=0.15$ fm. The corresponding
expression for $a_{\mu}^{\rm HVP}$ is obtained by removing the factor $W_{\rm
win}(t)$ from Eq. (3). Due to the presence of $W_{\rm win}(t)$ in Eq. (3) the
short- and long-distance contributions to the integral are strongly
suppressed.

Since we are concerned with the data-driven determination of $a_{\mu}^{{\rm
win,lqc}}$, we need the data-driven (or dispersive) counterpart to Eq. (3)
which is

$a_{\mu}^{\rm
win}=\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\,\widetilde{W}_{\rm
win}(s)\,\rho_{\rm EM}(s),$ (5)

where $\hat{K}(s)$ is a well known, slowly varying, kernel function [42, 43]
(see Ref. [7] for the explicit expression) and $\widetilde{W}_{\rm win}(s)$ is
the $s$-space representation of $W_{\rm win}(t)$,

$\widetilde{W}_{\rm win}(s)=\frac{\int_{0}^{\infty}dt\,W_{\rm
win}(t)\,w(t)\,e^{-\sqrt{s}t}}{\int_{0}^{\infty}dt\,w(t)\,e^{-\sqrt{s}t}}.$
(6)

Here, when evaluating the different contributions to Eq. (5) in the exclusive-
mode region, we employ the data compilation of Refs. [13, 15] (KNT19 in what
follows).

Our goal is to isolate the lqc contribution to the full result of Eq. (5).
This can be achieved employing an idea first implemented with sufficient
precision in Refs. [44, 45], where the foundations of our method are laid out
in detail. We start from the usual decomposition of the three-flavor EM
current into its $I=1$ and $I=0$ parts, which produces analogous
decompositions of $C(t)$ and $\rho_{\rm EM}$ into $I=1$, $I=0$ and mixed-
isospin (MI) parts. In the isospin limit, the contribution associated with the
$I=1$ current contains only light-quark connected contributions and one has

$\rho_{\rm EM}^{\rm lqc}=\frac{10}{9}\rho_{\rm EM}^{I=1}(s).$ (7)

The data-driven estimate of $a_{\mu}^{{\rm win,lqc}}$ thus requires the
identification of the $I=1$ component of $\rho_{\rm EM}$. This can be
accomplished, assuming isospin-symmetry and using KNT19 data, on a channel-by-
channel basis, in the KNT19 exclusive-mode region, $\sqrt{s}\leq 1.937$ GeV.

There are two classes of such contributions. The first, and dominant one,
consists of contributions from modes with well-defined, positive $G$ parity.
Such modes have $I=1$ and thus contribute to the lqc component in the isospin
limit. Contributions from such “unambiguous” modes constitute the main
ingredient in our determination of $a_{\mu}^{{\rm win,lqc}}$. The remaining
contributions come from higher-threshold modes with no well-defined $G$
parity. For these “ambiguous” modes, and especially for the dominant such
channels — $K\bar{K}$ and $K\bar{K}\pi$ — one resorts to external information,
whenever available, in order to identify, as accurately as possible, the $I=1$
component of the experimental $I=0+1$ sum. The third ingredient is
perturbative QCD supplemented with an estimate of Duality Violation
contributions, which we use in the inclusive region, $\sqrt{s}>1.937$ GeV.
Finally, to the sum of the results thus obtained one must apply isospin-
breaking (IB) corrections since the experimental data, inevitably, contain IB
contributions. Only after these corrections have been applied can the data-
based result be compared directly with isospin-limit lattice $a_{\mu}^{{\rm
win,lqc}}$ results. We now detail how we treat each of the four aforementioned
contributions.

We start with the unambiguous modes, which give the dominant contribution to
$a_{\mu}^{{\rm win,lqc}}$. The results are obtained with Eq. (5) using the
KNT19 spectra for the different exclusive-mode contributions to $\rho_{\rm
EM}(s)$. There are 13 such $I=1$ channels, with the largest contribution, by
far, arising from the $\pi^{+}\pi^{-}$ channel, which contributes
$144.15(49)\times 10^{-10}$ to $[a_{\mu}^{\rm win}]_{I=1}$. The results for
the different modes are given in Table 1. The sum over all $G$-parity-
unambiguous modes gives a total $I=1$ contribution to $a_{\mu}^{{\rm win}}$ of
$168.24(72)\times 10^{-10}$. From Eq. (7), the final contribution of all
unambiguous channels to $a_{\mu}^{\rm win,lqc}$ is then

$\displaystyle[a_{\mu}^{\rm win,lqc}]_{G-{\rm par}}$ $\displaystyle=$
$\displaystyle 186.93(80)\times 10^{-10}.$ (8)

Table 1:  Contributions from $G$-parity unambiguous modes to $a_{\mu}^{\rm win}$ for $\sqrt{s}\leq 1.937$ GeV obtained from KNT19 [15] exclusive-mode spectra. All entries in units of $10^{-10}$.  $I=1$ modes $X$ | $[a_{\mu}^{\rm win}]_{X}\times 10^{10}$
---|---
low-$s$ $\pi^{+}\pi^{-}$ | 0.02(00)
$\pi^{+}\pi^{-}$ | 144.13(49)
$2\pi^{+}2\pi^{-}$ | 9.29(13)
$\pi^{+}\pi^{-}2\pi^{0}$ | 11.94(48)
$3\pi^{+}3\pi^{-}$ (no $\omega$) | 0.14(01)
$2\pi^{+}2\pi^{-}2\pi^{0}$ (no $\eta$) | 0.83(11)
$\pi^{+}\pi^{-}4\pi^{0}$ (no $\eta$) | 0.13(13)
$\eta\pi^{+}\pi^{-}$ | 0.85(03)
$\eta 2\pi^{+}2\pi^{-}$ | 0.05(01)
$\eta\pi^{+}\pi^{-}2\pi^{0}$ | 0.07(01)
$\omega(\rightarrow\pi^{0}\gamma)\pi^{0}$ | 0.53(01)
$\omega(\rightarrow{\rm npp})3\pi$ | 0.10(02)
$\omega\eta\pi^{0}$ | 0.15(03)
TOTAL | 168.24(72)

We turn next to our treatment of ambiguous-mode contributions, which follows
the general strategy outlined in Sec. IV of Ref. [45]. For some of these
contributions, notably those of the numerically dominant $K\bar{K}$ and
$K\bar{K}\pi$ channels, external experimental information can be used in
separating the desired $I=1$ component from the experimental $I=0+1$ sum.
Modes for which external experimental information is not available have much
smaller contributions. For these modes, one employs a maximally conservative
$I=1/0$ separation, based on the observation that the $I=1$ part of the
mode-$X$ contribution to $\rho_{\rm EM}(s)$ must lie between 0 and the full
$I=0+1$ contribution obtained from the KNT19 spectrum for that mode. The
contribution of ambiguous mode $X$ to $a_{\mu}^{\rm win,lqc}$ lies, therefore,
in the following range

$[a_{\mu}^{\rm
win,lqc}]_{X}=\frac{10}{9}\left(\frac{1}{2}\pm\frac{1}{2}\right)[a_{\mu}^{\rm
win}]_{X}=\left(\frac{5}{9}\pm\frac{5}{9}\right)[a_{\mu}^{\rm win}]_{X}.$ (9)

Let us discuss in some detail the significant ambiguous-mode contribution
arising from the $K\bar{K}$ channels, $K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$.
Independent experimental information on the $K\bar{K}$ contribution to the
purely $I=1$ spectral function can be obtained from the BaBar spectrum for the
decay $\tau\to K^{-}K^{0}\nu_{\tau}$ [46]. Using the conserved vector current
(CVC) relation, these results can be used to determine the $I=1$ $K\bar{K}$
contribution to $\rho_{\rm EM}(s)$ up to $s=2.7556$ GeV2, and hence the
associated contribution to $\left[a_{\mu}^{\rm win,lqc}\right]_{K\bar{K}}$,
which, using Eq. (7), is found to be $10/9\times 0.465(29)\times 10^{-10}$.
For $s>2.7556$ GeV2, the $I=1$ part is found using KNT19 data and the
maximally conservative treatment of Eq. (9). We find, for $s>2.7556$ GeV2, a
contribution of $10/9\times 0.055(55)\times 10^{-10}$ to $\left[a_{\mu}^{\rm
win,lqc}\right]_{K\bar{K}}$. From these results one obtains, for the full
exclusive-region $K\bar{K}$ contribution,

$\left[a_{\mu}^{\rm win,lqc}\right]_{K\bar{K}}=0.578(69)\times 10^{-10}.$ (10)

A similar treatment of the $K\bar{K}\pi$ modes is possible thanks to the
Dalitz plot analysis of BaBar, which provides a separation of the $I=1$ and
$I=0$ contributions to the $K\bar{K}\pi$ cross sections[47]. Integrating the
BaBar $I=1$ result, we find

$\left[a_{\mu}^{\rm win,lqc}\right]_{K\bar{K}\pi}=0.521(86)\times 10^{-10}.$
(11)

For the $K\bar{K}2\pi$ modes, only a small improvement is possible over the
maximally conservative treatment. This is obtained by first subtracting the
small $I=0$ $\phi[\rightarrow K\bar{K}]\pi\pi$ contribution implied by BaBar
$e^{+}e^{-}\rightarrow\phi\pi\pi$ cross sections [48], and applying the
maximally conservative treatment only to the residual $I=0+1$ sum. This leads
to the result

$\left[a_{\mu}^{\rm win,lqc}\right]_{K\bar{K}2\pi}=0.60(60)\times 10^{-10}.$
(12)

The very small (often completely negligible) contributions of the remaining
ambiguous modes ($K\bar{K}3\pi$, $\omega(\rightarrow{\rm npp})K\bar{K}$,
$\eta(\rightarrow{\rm npp})K\bar{K}$ (no $\phi$), $p\bar{p}$, $n\bar{n}$, and
low-$s$ $\pi^{0}\gamma$ and $\eta\gamma$) (npp = non-purely pionic) are
obtained from the KNT19 spectra using the maximally conservative separation of
Eq. (9). The total contribution from all $G$-parity-ambiguous exclusive modes
is, finally,

$[a_{\mu}^{\rm win,lqc}]_{\rm amb.}=1.74(61)\times 10^{-10},$ (13)

with $1.70(61)\times 10^{-10}$ from $K\bar{K}$, $K\bar{K}\pi$ and
$K\bar{K}2\pi$.

In the inclusive region, $\sqrt{s}>1.937$ GeV, we use QCD perturbation theory,
which is known to $\mathcal{O}(\alpha_{s}^{4})$, supplemented with an estimate
for the $\mathcal{O}(\alpha_{s}^{5})$ coefficient, as described in Refs. [44,
45]. To this result, we add an estimate of the duality violation (DV)
contribution, obtained from our previous study of the $I=1$ hadronic spectral
function in $\tau\to{\rm hadrons}+\nu_{\tau}$ [49], using the parametrization
of DVs discussed in Refs. [50, 51, 52]. Since DVs represent a fundamental
limitation of perturbation theory, we use the resulting central value as the
total uncertainty on the perturbative contribution. This enlarges the
uncertainty of the perturbative contribution without DVs by a factor of about
10, and should provide a very conservative assessment. The resulting
inclusive-region contribution is then

$[a_{\mu}^{\rm win,lqc}]_{\rm pt.QCD+DVs}=11.06(16)\times 10^{-10}.$ (14)

The fourth and final ingredient in the determination of $a_{\mu}^{\rm
win,lqc}$ is an evaluation of the EM and SIB contributions to be subtracted
from the data-based results obtained above before comparison with isospin-
symmetric lattice-QCD results. The general strategy employed for this
subtraction is detailed in Refs. [44] and [45]. The main observation is that,
to first order in IB, SIB is present only in the MI component of $\rho_{\rm
EM}(s)$. EM IB, on the other hand, occurs in all of the pure $I=1/0$ and MI
components. The IB correction to $a_{\mu}^{\rm win,lqc}$ then contains two
parts. The first, which appears in the pure $I=1$ component, is of EM origin.
No breakdown of this correction into individual exclusive-mode contributions
is required; an inclusive determination is sufficient. The situation for the
MI contribution is different since we must estimate the MI contamination on a
channel-by-channel basis, removing from the “nominally” $I=1$ results above
the component that arises from $\rho^{\rm MI}_{\rm EM}(s)$. These
contributions are expected to be dominated by the $2\pi$ mode through
$\rho-\omega$ mixing in the process $e^{+}e^{-}\to\omega\to\rho\to 2\pi$.

At present, given the absence of complete data-driven estimates for some
potentially important components of the pure $I=1$ EM IB contribution (see,
e.g., the discussion in the Appendix of Ref. [44]), we are forced to rely on
the lattice, and employ for this correction the result obtained by BMW in Ref.
[32],

$\Delta_{\rm EM}a_{\mu}^{\rm win,lqc}=0.035(59)\times 10^{-10}\ .$ (15)

This correction is very small, given the size of other uncertainties, and we
will neglect it in what follows.

Figure 1: Comparison of our final result (BBGKMP 23), Eq. (17), with lattice
results for $a_{\mu}^{\rm win,lqc}$ from [32] (BMW 20), [53] (LM 20), [54]
($\chi$QCD 23), [55] (ABGP 22), [34] (Mainz/CLS 22), [35] (ETMC 22), [56] (FHM
23), and [36] (RBC/UKQCD 23)

The MI contamination to the $2\pi$ exclusive mode was obtained in Ref. [57]
from $2\pi$ electroproduction data fitting a dispersive representation of the
pion form factor that includes $\rho-\omega$ mixing. The $2\pi$ MI component
is found to be $\left[a_{\mu}^{\rm win}\right]_{\pi\pi}^{\rm MI}=0.83(6)\times
10^{-10}$, which is about $0.6\%$ of the total $2\pi$ contribution to
$a_{\mu}^{\rm win}$. Since the MI components of other nominally $I=1$ modes
have no analogous narrow-resonance enhancements, we consider it very safe to
assume that their MI total will not exceed $1\%$ of the sum, $25.68\times
10^{-10}$, of their contributions. To account for the total of the
non-$2\pi$-mode MI contaminations we thus add an uncertainty of $0.26\times
10^{-10}$ to the $2\pi$ results of Ref. [57]. Using Eq. (7), this leads to a
MI correction of

$\Delta^{\rm MI}a_{\mu}^{\rm win,lqc}=-0.92(7)_{2\pi}(29)_{{\rm
non-}2\pi}\times 10^{-10}.$ (16)

We are now in a position to obtain our final data-driven estimate for
$a_{\mu}^{\rm win,lqc}$. Adding the contributions from Eqs. (8), (13), (14),
and applying the IB correction of Eq. (16), we find, as our final result,

$a_{\mu}^{\rm win,lqc}=198.8(1.1)\times 10^{-10}.$ (17)

In Fig. 1, we compare our data-driven estimate with the lattice-QCD results
from 8 different collaborations. The tension between the data-driven and
lattice results is striking. Assuming, for simplicity, all errors to be
Gaussian, we find tensions ranging from $3.3\sigma$ to $6.1\sigma$. Our result
indicates that the discrepancy between data-driven and lattice-QCD results for
$a_{\mu}^{\rm win}$ is almost entirely due to the light-quark connected
contribution, which, in turn, is strongly dominated by the $2\pi$
channel—accounting for about 81% of the result of Eq. (17). Given this $2\pi$
dominance, it is relevant to note that recent CMD-3 results for the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross sections [58], which are in significant
tension with those of earlier experiments, and known to significantly increase
the $2\pi$ contribution to $a_{\mu}^{\rm HVP}$, would similarly increase our
result for $a_{\mu}^{\rm win,lqc}$, making it more compatible with lattice
determinations. Since the source of the disagreements between the previously
published and new CMD-3 $2\pi$ results is both presently unclear and the
subject of ongoing study, we refrain from addressing this issue more
quantitatively for now.

We note that our final result is based on the KNT19 data compilation. An
equivalent analysis using other dispersive evaluations (e.g. DHMZ data [12,
14]) would be desirable. We remark, however, that for the lqc contribution to
$a_{\mu}^{\rm HVP}$, which can be obtained based on publicly available
results, KNT19- and DHMZ-based estimates are in very good agreement [44].

In a companion paper [59] we present results for several other window
quantities, including both the light-quark-connected and strange-quark-plus-
all-disconnected contributions. The latter require the treatment of the $I=0$
sector. The impact of new phenomenological estimates of MI IB corrections in
the $3\pi$ channel [60] are discussed and we present a comparison between
lattice-QCD and phenomenological IB corrections. A preliminary estimate of the
potential impact of new CMD-3 results for $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross-
section [58] is also given.

###### Acknowledgements.

Acknowledgments: We would like to thank Martin Hoferichter and Peter Stoffer
for extensive discussions on isospin breaking. DB and KM thank San Francisco
State University where part of this work was carried out, for hospitality.
This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences Energy Frontier Research
Centers program under Award Number DE-SC-0013682 (GB and MG). DB’s work was
supported by the São Paulo Research Foundation (FAPESP) Grant No. 2021/06756-6
and by CNPq Grant No. 308979/2021-4. The work of AK is supported by The Royal
Society (URF\R1\231503), STFC (Consolidated Grant ST/S000925/) and the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No. 858199 (INTENSE). The work of KM is
supported by a grant from the Natural Sciences and Engineering Research
Council of Canada. SP is supported by the Spanish Ministry of Science,
Innovation and Universities (project
PID2020-112965GB-I00/AEI/10.13039/501100011033) and by Departament de Recerca
i Universitats de la Generalitat de Catalunya, Grant No 2021 SGR 00649. IFAE
is partially funded by the CERCA program of the Generalitat de Catalunya.

## References

  * Schwinger [1948] J. S. Schwinger, On Quantum electrodynamics and the magnetic moment of the electron, Phys. Rev. 73, 416 (1948).
  * Kusch and Foley [1948] P. Kusch and H. M. Foley, The Magnetic Moment of the Electron, Phys. Rev. 74, 250 (1948).
  * Albahri  _et al._ [2021] T. Albahri _et al._ (Muon $g$-2), Measurement of the anomalous precession frequency of the muon in the Fermilab Muon $g$-2 Experiment, Phys. Rev. D 103, 072002 (2021), arXiv:2104.03247 [hep-ex] .
  * Abi  _et al._ [2021] B. Abi _et al._ (Muon $g$-2), Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281 [hep-ex] .
  * Aguillard  _et al._ [2023] D. P. Aguillard _et al._ (Muon g-2), Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm, Phys. Rev. Lett. 131, 161802 (2023), arXiv:2308.06230 [hep-ex] .
  * Bennett  _et al._ [2006] G. W. Bennett _et al._ (Muon $g$-2), Final report of the muon E821 anomalous magnetic moment measurement at BNL, Phys. Rev. D 73, 072003 (2006), arXiv:hep-ex/0602035 .
  * Aoyama  _et al._ [2020] T. Aoyama _et al._ , The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887, 1 (2020), arXiv:2006.04822 [hep-ph] .
  * Aoyama  _et al._ [2012] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Complete Tenth-Order QED Contribution to the Muon g-2, Phys. Rev. Lett. 109, 111808 (2012), arXiv:1205.5370 [hep-ph] .
  * Aoyama  _et al._ [2019] T. Aoyama, T. Kinoshita, and M. Nio, Theory of the Anomalous Magnetic Moment of the Electron, Atoms 7, 28 (2019).
  * Czarnecki  _et al._ [2003] A. Czarnecki, W. J. Marciano, and A. Vainshtein, Refinements in electroweak contributions to the muon anomalous magnetic moment, Phys. Rev. D 67, 073006 (2003), [Erratum: Phys.Rev.D 73, 119901 (2006)], arXiv:hep-ph/0212229 .
  * Gnendiger  _et al._ [2013] C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement, Phys. Rev. D 88, 053005 (2013), arXiv:1306.5546 [hep-ph] .
  * Davier  _et al._ [2017] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data, Eur. Phys. J. C 77, 827 (2017), arXiv:1706.09436 [hep-ph] .
  * Keshavarzi  _et al._ [2018] A. Keshavarzi, D. Nomura, and T. Teubner, Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis, Phys. Rev. D 97, 114025 (2018), arXiv:1802.02995 [hep-ph] .
  * Davier  _et al._ [2020] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\alpha(m_{Z}^{2})$, Eur. Phys. J. C 80, 241 (2020), [Erratum: Eur.Phys.J.C 80, 410 (2020)], arXiv:1908.00921 [hep-ph] .
  * Keshavarzi  _et al._ [2020] A. Keshavarzi, D. Nomura, and T. Teubner, $g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium, Phys. Rev. D 101, 014029 (2020), arXiv:1911.00367 [hep-ph] .
  * Colangelo  _et al._ [2019] G. Colangelo, M. Hoferichter, and P. Stoffer, Two-pion contribution to hadronic vacuum polarization, JHEP 02, 006, arXiv:1810.00007 [hep-ph] .
  * Hoferichter  _et al._ [2019] M. Hoferichter, B.-L. Hoid, and B. Kubis, Three-pion contribution to hadronic vacuum polarization, JHEP 08, 137, arXiv:1907.01556 [hep-ph] .
  * Hoid  _et al._ [2020] B.-L. Hoid, M. Hoferichter, and B. Kubis, Hadronic vacuum polarization and vector-meson resonance parameters from $e^{+}e^{-}\rightarrow\pi^{0}\gamma$, Eur. Phys. J. C 80, 988 (2020), arXiv:2007.12696 [hep-ph] .
  * Kurz  _et al._ [2014] A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order, Phys. Lett. B 734, 144 (2014), arXiv:1403.6400 [hep-ph] .
  * Melnikov and Vainshtein [2004] K. Melnikov and A. Vainshtein, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D 70, 113006 (2004), arXiv:hep-ph/0312226 .
  * Masjuan and Sanchez-Puertas [2017] P. Masjuan and P. Sanchez-Puertas, Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach, Phys. Rev. D 95, 054026 (2017), arXiv:1701.05829 [hep-ph] .
  * Colangelo  _et al._ [2017a] G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Rescattering effects in the hadronic-light-by-light contribution to the anomalous magnetic moment of the muon, Phys. Rev. Lett. 118, 232001 (2017a), arXiv:1701.06554 [hep-ph] .
  * Colangelo  _et al._ [2017b] G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: two-pion contributions, JHEP 04, 161, arXiv:1702.07347 [hep-ph] .
  * Hoferichter  _et al._ [2018a] M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon, Phys. Rev. Lett. 121, 112002 (2018a), arXiv:1805.01471 [hep-ph] .
  * Hoferichter  _et al._ [2018b] M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Dispersion relation for hadronic light-by-light scattering: pion pole, JHEP 10, 141, arXiv:1808.04823 [hep-ph] .
  * Gérardin  _et al._ [2019] A. Gérardin, H. B. Meyer, and A. Nyffeler, Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks, Phys. Rev. D 100, 034520 (2019), arXiv:1903.09471 [hep-lat] .
  * Bijnens  _et al._ [2019] J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment, Phys. Lett. B 798, 134994 (2019), arXiv:1908.03331 [hep-ph] .
  * Colangelo  _et al._ [2020a] G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, Short-distance constraints on hadronic light-by-light scattering in the anomalous magnetic moment of the muon, Phys. Rev. D 101, 051501 (2020a), arXiv:1910.11881 [hep-ph] .
  * Colangelo  _et al._ [2020b] G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models, JHEP 03, 101, arXiv:1910.13432 [hep-ph] .
  * Colangelo  _et al._ [2014] G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, Remarks on higher-order hadronic corrections to the muon g$-$2, Phys. Lett. B 735, 90 (2014), arXiv:1403.7512 [hep-ph] .
  * Blum  _et al._ [2020] T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD, Phys. Rev. Lett. 124, 132002 (2020), arXiv:1911.08123 [hep-lat] .
  * Borsanyi  _et al._ [2021] S. Borsanyi _et al._ , Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593, 51 (2021), arXiv:2002.12347 [hep-lat] .
  * Blum  _et al._ [2018] T. Blum, P. A. Boyle, V. Gülpers, T. Izubuchi, L. Jin, C. Jung, A. Jüttner, C. Lehner, A. Portelli, and J. T. Tsang (RBC, UKQCD), Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, Phys. Rev. Lett. 121, 022003 (2018), arXiv:1801.07224 [hep-lat] .
  * Cè  _et al._ [2022] M. Cè _et al._ , Window observable for the hadronic vacuum polarization contribution to the muon g-2 from lattice QCD, Phys. Rev. D 106, 114502 (2022), arXiv:2206.06582 [hep-lat] .
  * Alexandrou  _et al._ [2023] C. Alexandrou _et al._ (Extended Twisted Mass), Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions, Phys. Rev. D 107, 074506 (2023), arXiv:2206.15084 [hep-lat] .
  * Blum  _et al._ [2023] T. Blum _et al._ , An update of Euclidean windows of the hadronic vacuum polarization, (2023), arXiv:2301.08696 [hep-lat] .
  * Colangelo  _et al._ [2022a] G. Colangelo, A. X. El-Khadra, M. Hoferichter, A. Keshavarzi, C. Lehner, P. Stoffer, and T. Teubner, Data-driven evaluations of Euclidean windows to scrutinize hadronic vacuum polarization, Phys. Lett. B 833, 137313 (2022a), arXiv:2205.12963 [hep-ph] .
  * Wittig [2023] H. Wittig, Progress on $(g-2)_{\mu}$ from Lattice QCD, in  _57th Rencontres de Moriond on Electroweak Interactions and Unified Theories_ (2023) arXiv:2306.04165 [hep-ph] .
  * Masjuan  _et al._ [2023] P. Masjuan, A. Miranda, and P. Roig, $\tau$ data-driven evaluation of Euclidean windows for the hadronic vacuum polarization, (2023), arXiv:2305.20005 [hep-ph] .
  * Miranda and Roig [2020] J. A. Miranda and P. Roig, New $\tau$-based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment, Phys. Rev. D 102, 114017 (2020), arXiv:2007.11019 [hep-ph] .
  * Bernecker and Meyer [2011] D. Bernecker and H. B. Meyer, Vector Correlators in Lattice QCD: Methods and applications, Eur. Phys. J. A 47, 148 (2011), arXiv:1107.4388 [hep-lat] .
  * Brodsky and De Rafael [1968] S. J. Brodsky and E. De Rafael, Suggested Boson-Lepton Pair Couplings and the Anomalous Magnetic Moment of the Muon, Phys. Rev. 168, 1620 (1968).
  * Lautrup and De Rafael [1968] B. E. Lautrup and E. De Rafael, Calculation of the sixth-order contribution from the fourth-order vacuum polarization to the difference of the anomalous magnetic moments of muon and electron, Phys. Rev. 174, 1835 (1968).
  * Boito  _et al._ [2023] D. Boito, M. Golterman, K. Maltman, and S. Peris, Data-based determination of the isospin-limit light-quark-connected contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 107, 074001 (2023), arXiv:2211.11055 [hep-ph] .
  * Boito  _et al._ [2022] D. Boito, M. Golterman, K. Maltman, and S. Peris, Evaluation of the three-flavor quark-disconnected contribution to the muon anomalous magnetic moment from experimental data, Phys. Rev. D 105, 093003 (2022), arXiv:2203.05070 [hep-ph] .
  * Lees  _et al._ [2018] J. P. Lees _et al._ (BaBar), Measurement of the spectral function for the $\tau^{-}\to K^{-}K_{S}\nu_{\tau}$ decay, Phys. Rev. D 98, 032010 (2018), arXiv:1806.10280 [hep-ex] .
  * Aubert  _et al._ [2008] B. Aubert _et al._ (BaBar), Measurements of $e^{+}e^{-}\to K^{+}K^{-}\eta$, $K^{+}K^{-}\pi^{0}$ and $K^{0}_{s}K^{\pm}\pi^{\mp}$ cross- sections using initial state radiation events, Phys. Rev. D 77, 092002 (2008), arXiv:0710.4451 [hep-ex] .
  * Lees  _et al._ [2012] J. P. Lees _et al._ (BaBar), Cross Sections for the Reactions $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$, $K^{+}K^{-}\pi^{0}\pi^{0}$, and $K^{+}K^{-}K^{+}K^{-}$ Measured Using Initial-State Radiation Events, Phys. Rev. D 86, 012008 (2012), arXiv:1103.3001 [hep-ex] .
  * Boito  _et al._ [2021] D. Boito, M. Golterman, K. Maltman, S. Peris, M. V. Rodrigues, and W. Schaaf, Strong coupling from an improved $\tau$ vector isovector spectral function, Phys. Rev. D 103, 034028 (2021), arXiv:2012.10440 [hep-ph] .
  * Boito  _et al._ [2018] D. Boito, I. Caprini, M. Golterman, K. Maltman, and S. Peris, Hyperasymptotics and quark-hadron duality violations in QCD, Phys. Rev. D 97, 054007 (2018), arXiv:1711.10316 [hep-ph] .
  * Catà  _et al._ [2005] O. Catà, M. Golterman, and S. Peris, Duality violations and spectral sum rules, JHEP 08, 076, arXiv:hep-ph/0506004 .
  * Catà  _et al._ [2008] O. Catà, M. Golterman, and S. Peris, Unraveling duality violations in hadronic tau decays, Phys. Rev. D 77, 093006 (2008), arXiv:0803.0246 [hep-ph] .
  * Lehner and Meyer [2020] C. Lehner and A. S. Meyer, Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio, Phys. Rev. D 101, 074515 (2020), arXiv:2003.04177 [hep-lat] .
  * Wang  _et al._ [2023] G. Wang, T. Draper, K.-F. Liu, and Y.-B. Yang (chiQCD), Muon g-2 with overlap valence fermions, Phys. Rev. D 107, 034513 (2023), arXiv:2204.01280 [hep-lat] .
  * Aubin  _et al._ [2022] C. Aubin, T. Blum, M. Golterman, and S. Peris, Muon anomalous magnetic moment with staggered fermions: Is the lattice spacing small enough?, Phys. Rev. D 106, 054503 (2022), arXiv:2204.12256 [hep-lat] .
  * Bazavov  _et al._ [2023] A. Bazavov _et al._ (Fermilab Lattice, HPQCD,, MILC), Light-quark connected intermediate-window contributions to the muon g-2 hadronic vacuum polarization from lattice QCD, Phys. Rev. D 107, 114514 (2023), arXiv:2301.08274 [hep-lat] .
  * Colangelo  _et al._ [2022b] G. Colangelo, M. Hoferichter, B. Kubis, and P. Stoffer, Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization, JHEP 10, 032, arXiv:2208.08993 [hep-ph] .
  * Ignatov  _et al._ [2023] F. V. Ignatov _et al._ (CMD-3), Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section from threshold to 1.2 GeV with the CMD-3 detector, (2023), arXiv:2302.08834 [hep-ex] .
  * Benton  _et al._ [2023] G. Benton, D. Boito, M. Golterman, A. Keshavarzi, K. Maltman, and S. Peris, Data-driven estimates for light-quark-connected and strange-plus-disconnected hadronic $g-2$ window quantities, (2023), arXiv:2311.09523 [hep-ph] .
  * Hoferichter  _et al._ [2023] M. Hoferichter, G. Colangelo, B.-L. Hoid, B. Kubis, J. R. de Elvira, D. Schuh, D. Stamen, and P. Stoffer, Phenomenological Estimate of Isospin Breaking in Hadronic Vacuum Polarization, Phys. Rev. Lett. 131, 161905 (2023), arXiv:2307.02532 [hep-ph] .