Data-driven estimates for light-quark-connected and strange-plus-disconnected
hadronic $g-2$ window quantities
Genessa Benton,a Diogo Boito,b Maarten Golterman,a Alex Keshavarzi,c Kim
Maltman,d,e and Santiago Perisa,f

aDepartment of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

bInstituto de Física de São Carlos, Universidade de São Paulo

CP 369, 13570-970, São Carlos, SP, Brazil

cDepartment of Physics and Astronomy, The University of Manchester,

Manchester M13 9PL, United Kingdom

dDepartment of Mathematics and Statistics, York University,

Toronto, ON Canada M3J 1P3

eCSSM, University of Adelaide, Adelaide, SA 5005 Australia

fDepartment of Physics and IFAE-BIST, Universitat Autònoma de Barcelona,

E-08193 Bellaterra, Barcelona, Spain

> A number of discrepancies have emerged between lattice computations and
> data-driven dispersive evaluations of the RBC/UKQCD intermediate-window-
> hadronic contribution to the muon anomalous magnetic moment. It is therefore
> interesting to obtain data-driven estimates for the light-quark-connected
> and strange-plus-disconnected components of this window quantity, allowing
> for a more detailed comparison between the lattice and data-driven
> approaches. The aim of this paper is to provide these estimates, extending
> the analysis to several other window quantities, including two windows
> designed to focus on the region in which the two-pion contribution is
> dominant. Clear discrepancies are observed for all light-quark-connected
> contributions considered, while good agreement with lattice results is found
> for strange-plus-disconnected contributions to the quantities for which
> corresponding lattice results exist. The largest of these discrepancies is
> that for the RBC/UKQCD intermediate window, where, as previously reported,
> our data-driven result, $a_{\mu}^{W1,{\rm lqc}}=198.9(1.1)\times 10^{-10}$,
> is in significant tension with the results of 8 different recent lattice
> determinations. Our strategy is the same as recently employed in obtaining
> data-driven estimates for the light-quark-connected and strange-plus-
> disconnected components of the full leading-order hadronic vacuum
> polarization contribution to the muon anomalous magnetic moment. Updated
> versions of those earlier results are also presented, for completeness.

##  I  Introduction

There have been many developments since the publication of the first of the
Fermilab E989 measurements of the muon anomalous magnetic moment [2, 3] and
the publication of the White-Paper (WP) Standard-Model (SM) estimate [4] that
preceded it. A lattice computation of the leading-order hadronic vacuum
polarization (HVP) contribution $a_{\mu}^{\rm HVP}$ by the BMW collaboration
resulted in a value that would bring the total SM expectation much closer to
the experimental value [5], if one assumes that the $5\sigma$ discrepancy
between the world average of the experimental value111Essentially that of the
Brookhaven E821 [6] and Fermilab E989 [2, 3] experiments. and the WP value is
due to the leading-order HVP contribution. In addition, there now exist a
large number of lattice computations [5, 7, 8, 9, 10, 11, 12, 13, 14] of the
RBC/UKQCD intermediate window quantity [15] that are in relatively good
agreement with each other, but not in agreement with data-driven estimates
[16].

The lattice computation of $a_{\mu}^{\rm HVP}$ is carried out by breaking down
the total contribution into several building blocks. The primary building
blocks are the isospin-symmetric light- and strange-quark connected and three-
flavor disconnected parts, together with smaller charm and bottom
contributions (both of which also have connected and disconnected components).
Electromagnetic (EM) and strong-isospin-breaking (SIB) effects (collectively,
isospin-breaking (IB) effects) are taken into account perturbatively, with
corrections linear in the fine-structure constant $\alpha$ and the up-down
quark mass difference $m_{u}-m_{d}$ sufficiently precise for the current
desired level of accuracy. In contrast, the data-driven dispersive approach is
based on an analysis of hadronic electro-production data on a channel-by-
channel basis ($\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}\pi^{0}$, etc.) up to squared
hadronic invariant masses, $s$, just below $4$ GeV2 and inclusive data above
that point. To gain a better understanding of the emerging discrepancies, it
is useful to find out in more detail which of the lattice components are most
strongly in tension with data-driven estimates. For this, it is necessary to
reorganize the data-driven approach such as to provide direct estimates for
the building blocks that constitute the lattice-based computation of
$a_{\mu}^{\rm HVP}$. It is, in addition, of interest to consider auxillary
lattice quantities that focus on the region in which the two-pion contribution
dominates, in order to further sharpen our understanding of the source(s) of
the lattice-dispersive discrepancy.

In recent work, we have obtained such data-driven estimates for the isospin-
symmetric light-quark connected (“lqc” for short) [17] and strange connected
plus (three-flavor-) disconnected contributions [18] to $a_{\mu}^{\rm HVP}$.
We refer to the latter as the strange plus disconnected contribution in what
follows, or “s+lqd” for short. While the strange plus disconnected
contribution was found to be in good agreement with most lattice computations
(with comparable errors), the light-quark connected contribution differed
significantly from the lattice result of Ref. [5].222At present, no other
lattice computations claim sufficient control over all systematic errors to
make a meaningful comparison. This state of affairs provides a strong
motivation for obtaining similar light-quark connected and strange plus
disconnected data-driven estimates for various window quantities such as the
RBC/UKQCD intermediate window, where, as mentioned above, a growing number of
lattice collaborations finds values not in agreement with data-driven
estimates.

The aim of this paper is to provide data-driven values for a number of light-
quark connected and strange plus disconnected window quantities, specifically,
the original RBC/UKQCD intermediate window quantity, the longer-distance
window quantity proposed in Ref. [10], and two window quantities introduced in
Ref. [19], and compare these with corresponding lattice values where
available. The result for the data-based light-quark connected contribution to
the intermediate window obtained with our method has already been reported in
a recent letter [20]. Here we extend the results to other windows, to the
strange plus disconnected contributions and provide several additional
details.

The analysis of Refs. [17, 18] was based on the tabulated exclusive-mode
hadronic contributions to $a_{\mu}^{\rm HVP}$ made available in Refs. [21]
(DHMZ) and [22] (KNT). Such a tabulation is not publicly available for any of
the window quantities considered in this paper. One therefore needs access to
the exclusive-mode spectral distributions for all hadronic channels
contributing to the HVP in order to evaluate the channel-by-channel
contributions to these window quantities, which then, following the strategy
of Refs. [17, 18], can be used to obtain the light-quark connected and strange
plus disconnected contributions to these window quantities. In this paper, we
will use the exclusive-mode spectral distributions and covariances from the
2019 KNT analysis (KNT19 in what follows). As analogous exclusive-mode
information from DHMZ is not available to us, we are unable to present
estimates based on DHMZ data. In this respect the analysis presented here
differs from that in Refs. [17, 18], in which both KNT19- and DHMZ-based
values for the light-quark connected and strange plus disconnected
contributions to $a_{\mu}^{\rm HVP}$ were obtained.

Of course, both the KNT and DHMZ data contain EM and SIB effects. These have
to be estimated and subtracted in order to arrive at purely hadronic and
isospin-symmetric estimates for the various window quantities. Both types of
corrections have been recently considered in Refs. [23, 24, 25, 26]. The
existence of as-yet-unquantified exclusive-mode contributions (see, e.g., the
discussions in Refs. [24, 25] and the appendix of Ref. [17]) means
insufficient experimental information is currently available to extract
inclusive EM corrections, and we thus choose to rely on available lattice
estimates for the EM corrections we employ; these will be the only lattice
data we will make use of in this paper. As we will see, these corrections are
very small. Additional IB corrections, in which a combination of EM and SIB
effects occur, are amenable to data-driven treatment. Useful information on
the sum of EM and SIB contributions in the two-pion channel, which is expected
to be dominated by its SIB component, is provided by Ref. [23]. We will use
these results in assessing and subtracting this sum. Other such EM+SIB
corrections will also be discussed below. We will use a definition of QCD in
the isospin limit in which the pion mass is equal to the physical $\pi^{0}$
mass. To first order in IB, i.e., to ${\cal O}(\alpha)$ and ${\cal
O}(m_{u}-m_{d})$, this is sufficient to define an unambiguous split between EM
and SIB corrections.

This paper is organized as follows. In Sec. II we define the window quantities
of interest in this paper (Sec. II.1), and remind the reader how to relate the
lqc and s+lqd parts of the EM spectral function to its isospin components. In
Sec. III we present our results, based on the KNT19 data, for the exclusive-
channel contributions to the lqc and s+lqd parts of our window quantities in
the region below $s=(1.937)^{2}$ GeV2, treating first the modes for which the
isospin can be identified through $G$-parity (Sec. III.1), and then the
isospin-ambiguous modes (Sec. III.2). Above $s=(1.937)^{2}$ GeV2 we employ
perturbation theory, which is discussed in Sec. III.3. IB corrections are
discussed in detail in Sec. IV. We then compare our results with recent
lattice results for the same quantities in Sec. VI, and conclude in Sec. VII.
There are two appendices, one tabulating intermediate results in addition to
those explained in the main text, and one with a more detailed discussion of
the $\pi^{0}\gamma$ and $\eta\gamma$ exclusive modes.

##  II  Review

We define the window quantities we will consider in this paper in Sec. II.1,
and recall how to write the light-quark connected and strange plus
disconnected spectral functions in terms of the $I=1$ and $I=0$ components.

###  II.1  Windows

The leading-order HVP contribution $a_{\mu}^{\rm HVP}$ to $a_{\mu}$ can be
written, in dispersive form, as [27, 28, 29]

$a_{\mu}^{\rm
HVP}=\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\rho_{\rm
EM}(s)\ ,$ (1)

where $m_{\pi}$ is the neutral pion mass. Here $\rho_{\rm EM}(s)$ is the
inclusive EM-current hadronic spectral function

$\displaystyle\rho_{\rm EM}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi^{2}}R(s)\ ,$ (2) $\displaystyle R(s)$
$\displaystyle=$
$\displaystyle\frac{3s}{4\pi\alpha^{2}}\,\sigma^{(0)}[e^{+}e^{-}\rightarrow\mbox{hadrons}(+\gamma)]\
,$

where $R(s)$ is the $R$ ratio obtained from the bare inclusive hadronic
electroproduction cross section
$\sigma^{(0)}[e^{+}e^{-}\rightarrow\mbox{hadrons}(+\gamma)]$, and $\hat{K}(s)$
is a known smoothly varying kernel with $\hat{K}(4m_{\pi}^{2})\approx 0.63$ at
the two-pion threshold and $\lim_{s\to\infty}\hat{K}(s)=1$.333For various
explicit expressions, see e.g. Ref. [4]. Equivalently, $a_{\mu}^{\rm HVP}$ can
be expressed in terms of the Euclidean-time two-point correlation function
[30]

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

of the EM current, $j_{i}^{\rm EM}(\vec{x},t)$, as

$a_{\mu}^{\rm HVP}=2\int_{0}^{\infty}dt\,w(t)C(t)\ ,$ (4)

where $w(t)$ is a known function related to $\hat{K}(s)$ by444For an explicit
expression as well as an approximation accurate to better than one part in
$3\times 10^{-6}$, see Ref. [31].

$\frac{\hat{K}(s)}{s^{2}}=\frac{3\sqrt{s}}{4\alpha^{2}m_{\mu}^{2}}\int_{0}^{\infty}dt\,w(t)\,e^{-\sqrt{s}t}\
.$ (5)

We note that $C(t)$ has a $\delta$-function singularity at $t=0$, but this
does not contribute to Eq. (4) since $w(t)\sim t^{4}$ for $t\to 0$.

Our first class of window quantities is defined by inserting the window
function [15]

$W(t;t_{0},t_{1};\Delta)=\frac{1}{2}\left(\tanh\frac{t-t_{0}}{\Delta}-\tanh\frac{t-t_{1}}{\Delta}\right)$
(6)

with $t_{1}>t_{0}>0$ into Eq. (4):

$\displaystyle a_{\mu}^{W}(t_{0},t_{1};\Delta)$ $\displaystyle=$
$\displaystyle 2\int_{0}^{\infty}dt\,W(t;t_{0},t_{1};\Delta)\,w(t)C(t)$
$\displaystyle=$ $\displaystyle\int_{m_{\pi}^{2}}^{\infty}ds\,\rho_{\rm
EM}(s)\,\sqrt{s}\int_{0}^{\infty}dt\,W(t;t_{0},t_{1};\Delta)\,w(t)\,e^{-\sqrt{s}t}$
$\displaystyle=$
$\displaystyle\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\,\widetilde{W}(s;t_{0},t_{1};\Delta)\,\rho_{\rm
EM}(s)\ ,$

where

$\widetilde{W}(s;t_{0},t_{1};\Delta)=\frac{\int_{0}^{\infty}dt\,W(t;t_{0},t_{1};\Delta)\,w(t)\,e^{-\sqrt{s}t}}{\int_{0}^{\infty}dt\,w(t)\,e^{-\sqrt{s}t}}$
(8)

is the window function in $s$-space.

In what follows, we will consider two window quantities of this type,
$a_{\mu}^{W1}$ and $a_{\mu}^{W2}$, corresponding to the window functions $W1$
and $W2$ obtained using the choices

$\displaystyle W1:\qquad$ $\displaystyle t_{0}=0.4~{}\mbox{fm}\ ,\quad
t_{1}=1.0~{}\mbox{fm}\ ,\quad\Delta=0.15~{}\mbox{fm}\ ,$ (9a) $\displaystyle
W2:\qquad$ $\displaystyle t_{0}=1.5~{}\mbox{fm}\ ,\quad t_{1}=1.9~{}\mbox{fm}\
,\quad\Delta=0.15~{}\mbox{fm}\ ,$ (9b)

for the external parameters, $t_{0}$, $t_{1}$ and $\Delta$, in Eq. (6). The
first of these, $a_{\mu}^{W1}$, is the RBC/UKQCD intermediate window quantity
introduced in Ref. [15]. The second, $a_{\mu}^{W2}$, is the alternate longer
distance “intermediate” window quantity introduced in Ref. [10] designed to be
better suited to treatment using chiral perturbation theory.

We will also consider window quantities $I_{\widehat{W}}$,

$I_{\widehat{W}}=\int_{m_{\pi}^{2}}^{\infty}ds\,\widehat{W}(s)\,\rho_{\rm
EM}(s)\ ,$ (10)

involving weights, $\widehat{W}(s)$, defined directly as functions of $s$ and
having the form555The design of such weights in Ref. [19] was inspired by the
work of Ref. [32].

$\widehat{W}(s;\\{t_{i}\\},\\{x_{i}\\})=\frac{1}{2}\sum_{i=1}^{n}x_{i}\sqrt{s}e^{-t_{i}\sqrt{s}}\
,$ (11)

with $\\{t_{i}\\}$ a set of positive numbers. Using Eq. (3) it follows that

$I_{\widehat{W}}=\sum_{i=1}^{n}x_{i}C(t_{i})\ ,$ (12)

thus providing a sum rule that allows for a comparison between the spectral
integral $I_{\widehat{W}}$ and the weighted sum of $n$ values of the
correlation function $C(t)$ that can be computed on the lattice [19]. By
choosing $n$ and the sets $\\{t_{i}\\}$ and $\\{x_{i}\\}$ judiciously, weights
can be constructed to focus on particular regions of interest in $s$. A key
advantage of the form Eq. (11) is that it turns out to be possible to obtain
such localization in $s$ using sets $\\{t_{i}\\}$ which entirely avoid the
large-$t$ region in which lattice $C(t)$ errors deteriorate [19], thus
simultaneously controlling errors on the lattice side of the sum rule Eq.
(12). The lattice errors corresponding to four such choices,
$W^{\prime}_{15}$, $W^{\prime}_{25}$, $\widehat{W}_{15}$ and
$\widehat{W}_{25}$, all focusing on the region around the $\rho$ peak, were
investigated in Ref. [19], using available light-quark connected $C(t)$
results, with $\widehat{W}_{15}$ and $\widehat{W}_{25}$ found to produce the
smallest relative lattice errors. We thus focus on these cases in what
follows. Both involve an $n=5$-fold sum, and the choice

$\\{t_{i}\\}=\\{3,\ 6,\ 9,\ 12,\ 15\\}\ \mbox{GeV}^{-1}$ (13)

for the set $\\{t_{i}\\}$. The weight $\widehat{W}_{15}$ is then obtained by
choosing

$\\{x_{i}^{(15)}\\}=\\{-78.8487,\ 5688.30,\ 2223.96,\ -36638.0,\ 8047.38\\}\
,$ (14)

and $\widehat{W}_{25}$ by choosing

$\\{x_{i}^{(25)}\\}=\\{44.8916,\ 590.933,\ -3373.53,\ 3716.86,\ 8047.38,\
879.149\\}\ .$ (15)

With the inclusive spectral function $\rho_{\rm EM}(s)$ one can obtain data-
driven values for $a_{\mu}^{W1}$, $a_{\mu}^{W2}$, $I_{\widehat{W}_{15}}$ and
$I_{\widehat{W}_{25}}$. For $a_{\mu}^{W1}$ (and a number of other weights
which we will not consider here) this was done in Ref. [16]. Here we are
interested in obtaining, for each of the weights we consider, the light-quark
connected and strange plus disconnected building blocks.

###  II.2  Light-quark connected and strange plus disconnected spectral
functions

First, we review the ingredients of the basic idea from Refs. [17, 18]. The
decomposition of the three-flavor EM current into its isospin $I=1$ and $I=0$
parts, produces related decompositions of $C(t)$ and $\rho_{\rm EM}(s)$, into
pure $I=1$, pure $I=0$ and mixed-isospin parts

$\displaystyle C(t)$ $\displaystyle=$ $\displaystyle C^{I=1}(t)+C^{\rm
MI}(t)+C^{I=0}(t)\ ,$ (16) $\displaystyle\rho_{\rm EM}(s)$ $\displaystyle=$
$\displaystyle\rho_{\rm EM}^{I=1}(s)+\rho_{\rm EM}^{\rm MI}(s)+\rho_{\rm
EM}^{I=0}(s)\ ,$

where in an isospin symmetric world the mixed-isospin (MI) components vanish.
Weighted integrals over $\rho_{\rm EM}$, of course, inherit this same
decomposition.

In the isospin limit, the $I=0$ contribution to the light-quark connected
(lqc) part of $C(t)$ is exactly $1/9$ times the corresponding $I=1$
contribution. The strange (connected plus disconnected) plus light-quark
disconnected (s+lqd) contribution is, similarly, the difference of the $I=0$
contribution and $1/9$ times the $I=1$ contribution. The lqc and s+lqd window
quantities considered in this paper, which are of the form (II.1) with $W=W1$
or $W=W2$, or of the form (10) with $\widehat{W}=\widehat{W}_{15}$ or
$\widehat{W}=\widehat{W}_{25}$, are thus given, in the isospin limit, by the
expressions

$\displaystyle a_{\mu}^{W,\rm lqc}$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\,\widetilde{W}(s;t_{0},t_{1};\Delta)\,\rho_{\rm
EM}^{\rm lqc}(s)\ ,$ (17) $\displaystyle a_{\mu}^{W,\rm s+lqd}$
$\displaystyle=$
$\displaystyle\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\,\widetilde{W}(s;t_{0},t_{1};\Delta)\,\rho_{\rm
EM}^{\rm s+lqd}(s)\ ,$

and

$\displaystyle{\cal{I}}_{\widehat{W}}^{\rm lqc}$ $\displaystyle=$
$\displaystyle\int_{m_{\pi}^{2}}^{\infty}ds\,\widehat{W}(s)\,\rho^{\rm
lqc}_{\rm EM}(s)\ ,$ (18) $\displaystyle{\cal{I}}_{\widehat{W}}^{\rm s+lqd}$
$\displaystyle=$
$\displaystyle\int_{m_{\pi}^{2}}^{\infty}ds\,\widehat{W}(s)\,\rho^{\rm
s+lqd}_{\rm EM}(s)\ ,$

where

$\rho^{\rm lqc}_{\rm EM}(s)=\frac{10}{9}\rho_{\rm EM}^{I=1}(s)$ (19)

and

$\rho^{\rm s+lqd}_{\rm EM}(s)=\rho_{\rm EM}^{I=0}(s)-\frac{1}{9}\rho_{\rm
EM}^{I=1}(s)=\rho_{\rm EM}(s)-\frac{10}{9}\rho_{\rm EM}^{I=1}(s)\ .$ (20)

The evaluation of the lqc and s+lqd parts of the four window quantities
defined in Sec. II.1 thus requires an identification of the separate $I=1$ and
$I=0$ components of $\rho_{\rm EM}(s)$. The separation of contributions from
all hadronic exclusive modes can, assuming isospin symmetry, be accomplished,
up to $\sqrt{s}=1.937$ GeV, using KNT19 exclusive-mode data, as described in
the following section. We will then, in Sec. IV, discuss the EM and SIB
corrections to our values for the lqc and s+lqd parts of our window
quantities.

##  III  Implementation

In this section, we obtain the lqc and s+lqd parts for all four window
quantities, postponing until Sec. IV a consideration of EM and SIB
corrections. In Sec. III.1 we collect exclusive-mode contributions from modes
which are $G$-parity eigenstates and hence have an unambiguous $I=0$ or $1$
assignment. The separation of contributions from modes which are not
$G$-parity eigenstates into separate $I=0$ and $I=1$ components is detailed in
Sec. III.2.

###  III.1  Modes with unambiguous isospin

As in Ref. [18], we take advantage of the fact that exclusive modes with
positive (negative) $G$-parity have $I=1$ ($I=0$). The contributions of such
modes to the $I=1$ and $I=0$ parts of $a_{\mu}^{W1}$ are shown in Table 1.
Analogous tables are provided for the other window quantities in App. A. From
these results, we obtain the following $G$-parity unambiguous contributions to
our window quantities:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{G\rm{\mathchar 45\relax
par}}$ $\displaystyle=$ $\displaystyle\frac{10}{9}\times 168.24(72)\times
10^{-10}=186.94(80)\times 10^{-10}\ ,$ $\displaystyle\left[a_{\mu}^{W2,\rm
lqc}\right]_{G\rm{\mathchar 45\relax par}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\times 85.05(33)\times 10^{-10}=94.50(36)\times
10^{-10}\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
lqc}\right]_{G\rm{\mathchar 45\relax par}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\times 39.37(14)\times 10^{-2}=43.75(15)\times
10^{-2}\ ,$ $\displaystyle\left[I_{\widehat{W}_{25}}^{\rm
lqc}\right]_{G\rm{\mathchar 45\relax par}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\times 67.29(30)\times 10^{-3}=74.76(34)\times
10^{-3}\ ,$ (21)

and

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{G\rm{\mathchar 45\relax
par}}$ $\displaystyle=$
$\displaystyle\left[20.69(37)-\frac{168.24(72)}{9}\right]\times
10^{-10}=1.99(38)\times 10^{-10}\ ,$ $\displaystyle\left[a_{\mu}^{W2,\rm
s+lqd}\right]_{G\rm{\mathchar 45\relax par}}$ $\displaystyle=$
$\displaystyle\left[6.23(14)-\frac{85.05(33)}{9}\right]\times
10^{-10}=-3.22(14)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{G\rm{\mathchar
45\relax par}}$ $\displaystyle=$
$\displaystyle\left[3.77(08)-\frac{39.37(14)}{9}\right]\times
10^{-2}=-0.609(84)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{G\rm{\mathchar
45\relax par}}$ $\displaystyle=$
$\displaystyle\left[8.92(16)-\frac{67.29(30)}{9}\right]\times
10^{-3}=1.45(16)\times 10^{-3}\ ,$ (22)

$I=1$ modes $X$ | $[a_{\mu}^{W1}]_{X}\times 10^{10}$ |  $I=0$ modes $X$ | $[a_{\mu}^{W1}]_{X}\times 10^{10}$
---|---|---|---
low-$s$ $\pi^{+}\pi^{-}$ | 0.02(00) | low-$s$ $3\pi$ | 0.00(00)
$\pi^{+}\pi^{-}$ | 144.13(49) | $3\pi$ | 18.69(35)
$2\pi^{+}2\pi^{-}$ | 9.29(13) |  $2\pi^{+}2\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.61(06)
$\pi^{+}\pi^{-}2\pi^{0}$ | 11.94(48) |  $\pi^{+}\pi^{-}3\pi^{0}$ (no $\eta$) | 0.39(07)
$3\pi^{+}3\pi^{-}$ (no $\omega$) | 0.14(01) |  $3\pi^{+}3\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.00(00)
$2\pi^{+}2\pi^{-}2\pi^{0}$ (no $\eta$) | 0.83(11) |  $\eta\pi^{+}\pi^{-}\pi^{0}$ (no $\omega$) | 0.44(05)
$\pi^{+}\pi^{-}4\pi^{0}$ (no $\eta$) | 0.13(13) | $\eta\omega$ | 0.19(01)
$\eta\pi^{+}\pi^{-}$ | 0.85(03) | $\omega(\rightarrow npp)2\pi$ | 0.08(01)
$\eta 2\pi^{+}2\pi^{-}$ | 0.05(01) | $\omega 2\pi^{+}2\pi^{-}$ | 0.00(00)
$\eta\pi^{+}\pi^{-}2\pi^{0}$ | 0.07(01) | $\eta\phi$ | 0.25(01)
$\omega(\rightarrow\pi^{0}\gamma)\pi^{0}$ | 0.53(01) | $\phi\rightarrow unaccounted$ | 0.02(02)
$\omega(\rightarrow npp)3\pi$ | 0.10(02) |  |
$\omega\eta\pi^{0}$ | 0.15(03) |  |
TOTAL: | 168.24(72) | TOTAL: | 20.69(37)
Table 1: $G$-parity-unambiguous exclusive-mode contributions to $a_{\mu}^{W1}$
for $\sqrt{s}\leq 1.937$ GeV using KNT19 exclusive-mode data. Entries in units
of $10^{-10}$. The notation “npp” is KNT shorthand for “non-purely-pionic.”

###  III.2  Modes with ambiguous isospin

We now consider those exclusive modes which are not $G$-parity eigenstates,
and thus have no definite isospin. Associated contributions to the various
window quantities thus, in general, have both $I=1$ and $I=0$ components,
which must be separated to complete determinations of the corresponding lqc
and s+lqd contributions. Modes of this type in the KNT19 exclusive-mode region
are $\pi^{0}\gamma$, $\eta\gamma$, $N\bar{N}$ and those containing a
$K\bar{K}$ pair.

For the numerically most important modes of the latter type, $K\bar{K}$ and
$K\bar{K}\pi$, the $I=0/I=1$ separation is facilitated by use of additional
experimental input ($\tau^{-}\rightarrow K^{-}K^{0}\nu_{\tau}$ data [33] for
$K\bar{K}$ and BaBar Dalitz plot analysis results [34] for $K\bar{K}\pi$),
following the strategy of Ref. [18], outlined briefly below.

For the modes $P\gamma$ with $P=\pi^{0},\,\eta$, the cross sections in the
KNT19 exclusive-mode region of relevance to the weighted integrals considered
in this paper are strongly dominated by contributions involving intermediate
$V=\rho$, $\omega$ and $\phi$ meson exchange. Vector-meson-dominance (VMD)
representations of the $e^{+}e^{-}\rightarrow P\gamma$ cross sections, which
require as input only $m_{P}$, the vector meson masses and widths and known
experimental $V\rightarrow e^{+}e^{-}$ and $V\rightarrow P\gamma$ decay
widths, turn out to saturate the corresponding KNT19 exclusive-mode integrals
considered here, confirming the reliability of the VMD representation. The
$I=0$, $I=1$ and MI contributions to those integrals can thus be determined
from the parts of the VMD representations involving the squared modulus of the
sum of $\omega$ and $\phi$ contributions to the amplitude, the squared modulus
of the $\rho$ contribution to the amplitude, and the interference terms
between those contributions, respectively. The VMD representation of the cross
sections, together with the numerical values of the required external inputs,
are detailed in Appendix B. As we will see below, the $I=0$ contribution
dominates these integrals for all four window quantities considered in this
paper.

For the (much smaller) contributions from other (“residual”) $G$-parity-
ambiguous exclusive-modes, $X$, where additional external experimental input
is not available, we follow Ref. [18] in employing a “maximally conservative”
assessment of the $I=1$/$I=0$ separation, based on the observation that the
$I=1$ part of the mode $X$ contribution to $\rho_{\rm EM}(s)$ necessarily lies
between $0$ and the full contribution. The mode $X$ lqc and s$+$lqd
contributions to $\rho_{\rm EM}(s)$ then necessarily lie in the ranges:

$\displaystyle\left[\rho^{\rm lqc}_{\rm EM}\right]_{X}$ $\displaystyle=$
$\displaystyle\left(\frac{5}{9}\pm\frac{5}{9}\right)\left[\rho_{\rm
EM}\right]_{X}\ ,$ (23) $\displaystyle\left[\rho^{\rm s+lqd}_{\rm
EM}\right]_{X}$ $\displaystyle=$
$\displaystyle\left(\frac{4}{9}\pm\frac{5}{9}\right)\left[\rho_{\rm
EM}\right]_{X}\ .$

These bounds produce related “maximally conservative” bounds for weighted
exclusive-mode lqc and s+lqd integrals involving weights, $W$, having fixed
sign in the region of nonzero $\left[\rho_{\rm EM}(s)\right]_{X}$:666The
weights $W1(s)$, $W2(s)$ and $\widehat{W}_{25}(s)$ are positive for all $s$.
$\widehat{W}_{15}(s)$, however, crosses zero, becoming small and negative
above $s\simeq 2.04$ GeV2. We have checked that non-negligible KNT19 results
for $\left[\rho_{\rm EM}(s)\right]_{X}$ for all modes $X$ for which the
maximally conservative assessment has been employed lie in regions with fixed
sign for $\widehat{W}_{15}(s)$, making the bounds of Eq. (24) valid for
$\widehat{W}_{15}$ as well.

$\displaystyle\left[a_{\mu}^{W,\rm lqc}\right]_{X}$ $\displaystyle=$
$\displaystyle\left(\frac{5}{9}\pm\frac{5}{9}\right)\left[a_{\mu}^{W}\right]_{X}\
,$ (24) $\displaystyle\left[a_{\mu}^{W,\rm s+lqd}\right]_{X}$ $\displaystyle=$
$\displaystyle\left(\frac{4}{9}\pm\frac{5}{9}\right)\left[a_{\mu}^{W}\right]_{X}\
.$

The maximally conservative $I=1$/$I=0$ separation bounds, while generally
valid, would, if applied to all $G$-parity-ambiguous modes, produce errors so
large that our estimates for the lqc and s+lqd parts of the window quantities
would be uninteresting. Fortunately, for those modes where this would be an
issue, more experimental information is available, allowing us to dramatically
reduce the uncertainty on the $I=1$/$I=0$ separation.

Let us first consider the $K\bar{K}$ modes, $K^{+}K^{-}$ and
$K^{0}\bar{K}^{0}$. Independent information on the $K\bar{K}$ contribution to
the $I=1$ spectral function is available from data on the differential
distribution for the decay $\tau\to K^{-}K^{0}\nu_{\tau}$ measured by BaBar
[33]. Using the CVC (conserved vector current) relation, these results can be
converted into the $I=1$ $e^{+}e^{-}\to K\bar{K}$ contribution to
$R(s)$.777Isospin-breaking corrections to the CVC relation are negligible for
our purposes. Using the results of Ref. [33] up to $s=2.7556$ GeV2, and a
KNT19-based maximally conservative assessment of $I=1$ contributions for $s$
from $2.7556$ GeV2 to $s=(1.937)^{2}$ GeV2, we find, following the steps of
Sec. IV.B of Ref. [18], the results

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{K\bar{K}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\left(0.465(29)+0.055(55)\right)\times
10^{-10}=0.578(69)\times 10^{-10}\ ,$ (25) $\displaystyle\left[a_{\mu}^{W2,\rm
lqc}\right]_{K\bar{K}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\left(0.0241(17)+0.00035(35)\right)\times
10^{-10}=0.0271(20)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{K\bar{K}}$
$\displaystyle=$
$\displaystyle\frac{10}{9}\left(0.0158(12)-0.0023(23)\right)\times
10^{-2}=0.0150(29)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{K\bar{K}}$
$\displaystyle=$
$\displaystyle\frac{10}{9}\left(0.209(13)+0.024(24)\right)\times
10^{-3}=0.259(30)\times 10^{-3}\ ,$

where in each case the first number in parentheses is the contribution from
the BaBar $\tau$ data in the region $s\leq 2.7556$ GeV2, and the second number
is the result of the maximally conservative assessment of the contribution
from $s=2.7556$ GeV2 to $(1.937)^{2}$ GeV2, obtained using KNT19 data.

For the s+lqd parts, using the second equation of Eq. (20), we find

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{K\bar{K}}$ $\displaystyle=$
$\displaystyle\left(19.13(15)-0.578(69)\right)\times 10^{-10}=18.55(17)\times
10^{-10}\ ,$ (26) $\displaystyle\left[a_{\mu}^{W2,\rm
s+lqd}\right]_{K\bar{K}}$ $\displaystyle=$
$\displaystyle\left(2.612(21)-0.0271(20)\right)\times 10^{-10}=2.585(21)\times
10^{-10}\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
s+lqd}\right]_{K\bar{K}}$ $\displaystyle=$
$\displaystyle\left(2.213(18)-0.015(29)\right)\times 10^{-2}=2.198(18)\times
10^{-2}\ ,$ $\displaystyle\left[I_{\widehat{W}_{25}}^{\rm
s+lqd}\right]_{K\bar{K}}$ $\displaystyle=$
$\displaystyle\left(8.642(70)-0.259(30)\right)\times 10^{-3}=8.383(76)\times
10^{-3}\ ,$

where the first number in parentheses in each case is the full $K^{+}K^{-}$
plus $K^{0}\bar{K}^{0}$ contribution from the KNT19 data, and the second
number is that of Eq. (25). By comparing Eq. (25), which is pure $I=1$, and
Eq. (26), which is dominated by $I=0$, we see that clearly the $K\bar{K}$
channels are dominated by $I=0$. This is expected because of the $I=0$ $\phi$
resonance.

Our next case to consider is that of the $K\bar{K}\pi$ modes. The separation
of the corresponding cross sections into their $I=0$ and $I=1$ parts was
carried out by BaBar in Ref. [34], and, as in Ref. [18], we obtain the $I=1$
(and hence the lqc) contributions, up to $s=(1.937)^{2}\ {\rm GeV}^{2}$,

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{K\bar{K}\pi}$
$\displaystyle=$
$\displaystyle\frac{10}{9}\left[a_{\mu}^{W1,I=1}\right]_{K\bar{K}\pi}=0.521(86)\times
10^{-10}\ ,$ (27) $\displaystyle\left[a_{\mu}^{W2,\rm
lqc}\right]_{K\bar{K}\pi}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\left[a_{\mu}^{W2,I=1}\right]_{K\bar{K}\pi}=0.0060(10)\times
10^{-10}\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
lqc}\right]_{K\bar{K}\pi}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\left[I_{\widehat{W}_{15}}^{I=1}\right]_{K\bar{K}\pi}=-0.0152(25)\times
10^{-2}\ ,$ $\displaystyle\left[I_{\widehat{W}_{25}}^{\rm
lqc}\right]_{K\bar{K}\pi}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\left[I_{\widehat{W}_{25}}^{I=1}\right]_{K\bar{K}\pi}=0.229(38)\times
10^{-3}\ .$

For the s+lqd $K\bar{K}\pi$ contributions from the same region, we find

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{K\bar{K}\pi}$
$\displaystyle=$ $\displaystyle\left(1.714(74)-0.521(86)\right)\times
10^{-10}=1.19(11)\times 10^{-10}\ ,$ (28) $\displaystyle\left[a_{\mu}^{W2,\rm
s+lqd}\right]_{K\bar{K}\pi}$ $\displaystyle=$
$\displaystyle\left(0.01846(82)-0.0060(10)\right)\times
10^{-10}=0.0124(13)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{K\bar{K}\pi}$
$\displaystyle=$ $\displaystyle\left(-0.0536(24)+0.0152(25)\right)\times
10^{-2}=-0.0384(35)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{K\bar{K}\pi}$
$\displaystyle=$ $\displaystyle\left(0.753(33)-0.229(38)\right)\times
10^{-3}=0.524(50)\times 10^{-3}\ ,$

where the first number in parentheses in each case is the full $K\bar{K}\pi$
contribution obtained using KNT19 data, and the second number is that of Eq.
(27).

A small improvement, relative to the maximally conservative assessment (24),
can also be obtained for contributions from the $K\bar{K}2\pi$ modes by making
use of the measured $e^{+}e^{-}\to\phi\pi\pi$ mode cross sections [35], which
allow the purely $I=0$ contribution resulting from
$e^{+}e^{-}\to\phi[\rightarrow K\bar{K}]\pi\pi$ to be subtracted from the
KNT19 $K\bar{K}2\pi$ total and the maximally conservative separation into
$I=1$ and $I=0$ components applied only to the remainder. Following Ref. [18],
we find888Note that the values, $49.2\%$ and $83.2\%$, for the $\phi\to
K^{+}K^{-}$ and two-mode $\phi\to K\bar{K}$ branching fractions, used in Ref.
[18] to update external input employed in the original BaBar analysis, have
been further updated to current PDG [36] values, $49.1\%$ and $83.0\%$,
respectively.

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle 0.60(60)\times 10^{-10}\ ,$ (29)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle 0.0031(31)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle-0.028(28)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle 0.26(26)\times 10^{-3}\ ,$

and

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle 0.58(60)\times 10^{-10}\ ,$ (30)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle 0.0032(31)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle-0.027(28)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{K\bar{K}2\pi}$
$\displaystyle=$ $\displaystyle 0.25(26)\times 10^{-3}\ .$

The final G-parity-ambiguous modes for which additional external experimental
input provides an improved isospin decomposition are the two radiative modes
$\pi^{0}\gamma$ and $\eta\gamma$. Using the accurate VMD representations of
the $e^{+}e^{-}\rightarrow\pi^{0}\gamma$ and $e^{+}e^{-}\rightarrow\eta\gamma$
cross sections detailed in Appendix B, and employing 2023 PDG input [36], we
find the following results for the lqc and s+lqd contributions from these
modes:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{\pi^{0}\gamma+\eta\gamma}$
$\displaystyle=$ $\displaystyle\left(0.074(13)+0.063(0)\right)\times
10^{-10}=0.137(13)\times 10^{-10}\ ,$ (31) $\displaystyle\left[a_{\mu}^{W2,\rm
lqc}\right]_{\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle\left(0.029(5)+0.021(0)\right)\times 10^{-10}=0.050(5)\times
10^{-10}\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
lqc}\right]_{\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle\left(0.018(3)+0.013(0)\right)\times 10^{-2}=0.031(3)\times
10^{-2}\ ,$ $\displaystyle\left[I_{\widehat{W}_{25}}^{\rm
lqc}\right]_{\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle\left(0.031(6)+0.028(0)\right)\times 10^{-3}=0.059(6)\times
10^{-3}\ ,$

and

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{\pi^{0}\gamma+\eta\gamma}$
$\displaystyle=$ $\displaystyle\left(1.25(6)+0.25(1)\right)\times
10^{-10}\,=\,1.50(6)\times 10^{-10}\ ,$ (32)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{\pi^{0}\gamma+\eta\gamma}$
$\displaystyle=$ $\displaystyle\left(0.54(3)+0.04(0)\right)\times
10^{-10}\,=\,0.58(3)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
s+lqd}\right]_{\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle\left(0.32(2)+0.03(0)\right)\times 10^{-2}\,=\,0.35(2)\times
10^{-2}\ ,$ $\displaystyle\left[I_{\widehat{W}_{25}}^{\rm
s+lqd}\right]_{\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle\left(0.53(3)+0.11(0)\right)\times 10^{-3}\,=\,0.64(3)\times
10^{-3}\ ,$

where the first term in each intermediate expression is the $\pi^{0}\gamma$
contribution and the second term the $\eta\gamma$ one.

Contributions from the remaining $G$-parity-ambiguous modes, $K\bar{K}3\pi$,
$\omega(\rightarrow npp)K\bar{K}$, $\eta(\rightarrow npp)K\bar{K}$ (no
$\phi$), $p\bar{p}$, $n\bar{n}$, and low-$s$ $\pi^{0}\gamma$ and $\eta\gamma$,
which are very small, are listed for completeness in Appendix A.

The sums of all exclusive-mode contributions below $s=(1.937)^{2}$ GeV2 for
the lqc window quantities are obtained from Eqs. (III.1), (25), (27), (29),
(31) and the further exclusive-mode contributions listed in Appendix A. The
results are

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{\rm excl}$ $\displaystyle=$
$\displaystyle 188.82(1.01)\times 10^{-10}\ ,$ (33)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{\rm excl}$ $\displaystyle=$
$\displaystyle 94.60(36)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{\rm excl}$
$\displaystyle=$ $\displaystyle 43.75(16)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{\rm excl}$
$\displaystyle=$ $\displaystyle 75.59(43)\times 10^{-3}\ .$

For the s+lqd window quantities, similarly, we obtain from Eqs. (III.1), (26),
(28), (30), (32) and the further exclusive-mode contributions listed in
Appendix A, the results

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{\rm excl}$ $\displaystyle=$
$\displaystyle 23.85(74)\times 10^{-10}\ ,$ (34)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{\rm excl}$ $\displaystyle=$
$\displaystyle-0.04(14)\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
s+lqd}\right]_{\rm excl}$ $\displaystyle=$ $\displaystyle 1.875(92)\times
10^{-2}\ ,$ $\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{\rm
excl}$ $\displaystyle=$ $\displaystyle 11.26(32)\times 10^{-3}\ .$

###  III.3 Perturbative contribution above $s=(1.937)^{2}$ GeV2

Above $s=(1.937)^{2}$ GeV2 we use perturbation theory to evaluate (three-
flavor) contributions to the window quantities. We employ massless
perturbation theory,999Mass corrections are negligibly small, see Ref. [18].
using the five-loop result of Ref. [37] for the Adler function, following the
steps outlined in Refs. [17, 18]. Above $s=(1.937)^{2}$ GeV2 and up to the
charm-quark threshold, perturbation theory agrees well with inclusive BES [38,
39] and KEDR [40] $R(s)$ measurements, although the agreement with recent,
more precise, BESIII results [41] is less good. We find for these perturbative
contributions, using $\alpha_{s}(m_{\tau}^{2})=0.3139(71)$ [36]

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle 10.896\times 10^{-10}\ ,$ (35)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle 0.00757\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle-0.574\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle 3.658\times 10^{-3}\ .$

Error estimates along the lines of Ref. [18] lead to errors of the order of
0.1% of the central values in Eq. (35), which are small enough that they can
be ignored in our final results. In order to obtain the s+lqd perturbative
contributions, the values in Eq. (35) have to be multiplied by
$(2/9)(9/10)=1/5$ [17]. For completeness, we provide the resulting values:

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle 2.179\times 10^{-10}\ ,$ (36)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle 0.00151\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle-0.115\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{\rm pert.th.}$
$\displaystyle=$ $\displaystyle 0.732\times 10^{-3}\ .$

Although the use of perturbation theory above $s=(1.937)^{2}$ GeV2 is
supported by the good agreement with the inclusive $R(s)$ data of Refs. [38,
39, 40], the slight tension with the recent BESIII data [41] hints at possible
residual duality violations (DVs) even in the inclusive region. While our
estimates for residual DV contributions to $a_{\mu}^{\rm HVP,lqc}$ and
$a_{\mu}^{\rm HVP,s+lqd}$ showed these to be small, DVs represent an intrinsic
limitation of perturbation theory. We will thus include the central values of
our DV estimates in our final results, assigning them an uncertainty of 100%.
The assigned DV uncertainty totally dominates our estimate of the uncertainty
associated with the use of perturbation theory in the inclusive region.

For lqc contributions, which involve only the $\rho_{\rm EM}^{I=1}(s)$
spectral function, we obtain our DV estimates using the results of finite-
energy sum-rule fits performed in Ref. [42] using an improved version of the
$I=1$ charged-current spectral function, $\rho_{ud;V}(s)$, obtained mainly
from $\tau$ decay data, and related by CVC to $\rho_{\rm EM}^{I=1}(s)$ by
$\rho_{\rm EM}^{I=1}(s)=\frac{1}{2}\rho_{ud;V}(s)$. Following the procedure
described in Sec. IIIA of Ref. [17], we find for our estimates of the DV
contributions to $a_{\mu}^{W,{\rm lqc}}$ and $I_{W}^{\rm lqc}$ the results

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{\rm DVs}$ $\displaystyle=$
$\displaystyle 0.162(77)\times 10^{-10}\ ,$ (37)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{\rm DVs}$ $\displaystyle=$
$\displaystyle 0.00034(20)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{\rm DVs}$
$\displaystyle=$ $\displaystyle-0.0092(44)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{\rm DVs}$
$\displaystyle=$ $\displaystyle 0.069(33)\times 10^{-3}\ .$

Possible residual DV corrections to the use of perturbation theory in the
inclusive region will be present in both the $I=1$ and $I=0$ contributions to
$a_{\mu}^{W,{\rm s+lqd}}$ and $I_{W}^{\rm s+lqd}$. To estimate the combination
of these effects, we will use the results of the finite-energy sum-rule fits
to KNT $\rho_{\rm EM}(s)$ data performed in Ref. [43]. We find

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{\rm DVs}$ $\displaystyle=$
$\displaystyle-0.17(5)\times 10^{-10}\ ,$ (38)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{\rm DVs}$ $\displaystyle=$
$\displaystyle-0.0004(1)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{\rm DVs}$
$\displaystyle=$ $\displaystyle 0.010(3)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{\rm DVs}$
$\displaystyle=$ $\displaystyle-0.07(2)\times 10^{-3}\ .$

The results of Eqs. (35) and (36), supplemented by the central values from
Eqs. (37) and Eqs. (38), which serve as our estimates for the DV-induced
perturbative uncertainties, have to be added to those of Eqs. (33) and (34),
respectively. As noted above, we take the total error on the perturbative plus
DV contributions to be equal to the (absolute value of the) central values of
the DV contributions, which are always larger than the DV errors quoted above.
This produces the following total lqc and s+lqd contributions, not yet
corrected for EM and SIB effects:

$\displaystyle a_{\mu}^{W1,\rm lqc}$ $\displaystyle=$ $\displaystyle
199.88(1.02)\times 10^{-10}\ ,$ (39) $\displaystyle a_{\mu}^{W2,\rm lqc}$
$\displaystyle=$ $\displaystyle 94.61(36)\times 10^{-10}\ ,$ $\displaystyle
I_{\widehat{W}_{15}}^{\rm lqc}$ $\displaystyle=$ $\displaystyle
43.17(16)\times 10^{-2}\ ,$ $\displaystyle I_{\widehat{W}_{25}}^{\rm lqc}$
$\displaystyle=$ $\displaystyle 79.32(43)\times 10^{-3}\ ,$

and

$\displaystyle a_{\mu}^{W1,\rm s+lqd}$ $\displaystyle=$ $\displaystyle
25.86(76)\times 10^{-10}\ ,$ (40) $\displaystyle a_{\mu}^{W2,\rm s+lqd}$
$\displaystyle=$ $\displaystyle-0.04(14)\times 10^{-10}\ ,$ $\displaystyle
I_{\widehat{W}_{15}}^{\rm s+lqd}$ $\displaystyle=$ $\displaystyle
1.770(92)\times 10^{-2}\ ,$ $\displaystyle I_{\widehat{W}_{25}}^{\rm s+lqd}$
$\displaystyle=$ $\displaystyle 11.92(32)\times 10^{-3}\ .$

##  IV  Electromagnetic and strong isospin-breaking effects

The results of Eqs. (39) and (40) were obtained using experimental data and
thus contain EM and SIB effects. These effects need to be estimated and
subtracted to obtain lqc and s+lqd results that can be compared directly to
those obtained from isospin-symmetric lattice QCD without QED. We employ the
same strategy to carry out these subtractions as that used in Refs. [17, 18],
which is predicated on the observation that, to first order in IB, SIB
contributions occur only in the MI component of $\rho_{\rm EM}(s)$, while EM
contributions are present in all of the pure $I=1$, pure $I=0$ and MI
components. IB corrections to the window quantities we consider are thus of
two types. Those present in the pure $I=1$ and pure $I=0$ contributions are,
to first order in IB, purely EM, and require only an estimate of the inclusive
combination of EM contributions from all exclusive modes, with no need for a
further breakdown of these corrections into those associated with individual
exclusive modes. The MI corrections, in contrast, require removing from the
nominal pure $I=1$ and pure $I=0$ sums obtained above exclusive-mode IB
contributions which, in fact, represent MI contaminations of those sums.
Examples of such MI contaminations are the two-pion and three-pion
contributions resulting from the $\rho-\omega$-mixing-induced processes
$e^{+}e^{-}\rightarrow\omega\rightarrow\rho\rightarrow 2\pi$ and
$e^{+}e^{-}\rightarrow\rho\rightarrow\omega\rightarrow 3\pi$. MI corrections
to the lqc and s+lqd combinations thus require identifying the combined EM+SIB
IB parts of the individual exclusive-mode contributions relevant to each, and
cannot be carried out using inclusive versions of the EM or SIB contributions,
or their sum. We do, however, expect the MI contaminations to be dominated by
contributions from the two- and three-pion modes, where the narrowness of the
$\omega$ peak and the small $\rho-\omega$ mass difference lead to a strong
enhancement of IB contributions from the $\rho-\omega$ region. We will
estimate MI two- and three-pion corrections from this region using fits to
data (which, of course, produce assessments of the EM+SIB sum) and employ a
generic $O(1\%)$ estimate for the size of MI contamination in the
contributions from other exclusive modes, which (i) lie higher in the
spectrum, and hence have contributions suppressed by the falloff in $s$ of the
weights considered here, and (ii) are not subject to any narrow-nearby-
resonance enhancements. Given the very small size of EM and SIB corrections to
the perturbative contribution to the lqc and s+lqd spectral functions, we will
ignore IB corrections from the $s\geq(1.937)^{2}\ {\rm GeV}^{2}$ (inclusive)
region.

In view of the above discussion, we treat separately the corrections for EM
effects in the nominally pure $I=1$ and $I=0$ contributions and those for
EM+SIB MI contamination, discussing the former in Sec. IV.1 and the latter in
Sec. IV.2. As mentioned above, we follow many lattice groups and define our
isospin limit of QCD as one in which all pions have a mass equal to the
physical neutral pion mass.

###  IV.1  $I=1$ and $I=0$ electromagnetic corrections

To quantify and subtract the EM contributions present in the pure $I=1$ and
$I=0$ parts of the window quantities of interest, we rely on information from
the lattice results obtained in Ref. [5]. While some EM effects have been
estimated from experimental data [23, 26, 25], additional potentially
significant EM effects have not101010For an expanded discussion of this point
see, e.g., the appendix of Ref. [17]. and we thus consider it unavoidable to
rely on lattice EM data for the EM corrections. The existence of significant
cancellations amongst the set of data-based EM contribution estimates detailed
in Refs. [24, 25] also argues in favor of using the inclusive lattice result
since that result necessarily includes contributions from all sources,
including those not currently amenable to data-based estimates, and whose
relative role might be enhanced by the strong cancellations amongst the
currently quantified contributions. This constitutes our only use of lattice
data in obtaining our estimates for the lqc and s+lqd window quantities; as we
will see, these corrections are very small.

For the RBC/UKQCD intermediate window $W1$, the lqc EM correction has been
obtained directly in Ref. [5]. The result is

$\Delta_{\rm EM}a_{\mu}^{W1,\rm lqc}=0.035(59)\times 10^{-10}\ .$ (41)

For the s+lqd EM correction for the same window, the relevant lattice data are
also given in Ref. [5]. Using exactly the same strategy as in Ref. [18], we
find

$\Delta_{\rm EM}a_{\mu}^{W1,\rm s+lqd}=0.012(11)\times 10^{-10}\ .$ (42)

As Ref. [5] did not obtain the relevant lattice estimates for EM contributions
to the other window quantities considered here, we do not have equivalent
estimates for these other quantities. However, if we compare the $W1$
corrections with Eqs. (39) and (40), we see that the central value of the lqc
EM correction is $\sim$30 times smaller than the error in Eqs. (39), while the
central value of the s+lqd correction is $\sim$60 times smaller than the error
in Eqs. (40). Since the relative errors on the other window quantities in Eqs.
(39) and (40) are of order the same size as or larger than those in the $W1$
quantities, we will assume that EM corrections to the pure $I=1$ and $I=0$
contributions to these quantities can be safely neglected, and ignore these EM
corrections in the rest of this paper. This assumption should be particularly
safe for the s+lqd contributions, where the diagrammatic analysis of Ref. [18]
shows the existence of generic strong cancellations, for example in the
numerically dominant light-quark EM valence-valence connected and disconnected
contributions.

###  IV.2  The mixed-isospin (MI) correction

As in Ref. [18], we expect the MI EM+SIB correction to be dominated by
contributions from the 2-pion and 3-pion exclusive modes, where there are
potentially strong enhancements due to $\rho-\omega$ interference in the
$\rho-\omega$ resonance region, and where contributions from that region are
more strongly weighted than are those of other modes, lying at higher $s$, in
all the window quantities considered in this paper. Such IB $\rho-\omega$
region $2\pi$ and $3\pi$ contributions can be estimated from the interference
terms in fits to the $e^{+}e^{-}\rightarrow 2\pi$ and $e^{+}e^{-}\rightarrow
3\pi$ electroproduction cross sections associated with the
$\rho-\omega$-mixing-induced IB processes
$e^{+}e^{-}\rightarrow\omega\rightarrow\rho\rightarrow 2\pi$ and
$e^{+}e^{-}\rightarrow\rho\rightarrow\omega\rightarrow 3\pi$, which, to first
order in $\alpha$ and $m_{u}-m_{d}$, produce contributions lying entirely in
the MI contribution, $\rho_{\rm EM}^{\rm MI}$, of Eq. (16).

For the window quantity $a_{\mu}^{W1}$, the $\rho-\omega$-mixing-enhanced, MI
two-pion exclusive-mode contribution has been obtained in Refs. [23, 25] from
fits to two-pion electroproduction data employing a dispersively constrained
representation of the pion form factor incorporating the effect of
$\rho-\omega$ mixing.111111The fits of Refs. [23, 25], of course, also provide
determinations of the $\rho$-$\omega$-mixing-enhanced, MI two-pion exclusive-
mode contribution to $a_{\mu}^{\rm HVP}$. The result of the most recent update
[25] in that case is $3.79(19)\times 10^{-10}$. The result,

$\left[a_{\mu}^{W1}\right]_{\pi\pi}^{\rm MI}=0.86(6)\times 10^{-10}\ ,$ (43)

represents a MI contamination to be subtracted from the nominal $I=1$
exclusive-mode sum, and hence produces a correction

$\Delta_{\pi\pi}^{\rm MI}a_{\mu}^{W1,{\rm lqc}}=-{\frac{10}{9}}\times
0.86(6)\times 10^{-10}=-0.96(7)\times 10^{-10}$ (44)

to the nominal $a_{\mu}^{W1,{\rm lqc}}$ result of Eqs. (39). In spite of its
enhancement by $\rho-\omega$ mixing, this correction is only $\sim 0.6\%$ of
the total two-pion contribution to $a_{\mu}^{W1}$. We thus consider it
extremely safe to assume that the magnitude of the sum of MI corrections to
the nominal $I=1$ sum from other exclusive modes having no analogous narrow-
nearby-resonance enhancements is less than $1\%$ of the sum, $25.68\times
10^{-10}$, of contributions to $a_{\mu}^{W1}$ from those modes. We thus add a
further uncertainty $\pm(10/9)\times 0.26\times 10^{-10}=\pm 0.29\times
10^{-10}$ to that in Eq. (44) and take as our final estimate for the MI
correction to the pre-IB-corrected $a_{\mu}^{W1,{\rm lqc}}$ value of Eq. (39),
the result

$\Delta^{\rm MI}a_{\mu}^{W1,{\rm lqc}}=-0.96(7)(29)\times 10^{-10}\ .$ (45)

This result is compatible within errors with the expectation,
$-0.753(43)\times 10^{-10}$, for the SIB part of the MI lqc correction implied
by the lattice result of Ref. [5] for the SIB contribution to
$a_{\mu}^{W1,{\rm lqc}}$, and hence with the expected dominance of the MI
EM+SIB lqc correction by its SIB component. Neglecting the very small pure
$I=1$ EM correction of Eq. (41), we obtain the IB-corrected result

$\displaystyle a_{\mu}^{W1,\rm lqc}$ $\displaystyle=$
$\displaystyle\left(199.88(1.02)-0.96(0.30)\right)\times
10^{-10}=198.9(1.1)\times 10^{-10}\ .$ (46)

Two-pion MI IB corrections due to $\rho-\omega$ mixing based on the analysis
of Ref. [23, 25] have also been made available to us for the other three
window quantities [44] considered here. In these cases, there are no lattice
results to compare with. The resulting MI contributions to the nominal $I=1$
sums are $0.767(31)\times 10^{-10}$ for the $W2$ window quantity, and
$0.331(13)\times 10^{-2}$ and $0.300(20)\times 10^{-3}$ for the
$\widehat{W}_{15}$ and $\widehat{W}_{25}$ window quantities. To obtain the
corresponding lqc IB corrections, these need to be multiplied by 10/9, and
subtracted from the totals in Eq. (39). The non-$2\pi$ exclusive-mode
contributions to these quantities are $0.9\times 10^{-10}$, vanishingly small,
and $13\times 10^{-3}$ respectively. Taking, as above, 1% of these
contributions as a further uncertainty induced by MI IB effects from nominally
$I=1$, non-$2\pi$ exclusive modes, leads to our final estimates

$\displaystyle a_{\mu}^{W2,\rm lqc}$ $\displaystyle=$
$\displaystyle\left(94.61(36)-\frac{10}{9}\times 0.767(32)\right)\times
10^{-10}=93.75(36)\times 10^{-10}\ ,$ $\displaystyle I_{\widehat{W}_{15}}^{\rm
lqc}$ $\displaystyle=$ $\displaystyle\left(43.17(16)-\frac{10}{9}\times
0.331(13)\right)\times 10^{-2}=42.80(16)\times 10^{-2}\ ,$ $\displaystyle
I_{\widehat{W}_{25}}^{\rm lqc}$ $\displaystyle=$
$\displaystyle\left(79.32(43)-\frac{10}{9}\times 0.30(13)\right)\times
10^{-3}=78.99(45)\times 10^{-3}\,.$ (47)

Unlike estimates for the MI corrections for the lqc components of the window
quantities considered in this paper, which, as explained above, can be
obtained using as key input the results of the dispersively constrained
analysis of experimental $2\pi$ electroproduction data detailed in Ref. [23],
estimates of the MI corrections for the analogous s+lqd components require
also input on what is expected to be the dominant nominally $I=0$ MI
correction, namely that from the $3\pi$ exclusive mode. While the fact that
$2\pi$ contributions to the window quantities listed in Tables 1, 4, 5 and 6
exceed the corresponding $3\pi$ contributions by factors of $7.0-13.6$, might
lead one to expect the $3\pi$ MI corrections to be similarly smaller than the
corresponding $2\pi$ MI ones, this is, in fact, unlikely to be the case due to
an countervailing numerical enhancement of the relative $\rho-\omega$-mixing-
induced $3\pi$ correction.

The existence of this enhancement can be understood as follows. The ratio of
the $\rho-\omega$-mixing-induced IB interference contribution to the dominant
isospin-conserving (IC) $\rho$ contribution to the $\rho-\omega$-resonance-
region $2\pi$ cross sections is proportional to the product
$P_{2\pi}\equiv\epsilon_{\rho\omega}f_{\omega}/f_{\rho}$, with
$\epsilon_{\rho\omega}$ the parameter characterizing the strength of
$\rho-\omega$ mixing and $f_{V}$, $V=\rho,\,\omega$ the vector-meson decay
constants, characterizing the strength of the couplings of the $\rho$ and
$\omega$ to the EM current. The analogous ratio, of $\rho-\omega$-mixing-
induced-IB-interference to $\omega$-dominated IC contributions to the
resonance-region 3-pion cross sections, is, in contrast, proportional to the
product $P_{3\pi}\equiv\epsilon_{\rho\omega}f_{\rho}/f_{\omega}$.
Experimentally (as expected for near-ideal mixing of the vector meson nonet),
$f_{\rho}\simeq 3f_{\omega}$. A natural enhancement, by a factor of
$P_{3\pi}/P_{2\pi}\simeq 9$, is thus present in the ratio of relative sizes of
$\rho-\omega$-mixing-induced resonance-region IB in the $3\pi$ versus $2\pi$
channels. It is thus unlikely that the MI $3\pi$ correction is safely
negligible on the scale of the MI $2\pi$ correction.

The $\rho-\omega$-mixing MI corrections in the $3\pi$ channel, relevant for
the $I=0$ contribution and thus the s+lqd contribution, have recently been
estimated in Refs. [26, 25], for $a_{\mu}^{\rm HVP}$ and $a_{\mu}^{W1}$. For
$a_{\mu}^{W1}$, their estimate, which is nominally a contribution to the $I=0$
part of $a_{\mu}^{W1}$, is

$\left[a_{\mu}^{W1}\right]_{3\pi}^{\rm MI}=-1.03(27)\times 10^{-10}.$ (48)

From Eqs. (33) and (34) and using Table 1 one finds that the other-than $3\pi$
exclusive-mode contribution for $I=0$ equals $22.53\times 10^{-10}$, 1% of
which is $0.23$. We add this as an additional error (in quadrature) to Eq.
(48), arriving at $-1.03(35)\times 10^{-10}$. To find the MI correction to the
s+lqd part of $a_{\mu}^{W1}$, we need to subtract this, while adding back in
$1/10$ times Eq. (45) (cf.  Eq. (20)). This leads to our estimate for the MI
correction

$\Delta^{\rm MI}a_{\mu}^{W1,{\rm s+lqd}}=1.13(36)\times 10^{-10}\ ,$ (49)

where we combined errors in Eqs. (48) and (45) ignoring correlations. We thus
obtain the IB-corrected result121212We again neglect EM corrections, which are
very small.

$\displaystyle a_{\mu}^{W1,\rm s+lqd}$ $\displaystyle=$
$\displaystyle\left(25.86(76)+1.13(36)\right)\times 10^{-10}=27.0(8)\times
10^{-10}\ .$ (50)

Three-pion MI IB corrections due to $\rho-\omega$ mixing based on the analysis
of Refs. [26, 25] have also been made available to us for the other three
window quantities [44] considered here. The resulting MI contributions to the
nominal $I=0$ sums are $-0.367(101)\times 10^{-10}$ for the $W2$ window
quantity, and $-0.230(61)\times 10^{-2}$ and $-0.44(12)\times 10^{-3}$ for the
$\widehat{W}_{15}$ and $\widehat{W}_{25}$ window quantities. The non-$3\pi$
exclusive-mode contributions to these quantities are $2.7\times 10^{-10}$,
$2.2\times 10^{-2}$ and $10.1\times 10^{-3}$, respectively.131313To be
conservative, we take the absolute value of all contributions in Table 5.
Taking, as above, 1% of these contributions as a further uncertainty induced
by MI IB effects from nominally $I=0$, non-$3\pi$ exclusive modes, leads to
our final estimates

$\displaystyle a_{\mu}^{W2,\rm s+lqd}$ $\displaystyle=$
$\displaystyle\left(-0.03(14)+0.37(11)+\frac{1}{9}\times
0.767(33)\right)\times 10^{-10}$ $\displaystyle=$ $\displaystyle
0.42(18)\times 10^{-10}\ ,$ $\displaystyle I_{\widehat{W}_{15}}^{\rm s+lqd}$
$\displaystyle=$ $\displaystyle\left(1.770(92)+0.230(65)+\frac{1}{9}\times
0.331(13)\right)\times 10^{-2}$ $\displaystyle=$ $\displaystyle 2.04(11)\times
10^{-2}\ ,$ $\displaystyle I_{\widehat{W}_{25}}^{\rm s+lqd}$ $\displaystyle=$
$\displaystyle\left(11.92(32)+0.44(16)+\frac{1}{9}\times 0.30(13)\right)\times
10^{-3}$ $\displaystyle=$ $\displaystyle 12.39(36)\times 10^{-3}\,.$

The terms between parentheses come from Eq. (40), and the results for the MI
$I=0$ $3\pi$ and MI $I=1$ $2\pi$ corrections quoted above, respectively.

It is possible to test the treatment of exclusive-mode MI contributions
described above by comparing the inclusive MI sums that treatment implies to
the corresponding inclusive MI lattice results, available for $a_{\mu}^{\rm
HVP}$ and $a_{\mu}^{W1}$ from Ref. [5]. The MI lattice results are obtained by
combining the SIB results from Ref. [5] with EM MI estimates obtained using
the EM results quoted in Ref. [5], following the diagrammatically based
analysis strategy employed in Ref. [18]. The latter analysis yields the
result, $-0.49(25)\times 10^{-10}$, obtained already in Ref. [18], for the
inclusive MI EM contribution to $a_{\mu}^{\rm HVP}$, and $-0.022(23)\times
10^{-10}$ for the MI EM contribution to $a_{\mu}^{W1}$. For the SIB
contributions, which, to first order in IB, are pure MI, we have, summing the
quoted connected and disconnected contributions, the lattice results [5]

$\displaystyle\left[a_{\mu}^{\rm HVP}\right]_{\rm SIB}$ $\displaystyle=$
$\displaystyle 1.93(1.20)\times 10^{-10}\ ,$ (52)
$\displaystyle\left[a_{\mu}^{W1}\right]_{\rm SIB}$ $\displaystyle=$
$\displaystyle 0.516(44)\times 10^{-10}\ .$

The inclusive MI lattice totals are thus

$\displaystyle\left[a_{\mu}^{\rm HVP}\right]_{\rm MI,latt}$ $\displaystyle=$
$\displaystyle 1.44(1.23)\times 10^{-10}\ ,$ (53)
$\displaystyle\left[a_{\mu}^{W1}\right]_{\rm MI,latt}$ $\displaystyle=$
$\displaystyle 0.494(50)\times 10^{-10}\ .$

In the treatment above, the inclusive MI total is, in contrast, obtained by
summing our estimates for the MI $\pi^{0}\gamma$, $\eta\gamma$, $2\pi$ and
$3\pi$ exclusive-mode contributions, with an additional uncertainty equal to
$1\%$ of the sums of the contributions for all other exclusive modes. The
$2\pi$ and $3\pi$ contributions are those detailed above, while the MI
$\pi^{0}\gamma$ and $\eta\gamma$ contributions are obtained using the same VMD
model used to determine the corresponding $I=0$ and $I=1$ contributions,
outlined in Appendix B. The results for the latter are

$\displaystyle\left[a_{\mu}^{\rm HVP}\right]^{\rm
MI}_{\pi^{0}\gamma+\eta\gamma}=\left(0.733(65)+0.067(24)\right)\times
10^{-10}=0.800(69)\times 10^{-10}$
$\displaystyle\left[a_{\mu}^{W1}\right]^{\rm
MI}_{\pi^{0}\gamma+\eta\gamma}=\left(0.268(24)+0.026(1)\right)\times
10^{-10}=0.294(24)\times 10^{-10}\,,$ (54)

where the first terms is the $\pi^{0}\gamma$ contribution and the second the
$\eta\gamma$ contribution. Combining these results with those from the other
modes, we find the following alternate data-driven (dd) results

$\displaystyle\left[a_{\mu}^{\rm HVP}\right]_{\rm MI,dd}$ $\displaystyle=$
$\displaystyle 1.91(73)(77)\times 10^{-10}\ ,$ (55)
$\displaystyle\left[a_{\mu}^{W1}\right]_{\rm MI,dd}$ $\displaystyle=$
$\displaystyle 0.12(28)(49)\times 10^{-10}\ ,$

where the first error is the quadrature sum of the errors on the MI
$\pi^{0}\gamma$, $\eta\gamma$, $2\pi$ and $3\pi$ contributions and the second
is our estimate of the uncertainty produced by neglecting MI contributions
from all other exclusive modes. The data-driven estimates are compatible
within errors with the lattice results in both cases.

##  V Updates of our previous determinations of $a_{\mu}^{\rm HVP,s+lqd}$,
$a_{\mu}^{\rm HVP,lqc}$ and $a_{\mu}^{\rm W1,lqc}$

In Refs. [17, 18, 20] we provided first data-driven estimates for
$a_{\mu}^{\rm HVP,lqc}$, $a_{\mu}^{\rm HVP,s+lqd}$ and $a_{\mu}^{\rm W1,lqc}$,
respectively. This section updates the results of those analyses, taking into
account (i) an improved treatment of the small contributions from the
$\pi^{0}\gamma$ and $\eta\gamma$ exclusive modes and (ii) changes in external
input for the MI $2\pi$ and $3\pi$ IB corrections. In those earlier analyses,
the exclusive-mode $\pi^{0}\gamma$ and $\eta\gamma$ contributions were, based
on the dominance of the experimental cross sections by the large $\omega$ and
$\phi$ peaks, assigned to the nominally pure $I=0$ category. The VMD
representation of those cross sections, outlined in Appendix B, allows for an
improved version of this treatment.141414We thank Martin Hoferichter for
bringing this possibility to our attention. Because the resulting pure $I=1$
contributions are very small, this improvement has only a small effect on the
previous lqc results. It has a larger (though still small) impact on the
$a_{\mu}^{\rm HVP,s+lqd}$ result. There have also been two small shifts in the
input for the MI $2\pi$ contribution since the preliminary version of the Ref.
[23] HVP result used in the $a_{\mu}^{\rm HVP,s+lqd}$ determination of Ref.
[18]. The second of these shifts also affects the $a_{\mu}^{\rm HVP,lqc}$ and
$a_{\mu}^{\rm W1,lqc}$ results of Refs. [17, 20]. The impacts of these shifts
on the lqc results are very small on the scale of the errors on those previous
results. Finally, the results of Ref. [26] for the MI $3\pi$ corrections
provide a significant improvement to the earlier treatment of those
corrections and hence to the reliability of the determination of $a_{\mu}^{\rm
HVP,s+lqd}$. The numerical impacts of these changes are quantified below.

Equations (31) and (32) provide the updated versions of the lqc and s+lqd
contributions from the $\pi^{0}\gamma$ and $\eta\gamma$ modes. The
contributions of these modes to $a_{\mu}^{\rm HVP,lqc}$ and $a_{\mu}^{\rm
W1,lqc}$, which were previously taken to be zero, are now $0.36(4)\times
10^{-10}$ and $0.14(1)\times 10^{-10}$, respectively. The updated version of
the contribution to $a_{\mu}^{\rm HVP,s+lqd}$ is $4.00(17)\times 10^{-10}$,
representing shifts of $-1.16(14)\times 10^{-10}$ and $-1.06(14)\times
10^{-10}$ relative to the old KNT19- and DHMZ-based results of Ref. [18].

The impact of the shift in the MI $2\pi$ contribution to $a_{\mu}^{\rm HVP}$
from the preliminary result, $3.65(67)\times 10^{-10}$, employed in Ref. [18]
to the most recent version, $3.79(19)\times 10^{-10}$, quoted in Ref. [25], is
a very small upward shift of $0.02\times 10^{-10}$ in the result for
$a_{\mu}^{\rm HVP,s+lqd}$ obtained in Ref. [18]. The increase from the
initially published result, $3.68(17)\times 10^{-10}$ [23], to the updated
$3.79(19)\times 10^{-10}$ version [25], similarly, produces a downward shift
of $0.12\times 10^{-10}$ in the result for $a_{\mu}^{\rm HVP,lqc}$ obtained in
Ref. [17]. The related increase of the MI $2\pi$ contribution to
$a_{\mu}^{W1}$ from the initially published $0.83(6)\times 10^{-10}$ result
[23] to the recently updated result, $0.86(6)\times 10^{-10}$ [25], similarly
produces a downward shift of $0.03\times 10^{-10}$ in the result for
$a_{\mu}^{\rm W1,lqc}$ obtained in Ref. [20].

We turn, finally, to the impact of the improved determination of the MI $3\pi$
contribution of Ref. [26] on the results for $a_{\mu}^{\rm HVP,s+lqd}$
obtained in Ref. [18]. In Ref. [18], the MI $3\pi$ correction was estimated
based on a VMD model fit by BaBar to BaBar $e^{+}e^{-}\rightarrow 3\pi$ cross
sections [45]. The model employed involved an amplitude consisting of a sum of
nominally isospin-conserving (IC) $\omega$, $\phi$ and excited $\omega$
contributions, each proportional to the corresponding propagator, and an IB
$\rho$ contribution proportional to the $\rho$ propagator. The MI $3\pi$
correction to $a_{\mu}^{\rm HVP,s+lqd}$ was estimated using results provided
by BaBar for the contributions to $a_{\mu}^{\rm HVP}$ obtained using the
fitted VMD form with and without the $\rho$ contribution included. The IB
$\rho$ contribution in the VMD model used by BaBar, however, does not have the
$\rho$-$\omega$ mixing form, and hence presumably represents the $\rho$ part
of the partial-fraction decomposition of the underlying IB mixing-induced
form. That partial-fraction decomposition would also produce a second IB
contribution, proportional to the $\omega$ propagator, the effect of which, in
the BaBar model, would be absorbed into the fitted strength of the nominally
IC $\omega$ contribution to the amplitude. The squared modulus of the $\omega$
amplitude contribution to BaBar’s fitted VMD representation of the cross
sections will thus contain a hidden IB part resulting from the interference of
this IB contribution with the corresponding IC part of the $\omega$
contribution to the amplitude. This hidden contribution is missing from the
BaBar-fit-based estimate of the MI $3\pi$ contribution to $a_{\mu}^{\rm HVP}$
employed in Ref. [18], but automatically taken into account in the form used
in determining that contribution in Refs. [26, 25]. We thus replace the BaBar-
fit-based estimate with that obtained in Refs. [26, 25]. This produces a shift
of $+2.12(69)\times 10^{-10}$ in the MI $3\pi$ correction to $a_{\mu}^{\rm
HVP,s+lqd}$.

Combining the effects of the updates above, we find that $a_{\mu}^{\rm
HVP,lqc}$ experiences only a very small $0.2\times 10^{-10}$ upward shift,
raising the KNT19- and DHMZ-based results of Ref. [17] from $635.0(2.7)\times
10^{-10}$ to $635.2(2.7)\times 10^{-10}$ and $638.1(4.1)\times 10^{-10}$ to
$638.3(4.1)\times 10^{-10}$, respectively. The effect on $a_{\mu}^{W1,\rm
lqc}$ is even smaller, shifting the KNT19-based result of Ref. [20],
$198.8(1.1)\times 10^{-10}$, to $198.9(1.1)\times 10^{-10}$. The updates have
a somewhat larger effect on $a_{\mu}^{\rm HVP,s+lqd}$, with the KNT19- and
DHMZ-based results of Ref. [18] shifted upward from $40.1(1.4)(1.3)\times
10^{-10}$ to $41.1(1.4)(0.4)$ and $38.7(1.4)(1.3)_{\rm lin}(0.4)\times
10^{-10}$ to $39.8(1.4)(1.3)_{\rm lin}(0.4)\times 10^{-10}$, respectively. The
corresponding results for the isospin-limit disconnected contribution to
$a_{\mu}^{\rm HVP}$, obtained by subtracting the lattice average for the
strange connected contribution from the s+lqd totals, are, using the same
notation for the errors as in Ref. [18], similarly shifted, from
$-13.3(1.4)(0.4)\times 10^{-10}$ to $-12.3(1.4)(0.4)\times 10^{-10}$ and
$-14.6(1.4)(1.3)_{\rm lin}(0.4)\times 10^{-10}$ to $-13.5(1.4)(1.3)_{\rm
lin}(0.4)\times 10^{-10}$.

| $a_{\mu}^{W1,\rm lqc}$ | tension
---|---|---
this work | 198.9(1.1) |
BMW 20 | 207.3(1.4) | 4.7$\sigma$
LM 20 | 206.0(1.2) | 4.4$\sigma$
$\chi$QCD 23 | 206.7(1.8) | 3.7$\sigma$
ABGP 22 | 206.8(2.2) | 3.2$\sigma$
Mainz/CLS 22 | 207.0(1.5) | 4.4$\sigma$
ETMC 22 | 206.5(1.3) | 4.5$\sigma$
FHM 23 | 206.6(1.0) | 5.2$\sigma$
RBC/UKQCD 23 | 206.36(0.61) | 5.9$\sigma$

> Table 2: Table of the result of Eq. (46) and lattice results for
> $a_{\mu}^{W1,\rm lqc}$ from Ref. [5] (BMW 20), Ref. [8] (LM 20), Ref. [9]
> ($\chi$QCD 23), Ref. [10] (ABGP 22), Ref. [11] (Mainz/CLS 22), Ref. [12]
> (ETMC 22), Ref. [13] (FHM 23), and Ref. [14] (RBC/UKQCD 23). Units of
> $10^{-10}$. The third column gives the tension between each of the lattice
> results and our data-based result, in units of the error on the difference.

##  VI  Comparison with other determinations

For the lqc $W1$ window quantity, $a_{\mu}^{W1,\rm lqc}$, we compare our
result, Eq. (46), with recent lattice computations in Table 2 and Fig. 1. We
refrain from quoting a lattice average for $a_{\mu}^{W1,\rm lqc}$,151515We
assume such an average to be forthcoming in an update of the WP, Ref. [4]. but
it is clear that there is a discrepancy of about $7\times 10^{-10}$ between
the data-based value and lattice results. In the table, we list the tensions
between each of the lattice results, and the value of Eq. (46). The tensions
are significant and range from $3.2\sigma$ up to $5.9\sigma$.

this work | 27.0(8)
---|---
RBC/UKQCD 18 | 26.0(2)
BMW 20 | 26.32(7)
Mainz/CLS 22 | 26.87(30)
ETMC 22 | 26.50(29)

> Table 3: Table of the result of Eq. (50) and lattice results for
> $a_{\mu}^{W1,\rm s+lqd}$ from Ref. [15] (RBC/UKQCD 18), Ref. [5] (BMW 20),
> Ref. [11] (Mainz/CLS 22), Ref. [12] (ETMC 22). Units of $10^{-10}$.

We also compare the s+lqd quantity $a_{\mu}^{W1,\rm s+lqd}$ of Eq. (50) with
results from those collaborations that have computed $a_{\mu}^{W1,\rm s+lqd}$
on the lattice as well, in Table 3 and Fig. 2. The lattice and dispersive
results are, in this case, seen to be compatible within errors, as was the
case for the related s+lqd quantity, $a_{\mu}^{\rm HVP,s+lqd}$.

Two lattice collaborations have computed $a_{\mu}^{W2,\rm lqc}$, with Ref.
[10] finding the value $102.1(2.4)\times 10^{-10}$, and Ref. [13] finding the
value $100.7(3.2)\times 10^{-10}$. This is to be compared with the data-based
value $93.75(36)\times 10^{-10}$ obtained in Eq. (IV.2), see Fig. 3. Our
result displays a tension of $3.4\sigma$ with the result of Ref. [10] and
$2.2\sigma$ with the result of Ref. [13].

Finally, up to the pure $I=0$ and $I=1$ EM corrections not included in Eq.
(IV.2) but expected to be very small, the lqc results of Eq. (IV.2) for
$I_{\widehat{W}_{15}}^{\rm lqc}$ and $I_{\widehat{W}_{25}}^{\rm lqc}$ can be
compared to the lattice results obtained from the data of Ref. [10] in Ref.
[19]:

$\displaystyle I_{\widehat{W}_{15}}^{\rm lqc}(\mbox{lattice})$
$\displaystyle=$ $\displaystyle 46.7(0.7)_{\rm stat\ only}\times 10^{-2}\ ,$
(56) $\displaystyle I_{\widehat{W}_{25}}^{\rm lqc}(\mbox{lattice})$
$\displaystyle=$ $\displaystyle 82.4(1.0)_{\rm stat\ only}\times 10^{-3}\ ,$

where the errors are statistical only. These comparison provide further
evidence of tension between lattice and data-driven results for lqc
contributions, though one should keep in mind that the lattice results were
obtained in Ref. [19] without a detailed investigation of systematic errors,
which was beyond the scope of that paper.

> Figure 1: Comparison of the result of Eq. (46) with lattice results for
> $a_{\mu}^{W1,\rm lqc}$ from Ref. [5] (BMW 20), Ref. [8] (LM 20), Ref. [9]
> ($\chi$QCD 23), Ref. [10] (ABGP 22), Ref. [11] (Mainz/CLS 22), Ref. [12]
> (ETMC 22), Ref. [13] (FHM 23), and Ref. [14] (RBC/UKQCD 23). Also shown is
> the data-based result if the 2-pion data in the interval between 0.33 and
> 1.2 GeV is replaced by the results from CMD-3 [46].

> Figure 2: Comparison of the result of Eq. (50) with lattice results for
> $a_{\mu}^{W1,\rm s+lqd}$ from Ref. [15] (RBC/UKQCD18), Ref. [5] (BMW 20),
> Ref. [11] (Mainz/CLS 22) and Ref. [12] (ETMC 22).

> Figure 3: Comparison of the result of Eq. (46) with lattice results for
> $a_{\mu}^{W2,\rm lqc}$ from Ref. [10] (ABGP 22) and Ref. [13] (FHM 23). Also
> shown is the data-based result if the 2-pion data in the interval between
> 0.33 and 1.2 GeV is replaced by the results from CMD-3 [46].

##  VII  Conclusion

In this paper we have obtained data-driven determinations of the lqc and s+lqd
contributions to a number of window quantities. Data-driven determinations of
such quantities require as input $s$-dependent exclusive-mode distributions,
and the results for those determinations reported here are based solely on
KNT19 results for those distributions. It would be of interest to repeat the
analysis with DHMZ exclusive-mode input, should results for those
distributions eventually become publicly available.

Our result for $a_{\mu}^{W1,\rm s+lqd}$ is in good agreement with lattice
determinations of this quantity. Similar agreement was found previously for
$a_{\mu}^{\rm HVP,s+lqd}$ in Ref. [18]. These are, at present, the only
quantities for which lattice s+lqd results exist. It would be of interest to
have lattice results, and carry out analogous comparisons, for the other s+lqd
quantities considered here.

In contrast to the s+lqd case, our results for the lqc contributions to all
four window quantities show tensions with corresponding lattice results. This
tension is particularly significant for $a_{\mu}^{W1,{\rm lqc}}$, where, for
example, our result differs by $5.9\sigma$ from that of Ref. [14]. Improved
lattice determinations of $a_{\mu}^{W2,{\rm lqc}}$, $I_{\widehat{W}_{15}}^{\rm
lqc}$ and $I_{\widehat{W}_{25}}^{\rm lqc}$ would be of interest for exploring
further the tensions in these cases, especially so for
$I_{\widehat{W}_{15}}^{\rm lqc}$ and $I_{\widehat{W}_{25}}^{\rm lqc}$ where,
at present, only statistical errors are available for the lattice results.

A final issue of relevance to assessing the significance of the observed lqc
discrepancies is the potential impact of recent CMD-3 results for the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ cross sections [14]. As is well known,
the results are significantly higher than those of earlier experiments in the
$\rho$ peak region and, were the CMD-3 results to be correct, the resulting
change in the dispersive evaluation of $a_{\mu}^{\rm HVP}$ would essentially
remove the current discrepancy between the Standard Model expectation and
experimental result for $a_{\mu}$. The discrepancies between the new CMD-3
results and those of the earlier experiments are, however, sufficiently large
that a convincing combination of all existing results does not, at present,
seem possible. Given the unsettled experimental situation, we can carry out
only a preliminary exploration of the potential impact of the new CMD-3
results. This has been done by replacing the KNT19 $2\pi$ contributions to
$R(s)$ in the region covered by CMD-3 data ($E_{\rm CM}$ from $0.327$ to
$1.199$ GeV) with the corresponding contributions implied by CMD-3 data alone.
This requires applying vacuum polarization (VP) corrections to the physical
cross sections implied by the results for the physical timelike pion form
factor quoted by CMD-3 and dressing the resulting bare cross sections with the
final state radiation (FSR) correction factors used by CMD-3 in their
evaluation of the contribution of their results to $a_{\mu}^{\rm HVP}$. We
have used the same VP corrections and same FSR dressing factors as those
employed by CMD-3.161616We thank Fedor Ignatov for providing a link to the
file containing the VP correction results. The lqc results produced by this
modification of the $2\pi$ distribution, of course, constitute only very
preliminary explorations, and should in no way be interpreted as resulting
from the use of some updated combination of the $2\pi$ data base, which no one
at present knows how to carry out. The results of this (we again emphasize
preliminary) exploration are shown for $a_{\mu}^{W1,{\rm lqc}}$ and
$a_{\mu}^{W2,{\rm lqc}}$ in Figs. 1 and 3. As found in the case of
$a_{\mu}^{\rm HVP}$, use of the CMD-3 $2\pi$ data alone in the region where it
exists removes essentially the entirety of the observed lqc lattice-data-
driven discrepancies.

While the experimental discrepancy between the CMD-3 data and other data sets
for $e^{+}e^{-}\to$ hadrons remains unresolved at present, we conclude that
there are significant discrepancies between the light-quark-connected parts of
all window quantities investigated in this paper as obtained from the KNT19
compilation of these other data sets and recent lattice results, with lattice
values pointing to a value for $a_{\mu}^{\rm HVP}$ that would bring the SM
expectation for $a_{\mu}$ much closer to the experimental value. Further
lattice computations of $a_{\mu}^{\rm W2,lqc}$ in particular would increase
our understanding of the discrepancy for this quantity discussed in Sec. VI.

Acknowledgments

We would like to thank Martin Hoferichter and Peter Stoffer for extensive
discussions on isospin breaking, as well as for providing us with the $2\pi$
mixed-isospin contributions to the $W2$, $\widehat{W}_{15}$ and
$\widehat{W}_{25}$ windows. We also thank Martin Hoferichter and collaborators
for providing the $3\pi$ mixed-isospin contributions to those window
quantities, and Martin Hoferichter for useful comments on the $\pi^{0}\gamma$
and $\eta\gamma$ channels. 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$\backslash$R1$\backslash$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.

##  Appendix A  Isospin tables for $a_{\mu}^{W2}$, $I_{\widehat{W}_{15}}$ and
$I_{\widehat{W}_{15}}$ and further G-parity-ambiguous exclusive-mode
contributions

In this appendix, we show, in Tables 4, 5 and 6, respectively, the G-parity-
unambiguous $I=1$ and $I=0$ exclusive-mode contributions to $a_{\mu}^{W2}$,
$I_{\widehat{W}_{15}}$ and $I_{\widehat{W}_{25}}$. We also list, in Eqs. (57)
to (66), the G-parity-ambiguous exclusive residual-mode contributions to the
lqc and s+lqd parts of $a_{\mu}^{W1}$, $a_{\mu}^{W2}$, $I_{\widehat{W}_{15}}$
and $I_{\widehat{W}_{25}}$, using the maximally conservative split
prescription of Eq. (24).

$I=1$ modes $X$ | $[a_{\mu}^{W2}]_{X}\times 10^{10}$ |  $I=0$ modes $X$ | $[a_{\mu}^{W2}]_{X}\times 10^{10}$
---|---|---|---
low-$s$ $\pi^{+}\pi^{-}$ | 0.09(02) | low-$s$ $3\pi$ | 0.00(00)
$\pi^{+}\pi^{-}$ | 84.20(33) | $3\pi$ | 6.20(14)
$2\pi^{+}2\pi^{-}$ | 0.26(00) |  $2\pi^{+}2\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.01(00)
$\pi^{+}\pi^{-}2\pi^{0}$ | 0.44(02) |  $\pi^{+}\pi^{-}3\pi^{0}$ (no $\eta$) | 0.00(00)
$3\pi^{+}3\pi^{-}$ (no $\omega$) | 0.00(00) |  $3\pi^{+}3\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.00(00)
$2\pi^{+}2\pi^{-}2\pi^{0}$ (no $\eta$) | 0.00(00) |  $\eta\pi^{+}\pi^{-}\pi^{0}$ (no $\omega$) | 0.00(00)
$\pi^{+}\pi^{-}4\pi^{0}$ (no $\eta$) | 0.00(00) | $\eta\omega$ | 0.00(00)
$\eta\pi^{+}\pi^{-}$ | 0.01(00) | $\omega(\rightarrow npp)2\pi$ | 0.00(00)
$\eta 2\pi^{+}2\pi^{-}$ | 0.00(00) | $\omega 2\pi^{+}2\pi^{-}$ | 0.00(00)
$\eta\pi^{+}\pi^{-}2\pi^{0}$ | 0.00(00) | $\eta\phi$ | 0.00(00)
$\omega(\rightarrow\pi^{0}\gamma)\pi^{0}$ | 0.03(00) | $\phi\rightarrow unaccounted$ | 0.00(00)
$\omega(\rightarrow npp)3\pi$ | 0.00(00) |  |
$\omega\eta\pi^{0}$ | 0.00(00) |  |
TOTAL: | 85.05(33) | TOTAL: | 6.23(14)
Table 4: $G$-parity-unambiguous exclusive-mode contributions to $a_{\mu}^{W2}$ for $\sqrt{s}\leq 1.937$ GeV using KNT19 exclusive-mode data. Entries in units of $10^{-10}$. The notation “npp” is KNT shorthand for “non-purely-pionic.”  $I=1$ modes $X$ | $[I_{\widehat{W}_{15}}]_{X}\times 10^{2}$ |  $I=0$ modes $X$ | $[I_{\widehat{W}_{15}}]_{X}\times 10^{2}$
---|---|---|---
low-$s$ $\pi^{+}\pi^{-}$ | 0.00(00) | low-$s$ $3\pi$ | 0.00(00)
$\pi^{+}\pi^{-}$ | 39.35(14) | $3\pi$ | 3.83(08)
$2\pi^{+}2\pi^{-}$ | -0.02(00) |  $2\pi^{+}2\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | -0.02(00)
$\pi^{+}\pi^{-}2\pi^{0}$ | 0.10(01) |  $\pi^{+}\pi^{-}3\pi^{0}$ (no $\eta$) | -0.01(00)
$3\pi^{+}3\pi^{-}$ (no $\omega$) | -0.01(00) |  $3\pi^{+}3\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.00(00)
$2\pi^{+}2\pi^{-}2\pi^{0}$ (no $\eta$) | -0.04(00) |  $\eta\pi^{+}\pi^{-}\pi^{0}$ (no $\omega$) | -0.02(00)
$\pi^{+}\pi^{-}4\pi^{0}$ (no $\eta$) | -0.01(01) | $\eta\omega$ | -0.01(00)
$\eta\pi^{+}\pi^{-}$ | -0.02(00) | $\omega(\rightarrow npp)2\pi$ | 0.00(00)
$\eta 2\pi^{+}2\pi^{-}$ | 0.00(00) | $\omega 2\pi^{+}2\pi^{-}$ | 0.00(00)
$\eta\pi^{+}\pi^{-}2\pi^{0}$ | 0.00(00) | $\eta\phi$ | -0.01(00)
$\omega(\rightarrow\pi^{0}\gamma)\pi^{0}$ | 0.02(00) | $\phi\rightarrow unaccounted$ | 0.00(00)
$\omega(\rightarrow npp)3\pi$ | 0.00(00) |  |
$\omega\eta\pi^{0}$ | -0.01(00) |  |
TOTAL: | 39.37(14) | TOTAL: | 3.77(08)
Table 5: $G$-parity-unambiguous exclusive-mode contributions to $I_{\widehat{W}_{15}}$ for $\sqrt{s}\leq 1.937$ GeV using KNT19 exclusive-mode data. Entries in units of $10^{-2}$. The notation “npp” is KNT shorthand for “non-purely-pionic.”  $I=1$ modes $X$ | $[I_{\widehat{W}_{25}}]_{X}\times 10^{3}$ |  $I=0$ modes $X$ | $[I_{\widehat{W}_{25}}]_{X}\times 10^{3}$
---|---|---|---
low-$s$ $\pi^{+}\pi^{-}$ | 0.00(00) | low-$s$ $3\pi$ | 0.00(00)
$\pi^{+}\pi^{-}$ | 56.60(19) | $3\pi$ | 8.05(15)
$2\pi^{+}2\pi^{-}$ | 4.13(06) |  $2\pi^{+}2\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.27(02)
$\pi^{+}\pi^{-}2\pi^{0}$ | 5.31(21) |  $\pi^{+}\pi^{-}3\pi^{0}$ (no $\eta$) | 0.17(03)
$3\pi^{+}3\pi^{-}$ (no $\omega$) | 0.06(00) |  $3\pi^{+}3\pi^{-}\pi^{0}$ (no $\omega$, $\eta$) | 0.00(00)
$2\pi^{+}2\pi^{-}2\pi^{0}$ (no $\eta$) | 0.36(05) |  $\eta\pi^{+}\pi^{-}\pi^{0}$ (no $\omega$) | 0.19(02)
$\pi^{+}\pi^{-}4\pi^{0}$ (no $\eta$) | 0.06(06) | $\eta\omega$ | 0.08(01)
$\eta\pi^{+}\pi^{-}$ | 0.37(01) | $\omega(\rightarrow npp)2\pi$ | 0.04(00)
$\eta 2\pi^{+}2\pi^{-}$ | 0.02(00) | $\omega 2\pi^{+}2\pi^{-}$ | 0.00(00)
$\eta\pi^{+}\pi^{-}2\pi^{0}$ | 0.03(01) | $\eta\phi$ | 0.11(01)
$\omega(\rightarrow\pi^{0}\gamma)\pi^{0}$ | 0.24(01) | $\phi\rightarrow unaccounted$ | 0.01(01)
$\omega(\rightarrow npp)3\pi$ | 0.05(01) |  |
$\omega\eta\pi^{0}$ | 0.06(01) |  |
TOTAL: | 67.29(30) | TOTAL: | 8.92(16)
Table 6: $G$-parity-unambiguous exclusive-mode contributions to
$I_{\widehat{W}_{25}}$ for $\sqrt{s}\leq 1.937$ GeV using KNT19 exclusive-mode
data. Entries in units of $10^{-3}$. The notation “npp” is KNT shorthand for
“non-purely-pionic.”

  * $\circ$

$K\bar{K}3\pi$ modes:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle 0.012(12)\times 10^{-10}\ ,$ (57)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle-0.00077(77)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle 0.0050(50)\times 10^{-3}\ .$

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle 0.010(12)\times 10^{-10}\ ,$ (58)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle-0.00062(77)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{K\bar{K}3\pi}$
$\displaystyle=$ $\displaystyle 0.0040(50)\times 10^{-3}\ .$

  * $\circ$

$\omega(\rightarrow npp)K\bar{K}$-modes:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0012(12)\times 10^{-10}\ ,$ (59)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle-0.000081(81)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.00052(52)\times 10^{-3}\ .$

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0010(12)\times 10^{-10}\ ,$ (60)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle-0.000065(81)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{\omega K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.00042(52)\times 10^{-3}\ .$

  * $\circ$

$\eta(\rightarrow npp)K\bar{K}$ modes:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0050(50)\times 10^{-10}\ ,$ (61)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle-0.00024(24)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0022(22)\times 10^{-3}\ .$

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0040(50)\times 10^{-10}\ ,$ (62)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm s+lqd}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle-0.00019(24)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{\eta K\bar{K}}$
$\displaystyle=$ $\displaystyle 0.0017(22)\times 10^{-3}\ .$

  * $\circ$

$p\bar{p}$ and $n\bar{n}$ modes:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{p\bar{p}+n\bar{n}}$
$\displaystyle=$ $\displaystyle 0.020(20)\times 10^{-10}\ ,$ (63)
$\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{p\bar{p}+n\bar{n}}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm lqc}\right]_{p\bar{p}+n\bar{n}}$
$\displaystyle=$ $\displaystyle-0.0011(11)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{p\bar{p}+n\bar{n}}$
$\displaystyle=$ $\displaystyle 0.0083(83)\times 10^{-3}\ .$

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{p\bar{p}+n\bar{n}}$
$\displaystyle=$ $\displaystyle 0.016(20)\times 10^{-10}\ ,$ (64)
$\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{p\bar{p}+n\bar{n}}$
$\displaystyle=$ $\displaystyle 0.0000(0)\times 10^{-10}\ ,$
$\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
s+lqd}\right]_{p\bar{p}+n\bar{n}}$ $\displaystyle=$
$\displaystyle-0.001(11)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm
s+lqd}\right]_{p\bar{p}+n\bar{n}}$ $\displaystyle=$ $\displaystyle
0.0066(83)\times 10^{-3}\ .$

  * $\circ$

Low-$s$ $\pi^{0}\gamma$ and $\eta\gamma$ modes:

$\displaystyle\left[a_{\mu}^{W1,\rm lqc}\right]_{{\rm low-}s\
\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$ $\displaystyle 0.0082(82)\times
10^{-10}\ ,$ (65) $\displaystyle\left[a_{\mu}^{W2,\rm lqc}\right]_{{\rm
low-}s\ \pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$ $\displaystyle
0.012(12)\times 10^{-10}\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
lqc}\right]_{{\rm low-}s\ \pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle 0.0030(30)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm lqc}\right]_{{\rm low-}s\
\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$ $\displaystyle 0.0020(20)\times
10^{-3}\ .$

$\displaystyle\left[a_{\mu}^{W1,\rm s+lqd}\right]_{{\rm low-}s\
\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$ $\displaystyle 0.0066(82)\times
10^{-10}\ ,$ (66) $\displaystyle\left[a_{\mu}^{W2,\rm s+lqd}\right]_{{\rm
low-}s\ \pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$ $\displaystyle
0.010(12)\times 10^{-10}\ ,$ $\displaystyle\left[I_{\widehat{W}_{15}}^{\rm
s+lqd}\right]_{{\rm low-}s\ \pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$
$\displaystyle 0.0024(30)\times 10^{-2}\ ,$
$\displaystyle\left[I_{\widehat{W}_{25}}^{\rm s+lqd}\right]_{{\rm low-}s\
\pi^{0}\gamma+\eta\gamma}$ $\displaystyle=$ $\displaystyle 0.0016(20)\times
10^{-3}\ .$

##  Appendix B  The VMD representations of the
$e^{+}e^{-}\rightarrow\pi^{0}\gamma$ and $e^{+}e^{-}\rightarrow\eta\gamma$
cross sections

The cross sections for $e^{+}e^{-}\rightarrow P\gamma$, $P=\pi^{0},\,\eta$,
$\sigma_{P\gamma}(s)$, are given by

$\sigma_{P\gamma}(s)={\frac{2\pi^{2}\alpha_{\rm
EM}^{3}}{3}}\,\left(1-{\frac{m_{P}^{2}}{s}}\right)^{3}\,|F_{P\gamma^{*}\gamma^{*}}(s,0)|^{2}\,,$
(67)

where $F_{P\gamma^{*}\gamma^{*}}(s,0)$ is the timelike $P=\pi^{0}$ or $\eta$
transition form factor. Here, and in what follows, we employ the notation of
Sec. 6 of the recent review, Ref. [47]. In this notation, the VMD
representation of the experimentally dominant isoscalar contribution to
$F_{P\gamma^{*}\gamma^{*}}(s,0)$ is

$F^{I=0}_{P\gamma\gamma}\sum_{V=\omega,\phi}{\frac{w_{PV\gamma}s}{m_{V}^{2}-s-im_{V}\Gamma_{V}(s)}}\,,$
(68)

where $F_{P\gamma\gamma}$ is related to $\Gamma[P\rightarrow\gamma\gamma]$ by

$\Gamma[P\rightarrow\gamma\gamma]={\frac{\pi\alpha_{\rm
EM}^{2}m_{P}^{3}}{4}}|F_{P\gamma\gamma}|^{2}\ ,$ (69)

and the weights $w_{PV\gamma}$ accompanying the $\omega$ and $\phi$
propagators are given by

$w^{2}_{PV\gamma}={\frac{9m_{V}^{2}m_{P}^{3}\Gamma[V\rightarrow
e^{+}e^{-}]\Gamma[V\rightarrow P\gamma]}{2\alpha_{\rm
EM}\left(m_{V}^{2}-m_{P}^{2}\right)^{3}\gamma[P\rightarrow\gamma\gamma]}}\ .$
(70)

In what follows we use $s$-dependent versions of the widths $\Gamma_{V}(s)$,
following the treatment of Refs. [48, 49, 50], which takes into account all
decay modes with branching fractions greater than $1\%$. A similar VMD form,

$F^{I=1}_{P\gamma\gamma}{\frac{w_{P\rho\gamma}s}{m_{\rho}^{2}-s-im_{\rho}\Gamma_{\rho}(s)}}$
(71)

again with $s$-dependent width, is used to approximate the smaller isovector
contribution to the transition form factor, with the weights $w_{P\rho\gamma}$
also given by the general expression in Eq. (70). The full VMD representation
of the cross section then becomes

$\sigma_{P\gamma}(s)={\frac{2\pi^{2}\alpha_{\rm
EM}^{3}}{3}}\,\left(1-{\frac{m_{P}^{2}}{s}}\right)^{3}|F_{P\gamma\gamma}|^{2}\,|\sum_{V=\rho,\omega,\phi}w_{PV\gamma}\,s/[m_{V}^{2}-s-im_{V}\Gamma_{V}(s)]|^{2}\,.$
(72)

An alternate form of this expression, obtained by pulling out the
$V$-independent factor $m_{P}/\left[2\alpha_{\rm
EM}\Gamma[P\rightarrow\gamma\gamma]\right]$ from the squared modulus in Eq.
(72) and substituting the RHS of Eq. (69) for
$\Gamma[P\rightarrow\gamma\gamma]$, is

$\sigma_{P\gamma}(s)={\frac{4\pi}{3m_{P}^{2}}}\,\left(1-{\frac{m_{P}^{2}}{s}}\right)^{3}\,|\sum_{V=\rho,\omega,\phi}\hat{w}_{PV\gamma}\,s/[m_{V}^{2}-s-im_{V}\Gamma_{V}(s)]|^{2}\,,$
(73)

where the alternate weights, $\hat{w}_{PV\gamma}$, are given by

$\hat{w}^{2}_{PV\gamma}=9m_{V}^{2}m_{P}^{2}\Gamma[V\rightarrow
e^{+}e^{-}]\Gamma[V\rightarrow
P\gamma]/\left(m_{V}^{2}-m_{P}^{2}\right)^{3}\,.$ (74)

This alternate form makes more explicit the fact that the contributions to the
amplitude are determined entirely by the vector meson masses and widths, and
the strengths of the corresponding $V\rightarrow e^{+}e^{-}$ and $V\rightarrow
P\gamma$ couplings.

Using PDG [36] input, we find the following values for the weights
$\hat{w}_{PV\gamma}$,

$\displaystyle\hat{w}_{\pi^{0}\rho\gamma}=1.559(133)\times 10^{-5}\ ,$ (75)
$\displaystyle\hat{w}_{\pi^{0}\omega\gamma}=1.491(36)\times 10^{-5}\ ,$
$\displaystyle\hat{w}_{\pi^{0}\phi\gamma}=-1.068(26)\times 10^{-6}\ ,$
$\displaystyle\hat{w}_{\eta\rho\gamma}=1.362(48)\times 10^{-4}\ ,$
$\displaystyle\hat{w}_{\eta\omega\gamma}=1.167(56)\times 10^{-4}\ ,$
$\displaystyle\hat{w}_{\eta\phi\gamma}=-2.210(41)\times 10^{-5}\ .$

As in Ref. [47], the sign of $w_{\eta\phi\gamma}$ is chosen negative, based on
the discussions of Ref. [51] and underlying arguments of the earlier review
Ref. [52]. While those arguments do not resolve the choice of sign for
$w_{\pi^{0}\phi\gamma}$, the results of the SND fit of Ref. [48] to the
$e^{+}e^{-}\rightarrow\pi^{0}\gamma$ cross sections clearly favor a relative
negative sign between the $\phi$ and $\rho$/$\omega$ amplitude contributions.
The choice of sign for $\hat{w}_{\pi^{0}\phi\gamma}$, in any case, turns out
to have only a very small effect on the $I=0$/$I=1$/MI decompositions of the
$\pi^{0}\gamma$ exclusive-mode contributions to the integrals we are
interested in.171717Explicitly, the $I=1$ contributions are unaffected, while
the $I=0$ and MI contributions to $a_{\mu}^{\rm HVP}$ shift from $79.5\%$ and
$16.6\%$ of the total when $w_{\pi^{0}\phi\gamma}$ is chosen $<0$ to $79.7\%$
and $16.3\%$ when it is chosen $>0$. Those same contributions, similarly,
shift from $79.1\%$ and $16.7\%$ to $79.3\%$ and $16.5\%$ for $a_{\mu}^{W1}$,
from $79.9\%$ and $16.2\%$ to $80.1\%$ and $16.0\%$ for $a_{\mu}^{W2}$, from
$79.8\%$ and $16.3\%$ to $80.0\%$ and $16.1\%$ for $I_{\widehat{W}_{15}}$ and
from $79.0\%$ and $16.8\%$ to $79.2\%$ an $16.6\%$ for $I_{\widehat{W}_{25}}$.

The above input produces the results quoted in the main text for the $I=0$,
$I=1$ and MI contributions to the exclusive-mode $\pi^{0}\gamma$ and
$\eta\gamma$ weighted integrals of interest in this paper. The associated
errors are those resulting from summing in quadrature the uncertainties
produced by those on the three input quantities, $\hat{w}_{PV\gamma}$,
$V=\rho,\,\omega,\,\phi$, for the $P=\pi^{0}$ and $\eta$ cases, respectively.
The resulting VMD-model totals from the region between $E_{\rm CM}=0.600$ GeV
(the lowest $E_{\rm CM}$ in the KNT19 $\pi^{0}\gamma$ mode data compilation)
and a point safely above the last of the enhanced VMD resonance contributions
(which we take to be $E_{\rm CM}=m_{\phi}+5\Gamma_{\phi}$), are
$4.44(21)\times 10^{-10}$ for the HVP case, $1.58(8)\times 10^{-10}$ for the
W1 case, $0.68(3)\times 10^{-10}$ for the W2 case, $0.40(2)\times 10^{-2}$ for
the $\widehat{W}_{15}$ case and $0.68(3)\times 10^{-3}$ for the
$\widehat{W}_{25}$. Comparing these to the corresponding KNT19 exclusive-mode
contributions, which are $4.46(10)\times 10^{-10}$, $1.58(4)\times 10^{-10}$,
$0.69(2)\times 10^{-10}$, $0.40(1)\times 10^{-2}$ and $0.66(2)\times 10^{-3}$,
respectively, we see that the VMD contributions, in all cases, saturate the
KNT19 $\pi^{0}\gamma$ mode results. A similar saturation of KNT19 results by
the corresponding VMD model results is observed for the $\eta\gamma$ mode as
well.

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