Spectral-weight sum rules for the
hadronic vacuum polarization
Diogo Boito,a Maarten Golterman,b,c Kim Maltmand,e and Santiago Peris,c

aInstituto de Física de São Carlos, Universidade de São Paulo

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

bDepartment of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

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

E-08193 Bellaterra, Barcelona, Spain

dDepartment of Mathematics and Statistics, York University

Toronto, ON Canada M3J 1P3

eCSSM, University of Adelaide, Adelaide, SA 5005 Australia

> We develop a number of sum rules comparing spectral integrals involving
> judiciously chosen weights to integrals over the corresponding Euclidean
> two-point function. The applications we have in mind are to the hadronic
> vacuum polarization that determines the most important hadronic correction
> $a_{\mu}^{\rm HVP}$ to the muon anomalous magnetic moment. First, we point
> out how spectral weights may be chosen that emphasize narrow regions in
> $\sqrt{s}$, providing a tool to investigate emerging discrepancies between
> data-driven and lattice determinations of $a_{\mu}^{\rm HVP}$.
> Alternatively, for a narrow region around the $\rho$ mass, they may allow
> for a comparison of the dispersive determination of $a_{\mu}^{\rm HVP}$ with
> lattice determinations zooming in on the region of the well-known BaBar–KLOE
> discrepancy. Second, we show how such sum rules can in principle be used for
> carrying out precision comparisons of hadronic-$\tau$-decay-based data and
> $e^{+}e^{-}\to\mbox{hadrons}(\gamma)$-based data, where lattice computations
> can provide the necessary isospin-breaking corrections.

##  I  Introduction

As is well known, the recent FNAL E989 experimental result for the muon
anomalous magnetic moment $a_{\mu}=(g-2)/2$ FNL  confirms the earlier BNL E821
result BNL  and produces an updated experimental world average $4.2\sigma$
larger than the $g-2$ Theory Initiative assessment review  of the Standard
Model (SM) prediction, based on the work of Ref. Aoyama:2012wk ;
Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ; Davier:2017zfy ;
Keshavarzi:2018mgv ; Davier:2019can ; KNT19 ; Colangelo:2018mtw ;
Hoferichter:2019mqg ; Hoid:2020xjs ; Kurz:2014wya ; Melnikov:2003xd ;
Masjuan:2017tvw ; Colangelo:2017qdm ; Colangelo:2017fiz ; Hoferichter:2018dmo
; Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ; Colangelo:2019lpu
; Colangelo:2019uex ; Colangelo:2014qya ; Blum:2019ugy . However, a lattice
result for the hadronic vacuum polarization (HVP) contribution $a_{\mu}^{\rm
HVP}$ BMW  comes out $2.1\sigma$ higher than the dispersive $R$-ratio-based
estimate which underlies the SM value of Ref. review . Replacing the
dispersive estimate for $a_{\mu}^{\rm HVP}$ with the value found in Ref. BMW ,
the SM-based estimate for $a_{\mu}$ is only $1.5\sigma$ lower than the world-
average experimental value. As there is no evidence for discrepancies in other
contributions to $a_{\mu}$, the recent focus has been on understanding the
discrepancy between the dispersive and lattice values for $a_{\mu}^{\rm
HVP}$.111For instance, there is good agreement between data-driven and lattice
values for the hadronic light-by-light contribution, within
$\,\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}\,20\%$ errors. This $20\%$ error
corresponds to an uncertainty of about $2\times 10^{-10}$ in $a_{\mu}$, which
is not sufficient to explain the discrepancy.

The dispersive vs. lattice discrepancy becomes even more pronounced if we
consider the “intermediate window” quantity introduced by RBC/UKQCD RBC , in
which the integral over Euclidean time, $t$, of the lattice correlator that
yields $a_{\mu}^{\rm HVP}$ is restricted to a “window” between $t=0.4$ and
$t=1$ fm (smeared by a width of $0.15$ fm on both boundaries to avoid lattice
artifacts) by multiplying the integrand with a (smoothed-out) double step
function in $t$. While the original RBC/UKQCD study found agreement between
the lattice-based result and the corresponding electro-production-based
dispersive estimate, Ref. ABGP19  found a significantly larger lattice value.
This larger value was subsequently confirmed in Ref. BMW , which produced a
lattice result $3.7\sigma$ above the dispersive estimate. The virtue of the
RBC/UKQD intermediate window quantity is that it can be computed with smaller
errors on the lattice than the quantity $a_{\mu}^{\rm HVP}$ itself, thus
allowing more stringent tests between different lattice computations, as well
as between lattice and dispersive results.

Meanwhile, the larger lattice values for the intermediate window quantity
found in Refs. ABGP19 ; BMW  have been confirmed, with different lattice
discretizations of the QCD action and the electromagnetic (EM) current, by a
number of other groups as well as in updates of the results of Refs. RBC ;
ABGP19 , in Refs. LM20 ; chiQCD ; ABGP22 ; Mainz ; ETMC ; RBC22 ; FHM22 .

Clearly, then, it is important to develop further tools to study the
discrepancies between dispersive and lattice estimates for $a_{\mu}^{\rm HVP}$
and closely related quantities such as the intermediate window. To obtain
dispersive estimates for the window quantity, which is defined as a function
of Euclidean time, the window function needs to be converted to a window in
$\sqrt{s}$, the center-of-mass energy in $e^{+}e^{-}\to$ hadrons. As a
function of $\sqrt{s}$, however, the weight which defines the intermediate
window is very broad, ranging from about 0.7 GeV to 3 GeV (taking the values
of $\sqrt{s}$ where it exceeds roughly half of its maximum value). To explore
the lattice vs. dispersive discrepancy in more detail, it would be useful to
have access to tools to compare data-driven dispersive results with lattice
results in narrower, more “custom-designed” windows in $\sqrt{s}$. In this
paper, we propose a class of sum rules designed with precisely this goal in
mind. Similar ideas were recently explored in Ref. Rwindow  by considering
linear combinations of a set of Euclidean windows, and in Ref. TdeG . Here,
instead, we define the weights we will employ in our weighted spectral
integrals directly as a function of $s$, and derive sum rules relating these
weighted spectral integrals to integrals in Euclidean time over correlation
functions which can be evaluated on the lattice.

In Sec. II we develop two sets of sum rules starting from weights defined as a
function of $s$. In Sec. II.1 we consider a class of rational weights that
allow us to define windows localized in $s$. They are similar to those used
recently in Ref. Kim  to obtain a lattice-based determination of $|V_{us}|$
from strange hadronic $\tau$-decay data. In Sec. II.2 we use these rational
weights as the starting point for defining a set of sum-of-exponential weights
with shapes very similar to those defined by the underlying rational weights,
following ideas proposed in Ref. HLT . In both cases exact sum rules exist
relating spectral integrals employing these weights to quantities that can be
directly computed on the lattice. We explain why the sum-of-exponential
weights may lead to smaller errors for the lattice side of the sum rules than
the corresponding rational weights, and provide examples of this reduction in
Sec. III.

Narrower windows in $\sqrt{s}$ are also potentially useful for comparing $I=1$
contributions to $a_{\mu}^{\rm HVP}$ inferred from $I=1$ hadronic $\tau$-decay
data with the corresponding contributions obtained using $R$-ratio data. An
example of the potential usefulness of narrower windows is the application to
the BaBar–KLOE discrepancy in the two-pion spectral distributions (for a
review, see Ref. review ), where the discrepancy occurs over a fairly narrow
range in energy around the $\rho$ peak. Attempts to use $\tau$-based data have
a long history Davier1 ; Davier2 ; Jegerlehner ; Zhang , but have been
abandoned more recently because of the increased precision of
electroproduction data, and the lack of a solid theoretical framework for
evaluating the isospin-breaking (IB) corrections that must be applied to the
$\tau$-based data. It would be interesting to revisit this possibility since
(i) a more precise $\tau$-based non-strange vector spectral function is now
available AO , and (ii) Belle II may provide improved $\tau$-decay-
distribution data for at least some of the most important vector-channel
exclusive modes BELLE ; BELLE2 . Moreover, we will argue in Sec. IV of this
paper that if the combined $2\pi$ and $4\pi$ channels are taken from $\tau$,
then, to good accuracy, the lattice can be used to compute the necessary IB
corrections from first principles. We illustrate the comparison of
electroproduction- and hadronic-$\tau$-decay-based $2\pi+4\pi$ data using two
rational weight choices, $W_{1,5}$ and $W_{2,5}$, defined in Sec. III.

We end the paper with a brief conclusion in Sec. V, and relegate some
technical details to two appendices.

##  II  Sum rules

In this section, we develop two types of sum rules, one, in Sec. II.1, based
on a set of weights used before in Ref. Kim , and one, in Sec. II.2, based on
the ideas advocated in Ref. HLT . They are closely related, and we will
explore their differences in examples in Sec. III.

We start from a Euclidean current-current correlator

$G_{\mu\nu}(x)=\left\langle
j_{\mu}(0)j^{\prime}_{\nu}(x)\right\rangle=\int\frac{d^{4}q}{(2\pi)^{4}}\,e^{iqx}\left(\delta_{\mu\nu}q^{2}-q_{\mu}q_{\nu}\right)\Pi(q^{2})\
,$ (1)

for two potentially different vector currents $j_{\mu}$ and
$j^{\prime}_{\nu}$, and define from this the time-momentum correlator $C(t)$
by

$C(t)=-\frac{1}{3}\sum_{k=1}^{3}\int d^{3}x\left\langle
j_{k}(0)j^{\prime}_{k}(\vec{x},t)\right\rangle=-\int\frac{dQ}{2\pi}\,e^{iQt}\,Q^{2}\Pi(Q^{2})\
,$ (2)

where $Q=q_{4}$, and we have assumed that a regulator (such as the lattice)
has been introduced, so that $C(t)$ and $\Pi(Q^{2})$ are finite. While in
straightforward applications to $a_{\mu}^{\rm HVP}$ the currents $j_{\mu}$ and
$j_{\nu}^{\prime}$ will both be the hadronic electromagnetic current, they can
also be chosen different, as will be done in the application described in Sec.
IV below. The corresponding subtracted polarization

$\hat{\Pi}(Q^{2})=\Pi(Q^{2})-\Pi(0)$ (3)

can be expressed in terms of $C(t)$ by

$\displaystyle\hat{\Pi}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{1}{Q^{2}}\int_{-\infty}^{\infty}dt\left(e^{iQt}-1\right)C(t)-\Pi(0)$
$\displaystyle=$
$\displaystyle-\frac{2}{Q^{2}}\int_{0}^{\infty}dt\left(\cos(Qt)-1\right)C(t)-\Pi(0)$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}dt\,\left(\frac{4\sin^{2}(Qt/2)}{Q^{2}}-t^{2}\right)\,C(t)\
,$

where we used $\int dt\,C(t)=0$ and, in the second step, that $C(t)$ is an
even function of $t$. We define the spectral function $\rho(s)$ as usual by

$\rho(s)=\frac{1}{\pi}\,\mbox{Im}\,\Pi(s)\ ,$ (5)

with $\hat{\Pi}(Q^{2})$ and $\rho(s)$ satisfying the subtracted dispersion
relation

$\hat{\Pi}(Q^{2})=-Q^{2}\int_{s_{\rm
th}}^{\infty}ds\,\frac{\rho(s)}{s(s+Q^{2})}\ ,$ (6)

with $s_{\rm th}$ the relevant threshold value for $\rho(s)$.

###  II.1  Rational-weight sum rules

> Figure 1: Contour $C$ used in Eq. (8); $z=q^{2}=-Q^{2}$. The black dot
> indicates the point $q^{2}=s_{\rm th}$.

We begin with a set of spectral weights of the form

$W_{m,n}(s;\\{Q_{\ell}^{2}\\})=\mu^{2(n-m-1)}\frac{(s-s_{\rm
th})^{m}}{\prod_{\ell=1}^{n}(s+Q_{\ell}^{2})}\ ,\qquad
Q_{n}^{2}>Q_{n-1}^{2}>\cdots>Q_{1}^{2}>0\ .$ (7)

Here the $Q_{\ell}^{2}$ are a set of fixed Euclidean squared momenta, and we
will always take $m$ sufficiently smaller than $n$ that the weighted spectral
integral with weight (7) is finite. We multiply by a generic mass scale
$\mu^{2(n-m-1)}$ to make the weighted spectral integrals considered below
dimensionless; any precisely known scale is suitable for this purpose and we
will employ the choice $\mu=m_{\tau}$ in what follows.

We then consider the integral

$\frac{1}{2\pi
i}\int_{C}dz\,W_{m,n}(z;\\{Q_{\ell}^{2}\\})\Pi(-z)=(-1)^{m}m_{\tau}^{2(n-m-1)}\sum_{k=1}^{n}\frac{(Q_{k}^{2}+s_{\rm
th})^{m}}{\prod_{\ell\neq k}(Q_{\ell}^{2}-Q_{k}^{2})}\,\Pi(Q_{k}^{2})\ ,$ (8)

with $C$ the contour in the complex $z=q^{2}=-Q^{2}$ plane shown in Fig. 1,
assumed to have a radius large enough that all the $z=-Q_{k}^{2}$ points lie
in its interior on the negative $z$-axis. The result on the right-hand side
follows from the fact that $\Pi(-z)$ is analytic in the complex plane except
for a cut starting at $s_{\rm th}$ on the positive real $z$-axis, as shown in
Fig. 1. If we now take the radius of the circular part of $C$ to infinity, we
obtain the sum rule

$I_{m,n}\equiv\int_{s_{\rm
th}}^{\infty}ds\,W_{m,n}(s;\\{Q_{\ell}^{2}\\})\rho(s)=(-1)^{m}m_{\tau}^{2(n-m-1)}\sum_{k=1}^{n}\frac{(Q_{k}^{2}+s_{\rm
th})^{m}}{\prod_{\ell\neq k}(Q_{\ell}^{2}-Q_{k}^{2})}\,\Pi(Q_{k}^{2})\ ,$ (9)

where the integral over the spectral function comes from the discontinuity
along the positive real $z$-axis. We will prove in App. A that

$\sum_{k=1}^{n}\frac{(Q_{k}^{2}+s_{\rm th})^{m}}{\prod_{\ell\neq
k}(Q_{\ell}^{2}-Q_{k}^{2})}=0\ ,$ (10)

from which it follows that we can replace $\Pi(Q_{k}^{2})$ by
$\hat{\Pi}(Q_{k}^{2})$ on the right-hand side of Eq. (9). This has to be the
case, as the left-hand side is finite by construction, and thus the term
proportional to $\Pi(0)$ on the right-hand side has to vanish.

A potentially useful application of this sum rule is to evaluate the spectral
integral using data, for instance obtained from $R$-ratio measurements, and
the sum on the right-hand side using data from lattice QCD. In fact, what is
usually computed in lattice QCD is a position-space correlator; in the case of
$a_{\mu}^{\rm HVP}$, Eq. (2) with $j_{\mu}=j_{\mu}^{\prime}$ the hadronic
electromagnetic current. Replacing $\Pi(Q_{k}^{2})$ with
$\hat{\Pi}(Q_{k}^{2})$ and using Eq. (II), the sum rule (9) can then be recast
as

$\int_{s_{\rm
th}}^{\infty}ds\,W_{m,n}(s;\\{Q_{\ell}^{2}\\})\rho(s)=\int_{0}^{\infty}dt\,c^{(m,n)}(t)\,C(t)\
,$ (11)

with

$c^{(m,n)}(t)=(-1)^{m}m_{\tau}^{2(n-m-1)}\,\sum_{k=1}^{n}\,\frac{(Q_{k}^{2}+s_{\rm
th})^{m}}{\prod_{\ell\neq
k}(Q_{\ell}^{2}-Q_{k}^{2})}\left(\frac{4\sin^{2}(Q_{k}t/2)}{Q_{k}^{2}}-t^{2}\right)\
.$ (12)

We will refer to the sum rules of Eq. (11) as “rational-weight sum rules”
(RWSRs). On the lattice, the integral over $t$ on the right-hand side of Eq.
(11) would be replaced by a sum over the discrete values of $t$ available on
the lattice, using, for example, the trapezoidal rule. We will discuss
examples in Sec. III.1 below.

###  II.2  Exponential-weight sum rules

The sum rules of Eq. (11) require the evaluation of $C(t)$ at all values of
$t$. In lattice QCD, the signal-to-noise ratio for $C(t)$ deteriorates when
$t$ gets large, and the contributions for these larger $t$ values will thus
“degrade” the precision with which the right-hand side of Eq. (11) can be
computed.

In this section, we therefore develop modified sum rules, involving new
weights constructed to require the evaluation of $C(t)$ at only a limited set
of $t$ values. While these new weights will, of necessity, differ from the old
weights $W_{m,n}$, we will see that, following the ideas of Ref. HLT , it is
possible to choose them to be remarkably similar to the $W_{m,n}$.

We begin with the observation that

$C(t)=\int_{E_{\rm
th}}^{\infty}dE\,\rho(E^{2})(E^{2}\,e^{-E|t|}-2E\,\delta(t))\ ,$ (13)

with $E_{\rm th}=\sqrt{s_{\rm th}}$ the threshold energy. The term
proportional to $\delta(t)$ ensures that $\int dt\,C(t)=0$.222This term is
formally divergent, but, as before, we assume that $C(t)$ has been regulated.
We then define new weights

$w_{n}(E;\\{t_{j}\\},\\{x_{j}\\})=\sum_{j=1}^{n}x_{j}(E^{2}e^{-|t_{j}|E}-2E\delta(t_{j}))\
,$ (14)

with $\\{t_{j}\\}$ a fixed set of $t$ values and $\\{x_{j}\\}$ a set of
coefficients to be determined below. In all our applications, the $t_{j}$ will
be chosen positive, and we can ignore the term with the $\delta$-function. The
$w_{n}(E;\\{t_{j}\\},\\{x_{j}\\})$-weighted spectral integrals then satisfy
sum rules,

$\int_{E_{\rm
th}}^{\infty}dE\,\rho(E^{2})w_{n}(E;\\{t_{j}\\},\\{x_{j}\\})=\sum_{j=1}^{n}x_{j}C(t_{j})\
,$ (15)

in which only the $n$ values $C(t_{j})$ occur on the right-hand side.

The goal then is to start with an initial weight, $W(s)$, having some desired
$s$ dependence (for example, one of the weights
$W_{m,n}(s,\\{Q^{2}_{\ell}\\})$) and find a set of $n$, $\\{t_{j}\\}$ and
$\\{x_{j}\\}$ such that the associated $w_{n}(E;\\{t_{j}\\},\\{x_{j}\\})$
represents a close approximation to the weight $2E\,W(s=E^{2})$ appearing in
the alternate, $E$-dependent expression

$\int_{s_{\rm th}}^{\infty}ds\,W(s)\rho(s)\,=\,\int_{E_{\rm
th}}^{\infty}dE\,2E\,W(E^{2})\,\rho(E^{2})$ (16)

for the $W(s)$-weighted spectral integral. If such a choice exists, the
associated sum rule, Eq. (15), will involve a spectral integral whose
weighting, by construction, is similar to that of the desired original
$W(s)$-weighted spectral integral, but whose right-hand side involves values
of $C(t)$ at only the finite number of chosen $t$.

The key point is that it is indeed possible, using the method of Ref. HLT , to
choose $n$, the $\\{t_{j}\\}$ and the $\\{x_{j}\\}$ such that
$w_{n}(E;\\{t_{j}\\},\\{x_{j}\\})$ provides a very good approximation to
$2E\,W(s=E^{2})$ for weight choices, like
$W(s)=W_{m,n}(s,\\{Q^{2}_{\ell}\\})$, of interest in exploring the
$a_{\mu}^{\rm HVP}$ problem. While in what follows we focus, to be specific,
on examples with $W(s)=W_{m,n}(s=E^{2};\\{Q_{\ell}^{2}\\})$, we stress that
the method is applicable to more general initial weight choices as well.

The construction of Ref. HLT  proceeds as follows. Starting with an initial
desired weight $W(s)$, to be referred to in what follows as the “pre-mold,” we
choose a set of (positive) $t$ values $\\{t_{j}\\}$, $j=1,\cdots,n$, and
minimize

$\int_{E_{\rm
th}}^{\infty}{\frac{dE}{E^{4}}}\,\biggl{|}w_{n}(E;\\{t_{j}\\},\\{x_{j}\\})-2E\,W(E^{2})\biggr{|}^{2}$
(17)

with respect to the parameters $x_{j}$, $j=1,\dots,n$. We denote the parameter
values which accomplish this minimization by $\\{x_{j}^{W}\\}$. The result is
a $w_{n}(E;\\{t_{j}\\},\\{x^{W}_{j}\\})$ which represents a (close)
approximation to $2E\,W(E^{2})$, one having the form of $E^{2}$ times a
weighted sum of exponentials. Following Ref. HLT , we will refer to the
product $2E\,W(E^{2})$ as the “mold,” and the approximation
$w_{n}(E;\\{t_{j}\\},\\{x^{W}_{j}\\})$ as the “cast.” The sum rule Eq. (15)
then takes the form

$I_{W^{\prime}}\,\equiv\,\int_{E_{\rm
th}}^{\infty}dE\,\rho(E^{2})w_{n}(E;\\{t_{j}\\},\\{x_{j}^{W}\\})=\sum_{j=1}^{n}x_{j}^{W}C(t_{j})\
,$ (18)

where the subscript/superscript $W$ emphasizes the role of the underlying pre-
mold function, $W(s)$, while the prime reminds us that the spectral integral
is evaluated with the derived cast $w_{n}$. We will refer to sum rules of the
form (18) as “exponential-weight sum rules” (EWSRs). Equation (17) is solved
by

$x_{i}=\sum_{j=1}^{n}A^{-1}_{ij}f_{j}\ ,$ (19)

with the matrix $A$ and the input vector $f$ defined by

$A_{ij}=\int_{E_{\rm th}}^{\infty}dE\,e^{-(t_{i}+t_{j})E}\ ,\qquad
f_{i}=2\int_{E_{\rm th}}^{\infty}dE\,e^{-t_{i}E}W(E^{2})/E\ .$ (20)

At this point, we depart from the philosophy of Ref. HLT : we throw away the
mold and keep the cast. If the cast is a good approximation of the mold, it
will serve equally well for comparing experimental data, represented by
$\rho(s)$, with lattice data, represented by $C(t)$, in the region of energies
characterized by the mold, and there is thus no need to keep the mold.
Instead, we work directly with the cast sum rule, Eq. (18).

Following this philosophy, there is also no need to keep the values of the
$x_{j}$ obtained from Eq. (19) to a very high precision. One can keep a fixed
number of digits, and declare the values chopped off after the last of these
digits as the exact values of the $x_{j}$ to be used. The only requirement is
that the thus-defined exponential weight still fullfills the goal for which it
was designed, i.e., that it probes the desired energy range. In our examples
in the next section, we will chop off the $x_{j}$ values at six digits, cf.
Eqs. (26) and (27).

The only role of the pre-mold and mold functions in this approach is to fix
the type of weighting one wants in the weighted spectral integral in cases
(such as exploring the BaBar–KLOE discrepancy) where the desired weighting is
naturally formulated in the $s$-space spectral integral representation. The
cast then provides a similarly weighted spectral integral having the advantage
that the corresponding weighted Euclidean integral is trivially written down
using the exact sum rule (18) above. Attempting to find a set of $x_{j}$ which
produce such a desired spectral integral weighting is a much more challenging
task without the intermediate step of first setting up the mold function and
using it to find a set of $x_{j}$ which implement the Euclidean representation
corresponding to the closely related $s$-dependent cast weighting. We will
discuss examples of this strategy in Sec. III.2, where the pre-molds will be
the weights we consider in Sec. III.1.

To close this subsection, we discuss one more variant. For the types of weight
functions we consider in this paper, the matrix $A$ defined in Eq. (20) may
have very small eigenvalues, and, correspondingly, the $x^{W}_{j}$ values may
span a rather large range (as in Eqs. (26) and (27) below), leading to
potentially sizeable cancellations and possible error inflation. We may reduce
this range by modifying the matrix $A$ to some extent, while keeping the cast
close to the mold, so that the modified sum rule still probes essentially the
same range in $\sqrt{s}$. The specific modification we will consider is to
replace the matrix $A$ with the related matrix $\hat{A}$, defined by

$\widehat{A}(\lambda)=(1-\lambda)A+\lambda{\bf 1}_{n}\ ,$ (21)

where ${\bf 1}_{n}$ is the $n\times n$ unit matrix. This simply replaces
eigenvalues of $A$ much smaller than $\lambda$ with new eigenvalues of order
$\lambda$, leaving eigenvalues much larger than $\lambda$ essentially
unchanged. We thus expect, and indeed find to be the case below, that if we
choose $\lambda$ larger than the smallest eigenvalue of $A$ the range of the
$x^{W}_{j}$ values, and hence the error on the lattice sides of the resulting
EWSRs, will be reduced. We will denote the analogue of the spectral integral
$I_{W^{\prime}}$ of Eq. (18) obtained using the modified set of $x^{W}_{j}$
resulting from this replacement by $I_{\widehat{W}}$ in what follows.

##  III  Example: application to comparison between experimental and lattice
data

> Figure 2: The weights $W_{1,5}$ (blue, solid), and $15\times W_{2,5}$ (red,
> dashed) with $Q^{2}_{i}$ defined in Eq. (22). ($W_{2,5}$ is multiplied by a
> factor 15 for clarity.) The black dotted curve shows the “intermediate
> window” weight of Ref. RBC , transformed to the equivalent
> $\sqrt{s}$-dependent form Rwindow .

We will now turn to two numerical examples, using two differents weights
$W_{m,n}$ of the type (7), both with $s_{\rm th}=4m_{\pi}^{2}$ and

$Q_{\ell}^{2}=0.25+0.075(\ell-1)\ \mbox{GeV}^{2}\ ,\qquad\ell=1,\dots,5\ .$
(22)

The first has $m=1$, $n=5$, and the second $m=2$, $n=5$. We choose
$m_{\pi}=134.977$ MeV and $m_{\tau}=1776.86$ MeV. These two weights are
plotted as a function of $\sqrt{s}$ in Fig. 2, where $W_{2,5}$ has been
multiplied by a factor 15 to show it on approximately the same scale as
$W_{1,5}$.333The absolute scale of the weights is arbitrary, as long as the
same scale is used on both sides of the sum rules. The weight $W_{2,5}$
overlaps better with the region dominated by the $\rho$, but falls off more
slowly for large $s$. The value of $n-m$ is smaller for $W_{2,5}$ than for
$W_{1,5}$, which leads us to expect a smaller error on the right-hand side of
the $W_{2,5}$ sum rule, according to App. A. For comparison, we also show the
Minkowski version Rwindow  of the “intermediate window” weight introduced as a
function of $t$ in Ref. RBC . These weights will be used for a study of the
sum rule (11) in Sec. III.1 and a study of the sum rule (18) in Sec. III.2.

> Figure 3: The weights $10^{-5}c^{(1,5)}(t)$ (blue), and $15\times
> 10^{-5}c^{(2,5)}(t)$ (red) with $Q^{2}_{\ell}$ as in Eq. (22).

In Fig. 3 we show the functions $c^{(1,5)}(t)$ and $c^{(2,5)}(t)$ defined in
Eq. (12), multiplied by $10^{-5}$ and $15\times 10^{-5}$, respectively. We see
that these functions have the bulk of their support toward larger $t$, while
being very small at values of $t$ smaller than approximately 1 fm. This is not
surprising, as the weights $W_{1,5}(s)$ and $W_{2,5}(s)$ have their support at
relatively small $\sqrt{s}$. We will see below to what extent this has an
effect on the size of the errors with which the right-hand side of Eq. (11)
can be evaluated.

In these examples, we will use the combined $R$-ratio data from Ref. KNT19  to
evaluate the weighted spectral integrals on the left-hand side (LHS) and the
light-quark-connected lattice data from Ref. ABGP22  to evaluate the weighted
integrals over $C(t)$ on the right-hand side (RHS). Since the data from Ref.
ABGP22  are for the light-quark-connected part only, this is a comparison
between apples and oranges as far as the central results are concerned. Here,
however, we are interested in seeing the size of typical errors on each side
of the sum rule, and using these immediately accessible recent data sources
suffices for this purpose. We emphasize that our intent is to investigate the
size of lattice and spectral integral errors in our proposed methodology, and
that, at present, no conclusions should be drawn from the level of numerical
agreement between the two sides of the sum rules considered throughout Sec.
III. We also neglect corrections to the lattice data of Ref. ABGP22  for
finite-volume, taste-breaking and pion-mass mistuning effects, having
convinced ourselves that these corrections are small, and do not lead to
qualitative changes in the conclusions we obtain below. This is consistent
with our intent to study the methodology of these sum rules, leaving concrete
applications to the future.

###  III.1  Examples using rational-weight sum rules

We begin with the values of the spectral integrals appearing on the LHS of the
sum rule (11) for the weights $W_{1,5}$ and $W_{2,5}$ obtained using the
$R$-ratio data from Ref. KNT19 . We find

$\displaystyle I_{W_{1,5}}({\rm LHS})$ $\displaystyle=$ $\displaystyle
0.4756(16)\ ,$ (23) $\displaystyle I_{W_{2,5}}({\rm LHS})$ $\displaystyle=$
$\displaystyle 0.09107(34)\ .$

Both spectral integrals are obtained with a precision better than $0.4\%$.

label |  $a$ (fm) | $L^{3}\times T$ |  $m_{\pi}$ (MeV) | $I_{W_{1,5}}({\rm RHS})$ | $I_{W_{2,5}}({\rm RHS})$
---|---|---|---|---|---
96 | 0.05684 | $96^{3}\times 192$ | 134.3 | 0.463(45) | 0.0860(74)
64 | 0.08787 | $64^{3}\times 96$ | 129.5 | 0.453(23) | 0.0826(28)
48I | 0.12121 | $48^{3}\times 64$ | 132.7 | 0.426(13) | 0.0831(13)
32 | 0.15148 | $32^{3}\times 48$ | 133.0 | 0.433(13) | 0.0801(17)
48II | 0.15099 | $48^{3}\times 64$ | 134.3 | 0.411(14) | 0.0824(18)

> Table 1: The right-hand sides of Eq. (11) for the lattice ensembles of Ref.
> ABGP22 . The columns to the left of the double vertical line contain the
> label of the ensemble, the lattice spacing $a$, the spatial/temporal extent,
> $L^{3}\times T$ (in lattice units), and the Nambu–Goldstone pion mass
> $m_{\pi}$. The columns to the right of the double line contain the integrals
> $I_{W_{1,5}}({\rm RHS})$ and $I_{W_{2,5}}({\rm RHS})$.

We next turn to the evaluation of the RHSs of the $W_{1,5}$ and $W_{2,5}$
RWSRs using lattice data for the light-quark connected contribution to the
correlator $C(t)$ obtained using the five ensembles of Ref. ABGP22 . Table 1
shows the ensemble parameters to the left of the double vertical line, and the
associated finite-lattice-spacing results for $I_{W_{1,5}}({\rm RHS})$ and
$I_{W_{2,5}}({\rm RHS})$ to the right of it. We note that the values of the
pion mass and the volume differ from ensemble to ensemble, and would, in
principle, need to be adjusted to common values before carrying out an
extrapolation to the continuum limit. Since, however, the current statistical
errors are large enough that these mistunings would make no difference in
practice, we have, instead, neglected such corrections, and carried out an
extrapolation linear in $a^{2}$ to the continuum limit. For the $a=0.15$ fm
lattice spacing we chose the larger of the two volumes, not using the 32
ensemble. The fits have excellent $p$-values, and we find the continuum-limit
values

$\displaystyle I_{W_{1,5}}({\rm RHS})$ $\displaystyle=$ $\displaystyle
0.468(26)\ ,$ (24) $\displaystyle I_{W_{2,5}}({\rm RHS})$ $\displaystyle=$
$\displaystyle 0.0838(33)\ .$

These integrals have a precision of $5.6\%$, respectively, $3.9\%$.444The
relatively large errors make the figures showing these fits not very
interesting, and we thus omit them. In the next subsection we will show the
equivalent fits for the EWSRs, for which the lattice values have smaller
errors. Based on these results we make the following observations. First, the
errors in Eq. (24) are large, and a significant reduction in errors would be
needed to make these sum rules useful. However, as we will see in the next
subsection, errors on the RHS, obtained using the same lattice data, are
smaller for sum rules of the type (18) than for those of the type (11).
Second, from the different size of the errors for the sum rules with $W_{1,5}$
and $W_{2,5}$, it is clear that a fine tuning of the rational weight may be
necessary to make optimal use of a certain set of lattice data. We emphasize
again that the numbers in Eqs. (23) and (24) should not be directly compared
in this example, because, in this pilot study, we used the fully inclusive
data on the LHS, but only light-quark-connected data on the RHS. Of course,
fully inclusive lattice data are becoming available, and with such data, a
direct comparison will be possible. It is, moreover, reasonable to expect such
lattice data to have significantly improved errors, allowing for significant
improvements to the relative errors in Eq. (24) as well.

###  III.2  Examples using exponential-weight sum rules

We now turn to sum rules of the type (18), using as examples of the pre-mold
the weights $W_{1,5}$ and $W_{2,5}$ of Sec. III.1. We still need to choose the
$t_{j}$ appearing in Eq. (18) and, in this section, will employ the following
values, given in units of GeV-1:

$t_{1}=3\ ,\quad t_{2}=6\ ,\quad t_{3}=9\ ,\quad t_{4}=12\ ,\quad t_{5}=15\ .$
(25)

Since these values do not exactly coincide with the values of $t$ available
for the lattice ensembles we employ, we obtain the associated $C(t_{j})$
values, in all cases, by linear interpolation from the nearest available $t$,
taking into account all correlations in the interpolation. The range of values
in Eq. (25) is one for which lattice computations of $C(t)$ generally have
small statistical errors.

With the $t_{j}$ defined, the values of the coefficients $x_{j}$ are fixed by
the pre-mold choice. For $W^{\prime}_{1,5}$ we find

$W^{\prime}_{1,5}:\quad x_{1}=37.4123\ ,\ x_{2}=2625.13\ ,\ x_{3}=25912.1\ ,\
x_{4}=-106707.\ ,\ x_{5}=78192.8\ ,$ (26)

while for $W^{\prime}_{2,5}$ we find

$W^{\prime}_{2,5}:\quad x_{1}=34.0249\ ,\ x_{2}=870.640\ ,\ x_{3}=-5501.14\ ,\
x_{4}=9933.01\ ,\ x_{5}=-5284.24\ .$ (27)

For $\widehat{W}_{1,5}$ and $\widehat{W}_{2,5}$, obtained by replacing $A$ by
the $\lambda=10^{-9}$ version of $\widehat{A}$ of Eq. (21) in Eq. (19), we
find

$\widehat{W}_{1,5}:\quad x_{1}=-78.8487\ ,\ x_{2}=5688.30\ ,\ x_{3}=2223.96\
,\ x_{4}=-36638.0\ ,\ x_{5}=8047.38\ ,$ (28)

and

$\widehat{W}_{2,5}:\quad x_{1}=44.8916\ ,\ x_{2}=590.933\ ,\ x_{3}=-3373.53\
,\ x_{4}=3716.86\ ,\ x_{5}=879.149\ .$ (29)

As already explained in Sec. II.2, we will use these exact values in the rest
of this subsection.

In the upper panels of Fig. 4 we show the molds $2\sqrt{s}\,W_{1,5}(s)$ and
$2\sqrt{s}\,W_{2,5}(s)$ together with their associated casts,
$W_{1,5}^{\prime}(s)=w_{5}(E;\\{t_{j}\\},\\{x_{j}^{W^{\prime}_{1,5}}\\})$ and
$W_{2,5}^{\prime}(s)=w_{5}(E;\\{t_{j}\\},\\{x_{j}^{W^{\prime}_{2,5}}\\})$,
determined by Eqs. (25) and (26), (27). We also show the casts
$\widehat{W}_{1,5}(s)=w_{5}(E;\\{t_{j}\\},\\{x_{j}^{\widehat{W}_{1,5}}\\})$
and
$\widehat{W}_{2,5}(s)=w_{5}(E;\\{t_{j}\\},\\{x_{j}^{\widehat{W}_{2,5}}\\})$,
determined with Eqs. (28) and (29) instead. On the scale of these figures, the
casts are almost indistinguishable from the underlying molds for the $A$-based
casts, and quite close for the $\widehat{A}$-based casts. The smaller-
vertical-scale lower panels show the corresponding cast–mold differences in
more visually evident form.

> Figure 4: Upper panels: molds (blue continuous curves) and casts (red dashed
> curves) for the molds $2\sqrt{s}\,W_{1,5}(s)$ (left panel) and
> $2\sqrt{s}\,W_{2,5}(s)$ (right panel). The black dot-dashed curves show the
> casts obtained with $\widehat{A}$ of Eq. (21) instead of $A$ of Eq. (20),
> choosing $\lambda=10^{-9}$. Lower panels: differences between mold and cast,
> for each case (with the dashed curves the difference for the
> $\widehat{A}$-based casts). Note the very different scales on the vertical
> axes in the upper and lower figures.

As for the RWSR cases, we first give the LHS values, $I_{W^{\prime}_{1,5}}$
and $I_{W^{\prime}_{2,5}}$, obtained by integrating over the $R$-ratio data:

$\displaystyle I_{W_{1,5}^{\prime}}({\rm LHS})$ $\displaystyle=$
$\displaystyle 0.4788(16)\ ,$ (30) $\displaystyle I_{W_{2,5}^{\prime}}({\rm
LHS})$ $\displaystyle=$ $\displaystyle 0.08922(34)\ .$

For $I_{\widehat{W}_{1,5}}$ and $I_{\widehat{W}_{{2,5}}}$ we find

$\displaystyle I_{\widehat{W}_{1,5}}({\rm LHS})$ $\displaystyle=$
$\displaystyle 0.4488(16)\ ,$ (31) $\displaystyle I_{\widehat{W}_{2,5}}({\rm
LHS})$ $\displaystyle=$ $\displaystyle 0.09214(37)\ .$

As before, the $R$-ratio data of Ref. KNT19  allow us to determine these
values with errors $\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}\,0.4\%$. We note that the values
in Eq. (30) or Eq. (31) are not the same as those in Eq. (23), even if they
are rather close. This reflects the fact that mold and cast weights are not
identical.

Because of the well-known even-odd time oscillation of staggered correlators,
we define smoothed-out correlators

$\widehat{C}(t)=\frac{1}{4}\left(C(t-1)+2C(t)+C(t+1)\right)\ .\\\ $ (32)

We will use the averaged correlator $\widehat{C}(t)$ when evaluating the RHS
of Eq. (18).555More sophisticated approaches are possible, but we consider
this average sufficient for the explorative nature of this section.

We show the results for $I_{W_{1,5}^{\prime}}({\rm RHS})$,
$I_{W_{2,5}^{\prime}}({\rm RHS})$, $I_{\widehat{W}_{1,5}}({\rm RHS})$ and
$I_{\widehat{W}_{2,5}}({\rm RHS})$ in Table 2. Using the values obtained with
the three smallest lattice spacings, i.e., the ensembles 96, 64 and 48I, and
performing fits linear in $a^{2}$, we find the continuum-limit values

$\displaystyle I_{W_{1,5}^{\prime}}({\rm RHS})$ $\displaystyle=$
$\displaystyle 0.496(17)\ ,$ (33) $\displaystyle I_{W_{2,5}^{\prime}}({\rm
RHS})$ $\displaystyle=$ $\displaystyle 0.0798(18)\ ,$

and

$\displaystyle I_{\widehat{W}_{1,5}}({\rm RHS})$ $\displaystyle=$
$\displaystyle 0.4669(68)\ ,$ (34) $\displaystyle I_{\widehat{W}_{2,5}}({\rm
RHS})$ $\displaystyle=$ $\displaystyle 0.0824(10)\ .$

These linear fit results are shown by the red lines in Fig. 5. The blue curves
in the same figure show the results of alternate fits quadratic in $a^{2}$
obtained using the four ensembles 96, 64, 48I and 48II. The linear and
quadratic fits give consistent continuum-limit results.

label | $I_{W_{1,5}^{\prime}}({\rm RHS})$ | $I_{W_{2,5}^{\prime}}({\rm RHS})$ | $I_{\widehat{W}_{1,5}}({\rm RHS})$ | $I_{\widehat{W}_{2,5}}({\rm RHS})$
---|---|---|---|---
96 | 0.504(20) | 0.0799(23) | 0.4625(75) | 0.0838(12)
64 | 0.4822(92) | 0.08703(94) | 0.4520(38) | 0.08994(54)
48I | 0.4790(72) | 0.09187(80) | 0.4401(40) | 0.09558(51)
32 | 0.498(12) | 0.10073(76) | 0.4398(68) | 0.10623(45)
48II | 0.4837(85) | 0.1009(11) | 0.4301(46) | 0.10605(64)
$a=0$ | 0.496(17) | 0.0798(18) | 0.4669(68) | 0.0824(10)
$p$-value | 0.44 | 0.15 | 0.78 | 0.10
rel. error | 3.4% | 2.2% | 1.5% | 1.2%

> Table 2: The right-hand side of Eq. (18) for the lattice ensembles of Ref.
> ABGP22 . The 2nd and 3rd columns give, respectively, the values for
> $I_{W_{1,5}^{\prime}}$ and $I_{W_{2,5}^{\prime}}$, obtained using Eq. (32).
> The 4th and 5th columns give, respectively, the values for
> $I_{\widehat{W}_{1,5}}$ and $I_{\widehat{W}_{2,5}}$, obtained using
> $\widehat{A}$ of Eq. (21) instead of $A$, with $\lambda=10^{-9}$. Below the
> double horizontal line we show (i) the continuum extrapolation for each
> column, obtained using fits linear in $a^{2}$ to the values for the first
> three ensembles (96, 64 and 48I), (ii) the $p$-value of that fit, and (iii)
> the relative statistical error of the continuum limit value.

As shown in the Table 2, the integrals, $I_{W_{1,5}^{\prime}}({\rm RHS})$ and
$I_{W_{2,5}^{\prime}}({\rm RHS})$ in Eqs. (33) have precisions of $3.4\%$ and
$2.2\%$, respectively, while $I_{\widehat{W}_{1,5}}({\rm RHS})$ and
$I_{\widehat{W}_{2,5}}({\rm RHS})$ in Eqs. (34) have precisions of $1.5\%$ and
$1.2\%$, respectively. We see that the modification (21) indeed pays off. We
emphasize again that the LHS and RHS central values should not be directly
compared in this study. What is of interest here is the size of the
statistical errors in Eqs. (30), (31), (33) and (34). While those in Eqs. (30)
and (31) are the same as those in Eq. (23), those in Eqs. (33) and (34) are
significantly smaller than those in Eq. (24). This reflects the freedom to
avoid, through the restricted set of $t_{j}$ values employed in Eqs. (33) and
(34), higher-error, large-$t$ contributions, and provides a concrete example
of the potential enhanced utility of EWSRs over their closely related RWSR
counterparts.

> Figure 5: Continuum extrapolations of $I_{W_{1,5}^{\prime}}({\rm RHS})$
> (left upper panel) and $I_{W_{2,5}^{\prime}}({\rm RHS})$ (right upper
> panel), and $I_{\widehat{W}_{1,5}}({\rm RHS})$ (left lower panel) and
> $I_{\widehat{W}_{2,5}}({\rm RHS})$ (right lower panel). The data points show
> the lattice results for the various ensembles. Linear fits to the three
> left-most points (omitting the gray filled circles) are shown by the red
> lines, and quadratic fits to all four data points (the black and gray filled
> circles) by the dashed blue curves. Results for the 32 ensemble, shown as
> orange diamonds, are not used in either of these extrapolations.

We add several comments. First, it is likely that lattice data in the near
future will have smaller statistical errors than does the lattice data of Ref.
ABGP22 , used in the current study. This makes it not unlikely that sub-
percent precision can be reached with EWSRs. Second, we did not discuss
systematic errors, which of course also need to be controlled. In particular,
for staggered fermions, smaller lattice spacings than the current smallest
value of about $0.06$ fm will be needed ABGP22 . We thus do not claim that our
linear fits to the results in Table 2 are the final word. However, similar
fits to improved data at smaller lattice spacings should allow these sum rules
to become of practical use. Finally, we note that the $W_{1,5}$ versions of
Eqs. (30) and (33) are closer together than the $W_{2,5}$ versions. This can
be qualitatively understood. The large-$t$ part of the full HVP correlator
$C(t)$ with $j_{\mu}=j^{\prime}_{\mu}$ the hadronic electromagnetic current is
dominated by the light-quark-connected part, more so than the smaller-$t$
part. As can be seen from Fig. 2, the weight $W_{1,5}$ has support at smaller
$s$. This translates into having support at larger $t$, as can be seen in Fig.
3. This explains the closer agreement for the $W_{1,5}$-based case. A similar
observation also holds for Eqs. (23) and (24) in Sec. III.1.

###  III.3  Comparison with BaBar–KLOE discrepancy

One would like to get an idea about what is needed to make the lattice errors
in Eq. (33) small enough for concrete applications of these sum rules. To this
end, we consider, as an example, the difference of the values of
$I_{W^{\prime}_{1,5}}$ and $I_{W^{\prime}_{2,5}}$ computed using the BaBar
BaBar ; BaBar:2012bdw  or KLOE KLOE  versions of the two-pion contribution to
the EM spectral function. As Ref. KLOE  provides the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ cross sections only up to $s=0.95$ GeV2,
we will assume that the full discrepancy originates from the difference in the
measured cross sections below $s=0.95$ GeV2. This allows us to evaluate the
two-pion contributions to $I_{W_{1/2,5}}$, $I_{W^{\prime}_{1/2,5}}$ and
$I_{\widehat{W}_{1/2,5}}$ on both the BaBar and KLOE data. We find

$\displaystyle
I_{W_{1,5}}^{\pi\pi}(\mbox{BaBar})-I_{W_{1,5}}^{\pi\pi}(\mbox{KLOE})$
$\displaystyle=$ $\displaystyle 0.0091(39)\ ,$ (35) $\displaystyle
I_{W_{2,5}}^{\pi\pi}(\mbox{BaBar})-I_{W_{2,5}}^{\pi\pi}(\mbox{KLOE})$
$\displaystyle=$ $\displaystyle 0.00152(51)\ ,$ $\displaystyle
I_{W^{\prime}_{1,5}}^{\pi\pi}(\mbox{BaBar})-I_{W^{\prime}_{1,5}}^{\pi\pi}(\mbox{KLOE})$
$\displaystyle=$ $\displaystyle 0.0091(39)\ ,$ $\displaystyle
I_{W^{\prime}_{2,5}}^{\pi\pi}(\mbox{BaBar})-I_{W^{\prime}_{2,5}}^{\pi\pi}(\mbox{KLOE})$
$\displaystyle=$ $\displaystyle 0.00153(52)\ ,$ $\displaystyle
I_{\widehat{W}_{1,5}}^{\pi\pi}(\mbox{BaBar})-I_{\widehat{W}_{1,5}}^{\pi\pi}(\mbox{KLOE})$
$\displaystyle=$ $\displaystyle 0.0094(40)\ ,$ $\displaystyle
I_{\widehat{W}_{2,5}}^{\pi\pi}(\mbox{BaBar})-I_{\widehat{W}_{2,5}}^{\pi\pi}(\mbox{KLOE})$
$\displaystyle=$ $\displaystyle 0.00150(51)\ .$

Comparing these values with the lattice values of Eqs. (33) and (34), we see
that the lattice errors are factors of about 1.9, 1.2, 0.7 and 0.7 times the
differences shown in Eq. (35) for $W^{\prime}_{1,5}$, $W^{\prime}_{2,5}$,
$\widehat{W}_{1,5}$ and $\widehat{W}_{2,5}$, respectively. This implies that
in order to use the EWSRs with the primed weights, an improvement of a factor
about 2 to 4 in the lattice errors would be needed, while for the hatted
weights, a very modest improvement would be sufficient for the EWSRs to weigh
in on the BaBar–KLOE discrepancy. A combination of more precise lattice data
and possibly a more fine-tuned design of the sum-rule weight functions would
help accomplishing this. We note that, for this goal, the weight
$W^{\prime}_{2,5}$ was better “designed” than the weight $W^{\prime}_{1,5}$,
and the weight $\widehat{W}_{2,5}$ similarly better “designed” than the weight
$\widehat{W}_{1,5}$. We also note that, in both cases, the hatted weights
outperform their primed counterparts.

##  IV  Using hadronic $\tau$-decay data for $a_{\mu}^{\rm HVP}$

We now turn to a different application of the sum rules introduced in Sec. II:
the use of spectral data obtained from non-strange vector-current-induced
hadronic $\tau$-decay data in the determination of $a_{\mu}^{\rm HVP}$.

The idea of using $\tau$ data is straightforward in the isospin limit. If
isospin were an exact symmetry, one could replace contributions to the
$R$-ratio from $I=1$ exclusive modes in $e^{+}e^{-}\to$ hadrons by the
corresponding contributions implied by CVC and the related vector-channel
($G$-parity positive) exclusive-mode hadronic $\tau$-decay distributions, for
$s\leq m_{\tau}^{2}$. Doing so is potentially useful for three reasons: (a)
if, for a certain energy region, the $\tau$ data are more precise, this would
help in improving the precision of the dispersive determination of
$a_{\mu}^{\rm HVP}$; (b) the $\tau$ data might shed light on the long-standing
discrepancy between BaBar and KLOE results for the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross sections observed in the $\rho$ region
discussed in Sec. III.3; and (c) likewise, the $\tau$ data might shed light on
the increasing evidence for discrepancies between $R$-ratio and lattice based
evaluations of $a_{\mu}^{\rm HVP}$ or closely related window quantities.

One of the reasons the current best dispersive determination of $a_{\mu}^{\rm
HVP}$ does not employ hadronic $\tau$-decay data is that, in the real world,
isospin is broken by electromagnetic corrections and the difference in up and
down quark masses. Typically, both isospin-breaking (IB) effects are expected
to be of order a percent or so (except possibly in the region of narrow
interfering resonances, where the effects may be larger). Given the precision
with which $a_{\mu}^{\rm HVP}$ is obtained in the dispersive approach, IB
effects thus need to be brought under quantitative control. Models for IB have
been employed in the past review , but these are not under full
phenomenological control, as evidenced by the fact that they fail to account
for the observed differences between the experimental $\tau$ and
electroproduction $\pi\pi$ distributions (see, for example, Figs. 20 and 22 of
Ref. review ). Because of the steady improvement in $R$-ratio data, and the
problem of achieving an understanding of the IB corrections one needs to apply
to the $\tau$ $\pi\pi$ data, recent high-precision dispersive estimates of
$a_{\mu}^{\rm HVP}$ do not involve $\tau$-decay data at all.

There are, nevertheless, several reasons to revisit this topic. First, lattice
QCD may be able to produce relevant, first-principle, estimates for IB effects
that are useful in the comparison between $e^{+}e^{-}$-based and $\tau$-based
data, as will be discussed below. Second, a new combination of the sum of the
$\pi^{-}\pi^{0}$, $2\pi^{-}\pi^{+}\pi^{0}$ and $\pi^{-}3\pi^{0}$ contributions
to the isovector, vector spectral function from ALEPH and OPAL data has
recently been obtained, producing an at-present-best publicly available result
for the sum of the contributions from these modes AO .666The reason only the
sum is available is that the ALEPH and OPAL covariance matrices for the two
$4\pi$ distributions are highly singular, which can lead to biased results in
the data-combination procedure. The sum of $2\pi$ and $4\pi$ contributions has
a much better behaved covariance matrix and avoids this issue. Significant
further improvements to the precision for this sum may, moreover, be
achievable using Belle II data.777An additional practical issue is the need to
ensure the radiative corrections applied to $\tau$ data match those employed
in the analysis of electroproduction data. For more on this issue, and work in
progress on evaluating the relevant corrections, see Ref. BrunoMeyer .

The basic idea is, using the weights $W_{m,n}$, $W_{m,n}^{\prime}$ or
$\widehat{W}_{m,n}$, to compare the weighted spectral integrals computed with
spectral data obtained from electroproduction and from hadronic $\tau$ decays.
For any (sum of) $I=1$ channels, the SM difference is caused by IB, and can be
expressed as the corresponding difference of the RHS of Eq. (11) (or (18)). In
principle, the RHS differences can be computed on the lattice, as we will
discuss in Sec. IV.1. The other important ingredient is the precision with
which the $W_{m,n}$-, $W_{m,n}^{\prime}$-, or $\widehat{W}_{m,n}$-weighted
spectral integrals can be evaluated. As we will see, choosing weights like
$W_{1,5}$ and $W_{2,5}$, which zoom in on the region of the BaBar–KLOE
discrepancy, the weighted spectral integrals of the electroproduction- and
$\tau$-based data are already sufficiently precise that results using current
$\tau$ data will allow one to weigh in on the BaBar–KLOE discrepancy, provided
the required IB corrections can be evaluated with comparable precision on the
lattice. We will discuss this point more quantitatively in Sec. IV.2.

###  IV.1  Isospin breaking

The main problem in comparing spectral data from electroproduction and
hadronic $\tau$ decays using lattice data to compute IB effects in the
hadronic vacuum polarization $\Pi(Q^{2})$ is that the lattice can only access
$\Pi(Q^{2})$ for Euclidean $Q^{2}$.888Or, equivalently, $C(t)$ for Euclidean
$t$. This is not a problem if one wants to compare fully inclusive results for
$a_{\mu}^{\rm HVP}$, since representations exist for $a_{\mu}^{\rm HVP}$ both
as a weighted integral of the $R$-ratio, $R(s)$, over $s$Brodsky:1967sr ;
Lautrup:1968tdb ; Gourdin:1969dm  and as a weighted integral of
$\hat{\Pi}(Q^{2})$ over $Q^{2}$Lautrup:1971jf ; deRafael:1993za ; Blum:2002ii
. It does, however, become a problem if one wants to restrict oneself to
contributions from a subset of exclusive modes, and/or to restrict $s$ to a
given region $s_{\rm min}\leq s\leq s_{\rm max}$. Because of the kinematic
restriction $s\leq m_{\tau}^{2}$ for $\tau$ decays, $s_{\rm max}$ by necessity
has to be chosen $\leq m_{\tau}^{2}$, whereas one may want to choose $s_{\rm
min}$ larger than threshold to reduce the relative statistical errors. Similar
observations apply to the sum rules of Sec. II, in which, as for $a_{\mu}^{\rm
HVP}$, the spectral integrals are evaluated from $s_{\rm th}$ to $\infty$.

The bulk of the dispersive contributions to $a_{\mu}^{\rm HVP}$ come from
electroproduction data below $\sqrt{s}=1.937$ GeV, with $e^{+}e^{-}\to 2\pi$,
$3\pi$, $4\pi$ and $K\bar{K}$ accounting for 72.8%, 6.7%, 4.9% and 5.2%,
respectively, for a total of 89.7%, according to Ref. KNT19 .

Contributions from all other exclusive modes below $\sqrt{s}=1.937$ GeV
represent a further 2.6%, those from narrow charm and bottom resonances a
further 1.1%, those from inclusive $R(s)$ data between $\sqrt{s}=1.937$ and
$11.199$ GeV a further 6.3%, and those (evaluated using pQCD) from $\sqrt{s}$
above $11.199$ GeV a further 0.3%. The contributions from the $2\pi$, $3\pi$,
$4\pi$ and $K\bar{K}$ exclusive modes below $\sqrt{s}=1.937$ GeV, similarly,
represent 98% and 89%, respectively, of the full spectral integrals
$I_{W_{1,5}}$ and $I_{W_{2,5}}$.

With contributions from the remaining exclusive modes below $\sqrt{s}=1.937$
GeV lying at $s$ far removed from narrow interfering resonances which might
enhance IB, it seems reasonable to expect IB in these residual exclusive-mode
contributions to be very small. We estimate these as of order 1% of the 2.6%
total contribution from these modes.

In the region above $\sqrt{s}=1.937$ GeV, away from the narrow charm and
bottom resonances, the experimental $R(s)$ data used in obtaining the 6.3%
inclusive-region contribution of Ref. KNT19  agrees with pQCD expectations to
significantly better than 10%. With IB corrections well below 1% in the OPE
representation of $\rho_{\rm EM}(s)$, estimating possible IB contributions
from the region above $\sqrt{s}=1.937$ GeV as less than 1% of the 7.7% total
from this region should represent a very conservative assessment.

With the 1% or less estimates discussed above for IB in both the contributions
from exclusive modes other than $2\pi$, $3\pi$, $4\pi$ and $K\bar{K}$ below
$\sqrt{s}=1.937$ GeV and all contributions in the region above
$\sqrt{s}=1.937$ GeV, we conclude that IB contributions from these sources
should represent less than 0.1% of the total for $a_{\mu}^{\rm HVP}$; a
similar conclusion holds for $I_{W_{1,5}}$ and $I_{W_{2,5}}$. In summary,
since the contributions from channels other than $2\pi$, $3\pi$, $4\pi$ and
$K\bar{K}$ below $\sqrt{s}=1.937$ GeV and all contributions in the region
above $\sqrt{s}=1.937$ GeV represent less than $11\%$ of the totals for all of
$a_{\mu}^{\rm HVP}$, $I_{W_{1,5}}$ and $I_{W_{2,5}}$, and it is safe to
estimate IB for these quantities as at most of order $1\%$, IB contributions
should represent of order $0.1\%$ or less of the total for all three
quantities. The uncertainty resulting from neglecting such contributions can
thus be safely neglected given the present state of the art.

In Ref. disc  it was shown that the IB effect in the $I=1$ vector $K\bar{K}$
channel is expected to be much smaller than the typical few percent of the
full $I=0+1$ sum, because $\tau$-decay data for the $I=1$ contribution allows
one to see that the $K\bar{K}$ electroproduction cross sections are strongly
dominated by the $I=0$ contribution from the $\phi$-peak region BaBarKKbar .
The IB effects on the small $I=1$ part of the $K\bar{K}$ channel should thus
also be negligibly small.

It follows from the analysis of Ref. disc  that, to first order in IB, IB
effects for all non-strange vector-channel $\tau$ exclusive-mode contributions
except those from the 2-pion and 4-pion modes are so small that for the
purpose of estimating IB corrections (apart from possible IB effects in the
$e^{+}e^{-}\rightarrow 3\pi$ distribution to be discussed in more detail
below) a fully inclusive lattice result for

$\Pi_{\rm
IB}(Q^{2})\equiv\frac{2}{\sqrt{3}}\,\Pi_{38}(Q^{2})+\left[\Pi_{33}(Q^{2})-{1\over
2}\Pi_{ud;V}(Q^{2})\right]$ (36)

can be used to correct the sum of $2\pi$ and $4\pi$ contributions obtained
from non-strange vector-channel hadronic $\tau$-decay data, producing an
alternate, combined $\tau$-plus-lattice-IB determination for the sum of
$\pi^{+}\pi^{-}$, $2\pi^{+}2\pi^{-}$ and $\pi^{+}\pi^{-}2\pi^{0}$
electroproduction contributions.999A preliminary lattice study of some of the
contributions to Eq. (36) already exists BILM ; BrunoMeyer .

From the above discussion, it follows, using Eq. (9), again up to possible IB
contributions in the $e^{+}e^{-}\rightarrow 3\pi$ distribution which, to first
order in IB, would produce unwanted 3-pion contributions to the mixed-isospin,
$ab=38$ part of $\rho_{\rm EM}(s)$, and hence also to $\Pi_{38}(Q^{2})$ in Eq.
(36), that

$\displaystyle\Delta_{m,n}$ $\displaystyle\equiv$ $\displaystyle\int_{s_{\rm
th}}^{s_{\rm
KNT}}ds\,W_{m,n}(s;\\{Q_{\ell}^{2}\\})\rho^{e^{+}e^{-}}_{2\pi+4\pi}(s)-\int_{s_{\rm
th}}^{s_{\rm
KNT}}ds\,W_{m,n}(s;\\{Q_{\ell}^{2}\\})\rho^{\tau}_{2\pi+4\pi}(s))$ (37)
$\displaystyle=$
$\displaystyle(-1)^{m}m_{\tau}^{2(n-m-1)}\sum_{k=1}^{n}\frac{(Q_{k}^{2}+s_{\rm
th})^{m}}{\prod_{\ell\neq k}(Q_{\ell}^{2}-Q_{k}^{2})}\,\Pi_{\rm
IB}(Q_{k}^{2})\ ,$

where $s_{\rm KNT}=(1.937\ {\rm GeV})^{2}$ is the upper edge of the exclusive-
mode data region of Ref. KNT19 , $\rho^{e^{+}e^{-}}_{2\pi+4\pi}$ is the sum of
the $2\pi$ and $4\pi$ contributions to the electromagnetic current spectral
function implied by $2\pi$ and $4\pi$ electroproduction cross sections and
$\rho^{\tau}_{2\pi+4\pi}$ is the sum of the $2\pi$ and $4\pi$ contributions to
the consistently normalized vector $I=1$ spectral function implied by non-
strange vector-current-induced hadronic $\tau$ decay distribution results.
Alternate versions of this sum rule in terms of $C_{\rm IB}(t)$ (cf.  Eqs.
(11) or (18)) can of course also be employed. We will discuss the feasibility
of this through concrete examples in Sec. IV.2, where we will evaluate the LHS
of Eq. (37) for the weights $W_{1,5}$ and $W_{2,5}$ using data from Refs. AO ;
KNT19 .

We now return to the issue of possible non-trivial IB effects in the
$e^{+}e^{-}\rightarrow 3\pi$ distribution, which complicates the process of
identifying the exclusive-mode contributions in the dispersive integral over
the electroproduction data to be replaced by lattice-IB-corrected $\tau$
results. This complication arises because there may be 3-pion exclusive-mode
contributions to the spectral function of the mixed-isospin $\Pi_{38}$
polarization. In the isospin limit, the $3\pi$ mode is pure $I=0$, and thus
not affected by the replacement of part of the $I=1$ component of the
electromagnetic spectral function by the corresponding part of the vector-
channel, $I=1$ $\tau$ spectral function. With IB, however, the $38$ part of
Eq. (36) will contain a 3-pion contribution, with $\rho$-$\omega$ mixing, for
example, inducing a contribution to the $38$ spectral function via
$e^{+}e^{-}\to\rho\to\omega\to 3\pi$. Applying the inclusive lattice IB
correction to the weighted $\tau$ $2\pi+4\pi$ integral and using the result to
replace the corresponding $2\pi+4\pi$ contribution to the weighted
electroproduction integral, the modified all-exclusive-mode sum would then
include this $3\pi$ IB component twice, once in the inclusive lattice IB
correction, and once in the $3\pi$ contribution produced by use of the
experimental $e^{+}e^{-}\rightarrow 3\pi$ cross sections. The result of Ref.
KNT19  for the contribution of the $3\pi$ channel to $a_{\mu}^{\rm HVP}$,
$46.73(94)\times 10^{-10}$, is sufficiently large that, were the associated IB
$3\pi$ contribution to be larger than naively expected, say at a level a few
to several percent of the full $3\pi$ contribution, a controlled estimate of
its size might be required to make an inclusive lattice $\Pi_{\rm IB}(Q^{2})$
determination, however precise, of actual numerical use in performing an IB
correction of $\tau$-decay data.

Fortunately, experimental information is now available on what should be the
dominant contribution to this $3\pi$ double counting, with recent BaBar
results BABAR:2021cde  providing $>6\sigma$ evidence for an IB
$e^{+}e^{-}\rightarrow\rho\rightarrow 3\pi$ contribution to the
$e^{+}e^{-}\rightarrow 3\pi$ amplitude. The interference of this contribution
with the isospin-conserving (IC) $e^{+}e^{-}\rightarrow\omega\rightarrow
3\pi$, $e^{+}e^{-}\rightarrow\phi\rightarrow 3\pi$,
$e^{+}e^{-}\rightarrow\omega^{\prime}\rightarrow 3\pi$ and
$e^{+}e^{-}\rightarrow\omega^{\prime\prime}\rightarrow 3\pi$ contributions
produces IB contributions to the cross section, and hence IB contributions to
the weighted integrals over the $3\pi$ distribution involving the weights
discussed in this paper. The normalization of these interference contributions
is fixed by the square root of the fitted $\rho\rightarrow 3\pi$ branching
fraction, $B(\rho\rightarrow 3\pi)$, to which these contributions are
proportional. Taking the integrals up to the upper edge, $s=(1.937\ {\rm
GeV})^{2}$, of the KNT19 exclusive-mode region to be specific, the IB
contributions implied by the preferred BaBar cross-section fit, as detailed in
App. B, turn out to represent $-1.2(1.2)\%$, $-1.1(1.1)\%$ and $-1.6(1.6)\%$
of the corresponding full $3\pi$ contributions to $a_{\mu}^{\rm HVP}$,
$I_{W_{1,5}}$ and $I_{W_{2,5}}$, respectively.

In the $a_{\mu}^{\rm HVP}$ case, the resulting $3\pi$-double-counting
correction, $+0.54(0.54)\times 10^{-10}$ (from Eq. (B)), represents
$+0.08(0.08)\%$ of the $a_{\mu}^{\rm HVP}$ total. At present, a $\pm
0.54\times 10^{-10}$ uncertainty on the $\tau$-modified alternate
determination of $a_{\mu}^{\rm HVP}$ is sufficiently small that replacing
electroproduction data by $\tau$-based data for the $2\pi$ and $4\pi$ channels
is potentially useful for the purposes of investigating, for example, the
BaBar–KLOE discrepancy, where the discrepancy between BaBar and KLOE $2\pi$
contributions from the region $0.6\ {\rm GeV}<\sqrt{s}<0.9$ GeV KNT19 ,
$(9.8\pm 3.5)\times 10^{-10}$, is much larger than the $3\pi$ double-counting-
correction uncertainty. In Sec. IV.2 we will show that a similar conclusion
holds for the $W_{1,5}$ and $W_{2,5}$ weight cases: the $\pm 0.00041$ and $\pm
0.00012$ uncertainties on the $W_{1,5}$ and $W_{2,5}$ $3\pi$-double-counting
corrections obtained in App. B are once more much smaller than the central
values and experimental errors on the corresponding electroproduction-$\tau$
spectral integral differences.

A more detailed discussion of the IB contributions to the weighted $3\pi$
integrals may be found in Appendix B.

###  IV.2  Comparison of $\tau$-based and $e^{+}e^{-}$-based spectral
integrals

We now turn to the evaluation of the $2\pi+4\pi$ contributions to the spectral
integrals appearing in Eq. (37), for the examples $W_{m,n}=W_{1,5}$ and
$W_{2,5}$. The $2\pi+4\pi$ data of Ref. KNT19  is used in the first integral
and that of Ref. AO  in the second integral. We also consider the $3\pi$
channel, in view of the discussion at the end of Sec. IV.1. The discussion is
restricted to contributions from the exclusive-mode region of Ref. KNT19 ,
$s\leq s_{\rm KNT}=(1.937\ {\rm GeV})^{2}$, since $2\pi$, $3\pi$ and $4\pi$
contributions above that point form part of the multi-mode inclusive
contribution, dealt with already in Sec. IV.1. Since no lattice data is
currently available to evaluate the RHS of either Eq. (37) or its EWSR
analogue, we restrict our attention to the $W_{1,5}$ and $W_{2,5}$ RWSR
examples, and do not consider the EWSR analogues. Once suitable IB lattice
data become available, the analogous EWSR cases may become of interest, for
the reasons explained in Sec. III.2.

Our goal here is to see whether the two integrals on the LHS of Eq. (37) can
be evaluated with sufficient precision to make this sum rule of potential use.
Of relevance to investigating this question is the size of the uncertainty on
the spectral integral differences appearing on the LHS of Eq. (37) relative to
(i) the uncertainties on estimates for the corresponding $3\pi$-double-
counting corrections, and (ii) variations in the spectral integral differences
themselves, induced, for example, by the BaBar–KLOE $\pi\pi$ discrepancy.

We start with the electroproduction-based $W_{1,5}$\- and $W_{2,5}$-weighted
integrals over the $2\pi$, $3\pi$ and $4\pi$ contributions to $\rho_{\rm
EM}(s)$, results for which are shown in Table 3. We use $s_{\rm
th}=4m_{\pi}^{2}$.

channel | $I_{W_{1,5}}(s_{\rm KNT})$ | $I_{W_{1,5}}(s_{\tau})$ | $\frac{I_{W_{1,5}}(s_{\tau})}{I_{W_{1,5}}(s_{\rm KNT})}$ | $I_{W_{2,5}}(s_{\rm KNT})$ | $I_{W_{2,5}}(s_{\tau})$ | $\frac{I_{W_{2,5}}(s_{\tau})}{I_{W_{2,5}}(s_{\rm KNT})}$
---|---|---|---|---|---|---
$\pi^{+}\pi^{-}$ | 0.3864(14) | 0.3863(14) | 1.00 | 0.05713(19) | 0.05710(19) | 1.00
$2\pi^{+}2\pi^{-}$ | 0.005743(81) | 0.005450(77) | 0.95 | 0.003568(49) | 0.003267(45) | 0.92
$\pi^{+}\pi^{-}2\pi^{0}$ | 0.00813(33) | 0.00772(32) | 0.95 | 0.00464(19) | 0.00422(17) | 0.91
$2\pi+4\pi$ | 0.4002(14) | 0.3995(14) | 1.00 | 0.06534(27) | 0.06459(26) | 0.99
$\pi^{+}\pi^{-}\pi^{0}$ | 0.03880(81) | 0.03877(81) | 1.00 | 0.00798(15) | 0.00794(15) | 1.00

> Table 3: Electroproduction-based exclusive-mode spectral integrals with
> weights $W_{1,5}$ (left of the double vertical line) and $W_{2,5}$ (right of
> the double vertical line). The 2nd and 5th columns show the integrals
> computed up to $s=s_{KNT}=(1.937\ {\rm GeV})^{2}$, the 3rd and 6th columns
> the same integrals computed up to $s=s_{\tau}=3.0574$ GeV2.

The data files containing the exclusive-mode contributions to $R(s)$ provided
by the authors of Ref. KNT19  extend up to $s=s_{\rm KNT}=3.7520$ GeV2.
Results for the exclusive-mode, $s_{\rm KNT}$-truncated $X=\pi^{+}\pi^{-}$,
$2\pi^{+}2\pi^{-}$, $\pi^{+}\pi^{-}2\pi^{0}$ and $\pi^{+}\pi^{-}\pi^{0}$,
$W_{1,5}$\- and $W_{2,5}$-weighted spectral integrals obtained using this
input, are denoted $I^{X}_{W_{1,5}}(s_{\rm KNT})$ and $I^{X}_{W_{2,5}}(s_{\rm
KNT})$ in what follows, and listed in Columns 2 and 5 of Table 3.

The upper endpoints of the $\tau$-based spectral integrals are, in contrast,
limited by the largest $s$ for which the $\tau$-based spectral function of
Ref. AO  is available, which is $s\equiv s_{\tau}=3.0574$ GeV2, slightly below
$m_{\tau}^{2}$. To compare to the resulting $s_{\tau}$-truncated $\tau$-based
integrals, we thus also require values for $s_{\tau}$-truncated versions of
the exclusive-mode electroproduction-based integrals. These are denoted
$I^{X}_{W_{1,5}}(s_{\tau})$ and $I^{X}_{W_{2,5}}(s_{\tau})$ and listed in
columns 3 and 6 of Table 3. As can be seen from the table, the
$s_{\tau}$-truncated integrals constitute more than $99\%$ of the
corresponding $s_{\rm KNT}$-truncated versions for the $2\pi$ and $3\pi$ modes
and more than 90% of the $s_{\rm KNT}$-truncated versions for the two $4\pi$
modes.

The $s_{\tau}$-truncated $2\pi+4\pi$ $\tau$-based integrals to be compared to
the $R$-ratio-based analogues of Table 3 are obtained using the $2\pi+4\pi$
$\tau$-based spectral function of Ref. AO . The results,

$\displaystyle I_{W_{1,5}}^{\tau,2\pi+4\pi}(s_{\tau})$ $\displaystyle=$
$\displaystyle 0.4103(22)\ ,$ (38) $\displaystyle
I_{W_{2,5}}^{\tau,2\pi+4\pi}(s_{\tau})$ $\displaystyle=$ $\displaystyle
0.06693(22)\ ,$

differ from the electroproduction results of Table 3,

$\displaystyle I_{W_{1,5}}^{2\pi+4\pi}(s_{\tau})$ $\displaystyle=$
$\displaystyle 0.3995(14)\ ,$ (39) $\displaystyle
I_{W_{2,5}}^{2\pi+4\pi}(s_{\tau})$ $\displaystyle=$ $\displaystyle
0.06459(26)\ ,$

by $4.1\sigma$ and $6.7\sigma$, respectively, for the $W_{1,5}$ and $W_{2,5}$
cases. The $s_{\tau}$-truncated contributions to the differences appearing on
the LHS of the $W_{1,5}$ and $W_{2,5}$ versions of Eq. (37),

$\displaystyle
I_{W_{1,5}}^{2\pi+4\pi}(s_{\tau})-I_{W_{1,5}}^{\tau,2\pi+4\pi}(s_{\tau})$
$\displaystyle=$ $\displaystyle\,-0.0108(26)\ ,$ (40) $\displaystyle
I_{W_{2,5}}^{2\pi+4\pi}(s_{\tau})-I_{W_{2,5}}^{\tau,2\pi+4\pi}(s_{\tau})$
$\displaystyle=$ $\displaystyle\,-0.00233(35)\ ,$

are thus determined with $\sim 24\%$ and $\sim 15\%$ precision. We note that
the errors in Eq. (40) are significantly smaller than the central values of
the BaBar–KLOE discrepancies in Eq. (35). The analogous values for
$a_{\mu}^{\rm HVP}$ itself are

$\displaystyle a_{\mu}^{2\pi+4\pi}(s_{\tau})$ $\displaystyle=$ $\displaystyle
535.0(2.0)\times 10^{-10}\ ,$ (41) $\displaystyle
a_{\mu}^{\tau,2\pi+4\pi}(s_{\tau})$ $\displaystyle=$ $\displaystyle
552.4(5.0)\times 10^{-10}\ ,$ $\displaystyle
a_{\mu}^{2\pi+4\pi}(s_{\tau})-a_{\mu}^{\tau,2\pi+4\pi}(s_{\tau})$
$\displaystyle=$ $\displaystyle-17.4(5.4)\times 10^{-10}\ .$

We comment on the $\tau$-based value in Eq. (41) (i.e., the second line of
this equation). The $\tau$-based data are very sparse near the 2-pion
threshold end of the spectral function, and a comparison with the $R$-ratio-
based data (which are much denser in the threshold region) suggests that the
trapezoidal interpolation of the $\tau$-based data may overestimate the near-
threshold, $\tau$-based contribution to $a_{\mu}^{\tau,2\pi+4\pi}(s_{\tau})$.
A rough estimate of this effect can be obtained by replacing the contribution
to $a_{\mu}^{\tau,2\pi+4\pi}(s_{\tau})$ between threshold and $s_{\rm
ChPT}\equiv s=(0.305$ GeV)2 by the ChPT-based 2-pion contribution from Ref.
KNT19 , and then employing trapezoidal integration above $s_{\rm ChPT}$. Doing
so would lead to a downward shift of $0.90(42)\times 10^{-10}$ of the value
$552.4(5.0)\times 10^{-10}$ in Eq. (41). This small additional near-threshold
uncertainty in the $\tau$ result for $a_{\mu}^{2\pi+4\pi}$ is a consequence of
the enhancement of low-$s$ contributions by the $a_{\mu}^{\rm HVP}$ kernel.
The weights $W_{1,5}$ and $W_{2,5}$, in contrast, produce no such low-$s$
enhancement (and in fact strongly suppress near-threshold contributions). The
uncertainty in the low-$s$ $\tau$-based $2\pi$ spectral distribution resulting
from the coarseness of the $\tau$ data near threshold thus has negligible
impact on the results in Eq. (38). Since weighted spectral integrals with the
weights $W_{1,5}$ and $W_{2,5}$ are the main focus of this paper, we do not
pursue this issue further here.

Of course, before we can meaningfully compare the $\tau$\- and
electroproduction-based integrals, we need to take the RHS, i.e., the
projected lattice-based IB correction, into account. Since this contribution
is inclusive, we have to deal with the fact that the $s_{\tau}$-truncated
spectral integrals whose differences we are actually able to determine from
data are smaller than the complete spectral integrals. While we have already
argued in Sec. IV.1 that IB contributions from modes other than $2\pi$ and
$4\pi$ below $s=s_{\rm KNT}$, and from all modes above $s=s_{\rm KNT}$, can be
safely neglected, we still need to address the size of possible IB
contributions from the $2\pi$ and $4\pi$ modes in the region
$s_{\tau}<s<s_{\rm KNT}$. For the $2\pi$ modes, this is not an issue, as the
integrals up to $s_{\tau}$ capture essentially the full contribution up to
$s_{\rm KNT}$, for both weights we consider here. The $s_{\tau}$-truncated
$4\pi$ integrals constitute $95\%$, respectively, $91$-$92\%$ of the full
$4\pi$ integrals, for the weights $W_{1,5}$ and $W_{2,5}$. In this high-$s$,
$s>s_{\tau}$ region, we expect IB to be of order $1\%$ of the corresponding
$4\pi$ totals. With this estimate, the IB contribution missed as a result of
truncating the $W_{1,5}$ and $W_{2,5}$ integrals at $s=s_{\tau}$ rather than
$s_{\rm KNT}$ are expected to be (i) in the $W_{1,5}$ case, of order $1\%$ of
$5\%$, or $0.05\%$, of the corresponding full $s_{\rm KNT}$-truncated $4\pi$
contribution, and hence of order $0.000007$, and (ii) in the $W_{2,5}$ case,
of order $1\%$ of $9\%$, or $0.09\%$, of the corresponding full $s_{\rm
KNT}$-truncated $4\pi$ contribution, and hence of order $0.000007$ as well.
These estimated higher-$s$ $2\pi+4\pi$ IB contribution effects, missed when
one considers the differences of electroproduction- and $\tau$-based
$2\pi+4\pi$ integrals only up to $s=s_{\tau}$, are thus more than an order of
magnitude smaller than the uncertainties, $0.00041$ and $0.00012$, on the
corresponding $3\pi$-double-counting corrections estimated in App. B, which
are, themselves, significantly smaller than both the central values and
experimental errors of the Eqs. (40) results, $-0.0108\pm 0.0026$ and
$-0.00233\pm 0.00035$, for the $W_{1,5}$\- and $W_{2,5}$-weighted,
$s=s_{\tau}$-truncated experimental electroproduction-$\tau$ $2\pi+4\pi$
spectral integral differences, indicating that a sufficiently precise lattice
determination of the corresponding RHSs of Eq. (37) will, indeed, make these
sum rules useful for investigating current electroproduction- and $\tau$-based
integral results. It remains of course crucial that a lattice estimate of the
inclusive IB correction reach a precision commensurate with the precision with
which the spectral integral differences in Eqs. (40) have been obtained.

##  V  Conclusions

At present, there are several discrepancies in the computation of
$a_{\mu}^{\rm HVP}$ that limit our ability to compare a SM-based estimate for
$a_{\mu}$ with the experimentally measured value. Most recently, a puzzling
discrepancy has emerged between data-driven and lattice evaluations of the
RBC/UKQCD intermediate window quantity, suggesting that a lattice-based value
for $a_{\mu}^{\rm HVP}$ may bring the SM value for $a_{\mu}$ much closer to
the experimental value. This may confirm the lattice result for $a_{\mu}^{\rm
HVP}$ found by the BMW collaboration, which is $2.1\sigma$ higher than the
data-driven value. The discrepancy for the intermediate window turns out to be
about half the total difference between the experimental and SM values for
$a_{\mu}$, when for the latter the data-driven white-paper value for
$a_{\mu}^{\rm HVP}$ of Ref. review  is used.

Another discrepancy results from the long-standing difference between the
BaBar and KLOE measured spectral distributions in the two-pion channel, in the
region around the $\rho$ mass. Taking this discrepancy at face value, i.e.,
considering the difference between the values obtained using either the BaBar
or the KLOE data, leads to a difference of about $10^{-9}$ in $a_{\mu}^{\rm
HVP}$, which is more than half the difference between the experimental and
white-paper values for $a_{\mu}$.

Given this puzzling state of affairs, it is important to develop methods that
allow for detailed comparisons, zooming in on specific regions in $s$. The sum
rules developed in this paper provide a tool for such investigations. The
RWSRs of Sec. II.1 are based on weighted spectral integrals with an adjustable
narrow-weight function defined directly as a function of $s$. The quantities
defined by such sum rules complement the window quantities of Ref. RBC , which
are defined as a function of Euclidean time, and translate into rather wide
windows as a function of $s$.101010Our method differs from that of Ref.
Rwindow , which attempts to narrow the window as a function of $s$ by taking
linear combinations of windows defined as a function of Euclidean time.

In Sec. II.2 we modified the RWSR approach by borrowing ideas from Ref. HLT ,
proposing a different set of sum rules, with weight functions that are linear
combinations of simple exponential functions of $\sqrt{s}$. This class of sum
rules has the advantage that the lattice side involves the lattice correlator
for the vacuum polarization at a set of Euclidean $t$ values which are chosen
by hand and which hence allow one to avoid contributions from the high-$t$
region, where lattice errors are large. This aids in the optimization of the
precision of the lattice side of the sum rule. A key observation is that the
weights used for the RWSRs can, moreover, be used as molds to cast the
exponential weights, thus retaining the advantage of RWSRs. We note that our
goal is not to reconstruct spectral data from the lattice, but rather to
compare appropriately weighted, moderately localized versions of existing
experimental spectral data with correspondingly weighted lattice data. Our
goal is thus simpler than the goal of Ref. HLT . In particular, the casts are
not used in any approximations; once a useful cast has been designed, it can
be used in an exact EWSR, and the underlying mold discarded. In fact, there is
considerable flexibility in designing EWSRs. An example of this flexibility is
provided by the hatted weight functions based on Eq. (21). Additional ideas
for designing practical exponential weights that are narrow as a function of
$s$, moreover, almost certainly remain to be explored.

At first glance, it is not obvious that these new methods will be practically
useful, as there are reasons to worry that the lattice sides of our sum rules
will typically have large errors. To investigate this worry, we studied, in
Sec. III, two rational weights, $W_{1,5}$ and $W_{2,5}$, and their exponential
cousins, numerically, using data from Ref. KNT19  on the data side, and from
Ref. ABGP22  on the lattice side. We found that especially the exponential
weights perform quite well for an investigation of, for example, the
BaBar–KLOE discrepancy. With the projected increase in the precision of data
from the lattice, and with more fine-tuning of the weights of Sec. II.2, we
believe that the new tools provided in this paper are likely to prove quite
useful for the investigation of currently existing discrepancies.

In Sec. IV we applied these ideas in a somewhat different context, the
comparison between contributions to $a_{\mu}^{\rm HVP}$ from $R$-ratio data
and hadronic $\tau$-decay data. In the past, such comparisons were difficult
to carry out because of the lack of a reliable method for estimating isospin-
breaking effects. We derived a sum rule allowing for the comparison between
$R$-ratio and hadronic $\tau$-decay based data in which the required IB
effects can, to a good approximation, be obtained from the lattice. In
particular, we showed that if the two- and four-pion channels are included in
the $R$-$\tau$ comparison, IB effects in other channels are small enough that
IB effects can be reliably incorporated by an inclusive lattice computation,
thus avoiding the difficulties associated with obtaining exclusive data from
the lattice. Using the weights of Sec. III, we demonstrated that this
comparison can become practical with sufficiently precise lattice data for the
inclusive IB correction.

We conclude by emphasizing that this paper is a method paper. The main reason
is that we do not have access to lattice data for the complete hadronic vacuum
polarization, or for the IB part of it needed in Sec. IV. We have not
optimized the concrete examples of Sec. III and Sec. IV for the specific goals
for which they may be used. We leave the optimization of these new classes of
sum rules to future work, and here just comment that this optimization will
depend on the specific application.

Acknowledgments

We thank Alex Keshavarzi for making all exclusive-mode data underlying Ref.
KNT19  available to us. We thank Mattia Bruno and Max Hansen for useful
discussions, and Marcus V. Rodrigues for his participation at an early stage
of this work. 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. MG is
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics, under Award DE-SC0013682. The work of KM is supported by a
grant from the Natural Sciences and Engineering 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 Grant 2017 SGR
1069. IFAE is partially funded by the CERCA program of the Generalitat de
Catalunya.

##  Appendix A  Relation between weights and precision

Consider the weights of Eq. (7), without the extra factors of $m_{\tau}$:

$W_{m,n}(s)={\frac{(s-s_{\rm th})^{m}}{\prod_{k=1}^{n}(s+Q_{k}^{2})}}\ ,$ (42)

where $0<Q_{1}^{2}<Q_{2}^{2}<\cdots<Q_{n}^{2}$. What we will discuss in this
appendix is the issue of the cancellations involved when this weight is used
in the sum-rule approach discussed in the main text.

The function $W_{m,n}(s)$ can be re-written in the alternate partial-fraction
representation form

$W_{m,n}(s)=\sum_{k=1}^{n}{\frac{c_{k}}{(s+Q_{k}^{2})}}\ ,$ (43)

where the $c_{k}$ are easily seen to be

$c_{k}={\frac{(-1)^{m}(Q_{k}^{2}+s_{\rm th})^{m}}{\prod_{\ell\neq
k}(Q_{\ell}^{2}-Q_{k}^{2})}}\ .$ (44)

By expanding in $1/s$ for large $s$, it is straightforward to work out a
number of relations satisfied by the $c_{k}$, as follows. In the form Eq.
(42), it is clear that

$W_{m,n}(s)={\frac{1}{s^{n-m}}}\,+{\cal O}\left({\frac{1}{s^{n-m+1}}}\right)\
,$ (45)

where the expansion is convergent for $s>Q_{N}^{2}$. In contrast, expanding
the form Eq. (43), the large $s$ behavior is

$W_{m,n}(s)=\sum_{k=1}^{n}c_{k}\,{\frac{1}{s}}\,\sum_{\ell=0}^{\infty}(-1)^{\ell}\left({\frac{Q_{k}^{2}}{s}}\right)^{\ell}\
.$ (46)

Comparing the two large $s$ expansions, it follows that the $c_{k}$ satisfy
the relations

$\sum_{k=1}^{n}c_{k}Q_{k}^{2\ell}=0\ ,\qquad\ell=0,\dots,n-m-2\ .$ (47)

The general dispersive sum rules involving these weights, assuming $n$ is
large enough that the weighted spectral integral converges, are of course Eq.
(9), without the factors of $m_{\tau}$:

$\int_{s_{\rm
th}}^{\infty}ds\,W_{m,n}(s)\,\rho(s)=\sum_{k=1}^{n}c_{k}\,\Pi(Q_{k}^{2})=(-1)^{m}\sum_{k=1}^{n}{\frac{(Q_{k}^{2}+s_{\rm
th})^{m}}{\prod_{\ell\neq k}(Q_{\ell}^{2}-Q_{k}^{2})}}\,\Pi(Q_{k}^{2})\ .$
(48)

The constraints of Eq. (47) allow us to understand the cancellations involved
in forming the sum that appears on the right-hand side of Eq. (48).
$\Pi(Q^{2})$ is analytic on the $Q^{2}>0$ axis, and hence has a convergent
Taylor series expansion around any point on that axis with radius of
convergence the distance from that point to the start of the cut at
$Q^{2}\,=\,-s_{\rm th}$. For illustration in what follows let us expand about
the midpoint of the interval containing all pole locations,
$\tilde{Q}^{2}\equiv(Q_{1}^{2}+Q_{n}^{2})/2$. All of the $Q_{k}^{2}$ then lie
in the region of convergence of the Taylor expansion, and we have

$\Pi(Q_{k}^{2})=\sum_{\ell=0}^{\infty}{\frac{1}{\ell!}}\,{\frac{d^{\ell}\Pi(Q^{2})}{d(Q^{2})^{\ell}}}\Bigg{|}_{Q^{2}=\tilde{Q}^{2}}\left(Q_{k}^{2}-\tilde{Q}^{2}\right)^{\ell}\
.$ (49)

The (first version of the) right-hand side of Eq. (48), then becomes

$\sum_{\ell=0}^{\infty}{\frac{1}{\ell!}}\,{\frac{d^{\ell}\Pi(Q^{2})}{d(Q^{2})^{\ell}}}\Bigg{|}_{Q^{2}=\tilde{Q}^{2}}\sum_{k=1}^{n}c_{k}\left(Q_{k}^{2}-\tilde{Q}^{2}\right)^{\ell}\
,$ (50)

and the constraints of Eq. (47) imply that terms in the last factor vanish for
$\ell=0,\dots,n-m-2$. The terms involving the derivatives of order $0$ through
$n-m-2$ of $\Pi(Q^{2})$ with respect to $Q^{2}$ at $Q^{2}=\tilde{Q}^{2}$ thus
vanish, and the first surviving terms are those involving the $(n-m-1)$-th
derivative.

There is thus significant cancellation on the right-hand side of the
dispersive sum rule for weights with the product-of-pole rational structure of
Eq. (42), and this cancellation gets stronger with increasing $n-m$. This will
lead to errors on a lattice evaluation of the right-hand side which will
typically increase with increasing $n-m$. This growth of lattice errors with
increasing $n-m$ was seen already in the dispersive analysis used to extract
$|V_{us}|$ from the experimental strange $\tau$-decay distribution, which
employed similar product-of-pole weights, though with a constant numerator.
There it was found that lattice errors could be kept under good control for
weights with 3, 4, or 5 poles, but the errors did grow as the number of pole
factors increased Kim .

##  Appendix B  Isopin breaking in the 3-pion channel

Experimental information on IB in the 3-pion channel is provided by BaBar’s
vector-meson-dominance (VMD) model fit to its recent high-precision
$e^{+}e^{-}\rightarrow 3\pi$ cross sections, detailed in Ref. BABAR:2021cde .
The VMD model for the amplitude is a sum of IC $\omega$, $\phi$,
$\omega^{\prime}$ and $\omega^{\prime\prime}$ resonance contributions,
supplemented by an IB $\rho$ contribution and provides an excellent fit (with
$\chi^{2}/{\rm dof}=136/129$) to the experimental cross-sections in the region
$E_{\rm CM}\leq 1.8$ GeV. The explicit model forms of the resonance
contributions are as specified in Refs. BABAR:2021cde ; Achasov:2003ir . The
fit confirms the necessity of including the IB $\rho$ contribution at the
$>6\sigma$ level. The model includes what should be the dominant (resonance-
enhanced) IB contributions to the cross section in the region from threshold
to slightly above the $\phi$ resonance peak, namely the effects of
$\rho$-$\omega$ and $\rho$-$\phi$ interference. It also includes sub-leading
IB effects in the form of $\rho$-$\omega^{\prime}$ and
$\rho$-$\omega^{\prime\prime}$ interference contributions from the region
above the $\phi$ peak. With no IB contributions to the amplitude from the
excited $\rho$ resonances, however, it will miss IB contributions in this
higher-$s$ region from, e.g., $\rho^{\prime}$-$\omega^{\prime}$ and
$\rho^{\prime}$-$\omega^{\prime\prime}$ interference. Such contributions will
be suppressed by the fall-off with $s$ of the weights considered in this paper
and are thus also expected to be numerically sub-dominant.

In what follows, we take, as our estimates for the IB contributions to
weighted, $s\leq s_{\rm KNT}$, $3\pi$ spectral integrals, the results produced
using the full BaBar VMD fit. As noted above, with the VMD model omitting
terms which would model, e.g., IB $\rho^{\prime}$ and $\rho^{\prime\prime}$
contributions to the $3\pi$ amplitude, such estimates will miss some
contributions from the region above the $\phi$ peak. We will discuss IB in
this region in more detail below and demonstrate that such missing
contributions are expected to be much smaller than the uncertainties on the
dominant low-$s$ $\rho$-$\omega$ plus $\rho$-$\phi$ interference
contributions, and hence that the estimates for the IB contributions to the
variously weighted, $s\leq s_{\rm KNT}$, $3\pi$ spectral integrals obtained
using the BaBar VMD model fit are expected to be reliable within their stated
errors.

The BaBar paper BABAR:2021cde  contains full results for the central values
and errors (though not the correlations) of the fit parameters governing the
$\omega$, $\phi$ and $\rho$ contributions to the amplitude, but not those
governing the $\omega^{\prime}$ and $\omega^{\prime\prime}$ contributions. It
is thus possible to determine the $s$-dependence of the combined
$\omega+\phi+\rho$ contribution to the VMD model representation of the cross
section, but not that of the full VMD model. BaBar has, however, provided a
table of central values of the full model representation at the midpoints of
the BaBar experimental bins, both with the $\rho$ contribution included and
with that contribution turned off thanksvdruzhinin . This provides us with the
$s$-dependence of the central fit values of the IB part of the VMD
representation of the cross section and allows us to evaluate the central
values of the variously weighted integrated versions of the IB $3\pi$ spectral
distribution of interest in this paper.

In the discussions which follow, we will denote the $3\pi$ spectral integrals
up to $s=s_{\rm max}$ with the $a_{\mu}^{\rm HVP}$, $W_{1,5}$ and $W_{2,5}$
weights by $a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm max}\right)$,
$I^{3\pi}_{W_{1,5}}\left(s_{\rm max}\right)$ and
$I^{3\pi}_{W_{2,5}}\left(s_{\rm max}\right)$. The corresponding IB
contributions obtained using BaBar’s fitted VMD model, again up to $s=s_{\rm
max}$, are similarly denoted $\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm
max}\right)\right]_{\rm IB}^{\rm VMD}$, $\left[I^{3\pi}_{W_{1,5}}\left(s_{\rm
max}\right)\right]_{\rm IB}^{\rm VMD}$ and
$\left[I^{3\pi}_{W_{2,5}}\left(s_{\rm max}\right)\right]_{\rm IB}^{\rm VMD}$.
Finally, the sums of the corresponding fully known IB VMD $\rho$-$\omega$ and
$\rho$-$\phi$ interference contributions from the region up to just above the
$\phi$ peak (which we characterize, to be specific, as $s\leq
s_{\phi}\equiv(m_{\phi}+4\Gamma_{\phi})^{2}$) are denoted by
$\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$,
$\left[I^{3\pi}_{W_{1,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$ and
$\left[I^{3\pi}_{W_{2,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$, respectively.

With this notation, we find the low-$s$ IB contributions, $\left[a_{\mu}^{{\rm
HVP};\,3\pi}\left(s_{\phi}\right)\right]_{\rm IB}^{\rho\omega\phi;\,{\rm
VMD}}$, $\left[I^{3\pi}_{W_{1,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$ and
$\left[I^{3\pi}_{W_{2,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$ to represent $80\%$, $87\%$ and $76\%$,
respectively, of the corresponding full IB estimates, $\left[a_{\mu}^{{\rm
HVP};\,3\pi}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm VMD}$,
$\left[I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm VMD}$
and $\left[I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm
VMD}$, in line with the expectation that the IB integrals in question will be
dominated by the contributions from the low-$s$ region. The full VMD-model-
based IB contributions, which provide our estimates for the IB contributions
to the $s\leq s_{\rm KNT}$, $3\pi$ integrals, represent $-1.2\%$, $-1.1\%$ and
$-1.6\%$ of the corresponding $s\leq s_{\rm KNT}$, $3\pi$ totals,
$a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm KNT}\right)$,
$I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)$ and
$I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)$, respectively.

As noted above, BaBar provides full information on the central values and
errors only for the fit parameters determining the $\rho$, $\omega$ and $\phi$
contributions to the $3\pi$ amplitude, and not for those determining the
$\omega^{\prime}$ and $\omega^{\prime\prime}$ contributions. The information
that is provided is, however, sufficient to allow us to study what should be
the dominant uncertainties on the estimated IB $3\pi$ contributions
$\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm
VMD}$, $\left[I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm
VMD}$ and $\left[I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)\right]_{\rm
IB}^{\rm VMD}$, namely those produced by the uncertainties on the parameters
entering the dominant, fully known, low-$s$ contributions,
$\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$,
$\left[I^{3\pi}_{W_{1,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$ and
$\left[I^{3\pi}_{W_{2,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$. We find that the dominant uncertainties on
these low-$s$ contributions come from the fit error in the relative phase,
$\phi_{\rho}$, of the $\rho$ and $\omega$ contributions to the amplitude,
which produces sizeable relative uncertainties of $87\%$, $92\%$ and $70\%$ on
$\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$,
$\left[I^{3\pi}_{W_{1,5}}\left(s_{\phi}\right)\right]_{\rm
IB}^{\rho\omega\phi;\,{\rm VMD}}$ and
$\left[I^{3\pi}_{W_{2,5}}\left(s_{\phi}\right)\right]^{\rho\omega\phi;\,{\rm
VMD}}_{\rm IB}$. There is a smaller $\sim 22\%$ uncertainty in all cases
associated with that on the square root of $B(\rho\rightarrow 3\pi)$, to which
the $\rho$-$\omega$ and $\rho$-$\phi$ interference contributions are
proportional. We expect the uncertainties in the full VMD model estimates for
the total IB contributions resulting from the absence of a representation of
excited $\rho$ resonance effects in the region above $s=s_{\phi}$ to be much
smaller than the dominant $\phi_{\rho}$-induced uncertainties, for the
following reason. First, the contributions to $a_{\mu}^{{\rm
HVP};\,3\pi}\left(s_{\rm KNT}\right)$, $I^{3\pi}_{W_{1,5}}\left(s_{\rm
KNT}\right)$ and $I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)$ from the region
between $s_{\phi}$ and $s_{\rm KNT}$ represent, respectively, only $6.6\%$,
$3.5\%$ and $9.4\%$ of these totals. Even if the omitted IB
$\rho^{\prime}$-$\omega^{\prime}$, $\rho^{\prime}$-$\omega^{\prime\prime}$,
etc. interference contributions represented anomalously high $3\%$ fractions
of the total contributions from this region, they would represent only
$0.20\%$, $0.11\%$ and $0.28\%$ of the corresponding $3\pi$ spectral integral
totals, $a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm KNT}\right)$,
$I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)$ and
$I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)$, and hence only $17\%$, $10\%$
and $18\%$, respectively, of the corresponding full VMD model IB estimates
$\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm
VMD}$, $\left[I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm
VMD}$ and $\left[I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)\right]_{\rm
IB}^{\rm VMD}$, a level well below that of the dominant low-$s$-region
uncertainties. The $s\leq s_{\phi}$, $\phi_{\rho}$-induced uncertainties thus
strongly dominate the uncertainties on the estimated IB $3\pi$ double-counting
correction, especially so in the quadrature combination of the above three
error contributions. Since BaBar does not provide the correlations between its
fitted VMD parameters, and a positive $\phi_{\rho}$-$B(\rho\rightarrow 3\pi)$
correlation would further increase the combined error, we assign, rather than
the quadrature combination, an expanded 100% uncertainty on our integrated
full VMD-model $3\pi$ IB estimates, $\left[a_{\mu}^{{\rm
HVP};\,3\pi}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm VMD}$,
$\left[I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm VMD}$
and $\left[I^{3\pi}_{W_{2,5}}\left(s_{\rm KNT}\right)\right]_{\rm IB}^{\rm
VMD}$.

From the results above, we conclude that it is possible to obtain an
experimentally constrained estimate of the IB $3\pi$ double-counting
correction, albeit with an uncertainty of order $100\%$. This estimate is of
use, despite the sizeable uncertainty, because of the small size of the
central values. Explicitly, the full VMD model IB results, including this
$100\%$ uncertainty, yield

$\displaystyle\left[a_{\mu}^{{\rm HVP};\,3\pi}\left(s_{\rm
KNT}\right)\right]_{\rm IB}=-0.54(54)\times 10^{-10}\ ,$
$\displaystyle\left[I^{3\pi}_{W_{1,5}}\left(s_{\rm KNT}\right)\right]_{\rm
IB}=-0.00041(41)\ ,$ $\displaystyle\left[I^{3\pi}_{W_{2,5}}\left(s_{\rm
KNT}\right)\right]_{\rm IB}=-0.00012(12)\ ,$ (51)

representing $-1.2\pm 1.2\,\%$, $-1.1\pm 1.1\,\%$ and $-1.6\pm\,1.6\,\%$,
respectively, of the full $3\pi$ VMD contributions. As shown in the main text,
the uncertainties on these corrections, even at 100%, are much smaller than
the experimental uncertainties on the differences of the correspondingly
weighted $s=s_{\tau}$, $2\pi+4\pi$ electroproduction and $\tau$ integrals. The
IB $3\pi$ double-counting correction is thus under good control for the
purposes envisioned in this paper.

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