* [43] Antonio Aurilia, Anais Smailagic, and Euro Spallucci, “Gauge theory of the string geodesic field,” Phys. Rev. D 47, 2536–2548 (1993), arXiv:hep-th/9301019
  * [44] Hamid Afshar, Erfan Esmaeili, and M. M. Sheikh-Jabbari, “Asymptotic Symmetries in $p$-Form Theories,” JHEP 05, 042 (2018), arXiv:1801.07752 [hep-th]
  * [45] Temple He and Prahar Mitra, “New magnetic symmetries in $(d+2)$-dimensional QED,” JHEP 01, 122 (2021), arXiv:1907.02808 [hep-th]
  * [46] Juan Maldacena, Douglas Stanford, and Zhenbin Yang, “Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,” PTEP 2016, 12C104 (2016), arXiv:1606.01857 [hep-th]
  * [47] Miguel Campiglia and Alok Laddha, “Asymptotic symmetries of QED and Weinberg’s soft photon theorem,” JHEP 07, 115 (2015), arXiv:1505.05346 [hep-th]
  * [48] Daniel Kapec, Vyacheslav Lysov, and Andrew Strominger, “Asymptotic Symmetries of Massless QED in Even Dimensions,” Adv. Theor. Math. Phys. 21, 1747–1767 (2017), arXiv:1412.2763 [hep-th]
  * [49] Daniel Kapec, Monica Pate, and Andrew Strominger, “New Symmetries of QED,” Adv. Theor. Math. Phys. 21, 1769–1785 (2017), arXiv:1506.02906 [hep-th]
  * [50] Miguel Campiglia and Alok Laddha, “Asymptotic charges in massless QED revisited: A view from Spatial Infinity,” JHEP 05, 207 (2019), arXiv:1810.04619 [hep-th]
  * [51] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A269, 21–52 (1962)
  * [52] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A270, 103–126 (1962)
  * [53] Daniel Kapec, Vyacheslav Lysov, Sabrina Pasterski, and Andrew Strominger, “Higher-dimensional supertranslations and Weinberg’s soft graviton theorem,” Ann. Math. Sci. Appl. 02, 69–94 (2017), arXiv:1502.07644 [gr-qc]
  * [54] Miguel Campiglia and Alok Laddha, “Asymptotic symmetries of gravity and soft theorems for massive particles,” JHEP 12, 094 (2015), arXiv:1509.01406 [hep-th]


#  $\tau$ data-driven evaluation of Euclidean windows for the hadronic vacuum
polarization

Pere Masjuan<EMAIL_ADDRESS>Grup de Física Teòrica, Departament de Física
Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
Institut de Física d’Altes Energies (IFAE) and The Barcelona Institute of
Science and Technology (BIST), Campus UAB, 08193 Bellaterra (Barcelona),
Spain.  Alejandro Miranda<EMAIL_ADDRESS>Institut de Física d’Altes
Energies (IFAE) and The Barcelona Institute of Science and Technology (BIST),
Campus UAB, 08193 Bellaterra (Barcelona), Spain.  Pablo Roig
<EMAIL_ADDRESS>Departamento de Física, Centro de Investigación y
de Estudios Avanzados del Instituto Politécnico Nacional
Apdo. Postal 14-740,07000 Ciudad de México, México

###### Abstract

We compute for the first time the $\tau$ data-driven Euclidean windows for the
hadronic vacuum polarization contribution to the muon $g-2$. We show that
$\tau$-based results agree with the available lattice window evaluations and
with the full result. On the intermediate window, where all lattice
evaluations are rather precise and agree, $\tau$-based results are compatible
with them. This is particularly interesting, given that the disagreement of
the $e^{+}e^{-}$ data-driven result with the lattice values in this window is
the main cause for their discrepancy, affecting the interpretation of the
$a_{\mu}$ measurement in terms of possible new physics.

##  1 Introduction

The first Fermilab measurement of the muon anomalous magnetic moment
($a_{\mu}=(g_{\mu}-2)/2$, with $g$ the gyromagnetic factor) confirmed [1] the
final result from Broohaven [2], yielding the world average

$a_{\mu}^{\mathrm{Exp}}\times 10^{11}=116592061(41)\,.$ (1)

This reaffirmed and strengthened the interest of the high-energy physics
community in this observable, given its $4.2\sigma$ tension with the Standard
Model prediction [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], obtained
within the Muon g-2 Theory Initiative [36] 444See later developments in e.g.
refs. [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62].

$a_{\mu}^{\mathrm{SM}}\times 10^{11}=116591810(43)\,.$ (2)

However, the BMWc lattice Collaboration published [63] a precise lattice-QCD
evaluation of the hadronic vacuum polarization (HVP) contribution with the
result

$a_{\mu}^{\mathrm{BMWc}}\times 10^{10}=707.5\pm 5.5\,,$ (3)

which yields an $a_{\mu}$ evaluation at $1.5\sigma$ from
$a_{\mu}^{\mathrm{Exp}}$, Eq. (1), and at $2.1\sigma$ from the data-driven
result, based on $\sigma(e^{+}e^{-}\to$hadrons$)$, that was employed to get
Eq. (2).

Recently, other three accurate lattice evaluations, by the Mainz CLS [64], the
ETMC [65] and the RBC/UKQCD [59] collaborations, have agreed within the so-
called intermediate window (defined below, in Eq. (6)) with the BMWc result,
having commensurate uncertainties.

Reference [66], which sets the bases to start this project, has translated the
data-driven results to the three different windows in Euclidean time used by
lattice practitioners, to scrutinize the root of the current discrepancy
between both groups of results. This is a crucial endeavour, since it
currently limits the implications on new physics of the $a_{\mu}$
measurements, as may help to pinpoint in what energy domain discrepancies
emerge. It is also a timely task, as we expect the soon release of the second
$a_{\mu}$ FNAL value, with improved precision, in the forthcoming months.

The situation has become even more puzzling with the recent accurate
$\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ 555This contribution yields a bit more
than $70\%$ of $a_{\mu}^{\text{HVP,LO}}$. CMD-3 measurement [67], which would
-on its own- reduce the discrepancy with $a_{\mu}^{\mathrm{Exp}}$ to less than
two standard deviations. Consequently, it is in tension with the previous
measurements of this reaction, by CMD-2 [68, 69, 70, 71], SND [72, 73], KLOE
[74, 75, 76, 77], BaBar [78], BES [79] and CLEO [80]. Dedicated studies
related to the radiative corrections employed in the Monte Carlo generators
used by these experiments [81] (see e.g. refs. [82, 83, 84]) may explain this
controversy.

Within the data-driven evaluations of $a_{\mu}^{\text{HVP}}$,
$\tau^{-}\to\nu_{\tau}$hadrons was proposed in 1997 to reduce the error (by
$\sim 37\%$ then) of the method when using only $e^{+}e^{-}\to$hadrons in LEP
times [85]. Through the years, this alternative data-driven method has always
been [86, 87, 88, 89, 90, 91] approximately $[2,2.5]\sigma$ away from the
$a_{\mu}^{\mathrm{Exp}}$ world average, a situation which seems now favored by
the lattice QCD evaluations of $a_{\mu}^{\text{HVP}}$ 666The difference
between both data-driven groups of results could also be due to non-standard
interactions modifying slightly -but noticeably at this precision- the di-pion
$\tau$ decays [92, 93, 94, 95].. Still, a direct comparison between lattice
QCD results and $\tau$ data-driven methodology is not straightforward either.
This motivates us to compute in this work Euclidean windows for
$a_{\mu}^{\text{HVP}}$ using $\tau$ data and compare results. We hope our
outcomes may shed some light on this puzzling situation and be useful for the
lattice effort [96] to compute the required isospin-breaking corrections (see
below Eq. (6)) entering the $\tau$-based method. In section 2 we explain the
needed formulae and apply them to obtain the results, which are summarized in
section 3.

##  2 Results

The HVP contribution at leading order (LO) in the data-driven approach is
given by [97, 98, 99, 100]

$a_{\mu}^{\text{HVP,LO}}=\frac{1}{4\pi^{3}}\int_{s_{\text{thr}}}^{\infty}ds\,K(s)\,\sigma_{e^{+}e^{-}\to\text{hadrons}(+\gamma)}^{0}(s),$
(4)

where $\sigma^{0}_{e^{+}e^{-}\to\text{hadrons}(+\gamma)}(s)$ is the bare
hadronic cross-section, which excludes effects from vacuum polarization (VP)
[101], and $K(s)$ is a smooth kernel concentrated at low energies [99]

$\begin{split}&K(s)=\frac{x^{2}}{2}(2-x^{2})+\frac{(1+x^{2})(1+x)^{2}}{x^{2}}\left(\log(1+x)-x+\frac{x^{2}}{2}\right)+\frac{(1+x)}{(1-x)}x^{2}\log(x),\end{split}$
(5)

which is written in terms of the variable
$x=\frac{1-\beta_{\mu}}{1+\beta_{\mu}}$,
$\beta_{\mu}=\sqrt{1-4m_{\mu}^{2}/s}$. This dispersive approach revolves
around the handiness of $e^{+}e^{-}$ hadronic cross-section measurements at
energies below a few GeV. However, as was pointed out early by Alemany,
Davier, and Höcker [85], it is also possible to use hadronic $\tau$ decays to
evaluate the HVP, LO contributions to $a_{\mu}$. For some time this approach
was competitive with the $e^{+}e^{-}$ data, although this is generally not
considered the case at the moment [36].

Recently, a new $\tau$ data-driven approach was performed in Ref. [91]. In
this section, we utilize their results to evaluate the leading HVP
contribution to the anomalous magnetic moment of the muon in the so-called
window quantities [10, 102]. For this enterprise, we make use of the weight
functions in center-of-mass energy $\tilde{\Theta}(s)$ from Eq. (12) in Ref.
[66] which are related to those in Euclidean time [10]

$\begin{split}\Theta_{SD}(t)&=1-\Theta(t,t_{0},\Delta),\\\
\Theta_{win}(t)&=\Theta(t,t_{0},\Delta)-\Theta(t,t_{1},\Delta),\\\
\Theta_{LD}(t)&=\Theta(t,t_{1},\Delta),\\\
\Theta(t,t^{\prime},\Delta)&=\frac{1}{2}\left(1+\tanh{\frac{t-t^{\prime}}{\Delta}}\right).\end{split}$
(6)

The subscripts in Eq. (6) refer to the short-distance ($SD$), intermediate
($win$, although we will use $int$ in the following) and long-distance ($LD$)
contributions with parameters

$t_{0}=0.4\text{ fm},\quad t_{1}=1.0\text{ fm},\quad\Delta=0.15\text{ fm},$
(7)

which correspond to inverse energies of the order of $500,200$ and $1300$ MeV,
respectively.

In what follows, we will focus on the dominant $2\pi$ contribution only.
Including isospin breaking (IB) corrections, i.e.,
$\mathcal{O}[(m_{u}-m_{d})p^{2}]$ and $\mathcal{O}(e^{2}p^{2})$ contributions,
the bare hadronic $e^{+}e^{-}$ cross-section $\sigma_{\pi\pi(\gamma)}^{0}$ is
related to the observed differential $\tau$ decay rate
$d\Gamma_{\pi\pi[\gamma]}$ through [86, 87]

$\sigma^{0}_{\pi\pi(\gamma)}=\left[\frac{K_{\sigma}(s)}{K_{\Gamma}(s)}\frac{d\Gamma_{\pi\pi[\gamma]}}{ds}\right]\times\frac{R_{\text{IB}}(s)}{S_{\text{EW}}},$
(8)

where

$\begin{split}K_{\Gamma}(s)&=\frac{G_{F}^{2}|V_{ud}|^{2}m_{\tau}^{3}}{384\pi^{3}}\left(1-\frac{s}{m_{\tau}^{2}}\right)^{2}\left(1+\frac{2s}{m_{\tau}^{2}}\right),\\\
K_{\sigma}(s)&=\frac{\pi\alpha^{2}}{3s},\end{split}$ (9)

and the IB corrections are encoded in

$R_{\text{IB}}(s)=\frac{\text{FSR}(s)}{G_{\text{EM}}(s)}\frac{\beta^{3}_{\pi^{+}\pi^{-}}}{\beta^{3}_{\pi^{+}\pi^{0}}}\left|\frac{F_{V}(s)}{f_{+}(s)}\right|^{2}.$
(10)

The $S_{\text{EW}}$ term in Eq. (8) includes the short-distance electroweak
corrections [103, 104, 105, 106, 107, 108, 109, 110]. FSR refers to the Final-
State-Radiation corrections to the $\pi^{+}\pi^{-}$ channel [111, 112], while
the $G_{\text{EM}}(s)$ factor includes the QED corrections to the
$\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decay with virtual plus real photon
radiation. $\beta^{3}_{\pi^{-}\pi^{+}}/\beta^{3}_{\pi^{-}\pi^{0}}$ is a phase
space correction owing to the $\pi^{\pm}-\pi^{0}$ mass difference. The last
term in $R_{\text{IB}}(s)$ corresponds to the ratio between the neutral
($F_{V}(s)$) and the charged ($f_{+}(s)$) pion form factor, which includes one
of the leading IB effects, the $\rho^{0}-\omega$ mixing correction.

The IB corrections to $a_{\mu}^{\text{HVP, LO}}$ using $\tau$ data in the
dominant $\pi\pi$ channel can be evaluated using the following expression [88]

$\Delta a_{\mu}^{\text{HVP,
LO}}[\pi\pi,\tau]=\frac{1}{4\pi^{3}}\int_{4m_{\pi}^{2}}^{m_{\tau}^{2}}ds\,K(s)\left[\frac{K_{\sigma}(s)}{K_{\Gamma}(s)}\frac{d\Gamma_{\pi\pi[\gamma]}}{ds}\right]\left(\frac{R_{\text{IB}}(s)}{S_{\text{EW}}}-1\right),$
(11)

which measures the difference between the correct expression for
$\sigma_{\pi\pi(\gamma)}^{0}$ and the naive Conserved Vector Current
approximation, with $S_{\text{EW}}=1$ and $R_{\text{IB}}(s)=1$.

These contributions are summarized in Table 1 for each IB correction.

  * •

The $S_{\text{EW}}$ factor ($S_{\text{EW}}=1.0201$, at the scale $m_{\tau}$)
contributes $3.0\%$, $28.7\%$ and $68.3\%$ to the complete $\Delta
a_{\mu}^{\text{HVP, LO}}=-10.31(0)\cdot 10^{-10}$ correction for the $SD$,
$int$ and $LD$ contributions, respectively.

  * •

The phase space (PS) correction yields $1.7\%$, $18.7\%$ and $79.6\%$ of a
total of $\Delta a_{\mu}^{\text{HVP, LO}}=-7.45(0)\cdot 10^{-10}$ for the
$SD$, $int$ and $LD$ contributions, respectively.

  * •

The final state radiation (FSR) induces $2.9\%$, $27.0\%$ and $70.1\%$ of a
total of $\Delta a_{\mu}^{\text{HVP, LO}}=+4.55(45)\cdot 10^{-10}$ for the
$SD$, $int$ and $LD$ contributions, respectively.

  * •

$G_{\text{EM}}(s)$ was originally computed in Ref. [87] including those
operators yielding resonance saturation of the $\mathcal{O}(p^{4})$ chiral
couplings in the frame of Resonance Chiral Theory (R$\chi$T) [113, 114, 115,
116] 777We will shorten this to ’$\mathcal{O}(p^{4})$’ for brevity, and
similarly at the next chiral order. 888Pseudoscalar pole (axial) contributions
to the hadronic light-by-light piece of $a_{\mu}$ have been computed within
R$\chi$T [117, 118] ([29]), as well as di-meson contributions to
$a_{\mu}^{\text{HVP}}$ [119, 62]., which comprises Goldstone bosons and
resonance fields extending the $\chi$PT framework [120, 121, 122] to higher
energies. A recalculation of $G_{\text{EM}}(s)$ was performed later in [123,
124] using a Vector Meson Dominance (VMD) model [125].
In Ref. [91] (see also [126]), two of us explored the impact of the R$\chi$T
operators contributing to resonance saturation at the next chiral order
($\mathcal{O}(p^{6})$) on the $G_{\text{EM}}(s)$, as well as estimating the
uncertainty of the original computation in [87]. These results [91] are
consistent with the earlier R$\chi$T and the VMD estimates [91]. Availing of
these results, $G_{\text{EM}}(s)$ produces a correction of $\sim-3.5\%$,
$\sim-17.1\%$ and $\sim+120.6\%$ of $\Delta a_{\mu}^{\text{HVP,
LO}}=-1.70(^{0.61}_{1.48})\cdot 10^{-10}$ at $\mathcal{O}(p^{4})$, and $\sim
0.4\%$, $\sim 7.6\%$ and $\sim 92.0\%$ of $\Delta a_{\mu}^{\text{HVP,
LO}}=-7.59(^{6.50}_{4.56})\cdot 10^{-10}$ at $\mathcal{O}(p^{6})$ for the
$SD$, $int$ and $LD$ contributions, respectively. Interestingly, the $SD$ and
$int$ contributions at $\mathcal{O}(p^{4})$ change in sign while this is not
the case at $\mathcal{O}(p^{6})$ where all the contributions are always
negative.

  * •

The ratio of the form factors (FF) gives an overall correction of $\Delta
a_{\mu}^{\text{HVP, LO}}=+7.10(1.48)(^{1.59}_{1.54})(^{0.85}_{0.80})\cdot
10^{-10}$, from which $\sim 2.2\%$, $\sim 26.8\%$ and $\sim 71.0\%$ stand for
the SD, int and LD corrections, respectively.
The errors quoted in this contribution correspond to the electromagnetic
shifts in the widths and masses of the $\rho$ meson, and to the
$\rho^{0}-\omega$ mixing parameter [127, 87], respectively. For this analysis,
we use the same numerical inputs as in [91]. The central value reported in
Table 1 corresponds to the weighted average between the FF1 and FF2 sets
999The main distinction between FF1 and FF2 comes from the width difference
between the $\rho^{\pm}$ and $\rho^{0}$ mesons
($\Delta\Gamma_{\rho}=\Gamma_{\rho^{0}}-\Gamma_{\rho^{\pm}}$), while the mass
difference ($\Delta M_{\rho}=M_{\rho^{0}}-M_{\rho^{\pm}}$) and the
$\theta_{\omega\rho}$ parameter are the same in both. .

The overall correction is also consistent with those in Refs. [87, 88, 91].

$\Delta a_{\mu}^{\text{HVP,LO}}$
---
| SD | int | LD | Total
| $\mathcal{O}(p^{4})$ | $\mathcal{O}(p^{6})$ | $\mathcal{O}(p^{4})$ | $\mathcal{O}(p^{6})$ | $\mathcal{O}(p^{4})$ | $\mathcal{O}(p^{6})$ | $\mathcal{O}(p^{4})$ | $\mathcal{O}(p^{6})$
$S_{\text{EW}}$ | $-0.31(0)$ | $-2.96(0)$ | $-7.04(0)$ | $-10.31(0)$
PS | $-0.13(0)$ | $-1.39(0)$ | $-5.93(0)$ | $-7.45(0)$
FSR | $+0.13(1)$ | $+1.23(12)$ | $+3.19(32)$ | $+4.55(45)$
$G_{\text{EM}}$ | $+0.06(^{0}_{2})$ | $-0.03(^{9}_{7})$ | $+0.29(^{6}_{19})$ | $-0.58(^{93}_{71})$ | $-2.05(^{0.55}_{1.27})$ | $-6.98(^{5.48}_{3.78})$ | $-1.70(^{0.61}_{1.48})$ | $-7.59(^{6.50}_{4.56})$
FF | $+0.16(5)(1)(^{2}_{1})$ | $+1.90(49)(^{29}_{27})(^{21}_{20})$ | $+5.04(94)(^{1.29}_{1.26})(^{62}_{59})$ | $+7.10(1.48)(^{1.59}_{1.54})(^{85}_{80})$
Total | $-0.09(6)$ | $-0.18(^{11}_{9})$ | $-0.93(^{62}_{64})$ | $-1.80(^{1.12}_{0.93})$ | $-6.79(^{1.83}_{2.13})$ | $-11.72(^{5.75}_{4.15})$ | $-7.81(^{2.51}_{2.83})$ | $-13.70(^{6.98}_{5.17})$

Table 1: $\Delta a_{\mu}^{\text{HVP,LO}}$ in units of $10^{-10}$. The
uncertainties in the fifth row (FF) are related to $\Delta\Gamma_{\rho}$,
$\Delta M_{\rho}$ and $\theta_{\omega\rho}$, respectively.

Using the $\tau$ spectral functions measured by ALEPH [128], Belle [129], CLEO
[130] and OPAL [131], we evaluate $a_{\mu}^{\text{HVP, LO}}[\pi\pi]$ using the
window parameters in Eq. (7). These results are outlined in Tables 2 and 3 for
$s\leq 1,\,2,\,3,\text{ and }3.125\,\text{ GeV}^{2}$, i.e., integrating Eq.
(4) with $\sigma_{\pi\pi(\gamma)}^{0}$ in Eq. (8) from
$s_{\text{thr}}=4m_{\pi}^{2}$ up to some given cut-off. In the aforementioned
tables 2 and 3, the first uncertainty is connected to the systematic errors on
the mass spectrum, and from the $\tau$-mass and $V_{ud}$ uncertainties; the
second error is associated to $B_{\pi\pi^{0}}$ and $B_{e}$; and the third one
is due to the IB corrections. The Mean value in the tables corresponds to the
weighted average from the different window contributions for each experiment,
the first error is related to the experimental measurements, while the second
one comes from the IB corrections.

An evaluation of $a_{\mu}^{\text{HVP}}$ in the windows in Euclidean time using
$e^{+}e^{-}$ data was performed in Ref. [66] using the parameters in Eq. (7).
A comparison between these window quantities for HVP in the $2\pi$ channel
below $1\,\text{GeV}$ amounts to a discrepancy of $4.3\,\sigma$, $3.2\,\sigma$
and $2.1\,\sigma$ between the $\tau$ and $e^{+}e^{-}$ evaluations applying the
$G_{\text{EM}}(s)$ correction at $\mathcal{O}(p^{4})$ in R$\chi$T to the
$\tau$ data for the $SD$, $int$ and $LD$ contributions, respectively. On the
other hand, when we include the corrections at $\mathcal{O}(p^{6})$, the
difference between these two evaluations decreases to $3.6\,\sigma$,
$2.6\,\sigma$ and $0.9\,\sigma$ for the $SD$, $int$ and $LD$ contributions,
respectively. These results are depicted in Figs. 1 and 2, where the blue band
corresponds to the experimental average using $\tau$ data. Fig. 3 shows a
zoomed comparison between $\tau$ (after IB corrections at
$\mathcal{O}(p^{6})$) and $e^{+}e^{-}\to\pi^{+}\pi^{-}$ spectral function
using the ISR measurements from BABAR [78] and KLOE [76] (left-hand panel) and
the energy-scan measurements from CMD-3 [67] (right-hand panel). Colored bands
correspond to the weighted average of the uncertainties coming from both sets
of data in each figure.

$a_{\mu}^{\text{HVP,LO}}[\pi\pi,\tau]$
---
SD
Experiment | $s\leq 1\,\mathrm{GeV}^{2}$ | $s\leq 2\,\mathrm{GeV}^{2}$ | $s\leq 3\,\mathrm{GeV}^{2}$ | $s\leq 3.125\,\mathrm{GeV}^{2}$
ALEPH | $14.29(4)(8)(6)$ | $15.51(2)(9)(6)$ | $15.57(3)(9)(6)$ | $15.57(3)(9)(6)$
Belle | $14.31(4)(22)(6)$ | $15.39(5)(24)(6)$ | $15.44(6)(24)(6)$ | $15.46(12)(24)(6)$
CLEO | $14.33(6)(25)(6)$ | $15.46(6)(27)(6)$ | $15.50(6)(27)(6)$ | $15.51(6)(27)(6)$
OPAL | $14.23(7)(19)(6)$ | $15.51(3)(21)(6)$ | $15.56(3)(21)(6)$ | $15.56(3)(21)(6)$
Mean | $14.29(7)(6)$ | $15.49(8)(6)$ | $15.55(8)(6)$ | $15.55(8)(6)$
Intermediate
Experiment | $s\leq 1\,\mathrm{GeV}^{2}$ | $s\leq 2\,\mathrm{GeV}^{2}$ | $s\leq 3\,\mathrm{GeV}^{2}$ | $s\leq 3.125\,\mathrm{GeV}^{2}$
ALEPH | $143.18(61)(79)(^{59}_{61})$ | $149.49(50)(83)(^{59}_{61})$ | $149.65(48)(83)(^{59}_{61})$ | $149.65(48)(83)(^{59}_{61})$
Belle | $143.70(45)(2.24)(^{59}_{61})$ | $149.30(48)(2.33)(^{60}_{62})$ | $149.44(49)(2.33)(^{60}_{62})$ | $149.47(54)(2.33)(^{60}_{62})$
CLEO | $143.86(60)(2.52)(^{59}_{61})$ | $149.70(61)(2.62)(^{60}_{62})$ | $149.81(61)(2.62)(^{60}_{62})$ | $149.84(61)(2.62)(^{60}_{62})$
OPAL | $143.43(1.17)(1.92)(^{58}_{60})$ | $150.03(97)(2.01)(^{58}_{61})$ | $150.16(97)(2.01)(^{58}_{61})$ | $150.16(97)(2.01)(^{58}_{61})$
Mean | $143.34(81)(^{59}_{61})$ | $149.56(80)(^{59}_{62})$ | $149.70(79)(^{59}_{62})$ | $149.71(79)(^{59}_{62})$
LD
Experiment | $s\leq 1\,\mathrm{GeV}^{2}$ | $s\leq 2\,\mathrm{GeV}^{2}$ | $s\leq 3\,\mathrm{GeV}^{2}$ | $s\leq 3.125\,\mathrm{GeV}^{2}$
ALEPH | $348.12(4.12)(1.93)(^{1.19}_{1.39})$ | $351.89(4.08)(1.95)(^{1.19}_{1.39})$ | $351.93(4.08)(1.95)(^{1.19}_{1.39})$ | $351.93(4.08)(1.95)(^{1.19}_{1.39})$
Belle | $351.65(1.36)(5.49)(^{1.19}_{1.40})$ | $355.01(1.37)(5.54)(^{1.20}_{1.42})$ | $355.04(1.37)(5.55)(^{1.20}_{1.42})$ | $355.04(1.37)(5.55)(^{1.19}_{1.41})$
CLEO | $350.59(2.88)(6.13)(^{1.19}_{1.39})$ | $354.11(2.88)(6.20)(^{1.19}_{1.39})$ | $354.13(2.88)(6.20)(^{1.19}_{1.40})$ | $354.14(2.88)(6.20)(^{1.19}_{1.39})$
OPAL | $362.04(9.33)(4.85)(^{1.18}_{1.44})$ | $365.96(9.26)(4.90)(^{1.18}_{1.45})$ | $365.98(9.26)(4.90)(^{1.18}_{1.45})$ | $365.98(9.26)(4.90)(^{1.18}_{1.45})$
Mean | $350.75(3.01)(^{1.19}_{1.41})$ | $354.36(3.01)(^{1.19}_{1.41})$ | $354.39(3.01)(^{1.19}_{1.42})$ | $354.39(3.01)(^{1.19}_{1.41})$

Table 2: IB-corrected $a_{\mu}^{\text{HVP,LO}}[\pi\pi,\tau]$ in units of
$10^{-10}$ at $\mathcal{O}(p^{4})$ in R$\chi T$ using the experimental
measurements from the ALEPH [128], Belle [129], CLEO [130] and OPAL [131]
Colls. The first error is related to the systematic uncertainties on the mass
spectrum and also includes contributions from the $\tau$-mass and $V_{ud}$
uncertainties. The second error arises from $B_{\pi\pi^{0}}$ and $B_{e}$, and
the third error comes from the isospin-breaking corrections. The uncertainties
in the mean value correspond to the experiment and to the IB corrections,
respectively.

$a_{\mu}^{\text{HVP,LO}}[\pi\pi,\tau]$
---
SD
Experiment | $s\leq 1\,\mathrm{GeV}^{2}$ | $s\leq 2\,\mathrm{GeV}$ | $s\leq 3\,\mathrm{GeV}$ | $s\leq 3.125\,\mathrm{GeV}^{2}$
ALEPH | $14.22(4)(8)(8)$ | $15.42(2)(9)(^{11}_{9})$ | $15.47(2)(9)(^{11}_{9})$ | $15.47(3)(9)(^{11}_{9})$
Belle | $14.24(4)(22)(8)$ | $15.30(5)(24)(^{10}_{8})$ | $15.35(5)(24)(^{10}_{8})$ | $15.36(12)(24)(^{10}_{8})$
CLEO | $14.27(6)(25)(8)$ | $15.37(6)(27)(^{10}_{8})$ | $15.41(6)(27)(^{10}_{9})$ | $15.42(6)(27)(^{10}_{9})$
OPAL | $14.16(7)(19)(8)$ | $15.42(3)(21)(^{11}_{9})$ | $15.46(3)(21)(^{11}_{9})$ | $15.46(3)(21)(^{11}_{9})$
Mean | $14.22(7)(8)$ | $15.40(8)(^{11}_{9})$ | $15.45(8)(^{11}_{9})$ | $15.46(8)(^{11}_{9})$
Intermediate
Experiment | $s\leq 1\,\mathrm{GeV}^{2}$ | $s\leq 2\,\mathrm{GeV}^{2}$ | $s\leq 3\,\mathrm{GeV}^{2}$ | $s\leq 3.125\,\mathrm{GeV}^{2}$
ALEPH | $142.49(60)(79)(^{92}_{81})$ | $148.67(50)(83)(^{1.02}_{87})$ | $148.82(48)(83)(^{1.02}_{87})$ | $148.82(48)(83)(^{1.02}_{87})$
Belle | $143.00(45)(2.23)(^{92}_{81})$ | $148.49(48)(2.32)(^{1.02}_{88})$ | $148.62(48)(2.32)(^{1.03}_{88})$ | $148.65(54)(2.32)(^{1.03}_{88})$
CLEO | $143.16(60)(2.50)(^{92}_{81})$ | $148.88(60)(2.60)(^{1.01}_{87})$ | $148.99(60)(2.61)(^{1.01}_{87})$ | $149.02(60)(2.61)(^{1.01}_{87})$
OPAL | $142.69(1.16)(1.91)(^{96}_{83})$ | $149.15(95)(2.00)(^{1.06}_{90})$ | $149.27(95)(2.00)(^{1.07}_{90})$ | $149.27(95)(2.00)(^{1.07}_{90})$
Mean | $142.64(80)(^{93}_{82})$ | $148.73(79)(^{1.03}_{0.88})$ | $148.87(79)(^{1.03}_{88})$ | $148.88(79)(^{1.03}_{0.88})$
LD
Experiment | $s\leq 1\,\mathrm{GeV}^{2}$ | $s\leq 2\,\mathrm{GeV}^{2}$ | $s\leq 3\,\mathrm{GeV}^{2}$ | $s\leq 3.125\,\mathrm{GeV}^{2}$
ALEPH | $344.17(3.99)(1.91)(^{3.63}_{2.74})$ | $347.87(3.95)(1.93)(^{3.69}_{2.79})$ | $347.90(3.95)(1.93)(^{3.69}_{2.79})$ | $347.90(3.95)(1.93)(^{3.69}_{2.79})$
Belle | $347.62(1.34)(5.43)(^{3.73}_{2.82})$ | $350.92(1.35)(5.48)(^{3.78}_{2.86})$ | $350.94(1.35)(5.48)(^{3.78}_{2.86})$ | $350.94(1.36)(5.48)(^{3.78}_{2.86})$
CLEO | $346.63(2.77)(6.06)(^{3.65}_{2.76})$ | $350.08(2.77)(6.12)(^{3.70}_{2.81})$ | $350.10(2.77)(6.12)(^{3.70}_{2.81})$ | $350.11(2.77)(6.13)(^{3.70}_{2.81})$
OPAL | $357.54(8.99)(4.79)(^{4.19}_{3.14})$ | $361.38(8.92)(4.84)(^{4.25}_{3.18})$ | $361.41(8.92)(4.84)(^{4.25}_{3.18})$ | $361.41(8.92)(4.84)(^{4.25}_{3.18})$
Mean | $346.73(2.95)(^{3.80}_{2.87})$ | $350.27(2.95)(^{3.86}_{2.91})$ | $350.29(2.95)(^{3.86}_{2.91})$ | $350.30(2.95)(^{3.86}_{2.91})$

Table 3: Same as Table 2, but the $\mathcal{O}(p^{6})$ contributions to
$G_{\text{EM}}(s)$ in Ref. [91] have been applied to the $\tau$ data.

Figure 1: Windows quantities for HVP at $\mathcal{O}(p^{4})$ for $2\pi$ below
$1.0\,\text{GeV}$ using the parameters in Eq. (7). The blue region corresponds
to the experimental average from $\tau$ data. The $e^{+}e^{-}$ number was
taken from Ref. [66].

Figure 2: Analog to Fig. 1 but at $\mathcal{O}(p^{6})$.

Figure 3: Comparison between the $\tau$ (after IB corrections) and
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ spectral function using the ISR measurements
from BABAR [78] and KLOE [76] (left-hand) and the energy-scan measurements
from CMD-3 [67] (right-hand).

A direct comparison between $a_{\mu}^{\text{HVP, LO}}[\pi\pi,\tau]$ and the
lattice results is not possible. For that endeavour, it is necessary to
supplement the $2\pi$ evaluation with the remaining contributions from all
other channels accounting for the hadronic cross-section. To illustrate the
impact of this contribution in $a_{\mu}^{\text{HVP, LO}}$, we follow two
approaches. Firstly, using the values reported in Table 1 of Ref. [66] we
subtract the contribution from the $2\pi$ channel below $1.0\,\text{GeV}$ (we
represent this procedure with ’$<1$ GeV’) and replace it by the corresponding
mean value in Tables 2 and 3. This way, we get

$a^{SD}_{\mu}=69.0(5)\times 10^{-10},\quad a^{int}_{\mu}=234.4(1.2)\times
10^{-10},\quad a^{LD}_{\mu}=403.6(3.3)\times 10^{-10},$ (12)

at $\mathcal{O}(p^{4})$, and

$a^{SD}_{\mu}=68.9(5)\times 10^{-10},\quad a^{int}_{\mu}=233.7(1.4)\times
10^{-10},\quad a^{LD}_{\mu}=399.5(^{4.9}_{4.2})\times 10^{-10},$ (13)

at $\mathcal{O}(p^{6})$.

Secondly, we rescale the contributions from the $2\pi$ channel using the
overall evaluation of $a_{\mu}^{\text{HVP, LO}}[\pi\pi,e^{+}e^{-}]$ in Refs.
[7, 8], then we subtract this value from the total contribution and replace it
by our results. Thus, we get

$a^{SD}_{\mu}=70.0(5)\times 10^{-10},\quad a^{int}_{\mu}=237.8(1.2)\times
10^{-10},\quad a^{LD}_{\mu}=399.7(^{3.3}_{3.4})\times 10^{-10},$ (14)

at $\mathcal{O}(p^{4})$, and

$a^{SD}_{\mu}=69.9(5)\times 10^{-10},\quad
a^{int}_{\mu}=237.0(^{1.5}_{1.4})\times 10^{-10},\quad
a^{LD}_{\mu}=395.6(^{4.9}_{4.2})\times 10^{-10},$ (15)

at $\mathcal{O}(p^{6})$. All these results are reasonably consistent with each
other.

We summarize these results in Table 4 along with the lattice results [10, 132,
63, 64, 65, 59] and other $e^{+}e^{-}$ data-driven evaluations [36, 66]. These
numbers are depicted in Fig. 4 for the intermediate window, where the blue
band represents the average of the lattice results excluding those from
RBC/UKQCD 2018 [10] and ETMC 2021 [132] collaborations. The contributions of
the intermediate window using $\tau$ data are slightly closer to the results
from lattice QCD than to the $e^{+}e^{-}$ values. Therefore, the $\sim
4.3\sigma$ discrepancy between the $e^{+}e^{-}$ data-driven and lattice
evaluations is reduced to $\sim 1.5\sigma$ when $\tau$ data is used for the
$2\pi$ channel. On the other hand, there is only one lattice result for the
short-distance window [65] which seems to be in agreement with both data-
driven HVP evaluations.

$a_{\mu}^{\text{HVP,LO}}$
---
| SD | int | LD | Total
$\tau$-data $\mathcal{O}(p^{4})$ $\leq 1\text{ GeV}$ | $69.0(5)$ | $234.4(1.2)$ | $403.6(3.3)$ | $707.0(5.0)$
$\tau$-data $\mathcal{O}(p^{6})$ $\leq 1\text{ GeV}$ | $68.9(5)$ | $233.7(1.4)$ | $399.5(^{4.9}_{4.2})$ | $702.2(^{6.8}_{6.1})$
$\tau$-data $\mathcal{O}(p^{4})$ | $70.0(5)$ | $237.8(1.2)$ | $399.7(^{3.3}_{3.4})$ | $707.5(^{5.0}_{5.1})$
$\tau$-data $\mathcal{O}(p^{6})$ | $69.9(5)$ | $237.0(^{1.5}_{1.4})$ | $395.6(^{4.9}_{4.2})$ | $702.4(^{6.9}_{6.1})$
RBC/UKQCD 2018 [10] | $-$ | $231.9(1.5)$ | $-$ | $715.4(18.7)$
ETMC 2021 [132] | $-$ | $231.7(2.8)$ | $-$ | $-$
BMW 2020 [63] | $-$ | $236.7(1.4)$ | $-$ | $707.5(5.5)$
Mainz/CLS 2022 [64] | $-$ | $237.30(1.46)$ | $-$ | $-$
ETMC 2022 [65] | $69.33(29)$ | $235.0(1.1)$ | $-$ | $-$
RBC/UKQCD 2023 [59] | $-$ | $235.56(82)$ | $-$ | $-$
WP [36] | $-$ | $-$ | $-$ | $693.1(4.0)$
BMW 2020/KNT [4, 63] | $-$ | $229.7(1.3)$ | $-$ | $-$
Colangelo et al. 2022 [66] | $68.4(5)$ | $229.4(1.4)$ | $395.1(2.4)$ | $693.0(3.9)$

Table 4: Window quantities for $a_{\mu}^{\text{HVP,LO}}$ in units of
$10^{-10}$. The first two rows correspond to the $\tau$ evaluation in the
first approach, while the rows 3 and 4 are the evaluations in the second one.
The rows 5-10 are the lattice results [10, 132, 63, 64, 65, 59]. The last
three rows are the evaluations obtained using $e^{+}e^{-}$ data. See Fig. 11
in Ref. [59] for more details. Figure 4: Comparison of the total intermediate
window contribution to $a_{\mu}^{\text{HVP, LO}}$. The blue band corresponds
to the lattice average excluding RBC/UKQCD 2018 [10] and ETMC 2021 [132].

##  3 Conclusions

While the BNL and FNAL measurements of $a_{\mu}$ agree nicely within errors,
the situation is not that clear for its SM prediction’s counterpart. On the
one hand, data-driven methods based on $e^{+}e^{-}\to$hadrons data have to
deal with the tensions between experiments (particularly among BaBar and KLOE,
and now with CMD-3), which makes the computing of the uncertainty in Eq. (2) a
non-trivial task, as will be its update. On the other side, there is still
only one lattice QCD evaluation (BMWc Coll.) of $a_{\mu}^{\text{HVP}}$ with
competitive uncertainties, that lies between $a_{\mu}^{\mathrm{Exp}}$ and its
SM data-driven prediction. However, the most recent Mainz/CLS, ETMC, and
RBC/UKQCD results have similar errors to BMWc in the intermediate window,
where all of them agree remarkably. It is long-known that alternative data-
driven evaluations are possible, utilizing this time semileptonic $\tau$-decay
data (and isospin-breaking corrections, with attached model-dependent
uncertainty), as we have done here.

In this context, we have applied the study from Ref. [59], that computed
window quantities in Euclidean time for data-driven evaluations of
$a_{\mu}^{\text{HVP}}$ using $e^{+}e^{-}\to$ hadrons data, to the semileptonic
$\tau$ decays case (focusing on the dominant two-pion contribution). Our main
results are collected in Table 4 and show that $\tau$-based results are
compatible with the lattice evaluations in the intermediate window, being the
$e^{+}e^{-}$-based values in tension with both of them. This difference is the
main cause for the larger discrepancy of the latter with
$a_{\mu}^{\mathrm{Exp}}$ and should be further scrutinized.

## Acknowledgements

We have benefited from discussions with Rafel Escribano on this topic. The
work of P. M. has been supported by the European Union’s Horizon 2020 Research
and Innovation Programme under grant 824093 (H2020-INFRAIA- 2018-1), the
Ministerio de Ciencia e Innovación under grant PID2020-112965GB-I00, and by
the Secretaria d’Universitats i Recerca del Departament d’Empresa i
Coneixement de la Generalitat de Catalunya under grant 2021 SGR 00649. IFAE is
partially funded by the CERCA program of the Generalitat de Catalunya. J. A.
M. is also supported by MICINN with funding from European Union
NextGenerationEU (PRTR-C17.I1) and by Generalitat de Catalunya. P. R. thanks
partial funding from Conacyt and Cátedras Marcos Moshinsky 2020 (Fundación
Marcos Moshinsky).

## References

  * [1] B. Abi et al.  Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm.  Phys. Rev. Lett., 126(14):141801, 2021.
  * [2] G. W. Bennett et al.  Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL.  Phys. Rev. D, 73:072003, 2006.
  * [3] Michel Davier, Andreas Hoecker, Bogdan Malaescu, and Zhiqing Zhang.  Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data.  Eur. Phys. J. C, 77(12):827, 2017.
  * [4] Alexander Keshavarzi, Daisuke Nomura, and Thomas Teubner.  Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis.  Phys. Rev. D, 97(11):114025, 2018.
  * [5] Gilberto Colangelo, Martin Hoferichter, and Peter Stoffer.  Two-pion contribution to hadronic vacuum polarization.  JHEP, 02:006, 2019.
  * [6] Martin Hoferichter, Bai-Long Hoid, and Bastian Kubis.  Three-pion contribution to hadronic vacuum polarization.  JHEP, 08:137, 2019.
  * [7] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang.  A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\boldsymbol{\alpha}(m_{Z}^{2})}$.  Eur. Phys. J. C, 80(3):241, 2020.  [Erratum: Eur.Phys.J.C 80, 410 (2020)].
  * [8] Alexander Keshavarzi, Daisuke Nomura, and Thomas Teubner.  $g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium.  Phys. Rev. D, 101(1):014029, 2020.
  * [9] Alexander Kurz, Tao Liu, Peter Marquard, and Matthias Steinhauser.  Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order.  Phys. Lett. B, 734:144–147, 2014.
  * [10] T. Blum, P. A. Boyle, V. Gülpers, T. Izubuchi, L. Jin, C. Jung, A. Jüttner, C. Lehner, A. Portelli, and J. T. Tsang.  Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.  Phys. Rev. Lett., 121(2):022003, 2018.
  * [11] D. Giusti, V. Lubicz, G. Martinelli, F. Sanfilippo, and S. Simula.  Electromagnetic and strong isospin-breaking corrections to the muon $g-2$ from Lattice QCD+QED.  Phys. Rev. D, 99(11):114502, 2019.
  * [12] Eigo Shintani and Yoshinobu Kuramashi.  Study of systematic uncertainties in hadronic vacuum polarization contribution to muon $g-2$ with 2+1 flavor lattice QCD.  Phys. Rev., D100(3):034517, 2019.
  * [13] C. T. H. Davies et al.  Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD.  Phys. Rev., D101(3):034512, 2020.
  * [14] Antoine Gérardin, Marco Cè, Georg von Hippel, Ben Hörz, Harvey B. Meyer, Daniel Mohler, Konstantin Ottnad, Jonas Wilhelm, and Hartmut Wittig.  The leading hadronic contribution to $(g-2)_{\mu}$ from lattice QCD with $N_{\rm f}=2+1$ flavours of O($a$) improved Wilson quarks.  Phys. Rev. D, 100(1):014510, 2019.
  * [15] Christopher Aubin, Thomas Blum, Cheng Tu, Maarten Golterman, Chulwoo Jung, and Santiago Peris.  Light quark vacuum polarization at the physical point and contribution to the muon $g-2$.  Phys. Rev., D101(1):014503, 2020.
  * [16] D. Giusti and S. Simula.  Lepton anomalous magnetic moments in Lattice QCD+QED.  PoS, LATTICE2019:104, 2019.
  * [17] Kirill Melnikov and Arkady Vainshtein.  Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited.  Phys. Rev., D70:113006, 2004.
  * [18] Pere Masjuan and Pablo Sánchez-Puertas.  Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach.  Phys. Rev., D95(5):054026, 2017.
  * [19] Gilberto Colangelo, Martin Hoferichter, Massimiliano Procura, and Peter Stoffer.  Dispersion relation for hadronic light-by-light scattering: two-pion contributions.  JHEP, 04:161, 2017.
  * [20] Martin Hoferichter, Bai-Long Hoid, Bastian Kubis, Stefan Leupold, and Sebastian P. Schneider.  Dispersion relation for hadronic light-by-light scattering: pion pole.  JHEP, 10:141, 2018.
  * [21] Antoine Gérardin, Harvey B. Meyer, and Andreas Nyffeler.  Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks.  Phys. Rev., D100(3):034520, 2019.
  * [22] Johan Bijnens, Nils Hermansson-Truedsson, and Antonio Rodríguez-Sánchez.  Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment.  Phys. Lett., B798:134994, 2019.
  * [23] Gilberto Colangelo, Franziska Hagelstein, Martin Hoferichter, Laetitia Laub, and Peter Stoffer.  Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models.  JHEP, 03:101, 2020.
  * [24] Vladyslav Pauk and Marc Vanderhaeghen.  Single meson contributions to the muon‘s anomalous magnetic moment.  Eur. Phys. J., C74(8):3008, 2014.
  * [25] Igor Danilkin and Marc Vanderhaeghen.  Light-by-light scattering sum rules in light of new data.  Phys. Rev., D95(1):014019, 2017.
  * [26] Friedrich Jegerlehner.  The Anomalous Magnetic Moment of the Muon.  Springer Tracts Mod. Phys., 274:pp.1–693, 2017.
  * [27] M. Knecht, S. Narison, A. Rabemananjara, and D. Rabetiarivony.  Scalar meson contributions to $a_{\mu}$ from hadronic light-by-light scattering.  Phys. Lett., B787:111–123, 2018.
  * [28] Gernot Eichmann, Christian S. Fischer, and Richard Williams.  Kaon-box contribution to the anomalous magnetic moment of the muon.  Phys. Rev. D, 101(5):054015, 2020.
  * [29] Pablo Roig and Pablo Sánchez-Puertas.  Axial-vector exchange contribution to the hadronic light-by-light piece of the muon anomalous magnetic moment.  Phys. Rev., D101(7):074019, 2020.
  * [30] Gilberto Colangelo, Martin Hoferichter, Andreas Nyffeler, Massimo Passera, and Peter Stoffer.  Remarks on higher-order hadronic corrections to the muon g$-$2\.  Phys. Lett. B, 735:90–91, 2014.
  * [31] Thomas Blum, Norman Christ, Masashi Hayakawa, Taku Izubuchi, Luchang Jin, Chulwoo Jung, and Christoph Lehner.  The hadronic light-by-light scattering contribution to the muon anomalous magnetic moment from lattice QCD.  Phys. Rev. Lett., 124(13):132002, 2020.
  * [32] Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, and Makiko Nio.  Complete Tenth-Order QED Contribution to the Muon g-2.  Phys. Rev. Lett., 109:111808, 2012.
  * [33] Tatsumi Aoyama, Toichiro Kinoshita, and Makiko Nio.  Theory of the Anomalous Magnetic Moment of the Electron.  Atoms, 7(1):28, 2019.
  * [34] Andrzej Czarnecki, William J. Marciano, and Arkady Vainshtein.  Refinements in electroweak contributions to the muon anomalous magnetic moment.  Phys. Rev. D, 67:073006, 2003.  [Erratum: Phys.Rev.D 73, 119901 (2006)].
  * [35] C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim.  The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement.  Phys. Rev. D, 88:053005, 2013.
  * [36] T. Aoyama et al.  The anomalous magnetic moment of the muon in the Standard Model.  Phys. Rept., 887:1–166, 2020.
  * [37] G. Colangelo et al.  Prospects for precise predictions of $a_{\mu}$ in the Standard Model.  3 2022.
  * [38] Johan Bijnens, Nils Hermansson-Truedsson, Laetitia Laub, and Antonio Rodríguez-Sánchez.  Short-distance HLbL contributions to the muon anomalous magnetic moment beyond perturbation theory.  JHEP, 10:203, 2020.
  * [39] En-Hung Chao, Antoine Gérardin, Jeremy R. Green, Renwick J. Hudspith, and Harvey B. Meyer.  Hadronic light-by-light contribution to $(g-2)_{\mu}$ from lattice QCD with SU(3) flavor symmetry.  Eur. Phys. J. C, 80(9):869, 2020.
  * [40] Gilberto Colangelo, Martin Hoferichter, and Peter Stoffer.  Constraints on the two-pion contribution to hadronic vacuum polarization.  Phys. Lett. B, 814:136073, 2021.
  * [41] Bai-Long Hoid, Martin Hoferichter, and Bastian Kubis.  Hadronic vacuum polarization and vector-meson resonance parameters from $e^{+}e^{-}\rightarrow\pi^{0}\gamma$.  Eur. Phys. J. C, 80(10):988, 2020.
  * [42] Pere Masjuan, Pablo Roig, and Pablo Sanchez-Puertas.  The interplay of transverse degrees of freedom and axial-vector mesons with short-distance constraints in g $-$ 2\.  J. Phys. G, 49(1):015002, 2022.
  * [43] Jan Lüdtke and Massimiliano Procura.  Effects of longitudinal short-distance constraints on the hadronic light-by-light contribution to the muon $\mathbf{g-2}$.  Eur. Phys. J. C, 80(12):1108, 2020.
  * [44] Johan Bijnens, Nils Hermansson-Truedsson, Laetitia Laub, and Antonio Rodríguez-Sánchez.  The two-loop perturbative correction to the $(g-2)_{\mu}$ HLbL at short distances.  JHEP, 04:240, 2021.
  * [45] Luigi Cappiello, Oscar Catà, and Giancarlo D’Ambrosio.  Scalar resonances in the hadronic light-by-light contribution to the muon (g-2).  Phys. Rev. D, 105(5):056020, 2022.
  * [46] En-Hung Chao, Renwick J. Hudspith, Antoine Gérardin, Jeremy R. Green, Harvey B. Meyer, and Konstantin Ottnad.  Hadronic light-by-light contribution to $(g-2)_{\mu}$ from lattice QCD: a complete calculation.  Eur. Phys. J. C, 81(7):651, 2021.
  * [47] Gilberto Colangelo, Martin Hoferichter, Bastian Kubis, Malwin Niehus, and Jacobo Ruiz de Elvira.  Chiral extrapolation of hadronic vacuum polarization.  Phys. Lett. B, 825:136852, 2022.
  * [48] Gilberto Colangelo, Franziska Hagelstein, Martin Hoferichter, Laetitia Laub, and Peter Stoffer.  Short-distance constraints for the longitudinal component of the hadronic light-by-light amplitude: an update.  Eur. Phys. J. C, 81(8):702, 2021.
  * [49] Igor Danilkin, Martin Hoferichter, and Peter Stoffer.  A dispersive estimate of scalar contributions to hadronic light-by-light scattering.  Phys. Lett. B, 820:136502, 2021.
  * [50] Martin Hoferichter and Thomas Teubner.  Mixed Leptonic and Hadronic Corrections to the Anomalous Magnetic Moment of the Muon.  Phys. Rev. Lett., 128(11):112002, 2022.
  * [51] Josef Leutgeb and Anton Rebhan.  Hadronic light-by-light contribution to the muon g-2 from holographic QCD with massive pions.  Phys. Rev. D, 104(9):094017, 2021.
  * [52] Ángel Miramontes, Adnan Bashir, Khépani Raya, and Pablo Roig.  Pion and Kaon box contribution to a$\mu$HLbL.  Phys. Rev. D, 105(7):074013, 2022.
  * [53] Marvin Zanke, Martin Hoferichter, and Bastian Kubis.  On the transition form factors of the axial-vector resonance $f_{1}(1285)$ and its decay into $e^{+}e^{-}$.  JHEP, 07:106, 2021.
  * [54] Volodymyr Biloshytskyi, En-Hung Chao, Antoine Gérardin, Jeremy R. Green, Franziska Hagelstein, Harvey B. Meyer, Julian Parrino, and Vladimir Pascalutsa.  Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization.  JHEP, 03:194, 2023.
  * [55] Gilberto Colangelo, Martin Hoferichter, Bastian Kubis, and Peter Stoffer.  Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization.  JHEP, 10:032, 2022.
  * [56] Johan Bijnens, Nils Hermansson-Truedsson, and Antonio Rodríguez-Sánchez.  Constraints on the hadronic light-by-light in the Melnikov-Vainshtein regime.  JHEP, 02:167, 2023.
  * [57] Diogo Boito, Maarten Golterman, Kim Maltman, and Santiago Peris.  Evaluation of the three-flavor quark-disconnected contribution to the muon anomalous magnetic moment from experimental data.  Phys. Rev. D, 105(9):093003, 2022.
  * [58] Dominik Stamen, Deepti Hariharan, Martin Hoferichter, Bastian Kubis, and Peter Stoffer.  Kaon electromagnetic form factors in dispersion theory.  Eur. Phys. J. C, 82(5):432, 2022.
  * [59] T. Blum et al.  An update of Euclidean windows of the hadronic vacuum polarization.  1 2023.
  * [60] Jan Lüdtke, Massimiliano Procura, and Peter Stoffer.  Dispersion relations for hadronic light-by-light scattering in triangle kinematics.  JHEP, 04:125, 2023.
  * [61] M. Davier, D. Díaz-Calderón, B. Malaescu, A. Pich, A. Rodríguez-Sánchez, and Z. Zhang.  The Euclidean Adler function and its interplay with $\Delta{\alpha}_{\textrm{QED}}^{\textrm{had}}$ and $\alpha$s.  JHEP, 04:067, 2023.
  * [62] Shi-Jia Wang, Zhen Fang, and Ling-Yun Dai.  Two body final states production in electron-positron annihilation and their contributions to $(g-2)_{\mu}$.  2 2023.
  * [63] Sz. Borsanyi et al.  Leading hadronic contribution to the muon magnetic moment from lattice QCD.  Nature, 593(7857):51–55, 2021.
  * [64] Marco Cè et al.  Window observable for the hadronic vacuum polarization contribution to the muon g-2 from lattice QCD.  Phys. Rev. D, 106(11):114502, 2022.
  * [65] C. Alexandrou et al.  Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions.  6 2022.
  * [66] G. Colangelo, A. X. El-Khadra, M. Hoferichter, A. Keshavarzi, C. Lehner, P. Stoffer, and T. Teubner.  Data-driven evaluations of Euclidean windows to scrutinize hadronic vacuum polarization.  Phys. Lett. B, 833:137313, 2022.
  * [67] F. V. Ignatov et al.  Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section from threshold to 1.2 GeV with the CMD-3 detector.  2 2023.
  * [68] R. R. Akhmetshin et al.  Reanalysis of hadronic cross-section measurements at CMD-2.  Phys. Lett. B, 578:285–289, 2004.
  * [69] V. M. Aul’chenko et al.  Measurement of the pion form-factor in the range 1.04-GeV to 1.38-GeV with the CMD-2 detector.  JETP Lett., 82:743–747, 2005.
  * [70] V. M. Aul’chenko et al.  Measurement of the e+ e- —$>$ pi+ pi- cross section with the CMD-2 detector in the 370 - 520-MeV c.m. energy range.  JETP Lett., 84:413–417, 2006.
  * [71] R. R. Akhmetshin et al.  High-statistics measurement of the pion form factor in the rho-meson energy range with the CMD-2 detector.  Phys. Lett. B, 648:28–38, 2007.
  * [72] M. N. Achasov et al.  Update of the e+ e- —$>$ pi+ pi- cross-section measured by SND detector in the energy region 400-MeV $<$ s**(1/2) $<$ file1000-MeV.  J. Exp. Theor. Phys., 103:380–384, 2006.
  * [73] M. N. Achasov et al.  Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ process cross section with the SND detector at the VEPP-2000 collider in the energy region $0.525<\sqrt{s}<0.883$ GeV.  JHEP, 01:113, 2021.
  * [74] F. Ambrosino et al.  Measurement of $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-}\gamma(\gamma)$ and the dipion contribution to the muon anomaly with the KLOE detector.  Phys. Lett. B, 670:285–291, 2009.
  * [75] F. Ambrosino et al.  Measurement of $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ from threshold to 0.85 GeV2 using Initial State Radiation with the KLOE detector.  Phys. Lett. B, 700:102–110, 2011.
  * [76] D. Babusci et al.  Precision measurement of $\sigma(e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\gamma)/\sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\gamma)$ and determination of the $\pi^{+}\pi^{-}$ contribution to the muon anomaly with the KLOE detector.  Phys. Lett. B, 720:336–343, 2013.
  * [77] A. Anastasi et al.  Combination of KLOE $\sigma\big{(}e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\gamma(\gamma)\big{)}$ measurements and determination of $a_{\mu}^{\pi^{+}\pi^{-}}$ in the energy range $0.10<s<0.95$ GeV2.  JHEP, 03:173, 2018.
  * [78] J. P. Lees et al.  Precise Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}(\gamma)$ Cross Section with the Initial-State Radiation Method at BABAR.  Phys. Rev. D, 86:032013, 2012.
  * [79] M. Ablikim et al.  Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section between 600 and 900 MeV using initial state radiation.  Phys. Lett. B, 753:629–638, 2016.  [Erratum: Phys.Lett.B 812, 135982 (2021)].
  * [80] T. K. Pedlar et al.  Precision measurements of the timelike electromagnetic form-factors of pion, kaon, and proton.  Phys. Rev. Lett., 95:261803, 2005.
  * [81] S. Actis et al.  Quest for precision in hadronic cross sections at low energy: Monte Carlo tools vs. experimental data.  Eur. Phys. J. C, 66:585–686, 2010.
  * [82] Francisco Campanario, Henryk Czyż, Janusz Gluza, Tomasz Jeliński, Germán Rodrigo, Szymon Tracz, and Dmitry Zhuridov.  Standard model radiative corrections in the pion form factor measurements do not explain the $a_{\mu}$ anomaly.  Phys. Rev. D, 100(7):076004, 2019.
  * [83] Fedor Ignatov and Roman N. Lee.  Charge asymmetry in e+e$-$ → $\pi$+$\pi$$-$ process.  Phys. Lett. B, 833:137283, 2022.
  * [84] Gilberto Colangelo, Martin Hoferichter, Joachim Monnard, and Jacobo Ruiz de Elvira.  Radiative corrections to the forward-backward asymmetry in e+e- → $\pi$+$\pi$-.  JHEP, 08:295, 2022.
  * [85] Ricard Alemany, Michel Davier, and Andreas Hocker.  Improved determination of the hadronic contribution to the muon (g-2) and to alpha (M(z)) using new data from hadronic tau decays.  Eur. Phys. J. C, 2:123–135, 1998.
  * [86] V. Cirigliano, G. Ecker, and H. Neufeld.  Isospin violation and the magnetic moment of the muon.  Phys. Lett. B, 513:361–370, 2001.
  * [87] V. Cirigliano, G. Ecker, and H. Neufeld.  Radiative tau decay and the magnetic moment of the muon.  JHEP, 08:002, 2002.
  * [88] M. Davier, A. Hoecker, G. Lopez Castro, B. Malaescu, X. H. Mo, G. Toledo Sanchez, P. Wang, C. Z. Yuan, and Z. Zhang.  The Discrepancy Between tau and e+e- Spectral Functions Revisited and the Consequences for the Muon Magnetic Anomaly.  Eur. Phys. J. C, 66:127–136, 2010.
  * [89] Michel Davier, Andreas Hoecker, Bogdan Malaescu, and Zhiqing Zhang.  Reevaluation of the Hadronic Contributions to the Muon g-2 and to alpha(MZ).  Eur. Phys. J. C, 71:1515, 2011.  [Erratum: Eur.Phys.J.C 72, 1874 (2012)].
  * [90] Michel Davier, Andreas Höcker, Bogdan Malaescu, Chang-Zheng Yuan, and Zhiqing Zhang.  Update of the ALEPH non-strange spectral functions from hadronic $\tau$ decays.  Eur. Phys. J. C, 74(3):2803, 2014.
  * [91] J. A. Miranda and P. Roig.  New $\tau$-based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment.  Phys. Rev. D, 102:114017, 2020.
  * [92] J. A. Miranda and P. Roig.  Effective-field theory analysis of the $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decays.  JHEP, 11:038, 2018.
  * [93] Vincenzo Cirigliano, Adam Falkowski, Martín González-Alonso, and Antonio Rodríguez-Sánchez.  Hadronic $\tau$ Decays as New Physics Probes in the LHC Era.  Phys. Rev. Lett., 122(22):221801, 2019.
  * [94] Sergi Gonzàlez-Solís, Alejandro Miranda, Javier Rendón, and Pablo Roig.  Exclusive hadronic tau decays as probes of non-SM interactions.  Phys. Lett. B, 804:135371, 2020.
  * [95] Vincenzo Cirigliano, David Díaz-Calderón, Adam Falkowski, Martín González-Alonso, and Antonio Rodríguez-Sánchez.  Semileptonic tau decays beyond the Standard Model.  JHEP, 04:152, 2022.
  * [96] Mattia Bruno, Taku Izubuchi, Christoph Lehner, and Aaron Meyer.  On isospin breaking in $\tau$ decays for $(g-2)_{\mu}$ from Lattice QCD.  PoS, LATTICE2018:135, 2018.
  * [97] Claude Bouchiat and Louis Michel.  La résonance dans la diffusion méson $\pi$— méson $\pi$ et le moment magnétique anormal du méson $\mu$.  J. Phys. Radium, 22(2):121–121, 1961.
  * [98] Loyal Durand.  Pionic Contributions to the Magnetic Moment of the Muon.  Phys. Rev., 128:441–448, 1962.
  * [99] Stanley J. Brodsky and Eduardo De Rafael.  SUGGESTED BOSON - LEPTON PAIR COUPLINGS AND THE ANOMALOUS MAGNETIC MOMENT OF THE MUON.  Phys. Rev., 168:1620–1622, 1968.
  * [100] M. Gourdin and E. De Rafael.  Hadronic contributions to the muon g-factor.  Nucl. Phys. B, 10:667–674, 1969.
  * [101] S. Eidelman and F. Jegerlehner.  Hadronic contributions to g-2 of the leptons and to the effective fine structure constant alpha (M(z)**2).  Z. Phys. C, 67:585–602, 1995.
  * [102] Christoph Lehner and Aaron S. Meyer.  Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio.  Phys. Rev. D, 101:074515, 2020.
  * [103] A. Sirlin.  Radiative corrections to g(v)/g(mu) in simple extensions of the su(2) x u(1) gauge model.  Nucl. Phys. B, 71:29–51, 1974.
  * [104] A. Sirlin.  Current Algebra Formulation of Radiative Corrections in Gauge Theories and the Universality of the Weak Interactions.  Rev. Mod. Phys., 50:573, 1978.  [Erratum: Rev.Mod.Phys. 50, 905 (1978)].
  * [105] A. Sirlin.  Large m(W), m(Z) Behavior of the O(alpha) Corrections to Semileptonic Processes Mediated by W.  Nucl. Phys. B, 196:83–92, 1982.
  * [106] W. J. Marciano and A. Sirlin.  Radiative Corrections to beta Decay and the Possibility of a Fourth Generation.  Phys. Rev. Lett., 56:22, 1986.
  * [107] W. J. Marciano and A. Sirlin.  Electroweak Radiative Corrections to tau Decay.  Phys. Rev. Lett., 61:1815–1818, 1988.
  * [108] William J. Marciano and A. Sirlin.  Radiative corrections to pi(lepton 2) decays.  Phys. Rev. Lett., 71:3629–3632, 1993.
  * [109] Eric Braaten and Chong-Sheng Li.  Electroweak radiative corrections to the semihadronic decay rate of the tau lepton.  Phys. Rev. D, 42:3888–3891, 1990.
  * [110] Jens Erler.  Electroweak radiative corrections to semileptonic tau decays.  Rev. Mex. Fis., 50:200–202, 2004.
  * [111] Julian S. Schwinger.  PARTICLES, SOURCES, AND FIELDS. VOL. 3.  1989\.
  * [112] Manuel Drees and Ken-ichi Hikasa.  Scalar top production in e+ e- annihilation.  Phys. Lett. B, 252:127–134, 1990.
  * [113] G. Ecker, J. Gasser, A. Pich, and E. de Rafael.  The Role of Resonances in Chiral Perturbation Theory.  Nucl. Phys. B, 321:311–342, 1989.
  * [114] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, and E. de Rafael.  Chiral Lagrangians for Massive Spin 1 Fields.  Phys. Lett. B, 223:425–432, 1989.
  * [115] V. Cirigliano, G. Ecker, M. Eidemuller, Roland Kaiser, A. Pich, and J. Portoles.  Towards a consistent estimate of the chiral low-energy constants.  Nucl. Phys. B, 753:139–177, 2006.
  * [116] Karol Kampf and Jiri Novotny.  Resonance saturation in the odd-intrinsic parity sector of low-energy QCD.  Phys. Rev. D, 84:014036, 2011.
  * [117] P. Roig, A. Guevara, and G. López Castro.  $VV^{\prime}P$ form factors in resonance chiral theory and the $\pi-\eta-\eta^{\prime}$ light-by-light contribution to the muon $g-2$.  Phys. Rev. D, 89(7):073016, 2014.
  * [118] A. Guevara, P. Roig, and J.J. Sanz-Cillero.  Pseudoscalar pole light-by-light contributions to the muon $(g-2)$ in Resonance Chiral Theory.  JHEP, 06:160, 2018.
  * [119] Wen Qin, Ling-Yun Dai, and Jorge Portoles.  Two and three pseudoscalar production in e+e- annihilation and their contributions to (g $-$ 2)μ.  JHEP, 03:092, 2021.
  * [120] J. Gasser and H. Leutwyler.  Chiral Perturbation Theory to One Loop.  Annals Phys., 158:142, 1984.
  * [121] J. Gasser and H. Leutwyler.  Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark.  Nucl. Phys. B, 250:465–516, 1985.
  * [122] H. Leutwyler.  On the foundations of chiral perturbation theory.  Annals Phys., 235:165–203, 1994.
  * [123] F. Flores-Baez, A. Flores-Tlalpa, G. Lopez Castro, and G. Toledo Sanchez.  Long-distance radiative corrections to the di-pion tau lepton decay.  Phys. Rev. D, 74:071301, 2006.
  * [124] A. Flores-Tlalpa, F. Flores-Baez, G. Lopez Castro, and G. Toledo Sanchez.  Model-dependent radiative corrections to $\tau^{-}\to\pi^{-}\pi^{0}\nu$ revisited.  Nucl. Phys. B Proc. Suppl., 169:250–254, 2007.
  * [125] J. J. Sakurai.  Theory of strong interactions.  Annals Phys., 11:1–48, 1960.
  * [126] Rafel Escribano, Alejandro Miranda, and Pablo Roig.  Radiative corrections to the $\tau^{-}\to(P_{1}P_{2})^{-}\nu_{\tau}$ ($P_{1,2}=\pi,K$) decays.  3 2023.
  * [127] Res Urech.  rho0 - omega mixing in chiral perturbation theory.  Phys. Lett. B, 355:308–312, 1995.
  * [128] S. Schael et al.  Branching ratios and spectral functions of tau decays: Final ALEPH measurements and physics implications.  Phys. Rept., 421:191–284, 2005.
  * [129] M. Fujikawa et al.  High-Statistics Study of the tau- —$>$ pi- pi0 nu(tau) Decay.  Phys. Rev. D, 78:072006, 2008.
  * [130] S. Anderson et al.  Hadronic structure in the decay tau- —$>$ pi- pi0 neutrino(tau).  Phys. Rev. D, 61:112002, 2000.
  * [131] K. Ackerstaff et al.  Measurement of the strong coupling constant alpha(s) and the vector and axial vector spectral functions in hadronic tau decays.  Eur. Phys. J. C, 7:571–593, 1999.
  * [132] Davide Giusti and Silvano Simula.  Window contributions to the muon hadronic vacuum polarization with twisted-mass fermions.  PoS, LATTICE2021:189, 2022.