# Study of pion vector form factor and its contribution to the muon $(g-2)$

Jing-Yu Yi  School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China
Zhong-Yu Wang  School of Physics and Electronics, Hunan Key Laboratory of
Nanophotonics and Devices, Central South University, Changsha 410083, China
C. W. Xiao<EMAIL_ADDRESS>School of Physics and Electronics, Hunan Key
Laboratory of Nanophotonics and Devices, Central South University, Changsha
410083, China

###### Abstract

In the present work, we investigate several theoretical models of the pion
vector form factor and aim at getting the best fit for the two-pion cross
sections to reduce the uncertainties of the calculation of two-pion
contribution to the muon anomalous magnetic moment. Combined with a polynomial
description to the pion vector form factor, we obtain the best fit from the
Gounaris-Sakurai (or Kühn-Santamaria) model for the experimental data up to 1
GeV. By product, the branching ratio of $\omega\to\pi\pi$ can be extracted as
$\text{Br}(\omega\to\pi\pi)=(1.52\pm 0.06)\%$, which is consistent with the
one of Particle Data Group. With the best fit of the data, we obtain the muon
anomalous magnetic moment from two-pion contribution as
$a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-}\leq 1\ \text{GeV})=(497.76\pm
3.15)\times 10^{-10}$. Our results are consistent with the other works.

##  I Introduction

The muon magnetic moment is an important and long historical issue in particle
physics Garwin:1957hc ; Garwin:1960zz ; Terazawa:1968jh ; Terazawa:1968mx ;
Terazawa:1969ih ; Kinoshita:1970js ; Terazawa:2018pdc . Using Dirac theory,
the gyromagnetic ratio $g_{\mu}$ is predicted as $g_{\mu}=2$ for the
structureless and spin $\frac{1}{2}$ muon. In fact, due to the developments of
the experiments and theories, it is found that $g_{\mu}$ is slightly greater
than 2, which can be referred to as the anomalous magnetic moment
$a_{\mu}=(g_{\mu}-2)/2$. As already known, it is Schwinger’s value of
$a_{\mu}=\alpha/(2\pi)\simeq 0.00116$ from one loop QED radiative corrections,
which is universal for all leptons. More discussions for that can be found in
the reviews of Jegerlehner:2009ry ; Miller:2012opa . The anomalous magnetic
moment of the muon $a_{\mu}$ is experimentally and theoretically known to very
high accuracy. Its measurement at Brookhaven National Laboratory was reported
as Bennett:2006fi ; Mohr:2012tt

$a_{\mu}^{\mathrm{exp}}=(11659209.1\pm 5.4\pm 3.3)\times 10^{-10},$ (1)

where the errors were given by statistical and systematic uncertainties,
respectively. The Standard Model (SM) prediction was given by Davier:2019can

$a_{\mu}^{\mathrm{SM}}=(11659183.1\pm 4.0\pm 2.6\pm 0.1)\times 10^{-10}.$ (2)

And thus, the difference between the experiment and theory is

$\Delta a_{\mu}=a_{\mu}^{\exp}-a_{\mu}^{\mathrm{SM}}=(26.0\pm 7.9)\times
10^{-10},$ (3)

where one can see that there is a discrepancy of about 3.3$\sigma$ between the
measured value and the full Standard Model prediction. But, this discrepancy
has been updated with the recent results both in theory calculations and
experimental measurements. The latest measurement of the anomalous magnetic
moment of the muon was performed at Fermilab National Accelerator Laboratory
Muon $g-2$ Experiment Muong-2:2021hlp , gotten

$a_{\mu}^{\mathrm{exp}}=(11659204.0\pm 5.4)\times 10^{-10}.$ (4)

Combined with the measurement at Brookhaven above, one can easily get the
experimental average of

$a_{\mu}^{\mathrm{exp}}=(11659206.1\pm 4.1)\times 10^{-10}.$ (5)

Note that the reported results of Ref. Bennett:2006fi  were used for this
average. The latest SM prediction was given by the recent review of the White
Paper (WP) Aoyama:2020ynm

$a_{\mu}^{\mathrm{SM}}=(11659181.0\pm 4.3)\times 10^{-10}.$ (6)

Therefore, the difference between the experiment and the theory is updated as

$\Delta a_{\mu}^{\mathrm{new}}=a_{\mu}^{\exp}-a_{\mu}^{\mathrm{SM}}=(25.1\pm
5.9)\times 10^{-10},$ (7)

which leads to a discrepancy of $4.2\sigma$. This discrepancy possibly hints
the new physics beyond SM and draws much theoretical attention Chiang:2021pma
; CarcamoHernandez:2021iat ; Arcadi:2021cwg ; Zhu:2021vlz ; Endo:2021zal ;
Han:2021gfu ; Das:2021zea ; Ge:2021cjz ; Baum:2021qzx ; Zhang:2021gun ;
Ahmed:2021htr ; Cao:2021tuh ; Crivellin:2021rbq ; Athron:2021iuf ; Yin:2020afe
; Yin:2021yqy ; Yin:2021mls . More discussions can be referred to Refs.
Athron:2021iuf ; Terazawa  for new physics beyond SM and the latest review of
Keshavarzi:2021eqa  for recent status, and references therein.

So it is crucial to know the prediction of the SM as precisely as possible.
The prediction of the SM $a_{\mu}^{\mathrm{SM}}$ can be divided into several
different contributions Aoyama:2020ynm ; Zyla:2020zbs ,

$a_{\mu}^{\mathrm{SM}}=a_{\mu}^{\mathrm{QED}}+a_{\mu}^{\mathrm{had}}+a_{\mu}^{\mathrm{weak}},$
(8)

where $a_{\mu}^{\mathrm{QED}}$ is the pure electromagnetic contribution,
$a_{\mu}^{\mathrm{had}}$ is the hadronic contribution, and
$a_{\mu}^{\text{weak}}$ accounts for the electroweak corrections due to the
exchange of the weak interacting bosons. At present, $a_{\mu}^{\mathrm{QED}}$
was calculated with high accuracy up to five-loop order Aoyama:2012wj ;
Aoyama:2012wk ; Baikov:2013ula ; Volkov:2019phy , and $a_{\mu}^{\text{weak }}$
was also done up to two-loop order Czarnecki:2002nt ; Gnendiger:2013pva ;
Ishikawa:2018rlv , which were given by

$\displaystyle a_{\mu}^{\mathrm{QED}}$ $\displaystyle=(11658471.8931\pm
0.0104)\times 10^{-10},$ (9) $\displaystyle a_{\mu}^{\text{weak }}$
$\displaystyle=(15.36\pm 0.10)\times 10^{-10},$ (10)

where one can see the reviews of Aoyama:2020ynm ; Keshavarzi:2021eqa ;
Zyla:2020zbs  for more details. Thus, the large uncertainties of
$a_{\mu}^{\mathrm{SM}}$ mainly come from the hadronic part of
$a_{\mu}^{\mathrm{had}}$ due to the confinement and non-perturbative
properties in the low energy region, which can be divided into two parts, one
part from hadronic light-by-light (HLbL) scattering
($a_{\mu}^{\mathrm{HLbL}}$) and the other one from the hadronic vacuum
polarization (HVP) contribution ($a_{\mu}^{\mathrm{HVP}}$).

For the HLbL contribution Melnikov:2003xd ; Masjuan:2017tvw ;
Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ,
the phenomenological estimation was given by the WP Aoyama:2020ynm

$a_{\mu}^{\mathrm{HLbL}}=(9.2\pm 1.9)\times 10^{-10},$ (11)

which was consistent with Lattice QCD calculations
$a_{\mu}^{\mathrm{HLbL}}=(7.87\pm 3.06\pm 1.77)\times 10^{-10}$ Blum:2019ugy
and $a_{\mu}^{\mathrm{HLbL}}=(10.68\pm 1.47)\times 10^{-10}$ Chao:2021tvp
within the uncertainties. Recently (after the WP), with a model-independent
method, the effects of short-distance constraints on the
$a_{\mu}^{\mathrm{HLbL}}$ were evaluated in Ref. Ludtke:2020moa  by
considering the known states below 1 GeV, which obtained
$a_{\mu}^{\mathrm{HLbL}}=(0.91\pm 0.50)\times 10^{-10}$ for the contribution
of pseudoscalar ground-states and $a_{\mu}^{\mathrm{HLbL}}=(0.26\pm
0.15)\times 10^{-10}$ for the one of isovector parts. The short-distance
expansion for the four-point function was derived in Ref. Bijnens:2020xnl  via
a systematic operator product expansion, where it was found that the
contribution of the massless quark loop in leading order to the
$a_{\mu}^{\mathrm{HLbL}}$ was dominant and the ones from higher order were
estimated to be small. In a further work of Bijnens:2021jqo , the perturbative
QCD correction to the massless quark loop was computed, and they found that
the correction up to two loops was a quite small contribution to the
$a_{\mu}^{\mathrm{HLbL}}$. Ref. Masjuan:2020jsf  discussed the short-distance
constraints to the calculation of $a_{\mu}^{\mathrm{HLbL}}$ from the
contribution of the axial-vector mesons. Employing resonance chiral theory,
the axial-vector contribution to the $a_{\mu}^{\mathrm{HLbL}}$ was discussed
in Ref. Roig:2019reh  with a small result of
$a_{\mu}^{\mathrm{HLbL;\;A}}=(0.8^{+3.5}_{-0.8})\times 10^{-11}$. On the other
hand, with a warped five-dimensional model, Ref. Cappiello:2019hwh  considered
the contributions of pseudoscalar and axial-vector resonances to the
$a_{\mu}^{\mathrm{HLbL}}$ and obtained a value of
$a_{\mu}^{\mathrm{HLbL;\;P+A}}=(12.5\pm 1.5)\times 10^{-10}$ with much larger
role for the axial-vector contribution. In Ref. Zanke:2021wiq , the transition
form factors of the resonance $f_{1}(1285)$ were analyzed in detail with the
framework of vector meson dominance due to its contribution to the HLbL
scattering, see more details in Ref. Leutgeb:2019gbz . Using dispersion
relations, Ref. Danilkin:2021icn  considered the contribution of scalar
resonances to the $a_{\mu}^{\mathrm{HLbL}}$ and obtained an estimate of
$a_{\mu}^{\mathrm{HLbL}}[\mathrm{scalar}]=(-0.9\pm 0.1)\times 10^{-10}$.
Moreover, several models for the short-distance constraints to the calculation
of $a_{\mu}^{\mathrm{HLbL}}$ were investigated detailedly in Ref.
Colangelo:2021nkr , where the perturbative QCD correction was also taken into
account and the result of the perturbative corrections to the operator product
expansion was updated as $a_{\mu}^{\mathrm{HLbL}}=(1.3\pm 0.5)\times
10^{-10}$.

On the other hand, at the current status, the total HVP contribution was
estimated as Aoyama:2020ynm ; Keshavarzi:2021eqa

$a_{\mu}^{\mathrm{HVP}}=(684.5\pm 4.0)\times 10^{-10},$ (12)

which included the leading order (LO) Davier:2019can ; Keshavarzi:2019abf ,
next-to-leading order Keshavarzi:2019abf  and next-next-to-leading order
Kurz:2014wya  contributions. In fact, the dominant one is the LO part, given
by the data-driven calculations Aoyama:2020ynm ; Davier:2017zfy ;
Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg ; Davier:2019can
; Keshavarzi:2019abf

$a_{\mu\quad\mathrm{DaDr}}^{\mathrm{HVP,\;LO}}=(693.1\pm 4.0)\times 10^{-10},$
(13)

which is overlapped with the lattice world average for the total LO HVP
contribution Aoyama:2020ynm ,

$a_{\mu\quad\mathrm{lat}}^{\mathrm{HVP,\;LO}}=(711.6\pm 18.4)\times 10^{-10}.$
(14)

But, the recent calculation of lattice QCD for the LO HVP contribution was
reported as Borsanyi:2020mff

$a_{\mu\quad\mathrm{lat}}^{\mathrm{HVP,\;LO}}=(707.5\pm 5.5)\times 10^{-10},$
(15)

with high accuracy, which is a bit smaller than the one obtained in Ref.
Lehner:2020crt , $(714\pm 27\pm 13)\times 10^{-10}$. These new lattice results
lead the SM prediction to be in agreement with the current experimental
measurement and the new physics to be questionable.

One thing should be mentioned that, the LO part from the data-driven
calculations in Eq. (13) is only taken the $e^{+}e^{-}$ annihilation data into
account, since the one from $\tau$ decay data is not precise enough at
present. With the results of Ref. Davier:2013sfa , there was still
$2.2\;\sigma$ discrepancy between the $e^{+}e^{-}$ based and $\tau$ based
results. Recently, with resonance chiral theory supplemented by dispersion
relations, Ref. Gonzalez-Solis:2019iod  studied the pion vector form factor
using the experimental data of $\tau$ decay from Belle and recent BaBar
measurements. Based on these results, the $a_{\mu}^{\mathrm{HVP,\;LO}}$ was
extracted from the $\tau$ decay data of $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$
in a further work of Miranda:2020wdg , obtained
$a_{\mu\quad\tau\;\mathrm{Dat}}^{\mathrm{HVP,\;LO}}=(705.7^{+4.0}_{-4.1})\times
10^{-10}$ and
$a_{\mu\quad\tau\;\mathrm{Dat}}^{\mathrm{HVP,\;LO}}=(700.7^{+6.1}_{-5.2})\times
10^{-10}$ for different strategies. Using a framework of the hidden local
symmetry model combined with appropriate symmetry breaking mechanisms, both
the $e^{+}e^{-}$ annihilation and $\tau$ decay data were analysed in Ref.
Benayoun:2019zwh , where a value of $a_{\mu}^{\mathrm{HVP,\;LO}}=(686.65\pm
3.01)\times 10^{-10}$ with the uncertainties of $\rho-\gamma$ mixing was
reported and a further result was updated in the recent work of
Benayoun:2021ody .

As one can see that, for the SM prediction $a_{\mu}^{\mathrm{SM}}$, the
hadronic part $a_{\mu}^{\mathrm{had}}$ still has large uncertainties,
especially for the one $a_{\mu}^{\mathrm{HVP}}$. Recently, to solve the
inverse problem to the dispersion relation, a value for the HVP contribution
was obtained as $a_{\mu}^{\mathrm{HVP}}=(641^{+65}_{-63})\times 10^{-10}$ in
Ref. Li:2020fiz . Based on the chiral perturbation theory, Ref. Aubin:2020scy
discussed that the finite-volume corrections to $a_{\mu}^{\mathrm{HVP}}$ could
be precisely evaluated, where once all low-energy constants were already
known. Ref. Malaescu:2020zuc  investigated the potential impact on the
electroweak fits of the tensions between the current determinations of the HVP
contributions to the $a_{\mu}$, based on either phenomenological calculations
or Lattice QCD calculations. Note that, taking into account the measurement of
the Higgs mass, the impact of HVP on $a_{\mu}^{\mathrm{SM}}$ and the global
fits to electroweak precision data was studied in Ref. Crivellin:2020zul ,
where some options for physics beyond SM were discussed.

To reduce the uncertainties of the part $a_{\mu}^{\mathrm{HVP}}$, indeed, the
accurate evaluations must rely on the corresponding cross section measurements
for the normal calculations of data-driven. Combined with the Effective
Lagrangian, an iterated global fit scheme was adopted in Ref. Benayoun:2015gxa
to reduce the uncertainties for the description of the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ annihilation data up to 1.05 GeV. With a
dispersive representation of the pion vector form factor (PVFF), different
constraints on the two-pion contribution were examined for its effects on the
$a_{\mu}^{\mathrm{HVP}}$ in Ref. Colangelo:2020lcg , where a value of
$a_{\mu}^{\mathrm{HVP}}(\pi\pi\leq 1\text{GeV})=497.0(1.4)\times 10^{-10}$ was
gotten in one case of their fits. Using a parametrization-free formalism based
on analyticity and unitarity, the PVFF and its contribution to the
$a_{\mu}^{\mathrm{HVP}}$ were investigated in Refs. Ananthanarayan:2016mns ;
Ananthanarayan:2018nyx ; Ananthanarayan:2020vum  for the energy range around
the $\rho(770)$ resonance. As already known, about $73\%$ of the LO hadronic
contribution and about $60\%$ of the total uncertainty are given by
$e^{+}e^{-}$ annihilated to the $\pi^{+}\pi^{-}(\gamma)$ final states, which
are dominated by the $\rho(770)$ resonance. Therefore, it is important to
study the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ annihilation, which always relates to
the PVFF. Thus, in the present work, we study several theoretical models of
the PVFF, and aim at finding out the best fit of the $\pi^{+}\pi^{-}$
scattering cross sections. In the next section, we first introduce the
calculation of $a_{\mu}^{\mathrm{HVP,\;LO}}$ with data-driven briefly.
Following, we discuss several phenomenological models of the PVFF, combined
with a polynomial description, or equivalently how to take into account the
contribution of the $\rho(770)$ resonance. Then, we obtain the results from
fitting the PVFF data of the collaborations Orsay, DM1, OLYA, CMD1, CMD2,
BABAR, BESIII, KLOE, SND, and so on. And thus, we perform a calculation of
$a_{\mu}^{\mathrm{HVP,\;LO}}$ up to 1 GeV. At the end, it is our conclusion.

##  II Muon ($g-2$) calculation with dispersion relation

As we discussed above, the theoretical prediction of $a_{\mu}^{\mathrm{SM}}$
has large uncertainties from the parts of hadronic contribution
$a_{\mu}^{\mathrm{had}}$. Due to the confinement, the non-perturbative
properties become dominant in the low energy region, where the quarks are
confined inside hadrons. Therefore, perturbative QCD fails to evaluate the
hadronic (quark and gluon) loop contributions to the $a_{\mu}^{\mathrm{SM}}$
precisely. In principle, one can do the calculation of
$a_{\mu}^{\mathrm{had}}$ from first principle calculation in lattice QCD. But,
most of the evaluations in lattice QCD are still not precise enough, except
for the recent one of Borsanyi:2020mff . Alternatively, the HVP contributions
$a_{\mu}^{\mathrm{HVP}}$ can be calculated with the data-driven approach,
which uses the dispersion relation together with the optical theorem and
experimental data. Thus, the LO HVP contribution to the
$a_{\mu}^{\mathrm{SM}}$ can be calculated via a dispersion relation using the
measured cross sections of $e^{+}e^{-}\rightarrow$ hadrons Gourdin:1969dm

$a_{\mu}^{\mathrm{HVP,\;LO}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}s}{s}\operatorname{Im}\Pi^{(\mathrm{H})}(s)K(s),$
(16)

where the kernel function is given by

$K(s)=\left(\frac{\alpha}{\pi}\right)\left\\{\frac{1}{2}x^{2}\left(2-x^{2}\right)+(1+x)^{2}\left(1+x^{2}\right)\frac{\ln(1+x)-x+\frac{1}{2}x^{2}}{x^{2}}+\frac{1+x}{1-x}x^{2}\ln
x\right\\},$ (17)

with the definitions

$\beta_{\mu}=\sqrt{1-\left(4m_{\mu}^{2}/s\right)}\text{ ,
}x=\left(1-\beta_{\mu}\right)/\left(1+\beta_{\mu}\right),$ (18)

and the electromagnetic coupling taking as $\alpha=e^{2}/(4\pi)\approx
1/137.036$ from Particle Data Group (PDG) Zyla:2020zbs . Note that $s$ is the
total energy of two-body system, $s\equiv(p_{1}+p_{2})^{2}$. With the optical
theorem, the imaginary part of the vacuum polarization amplitude
$\operatorname{Im}\mathrm{\Pi}^{(\mathrm{H})}(s)$ can be expressed in terms of
the total cross section of the electron-positron annihilation into hadrons

$\sigma_{\text{tot}}\left(\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow\text{hadrons
}\right)=\frac{4\pi^{2}\alpha}{s}\frac{1}{\pi}\operatorname{Im}\Pi^{(\mathrm{H})}(s).$
(19)

Thus, one can deduce

$a_{\mu}^{\mathrm{HVP,\;LO}}=\frac{1}{4\pi^{2}\alpha}\int_{m_{\pi}^{2}}^{\infty}\mathrm{d}s\;\sigma_{\text{tot}}\left(\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow\text{hadrons
}\right)K(s),$ (20)

which uses the measured bare cross sections for the annihilation
$\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow$ hadrons as inputs, and where the
lower limit of the dispersion integral is in fact the $\pi^{0}\,\gamma$ cut.
One should keep in mind that the experimentally measured cross sections are
the dressed one, where the bare cross sections can be corrected by the running
of the coupling constant $\alpha(s)$. In fact, this correction has always been
done in the experimental data reported. Thus, the corresponding cross section
measurements play a key role in the accurate evaluation of
$a_{\mu}^{\mathrm{had,\;LO}}$.

Note that, the kernel function $K(s)$ decreases monotonically with increasing
$s$, so it gives strong weight to the low-energy part of the integral, where
about $73\%$ of the LO hadronic contributions are given by the
$\pi^{+}\pi^{-}(\gamma)$ final states, dominated by the $\rho(770)$ resonance.
In the present work, we focus on the energy region of about 1 GeV, which is
mainly contributed by the $\pi^{+}\pi^{-}(\gamma)$ final states. The total
cross section contributed by two-pion final states is given by 111In fact, the
two-pion cross section is inclusive of final state radiation effects and
exclusive of all vacuum polarization effects in the experimental measurements.

$\sigma_{\mathrm{tot}}\left(\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow\pi^{+}\pi^{-}\right)=\frac{1}{3}\pi\alpha^{2}\frac{1}{s}\beta_{\pi}^{3}(s)\mid
F_{\pi}(s)\mid^{2},$ (21)

where $F_{\pi}(s)$ is the PVFF and the pion phase is defined as
$\beta_{\pi}(s)=\sqrt{1-4m_{\pi}^{2}/s}$. Then, it is important to study the
model of PVFF, see the discussion in the next section. Thus, the two-pion
contribution to the anomalous magnetic moment of the muon can be written as

$a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-})=\frac{\alpha}{12\pi}\int_{4m_{\pi}^{2}}^{\infty}\frac{\mathrm{d}s}{s}\beta_{\pi}^{3}(s)\left|F_{\pi}(s)\right|^{2}K(s).$
(22)

##  III The model for the pion form factor

In the present work, we are interested in the experimental data at a centre-
of-mass energy below 1 GeV, which is around the energy region corresponding to
the $\rho$ resonance. For the cross section of
$\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow\pi^{+}\pi^{-}$, it can be associated
with the PVFF, see Eq. (21), which can be defined as

$\langle\pi^{+}(p^{\prime})\pi^{-}(p)|J_{\mu}(0)|0\rangle=(p^{\prime}-p)_{\mu}F_{\pi}(s),$
(23)

with $s=(p^{\prime}+p)^{2}$ and $J_{\mu}$ the vector-isovector current. For
our case of energy range below 1 GeV, in terms of the pion $P$-wave phase
shift $\delta_{11}(s)$, the PVFF fulfills the following discontinuity
condition,

$\text{Im}\;F_{\pi}(s)=F_{\pi}(s)\sin\delta_{11}(s)e^{-i\delta_{11}(s)}\Theta(s-4m_{\pi}^{2}).$
(24)

With a once subtracted dispersion relation, the solution of Eq. (24) can be
written into a general ansatz Roos:1975zf ; Lang:1975ge

$F_{\pi}(s)=P(s)\Omega(s),$ (25)

where $P(s)$ is a polynomial and $\Omega(s)$ is the Omnés function
Omnes:1958hv . Note that the solution for higher subtracted dispersion
relation can be referred to Refs. Pich:2001pj ; Hoferichter:2014vra ;
Isken:2017dkw  for more discussions and applications, where especially in Ref.
Pich:2001pj  the experimental data of the PVFF up to $\sqrt{s}\simeq 1.2$ GeV
was in a good description with two subtraction constants of thrice-subtracted
dispersion relation and the two-pion contribution to the
$a_{\mu}^{\mathrm{HVP,\;LO}}$ was evaluated. Furthermore, this ansatz was
extrapolated to the radiative decays of
$\eta^{(\prime)}\to\pi^{+}\pi^{-}\gamma$ Stollenwerk:2011zz ; Hanhart:2013vba
, where a linear polynomial was used,

$P(s)=1+\alpha s,$ (26)

with $\alpha$ a free parameter, determined from the data. Indeed, the linear
behaviour was clearly shown in the results of Refs. Stollenwerk:2011zz ;
Hanhart:2013vba  for the data of the PVFF below 1 GeV. A new parameterization
to the PVFF for a full energy range can be referred to Ref. Hanhart:2012wi ,
where the isospin violation mechanism was also considered, such as the mixing
effects of $\rho-\omega$ and $\omega-\phi$. Besides, the Omnés function is
given by

$\Omega(s)=\exp\left(\frac{s}{\pi}\int_{4m_{\pi}^{2}}^{\infty}\frac{ds^{\prime}}{s^{\prime}}\frac{\delta_{11}\left(s^{\prime}\right)}{s^{\prime}-s-i\epsilon}\right),$
(27)

where the phase shift ${\delta_{11}\left(s\right)}$ can be taken from the
Madrid model’s results GarciaMartin:2011cn . In Ref. GarciaMartin:2011cn , the
phase shift for $s^{1/2}\leq 2m_{K}$ fulfilled

$\displaystyle\cot\delta_{11}(s)$
$\displaystyle=\frac{s^{1/2}}{2k^{3}}\left(m_{\rho}^{2}-s\right)\left\\{\frac{2m_{\pi}^{3}}{m_{\rho}^{2}\sqrt{s}}+B_{0}+B_{1}w(s)\right\\},$
(28) $\displaystyle w(s)$
$\displaystyle=\frac{\sqrt{s}-\sqrt{s_{0}-s}}{\sqrt{s}+\sqrt{s_{0}-s}},\quad
s_{0}^{1/2}=1.05\mathrm{GeV},$

where the $\rho$ mass was fixed to ${m_{\rho}}=773.6$ MeV, the other masses
were taken as ${m_{\pi}}=139.57$ MeV, ${m_{K}}=496$ MeV, ${m_{\eta}}=547.51$
MeV, and the central-mass momentum in the two-pion final states
$k=\sqrt{s-4m_{\pi}^{2}}/2$. For $2m_{K}\leq s^{1/2}\leq 1420$ MeV, one can
have

$\delta_{11}(s)=\lambda_{0}+\lambda_{1}\left(\sqrt{s}/2m_{K}-1\right)+\lambda_{2}\left(\sqrt{s}/2m_{K}-1\right)^{2},$
(29)

where $\lambda_{0}$ is fixed from the value of $\delta_{11}(4m_{K}^{2})$
obtained from the low energy parametrization, so that the phase shift is
continuous. Besides, the parameters of $B_{0},B_{1},\lambda_{1},\lambda_{2}$
were determined with UFD set or CFD set in Ref. GarciaMartin:2011cn . For
higher energy, we choose the phase shifts close to $\pi$ smoothly, written as

$\delta_{11}(s)=\pi-\frac{a}{b-s},$ (30)

where the coefficients $a$ and $b$ are determined with the value of
$\delta_{11}(s_{0})$ ($\sqrt{s_{0}}=1420$ MeV for example, in fact, we take
$\sqrt{s_{0}}=1300$ MeV for the best results) to make it continuous, and also
keep the derivative at $s=s_{0}$.

In fact, as discussed above, the data for the cross section of two pions below
1 GeV is dominated by the contribution of the $\rho$ resonance 222Note that a
new expression for the PVFF was proposed in Ref. Achasov:2012bz , where the
contributions from the loops of $\pi^{+}\pi^{-}$ and $K\bar{K}$ and higher
$\rho$ resonances were considered, and which described the data well in the
range from -10 GeV to 1 GeV and was extrapolated to the energy up to 3 GeV in
a further work of Achasov:2013usa .. Indeed, the Omnés function $\Omega(s)$ in
Eq. (25) is mainly contributed by the pion p-wave phase shift, see Eq. (27).
In the present work, we investigate how to include the contribution of $\rho$
to get better description of the experimental data. Thus, we want to know the
effects of different models for the part of Omnés function. Note that the
energy region of 1 GeV is safely below the inelastic threshold, see the
discussions in Ref. Danilkin:2014cra . First, we use the model of Heyn and
Lang (HL) Heyn:1980bh ,

$\Omega(s)=\frac{c+m_{\pi}^{2}g(0)}{\widetilde{s}_{p}}\frac{\tilde{s}_{p}-s}{as^{2}+bs+c-\left(s-4m_{\pi}^{2}\right)g(s)/4},$
(31)

where the function $g(s)$ is given by the one-pion-loop diagram in the self-
energy of $\rho$ resonance,

$\displaystyle g(s)$ $\displaystyle=-\frac{1}{\pi}u\ln\frac{1+u}{1-u}+iu,\quad
u=\sqrt{1-4m_{\pi}^{2}/s},\quad s\geqslant 4m_{\pi}^{2}$ (32)
$\displaystyle=-\frac{2}{\pi}u\arctan\frac{1}{u},\quad
u=\sqrt{4m_{\pi}^{2}/s-1},\quad 0\leqslant s\leqslant 4m_{\pi}^{2}$
$\displaystyle=-\frac{1}{\pi}u\ln\frac{u+1}{u-1},\quad
u=\sqrt{1-4m_{\pi}^{2}/s},\quad s<0,$ $\displaystyle g(0)$
$\displaystyle=-2/\pi,$

and the parameters $a,\ b,\ c$ are free. Besides, $\tilde{s}_{p}$ is the value
of the zero of the denominator and can be determined from $a,\ b,\ c$, using
the condition $f(s)=0$, reading

$f(s)=as^{2}+bs+c-\left(s-4m_{\pi}^{2}\right)g(s)/4=0.$ (33)

Second, as discussed above, the Omnés function is mainly considered the
resonance contribution of $\rho$. Thus, using the vector meson dominance
approach, one can replace the Omnés function with the simple Breit-Wigner (BW)
form (BW1),

$\Omega(s)\rightarrow
BW1=-\frac{M_{\rho}^{2}}{s-M_{\rho}^{2}+iM_{\rho}\Gamma_{\rho}},$ (34)

with $M_{\rho}$ and $\Gamma_{\rho}$ as free parameters for the $\rho$ meson,
which are also fitted by the experimental data. Furthermore, due to the large
$\rho$ width, one can use the more common one (BW2) Danilkin:2014cra ,

$\Omega(s)\rightarrow
BW2=-\frac{M_{\rho}^{2}}{s-M_{\rho}^{2}-i\sqrt{s}\Gamma(s)},$ (35)

where the energy dependent decay width is given by

$\Gamma(s)=\Gamma_{\rho}\left[\frac{p(s)}{p\left(M_{\rho}^{2}\right)}\right]^{3}\frac{M_{\rho}^{2}}{s},$
(36)

with the pion momentum

$p(s)=\frac{1}{2}\sqrt{s}\beta_{\pi}(s)=\frac{\sqrt{s-4m_{\pi}^{2}}}{2}.$ (37)

Third, for taking into account more precisely the finite-width corrections,
one can also use the method of Gounaris and Sakurai (GS) Gounaris:1968mw ,
written as

$\Omega(s)\rightarrow
BW^{\mathrm{GS}}(s)=\frac{-\left(M_{\rho}^{2}+dM_{\rho}\Gamma_{\rho}\right)}{s-M_{\rho}^{2}-\Gamma_{\rho}\left(M_{\rho}^{2}/p_{\rho}^{3}\right)\left[p^{2}\left(h-h_{\rho}\right)-\left(s-M_{\rho}^{2}\right)p_{\rho}^{2}h_{\rho}^{\prime}\right]+iM_{\rho}\Gamma_{\rho}(s)},$
(38)

where

$\begin{array}[]{l}p=\left(s/4-m_{\pi}^{2}\right)^{1/2},\quad
h(s)=\frac{2p}{\pi\sqrt{s}}\ln\left(\frac{\sqrt{s}+2p}{2m_{\pi}}\right),\quad
s\geqslant 4m_{\pi}^{2},\\\ p=i\left(m_{\pi}^{2}-s/4\right)^{1/2},\quad
h(s)=\frac{2pi}{\pi\sqrt{s}}\operatorname{arccot}\left(\frac{s}{4m_{\pi}^{2}-s}\right)^{1/2},\quad
0\leqslant s<4m_{\pi}^{2},\end{array}$ (39)
$p_{\rho}=p\left(M_{\rho}^{2}\right),\quad
h_{\rho}=h\left(M_{\rho}^{2}\right),\quad\Gamma_{\rho}(s)=\Gamma_{\rho}\left(\frac{p}{p_{\rho}}\right)^{3}\frac{M_{\rho}}{\sqrt{s}},h_{\rho}^{\prime}=\left.h^{\prime}(s)\right|_{s=M_{\rho}^{2}},$
(40)

and $d$ is fixed in terms of the masses $M_{\rho}$ and $m_{\pi}$,

$d=\frac{3}{\pi}\frac{m_{\pi}^{2}}{p_{\rho}^{2}}\ln\left(\frac{M_{\rho}+2p_{\rho}}{2m_{\pi}}\right)+\frac{M_{\rho}}{2\pi
p_{\rho}}-\frac{m_{\pi}^{2}M_{\rho}}{\pi p_{\rho}^{3}}.$ (41)

Note that, in this GS model, one can also determine the $M_{\rho}$ and
$\Gamma_{\rho}$ from the fits. Thus, in BaBar’s paper Lees:2012cj , its form
was changed equivalently as

$BW^{\mathrm{GS}}\left(s,M_{\rho},\Gamma_{\rho}\right)=\frac{M_{\rho}^{2}\left[1+d(M_{\rho})\Gamma_{\rho}/M_{\rho}\right]}{M_{\rho}^{2}-s+f\left(s,M_{\rho},\Gamma_{\rho}\right)-iM_{\rho}\Gamma\left(s,M_{\rho},\Gamma_{\rho}\right)},$
(42)

where

$\Gamma\left(s,M_{\rho},\Gamma_{\rho}\right)=\Gamma_{\rho}\frac{s}{M_{\rho}^{2}}\left[\frac{\beta_{\pi}(s)}{\beta_{\pi}\left(M_{\rho}^{2}\right)}\right]^{3},$
(43)
$d\left(M_{\rho}\right)=\frac{3}{\pi}\frac{m_{\pi}^{2}}{k^{2}\left(M_{\rho}^{2}\right)}\ln\left[\frac{M_{\rho}+2k\left(M_{\rho}^{2}\right)}{2m_{\pi}}\right]+\frac{M_{\rho}}{2\pi
k\left(M_{\rho}^{2}\right)}-\frac{m_{\pi}^{2}M_{\rho}}{\pi
k^{3}\left(M_{\rho}^{2}\right)},$ (44)
$f\left(s,M_{\rho},\Gamma_{\rho}\right)=\frac{\Gamma_{\rho}M_{\rho}^{2}}{k^{3}\left(M_{\rho}^{2}\right)}\left[k^{2}(s)\left(h(s)-h\left(M_{\rho}^{2}\right)\right)+\left(M_{\rho}^{2}-s\right)k^{2}\left(M_{\rho}^{2}\right)h^{\prime}\left(M_{\rho}^{2}\right)\right],$
(45)

with the pion phase $\beta_{\pi}(s)$ defined as in the last section, and

$k(s)=\frac{1}{2}\sqrt{s}\beta_{\pi}(s),$ (46)
$h(s)=\frac{2}{\pi}\frac{k(s)}{\sqrt{s}}\ln\left(\frac{\sqrt{s}+2k(s)}{2m_{\pi}}\right),$
(47)

and $h^{\prime}(s)$ is the derivative of $h(s)$.

Moreover, similar to GS model, there is another form presented by Kühn and
Santamaria (KS) Kuhn:1990ad ,

$BW^{KS}(s)=\frac{M_{\rho}^{2}+\Gamma_{\rho}M_{\rho}d^{\prime}}{M_{\rho}^{2}-s+H(s)-i\sqrt{s}\Gamma_{\rho}(s)},$
(48)

where

$H(s)=\hat{H}(s)-\hat{H}\left(M_{\rho}^{2}\right)-\left(s-M_{\rho}^{2}\right)\hat{H}^{\prime}\left(M_{\rho}^{2}\right),$
(49)
$\hat{H}(s)=\frac{\Gamma_{\rho}M_{\rho}^{2}}{p_{\rho}^{3}}\left(s/4-m_{\pi}^{2}\right)h(s),$
(50)
$h\left(s\right)=\left\\{\begin{array}[]{l}\frac{1}{2\pi}\left(1-\frac{4m_{\pi}^{2}}{s}\right)^{1/2}\ln\left(\frac{1+\left(1-\frac{4m_{\pi}^{2}}{s}\right)^{1/2}}{1-\left(1-\frac{4m_{\pi}^{2}}{s}\right)^{1/2}}\right),4m_{\pi}^{2}\leq
s\\\
\frac{i}{2\pi}\left(\frac{4m_{\pi}^{2}}{s}-1\right)^{1/2}\ln\left(\frac{i\left(\frac{4m_{\pi}^{2}}{s}-1\right)^{1/2}+1}{i\left(\frac{4m_{\pi}^{2}}{s}-1\right)^{1/2}-1}\right),0\leq
s\leq 4m_{\pi}^{2}\end{array}\right.$ (51)
$d^{\prime}=\frac{3}{2\pi}\frac{m_{\pi}^{2}}{p_{\rho}^{2}}\ln\left(\frac{M_{\rho}+2p_{\rho}}{M_{\rho}-2p_{\rho}}\right)+\frac{M_{\rho}}{2\pi
p_{\rho}}-\frac{m_{\pi}^{2}M_{\rho}}{\pi p_{\rho}^{3}},$ (52)
$p(s)=\frac{1}{2}\left(s-4m_{\pi}^{2}\right)^{1/2},\quad
p_{\rho}=\left(M_{\rho}^{2}-4m_{\pi}^{2}\right)^{1/2}/2.$ (53)

Finally, since $e^{+}e^{-}$ annihilation data has the $\rho-\omega$ mixing
effects, we should take into account this effect as done in Refs.
Davier:2019can ; Hanhart:2016pcd

$P(s)=1+\alpha s+\frac{\kappa
s}{m_{\omega}^{2}-s-im_{\omega}\Gamma_{\omega}},$ (54)

where we take $m_{\omega}=782.65$ MeV and $\Gamma_{\omega}=8.49$ MeV from PDG
Zyla:2020zbs , and $\kappa$ is a free parameter containing the information of
$\omega\pi\pi$ coupling, see our results later, where more discussions can be
referred to Ref. Hanhart:2016pcd . Note that, Eq. (54) fulfils $P(0)=1$, which
guarantees the condition $F_{\pi}(0)=1$, except for the BW1 model. Indeed, the
BW1 model is the typical one of the vector meson dominance, which is known to
violate the condition OConnell:1995nse . Furthermore, in general, one can also
float the parameters $m_{\omega}$ and $\Gamma_{\omega}$ in Eq. (54) as done in
Refs. Hanhart:2016pcd ; Davier:2019can . But, as found in Ref. Davier:2019can
, a value of $m_{\omega}=(782.0\pm 0.1)$ MeV was obtained in the fitting
results with all the experimental data, which is not much different from the
PDG one we used. Due to the small width of the $\omega$ meson and the narrow
energy region of $\rho-\omega$ mixing, we fixed them, where in fact the errors
of $m_{\omega}$ and $\Gamma_{\omega}$ only contributed tiny influences to the
uncertainties of final results as found later. Another thing should be
mentioned that, the parameters $M_{\rho}$ and $\Gamma_{\rho}$ are model
dependent. Thus, the differences between these models can be seen from our
results in the next section, see more discussions later.

##  IV Results

For the data of the PVFF below the energy region of 1 GeV, we take them from
the experimental collaborations of Orsay, DM1, OLYA, CMD1, CMD2, BABAR,
BESIII, KLOE and SND Augustin:1969kn ; Quenzer:1978qt ; Barkov:1985ac ;
Akhmetshin:2001ig ; Akhmetshin:2003zn ; Aulchenko:2006na ; Akhmetshin:2006wh ;
Akhmetshin:2006bx ; Aubert:2009ad ; Ablikim:2015orh ; Ablikim:2020bah ;
Aloisio:2004bu ; Ambrosino:2008aa ; Ambrosino:2010bv ; Babusci:2012rp ;
Anastasi:2017eio ; Achasov:2005rg ; Achasov:2006vp ; Achasov:2020iys . Our
fitting results for each set of experimental data are given in Table I, where
the details of $\chi^{2}/dof$ for the combined data (Com. Dat.) are given in
the last line. In fact, there are two general parameters for all the models,
$\alpha$ and $\kappa$ appearing in $P(s)$, see Eq. (54). For the Omnés model,
there is no extra parameter. In the one of HL, there are three more, $a,\ b,\
c$, as discussed above. Besides, the other BW types, including BW1, BW2, GS
and KS models, have another two parameters for the $\rho$ meson, $M_{\rho}$
and $\Gamma_{\rho}$. From the results of Table I, one can see that
$\chi^{2}/dof$ for Orsay1969 data are all too small due to only a few data
points, see Fig. 1, and the ones for KLOE2005 are much larger than the others
owing to the fitted discrepancy around the $\rho-\omega$ mixing region. With
the summarized results in Table I and a systematic analysis of all the fitting
results, one can easily find that the results with the GS and KS models are
better than the others, which are compatible with each other. Indeed, the
Omnés model has only two free parameters, but it needs the $P$-wave phase
shift as input, which is constrained by analyticity and unitarity, as well as
crossing symmetry, and depends on the accuracy of the measured $\pi\pi$
$P$-wave phase shift. The HL model is in fact used $N/D$ method for the
elastic $P$-wave $\pi\pi$ scattering amplitude with a simple one-pole
contribution without considering the detail of the pole width, which can be
matched with the simple BW pole ansatz, see the results later. For the BW1
model, it is only a simple BW pole ansatz from the vector meson dominance,
which violates the charge normalization condition as discussed in the last
section and is improved by the BW2 model with an energy dependent decay width.
Utilized the unitarity condition and considered the detail of the energy
dependence of the resonance width for the $\rho$ propagator, this is done in
the GS and KS models. Therefore, it is not surprising that the fitting results
of the GS and KS models are the best ones. Thus, our final results are
favoured with the one of GS (or KS) model with the fit of the Com. Dat.. In
Figs. 1 and 2, we show the results fitted with GS model for each set of
experimental data and the Com. Dat., respectively. Note that some data above 1
GeV in the sets of DM1-1978, OLYA1985 and BABAR has been ignored.

Table I: Results of $\chi^{2}/dof$ for each fit in different sets of experimental data. Data set | Omnés | HL | BW1 | BW2 | GS | KS
---|---|---|---|---|---|---
Orsay1969 Augustin:1969kn  | 0.63 | \- 333This is due to $dof=0$ with only five data points available. | 0.02 | 0.02 | 0.01 | 0.01
DM1-1978 Quenzer:1978qt  | 0.74 | 1.99 | 0.96 | 0.92 | 0.81 | 0.81
OLYA1985 Barkov:1985ac  | 0.57 | 0.59 | 0.54 | 0.54 | 0.58 | 0.58
CMD1985 Barkov:1985ac  | 1.72 | 1.31 | 1.92 | 1.78 | 1.68 | 1.68
CMD2-2002 Akhmetshin:2001ig  | 1.14 | 1.32 | 1.11 | 1.11 | 1.14 | 1.14
CMD2-2004Akhmetshin:2003zn  | 1.14 | 1.20 | 1.15 | 1.14 | 1.17 | 1.17
CMD2-2006Akhmetshin:2006wh  | 1.43 | 12.72 | 1.7 | 1.72 | 1.77 | 1.79
CMD2-2007 Akhmetshin:2006bx  | 2.33 | 2.71 | 2.13 | 2.04 | 1.86 | 1.86
BaBar2009Aubert:2009ad  | 1.83 | 1.06 | 1.34 | 1.08 | 1.05 | 1.05
BESIII2020 Ablikim:2015orh ; Ablikim:2020bah  | 1.05 | 1.10 | 0.84 | 0.86 | 0.95 | 0.95
KLOE2005Aloisio:2004bu  | 68.42 | 19.18 | 21.34 | 20.52 | 18.84 | 18.84
KLOE2009 Ambrosino:2008aa  | 4.74 | 2.28 | 6.92 | 5.3 | 2.24 | 2.24
KLOE2011 Ambrosino:2010bv  | 1.09 | 1.09 | 1.31 | 1.15 | 1.08 | 1.08
KLOE2013 Babusci:2012rp  | 1.27 | 1.13 | 1.53 | 1.35 | 1.11 | 1.11
KLOE2018 Anastasi:2017eio  | 1.26 | 0.77 | 2.08 | 1.53 | 0.77 | 0.77
SND2005 Achasov:2005rg  | 4.05 | 3.52 | 3.54 | 3.45 | 3.43 | 3.43
SND2006 Achasov:2006vp  | 4.04 | 3.61 | 3.38 | 3.36 | 3.52 | 3.52
SND2020 Achasov:2020iys  | 3.55 | 3.93 | 3.39 | 3.43 | 3.68 | 3.68
Com. Dat. | 11.29 | 10.20 | 11.01 | 10.65 | 10.19 | 10.19
$\frac{\chi^{2}}{dof}$ (Com. Dat.) | $\frac{11337.26}{1006-2}$ | $\frac{10214.46}{1006-5}$ | $\frac{11032.14}{1006-4}$ | $\frac{10671.72}{1006-4}$ | $\frac{10214.59}{1006-4}$ | $\frac{10214.59}{1006-4}$
Figure 1: Fitting results with the GS model for each set of experimental data.
Figure 2: Fitting results with the GS model for the Com. Dat..

Furthermore, using the BW types of BW1, BW2, GS and KS models, one can also
determine the parameters of $\rho$ meson, $M_{\rho}$ and $\Gamma_{\rho}$, from
the fits, see the results of Table II. Note that in the first column, the ones
for the HL model are not obtained directly from the fits. As discussed in
Refs. Heyn:1980bh ; Gardner:1997ie , once the parameters $a,\ b,\ c$ were
determined from the fit, the $M_{\rho}$ and $\Gamma_{\rho}$ could also be
determined, since the function $f(s)$, see Eq. (33), should be matched with
the BW form when $s\to M_{\rho}^{2}$. And thus, one can have
$\mathrm{Re}[f(M_{\rho}^{2})]=0$ and
$\mathrm{Im}[f(M_{\rho}^{2})]=-M_{\rho}\Gamma_{\rho}$. Since the small
uncertainties are obtained from the fits for the parameters of $a$, $b$, and
$c$, the uncertainties for their results of determining $M_{\rho}$ and
$\Gamma_{\rho}$ are small too, also for the results of
$a_{\mu}^{\mathrm{HVP},\mathrm{LO}}$ later. For the data of CMD2-2006, because
the data is close to the $\pi\pi$ threshold, it is not possible to get the
results for $M_{\rho}$ and $\Gamma_{\rho}$ correctly. The results for
Orsay1969 are bigger than the others, whereas the ones for KLOE2005 are much
smaller. And compared with different models, the results with the BW2 model
are bigger than the ones with the BW1 model, and also larger than the numbers
obtained with the GS (or KS) model. Finally, we obtain the results of the
$\rho$ meson parameters, $M_{\rho}$ and $\Gamma_{\rho}$ 444In fact, these
results are for the neutral one of $\rho^{0}$ meson in the $e^{+}e^{-}$
annihilation processes., from the fit with GS (or KS) model and the Com. Dat.,

$M_{\rho}=(774.07\pm 0.04)\ \text{MeV},\quad\Gamma_{\rho}=(149.54\pm 0.08)\
\text{MeV},$ (55)

which are 1 MeV smaller than the one reported in PDG Zyla:2020zbs ,
$(775.26\pm 0.23)$ MeV, for the mass, and 2 MeV bigger than the one $(147.4\pm
0.8)$ MeV for the width, and a bit smaller than the other one
$M_{\rho}=(775.13\pm 0.02)$ MeV obtained in Ref. Pich:2001pj . Our results are
also consistent with the one obtained in Ref. Davier:2019can ,
$M_{\rho}=(774.5\pm 0.8)$ MeV. Note that, the small errors for the pole
parameters are due to more constraints in the Com. Dat.. One thing should be
remarked that, as discussed above and shown in Table II, the $\rho$ meson
parameters, $M_{\rho}$ and $\Gamma_{\rho}$, are model dependent and do not
correspond to the physical resonance’s mass and width, which should be looked
for the pole in the second Riemann sheet. It is complicated to extrapolate the
form factor to the second Riemann sheet GBarton , which is out of our concern
in the present work and where one can refer to Refs. Gonzalez-Solis:2019iod ;
GomezDumm:2013sib  for more discussions 555In their model, a pole $(762.0\pm
0.3,\frac{i}{2}(143.0\pm 0.2)\,)$ MeV was found for a mass parameter
$M_{\rho}=(775.2\pm 0.4)$ MeV from the fitting of $\tau$ decay data. and Ref.
LHCb:2021auc  for the application in the unitarised three-body BW function.

Besides, for the other parameters, we show the results in Table IV in Appendix
A, where the details for $\alpha$ and $\kappa$ are given and the results with
Omnés model are consistent with the ones of Ref. Hanhart:2016pcd  within the
uncertainties. As done in Ref. Hanhart:2016pcd , one can extract the
$\omega\pi\pi$ coupling and the branching ratio of $\omega\to\pi\pi$ from the
results of $\kappa$ parameter in Table IV. Thus, following the same way and
using the favoured results of GS (or KS) model for the Com. Dat.,
$\kappa=(1.77\pm 0.01)\times 10^{-3}$, we have

$g_{\omega\pi\pi}=(3.02\pm 0.05)\times
10^{-2},\quad\text{Br}(\omega\to\pi\pi)=(1.52\pm 0.06)\%,$ (56)

which are consistent with the ones obtained in Ref. Hanhart:2016pcd  within
the uncertainties. Our result for the branching ratio is also in good
agreement with the one in PDG Zyla:2020zbs ,
$\text{Br}(\omega\to\pi\pi)=(1.53^{+0.11}_{-0.13})\%$.

Table II: Results for the $\rho$ meson parameters $M_{\rho}$ and $\Gamma_{\rho}$ from the fits. Data set | HL | BW1 | BW2 | GS | KS
---|---|---|---|---|---
Orsay1969 |  | $779.00\pm 0.00$
---
$158.47\pm 0.00$
| $762.10\pm 8.99$
---
$178.58\pm 32.87$
| $793.08\pm 13.13$
---
$193.37\pm 40.43$
| $790.76\pm 12.59$
---
$190.88\pm 38.85$
| $790.72\pm 12.59$
---
$190.59\pm 38.78$
DM1-1978 |  | $767.53\pm 0.00$
---
$155.10\pm 0.00$
| $757.41\pm 3.52$
---
$144.63\pm 5.11$
| $778.05\pm 3.44$
---
$153.03\pm 6.01$
| $776.83\pm 3.49$
---
$152.50\pm 5.83$
| $776.83\pm 3.49$
---
$152.50\pm 5.83$
OLYA1985 |  | $775.81\pm 5.75$
---
$157.54\pm 1.69$
| $757.41\pm 0.87$
---
$143.43\pm 1.32$
| $777.54\pm 0.77$
---
$151.72\pm 1.56$
| $775.87\pm 0.78$
---
$151.71\pm 1.53$
| $775.87\pm 0.78$
---
$151.71\pm 1.53$
CMD1985 |  | $761.57\pm 102.54$
---
$153.34\pm 30.26$
| $756.54\pm 2.09$
---
$134.54\pm 6.51$
| $774.29\pm 3.33$
---
$140.22\pm 7.19$
| $772.47\pm 3.15$
---
$136.14\pm 6.75$
| $772.47\pm 3.14$
---
$136.12\pm 6.73$
CMD2-2002 |  | $775.38\pm 0.00$
---
$157.41\pm 0.00$
| $758.39\pm 0.63$
---
$140.96\pm 1.22$
| $777.83\pm 0.62$
---
$148.65\pm 1.43$
| $776.40\pm 0.62$
---
$148.17\pm 1.39$
| $776.40\pm 0.62$
---
$148.17\pm 1.39$
CMD2-2004 |  | $776.02\pm 4.92$
---
$157.60\pm 1.45$
| $758.14\pm 0.62$
---
$140.78\pm 1.22$
| $777.53\pm 0.61$
---
$148.43\pm 1.42$
| $776.07\pm 0.62$
---
$147.90\pm 1.38$
| $776.07\pm 0.62$
---
$147.90\pm 1.38$
CMD2-2006 | $-$ | $-$ | $-$ | $-$ | $-$
CMD2-2007 |  | $775.70\pm 0.00$
---
$157.50\pm 0.00$
| $757.82\pm 0.44$
---
$143.26\pm 0.57$
| $777.91\pm 0.44$
---
$151.46\pm 0.68$
| $776.35\pm 0.44$
---
$151.21\pm 0.66$
| $776.35\pm 0.44$
---
$151.21\pm 0.66$
BaBar2009 |  | $774.53\pm 0.00$
---
$157.16\pm 0.00$
| $755.14\pm 0.11$
---
$144.60\pm 0.24$
| $775.85\pm 0.11$
---
$152.70\pm 0.28$
| $774.58\pm 0.11$
---
$151.38\pm 0.27$
| $774.58\pm 0.11$
---
$151.38\pm 0.27$
BESIII2020 |  | $775.86\pm 0.00$
---
$157.55\pm 0.00$
| $757.87\pm 0.36$
---
$142.52\pm 1.05$
| $777.73\pm 0.40$
---
$149.93\pm 1.21$
| $776.22\pm 0.39$
---
$147.87\pm 1.17$
| $776.22\pm 0.39$
---
$147.87\pm 1.17$
KLOE2005 |  | $768.79\pm 2.09$
---
$155.47\pm 0.62$
| $750.52\pm 0.13$
---
$141.37\pm 0.19$
| $770.43\pm 0.12$
---
$149.63\pm 0.22$
| $768.87\pm 0.12$
---
$149.67\pm 0.22$
| $768.87\pm 0.12$
---
$149.67\pm 0.22$
KLOE2009 |  | $774.86\pm 2.06$
---
$157.26\pm 0.61$
| $756.58\pm 0.09$
---
$142.07\pm 0.13$
| $776.41\pm 0.08$
---
$150.17\pm 0.15$
| $774.93\pm 0.08$
---
$150.18\pm 0.14$
| $774.93\pm 0.08$
---
$150.18\pm 0.14$
KLOE2011 |  | $775.61\pm 0.00$
---
$157.48\pm 0.00$
| $757.25\pm 0.19$
---
$142.24\pm 0.32$
| $777.11\pm 0.16$
---
$150.09\pm 0.38$
| $775.66\pm 0.17$
---
$149.29\pm 0.37$
| $775.66\pm 0.17$
---
$149.29\pm 0.37$
KLOE2013 |  | $775.69\pm 2.38$
---
$157.50\pm 0.70$
| $757.30\pm 0.28$
---
$141.34\pm 0.41$
| $776.99\pm 0.26$
---
$149.40\pm 0.49$
| $775.76\pm 0.27$
---
$149.86\pm 0.48$
| $775.76\pm 0.27$
---
$149.86\pm 0.48$
KLOE2018 |  | $775.15\pm 2.67$
---
$157.34\pm 0.78$
| $756.63\pm 0.14$
---
$142.29\pm 0.20$
| $776.57\pm 0.12$
---
$150.37\pm 0.24$
| $775.22\pm 0.13$
---
$150.21\pm 0.23$
| $775.22\pm 0.13$
---
$150.21\pm 0.23$
SND2005 |  | $775.27\pm 0.00$
---
$157.38\pm 0.00$
| $756.46\pm 0.26$
---
$144.34\pm 0.49$
| $776.92\pm 0.27$
---
$152.62\pm 0.58$
| $775.33\pm 0.27$
---
$151.88\pm 0.56$
| $775.33\pm 0.27$
---
$151.88\pm 0.56$
SND2006 |  | $775.54\pm 1.92$
---
$157.46\pm 0.56$
| $756.70\pm 0.27$
---
$144.27\pm 0.49$
| $777.14\pm 0.27$
---
$152.55\pm 0.57$
| $775.60\pm 0.27$
---
$151.91\pm 0.56$
| $775.60\pm 0.27$
---
$151.91\pm 0.56$
SND2020 |  | $775.59\pm 0.00$
---
$157.47\pm 0.00$
| $757.33\pm 0.29$
---
$144.33\pm 0.81$
| $777.67\pm 0.33$
---
$152.08\pm 0.94$
| $775.92\pm 0.32$
---
$149.98\pm 0.90$
| $775.92\pm 0.32$
---
$149.98\pm 0.90$
Com. Dat. |  | $774.00\pm 0.79$
---
$157.00\pm 0.23$
| $755.62\pm 0.05$
---
$141.74\pm 0.07$
| $775.44\pm 0.04$
---
$149.76\pm 0.08$
| $774.07\pm 0.04$
---
$149.54\pm 0.08$
| $774.07\pm 0.04$
---
$149.54\pm 0.08$

With the fitting results obtained above with different models of the PVFF for
the data below 1 GeV, we can evaluate the value of
$a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-})$ from the two-pion contribution
with the dispersion integral defined in Eq. (22). Our results are given in
Table III, using different PVFF models for each set of experimental data up to
1 GeV, except for the one of CMD2-2006. We also show the ones with GS model
for the Com. Dat. in Fig. 3 clearly. As one can find in Table III that the
results for Orsay1969 are bigger than the others, and conversely the ones for
CMD1985 are smaller than the others. The uncertainties for the results of
Omnés model are smaller than the others, whereas the ones with BW1 model are
the biggest, see Fig. 3. Indeed, from Fig. 3, compared to the one with GS or
KS model, there is a 1% difference from the smallest one with the BW1 model.
At the end, our final results are taken from the one with the GS (or KS) model
using the Com. Dat., given as

$a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-}\leq 1\ \text{GeV})=(497.76\pm
3.15)\times 10^{-10},$ (57)

which are consistent with the one obtained in Ref. Colangelo:2020lcg  very
well, $a_{\mu}^{\mathrm{HVP}}(\pi\pi\leq 1\ \text{GeV})=(497.0\pm 1.4)\times
10^{-10}$, an updated result of Ref. Colangelo:2018mtw  by considering the
inelastic effects from the constraints of the Eidelman-Łukaszuk bound.
Furthermore, using the framework of resonance chiral theory, two similar
values of $a_{\mu}^{\mathrm{HVP}}(\pi\pi\leq 1\ \text{GeV})=(498.48\pm
2.34)\times 10^{-10}$ (Fit I) and $(498.47\pm 2.33)\times 10^{-10}$ (Fit II)
were obtained in Ref. Qin:2020udp , which are consistent with ours within the
uncertainties. One thing should be mentioned that the obtained error is
estimated by the average of the reasonable ones in Table III for different
sets of data, since one can find that the errors for different sets of data
are mainly contributed by the errors of the pole parameters, see Table II, and
the errors for four sets of data are quite large due to the fewer data points
in the $\rho$ region. This is why the error is so small for the one of the
Omnés model with no pole parameter, see the results of Table III.

Table III: Results of $a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-})$ ($\times 10^{-10}$) from two-pion contribution up to 1 GeV. Data set | Omnés | HL | BW1 | BW2 | GS | KS
---|---|---|---|---|---|---
Orsay1969 | $555.48\pm 29.98$ | $553.24\pm 0.16$ | $562.66\pm 174.98$ | $565.56\pm 165.17$ | $568.00\pm 139.17$ | $567.72\pm 138.87$
DM1-1978 | $468.24\pm 8.97$ | $491.35\pm 0.01$ | $463.46\pm 30.03$ | $465.12\pm 29.31$ | $467.90\pm 25.56$ | $467.90\pm 25.59$
OLYA1985 | $490.73\pm 2.32$ | $490.64\pm 9.98$ | $483.72\pm 7.80$ | $486.07\pm 7.68$ | $490.64\pm 6.68$ | $490.64\pm 6.69$
CMD1985 | $475.48\pm 6.12$ | $461.50\pm 76.32$ | $470.32\pm 34.68$ | $469.71\pm 32.50$ | $467.19\pm 27.65$ | $467.20\pm 27.64$
CMD2-2002 | $492.46\pm 2.06$ | $495.36\pm 0.01$ | $483.93\pm 7.31$ | $485.80\pm 7.15$ | $489.02\pm 6.24$ | $489.02\pm 6.23$
CMD2-2004 | $503.86\pm 2.11$ | $501.27\pm 9.02$ | $496.16\pm 7.49$ | $498.06\pm 7.33$ | $501.26\pm 6.40$ | $501.26\pm 6.41$
CMD2-2006 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
CMD2-2007 | $501.45\pm 0.99$ | $501.58\pm 0.01$ | $494.63\pm 3.86$ | $496.76\pm 3.80$ | $500.57\pm 3.36$ | $500.57\pm 3.36$
BaBar2009 | $501.38\pm 0.41$ | $503.35\pm 0.01$ | $500.01\pm 1.28$ | $501.45\pm 1.24$ | $503.36\pm 1.06$ | $503.36\pm 1.06$
BESIII2020 | $498.39\pm 1.56$ | $500.35\pm 0.01$ | $493.96\pm 5.82$ | $495.35\pm 5.63$ | $496.93\pm 4.83$ | $496.93\pm 4.83$
KLOE2005 | $473.67\pm 0.32$ | $496.27\pm 7.30$ | $490.14\pm 1.08$ | $492.27\pm 1.06$ | $496.25\pm 0.92$ | $496.25\pm 0.92$
KLOE2009 | $495.06\pm 0.22$ | $497.11\pm 7.44$ | $491.06\pm 0.72$ | $493.19\pm 0.71$ | $497.09\pm 0.62$ | $497.09\pm 0.61$
KLOE2011 | $494.60\pm 0.52$ | $493.81\pm 0.01$ | $489.13\pm 1.89$ | $490.92\pm 1.85$ | $493.77\pm 1.59$ | $493.77\pm 1.59$
KLOE2013 | $495.90\pm 0.62$ | $495.57\pm 6.02$ | $490.64\pm 2.10$ | $492.45\pm 2.09$ | $495.56\pm 1.80$ | $495.56\pm 1.80$
KLOE2018 | $495.21\pm 0.35$ | $495.95\pm 7.13$ | $490.53\pm 1.13$ | $492.49\pm 1.12$ | $495.93\pm 0.96$ | $495.93\pm 0.96$
SND2005 | $508.68\pm 0.77$ | $513.28\pm 0.01$ | $508.34\pm 3.19$ | $510.25\pm 3.12$ | $513.28\pm 2.73$ | $513.28\pm 2.75$
SND2006 | $497.75\pm 0.76$ | $501.18\pm 3.61$ | $496.16\pm 3.07$ | $498.05\pm 3.01$ | $501.17\pm 2.63$ | $501.17\pm 2.65$
SND2020 | $500.47\pm 0.99$ | $499.04\pm 0.01$ | $498.29\pm 5.04$ | $499.63\pm 4.89$ | $501.04\pm 4.27$ | $501.04\pm 4.29$
Com. Dat. | $492.81\pm 1.00$ | $497.78\pm 3.61$ | $492.57\pm 3.70$ | $494.46\pm 3.62$ | $497.76\pm 3.15$ | $497.76\pm 3.15$
Figure 3: Summarized results of $a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-})$
from two-pion contribution up to 1 GeV using the GS model and fitting with the
Com. Dat..

##  V Conclusion

In the present work, in order to reduce the uncertainties of the calculation
of two-pion contribution to the muon anomalous magnetic moment, we try to get
the best fit for the two-pion cross sections with several theoretical models
of the pion vector form factor, combined with a polynomial description. Since
the polynomial description is valid up to 1 GeV, we only take into account all
the experimental data below 1 GeV, which is below the significant inelastic
threshold and contributes almost more than 70% of the hadronic contribution to
the muon anomalous magnetic moment. From our results, we find that the fit
with the Gounaris-Sakurai (or Kühn-Santamaria) model is the best one. From the
best fit to the pion vector form factor, one can also extract the branching
ratio of $\omega\to\pi\pi$, given by

$\displaystyle\text{Br}(\omega\to\pi\pi)=(1.52\pm 0.06)\%,$

which are compatible with the results reported in Particle Data Group. Based
on the best fit to data, we calculate the two-pion contribution to the muon
anomalous magnetic moment, obtaining

$a_{\mu}^{\mathrm{HVP,\;LO}}(\pi^{+}\pi^{-}\leq 1\ \text{GeV})=(497.76\pm
3.15)\times 10^{-10},$

which is in good agreement with the recent theoretical evaluations
Colangelo:2020lcg ; Qin:2020udp . Our results for two-pion contribution are
helpful to pin down the uncertainties of the calculation for the hadronic
vacuum polarization contribution to the muon anomalous magnetic moment.

## Acknowledgments

We thank Profs. Martin Hoferichter, Irinel Caprini, Hidezumi Terazawa, Peter
Athron, Yusi Pan, Wen Yin and Nikolay N. Achasov for valuable comments and
useful information, and acknowledge the referee for helpful suggestions.

##  Appendix A The other parameters

We give the details of some other parameters in Table IV, where the values of
$\alpha$ and $\kappa$ are shown for each fit with different PVFF models, and
the ones with Omnés model are consistent with the results obtained in Ref.
Hanhart:2016pcd  within the uncertainties.

Table IV: Parameters of $\alpha(GeV^{-2})$ and $\kappa\times 10^{3}$ for each fit. Data set | Omnés | HL | BW1 | BW2 | GS | KS
---|---|---|---|---|---|---
Orsay1969 |  | $0.15\pm 0.05$
---
$7.68\pm 2.75$
| $0.23\pm 0.00$
---
$9.26\pm 0.00$
| $0.73\pm 0.32$
---
$18.72\pm 8.63$
| $0.69\pm 0.29$
---
$18.52\pm 8.33$
| $0.45\pm 0.22$
---
$16.79\pm 7.21$
| $0.45\pm 0.22$
---
$16.74\pm 7.20$
DM1-1978 |  | $0.01\pm 0.02$
---
$2.14\pm 0.47$
| $0.12\pm 0.00$
---
$1.64\pm 0.00$
| $0.30\pm 0.06$
---
$2.61\pm 0.61$
| $0.28\pm 0.06$
---
$2.59\pm 0.60$
| $0.12\pm 0.05$
---
$2.41\pm 0.55$
| $0.12\pm 0.05$
---
$2.41\pm 0.55$
OLYA1985 |  | $0.06\pm 0.00$
---
$1.68\pm 0.17$
| $0.16\pm 0.01$
---
$1.93\pm 0.21$
| $0.34\pm 0.02$
---
$2.06\pm 0.23$
| $0.32\pm 0.01$
---
$2.05\pm 0.23$
| $0.17\pm 0.01$
---
$1.94\pm 0.21$
| $0.17\pm 0.01$
---
$1.94\pm 0.21$
CMD1985 |  | $0.02\pm 0.01$
---
$2.03\pm 0.26$
| $-0.07\pm 0.05$
---
$1.52\pm 0.30$
| $0.23\pm 0.06$
---
$2.05\pm 0.38$
| $0.20\pm 0.06$
---
$2.02\pm 0.37$
| $0.02\pm 0.05$
---
$1.79\pm 0.33$
| $0.02\pm 0.05$
---
$1.79\pm 0.33$
CMD2-2002 |  | $0.06\pm 0.00$
---
$1.48\pm 0.13$
| $0.15\pm 0.00$
---
$1.50\pm 0.00$
| $0.32\pm 0.01$
---
$1.72\pm 0.16$
| $0.30\pm 0.01$
---
$1.72\pm 0.16$
| $0.14\pm 0.01$
---
$1.62\pm 0.15$
| $0.14\pm 0.01$
---
$1.62\pm 0.15$
CMD2-2004 |  | $0.08\pm 0.00$
---
$1.50\pm 0.13$
| $0.16\pm 0.01$
---
$1.60\pm 0.15$
| $0.35\pm 0.01$
---
$1.71\pm 0.16$
| $0.33\pm 0.01$
---
$1.71\pm 0.16$
| $0.17\pm 0.01$
---
$1.60\pm 0.15$
| $0.17\pm 0.01$
---
$1.60\pm 0.15$
CMD2-2006 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
CMD2-2007 |  | $0.08\pm 0.00$
---
$1.54\pm 0.05$
| $0.17\pm 0.00$
---
$1.61\pm 0.00$
| $0.37\pm 0.01$
---
$1.87\pm 0.07$
| $0.35\pm 0.01$
---
$1.86\pm 0.07$
| $0.19\pm 0.01$
---
$1.74\pm 0.06$
| $0.19\pm 0.01$
---
$1.74\pm 0.06$
BaBar2009 |  | $0.08\pm 0.00$
---
$2.16\pm 0.03$
| $0.19\pm 0.00$
---
$2.25\pm 0.00$
| $0.39\pm 0.00$
---
$2.39\pm 0.05$
| $0.37\pm 0.00$
---
$2.39\pm 0.05$
| $0.19\pm 0.00$
---
$2.26\pm 0.04$
| $0.19\pm 0.00$
---
$2.26\pm 0.04$
BESIII2020 |  | $0.07\pm 0.00$
---
$1.60\pm 0.13$
| $0.16\pm 0.00$
---
$1.60\pm 0.00$
| $0.36\pm 0.01$
---
$1.98\pm 0.18$
| $0.33\pm 0.01$
---
$1.96\pm 0.17$
| $0.16\pm 0.01$
---
$1.82\pm 0.16$
| $0.16\pm 0.01$
---
$1.82\pm 0.16$
KLOE2005 |  | $0.02\pm 0.00$
---
$1.78\pm 0.03$
| $0.16\pm 0.00$
---
$1.31\pm 0.04$
| $0.35\pm 0.00$
---
$1.36\pm 0.04$
| $0.33\pm 0.00$
---
$1.37\pm 0.04$
| $0.17\pm 0.00$
---
$1.32\pm 0.04$
| $0.17\pm 0.00$
---
$1.32\pm 0.04$
KLOE2009 |  | $0.07\pm 0.00$
---
$1.66\pm 0.02$
| $0.17\pm 0.00$
---
$1.75\pm 0.03$
| $0.35\pm 0.00$
---
$1.81\pm 0.03$
| $0.33\pm 0.00$
---
$1.82\pm 0.03$
| $0.17\pm 0.00$
---
$1.75\pm 0.03$
| $0.17\pm 0.00$
---
$1.75\pm 0.03$
KLOE2011 |  | $0.06\pm 0.00$
---
$1.57\pm 0.05$
| $0.15\pm 0.00$
---
$1.73\pm 0.00$
| $0.35\pm 0.00$
---
$1.82\pm 0.06$
| $0.32\pm 0.00$
---
$1.83\pm 0.06$
| $0.16\pm 0.00$
---
$1.74\pm 0.06$
| $0.16\pm 0.00$
---
$1.74\pm 0.06$
KLOE2013 |  | $0.07\pm 0.00$
---
$1.39\pm 0.10$
| $0.16\pm 0.00$
---
$1.58\pm 0.11$
| $0.34\pm 0.00$
---
$1.57\pm 0.12$
| $0.32\pm 0.00$
---
$1.59\pm 0.12$
| $0.17\pm 0.00$
---
$1.59\pm 0.11$
| $0.17\pm 0.00$
---
$1.59\pm 0.11$
KLOE2018 |  | $0.07\pm 0.00$
---
$1.58\pm 0.04$
| $0.16\pm 0.00$
---
$1.69\pm 0.05$
| $0.35\pm 0.00$
---
$1.70\pm 0.05$
| $0.33\pm 0.00$
---
$1.72\pm 0.05$
| $0.17\pm 0.00$
---
$1.69\pm 0.05$
| $0.17\pm 0.00$
---
$1.69\pm 0.05$
SND2005 |  | $0.09\pm 0.00$
---
$1.71\pm 0.04$
| $0.21\pm 0.00$
---
$1.89\pm 0.00$
| $0.41\pm 0.01$
---
$2.06\pm 0.05$
| $0.39\pm 0.01$
---
$2.05\pm 0.05$
| $0.22\pm 0.01$
---
$1.90\pm 0.04$
| $0.22\pm 0.01$
---
$1.90\pm 0.04$
SND2006 |  | $0.07\pm 0.00$
---
$1.68\pm 0.04$
| $0.19\pm 0.00$
---
$1.87\pm 0.04$
| $0.38\pm 0.01$
---
$2.02\pm 0.05$
| $0.36\pm 0.01$
---
$2.01\pm 0.05$
| $0.19\pm 0.00$
---
$1.87\pm 0.04$
| $0.19\pm 0.00$
---
$1.87\pm 0.04$
SND2020 |  | $0.08\pm 0.00$
---
$1.74\pm 0.05$
| $0.16\pm 0.00$
---
$1.86\pm 0.00$
| $0.39\pm 0.01$
---
$2.11\pm 0.07$
| $0.36\pm 0.01$
---
$2.08\pm 0.07$
| $0.18\pm 0.01$
---
$1.90\pm 0.06$
| $0.18\pm 0.01$
---
$1.90\pm 0.06$
Com. Dat. |  | $0.06\pm 0.00$
---
$1.76\pm 0.01$
| $0.16\pm 0.00$
---
$1.76\pm 0.01$
| $0.35\pm 0.00$
---
$1.84\pm 0.01$
| $0.33\pm 0.00$
---
$1.85\pm 0.01$
| $0.17\pm 0.00$
---
$1.77\pm 0.01$
| $0.17\pm 0.00$
---
$1.77\pm 0.01$

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