# Momentum dependence of $\rho-\omega$ mixing in the pion vector form factor
and its effect on $(g-2)_{\mu}$

Yun-Hua Chen<EMAIL_ADDRESS>Meng-Ge Qin  School of Mathematics and
Physics, University of Science and Technology Beijing, Beijing 100083, China

###### Abstract

The inclusion of the $\rho-\omega$ mixing effect is essential for a precise
description of the pion electromagnetic form factor in the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ process, which quantifies the two-pion
contribution to the anomalous magnetic moment of the muon $a_{\mu}$. In this
paper, we analyse the momentum dependence of the $\rho-\omega$ mixing by
considering loop contributions at the next-to-leading order in the $1/N_{C}$
expansion within the framework of resonance chiral theory. We revisit the work
[Y. H. Chen, D. L. Yao, and H. Q. Zheng, Commun. Theor. Phys. 69 (2018) 1],
considering the contribution arising from the kaon mass splitting in the kaon
loops and the latest experimental data. We perform two kinds of fits (with
momentum-independent or momentum-dependent $\rho-\omega$ mixing amplitude) to
describe the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ and
$\tau\rightarrow\nu_{\tau}2\pi$ data within the energy region of 600$\sim$900
MeV and the decay width of $\omega\rightarrow\pi^{+}\pi^{-}$, and compare
their results. Our findings indicate that both the momentum-independent and
momentum-dependent $\rho-\omega$ mixing schemes provide appropriate
descriptions of the data. However, the momentum-dependent scheme exhibits
greater self-consistency, considering the reasonable imaginary part of the
mixing matrix element $\Pi_{\rho\omega}$ obtained. Regarding the contribution
to the anomalous magnetic moment of the muon,
$a_{\mu}^{\pi\pi}|_{[0.6,0.9]\text{GeV}}$, the results obtained from the fits
considering the momentum-dependent $\rho-\omega$ mixing amplitude agree well
with those obtained without incorporating the momentum dependence of the
$\rho-\omega$ mixing, within the margin of errors. Furthermore, based on the
fitted values of the relevant parameters, we observe that the decay width of
$\omega\rightarrow\pi^{+}\pi^{-}$ is predominantly influenced by the
$\rho-\omega$ mixing effect.

##  I Introduction

The anomalous magnetic moment of the muon, denoted as $a_{\mu}=(g_{\mu}-2)/2$,
plays an crucial role in the precision tests of the Standard Model (SM)
Aoyama:2020ynm ; Colangelo:2022jxc . The long-standing discrepancy between the
SM prediction of $a_{\mu}$ and its experimental measurement recently has been
updated to 4.2 standard deviations Muong-2:2006rrc ; Muong-2:2021ojo  and it
has sparked numerous theoretical investigations. The SM uncertainty on
$a_{\mu}$ is dominated by hadronic vacuum polarization (HVP), with the largest
contribution originating from the $\pi\pi$ intermediate states, accounting for
over 70% of the HVP contribution. Theoretically, the two-pion low-energy
contribution to $a_{\mu}$ is expressed as an integral over the modulus squared
of the pion electromagnetic form factor, which can be extracted from the
$e^{+}e^{-}-$annihilation experiments. In principle, the two-pion contribution
to $a_{\mu}$ can be evaluated accurately as long as the experimental data of
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ are available everywhere at the required level
of precision. While it is known that a tension exists between the two most
precise measurements by BaBar and KLOE Collaborations: the BaBar data lie
systematically above the KLOE results in the dominant $\rho$ region.
Consequently, considerable efforts have been dedicated to finely describing
the pion electromagnetic form factor Colangelo:2018mtw ; Colangelo:2020lcg ;
Colangelo:2022prz ; Davier:2019can ; Qin:2020udp ; Yi:2021ccc ;
Ananthanarayan:2016mns . In the dominant $\rho$ region of the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ process, the isospin-breaking effect due to
$\rho-\omega$ mixing, which becomes enhanced by the small mass difference
between the $\rho$ and $\omega$ mesons, plays a significant role and needs to
be considered properly.

Usually, the momentum dependence of the $\rho-\omega$ mixing amplitude is
neglected, and a constant mixing amplitude is used to describe the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ data due to the narrowness of the $\omega$
resonance. The first study of the momentum dependence of the $\rho-\omega$
mixing amplitude was conducted by Ref. Goldman . Based on a quark loop
mechanism of the $\rho-\omega$ mixing, it was found that the mixing amplitude
significantly depends on momentum. Subsequently, the investigation of various
loop mechanisms for $\rho-\omega$ mixing was initiated in different models,
such as the global color model Mitchell , extended Nambu-Jona-Lasinio (NJL)
model Shakin ; Braghin:2020enw , the chiral constituent quark model MLYan98 ;
MLYan00 , and the hidden local symmetry model Benayoun00 ; Benayoun01 ;
Benayoun08 . In our pervious work Chen:2017jcw , we studied the $\rho-\omega$
mixing in a model independent way by invoking Resonance Chiral Theory
(R$\chi$T) Ecker . Guided by chiral symmetry and large $N_{C}$ expansion,
R$\chi$T provides us a reliable theoretical framework to study the dynamics
with both light flavor resonances and pseudo-Goldstone mesons in the
intermediate energy region Guo:2011pa ; Jamins ; Roig:2013baa ; chen2012 ;
chen2014 ; Chen:2014yta , and it has been successfully applied in the
calculation of $a_{\mu}$ in the SM Pich:2001pj ; Cirigliano:2002pv ;
Kampf:2011ty ; Roig:2014uja ; Guevara:2018rhj ; Roig:2019reh ; Miranda:2020wdg
; Qin:2020udp ; Arroyo-Urena:2021nil . In Ref. Chen:2017jcw , we calculated
the one-loop contributions to the $\rho-\omega$ mixing, which are at the next-
to-leading order (NLO) in the $1/N_{C}$ expansion Hooft ; Rosell04 ;
Cirigliano:2003yq ; Guo:2014yva ; Roig:2013baa . In this article, we update
the previous work of Ref. Chen:2017jcw  by incorporating the contribution
arising from the kaon mass splitting in the kaon loops.

Moreover, we focus on analysing the impact of the momentum dependence of
$\rho-\omega$ mixing on describing the pion vector form factor data and its
contribution to $a_{\mu}$. Specifically, we perform two types of fits (with
momentum-independent or momentum-dependent $\rho-\omega$ mixing amplitude)
describing the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ and and
$\tau\rightarrow\nu_{\tau}2\pi$ data in the energy region of 600$\sim$900 MeV,
the decay width of $\omega\rightarrow\pi^{+}\pi^{-}$, and compare their
results. Our fit results demonstrate that both the momentum-independent and
momentum-dependent $\rho-\omega$ mixing schemes can effectively describe the
data, while the momentum-dependent scheme exhibits greater self-consistency
due to the reasonable imaginary part of the extracted mixing matrix element
$\Pi_{\rho\omega}$. Regarding the contribution to the anomalous magnetic
moment of the muon, $a_{\mu}^{\text{HVP,LO}}[\pi^{+}\pi^{-}]$, evaluated
between 0.6 GeV and 0.9 GeV, the results obtained from fits considering the
momentum-dependent $\rho-\omega$ mixing amplitude are in good agreement with
those from fits that do not include the momentum dependence of $\rho-\omega$
mixing, within the margin of errors.

This paper is organized as follows. In Sec. II, we introduce the description
of $\rho-\omega$ mixing and elaborate on the calculation of $\rho-\omega$
mixing amplitude up to the next-to-leading order in the $1/N_{C}$ expansion.
In Sec. III, the fit results are shown and the related phenomenologies are
discussed. A summary is given in Sec. IV.

##  II Calculations in resonance chiral theory

In the isospin basis $|I,I_{3}\rangle$, we define the pure isospin states
$|\rho_{I}\rangle\equiv|1,0\rangle$ and $|\omega_{I}\rangle\equiv|0,0\rangle$.
The mixing between the isospin states of $|\rho_{I}\rangle$ and
$|\omega_{I}\rangle$ can be implemented by considering the self-energy matrix

$\displaystyle\Pi_{\mu\nu}=T_{\mu\nu}\left(\begin{array}[]{cc}\Pi_{\rho\rho}(s)&\Pi_{\rho\omega}(s)\\\
\Pi_{\rho\omega}(s)&\Pi_{\omega\omega}(s)\\\ \end{array}\right)\,,$ (3)

with $T_{\mu\nu}\equiv g_{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}$ and $s\equiv
p^{2}$. The none-zero off-diagonal matrix element $\Pi_{\rho\omega}(s)$
contains the information of $\rho-\omega$ mixing. The mixing between the
physical states of $\rho^{0}$ and $\omega$, is obtainable by introducing the
following relation

$\displaystyle\left(\begin{array}[]{c}\rho^{0}\\\
\omega\end{array}\right)=C\left(\begin{array}[]{c}\rho_{I}\\\
\omega_{I}\end{array}\right)\,,\qquad
C=\left(\begin{array}[]{cc}1&-\epsilon_{1}\\\
\epsilon_{2}&1\end{array}\right),$ (10)

where $\epsilon_{1}$ and $\epsilon_{2}$ are the mixing parameters. The matrix
of dressed propagators corresponding to physical states is diagonal Connell97
,

$\displaystyle\left(\begin{array}[]{cc}1/s_{\rho}&0\\\
0&1/s_{\omega}\end{array}\right)=C\left(\begin{array}[]{cc}1/s_{\rho}&\Pi_{\rho\omega}/s_{\rho}s_{\omega}\\\
\Pi_{\rho\omega}/s_{\rho}s_{\omega}&1/s_{\omega}\end{array}\right)C^{-1},$
(15)

where the abbreviations $s_{\rho}$ and $s_{\omega}$ are defined by the
following:

$\displaystyle s_{\rho}$ $\displaystyle\equiv$ $\displaystyle
s-\Pi_{\rho\rho}(s)-m_{\rho}^{2},$ $\displaystyle s_{\omega}$
$\displaystyle\equiv$ $\displaystyle s-\Pi_{\omega\omega}(s)-m_{\omega}^{2}.$
(16)

The information of $\rho-\omega$ mixing is encoded in the off-diagonal element
of the self-energy matrix, decomposed as follows:

$\displaystyle\Pi_{\rho\omega}(s)=\Delta_{ud}\,S_{\rho\omega}(s)+4\pi\alpha\,E_{\rho\omega}(s)\
,$ (17)

where $\Delta_{ud}=m_{u}-m_{d}$ is the mass difference between $u$ and $d$
quarks, and $\alpha$ is the fine-structure constant. $S_{\rho\omega}(s)$ and
$E_{\rho\omega}(s)$ represent the structure functions of the strong and
electromagnetic interactions, respectively. In this work, the diagrams in Fig.
1 are calculated in R$\chi$T up to NLO in $1/N_{C}$ expansion.

In R$\chi$T, the vector resonances can be described in terms of antisymmetric
tensor fields with the normalization

$\displaystyle\langle
0|V_{\mu\nu}|V,p\rangle=iM_{V}^{-1}\\{p_{\mu}\epsilon_{\nu}(p)-p_{\nu}\epsilon_{\mu}(p)\\}\,$
(18)

where $\epsilon_{\mu}$ denotes the polarization vector. Here the vector mesons
are collected in a $3\times 3$ matrix

$V_{\mu\nu}=\left({\begin{array}[]{*{3}c}\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&{\rho^{+}}&{K^{\ast+}}\\\
{\rho^{-}}&-\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&{K^{\ast
0}}\\\ {K^{\ast-}}&{\overline{K}^{\ast 0}}&-\phi\\\
\end{array}}\right)_{\mu\nu}\,.$ (19)

The effective Lagrangian for the leading order strong isospin-breaking effect,
corresponding to the tree-level contribution diagram (a) in Fig. 1, reads
Urech ; Cirigliano:2006hb

$\displaystyle\mathcal{L}_{2}^{\rho\omega}=\lambda_{6}^{VV}\langle
V_{\mu\nu}V^{\mu\nu}\chi_{+}\rangle\,,$ (20)

with $\chi_{+}=u^{+}\chi u^{+}+u\chi^{+}u$ and $\chi=2B_{0}(s+ip)$. The
pseudo-Goldstone bosons originating from the spontaneous breaking of chiral
symmetry, can be filled nonlinearly into

$u_{\mu}=i\left(u^{\dagger}\partial_{\mu}u\,-\,u\partial_{\mu}u^{\dagger}\right)\,,\qquad
u=\exp\Big{(}\frac{i\Phi}{\sqrt{2}F}\Big{)}\,,$ (21)

with the Goldstone fields

$\displaystyle\Phi$
$\displaystyle=\begin{pmatrix}{\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta}&{\pi^{+}}&{K^{+}}\\\
{\pi^{-}}&{-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta}&{K^{0}}\\\
{K^{-}}&{\bar{K}^{0}}&{-\frac{2}{\sqrt{6}}\eta}\\\ \end{pmatrix}.$ (22)

Here $F$ is the pion decay constant. Considering the mass relations of the
vector mesons at $O(p^{2})$ in terms of the quark counting rule, the value of
the coupling constant is determined: $\lambda_{6}^{VV}=1/8$ Urech ;
Cirigliano:2006hb . Thus the tree-level strong contribution turns out to be

$\displaystyle S_{\rho\omega}^{(a)}=2\,M_{V}\ .$ (23)

The Lagrangian describing the interactions between $V_{\mu\nu}$ and
electromagnetic fields or Goldstone bosons are given by

$\displaystyle\mathcal{L}_{2}(V)$ $\displaystyle=$
$\displaystyle\frac{F_{V}}{2\sqrt{2}}\langle
V_{\mu\nu}f_{+}^{\mu\nu}\rangle+\frac{iG_{V}}{\sqrt{2}}\langle
V_{\mu\nu}u^{\mu}u^{\nu}\rangle\ ,$ (24)

with the relevant building blocks defined by

$\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle
i[u^{+}(\partial_{\mu}\ -ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{+}]\,,$
$\displaystyle f_{\pm}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
uF_{L}^{\mu\nu}u^{+}\pm u^{+}F_{R}^{\mu\nu}u\,.$ (25)

Here $F_{L,R}^{\mu\nu}$ are field strength tensors composed of the left and
right external sources $l_{\mu}$ and $r_{\mu}$, and $F_{V}$ and $G_{V}$ are
real resonance couplings constants. The tree-level electromagnetic
contribution from diagram (b) in Fig. 1 can be calculated by using the
Lagrangian in Eq. (24):

$\displaystyle E_{\rho\omega}^{(b)}=\frac{F_{\rho}F_{\omega}}{3}\,.$ (26)

The physical decay constants $F_{\rho}$ and $F_{\omega}$ have been employed in
the amplitude, and are differentiated by means of isospin breaking.

The loop contributions of diagrams (d)-(i) in Fig. 1 have been extensively
discussed in our previous work Chen:2017jcw . However, a noteworthy
distinction in our current study is the inclusion of the contribution from
diagram (c), which arises from the kaon mass splitting within the kaon loops.
To ensure comprehensiveness, we present the expressions for the loop
contributions in the Appendix A. Furthermore, it should be noted that the
ultimate expression for the renormalised mixing amplitude
${\Pi}_{\rho\omega}(p^{2})$ is presented in Equation (A23).

Figure 1:  Feynman diagrams contributing to $\rho-\omega$ mixing.

##  III Phenomenological discussion

The mass and width of the $\rho$ meson are conventionally determined by
fitting to the experimental data of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ and
$\tau\rightarrow\nu_{\tau}2\pi$ ParticleDataGroup:2020ssz , where various
mechanisms are used to describe the $\rho-\omega$ mixing effect. To prevent
interference caused by their $\rho-\omega$ mixing mechanisms, we treat the
mass $M_{\rho}$, the relevant couplings $G_{\rho}$ and $F_{\rho}$ as free
parameters in our fit. Regarding its width, the energy-dependent form is
constructed in the way introduced in Dumm :

$\displaystyle\Gamma_{\rho}(s)=\frac{sM_{\rho}}{96\pi
F^{2}}\left[\sigma_{\pi}^{3}\theta(s-4m_{\pi}^{2})+\frac{1}{2}\sigma_{K}^{3}\theta(s-4m_{K}^{2})\right]\,,$
(27)

where $\sigma_{P}\equiv\sqrt{1-4m_{P}^{2}/s}$, and $\theta(s)$ is the step
function.

For the $\omega$ mass, it has been pointed out in Refs. Colangelo:2018mtw ;
Colangelo:2022prz  that the result determined from
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ is inconsistent with that from particle
data group (PDG) ParticleDataGroup:2020ssz , primarily determined by
experiments involving $e^{+}e^{-}\rightarrow 3\pi$ and
$e^{+}e^{-}\rightarrow\pi^{0}\gamma$. Therefore, we treated the $\omega$ mass
and width as free parameters and estimated them by fitting in our programme.
The physical coupling $F_{\omega}$ can be determined from the decay width of
$\omega\rightarrow e^{+}e^{-}$. Using the Lagrangian formula in Eq. (24), one
can derive the decay width

$\displaystyle\Gamma_{\omega}^{e^{+}e^{-}}=\frac{4\alpha^{2}\pi
F_{\omega}^{2}(2m_{e}^{2}+M_{\omega}^{2})\sqrt{M_{\omega}^{2}-4m_{e}^{2}}}{27M_{\omega}^{4}}\,,$
(28)

and obtain the expression of $F_{\omega}$. With the decay widths given above,
$s_{\rho}$ and $s_{\omega}$ in Eq. (II) can be rewritten as

$\displaystyle s_{\rho}$ $\displaystyle\simeq$ $\displaystyle
s-M_{\rho}^{2}+iM_{\rho}\Gamma_{\rho}(s)\ ,$ $\displaystyle s_{\omega}$
$\displaystyle\simeq$ $\displaystyle
s-M_{\omega}^{2}+iM_{\omega}\Gamma_{\omega}\ .$ (29)

The pion form-factor in $\tau\rightarrow\nu_{\tau}2\pi$ decay, irrelevant to
$\rho-\omega$ mixing effect, were thoroughly studied in Refs. (Arganda:2008jj,
; Guerrero:1997ku, ; Pich:2001pj, ; Miranda:2018cpf, )

$\displaystyle
F_{\pi}^{\tau}(s)=\Big{(}1-\frac{G_{\rho}F_{\rho}s}{F^{2}}\frac{1}{s_{\rho}}\Big{)}\times\exp\left[\frac{-s}{96\pi^{2}F^{2}}\left({\rm
Re}\left[A[m_{\pi},M_{\rho},s]+\frac{1}{2}A[m_{K},M_{\rho},s]\right]\right)\right]\,.$
(30)

The function

$\displaystyle
A\left(m_{P},\mu,s\right)=\ln\left(m_{P}^{2}/\mu^{2}\right)+\frac{8m_{P}^{2}}{s}-\frac{5}{3}+\sigma_{P}^{3}\ln\left(\frac{\sigma_{P}+1}{\sigma_{P}-1}\right)\,.$
(31)

To incorporate isospin-breaking effects, one approach is to multiply
$|F_{\pi}^{\tau}(s)|^{2}$ by the factor $S_{EW}G_{EM}(s)$, where
$S_{EW}=1.0233$ corresponds to the short distance correction Davier03 .
Additionally, $G_{EM}(s)$ accounts for the long-distance radiative correction,
as described in Flores . Specifically, in our fit of the
$\tau\rightarrow\nu_{\tau}2\pi$ decay data we perform the following
substitution

$\displaystyle|F_{\pi}^{\tau}(s)|^{2}\Rightarrow
S_{EW}G_{EM}(s)|F_{\pi}^{\tau}(s)|^{2}.$ (32)

The pion form-factor in $e^{+}e^{-}$ annihilation reads

$\displaystyle F_{\pi}^{ee}(s)$
$\displaystyle=\Big{(}1-\frac{G_{\rho}F_{\rho}s}{F^{2}}\frac{1}{s_{\rho}}-\frac{G_{\rho}F_{\omega}s}{3F^{2}}\frac{1}{s_{\omega}}\Pi_{\rho\omega}\frac{1}{s_{\rho}}-\frac{4\sqrt{2}aB_{0}F_{\omega}(m_{u}-m_{d})s}{3F^{2}}\frac{1}{s_{\omega}}\Big{)}$
(33) $\displaystyle\times\exp\left[\frac{-s}{96\pi^{2}F^{2}}\left({\rm
Re}\left[A[m_{\pi},M_{\rho},s]+\frac{1}{2}A[m_{K},M_{\rho},s]\right]\right)\right]\,.$

As defined in the appendix A.2, the parameter $a$ is associated with the
combined coupling constant of the direct $\omega\pi\pi$ interaction. In the
first bracket of Eq. (33), the second term corresponds to the contribution
from the $\rho\pi\pi$ coupling, the third term represents the contribution of
$\rho-\omega$ mixing, and the fourth term corresponds to contribution from the
direct isospin-breaking coupling of $\omega$ to the pion pair.

The leading order contribution of the $\pi\pi(\gamma)$ intermediate state to
the anomalous magnetic moment of the muon is given by Gourdin:1969dm

$a_{\mu}^{\pi\pi(\gamma),\text{LO}}=\left(\frac{\alpha
m_{\mu}}{3\pi}\right)^{2}\int_{4m_{\pi}^{2}}^{\infty}\mathrm{d}s\frac{\hat{K}(s)}{s^{2}}R_{\pi\pi(\gamma)}(s)\text{\
,}$ (34)

where

$\displaystyle
R_{\pi\pi(\gamma)}(s)=\frac{3s}{4\pi\alpha^{2}}\,\sigma^{(0)}\left(e^{+}e^{-}\rightarrow\pi\pi(\gamma)\right)\
,$ (35)

and the kernel function is defined as follows:

$\hat{K}(s)=\frac{3s}{m_{\mu}^{2}}\left.\left[\frac{\left(1+x^{2}\right)(1+x)^{2}}{x^{2}}\left(\ln(1+x)-x+\frac{x^{2}}{2}\right)\right.\left.+\frac{x^{2}}{2}\left(2-x^{2}\right)+\frac{1+x}{1-x}x^{2}\ln
x\right]\text{\ ,}\right.$ (36)

with

$\displaystyle x=\frac{1-\beta_{\mu}(s)}{1+\beta_{\mu}(s)},$
$\displaystyle\qquad\beta_{\mu}(s)=\sqrt{1-\frac{4m_{\mu}^{2}}{s}}\ .$ (37)

Note that in the formula for $a_{\mu}^{\pi\pi(\gamma),\text{LO}}$ in Eq. (34),
the integration is performed from 4$m_{\pi}^{2}$ to $\infty$. In this work, we
focus on the momentum dependence of $\rho-\omega$ mixing. Therefore, we only
describe the pion vector form factor up to 900 MeV. To extend the study by
considering higher energies, we must consider the effects of the excited
resonances, such as $\rho^{\prime}(1450)$ and $\rho^{\prime\prime}(1700)$,
etc. However, these are beyond the scope of this work. It is interesting to
note that the $1/s^{2}$ enhancement factor in Eq. (34) gives higher weight to
the lowest lying resonance $\rho(770)$ that couples strongly to
$\pi^{+}\pi^{-}$.

The bare cross section including final-state radiation takes the following
form Gluza:2002ui ; Czyz:2004rj ; Bystritskiy:2005ib ; Colangelo:2018mtw

$\displaystyle\sigma^{(0)}(e^{+}e^{-}\to\gamma^{*}\to\pi^{+}\pi^{-}(\gamma))=\Big{[}1+\frac{\alpha}{\pi}\eta(s)\Big{]}\frac{\pi\big{|}\alpha(s)\big{|}^{2}}{3s}\sigma_{\pi}^{3}(s)\big{|}F_{\pi}^{ee}(s)\big{|}^{2}\frac{s+2m_{e}^{2}}{s\sigma_{e}(s)},$
(38)

where

$\displaystyle\begin{split}\eta(s)&=\frac{3(1+\sigma_{\pi}^{2}(s))}{2\sigma_{\pi}^{2}(s)}-4\log\sigma_{\pi}(s)+6\log\frac{1+\sigma_{\pi}(s)}{2}+\frac{1+\sigma_{\pi}^{2}(s)}{\sigma_{\pi}(s)}F(\sigma_{\pi}(s))\\\
&\quad-\frac{(1-\sigma_{\pi}(s))\big{(}3+3\sigma_{\pi}(s)-7\sigma_{\pi}^{2}(s)+5\sigma_{\pi}^{3}(s)\big{)}}{4\sigma_{\pi}^{3}(s)}\log\frac{1+\sigma_{\pi}(s)}{1-\sigma_{\pi}(s)},\\\
F(x)&=-4\mathrm{Li}_{2}(x)+4\mathrm{Li}_{2}(-x)+2\log
x\log\frac{1+x}{1-x}+3\mathrm{Li}_{2}\Big{(}\frac{1+x}{2}\Big{)}-3\mathrm{Li}_{2}\Big{(}\frac{1-x}{2}\Big{)}+\frac{\pi^{2}}{2},\\\
\mathrm{Li}_{2}(x)&=-\int_{0}^{x}dt\frac{\log(1-t)}{t}.\end{split}$ (39)

The experimental data considered in this work are the pion form factor
$F_{\pi}^{ee}(s)$ of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ process
measured by the OLYA OLYACMD , CMD CMD2007 , BaBar BaBar:2012bdw , BESIII
BESIII , KLOE KLOE-2:2017fda , CLEO Xiao:2017dqv , and SND SND:2020nwa
Collaborations, the form factor $F_{\pi}^{\tau}(s)$ of
$\tau\rightarrow\nu_{\tau}2\pi$ decay measured by the ALEPH ALEPH  and CLEO
CLEO  Collaborations, and the decay width of $\omega\rightarrow\pi^{+}\pi^{-}$
ParticleDataGroup:2020ssz . Note that in the experimentally published form
factor data $F_{\pi}^{ee}(s)$, the vacuum polarization effects have been
excluded through the subtraction of the hadronic running of $\alpha(s)$, and
thus in our fitting of the form factor data $F_{\pi}^{ee}(s)$ the one-photon-
reducible Fig. 1(b) should not be considered. Since we focus on the analysis
of the $\rho-\omega$ mixing effect, we only take into account the form factors
$F_{\pi}^{ee}(s)$ and $F_{\pi}^{\tau}(s)$ data in the energy region of
600$\sim$900 MeV. Note that for the pion form factor $F_{\pi}^{ee}(s)$, a
tension between the two most precise measurements by BaBar and KLOE is
observed in at and above the $\rho$ peak region, while the other measurements
are consistent with theirs within the given uncertainty. To demonstrate the
impact of the momentum dependence of $\rho-\omega$ mixing and to avoid the
tension between the BaBar and KLOE data, we conduct four separate fits.
Specifically, in Fits Ia and Ib, we fit all data sets but BaBar with momentum-
independent $\Pi_{\rho\omega}$ and momentum-dependent $\Pi_{\rho\omega}$,
respectively. In Fits IIa and IIb, we fit all data sets but KLOE with
momentum-independent $\Pi_{\rho\omega}$ and momentum-dependent
$\Pi_{\rho\omega}$, respectively. Fits Ia and IIa involve eight free
parameters: $M_{\rho}$, $G_{\rho}$, $F_{\rho}$, $M_{\omega}$,
$\Gamma_{\omega}$, $a$, the real and imaginary part of constant
$\Pi_{\rho\omega}$. There are nine free parameters in Fits Ib and IIb:
$M_{\rho}$, $G_{\rho}$, $F_{\rho}$, $M_{\omega}$, $\Gamma_{\omega}$, $a$,
$X_{W}^{r}$, $X_{Z}^{r}$, and $X_{R}^{r}$. As defined in the appendix A,
$X_{W}^{r}$, $X_{Z}^{r}$, and $X_{R}^{r}$ are the corresponding parameters for
the counterterms.

Figure 2:  Fit results of the pion form factor in the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ process (left panel) and the $\tau\rightarrow\nu_{\tau}2\pi$ process (right panel) in the energy region of 600$\sim$900 MeV. The data of $e^{+}e^{-}$ annihilation are taken from the the OLYA (Gray) OLYACMD , CMD (Yellow) CMD2007 , BaBar (Blue) BaBar:2012bdw , BESIII (Green) BESIII , KLOE (Cyan) KLOE-2:2017fda , CLEO (Magenta) Xiao:2017dqv , and SND (Orange) SND:2020nwa  Collaborations. The $\tau$ decay data are taken from the ALEPH (Orange) ALEPH  and CLEO (Green) CLEO  collaborations. Fits Ia and Ib fit all data sets but BaBar (top), Fits IIa and IIb fit all data sets but KLOE (bottom). Fits Ia and IIa use momentum-independent $\Pi_{\rho\omega}$ and are shown as the red dashed lines. Fits Ib and IIb use momentum-dependent $\Pi_{\rho\omega}$ and are shown as the black solid lines. The vertical lines lie at $M_{\rho}$, $M_{\omega}$, and $2M_{\omega}-M_{\rho}$ (from left to right).  Table 1:  The fit results of the parameters. Fits Ia and Ib fit all data sets but BaBar, Fits IIa and IIb fit all data sets but KLOE. Fits Ia and IIa use momentum-independent $\Pi_{\rho\omega}$, while Fits Ib and IIb use momentum-dependent $\Pi_{\rho\omega}$. | Fit Ia | Fit Ib | Fit IIa | Fit IIb
---|---|---|---|---
$M_{\rho}$ [MeV] | $775.35\pm 0.10$ | $775.68\pm 0.12$ | $775.45\pm 0.10$ | $775.55\pm 0.11$
$G_{\rho}$ [MeV] | $55.25\pm 0.09$ | $55.74\pm 0.08$ | $54.21\pm 0.09$ | $55.03\pm 0.07$
$F_{\rho}$ [MeV] | $152.65\pm 0.29$ | $151.40\pm 0.21$ | $155.65\pm 0.23$ | $153.38\pm 0.31$
$M_{\omega}$ [MeV] | $782.59\pm 0.13$ | $782.68\pm 0.12$ | $782.39\pm 0.11$ | $782.45\pm 0.11$
$\Gamma_{\omega}$ [MeV] | $8.97\pm 0.27$ | $9.03\pm 0.26$ | $8.04\pm 0.16$ | $8.16\pm 0.17$
$a~{}[\text{GeV}^{-1}]$ | $-0.0020\pm 0.0150$ | $-0.0054\pm 0.0010$ | $-0.1066\pm 0.0152$ | $-0.0067\pm 0.0009$
$\text{Re}(\Pi_{\rho\omega})[\text{MeV}^{2}]$ | $-3372\pm 112$ | - | $-3799\pm 85$ | -
$\text{Im}(\Pi_{\rho\omega})[\text{MeV}^{2}]$ | $296\pm 669$ | - | $-4544\pm 704$ | -
$X_{W}^{r}~{}[\text{GeV}^{-6}]$ | - | $-0.141\pm 0.013$ | - | $-0.177\pm 0.008$
$X_{Z}^{r}~{}[\text{GeV}^{-4}]$ | - | $0.195\pm 0.016$ | - | $0.303\pm 0.007$
$X_{R}^{r}~{}[\text{GeV}^{-2}]$ | - | $-0.081\pm 0.006$ | - | $-0.133\pm 0.003$
${\chi^{2}}/{\rm d.o.f.}$ | $\frac{410.7}{(238-8)}=1.79$ | $\frac{405.8}{(238-9)}=1.77$ | $\frac{392.2}{(341-8)}=1.18$ | $\frac{394.6}{(341-9)}=1.19$
$a_{\mu}^{\pi\pi}|_{[0.6,0.9]\text{GeV}}[\times 10^{10}]$ | $367.72\pm 1.07$ | $367.80\pm 2.92$ | $375.41\pm 1.03$ | $375.29\pm 2.21$

In Fig. 2, the fitted results of the fits using momentum-independent
$\Pi_{\rho\omega}$ (Fits Ia and IIa) and momentum-dependent $\Pi_{\rho\omega}$
(Fits Ib and IIb) are shown as red dotted lines and black solid lines,
respectively. The fitted parameters as well as the $\chi^{2}/\text{d.o.f.}$
are given in Table 1. It is intriguing to compare the results obtained from
fits utilising momentum-independent $\Pi_{\rho\omega}$ and momentum-dependent
$\Pi_{\rho\omega}$ for the same datasets. Specifically, we compare Fit Ia and
Ib, Fit IIa and IIb, respectively. When examining the pion form factors
$|F_{\pi}^{ee}(s)|^{2}$ and $|F_{\pi}^{\tau}(s)|^{2}$, we observe that the
differences between the theoretical predictions of the fits using momentum-
independent $\Pi_{\rho\omega}$ and the corresponding ones using momentum-
dependent $\Pi_{\rho\omega}$ are tiny. Furthermore, one notes that for the
pion form factor $|F_{\pi}^{ee}(s)|^{2}$ in Fits Ia and Ib, the theoretical
predictions are much higher than the KLOE data at the $\rho$ peak, and these
deviations contribute a lot to their value of $\chi^{2}$. Thus we conclude
that both the momentum-independent $\Pi_{\rho\omega}$ and momentum-dependent
$\Pi_{\rho\omega}$ can describe the data well, and the discordances among
different collaborations contribute significantly to $\chi^{2}$ values in the
fits.

In the last line of Table 1, we give the results of
$a_{\mu}^{\text{HVP,LO}}[\pi^{+}\pi^{-}]$, evaluated between 0.6 GeV and 0.9
GeV. The differences between the results using the momentum-independent
$\Pi_{\rho\omega}$ and the results using the momentum-dependent
$\Pi_{\rho\omega}$ for the same datasets, namely the differences between Fits
Ia and Ib, Fits IIa and IIb, respectively, are negligible.

Figure 3:  The momentum dependence of the mixing amplitudes
$\Pi_{\rho\omega}(s)$. The black solid and red dot-dashed lines correspond to
the real part of $\Pi_{\rho\omega}(s)$ in Fits Ib and IIb, respectively. The
and blue dashed and magenta dash-dot-dotted lines correspond to the imaginary
part of $\Pi_{\rho\omega}(s)$ in Fits Ib and IIb, respectively. The vertical
line lies at $M_{\omega}$.

In Fig. 3, we plot the real and imaginary parts of the mixing amplitudes
$\Pi_{\rho\omega}(s)$ in Fits Ib and IIb. It is found that the real part is
dominant within the $\rho-\omega$ mixing region. The real part in Fit IIb
demonstrates a significant momentum dependence, whereas the real part in Fit
Ib displays a smooth momentum dependence. Additionally, it should be noted
that the real parts of the two fits nearly reach the same point at
$s^{1/2}=M_{\omega}$. In comparison to the real part, the imaginary part is
rather small. At $s=M_{\omega}^{2}$, in Fit Ib the mixing amplitude
$\Pi_{\rho\omega}(M_{\omega}^{2})=(-3405.0+62.1i)$ MeV2, and in Fit IIb
$\Pi_{\rho\omega}(M_{\omega}^{2})=(-3316.3+113.7i)$ MeV2. The smallness of the
imaginary part is consistent with the findings reported in Refs. Renard ;
Connell98 , though therein the effect of direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ was not considered. It is worth
mentioning that larger imaginary part is obtained in Mitchell ; MLYan00  by
using global color model and a chiral constituent quark model, respectively.
By utilising our fitted parameter results, we proceed to calculate the ratio
of the two-pion couplings associated with the isospin-pure $\omega$ and $\rho$

$\displaystyle
G=\frac{g_{\omega_{I}\pi\pi}}{g_{\rho_{I}\pi\pi}}=\frac{4\sqrt{2}aB_{0}(m_{u}-m_{d})}{G_{\rho}}.$
(40)

The results are $G=(2.1\pm 1.1)\times 10^{-3}$ in Fit Ib, and $G=(2.6\pm
1.2)\times 10^{-3}$ in Fit IIb. Note the value of $G$ is expected to be of
order $\alpha=1/137$ in Ref. Renard . The central values of our results of $G$
agree with the expectation in Ref. Renard , while they are lower than other
two estimations, namely $G=(5.0\pm 1.7_{\text{model}}\pm
1.0_{\text{data}})\times 10^{-2}$ in Wolfe:2010gf , and $G=(3.47\pm
0.64)\times 10^{-2}$ in Benayoun:2012etq . As shown in Table 1, the
differences of ${\chi^{2}}/{\rm d.o.f.}$ between the momentum-dependent fits
and the momentum-independent fits for the same data sets are tiny. The
$\chi^{2}$ of Fit IIa is slightly lower than the $\chi^{2}$ of Fit IIb, though
Fit IIa contains one less fitting parameter than Fit IIb. One observes that
the magnitude of the imaginary part of $\Pi_{\rho\omega}$ in Fit IIa is
significantly greater than those in other three fits. In our framework, the
imaginary part of $\Pi_{\rho\omega}$ arises from the $\pi^{0}\gamma$ and
$\pi\pi$ real intermediate states. By considering the decay widths of
$\omega\rightarrow\pi^{0}\gamma$ and $\rho\rightarrow\pi^{0}\gamma$, the
imaginary part of $\Pi_{\rho\omega}$ contributed from the $\pi^{0}\gamma$
intermediate state can be estimated to be approximately $-$150 MeV2 Renard ;
Colangelo:2022prz . If one use the estimated ratio of the two-pion couplings
of the isospin-pure $\omega$ and $\rho$: $G\sim\alpha=1/137$ Renard , one can
obtain that the $\pi\pi$ intermediate state contribution to the imaginary part
of $\Pi_{\rho\omega}$ to be in the order of several hundred MeV2. In our
momentum-dependent scheme, the imaginary part of $\Pi_{\rho\omega}$ due to the
$\pi^{0}\gamma$ and $\pi\pi$ intermediate states are explicitly computed, and
both the numerical results of Im$\Pi_{\rho\omega}$ in Fits Ib and IIb are of
the order of one hundred MeV2 as expected. While in the momentum-independent
Fits Ia and IIa, the imaginary part of $\Pi_{\rho\omega}$ is a free fitting
parameter. As can be seen in Table 1, the fitted results of
Im$\Pi_{\rho\omega}$ and the parameter “$a$” in Fit IIa are unreasonably
large. The fitted results of Im$\Pi_{\rho\omega}$ and the parameter “$a$” in
Fit Ia exhibit large error bars. Consequently, we conclude that though both
the momentum-independent and momentum-dependent $\rho-\omega$ mixing schemes
can describe the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ data well, the
momentum-dependent $\rho-\omega$ mixing scheme is more self-consistent,
especially given the reasonable imaginary part of $\Pi_{\rho\omega}$
extracted.

We would like to point out that the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling is generally an unknown
quantity, and it impacts $F_{\pi}^{ee}(s)$ in two ways, both through the third
term in the first bracket of Eq. (33) as it appearing as real intermediate
state in the contributions to $\Pi_{\rho\omega}$ and through the fourth term
in that bracket. Conventionally, the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ is assumed to cancel out in
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ due to the fact that $\omega$ and $\rho$
are quasidegenerate and that the 2$\pi$ channel dominates the $\rho$ decay
Renard . While theoretical models that do not neglect the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling may be more comprehensive,
especially given the availability of high-precision data nowadays. Note that
in Refs. Colangelo:2018mtw ; Colangelo:2022prz  the pion form factor has been
studied in a model-independent way using dispersion theory. The $\rho-\omega$
mixing is subsumed in one parameter $\epsilon_{\omega}$, which should contain
a small imaginary part originating from the radiative intermediate states
(with an estimated phase of approximately 4 degrees). Note since the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling is not considered in Refs.
Colangelo:2018mtw ; Colangelo:2022prz , the $\epsilon_{\omega}$ term there is
actually a combination of the $\rho-\omega$ mixing and direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$. Therefore, it cannot be directly
compared to the $\Pi_{\rho\omega}$ discussed in this context. (At
$s=M_{\omega}^{2}$, our $\Pi_{\rho\omega}(M_{\omega}^{2})$ in Fits Ib and IIb
contains negative phase.) Nevertheless, the ratio between the
on-$\omega$-mass-shell $\gamma^{\ast}\rightarrow\omega\rightarrow\pi\pi$
transition amplitude and $\gamma^{\ast}\rightarrow\rho\rightarrow\pi\pi$
transition amplitude (without $\pi\pi$ final state interaction) should be
model independent. With $s=M_{\omega}^{2}$, the ratio between the second term
and the first term in Eq. (2.5) of Colangelo:2022prz  yields
$R_{\omega\rho}=\text{Amplitude}({\gamma^{\ast}\rightarrow\omega\rightarrow\pi\pi})/\text{Amplitude}({\gamma^{\ast}\rightarrow\rho\rightarrow\pi\pi})=(0.178\pm
0.003)\times e^{i(4.66\pm 1.13)^{\circ}}$, using Re$\epsilon_{\omega}=(1.97\pm
0.03)\times 10^{-3}$ and $\delta_{\epsilon}=(4.5\pm 1.2)^{\circ}$ obtained
therein. One observes that the difference between the phase of
$R_{\rho\omega}(M_{\omega}^{2})$ and $\delta_{\epsilon}$ is tiny. In the
present work, the ratio between the sum of the third term and fourth terms and
the sum of the first and second terms in the first bracket of Eq. (33)
predicts $R_{\omega\rho}=(0.155\pm 0.002)\times e^{i(5.80\pm 1.71)^{\circ}}$
and $R_{\omega\rho}=(0.150\pm 0.002)\times e^{i(3.67\pm 1.71)^{\circ}}$ in Fit
Ib and IIb, respectively. One can see that our results of $R_{\omega\rho}$
roughly agree with that in Colangelo:2022prz .

Using the central values of the fitted parameters of our best fit (Fit IIb) in
Table 1, we calculate the decay width of $\omega\rightarrow\pi^{+}\pi^{-}$

$\displaystyle\Gamma_{\omega\rightarrow\pi^{+}\pi^{-}}$ $\displaystyle=$
$\displaystyle\frac{1}{192\pi
F^{4}}(M_{\omega}^{2}-4m_{\pi}^{2})^{\frac{3}{2}}\times\Big{(}\exp\left[\frac{-M_{\omega}^{2}}{96\pi^{2}F^{2}}\left({\rm
Re}\left[A[m_{\pi},M_{\rho},M_{\omega}^{2}]+\frac{1}{2}A[m_{K},M_{\rho},M_{\omega}^{2}]\right]\right)\right]\Big{)}^{2}$
(41)
$\displaystyle\times\Big{|}8\sqrt{2}B_{0}(m_{u}-m_{d})a+\frac{2G_{\rho}\Pi_{\rho\omega}(M_{\omega}^{2})}{M_{\omega}^{2}-M_{\rho}^{2}-i(M_{\omega}\Gamma_{\omega}-M_{\rho}\Gamma_{\rho})}\Big{|}^{2}$
$\displaystyle=$ $\displaystyle 0.013\mid(0.29)+(-0.22+3.35i)\mid^{2}\,.$

From Eq. (41), we can find that the first term due to the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ is smaller than the second term due to
the $\rho-\omega$ mixing by one order of magnitude. Within 1$\sigma$
uncertainties, our theoretical value of the branching fraction is
$\mathscr{B}(\omega\rightarrow\pi^{+}\pi^{-})=(1.48\pm 0.10)\times 10^{-2}$,
which is consistent with the values provided in PDG ParticleDataGroup:2020ssz
and by the recent dispersive analysis Kubis2017 .

Regarding the mass of $\omega$ meson, previous studies BaBar:2012bdw ;
Colangelo:2018mtw ; Colangelo:2022prz  have pointed out that the result
extracted from $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ is substantially lower
than the current PDG average ParticleDataGroup:2020ssz , which primarily
relies on $e^{+}e^{-}\rightarrow 3\pi$ and
$e^{+}e^{-}\rightarrow\pi^{0}\gamma$ experiments. The discrepancy amounts to
approximately 1 MeV, corresponding to around 5 $\sigma$ considering the
current precision. It has been observed that the fitted value for $M_{\omega}$
and the phase of $\epsilon_{\omega}$ are strongly correlated BaBar:2012bdw ;
Colangelo:2018mtw ; Colangelo:2022prz . Note that the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling has not been considered in
BaBar:2012bdw ; Colangelo:2018mtw ; Colangelo:2022prz . As indicated in Table
1 above, our fitted results for the mass of $\omega$ agree well with the value
in PDG: $M_{\omega}=782.66\pm 0.13$ MeV, and this agreement remains unaffected
by the inclusion or exclusion of the momentum dependence of
$\Pi_{\rho\omega}$. We also find that strong correlation (80%) exists between
the parameter “$a$”, which quantify the direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling, and the mass of $\omega$. As
mentioned earlier, the direct $\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling
influences both the imaginary part and real part of the amplitude and thus
affects the phase of $R_{\rho\omega}(M_{\omega}^{2})$. Note that the phase of
$R_{\rho\omega}(M_{\omega}^{2})$ roughly agrees with the phase of
$\epsilon_{\omega}$. Thus our observations align with with those in Refs.
BaBar:2012bdw ; Colangelo:2018mtw ; Colangelo:2022prz , namely strong
correlation exists between the mass of the omega meson and the phase of
$\epsilon_{\omega}$. Our findings suggest that the inclusion of direct
$\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling is likely crucial in the
analysis aimed at extracting the $\omega$ mass from the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ process.

##  IV Conclusions

We have used the resonance chiral theory to study the $\rho-\omega$ mixing. In
particular, we have analyzed the impact of the momentum dependence of the
$\rho-\omega$ mixing on the describing the pion vector form factor in the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ process and its contribution to the
anomalous magnetic moment of the muon $a_{\mu}$. The incorporation of momentum
dependence arises from the calculation of loop contributions, which
corresponds to the next-to-leading orders in $1/N_{C}$ expansion. Through
fitting to the data of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ and and
$\tau\rightarrow\nu_{\tau}2\pi$ processes within the energy range of
600$\sim$900 MeV and the decay width of $\omega\rightarrow\pi^{+}\pi^{-}$, we
find that the $\rho-\omega$ mixing amplitude is dominated by its real part,
and its imaginary part is relatively small. Although both momentum-independent
and momentum-dependent $\rho-\omega$ mixing schemes yield satisfactory data
descriptions, the latter proves to be more self-consistent due to the
reasonable imaginary part of the mixing matrix element $\Pi_{\rho\omega}$.
Regarding the contribution to anomalous magnetic moment of the muon
$a_{\mu}^{\pi\pi}|_{[0.6,0.9]\text{GeV}}$, the results obtained from fits
considering the momentum-dependent $\rho-\omega$ mixing amplitude align well
with those from corresponding fits that exclude the momentum dependence of
$\rho-\omega$ mixing, within the margin of error. Additionally, we provide the
ratio of the isospin-pure $\omega$ and $\rho$ two-pion couplings, denoted as
$G=g_{\omega_{I}\pi\pi}/g_{\rho_{I}\pi\pi}$, and observe that $\rho-\omega$
mixing plays a crucial role in the decay width of
$\omega\rightarrow\pi^{+}\pi^{-}$. Furthermore, we ascertain that including
the direct $\omega_{I}\rightarrow\pi^{+}\pi^{-}$ coupling is essential in
analysing the extraction of the mass of the $\omega$ meson from the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ process.

## Acknowledgments

We are grateful to Pablo Roig for helpful discussions and valuable
suggestions. This work is supported in part by the Fundamental Research Funds
for the Central Universities under Grants No. FRF-BR-19-001A, by the National
Natural Science Foundation of China (NSFC) under Grants No. 11975028, No.
11974043.

##  Appendix A  Loop contributions

###  A.1 Diagram (c): kaon loops

Using the $\rho K\bar{K}$ and $\omega K\bar{K}$ vertexes given by the
Lagrangian in Eq. (24): $iG_{V}/\sqrt{2}\langle
V_{\mu\nu}u^{\mu}u^{\nu}\rangle=iG_{V}/F^{2}\rho^{0}_{\mu\nu}(\partial^{\mu}K^{+}\partial^{\nu}K^{-}-\partial^{\mu}K^{0}\partial^{\nu}\bar{K}^{0})+iG_{V}/F^{2}\omega_{\mu\nu}(\partial^{\mu}K^{+}\partial^{\nu}K^{-}+\partial^{\mu}K^{0}\partial^{\nu}\bar{K}^{0})+...$,
we can calculate the charged and the neutral kaon loops contribution to the
amplitude

$\displaystyle\Pi_{\rho\omega}^{\text{kaon,charged}}$ $\displaystyle=$
$\displaystyle\frac{G_{V}^{2}p^{4}}{192F^{4}\pi^{2}}\bigg{\\{}(1-\frac{6m_{K^{+}}^{2}}{p^{2}})(\lambda_{\infty}-\ln\frac{m_{K^{+}}^{2}}{\mu^{2}})$
(42)
$\displaystyle+\frac{5}{3}-\frac{8m_{K^{+}}^{2}}{p^{2}}-\sigma_{K^{+}}^{3}\ln(\frac{\sigma_{K^{+}}+1}{\sigma_{K^{+}}-1})\bigg{\\}}\,,$

and

$\displaystyle\Pi_{\rho\omega}^{\text{kaon,neutral}}$ $\displaystyle=$
$\displaystyle-\frac{G_{V}^{2}p^{4}}{192F^{4}\pi^{2}}\bigg{\\{}(1-\frac{6m_{K^{0}}^{2}}{p^{2}})(\lambda_{\infty}-\ln\frac{m_{K^{0}}^{2}}{\mu^{2}})$
(43)
$\displaystyle+\frac{5}{3}-\frac{8m_{K^{0}}^{2}}{p^{2}}-\sigma_{K^{0}}^{3}\ln(\frac{\sigma_{K^{0}}+1}{\sigma_{K^{0}}-1})\bigg{\\}}\,,$

where $\sigma_{P}\equiv\sqrt{1-4m_{P}^{2}/p^{2}}$ and
$\lambda_{\infty}\equiv\frac{1}{\epsilon}-\gamma_{E}+1+\ln 4\pi$ with
$\epsilon=2-\frac{d}{2}$ and $\gamma_{E}$ being the Euler constant.

The non-vanishing of the structure function proceeds from the mass difference
between the charged and neutral kaons as follows

$\displaystyle
S_{\rho\omega}^{(c)}=\frac{1}{m_{u}-m_{d}}(\Pi_{\rho\omega}^{\text{kaon,charged}}+\Pi_{\rho\omega}^{\text{kaon,neutral}})\,.$
(44)

###  A.2 Diagram (d): $\pi\pi$ loop

For the isospin-violating vertex of $\omega_{I}\rightarrow\pi^{+}\pi^{-}$, we
construct the Lagrangian

$\displaystyle\mathcal{L}_{\omega_{I}\rightarrow\pi^{+}\pi^{-}}$
$\displaystyle=$ $\displaystyle a_{1}i\langle
V_{\mu\nu}\\{\chi_{+},u^{\mu}u^{\nu}\\}\rangle+a_{2}i\langle
V_{\mu\nu}u^{\mu}\chi_{+}u^{\nu}\rangle$ (45) $\displaystyle=$
$\displaystyle(a_{1}-\frac{1}{2}a_{2})\frac{8\sqrt{2}B_{0}i}{F^{2}}\Delta_{ud}\,\omega_{\alpha\beta}\pi^{+\alpha}\pi^{-\beta}.$

For convenience, we define the combination $a\equiv a_{1}-\frac{1}{2}a_{2}.$
The $\pi\pi$-loop contribution to the structure function can be calculated,
which reads

$\displaystyle S_{\rho\omega}^{(d)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}G_{V}B_{0}a}{12F^{4}\pi^{2}}p^{4}\bigg{\\{}(1-\frac{6m_{\pi}^{2}}{p^{2}})(\lambda_{\infty}-\ln\frac{m_{\pi}^{2}}{\mu^{2}})$
(46) $\displaystyle\hskip
42.67912pt+\frac{5}{3}-\frac{8m_{\pi}^{2}}{p^{2}}-\sigma_{\pi}^{3}\ln(\frac{\sigma_{\pi}+1}{\sigma_{\pi}-1})\bigg{\\}}\,.$

###  A.3 Diagram (e): $\pi$-tadpole loop

According to the Lorentz, $P$ and $C$ invariances, the Lagrangian
corresponding to the interaction of $\omega_{I}\rho_{I}\pi\pi$ can be
constructed as follows:

$\displaystyle\mathcal{L}_{\omega_{I}\rho_{I}PP}$ $\displaystyle=$
$\displaystyle b_{1}\langle
V_{\mu\nu}V^{\mu\nu}(u^{\alpha}u_{\alpha}\chi_{+}+\chi_{+}u^{\alpha}u_{\alpha})\rangle$
(47) $\displaystyle+$ $\displaystyle b_{2}\langle
V_{\mu\nu}V^{\mu\nu}u^{\alpha}\chi_{+}u_{\alpha}\rangle+b_{3}\langle
V_{\mu\nu}\chi_{+}V^{\mu\nu}u^{\alpha}u_{\alpha}\rangle$ $\displaystyle+$
$\displaystyle b_{4}\langle
V_{\mu\nu}u^{\alpha}V^{\mu\nu}(\chi_{+}u_{\alpha}+u_{\alpha}\chi_{+})\rangle$
$\displaystyle+$ $\displaystyle b_{5}\langle
V_{\mu\alpha}V^{\nu\alpha}u^{\mu}u_{\nu}\chi_{+}+V^{\nu\alpha}V_{\mu\alpha}\chi_{+}u_{\nu}u^{\mu}\rangle$
$\displaystyle+$ $\displaystyle b_{6}\langle
V_{\mu\alpha}V^{\nu\alpha}u^{\mu}\chi_{+}u_{\nu}\rangle+b_{7}\langle
V_{\mu\alpha}\chi_{+}V^{\nu\alpha}u^{\mu}u_{\nu}\rangle$ $\displaystyle+$
$\displaystyle b_{8}\langle
V_{\mu\alpha}V^{\nu\alpha}u_{\nu}u^{\mu}\chi_{+}+V^{\nu\alpha}V_{\mu\alpha}\chi_{+}u^{\mu}u_{\nu}\rangle$
$\displaystyle+$ $\displaystyle b_{9}\langle
V_{\mu\alpha}V^{\nu\alpha}u_{\nu}\chi_{+}u^{\mu}\rangle+b_{10}\langle
V_{\mu\alpha}\chi_{+}V^{\nu\alpha}u_{\nu}u^{\mu}\rangle$ $\displaystyle+$
$\displaystyle b_{11}\langle
V_{\mu\alpha}u^{\alpha}V^{\mu\beta}u_{\beta}\chi_{+}+V^{\mu\beta}u^{\alpha}V_{\mu\alpha}\chi_{+}u_{\beta}\rangle$
$\displaystyle+$ $\displaystyle b_{12}\langle
V_{\mu\alpha}u^{\alpha}V^{\mu\beta}\chi_{+}u_{\beta}+V^{\mu\beta}u^{\alpha}V_{\mu\alpha}u_{\beta}\chi_{+}\rangle$
$\displaystyle+$ $\displaystyle b_{13}\langle
V_{\mu\alpha}u_{\beta}V^{\mu\beta}u^{\alpha}\chi_{+}+V^{\mu\beta}u_{\beta}V_{\mu\alpha}\chi_{+}u^{\alpha}\rangle$
$\displaystyle+$ $\displaystyle b_{14}\langle
V_{\mu\alpha}u_{\beta}V^{\mu\beta}\chi_{+}u^{\alpha}+V^{\mu\beta}u_{\beta}V_{\mu\alpha}u^{\alpha}\chi_{+}\rangle$
$\displaystyle+$ $\displaystyle g_{1}i\langle
V_{\mu\nu}V^{\mu\nu}(u^{\alpha}\nabla_{\alpha}\chi_{-}+\nabla_{\alpha}\chi_{-}u^{\alpha})\rangle$
$\displaystyle+$ $\displaystyle g_{2}i\langle
V_{\mu\nu}u^{\alpha}V^{\mu\nu}\nabla_{\alpha}\chi_{-}\rangle$ $\displaystyle+$
$\displaystyle g_{3}i\langle
V_{\mu\beta}V^{\mu\alpha}u^{\beta}\nabla_{\alpha}\chi_{-}+V^{\mu\alpha}V_{\mu\beta}\nabla_{\alpha}\chi_{-}u^{\beta}\rangle$
$\displaystyle+$ $\displaystyle g_{4}i\langle
V_{\mu\beta}V^{\mu\alpha}\nabla_{\alpha}\chi_{-}u^{\beta}+V^{\mu\alpha}V_{\mu\beta}u^{\beta}\nabla_{\alpha}\chi_{-}\rangle$
$\displaystyle+$ $\displaystyle g_{5}i\langle
V_{\mu\beta}u^{\beta}V^{\mu\alpha}\nabla_{\alpha}\chi_{-}+V^{\mu\alpha}u^{\beta}V_{\mu\beta}\nabla_{\alpha}\chi_{-}\rangle$
$\displaystyle+$ $\displaystyle\lambda_{6}^{VV}\langle
V_{\mu\nu}V^{\mu\nu}\chi_{+}\rangle\,.$

For simplicity, we define the combinations,

$\displaystyle h_{1}$ $\displaystyle\equiv$ $\displaystyle
6b_{1}-b_{2}+3b_{3}+b_{4}-2g_{1}-g_{2},$ $\displaystyle h_{2}$
$\displaystyle\equiv$ $\displaystyle
4b_{5}-b_{6}+3b_{7}+4b_{8}-b_{9}+3b_{10}+2b_{11}+2b_{12}$ (48)
$\displaystyle+2b_{13}+2b_{14}-2g_{3}-2g_{4}-2g_{5}\,.$

The mass difference between the charged and neutral pions in the internal
lines of loops can be disregarded due to its higher-order magnitude beyond our
scope of consideration. Consequently, the expanded expression of Lagrangian
(47) can be simplified as follows:

$\displaystyle\mathcal{L}_{\omega_{I}\rho_{I}\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{4B_{0}}{F^{2}}h_{1}(m_{u}-m_{d})\rho_{I\mu\nu}\omega^{\mu\nu}\pi_{\alpha}{\pi}^{\alpha}$
(49) $\displaystyle-$
$\displaystyle\frac{2B_{0}}{F^{2}}\lambda_{6}^{VV}(m_{u}-m_{d})\rho_{I\mu\nu}\omega^{\mu\nu}{\pi}^{2}$
$\displaystyle+$
$\displaystyle\frac{4B_{0}}{F^{2}}h_{2}(m_{u}-m_{d})\rho_{I\mu\alpha}\omega^{\nu\alpha}\pi_{\mu}{\pi}^{\nu}\,.$

With the above Lagrangian, the $\pi$-tadpole contribution to the $\rho-\omega$
mixing can be derived:

$\displaystyle S^{(e)}_{\rho\omega}$ $\displaystyle=$
$\displaystyle-\frac{m_{\pi}^{2}B_{0}}{8\pi^{2}F^{2}}\bigg{\\{}(-16\lambda_{6}^{VV}+4h_{1}m_{\pi}^{2}+h_{2}m_{\pi}^{2})$
(50) $\displaystyle\hskip
56.9055pt\times(\lambda_{\infty}-\ln\frac{m_{\pi}^{2}}{\mu^{2}})+\frac{h_{2}}{2}m_{\pi}^{2}\bigg{\\}}\,.$

###  A.4 Diagrams (f)-(i): $\pi^{0}\gamma$ loops

In the loop diagrams (f)-(i), the resonance chiral effective Lagrangian
describing vector-photon-pseudoscalar (VJP) and vector-vector-pseudoscalar
(VVP) vertices has been given in Ref. Femenia :

$\displaystyle\mathcal{L}_{VJP}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle\\{V^{\mu\nu},f_{+}^{\rho\alpha}\\}\nabla_{\alpha}u^{\sigma}\rangle$
(51) $\displaystyle+$
$\displaystyle\frac{c_{2}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle\\{V^{\mu\alpha},f_{+}^{\rho\sigma}\\}\nabla_{\alpha}u^{\nu}\rangle$
$\displaystyle+$
$\displaystyle\frac{ic_{3}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle\\{V^{\mu\nu},f_{+}^{\rho\sigma}\\}\chi_{-}\rangle$
$\displaystyle+$
$\displaystyle\frac{ic_{4}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle
V^{\mu\nu}[f_{-}^{\rho\sigma},\chi_{+}]\rangle$ $\displaystyle+$
$\displaystyle\frac{c_{5}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle\\{\nabla_{\alpha}V^{\mu\nu},f_{+}^{\rho\alpha}\\}u^{\sigma}\rangle$
$\displaystyle+$
$\displaystyle\frac{c_{6}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle\\{\nabla_{\alpha}V^{\mu\alpha},f_{+}^{\rho\sigma}\\}u^{\nu}\rangle$
$\displaystyle+$
$\displaystyle\frac{c_{7}}{M_{V}}\epsilon_{\mu\nu\rho\sigma}\langle\\{\nabla^{\sigma}V^{\mu\nu},f_{+}^{\rho\alpha}\\}u_{\alpha}\rangle\,,$

and

$\displaystyle\mathcal{L}_{VVP}$ $\displaystyle=$ $\displaystyle
d_{1}\epsilon_{\mu\nu\rho\sigma}\langle\\{V^{\mu\nu},V^{\rho\alpha}\\}\nabla_{\alpha}u^{\sigma}\rangle$
(52) $\displaystyle+$ $\displaystyle
id_{2}\epsilon_{\mu\nu\rho\sigma}\langle\\{V^{\mu\nu},V^{\rho\sigma}\\}\chi_{-}\rangle$
$\displaystyle+$ $\displaystyle
d_{3}\epsilon_{\mu\nu\rho\sigma}\langle\\{\nabla_{\alpha}V^{\mu\nu},V^{\rho\alpha}\\}u^{\sigma}\rangle$
$\displaystyle+$ $\displaystyle
d_{4}\epsilon_{\mu\nu\rho\sigma}\langle\\{\nabla^{\sigma}V^{\mu\nu},V^{\rho\alpha}\\}u_{\alpha}\rangle\,.$

The couplings involved, or their combinations, can be estimated by matching
the leading operator product expansion of the $\langle VVP\rangle$ Green
function to the same quantity evaluated within R$\chi$T. This procedure leads
to high energy constraints on the resonance couplings Femenia :

$\displaystyle 4c_{3}+c_{1}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle c_{1}-c_{2}+c_{5}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle c_{5}-c_{6}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{64\pi^{2}}\frac{M_{V}}{\sqrt{2}F_{V}},$
$\displaystyle d_{1}+8d_{2}$ $\displaystyle=$
$\displaystyle-\frac{N_{c}}{64\pi^{2}}\frac{M_{V}^{2}}{F_{V}^{2}}+\frac{F^{2}}{4F_{V}^{2}},$
$\displaystyle d_{3}$ $\displaystyle=$
$\displaystyle-\frac{N_{c}}{64\pi^{2}}\frac{M_{V}^{2}}{F_{V}^{2}}+\frac{F^{2}}{8F_{V}^{2}}.$
(53)

Using the the effective vertices stated in Eqs. (51) and (52), the $\pi\gamma$
loop contribution, i.e., the summation of the loops diagrams (f)-(i), can be
expressed as:

$\displaystyle i\Pi_{\rho\omega}\epsilon_{\rho\mu}\epsilon_{\omega}^{\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\int\frac{d^{n}k}{(2\pi)^{n}}\frac{-i}{k^{2}}\frac{i}{(p-k)^{2}-m_{\pi}^{2}}$
(54) $\displaystyle\times$ $\displaystyle[(k\cdot
p)^{2}\epsilon_{\rho}^{\mu}\epsilon_{\omega\mu}-k^{2}p^{2}\epsilon_{\rho}^{\mu}\epsilon_{\omega\mu}+p^{2}k\cdot\epsilon_{\rho}k\cdot\epsilon_{\omega}]$
$\displaystyle\times$
$\displaystyle\bigg{\\{}\frac{-32e^{2}}{3M_{V}^{2}F^{2}}\big{[}c_{1}(p-k)\cdot
k-c_{2}(p-k)\cdot p$ $\displaystyle\hskip
5.69046pt-4c_{3}m_{\pi}^{2}-c_{5}p\cdot k+c_{6}p^{2}\big{]}^{2}$
$\displaystyle-$
$\displaystyle\frac{16\sqrt{2}F_{V}e^{2}}{3M_{V}F^{2}}\bigg{[}\frac{1}{M_{\omega}^{2}-k^{2}}+\frac{1}{M_{\rho}^{2}-k^{2}}\bigg{]}$
$\displaystyle\times$
$\displaystyle\big{[}d_{1}(p-k)^{2}+8d_{2}m_{\pi}^{2}+2d_{3}p\cdot k\big{]}$
$\displaystyle\times$ $\displaystyle\big{[}c_{1}(p-k)\cdot k-c_{2}(p-k)\cdot
p-4c_{3}m_{\pi}^{2}$ $\displaystyle-$ $\displaystyle c_{5}p\cdot
k+c_{6}p^{2}\big{]}-\frac{16F_{V}^{2}e^{2}}{3F^{2}(M_{\rho}^{2}-k^{2})(M_{\omega}^{2}-k^{2})}$
$\displaystyle\times$
$\displaystyle\big{[}d_{1}(p-k)^{2}+8d_{2}m_{\pi}^{2}+2d_{3}p\cdot
k\big{]}^{2}\bigg{\\}}\,.\hskip 298.75394pt$

The subsequent calculation is straightforward, while the result of the
extracted electromagnetic structure function $E^{\pi\gamma}_{\rho\omega}\equiv
E_{\rho\omega}^{(f)}+E_{\rho\omega}^{(g)}+E_{\rho\omega}^{(h)}+E_{\rho\omega}^{(i)}$
is too lengthy to be given here. Notes that in our numerical computation we
employ the high energy constraints in Eq. (A.4) together with the fitted
parameters given in Ref. chen2012 , and therefore all the parameters involved
in $E^{\pi\gamma}_{\rho\omega}$ are known.

###  A.5  Counterterms and renormalized amplitude

Since the $\omega$ meson predominantly decays into the three-pion state, its
two-loop self energy diagram contributes beyond the NLO in $1/N_{C}$ and is
not relevant for our current consideration. The self-energy diagrams for the
$\rho$ meson are depicted in Fig. 4. The Lagrangian needed to renormalize the
$\rho$ meson one-loop self-energy has been given in Ref. Rosell04 ,

$\displaystyle{\cal L}_{4Y}$ $\displaystyle=$
$\displaystyle\frac{X_{Y_{1}}}{2}\,\langle\nabla^{2}V^{\mu\nu}\left\\{\nabla_{\nu},\nabla^{\sigma}\right\\}V_{\mu\sigma}\rangle+\frac{X_{Y_{2}}}{4}\,\langle\left\\{\nabla_{\nu},\nabla_{\alpha}\right\\}V^{\mu\nu}\left\\{\nabla^{\sigma},\nabla^{\alpha}\right\\}V_{\mu\sigma}\rangle$
(55) $\displaystyle+$
$\displaystyle\frac{X_{Y_{3}}}{4}\,\langle\left\\{\nabla^{\sigma},\nabla^{\alpha}\right\\}V^{\mu\nu}\left\\{\nabla_{\nu},\nabla_{\alpha}\right\\}V_{\mu\sigma}\rangle\,.$

Actually, only the combination of couplings $X_{Y}\equiv
X_{Y_{1}}+X_{Y_{2}}+X_{Y_{3}}\equiv X_{Y}^{r}+\delta X_{Y}$ is relevant for
this purpose. Using the Lagrangians in Eqs. (24) and (55), the $\rho$ self-
energy takes the form

$\displaystyle\Sigma_{\rho}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{G_{V}^{2}}{48F^{4}\pi^{2}}p^{4}\bigg{\\{}(1-\frac{6m_{\pi}^{2}}{p^{2}})(\lambda_{\infty}-\ln\frac{m_{\pi}^{2}}{\mu^{2}})-\frac{8m_{\pi}^{2}}{p^{2}}-\sigma_{\pi}^{3}\ln(\frac{\sigma_{\pi}+1}{\sigma_{\pi}-1})$
(56)
$\displaystyle+(1-\frac{6m_{K}^{2}}{p^{2}})(\lambda_{\infty}-\ln\frac{m_{K}^{2}}{\mu^{2}})-\frac{8m_{K}^{2}}{p^{2}}-\sigma_{K}^{3}\ln(\frac{\sigma_{K}+1}{\sigma_{K}-1})+\frac{10}{3}\bigg{\\}}-X_{Y}p^{4}\,.$

The renormalized $\rho$ mass fulfills

$\displaystyle M_{\rho}^{2}=M_{V}^{2}+\Sigma_{\rho}(M_{\rho}^{2})\,.$ (57)

Since the physical $M_{\rho}$ is finite, one gets

$\displaystyle\delta
X_{Y}=-\frac{G_{V}^{2}}{48F^{4}M_{\rho}^{2}\pi^{2}}(1-6\frac{m_{\pi}^{2}}{M_{\rho}^{2}}-6\frac{m_{K}^{2}}{M_{\rho}^{2}})\lambda_{\infty}\,.$
(58)

The wave-function renormalization constant of the $\rho$ meson is obtained
from

$\displaystyle Z_{\rho}=1+\frac{\partial\Sigma_{\rho}(p^{2})}{\partial
p^{2}}|_{p^{2}=M_{\rho}^{2}}\,.$ (59)

Figure 4:  Feynman diagrams contributing to $\rho$ self-energy.

In our calculation of the $\rho-\omega$ mixing, the tree amplitudes can only
absorb the ultraviolet divergence that is proportional to $p^{0}$. To cancel
the $O(p^{2})$, $O(p^{4})$ and $O(p^{6})$ ultraviolet divergence originating
from the loop contribution $S_{\rho\omega}^{(c)}$, $S_{\rho\omega}^{(d)}$,
$S_{\rho\omega}^{(e)}$, and $E_{\rho\omega}^{\pi\gamma}$, we construct the
counterterms as follows:

$\displaystyle{\cal L}_{ct}$ $\displaystyle=$ $\displaystyle Y_{A}\langle
V_{\mu\nu}V^{\mu\nu}\chi_{+}\rangle-\frac{1}{2}Y_{B}\langle\nabla^{\lambda}V_{\lambda\mu}\nabla_{\nu}V^{\nu\mu}\chi_{+}\rangle$
(60) $\displaystyle+$
$\displaystyle\frac{Y_{C_{1}}}{2}\langle\nabla^{2}V^{\mu\nu}\\{\chi_{+},\\{\nabla_{\nu},\nabla^{\sigma}\\}V_{\mu\sigma}\\}\rangle$
$\displaystyle+$
$\displaystyle\frac{Y_{C_{2}}}{4}\langle\\{\nabla_{\nu},\nabla_{\alpha}\\}V^{\mu\nu}\\{\chi_{+},\\{\nabla^{\sigma},\nabla^{\alpha}\\}V_{\mu\sigma}\\}\rangle$
$\displaystyle+$
$\displaystyle\frac{Y_{C_{3}}}{4}\langle\\{\nabla^{\sigma},\nabla^{\alpha}\\}V^{\mu\nu}\\{\chi_{+},\\{\nabla_{\nu},\nabla_{\alpha}\\}V_{\mu\sigma}\\}\rangle$
$\displaystyle+$ $\displaystyle\frac{Z_{A}F_{V}}{2\sqrt{2}}\langle
V_{\mu\nu}f_{+}^{\mu\nu}\rangle+\frac{Z_{B}F_{V}}{2\sqrt{2}}\langle
V_{\mu\nu}\nabla^{2}f_{+}^{\mu\nu}\rangle$ $\displaystyle+$
$\displaystyle\frac{Z_{C}F_{V}}{2\sqrt{2}}\langle
V_{\mu\nu}\nabla^{4}f_{+}^{\mu\nu}\rangle+\frac{Z_{D}F_{V}}{2\sqrt{2}}\langle
V_{\mu\nu}\nabla^{6}f_{+}^{\mu\nu}\rangle\,.$

We adopt the $\overline{\rm MS}-1$ subtraction scheme and absorb the divergent
pieces proportional to $\lambda_{\infty}$ by the bare couplings in the
counterterms. Consequently, the remanent finite pieces of counterterms can be
written as:

$\displaystyle\Pi_{\rho\omega}^{ct}=X_{W}^{r}\,p^{6}+X_{Z}^{r}\,p^{4}+X_{R}^{r}\,p^{2}\
,$ (61)

with

$\displaystyle X_{W}^{r}$ $\displaystyle\equiv$ $\displaystyle\frac{8\pi\alpha
F_{\rho}F_{\omega}}{3}(Z_{D}^{r}+Z_{B}^{r}Z_{C}^{r})\ ,$ $\displaystyle
X_{Z}^{r}$ $\displaystyle\equiv$ $\displaystyle\frac{4\pi\alpha
F_{\rho}F_{\omega}}{3}(2Z_{C}^{r}+{Z_{B}^{r}}^{2})$
$\displaystyle+16M_{\rho}(m_{u}-m_{d})(Y_{C_{1}}^{r}+Y_{C_{2}}^{r}+Y_{C_{3}}^{r})\
,$ $\displaystyle X_{R}^{r}$ $\displaystyle\equiv$
$\displaystyle\frac{8\pi\alpha
F_{\rho}F_{\omega}}{3}Z_{B}^{r}-4M_{\rho}(m_{u}-m_{d})Y_{B}^{r}\ .$ (62)

In summary, at the NLO in $1/N_{C}$, the UV-renormalized mixing amplitude
reads

$\displaystyle{\Pi}^{r}_{\rho\omega}(p^{2})$ $\displaystyle=$ $\displaystyle
S_{\rho\omega}^{(a)}\sqrt{Z_{\rho}}+\bar{S}_{\rho\omega}^{(c)}+\bar{S}_{\rho\omega}^{(d)}+\bar{S}_{\rho\omega}^{(e)}+\overline{E}_{\rho\omega}^{\pi\gamma}(p^{2})+X_{W}^{r}p^{6}+X_{Z}^{r}p^{4}+X_{R}^{r}p^{2}\,,$
(63)

where a bar indicates that the divergences are subtracted.

As discussed in Ref. Connell97 , the mixing amplitude should vanish as
$p^{2}\to 0$. Thus, the final expression of the renormalized mixing amplitude
is obtained as follows:

$\displaystyle{\Pi}_{\rho\omega}(p^{2})={\Pi}^{r}_{\rho\omega}(p^{2})-{\Pi}^{r}_{\rho\omega}(0)\
,$ (64)

where an additional finite shift is imposed to guarantee that the constraint
${\Pi}_{\rho\omega}(0)=0$ is satisfied. Note that due to the finite shift
performed in Eq. (64), our numerical calculation is actually independent of
the coupling $X_{Y}^{r}$. In our numerical computation, the scale $\mu$ will
be set to $M_{\rho}$ and we use $(m_{u}-m_{d})=-2.49$ MeV provided by PDG
ParticleDataGroup:2020ssz .

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#  Quantifying the sustainability impact of Google Maps: A case study of Salt
Lake City111authors ordered alphabetically.

Neha Arora
Google Research
<EMAIL_ADDRESS>Theophile Cabannes
Google Research222On leave from UC Berkeley.
<EMAIL_ADDRESS>Sanjay Ganapathy
Google Research
<EMAIL_ADDRESS>Yechen Li
Google Research
<EMAIL_ADDRESS>Preston McAfee
Google Research
<EMAIL_ADDRESS>Marc Nunkesser
Google Geo
<EMAIL_ADDRESS>Carolina Osorio
Google Research
HEC Montreal
<EMAIL_ADDRESS>Andrew Tomkins
Google Research
<EMAIL_ADDRESS>Iveel Tsogsuren
Google Research
<EMAIL_ADDRESS>
###### Abstract

Google Maps uses current and historical traffic trends to provide routes to
drivers. In this paper, we use microscopic traffic simulation to quantify the
improvements to both travel time and CO2 emissions from Google Maps real-time
navigation. A case study in Salt Lake City shows that Google Maps users are,
on average, saving 1.7% of CO2 emissions and 6.5% travel time. If we restrict
to the users for which Google Maps finds a different route than their original
route, the average savings are 3.4% of CO2 emissions and 12.5% of travel time.
These results are based on traffic conditions observed during the Covid-19
pandemic. As congestion gradually builds back up to pre-pandemic levels, it is
expected to lead to even greater savings in emissions.

##  1 Approach Overview

We build a microscopic traffic simulation model of Salt Lake City, Utah, USA.
Various well established traveler behavioral models for car-following
behavior, and lane-changing behavior are used to model travel patterns.
Metrics presented in this paper are computed from aggregated and anonymized
data of Google users who have opted-in to Location History [4], a feature
which is off by default. Travel demand is sourced from a combination of
Location History data and city sensor vehicular count data [14]. To validate
that trip durations in our simulation match the observed durations for those
same trips in the real world, we compute aggregated and anonymized metrics
based on Google Maps navigation data [6]. The simulation is validated by
comparing simulated trip durations with aggregated and anonymized trip
durations from Google Maps navigation data [6]. The traffic simulator is
combined with an emissions model developed by the National Renewable Energy
Lab (NREL) to yield emissions estimates [7, 12].

To estimate travel time and emissions impacts of Google Maps real-time
navigation, we begin with a baseline scenario in which vehicles follow the
Location History based routes used in our high-fidelity simulation described
above. We then consider a routed scenario in which a fraction of vehicles,
denoted the routed vehicles, are routed using real-time shortest paths as in
Google Maps real-time navigation [5]. The number of routed vehicles is equal
to the fraction of Google Maps users. We then evaluate the impact of real-time
routing on emissions and trip duration for the routed vehicles by comparing
the baseline and the routed scenarios.

###  1.1 Preserving Data Privacy

As stated above, insights and metrics in this paper are derived from
aggregated and anonymized sets of data from users who have turned on the
Location History [4] setting, which is off by default. People who have
Location History turned on can choose to turn it off at any time from their
Google Account [2] and can always delete Location History data directly from
their Timeline [9]. Simulation validation metrics computed from Maps
navigation data are also aggregated and anonymized. Access to this data is
limited and audited. The non-anonymized data is encrypted at rest and models
trained on this are not kept past the lifetime of the source data.

To provide strong privacy guarantees, all reported metrics are anonymized and
aggregated using a differentially private mechanism [1]. The automated Laplace
mechanism adds random noise drawn from a zero mean Laplace distribution and
yields $(\epsilon,\delta)$-differential privacy guarantee of
$\epsilon=1.09861$ and $\delta=1e-5$ [3].

To compute emission estimates the NREL model [7, 12] was integrated into the
simulator and the data never left Google servers.

##  2 Simulation Framework

The below figure summarizes the simulation framework. The main components are
the traffic simulation combined with NREL’s emissions model [7, 12].

###  2.1 Traffic Simulation

With the traffic simulation model we want to replay, via simulation, the
traffic situation for a specified day as closely as possible. We use an open-
source simulation software, SUMO (Simulation of Urban Mobility) [13]. We
consider a single travel mode, which is a car, and a single vehicle type,
which is SUMO’s default passenger car. To describe traffic dynamics, we use
travel behavioral models available in SUMO. These describe, for instance, how
drivers make decisions such as car-following, lane changing and speed limit
compliance. Google Maps data is used to define the road network supply (e.g.
network topology, connectivity) and the various attributes of each static road
segment such as number of lanes, speed limit, and so forth. In this paper, we
use the terms road and segment interchangeably.

Figure 1: Simulation Framework.

Calibration (i.e., input parameter estimation) and validation of the traffic
simulator is carried out considering the baseline scenario. The vehicles (or
trips) to be simulated are determined by a travel demand model. The road
network of a metropolitan area is decomposed into zones defined by S2 cells of
level 13 [8], with an average area of 1.27 km2. Travel demand for a given time
period (e.g., one hour) is defined by a coarsened origin-destination matrix
(or OD matrix) which gives the expected number of vehicles traveling from each
origin zone to each destination zone. For a given OD pair, the set of feasible
routes (i.e., the route choice set) in the baseline scenario is determined by
the set of all routes derived from Location History data for the considered
time period over several weeks.

For a given time period, the entries in the OD matrix are obtained as a
combination of Location History trajectories plus additional trajectories as
required to match segment vehicular count data from city sensors [14]. The
additional trajectories are produced using the OD calibration algorithm
described in [11]. This algorithm generates an OD matrix that induces
simulated count data that is consistent with the overall city sensor segment
count data.

For the Salt Lake City case study described here, the city sensor count data
is obtained from the Utah Department of Transportation (UDOT) Automated
Traffic Signal Performance Metrics (ATSPM) portal [14]. We measure the error
between ground truth data from the portal versus a particular OD matrix
through the count-normalized root mean square error (nRMSE):

$nRMSE=100*\frac{\sqrt{\frac{1}{|I|}\sum_{i\in
I}{(y_{i}-\hat{y}_{i})^{2}}}}{\frac{1}{|I|}\sum_{i\in I}{y_{i}}}$ (1)

where $I$ denotes the set of segments where we have ground truth, $y_{i}$ is
the ground truth count of segment $i$, and $\hat{y}_{i}$ denotes the
calibration-based estimate of $y_{i}$. For this case study, calibrated OD
matrices are obtained by considering 415 segments with ground truth data. They
yield nRMSE’s of $22\pm 2\%$ for off-peak hours and $35\pm 2\%$ for peak
hours.

To validate the quality of the calibrated ODs, and more generally the ability
of the traffic simulator to accurately represent real traffic patterns, we
compare the simulated trip travel times (i.e., simulated ETAs) to those
observed from Google Maps navigation logs. For the current case study, the
nRMSE of the simulated ETAs is $25\pm 1\%$ for peak hours and $17\pm 1\%$ for
off-peak hours. The definition of nRMSE is similar to Equation (1), where we
replace the set of ground truth counts with ground truth trajectory durations.

Hour of Day | ETA nRMSE (%) | Segment Counts nRMSE (%)
---|---|---
11 am (Off peak) | $17\pm 1\%$ | $22\pm 2\%$
5 pm (Peak) | $25\pm 1\%$ | $35\pm 2\%$
Table 1: Summary of simulation evaluation in Salt Lake City

###  2.2 Emissions Model

We use NREL’s Route Energy Prediction model, RouteE [7, 12], that predicts the
energy consumption and emissions for any given route. This model accounts for
driving conditions such as traffic speed, road grade and turns. For every ten
minutes of simulation, we produce a snapshot of the current traffic on each
road segment. Based on this snapshot, we then estimate the fuel consumption
and emissions for each simulated trip using the RouteE model.

###  2.3 Experimental setup

To evaluate the impact of Google Maps route guidance on CO2 emissions, we
compare the baseline scenario with the routed scenario. The baseline scenario
is intended to capture vehicles who are making their own route choices, as
described above in Section 2.1.

The routed scenario considers the same set of vehicles, but replaces the route
choice of a subset of those with the shortest path based on real-time traffic
information, as in Google Maps with real-time navigation. The vehicles re-
routed in this way (called the _routed vehicles_) are chosen uniformly at
random from all the vehicles. The rest of the vehicles are referred to as
_background vehicles_. The background vehicles and their corresponding routes
remain identical to those of the baseline scenario. The number of routed
vehicles corresponds to the number of navigation requests received for Google
Maps travel guidance during the same time period.

As described in Section 2.1, the route choice set of the baseline scenario
consists of routes observed from Location History data. If a baseline vehicle
is already using a navigation service for real-time routing, that vehicle may
already be on the best route, and our simulation analysis may understate the
improved outcomes that are possible using real-time traffic information.

For the routed scenario, the real-time routes provided to routed vehicles are
selected as follows. For each origin-destination pair, we query Google Maps to
obtain a set of plausible routes for the given source/destination pair. Next,
we maintain a snapshot of current simulated travel time on each road segment,
updated every ten minutes, and use the most recent snapshot to score and rank
the routes based on travel time. Finally, we select the fastest route, and do
not allow any changes to the route during driving. In other words, only pre-
trip real-time guidance is considered, en-route guidance is not accounted for.
Again, this may underestimate the congestion impacts of the case study. To
estimate the impact of Google Maps real-time navigation we compare the travel
time and CO2 emissions for each routed vehicle in the baseline versus the
routed scenario.

##  3 Case Study Results

We build the network for a region of Salt Lake City with 45K road segments. We
consider a representative day in December 2020, during the Covid-19 pandemic
and analyze hourly simulation runs for the hours from 7am to 7pm. For each
hour of the day we simulate between 82,500 to 137,000 trajectories per hour
across 12,434 OD pairs distributed throughout the network. The average driving
distance is approximately 6km.

For each hour, we run 6 simulation replications. Sources of stochasticity
across the replications include sampling from the demand distribution, and
sampling the set of routed trips. In the analysis of travel time and CO2, we
exclude unrealistic trips that have a duration that is more than twice that of
the shortest path or is longer than 20 minutes. We do this in order to limit
the potential to overestimate the travel time and CO2 impacts.

The plot on the left of Figure 2 shows hourly total demand from 7am to 7pm in
Salt Lake City for that day. There are error bars that have a half-width equal
to the standard deviation. However, the error bars are barely visible in this
plot. Demand gradually increases throughout the day, with a peak period
between 4pm and 6pm. Interestingly, we observe no morning peak period but only
afternoon peak period. This is consistent with recent analysis of other cities
carried out during the pandemic [10].

The plot on the right of Figure 2 shows the average, across all routed
vehicles, of the percentage difference between the baseline-scenario trip
duration and the routed-scenario trip duration. A positive percentage change
indicates a reduction in trip duration after routing using real-time traffic
information. The error bars are computed in the same way as for the left plot.
The average saving varies between $2.5\%$ and $7.7\%$ over the day.

Figure 2: Total hourly demand (left plot) and percentage savings in duration
(right plot). Error bars show standard deviation across replications.

Figure 3 shows trip emissions (left) and trip distance (right). As for
duration, the $y$ axis shows the average across all routed vehicles of the
percentage difference between the baseline-scenario and the routed-scenario
trip. The left plot indicates that the average percentage emissions savings
vary, throughout the day, between $1\%$ and $2.2\%$. Similarly, the right plot
indicates that the average percentage distance savings vary, throughout the
day, between $0.8\%$ to $1.4\%$. For most of the day, emissions savings are
positively correlated with distance savings. Routed vehicles save $6.5\%$
travel duration on average and $2.6\%$ to $8.3\%$ depending on the time of
day. Conditioning on routed vehicles for whom the route actually changed, so
they were not already on the shortest path, these vehicles save $3.4\%$ of
emissions and $12.5\%$ of travel time on average.

Figure 3: Percentage savings in emissions (left plot) and distance (right
plot). Error bars show standard deviation across replications.

Our analysis shows that the emission savings are mostly coming from routing
users more efficiently to an optimal route with less travel duration and
travel distance, both for long trips and short trips. While in principle the
shortest duration path could result in higher travel distance and emissions,
our analysis shows that in practice, careful routing is a win-win-win, saving
simultaneously on time, distance, and emissions.

##  4 Discussion

This paper develops a high-resolution disaggregate traffic and emissions
simulation framework for metropolitan vehicular traffic. The framework is used
to quantify the travel time and CO2 emissions impacts of real-time navigation
with Google Maps. Location History, Google Maps and UDOT data are combined to
calibrate and validate the vehicular traffic model. The simulation is combined
with NREL’s RouteE model to estimate vehicular CO2 emissions. The analysis
focuses on Salt Lake City. It compares two scenarios that differ in whether or
not a subset of vehicles are routed with Google Maps. Routed drivers save
$6.5\%$ travel time and $1.7\%$ of emission by Google Maps navigation. The
results are based on traffic conditions observed during the Covid-19 pandemic,
where lower levels of congestion are observed. The impact of real-time
navigation is expected to be greater for higher levels of congestion. The
routing mechanism of this paper does not account for en-route trip guidance,
which is commonly used in Google Maps. We expect en-route guidance to be
particularly important and impactful for congested scenarios. This work is
part of ongoing work that quantifies the sustainability impact of real-time
route guidance. This paper focuses on the common policy of routing based on
shortest travel time. Ongoing work considers guidance based on both travel
time and emissions costs. It contributes to advancing our understanding of the
tradeoffs between travel time and emissions. Moreover, it informs the design
of future real-time navigation services that improve both network efficiency
and sustainability.

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