Two and Three Pseudoscalar Production in $e^{+}e^{-}$ annihilation and their contributions to $(g-2)_{\mu}$

Massless QCD exhibits a chiral symmetry that rules its effective field theory at low energy. $\chi$PT, valid at $E\ll M_{\rho}$, provides the interaction between the lightest octet of pseudoscalar mesons, and of these with external currents. At higher energies we need to take into account the hadronic resonance states, and a successful phenomenological approach is provided by R$\chi$T. Only the aspects of interest for our case are collected here. We follow the language and notation of Ref. Cirigliano et al. (2006).

The structure of the lagrangian has, essentially, three pieces:

$${\cal L}_{\mathrm{R}\chi\mathrm{T}}\,=\,{\cal L}_{\mathrm{GB}}\,+\,{\cal L}^{\mathrm{V}}_{\mathrm{kin}}\,+\,{\cal L}_{\mathrm{V-GB}}\,.$$

(1)

The first piece involves interaction terms with Goldstone bosons that cannot be generated by integrating out the vector resonance states. They are characterized by a perturbative expansion in terms of momenta (and masses), as in $\chi$PT. ${\cal L}^{\mathrm{V}}_{\mathrm{kin}}$ involves the kinetic term of the vector resonance states and ${\cal L}_{\mathrm{V-GB}}$ the interaction between Goldstone bosons and vector resonance fields. For the processes that we study in this work only the vector resonance fields will be needed. All of these lagrangians include also external fields coupled to scalar, pseudoscalar, vector, axial-vector or tensor currents. The lowest even-intrinsic-parity ${\cal O}(p^{2})$ of the ${\cal L}_{\mathrm{GB}}$ Lagrangian is given by

$$\mathcal{L}_{(2)}^{\mathrm{GB}}\equiv\mathcal{L}_{(2)}^{\chi\mathrm{PT}}=\frac{F^{2}}{4}\left\langle u_{\mu}u^{\mu}+\chi_{+}\right\rangle\ ,$$

(2)

being $F$ the decay constant of the pion and $\langle...\rangle$ indicates the trace in the $SU(3)$ space. The leading Wess-Zumino-Witten term describing the anomaly with odd-intrinsic-parity is of ${\cal O}(p^{4})$ Witten (1983); Wess and Zumino (1971). The explicit expression of interest for our work is given by

$$\mathcal{L}_{(4)}^{\mathrm{GB}}\,=\,i\frac{N_{C}\sqrt{2}}{12\pi^{2}F^{3}}\,\varepsilon_{\mu\nu\rho\sigma}\,\left\langle\partial^{\mu}\Phi\partial^{\nu}\Phi\partial^{\rho}\Phi v^{\sigma}\right\rangle+\cdots\ \ ,$$

(3)

where $v^{\sigma}$ is the external vector current and $\Phi$ the multiplet of Goldstone bosons. Higher orders of the ${\cal L}_{\mathrm{GB}}$ lagrangian will not be considered, as we assume that their couplings are dominated by resonance contributions ²²2Up to ${\cal O}(p^{4})$ at least, this setting depends on the realization of the spin-1 resonance fields. In Ref. Ecker et al. (1989b), it was proven that this assumption is correct if one uses the antisymmetric formulation for those fields, as we do..

The kinetic term of the vector resonance field is given by

$$\mathcal{L}_{\mathrm{kin}}^{\mathrm{V}}=-\frac{1}{2}\left\langle\nabla^{\lambda}V_{\lambda\mu}\nabla_{\nu}V^{\nu\mu}\right\rangle+\frac{1}{4}M_{V}^{2}\left\langle V_{\mu\nu}V^{\mu\nu}\right\rangle\ ,$$

(4)

Here the resonances are collected as $SU(3)_{V}$ octets and have the corresponding properties under chiral transformations. The Lagrangian that involves the interaction between Goldstone bosons and vector resonances, ${\cal L}_{\mathrm{V-GB}}$, couples the later octets with a chiral tensor constituted by the pseudoscalar nonet and external fields. Hence these chiral tensors obey a chiral counting ${\cal O}(p^{n})$. This allows us to assign a label $n$ to the different pieces as ${\cal L}_{(n)}^{V...}$, where the numerator indicates the resonance fields in the interaction terms. We will consider

$$\mathcal{L}_{\mathrm{V-GB}}=\mathcal{L}_{(2)}^{\mathrm{V}}+\mathcal{L}_{(4)}^{\mathrm{V}}+\mathcal{L}_{(2)}^{\mathrm{VV}}\ .$$

(5)

For instance, in the antisymmetric formulation for the spin-one vector resonances that we use,

	$\displaystyle{\cal L}_{(2)}^{\mathrm{V}}$	$\displaystyle=$	$\displaystyle\langle\,V_{\mu\nu}\,\chi_{(2)}^{\mu\nu}\,\rangle\,,$
	$\displaystyle\chi_{(2)}^{\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{F_{V}}{2\sqrt{2}}\,f_{+}^{\mu\nu}\,+\,i\,\frac{G_{V}}{\sqrt{2}}\,u^{\mu}\,u^{\nu}\,$		(6)

where $F_{V}$ and $G_{V}$ are coupling constants not determined by the symmetry. The rest of terms in Eq. (5) are collected in Ref. Cirigliano et al. (2006) for the even-intrinsic-parity terms and Refs. Ruiz-Femenia et al. (2003); Dumm et al. (2010a); Dai et al. (2013) for those of odd-intrinsic parity. The coupling constants of the interaction terms of $\mathcal{L}_{\mathrm{V-GB}}$ could be extracted from the phenomenology involving those states. As commented in the introduction the matching between the leading order in the OPE expansion of specific Green functions of QCD and their expressions within R$\chi$T is also a useful tool that has been employed in the bibliography Knecht and Nyffeler (2001); Ruiz-Femenia et al. (2003); Cirigliano et al. (2004, 2005); Husek and Leupold (2015); Dai et al. (2019); Kadavy et al. (2020). We will implement this procedure as far as it helps in our task. In particular we will use the relations between couplings specified in Ref. Dai et al. (2013).

However, the large energy region of study cannot be described fully with only one multiplet of vector resonances $V_{\mu\nu}$. The lightest one is situated around $M_{\rho}$, i.e. under $1\,\mbox{GeV}$. Two other vector multiplets populate the interval $1\,\mbox{GeV}\lesssim E\lesssim 2\,\mbox{GeV}$, that we will call $V^{\prime}_{\mu\nu}$ and $V^{\prime\prime}_{\mu\nu}$. Their couplings to the pseudoscalar mesons will be defined with respect to the ones of the lightest multiplet as $\beta_{\pi\pi}^{\prime}$,$\beta_{\pi\pi}^{\prime\prime}$,$\beta_{KK}^{\prime}$,$\beta_{KK}^{\prime\prime}$, through their poles, as

$$\frac{1}{M_{V}^{2}-x}\rightarrow\frac{1}{M_{V}^{2}-x}+\frac{\beta_{\pi\pi,KK}^{\prime}}{M_{V^{\prime}}^{2}-x}+\frac{\beta_{\pi\pi,KK}^{\prime\prime}}{M_{V^{\prime\prime}}^{2}-x}\ .$$

(7)

The $\rho-\omega$ mixing, required by the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ process, is reconsidered. While a constant mixing angle $\delta_{0}$ is enough to describe mixing in the three pseudoscalar case as discussed in Ref. (Dai et al., 2013):

$$\left(\begin{array}[]{c}|\bar{\rho}^{0}\rangle\\ |\bar{\omega}\rangle\end{array}\right)=\left(\begin{array}[]{cc}\cos\delta_{0}&-\sin\delta_{0}\\ \sin\delta_{0}&\cos\delta_{0}\end{array}\right)\left(\begin{array}[]{c}|\rho^{0}\rangle\\ |\omega\rangle\end{array}\right)\ ,$$

(8)

an energy dependent mixing angle is discussed in Ref. (Gasser and Leutwyler, 1982), although in the non-relativistic limit and we need to generalize it to the relativistic case. The energy dependent mixing angle could be parameterized as

	$\displaystyle\left(\begin{array}[]{c}|\bar{\rho}^{0}\rangle\\ |\bar{\omega}\rangle\end{array}\right)$	$\displaystyle=\left(\begin{array}[]{cc}{\cos\delta}&{\frac{M_{V}\Gamma_{\rho}\sin\delta}{-(M_{V}^{2}-s)+iM_{V}(\Gamma_{\rho}-\Gamma_{\omega})}}\\ {\frac{M_{V}\Gamma_{\rho}\sin\delta}{-(M_{V}^{2}-s)-iM_{V}(\Gamma_{\rho}-\Gamma_{\omega})}}&{\cos\delta}\end{array}\right)\left(\begin{array}[]{c}\left|\rho^{0}\right\rangle\\ \left|\omega\right\rangle\end{array}\right)$		(15)
		$\displaystyle\equiv\left(\begin{array}[]{cc}{\cos\delta}&{-\sin\delta^{\omega}(s)}\\ {\sin\delta^{\rho}(s)}&{\cos\delta}\end{array}\right)\left(\begin{array}[]{c}\left|\rho^{0}\right\rangle\\ \left|\omega\right\rangle\end{array}\right)\ ,$		(20)

where $\left|\rho^{0}\right\rangle,|\omega\rangle$ denote the physical states. Hence the energy dependence of the mixing angle is driven by the resonance propagators. Here $M_{V}$ is the mass of the nonet of vector resonances in the $SU(3)$ limit. We will take $M_{V}=M_{\rho}$. For the two body final state processes $e^{+}e^{-}\to\pi^{+}\pi^{-},K^{+}K^{-}$, we always take energy dependent mixing mechanism according to Eq. (20). For the three body cases, we adopt two ways. One is to take the same energy dependent $\rho-\omega$ mixing mechanism as that of the two body case. This will be Fit I. The other is to use the constant mixing angle $\delta_{0}$. This will be our Fit II. Comparison of both fits will unveil the influence of $\rho-\omega$ mixing in the analysis of data.

The amplitude for three-meson production in $e^{+}e^{-}$ collisions is driven by the hadronization of the electromagnetic current, in terms of one vector form factor only:

$$\langle\pi^{+}(p_{1})\pi^{-}(p_{2})P(p_{3})|\left({\cal V}_{\mu}^{3}+{\cal V}_{\mu}^{8}/\sqrt{3}\right)\,e^{i{\cal L}_{\mbox{\tiny{QCD}}}}|0\rangle\,=\,i\,F_{V}^{P}(Q^{2},s,t)\,\varepsilon_{\mu\nu\alpha\beta}\,p_{1}^{\nu}p_{2}^{\alpha}p_{3}^{\beta}\,,$$

(21)

being ${\cal V}_{\mu}^{i}=\overline{q}\gamma_{\mu}(\lambda^{i}/2)q$ and $P=\pi,\eta$. The Mandelstam variables are defined as $s=(Q-p_{3})^{2}$, $t=(Q-p_{1})^{2}$, with $Q=p_{1}+p_{2}+p_{3}$. The cross section and amplitudes for the three pseudoscalar cases that we are considering, namely $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\eta$, are quite the same as specified in Ref. (Dai et al., 2013), except for a small change in the treatment of $\rho-\omega$ mixing, as illustrated in Sect. II.1. The corresponding expressions for the cross-section and the modified form factors for the three pseudoscalar cases are collected in Appendix A.

These form factors depend on several couplings of the R$\chi$T lagrangian that are not determined by the symmetry. However, some of them or, at least, relations between them can be established by matching Green functions calculated in this framework with their expressions at leading order OPE expansion of QCD, as it has been commented before. By implementing these short-distance relations our form factors satisfy both the chiral constraints in the low-energy region and the asymptotic constraints at the high energy limit ($Q^{2}\rightarrow\infty$). Hence the only unknown couplings in these form factors will be $F_{V}$, $2g_{4}+g_{5}$, $d_{2}$, $c_{3}$ and $\alpha_{V}$ Dai et al. (2013), to be added to the $\beta_{\pi\pi,KK}^{\prime}$ and $\beta_{\pi\pi,KK}^{\prime\prime}$ from Eq. (7) and the mixing angles between the octet and singlet pseudoscalar ($\theta_{P}$) and vector ($\theta_{V}$) components, defined also in Dai et al. (2013).

Two-pseudoscalar final states in $e^{+}e^{-}$ annihilation are given by the corresponding vector form factor

$$\langle P^{+}(p_{1})P^{-}(p_{2})|\left({\cal V}_{\mu}^{3}+{\cal V}_{\mu}^{8}/\sqrt{3}\right)\,e^{i{\cal L}_{\mbox{\tiny{QCD}}}}|0\rangle\,=\,(p_{1}-p_{2})_{\mu}\,F_{V}^{P}(Q^{2})\,,$$

(22)

with $Q=p_{1}+p_{2}$ and $P=\pi,K$. The energy in the center of mass frame is given by $E_{cm}\equiv\sqrt{Q^{2}}$. The cross sections $\sigma_{\pi\pi}\equiv\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ and $\sigma_{KK}\equiv\sigma(e^{+}e^{-}\to K^{+}K^{-})$ are given by

$$\sigma_{PP}=\alpha_{e}^{2}\,\frac{\pi}{3\,Q^{2}}\,\left(1-4\frac{m_{P}^{2}}{Q^{2}}\right)^{3/2}\,|F_{V}^{P}(Q^{2})|^{2}\,.$$

(23)

The form factors $F_{V}^{\pi}(Q^{2})$ and $F_{V}^{K}(Q^{2})$ were thoroughly studied in Ref. (Arganda et al., 2008) (see also Guerrero and Pich (1997); Pich and Portolés (2001); Miranda and Roig (2018) for alternative parameterizations) in the case of tau decays. Hence we need to include now the new $\rho-\omega$ mixing mechanism, present in $e^{+}e^{-}$ into hadrons. We also extend the described energy region by adding heavier vector multiplets, as commented above. Their expressions are:

	$\displaystyle F_{V}^{\pi}$	$\displaystyle=\left(1+\frac{F_{V}G_{V}}{F^{2}}Q^{2}\left(BW(M_{\rho},\Gamma_{\rho,},Q^{2})\right.\right.$
		$\displaystyle\left.\ \ +\beta_{\pi\pi}^{{}^{\prime}}BW(M_{\rho^{{}^{\prime}}},\Gamma_{\rho^{{}^{\prime}},},Q^{2})+\beta_{\pi\pi}^{{}^{\prime\prime}}BW(M_{\rho^{{}^{\prime\prime}}},\Gamma_{\rho^{{}^{\prime\prime}},},Q^{2})\right)$
		$\displaystyle\ \ (\frac{1}{\text{$\sqrt{3}$}}\sin\theta_{V}\sin\delta^{\rho}+\cos\delta)\cos\delta$
		$\displaystyle-\frac{F_{V}G_{V}}{F^{2}}Q^{2}\left(BW(M_{\omega},\Gamma_{\omega,},Q^{2})+\beta_{\pi\pi}^{{}^{\prime}}BW(M_{\omega^{{}^{\prime}}},\Gamma_{\omega^{{}^{\prime}},},Q^{2})\right.$
		$\displaystyle\left.\left.+\beta_{\pi\pi}^{{}^{\prime}"}BW(M_{\omega^{{}^{\prime\prime}}},\Gamma_{\omega^{{}^{\prime\prime}},},Q^{2})\right)(\frac{1}{\text{$\sqrt{3}$}}\sin\theta_{V}\cos\delta-\sin\delta^{\omega})\sin\delta^{\omega}\right)$
		$\displaystyle\ \ \exp\left[\frac{-s}{96\pi^{2}F^{2}}\left({\rm Re}\left[A[m_{\pi},M_{\rho},Q^{2}]+\frac{1}{2}A[m_{K},M_{\rho},Q^{2}]\right]\right)\right]\ ,$		(24)

	$\displaystyle F_{V}^{K}$	$\displaystyle=\Bigm{(}\frac{\cos\theta_{V}{}^{2}}{2}\frac{F_{V}G_{V}}{F^{2}}(1+8\sqrt{2}\alpha_{V}\frac{2m_{K}^{2}-m_{\pi}^{2}}{M_{V}^{2}})M_{\phi}^{2}\bigm{(}BW(M_{\text{$\phi$}},\Gamma_{\phi},Q^{2})+\beta_{KK}^{{}^{\prime}}BW(M_{\phi^{{}^{\prime}}},\Gamma_{\phi^{{}^{\prime}},},Q^{2})$
		$\displaystyle\ \ +\beta_{KK}^{{}^{\prime\prime}}BW(M_{\phi^{{}^{\prime\prime}}},\Gamma_{\phi^{{}^{\prime\prime}},},Q^{2}))+\frac{X_{1}}{24}\frac{F_{V}G_{V}}{F^{2}}(1+8\sqrt{2}\alpha_{V}\frac{m_{\pi}^{2}}{M_{V}^{2}})M_{\omega}^{2}\bigm{(}BW(M_{\omega},\Gamma_{\omega},Q^{2})$
		$\displaystyle\ \ +\beta_{KK}^{{}^{\prime}}BW(M_{\omega^{{}^{\prime}}},\Gamma_{\omega^{{}^{\prime}},},Q^{2})+\beta_{KK}^{{}^{\prime\prime}}BW(M_{\omega^{{}^{\prime\prime}}},\Gamma_{\omega^{{}^{\prime\prime}},},Q^{2})\Bigm{)}\exp\left[\frac{-q^{2}}{96\pi^{2}F^{2}}\left(\frac{3}{2}{\rm Re}(A[m_{K},M_{\rho},Q^{2}])\right)\right]$
		$\displaystyle\ \ +\frac{X_{2}}{24}\frac{F_{V}G_{V}}{F^{2}}M_{\rho}^{2}(1+8\sqrt{2}\alpha_{V}\frac{m_{\pi}^{2}}{M_{V}^{2}})\Bigm{(}BW(M_{\rho},\Gamma_{\rho},Q^{2})+\beta_{KK}^{{}^{\prime}}BW(M_{\rho^{{}^{\prime}}},\Gamma_{\rho^{{}^{\prime}},},Q^{2})$
		$\displaystyle\ \ +\beta_{KK}^{{}^{\prime\prime}}BW(M_{\rho^{{}^{\prime\prime}}},\Gamma_{\rho^{{}^{\prime\prime}},},Q^{2})\Big{)}\exp\left[\frac{-q^{2}}{96\pi^{2}F^{2}}\left({\rm Re}\left[A[m_{\pi},M_{\rho},Q^{2}]+\frac{1}{2}A[m_{K},M_{\rho},Q^{2}]\right]\right)\right]\text{\ .}$		(25)

The functions in Eqs. (24,25) are given by:

	$\displaystyle\left[BW(M_{V},\Gamma_{V},Q^{2})\right]^{-1}$	$\displaystyle=$	$\displaystyle M_{V}^{2}-iM_{V}\Gamma_{V}(Q^{2})-Q^{2}\,,$
	$\displaystyle A\left(m_{P},\mu,Q^{2}\right)$	$\displaystyle=$	$\displaystyle\ln\left(m_{P}^{2}/\mu^{2}\right)+\frac{8m_{P}^{2}}{Q^{2}}-\frac{5}{3}+\sigma_{P}^{3}\ln\left(\frac{\sigma_{P}+1}{\sigma_{P}-1}\right)\,,$
	$\displaystyle\sigma_{P}$	$\displaystyle\equiv$	$\displaystyle\sqrt{1-4m_{P}^{2}/Q^{2}}\,,$		(26)

and

	$\displaystyle X_{1}$	$\displaystyle=$	$\displaystyle-\,16\sqrt{3}\cos\delta\sin\theta_{V}\sin\delta^{\omega}(Q^{2})-6\cos^{2}\delta\cos 2\theta_{V}+12\sin^{2}\delta_{\omega}(Q^{2})+3\cos 2\delta+3\ ,$
	$\displaystyle X_{2}$	$\displaystyle=$	$\displaystyle-\,6\cos 2\theta_{V}\sin^{2}\delta^{\rho}(Q^{2})+16\sqrt{3}\cos\delta\sin\theta_{V}\sin\delta^{\rho}(Q^{2})$		(27)
			$\displaystyle+\,6\sin^{2}\delta^{\rho}(Q^{2})+6\cos 2\delta+6\ ,$

Notice that $X_{1}=12\sin^{2}\theta_{V}$ and $X_{2}=12$ in the isospin limit. The angles $\sin\delta^{\rho,\omega}$ related with the $\rho-\omega$ mixing are defined in Eq. (20). The $Q^{2}$ dependence of resonance widths are a debated issue. A thorough proposal within the chiral framework was proposed in Ref. Gómez Dumm et al. (2000) for wide resonances. We will use this result for $\Gamma_{\rho}(Q^{2})$, while a parameterization in terms of the on-shell widths, driven by the two-body phase-space decay will be employed for $\Gamma_{\rho^{\prime},\rho^{\prime\prime}}(Q^{2})$. The precise expressions are collected in Ref. Dai et al. (2013). Meanwhile the rest of resonances, that are quite narrow, will be taken constant. Notice that the two-body vector form factors do not include more unknown couplings to those of three-body form factors.

A comparison between our fitted parameters and those of Fit 4 in Ref. (Dai et al., 2013) is shown in Table 2. We also compare the masses and widths of the resonances with those listed in PDG (Zyla et al., 2020). The fitting procedure is carried out with MINUIT (James and Roos, 1975).

The quoted errors in the fitted parameters are provided by the Bootstrap method. In general, the parameters in Fit I and Fit II are consistent with those of Fit 4 in Ref. (Dai et al., 2013), within a deviation of about $10\%$. $F_{V}$, $2g_{4}+g_{5}$, $\theta_{V}$, $\delta_{0}$ and/or $\delta$ are mainly determined by the experimental data under $1.05\,\mbox{GeV}$, where it has higher statistics and precision. However, the joint fit including the $e^{+}e^{-}\to K^{+}K^{-}$ process constrain $\theta_{V}$ strongly. This can be understood from the form factor in Eq. (25), where the cross section around the $\phi$ peak increases with the descent of $\theta_{V}$. In contrast, the cross section of $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ around the $\phi$ peak decreases when $\theta_{V}$ goes down, which could be deduced from the expressions in Appendix A. As a consequence, $\theta_{V}$ is about $1^{\circ}$ larger than that of Ref. (Dai et al., 2013). The inclusion of $e^{+}e^{-}\to K^{+}K^{-}$ process also constrains $\alpha_{V}$, the higher order correction to the $F_{V}$ coupling arising from $SU(3)$ symmetry breaking. The cross section of $e^{+}e^{-}\to K^{+}K^{-}$ increases with rising $\alpha_{V}$. To confront the theoretical predictions to the experimental data of the cross section of $e^{+}e^{-}\to K^{+}K^{-}$, $\alpha_{V}$ is fixed to be negative. Notice that $\alpha_{V}$ is small as it is higher order correction.

The energy dependent $\rho-\omega$ mixing angle $\delta$ is determined by the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ process. From Eq. (24), the cross section of $e^{+}e^{-}\to\pi^{+}\pi^{-}$ is mainly determined by $\delta$, since $F_{V}G_{V}/F^{2}=1$ is constrained by the high energy behaviour and $\theta_{V}$ could be determined as above. The two mechanisms of $\rho-\omega$ mixing adopted in Fit I and II have almost no effects on the three body final state case. There is only a very little difference reflected around the $\rho$ peak in the $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ process. In the energy region around their masses, $\rho$ and $\omega$ mix with a relative phase that results in a larger mode of $|F_{V}^{\pi}|^{2}$. Hence the magnitude of $F_{V}$ and $2g_{4}+g_{5}$ are smaller in Fit I in comparison to Fit II and the results in Ref. (Dai et al., 2013).

The parameters related with the resonance multiplets are almost the same in Fit I and Fit II, but some of them are different from those of Ref. (Dai et al., 2013). They are mainly determined by the energy region above $1.0\,\mbox{GeV}$. Both $e^{+}e^{-}\to\pi^{+}\pi^{-}$ and $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ processes are sensitive to the masses and widths of $\rho^{{}^{\prime}}$ and $\rho^{{}^{\prime\prime}}$ in this energy region. The $e^{+}e^{-}\to\pi^{+}\pi^{-}$ data gives relative smaller masses and larger widths of $\rho^{{}^{\prime}}$, compared with those provided by the $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ process. Hence the combined fitted $\rho^{{}^{\prime}}$ mass is about $30\,\mbox{MeV}$ smaller and the $\rho^{{}^{\prime}}$ width is about $100\,\mbox{MeV}$ larger than those in Ref. (Dai et al., 2013). The mass and width of $\rho^{{}^{\prime\prime}}$ also changes slightly. Consequently, the relative weights of the $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ process $\beta^{\prime}_{\eta}$ and $\beta^{\prime\prime}_{\eta}$ have sizable changes compared with those in Ref. (Dai et al., 2013). Meanwhile, the strengths of the $e^{+}e^{-}\to\pi^{0}\pi^{+}\pi^{-}$ process $\beta^{\prime}_{\pi}$ and $\beta^{\prime\prime}_{\pi}$ are similar. Notice that in the two-body processes $e^{+}e^{-}\to\pi^{+}\pi^{-}$ and $e^{+}e^{-}\to K^{+}K^{-}$, the parameters $\beta_{\pi\pi}^{{}^{\prime}(^{\prime\prime})}$ and $\beta_{KK}^{{}^{\prime}(^{\prime\prime})}$ turn out to be very small with magnitudes $\lesssim\,0.2$, as expected by lowest meson dominance Moussallam (1995, 1997); Knecht et al. (1999); Knecht and Nyffeler (2001); Bijnens et al. (2003).

Since $d_{2}$, $c_{3}$ and $\theta_{P}$ are mainly correlated with the $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ process, they also have sizable changes, while masses and widths of other resonance multiplets are quite the same. In summary, and as shown in Table 2, the fitted masses and widths of heavier multiplets are closer to the experimental average values in PDG (Zyla et al., 2020), due to a combination of updated experimental measurements and the constraints from $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ processes.

Notice however, that the masses and widths of $\rho^{\prime}$ and $\rho^{\prime\prime}$ obtained here correspond to the specific definition of the energy dependent width propagator shown in Eq. (40) of Ref. (Dai et al., 2013), which may not be used by the experimentalists. Hence a precise comparison with the experimental determinations is not straightforward.

Finally $\theta_{V}$ and $\alpha_{V}$ change sizeably with respect to the results of Ref. (Dai et al., 2013) due to the inclusion of the process $e^{+}e^{-}\to K^{+}K^{-}$, so that the partial widths sensitive to $\theta_{V}$ and $\alpha_{V}$ become worse. Nevertheless, these partial widths turn out to be bearable with the experimental data from PDG (Zyla et al., 2020), considering the incertitude associated with the theoretical framework of large-$N_{C}$ expansion implemented in the framework of R$\chi$T. In addition, the difference of partial widths of $\Gamma_{\rho^{0}\to\pi\pi\pi}$ in Fit I and Fit II are caused by the different parametrization of the $\rho-\omega$ mixing. The $\rho^{0}$ decays in Fits. I and II have different mixing angles and also the former one is energy dependent, see eq.(2.9).

The comparison of our solutions for the three pseudoscalar case with experimental data is shown in Figure 1, and that of the two pseudoscalar case is shown in Figur 2. The results of Fit I are shown in blue dashed lines and those of Fit II are shown in solid black lines. In general, Fit I and Fit II are almost indistinguishable. There is slight difference shown around the $\rho$ peak in $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ process at $0.6\,<\,E\,<\,1$ (GeV), due to the different parametrization of $\rho-\omega$ mixing adopted. Noted that Fit II is a little better in this region, since there is one more parameter $\delta_{0}$ and the energy dependent mixing mechanism designed for the $\pi\pi$ scattering may not be suitable for the three pion case, where the three body re-scattering needs to be considered.. Fit II seems also a little better at the $\phi$ peak in the $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ process. This is because that, $F_{V}$ and $2g_{4}+g_{5}$ in Fit II are allowed to have larger values than in Fit I, which can slightly compensate the $\phi$ peak in $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$. As illustrated above, the $\theta_{V}$ and $\alpha_{V}$ constrained by the $e^{+}e^{-}\to K^{+}K^{-}$ will suppress the $\phi$ peak in $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$. The high energy behaviour of $e^{+}e^{-}\to\pi^{+}\pi^{-}$, as shown in Figure 2, is balanced with the $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ process through the mass and width of $\rho^{{}^{\prime}}$.