# Simulation of $1/f$ charge noise affecting a quantum dot in a Si/SiGe
structure

M. Kępa  Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46,
PL 02-668 Warsaw, Poland  N. Focke  JARA-FIT Institute for Quantum
Information, Forschungszentrum Jülich GmbH and RWTH Aachen University, Aachen,
Germany  Ł. Cywiński  Institute of Physics, Polish Academy of Sciences, al.
Lotników 32/46, PL 02-668 Warsaw, Poland  J. A. Krzywda<EMAIL_ADDRESS>Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, PL
02-668 Warsaw, Poland  Lorentz Institute and Leiden Institute of Advanced
Computer Science, Leiden University, P.O. Box 9506, 2300 RA Leiden, The
Netherlands

###### Abstract

Due to presence of magnetic field gradient needed for coherent spin control,
dephasing of single-electron spin qubits in silicon quantum dots is often
dominated by $1/f$ charge noise. We investigate theoretically fluctuations of
ground state energy of an electron in gated quantum dot in realistic Si/SiGe
structure. We assume that the charge noise is caused by motion of charges
trapped at the semiconductor-oxide interface. We consider a realistic range of
trapped charge densities, $\rho\\!\sim\\!10^{10}$ cm-2, and typical
lenghtscales of isotropically distributed displacements of these charges,
$\delta r\\!\leq\\!1$ nm, and identify pairs $(\rho,\delta r)$ for which the
amplitude and shape of the noise spectrum is in good agreement with spectra
reconstructed in recent experiments on similar structures.

Charge noise with $1/f$ spectrum is ubiquitous in semiconductor
nanostructures. Dutta and Horn (1981); Paladino  _et al._ (2014); Fleetwood
(2015); Islam  _et al._ (2022) In gate-defined quantum dots (QDs) Wuetz  _et
al._ (2022) this noise is probably caused by bistable sources of electric
fields localized at the semiconductor/insulator interface, or within the
insulator separating semiconducting material from the gate.Sakamoto  _et al._
(1995); Liefrink  _et al._ (1994); Ramon and Hu (2010); Bermeister  _et al._
(2014) Spins of electrons or holes confined in QDs have been demonstrated to
be viable qubits,Hanson  _et al._ (2007); Burkard  _et al._ (2021) with the
gated QD architectures showing constant progress towards scalable multi-qubit
devices.Philips  _et al._ (2022); Borsoi  _et al._ (2022) While using the spin
of a single electron as the qubit makes its quantum state insensitive, to the
first order, to charge noise, decoherence caused by that noise is relevant for
operation of two-qubit gates Huang  _et al._ (2018); Veldhorst  _et al._
(2015); Zajac  _et al._ (2018); Watson  _et al._ (2018) and evolution of
singlet-triplet qubits based on two electrons in two QDs. Coish and Loss
(2005); Hu and Das Sarma (2006); Dial  _et al._ (2013); Takeda  _et al._
(2020); Connors  _et al._ (2022); Chan  _et al._ (2018); Qi  _et al._ (2017);
Culcer and Zimmerman (2013) Moreover, charge noise has been recently
recognized as the factor limiting the quality of single-spin qubits in
silicon-based QDs. Yoneda  _et al._ (2018); Struck  _et al._ (2029); Saraiva
_et al._ (2021) This is due to presence of either real, Gilbert  _et al._
(2023); Harvey-Collard  _et al._ (2019); Tanttu  _et al._ (2019) or artificial
spin-orbit couplings caused by presence of magnetic field gradients, which are
necessary for single-qubit operations.Takeda  _et al._ (2022, 2016); Brunner
_et al._ (2011); Zajac  _et al._ (2018); Yoneda  _et al._ (2018)

The microscopic sources of charge noise in gated QD architectures have not
been fully characterized yet. However, recent progress in reconstruction of
spectra of $1/f$ charge noise affecting the QDs Struck  _et al._ (2029);
Connors  _et al._ (2019, 2022); Yoneda  _et al._ (2022) has provided a lot of
experimental data against which one could test various models of these
sources. It is now known that the power spectral density (PSD) of noise of
electronic energy levels in a typical QD has $1/f$ form for
$f\in[10^{-3},10^{6}]$ Hz (and possibly in an even broader range), albeit with
visible deviations from ideal $1/f$ scaling, and the typical amplitude of
noise at 1 Hz is of the order of $1$ $\mu$eV/$\sqrt{\mathrm{Hz}}$ for
electrons in Si/SiGe,Connors  _et al._ (2022); Struck  _et al._ (2029); Wuetz
_et al._ (2022); Petersson  _et al._ (2010); Mi  _et al._ (2018) while being
within one order of magnitude from this value for any other characterized spin
qubit, see Ref. Wuetz  _et al._ , 2022 and references therein.

In recent papers modeling of charge noise was performed for electronShehata
_et al._ (2022) and holeShalak  _et al._ (2022) spin qubits in SiMOS devices.
Here we focus on electron spin qubits in Si/SiGe devices of the type used in
recent experiments on electron spin qubit shuttling.Seidler  _et al._ (2022);
Langrock  _et al._ (2022) We assume that the noise is caused by sub-nm motion
of charges trapped at the Si/SiO2 interface Klos  _et al._ (2018, 2019);
Spruijtenburg  _et al._ (2016); Thoan  _et al._ (2011); Campbell  _et al._
(2005); Langrock  _et al._ (2022) that have density
$\rho_{\mathrm{ic}}\\!\sim\\!10^{10}$ cm-2. Simulations in Ref. Langrock  _et
al._ , 2022 showed that using $\rho_{\mathrm{ic}}\\!=\\!5\times 10^{10}$ cm-2
reproduces typically observed electrostatic-disorder-induced energy detunings
between neighboring QDs. Focusing on $\rho_{\mathrm{ic}}$ in range from
$5\times 10^{9}$cm-2 to $10^{11}$cm-2, we calculate the influence of these
charges on the ground-state (GS) orbital energy of an electron in a QD in the
considered nanostructure. We then investigate what typical length scale of
isotropic charge motion and charge densities lead to reproduction of measured
shape and amplitude of $1/f$-like noise spectra.

We employ the simplest model leading to $1/f$ noise spectrum: that of many
independent sources of Random Telegraph Noise (RTN) Machlup (1954); Ramon and
Hu (2010); Constantin  _et al._ (2009) \- the so-called two-level fluctuators
(TLFs)- having switching rates $\gamma$ drawn from a log-normal distribution
spanning over many orders of magnitude of $\gamma$. Schriefl  _et al._ (2006);
You  _et al._ (2021); Shnirman  _et al._ (2005); Müller  _et al._ (2019)
Distinct electric fields generated by the TLF in each of its states cause
state-dependent shifts of ground state energy of the nearby QD. We define the
coupling of the QD to the $n$-th TLF, $\eta_{n}$, as the shift of the energy
of the GS orbital of an electron in the QD caused by a switching of the TLF.
We focus on switching events that correspond to a sub-nm motion of a defect
charge in the insulating layer between the semiconductor and the metallic gate
(the blue region in Fig. 1). The two locations of the charge corresponding to
the two states of the $n$-th TLF are $\mathbf{r}_{n}$ and
$\mathbf{r}_{n}+\delta\mathbf{r}_{n}$. We have checked that charge tunneling
between the insulator and the metal leads to shifts of energy levels of the QD
that are too large to be reconciled with the experimentally observed noise
levels.Shehata  _et al._ (2022)

The PSD of the noise in GS energy of the electron in the QD is given by a sum
of Lorentzian PSDs of the RTNs:

$S(f)=\sum_{n=1}^{N}\frac{2\eta_{n}^{2}}{\gamma_{n}+(2\pi
f)^{2}/\gamma_{n}}\,\,.$ (1)

For a large number of TLFs, a flat distribution of $\eta_{n}$, and a log-
normal distribution of $\gamma_{n}$, i.e. $p(\gamma_{n})\propto 1/\gamma_{n}$,
results in the PSD that is well approximated by the 1/f noise spectrum, i.e.
$S(f)\\!\approx\\!S_{1}f_{1}/f$, where $S_{1}$ is the power of the noise at
$f_{1}=1$Hz, which is of the order of $1$ $\mu$eV2/Hz in state-of-the-art
Si/SiGe QDs.Kranz  _et al._ (2020)

Figure 1: a) CAD model of the device. Electrodes that define the potential
located on top of the device are separated from the bulk structure with
conductive and insulating oxide layers. $d_{Gate}=50\ \text{nm}$,
$d_{Channel}=60\ \text{nm}$, $h_{Top}=60\ \text{nm}$, $h_{Si}=10\ \text{nm}$,
$h_{Bottom}=30\ \text{nm}$. b) Orbital gap $\omega=E_{1}-E_{2}$ as a function
of applied voltage with interpolated trend line. Grey dashed line marks
$V=0.25\ \text{V}$ that corresponds to $\omega=1.99\approx 2\ \text{meV}$,
which is used in all further simulations. c) The dependence of individual
energy levels along with lever arm fits with $\alpha_{0}=-148.59\
\text{meV/V}$ and $\alpha_{1}=-144.32\ \text{meV/V}$. Figure 2: Results of the
finite-element method implemented in COMSOL. a) Shift of ground state energy
$\delta E(\mathrm{r})$ due to charge defect located at
$\mathrm{r}=(x,y,z_{0})$, where $z_{0}=102$nm (see Fig. 1a). b) Illustration
of the energy shift $\delta E$ (left) caused by the presence of a point
charge, and additional shift $\eta$ (right) caused by a displacement of this
charge from red to blue point in panel a). We exaggerate the displacement size
for illustration purposes. c) Interpolated spatial derivatives of $\delta
E(\mathbf{r})$ along all three dimensions.

We consider a realistic Si/SiGe device based on Refs. Langrock  _et al._ ,
2022; Seidler  _et al._ , 2022, the CAD model which is shown in Fig. 1a. The
structure consists of a quantum well (pink) that has two
$\text{Si}_{0.7}\text{Ge}_{0.3}$ layers with a 10 nm thick Si layer in
between. On top, there are 13 electrodes (yellow, red),to which voltages are
applied. We use a gate pitch of $60\,\mathrm{nm}$ and the width of each gate
is set to $50\,\mathrm{nm}$, which are within the range considered in Ref.
Langrock  _et al._ , 2022. In this design, we have a 10 nm screening layer
(green) that is insulated from the quantum well and the electrodes by another
10 nm of the oxide layer (blue), followed by a 1.5 nm Si Cap layer. We use
$\epsilon_{r}\\!=\\!3.9$ for the relative permittivity of the SiO2 oxide. We
define a channel as a gap in the screening layer, which allows the electric
field to penetrate the structure and define the QDs about 80 nm below the
terminal. We set $d_{\text{Channel}}=200\,\mathrm{nm}$ as in Refs. Langrock
_et al._ , 2022; Mills  _et al._ , 2022. The remaining dimensions are given in
the caption of Fig. 1a. The device structure and dimensions are similar to
those of experimentally studied QDs,Zajac  _et al._ (2015); Connors  _et al._
(2022); Borselli  _et al._ (2015); Paquelet Wuetz  _et al._ (2022); Holman
_et al._ (2021), although except for Ref. Holman  _et al._ , 2021 plunger and
barrier gates tend to be located on separate layers.

We condsider a single QD defined under the central gate (red). We solve for
electrostatic potential with voltages set on the electrodes, and we then find
the electron GS energy, $E_{0}$, by solving the 3-dimensional Schroedinger
equation in the 10 nm Si layer. We characterize the QD in Fig. 1bc, where the
orbital gap and energies of the two lowest orbitals are shown as a function of
the voltage applied to central gate, $V$, while keeping the other gates
grounded. We choose $V=0.25$ V giving a realistic orbital gap
$\omega=E_{1}-E_{0}\approx 2$ meV.

We assume that the charge noise is generated by the motion of negative charges
close to the semiconductor/oxide interface about 60 nm above the Si quantum
well, i.e. on the upper surface of pink block at $z_{0}\approx 100$ nm in Fig.
1a. A charge at $\mathbf{r}_{n}=(x_{n},y_{n},z_{0})$ shifts the GS of the
electron by $\delta E(\mathbf{r}_{n})=E_{0}^{\prime}-E_{0}$, where
$E_{0}^{\prime}$, $E_{0}$ are the GS energies with, and without the charge
present, respectively. To compute this shift we insert a point charge $(-|e|)$
at $\mathbf{r}_{n}=(x_{n},y_{n},z_{0})$, and repeat the procedure described
above: compute the electrostatic potential, and find the new GS energy
$E_{0}^{\prime}=E_{0}+\delta E(\mathbf{r}_{n})$. In Fig. 2a we plot $\delta
E(\mathbf{r}_{n})$ for the device shown in Fig. 1. We assume that the charges
are located near the $z_{0}=102$ nm plane, i.e. $0.5$ nm above the interface
between the oxide and the Si cap layer. The results for $A=600$ nm x $600$ nm
region are obtained using COMSOL software, with a regular grid of 31x31 charge
positions in the plane. We have checked that shifts from two charges localized
at $\mathbf{r}_{n}$ and $\mathbf{r}_{m}$ are approximately additive, i.e.
$\delta E(\mathbf{r}_{n},\mathbf{r}_{m})\approx\delta E(\mathbf{r}_{n})+\delta
E(\mathbf{r}_{m})$ within a $2\%$ relative error.

Figure 3: Statistics of the amplitude of the noise, for different charge
densities $\rho$ and magnitudes of charge displacement, $\delta r$. a) The
distribution of noise amplitude from 1000 realizations for $\delta
r=0.1,0.2,0.3,0.4,0.5$nm (x-axis) and $\rho=5\times 10^{9},10^{10},2\times
10^{10}$cm-2 (colors). Black dashed line corresponds to the targeted
$\sigma_{0}=8\mu$eV. b) Ensemble averaged noise amplitude,
$\langle\sigma\rangle$ and c) standard deviation of noise amplitude over the
ensemble
$\sqrt{\text{Var}(\sigma)}=\sqrt{\langle\sigma^{2}\rangle-\langle\sigma\rangle^{2}}$,
both as a function of $\delta r$ for three charge densities from a), with
linear trends marked by dashed lines).

To model a single TLF we add a random displacement of the charge position,
$\delta\mathbf{r}_{n}$ that causes an additional shift in the GS energy (see
Fig. 2), given by

$\eta_{n}=\delta E(\mathbf{r}_{n}+\delta\mathbf{r}_{n})-\delta
E(\mathbf{r}_{n})\,\,,$ (2)

defining the coupling between the TLF and the QD. As the relative energy
shifts due to $|\delta\mathbf{r}_{n}|\\!<\\!1$ nm are small, we approximate
the coupling by

$\eta_{n}\approx\nabla\delta E(\mathbf{r}_{n})\cdot\delta\mathbf{r}_{n},$ (3)

where $\nabla=(\partial_{x},\partial_{y},\partial_{z})$. In Fig. 2c we show
the components of $\nabla\delta E(\mathbf{r})$, generated from interpolated
$\delta E(\mathbf{r})$. For slowly varying $x$ and $y$ derivatives we
interpolate results from Fig. 2a. For $z$ derivative we interpolate results
obtained at $z_{0}=101.75,102.0,102.25,102.5$nm planes.

The motion of the charge away from the dot leads to decrease of $\delta
E(\mathbf{r})$, i.e. $\eta<0$. In the $xy$ plane this is due to change of a
distance between the defect and a QD, which leads to $|\partial_{x,y}\delta
E(\mathbf{r})|\approx 10\mu$eV/nm in the central region of $<\\!100$ nm from
the dot. The motion towards and away from the dot results in the opposite sign
of $x$ and $y$ derivatives on the two sides of the QD center. On the other
hand, the derivative in $z$ direction is always negative, and about an order
of magnitude larger, as it is related to a decrease of the distance between
the defect and metallic layer, i.e. a decrease of a dipole moment of the
electron and its mirror charge in the metallic gate. As a result,
$\partial_{z}\delta E(\mathbf{r})\approx 50\mu$eV/nm in the central region of
$\approx 100$ nm radius, in which $\delta E(\mathbf{r})$ is significant.
Outside of this region $\partial_{z}\delta E(\mathbf{r})$ decays towards
$\approx 0.5\mu$eV/nm at the edge of the considered region $A$. This shows
that the TLFs outside of $A$ do not contribute significantly to the energy
shift.

We compute the PSD of total noise by adding contributions of all the TLFs,
each given by Eq. (1). This approach is consistent with $\eta_{n}\ll\delta
E(\mathbf{r}_{n})$, and additivity of $\delta E(\\{\mathbf{r}\\})$, which has
been tested for two charges. We define $N$ as the number of TLFs that are
active, i.e. have their $\gamma_{n}$ in the range of frequencies
$[f_{\mathrm{min}},f_{\mathrm{max}}]$. We choose this range as $12$ order of
magnitude in frequency, with $f_{\mathrm{min}}=10^{-6}\,$Hz and
$f_{\mathrm{max}}=10^{6}\,$Hz. This safely covers the range in which
$1/f$-like charge noise was observed in Si/SiGe QDs. Connors  _et al._ (2019);
Struck  _et al._ (2029); Connors  _et al._ (2022); Yoneda  _et al._ (2022) It
is also roughly the range of frequencies that contribute to inhomogeneous
broadening of splittings of QD-based qubits, for which $f_{\mathrm{min}}$ and
$f_{\mathrm{max}}$ are given, roughly, by inverse of total data acquisition
time, and single data point measurement time, respectively.

As a result the density of active TLFs, $\rho$, is upper-bounded by density of
all the interface charges, $\rho_{\mathrm{ic}}\\!\sim\\!10^{11}$ cm-2 (see
Ref. Zimmermann and Weber, 1981), as it is not known how many TLFs have
$\gamma_{n}$ outside of the considered range. Note, however, that the log-
normal distribution of $\gamma_{n}$ implies a constant number of TLFs per
frequency decade, defined as
$\text{fpd}=\ln(f_{\mathrm{max}}/f_{\mathrm{min}})/N$. Finally, for a given
$\rho$ we have $N\\!=\\!\rho A$, where $A\\!=\\!600\text{nm}\times
600\text{nm}\approx 3.6\times 10^{-10}\text{cm}^{2}$ is the considered area
near of the oxide-semiconductor interface, see Fig. 1.

The initial positions of the TLFs $\\{\mathbf{r}_{n}\\}$ are drawn from a
uniform spatial distribution. As we assume the displacement vector of each RTN
to be independent and isotropic, for each realization we draw components of
$\delta\mathbf{r}_{n}$ from independent normal distributions,
$(\delta\mathbf{r}_{n})_{i}=\mathcal{N}(0,\delta r^{2}/3)$ for
$i\\!=\\!x,y,z$, so that the displacement size is $\delta r$.

To gauge our results against a measurable quantity, we first concentrate on
the variance of the fluctuations of the GS energy, defined as:

$\sigma^{2}=2\int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\text{d}fS(f)\approx
2\sum_{n}^{N}\eta_{n}^{2}.$ (4)

This quantity is related to the cohererence time of the qubit
$T_{2}^{*}\\!=\\!\alpha\sqrt{2}/\sigma$, where for spin qubits $\alpha$ is the
appropriate lever arm converting electric noise to spin splitting noise due
to, e.g., presence of a magnetic field gradient.Yoneda  _et al._ (2018);
Struck  _et al._ (2029); Yoneda  _et al._ (2022) For charge qubits,
$\alpha\\!\approx\\!1$, and typically measuredShi  _et al._ (2013)
$T_{2}^{*}\approx 0.1$ns gives $\sigma_{0}\approx 8$ $\mu$eV. This is
confirmed by measured $1/f$ spectrum, with its value at $f_{1}=1$Hz given by
$S_{1}\approx 1\mu$eV2/Hz Kranz  _et al._ (2020), as we have the relation

$\sigma^{2}=2\int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\text{d}fS(f)=2S_{1}f_{1}\ln(\tfrac{f_{\mathrm{max}}}{f_{\mathrm{min}}})\approx(8\mu\text{eV})^{2},$
(5)

where we have used $f_{\mathrm{max}}/f_{\mathrm{min}}=10^{12}$.

Let us now narrow down the range of possible combinations of $\rho$ and
$\delta$r that give $\sigma_{0}=8$ $\mu$eV. In Fig. 3a we show numerically
generated distributions of $\sigma$ as a function of $\delta r$ for different
values of $\rho$. The considered range of $\delta r$ covers the values in the
few $\AA$ range considered in Refs. Shehata  _et al._ , 2022; Culcer and
Zimmerman, 2013; Reinisch and Heuer, 2006; Biswas and Li, 1999. The values of
$\rho$ correspond to $N\approx 18,36,72$ charges in the considered region. In
Fig. 3b we show that the typical noise amplitude, defined as the ensemble
average of $\sigma$, scales linearly with $\delta r$, i.e.
$\langle\sigma\rangle=a_{\text{avg}}(\rho)\delta r$. We highlight that larger
$\delta r$ increases the ensemble variance of noise amplitude
$\text{Var}(\sigma)=\langle\sigma^{2}\rangle-\langle\sigma\rangle^{2}$. From
Fig. 3c we conclude that $\sqrt{\text{Var}(\sigma)}$ also scales linearly with
$\delta r$, i.e. $\sqrt{\text{Var}(\sigma)}\approx a_{\text{std}}(\rho)\delta
r$, however its dependence on $\rho$ is weak.

Figure 4: Noise amplitude as a function of $\rho$ and $\delta r$. a)
Proportionality factor of the ensemble average of noise amplitude
$\langle\sigma\rangle\propto a_{\text{avg}}(\rho)\delta r$ (black) and
corresponding ensemble variance
$\sqrt{\langle\sigma^{2}\rangle-\langle\sigma\rangle^{2}}\propto
a_{\text{std}}(\rho)\delta r$ (red). Points corresponds to the numerical
results obtained by averaging 1000 realizations, dashed-lines are the linear
fit on the log scale. From the fit we identify the corresponding exponents
$a(\rho)\propto\rho^{\gamma}$ (see the legend for their values). b) The fitted
formula, Eq. (6), as a function of $\rho$ and $\delta r$. Black line
corresponds to $\langle\sigma\rangle\\!=\\!\sigma_{0}\\!=\\!8\mu$eV, and dots
denote sets of parameters used to plot spectral densities in Fig. 5.

We investigate the $\rho$-dependence of $a_{\text{avg}}(\rho)$ and
$a_{\text{std}}(\rho)$ in Fig. 4a, where we plot them for five values of
$\rho$. The match between the results and the linear trend on the log-log
scale allows us to extract the exponents $\gamma$ in
$a(\rho)\propto\rho^{\gamma}$. Fitting gives an approximate expression for the
ensemble-averaged amplitude:

$\langle\sigma\rangle=(c_{0}\rho^{\gamma_{\text{avg}}}+a_{\text{avg},0})\delta
r+b_{0},$ (6)

where $c_{0}=2.77\times 10^{-4}\tfrac{\text{cm}}{\text{nm}}\mu$eV,
$\gamma_{\text{avg}}=0.533$, $a_{\text{avg},0}=0.951\mu$eV/nm and $b_{0}$ is
negligibly small. In Fig. 4b we plot Eq. (6) as a function of $\varrho$ and
$\delta r$. Black line shows the set of parameters resulting in
$\langle\sigma\rangle\\!=\\!\sigma_{0}=8\mu$eV. Close to this line we mark
four points which are used to generate the PSDs in Fig. 5.

In Fig. 4 we also prove that $\text{Var}(\sigma)$ does not depend visibly on
$\rho$, see an almost horizontal red line in Fig. 4). Consequently, in the
relevant range of $\rho$ it is given by $\sqrt{\text{Var}(\sigma)}\approx
a_{\text{std},0}\delta r,$ where $a_{\text{std},0}\approx 20$
$\mu\text{eV}$/nm. This Var$(\sigma)$ is related to variability between
statistically equivalent devices, which determines the reproducibility of QD
properties and thus the yield of usable devices.Zwerver  _et al._ (2022)

Figure 5: Random realizations of power spectral densities for a selection of
displacement magnitudes, $\delta r$, and charge densities, $\rho$, which
result in a typical noise amplitude $\sigma_{0}$. The dashed lines are the
ideal $1/f$ noise with $S_{1}=1\mu$eV2/Hz, and their color is related to
points marked in Fig. 4 b. The solid lines are used to plot 10 realizations of
TLF positions and their displacement for each set of parameters.

Finally, we use the above analysis and the obtained couplings $\\{\eta_{n}\\}$
to generate the noise PSDs, defined via Eq. (1). We concentrate on pairs
($\rho,\delta r$) that reproduce $\langle\sigma\rangle\approx 8$ $\mu$eV (the
points in Fig. 4b). In Fig. 5 we plot 10 realizations of the PSD and compare
them with a model $1/f$ noise spectrum with $S_{1}\\!=\\!1\mu$eV$/$Hz (dashed
lines) for each point marked in Fig. 4b. Unsurprisingly, for all the sets we
recover the PSD at 1 Hz $\approx\\!1$ $\mu$eV/Hz. However, for smallest
$\rho=5\times 10^{9}$cm-2 ($\mathrm{fpd}\\!\approx\\!1.5$) and largest
considered $\delta r\\!=\\!0.2\,\mathrm{nm}$, in the uppermost panel of Fig. 4
we see very strong effects of discrete number of TLFs, reflected in a locally
Lorentzian PSD. While such strong single-TLF effects were seen in some
experiments, Kafanov  _et al._ (2008); Struck  _et al._ (2029); Connors  _et
al._ (2019) the PSD recently observed in Ref. Rojas-Arias  _et al._ , 2023 is
more closely matched by the results of the second panel, where
$\rho\\!\approx\\!10^{10}\,$cm-2 and $\mathrm{fpd}\approx 2.7$, for which flat
regions of PSD are less likely. The third panel ($\mathrm{fpd}\approx 6$)
typically shows $S(f)\\!\sim\\!1/f^{\alpha}$ with non-constant exponent
$\alpha<2$, and resembles other experimentally measured PSDs.Struck  _et al._
(2029); Yoneda  _et al._ (2022); Connors  _et al._ (2022) Finally, in the
fourth panel, we have results with $\alpha\approx 1$, i.e. spectra close to
the ideal $1/f$ behavior.

Summarizing, we have calculated electrostatic couplings between a quantum dot
defined in realistic Si/SiGe nanostructure, and the charges trapped in the
oxide above the semiconductor layer. Starting with reasonable initial ranges
parameters for isotropic displacements of these charges, $\delta r$ and their
densities, $\rho$, we have identified a set of $(\rho,\delta r)$ pairs, for
which in the typically observed amplitude of $1/f$ charge noise at $1$ Hz, and
its total power, can be reproduced. Deviations from ideal $1/f$ shape of
charge noise PSDs measured in various Si/SiGe structuresKafanov  _et al._
(2008); Struck  _et al._ (2029); Connors  _et al._ (2019); Rojas-Arias  _et
al._ (2023); Struck  _et al._ (2029); Yoneda  _et al._ (2022); Connors  _et
al._ (2022) can be reproduced with $\rho\geq 10^{10}$ cm-2 and $\delta
r\\!\leq\\!0.15\,\mathrm{nm}$. Note that these numbers might differ from
actual values of $\rho$ and $\delta r$, as the QDs that work well as spin
qubits could be subjected to selection bias, i.e. focusing on QDs with lower-
than typical amplitude of noise. The ranges of parameters found here to be
consistent with experiments can be further narrowed down if more information
on noise in a given nanostructure - for example on cross-correlations of
noises in two QDs Yoneda  _et al._ (2022) \- is available. Note that a recent
paperRojas-Arias  _et al._ (2023) shows that those correlations can be
obtained in a model with $\rho$ similar to the one obtained here. Using the
model and the parameter ranges identified here to calculation of PSD of spin
splitting noise for a given direction of magnetic field gradient, is left for
future work.

###### Acknowledgements.

This work was supported by funds from PRELUDIUM grant of the Polish National
Science Centre (NCN), Grant No. 2021/41/N/ST3/02758.

## AUTHOR DECLARATIONS

### Conflict of interest

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Identifying the contribution of higher $\rho$ mesons around 2 GeV in the
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$ processes

Qin-Song Zhou1,2<EMAIL_ADDRESS>Jun-Zhang Wang1,2<EMAIL_ADDRESS>Xiang Liu1,2,3,4<EMAIL_ADDRESS>Takayuki Matsuki5 matsuki@tokyo-
kasei.ac.jp 1School of Physical Science and Technology, Lanzhou University,
Lanzhou 730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University $\&$ Institute
of Modern Physics of CAS, Lanzhou 730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China
4Joint Research Center for Physics, Lanzhou University and Qinghai Normal
University, Xining 810000, China
5Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan

###### Abstract

The properties of the light vector meson states around 2.0 GeV have been
poorly understood for a long time, which has become a barrier to the expansion
to higher light vector meson spectrum. Recently, the BESIII collaboration
released the measurements of the $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$ reactions above 2.0 GeV, both of which are
ideal processes to study the isovector $\rho$ meson family. In this work,
through carrying out a combined analysis of the Born cross section data for
the above two processes with the theoretical support on mass spectrum, and
production and strong decay behaviors of the $\rho$ meson family around 2.0
GeV, we identify the enhancement structure near 2034 MeV observed in
$e^{+}e^{-}\to\omega\pi^{0}$ to be the interference contribution from two
resonances $\rho(1900)$ and $\rho(2150)$, and another enhancement structure at
2111 MeV reported in $e^{+}e^{-}\to\rho\eta^{\prime}$ to be the contributions
from $\rho(2000)$ and $\rho(2150)$. This conclusion means that the
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$ are the
excellent golden channels to establish $\rho(1900)$ and $\rho(2000)$,
especially for a $D$-wave state $\rho(2000)$, whose experimental search in the
$e^{+}e^{-}$ collision should be quite challenging. The relevant cross section
measurements with higher precision are expected in the future BESIII and Belle
II experiments.

##  I Introduction

The $e^{+}e^{-}$ annihilation is an ideal platform to produce vector particles
like $\rho$, $\omega$, $\phi$, and $J/\psi$. Recently, the Born cross sections
of the $e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ BESIII:2020kpr  processes at
different center-of-mass energies between 2.0 and 3.08 GeV were measured by
the BESIII Collaboration. One resonance with a mass of $2034\pm 13$ MeV and a
width of $234\pm 30$ MeV was observed in $e^{+}e^{-}\to\omega\pi^{0}$, while
another resonance with a mass of $2111\pm 43\pm 25$ MeV and a width of $135\pm
35\pm 30$ MeV was observed in $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$.
Since the final state of $\pi^{+}\pi^{-}$ is found to be dominant in the
$\rho(770)$ decay, and the non-$\rho(770)$ contribution is less than $10\%$ by
analyzing the Dalitz plots BESIII:2020kpr , the three-body reaction
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ can be treated as
$e^{+}e^{-}\to\rho\eta^{\prime}\to\eta^{\prime}\pi^{+}\pi^{-}$. Obviously, the
$\omega\pi^{0}$ and $\rho\eta^{\prime}$ with quantum number $I^{G}=1^{+}$ are
clean channels to explore the $\rho$ meson family, where the intermediate
$\omega$ and $\phi$ states are forbidden. Thus, the experimental results of
these two processes measured by the BESIII Collaboration may provide a good
opportunity to study the $\rho$ meson family around 2 GeV.

The $\rho$ meson family around 2 GeV is far from being established. There are
many $\rho$ meson states collected in the Particle Data Group (PDG)
ParticleDataGroup:2020ssz . In the mass region around 2 GeV, there are three
$\rho$ meson states, which are $\rho(1900)$, $\rho(2000)$, and $\rho(2150)$ .
The $\rho(1900)$ was first observed in the measurement of
$e^{+}e^{-}\to\rm{hadrons}$ by the FENICE Collaboration FENICE:1996xlc , which
shows a very narrow dip with a width of 10 MeV near 1.87 GeV. Thereafter, the
$\rho(1900)$ was also reported in $\gamma p\to 3\pi^{+}3\pi^{-}p$ by E687
Frabetti:2001ah ; Frabetti:2003pw  and $e^{+}e^{-}\to\phi\pi^{0}\gamma$ by
BaBar BaBar:2007ceh  as a narrow state with width of 24 to 65 MeV. However,
the measurements of both $e^{+}e^{-}\to 2(\pi^{+}\pi^{-}\pi^{0})\gamma$ and
$e^{+}e^{-}\to 3\pi^{+}3\pi^{-}\gamma$ by BaBar BaBar:2006vzy  indicate that
the width of $\rho(1900)$ is more than 100 MeV. In fact, the resonance
parameters of the $\rho(1900)$ is not well determined. Referring to PDG
ParticleDataGroup:2020ssz , we can find abundant experimental results about
$\rho(2150)$. However, the resonance parameters of the $\rho(2150)$ are very
diverse among different experiments. If considering the experimental
uncertainties among different measurements, the mass and width of the
$\rho(2150)$ are in the range of 1910-2310 MeV and 32-630 MeV, respectively.
The $\rho(2000)$ has been collected in PDG as the “further state”
ParticleDataGroup:2020ssz . The reason for this is that $\rho(2000)$ was first
reported by an amplitude analysis of the data of $p\bar{p}\to\pi\pi$
Hasan:1994he , and then its existence was confirmed by a combined analysis for
the $e^{+}e^{-}\to\omega\eta\pi^{0}$ and $\omega\pi$ Bugg:2004xu . In Fig. 1,
we show the messy situation of the measurements of the resonance parameters of
these $\rho$ meson states around 2 GeV, where we take $\rho(1900)$ and
$\rho(2150)$ as an example. We hope that experimentalists make much more
effort to establish these $\rho$ states around 2 GeV.

---
Figure 1: The comparison of the resonance parameters of the $\rho(1900)$ and
$\rho(2150)$ measured by different experiments collected in PDG
ParticleDataGroup:2020ssz .

Until now, there have been some theoretical studies on these $\rho$ states
around 2 GeV. The analysis of mass spectrum of the $\rho$ meson family has
been done in He:2013ttg ; Li:2021qgz ; Feng:2021igh ; Anisovich:2000kxa ;
Masjuan:2012gc ; Masjuan:2013xta ; Bugg:2012yt ; Wang:2021gle , which
suggested that three $\rho$ meson states $\rho(1900)$, $\rho(2000)$, and
$\rho(2150)$ are generally assigned to the $\rho(3^{3}S_{1})$,
$\rho(2^{3}D_{1})$, and $\rho(4^{3}S_{1})$ states, respectively. For the
observed two enhancement structures reported in $e^{+}e^{-}\to\omega\pi^{0}$
and $e^{+}e^{-}\to\rho\eta^{\prime}$, the authors of Ref. Li:2021qgz  adopted
the modified Godfrey-Isgur model to study the $\rho$ meson spectroscopy, and
suggested the newly observed structure in $e^{+}e^{-}\to\omega\pi^{0}$ to be
the same state as the $\rho(2000)$ with the $2^{3}D_{1}$ quantum number. In
Ref. Yu:2021ggd , Yu et al. estimated the masses of the $D$-wave vector $\rho$
mesons by the QCD sum rule, which also supports the above interpretation. In
these theoretical studies, one usually focuses on whether or not the resonance
parameters of these observed structures can be reproduced, and then comes to a
conclusion of the properties of these $\rho$ states. In fact, besides the
measurement of resonance parameters for the reported enhancement structures
$e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\rho\eta^{\prime}$ BESIII:2020kpr , there exist abundant data of
the cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\rho\eta^{\prime}$ BESIII:2020kpr , which should be paid more
attention since this information has a close relationship with the mass
spectrum and decay behavior of these discussed states and is a crucial step to
decode their properties Wang:2020kte ; Wang:2021gle .

Along this line, in this paper, we perform the study of the cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$. Checking
the experimental results of the $e^{+}e^{-}\to\omega\pi^{0}$ process, we
notice that it is not suitable to simply consider the observed enhancement
structure existing in $e^{+}e^{-}\to\omega\pi^{0}$ due to the $\rho(2000)$
contribution. Usually, the dilepton width of the $D$-wave vector meson states
should be suppressed compared to the case of the corresponding $S$-wave vector
mesons, where the $\rho(2000)$ is a typical $D$-wave state. We want to ask a
natural question: why is the $S$-wave meson state near the $\rho(2000)$ absent
in the cross section data? Here, we only take the $e^{+}e^{-}\to\omega\pi^{0}$
process as an example. Facing these data of the cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\rho\eta^{\prime}$ BESIII:2020kpr , we should identify the
contribution from different $\rho$ meson states to these obtained cross
sections, which is a good opportunity to establish the $\rho$ meson states
around 2 GeV.

In order to solve these problems pointed above, in this paper, we perform a
combined analysis to the experimental data of Born cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$ with the
theoretical support from spectroscopy He:2013ttg ; Wang:2020kte .
Subsequently, we show that the enhancement structure near 2034 MeV observed in
$e^{+}e^{-}\to\omega\pi^{0}$ cannot be explained as a single $\rho(2000)$
state, where the $\rho(1900)$ and $\rho(2150)$ are the main source of the
enhancement structure near 2034 MeV of $e^{+}e^{-}\to\omega\pi^{0}$. Our study
indicates that another enhancement structure near 2111 MeV existing in
$e^{+}e^{-}\to\rho\eta^{\prime}$ can explain it due to the contributions from
the $\rho(2000)$ and $\rho(2150)$. With the accumulation of more experimental
data, we believe the above observations made in this work can be further
tested.

This paper is organized as follows. After the introduction, we present our
theoretical framework of analyzing $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$ by considering the higher $\rho$ mesons as
the intermediate states in Sec. II. In Sec. III, we study the experimental
data of the Born cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$. Finally, this paper ends with a short
summary in Sec. IV.

##  II $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}\to\eta^{\prime}\pi^{+}\pi^{-}$ processes

As mentioned in the introduction, due to the conservation of isospin and $G$
parity, both of the $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$ reactions are the clean processes for
studying $\rho$ meson states. For the discussed $e^{+}e^{-}\to\omega\pi^{0}$
and $e^{+}e^{-}\to\rho\eta^{\prime}$ reactions, there exist two mechanisms as
shown in Fig. 2. Here, not only does the virtual photon from the $e^{+}e^{-}$
annihilation directly couple with final meson states, but also it can first
interact with the intermediate $\rho$ mesons, which decay into the final meson
states.

For calculating the cross section of these reactions, we adopt the effective
Lagrangian approach. The involved effective Lagrangians include Bauer:1975bv ;
Bauer:1975bw ; Kaymakcalan:1983qq ; Lin:1999ad ; Oh:2000qr ; Chen:2011cj

$\displaystyle\mathcal{L}_{\rho^{*}\mathcal{V}\mathcal{P}}$ $\displaystyle=$
$\displaystyle
g_{\rho^{*}\mathcal{V}\mathcal{P}}\epsilon_{\mu\nu\alpha\beta}\partial^{\mu}\rho^{*\nu}\partial^{\alpha}\mathcal{V}^{\beta}\mathcal{P},$
(1) $\displaystyle\mathcal{L}_{\mathcal{V}\mathcal{P}\mathcal{P}}$
$\displaystyle=$ $\displaystyle
g_{\mathcal{V}\mathcal{P}\mathcal{P}}\mathcal{V}^{\mu}\mathcal{P}\overleftrightarrow{\partial}_{\mu}\mathcal{P},$
(2) $\displaystyle\mathcal{L}_{\gamma\mathcal{V}\mathcal{P}}$ $\displaystyle=$
$\displaystyle
e\epsilon_{\mu\nu\alpha\beta}\partial^{\mu}A^{\nu}\partial^{\alpha}\mathcal{V}^{\beta}\mathcal{P},$
(3) $\displaystyle\mathcal{L}_{\gamma\rho^{*}}$ $\displaystyle=$
$\displaystyle-e\frac{m_{\rho^{*}}^{2}}{f_{\rho^{*}}}\rho^{*\mu}A_{\mu},$ (4)

where the $\rho^{*}$, $\mathcal{V}$, and $\mathcal{P}$ denote the fields of
excited $\rho$ meson states, the vector field, and pseudoscalar field,
respectively. With the above effective Lagrangians, relevant amplitudes of
$e^{+}e^{-}\to\omega\pi^{0}$ corresponding to diagrams in Fig. 2 are written
as

$\displaystyle\mathcal{M}_{\rho^{*}_{i}}$ $\displaystyle=$
$\displaystyle\bar{v}(p_{2})(ie\gamma^{\mu})u(p_{1})\frac{-g_{\mu\nu}}{q^{2}}(-e\frac{m_{\rho^{*}_{i}}^{2}}{f_{\rho^{*}_{i}}})\frac{-g_{\nu\delta}+q_{\nu}q_{\delta}/q^{2}}{q^{2}-m_{\rho^{*}_{i}}^{2}+im_{\rho^{*}_{i}}\Gamma_{\rho^{*}_{i}}}$
(5)
$\displaystyle\times\left[g_{\rho^{*}_{i}\omega\pi^{0}}\epsilon^{\tau\delta\alpha\beta}(-iq_{\tau})(ip_{3\alpha})\right]\varepsilon_{\beta}(p_{3}),$
$\displaystyle\mathcal{M}_{\rm{Dir}}$ $\displaystyle=$
$\displaystyle\bar{v}(p_{2})(ie\gamma^{\mu})u(p_{1})\frac{-g_{\mu\nu}}{q^{2}}\left[e\epsilon_{\tau\nu\alpha\beta}(-iq^{\tau})(ip_{3}^{\alpha})\right]$
$\displaystyle\times\varepsilon_{\beta}(p_{3})\mathcal{F}(s),$

where $p_{1}$, $p_{2}$, $p_{3}$, and $p_{4}$ are the four-momentum of $e^{+}$,
$e^{-}$, $\omega$, and $\pi^{0}$, respectively, and
$q=p_{1}+p_{2}=p_{3}+p_{4}$. The
$\mathcal{F}(s)=a\exp\left({-b(\sqrt{s}-\sum_{f}m_{f})}\right)/s$ denotes the
form factor, where the $a$ and $b$ are free parameters, which can be
determined by fitting experimental data, and the $\sum_{f}m_{f}$ is the sum
for the masses of the final particles. The total amplitude of
$e^{+}e^{-}\to\omega\pi^{0}$ is

$\displaystyle\mathcal{M}_{\rm{Total}}=\mathcal{M}_{\rm{Dir}}+\sum_{\rho^{*}_{i}}\mathcal{M}_{\rho^{*}_{i}}e^{i\phi^{\rho^{*}_{i}}},$
(7)

where the $\phi^{\rho^{*}_{i}}$ is the phase angle between the amplitudes from
the direct annihilation and intermediate $\rho$ meson state contributions.
With the above amplitudes, the Born cross section of
$e^{+}e^{-}\to\omega\pi^{0}$ can be calculated by

$\displaystyle d\sigma=\frac{1}{32\pi
s}\frac{|\vec{p}_{3cm}|}{|\vec{p}_{\rm{1cm}}|}\overline{|\mathcal{M}_{\rm{Total}}|^{2}}d\rm{cos}\theta.$
(8)

Here, $\theta$ is the scattering angle of an outgoing $\omega$ relative to the
direction of the electron beam in the center-of-mass frame, while
$\vec{p}_{1cm}$ and $\vec{p}_{\rm{3cm}}$ are the three-momentum of the
electron and $\omega$ in the center-of-mass frame, respectively. The overline
in $\overline{|\mathcal{M}_{\rm{Total}}|^{2}}$ indicates the average over the
polarizations of $e^{+}e^{-}$ in the initial states and the sum over the
polarizations of $\omega\pi^{0}$ in the final states.

|
---|---
(a) | (b)
Figure 2: The Feynman diagrams for depicting the $e^{+}e^{-}\to\omega\pi^{0}$
and $\rho\eta^{\prime}$ reactions. Here, diagram (a) is from the contribution
of the intermediate $\rho$ meson states, while diagram (b) reflects the
background contribution.

Since experiments usually measure
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho^{*}\to\omega\pi^{0})$ rather than the
coupling constant $g_{\rho^{*}\omega\pi^{0}}/f_{\rho^{*}}$, we replace
$g_{\rho^{*}\omega\pi^{0}}/f_{\rho^{*}}$ in Eq. (5) with the function of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho^{*}\to\omega\pi^{0})$ for the sake of
convenience. With the effective Lagrangians given above, the dilepton and
$\omega\pi^{0}$ decay widths of the discussed $\rho$ mesons can be written as

$\displaystyle\Gamma_{e^{+}e^{-}}=\frac{e^{4}m_{\rho^{*}_{i}}}{12\pi
f_{\rho^{*}_{i}}^{2}}\quad{\rm
and}\quad\Gamma_{\omega\pi^{0}}=\frac{g_{\rho^{*}_{i}\omega\pi^{0}}^{2}|\vec{p}_{3\rm{cm}}|^{3}}{12\pi},$
(9)

where the $\vec{p}_{3\rm{cm}}$ is the three-momentum of $\omega$ in the
center-of-mass frame, which can be expressed as
$|\vec{p}_{3\rm{cm}}|=\sqrt{[m_{\rho_{i}^{*}}^{2}-(m_{\omega}-m_{\pi^{0}})^{2}][m_{\rho_{i}^{*}}^{2}-(m_{\omega}+m_{\pi^{0}})^{2}]}/(2m_{\rho_{i}^{*}})$.
So the absolute value of the coupling constant
$g_{\rho^{*}\omega\pi^{0}}/f_{\rho^{*}}$ can be expressed as

$\displaystyle\left|\frac{g_{\rho_{i}^{*}\omega\pi^{0}}}{f_{\rho_{i}^{*}}}\right|=\sqrt{\frac{144\pi^{2}\Gamma_{\rho_{i}^{*}}\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho_{i}^{*}\to\omega\pi^{0})}{m_{\rho_{i}^{*}}|\vec{p}_{3\rm{cm}}|^{3}}}.$
(10)

For the reaction
$e^{+}e^{-}\to\rho\eta^{\prime}\to\eta^{\prime}\pi^{-}\pi^{-}$, we can
calculate its cross section by replacing the masses of the involved $\omega$
and $\pi^{0}$ mesons in Eqs. (8)-(10) with the corresponding masses of $\rho$
and $\eta^{\prime}$ mesons, and then replacing the
$\varepsilon_{\beta}(p_{3})$ to

$\frac{-g_{\beta\kappa}+p_{3\beta}p3_{3\kappa}/m_{\rho}^{2}}{p_{3}^{2}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}}g_{\rho\pi\pi}(ip_{5}^{\kappa}-ip_{6}^{\kappa}),$

where the $p_{5}$ and $p_{6}$ are the four momenta of $\pi^{+}$ and $\pi^{-}$,
respectively. By utilizing the partial decay widths of $\rho\to\pi\pi$ listed
in PDG ParticleDataGroup:2020ssz , we obtain the $g_{\rho\pi\pi}$=6.0
$\rm{GeV}^{2}$.

##  III numerical results

In this section, we study the Born cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ BESIII:2020kpr . As a first step,
we need to identify the contributions of different intermediate $\rho$ meson
states to these two reactions with theoretical support on the spectrum and
decay behaviors of these $\rho$ meson states around 2 GeV. In Table 1, we
present the information of masses and decay behaviors of these three $\rho$
mesons around 2 GeV. Here, the strong decay behaviors were estimated by the
quark pair creation model He:2013ttg , in which the $\rho(1900)$,
$\rho(2000)$, and $\rho(2150)$ are assigned as $\rho(3^{3}S_{1})$,
$\rho(2^{3}D_{1})$, and $\rho(4^{3}S_{1})$, respectively. The dilepton widths
of vector mesons can be estimated according to the zero-point behavior of
their radiative wave functions, which are given in Refs. He:2013ttg ;
Wang:2020kte ; Godfrey:1985xj . The mass of $\rho(1900)$ shown in Table 1 is
adopted from the average value of experimental measurements collected in PDG
ParticleDataGroup:2020ssz  because the measurements of different experiments
are almost similar within the range of errors. However, the $\rho(2000)$ lacks
direct experimental measurements, and the different experimental measurements
of the mass of $\rho(2150)$ are in a large range from 1910 to 2310 MeV. In
these cases, it becomes meaningless to take the average of the measurements
collected in PDG. Therefore, the masses of $\rho(2000)$ and $\rho(2150)$ are
adopted from the values obtained from an analysis of the Regge trajectory
He:2013ttg  in this work. For the $e^{+}e^{-}\to\omega\pi^{0}$, the
contribution of $\rho(2150)$ must be taken into account. But it is difficult
for us to identify the contributions of $\rho(1900)$ and $\rho(2000)$ only
from the values of $\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho^{*}\to\omega\pi^{0})$
because the value of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ is comparable
with lower limit of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(1900)\to\omega\pi^{0})$. Here, we adopt
four different schemes to analyze the contributions of $\rho(1900)$ and
$\rho(2000)$ in $e^{+}e^{-}\to\omega\pi^{0}$.

Table 1: The information of $\rho$ meson states involved in the $e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$ processes around 2.0 GeV. The $R$ is the parameter in a simple harmonic oscillator wave function. $\Gamma_{e^{+}e^{-}}$ is the dilepton decay width that was estimated by Ref. Wang:2020kte . $\mathcal{B}(\omega\pi^{0})$ and $\mathcal{B}(\rho\eta^{\prime})$ are the branching ratios of $\omega\pi^{0}$ and $\rho\eta^{\prime}$ modes estimated via the quark pair creation model He:2013ttg , respectively . In the last two lines, we give the values of a product of $\Gamma_{e^{+}e^{-}}$ and $\mathcal{B}(\omega\pi^{0})$, and $\Gamma_{e^{+}e^{-}}$ and $\mathcal{B}(\rho\eta^{\prime})$, respectively. State |  | $\rho(1900)$ |  | $\rho(2150)$ |  | $\rho(2000)$
---|---|---|---|---|---|---
Mass (MeV) |  |  $1890\pm 20$ ParticleDataGroup:2020ssz  |  |  $2160$ He:2013ttg  |  |  $2040$ He:2013ttg
$R$ (GeV-1) He:2013ttg ; Wang:2020kte  |  | $3.8\sim 4.3$ |  | $4.5\sim 5.0$ |  | $4.3\sim 4.8$
$\Gamma_{e^{+}e^{-}}$ (eV) Wang:2020kte  |  | $166.31\sim 213.91$ |  | $78.72\sim 96.80$ |  | $28.74\sim 16.74$
$\mathcal{B}(\omega\pi^{0})$ ($10^{-3}$) He:2013ttg  |  | $0.18\sim 24.79$ |  | $18.22\sim 88.76$ |  | $15.01\sim 19.65$
$\mathcal{B}(\rho\eta^{\prime})$ ($10^{-3}$) He:2013ttg  |  | $6.85\times 10^{-4}\sim 2.73$ |  | $0.94\sim 2.65$ |  | $5.55\sim 15.78$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\omega\pi^{0})$ (eV) |  | $0.03\sim 5.31$ |  | $1.76\sim 6.99$ |  | $0.23\sim 0.38$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho\eta^{\prime})$ (eV) |  | $0.01\times 10^{-2}\sim 0.45$ |  | $0.10\sim 0.20$ |  | $0.11\sim 0.24$

In our fitting process, we take the input masses of the $\rho(1900)$,
$\rho(2000)$, and $\rho(2150)$ from Table 1, and we regard the widths, the
relevant combined branching ratios
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho^{*}_{i}\to{\omega\pi^{0}})$, the phases
$\phi^{\rho^{*}_{i}}_{1}$, and the parameters in the form factor $a_{1}$,
$b_{1}$ as free parameters, which are determined by fitting experimental data
of the Born cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ measured by the
BESIII BESIII:2020xmw  and SND Achasov:2016zvn  collaborations. It needs to be
explained here that the experimental data given by the SND Collaboration
Achasov:2016zvn  are the cross sections of
$e^{+}e^{-}\to\omega\pi^{0}\to\gamma\pi^{0}\pi^{0}$, and then the cross
sections of $e^{+}e^{-}\to\omega\pi^{0}$ can be obtained by dividing by the
branching ratio of the $\omega$ decay to $\gamma\pi^{0}$. Here, the branching
ratio of $e^{+}e^{-}\to\omega\pi^{0}$ is adopted by the value $8.40\%$, which
is given in PDG ParticleDataGroup:2020ssz . The fitting parameters of
different schemes are presented in Table 2. Applying the central values of
fitted parameters, we can plot the Born cross section of
$e^{+}e^{-}\to\omega\pi^{0}$ as a function of center-of-mass energy, and show
them in Fig. 3.

Table 2: The parameters obtained by fitting the Born cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ measured by the BESIII BESIII:2020xmw  and SND Achasov:2016zvn  collaboration . Parameters |  | Scheme 1 |  | Scheme 2 |  | Scheme 3 |  | Scheme 4
---|---|---|---|---|---|---|---|---
$\Gamma_{\rho(1900)}$ (MeV) |  | … |  | … |  | … |  | $169\pm 15$
$\Gamma_{\rho(2000)}$ (MeV) |  | $238\pm 22$ |  | … |  | $206\pm 29$ |  | …
$\Gamma_{\rho(2150)}$ (MeV) |  | … |  | $171\pm 23$ |  | $169\pm 23$ |  | $177\pm 19$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(1900)\to\omega\pi^{0})$ (eV) |  | … |  | … |  | … |  | $6.59\pm 1.91$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ (eV) |  |  |  $7.37\pm 1.52$ 111Solution of constructive interference in scheme 1.
---
$233.52\pm 4.87$ 222Solution of destructive interference in scheme 1.
| … |  | $4.15\pm 0.33$ |  | …
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2150)\to\omega\pi^{0})$ (eV) |  | … |  |  $3.65\pm 0.44$ |  | $1.92\pm 0.42$ |  | $3.40\pm 0.97$
$a_{1}$ ($\rm{GeV}^{-2}$) |  | $2.93\pm 0.21$ |  | $2.58\pm 0.02$ |  | $3.36\pm 0.28$ |  | $3.38\pm 0.29$
$b_{1}$ ($\rm{GeV}^{-1}$) |  | $0.68\pm 0.01$ |  | $0.53\pm 0.01$ |  | $0.69\pm 0.05$ |  | $0.63\pm 0.05$
$\phi^{\rho(1900)}_{1}$ (rad) |  | … |  | … |  | … |  | $4.31\pm 0.11$
$\phi^{\rho(2000)}_{1}$ (rad) |  |  |  $0.52\pm 0.09$ a
---
$4.63\pm 0.03$ b
| … |  | $5.96\pm 0.17$ |  | …
$\phi^{\rho(2150)}_{1}$ (rad) |  | … |  | $1.78\pm 0.08$ |  | $2.62\pm 0.11$ |  | $2.10\pm 0.07$
$\chi^{2}/\rm{d.o.f.}$ |  | 1.25 |  | 1.13 |  | 0.72 |  | 0.67
|  |
---|---|---
|  |
Figure 3: The fitted results of the Born cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ measured by the BESIII BESIII:2020xmw  (red dots
with error bars) and SND Achasov:2016zvn  (green dots with error bars)
collaborations with different schemes. For comparison, we also present the
experimental data measured by the BaBar BaBar:2017zmc  (violet dots with error
bars) Collaboration. In the scheme 1 of $e^{+}e^{-}\to\omega\pi^{0}$, the main
figure is a result of constructive interference of $\rho(2000)$, and the
insert is a result of destructive interference.

From the measured resonance parameters of the enhancement structure near 2034
MeV observed in $e^{+}e^{-}\to\omega\pi^{0}$, it seems natural to interpret
this enhancement structure as a contribution from $\rho(2000)$. In Refs.
Li:2021qgz ; Yu:2021ggd , the authors interpreted this structure as
$\rho(2000)$ based on a study of the mass spectrum and strong decay width of
$\rho$ meson states. Thus, we try to reproduce the experimental data of the
Born cross section of the reaction $e^{+}e^{-}\to\omega\pi^{0}$ only by using
$\rho(2000)$ as the intermediate resonance state in the scheme 1. As seen in
Fig. 3, we can find that the experimental data measured by the BESIII
Collaboration can barely be reproduced in the scheme 1. However, the fitting
solution of constructive interference of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ is $7.37\pm 1.52$
eV, which is about 20 times as large as theoretical estimate $0.23\sim 0.38$
eV presented in Table 1, and another solution of destructive interference is
$233.52\pm 4.87$, which is about three orders of magnitude larger than
theoretical estimate. Therefore, our conclusion from the scheme 1 is that the
enhancement structure observed near 2034 MeV in $e^{+}e^{-}\to\omega\pi^{0}$
cannot be interpreted as a contribution only from the $\rho(2000)$.

In the scheme 2, we only consider the $\rho(2150)$ as an intermediate
resonance in the $e^{+}e^{-}\to\omega\pi^{0}$. Here, we need to point out that
all the schemes virtually have multiple solutions, but the solutions of
destructive interference deviate greatly from the theoretical estimates, so we
do not mention multiple solutions in the following discussion. As shown in
Fig. 3, this scheme not only cannot well describe the experimental data above
2 GeV measured by the BESIII Collaboration, but also fails to describe the
experimental data measured by SND Achasov:2016zvn  and BaBar BaBar:2017zmc .
Combining the results of the schemes 1 and 2, we can conclude that we cannot
reproduce the experimental data of the Born reaction cross section of
$e^{+}e^{-}\to\omega\pi^{0}$ well with a single resonance fitting.

In the scheme 3, we further assume that the enhancement structure observed in
$e^{+}e^{-}\to\omega\pi^{0}$ process is the contribution from $\rho(2000)$ and
$\rho(2150)$. Although the experimental data above 2 GeV measured by the
BESIII Collaboration can be reproduced by the scheme 3, the fitting value of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ is $4.15\pm 0.33$
eV, which is still about one order of magnitude larger than the theoretical
estimate. On the other hand, the line shape in the energy range from 1.8 GeV
to 2 GeV is not consistent with the experimental data measured by SND
Achasov:2016zvn  and BaBar BaBar:2017zmc .

In the scheme 4, we consider the contributions from $\rho(1900)$ and
$\rho(2150)$ in $e^{+}e^{-}\to\omega\pi^{0}$ process to reproduce experimental
data. From the fitting result of the scheme 4 in Fig. 3, we can see that this
scheme not only can well reproduce the experimental data above 2 GeV measured
by the BESIII Collaboration, but also can well describe the experimental data
measured by SND Achasov:2016zvn  and BaBar BaBar:2017zmc  in the range of 1.8
to 2 GeV. Comparing all the schemes, we have reason to guess that the dip near
1.9 GeV shown in SND Achasov:2016zvn  and BaBar BaBar:2017zmc  possibly
corresponds to the contribution of $\rho(1900)$. This conclusion can be tested
by more precise measurement in the future with more accurate and richer
experimental data. As seen in the fitted parameters of the scheme 4 shown in
Table 2, the values of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(1900)\to\omega\pi^{0})$ and
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2150)\to\omega\pi^{0})$ are $6.59\pm
1.91$ eV and $3.40\pm 0.97$ eV, respectively, both of which are within the
error range of theoretical estimates shown in Table 1. Obviously, there is no
clear evidence of $\rho(2000)$ in $e^{+}e^{-}\to\omega\pi^{0}$ based on our
analysis, and the theoretical estimate of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho^{*}\to\omega\pi^{0})$ also implies that
$\rho(2000)$ does not make a dominant contribution in
$e^{+}e^{-}\to\omega\pi^{0}$. Finally, we find that the $\chi^{2}/\rm{d.o.f.}$
of scheme 4 is the smallest of all the schemes. Based on the analysis of the
above four schemes, we can conclude that the enhancement structure near 2034
MeV observed in $e^{+}e^{-}\to\omega\pi^{0}$ by the BESIII Collaboration
cannot be interpreted as a single resonance contribution from the
$\rho(2000)$, but it is dominated by the contributions of the $\rho(1900)$ and
$\rho(2150)$.

In order to further test the role of $\rho(2000)$ in
$e^{+}e^{-}\to\omega\pi^{0}$, we add the contribution of the $\rho(2000)$ in a
$3R$ fitting scheme to fit the experimental data. The fitting results are
shown in Fig. 4, and the corresponding parameters are summarized in Table III.
It can be seen that the $\chi^{2}/\rm{d.o.f.}=0.66$ in a $3R$ fitting scheme
can be obtained. Compared with the value of $\chi^{2}/\rm{d.o.f.}=0.67$ in
scheme 4, the fitting quality is not significantly improved after adding the
contribution of $\rho(2000)$. On the other hand, the fitting value of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ is $0.48\pm 0.28$
eV, which is still about one order of magnitude smaller than the values of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(1900)\to\omega\pi^{0})$ and
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2150)\to\omega\pi^{0})$. It is worth
noting that this fitted
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$=$0.48\pm 0.28$ eV
is also consistent with our theoretical estimates of $0.23\sim 0.38$ eV.
Therefore, it proves that the observed enhancement structure near 2.0 GeV in
$e^{+}e^{-}\to\omega\pi^{0}$ is dominantly produced by the contributions of
the $\rho(1900)$ and $\rho(2000)$ again, while the contribution of the
$\rho(2000)$ is insignificant.

Table 3: The parameters obtained by fitting the Born cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ measured by the BESIII BESIII:2020xmw  and SND Achasov:2016zvn  Collaborations with the 3R fitting scheme. Parameters |  | Values
---|---|---
$\Gamma_{\rho(1900)}$ (MeV) |  | $175\pm 14$
$\Gamma_{\rho(2000)}$ (MeV) |  | $202\pm 16$
$\Gamma_{\rho(2150)}$ (MeV) |  | $183\pm 16$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(1900)\to\omega\pi^{0})$ (eV) |  | $0.48\pm 0.28$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2150)\to\omega\pi^{0})$ (eV) |  | $2.82\pm 0.61$
$a_{1}$ ($\rm{GeV}^{-2}$) |  | $3.40\pm 0.30$
$b_{1}$ ($\rm{GeV}^{-1}$) |  | $0.64\pm 0.02$
$\phi^{\rho(1900)}_{1}$ (rad) |  | $4.19\pm 0.20$
$\phi^{\rho(2000)}_{1}$ (rad) |  | $0.23\pm 0.12$
$\chi^{2}/\rm{d.o.f.}$ |  | 0.66
---
Figure 4: The fitted results of the Born cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ measured by the BESIII BESIII:2020xmw  (red dots
with error bars) and SND Achasov:2016zvn  (green dots with error bars)
Collaborations with the 3R fitting scheme.

Following the analysis for $e^{+}e^{-}\to\omega\pi^{0}$, we search for the
possibility of multi-$\rho$-state dominance in the reaction
$e^{+}e^{-}\to\rho\eta^{\prime}$ too by considering the value of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho^{*}\to\rho\eta^{\prime})$. As seen in
Table 1, the contribution of both $\rho(2000)$ and $\rho(2150)$ are
significant and although the contribution from $\rho(1900)$ may be comparable
to $\rho(2000)$ and $\rho(2150)$ in $e^{+}e^{-}\to\rho\eta^{\prime}$, its
contribution is obviously suppressed in the energy region of the enhancement
structure because its mass is far from the enhancement structure position of
2111 MeV. Hence, we can conclude that $\rho(2000)$ and $\rho(2150)$ are
dominant for $e^{+}e^{-}\to\rho\eta^{\prime}$. In order to verify this
assumption and understand the enhancement structure near 2111 MeV observed in
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ by the BESIII Collaboration, we
perform a simultaneous fit to the experimental data of the Born cross sections
of $e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$
above 2 GeV measured by the BESIII Collaboration. The reason why we use the
combined fitting scheme is that it can restrict the free resonance parameters
more strongly. From Fig. 5, we can see that the enhancement structure near
2111 MeV observed in $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ by the BESIII
Collaboration can be reproduced by the resonance contributions from
$\rho(2000)$ and $\rho(2150)$. From the values of fitted parameters shown in
Table 4, we can see that both of the fitted results of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\rho\eta^{\prime})$ and
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2150)\to\rho\eta^{\prime})$ are within
the theoretical prediction ranges of $0.10\sim 0.20$ and $0.11\sim 0.24$,
respectively. On the other hand, current experimental data show no signal of
enhancement structure in the range of 1.8 to 2.0 GeV, which indicates that it
is reasonable to ignore the contribution from $\rho(1900)$ in the process of
$e^{+}e^{-}\to\rho\eta^{\prime}$. According to our analysis, we can draw a
conclusion that the enhancement structure near 2111 MeV observed in
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ by the BESIII Collaboration can be
interpreted as an interference structure from the $\rho(2000)$ and
$\rho(2150)$. As a $D$-wave $\rho$ meson state, the dilepton width of the
$\rho(2000)$ is generally suppressed, which leads to the difficulty of
searching for the $\rho(2000)$ in $e^{+}e^{-}$ collision experiments. However,
our analysis shows that the contributions of the $\rho(2000)$ and $\rho(2150)$
are comparable in the $e^{+}e^{-}\to\rho\eta^{\prime}$ reaction. Therefore,
$e^{+}e^{-}\to\rho\eta^{\prime}$ may be a golden reaction to study
$\rho(2000)$. According to the values of
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ and
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\rho\eta^{\prime})$, although the
absolute contribution of the $\rho(2000)$ in $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$ are similar to each other, the total cross
section near 2 GeV of $e^{+}e^{-}\to\omega\pi^{0}$ is about one order of
magnitude larger than $e^{+}e^{-}\to\rho\eta^{\prime}$, which means that the
relative contribution of $\rho(2000)$ in
$e^{+}e^{-}\to\rho\eta^{\prime}\to\eta^{\prime}\pi^{+}\pi^{-}$ is larger than
that in $e^{+}e^{-}\to\omega\pi^{0}$. Therefore, we conclude that
$e^{+}e^{-}\to\rho\eta^{\prime}\to\eta^{\prime}\pi^{+}\pi^{-}$ is a more
appropriate reaction to study $\rho(2000)$.

Table 4: The parameters of the combined fit to the experimental data of the Born cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw ; Achasov:2016zvn  and $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ BESIII:2020kpr  processes. Here, subscripts 1 and 2 indicate the corresponding parameters belonging to $e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$, respectively. Parameters |  | Values
---|---|---
$\Gamma_{\rho(1900)}$ (MeV) |  | $173\pm 18$
$\Gamma_{\rho(2000)}$ (MeV) |  | $194\pm 38$
$\Gamma_{\rho(2150)}$ (MeV) |  | $175\pm 22$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(1900)\to\omega\pi^{0})$ (eV) |  | $6.88\pm 1.92$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\omega\pi^{0})$ (eV) |  | $3.37\pm 0.79$
$a_{1}$ ($\rm{GeV}^{-2}$) |  | $3.41\pm 0.33$
$b_{1}$ ($\rm{GeV}^{-1}$) |  | $0.64\pm 0.04$
$\phi^{\rho(1900)}_{1}$ (rad) |  | $4.34\pm 0.14$
$\phi^{\rho(2150)}_{1}$ (rad) |  | $2.11\pm 0.14$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2000)\to\rho\eta^{\prime})$ (eV) |  | $0.44\pm 0.21$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(2150)\to\rho\eta^{\prime})$ (eV) |  | $0.51\pm 0.27$
$a_{2}$ ($\rm{GeV}^{-2}$) |  | $1.78\pm 0.15$
$b_{2}$ ($\rm{GeV}^{-1}$) |  | $0.64\pm 0.06$
$\phi^{\rho(2000)}_{2}$ (rad) |  | $6.17\pm 0.29$
$\phi^{\rho(2150)}_{2}$ (rad) |  | $4.58\pm 0.42$
$\chi^{2}/\rm{d.o.f.}$ |  | 0.74
|  |
---|---|---
Figure 5: The combined fit to the precise Born cross sections of
$e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw ; Achasov:2016zvn  and
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ BESIII:2020kpr . For comparison, we
also present the experimental data measured by the BaBar BaBar:2017zmc ;
BaBar:2007qju  Collaboration.

We also present the individual contributions to the Born cross sections in
Figs. 3, 4 and 5, we can find that the resonance contribution is obviously
smaller than the direct term, but the interference effect is very important to
reproduce the experimental data. Because the direct process provides a
dominant contribution in both of $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$, one expect that it may enhance the
contribution from reaction
$e^{+}e^{-}\to\gamma^{*}\to\rho^{*}\to\gamma^{*}\to\omega\pi^{0}$ or
$\rho\eta^{\prime}$. As an example, we have calculated the cross sections of
$e^{+}e^{-}\to\gamma^{*}\to\omega\pi^{0}$ (reaction I),
$e^{+}e^{-}\to\gamma^{*}\to\rho(2150)\to\omega\pi^{0}$ (reaction II) and
$e^{+}e^{-}\to\gamma^{*}\to\rho(2150)\to\gamma^{*}\to\omega\pi^{0}$ (reaction
III) by using the parameters in Table 4, whose results are shown in Fig. 6. It
can be seen from Fig. 6 that the cross section of
$e^{+}e^{-}\to\gamma^{*}\to\rho(2150)\to\gamma^{*}\to\omega\pi^{0}$ is much
smaller than those of two other mechanisms. The reason for this is that there
exist a suppression vertex of electromagnetic interaction. Therefore, it is
reasonable to not consider the contribution of
$e^{+}e^{-}\to\gamma^{*}\to\rho(2150)\to\gamma^{*}\to\omega\pi^{0}$ or
$\rho\eta^{\prime}$ in our calculation. The interference effect considerably
changes the line shape of the Born cross sections, which makes the original
two resonance states to appear as one structure in the experimental data. This
is the important reason why the resonance parameters of $\rho$ meson states
around 2.0 GeV are so poorly measured. In this case, it is not sufficient to
understand a structure observed by experiments only by the resonance
parameters, and their production and decay behaviors provide more information
for us. On the other hand, the theoretical studies of the spectrum and decay
behaviors of $\rho$ meson states around 2 GeV greatly help us identify the
contributions of the $\rho$ meson states to these two processes of
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$. We believe
that the resonance parameters of $\rho(1900)$, $\rho(2000)$, and $\rho(2150)$
can be established by more abundant experimental data of
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$ and more
theoretical researches in the future.

---
Figure 6: Cross section sizes of $e^{+}e^{-}\to\omega\pi^{0}$ are from
different reactions, where reaction I, reaction II, and reaction III represent
$e^{+}e^{-}\to\gamma^{*}\to\omega\pi^{0}$,
$e^{+}e^{-}\to\gamma^{*}\to\rho(2150)\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\gamma^{*}\to\rho(2150)\to\gamma^{*}\to\omega\pi^{0}$,
respectively.

##  IV summary

Recently, the BESIII Collaboration measured the Born cross sections for the
processes $e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\rho\eta^{\prime}$ BESIII:2020kpr  at different center-of-mass
energies between 2.0 and 3.08 GeV, where an enhancement structure near 2034
MeV was observed in $e^{+}e^{-}\to\omega\pi^{0}$, and another enhancement
structure near 2111 MeV in $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$. Both of
these two processes are ideal processes to study $\rho$ meson states, which
provide a good and precious opportunity to study the exited $\rho$ mesons
around 2 GeV. There are three $\rho$ meson states $\rho(1900)$, $\rho(2000)$,
and $\rho(2150)$ included in PDG ParticleDataGroup:2020ssz  around 2 GeV,
which are usually assigned as $\rho(3^{3}S_{1})$, $\rho(2^{3}D_{1})$, and
$\rho(4^{3}S_{1})$ based on mass spectrum studies, respectively. However, as
pointed out in the introduction, there are some problems in establishing the
$\rho$ meson family around 2 GeV and understand these two structures observed
in $e^{+}e^{-}\to\omega\pi^{0}$ BESIII:2020xmw  and
$e^{+}e^{-}\to\rho\eta^{\prime}$ BESIII:2020kpr .

In order to solve these problems, we have analyzed the experimental data of
the Born cross sections of $e^{+}e^{-}\to\omega\pi^{0}$ and
$e^{+}e^{-}\to\rho\eta^{\prime}$ with the theoretical support on mass
spectrum, and decay behaviors of $\rho$ meson states around 2 GeV. Our
analysis has shown that the enhancement structure near 2034 MeV observed in
$e^{+}e^{-}\to\omega\pi^{0}$ cannot be explained as a single state of
$\rho(2000)$, which is dominated by the contributions of $\rho(1900)$ and
$\rho(2150)$. Another enhancement structure at 2111 MeV observed in
$e^{+}e^{-}\to\rho\eta^{\prime}$ can be reproduced by the contributions from
$\rho(2000)$ and $\rho(2150)$. As a $D$-wave $\rho$ meson state, the
$\rho(2000)$ decay to electron and positron should be suppressed, which makes
it difficult to observe it in the $e^{+}e^{-}$ collision experiment. However,
our research has shown that the contribution of $\rho(2000)$ to
$e^{+}e^{-}\to\rho\eta^{\prime}$ is comparable to that of $\rho(2150)$ because
the $\rho(2000)$ has a strong coupling with the $\rho\eta^{\prime}$ channel.
Therefore, the $e^{+}e^{-}\to\rho\eta^{\prime}$ may be a golden process to
look for the $\rho(2000)$. On the other hand, we have found that the
interference effect is very important to reproduce the experimental data,
which may dramatically change the line shape of the Born cross sections. The
interference effect makes it difficult to accurately measure the resonance
parameters in the experiments, especially when the statistics of experimental
data are not abundant enough and multiple resonant states exist in a very
small energy region. This is one of the reasons why the resonance parameters
of the $\rho$ meson states around 2 GeV are so poorly measured.

Establishing the $\rho$ meson states around 2 GeV is not only necessary for
the study of $\rho$ meson states in the higher energy region, but also very
helpful for the study of $\omega$ and $\phi$ meson states around 2 GeV. The
$e^{+}e^{-}\to\omega\pi^{0}$ and $e^{+}e^{-}\to\rho\eta^{\prime}$ are precious
and clean two-body processes to study $\rho$ meson states. We strongly urge
researchers to focus on these two processes in the future, which could verify
our analysis in this work, and more importantly, help establish the $\rho$
meson family around 2 GeV.

## ACKNOWLEDGEMENTS

This work is supported by the China National Funds for Distinguished Young
Scientists under Grant No. 11825503, National Key Research and Development
Program of China under Contract No. 2020YFA0406400, the 111 Project under
Grant No. B20063, the National Natural Science Foundation of China under Grant
No. 12047501, and by the Fundamental Research Funds for the Central
Universities.

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