Current address: ]Department of Physics, University of Surrey, Guildford GU2
7XH, United Kingdom

# Searching for resonance states in 22Ne($p,\gamma$)23Na

D. P. Carrasco-Rojas<EMAIL_ADDRESS>Department of Physics, The
University of Texas at El Paso, El Paso, TX 79968-0515, USA  M. Williams  [
<EMAIL_ADDRESS>TRIUMF, Vancouver, BC V6T 2A3, Canada
Department of Physics, University of York, Heslington, York, YO10 5DD, United
Kingdom  P. Adsley<EMAIL_ADDRESS>Department of Physics and Astronomy,
Texas A&M University, College Station, Texas 77843-4242, USA  Cyclotron
Institute, Texas A&M University, College Station, Texas 77843-3636, USA
iThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South
Africa  School of Physics, University of the Witwatersrand, Johannesburg 2050,
South Africa  L. Lamia  Laboratori Nazionali del Sud - Istituto Nazionale di
Fisica Nucleare, Via Santa Sofia 62, 95123 Catania, Italy  Dipartimento di
Fisica e Astronomia “E.Majorana”, Università di Catania, Italy  Centro
Siciliano di Fisica Nucleare e Struttura della Materia (CSFNSM), Catania,
Italy  B. Bastin  GANIL, CEA/DRF-CNRS/IN2P3, Bvd Henri Becquerel, 14076 Caen,
France  T. Faestermann  Physik Department E12, Technische Universität München,
D-85748 Garching, Germany  C. Fougères  GANIL, CEA/DRF-CNRS/IN2P3, Bvd Henri
Becquerel, 14076 Caen, France  F. Hammache  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  D. S. Harrouz  Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  R. Hertenberger  Fakultät für
Physik, Ludwig-Maximilians-Universität München, D-85748 Garching, Germany  M.
La Cognata  Laboratori Nazionali del Sud - Istituto Nazionale di Fisica
Nucleare, Via Santa Sofia 62, 95123 Catania, Italy  A. Meyer  Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  F. de Oliveira Santos
GANIL, CEA/DRF-CNRS/IN2P3, Bvd Henri Becquerel, 14076 Caen, France  S.
Palmerini  Dipartimento di Fisica e Geologia, Università degli Studi di
Perugia, via A. Pascoli s/n, 06125 Perugia, Italy  Istituto Nazionale di
Fisica Nucleare - Sezione di Perugia, via A. Pascoli s/n, 06125 Perugia, Italy
R. G. Pizzone  Laboratori Nazionali del Sud - Istituto Nazionale di Fisica
Nucleare, Via Santa Sofia 62, 95123 Catania, Italy  S. Romano  Laboratori
Nazionali del Sud - Istituto Nazionale di Fisica Nucleare, Via Santa Sofia 62,
95123 Catania, Italy  N. de Séréville  Université Paris-Saclay, CNRS/IN2P3,
IJCLab, 91405 Orsay, France  A. Tumino  Laboratori Nazionali del Sud -
Istituto Nazionale di Fisica Nucleare, Via Santa Sofia 62, 95123 Catania,
Italy  Facoltà di Ingegneria e Architettura, Università degli Studi di Enna
“Kore”, Cittadella Universitaria, 94100 Enna, Italy  H.-F. Wirth  Fakultät für
Physik, Ludwig-Maximilians-Universität München, D-85748 Garching, Germany

###### Abstract

Background

Globular clusters show strong correlations between different elements, such as
the well-known sodium-oxygen anticorrelation. One of the main sources of
uncertainty in this anticorrelation is the 22Ne($p,\gamma$)23Na reaction rate,
due to the possible influence of an unobserved resonance state at
$E_{\mathrm{x}}=8862$ keV ($E_{\mathrm{r,c.m.}}=68$ keV). The influence of two
higher-lying resonance states at $E_{\mathrm{x}}=8894$ and $9000$ keV has
already been ruled out by direct 22Ne($p,\gamma$)23Na measurements.

Purpose

To study excited states in 23Na above the proton threshold to determine if the
unconfirmed resonance states in 23Na exist.

Methods

The non-selective proton inelastic scattering reaction at low energies was
used to search for excited states in 23Na above the proton threshold. Protons
scattered from various targets were momentum-analysed in the Q3D magnetic
spectrograph at the Maier-Leibnitz Laboratorium, Munich, Germany.

Results

The resonance states previously reported at $E_{\mathrm{x}}=8862$, $8894$ and
$9000$ keV in other experiments were not observed in the present experiment at
any angle. This result, combined with other non-observations of these
resonance states in most other experiments, results in a strong presumption
against the existence of these resonance states.

Conclusions

The previously reported resonance states at $E_{\mathrm{x}}=8862$, $8894$ and
$9000$ keV are unlikely to exist and should be omitted from future evaluations
of the 22Ne($p,\gamma$)23Na reaction rates. Indirect studies using low-energy
proton inelastic scattering are a simple and yet exceptionally powerful tool
in helping to constrain astrophysical reaction rates by providing non-
selective information of the excited states of nuclei.

††preprint: APS/123-QED

##  I Astrophysical Background

Globular clusters (GCs) are large populations of stars bound together by
gravity. Since GCs are considered to be formed and evolved without external
factors, they are one of the most important laboratories for understanding
stellar evolution. In contrast to what was believed years ago, they are
composed of multiple populations of stars [1, 2] and are observed to exhibit
anomalous abundance trends in elements from C to Al that are not seen in field
stars. These abundance anomalies challenge our understanding of how GCs have
formed and evolved, with some unknown polluting site or sites [3, 4, 5, 6, 7,
8, 9, 10, 11, 12] within the GCs causing significant star-to-star variations.
Suggested polluting sites include low-mass stars, AGB and super-AGB stars,
massive and supermassive stars, and classical novae based on carbon-oxygen or
oxygen-neon white dwarfs (see Refs. [1, 13] for evaluations of possible
sites).

One of the most uncertain abundance patterns is the sodium-oxygen anti-
correlation [14, 2]. The abundance of 23Na has a significant impact on this
anti-correlation; the 22Ne($p,\gamma$)23Na reaction, found in the NeNa cycle
of hydrogen burning, has a large effect on the abundance of 23Na along with
other reactions such as 23Na($p,\gamma$)24Mg, 23Na($p,\alpha$)20Ne [15, 16,
17].

Different facilities have studied the 22Ne($p,\gamma$)23Na reaction and 23Na
resonance states. Direct measurements of resonance strengths within the
astrophysical region of interest are challenging. Indirect measurements, such
as $\gamma$-ray spectroscopy and single-proton transfer reactions have
provided valuable constraints on the reaction rate. Significant uncertainties
of up to an order of magnitude in the reaction rate at temperatures relevant
to AGB stars and Hot-Bottom Burning ($T<0.1$ GK) remain due to unclear nuclear
data; the reaction rate at temperatures relevant to classical novae is rather
well constrained [14, 18, 19].

Direct measurements of resonance strengths have extended down to
$E_{\mathrm{c.m.}}=65$ keV. The direct measurements performed at the
Laboratory for Undeground Nuclear Astrophysics [2, 18], the Laboratory for
Experimental Nuclear Astrophysics [20, 21, 22] and the DRAGON recoil separator
at TRIUMF [23, 14] have provided resonance strengths or upper limits of
resonance strengths for most known resonance states [24, 25, 26]. However,
three resonances corresponding to states reported by Powers et al. [27] at
$E_{\mathrm{x}}=8862$, $8892$ and $9000$ keV ($E_{\mathrm{r,c.m.}}=68$, $100$
and $206$ keV, $E_{\mathrm{r,lab}}=71$, $105$ and $215$ keV) have not been
observed. In fact, the LUNA direct measurements have ruled out the
$E_{\mathrm{x}}=8892$\- and $9000$-keV resonance strengths as being
astrophysically important. However, even with the stringent limit of
$\omega\gamma_{\mathrm{68}}<6\times 10^{-11}$ eV (at 90% confidence) on the
strength of the $E_{\mathrm{r,c.m.}}=68$-keV resonance provided by Ferraro et
al. [18], the 22Ne($p,\gamma$)23Na reaction rate is uncertain by a factor of
ten depending on the existence of this state.

This paper reports an experimental study of the 23Na($p,p^{\prime}$)23Na
proton inelastic-scattering reaction with a beam energy of $E_{p}=14$ MeV
using the Q3D magnetic spectrometer at Munich [28] to momentum analyse the
scattered particles. At this energy, the proton inelastic scattering reaction
is rather insensitive to the structure of the excited states enabling a
stringent test of the existence of possible resonance states at
$E_{\mathrm{x}}=8862$, $8892$ and $9000$ keV. Section II discusses the
applicability of the ($p,p^{\prime}$) reaction as an unselective probe to
search for 23Na states of interest. Section III details the experimental
methodology and data analysis procedures. Our results are then presented and
discussed in Section IV, with concluding remarks given in Section V.

##  II The ($p,p^{\prime}$) reaction

The ($p,p^{\prime}$) reaction has a long history of being used to perform
simple spectroscopy to identify states at the energy used in this experiment
($E_{p}=14$ MeV) and in similar past experiments [29, 25, 30, 31, 32]. One of
the primary features of this reaction is that it appears to be non-selective,
unlike many of the other reactions used for these experiments such as, e.g.
22Ne(${}^{3}\mathrm{He},d$)23Na which is sensitive to the 22Ne$+p$ structure
of the 23Na states. The evidence for the selectivity of the reaction is
empirical; the outline of why this reaction is believed to have weak or no
selectivity to the structure of the excited states is summarised below.

Measurements of the 27Al($p,p^{\prime}$)27Al reaction have with the Orsay
SplitPole and the Munich Q3D [30, 31] observed every known state in 27Al
between the ground state and the 23Na$+\alpha$ threshold at
$S_{\alpha}=10091.8(1)$ keV. The Q3D measurement, which will be the subject of
a future publication reports angular distributions which are flat and
featureless, implying that the reaction proceeds through a compound process.
This is possibly due to the outgoing protons having energies ($4.3$ and $5.1$
MeV) just above the Coulomb barrier height ($V_{\mathrm{Coulomb}}=3.3$ MeV)
meaning that compound processes dominate over direct ones. One result of this
is that no information may be obtained about the spins and parities of the
states populated since the characteristic diffraction patterns obtained in
higher-energy inelastic scattering, dominated by direct processes, are absent.

All known isolated states between the ground state and the 23Na$+\alpha$
threshold at $S_{\alpha}=10091.8(1)$ keV were observed in Ref. [30]. This is
in contrast to the suggestion of Moss and Sherman [29] that there is a strong
antiselectivity to $J=\frac{1}{2}$ states in the ($p,p^{\prime}$) reaction
based on the non-observation of a number of levels in that study. The cause of
this disagreement is unclear though we note that the studies of Benamara et
al. [30] and Adsley et al. [32] observed a number of additional states in 27Al
(Benamara) and 26Mg (Adsley) which were not observed by Moss and Sherman [29]
and Moss [25], suggesting that the experimental conditions may have played a
role in their conclusion regarding the antiselection to $J=1/2$ states.

Finally, we note that this proton scattering reaction is not a resonance
reaction of the form 22Ne$+p$ and this means that the cross sections for
populating 23Na states are not thought to be proportional to the proton widths
for those 23Na states. This means that the 22Ne$+p$ widths required for the
22Ne($p,\gamma$)23Na resonance strengths and subsequently reaction rates
cannot be deduced from these data.

##  III Experimental details and data analysis

A 14-MeV proton beam from the tandem accelerator at the Maier-Leibnitz
Laboratorium was transported to the target of the Q3D spectrograph [28]. The
target used to probe 23Na states consisted of around 50 $\mu$g/cm2 of NaF on a
20 $\mu$g/cm2 carbon backing. In order to quantify the background from
instrumental effects and target contamination, scattering from a carbon foil
similar to the backing of the NaF target was also measured. A LiF target was
used to identify 19F states but only for the 70-degree data.

Reaction products were momentum analyzed by the Munich Q3D and detected at the
focal plane in a detector consisting of two gas-proportional counters backed
by a plastic scintillator (for details about the focal-plane detector and
similar devices see Refs. [33]). Data were collected at five scattering
angles: 25, 35, 40, 50, and 70 degrees. Data from different angles are used to
identify peaks in the focal plane coming from target contaminants. Target
contaminants appear to move in excitation energy with changing angle due to
different kinematic shifts. This shift technique has been used to identify
contaminating states in other reactions such as the 26Mg($p,p^{\prime}$)26Mg
reaction where a small amount of 24Mg contamination was present in the target
and the resulting peaks could be identified and rejected through the kinematic
shift [32].

Figure 1: The focal-plane position to magnetic rigidity ($B\rho$) calibration
for the data taken at a scattering angle of $\theta_{\mathrm{Q3D}}=70$
degrees. The black points correspond to the known levels in 19F at
$E_{\mathrm{x}}=8629(4)$, $8864(4)$ and $8953(3)$ keV, and in 23Na at
$E_{\mathrm{x}}=8798.7(8)$ and $8945.1(8)$ keV used for the calibration. Both
the known uncertainties in the excitation energies listed in the ENSDF [34]
and the fitting uncertainties from the peaks are included in the plot, though
the error bars are frequently smaller than the points. The red line represents
a best fit with a quadratic function and the blue region shows the $3\sigma$
uncertainty band.

Figure 2: (Top) Excitation-energy spectrum obtained at scattering angle of
$\theta_{\mathrm{Q3D}}=70$ degrees and assuming 23Na kinematics. The hollow
black-lined spectrum is taken with the NaF target, the solid red-filled
spectrum is taken with the LiF target. The solid black vertical lines show the
listed energies of levels in 23Na from the ENSDF [34] with the associated grey
box showing the listed $E_{\mathrm{x}}$ uncertainty and a blue box showing the
$E_{\mathrm{x}}$ uncertainty from the present measurement. The green dotted
vertical lines show the energies of the tentative resonances at
$E_{\mathrm{x}}=8862$, $8894$ and $9000$ keV from Ref. [27]. The centre-of-
mass energy for the 22Ne$+p$ colliding system is shown on the top axis.
(Bottom) The corresponding $1\sigma$, $2\sigma$ and $3\sigma$ uncertainty
bands for the excitation-energy calibration relative to the central value for
the 70-degree data.

The focal-plane position was related to the magnetic rigidity using known
states in 16O, 19F and 23Na for calibration. For the 70-degree data, the
focal-plane to rigidity calibration for these states is shown in Fig. 1. The
corresponding 23Na excitation-energy spectrum is shown in Fig. 2 along with
the $3\sigma$ uncertainty bands for the excitation energy. Similar
calibrations to 23Na, 19F and 16O states were performed for other angles. The
resulting $E_{\mathrm{x}}$ spectra and uncertainty bands are shown in Figs.
3-6. Each of the calibrations included at least one state from a different
nucleus (19F from the NaF targets and oxygen contamination in the carbon
targets to verify the assigned 23Na levels in the calibrations). Information
on the states used to calibrate at each angles is given in the figure
captions.

The increase in the excitation-energy uncertainty at higher excitation
energies is due to the increased uncertainty in the excitation energies of the
states used in the calibration at the low-rigidity (high-excitation energy)
end of the focal plane. At these higher excitation energies, the
$E_{\mathrm{x}}=9072(3)$-keV state in 23Na has been used for the calibration
when possible. However, since the excitation energy of this state has a
$3$-keV uncertainty, the corresponding uncertainty band in the present
experiment is of the same size.

Figure 3: Same as Fig. 2 at $\theta_{\mathrm{Q3D}}=50$ degrees except that the
red filled spectrum is taken with the carbon target; the peak in the spectrum
is from a contaminating 16O state, likely from water or oil on the carbon and
NaF targets (or the carbon backing of the NaF target). The spectrum was
calibrated using the 23Na states at $E_{\mathrm{x}}=8798.7(8)$, $8975.3(7)$
and $9072(3)$ keV, and the 16O state at $E_{\mathrm{x}}=8871.9(5)$ keV, which
is the only peak in the red filled spectrum with the carbon target. The peak
at around $E_{\mathrm{x}}=8900$ keV is an unidentified contaminant since it
does not appear at other angles and is broader than the spectrometer
resolution, which may be due to the state itself being broad or the kinematics
being out of focus.

Figure 4: As Fig. 3 at $\theta_{\mathrm{Q3D}}=40$ degrees. The spectrum was
calibrated using the 23Na states at $E_{\mathrm{x}}=8798.7(8)$, $9072(3)$ and
$9113(3)$ keV, the 19F state at $E_{\mathrm{x}}=8864(4)$ keV, and the 16O
state at $E_{\mathrm{x}}=8871.9(5)$ keV.

Figure 5: Same as Fig. 3 at $\theta_{\mathrm{Q3D}}=35$ degrees. The 23Na
states at $E_{\mathrm{x}}=8798.7(8)$ and $9113(3)$ keV, the 19F state at
$E_{\mathrm{x}}=8864(4)$ keV and the 16O state at $E_{\mathrm{x}}=8871.9(5)$
keV were used to calibrate at this angle.

Figure 6: Same as Fig. 3 but for $\theta_{\mathrm{Q3D}}=25$ degrees. The 23Na
states at $E_{\mathrm{x}}=8798.7(8)$, $8975.3(7)$ and $9113(3)$ keV and the
19F state at $E_{\mathrm{x}}=8864(4)$ keV were used to calibrate at this
angle.

##  IV Discussion

There are three tentative resonance states listed in this excitation-energy
region in 23Na: at $E_{\mathrm{x}}=8862$, $8894$ and $9000$ keV, note that no
uncertainties were given in the original study which claimed these tentative
resonances. Around the tentative $E_{\mathrm{x}}=8862$\- and $8894$-keV states
the spectrum is notably featureless. Both of the tentative states are well
resolved from other known 23Na levels. It is unlikely that these resonance
states exist.

The $E_{\mathrm{x}}=9000$-keV state is located in a region with multiple
populated levels, including contaminant states from other nuclei including
19F. Since there is no state populated at multiple angles with consistent
excitation energy, we also conclude that this state has not been populated in
this experiment and that it is unlikely to exist.

In this experiment, we have not observed a state at
$E_{\mathrm{x}}=8963.9(11)$ keV which has been assigned in $\gamma$-ray decays
following the 12C(12C$,p$)23Na fusion-evaporation reaction in coincidence with
the $E_{\gamma}=440$-keV transition from the decay of the first excited state
in 23Na [26, 35]. It is possible that the $\gamma$-ray is feeding the
$J^{\pi}=7/2^{+}$ state at $E_{\mathrm{x}}=2075.9(4)$ keV and subsequent the
$E_{\gamma}=1636$-keV transition was not observed and so the excitation energy
of this state has been misplaced in those $\gamma$-ray studies, though this is
not confirmed [36]. Alternatively, if the state is weakly populated in the
present experiment then its close proximity to the
$E_{\mathrm{x}}=8946.8(6)$-keV may make it hard to observe.

##  V Conclusions

The major remaining uncertainty in the 22Ne($p,\gamma$)23Na reaction rate
contributing to the Na-O anticorrelation observed in globular clusters is the
existence of a state at $E_{\mathrm{x}}=8862$ keV, corresponding to a
resonance energy of $E_{\mathrm{r,c.m.}}=68$ keV. A strong contribution of the
potential $E_{\mathrm{r,c.m.}}=100$-keV and $E_{\mathrm{c.m.}}=206$-keV
resonances (from purported states at $E_{\mathrm{x}}=8894$ and $9000$ keV) to
the reaction rate had previously been ruled out by previous direct
measurements which provide a tight upper limit on the resonance strength of
these resonances (see e.g. Ref. [18]). The influence of the
$E_{\mathrm{r,c.m.}}=68$-keV resonance on the reaction rate remained an open
question largely due to the lower resonance energy. In the present
measurement, which used the non-selective 23Na($p,p^{\prime}$)23Na reaction to
populate excited states in 23Na, none of the potential resonance states are
observed, in agreement with other direct and indirect studies of the excited
levels of 23Na. Only one study reports these tentative resonance states [27]
and multiple studies using different experimental probes [24, 2, 18] including
the present one see no evidence for them. These results strongly indicate that
the potential resonance states do not exist (though it is not possible to
prove that) and should be omitted from evaluations of the reaction rates until
and unless conclusive evidence may be obtained for the existence of these
states. Efforts to directly measure these resonances are unlikely to observe
any signal above the direct-capture contribution. Finally, we recommend using
the reaction-rate evaluation of Ref. [14] and its associated uncertainties for
future astrophysical models, reducing the reaction-rate uncertainty in the
temperature range of hot-bottom burning from an order of magnitude to 40%.

###### Acknowledgements.

The authors thank the beam operators at the Maier-Leibnitz Laboratorium for
their work and support and the target preparation laboratory of INFN-LNS for
sample and target preparation and characterisation. DPCR thanks the Department
of Energy Nuclear Physics grant number DE-FG02-93ER40773 and DE-SC0022469, the
Texas Research Expanding Nuclear Diversity (TREND) program, the Cyclotron
Institute at Texas A&M University. MW acknowledges support provided by the
Natural Sciences & Engineering Research Council of Canada grant
SAPPJ-2019-00039. PA thanks the Claude Leon Foundation for support during his
time at the University of the Witwatersrand and iThemba LABS in the form of a
postdoctoral fellowship. PA also thanks David Jenkins of the University of
York for useful comments relating to the 12C(${}^{12}\mathrm{C},p$)23Na
Gammasphere data and Richard Longland of North Carolina State University for
useful comments and references regarding 22Ne($p,\gamma$)23Na direct
measurements.

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* ${\bf G} \setminus  \av_{\Gc_{2,k}}$ has rank equal to $\Usf-1$ for each $k\in (\Ac \cap [\Ksf-\Usf])$  (such that $\sv_k$ exists   by using the    Cramer's rule and thus  $Q_{\Ac}$ is not zero);

* the $\Usf$ rows of the matrix in (<ref>), including $\sv_{k}$ where $k\in (\Ac \cap [\Ksf-\Usf])$ and $\ev^{\text{T}}_{\Usf,j-\Ksf+\Usf}$ where $j\in (\Ac \cap [\Ksf-\Usf+1:\Ksf])$, are linearly independent (such that $D_{\Ac}$ is not zero).

We divide the set  $\Ac \cap [\Ksf-\Usf+1:\Ksf]$ into two subsets, $\Sc_1=  \Ac \cap    [\Ksf-\Usf+1:2\Ksf-2\Usf]$  where

\begin{align}
x=|\Sc_1|=|\Ac \cap    [\Ksf-\Usf+1:2\Ksf-2\Usf]| \leq \Ksf-\Usf, \label{eq:def of x}
\end{align}

and $\Sc_2=  \Ac   \cap [ 2\Ksf-2\Usf+1:\Ksf]$ where

\begin{align}
y=|\Sc_2|=|\Ac   \cap [ 2\Ksf-2\Usf+1:\Ksf]| \leq 2\Usf-\Ksf. \label{eq:def of y}
\end{align}

For each user $j_1 \in \Sc_1$, we have $j_1-\Ksf+\Usf \in [\Ksf-\Usf]$; for each user  $j_2 \in \Sc_2$, we have   $j_2-\Ksf+\Usf \in [\Ksf-\Usf+1:\Usf]$.
Since $x+y= |\Ac \cap [\Ksf-\Usf+1:\Ksf]|$ and $|\Ac|=\Usf$, we have

\begin{align}
\Usf-(\Ksf-\Usf)\leq x+y\leq  \Usf. \label{eq:x+y range}
\end{align}

If $x+y=\Usf$, we can see that the matrix in (<ref>) is the identity matrix $\mathbf{I}_{\Usf}$ which is full rank. Hence, in the rest of the proof, we  focus on the case where  $2\Usf-\Ksf\leq x+y<\Usf$.

By  symmetry,  we only need to consider the case where $\Ac \cap [\Ksf-\Usf]=[\Usf-x-y]$,
$\Sc_1 =\Ac \cap    [\Ksf-\Usf+1:2\Ksf-2\Usf]= [\Ksf-\Usf+1:\Ksf-\Usf+x]$ and $\Sc_2= \Ac   \cap [ 2\Ksf-2\Usf+1:\Ksf]=[ 2\Ksf-2\Usf+1:  2\Ksf-2\Usf+y]$,
and find     one realization  of  the `$*$'s in ${\bf G}$ satisfying
the   constraints 1) and 2).

Thus the last $|\Ac \setminus [\Ksf-\Usf]|=x+y$ rows
of the matrix in (<ref>) includes $\ev^{\text{T}}_{\Usf,i}$ where $i\in ([x] \cup [\Ksf-\Usf+1 :\Ksf-\Usf+y])$.
To determine the first $\Usf-x-y$ rows of the matrix in (<ref>),
we select a realization of ${\bf G}$ as follows  (recall that ${\bf 0}_{m,n}$ and ${\bf 1}_{m,n}$ represents all-zero matrix and all-one matrix  of  dimension $m \times n$, respectively)

\begin{align}
&{\bf G} = \begin{bmatrix}
\av_{\Gc_{2,1}},\ldots, \av_{\Gc_{2,\Ksf-\Usf}}, \av_{\Gc_{2,2\Ksf-2\Usf+1}},\ldots, \av_{\Gc_{2,\Ksf}}
\end{bmatrix}= \nonumber\\
& c_{[g_1]} & c_{[g_1+1: \Usf-x-y]} & c_{[\Usf-x-y+1 : \Ksf-\Usf]} & c_{[\Ksf-\Usf+1 : \Ksf-\Usf+y]} &c_{[\Ksf-\Usf+y+1 : \Usf]}\\
\begin{block}{c[ccccc]}
r_{[2\Usf-\Ksf-y]} & {\bf 0}_{2\Usf-\Ksf-y,g_1} &{\bf I}_{2\Usf-\Ksf-y}&{\bf 0}_{2\Usf-\Ksf-y, g_2}&{\bf 0}_{2\Usf-\Ksf-y, y} & -{\bf I}_{2\Usf-\Ksf-y}\\
r_{[2\Usf-\Ksf-y+1:x]}&{\bf 0}_{g_2,g_1}& {\bf 0}_{g_2,2\Usf-\Ksf-y}& {\bf I}_{g_2}& {\bf 0}_{g_2, y}&{\bf 0}_{g_2, 2\Usf-\Ksf-y}\\
r_{[x+1:\Ksf-\Usf]} & {\bf I}_{g_1} & -{\bf 1}_{g_1,2\Usf-\Ksf-y} &  -{\bf 1}_{g_1,g_2} & {\bf 1}_{g_1,y} & {\bf 1}_{g_1,2\Usf-\Ksf-y}\\
r_{[\Ksf-\Usf+1:\Ksf-\Usf+y]} & {\bf 0}_{y,g_1} &  {\bf 0}_{y,2\Usf-\Ksf-y} & {\bf 0}_{y,g_2} & {\bf I}_{y} & {\bf 0}_{y,2\Usf-\Ksf-y}\\
r_{[\Ksf-\Usf+y+1:\Usf]} &  {\bf 0}_{2\Usf-\Ksf-y,g_1} & {\bf 0}_{2\Usf-\Ksf-y,2\Usf-\Ksf-y} & {\bf 0}_{2\Usf-\Ksf-y,g_2}& {\bf 0}_{2\Usf-\Ksf-y,y}& {\bf I}_{2\Usf-\Ksf-y}\\
\end{block}
\end{blockarray}
\label{eq:choice of G}
\end{align}

where  $g_1:=\Ksf-\Usf-x$, $g_2:=x+y-2\Usf+\Ksf$,  $r_{[i:j]}$ represents $r_i,r_{i+1},\ldots,r_j$, and  $c_{[i:j]}$ represents $c_i,c_{i+1},\ldots,c_j$.
Let us then derive $\sv_k$ for each user  $k\in (\Ac \cap [\Ksf-\Usf]) =[\Usf-x-y]$.

For each user $k\in [g_1]$, the matrix ${\bf G} \setminus \av_{\Gc_{2,k}}$  has rank equal to $\Usf-1$, since one can easily check that the columns in ${\bf G}$
are linearly independent.  Thus ${\bf G} \setminus \av_{\Gc_{2,k}}$ contains exactly one linearly independent left null space vector. We can check that this vector could be   (recall that  $\mathbf{1}_{n}$  and $\mathbf{0}_{n}$ represent  the vertical $n$-dimensional   vector  whose elements are all $1$ and all $0$, respectively)

\begin{align}
&\sv_{k}= [{\bf 1}^{\text{T}}_{2\Usf-\Ksf-y}, \ {\bf 1}^{\text{T}}_{g_2}, \ \ev^{\text{T}}_{g_1,k}, \ -{\bf 1}^{\text{T}}_{y}, \  {\bf 0}^{\text{T}}_{2\Usf-\Ksf-y}   ], \ \forall k\in [g_1].
\label{eq:sk of first set first group}
\end{align}

\begin{align}
\begin{blockarray}{cccccccccccccccc}
r_{1} & \cdots &  r_{x} & r_{x+1} &  r_{x+2} &\cdots & r_{k} &r_{k+1} &\cdots &  r_{\Ksf-\Usf} &  r_{\Ksf-\Usf+1} &\cdots & r_{\Ksf-\Usf+y} & r_{\Ksf-\Usf+y+1}&\cdots & r_{\Usf}  \\
\begin{block}{[cccccccccccccccc]}
1& \cdots & 1 & 0 & 0 &\cdots & 1& 0 &\cdots &0 &-1  & \cdots &-1 &0 &\cdots &0\\
\end{block}
\end{blockarray}
\end{align}

For each user $k\in [g_1+1:\Usf-x-y]$,  since  the columns in ${\bf G}$
are linearly independent, the matrix ${\bf G} \setminus \av_{\Gc_{2,k}}$  has rank equal to $\Usf-1$.
Thus ${\bf G} \setminus \av_{\Gc_{2,k}}$ contains exactly one linearly independent left null space vector. We can check that this vector could be

\begin{align}
&\sv_{k}= [\ev^{\text{T}}_{2\Usf-\Ksf-y,k-g_1}, \ {\bf 0}^{\text{T}}_{g_2}, \ {\bf 0}^{\text{T}}_{g_1}, \  {\bf 0}^{\text{T}}_{y},  \  \ev^{\text{T}}_{2\Usf-\Ksf-y,k-g_1}], \ \forall k\in [g_1+1:\Usf-x-y].
\label{eq:sk of second set first group}
\end{align}

Recall that the last $ x+y$ rows
of the matrix in (<ref>) include  $\ev^{\text{T}}_{\Usf,i}$ where $i\in ([x] \cup [\Ksf-\Usf+1 :\Ksf-\Usf+y])$. Hence, together with the first $\Usf-x-y$ rows as shown in (<ref>) and (<ref>), we can see that the matrix in (<ref>)  is

\begin{align}
\begin{blockarray}{ccccc}
& c_{[x]} & c_{[x+1: \Ksf-\Usf]} & c_{[\Ksf-\Usf+1 : \Ksf-\Usf+y]} & c_{[\Ksf-\Usf+y+1 : \Usf]} \\
\begin{block}{c[cccc]}
r_{[g_1]} &  {\bf 1}_{g_1,x}  &{\bf I}_{g_1} &-{\bf 1}_{g_1,y} & {\bf 0}_{g_1,2\Usf-\Ksf-y}\\
r_{[g_1+1:\Usf-x-y]} & ({\bf I}_{2\Usf-\Ksf-y}, \ {\bf 0}_{2\Usf-\Ksf-y,g_2})&{\bf 0}_{2\Usf-\Ksf-y,g_1}& {\bf 0}_{2\Usf-\Ksf-y,y} & {\bf I}_{2\Usf-\Ksf-y}\\
%r_{1} & ({\bf 1}^{\text{T}}_{2\Usf-\Ksf-y}, \ {\bf 1}^{\text{T}}_{g_2}) &\ev^{\text{T}}_{g_1,1} &-{\bf 1}^{\text{T}}_{y} & {\bf 0}^{\text{T}}_{2\Usf-\Ksf-y}\\
%r_{2} & ({\bf 1}^{\text{T}}_{2\Usf-\Ksf-y}, \ {\bf 1}^{\text{T}}_{g_2}) &\ev^{\text{T}}_{g_1,2} &-{\bf 1}^{\text{T}}_{y} & {\bf 0}^{\text{T}}_{2\Usf-\Ksf-y}\\
%\vdots & \vdots & \vdots &\vdots &\vdots  \\
%r_{g_1} & ({\bf 1}^{\text{T}}_{2\Usf-\Ksf-y}, \ {\bf 1}^{\text{T}}_{g_2}) &\ev^{\text{T}}_{g_1,g_1} &-{\bf 1}^{\text{T}}_{y} & {\bf 0}^{\text{T}}_{2\Usf-\Ksf-y}\\
%r_{g_1+1} & (\ev^{\text{T}}_{2\Usf-\Ksf-y,1}, \ {\bf 0}^{\text{T}}_{g_2}) &{\bf 0}^{\text{T}}_{g_1} &  {\bf 0}^{\text{T}}_{y} & \ev^{\text{T}}_{2\Usf-\Ksf-y,1}\\
%r_{g_1+2} & (\ev^{\text{T}}_{2\Usf-\Ksf-y,2}, \ {\bf 0}^{\text{T}}_{g_2}) &{\bf 0}^{\text{T}}_{g_1} &  {\bf 0}^{\text{T}}_{y} & \ev^{\text{T}}_{2\Usf-\Ksf-y,2}\\
%\vdots & \vdots & \vdots &\vdots &\vdots  \\
%r_{\Usf-x-y} & (\ev^{\text{T}}_{2\Usf-\Ksf-y,2\Usf-\Ksf-y}, \ {\bf 0}^{\text{T}}_{g_2}) &{\bf 0}^{\text{T}}_{g_1} &  {\bf 0}^{\text{T}}_{y} & \ev^{\text{T}}_{2\Usf-\Ksf-y,2\Usf-\Ksf-y}\\
r_{[\Usf-x-y+1:\Usf-y]} & {\bf I}_{x} & {\bf 0}_{x,\Ksf-\Usf-x} & {\bf 0}_{x,y} & {\bf 0}_{x,2\Usf-\Ksf-y}\\
r_{[\Usf-y+1:\Usf]} & {\bf 0}_{y,x} & {\bf 0}_{y,\Ksf-\Usf-x} & {\bf I}_{y} & {\bf 0}_{y,2\Usf-\Ksf-y}\\
\end{block}
\end{blockarray} ,
\end{align}

which is full rank. Thus we proved that with the choice of ${\bf G}$ in (<ref>), the constraints 1) and 2) are satisfied; thus $P_{\Ac}$ is a non-zero polynomial. This completes the proof of Lemma <ref>.

\begin{align}
&{\bf G} = \begin{bmatrix}
\av_{\Gc_{2,1}},\ldots, \av_{\Gc_{2,\Ksf-\Usf}}, \av_{\Gc_{2,2\Ksf-2\Usf+1}},\ldots, \av_{\Gc_{2,\Ksf}}
\end{bmatrix}= \nonumber\\
& r_1  & \cdots & r_{t_1} & r_{t_1+1}  & \cdots & r_{t_1+t_3} &r_{t_1+t_3+1} &\cdots  & r_{t_2} &r_{t_2+1}   & \cdots & r_{  t_2+y } & r_{t_2+y+1}& \cdots& r_{\Usf} \\
\begin{block}{c[cccccccccccccccc]}
c_1 & 0   & \cdots & 0 & 1  & \cdots & 0 &0 & \cdots & 0  &0 &  \cdots & 0&-1&  \cdots&0\\
\vdots &  \vdots  & \ddots & \vdots & \vdots  & \ddots &  \vdots&  \vdots&    \ddots&  \vdots&  \vdots&  \ddots&  \vdots&  \vdots&  \ddots&  \vdots\\
c_{ t_3} & 0   & \cdots & 0  &0 & \cdots & 1& 0 &\cdots&0&0&\cdots&0&0&\cdots&-1\\
c_{ t_3+1} & 0   & \cdots & 0  &0 & \cdots & 0& 1 &\cdots&0&0&\cdots&0&0&\cdots&0\\
\vdots &  \vdots  & \ddots & \vdots & \vdots  & \ddots &  \vdots&  \vdots&    \ddots&  \vdots&  \vdots&  \ddots&  \vdots&  \vdots&  \ddots&  \vdots\\
c_{ x} & 0   & \cdots & 0  &0 & \cdots & 0& 0 &\cdots&1&0&\cdots&0&0&\cdots&0\\
c_{ x+1} & 1   & \cdots & 0  &-1  & \cdots & -1 &-1 &\cdots &-1 & 1&\cdots&1&1&\cdots&1\\
\vdots &\vdots  & \ddots & \vdots & \vdots  & \ddots &  \vdots&  \vdots &    \ddots &  \vdots&  \vdots&\ddots&  \vdots&  \vdots&\ddots&  \vdots\\
c_{t_2} &  0   & \cdots & 1&-1  & \cdots & -1&-1 &\cdots &-1&1 &\cdots&1&1&\cdots&1\\
c_{t_2+1} &  0   & \cdots & 0&0 & \cdots & 0&0 & \cdots & 0 &1& \cdots & 0& 0&\cdots&0\\
\vdots &   \vdots   & \ddots & \vdots&\vdots  & \ddots & \vdots& \vdots &\ddots &  \vdots&  \vdots&\ddots&  \vdots&  \vdots&\ddots&  \vdots\\
c_{  t_2+y } &    0   & \cdots & 0&0  & \cdots & 0&0 & \cdots & 0&0& \cdots & 1& 0&\cdots&0\\
c_{t_2+y+1} &    0   & \cdots & 0&0 & \cdots & 0&0 & \cdots & 0&0& \cdots & 0& 1&\cdots&0\\
\vdots &   \vdots  & \ddots & \vdots&\vdots& \ddots & \vdots& \vdots &\ddots &  \vdots&  \vdots&\ddots&  \vdots&  \vdots&\ddots&  \vdots\\
c_{\Usf} &  0  & \cdots & 0&0  & \cdots & 0& 0&  \cdots & 0&0 & \cdots & 0& 0&\cdots&1\\
\end{block}
\end{blockarray}
\end{align}

where $ t_1:=\Ksf-\Usf-x$,
$t_2:=\Ksf-\Usf$, $t_3:=2\Usf-\Ksf-y$, $t_4=x+y-2\Usf+\Ksf$.

§ DATA TABLES OF THE EXPERIMENTAL RESULTS IN SECTION <REF>

In the following, we consider the cases where $\Usf=(\Ksf+1)/2$ and $\Usf=\Ksf-1$, and list the running times  of each procedure in our experiments. In the   tables provided in this Section, we use “IPS” to represent  the size of each input vector; use “KST” to represent  key sharing time; use  “R1AT” and “R2AT” to represent the running times   in the first and second rounds of model aggregation, respectively; use `R3AT” and “R4AT” to represent the running times   in the   third and fourth rounds of model aggregation (only needed by  ), respectively; use “TMA” to represent the total model aggregation time.

§.§.§ Case $\Usf=(\Ksf+1)/2$:  vs.  vs.

6|c|Running Times (ms): $\Ksf = 5, \Usf = 3$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     622.6968     19.4242     4.4532     23.8774

100,000     707.118     19.2629     4.156     23.4189

200,000     1232.1053     31.6589     9.8977     41.5566

200,000     1313.3104    32.7853     9.7137     42.4991

300,000     1854.6576     42.6247     19.1066     61.7313

300,000     2024.4369     41.3044     23.4556     64.76

7|c|Running Times (ms): $\Ksf = 5, \Usf = 3$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     7.4139     3.53     38.5121     0.936     47.752

200,000     13.7358     3.1454     77.5049     0.8713     84.845

300,000     27.0172     3.1587     116.5683     0.822     124.073

6|c|Running Times (ms): $\Ksf = 10, \Usf = 5$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     1268.6858     27.843     7.2585     35.1015

100,000     1422.048    29.9577     7.3137     37.2714

200,000     2536.0957      49.1541     12.4002     61.5543

200,000     2911.8499    54.2802     11.752     66.0322

300,000     3810.1522     74.5584     18.7861     93.3445

300,000     4729.238     73.5346     18.6304     92.165

7|c|Running Times (ms): $\Ksf = 10, \Usf = 5$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     10.6461     4.1311     59.2228     0.8841     71.911

200,000     12.1471     4.5201     118.2981     1.2305     133.9134

300,000     9.471     4.0999     179.6695     1.1038     191.9444

6|c|Running Times (ms): $\Ksf = 15, \Usf = 8$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     1853.2537     45.9662     10.5245     56.4906

100,000     2061.428     40.4577     10.4448     50.9025

200,000     3704.3875     90.7609     18.5824     109.3433

200,000     4650.0179     92.1633     16.3515     108.5148

300,000     5556.7188     140.9258     22.2047     163.1305

300,000     7346.741     130.8297     22.7997     153.6293

7|c|Running Times (ms): $\Ksf = 15, \Usf = 8$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     12.7749     6.6379     90.3237     1.4144     107.5878

200,000     12.0194     7.2726     182.6603     1.4009     199.3108

300,000     13.6192     6.5942     279.8947     1.3217     298.5508

6|c|Running Times (ms): $\Ksf = 20, \Usf = 10$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     2510.2112     51.5425     14.4271     65.9696

100,000     2925.3894    53.7125     13.9064     67.6189

200,000     5038.559     110.6973     20.6763     131.3736

200,000     6314.3566     111.0032     20.7812     131.7844

300,000     7501.9033     165.4     29.3871     194.787

300,000     9878.072     158.9753     28.1264     187.1017

7|c|Running Times (ms): $\Ksf = 20, \Usf = 10$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     18.0595     8.7712     112.2543     1.5265     133.58

200,000     20.297     8.6479     236.0381     1.6135     260.7115

300,000     34.5755     8.0978     330.9472     1.8907     359.9537

§.§.§ Case $\Usf=(\Ksf+1)/2$:  vs.

5|c|Running Times (ms): $\Ksf = 5, \Usf = 3$

Scheme     IPS     R1AT     R2AT     Total

100,000     19.4242     4.4532     23.8774

200,000     31.6589     9.8977     41.5566

300,000     42.6247     19.1066     61.7313

5|c|Running Times (ms): $\Ksf = 10, \Usf = 5$

Scheme     IPS     R1AT     R2AT     Total

100,000     27.843     7.2585     35.1015

200,000     49.1541     12.4002     61.5543

300,000     74.5584     18.7861     93.3445

5|c|Running Times (ms): $\Ksf = 15, \Usf = 8$

Scheme     IPS     R1AT     R2AT     Total

100,000     45.9662     10.5245     56.4906

200,000     90.7609     18.5824     109.3433

300,000     140.9258     22.2047     163.1305

5|c|Running Times (ms): $\Ksf = 20, \Usf = 10$

Scheme     IPS     R1AT     R2AT     Total

100,000     51.5425     14.4271     65.9696

200,000     110.6973     20.6763     131.3736

300,000     165.4     29.3871     194.787

§.§.§ Case $\Usf=\Ksf-1$:  () vs.  vs.

6|c|Running Times (ms): $\Ksf = 5, \Usf = 4$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     573.5159     32.0896     4.5524     36.642

100,000     565.963     32.0896     4.5524     36.642

100,000     571.285     31.4279     4.5067     35.9346

200,000     1212.9158     61.3236     9.8175     71.1411

200,000     1180.7742     61.3236     9.8175     71.1411

200,000     1215.373     61.6626     9.8922     71.5548

300,000     1808.522     92.8807     14.3178     107.1984

300,000     1674.8969     92.8807     14.3178     107.1984

300,000     1714.3089     90.6547     15.0228     105.6774

7|c|Running Times (ms): $\Ksf = 5, \Usf = 4$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     8.0442     2.3905     52.2406     0.8506     63.5258

200,000     6.0251     4.5155     99.1241     0.9645     110.629

300,000     13.3259     2.4766     148.4597     0.8532     165.1157

6|c|Running Times (ms): $\Ksf = 10, \Usf = 9$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     1290.1098     60.1389     6.5752     66.7142

100,000     1090.2839     60.1389     6.5752     66.7142

100,000     1165.683     61.817     6.1287     67.9457

200,000     2590.862      119.5961     13.5527     133.1488

200,000     2231.250      119.5961     13.5527     133.1488

200,000     2372.629     122.3105     12.7634     135.0739

300,000     4000.3806      189.2828     16.931     206.2138

300,000     3122.4385      189.2828     16.931     206.2138

300,000     3159.0967     192.4548     17.065     209.5198

7|c|Running Times (ms): $\Ksf = 10, \Usf = 9$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     6.501     5.3933     101.6094     1.2792     114.7829

200,000     6.1388     5.6953     202.9011     1.3435     216.0788

300,000     5.5022     4.9979     303.9231     1.1925     315.6156

6|c|Running Times (ms): $\Ksf = 15, \Usf = 14$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     2003.5729     99.9402     2.6843     102.6245

100,000     1699.1491     99.9402     2.6843     102.6245

100,000     1748.1338     98.909     2.8935     101.8025

200,000     4017.3969     201.3512     13.6942     215.0454

200,000     3293.6826    201.3512     13.6942     215.0454

200,000     3514.3759     196.7829     14.7847     211.5676

300,000     6020.5340     285.3097     21.7039     307.0137

300,000     4605.5441     285.3097     21.7039     307.0137

300,000     4657.3419     299.4586     19.5312     318.9898

7|c|Running Times (ms): $\Ksf = 15, \Usf = 14$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     17.8189     8.3383     153.163     1.5803     180.9006

200,000     10.6163     7.6086     303.9652     1.6801     323.87

300,000     9.1441     8.3122     459.9635     1.7766     479.196

6|c|Running Times (ms): $\Ksf = 20, \Usf = 19$

Scheme     IPS     KST     R1AT     R2AT     TMA

100,000     2724.8058      132.5762     4.0336     136.6098

100,000     2387.7981     132.5762     4.0336     136.6098

100,000     2419.7531     128.7957     3.7095     132.5053

200,000     5460.64     263.683     16.1745     279.8575

200,000     4466.254     263.683     16.1745     279.8575

200,000     4680.175     260.3788     14.6445     275.0233

300,000     8230.2930     404.7795     24.8559     429.6354

300,000     6114.4559     404.7795     24.8559     429.6354

300,000     6277.1279     400.4311     25.4461     425.8771

7|c|Running Times (ms): $\Ksf = 20, \Usf = 19$

Scheme     IPS       R1AT     R2AT     R3AT     R4AT     TMA

100,000     10.1681     11.6254     204.3118     2.0399     228.1454

200,000     11.1408     11.8139     406.3665     2.5097     431.831

300,000     10.2389     12.2843     611.3178     2.1002     635.9413

§.§.§ Case $\Usf=\Ksf-1$:  vs.

5|c|Running Times (ms): $\Ksf = 5, \Usf = 4$

Scheme     IPS     R1AT     R2AT     Total

100,000     32.0896     4.5524     36.642

200,000     1.3236     9.8175     71.1411

300,000     92.8807     14.3178     107.1984

5|c|Running Times (ms): $\Ksf = 10, \Usf = 9$

Scheme     IPS     R1AT     R2AT     Total

100,000     60.1389     6.5752     66.7142

200,000     119.5961     13.5527     133.1488

300,000     189.2828     16.931     206.2138

5|c|Running Times (ms): $\Ksf = 15, \Usf = 14$

Scheme     IPS     R1AT     R2AT     Total

100,000     99.9402     2.6843     102.6245

200,000     201.3512     13.6942     215.0454

300,000     285.3097     21.7039     307.0137

5|c|Running Times (ms): $\Ksf = 20, \Usf = 19$

Scheme     IPS     R1AT     R2AT     Total

100,000     132.5762     4.0336     136.6098

200,000     263.683     16.1745     279.8575

300,000     404.7795     24.8559     429.6354