EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-081 LHCb-PAPER-2016-008 April 5, 2016

Measurement of the mass and lifetime of the ${{\mathchar
28938\relax}^{-}_{b}}$ baryon

The LHCb collaboration†††Authors are listed at the end of this paper.

A proton-proton collision data sample, corresponding to an integrated
luminosity of 3$\mbox{\,fb}^{-1}$ collected by LHCb at $\sqrt{s}=7$ and 8
$\mathrm{\,Te\kern-1.00006ptV}$, is used to reconstruct $63\pm 9$ ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$,
${{\mathchar 28938\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
decays. Using the ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}$, ${{\mathchar
28932\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ decay mode
for calibration, the lifetime ratio and absolute lifetime of the ${{\mathchar
28938\relax}^{-}_{b}}$ baryon are measured to be

$\displaystyle\frac{\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}}{\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}}$
$\displaystyle=1.11\pm 0.16\pm 0.03,$ $\displaystyle\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}$
$\displaystyle=1.78\pm 0.26\pm 0.05\pm 0.06~{}{\rm ps},$

where the uncertainties are statistical, systematic and from the calibration
mode (for $\tau_{{{\mathchar 28938\relax}^{-}_{b}}}$ only). A measurement is
also made of the mass difference, $m_{{{\mathchar
28938\relax}^{-}_{b}}}-m_{{{\mathchar 28932\relax}^{-}_{b}}}$, and the
corresponding ${{\mathchar 28938\relax}^{-}_{b}}$ mass, which yields

$\displaystyle m_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}-m_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}$ $\displaystyle=247.4\pm 3.2\pm
0.5~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle m_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}$
$\displaystyle=6045.1\pm 3.2\pm 0.5\pm
0.6~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$

These results are consistent with previous measurements.

Published in Phys. Rev. D93, 092007 (2016)

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

Measurements of the lifetimes of beauty baryons provide an important test of
Heavy Quark Effective Theory (HQET) [1, 2, 3, 4, 5, 6, 7, 8], in which it is
predicted that the decay width is dominated by the weak decay of the heavy $b$
quark. The large samples of $b$ baryons collected by LHCb have led to greatly
improved measurements of their lifetimes [9, 10, 11, 12], which are in good
agreement with HQET predictions. In particular, the lifetime of the
${{\mathchar 28931\relax}^{0}_{b}}$ baryon is now measured to a precision of
better than 1% [13], and those of the ${{\mathchar 28932\relax}^{0}_{b}}$ and
${{\mathchar 28932\relax}^{-}_{b}}$ to about 3% [13, 12]. Within HQET it is
expected that the lifetimes of weakly-decaying $b$ baryons follow the
hierarchy $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}\simeq\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}>\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{0}_{b}$}}}}\approx\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28931\relax}^{0}_{b}$}}}}$ [14, 15,
16], and thus far, the measured lifetimes respect this pattern within the
uncertainties. However, the uncertainty on the measured lifetime of the
${{\mathchar 28938\relax}^{-}_{b}}$ baryon is too large to fully verify this
prediction. The single best measurement to date of the ${{\mathchar
28938\relax}^{-}_{b}}$ lifetime is $1.54^{+0.26}_{-0.21}\pm 0.05$ ps [10] by
the LHCb experiment, based on a sample of $58\pm 8$ reconstructed ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\mathchar 28938\relax}^{-}}$ decays, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$,
${{\mathchar 28938\relax}^{-}}\rightarrow{\mathchar 28931\relax}{{K}^{-}}$ and
${\mathchar 28931\relax}\rightarrow p{{\pi}^{-}}$. Larger samples are needed
to reduce the statistical uncertainty.

Improved knowledge of the ${{\mathchar 28938\relax}^{-}_{b}}$ mass would
provide tighter experimental constraints for tests of lattice quantum
chromodynamics (QCD) and QCD-inspired models, which aim to accurately predict
the masses of hadrons [17]. The two most recent measurements of the
${{\mathchar 28938\relax}^{-}_{b}}$ mass, by the LHCb [18] and CDF [19]
collaborations are in agreement, but an earlier measurement by the D0
collaboration [20] is larger by about 10 standard deviations.

In this paper, we report measurements of the mass and lifetime of the
${{\mathchar 28938\relax}^{-}_{b}}$ baryon using the decay mode ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$,
where ${{\mathchar 28938\relax}^{0}_{c}}\rightarrow
p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$. (Charge-conjugate processes are implied
throughout.) The only prior evidence of the ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$
decay has been in the ${{\mathchar 28938\relax}^{0}_{c}}\rightarrow{{\mathchar
28938\relax}^{-}}{{\pi}^{+}}$ mode, with a signal of 4 events (3.3$\sigma$
significance) [19]. The ${\mathchar 28938\relax}^{0}_{c}$ $\rightarrow$ $p$
${K}^{-}$ ${K}^{-}$ ${\pi}^{+}$ decay mode is Cabibbo suppressed and is yet to
be observed. However, it has the advantage of a larger acceptance in the LHCb
detector compared to decay modes with hyperons in the final state. For
example, the yield of ${{\mathchar 28932\relax}^{-}_{b}}$ decays reconstructed
using ${\mathchar 28932\relax}^{-}_{b}$ $\rightarrow$ ${\mathchar
28932\relax}^{0}_{c}$ ${\pi}^{-}$, ${\mathchar 28932\relax}^{0}_{c}$
$\rightarrow$ $p$ ${K}^{-}$ ${K}^{-}$ ${\pi}^{+}$ decays [12] is about six
times larger than that obtained using ${\mathchar 28932\relax}^{-}_{b}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ${\mathchar
28932\relax}^{-}$ decays [10], where ${\mathchar 28932\relax}^{-}$
$\rightarrow$ $\mathchar 28931\relax$ ${\pi}^{-}$ and $\mathchar 28931\relax$
$\rightarrow$ $p$ ${\pi}^{-}$.

The mass and lifetime measurements are calibrated with respect to those of the
${{\mathchar 28932\relax}^{-}_{b}}$ baryon, reconstructed in the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$,
${{\mathchar 28932\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
decay mode. The mass and lifetime of the ${{\mathchar 28932\relax}^{-}_{b}}$
are measured to be $m_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}=5797.72\pm
0.55{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}=1.599\pm
0.041\pm 0.022$ ps [12], respectively; the measurements are of sufficiently
high precision that they do not represent a limiting uncertainty in the
${{\mathchar 28938\relax}^{-}_{b}}$ measurements presented here. The two
quantities that are measured are the mass difference, $\delta m=m_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}-m_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}$, and the
lifetime ratio $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}/\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}$. The identical final states and similar energy
release in the $b$ and $c$ baryon decays lead to a high degree of cancellation
of the systematic uncertainties on these quantities. Throughout this article,
we use $X_{b}$ ($X_{c}$) to refer to either a ${{\mathchar
28932\relax}^{-}_{b}}$ (${{\mathchar 28932\relax}^{0}_{c}}$) or ${{\mathchar
28938\relax}^{-}_{b}}$ (${{\mathchar 28938\relax}^{0}_{c}}$) baryon.

##  2 Detector and simulation

The measurements use proton-proton ($pp$) collision data samples, collected by
the LHCb experiment, corresponding to an integrated luminosity of
3.0$\mbox{\,fb}^{-1}$, of which 1.0$\mbox{\,fb}^{-1}$ was recorded at a
center-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$ and
2.0$\mbox{\,fb}^{-1}$ at 8$\mathrm{\,Te\kern-1.00006ptV}$. The LHCb detector
[21, 22] is a single-arm forward spectrometer covering the pseudorapidity
range $2<\eta<5$, designed for the study of particles containing $b$ or $c$
quarks. The detector includes a high-precision tracking system consisting of a
silicon-strip vertex detector surrounding the $pp$ interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet with a
bending power of about $4{\mathrm{\,Tm}}$, and three stations of silicon-strip
detectors and straw drift tubes placed downstream of the magnet. The tracking
system provides a measurement of momentum of charged particles with a relative
uncertainty that varies from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.

The online event selection is performed by a trigger [23], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. The
software trigger requires a two-, three- or four-track secondary vertex with a
large $p_{\mathrm{T}}$ sum of the tracks and a significant displacement from
the primary $pp$ interaction vertices. At least one particle should have
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be
inconsistent with coming from any of the PVs. The signal candidates are
required to pass a multivariate software trigger selection algorithm [24].

Proton-proton collisions are simulated using Pythia [25, *Sjostrand:2007gs]
with a specific LHCb configuration [27]. Decays of hadronic particles are
described by EvtGen [28], in which final-state radiation is generated using
Photos [29]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [30,
*Agostinelli:2002hh] as described in Ref. [32]. The ${{\mathchar
28932\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ and
${{\mathchar 28938\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
decays are modeled as an equal mixture of $X_{c}\rightarrow p{{K}^{-}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}},~{}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
and $X_{c}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ (nonresonant) decays;
this composition reproduces well the only clear structure in these decays, a
${{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ peak in the
${{K}^{-}}{{\pi}^{+}}$ mass distribution.

##  3 Candidate selection

Candidate $X_{c}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ decays are formed
by combining four tracks consistent with this decay chain, and requiring a
good quality vertex fit. In forming the $X_{c}$ candidate, each particle must
be significantly detached from all PVs in the event, have $p_{\mathrm{T}}$
greater than 100${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and have particle
identification (PID) information consistent with the decay hypothesis. The PID
requirements on the proton and the kaon candidates have a combined efficiency
of 70% on signal, while reducing the combinatorial background by a factor of
3.5.

Candidate $X_{b}$ baryons are formed by combining an $X_{c}$ candidate with a
${{\pi}^{-}}$ candidate. For each $X_{b}$ and PV pair in an event, a quantity
$\chi^{2}_{\text{IP}}(X_{b})$ is computed, defined as the increase in
$\chi^{2}$ when the $X_{b}$ candidate is included as an additional particle in
the PV fit. The $X_{b}$ candidate is assigned to the PV with the smallest
value of $\chi^{2}_{\text{IP}}(X_{b})$, and it is required to be significantly
displaced from that PV. The invariant mass $M(p{{K}^{-}}{{K}^{-}}{{\pi}^{+}})$
is required to lie in the range
2461–2481${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
2685–2705${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for ${{\mathchar
28932\relax}^{0}_{c}}$ and ${{\mathchar 28938\relax}^{0}_{c}}$ signal
candidates, respectively; these intervals cover a mass region that represents
about $\pm$2.5 and $\pm$2.0 times the expected mass resolution. The tighter
requirement on the ${{\mathchar 28938\relax}^{0}_{c}}$ candidates is used
because of a lower signal-to-background ratio. Candidates for which the
$p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ mass is outside the signal region are also
used to model the $X_{c}$ combinatorial background contribution to the signal
sample. To suppress combinatorial background, candidate $X_{b}$ decays are
required to have a reconstructed decay time larger than 0.2 ps, which is about
five times the decay time resolution for these decays.

To further improve the signal-to-background ratio, a multivariate analysis is
employed, based on a boosted decision tree (BDT) algorithm [33, 34]
implemented within the TMVA package [35]. Simulated ${{\mathchar
28932\relax}^{-}_{b}}$ and ${{\mathchar 28938\relax}^{-}_{b}}$ decays are used
to represent the signal distributions, and background events are taken from
the signal sidebands in data. The sidebands consist of events that are close
in mass to the $X_{b}$ signal region, but have either the
$p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ or $X_{c}{{\pi}^{-}}$ mass inconsistent with
the known $X_{c}$ or $X_{b}$ masses. Independent training and test samples are
used to ensure that the BDT is not overtrained.

A total of 18 discriminating variables are used to help differentiate signal
and background candidates, including: the $X_{b}$ decay vertex fit $\chi^{2}$;
the $\chi^{2}_{\text{IP}}$ of the $X_{b}$, $X_{c}$ and final-state decay
products; the consistency of the candidate with being produced at one of the
PVs in the event; the $p_{\mathrm{T}}$ of the decay products; and the PID
information on the proton and two kaons. Due to differences in the PID
information between simulation and data, the distributions of PID variables
for signal are taken from ${D}^{*+}$ $\rightarrow$ ${D}^{0}$ ${\pi}^{+}$ with
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$, ${\mathchar
28931\relax}\rightarrow p{{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decays in data [36],
and are reweighted to account for differences in kinematics between the
control and signal samples. The output of the training is a single
discriminating variable that ranges from $-1$ to $1$. For convenience, the
output value is also referred to as BDT.

The BDT requirement is chosen to maximize the figure of merit
$N_{S}/\sqrt{N_{S}+N_{B}}$ for the ${{\mathchar 28938\relax}^{-}_{b}}$ signal.
Here, $N_{S}$ and $N_{B}$ are the expected signal and background yields as a
function of the BDT requirement. The chosen requirement of BDT$>$0.3 provides
an expected signal (background) efficiency of about 90% (10%).

##  4 Mass spectra and fits

The $X_{c}$ invariant mass spectra for $X_{b}$ signal candidates are shown in
Fig. 1. All candidates within the regions contributing to the ${{\mathchar
28938\relax}^{-}_{b}}$ mass fit,
5420–6380${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the ${{\mathchar
28932\relax}^{-}_{b}}$ mass fit,
5630–6590${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, are included. The
simulated distributions, normalized to the fitted number of $X_{c}$ signal
decays in data, are overlaid. The vertical and horizontal arrows indicate the
signal and sideband regions.

Figure 1: Invariant mass distribution for (left) ${{\mathchar
28932\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ and (right)
${{\mathchar 28938\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
candidates over the full $X_{b}$ fit regions. The corresponding simulations
(sim.) are overlaid. The vertical arrows indicate the signal regions, and the
horizontal ones show the sideband regions.

While the overall background yields in these spectra are comparable, the
signal-to-background ratio is much lower within the ${{\mathchar
28938\relax}^{0}_{c}}$ candidate sample due to the lower production rate of
${{\mathchar 28938\relax}^{-}_{b}}$ relative to ${{\mathchar
28932\relax}^{-}_{b}}$ baryons, and likely a smaller $X_{c}\rightarrow
p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ branching fraction. Due to the very different
$X_{c}$ background levels for the signal and calibration mode, we use the
$X_{c}$ sidebands to model the $X_{c}$ combinatorial background in the $X_{b}$
invariant mass spectra.

To measure the ${{\mathchar 28938\relax}^{-}_{b}}$ mass and yield, the data
are fitted using a simultaneous extended unbinned maximum likelihood fit to
four $X_{b}$ invariant mass distributions; one pair is formed from the $X_{c}$
signal regions, and the second pair comprises events taken from the $X_{c}$
sidebands, as indicated in Fig. 1.

The signal shapes, determined from ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$
and ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}{{\pi}^{-}}$ simulated events, are each modeled by the
sum of two Crystal Ball (CB) functions [37] which have a common mean value.
The general forms of the two signal shapes are

$\displaystyle\mathcal{F}_{\rm sig}^{{{\mathchar 28932\relax}^{-}_{b}}}$
$\displaystyle=f_{\rm low}{\rm
CB}_{-}(m_{0},f_{\sigma}r_{\sigma}\sigma,\alpha_{-},N_{-})+(1-f_{\rm low}){\rm
CB}_{+}(m_{0},f_{\sigma}\sigma,\alpha_{+},N_{+}),$ (1)
$\displaystyle\mathcal{F}_{\rm sig}^{{{\mathchar 28938\relax}^{-}_{b}}}$
$\displaystyle=f_{\rm low}{\rm CB}_{-}(m_{0}+\delta
m,r_{\sigma}\sigma,\alpha_{-},N_{-})+(1-f_{\rm low}){\rm CB}_{+}(m_{0}+\delta
m,\sigma,\alpha_{+},N_{+}).$ (2)

Several of the parameters are common in the two signal shapes, and are
determined from a simultaneous fit to the mass spectra from simulated samples
of ${{\mathchar 28938\relax}^{-}_{b}}$ and ${{\mathchar 28932\relax}^{-}_{b}}$
decays. The ${\rm CB}_{\pm}$ function represents the signal contribution with
a tail toward low ($-$) or high ($+$) invariant mass. The parameters $m_{0}$
and $m_{0}+\delta m$ represent the fitted peak mass values of the ${{\mathchar
28932\relax}^{-}_{b}}$ and ${{\mathchar 28938\relax}^{-}_{b}}$ baryons,
respectively; $r_{\sigma}$ relates the lower CB width to the upper one; and
$f_{\sigma}$ allows for a small difference in the mass resolution for the
signal and calibration modes. The exponential tail parameters $\alpha_{\pm}$
are common to the signal and calibration modes. We fix the power-law tail
parameters $N_{-}=N_{+}=10$, and the fraction $f_{\rm low}=0.5$, as the
simulated signal shapes are well described without these parameters freely
varied. In fits to the data, $m_{0}$, $\delta m$ and $\sigma$ are left free to
vary, and all other shape parameters are fixed to the values from the
simulation.

Several sources of background contribute to the invariant mass spectrum for
both the signal and the calibration modes. These include: (i) partially-
reconstructed $X_{b}\rightarrow X_{c}\rho^{-}$ decays; (ii) misidentified
$X_{b}\rightarrow X_{c}{{K}^{-}}$ decays; (iii) partially-reconstructed
${{\mathchar 28938\relax}^{-}_{b}}\rightarrow{{\mathchar
28938\relax}^{*0}_{c}}{{\pi}^{-}}$ decays (${{\mathchar 28938\relax}^{-}_{b}}$
only); (iv) random $X_{c}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
combinations; and (v) $X_{b}\rightarrow X_{c}{{\pi}^{-}}$ combinatorial
background. The $X_{b}\rightarrow X_{c}\rho^{-}$ background shape is based on
simulated decays, and is parameterized by an ARGUS distribution [38] convolved
with a Gaussian resolution function of
16.4${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ fixed width, the value obtained
from fully reconstructed ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$
decays in data. The ARGUS shape parameters are left free to vary in the fit,
as is the yield, expressed as a fraction of the $X_{b}\rightarrow
X_{c}{{\pi}^{-}}$ yield. The $X_{b}\rightarrow X_{c}{{K}^{-}}$ background
shape is fixed based on simulation. The yield fraction $N(X_{b}\rightarrow
X_{c}{{K}^{-}})/N(X_{b}\rightarrow X_{c}{{\pi}^{-}})$ is fixed to 3.1%, which
is the product of an assumed ratio of branching fractions
${\cal{B}}(X_{b}\rightarrow X_{c}{{K}^{-}})/{\cal{B}}(X_{b}\rightarrow
X_{c}{{\pi}^{-}})=7\%$, based on the value from ${{\mathchar
28931\relax}^{0}_{b}}$ decays [39], and the efficiency of the PID requirements
on the ${{K}^{-}}$ and ${{\pi}^{-}}$. The shape parameters used to describe
these two backgrounds are common to the signal and calibration modes, apart
from an overall mass offset, which is fixed to be equal to $\delta m$. The
invariant mass distribution of the ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{*0}_{c}}{{\pi}^{-}}$
background is taken from a parametrization of the mass distribution obtained
from a phase-space simulation [40], combined with a Gaussian smearing based on
the measured mass resolution. The yield fraction $N({{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar
28938\relax}^{0}_{c}}{{\pi}^{-}})/N({{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar
28938\relax}^{*0}_{c}}{{\pi}^{-}})$ is freely varied in the fit to data.

The $X_{c}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ combinatorial background
contribution is constrained by including the $X_{c}$ sidebands in the
simultaneous fit, as discussed above. The shape of this background is modeled
by the sum of a broad Gaussian function and an exponential shape. In the
$X_{c}$ sidebands there is no indication of any ${{\mathchar
28932\relax}^{-}_{b}}$ or ${{\mathchar 28938\relax}^{-}_{b}}$ contributions,
which might result from nonresonant $X_{b}\rightarrow
p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays. The shape parameters and
yields of this background component are freely varied in the fit, but their
values are common for the $X_{c}$ signal and sideband data samples. A
different set of parameters is used for the ${{\mathchar
28938\relax}^{-}_{b}}$ and ${{\mathchar 28932\relax}^{-}_{b}}$ decay modes.
Random $X_{c}{{\pi}^{-}}$ combinatorial background is described by a single
exponential function with variable slope and yield.

The $X_{b}$ invariant mass spectra with the fits overlaid are shown in Fig. 2
for the $X_{c}$ signal regions. The fitted yields are $62.6\pm 9.0$ and
$1384\pm 39$ for the ${{\mathchar 28938\relax}^{-}_{b}}\rightarrow{{\mathchar
28938\relax}^{0}_{c}}{{\pi}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{c}}{{\pi}^{-}}$
modes, respectively. The ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$,
${{\mathchar 28938\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
decay is observed for the first time with large significance, about 10
standard deviations based on Wilks’s theorem [41]. The yield of ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$
decays is comparable to that obtained in ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\mathchar 28938\relax}^{-}}$ decays [10]. The mass difference is
measured to be $\delta m=247.7\pm
3.0$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the uncertainty is
statistical only.

Figure 2: Results of the simultaneous mass fit to the signal and calibration
modes. The fitted ${{\mathchar 28938\relax}^{-}_{b}}$ combinatorial (comb.)
background yield is very small, and not clearly visible.

##  5 ${{\mathchar 28938\relax}^{-}_{b}}$ lifetime

To measure the ${{\mathchar 28938\relax}^{-}_{b}}$ lifetime, the data from the
signal and calibration modes are divided into four bins of $X_{b}$ decay time:
0.0–1.5 ps, 1.5–2.5 ps, 2.5–4.0 ps, and 4.0–12.0 ps. The decay time binning
was chosen based on pseudoexperiments which replicate the yields of events in
data as a function of decay time for the signal and calibration modes. Several
binning schemes were investigated and the one above minimizes the systematic
uncertainty on the lifetime due to the small ${{\mathchar
28938\relax}^{-}_{b}}$ sample size.

The yields in each decay time bin in data are determined by repeating the mass
fit for each decay time bin, allowing the signal and background yields to vary
freely. All shape parameters are fixed to the values obtained from the fit to
the whole data sample, since simulations show that they do not depend on the
decay time. The results of the fits to the individual decay time bins are
shown in Figs. 3 and 4 for the signal and calibration modes. The yields are
presented in Table 1.

Table 1: Results of the fit to data for each decay time bin, and the relative efficiency. The uncertainties are statistical only. Decay time bin (ps) |  ${{\mathchar 28938\relax}^{-}_{b}}$ yield |  ${{\mathchar 28932\relax}^{-}_{b}}$ yield | $\epsilon({{\mathchar 28932\relax}^{-}_{b}})/\epsilon({{\mathchar 28938\relax}^{-}_{b}})$
---|---|---|---
0.0–1.5 | $20.8\pm 4.8$ | $450\pm 21$ | $1.10\pm 0.03$
1.5–2.5 | $12.0\pm 3.7$ | $427\pm 21$ | $1.11\pm 0.04$
2.5–4.0 | $17.7\pm 4.2$ | $305\pm 17$ | $1.02\pm 0.04$
4.0–12.0 | $10.5\pm 3.3$ | $201\pm 14$ | $1.03\pm 0.05$

The relative efficiency in each bin is determined using simulated events. The
efficiency-corrected yield ratio is then

$\displaystyle\frac{N_{{{\mathchar 28938\relax}^{-}_{b}}\rightarrow{{\mathchar
28938\relax}^{0}_{c}}{{\pi}^{-}}}(t)}{N_{{{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}}{{\pi}^{-}}}(t)}=A\exp{(\kappa t)},$ (3)

where $A$ is a calibration factor, and

$\displaystyle\kappa\equiv 1/\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}-1/\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}.$ (4)

The value of $\kappa$ is obtained by fitting an exponential function to the
efficiency-corrected ratio of yields, which in turn allows $\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}$ to be
determined. The efficiencies for the signal and normalization modes are
expressed as the fraction of generated signal decays with true decay time in
bin $i$, which have a reconstructed decay time also in bin $i$. When defined
in this way, effects of time resolution and selection requirements are
accounted for, and the corrected signal and calibration mode yields are
exponential in nature. The relative efficiencies after all selection
requirements are given in Table 1.

The efficiency ratio is consistent with having no dependence on the decay
time, as expected from the similarity of the two decay modes.

Figure 3: Results of the simultaneous mass fit to the ${{\mathchar
28938\relax}^{-}_{b}}$ signal in the four decay time bins, as indicated in
each plot.

Figure 4: Results of the simultaneous mass fit to the ${{\mathchar
28932\relax}^{-}_{b}}$ signal in the four decay time bins, as indicated in
each plot.

The efficiency-corrected yield ratio as a function of decay time is shown in
Fig. 5, along with a $\chi^{2}$ fit to the data using an exponential function.
The position of the points along the decay time axis is determined by taking
the average value within the bin, assuming an exponential decay time
distribution with $\tau=1.60$ ps. From the fitted value of $\kappa=0.053\pm
0.085$ ps-1 and the measured value of the ${{\mathchar 28932\relax}^{-}_{b}}$
lifetime, the lifetime ratio is found to be

$\displaystyle\frac{\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}}{\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}}$
$\displaystyle=\frac{1}{1-\kappa\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}}=1.09\pm
0.16,$ (5)

where the uncertainty is statistical only.

Figure 5: Corrected signal yield ratio as a function of decay time, along with
a fit to an exponential function. The horizontal bars indicate the bin sizes,
and are not an indication of the uncertainty.

##  6 Systematic uncertainties

A number of systematic uncertainties are evaluated, and are summarized in
Table 2. Most of the systematic uncertainties are estimated by modifying each
fixed input or function, and taking the difference with respect to the nominal
value as the systematic uncertainty. The signal shape uncertainty is
determined by changing the description to the sum of two Gaussian functions
and repeating the analysis. The nominal $X_{c}$ combinatorial background shape
is changed from the sum of a Gaussian shape and an exponential function to a
single exponential distribution. The sensitivity to the ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{*0}_{c}}{{\pi}^{-}}$
shape description is investigated by varying the shape parameters obtained
from the simulation to account for the uncertainty on the mass resolution, as
well as using a different function to parametrize the simulation. The
uncertainty on the yield of misidentified $X_{b}\rightarrow X_{c}{{K}^{-}}$
decays is quantified by varying the fractional contribution by $\pm$30%
relative to the nominal value, to allow for uncertainty in the
$X_{b}\rightarrow X_{c}{{K}^{-}}$ branching fractions amongst these modes and
for uncertainty in the PID efficiencies. The relative efficiency is obtained
from simulation. However, the BDT performance in data is slightly worse than
in simulation, so to estimate a potential bias in the lifetime ratio, we re-
evaluate the relative efficiency with a BDT$>$0.6 requirement, while keeping
the nominal requirement on the data. This larger value was chosen since it
provides equal efficiency of the BDT requirement on ${{\mathchar
28932\relax}^{-}_{b}}$ simulation as in data. To test the sensitivity to the
position of the points along the decay time axis (in Fig. 5), the fit is
repeated assuming an exponential distribution with $\tau=1.80$ ps. Bias due to
the small signal size has been studied using pseudoexperiments, and we find a
small fit bias in $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}/\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}$, which pulls the value down by 10% of the
statistical uncertainty. We correct the data for this bias, and assign half
the shift as a systematic uncertainty. The simulated samples used to determine
the relative efficiency are of finite size, and those uncertainties are
propagated to the final result.

Table 2: Summary of systematic uncertainties in $\delta m$ and the lifetime ratio. When two values are indicated, the first is a correction, and the second is the uncertainty. Source | $\delta m$ | $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}/\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}$
---|---|---
| (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) |
Signal shape |  $\pm$0.3 |  $\pm$0.005
Background shape |  $\pm$0.1 |  $\pm$0.009
${{\mathchar 28938\relax}^{*0}_{c}}$ shape |  $\pm$0.1 |  $\pm$0.003
$X_{b}\rightarrow X_{c}{{K}^{-}}$ background |  $\pm$0.2 |  $\pm$0.002
Relative efficiency | – |  $\pm$0.018
Average time in bin | – |  $\pm$0.002
Lifetime fit | – | $+0.016\pm 0.008$
Simulated sample size | $-0.38\pm 0.28$ |  $\pm$0.017
Momentum scale |  $\pm$0.1 | –
${{\mathchar 28932\relax}^{-}_{b}}$ lifetime | – |  $\pm$0.004
Total systematic | $-0.4\pm 0.5$ | $+0.016\pm 0.029$
Total statistical |  $\pm$3.2 |  $\pm$0.16

For the $\delta m$ measurement, the fitted value of $\delta m_{\rm
meas}-\delta m_{\rm true}$ in simulation is $-0.38\pm
0.28$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. We apply this value as a
correction, and assign the 0.28${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ as a
systematic uncertainty. The momentum scale has a fractional uncertainty of
$\pm 0.0003$ [42]. Its effect is evaluated by shifting all momentum components
of the final-state particles by this amount in simulated decays, and comparing
to the case when no shift is applied. Lastly, the uncertainty in the
${{\mathchar 28932\relax}^{-}_{b}}$ lifetime enters weakly into the lifetime
ratio (see Eq. 5), and is also included as a source of uncertainty. All
sources of systematic uncertainty are added in quadrature to obtain the
corrections and systematic uncertainties of $-0.4\pm
0.5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ on $\delta m$ and $+0.016\pm
0.029$ on $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}/\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}$.

##  7 Summary

In summary, a 3.0$\mbox{\,fb}^{-1}$ $pp$ collision data sample is used to
reconstruct a sample of $63\pm 9$ ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{{\mathchar 28938\relax}^{0}_{c}}{{\pi}^{-}}$,
${{\mathchar 28938\relax}^{0}_{c}}\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$
decays. This is the first observation of these ${{\mathchar
28938\relax}^{-}_{b}}$ and ${{\mathchar 28938\relax}^{0}_{c}}$ decay modes,
with well over 5$\sigma$ significance. Using these signals, the mass
difference and mass are measured to be

$\displaystyle m_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}-m_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}$ $\displaystyle=247.3\pm 3.2\pm
0.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle m_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}$
$\displaystyle=6045.1\pm 3.2\pm 0.5\pm
0.6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$

where the uncertainties are statistical, systematic, and from knowledge of the
${{\mathchar 28932\relax}^{-}_{b}}$ mass [12] ($m_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}$ only). The
measured ${{\mathchar 28938\relax}^{-}_{b}}$ mass is consistent with previous
measurements from LHCb, $6046.0\pm 2.2\pm
0.5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [18], and CDF, $6047.5\pm
3.8\pm 0.6$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [19], but inconsistent
with the value of $6165\pm 10\pm 13$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
obtained by the D0 experiment [20]. An average of the two LHCb measurements
yields $m_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}=6045.7\pm
1.9{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the momentum scale
uncertainty is taken as 100% correlated, and the rest of the uncertainties are
uncorrelated.

The lifetime ratio and absolute lifetime of the ${{\mathchar
28938\relax}^{-}_{b}}$ baryon are also measured to be

$\displaystyle\frac{\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}}{\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28932\relax}^{-}_{b}$}}}}}$
$\displaystyle=1.11\pm 0.16\pm 0.03,$ $\displaystyle\tau_{\lower
0.85pt\hbox{\scalebox{0.6}{{${\mathchar 28938\relax}^{-}_{b}$}}}}$
$\displaystyle=1.78\pm 0.26\pm 0.05\pm 0.06~{}{\rm ps},$

using $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}=1.599\pm 0.041\pm 0.022$ ps [12]. The first
uncertainty in each case is statistical. The second uncertainty on
$\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}/\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28932\relax}^{-}_{b}$}}}}$ is the total systematic uncertainty, as given in
Table 2. For $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}$, the second uncertainty is from all sources in
Table 2 except the ${{\mathchar 28932\relax}^{-}_{b}}$ lifetime, and the third
uncertainty stems from the uncertainty in the ${{\mathchar
28932\relax}^{-}_{b}}$ lifetime. The lifetime is consistent with the previous
measurements of $\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}=1.54^{+0.26}_{-0.21}\pm 0.05$ ps [10] and
$\tau_{\lower 0.85pt\hbox{\scalebox{0.6}{{${\mathchar
28938\relax}^{-}_{b}$}}}}=1.66^{+0.53}_{-0.40}$ ps [19] by the LHCb and CDF
collaborations, respectively. The average of the LHCb measurements, assuming
no correlation among the uncertainties, yields an ${{\mathchar
28938\relax}^{-}_{b}}$ lifetime of $1.66^{+0.19}_{-0.18}$ ps. These
measurements improve our knowledge of the mass and the lifetime of the
${{\mathchar 28938\relax}^{-}_{b}}$ baryon. Due to the similarity of the
signal and calibration modes, this pair of decay modes is very promising for
future studies of the ${{\mathchar 28938\relax}^{-}_{b}}$ baryon.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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Akar6, J. Albrecht10, F. Alessio39, M. Alexander52, S. Ali42, G. Alkhazov31,
P. Alvarez Cartelle54, A.A. Alves Jr58, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3,40, L. Anderlini18, G. Andreassi40, M. Andreotti17,g, J.E. Andrews59, R.B.
Appleby55, O. Aquines Gutierrez11, F. Archilli39, P. d’Argent12, A.
Artamonov36, M. Artuso60, E. Aslanides6, G. Auriemma26,n, M. Baalouch5, S.
Bachmann12, J.J. Back49, A. Badalov37, C. Baesso61, S. Baker54, W. Baldini17,
R.J. Barlow55, C. Barschel39, S. Barsuk7, W. Barter39, V. Batozskaya29, V.
Battista40, A. Bay40, L. Beaucourt4, J. Beddow52, F. Bedeschi24, I. Bediaga1,
L.J. Bel42, V. Bellee40, N. Belloli21,k, I. Belyaev32, E. Ben-Haim8, G.
Bencivenni19, S. Benson39, J. Benton47, A. Berezhnoy33, R. Bernet41, A.
Bertolin23, F. Betti15, M.-O. Bettler39, M. van Beuzekom42, S. Bifani46, P.
Billoir8, T. Bird55, A. Birnkraut10, A. Bizzeti18,i, T. Blake49, F. Blanc40,
J. Blouw11, S. Blusk60, V. Bocci26, A. Bondar35, N. Bondar31,39, W.
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Boubdir9, T.J.V. Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M.
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Hall54, B. Hamilton59, X. Han12, S. Hansmann-Menzemer12, N. Harnew56, S.T.
Harnew47, J. Harrison55, J. He39, T. Head40, A. Heister9, K. Hennessy53, P.
Henrard5, L. Henry8, J.A. Hernando Morata38, E. van Herwijnen39, M. Heß65, A.
Hicheur2, D. Hill56, M. Hoballah5, C. Hombach55, L. Hongming40, W.
Hulsbergen42, T. Humair54, M. Hushchyn67, N. Hussain56, D. Hutchcroft53, M.
Idzik28, P. Ilten57, R. Jacobsson39, A. Jaeger12, J. Jalocha56, E. Jans42, A.
Jawahery59, M. John56, D. Johnson39, C.R. Jones48, C. Joram39, B. Jost39, N.
Jurik60, S. Kandybei44, W. Kanso6, M. Karacson39, T.M. Karbach39,†, S.
Karodia52, M. Kecke12, M. Kelsey60, I.R. Kenyon46, M. Kenzie39, T. Ketel43, E.
Khairullin67, B. Khanji21,39,k, C. Khurewathanakul40, T. Kirn9, S. Klaver55,
K. Klimaszewski29, M. Kolpin12, I. Komarov40, R.F. Koopman43, P. Koppenburg42,
M. Kozeiha5, L. Kravchuk34, K. Kreplin12, M. Kreps49, P. Krokovny35, F.
Kruse10, W. Krzemien29, W. Kucewicz27,o, M. Kucharczyk27, V. Kudryavtsev35, A.
K. Kuonen40, K. Kurek29, T. Kvaratskheliya32, D. Lacarrere39, G.
Lafferty55,39, A. Lai16, D. Lambert51, G. Lanfranchi19, C. Langenbruch49, B.
Langhans39, T. Latham49, C. Lazzeroni46, R. Le Gac6, J. van Leerdam42, J.-P.
Lees4, R. Lefèvre5, A. Leflat33,39, J. Lefrançois7, E. Lemos Cid38, O. Leroy6,
T. Lesiak27, B. Leverington12, Y. Li7, T. Likhomanenko67,66, R. Lindner39, C.
Linn39, F. Lionetto41, B. Liu16, X. Liu3, D. Loh49, I. Longstaff52, J.H.
Lopes2, D. Lucchesi23,r, M. Lucio Martinez38, H. Luo51, A. Lupato23, E.
Luppi17,g, O. Lupton56, N. Lusardi22, A. Lusiani24, X. Lyu62, F. Machefert7,
F. Maciuc30, O. Maev31, K. Maguire55, S. Malde56, A. Malinin66, G. Manca7, G.
Mancinelli6, P. Manning60, A. Mapelli39, J. Maratas5, J.F. Marchand4, U.
Marconi15, C. Marin Benito37, P. Marino24,t, J. Marks12, G. Martellotti26, M.
Martin6, M. Martinelli40, D. Martinez Santos38, F. Martinez Vidal68, D.
Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R. Matev39, A. Mathad49,
Z. Mathe39, C. Matteuzzi21, A. Mauri41, B. Maurin40, A. Mazurov46, M.
McCann54, J. McCarthy46, A. McNab55, R. McNulty13, B. Meadows58, F. Meier10,
M. Meissner12, D. Melnychuk29, M. Merk42, A Merli22,u, E Michielin23, D.A.
Milanes64, M.-N. Minard4, D.S. Mitzel12, J. Molina Rodriguez61, I.A. Monroy64,
S. Monteil5, M. Morandin23, P. Morawski28, A. Mordà6, M.J. Morello24,t, J.
Moron28, A.B. Morris51, R. Mountain60, F. Muheim51, D. Müller55, J. Müller10,
K. Müller41, V. Müller10, M. Mussini15, B. Muster40, P. Naik47, T. Nakada40,
R. Nandakumar50, A. Nandi56, I. Nasteva2, M. Needham51, N. Neri22, S.
Neubert12, N. Neufeld39, M. Neuner12, A.D. Nguyen40, C. Nguyen-Mau40,q, V.
Niess5, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A. Novoselov36,
D.P. O’Hanlon49, A. Oblakowska-Mucha28, V. Obraztsov36, S. Ogilvy52, O.
Okhrimenko45, R. Oldeman16,48,f, C.J.G. Onderwater69, B. Osorio Rodrigues1,
J.M. Otalora Goicochea2, A. Otto39, P. Owen54, A. Oyanguren68, A. Palano14,d,
F. Palombo22,u, M. Palutan19, J. Panman39, A. Papanestis50, M. Pappagallo52,
L.L. Pappalardo17,g, C. Pappenheimer58, W. Parker59, C. Parkes55, G.
Passaleva18, G.D. Patel53, M. Patel54, C. Patrignani20,j, A. Pearce55,50, A.
Pellegrino42, G. Penso26,m, M. Pepe Altarelli39, S. Perazzini15,e, P. Perret5,
L. Pescatore46, K. Petridis47, A. Petrolini20,j, M. Petruzzo22, E. Picatoste
Olloqui37, B. Pietrzyk4, M. Pikies27, D. Pinci26, A. Pistone20, A. Piucci12,
S. Playfer51, M. Plo Casasus38, T. Poikela39, F. Polci8, A. Poluektov49,35, I.
Polyakov32, E. Polycarpo2, A. Popov36, D. Popov11,39, B. Popovici30, C.
Potterat2, E. Price47, J.D. Price53, J. Prisciandaro38, A. Pritchard53, C.
Prouve47, V. Pugatch45, A. Puig Navarro40, G. Punzi24,s, W. Qian56, R.
Quagliani7,47, B. Rachwal27, J.H. Rademacker47, M. Rama24, M. Ramos Pernas38,
M.S. Rangel2, I. Raniuk44, G. Raven43, F. Redi54, S. Reichert10, A.C. dos
Reis1, V. Renaudin7, S. Ricciardi50, S. Richards47, M. Rihl39, K.
Rinnert53,39, V. Rives Molina37, P. Robbe7, A.B. Rodrigues1, E. Rodrigues58,
J.A. Rodriguez Lopez64, P. Rodriguez Perez55, A. Rogozhnikov67, S. Roiser39,
V. Romanovsky36, A. Romero Vidal38, J. W. Ronayne13, M. Rotondo23, T. Ruf39,
P. Ruiz Valls68, J.J. Saborido Silva38, N. Sagidova31, B. Saitta16,f, V.
Salustino Guimaraes2, C. Sanchez Mayordomo68, B. Sanmartin Sedes38, R.
Santacesaria26, C. Santamarina Rios38, M. Santimaria19, E. Santovetti25,l, A.
Sarti19,m, C. Satriano26,n, A. Satta25, D.M. Saunders47, D. Savrina32,33, S.
Schael9, M. Schiller39, H. Schindler39, M. Schlupp10, M. Schmelling11, T.
Schmelzer10, B. Schmidt39, O. Schneider40, A. Schopper39, M. Schubiger40,
M.-H. Schune7, R. Schwemmer39, B. Sciascia19, A. Sciubba26,m, A. Semennikov32,
A. Sergi46, N. Serra41, J. Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin36,
I. Shapoval17,44,g, Y. Shcheglov31, T. Shears53, L. Shekhtman35, V.
Shevchenko66, A. Shires10, B.G. Siddi17, R. Silva Coutinho41, L. Silva de
Oliveira2, G. Simi23,s, M. Sirendi48, N. Skidmore47, T. Skwarnicki60, E.
Smith54, I.T. Smith51, J. Smith48, M. Smith55, H. Snoek42, M.D. Sokoloff58,
F.J.P. Soler52, F. Soomro40, D. Souza47, B. Souza De Paula2, B. Spaan10, P.
Spradlin52, S. Sridharan39, F. Stagni39, M. Stahl12, S. Stahl39, S.
Stefkova54, O. Steinkamp41, O. Stenyakin36, S. Stevenson56, S. Stoica30, S.
Stone60, B. Storaci41, S. Stracka24,t, M. Straticiuc30, U. Straumann41, L.
Sun58, W. Sutcliffe54, K. Swientek28, S. Swientek10, V. Syropoulos43, M.
Szczekowski29, T. Szumlak28, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G.
Tellarini17,g, F. Teubert39, C. Thomas56, E. Thomas39, J. van Tilburg42, V.
Tisserand4, M. Tobin40, S. Tolk43, L. Tomassetti17,g, D. Tonelli39, S. Topp-
Joergensen56, E. Tournefier4, S. Tourneur40, K. Trabelsi40, M. Traill52, M.T.
Tran40, M. Tresch41, A. Trisovic39, A. Tsaregorodtsev6, P. Tsopelas42, N.
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Vagnoni15,39, S. Valat39, G. Valenti15, A. Vallier7, R. Vazquez Gomez19, P.
Vazquez Regueiro38, C. Vázquez Sierra38, S. Vecchi17, M. van Veghel42, J.J.
Velthuis47, M. Veltri18,h, G. Veneziano40, M. Vesterinen12, B. Viaud7, D.
Vieira2, M. Vieites Diaz38, X. Vilasis-Cardona37,p, V. Volkov33, A.
Vollhardt41, D. Voong47, A. Vorobyev31, V. Vorobyev35, C. Voß65, J.A. de
Vries42, R. Waldi65, C. Wallace49, R. Wallace13, J. Walsh24, J. Wang60, D.R.
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G. Wilkinson56,39, M. Wilkinson60, M. Williams39, M.P. Williams46, M.
Williams57, T. Williams46, F.F. Wilson50, J. Wimberley59, J. Wishahi10, W.
Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton48, K. Wraight52, S. Wright48,
K. Wyllie39, Y. Xie63, Z. Xu40, Z. Yang3, H. Yin63, J. Yu63, X. Yuan35, O.
Yushchenko36, M. Zangoli15, M. Zavertyaev11,c, L. Zhang3, Y. Zhang3, A.
Zhelezov12, Y. Zheng62, A. Zhokhov32, L. Zhong3, V. Zhukov9, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62University of Chinese Academy of Sciences, Beijing, China, associated to 3
63Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
64Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
65Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
66National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
67Yandex School of Data Analysis, Moscow, Russia, associated to 32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
†Deceased


# Tidal capture of an asteroid by a magnetar: FRB-like bursts, glitch and
anti-glitch

Qin Wu,1 Zhen-Yin Zhao,1 Fa-Yin Wang1,2,3
1School of Astronomy and Space Science, Nanjing University, Nanjing 210093,
China
2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, China
3 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210023,
China E-mail<EMAIL_ADDRESS>0000-0001-6021-5933 0000-0002-2171-9861
0000-0003-4157-7714

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Recently, remarkable anti-glitch and glitch accompanied by bright radio bursts
of the Galactic magnetar SGR J1935+2154 were discovered. These two infrequent
temporal coincidences between the glitch/anti-glitch and the fast radio burst
(FRB)-like bursts reveal their physical connection of them. Here we propose
that the anti-glitch/glitch and FRB-like bursts can be well understood by an
asteroid tidally captured by a magnetar. In this model, an asteroid is tidally
captured and disrupted by a magnetar. Then, the disrupted asteroid will
transfer the angular momentum to the magnetar producing a sudden change in the
magnetar rotational frequency at the magnetosphere radius. If the orbital
angular momentum of the asteroid is parallel (or anti-parallel) to that of the
spinning magnetar, a glitch (or anti-glitch) will occur. Subsequently, the
bound asteroid materials fall back to the pericenter and eventually are
accreted to the surface of the magnetar. Massive fragments of the asteroid
cross magnetic field lines and produce bright radio bursts through coherent
curvature radiation. Our model can explain the sudden magnetar spin changes
and FRB-like bursts in a unified way.

###### keywords:

stars: magnetars - transients: fast radio bursts

††pubyear: 2023††pagerange: Tidal capture of an asteroid by a magnetar: FRB-
like bursts, glitch and anti-glitch–References

##  1 Introduction

SGR J1935+2154 is a Galactic magnetar with a spin period of $P\sim 3.24\
\rm{s}$ and a magnetic field strength of $B\sim 2.2\times 10^{14}\ \rm{G}$
(Israel et al., 2016). On 28 April 2020, a bright millisecond radio burst FRB
20200428 from SGR J1935+2154 was detected by CHIME and STARE2 (CHIME/FRB
Collaboration et al., 2020; Bochenek et al., 2020). Meanwhile, a bright X-ray
burst was temporally coincident with FRB 20200428 (Mereghetti et al., 2020; Li
et al., 2021a; Ridnaia et al., 2021; Tavani et al., 2021). The detection of
FRB 20200428 from a Galactic magnetar indicates that at least part of FRBs
originates from magnetars (Wang et al., 2022b). Some models have been proposed
to explicate the simultaneous production of FRB 20200428 and X-ray burst from
SGR J1935+2154 (Dai, 2020; Geng et al., 2020; Wu et al., 2020; Margalit et
al., 2020; Lu et al., 2020; Shen et al., 2023).

From 1 October to 27 November 2020, the NICER and XMM-Newton telescopes
monitored SGR J1935+2154 regularly in the 1-3 keV band. A timing analysis of
the X-ray pulses was employed and an anti-glitch with
$\Delta\nu=1.8^{+0.7}_{-0.5}\times 10^{-6}\ \rm{Hz}$ was found to occur on 5
October 2020 (+/-1 day) (Younes et al., 2023; Zhu et al., 2020). Subsequently,
three bright radio bursts were detected by CHIME on 8 October 2020 (Good &
Chime/Frb Collaboration, 2020). The fluence of three FRB-like bursts is
900$\pm$160 Jy ms, 9.2$\pm$1.6 Jy ms, and 6.4$\pm$1.1 Jy ms, respectively.
Considering the distance of SGR J1935+2154 is about $D=9\ \rm{kpc}$ (Zhong et
al., 2020), the energies of radio bursts are $3.5\times 10^{31}$ erg,
$3.6\times 10^{29}$ erg and $2.5\times 10^{29}$ erg. More recently, Ge et al.
(2022) studied the timing properties of SGR J1935+2154 using the archived
NICER, NuSTAR, Chandra, and XMM-Newton observations, and reported that a giant
spin-up glitch with $\Delta\nu=1.98\pm 0.14\times 10^{-6}\ \rm{Hz}$ happened
$3.1\pm 2.5$ days before FRB 20200428.

Glitch is a frequent phenomenon observed in many pulsars, which is defined as
a sudden increase in rotational frequency. The origin of glitches is still
unknown. They can be caused by internal processes within the pulsar or
external processes outside the pulsar (Haskell & Melatos, 2015). Anti-glitch
is characterized by a sudden decrease in rotational frequency, which is not
common as a glitch. Some models have been proposed to explain the origin of
anti-glitches (Thompson et al., 2000; Lyutikov, 2013; Tong, 2014; Katz, 2014;
Huang & Geng, 2014; Kantor & Gusakov, 2014). The accetion of planetesimals by
a magnetar and the collision of a solid body with a magnetar was used to
explain the anti-glitch from magnetar 1E 2259+586 (Katz, 2014; Huang & Geng,
2014).

Different from normal glitches and anti-glitches of pulsars, the glitch, and
anti-glitch of SGR J1935+2154 are accompanied by FRB-like bursts. The temporal
coincidence between the glitch/anti-glitch and the FRB-like bursts of SGR
J1935+2154 indicate a physical connection between them (Younes et al., 2023;
Ge et al., 2022). Younes et al. (2023) proposed that the rapid reduction of
the angular momentum of magnetar comes from the escape of the particle wind
along the open magnetic field lines. While the subsequent radio bursts arise
from the alteration of the magnetospheric field geometry. Ge et al. (2022)
supplied an interpretation that the spin-up glitch before FRB 20200428 is
produced by the violent movement of the magnetosphere. While the crustal
cracking and the propagation of Alfvén wave produced the following FRB
20200428 and the associated X-ray bursts. There are also other discussions on
the glitch (Wang et al., 2022a), the anti-glitch (Wang et al., 2021; Tong,
2023), and the glitch-associated activities (Du, 2023) of SGR J1935+2154.
However, neither of them provides a self-consistent model that can explain the
anti-glitch/glitch and FRB-like bursts simultaneously.

In this paper, we propose that an asteroid captured by a magnetar (hereafter
the capture model) can explain the glitch and anti-glitch of SGR J1935+2154
and the subsequent FRB-like bursts simultaneously. The capture model and the
parameter constraints are given in Section 2. In the last, the discussion and
conclusion are shown in Section 3.

Figure 1:  Schematic diagram of the capture model. (a) An iron-nickel asteroid
with mass $m$ is tidally captured and disrupted by a magnetar at the tidal
disruption radius $R_{\rm t}$. (b) The orbital angular momentum of the
asteroid is transferred to the magnetar at the magnetosphere radius $R_{\rm
m}$, which will cause the observed anti-glitch or glitch. The blue arrow
represents the direction of the spin of the magnetar. A fraction of the
disrupted material will escape the magnetar at high speed, while the rest will
be bound to the magnetar and experience a process of orbital circularization.
The bound material will fall back to the pericenter and eventually be accreted
to the magnetar. (c) The broken fragment with the mass of $m_{0}$ crosses the
magnetic field lines and induces the FRB-like burst through coherent curvature
radiation at the interaction radius $R_{\rm I}$. This capture model can
explain the glitch/anti-glitch and the associated FRB-like bursts in a unified
way.

##  2 The Capture Model

We consider that an asteroid is tidally captured and disrupted by the strong
gravitational force of a magnetar when it approaches the tidal disruption
radius (see Figure 1). A part of the disrupted material will be accreted and
experience orbital circularization, while the rest material will be thrown out
at a high speed. The broken and accreted asteroid transfers the angular
momentum to the magnetar at the magnetosphere radius, which causes a sudden
change in the rotational frequency of the magnetar (Ghosh & Lamb, 1979). When
the orbital angular momentum of the asteroid is parallel to the spin of the
magnetar, an increase in the rotational frequency naturally happens. On the
contrary, the anti-parallel orientation will cause a decrease in the
rotational frequency of the magnetar. Then the bound asteroid will fall back
to the magnetar due to dissipative processes such as collision and viscous
dissipation and eventually be accreted onto the surface of the magnetar (Rees,
1988; Perets et al., 2016). The FRB-like bursts are naturally produced when
the broken fragments are across the magnetic field lines through coherent
curvature radiation processes (Dai et al., 2016; Dai, 2020).

###  2.1 Glitch and anti-glitch

Here we suppose an asteroid is disrupted tidally by the magnetar with typical
mass $M_{\rm NS}={\rm 1.4\ M_{\odot}}$ and radius $R_{\rm NS}=10$ km at the
tidal disruption radius (Hills, 1975)

$R_{\mathrm{t}}\approx(6M_{\rm NS}/\pi\rho)^{1/3}\sim 8.7\times
10^{10}(M_{\rm{NS}}/1.4M_{\odot})^{1/3}\rho_{0.9}^{-1/3}\ \rm{cm},$ (1)

where the expression $Q_{x}$ represents $Q/10^{x}$ in cgs unit system and
$\rho={\rm 8\ g\ cm^{-3}}$ is the density of a homogeneous iron-nickel
asteroid (Colgate & Petschek, 1981). Then a fraction of the disrupted material
is tidally captured by the magnetar while the rest will be thrown out. The
bound material will enter an elliptical orbit, which length of the semi-major
axis can be estimated as ${R_{\rm orb}\sim 4.4\times 10^{14}}$ cm (Zubovas et
al., 2012). The orbital period is calculated as 135 years using Kepler’s third
law. After a few orbits, the circularization can occur (Cannizzo et al.,
1990). The material will fall back to the magnetar vicinity within this
timescale (Rees, 1988). The ram pressure is generated during the falling of
the bound material. Considering the balance of the ram pressure and the
magnetic pressure, we can calculate the magnetosphere radius $R_{\rm m}$ of
the disrupted material (Ghosh & Lamb, 1979)

$R_{\mathrm{m}}=\left(\frac{\mu^{4}}{2GM_{\rm NS}\dot{m}^{2}}\right)^{1/7},$
(2)

where $\mu=BR_{\rm NS}^{3}$ is the magnetic dipole moment and the surface
magnetic filed is $B=3.52\times 10^{14}\ G$ for SGR J1935+2154.

From the magnetosphere radius $R_{\rm m}$, the orbital angular momentum of the
asteroid will transfer to the magnetar, which will cause the observed glitch
or anti-glitch. The velocity of the asteroid with mass $m$ is $V_{m}=(2GM_{\rm
NS}/R_{\rm m})^{1/2}$ and the orbital angular momentum of the asteroid can be
calculated as $-mV_{\rm m}R_{\rm m}$. Conservation of angular momentum for the
system composed of a magnetar and an asteroid can be written as

$2\pi\nu I-mV_{\rm m}R_{\rm m}=2\pi(\nu-\Delta\nu)I,$ (3)

where $I$ is the moment of inertia of the magnetar and $\Delta\nu$ is the
change of frequency. Considering a magnetar with mass $M_{\rm
NS}=1.4M_{\odot}$ and radius $R_{\rm NS}=10$ km, the moment of inertia of the
magnetar $I=\frac{2}{5}M_{\rm NS}R_{\rm NS}^{2}=1.12\times 10^{45}\ {\rm g}\
{\rm cm}^{2}$ (Pizzochero, 2011; Hooker et al., 2015).

In our model, the disrupted material is assumed to be accreted and adheres to
the central magnetar from the magnetosphere radius (Ghosh & Lamb, 1979). The
average mass accretion rate is

$\dot{m}=\frac{m_{\rm acc}}{\Delta t},$ (4)

where $m_{\rm acc}$ is the mass of accreted material and $\Delta t$ is the
accretion timescale, which is the time from the initial accretion of the
asteroid to the magnetosphere radius. The value of ${\Delta t}$ is hard to
determine from observations. Here we suppose that the time of accretion is
about ${\Delta t\sim 1}$ hour, which is shorter than the time lag between the
glitch/anti-glitch and the radio burst (Younes et al., 2023; Ge et al., 2022).

Utilizing equations (2), (3) and (4), we can derive the mass accretion rate
$\dot{m}$. For the anti-glitch with $\Delta\nu_{1}=1.8^{+0.7}_{-0.5}\times
10^{-6}$ Hz (Younes et al., 2023), the mass accretion rate $\dot{m_{1}}$ is
given

$\begin{split}\dot{m_{1}}&=\frac{(2\pi
I\Delta\nu_{1})^{7/6}}{\mu^{1/3}(2GM_{\rm NS})^{1/2}\Delta t^{7/6}}\\\
&\sim{7.82\times
10^{16}}I_{45}^{7/6}\Delta\nu_{1,-5.7}^{7/6}\mu_{32.5}^{-1/3}\Delta
t_{4.9}^{-7/6}\ \rm{g\ s^{-1}}.\end{split}$ (5)

Then the accretion mass of the anti-glitch $m_{1}$ is

$m_{1}=\dot{m_{1}}\Delta t={6.76\times
10^{21}}I_{45}^{7/6}\Delta\nu_{1,-5.7}^{7/6}\mu_{32.5}^{-1/3}\Delta
t_{4.9}^{-1/6}\ \rm{g},$ (6)

which is the lower limit of the mass of the asteroid. The magnetosphere radius
$R_{\rm m1}$ of the anti-glitch is estimated as

$R_{\rm m1}=\left(\frac{\mu^{2}\Delta t}{2\pi
I_{c}\Delta\nu_{1}}\right)^{1/3}={9.45\times 10^{9}}\mu_{32.5}^{2/3}\Delta
t_{4.9}^{1/3}I_{45}^{-1/3}\Delta\nu_{1,-5.7}^{-1/3}\ \rm{cm},$ (7)

which is smaller than the tidal disruption radius $R_{\rm t}$ (equation (1)).

If the orbital angular momentum of the asteroid is parallel to the spin of the
magnetar, a spin-up glitch naturally occurs. For the spin-up glitch with
${\Delta\nu_{2}=1.98\times 10^{-5}}$ Hz (Ge et al., 2022), the mass accretion
rate is ${\dot{m_{2}}\simeq 1.28\times 10^{18}}\ \rm{{g\ s^{-1}}}$, the
accretion mass is ${m_{2}\simeq 1.11\times 10^{23}}$ g and the corresponding
magnetosphere radius is ${R_{\rm m2}\simeq 4.25\times 10^{9}}$ cm.

The radius of the light-cylinder of SGR J1935+2154 is calculated as

${R_{\rm L}=\frac{c}{2\pi\nu}=1.55\times 10^{10}}\ \rm{cm},$ (8)

where ${\nu=0.307896}$ Hz is the spin frequency of SGR J1935+2154. The
magnetic field lines are connected to the magnetar inside the light cylinder.
The magnetosphere radius ${R_{\rm{m}}}$ for the glitch/anti-glitch is well
inside the light cylinder. So the angular momentum of the asteroid can be
transferred to the magnetar.

After the tidal disruption, a fraction of the disrupted and bound debris may
fall back and self-interact. Then the orbiting debris with high eccentricity
will experience orbital circularization and a transient disk is formed around
the magnetar. As the corotation radius ${R_{\rm co}\sim 4.6\times 10^{8}}$ cm
is smaller than the magnetosphere radius ${R_{\rm{m}}}$, the bound material
will accumulate at the magnetosphere radius and eventually fall back to the
magnetar once the ram pressure exceeds the magnetic pressure. The glitch/anti-
glitch happens due to the angular momentum transfer between the bound material
and the magnetar at the magnetosphere radius ${R_{\rm{m}}}$. Then the bound
material will fall back to the vicinity of the magnetar and produce bright
radio bursts. The fallback timescale that the bound asteroid from the
magnetosphere radius returns to the vicinity of the magnetar can be calculated
as (Perets et al., 2016)

${t_{\mathrm{fb}}=\frac{2\pi R_{\mathrm{m}}^{3}}{(GM_{\rm
NS})^{1/2}(2r)^{3/2}}\sim 3.6\ r_{6.8}^{-3/2}\mu_{32.5}^{2}\Delta
t_{4.9}I_{45}^{-1}\Delta\nu_{1,-5.7}^{-1}}\ \rm{{days}},$ (9)

where ${r}$ is the radius of the asteroid. The fallback time is comparable to
the observed time delay ($\sim$ 3 days) between the glitch/anti-glitch and the
FRB-like bursts (Younes et al., 2023; Ge et al., 2022), which means our model
is self-consistent. The bound disk formed by the fallback material will be
accreted by the magnetar and produce bright radio bursts when the large
fragments cross the magnetic lines. Dai (2020) gave an appropriate explanation
for the production of FRB 20200428 and the associated X-ray burst, which are
temporal coincident with the spin-up glitch (Ge et al., 2022). In the
following section, we discuss the generation of the FRB-like bursts that
occurred on 8 October 2020 after the anti-glitch (Good & Chime/Frb
Collaboration, 2020; Younes et al., 2023).

###  2.2 FRB-like bursts

The asteroid is firstly tidally disrupted into a great number of fragments at
the tidal disruption radius $R_{\rm t}$. The broken fragments will slow down
in a strong magnetic field of the magnetar. Three massive fragments are
distorted by the magnetar at the broken-up radius $R_{b}=(\rho
r_{0}^{2}GM_{\rm NS}/s)^{1/3}$ with the tensile strength $s$ and the original
radius $r_{0}$ of the fragment (Colgate & Petschek, 1981). The broken fragment
will be elongated longitudinally as a cylinder radius $r_{0}$ and length
$l_{0}=2r_{0}$ and will be further disrupted into a train of small chunks.
Bright FRB-like bursts are produced when these disrupted fragments cross the
magnetic lines through curvature radiation. The time difference between the
leading and lagging parts of a train of small fragments reaching the surface
of the magnetar is

$\delta t\simeq\frac{12r_{0}}{5}\left(\frac{2GM_{\rm
NS}}{R_{b}}\right)^{-1/2},$ (10)

where $s\sim 10^{10}\ \rm{dyn\ cm^{-2}}$ for the iron-nickel asteroid. This
time difference is comparable to the duration of an FRB-like burst, which is
independent of the position of the fragments (Geng & Huang, 2015). The
duration of the first and the brightest burst of three FRB-like bursts is
$\delta t\sim 0.26\pm 0.01$ ms (Good & Chime/Frb Collaboration, 2020), thus
the collective mass of a train of fragments $m_{0}$ can be calculated as

$\begin{split}m_{0}=2\pi\left(\frac{5\delta
t}{12}\right)^{3}\left(\frac{2GM_{\rm NS}}{R_{b}}\right)^{3/2}=1.7\times
10^{16}\delta t_{-3.6}^{3}\rho_{0.9}R_{\rm b,9.8}^{-3/2}\ \rm{g}.\end{split}$
(11)

The break-up radius $R_{b}\sim 5.9\times 10^{9}\ \rm{cm}$ is smaller than the
tidal disruption radius $R_{\rm t}\sim 8.7\times 10^{10}\ \rm{cm}$ and the
magnetosphere radius of the anti-glitch ${R_{\rm m1}\sim 9.45\times 10^{9}}\
\rm{cm}$, implying that our calculations are self-consistent.

The disrupted fragments are apparently influenced by the strong magnetic field
of the magnetar at the interaction radius ($R_{\rm I}$) where the kinetic
energy density of the fragment $\rho v_{\rm ff}^{2}/2$ is equal to the
magnetic energy density $\mu^{2}/8\pi R_{\rm I}^{6}$. Here we give the
interaction radius $R_{\rm I}$ of the broken-up fragments

$R_{\rm I}=3.05\times 10^{7}\rho_{0.9}^{-1/5}B_{14.5}^{2/5}R_{\rm NS,6}^{6/5}\
\rm{cm},$ (12)

where $v_{\rm ff}=\sqrt{2GM_{\rm NS}/R_{\rm I}}$ and the magnetic dipole
moment of the magnetar $\mu=BR_{\rm NS}^{3}$. The disrupted bound material
will induce a strong electric field (${E}=-v_{\rm ff}\times B/c$) when it
penetrates through the stellar magnetosphere (Lamb et al., 1973; Burnard et
al., 1983; Hameury et al., 1986; Litwin & Rosner, 2001). The electrons are
accelerated to ultra-relativistic energies and move along magnetic field
lines. Movement of electrons along magnetic field line with curvature radius
$\rho_{c}$ generates coherent curvature radiation and leads to FRB-like bursts
at the radius $R_{\rm I}$ (Dai et al., 2016; Dai, 2020). The typical Lorentz
factor $\gamma$ of an accelerated electron is given by

$\gamma\equiv\chi\gamma_{\rm max}\simeq\chi\left(\frac{6\pi eE}{\sigma_{\rm
T}B^{2}}\right)^{1/2}=216.07\chi R_{\rm I,7.5}^{5/4}\mu_{32.5}^{-1/2},$ (13)

where $\sigma_{\rm T}$ is the Thomson scattering cross section and $\chi$ is
defined as the ratio of $\gamma$ and the maximum Lorentz factor $\gamma_{\rm
max}$ (Dai et al., 2016). Condition $\chi\lesssim 1$ needs to be satisfied and
we generally assume $\chi\simeq 0.5$, then $\gamma\sim 108$.

The characteristic frequency of the curvature radiation is

$\nu_{\rm c}=\frac{3c\Gamma\gamma^{3}}{4\pi\rho_{c}}=465.49\delta R_{\rm
I,7.5}^{11/4}\mu_{32.5}^{-3/2}\ \rm{MHz},$ (14)

where curvature radius $\rho_{c}\sim 0.635R_{\rm I}$ is assumed (Yang & Zhang,
2018) and ${\Gamma}$ is the factor related to the Doppler effect (Dai, 2020).
This characteristic frequency is consistent with the FRB-like bursts detected
by CHIME in the range 400-600 MHz (Good & Chime/Frb Collaboration, 2020).

The disrupted and bound fragments are radially elongated and transversely
compressed to a cylindrical shape from the break-up radius ${R_{\rm b}}$. Some
small fragments will undergo further disruption into smaller chunks and move
with the closed magnetic field lines (Metzger et al., 2018). A train of chunks
will cross the magnetic field lines and produce the bright radio bursts
through the coherent curvature radiation at the interaction radius ${R_{\rm
I}}$. And an openmouthed-clam-shaped radiating region is naturally formed with
an inclination angle of ${\theta_{\rm i}\simeq r(R_{\rm I})/R_{\rm I}}$ (Dai,
2020). The total luminosity of an FRB-like burst is derived by considering an
openmouthed-clam-shaped radiating region (Dai, 2020),

$L_{\rm tot}=3.32\times 10^{32}R_{\rm
I,7.5}^{-27/4}\mu_{32.5}^{3/2}m_{0,16.2}^{8/9}\rho_{0.9}^{-14/9}s_{10}^{2/3}\
\rm{erg\ s^{-1}}.$ (15)

In the observer’s rest frame, the isotropic-equivalent emission energy of the
FRB-like burst is given by

$E_{\rm radio}\simeq\frac{L_{\rm tot}\delta t\Gamma^{3}}{f}\sim 1.86\times
10^{31}\Gamma^{3}\ \rm{erg},$ (16)

where $f\simeq 1/2\gamma$ is the beaming factor and ${\delta t\sim 0.26\pm
0.01}$ ms is the duration of the radio burst. This is comparable to the energy
of three FRB-like bursts observed by CHIME on 8 October (Good & Chime/Frb
Collaboration, 2020), which indicates the FRB-like burst can be naturally
produced in our model.

When the asteroid gets closer to the magnetar, evaporation caused by the
heating of the asteroid should be considered. Considering the thermal
equilibrium with the magnetar surface radiation, the temperature of the
asteroid is ${T_{\rm e}=T_{\rm sur}\sqrt{R_{\rm NS}/2r}}$ (Cordes & Shannon,
2008). Here we assume that the surface temperature of the asteroid is ${T_{\rm
sur}\sim 10^{5}}$ K regarding the age and the mass of this relatively old
magnetar. At the broken-up radius ${R_{\rm b}\sim 5.9\times 10^{9}}$ cm, the
temperature of the asteroid is ${T_{\rm e}\sim 920}$ K, which is smaller than
the evaporation temperature of the iron ($\sim$ 2000 K). Thus, the curvature
radiation will not be significantly affected by evaporation before the tidal
break-up. Heating by high-energy particles by the non-thermal radiation from
the magnetar will not significantly evaporate the asteroid, because of the
decreasing kinetic energy of the high-energy particles when the bound material
approaches the magnetar (Geng et al., 2020). However, the evaporation can be
significant for the small fragment because the evaporation happens faster for
the smaller asteroid (Zubovas et al., 2012). The study on the evaporation near
supermassive black holes had been performed by Zubovas et al. (2012).

##  3 Discussion and Conclusions

Observations show that X-ray bursts are temporal coincident with FRB 20200428
about 3 days after the glitch. Ge et al. (2022) proposed that FRB 20200428 and
the associated X-ray bursts originate from the violent alteration of
magnetosphere structure. However, due to the lack of model details, whether it
can explain the observation of the anti-glitch is unclear. Younes et al.
(2023) postulated that an ephemeral and strong wind that extracts angular
momentum from the magnetar leads to an abrupt spin-down of SGR J1935+2154.
Obviously, the outward wind is difficult to explain the sudden spin-up of SGR
J1935+2154 (Ge et al., 2022). The energy ratio between FRB 20200428 and the
X-ray burst from SGR J1935+2154 is $\eta=2.13\times 10^{-5}$ (Wu et al.,
2020). If a similar ratio is assumed for the radio bursts observed in October
2020, the luminosity of the X-ray burst associated with the FRB-like burst can
be estimated as

$L_{\rm X}=\frac{L_{\rm radio}}{\eta}\sim 8.74\times
10^{35}\eta_{-4.7}^{-1}L_{\rm tot,32.5}\ \rm{erg\ s^{-1}}.$ (17)

This is comparable to the 5$\sigma$ detection upper-limit of the X-ray
luminosity, i.e., $\rm 10^{36}\ erg\ s^{-1}$ (Younes et al., 2023). Thus the
absence of X-ray bursts after the anti-glitch can be naturally explained as
being too faint to be observed.

The temporal coincidence between the infrequent glitch/anti-glitch and bright
radio bursts implies some physical association between them, which is also
supported by statistical similarities. The size distribution of glitches in
individual pulsars is power-law form (Morley & Garcia-Pelayo, 1993; Melatos et
al., 2008). The waiting times of glitches also show power-law distribution
Carlin & Melatos (2021), which is consistent with a non-stationary Poisson
process. Power-law distributions of energy and waiting time are found in some
repeating FRBs (Wang & Yu, 2017; Katz, 2018; Cheng et al., 2020; Hewitt et
al., 2022; Jahns et al., 2023; Wang et al., 2023). A large sample of FRB
121102 observed by the Five-hundred-meter Aperture Spherical radio Telescope
(Li et al., 2021b) also confirmed the power-law distributions of energy and
waiting time (Zhang et al., 2021).

In this paper, we propose a capture model to explicate the glitch/anti-glitch
and the associated FRB-like bursts in a unified way. In this model, both the
glitch and the anti-glitch can be naturally explained. The FRB-like bursts are
produced when the disrupted fragments penetrate through the stellar
magnetosphere. The time interval between the glitch/anti-glitch and the FRB-
like bursts can be interpreted as the fallback process of the bound material.
Our main conclusions are as follows.

1\. A iron-nickel asteroid is tidally disrupted by the magnetar at the tidal
disruption radius $R_{\rm t}\sim 8.7\times 10^{10}\ \rm{cm}$. Then the angular
momentum of the captured asteroid transfer to the magnetar at the
magnetosphere radius ${R_{\rm m1}\sim 9.45\times 10^{9}}\ \rm{cm}$ for the
anti-glitch and ${R_{\rm m2}\sim 4.25\times 10^{9}}\ \rm{cm}$ for the glitch,
respectively. The lower-limit mass of the asteroid is estimated as ${m_{1}\sim
6.76\times 10^{21}}\ \rm{g}$ and ${m_{2}\sim 1.11\times 10^{23}}\ \rm{g}$ for
the anti-glitch and glitch, respectively. These results are consistent with
the mass range of asteroids $1.0\times 10^{10}\ \rm{g}-1.0\times 10^{24}\
\rm{g}$ from observations.

2\. The disrupted and bound asteroid falls back to the magnetar from the
magnetosphere radius ${R_{\rm m}}$ and eventually is accreted onto the surface
of the magnetar. The fallback time is consistent with the observed time delay
between the glitch/anti-glitch and the radio burst. In the case of anti-
glitch, a train of fragments with collective mass $m_{0}\sim 1.7\times
10^{16}\ \rm{g}$ produces bright radio burst with energy $E_{\rm radio}\sim
1.86\times 10^{31}\ \rm{erg}$ at the interaction radius $R_{\rm I}\sim
3.05\times 10^{7}\ \rm{cm}$.

3\. The luminosity of the possible associated X-ray burst is about $8.74\times
10^{35}\ \rm{erg\ s^{-1}}$, which is comparable to the 5$\sigma$ detection
upper-limit of the X-ray luminosity. This is consistent with the non-detection
of X-ray bursts during the anti-glitch and FRB-like bursts.

4\. The time interval between the glitch and the anti-glitch of SGR J1935+2154
is about 158 days. According to Dai et al. (2016), the edge-on collision
between the asteroid belt and the pulsar implies two travels through the belt,
and the pulsar captures the asteroid twice resulting in glitch/anti-glitch.
The time interval between them is $\sim R_{\rm a}/v_{\odot}\sim 170(R_{\rm
a}/20\ \rm{au})(v_{\odot,7}/2)^{-1}$ days, where $R_{\rm a}\sim 20$ au is the
inner radius of an asteroid belt (DeMeo & Carry, 2014) and $v_{\odot}\sim
2\times 10^{7}\ \rm{cm}\ \rm{s^{-1}}$ is the average proper velocity of the
magnetar. The parameters used here are reasonable for the Solar System. This
time interval is consistent with the observed time interval between the glitch
and the anti-glitch. However, the rate of capture of an asteroid to produce
the observed glitch/anti-glitch is difficult to determine, cause the angular
momentum of the asteroid parallel/anti-parallel to the spinning magnetar is a
rare event. Glitch/anti-glitch accompanied with bright radio bursts are so
rare in observations.

Table 1: Definitions of Some Parameters of the Capture Model

Description | Symbol | Dependence
---|---|---
Tidal disruption radius | $R_{\rm t}$ | $\sim 8.7\times 10^{10}\rho_{0.9}^{-1/3}\ \rm{cm}$
Magnetosphere radiusa | $R_{\rm m}$ | $\sim 9.45\times 10^{9}\mu_{32.5}^{2/3}\Delta t_{4.9}^{1/3}I_{45}^{-1/3}\Delta\nu_{1,-5.7}^{-1/3}\ \rm{cm}$
Interaction radiusa | $R_{\rm I}$ | $\sim 3.05\times 10^{7}\rho_{0.9}^{-1/5}B_{14.5}^{2/5}\ \rm{cm}$
Mass accretion ratea | $\dot{m}$ | $\sim 7.82\times 10^{16}I_{45}^{7/6}\Delta\nu_{1,-5.7}^{7/6}\mu_{32.5}^{-1/3}\Delta t_{4.9}^{-7/6}\ \rm{g\ s^{-1}}$
Mass of the asteroida | $m$ | $\sim 6.76\times 10^{21}I_{45}^{7/6}\Delta\nu_{1,-5.7}^{7/6}\mu_{32.5}^{-1/3}\Delta t_{4.9}^{-1/6}\ \rm{g}$
Mass of the heaviest broken fragmenta | $m_{0}$ | $\sim 1.7\times 10^{16}\delta t_{-3.6}^{3}\rho_{0.9}R_{\rm b,9.8}^{-3/2}\ \rm{g}$
Characteristic frequency of curvature radiationa | $\nu_{\rm c}$ | $\sim 465.49R_{\rm I,7.5}^{11/4}\mu_{32.5}^{-3/2}\ \rm{MHz}$
Luminosity of the FRB-like bursta | $L_{\rm tot}$ | $\sim 3.32\times 10^{32}R_{\rm I,7.5}^{-27/4}\mu_{32.5}^{3/2}m_{0,16.2}^{8/9}\rho_{0.9}^{-14/9}\ \rm{erg\ s^{-1}}$
Luminosity of the X-ray bursta | $L_{\rm X}$ | $\sim 8.74\times 10^{35}\eta_{-4.7}^{-1}L_{\rm tot,32.5}\ \rm{erg\ s^{-1}}$

  * a

This calculation is for the case of the anti-glitch.

## Acknowledgements

We appreciate the referee for valuable comments and suggestions, which have
helped to improve this manuscript. We thank Zigao Dai, Chen Deng, Guoqiang
Zhang, Ziqian Hua, Yongfeng Huang, Jinjun Geng, and Zenan Liu for their
helpful discussions. This work was supported by the National Natural Science
Foundation of China (grant No. 12273009), the National SKA Program of China
(grant No. 2022SKA0130100), and the China Manned Spaced Project (CMS-
CSST-2021-A12).

## Data Availability

No new data were generated in support of this theoretical research.

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