EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-027 LHCb-PAPER-2019-002 March 5, 2019

Search for $C\\!P$ violation in
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ decays

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

A search for charge-parity ($C\\!P$) violation in Cabibbo-suppressed
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ decays is reported using proton-proton
collision data, corresponding to an integrated luminosity of
3.8$\text{\,fb}^{-1}$, collected at a center-of-mass energy of 13 TeV with the
LHCb detector. High-yield samples of kinematically and topologically similar
Cabibbo-favored ${D}_{({s})}^{+}$ decays are analyzed to subtract nuisance
asymmetries due to production and detection effects, including those induced
by $C\\!P$ violation in the neutral kaon system. The results are

$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$
$\displaystyle=\left(\phantom{-}1.3\phantom{0}\pm 1.9\phantom{0}\pm
0.5\phantom{0}\right)\times 10^{-3},$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$
$\displaystyle=\left(-0.09\pm 0.65\pm 0.48\right)\times 10^{-3},$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$
$\displaystyle=\left(\phantom{-}0.05\pm 0.42\pm 0.29\right)\times 10^{-3},$

where the first uncertainties are statistical and the second systematic. They
are the most precise measurements of these quantities to date, and are
consistent with $C\\!P$ symmetry. A combination with previous LHCb
measurements, based on data collected at 7 and 8 TeV, is also reported.

Published in Phys. Rev. Lett. 122 (2019) 191803

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

Violation of charge-parity ($C\\!P$) symmetry arises in the Standard Model
(SM) of particle physics through the complex phase of the
Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix [1, 2]. $C\\!P$ violation
is well established in $K$\- and $B$-meson systems [3, 4, 5, 6, 7], and has
been observed only recently in charm decays [8]. $C\\!P$ violation in charm
decays can arise from the interference between tree- and loop-level diagrams
through Cabibbo-suppressed ${c}\rightarrow{d}{\overline{d}}{u}$ and
${c}\rightarrow{s}{\overline{s}}{u}$ transition amplitudes. In the loop-level
processes, contributions from physics beyond the SM may arise that can lead to
additional sources of $C\\!P$ violation [9]. However, the expected SM
contribution is difficult to compute due to the presence of low-energy strong-
interaction effects, with current predictions spanning several orders of
magnitude [10, 11, 12, 9, 13]. A promising handle to determine the origin of
possible $C\\!P$-violation signals are correlations between $C\\!P$
asymmetries in flavor-$SU(3)$ related decays [14, 15, 16, 17, 18, 19, 20, 21,
22]. Particularly interesting in this respect are ${D}^{+}_{s}$ and ${D}^{+}$
decays to two-body (or quasi two-body) final states, such as
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$.111The inclusion of charge-conjugate
processes is implied throughout this Letter, unless stated otherwise. Searches
for $C\\!P$ violation in these modes have been performed by the CLEO [23],
BaBar [24, 25], Belle [26, 27, 28] and LHCb [29, 30] collaborations. No
evidence for $C\\!P$ violation has been found within a precision of a few per
mille.

This Letter presents measurements of $C\\!P$ asymmetries in
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ decays performed using proton-proton
collision data collected with the LHCb detector between 2015 and 2017 at a
center-of-mass energy of 13 TeV, and corresponding to an integrated luminosity
of $3.8\text{\,fb}^{-1}$. In the presence of a ${K}^{0}_{\mathrm{S}}$ meson in
the final state, a $C\\!P$ asymmetry is expected to be induced by
${K}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ mixing [31].
This effect is well known and predictable, allowing for a precise measurement
of $C\\!P$ violation in the charm-quark transition. The
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ decay is reconstructed with the
$\phi\rightarrow{{K}^{+}}{{K}^{-}}$ mode. Several intermediate states
contribute to the ${{D}^{+}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ decay
amplitude [32]. In this Letter, no attempt is made to separate them through an
amplitude analysis and the measurement is performed by simply restricting the
${{K}^{+}}{{K}^{-}}$ pair to the mass region around the $\phi(1020)$
resonance.

The $C\\!P$ asymmetry of a ${D}_{({s})}^{+}$ meson decaying to the final state
$f^{+}$ is defined as

$\mathcal{A}_{{C\\!P}}({D}_{({s})}^{+}\rightarrow
f^{+})\equiv\frac{\Gamma({D}_{({s})}^{+}\rightarrow
f^{+})-\Gamma({D}_{({s})}^{-}\rightarrow
f^{-})}{\Gamma({D}_{({s})}^{+}\rightarrow
f^{+})+\Gamma({D}_{({s})}^{-}\rightarrow f^{-})},$ (1)

where $\Gamma$ is the partial decay rate. If $C\\!P$ symmetry is violated in
the decay, $\mathcal{A}_{{C\\!P}}\neq 0$. An experimentally convenient
quantity to measure is the “raw” asymmetry of the observed yields $N$,

$A({D}_{({s})}^{+}\rightarrow f^{+})\equiv\frac{N({D}_{({s})}^{+}\rightarrow
f^{+})-N({D}_{({s})}^{-}\rightarrow f^{-})}{N({D}_{({s})}^{+}\rightarrow
f^{+})+N({D}_{({s})}^{-}\rightarrow f^{-})}.$ (2)

The raw asymmetry can be approximated as

$A({D}_{({s})}^{+}\rightarrow
f^{+})\approx\mathcal{A}_{{C\\!P}}({D}_{({s})}^{+}\rightarrow
f^{+})+A_{P}({D}_{({s})}^{+})+A_{D}(f^{+}),\\\ $ (3)

where $A_{P}({D}_{({s})}^{+})$ is the asymmetry of the ${D}_{({s})}^{+}$-meson
production cross-section [33, 34] and $A_{D}(f^{+})$ is the asymmetry of the
reconstruction efficiency for the final state $f^{+}$. When
$f^{+}={{K}^{0}_{\mathrm{S}}}h^{+}$ (with $h=K,\pi$), the detection asymmetry
receives contributions from the $h^{+}$ hadron (indicated as companion hadron
in the following), $A_{D}(h^{+})$, and from the neutral kaon, $A_{D}({{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}})$. Relevant instrumental
effects contributing to $A_{D}(h^{+})$ may include differences in interaction
cross-sections with matter between positive and negative hadrons and the
slightly charge-asymmetric performance of the reconstruction algorithms. The
contribution to $A_{D}({{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}})$ arises from ${K}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ mesons having different
interaction cross-sections with matter and from their propagation in the
detector being affected by the presence of $C\\!P$ violation in the
${K}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ system. When
$f^{+}=\phi(\rightarrow{{K}^{+}}{{K}^{-}}){{\pi}^{+}}$, the detection
asymmetry is mostly due to the charged pion, as the contributions from the
oppositely charged kaons cancel to a good precision.

The detection and production asymmetries are canceled by using the decays
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$, which proceed through the Cabibbo-
favored ${c}\rightarrow{s}{\overline{d}}{u}$ transition. In the SM, these
decays are expected to have $C\\!P$ asymmetries that are negligibly small
compared to the Cabibbo-suppressed modes, when effects induced by the neutral
kaons are excluded [31, 35]. Hence, their raw asymmetries can be approximated
as in Eq. (3), but with $\mathcal{A}_{{C\\!P}}=0$. The $C\\!P$ asymmetries of
the decay modes of interest are determined by combining the raw asymmetries as
follows:

$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$
$\displaystyle\approx
A({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})-A({{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}),$
(4)
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$
$\displaystyle\approx
A({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})-A({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$
$\displaystyle\qquad-A({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})+A({{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}),$
(5) $\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$
$\displaystyle\approx
A({{D}^{+}}\rightarrow\phi{{\pi}^{+}})-A({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}),$
(6)

where the contribution from $A_{D}({{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}})$ is omitted and should be
subtracted from any of the measured asymmetries where it is present.

The LHCb detector [36, 37] is a single-arm forward spectrometer designed for
the study of particles containing $b$ or $c$ quarks. The detector elements
that are particularly relevant to this analysis are: a silicon-strip vertex
detector that allows for a precise measurement of the impact parameter, i.e.,
the minimum distance of a charged-particle trajectory to a $p$ $p$ interaction
point (primary vertex); a tracking system that provides a measurement of the
momentum of charged particles; two ring-imaging Cherenkov detectors that are
able to discriminate between different species of charged hadrons; and a
calorimeter system that is used for the identification of photons, electrons
and hadrons. The polarity of the magnetic field is periodically reversed
during data-taking to mitigate the differences between reconstruction
efficiencies of oppositely charged particles.

The online event selection is performed by a trigger, which consists of a
hardware stage followed by a two-level software stage. In between the two
software stages, an alignment and calibration of the detector is performed in
near real-time and their results are used in the trigger [38]. Events with
candidate ${D}_{({s})}^{+}$ decays are selected by the hardware trigger by
imposing either that one or more ${D}_{({s})}^{+}$ decay products are
associated with large transverse energy deposits in the calorimeter or that
the accept decision is independent of the ${D}_{({s})}^{+}$ decay products
(i.e., it is caused by other particles in the event). In the first level of
the software trigger, one or more ${D}_{({s})}^{+}$ decay products must have
large transverse momentum and be inconsistent with originating from any
primary vertex. In the second level, the candidate decays are fully
reconstructed using kinematic, topological and particle-identification
criteria. The ${D}_{({s})}^{+}\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}$
candidates are made by combining charged hadrons with
${{K}^{0}_{\mathrm{S}}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ candidates that
decay early enough for the final-state pions to be reconstructed in the vertex
detector. This requirement suppresses to a negligible level possible
$C\\!P$-violation effects due to interference between Cabibbo-favored and
doubly Cabibbo-suppressed amplitudes with neutral-kaon mixing in the control-
sample decays ${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ and
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ [35].

The ${D}_{({s})}^{+}$ candidates reconstructed in the trigger are used
directly in the offline analysis [39, 40]. The candidates with a
${K}^{0}_{\mathrm{S}}$ meson in the final state are further selected offline
using an artificial neural network (NN), based on the multilayer perceptron
algorithm [41], to suppress background due to random combinations of
${K}^{0}_{\mathrm{S}}$ mesons and hadrons not originating from a
${D}_{({s})}^{+}\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}$ decay. The quantities
used in the NN to discriminate signal from combinatorial background are: the
${K}^{0}_{\mathrm{S}}$ candidate momentum; the transverse momenta of the
${D}_{({s})}^{+}$ candidate and of the companion hadron; the angle between the
${D}_{({s})}^{+}$ candidate momentum and the vector connecting the primary and
secondary vertices; the quality of the secondary vertex; and the track quality
of the companion hadron. The NN is trained using signal and background data
samples, obtained with the sPlot method [42], from a ${\mathcal{O}}(1\%)$
fraction of candidates randomly sampled. In the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ case, thanks to
similar kinematics, background-subtracted
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ decays are exploited
as a signal proxy to profit from larger yields. The thresholds on the NN
response are optimized for the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ decays by maximizing the
value of $S/\sqrt{S+B}$, where $S$ and $B$ stands for the signal and
background yield observed in the mass ranges
$1.93<m({{K}^{0}_{\mathrm{S}}}\pi^{+})<2.01$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$
and
$1.83<m({{K}^{0}_{\mathrm{S}}}K^{+})<1.91$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
respectively. Candidate
${D}_{({s})}^{+}\rightarrow\phi(\rightarrow{{K}^{+}}{{K}^{-}}){{\pi}^{+}}$
decays are selected offline with requirements on the transverse momenta of the
${D}_{({s})}^{+}$ candidate and of the companion hadron, on the quality of the
secondary vertex, and on the ${{K}^{+}}{{K}^{-}}$ mass to be within
$10$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the nominal $\phi(1020)$-meson
mass [32]. The mass window is chosen considering that the observed width is
dominated by the $\phi(1020)$-meson natural width of
$4.2\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [32] and is only marginally affected
by the experimental resolution of $1.3\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.

The contribution of ${D}_{({s})}^{+}$ mesons produced through decays of $b$
hadrons, referred to as secondaries throughout, is suppressed by requiring
that the ${D}_{({s})}^{+}$ impact parameter in the plane transverse to the
beam (TIP) is smaller than $40\,\upmu\text{m}$. The remaining percent-level
contribution is evaluated by means of a fit to the TIP distribution when such
requirement is released, as shown in Fig. 1 for the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ decay. The impact
of the secondary background on the results is accounted for in the systematic
uncertainties.

Typical sources of background from ${D}_{({s})}^{+}$ meson and ${\mathchar
28931\relax}^{+}_{c}$ baryon decays are: the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{{K}^{0}_{\mathrm{S}}}{p}$ decays, where the
kaon and the proton are misidentified as a pion, when the signal is the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ decay; the
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{{K}^{0}_{\mathrm{S}}}{p}$ decays, where the
pion and the proton are misidentified as a kaon, in the
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ case; and the
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow\phi{p}$ decay, where the proton
is misidentified as a pion, when the signal is the
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ decay. These are all reduced to a
negligible level using particle-identification requirements and kinematic
vetos.

Fiducial requirements are imposed to exclude kinematic regions that induce a
large asymmetry in the companion-hadron reconstruction efficiency. These
regions occur because low momentum particles of one charge at large (small)
angles in the bending plane may be deflected out of the detector acceptance
(into the noninstrumented beam pipe region), whereas particles with the other
charge are more likely to remain within the acceptance. About 78%, 93% and 94%
of the selected candidates are retained by these fiducial requirements for
${D}_{({s})}^{+}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
${D}_{({s})}^{+}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${D}_{({s})}^{+}\rightarrow\phi{{\pi}^{+}}$ decays, respectively.

Figure 1: Distribution of the transverse impact parameter (TIP) for
background-subtracted
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ candidates with
fit projections overlaid.

Detection and production asymmetries may depend on the kinematics of the
involved particles. Therefore, the cancellation provided by the control decays
is accurate only if the kinematic distributions agree between any pair of
signal and control modes, or pair of control modes entering Eqs. (4)–(6).
Differences are observed, and the ratio between background-subtracted [42]
signal and control sample distributions of transverse momentum, azimuthal
angle and pseudorapidity are used to define candidate-by-candidate weights.
The background-subtracted candidates of the control decays are weighted such
that their distributions agree with those of the signal using an iterative
procedure. The process consists of calculating the weights in each one-
dimensional distribution of the weighting variables and repeating the
procedure until good agreement is achieved among all the distributions. For
the measurements of the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ $C\\!P$ asymmetries, the
${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ control samples are
weighted so that the ${D}_{({s})}^{+}$ meson and companion-pion kinematic
distributions agree with their respective signal samples to cancel the
${D}_{({s})}^{+}$ production and companion-pion detection asymmetries. In the
case of the
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$
measurement, the ${D}^{+}$ kinematic distributions of the
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ sample are weighted to
those of the ${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ signal to
cancel the ${D}^{+}$ production asymmetry, and the ${{K}^{+}}$ distributions
of the ${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ decays are
weighted to those of the ${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$
signal to cancel the kaon detection asymmetry. The
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ and
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ control decays then
introduce their own additional nuisance asymmetries, which need to be
corrected for using the ${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$ control
decay. Hence, the ${D}^{+}_{s}$ and companion-pion kinematic distributions of
the ${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$ sample are made to agree with
those of the ${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ samples, respectively,
to cancel the ${D}^{+}_{s}$ production and companion-pion detection
asymmetries.

\begin{overpic}[width=216.81pt]{Fig2a}
\put(50.0,22.0){\includegraphics[scale={0.18}]{Fig2a_inset}} \end{overpic}

Figure 2: Mass distributions of the selected (top)
${D}_{({s})}^{+}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$, (middle)
${D}_{({s})}^{+}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$ and (bottom)
${D}_{({s})}^{+}\rightarrow\phi{{\pi}^{+}}$ candidates with fit projections
overlaid. The inset in the top plot shows the mass distribution around the
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ signal region.

Simultaneous least-squares fits to the mass distributions of weighted
${D}_{({s})}^{+}$ and ${D}_{({s})}^{-}$ candidates determine the raw
asymmetries for each decay mode considered. To avoid experimenter bias, the
raw asymmetries of the Cabibbo-suppressed signals were shifted by unknown
offsets sampled uniformly between $-1\%$ and $1\%$, such that the results
remained blind until the analysis procedure was finalized. In the fits, the
signal and control decays are modeled as the sum of a Gaussian function to
describe the core of the peaks, and a Johnson $S_{U}$ distribution [43], which
accounts for the asymmetric tails. The combinatorial background is described
by the sum of two exponential functions. All shape parameters are determined
from the data. In each fit, signal and control decays share the same shape
parameters apart from a mass shift, which accounts for the known difference
between the ${D}^{+}_{s}$ and ${D}^{+}$ masses [32], and a relative scale
factor between the peak widths, which is also determined from the data. The
means and widths of the peaks, as well as all background shape parameters, are
allowed to differ between ${D}_{({s})}^{+}$ and ${D}_{({s})}^{-}$ decays. The
projections of the fits to the combined ${D}_{({s})}^{+}$ and
${D}_{({s})}^{-}$ data are shown in Fig. 2. The samples contain approximately
$600$ thousand ${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$,
$5.1$ million ${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$, and
$53.3$ million ${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ signal candidates,
together with approximately $30.5$ million
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$, $6.5$ million
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$, and $107$ million
${{D}^{+}_{s}}\rightarrow\phi{{\pi}^{+}}$ control decays.

The raw asymmetries are, where relevant, corrected for the neutral-kaon
detection asymmetry. The net correction is estimated following Ref. [44] to be
$(+0.084\pm 0.005)\%$ for
$\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$,
$(-0.086\pm 0.005)\%$ for
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$,
and $(-0.068\pm 0.004)\%$ for
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$, where the
uncertainty is dominated by the accuracy of the detector modeling in the
simulation. The asymmetries are combined following Eqs. (4)–(6) to obtain
$\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})=\left(1.3\pm
1.9\right)\times 10^{-3}$,
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})=\left(-0.09\pm
0.65\right)\times 10^{-3}$,
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})=\left(0.05\pm
0.42\right)\times 10^{-3}$, where the uncertainties are only statistical.

Several sources of systematic uncertainty affecting the measurement are
considered as reported in Table 1. The dominant contribution is due to the
assumed shapes in the mass fits. This is evaluated by fitting with the default
model large sets of pseudoexperiments where alternative models that describe
data equally well are used in generation. For
$\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$
and
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$,
the second leading contribution is due to the residual contamination from
secondary ${D}_{({s})}^{+}$ decays, which introduces a small difference
between the asymmetry of ${D}_{({s})}^{+}$-meson production cross-sections of
the signal and control modes. For
$\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$, instead, the
second leading systematic uncertainty arises from neglected kinematic
differences between the $\phi$-meson decay products. These differences, mainly
caused by the interference between the $S$-wave and $\phi{{\pi}^{+}}$ decay
amplitudes in the ${{K}^{+}}{{K}^{-}}$-mass region under study, result in an
imperfect cancelation of the charged-kaon detection asymmetry. Other
subleading contributions are due to the inaccuracy in the equalization of the
kinematic distributions between signal and control samples, and to the
uncertainty in the neutral-kaon detection asymmetry.

In addition, several consistency checks are performed to investigate possible
unexpected biases by comparing results obtained in subsamples of the data
defined according to the data-taking year and magnetic-field polarity, the
per-event track multiplicity, the configurations of the hardware- and
software-level triggers, and the ${D}_{({s})}^{+}$ momentum. A $\chi^{2}$ test
has been performed for each cross-check and the corresponding $p$ values are
consistent with being uniformly distributed; the lowest (largest) $p$ value is
$4\%$ ($86\%$). Therefore, the observed variations in results are consistent
with statistical fluctuations and no additional sources of systematic
uncertainties are considered.

Table 1: Summary of the systematic uncertainties (in units of $10^{-3}$) on
the measured quantities. The total is the sum in quadrature of the different
contributions.

Source | $\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$ | $\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$ | $\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$
---|---|---|---
Fit model | 0.39 | 0.44 | 0.24
Secondary decays | 0.30 | 0.12 | 0.03
Kinematic differences | 0.09 | 0.09 | 0.04
Neutral kaon asymmetry | 0.05 | 0.05 | 0.04
Charged kaon asymmetry | 0.08 | 0.09 | 0.15
Total | 0.51 | 0.48 | 0.29

In summary, using proton-proton collision data collected with the LHCb
detector at a center-of-mass energy of 13 TeV, and corresponding to
$3.8\text{\,fb}^{-1}$ of integrated luminosity, the following $C\\!P$
asymmetries are measured:

$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$
$\displaystyle=\left(\phantom{-}1.3\phantom{0}\pm 1.9\phantom{0}\pm
0.5\phantom{0}\right)\times 10^{-3},$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$
$\displaystyle=\left(-0.09\pm 0.65\pm 0.48\right)\times 10^{-3},$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$
$\displaystyle=\left(\phantom{-}0.05\pm 0.42\pm 0.29\right)\times 10^{-3},$

where the first uncertainties are statistical and the second systematic.
Effects induced by $C\\!P$ violation in the neutral kaon system are subtracted
from the measured asymmetries. The results represent the most precise
determination of these quantities to date and are consistent with $C\\!P$
symmetry. They are in agreement with previous LHCb determinations based on
independent data samples collected at center-of-mass energies of 7 and 8 TeV
[29, 30], as well as with measurements from other experiments [23, 26, 24, 27,
25, 28]. The results are combined with previous LHCb measurements using the
BLUE method [45]. The systematic uncertainties are considered uncorrelated,
apart from those due to the neutral- and charged-kaon detection asymmetries
that are fully correlated. The combination yields

$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}})$
$\displaystyle=\left(\phantom{-}1.6\phantom{0}\pm 1.7\phantom{0}\pm
0.5\phantom{0}\right)\times 10^{-3},$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}})$
$\displaystyle=\left(-0.04\pm 0.61\pm 0.45\right)\times 10^{-3},$
$\displaystyle\mathcal{A}_{{C\\!P}}({{D}^{+}}\rightarrow\phi{{\pi}^{+}})$
$\displaystyle=\left(\phantom{-}0.03\pm 0.40\pm 0.29\right)\times 10^{-3},$

where the first uncertainties are statistical and the second systematic. No
evidence for $C\\!P$ violation in these decays is found. More precise
measurements of these asymmetries can be expected when the data already
collected by LHCb in 2018 are included in a future analysis, and when much
larger samples will become available at the upgraded LHCb detector [46].

## 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); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(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
(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); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and
Development program of LANL (USA).

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LHCb collaboration

R. Aaij28, C. Abellán Beteta46, B. Adeva43, M. Adinolfi50, C.A. Aidala77, Z.
Ajaltouni6, S. Akar61, P. Albicocco19, J. Albrecht11, F. Alessio44, M.
Alexander55, A. Alfonso Albero42, G. Alkhazov41, P. Alvarez Cartelle57, A.A.
Alves Jr43, S. Amato2, Y. Amhis8, L. An18, L. Anderlini18, G. Andreassi45, M.
Andreotti17, J.E. Andrews62, F. Archilli28, P. d’Argent13, J. Arnau Romeu7, A.
Artamonov40, M. Artuso63, K. Arzymatov37, E. Aslanides7, M. Atzeni46, B.
Audurier23, S. Bachmann13, J.J. Back52, S. Baker57, V. Balagura8,b, W.
Baldini17,44, A. Baranov37, R.J. Barlow58, G.C. Barrand8, S. Barsuk8, W.
Barter57, M. Bartolini20, F. Baryshnikov74, V. Batozskaya32, B. Batsukh63, A.
Battig11, V. Battista45, A. Bay45, F. Bedeschi25, I. Bediaga1, A. Beiter63,
L.J. Bel28, S. Belin23, N. Beliy66, V. Bellee45, N. Belloli21,i, K. Belous40,
I. Belyaev34, E. Ben-Haim9, G. Bencivenni19, S. Benson28, S. Beranek10, A.
Berezhnoy35, R. Bernet46, D. Berninghoff13, E. Bertholet9, A. Bertolin24, C.
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Bezshyiko46, S. Bhasin50, J. Bhom30, M.S. Bieker11, S. Bifani49, P. Billoir9,
A. Birnkraut11, A. Bizzeti18,u, M. Bjørn59, M.P. Blago44, T. Blake52, F.
Blanc45, S. Blusk63, D. Bobulska55, V. Bocci27, O. Boente Garcia43, T.
Boettcher60, A. Bondar39,x, N. Bondar41, S. Borghi58,44, M. Borisyak37, M.
Borsato13, M. Boubdir10, T.J.V. Bowcock56, C. Bozzi17,44, S. Braun13, M.
Brodski44, J. Brodzicka30, A. Brossa Gonzalo52, D. Brundu23,44, E. Buchanan50,
A. Buonaura46, C. Burr58, A. Bursche23, J. Buytaert44, W. Byczynski44, S.
Cadeddu23, H. Cai68, R. Calabrese17,g, R. Calladine49, M. Calvi21,i, M. Calvo
Gomez42,m, A. Camboni42,m, P. Campana19, D.H. Campora Perez44, L.
Capriotti16,e, A. Carbone16,e, G. Carboni26, R. Cardinale20, A. Cardini23, P.
Carniti21,i, K. Carvalho Akiba2, G. Casse56, M. Cattaneo44, G. Cavallero20, R.
Cenci25,p, D. Chamont8, M.G. Chapman50, M. Charles9,44, Ph. Charpentier44, G.
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S.-G. Chitic44, V. Chobanova43, M. Chrzaszcz44, A. Chubykin41, P. Ciambrone19,
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H.V. Cliff51, J. Closier44, V. Coco44, J.A.B. Coelho8, J. Cogan7, E.
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Alepuz76, Z. Ren3, V. Renaudin59, S. Ricciardi53, S. Richards50, K. Rinnert56,
P. Robbe8, A. Robert9, A.B. Rodrigues45, E. Rodrigues61, J.A. Rodriguez
Lopez70, M. Roehrken44, S. Roiser44, A. Rollings59, V. Romanovskiy40, A.
Romero Vidal43, J.D. Roth77, M. Rotondo19, M.S. Rudolph63, T. Ruf44, J. Ruiz
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R. Santacesaria27, C. Santamarina Rios43, M. Santimaria19,44, E.
Santovetti26,j, G. Sarpis58, A. Sarti19,k, C. Satriano27,s, A. Satta26, M.
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Seyfert44, M. Shapkin40, T. Shears56, L. Shekhtman39,x, V. Shevchenko73, E.
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Simi24,o, S. Simone15,d, I. Skiba17, N. Skidmore13, T. Skwarnicki63, M.W.
Slater49, J.G. Smeaton51, E. Smith10, I.T. Smith54, M. Smith57, M. Soares16,
l. Soares Lavra1, M.D. Sokoloff61, F.J.P. Soler55, B. Souza De Paula2, B.
Spaan11, E. Spadaro Norella22,q, P. Spradlin55, F. Stagni44, M. Stahl13, S.
Stahl44, P. Stefko45, S. Stefkova57, O. Steinkamp46, S. Stemmle13, O.
Stenyakin40, M. Stepanova41, H. Stevens11, A. Stocchi8, S. Stone63, S.
Stracka25, M.E. Stramaglia45, M. Straticiuc33, U. Straumann46, S. Strokov75,
J. Sun3, L. Sun68, Y. Sun62, K. Swientek31, A. Szabelski32, T. Szumlak31, M.
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Yeomans56, H. Yin69, J. Yu69,aa, X. Yuan63, O. Yushchenko40, K.A. Zarebski49,
M. Zavertyaev12,c, M. Zeng3, D. Zhang69, L. Zhang3, W.C. Zhang3,z, Y. Zhang44,
A. Zhelezov13, Y. Zheng66, X. Zhu3, V. Zhukov10,35, J.B. Zonneveld54, S.
Zucchelli16,e.

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
4Institute Of High Energy Physics (ihep), Beijing, China
5Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
6Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
7Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
8LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
9LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
10I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
11Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
12Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
13Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
14School of Physics, University College Dublin, Dublin, Ireland
15INFN Sezione di Bari, Bari, Italy
16INFN Sezione di Bologna, Bologna, Italy
17INFN Sezione di Ferrara, Ferrara, Italy
18INFN Sezione di Firenze, Firenze, Italy
19INFN Laboratori Nazionali di Frascati, Frascati, Italy
20INFN Sezione di Genova, Genova, Italy
21INFN Sezione di Milano-Bicocca, Milano, Italy
22INFN Sezione di Milano, Milano, Italy
23INFN Sezione di Cagliari, Monserrato, Italy
24INFN Sezione di Padova, Padova, Italy
25INFN Sezione di Pisa, Pisa, Italy
26INFN Sezione di Roma Tor Vergata, Roma, Italy
27INFN Sezione di Roma La Sapienza, Roma, Italy
28Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
29Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
30Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
31AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
32National Center for Nuclear Research (NCBJ), Warsaw, Poland
33Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
34Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
35Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
36Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
37Yandex School of Data Analysis, Moscow, Russia
38National Research University Higher School of Economics, Moscow, Russia
39Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
40Institute for High Energy Physics (IHEP), Protvino, Russia
41Konstantinov Nuclear Physics Institute of National Research Centre
”Kurchatov Institute”, PNPI, St.Petersburg, Russia
42ICCUB, Universitat de Barcelona, Barcelona, Spain
43Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
44European Organization for Nuclear Research (CERN), Geneva, Switzerland
45Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
46Physik-Institut, Universität Zürich, Zürich, Switzerland
47NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
48Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
49University of Birmingham, Birmingham, United Kingdom
50H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
51Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
52Department of Physics, University of Warwick, Coventry, United Kingdom
53STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
54School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
55School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
56Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
57Imperial College London, London, United Kingdom
58School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
59Department of Physics, University of Oxford, Oxford, United Kingdom
60Massachusetts Institute of Technology, Cambridge, MA, United States
61University of Cincinnati, Cincinnati, OH, United States
62University of Maryland, College Park, MD, United States
63Syracuse University, Syracuse, NY, United States
64Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
65Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
66University of Chinese Academy of Sciences, Beijing, China, associated to 3
67South China Normal University, Guangzhou, China, associated to 3
68School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
69Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
70Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 9
71Institut für Physik, Universität Rostock, Rostock, Germany, associated to 13
72Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 28
73National Research Centre Kurchatov Institute, Moscow, Russia, associated to
34
74National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 34
75National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
34
76Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 42
77University of Michigan, Ann Arbor, United States, associated to 63
78Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 63

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 Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vH.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
ySezione INFN di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abLanzhou University, Lanzhou, China
†Deceased


# 6-Body Central Configurations Formed by Tow Isosceles Triangles111This work
is partially supported by NSF of China and Youth found of Mianyang Normal
University.

Furong Zhao1,2 and Shiqing Zhang1

1Department of Mathematics, Sichuan University, Chengdu, 610064,P.R.China

2Department of Mathematics and Computer Science, Mianyang Normal University,
Mianyang, Sichuan,621000,P.R.China

Abstract: In this paper,we show the existence of a class of 6-body central
configurations with two isosceles triangles;which are congruent to each other
and keep some distance.We also study the necessary conditions about masses for
the bodies which can form a central configuration.
Keywords :6-body problems,central configurations,isosceles triangles.
MSC: 34C15,34C25.

##  1 Introduction and Main Results

The Newtonian N-body problem concerns the motion of N particles with masses
$m_{j}\in R^{+}$ and positions $q_{j}\in R^{3}$$(j=1,2,...,N)$ ,the motion is
governed by Newton’s second law and the Universal law:

$m_{j}\ddot{q}_{j}=\frac{\partial U(q)}{\partial{q}_{j}},$ (1.1)

where $q=(q_{1},q_{2},\cdots,q_{N})$ and $U(q)$ is Newtonian potential:

$U(q)=\sum_{1\leqslant j<k\leqslant N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|},$ (1.2)

Consider the space

$X=\\{q=(q_{1},q_{2},\cdots,q_{N})\in R^{3N}:\sum_{j=1}^{N}m_{j}q_{j}=0\\},$
(1.3)

i.e,suppose that the center of mass is fixed at the origin of the space.
Because the potential is singular when two particles have same position, it is
natural to assume that the configuration avoids the collision set
$\triangle=\\{q=(q_{1},\cdots,q_{N}):q_{j}=q_{k}$ for some $k\neq j\\}$.The
set $X\backslash\triangle$ is called the configuration space.
Definition 1.1([17,22]):A configuration $q=(q_{1},q_{2},\cdots,q_{N})\in
X\backslash\triangle$ is called a central configuration if there exists a
constant $\lambda$ such that

$\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leqslant k\leqslant N.$ (1.4)

The value of constant $\lambda$ in (1.4) is uniquely determined by

$\lambda=\frac{U}{I},$ (1.5)

where

$I=\sum_{k=1}^{N}m_{k}|q_{k}|^{2}.$ (1.6)

Since the general solution of the N-body problem can’t be given, great
importance has been attached to search for particular solutions from the very
beginning. A homographic solution is that a configuration is preserved for all
time. Central configurations and homographic solutions are linked by the
Laplace theorem ([17,22]). Collapse orbits and parabolic orbits have relations
with the central configurations([15,16]).So finding central configurations
becomes very important. The main general open problem for the cental
configurations is due to Winter[22]and Smale[20]:Is the number of planar
central configurations finite for any choice of positive masses
$m_{1},...,m_{N}$?Hampton and Moeckel([6]) have proved this conjecture for
four any given positive masses.
In 1941, Wintner([22]) have studied regular polygon central configurations.
Moeckel ([11]),Zhang and Zhou([23]) have studied highly symmetrical central
configuration of Newtonian N-body problems.Llibre and Mello ([8]) have studied
a class of 6-body central configurations.
Based the above works,we find a classes of central configurations in the
6-body problems, for which three bodies are at the vertices of an isosceles
triangles , the others are located at the vertices of another isosceles
triangles and the two triangles are congruent to each other;
Related assumptions will be interpreted more precisely in the following.
Assume $m_{1},m_{2}$ and $m_{5}$ are located the vertices of a isosceles
triangles $\Delta_{1}$;$m_{3},m_{4}$ and $m_{6}$ are located at the vertices
of another isosceles triangles $\Delta_{2}$. $\Delta_{1}$ and $\Delta_{2}$ are
coplanar and are congruent to each other;$q_{1}-q_{2}$ is parallel to
$q_{4}-q_{3}$;$|q_{1}-q_{4}|<|q_{5}-q_{6}|$;$q_{5}$ and $q_{6}$ are located at
the common perpendicular bisector for $q_{1}q_{2}$ and $q_{3}q_{4}$.Without
loss of generality we can take a coordinate system such that
$q_{1}=(-1,y)$,$q_{2}=(-1,-y)$,$q_{3}=(1,-y)$,
$q_{4}=(1,y)$,$q_{5}=(-1-x,0)$,$q_{6}=(1+x,0)$.(See Fig).

$m_{1}$$m_{2}$$m_{5}$$m_{6}$$m_{3}$$m_{4}$$x$$y$2

We have:
Theorem1.1:If $m_{1},m_{2},m_{3},m_{4},m_{5}$ and $m_{6}$ form a central
configuration,then $m_{1}=m_{2}=m_{3}=m_{4}$ and $m_{5}=m_{6}$.
Theorem1.2:Assume $m_{1}=m_{2}=m_{3}=m_{4}=1$,$m_{5}=m_{6}=m$, then there
exists exists a non-empty open set $U\subset(1,+\infty),$ $\varphi(y)\in C(U)$
such that $\varphi(\sqrt{3})=1$ and $m=m(x,y)$=$m(\varphi(y),y)$, so that
$(q_{1},q_{2},q_{3},q_{4},q_{5},q_{6})$ form a central configuration.
Remark:When $x=1$ and $y=\sqrt{3}$, $q_{i}$ is the vertex of a regular 6-gons
$(i=1,\cdots,6)$.

##  2 The Proofs of Theorems

###  2.1 The Proof of Theorem 1.1

Note that

$\begin{split}\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k}=-\lambda m_{k}(q_{k}-0)\\\ =-\lambda
m_{k}(q_{k}-\frac{\sum_{j=1}^{N}m_{j}q_{j}}{M})=-m_{k}\frac{\lambda}{M}\sum_{j=1}^{N}m_{j}(q_{k}-q_{j})\end{split}$
(2.1)

where $M=\sum_{i=1}^{N}m_{i}$.
So (1.4) is also equivalent to

$\sum_{j=1,j\neq
k}^{N}m_{j}(\frac{1}{|q_{j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{k})=0$
(2.2)

By (2.2) we have

$\begin{split}0m_{1}+0m_{2}+2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{3}+2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{4}\\\
-x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}+(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.3)
$\begin{split}0m_{1}-2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{2}-2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{3}+0m_{4}\\\
-y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}-y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.4)
$\begin{split}0m_{1}+0m_{2}+2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{3}+2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{4}\\\
-x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}+(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.5)
$\begin{split}2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{1}+0m_{2}+0m_{3}+2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{4}\\\
+y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{5}+y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.6)
$\begin{split}-2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{1}-2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{2}+0m_{3}+0m_{4}\\\
-(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}+x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.7)
$\begin{split}2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{1}+0m_{2}+0m_{3}+2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{4}\\\
+y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}+y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.8)
$\begin{split}-2(\frac{1}{2^{3}}-\frac{\lambda}{M})m_{1}-2(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{2}+0m_{3}+0m_{4}\\\
-(2+x)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}+x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.9)
$\begin{split}0m_{1}-2y(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{\lambda}{M})m_{2}-2y(\frac{1}{|2y|^{3}}-\frac{\lambda}{M})m_{3}+0m_{4}\\\
-y(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})m_{5}-y(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.10)
$\begin{split}x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})(m_{1}+m_{2})+\\\
(x+2)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})(m_{3}+m_{4})\\\
+0m_{5}+2(1+x)(\frac{1}{|2(1+x)|^{3}}-\frac{\lambda}{M})m_{6}=0,\end{split}$
(2.11)
$\begin{split}(x+2)(\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{\lambda}{M})(m_{1}+m_{2})\\\
+x(\frac{1}{|x^{2}+y^{2}|^{3/2}}-\frac{\lambda}{M})(m_{3}+m_{4})+\\\
+2(1+x)(\frac{1}{|2(1+x)|^{3}}-\frac{\lambda}{M})m_{5}+0m_{6}=0,\end{split}$
(2.12)

By(2.3),(2.5),(2.7) and (2.9),we have:

$\begin{split}(m_{3}-m_{4})(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})=0,\\\
(m_{1}-m_{2})(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})=0.\end{split}$
(2.13)

By(2.4),(2.6),(2.8) and (2.10),we have:

$\begin{split}(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})(m_{1}-m_{3})+(\frac{1}{|2y|^{3}}-\frac{1}{2^{3}})(m_{4}-m_{2})=0,\\\
(\frac{1}{|2y|^{3}}-\frac{1}{2^{3}})(m_{1}-m_{3})+(\frac{1}{|4+4y^{2}|^{3/2}}-\frac{1}{2^{3}})(m_{4}-m_{2})=0.\end{split}$
(2.14)

By(2.13) and (2.14),we have:

$m_{1}=m_{2}=m_{3}=m_{4}.$ (2.15)

By (2.4),(2.6),(2.11), (2.12) and (2.15),we have

$m_{5}=m_{6}.$ (2.16)

The proof of Theorem1.1 is completed.

###  2.2 The Proof of Theorem 1.2

Notice that $(q_{1},\cdots,q_{6})$ is a central configuration if and only if

$\sum_{j=1,j\neq
k}^{6}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leqslant k\leqslant 6.$ (2.17)

Since the symmetries,(2.17) is equivalent to

$\sum_{j=1,j\neq
k}^{6}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},k=2,5.$ (2.18)

Now (2.18) is equivalent to

$\lambda=\frac{1}{4}+\frac{1}{4|1+y^{2}|^{3/2}}-\frac{xm}{|x^{2}+y^{2}|^{3/2}}+\frac{(2+x)m}{|x^{2}+y^{2}+4x+4|^{3/2}},$
(2.19)
$\lambda=\frac{1}{4y^{3}}+\frac{1}{4|1+y^{2}|^{3/2}}+\frac{m}{|x^{2}+y^{2}|^{3/2}}+\frac{m}{|x^{2}+y^{2}+4x+4|^{3/2}},$
(2.20)
$\lambda=\frac{2x}{|x^{2}+y^{2}|^{3/2}(1+x)}+\frac{2(2+x)}{|x^{2}+y^{2}+4x+4|^{3/2}(1+x)}+\frac{m}{4|1+x|^{3}},$
(2.21)

(2.19) ,(2.20) and (2.21) are equivalent to

$(\frac{1+x}{|x^{2}+y^{2}|^{3/2}}-\frac{1+x}{|x^{2}+y^{2}+4x+4|^{3/2}})m=\frac{1}{4}(1-\frac{1}{y^{3}}),$
(2.22)
$\begin{split}(\frac{1}{|x^{2}+y^{2}|^{3/2}}+\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{1}{4|1+x|^{3}})m=\\\
\frac{2x}{|x^{2}+y^{2}|^{3/2}(1+x)}+\frac{2(2+x)}{|x^{2}+y^{2}+4x+4|^{3/2}(1+x)}-\frac{1}{4y^{3}}-\frac{1}{4|1+y^{2}|^{3/2}},\end{split}$
(2.23)

By (2.22) we have

$m=m_{1}(x,y)=\frac{1}{4}(1-\frac{1}{y^{3}})\frac{|x^{2}+y^{2}|^{3/2}|x^{2}+y^{2}+4x+4|^{3/2}}{(1+x)(|x^{2}+y^{2}+4x+4|^{3/2}-|x^{2}+y^{2}|^{3/2})}$
(2.24)

$m_{1}(x,y)>0$ if and only if $y>1$.
By (2.23) we have

$\begin{split}m=m_{2}(x,y)=[\frac{2x}{|x^{2}+y^{2}|^{3/2}(1+x)}+\frac{2(2+x)}{|x^{2}+y^{2}+4x+4|^{3/2}(1+x)}\\\
-\frac{1}{4y^{3}}-\frac{1}{4|1+y^{2}|^{3/2}}]\times[\frac{1}{|x^{2}+y^{2}|^{3/2}}+\frac{1}{|x^{2}+y^{2}+4x+4|^{3/2}}-\frac{1}{4|1+x|^{3}}]^{-1}\end{split}$
(2.25)

Then $(q_{1},\cdots,q_{6})$ is a central configuration if and only if

$m_{1}(x,y)=m_{2}(x,y)>0$ (2.26)

It is obvious that

$m_{1}(1,\sqrt{3})=m_{2}(1,\sqrt{3})=1.$ (2.27) $\frac{\partial
m_{1}(1,\sqrt{3})}{\partial x}=\frac{1}{4}.$ (2.28) $\frac{\partial
m_{2}(1,\sqrt{3})}{\partial
x}=\frac{1}{2}\frac{(9-16\sqrt{3})}{(27+4\sqrt{3})}\neq\frac{1}{4}.$ (2.29)

By implicit function theorem,there exists exists a non-empty open set $U$ and
$\varphi(y)\in C(U)$ such that,$\sqrt{3}\in U$ , $\varphi(\sqrt{3})=1$ and
$\forall y\in U$, $m_{1}(\varphi(y),y)=m_{2}(\varphi(y),y)$.
The proof of Theorem1.2 is completed.

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