EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-322 LHCb-PAPER-2015-052 26 January 2016

Study of $D^{(*)+}_{sJ}$ mesons decaying to
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ final states

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

A search is performed for $D^{(*)+}_{sJ}$ mesons in the reactions
$pp\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}X$ and
$pp\rightarrow{{D}^{*0}}{{K}^{+}}X$ using data collected at centre-of-mass
energies of 7 and $8\,\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector.
For the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final state, the
decays ${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ are used.
For ${{D}^{*0}}{{K}^{+}}$, the decay
${{D}^{*0}}\\!\rightarrow{{D}^{0}}{{\pi}^{0}}$ with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ is used. A prominent
$D_{s1}(2536)^{+}$ signal is observed in both
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ final states. The resonances $D^{*}_{s1}(2700)^{+}$ and
$D_{s3}^{*}(2860)^{+}$ are also observed, yielding information on their
properties, including spin-parity assignments. The decay
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ is observed for the first time, at a significance of $6.9\,\sigma$, and
its branching fraction relative to the
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ decay mode is measured.

Submitted to JHEP

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

##  1 Introduction

The discovery by the BaBar collaboration of a narrow state
$D_{s0}^{*}(2317)^{+}$ in the decay to ${{D}^{+}_{s}}{{\pi}^{0}}$ [1], and the
subsequent discovery of a second narrow particle, $D_{s1}(2460)^{+}$ in the
decay to $D_{s}^{*+}{{\pi}^{0}}$ [2, 3, 4], raised considerable interest in
the spectroscopy of heavy mesons.111The inclusion of charge-conjugate
processes is implied, unless stated otherwise. These discoveries were a
surprise because quark model calculations based on heavy quark effective
theory (HQET) [5] predicted the masses of these resonances to be above the
$DK$ and $D^{*}K$ thresholds, respectively.222In the following $D^{*}$ is a
generic label to indicate the ground state $D^{*}(2010)^{+}$ or
$D^{*}(2007)^{0}$ resonances. Consequently their widths were expected to be
very large, as for the corresponding $J^{P}=0^{+}$ and $J^{P}=1^{+}$
resonances in the $D_{J}$ spectrum.

The ${D}_{sJ}^{+}$ mesons are expected to decay into the $DK$ and $D^{*}K$
final states if they are above threshold. The BaBar collaboration has explored
the $DK$ and $D^{*}K$ mass spectra [6, 7] observing two states,
$D^{*}_{sJ}(2700)^{+}$ and $D_{sJ}^{*}(2860)^{+}$, both decaying to $DK$ and
$D^{*}K$ with a natural parity (NP) assignment.333States having $P=(-1)^{J}$
and therefore $J^{P}=0^{+},1^{-},2^{+},...$ are referred as natural parity
states and are labelled as $D^{*}$, while unnatural parity indicates the
series $J^{P}=0^{-},1^{+},2^{-},...$. A third structure, $D_{sJ}(3040)^{+}$,
is observed only in the $D^{*}K$ decay mode with a preferred unnatural parity
(UP) assignment. The $D^{*}_{sJ}(2700)^{+}$ resonance was also observed by the
Belle and BaBar collaborations in a study of $B$ decays to ${D}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ [8, 9]. Both collaborations obtain
a spin-parity assignment $J^{P}=1^{-}$ for this state, and so it is labelled
as $D^{*}_{s1}(2700)^{+}$.

The LHCb experiment has performed studies of the $DK$ final states in the
inclusive process, $pp\rightarrow DKX$ [10], and in the Dalitz plot analysis
of $B_{s}^{0}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-}\pi^{+}$ decays [11, 12].
In the inclusive analysis, the $D^{*}_{s1}(2700)^{+}$ and
$D_{sJ}^{*}(2860)^{+}$ are observed with large statistical significance and
their properties are found to be in agreement with previous measurements. In
the exclusive Dalitz plot analysis of the $B_{s}^{0}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-}\pi^{+}$ decays, the
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-}$ mass spectrum
shows a complex resonant structure in the
$2860\,\mathrm{\,Me\kern-1.00006ptV}$ mass region.444Natural units are used
throughout the paper. This is described by a superposition of a broad
$J^{P}=1^{-}$ resonance and a narrow $J^{P}=3^{-}$ resonance with no evidence
for the production of $D_{s1}^{*}(2700)^{-}$. Since the narrow structure at
$2860\,\mathrm{\,Me\kern-1.00006ptV}$ seen in inclusive $DK$ and $D^{*}K$
analyses could contain contributions from various resonances with different
spins, it is labelled as $D_{sJ}^{*}(2860)^{+}$.

In references [13, 14, 15, 16, 17, 18] attempts are made to identify these
states within the quark model and in Ref. [19] within molecular models. The
expected spectrum for $D_{s}^{+}$ mesons has recently been recomputed in Refs.
[20, 21]. In particular, Ref. [20] points out that six states are expected in
the mass region between 2.7 and $3.0\,\mathrm{\,Ge\kern-1.00006ptV}$. To date,
evidence has been found for three of the states; hence finding the rest would
provide an important test of these models. In this paper we study the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ systems using $pp$ collision data, collected at centre-
of-mass energies of 7 and $8\,\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb
detector.

##  2 Detector and simulation

The LHCb detector [22, 23] 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, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at $200\,\mathrm{\,Ge\kern-1.00006ptV}$. The minimum distance
of a track to a primary vertex (PV), the impact parameter, 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
$\mathrm{\,Ge\kern-1.00006ptV}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
(RICH). 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, 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.

In the simulation, $pp$ collisions are generated using Pythia [24] with a
specific LHCb configuration [25]. Decays of hadronic particles are described
by EvtGen [26], in which final-state radiation is generated using Photos [27].
The interaction of the generated particles with the detector, and its
response, are implemented using the Geant4 toolkit [28] as described in Ref.
[29]. We also make use of simple generator-level simulations [30] to study
kinematic effects.

##  3 Event selection

We search for $D^{(*)+}_{sJ}$ mesons using the inclusive reactions

$pp\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}X$ (1)

and

$pp\rightarrow{{D}^{*0}}{{K}^{+}}X,$ (2)

where $X$ represents a system composed of any collection of charged and
neutral particles. Use is made of both 7 and
$8\,\mathrm{\,Te\kern-1.00006ptV}$ data for reaction (1), corresponding to an
integrated luminosity of $3\,\mbox{\,fb}^{-1}$, and
$8\,\mathrm{\,Te\kern-1.00006ptV}$ data only for reaction (2) which
corresponds to an integrated luminosity of $2\,\mbox{\,fb}^{-1}$.

The charmed mesons in the final state are reconstructed in the decay modes
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$, with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, and
${{D}^{*0}}\rightarrow{{D}^{0}}{{\pi}^{0}}$, with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and
${{\pi}^{0}}\rightarrow\gamma\gamma$. The ${K}^{0}_{\mathrm{\scriptscriptstyle
S}}$ mesons are reconstructed in their ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\rightarrow\pi^{+}\pi^{-}$ decay mode. Because of their long lifetime,
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ mesons may decay inside or outside
the vertex detector. Candidate ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
mesons that are reconstructed using vertex detector information are referred
to as “long” while those reconstructed without vertex detector information are
called “downstream”. Those that decay within the vertex detector acceptance
have a mass resolution about half as large as those that decay outside of its
acceptance. Reaction (1) with ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
serves as the primary channel for studying the $D^{(*)+}_{sJ}$ resonance
structures and their parameters, while reaction (1) with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
reaction (2) are used for cross-checks and to confirm the observed signatures.

Charged tracks are required to have good track fit quality, momentum
$p>3\,\mathrm{\,Ge\kern-1.00006ptV}$ and
$\mbox{$p_{\mathrm{T}}$}>250\,\mathrm{\,Me\kern-1.00006ptV}$. These conditions
are relaxed to $p>1\,\mathrm{\,Ge\kern-1.00006ptV}$ and
$\mbox{$p_{\mathrm{T}}$}>150\,\mathrm{\,Me\kern-1.00006ptV}$ for the “soft”
pion originating directly from the ${D}^{*+}$ decay. In the reconstruction of
the ${D}^{0}$ candidates we remove candidate tracks pointing to a PV, using an
impact parameter requirement. All tracks used to reconstruct the $D$ mesons
are required to be consistent with forming a common vertex and the $D$ meson
candidate must be consistent with being produced at a PV. The ${{D}^{*+}}$ and
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$, and similarly the $D^{0}$ and
$K^{+}$ candidates, are fitted to a common vertex, for which a good quality
fit is required. The purity of the charmed meson sample is enhanced by
requiring the decay products to be identified by the particle identification
system, using the difference in the log-likelihood between the kaon and pion
hypotheses $\Delta\ln\mathcal{L}_{K\pi}$ [31]. We impose a tight requirement
of $\Delta\ln\mathcal{L}_{K\pi}>3$ for kaon tracks and a loose requirement of
$\Delta\ln\mathcal{L}_{K\pi}<10$ for pions. The overlap region in the particle
identification definition of a kaon and a pion is small and does not affect
the measured yields, given the small number of multiple candidates per event.

Candidate ${D}^{0}$ mesons are required to be within $\pm 2.5\,\sigma$ of the
fitted ${D}^{0}$ mass where the mass resolution $\sigma$ is
$8.3\,\mathrm{\,Me\kern-1.00006ptV}$. The ${{D}^{0}}{{\pi}^{+}}$ invariant
mass is computed as

$m({{D}^{0}}{{\pi}^{+}})=m({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})-m({{K}^{-}}{{\pi}^{+}})+m_{{{D}^{0}}},$
(3)

where $m_{{{D}^{0}}}$ is the world average value of the ${D}^{0}$ mass [32].
For the channel
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, the
invariant mass $m({{D}^{0}}{{\pi}^{+}})$ is defined similarly.

Figure 1 shows the ${{D}^{0}}{{\pi}^{+}}$ invariant mass spectrum for (a)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and (b)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$. Clean
${{D}^{*+}}$ signals for both ${D}^{0}$ decay modes are observed. We fit the
mass spectra using the sum of a Gaussian function for the signal and a second-
order polynomial for the background. The signal regions are defined to be
within $\pm 2.5\,\sigma$ of the peak values, where
$\sigma=0.7\,\mathrm{\,Me\kern-1.00006ptV}$ for both channels.

The ${{\pi}^{+}}{{\pi}^{-}}$ mass spectra for the two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ types, the long
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ and downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, are shown in Figs. 1(c) and 1(d)
and are fitted using the same model as for the ${{D}^{0}}{{\pi}^{+}}$
invariant masses. The signal regions are similarly defined within $\pm
2.5\,\sigma$ of the peak, with $\sigma=4.1\,\mathrm{\,Me\kern-1.00006ptV}$ and
$8.7\,\mathrm{\,Me\kern-1.00006ptV}$ for long and downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, respectively.

Figure 1:  Distributions of ${{D}^{0}}{{\pi}^{+}}$ invariant mass for (a)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and (b)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$.
${{\pi}^{+}}{{\pi}^{-}}$ mass spectrum for (c) long and (d) downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$. The full (red) lines describe the
fitting function. The dashed lines show the background contributions and the
vertical dotted lines indicate the signal regions.

The ${\pi}^{0}$ candidates are obtained by kinematically fitting to a
$\pi^{0}$ hypothesis each pair of photon candidates with energy greater than
$600\,\mathrm{\,Me\kern-1.00006ptV}$, with the diphoton mass constrained to
the nominal $\pi^{0}$ mass [32]. Candidate ${D}^{*0}$ mesons are formed by
combining ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays with all
${\pi}^{0}$ candidates in the event that have
$\mbox{$p_{\mathrm{T}}$}>450\,\mathrm{\,Me\kern-1.00006ptV}$. The resulting
${D}^{*0}$ candidate is required to have
$\mbox{$p_{\mathrm{T}}$}>6000\,\mathrm{\,Me\kern-1.00006ptV}$. Figure 2 shows
the $\Delta
m({{D}^{0}}{{\pi}^{0}})=m({{K}^{-}}{{\pi}^{+}}{{\pi}^{0}})-m({{K}^{-}}{{\pi}^{+}})$
distribution, where a clear ${D}^{*0}$ signal can be seen. The mass spectrum
is fitted using for background the threshold function

$B(m)=P(m)(m-m_{th})^{\alpha}e^{-\beta m-\gamma m^{2}},$ (4)

where in this case $m=\Delta m({{D}^{0}}{{\pi}^{0}})$, $m_{th}$ is the $\Delta
m({{D}^{0}}{{\pi}^{0}})$ threshold mass and $\alpha$, $\beta$ and $\gamma$ are
free parameters. In Eq. (4) $P(m)$ is the center of mass momentum of the two-
body decay of a particle of mass $m$ into two particles with masses $m_{1}$
and $m_{2}$,

$P(m)=\frac{1}{2m}\sqrt{[m^{2}-(m_{1}+m_{2})^{2}][m^{2}-(m_{1}-m_{2})^{2}]}.$
(5)

The function $B(m)$ gives the correct behaviour of the fit at threshold. The
${D}^{*0}$ signal is modelled using the sum of two Gaussian functions. We
select the candidates in the $\pm 2\,\sigma$ window around the peak, where
$\sigma=1.72\,\mathrm{\,Me\kern-1.00006ptV}$ is the width of the dominant
Gaussian fitting function, and we form $D^{*}K$ pairings by combining
${D}^{*+}$ and ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates for
reaction (1), and ${D}^{*0}$ and ${K}^{+}$ candidates for reaction (2).

Figure 2:  Distribution of $\Delta m({{D}^{0}}{{\pi}^{0}})$ invariant mass.
The full (red) line describes the fitting function. The dashed line shows the
background contribution and the dotted vertical lines define the $D^{*0}$
signal region.

To suppress the large combinatorial background, a set of additional criteria
is applied. We define $\theta$ as the angle between the momentum direction of
the kaon in the $D^{*}K$ rest frame and the momentum direction of the $D^{*}K$
system in the laboratory frame. Whereas the signal events are expected to be
symmetrically distributed in the variable $\cos\theta$, after correcting for
efficiency, more than 90% of the combinatorial background is found in the
negative $\cos\theta$ region. The $\cos\theta$ requirements are optimized
using the $D^{*}_{s1}(2700)^{+}$ signal, an established resonance. We fit the
$D^{*}K$ mass spectra (using the model described below) with different
$\cos\theta$ selections and obtain the yields for $D^{*}_{s1}(2700)^{+}$
signal ($N_{\rm S}$) and background events ($N_{\rm B}$) in the
$D^{*}_{s1}(2700)^{+}$ signal region (defined in the window
$|m(D^{*}K)-m(D^{*}_{s1}(2700)^{+})|<\Gamma(D^{*}_{s1}(2700)^{+})/2$). We
compute the signal significance $S=N_{\rm S}/\sqrt{N_{\rm S}+N_{\rm B}}$ and
signal purity $P=N_{\rm S}/(N_{\rm S}+N_{\rm B})$ and find that the
requirements $\cos\theta>0$ (for
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$), $\cos\theta>-0.15$ (for
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$) and
$\cos\theta>-0.1$ (for ${{D}^{*0}}{{K}^{+}}$) each provide a good compromise
between significance and purity in the respective channel. With the same
method it is also found that it is optimal to require
$\mbox{$p_{\mathrm{T}}$}>4000\,\mathrm{\,Me\kern-1.00006ptV}$ for all three
final states. Simulations show that the mass resolution is much smaller than
the natural widths of the resonances.

The analysis of the $D^{*}K$ system, with $D^{*}\rightarrow D\pi$, is a three-
body decay and therefore allows a spin analysis of the produced resonances and
a separation of the different spin-parity components. We define the helicity
angle $\theta_{\rm H}$ as the angle between the
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and the ${{\pi}^{+}}$ from the
${{D}^{*+}}$ decay, in the rest frame of the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ system. Simulated events
are used to determine the efficiency as a function of $\cos\theta_{\rm H}$,
which is found to be uniform only for the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ candidates formed from
the downstream ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ sample. Therefore,
for studying the angular distributions we do not use the long
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ sample, which removes approximately
30% of the data.

##  4 Mass spectra

In order to improve the mass resolution on the $D^{*}K$ mass spectra, we
compute the ${D}^{*+}$, ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ and
${D}^{*0}$ energies using the world average mass measurements [32]. The
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ is shown in Fig. 3 and contains
5.72$\times 10^{5}$ combinations. We observe a strong $D_{s1}(2536)^{+}$
signal and weaker resonant contributions due to $D^{*}_{s2}(2573)^{+}$,
$D^{*}_{s1}(2700)^{+}$, and $D_{sJ}^{*}(2860)^{+}$ states. The
$D^{*}_{s2}(2573)^{+}$ decay to
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ is observed for the
first time. A binned $\chi^{2}$ fit to the mass spectrum is performed in which
the narrow $D_{s1}(2536)^{+}$ is described by a Gaussian function with free
parameters. Other resonances are described by relativistic Breit-Wigner (BW)
functions (in $D$-, $P$\- and $F$-wave for $D^{*}_{s2}(2573)^{+}$,
$D^{*}_{s1}(2700)^{+}$, and $D_{s3}^{*}(2860)^{+}$ respectively).

Using the definition of the center-of-mass momentum $P(m)$ given in Eq. (5),
we parameterize the BW function for a resonance of mass $M$ as

$BW(m)={{P(m)\left({P(m)\over P(M)}\right)^{2L}{D^{2}(P(M))\over
D^{2}(P(m))}}\over{(m^{2}-M^{2})^{2}+M^{2}\Gamma^{2}(m)}},$ (6)

where

$\Gamma(m)=\Gamma{M\over m}\left({P(m)\over
P(M)}\right)^{2L+1}{D^{2}(P(M))\over D^{2}(P(m))},$ (7)

and

$D(P)=\left\\{\begin{array}[]{l}\sqrt{1+(PR)^{2}}\ {\rm for}\ L=1,\\\
\sqrt{9+3(PR)^{2}+(PR)^{4}}\ {\rm for}\ L=2,\\\
\sqrt{225+45(PR)^{2}+6(PR)^{4}+(PR)^{6}}\ {\rm for}\ L=3,\\\
\end{array}\right.$ (8)

are the Blatt-Weisskopf form factors [33]. No dependence of the resonance
parameters on the Blatt-Weisskopf radius $R$ is found and it is therefore
fixed to $2.5\,\mathrm{\,Ge\kern-1.00006ptV}^{-1}$. The quantity $L$ is the
angular momentum between the two decay fragments: $L=1$ for $P$-wave, $L=2$
for $D$-wave and $L=3$ for $F$-wave resonances. The $D_{sJ}(3040)^{+}$
resonance is described by a nonrelativistic BW function multiplied by $P(m)$.
The $D^{*}_{s2}(2573)^{+}$ parameters are fixed to the values obtained in the
fit to the $DK$ mass spectra [10].

Figure 3: Distribution of the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ invariant mass for ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay.
The full (red) line describes the fitting function. The dashed line displays
the fitted background and the dotted lines the $D_{s1}(2536)^{+}$,
$D^{*}_{s2}(2573)^{+}$, $D^{*}_{s1}(2700)^{+}$, $D_{sJ}^{*}(2860)^{+}$ and
$D_{sJ}(3040)^{+}$ contributions. The inset displays the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum after
subtracting the fitted background.

The background is described by an empirical model [34],

$B(m)=\left\\{\begin{array}[]{l}P(m)e^{a_{1}m+a_{2}m^{2}}\ {\rm for}\
m<m_{0},\\\ P(m)e^{b_{0}+b_{1}m+b_{2}m^{2}}\ {\rm for}\
m>m_{0},\end{array}\right.$ (9)

where $P(m)$ is described in Eq. (5) and $m_{0}$, $a_{i=1,2}$ and
$b_{i=0,1,2}$ are free parameters. In Eq.(9) we impose continuity to $B(m)$
and to its first derivative at the mass $m_{0}$ and therefore the number of
free parameters is reduced by two. Resonances are included sequentially in
order to test the $\chi^{2}$ improvement when a new contribution is added. A
better fit is obtained if a broad resonance in the
$3000\,\mathrm{\,Me\kern-1.00006ptV}$ mass region is included. We find strong
correlation between the parameters of this structure and the background and
therefore we add the $D_{sJ}(3040)^{+}$ resonance in the fit with parameters
fixed to the values obtained by BaBar [7].555$m(D_{sJ}(3040)^{+})=3044\pm
8\mathrm{\,(stat)}^{+30}_{-5}\mathrm{\,(syst)}\,\mathrm{\,Me\kern-1.00006ptV}$,
$\Gamma(D_{sJ}(3040)^{+})=239\pm
35\mathrm{\,(stat)}^{+46}_{-42}\mathrm{\,(syst)}\,\mathrm{\,Me\kern-1.00006ptV}$.

We also study the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ in the
${D}^{*+}$ sideband region, defined as
$2014.0<m({{D}^{0}}{{\pi}^{+}})<2018.1\,\mathrm{\,Me\kern-1.00006ptV}$. A
smooth mass spectrum is obtained, well fitted by the above background model
with no evidence for additional structures.

Table 1(a) gives the resulting $D^{*}_{s1}(2700)^{+}$ and
$D_{sJ}^{*}(2860)^{+}$ fitted parameters. Statistical significances are
computed as $S=\sqrt{\Delta\chi^{2}}$, where $\Delta\chi^{2}$ is the
difference in $\chi^{2}$ between fits with the resonance included and excluded
from the fitting model. Large significances for $D^{*}_{s1}(2700)^{+}$ and
$D_{sJ}^{*}(2860)^{+}$ are obtained, especially for the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay mode. The significance of
the $D_{sJ}(3040)^{+}$ enhancement is $2.4\,\sigma$.

A search is performed for the $D_{s1}^{*}(2860)^{+}$ resonance previously
observed in the $B_{s}^{0}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$ Dalitz
plot analysis [11, 12]. We first introduce in the fit an incoherent BW
function with parameters free to vary within their statistical uncertainties
around the reported values in Ref. [11], but the fit returns a negligible
contribution for this state. Since two $J^{P}=1^{-}$ overlapping resonances
may be present in the mass spectrum, interference is allowed between the
$D_{s1}^{*}(2860)^{+}$ and the $D^{*}_{s1}(2700)^{+}$ resonance by including
the amplitude

$A_{1^{-}}=|{\rm BW}_{D^{*}_{s1}(2700)^{+}}+ce^{i\phi}{\rm
BW}_{D_{s1}^{*}(2860)^{+}}|^{2}$ (10)

where $c$ and $\phi$ are free parameters. In this fit we also add the
$D_{s3}^{*}(2860)^{+}$ resonance with parameters fixed to those from Refs.
[11, 12] and the $D^{*}_{s1}(2700)^{+}$ with parameters fixed to those from
the $DK$ analysis [10]. The resulting fit quality is similar to that obtained
without the presence of the $D_{s1}^{*}(2860)^{+}$ resonance ($\chi^{2}/{\rm
ndf}=92/103$). However it is found that the $D_{s1}^{*}(2860)^{+}$ is
accommodated by the fit with strong destructive interference. We conclude that
the data are not sensitive to the $D_{s1}^{*}(2860)^{+}$ resonance.

Table 1: Results from the fits to the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ mass spectra. Resonances parameters are expressed in
$\mathrm{\,Me\kern-1.00006ptV}$. When two uncertainties are presented, the
first is statistical and the second systematic. The symbol ndf indicates the
number of degrees of freedom.

Data |  | $D^{*}_{s1}(2700)^{+}$ | $D_{sJ}^{*}(2860)^{+}$ | $\chi^{2}/{\rm ndf}$
---|---|---|---|---
(a) ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | Mass | $2732.3\pm 4.3\pm 5.8$ | $2867.1\pm 4.3\pm 1.9$ |
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ | Width | $136\pm 19\pm 24$ | $50\pm 11\pm 13$ |
| Yield | $(1.57\pm 0.28)\times 10^{4}$ | $(3.1\pm 0.8)\times 10^{3}$ | 94/103
| Significance | 8.3 | 6.3 |
(b) ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | Mass | $2729.3\pm 3.3$ | $2861.2\pm 4.3$ |
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ | Width | 136 (fixed) | $57\pm 14$ |
NP sample | Yield | $(1.50\pm 0.11)\times 10^{4}$ | $(2.50\pm 0.60)\times 10^{3}$ | 90/104
| Significance | 7.6 | 7.1 |
(c) ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | Mass | 2732.3 (fixed) | $2876.7\pm 6.4$ |
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ | Width | 136 (fixed) | $50\pm 19$ |
UP sample | Yield | $(0\pm 0.8)\times 10^{3}$ | $(1.0\pm 0.4)\times 10^{3}$ | 100/105
| Significance | 0.0 | 3.6 |
(d) ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | Mass | $2725.5\pm 6.0$ | $2844.0\pm 6.5$ |
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ | Width | 136 (fixed) | $50\pm 15$ |
| Yield | $(2.6\pm 0.4)\times 10^{3}$ | $490\pm 180$ | 89/97
| Significance | 4.7 | 3.8 |
(e) ${{D}^{*0}}{{K}^{+}}$ | Mass | $2728.3\pm 6.5$ | $2860.9\pm 6.0$ |
| Width | 136 (fixed) |  $50$ (fixed) |
| Yield | $(1.89\pm 0.30)\times 10^{3}$ | $290\pm 90$ | 79/99
| Significance | 6.6 | 3.1 |

Systematic uncertainties on the resonance parameters are computed as quadratic
sums of the differences between the nominal fit and fits in which the
following changes are made.

  * •

The alternative background function Eq. (4) is used.

  * •

The fit bias is evaluated by generating and fitting pseudoexperiments obtained
using the parameters from the best fit. The deviations of the mean values of
the distributions from the generated ones are taken as systematic
uncertainties.

  * •

The parameters of the $D_{sJ}(3040)^{+}$ state, fixed to the values of Ref.
[7] in all the fits, have been varied according to their total uncertainties.

  * •

From the study of high-statistics control samples, a systematic uncertainty of
$0.0015\,Q$ on the mass scale is added, where $Q$ is the $Q$-value involved in
the resonance decay.

  * •

The fitting model that includes the $D_{s1}^{*}(2860)^{+}$ resonance is tested
with $D_{s3}^{*}(2860)^{+}$ parameters fixed and the $D^{*}_{s1}(2700)^{+}$
parameters left free.

The different contributions to the systematic uncertainties are summed in
quadrature and are summarized in Table 2. It can be noted that, combining
statistical and systematic uncertainties, the resulting $D^{*}_{s1}(2700)^{+}$
mass is about $3\,\sigma$ higher than previous measurements while the
$D_{sJ}^{*}(2860)^{+}$ parameters are consistent with those of the
$D_{s3}^{*}(2860)^{+}$ resonance [32].

Table 2: Contributions (in $\mathrm{\,Me\kern-1.00006ptV}$) to the systematic
uncertainties on the $D^{*}_{s1}(2700)^{+}$ and $D_{sJ}^{*}(2860)^{+}$
resonances parameters.

Source | m($D^{*}_{s1}(2700)^{+}$) |  $\Gamma(D^{*}_{s1}(2700)^{+}$) | m($D_{sJ}^{*}(2860)^{+}$) |  $\Gamma(D_{sJ}^{*}(2860)^{+}$)
---|---|---|---|---
Background function | 5.0 | 19.4 | 1.7 | 12.7
Fit bias | 0.2 | 1.6 | 0.2 | 1.5
$D_{sJ}(3040)^{+}$ parameters | 1.3 | 5.7 | 0.5 | 3.2
Mass scale | 0.3 |  | 0.5 |
Fit model | 2.6 | 12.0 |  |
Total | 5.8 | 23.6 | 1.9 | 13.2

The angular distributions are expected to be proportional to
$\sin^{2}\mbox{$\theta_{\rm H}$}$ for NP resonances and proportional to
$1+h\cos^{2}\mbox{$\theta_{\rm H}$}$ for UP resonances, where $h$ is a free
parameter. The ${{D}^{*}}K$ decay is forbidden for a $J^{P}=0^{+}$ resonance.
Therefore the selection of candidates in different ranges of $\cos\theta_{\rm
H}$ can enhance or suppress different spin-parity contributions. We separate
the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ data into two
different categories, the NP sample, obtained with the selection
$|\cos\theta_{\rm H}|<0.5$ and the UP sample, with the selection
$|\cos\theta_{\rm H}|>0.5$.

The ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectra for the
NP sample is shown in Fig. 4(a), while the corresponding mass spectrum for the
UP sample is shown in Fig. 4(b). Most resonant structures are in the NP
sample. An enhancement in the $2860\,\mathrm{\,Me\kern-1.00006ptV}$ mass
region in Fig. 4(b) indicates the possible presence of additional UP
contributions. The fitted parameters are given in Tables 1(b) and 1(c).

Figure 4: Mass spectrum of ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ candidates for ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ in (a) the
NP sample, and (b) the UP sample. The full (red) lines describe the fitting
function. The dashed lines show the fitted background and the dotted lines the
$D^{*}_{s2}(2573)^{+}$, $D^{*}_{s1}(2700)^{+}$, $D_{sJ}^{*}(2860)^{+}$ and
$D_{sJ}(3040)^{+}$ contributions. The insets display the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum after
subtracting the fitted background.

Figure 5(a) shows the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$
mass spectrum for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, which
contains 3.92$\times 10^{4}$ combinations. Similar resonant structures to
those seen for the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final
state with ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ are observed, albeit
at lower significance. Table 1(d) provides the fitted resonance parameters.
Due to the limited data samples, some parameters have been fixed to the values
obtained from the fit to the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ sample with ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$. The mass
values are found to be consistent with the results from the other
measurements.

Figure 5: Mass spectrum of (a) ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ candidates for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, and (b)
${{D}^{*0}}{{K}^{+}}$ candidates for the NP sample. The full (red) lines
describe the fitting function. The dashed lines show the fitted background and
the dotted lines the $D^{*}_{s2}(2573)^{+}$, $D^{*}_{s1}(2700)^{+}$,
$D_{sJ}^{*}(2860)^{+}$ and $D_{sJ}(3040)^{+}$ contributions. The insets
display the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ mass spectra after subtracting the fitted background.

The ${{D}^{*0}}{{K}^{+}}$ mass spectrum is affected by a high level of
combinatorial background, mostly due to the ${D}^{*0}$ reconstruction (see
Fig. 2). As observed previously, the ${{D}^{*}}K$ mass spectra are dominated
by NP resonances and therefore in Fig. 5(b) we show the ${{D}^{*0}}{{K}^{+}}$
mass spectrum for the NP sample. The mass spectrum contains 2.53$\times
10^{4}$ combinations. We observe similar resonant structures as seen in the
study of the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass
spectra. The fitted resonance parameters are given in Table 1(e); mass values
are consistent with the results from the fits to the other mass spectra. We do
not have the sensitivity to the parameters of the $D^{*}_{s1}(2700)^{+}$ and
$D_{s3}^{*}(2860)^{+}$ resonances in the fits to the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, and
${{D}^{*0}}{{K}^{+}}$ mass spectra due to the low statistical significance of
the signals.

##  5 Measurement of the branching fraction of the decay
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$

We measure the branching fraction of the decay
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$, ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ relative to that of the
decay
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$. For this purpose the ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ mass spectrum from Ref. [10], collected at 7
$\mathrm{\,Te\kern-1.00006ptV}$ with an integrated luminosity of
$1\,\mbox{\,fb}^{-1}$, is re-fitted. In this study both long and downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidate types are used. The final
states ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$, with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{D}^{*0}}{{K}^{+}}$ are used as cross checks and to aid in determining the
significance of the signal.

Figure 6 shows the ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass
spectrum from Ref. [10] along with the results of the fit described below. A
narrow structure is seen near threshold, due to the cross-feed from the decay

$D_{s1}(2536)^{+}\ \rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{D}^{*+}}(\rightarrow{{D}^{+}}{{\pi}^{0}}/\gamma),$ (11)

where the ${{\pi}^{0}}/\gamma$ are not reconstructed. In the higher mass
region, a strong $D^{*}_{s2}(2573)^{+}$ signal and a weak signal due to the
$D^{*}_{s1}(2700)^{+}$ resonance are observed. Due to the difficulty of
controlling the systematic uncertainties related to the determination of the
relative efficiencies of the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ and ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final states,
we normalize the two mass spectra using the $D_{s1}(2536)^{+}$ signal which is
observed as a peak in the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ and as cross-feed in the ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ final states.

Figure 6: Distribution of the invariant mass of
${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ candidates from Ref.
[10]. The full (red) line is the result from the fit described in the text.
The dashed line indicates the fitted background and the dotted line shows the
fitted $D^{*}_{s2}(2573)^{+}$ contribution.

The $D^{*}_{s2}(2573)^{+}$ resonance is a well known NP $J^{P}=2^{+}$ state.
To enhance the signal to background ratio, we plot in Fig. 7 the ${{D}^{*}}K$
mass spectra for the NP sample of the three final states. All three
distributions show a strong $D_{s1}(2536)^{+}$ signal and an enhancement at
the $D^{*}_{s2}(2573)^{+}$ mass.

Figure 7: Mass spectra, in the $D_{s1}(2536)^{+}$ mass region, for the NP
sample of (a) ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, (b)
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, and (c)
${{D}^{*0}}{{K}^{+}}$ final states. The full (red) lines describe the fitting
function. The dashed lines show the fitted background and the dotted lines the
$D^{*}_{s2}(2573)^{+}$ contributions. The insets display the ${{D}^{*}}K$ mass
spectra after subtracting the fitted background.

The ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum and the
three ${{D}^{*}}K$ mass spectra are fitted using the background function

$B(m)=P(m)e^{\beta m+\gamma m^{2}},$ (12)

where $P(m)$ is given in Eq. (5) and $\beta$ and $\gamma$ are free parameters.
The $D_{s1}(2536)^{+}$ cross-feed into
${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ is modelled using the sum
of two Gaussian functions with the same mean, and the $D^{*}_{s2}(2573)^{+}$
resonance is modelled as a relativistic BW function convolved with a Gaussian
function describing the experimental resolution
($\sigma=3.5\,\mathrm{\,Me\kern-1.00006ptV}$). Since the intrinsic width of
the $D_{s1}(2536)^{+}$ state in the ${{D}^{*}}K$ spectra is much smaller than
the experimental resolution, the $D_{s1}(2536)^{+}$ is modelled using the sum
of two Gaussian functions with the same mean. We obtain
$m(D_{s1}(2536)^{+})=2535.00\pm 0.01\,\mathrm{\,Me\kern-1.00006ptV}$, in good
agreement with the PDG average. The $D^{*}_{s2}(2573)^{+}$ resonance is
modelled as a relativistic BW function convolved with the experimental
resolution ($\sigma=2.5\,\mathrm{\,Me\kern-1.00006ptV}$ for
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$, ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$) taking the mass value as
a free parameter and with the full width constrained to the value obtained
from the fit to the ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass
spectrum ($\Gamma=17.5\pm 0.4\,\mathrm{\,Me\kern-1.00006ptV}$).

Table 3: Results from the fits to the ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectra for the evaluation of the $D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ relative branching fraction. Final state | Mass (MeV) |  $\Gamma$ (MeV) | Yield | Significance
---|---|---|---|---
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | 2566.9 $\pm$ 0.1 | 17.5 $\pm$ 0.4 | (2.55 $\pm$ |  0.38)$\times 10^{4}$ |
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | 2568.0 $\pm$ 1.0 | 17.5 (fixed) | (2.04 $\pm$ |  0.26)$\times 10^{3}$ | $6.9\,\sigma$
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, NP |  |  |  |  |
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | 2572.0 $\pm$ 1.3 | 17.5 (fixed) | (5.0 $\pm$ |  1.0)$\times 10^{2}$ | $4.6\,\sigma$
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, NP |  |  |  |  |
$D^{*}_{s2}(2573)^{+}\\!\rightarrow{{D}^{*0}}{{K}^{+}}$ | 2567.3 $\pm$ 4.7 | 17.5 (fixed) | (1.1 $\pm$ |  0.7)$\times 10^{2}$ | $1.2\,\sigma$
$D_{s1}(2536)^{+}\\!\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ | 2535.00 $\pm$ 0.01 |  | (3.59 $\pm$ |  0.15)$\times 10^{4}$ |
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, Total |  |  |  |  |

Table 3 summarizes the fit results. We note the large statistical significance
of the $D^{*}_{s2}(2573)^{+}$ in the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final states, especially
for the sample with ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$. Consistency
is found, within the uncertainties, in the $D^{*}_{s2}(2573)^{+}$ mass
measurements for the different final states. We therefore identify the
observed structure as the first observation of the
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ decay.

The relative branching fraction

${\cal R}=\frac{{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}{{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}$ (13)

is determined using the results of fits to the
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$, $D^{0}\rightarrow K\pi$ data shown in Fig. 7(a) and the
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ data shown in Fig. 6, summarized in Table 3.

Using the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final state,
we verify that the $D_{s1}(2536)^{+}$ cross-feed into the
${{D}^{0}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum, when the
pion from the ${D}^{*+}$ decay is ignored, contains all the $D_{s1}(2536)^{+}$
signal. Similarly, using the ${{D}^{*0}}{{K}^{+}}$ data, we ignore the
${\pi}^{0}$ from the ${D}^{*0}$ decay and plot the ${{D}^{0}}{{K}^{+}}$ mass
spectrum. Also in this case, it is found that the $D_{s1}(2536)^{+}$ cross-
feed contains all the decays in the $D_{s1}(2536)^{+}$ signal region. It is
assumed that the $D_{s1}(2536)^{+}$ meson decay to ${{D}^{*}}K$ is dominant.
We test this hypothesis by studying the ${{D}^{0}}{{\pi}^{0}}{{K}^{+}}$ mass
spectrum and find that no $D_{s1}(2536)^{+}$ signal is present outside the
${{D}^{*0}}\\!\rightarrow{{D}^{0}}{{\pi}^{0}}$ signal region.

Indicating explicitly in brackets the ${{D}^{*+}}$ decay modes, we define

$R_{1}=\frac{N(D^{*}_{s2}(2573)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}{N(D_{s1}(2536)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}$ (14)

and

$R_{2}=\frac{N(D_{s1}(2536)^{+}\rightarrow({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})_{f})}{N(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})},$ (15)

where $N$ indicates the yields and
$D_{s1}(2536)^{+}\rightarrow({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})_{f}$ indicates the cross-feed from
$D_{s1}(2536)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ where ${{D}^{*+}}\rightarrow{{D}^{+}}({{\pi}^{0}}/\gamma)$ and the
${{\pi}^{0}}/\gamma$ are undetected.

We measure the $D^{*}_{s2}(2573)^{+}$ relative branching ratio as

${\cal
R}=R_{1}\frac{\epsilon(D_{s1}(2536)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}{\epsilon(D^{*}_{s2}(2573)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}R_{2}\frac{\epsilon(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}{\epsilon(D_{s1}(2536)^{+}\rightarrow({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})_{f})}B_{D}f_{\rm NP},$ (16)

where $\epsilon$ indicates the efficiency for each final state. The ratio
$B_{D}$, defined below, is taken from Ref. [32],

$B_{D}=\frac{{\cal B}({{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}})}{{\cal
B}({{D}^{*+}}\rightarrow{{D}^{+}}({{\pi}^{0}}/\gamma))}=2.10\pm 0.05,$ (17)

where ${{D}^{+}}({{\pi}^{0}}/\gamma)$ indicates both ${{D}^{+}}{{\pi}^{0}}$
and ${{D}^{+}}\gamma$ decays and $f_{\rm NP}$ is defined below.

In the evaluation of the $D^{*}_{s2}(2573)^{+}$ relative branching fraction,
we make use of the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ NP
sample. This selection is used to improve the signal to background ratio for
the $D^{*}_{s2}(2573)^{+}$ resonance in the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final state. We also fit
the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum using
the full dataset and we report the $D_{s1}(2536)^{+}$ yield indicated as Total
in Table 3. In Eq. (16) the total $D_{s1}(2536)^{+}$ yield is used because of
the unnatural parity of this state, and this requires a correction to the
$D^{*}_{s2}(2573)^{+}$ yield for the effects of the NP sample selection. The
angular distribution for a NP resonance is expected to be proportional to
$\sin^{2}\theta_{H}$ and therefore the requirement $|\cos\theta_{H}|<0.5$
selects 69% of the candidates. This correction in Eq. (16) is included through
the factor $f_{\rm NP}=1.45$.

In Eq. (16) it can be noted that the efficiencies
$\epsilon(D^{*}_{s2}(2573)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ and
$\epsilon(D_{s1}(2536)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$ involve the same final state. They are determined from simulation and
are found to be the same within uncertainties. Similarly, the efficiencies
$\epsilon(D_{s1}(2536)^{+}\rightarrow({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})_{f})$ and
$\epsilon(D^{*}_{s2}(2573)^{+}\rightarrow({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}))$ are also found to be the same within uncertainties. Therefore, the
efficiency ratios are set to unity.

Table 4: Measurements used to evaluate the $D^{*}_{s2}(2573)^{+}$ relative
branching fraction ${\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})/{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})$.

Quantities | Value
---|---
$N(D^{*}_{s2}(2573)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ | (2.04 $\pm$ |  0.26$\mathrm{\,(stat)}$ $\pm$ |  0.14$\mathrm{\,(syst)}$) $\times 10^{3}$
$N(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ | (2.55 $\pm$ |  0.04$\mathrm{\,(stat)}$ $\pm$ |  0.08$\mathrm{\,(syst)}$) $\times 10^{4}$
$N(D_{s1}(2536)^{+}\rightarrow({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})_{f})$ | (6.54 $\pm$ |  0.12$\mathrm{\,(stat)}$ $\pm$ |  0.05$\mathrm{\,(syst)}$) $\times 10^{3}$
$N(D_{s1}(2536)^{+}\rightarrow({{D}^{0}}{{\pi}^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ | (3.59 $\pm$ |  0.15$\mathrm{\,(stat)}$ $\pm$ |  0.02$\mathrm{\,(syst)}$) $\times 10^{4}$

$R_{1}\qquad\qquad\qquad\qquad\qquad\qquad$ | 0.057 $\pm$ 0.006$\mathrm{\,(stat)}$ $\pm$ 0.004$\mathrm{\,(syst)}$
---|---
$R_{2}\qquad\qquad\qquad\qquad\qquad\qquad$ | 0.256 $\pm$ 0.006$\mathrm{\,(stat)}$ $\pm$ 0.008$\mathrm{\,(syst)}$
$f_{\rm NP}\qquad\qquad\qquad\qquad\qquad\qquad$ | 1.45
$B_{D}\qquad\qquad\qquad\qquad\qquad\qquad$ | 2.10 $\pm$ 0.05 $\mathrm{\,(stat)}$

Table 4 summarizes the measurements used to estimate the
$D^{*}_{s2}(2573)^{+}$ relative branching fraction. We obtain

${\cal R}=\frac{{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}{{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}=0.044\pm 0.005\mathrm{\,(stat)}\pm 0.011\mathrm{\,(syst)}.$ (18)

The systematic uncertainty on the $D^{*}_{s2}(2573)^{+}$ relative branching
fraction is computed as the quadratic sum of the differences between the
reference values and those obtained when the following changes are made.

  * •

The ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ data are collected at
7 $\mathrm{\,Te\kern-1.00006ptV}$, while the
${{D}^{*}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ data include 7 TeV and 8
TeV data samples. We compute systematic uncertainties on the $R_{1}$ and
$R_{2}$ ratios using the ${{D}^{*}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$
at 7 $\mathrm{\,Te\kern-1.00006ptV}$ only and include the deviation in the
systematic uncertainty.

  * •

The uncertainty on the $B_{D}$ parameter is propagated as a systematic
uncertainty.

  * •

Using simulation, we compute efficiency distributions as functions of
$m({{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ and
$m({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ and observe that they
have weak variations in the regions used to evaluate the relative branching
fraction. We assign a 10% systematic uncertainty to cover the assumptions that
the efficiencies as functions of
$m({{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ and
$m({{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}})$ in Eq. (16) are the
same.

  * •

We vary the shape of the background function using Eq. (4) in the fits to the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectra and obtain
new estimates for the resonance yields. We also remove the convolution with
the resolution function or replace the relativistic BW functions with simple
BW functions and include an additional Gaussian function to describe the
$D_{s1}(2536)^{+}$ signal.

  * •

We vary the $D^{*}_{s2}(2573)^{+}$ width by its statistical uncertainty
($0.4\,\mathrm{\,Me\kern-1.00006ptV}$) simultaneously in the fits to the
${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectra.

The contributions to the systematic uncertainty are summarized in Table 5 with
the dominant component arising from the use of different datasets collected at
different centre-of-mass energies.

Table 5: Relative systematic uncertainties in the evaluation of the ratio of branching fractions $\cal R$. Source | Value (%)
---|---
Datasets | 22.2
Error on $B_{D}$ | 2.1
Efficiency | 10.0
Resonance parameters and backgrounds | 7.5
$D^{*}_{s2}(2573)^{+}$ width | 0.3
Total | 25.6

We also perform a new estimate of the $D^{*}_{s2}(2573)^{+}$ significance in
the ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final state by
combining in quadrature the statistical and systematic uncertainties on the
yield (see Table 4) and obtain $S=N_{\rm signal}/\sigma_{\rm tot}=6.9$, where
$\sigma_{\rm tot}$ is the total error. This estimate is in good agreement with
that reported in Table 3.

##  6 Spin-parity analysis of the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ system

We obtain information on the spin-parity of the states observed in the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ mass spectrum. The data
for ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ are first divided into five
equally spaced bins in $\cos\theta_{\rm H}$. The five mass spectra in the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ threshold region
($m({{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})<2650\,\mathrm{\,Me\kern-1.00006ptV}$) are fitted using the model
described in Sec. 5 with fixed $D_{s1}(2536)^{+}$ and $D^{*}_{s2}(2573)^{+}$
resonance parameters, to obtain the signal yields as functions of
$\cos\theta_{\rm H}$ for each resonance.

Figure 8:  Distributions of the measured signal yields for (a)
$D_{s1}(2536)^{+}$ and (b) $D^{*}_{s2}(2573)^{+}$ as a function of the
helicity angle $\cos\theta_{\rm H}$. The distributions are fitted with UP (a)
and NP (b) functions.

As stated previously, we determine from simulations that the efficiency as a
function of $\cos\theta_{H}$ is consistent with being uniform; therefore we
plot uncorrected angular distributions. The resulting distributions for
$D_{s1}(2536)^{+}$ and $D^{*}_{s2}(2573)^{+}$ are shown in Fig. 8(a) and Fig.
8(b), and are fitted using the functions described in Table 6. A good
description of the data is obtained in terms of the expected angular
distributions for $J^{P}=1^{+}$ and $J^{P}=2^{+}$ resonances. We note that the
shape of the $D_{s1}(2536)^{+}$ angular distribution is in agreement with that
measured in Ref. [35].

Table 6: Values of $\chi^{2}/{\rm ndf}$ from the fits to the helicity angles distributions. Resonance | $J^{P}$ | Function | $\chi^{2}/{\rm ndf}$
---|---|---|---
$D_{s1}(2536)^{+}$ | $1^{+}$ | $1+h\cos^{2}\theta_{\rm H}$ | 0.1/3
$D^{*}_{s2}(2573)^{+}$ | $2^{+}$ | $\sin^{2}\theta_{\rm H}$ | 2.2/4
$D^{*}_{s1}(2700)^{+}$ | $1^{-}$ | $\sin^{2}\theta_{\rm H}$ | 11.4/7
$D_{s3}^{*}(2860)^{+}$ | $3^{-}$ | $\sin^{2}\theta_{\rm H}$ | 13.4/7
$D_{sJ}(3040)^{+}$ | UP | $1+h\cos^{2}\theta_{\rm H}$ | 8.0/6

Figure 9:  Distributions of the measured signal yields for (a)
$D^{*}_{s1}(2700)^{+}$, (b) $D_{sJ}^{*}(2860)^{+}$ and (c) $D_{sJ}(3040)^{+}$
as a function of the helicity angle $\cos\theta_{\rm H}$. The distributions
are fitted with NP (a,b) and UP (c) functions.

The ${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ data, with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, are then divided into eight
equally spaced bins in $\cos\theta_{\rm H}$. The mass spectra are fitted (for
$m({{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})<3400\,\mathrm{\,Me\kern-1.00006ptV}$) with the model described in Sec. 4
with fixed resonance parameters, to obtain the yields as functions of
$\cos\theta_{\rm H}$ for each resonance. The resulting distributions are shown
in Fig. 9 and details of the fit results are given in Table 6.

We observe that the $D^{*}_{s1}(2700)^{+}$ state is reasonably well described
by the expected NP function ($\chi^{2}/{\rm ndf}=11.4/7$ with p-value 12.2%).
The fit to the $D_{s3}^{*}(2860)^{+}$ angular distribution has a slightly
lower p-value (6.3%). Reference [18] suggests the possibility of the presence
of UP state contributions in this mass range, which cannot be excluded in this
fit: there is evidence for the presence of a small signal in the
$2860\,\mathrm{\,Me\kern-1.00006ptV}$ mass region for the UP sample shown in
Fig. 4(b). The consistency with the NP assignment confirms the presence of the
decay
$D_{s3}^{*}(2860)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$. We also show in Fig. 9(c) the $\cos\theta_{\rm H}$ distribution for the
enhancement at the $D_{sJ}(3040)^{+}$ position and find it consistent with a
UP assignment.

##  7 Summary

A study of the resonant structures in the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ systems is performed using $pp$ collision data,
collected at centre-of-mass energies of 7 and 8
$\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector. For the
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ final state, the decay
chains ${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ with
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ are used,
with an integrated luminosity of $3.0\,\mbox{\,fb}^{-1}$. For
${{D}^{*0}}{{K}^{+}}$, the decay chain
${{D}^{*0}}\\!\rightarrow{{D}^{0}}{{\pi}^{0}}$,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ is used, with an integrated
luminosity of $2.0\,\mbox{\,fb}^{-1}$.

A prominent $D_{s1}(2536)^{+}$ resonance is observed in both
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{D}^{*0}}{{K}^{+}}$ final states. Resonances $D^{*}_{s1}(2700)^{+}$ and
$D_{s3}^{*}(2860)^{+}$ are also observed and their parameters are measured to
be

$\displaystyle m(D^{*}_{s1}(2700)^{+})$ $\displaystyle=2732.3\pm
4.3\mathrm{\,(stat)}\pm 5.8\mathrm{\,(syst)}\,\mathrm{\,Me\kern-1.00006ptV},$
$\displaystyle\Gamma(D^{*}_{s1}(2700)^{+})$ $\displaystyle=136\pm
19\mathrm{\,(stat)}\pm 24\mathrm{\,(syst)}\,\mathrm{\,Me\kern-1.00006ptV},$

and

$\displaystyle m(D_{sJ}^{*}(2860)^{+})$ $\displaystyle=2867.1\pm
4.3\mathrm{\,(stat)}\pm 1.9\mathrm{\,(syst)}\,\mathrm{\,Me\kern-1.00006ptV},$
$\displaystyle\Gamma(D_{sJ}^{*}(2860)^{+})$ $\displaystyle=50\pm
11\mathrm{\,(stat)}\pm 13\mathrm{\,(syst)}\,\mathrm{\,Me\kern-1.00006ptV}.$

Study of the angular distributions supports natural parity assignments for
both resonances, although the presence of an additional unnatural parity
contribution in the $2860\,\mathrm{\,Me\kern-1.00006ptV}$ mass range cannot be
excluded. The data are not sensitive to the presence of an additional
$D_{s1}^{*}(2860)^{+}$ resonance.

The $D^{*}_{s2}(2573)^{+}$ decay to
${{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ is also observed for the
first time, at a significance of $6.9\,\sigma$, with a branching fraction
relative to the ${{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decay
mode of

$\frac{{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}{{\cal
B}(D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}})}=0.044\pm 0.005\mathrm{\,(stat)}\pm 0.011\mathrm{\,(syst)}.$ (19)

This measurement is in agreement with expectations from recent calculations of
the charm and charm-strange mesons spectra [21] which predict a value of 0.058
for this ratio. A spin-parity analysis of the decay
$D^{*}_{s2}(2573)^{+}\rightarrow{{D}^{*+}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ supports the natural parity assignment. The data also show weak evidence
for further structure in the region around
$3040\,\mathrm{\,Me\kern-1.00006ptV}$ consistent with contributions from
unnatural parity states.

## 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), The Royal Society, Royal
Commission for the Exhibition of 1851 and the Leverhulme Trust (United
Kingdom).

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LHCb collaboration

R. Aaij39, C. Abellán Beteta41, B. Adeva38, M. Adinolfi47, A. Affolder53, Z.
Ajaltouni5, S. 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, W.
Baldini17,39, 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, 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. Bonivento16, S. Borghi55, M. Borisyak66, M. Borsato38, T.J.V.
Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M. Britsch12, T. Britton60,
J. Brodzicka55, N.H. Brook47, E. Buchanan47, C. Burr55, A. Bursche41, J.
Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,k, M. Calvo Gomez37,p,
P. Campana19, D. Campora Perez39, L. Capriotti55, A. Carbone15,e, G.
Carboni25,l, R. Cardinale20,j, A. Cardini16, P. Carniti21,k, L. Carson51, K.
Carvalho Akiba2, G. Casse53, L. Cassina21,k, L. Castillo Garcia40, M.
Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M. Charles8, Ph.
Charpentier39, M. Chefdeville4, S. Chen55, S.-F. Cheung56, N. Chiapolini41, M.
Chrzaszcz41,27, X. Cid Vidal39, G. Ciezarek42, P.E.L. Clarke51, M.
Clemencic39, H.V. Cliff48, J. Closier39, V. Coco39, J. Cogan6, E. Cogneras5,
V. Cogoni16,f, L. Cojocariu30, G. Collazuol23,r, P. Collins39, A. Comerma-
Montells12, A. Contu39, A. Cook47, M. Coombes47, S. Coquereau8, G. Corti39, M.
Corvo17,g, B. Couturier39, G.A. Cowan51, D.C. Craik51, A. Crocombe49, M. Cruz
Torres61, S. Cunliffe54, R. Currie54, C. D’Ambrosio39, E. Dall’Occo42, J.
Dalseno47, P.N.Y. David42, A. Davis58, O. De Aguiar Francisco2, K. De Bruyn6,
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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. Qian4, R.
Quagliani7,47, B. Rachwal27, J.H. Rademacker47, M. Rama24, M. Ramos Pernas38,
M.S. Rangel2, I. Raniuk44, N. Rauschmayr39, G. Raven43, F. Redi54, S.
Reichert55, A.C. dos Reis1, V. Renaudin7, S. Ricciardi50, S. Richards47, M.
Rihl39, K. Rinnert53,39, V. Rives Molina37, P. Robbe7,39, A.B. Rodrigues1, E.
Rodrigues55, J.A. Rodriguez Lopez63, P. Rodriguez Perez55, S. Roiser39, V.
Romanovsky36, A. Romero Vidal38, J. W. Ronayne13, M. Rotondo23, T. Ruf39, P.
Ruiz Valls67, J.J. Saborido Silva38, N. Sagidova31, B. Saitta16,f, V.
Salustino Guimaraes2, C. Sanchez Mayordomo67, 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.
Shevchenko65, A. Shires10, B.G. Siddi17, R. Silva Coutinho41, L. Silva de
Oliveira2, G. Simi23,s, M. Sirendi48, N. Skidmore47, T. Skwarnicki60, E.
Smith56,50, E. Smith54, I.T. Smith51, J. Smith48, M. Smith55, H. Snoek42, M.D.
Sokoloff58,39, 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, J. Todd58, S. Tolk43, L. Tomassetti17,g, D. Tonelli39,
S. Topp-Joergensen56, N. Torr56, E. Tournefier4, S. Tourneur40, K. Trabelsi40,
M. Traill52, M.T. Tran40, M. Tresch41, A. Trisovic39, A. Tsaregorodtsev6, P.
Tsopelas42, N. Tuning42,39, A. Ukleja29, A. Ustyuzhanin66,65, U. Uwer12, C.
Vacca16,39,f, V. Vagnoni15, G. Valenti15, A. Vallier7, R. Vazquez Gomez19, P.
Vazquez Regueiro38, C. Vázquez Sierra38, S. Vecchi17, M. van Veghel43, 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. Voß64, J.A. de
Vries42, R. Waldi64, C. Wallace49, R. Wallace13, J. Walsh24, J. Wang60, D.R.
Ward48, N.K. Watson46, D. Websdale54, A. Weiden41, M. Whitehead39, J. Wicht49,
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. Xie62, Z. Xu40, Z. Yang3, J. Yu62, X. Yuan35, O. Yushchenko36,
M. Zangoli15, M. Zavertyaev11,c, L. Zhang3, Y. Zhang3, A. Zhelezov12, 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
62Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
63Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
64Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
65National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
66Yandex School of Data Analysis, Moscow, Russia, associated to 32
67Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
68Van 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