Measurement of the Born Cross Sections for $e^+e^-\to D_s^+ D_{s1}(2460)^- +c.c.$ and $e^+e^-\to D_s^{\ast +} D_{s1}(2460)^- +c.c.$

The processes $e^+e^-\to D_s^+ D_{s1}(2460)^- +c.c.$ and $e^+e^-\to D_s^{\ast +} D_{s1}(2460)^- +c.c.$ are studied for the first time using data samples collected with the BESIII detector at the BEPCII collider. The Born cross sections of $e^+e^-\to D_s^+ D_{s1}(2460)^- +c.c.$ at nine center-of-mass energies between 4.467\,GeV and 4.600\,GeV and those of $e^+e^-\to D_s^{\ast +} D_{s1}(2460)^- +c.c.$ at ${\sqrt s}=$ 4.590\,GeV and 4.600\,GeV are measured. No obvious charmonium or charmonium-like structure is seen in the measured cross sections.


Abstract
The processes e + e − → D + s D s1 (2460) − +c.c. and e + e − → D * + s D s1 (2460) − +c.c. are studied for the first time using data samples collected with the BESIII detector at the BEPCII collider. The Born cross sections of e + e − → D + s D s1 (2460) − + c.c. at nine center-of-mass energies between 4.467 GeV and 4.600 GeV and those of e + e − → D * + s D s1 (2460) − + c.c. at √ s = 4.590 GeV and 4.600 GeV are measured. No obvious charmonium or charmonium-like structure is seen in the measured cross sections.

I. INTRODUCTION
The charmed-strange mesons, known as D s , are made up of cs orcs quarks. The D s1 (2460) meson was first observed in 2003 by the CLEO experiment via its decay into D * + s π 0 [1]. It was subsequently confirmed by the Belle [2] and BABAR [3] experiments. The experimental results favor a J P = 1 + quantum number assignment for D s1 (2460) as a P -wave state. However, its measured mass (2459.5 ± 0.6) MeV/c 2 is at least 70 MeV/c 2 lower than the quark model predictions [4,5], leading to an unexpectedly narrow width. It has also been proposed to be a good candidate for a D * K molecule state [6][7][8], or a mixture of the cs and D * K state [9].
The D s1 (2460) can be produced in the processes e + e − → D + s D s1 (2460) − +c.c. and e + e − → D * + s D s1 (2460) − + c.c.. Following the excitation behavior of S-wave production, Ref. [10] is the center-of-mass (c.m.) energy and E 0 ≈ 4.43 GeV is the mass threshold of both channels.
In this paper, we report the first measurement of the Born cross sections for e + e − → D + s D s1 (2460) − + c.c. and e + e − → D * + s D s1 (2460) − + c.c., and the search for possible vector charmonium-like states. Throughout the paper, charged-conjugate modes are always implied.

II. DETECTOR, DATA SAMPLES AND MONTE CARLO SIMULATIONS
BESIII [22] and BEPCII are major upgrades of the BESII detector [23] and the BEPC accelerator. The cylindrical core of the BESIII detector consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal fluxreturn yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over 4π solid angle. The charged particle momentum resolution at 1 GeV/c is 0.5%, and the energy loss (dE/dx) resolution is 6% for the electrons from Bhabha scattering. The EMC photon energy resolution is 2.5% (5%) at 1 GeV in the barrel (end cap) region. The time resolution of the TOF barrel (end cap) is 68 ps (110 ps). Our particle identification (PID) methods combine the TOF information with the dE/dx measured in the MDC to calculate the probability Prob(h), h = π, K, for a track to be a pion or a kaon.
In this paper, the Born cross sections of the processes e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − are measured for the first time at nine energy points between 4.467 and 4.600 GeV, and at 4.590 and 4.600 GeV, respectively. Table I lists the data samples used in this analysis and their integrated luminosities. The c.m. energies are measured using the process e + e − → µ + µ − with an uncertainty of 0.8 MeV [24]. The integrated luminosities are measured with an uncertainty of 1.0% using large-angle Bhabha scattering events [25,26].
The geant4-based [27] Monte Carlo (MC) simulation framework boost [28], which consists of event generators and the description of the detector geometry and response, is used to produce large simulated event samples. These are used to optimize the event selection criteria, determine the detection efficiency, evaluate the initial state radiation (ISR) correction factor (1 + δ), and estimate background contributions. The simulation includes the beam energy spread and ISR modeled with kkmc [29][30][31] and besevtgen [32,33]. The final state radiation (FSR) effects are simulated by the photos [34] package. For each energy point, we generate MC samples of the signal processes e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − with a uniform distribution in phase space (PHSP). The signal process e + e − → D + s D s1 (2460) − is simulated with D + s decaying into K + K − π + , and the D s1 (2460) − decaying into all possible final states. The signal process e + e − → D * + s D s1 (2460) − is simulated with D * + s decaying into γD + s and the D s1 (2460) − decaying into all possible final states. A P -wave model and a Dalitz plot decay model [35] are used to simulate D * + s → γD + s and D + s → K + K − π + , respectively. Two generic MC simulated samples at √ s = 4.575 GeV and 4.600 GeV, equivalent to the respective integrated luminosity of each data set, are produced to investigate potential peaking background channels. Known processes and decay modes are generated by besevtgen with cross sections and branching fractions obtained from the Particle Data Group (PDG) [36]. The remaining unmeasured phenomena associated with charmonium decays or open charm processes are simulated with lundcharm [32,37], while continuum light hadronic events are produced with pythia [38].

III. COMMON SELECTION CRITERIA
The candidate events for e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − are selected with a partial reconstruction method to obtain higher efficiencies. The D + s candidates The D s1 (2460) − signals are identified with the mass recoiling against the reconstructed D + s and D * + s . There are three charged tracks in D + s → K + K − π + , and one additional photon candidate in D * + s → γD + s . For each charged track candidate, the polar angle θ in the MDC with respect to the detector axis must satisfy | cos θ| < 0.93, and the point of closest approach to the e + e − interaction point must be within ±10 cm in the beam direction and within 1 cm in the plane perpendicular to the beam direction. Pion candidates are required to satisfy Prob(π) > Prob(K) and Prob(π) > 0.001. Kaon candidates are required to satisfy Prob(K) > Prob(π) and Prob(K) > 0.001.
The photon candidates are selected from showers in the EMC. The deposited energy in the EMC is required to be larger than 25 MeV in the barrel region (|cos θ| < 0.80) or greater than 50 MeV in the endcap region (0.86 < |cos θ| < 0.92). To eliminate the showers produced by charged tracks, photon candidates must be separated by at least 20 • from the extrapolated position of all charged tracks in the EMC. The timing of the shower is required to be within 700 ns from the reconstructed event start time to suppress noise and energy deposits unrelated to the event.
The candidate events of both e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − are required to contain at least two kaons and one pion. One additional photon candidate is required for e + e − → D * + s D s1 (2460) − . All combinations of K + K − π + that pass the vertex fit are kept.
is the nominal mass of the φ (K * 0 ) meson taken from the PDG [36].
To improve the resolution of the D + s recoil mass, we define M rec , P K + , P K − , and P π + are the four-momenta of the initial e + e − system, the selected K + , K − , and π + , respectively, M(K + K − π + ) is the invariant mass of the K + K − π + system, and m D + s is the nominal mass of the D + s meson [36]. We separate the M rec   Table I. signal significances are summarized in Table I.  Table I.
Since the statistical significances of the D s1 (2460) − signal at some energy points are less than 3σ, the upper limits on the numbers of D s1 (2460) − signal events (N U.L. ) are determined at the 90% confidence level (C.L.) by solving the following equation: where x is the assumed yield of D s1 (2460) − signal, and L(x) is the corresponding maximum likelihood from the data. The resulting N U.L. obtained using the above method are listed in Table I. The Born cross section of e + e − → D + s D s1 (2460) − is calculated using the formula: where N fit is the D s1 (2460) − signal yield, 1+δ is the radiative correction factor obtained from a QED calculation with 1% accuracy [40] using the kkmc generator, 1 + δ vp is the vacuum polarization factor, whose calculations are from Ref. [41] (δ vp = 0.055 for all studied energy points), and L int is the integrated luminosity at each energy point. The product of the D s efficiency and branching fraction is ǫ Ds = ǫB(D + s → K + K − π + ) where ǫ is the detection efficiency and B(D + s → K + K − π + ) is the branching fraction for D + s → K + K − π + [36]. The calculation of the upper limits for Born cross sections at the 90% C.L. is performed analogously, replacing N fit with N U.L. . The measured Born cross sections of e + e − → D + s D s1 (2460) − and the corresponding upper limits at the 90% C.L. (with systematic uncertainties included) for the energy points with signal significances less than 3σ are summarized in Table I is listed in Table I. The Born cross section of e + e − → D * + s D s1 (2460) − is calculated using the formula (3) Here, the parameters have the same meaning as in Eq. 2, except that ǫ D *  Table I. The systematic uncertainties are discussed in Sec. VI. integrated luminosity L int , the signal efficiency ǫ (ǫ * ) from signal MC samples, the number of fitted D s1 (2460) − signal events N fit , the 90% C.L. upper limit on the number of fitted D s1 (2460) − signal yields N U.L. , the ISR radiative correction factor (1 + δ), the statistical signal significance, and the measured Born cross section σ B and its 90% C.L. upper limit σ U.L.

B
(with systematic uncertainties included).

VI. SYSTEMATIC UNCERTAINTIES
The systematic uncertainties on the measured cross sections of e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − come from tracking and PID efficiencies, photon detection efficiency, and MC statistics. We also consider the uncertainties from ISR and vacuum polarization corrections, the luminosity measurement, branching fractions of intermediate states, the kinematic fit, MC generator, D + s mass resolution, M rec D + s bin width, D s1 (2460) − mass, the background shape, and the fit range. These contributions to the systematic uncertainty are divided below into two categories: multiplicative systematic uncertainties and additive systematic uncertainties.
Multiplicative systematic uncertainties are analyzed as follows. The uncertainties of tracking and PID are determined to be 1.5%, 1.0%, and 1.0% for K + , K − , and π + , respectively, using the control samples of J/ψ → ppπ + π − and J/ψ → K 0 S K + π − , where the transverse momentum and angular region of the signal channels are taken into account. The uncertainty of the photon reconstruction efficiency is 1.0% per photon, which is derived from the study of J/ψ → ρ 0 (→ π + π − )π 0 (→ γγ) [44]. The uncertainties due to MC statistics are determined to be 1.1% at each energy point. The shapes of the cross section of the processes e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − affect the radiative correction factor and the detection efficiency. Due to the small number of data points with low statistics, a detailed determination of the energy dependence ("line shape"), which would allow for an iterative determination of radiative correction factors, is not possible. Therefore, we change the input line shapes to a simple polynomial form, and the differences in ε(1 + δ) are taken as the systematic uncertainties. The uncertainty from the vacuum polarization factor is less than 0.1% [41], which is negligible compared to other sources of uncertainties. The integrated luminosities of the data samples are measured using large angle Bhabha scattering events with an uncertainty less than 1.0%. The uncertainties of B(D + s → K + K − π + ) and B(D * + s → γD + s ) are 3.2% and 0.7%, respectively [36]. The uncertainty of the 2C kinematic fit is estimated using the control samples of e + e − → D * + s D * − s at √ s = 4.420 GeV and 4.600 GeV. The difference in the data and MC efficiencies due to the addition of the 2C kinematic fit requirement is 1.7%, which is taken as the systematic uncertainty. Signal MC samples are generated with a PHSP model. We also generate signal MC samples with a polar angle distribution of 1 + cos 2 θ or 1 − cos 2 θ for the D + s /D * + s meson. The maximum differences in detection efficiencies are 1.3% and 1.7% for the reconstructed D + s and D * + s candidates. Additive systematic uncertainties due to the fit are analyzed as follows. The uncertainty due to the D + s mass resolution is estimated by varying this mass resolution by ±1σ when fitting the K + K − π + invariant mass distributions in M rec For those energy points with a statistical significance larger than 3σ, the central values of the cross section with statistical and systematic uncertainties are reported, and all of the systematic uncertainties are summarized in Table II. For the other energy points with D s1 (2460) − signal significance less than 3σ, the upper limits on the cross section at the 90% C.L. are reported and the systematic uncertainties are taken into account in two steps. First, when we study the additive systematic uncertainties described above, we take the most conservative upper limit at the 90% C.L. on the number of D s1 (2460) − signal yields. Then, to take into account the multiplicative systematic uncertainty, the likelihood with the most conservative upper limit is convolved with a Gaussian function, with a width equal to the corresponding total multiplicative systematic uncertainty. All of the multiplicative systematic uncertainties for the energy points with D s1 (2460) − signal significance less than 3σ are summarized in Table III. Assuming that all the sources are independent, the total systematic uncertainty is obtained by adding them in quadrature. The final results of the Born cross section with systematic uncertainties considered are listed in Table I. The com-parison of the Born cross sections of e + e − → D + s D s1 (2460) − and e + e − → D * + s D s1 (2460) − is shown in Fig. 5 with statistical error bars only.   .590 Tracking, PID and photon 3.5% 3.5% 3.5% 3.5% 3.5% 3.5% 3.7% MC statistics 0.5% 0.5% 0.5% 0.5% 0.5% 0.5% 1.1% ISR correction 13.1% 7.6% 8.1% 2.8% 7.6% 7.0% 1.6% Luminosity 0.7% 0.8% 0.8% 0.8% 0.7% 0.7% 0.7% Branching fraction 3.2% 3.2% 3.2% 3.2% 3.2% 3.2% 3.3% Kinematic fit · · · · · · · · · · · · · · · · · · 1.7% MC generator 1.3% 1.3% 1.3% 1.3% 1.3% 1.3% 1.7% Total 14.0% 9.1% 9.5% 5.7% 9.1% 8.6% 5.7%

VII. SUMMARY
In summary, we observe D s1 (2460) − signals with statistical significances larger than 3σ in the processes e + e − → D