Measurement of branching fraction of $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ relative to $D^{*+}_{s}\to D^{+}_{s}\gamma$

The excited strange charmed meson, $D_{s}^{*+}$, is formed from $c\bar{s}$ quark-antiquark pair. Throughout this paper, charge-conjugate states are always included. The $D_{s}^{*+}$ decays are dominated by the radiative process $D_{s}^{*+}\to D_{s}^{+}\gamma$ and the isospin-violating hadronic process $D_{s}^{*+}\to D_{s}^{+}\pi^{0}$ due to the quark SU(2) flavor breaking and isospin violating effects. Measurements of the branching fractions (BFs) of the $D_{s}^{*+}$ decays are important to explore quantum chromodynamics (QCD) Fritzsch:1973pi describing the strong interaction. The decay widths of $D_{s}^{*+}\to D^{+}_{s}\gamma$ and/or $D_{s}^{*+}\to D^{+}_{s}\pi^{0}$ have been theoretically predicted based on effective models, e.g. chiral perturbation theory ($\chi$PT) Yang:2019cat ; Cheng:1993kp ; Cho:1994zu ; Wang:2019mhm , the light-front quark model (LFQM) Choi:2007us , the relativistic quark model (RQM)Goity:2000dk , QCD sum rules (QCDSR) Aliev:1994nq ; Yu:2015xwa , the Nambu-Jona-Lasinio model (NJLM) Deng:2013uca , lattice QCD (LQCD) Donald:2013sra , the non-relativistic quark model (NRQM) Kamal:1992uv ; Fayyazuddin:1993eb , and the covariant model (CM) Cheung:2015rya . The BF of $D_{s}^{*+}\to D^{+}_{s}\pi^{0}$ relative to that of $D_{s}^{*+}\to D^{+}_{s}\gamma$ has been measured by using $e^{+}e^{-}$ collision data accumulated at the $\Upsilon(3S)$ and $\Upsilon(4S)$ by the CLEO cleo2 and BaBar babar2 experiments. The precision of the world average of the BF of $D_{s}^{*+}\to D^{+}_{s}\gamma$ is about 0.7% pdg2022 . Precision measurements of these BFs help to constrain the model parameters, thereby improving the effective models. In addition, the BFs are important inputs in the precise determination of the $D^{+}_{s}$ decay constant $f_{D^{+}_{s}}$ and the $c\to s$ CKM matrix element $|V_{cs}|$ via the $e^{+}e^{-}\to D_{s}^{*\pm}D_{s}^{\mp}$ processes.

In this paper, we report an improved measurement of the BF of $D_{s}^{*+}\to D^{+}_{s}\pi^{0}$ relative to $D_{s}^{*+}\to D^{+}_{s}\gamma$ and then determine the BFs of $D_{s}^{*+}\to D^{+}_{s}\gamma$ and $D_{s}^{*+}\to D^{+}_{s}\pi^{0}$. This analysis is carried out by using 7.33 fb^-1 of $e^{+}e^{-}$ collision data taken at center-of-mass energies $E_{\rm cm}$ between 4.128 and 4.226 GeV with the BESIII detector.

The BESIII detector Ablikim:2009aa records symmetric $e^{+}e^{-}$ collisions provided by the BEPCII storage ring Yu:IPAC2016-TUYA01 in the center-of-mass energy range from 2.0 to 4.95 GeV, with a peak luminosity of $1\times 10^{33}$ cm^-2s^-1 achieved at $\sqrt{s}=3.773~{}\text{GeV}$. BESIII has collected large data samples in this energy region Ablikim:2019hff . BESIII is a cylindrical spectrometer with a geometrical acceptance of $93\%$ over the $4\pi$ solid angle. It 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 flux-return yoke with resistive plate counter muon identifier modules interleaved with steel Huang:geometry . The charged particle momentum resolution is $0.5\%$ at 1 GeV/$c$, and the specific energy loss (d$E$/d$x$) resolution is $6\%$ for the electrons from Bhabha scattering. The EMC measures photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel (end-cap) region. The time resolution in the TOF barrel region is 68 ps. The end-cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps etof . Approximately 83% of the data used here was collected after this upgrade; luminosities lumi at each energy are given in Table 1.

Simulated data samples are produced with a geant4-based geant4 Monte Carlo (MC) toolkit including the geometric description of the BESIII detector and the detector response. The simulation includes the beam energy spread and initial state radiation (ISR) in the $e^{+}e^{-}$ annihilations with the generator kkmc ref:kkmc . In the MC simulation, the production of open-charm processes directly produced via $e^{+}e^{-}$ annihilations are modelled with the generator conexc ref:conexc . The ISR production of vector charmonium(-like) states and the continuum processes are incorporated in kkmc ref:kkmc . All particle decays are modelled with evtgen ref:evtgen using BFs either taken from the Particle Data Group pdg2022 , when available, or otherwise estimated with lundcharm ref:lundcharm . Final state radiation (FSR) from charged final state particles is incorporated using photos photos .

The input cross section line shape of $e^{+}e^{-}\to D^{*\pm}_{s}D^{\mp}_{s}$ is based on the results in Ref. crsDsDss . In this analysis, the inclusive MC sample, which is generated at various energy points and has an integrated luminosity of 40 times individual data sets, is used to determine detection efficiencies and to estimate background contributions.

At the center-of-mass energies between 4.128 and 4.226 GeV, $D^{*+}_{s}D^{-}_{s}$ pairs are produced copiously by $e^{+}e^{-}$ collisions. The $D^{*+}_{s}$ mesons decay predominantly via $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$. Candidate events are selected by reconstructing $D^{+}_{s}$ and $D^{-}_{s}$ mesons via hadronic decay modes. To obtain better momentum resolution and lower background contamination, we use three modes of $D^{+}_{s}\to K^{+}K^{-}\pi^{+}$ versus $D^{-}_{s}\to K^{+}K^{-}\pi^{-}$, $D^{+}_{s}\to K^{+}K^{-}\pi^{+}$ versus $D^{-}_{s}\to K^{0}_{S}K^{-}$, and $D^{+}_{s}\to K^{0}_{S}K^{+}$ versus $D^{-}_{s}\to K^{0}_{S}K^{-}$, which are labelled as modes I, II, and III, respectively. In order to improve detection efficiencies, no transition photon or $\pi^{0}$ from the $D_{s}^{*+}$ decay is required.

Charged tracks detected in the MDC are required to be within a polar angle ($\theta$) range of $|\rm{cos\theta}|<0.93$, where $\theta$ is defined with respect to the $z$-axis, which is the symmetry axis of the MDC. For charged tracks not originating from $K_{S}^{0}$ decays, the distance of closest approach to the interaction point (IP) must be less than 10 cm along the $z$-axis, $|V_{z}|$, and less than 1 cm in the transverse plane, $|V_{xy}|$. No additional charged track passing the $\rm{cos\theta}$ and IP cuts is allowed for selected candidates. Particle identification (PID) for charged tracks combines measurements of the d$E$/d$x$ in the MDC and the flight time in the TOF to form likelihoods for charged pion and kaon hypotheses, $\mathcal{L}(\pi)$ and $\mathcal{L}(K)$. Pion candidates are required to satisfy $\mathcal{L}(\pi)$ $>$ $\mathcal{L}(K)$ and $\mathcal{L}(\pi)>0$, and kaon candidates are required to satisfy $\mathcal{L}(K)$ $>$ $\mathcal{L}(\pi)$ and $\mathcal{L}(K)>0$.

The $K_{S}^{0}$ candidates are reconstructed via the decay $K^{0}_{S}\to\pi^{+}\pi^{-}$. The two charged pions are required to satisfy $|V_{z}|<20$ cm and $|\!\cos\theta|<0.93$ but no particle identification is applied. The $\pi^{+}\pi^{-}$ invariant mass is required to be within the interval $(0.487,0.511)$ GeV/$c^{2}$. A vertex fit is performed, constraining the two tracks to originate from a common vertex, and the decay length of $K_{S}^{0}$ candidates is required to be greater than twice the resolution.

To suppress non-$D_{s}^{\pm}D_{s}^{*\mp}$ events, the beam-constrained mass of the $D_{s}^{-}$ candidate

$$M_{\rm BC}\equiv\sqrt{E_{\rm beam}^{2}-|\vec{p}_{\rm tag}|^{2}},$$

(1)

is required to be within the intervals as shown in Table 1. Here, $E_{\rm beam}$ is the beam energy and $\vec{p}_{\rm tag}$ is the three-momentum of the reconstructed $D_{s}^{-}$ candidate in the $e^{+}e^{-}$ center-of-mass frame. In each event, we only keep one candidate per tag mode per charge, selecting the one with the $D_{s}^{-}$ recoil mass

$$M_{\rm rec}\equiv\sqrt{(E_{\rm cm}-\sqrt{|\vec{p}_{\rm tag}|^{2}+m^{2}_{D_{s}}})^{2}-|\vec{p}_{\rm tag}|^{2}},$$

(2)

closest to the nominal $D_{s}^{*+}$ mass pdg2022 . The $D^{+}_{s}$ candidate is selected in the presence of the tag $D^{-}_{s}$. If there are multiple $D^{+}_{s}$ combinations in an event, the one giving the minimum $|M_{D_{s}^{+}}+M_{D_{s}^{-}}-2m_{D_{s}}|$ is retained for further analysis. Here $M_{D^{\pm}_{s}}$ is the invariant mass of the $D^{\pm}_{s}$ candidate and $m_{D_{s}}$ is the nominal $D_{s}$ mass pdg2022 . Figure 1 shows the distribution of $M_{D^{-}_{s}}$ vs. $M_{D^{+}_{s}}$ of the accepted candidates in data. To suppress background, the invariant masses of $K^{+}K^{-}\pi^{\pm}$ and $K^{0}_{S}K^{\pm}$ combinations are required to be within the interval $M_{D^{\pm}_{s}}\in(1.958,1.978)$ GeV/$c^{2}$.

To improve momentum resolution, a two-constraint (2C) kinematic fit, in which the invariant mass of the $K^{+}K^{-}\pi^{\pm}$ or $K^{0}_{S}K^{\pm}$ combination is constrained to the known $D_{s}$ mass pdg2022 is performed. The momenta updated by the kinematic fit are kept for further analysis.

To separate the $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ candidates, we define the missing mass squared of the reconstructed $D^{+}_{s}D^{-}_{s}$ combination as

$$M^{2}_{\rm miss}\equiv\left(E_{\rm cm}-E_{D^{+}_{s}}-E_{D^{-}_{s}}\right)^{2}-|-\vec{p}_{D^{+}_{s}}-\vec{p}_{D^{-}_{s}}|^{2},$$

(3)

where $E_{D^{\pm}_{s}}$ and $\vec{p}_{D^{\pm}_{s}}$ are the energy and momentum of $D^{\pm}_{s}$ in the $e^{+}e^{-}$ center-of-mass system, respectively. The resultant $M^{2}_{\rm miss}$ distribution of the accepted $D^{+}_{s}D^{-}_{s}$ candidate combinations is shown in Fig. 2, where the peak near to zero and its right-side peak correspond to $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ candidates, respectively.

Following Ref. songwm , the BF of $D^{*+}_{s}\to D^{+}_{s}\gamma$ relative to the sum of $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ is determined by

$$f_{\gamma}=\frac{{\cal B}_{D^{*+}_{s}\to D^{+}_{s}\gamma}}{{\cal B}_{D^{*+}_{s}\to D^{+}_{s}\gamma}+{\cal B}_{D^{*+}_{s}\to D^{+}_{s}\pi^{0}}}=\frac{N_{\gamma}^{\rm prod}}{N_{\gamma}^{\rm prod}+N_{\pi^{0}}^{\rm prod}},$$

(4)

where $N_{\gamma}^{\rm prod}$ and $N_{\pi^{0}}^{\rm prod}$ are the numbers of produced $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ events, respectively. This ratio captures the binomial nature of the separation of the low-background signal into the two decays under study.

As shown in Fig. 2, the individual signal regions of $M_{\rm miss}^{2}$ are defined as $[-0.020,0.013]$ and $[0.013,0.040]$ GeV²/$c^{4}$ for $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$, respectively. The dividing line accepts about 99.0% of the $D^{*+}\to D_{s}^{+}\gamma$ signal and about 98.5% of the $D_{s}^{*+}\to D_{s}^{+}\pi^{0}$ signal. Due to the overlapping $M_{\rm miss}^{2}$ distributions, some $D_{s}^{*+}\to D_{s}^{+}\gamma$ events can be misidentified as $D_{s}^{*+}\to D_{s}^{+}\pi^{0}$, and vice versa. To account for this effect, the yields of $N_{\gamma}^{\rm prod}$ and $N_{\pi^{0}}^{\rm prod}$ are obtained by solving the following equation

$$\left(\begin{array}[]{l}N_{\gamma}^{\rm obs}-N_{\gamma}^{\rm bkg}\\ N_{\pi^{0}}^{\rm obs}-N_{\pi^{0}}^{\rm bkg}\end{array}\right)=\left(\begin{array}[]{cc}\epsilon_{\gamma\gamma}&\epsilon_{\pi^{0}\gamma}\\ \epsilon_{\gamma\pi^{0}}&\epsilon_{\pi^{0}\pi^{0}}\end{array}\right)\left(\begin{array}[]{l}N_{\gamma}^{\rm prod}\\ N_{\pi^{0}}^{\rm prod}\end{array}\right),$$

(5)

where $N_{i}^{\rm obs}$ is the number of selected events in data by counting, $N_{i}^{\rm bkg}$ is the number of background events estimated from the inclusive MC sample; $\epsilon_{ij}$ is the efficiency of the generated $D^{*+}_{s}\to D^{+}_{s}+i$ events selected as $D^{*+}_{s}\to D^{+}_{s}+j$, where $i$ and $j$ denote $\gamma$ or $\pi^{0}$. Both $D^{*+}_{s}\to D_{s}^{+}\pi^{0},D_{s}^{+}\gamma$ are simulated. The background rates estimated from the inclusive MC sample for modes I, II, and III are all less than 1.5%.

To consider different detection efficiencies for ISR and FSR effects, the detection efficiencies at various energy points have been weighted by individual single tag $D^{+}_{s}$ yields in data.

Table 2 lists the quantities used for the $f_{\gamma}$ measurements and the results obtained. Weighting the $f_{\gamma}$ results for modes I, II, and III by their inverse statistical uncertainties squared, we obtain their average $f_{\gamma}=(94.20\pm 0.38)\%$.

The systematic uncertainties in the BF measurements are discussed below. The systematic uncertainty due to $M_{{\rm miss}}^{2}$ resolution is examined in the following procedure. We perform a fit to $M_{{\rm miss}}^{2}$ distribution of data. To take into account the resolution difference between data and MC, a signal MC shape smeared with a Gaussian function is used. From the fit, we obtain the parameters (means, widths) of the Gaussian resolution functions, which are $(1.0\pm 0.1,1.0\pm 0.2)$, $(1.1\pm 0.1,1.0\pm 0.2)$, and $(0.4\pm 0.3,1.7\pm 0.4)$ MeV²/$c^{4}$ for modes I, II, and III, respectively. The change of BF before and after smearing the Gaussian resolution function to the $M_{\rm miss}^{2}$ distribution of the signal MC events, 0.07%, is taken as the associated systematic uncertainty.

The systematic uncertainty caused by the statistical uncertainty of the MC efficiencies is estimated by varying each of the efficiency matrix elements by $\pm 1\sigma$. The largest change of the BF is taken as the systematic uncertainty.

The systematic uncertainty from background estimation is considered in two parts. The number of background events is calculated from the inclusive MC sample. The corresponding systematic uncertainty is estimated from the uncertainties of the cross sections used in generating this sample. The dominant background events are from open charm processes of $e^{+}e^{-}\to D^{+}_{s}D^{-}_{s}$ and $e^{+}e^{-}\to D^{*+}_{s}D^{-}_{s}$. The systematic uncertainty is estimated by varying the cross sections and BFs of the hadronic $D^{+}_{s}$ decays by $\pm 1\sigma$. This effect on the BF measurement is negligible. In addition, we have also varied the simulated background events by the ratio of the background events observed in the $D^{+}_{s}D^{-}_{s}$ sideband regions between data and the inclusive MC sample. The change of the BF, 0.10%, is taken as the corresponding systematic uncertainty.

Other possible systematic uncertainty sources, such as the ISR simulation, the kinematic fit, the tracking and the particle identification efficiencies between the two decay modes of $D_{s}^{*+}$, the $M_{\rm BC}$ requirement and the $M^{2}_{\rm miss}$ range, have also been investigated. All of them are negligible.

All systematic uncertainties are summarized in Table 3. Assuming the systematic uncertainties from different sources are independent, the total systematic uncertainty is obtained to be 0.17% by adding all the sources quadratically.

By analyzing 7.33 fb^-1 of $e^{+}e^{-}$ collision data taken at center-of-mass energies between 4.128 and 4.226 GeV, we measure the BF of $D^{*+}_{s}\to D^{+}_{s}\gamma$ relative to the sum of $D^{*+}_{s}\to D^{+}_{s}\gamma$ and $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ to be $f_{\gamma}=(94.20\pm 0.38\pm 0.17)\%$. This gives the BF of $D^{*+}_{s}\to D^{+}_{s}\pi^{0}$ relative to that of $D^{*+}_{s}\to D^{+}_{s}\gamma$ to be ${\cal B}_{D^{*+}_{s}\to D^{+}_{s}\pi^{0}}/{\cal B}_{D^{*+}_{s}\to D^{+}_{s}\gamma}=\frac{1}{f_{\gamma}}-1=(6.16\pm 0.43\pm 0.19)\%$. The $D^{*+}_{s}$ is known to decay dominantly into three final states of $D^{+}_{s}\gamma$, $D^{+}_{s}\pi^{0}$ and $D^{+}_{s}e^{+}e^{-}$ Cheung:2015rya . Combining the world average of ${\cal B}_{D^{*+}_{s}\to D^{+}_{s}e^{+}e^{-}}=(0.67\pm 0.16)\%$ pdg2022 , we obtain ${\cal B}_{D^{*+}_{s}\to D^{+}_{s}\gamma}=(93.57\pm 0.44\pm 0.19)\%$ and ${\cal B}_{D^{*+}_{s}\to D^{+}_{s}\pi^{0}}=(5.76\pm 0.44\pm 0.19)\%$.

Figure 3 shows the comparison of the measured BF of ${\cal B}_{D^{*+}_{s}\to D^{+}_{s}\pi^{0}}/{\cal B}_{D^{*+}_{s}\to D^{+}_{s}\gamma}$ with other experiments and the world average value pdg2022 . Our measurement is well consistent with the previous ones but with better precision. Table 4 shows comparisons of the BFs measured in this work with the world average values and the decay widths or BFs predicted by various theories. Our results of ${\mathcal{B}}_{D^{*}_{s}\to D_{s}^{+}\gamma}$ and ${\mathcal{B}}_{D^{*}_{s}\to D_{s}^{+}\pi^{0}}$ are consistent with those predicted in Ref. Cheung:2015rya . At present, only limits on the $D^{*+}_{s}$ width have been reported. More experimental measurements and theoretical calculations of the $D^{*+}_{s}$ decays will be beneficial to give quantitative tests on the predicted partial decay widths, thereby better understand the radiative and strong decays of $D^{*+}_{s}$. As necessary inputs, the reported BFs with much improved precision are also important for the precise measurements of $f_{D^{+}_{s}}$ and $|V_{cs}|$ by using the reactions of $e^{+}e^{-}\to D_{s}^{*\pm}D_{s}^{\mp}$.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207, U1932102; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement under Contract No. 894790; German Research Foundation DFG under Contracts Nos. 443159800, Collaborative Research Center CRC 1044, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation under Contract No. B16F640076; STFC (United Kingdom); Suranaree University of Technology (SUT), Thailand Science Research and Innovation (TSRI), and National Science Research and Innovation Fund (NSRF) under Contract No. 160355; The Royal Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research Council; U. S. Department of Energy under Contract No. DE-FG02-05ER41374.