Search for a massless dark photon in $\Lambda^{+}_{c}\to p \gamma^{\prime}$ decay

A search for a massless dark photon $\gamma^{\prime}$ is conducted using 4.5 $\mathrm{fb}^{-1}$ of $e^+e^-$ collision data collected at center-of-mass energies between 4.600 and 4.699 $\mathrm{GeV}$ with the BESIII detector at BEPCII. No significant signal is observed, and the upper limit on the branching fraction $\mathcal{B}(\Lambda^{+}_{c}\to p \gamma^{\prime})$ is determined to be $8.0\times 10^{-5}$ at 90% confidence level.


I. INTRODUCTION
The flavor changing neutral current (FCNC) transitions of the charmed baryon Λ + c are of great interest since they can provide indications for physics beyond the Standard Model (SM). In the framework of the SM, FCNC transitions are strongly suppressed by the Glashow, Iliopoulos and Maiani (GIM) mechanism [1] in the charm sector. The SM predictions for the branching fractions (BFs) of FCNC decays in the charm sector are less than 10 −9 [2]. The minimal super-symmetric SM with Rparity violation [3] and the two-Higgs-doublet model [4] predict the BFs of the same FCNC decays to be two to three orders of magnitude larger. Observation of a FCNC decay with the current experimental sensitivity would imply new physics beyond the SM.
Models of new physics beyond the SM may have a dark sector containing an extra Abelian gauge group, U (1) D , under which all the SM fields are singlets. This symmetry may be broken spontaneously or may remain unbroken, causing the associated gauge boson, the dark photon, to acquire a mass or remain massless. These possibilities have received a great deal of attention in recent decades [5][6][7][8][9][10][11][12][13][14][15]. If U (1) D remains unbroken, there is always a linear combination of the dark and SM Abelian gauge fields which does not have renormalizable cou-plings to SM members and which can be identified with the massless dark photon (γ ) [5,6]. While it has no direct interactions with SM fermions, the γ can still exert influence on the SM via higher-dimensional operators generated by loop diagrams involving particles that are charged under U (1) D and also coupled to SM fields [6][7][8].
In experiment, LHCb reported the evidence for the breaking of lepton universality in bottom-quark FCNC decays to charged lepton pairs with a significance of 3.1σ [16]. As a complementary study, we concentrate on FCNC effects arising from the dark photon with the c and u quarks, where the missing energy due to the dark photon is the feature of the signal processes. BESIII has searched for the invisible signals within various hadron decays, including η/η → invisible [17], ω/φ → invisible [18], Λ → invisible [19] and J/ψ → γ + invisible [20], and no significant signals are observed. However, this has never been probed in the charmed baryon sector. The two-body charmed baryon decay potentially offers a competitive window to access c → uγ , which gives rise to the FCNC decays of charmed baryon into a lighter baryon plus missing energy carried away by the massless dark photon. Figure 1 presents a typical Feynman diagram of Λ + c → pγ . It is found that the BFs of some charmed baryon decays are allowed to be as high as a few times 10 −5 [21] . Such BFs are likely to be within the sensitivity reaches of some ongoing experiments like BESIII. This paper presents an experimental search for a massless dark photon in Λ + c → pγ decay using 4.5 fb −1 of e + e − collision data collected with the BESIII detector at seven center-of-mass (CM) energies between 4.600 and 4.699 GeV. The CM energies and the integrated luminosities for each energy point are listed in Table 1 [26][27][28]. Taking advantage of the Λ + cΛ − c production just above the mass threshold 4572.92 MeV, a double-tag (DT) approach [22] is implemented. Throughout the text, the charge conjugate states are always implied.

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector [23] records symmetric e + e − collisions provided by the BEPCII storage ring, which operates with a peak luminosity of 1 × 10 33 cm −2 s −1 in the CM energy range from 2.0 GeV to 4.9 GeV. BESIII has collected large data samples at these energy regions [24]. The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator timeof-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 based muon identification modules interleaved with steel. The charged-particle momentum resolution at 1 GeV/c is 0.5%, and resolution of the ionization energy loss in the MDC (dE/dx) is 6% for 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, while that in the end cap region is 110 ps. The end cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [25]. About 13% of the data (the 4.600 GeV sample) used in the current analysis predates this upgrade.
Monte Carlo (MC) simulated data samples are produced with a geant4-based [29] MC package, which includes the geometric description and response of the BESIII detector. The signal MC samples of e + e − → Λ + cΛ − c withΛ − c decaying into ten specific tag modes (as described below and listed in Table 2) and Λ + c → pγ , which are used to determine the detection efficiencies, are generated for each CM energy using the generator kkmc [30] incorporating initial-state radiation (ISR) effects and the beam energy spread. The inclusive MC samples, which consist of Λ + cΛ − c events, charmed meson D ( * ) (s) pair production, ISR return to the charmonium(like) ψ states at lower masses, and continuum processes e + e − → qq (q = u, d, s), are generated to estimate the potential background. Decay modes as specified in the Particle Data Group (PDG) [33] are modeled with evtgen [31,32], and the remaining unknown decays are modeled with lundcharm [34,35]. Final state radiation (FSR) from charged final state particles is incorporated using photos [36].

III. METHODOLOGY
A DT approach [22] is implemented to search for Λ + c → pγ . A data sample ofΛ − c baryon, referred to as the single-tag (ST) sample, is reconstructed with ten exclusive hadronic decay modes, as listed in Table 2. The subset of those events in which a signal decay Λ + c → pγ candidate is reconstructed in the system recoiling against theΛ − c candidate are denoted as DT candidates. The Λ + c decay BF is determined as where the N obs is the number of observed events in the signal region from data, and N bkg is the number of estimated background as explained explicitly in Section VI. The subscripts i and j represent the ST modes and the data samples at different CM energies, respectively. The parameters N ST ij , ST ij and DT ij are the ST yields, ST and DT detection efficiencies, respectively.

IV. ST EVENT SELECTIONS
Charged tracks detected in the MDC are required to be within a polar angle (θ) range of |cos θ| < 0.93, where θ is defined with respect to the positron beam direction. For prompt tracks not from K 0 S andΛ decays, the distances of the closest approach to the interaction point (IP) are required to be within ±10 cm along the beam direction and 1 cm in the plane perpendicular to the beam (referred to as tight track hereafter). The particle identification (PID) is implemented by combining measurements of the dE/dx and the flight time in the TOF. Every charged track is assigned a particle type of pion, kaon or proton, by choosing the type with the highest probability.
Photon candidates are selected from showers reconstructed in the EMC. The deposited energy of each shower must be more than 25 MeV in the barrel region (|cos θ| ≤ 0.80) or more than 50 MeV in the end cap region (0.86 ≤ |cos θ| ≤ 0.92). To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within (0, 700) ns. The π 0 candidates are reconstructed from photon pairs with an invariant mass in the range (0.115, 0.150) GeV/c 2 . To improve the resolution, a kinematic fit is performed constraining the invariant mass of the photon pair to the known π 0 mass [33]. The corresponding χ 2 of the fit must be less than 200. The momenta updated by the kinematic fit are used in further analysis.
Candidates for K 0 S andΛ are reconstructed in their decays to π + π − andpπ + , respectively. Each charged track must have a distance of closest approach to the IP within ±20 cm along the beam direction (referred to as loose track hereafter). To improve the signal purity, PID is applied to the proton candidates, but not the pion candidates. A secondary vertex fit is performed to each K 0 S orΛ candidate, and the momenta updated by the fit are used in the further analysis. To keep a high signal efficiency, a K 0 S orΛ candidate is accepted if the χ 2 of this fit is less than 100. Furthermore, the decay vertex is required to be separated from the IP by a distance of at least twice the fitted vertex resolution, and the invariant mass must be within (0.487, 0.511) GeV/c 2 for π + π − or (1. c candidates are required to satisfy the tag-mode dependent ∆E requirements, the asymmetric intervals of which take into account the effects of ISR and correspond to three times the resolution around the peak, as summarized in Table 2. If there are more than one candidate satisfying the above requirements for a specific tag mode, the one with the minimum |∆E| is kept. For theΛ − c →pK 0 S π 0 ST mode, candidate events with Mp π + ∈ (1.100, 1.125) GeV/c 2 and Mp π 0 ∈ (1.170, 1.200) GeV/c 2 are vetoed to avoid double counting with theΛ − c →Λπ − π 0 orΛ − c →Σ − π + π − ST modes, respectively. For theΛ − c →Σ − π + π − ST mode, candidate events with M π + π − ∈ (0.490, 0.510) GeV/c 2 and Mp π + ∈ (1.110, 1.120) GeV/c 2 are rejected to avoid double counting with theΛ − c →pK 0 S π 0 or Λ − c →Λπ − π 0 ST modes, respectively. In theΛ − c → pK 0 S π + π − andΛπ − π + π − selections, candidate events with Mp π + ∈ (1.100, 1.125) GeV/c 2 and M π + π − ∈ (0.490, 0.510) GeV/c 2 are rejected, respectively.
The M BC distributions of candidates for the ten ST modes with the data sample at √ s = 4.600 GeV are illustrated in Fig. 2, where clearΛ − c signals are observed in each mode. No peaking background is found using the inclusive MC samples. To obtain the ST yields, unbinned maximum likelihood fits on these M BC distributions are performed, where the signal shape is modeled with the MC-simulated shape convolved with a Gaussian function representing for the resolution difference between data and MC simulation, and the background shape is described by the ARGUS function [37]. The candidates with M BC ∈ (2.275, 2.310) GeV/c 2 are retained for further analysis, and the signal yields for the individual ST modes are summarized in Table 2. The same procedure is performed for the other six data samples at different CM energies, the results can be found in Ref. [

V. RECONSTRUCTION OF pγ CANDIDATES
The decay Λ + c → pγ is searched for among the remaining tracks and showers recoiling against theΛ − c candidates. Only one tight track is allowed, and it must satisfy the PID criteria L(p) > L(K) and L(p) > L(π). To suppress contamination from long-lifetime particles in the final state, the candidate events are further required to be without any unused loose tracks. The γ does not interact directly with SM fermions and thus it deposits no energy in the EMC. Backgrounds containing a π 0 are vetoed with the requirement of E max < 0.3 GeV and E sum < 0.5 GeV, where E max and E sum are the maximum energy and the energy sum of the unused showers, respectively. The γ signal is selected using the square of the recoil mass, M 2 rec(Λ − c p) , against the STΛ − c and p. After imposing all selection conditions mentioned above, the distribution of M 2 rec(Λ − c p) of the accepted DT candidate events from the combined seven data samples at different CM energies is shown in Fig. 3(a). There is a peaking structure at the K 0 L mass position, from the process Λ + c → pK 0 L .

VI. BACKGROUND ANALYSIS
The potential background can be classified into two categories: those directly originated from continuum hadron production in the e + e − annihilation, denoted as qq background, and those from the e + e − → Λ +  Fig. 3(a).

VII. UPPER LIMIT SETTING
In order to extract the signal yield, a signal region is defined as (0.0, 0.1) GeV 2 /c 4 in the M 2 rec(Λ − c p) distribution, corresponding to a 96% signal detection efficiency after imposing all the selection criteria. The distribution of M 2 rec(Λ − c p) in the signal region is shown in Fig. 3(b). Thirteen candidate events are observed in the signal region, while the background events are estimated to be 14.6 ± 1.5, where the uncertainty is statistical only. The Λ + c decay BF is determined as in Eq. (1). The detec-  c → pK 0 L process. The brown hatched histogram represents the qq background. The yellow hatched histogram is the Λ + c → pγ signal, which is normalized to data luminosity with the upper limit of the BF. tion efficiency ST ij is obtained with the same procedure as in Ref. [38], and the DT ij is derived with the signal MC samples. The DT efficiencies are summarized in Table 3. Since no significant signal is observed, the profilelikelihood approach [39] is used to determine the upper limit on the BF of Λ + c → pγ . The likelihood function which depends on the parameter of interest B(Λ + c → pγ ) and the nuisance parameters θ = ( eff , N bkg ) is defined as: where the observed events are assumed to follow a Poisson distribution (P ). The N exp is the expected number of events; it is defined as the sum of the number of background events and number of signal events estimated in the signal region, corresponding to Eq. (1). The detection efficiency eff and N bkg follow Gaussian distributions (G). The upper limit on the BF of Λ + c → pγ is determined by scanning the parameter of interest. The resultant profilelikelihood scan distribution is presented in Fig. 4. The upper limit is calculated to be B(Λ + c → pγ ) < 8.0 × 10 −5 at 90% confidence level (CL), where the statistical and systematic uncertainties are all incorporated. The systematic uncertainties associated with the detection efficiency and the background estimation are performed with the two nuisance parameters in a profile-likelihood fit.

VIII. SYSTEMATIC UNCERTAINTY
The systematic uncertainties for the BF measurement include those associated with the ST yields (N ST ij ), reconstruction efficiencies of the STΛ − c ( ST ij ) and reconstruction efficiencies of the DT ( DT ij ). As the DT technique is adopted, the systematic uncertainties originating from reconstructing the ST side largely cancel. Table 4 summarizes the possible sources of systematic uncertainties. Each of them is evaluated relative to the measured BF. The details are described below.
The uncertainties associated with the proton tracking and PID efficiencies are determined with the control sample J/ψ → ppπ + π − [40]. The systematic uncertainties in proton tracking and PID are assigned to be 1.0% each.
The uncertainty in the ST yields is 0.5%, which arises from the statistical uncertainty and fitting M BC distributions. The uncertainty in the fitting procedure is evaluated by varying the ARGUS background parameter and changing the Gaussian function to a sum of two Gaussian functions.
The systematic uncertainty for the requirements of E max and E sum is studied using the control sample of Λ + c → pK − π + . The systematic uncertainty, 3.0%, is determined by comparing the efficiencies between data and MC simulation.
According to Eq. (1), the uncertainty related to the ST efficiency is expected to be canceled. However, due to the different multiplicities of tracks and showers in the rest of the event, the ST efficiencies estimated with the generic and the signal MC samples are expected to differ slightly. Thus, the uncertainty associated with the ST efficiency is not canceled fully, which results in a so called "tag bias" uncertainty. The difference of ST efficiency between generic and the signal MC samples, 0.9%, is assigned as the corresponding uncertainty.
The uncertainty in the qq background estimation is obtained by widening the M BC sideband range by 5 MeV/c 2 in comparison to the nominal one, yielding 4.1%. In addition, the statistical uncertainty of 5.7% on the data yield in the M BC sideband range (2.21,2.26) GeV/c 2 is taken into account. The total qq background systematic uncertainty of 7.0% is the quadratic sum of these two effects.
The main Λ + cΛ − c background is the Λ + c → pK 0 L process. The uncertainty of its BF quoted from the PDG on B(Λ + c → pK 0 S ) [33] is 5.0%, and the fraction of pK 0 L in the signal region is 36%. The net systematic uncertainty in Λ + cΛ − c background estimation is thus 1.8%. Other uncertainties are negligible. Assuming that all the sources of the uncertainties are uncorrelated, the total systematic uncertainties associated with the detection efficiency and the background estimation are 3.5% and 7.2%, respectively.

IX. SUMMARY
In summary, with a sample of 4.5 fb −1 collected at CM energies between 4.600 and 4.699 GeV with the BESIII detector, the first investigation for a massless dark photon in Λ + c → pγ decay is carried out. No significant signal is observed with respect to the expected background. The upper limit on the BF of Λ + c → pγ is measured to be B(Λ + c → pγ ) < 8.0 × 10 −5 at 90% CL. It is below the sensitivity of theory prediction in Ref. [21], which predicts the BF to be 1.6 × 10 −5 or 9.1 × 10 −6 with different inputs of form factors. A more stringent constrain on B(Λ + c → pγ ) is expected in the near future with larger Λ + c samples at BESIII [24].