Measurement of cross sections for $e^{+}e^{-} \rightarrow \mu^+\mu^-$ at center-of-mass energies from 3.80 to 4.60 GeV

The observed cross sections for $e^+e^-\rightarrow \mu^+\mu^-$ at energies from 3.8 to 4.6 GeV are measured using data samples taken with the BESIII detector operated at the BEPCII collider. We measure the muonic widths and determine the branching fractions of the charmonium states $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$ decaying to $\mu^+\mu^-$, as well as making a first determination of the phase of the amplitudes. In addition, we observe evidence for a structure in the dimuon cross section near 4.220 GeV/$c^2$, which we denote as $S(4220)$. Analyzing a coherent sum of amplitudes yields eight solutions, one of which gives a mass of ${M}_{S(4220)}=4216.7 \pm 8.9 \pm 4.1$~MeV/$c^2$, a total width of ${\rm \Gamma^{\rm tot}_{S(4220)}}=47.2 \pm 22.8 \pm 10.5$~MeV, and a muonic width of ${\rm \Gamma}^{\mu\mu}_{S(4220)}=1.53\pm1.26\pm0.54$~keV, where the first uncertainties are statistical and the second systematic. The eight solutions give the central values of the mass, total width, muonic width to be, respectively, in the range from 4212.8 to 4219.4 MeV/$c^2$, from 36.4 to 49.6 MeV, and from 1.09 to 1.53 keV. The statistical significance of the $S(4220)$ signal is $3.9\sigma$. Correcting the total dimuon cross section for radiative effects yields a statistical significance for this structure of more than $7\sigma$.

The observed cross sections for e + e − → µ + µ − at energies from 3.8 to 4.6 GeV are measured using data samples taken with the BESIII detector operated at the BEPCII collider. We measure the muonic widths and determine the branching fractions of the charmonium states ψ(4040), ψ(4160), and ψ(4415) decaying to µ + µ − , as well as making a first determination of the phase of the amplitudes. In addition, we observe evidence for a structure in the dimuon cross section near 4.220 GeV/c 2 , which we denote as S(4220). Analyzing a coherent sum of amplitudes yields eight solutions, one of which gives a mass of M S(4220) = 4216.7±8.9±4.1 MeV/c 2 , a total width of Γ tot S(4220) = 47.2±22.8±10.5 MeV, and a muonic width of Γ µµ S(4220) = 1.53±1.26±0.54 keV, where the first uncertainties are statistical and the second systematic. The eight solutions give the central values of the mass, total width, muonic width to be, respectively, in the range from 4212.8 to 4219.4 MeV/c 2 , from 36.4 to 49.6 MeV, and from 1.09 to 1.53 keV. The statistical significance of the S(4220) signal is 3.9σ. Correcting the total dimuon cross section for radiative effects yields a statistical significance for this structure of more than 7σ.
For a long time the meson resonances produced in e + e − collisions above the open-charm (OC) and below the openbottom thresholds had been thought to decay entirely to OC final states through the strong interaction. Consequently, the possibility of non-open-charm (NOC) decays attracted little experimental interest until the early years of the millennium. For convenience, in this Letter we denote these resonances X above DD , which encompasses both heavy cc states, i.e. ψ(3770), ψ(4040), ψ(4160), and ψ(4415), and non-cc states, such as four-quark composites, hybrid charmonium states, and open-charm molecule states [1][2][3] that are expected by QCD. Finding these non-cc states would be a crucial validation of the QCD predictions.
Since non-cc states may easily decay to non-open-charm final states, such decays of X above DD mesons were searched for by the BES collaboration using the data collected with the BES-I detector at energies of 4.04 and 4.14 GeV, and the BES-II detector at energies around 3.773 GeV. The first evidence for such decays was reported in the J/ψπ + π − final state by BES in 2003 [4]. This final state could come from the decay of a cc or a non-cc state, or even both of these states. On the assumption that there is no other resonance at energies near 3.773 GeV, the signal was interpreted to be ψ(3770) → J/ψπ + π − [5]. This first NOC decay was confirmed by the CLEO collaboration [6] two year after the BES analysis. This discovery overturned the conventional understanding that X above DD decay into open-charm final states through the strong interaction with branching fractions of almost 100%. It stimulated strong interest in searching for other non-open-charm decays of X above DD mesons [7], in particu-lar into J/ψπ + π − and similar final states, and led to the discovery of several new resonances [8][9][10].
In the last 17 years, several new states [8][9][10], and new di-structures, such as the Rs(3770) [11] and R(4220) and R(4320) [12], as well as structures lying above 4.2 GeV [13,14] have been observed in e + e − annihilation at energies above the open-charm threshold. The X(3872) [8], Y (4260) [9], and R(4220) and R(4320) [12] resonances were observed in the J/ψπ + π − final state, while the Y (4360) [10] and Y (4660) [10] were observed in the ψ(3686)π + π − final state. In addition, the Y (4220) [13] was observed in the final state ωχ c0 , and the Y (4220) and Y (4390) [12] were observed in the final state h c π + π − . All of these resonances were observed in final states of inclusive hadrons, where no attempt was made to identify the hadron species, and in final states of M cc X LH , where M cc is a hidden-charm meson and X LH is a light hadron. In Ref. [15] it was suggested to search for new vector states at BESIII by means of analyzing the lineshape of cross sections for e + e − → f NOC , where f NOC can be J/ψX, ψ(3686)X (X =light hadrons or photons), light hadrons only, or inclusive hadrons.
In addition to studying the final states J/ψX or ψ(3686)X produced in e + e − annihilations, searches for new vector states may be performed by analyzing the cross section for e + e − → µ + µ − , in which the contribution from the decays of heavy cc resonances are strongly suppressed, and consequently the production and decay of the non-cc states can be significantly enhanced.
In this Letter, we report measurements of the cross section for e + e − → µ + µ − at center-of-mass (c.m.) energies from 3.8 to 4.6 GeV, and studies of the known cc resonances and searches for new structures in this regime by performing an analysis of a coherent sum of amplitudes contributing of this cross section. The data samples used in measuring the cross section were collected at 133 c.m. energies with the BESIII detector operated at the BEPCII collider from 2011 to 2017. The total integrated luminosity of the data sets used in the analysis is 13.2 fb −1 , determined from large-angle Bhabha scattering events [16]. The c.m. energy of each data set is measured using dimuon events, with an uncertainty of ±0.8 MeV [17].
The BESIII detector is described in detail in Ref. [18]. The detector response is studied using samples of Monte Carlo (MC) events which are simulated with the GEANT4based [19] detector simulation software package BOOST. Simulated samples for all vector qq states (i.e. uū, dd, ss, and cc resonances) and their decays to µ + µ − are generated with the MC event generator BABAYAGA [20]. Possible background sources are estimated with Monte Carlo simulated events generated with the event generator KKMC [21]. The detection efficiency is determined with Monte Carlo simulated e + e − → µ + µ − events generated with BABAYAGA, which includes initial and final state radiation, as well as vacuum polarization effects.
The observed cross section for e + e − → µ + µ − at a certain c.m. energy √ s is determined by where N obs is the background-subtracted number of observed events for e + e − → µ + µ − , L is the integrated luminosity, and ǫ is the detection efficiency.
Each candidate for e + e − → µ + µ − is required to have two tracks of opposite charge. Each of the two charged tracks must satisfy | cos θ| < 0.8, where θ is the polar angle of the tracks. In addition, the charged tracks are required to satisfy V r < 1.0 cm and |V z | < 10.0 cm, where V r is the distance of closest approach to the interaction point in the r-φ plane, and |V z | is the distance between the point of the closest approach and the interaction point along the beam axis. Furthermore, the total momentum | p + | + | p − | of the two charged tracks is required to be greater than 90% of the nominal collision energy √ s. To reject Bhabha scattering events, we require the ratio of the energy E ± deposited in the electromagnetic calorimeter to the momentum p ± of the charged track to satisfy 0.05 < E ± /p ± < 0.40. This criterion also rejects π + π − pairs. The rejection fraction for π + π − events is energy dependent, ranging from 41.5% at 3.8 GeV to 37.5% at 4.6 GeV. The remaining π + π − background is subtracted using the extrapolation of the e + e − → π + π − cross section measured by the CLEO collaboration [22] and the rate of misidentifying π + π − as µ + µ − obtained from the MC simulation. In order to reduce the K + K − and pp background, the event is subjected to a four-constraint kinematic fit with the hypothesis e + e − → µ + µ − , constraining the total four-momentum of the µ + µ − to that of the colliding beams, and the fit χ 2 4C is required to be less than 60.
The number of e + e − → µ + µ − candidates is determined by analyzing the ratio E µ + µ − / √ s, where E µ + µ − is the total energy of µ + and µ − determined from the measured track momenta. As an example, Fig. 1 (left) shows the distribution of E µ + µ − / √ s for the events selected from the data collected at √ s = 4.420 GeV. A fit to the distribution with a double-Gaussian function for the signal shape and a first order polynomial to describe the background yields N fit =(2500.2 ± 1.6) × 10 3 e + e − → µ + µ − candidates, where the uncertainty is statistical. The systematic uncertainty due to the non-peaking background (mainly cosmic rays and beamgas events) is estimated to be less than 0.01%, and therefore negligible. The imperfection of the signal peak description is taken into account as a systematic uncertainty (see below). The signal yield N fit is still contaminated by peaking background from several sources, e.g. e + e − → (γ)e + e − , e + e − → π + π − , and e + e − → K + K − . Using the highstatistics samples of MC simulated events and the extrapolated cross sections for these processes, the number of the background events is estimated to be N b = 4764 ± 18, where the uncertainty is mostly due to the cross-section extrapolation. Subtracting N b from N fit yields N obs =(2495.4 ± 1.6) × 10 3 signal events.
The integrated luminosity of the data sample taken at 4.420 GeV was previously measured to be L = 1043.9 ± 0.1 ± 6.9 pb −1 [16], where the first uncertainty is statistical and the second one is systematic. At 4.420 GeV, the detection efficiency of e + e − → µ + µ − is ǫ = (41.09 ± 0.01)%, as determined from the MC. Using these values in Eq. (1) yields the observed cross section of σ obs (e + e − → µ + µ − ) = 5.818 ± 0.010 ± 0.169 nb. The first error includes the uncertainties of statistical origin (signal sample size, MC event statistics and the statistical uncertainty of the luminosity measurement). The second error represents the remaining systematic uncertainties (see below). Similarly, we determine the observed cross sections for e + e − → µ + µ − at the other 132 energies from 3.81 to 4.6 GeV.
The systematic uncertainty for the observed cross section originates from several sources. They are 1% due to the luminosity measurement, 1% per track associated with the knowledge of the tracking efficiency, 0.64% due to requiring | cos θ| < 0.8, 0.59% due to requiring | p + | + | p − | > 0.9 √ s, 0.12% due to the selection on E ± /| p ± |, 0.41% due to the four-constraint kinematic fit, 1.23% due to the fit to the E µ + µ − / √ s distribution, 0.03% due to the background subtraction, and 1% due to the theoretical uncertainty associated with the Monte Carlo generator. An additional uncertainty arises from the imperfect description of the signal shape by the fit (see Figure 1). This effect is only partially compensated by the MC, and the residual uncertainty is 0.03%. Adding these uncertainties in quadrature yields a total systematic uncertainty of 2.91%.
To search for new vector states in e + e − → µ + µ − , a χ 2 fit is performed to the measured cross section. In the fit, the expected cross section is given by [23,24] where m µ is the mass of muon and F (x, s) is a sampling function [25] for the radiative photon energy fraction x. σ D (s(1 − x)) is the dressed cross section including vacuum-polarization effects, where A cnt , A R k and A S are, respectively, the amplitude for continuum e + e − → µ + µ − production, the Breit-Wigner (BW) amplitude describing nine vector resonances (ρ(770), ω(782), φ(1020), J/ψ, ψ(3686), ψ(3770), ψ(4040), ψ(4160), and ψ(4415)), and a new vector structure S decaying into µ + µ − , while φ R k and φ S are the corresponding phases of the amplitudes. The continuum amplitude can be parameterized as A cnt = f cnt /s ′ , where f cnt is a free parameter, and s ′ = s(1 − x). The decay amplitude of resonance R, being either one of the known vector states or the new structure S, is written as R , Γ µµ R and Γ tot R are the mass, electron width, muonic width, and total width, respectively.
In the fit the observed cross-section values are assumed to be influenced only by the uncertainties of statistical origin. The uncertainties on the parameters returned by the fit are referred to as statistical uncertainties in the subsequent discussion. The remaining cross-section uncertainties (assumed to be fully correlated between different energies) are taken into account using the "offset method" [26]: the cross-section values are changed for all energies simultaneously by the size of the uncertainty and the resulting change in the fit parameter is taken as a systematic uncertainty.
Since the analysis does not include the observed cross section at energies below 3.8 GeV, the parameters of the first six lower mass vector resonances are all fixed to the values given by the particle data group (PDG) [27], and the phases are fixed to zero. For the three heavy cc states, i.e. ψ(4040), ψ(4160), and ψ(4415), the masses and the total widths are also fixed to the values given by the PDG. The partial widths Γ µµ and the phases are left free, and lepton universality is assumed (i.e. Γ ee R = Γ µµ R ). It is noted that the earlier studies contributing to the values for Γ ee R reported in Ref. [27] did not consider the contributions from non-cc states in the calculations of the Initial State Radiative (ISR) correction factors; furthermore they assumed a selection efficiency for e + e − → hadrons that is a smooth curve, increasing as the c.m. energy increases, rather than the BW-like function observed in e.g. Fig. 1(b) of Ref. [28]. Neglecting these effects may lead to bias, as may the difficulties of accounting for interference effects between the continuum e + e − → hadrons amplitude and the resonance decay amplitudes. Following these considerations we leave these partial widths as free parameters in our fit.
The fit returns eight acceptable solutions with distinct results for the four free phases. Table I shows the results from the fit. All solutions include a result for a new structure with mass close to 4220 MeV, and so we denote this possible state as S(4220). For Solution I, the fit returns f cnt = 88.51 ± 0.11 nb/GeV 2 and χ 2 = 135.47 for 121 degrees of freedom. Taking Γ µµ S(4220) = Γ tot S(4220) B(S(4220) → µ + µ − ), where B(S(4220) → µ + µ − ) is the branching fraction for the decay of S(4220) → µ + µ − , the fit yields Γ ee S(4220) B(S(4220) → µ + µ − ) = 0.05 ± 0.06 ± 0.03 eV, where the first uncertainty is statistical and the second is systematic. Figure 2 (left) shows the observed cross sections with a fit to the sum of eleven contributions: continuum e + e − → µ + µ − , the nine known vector states and the S(4220) decay into µ + µ − . The black empty circles in Fig. 2 are for the lower luminosity data (integrated luminosity less than 12 pb −1 ), the filled red circles are for the higher luminosity data, the solid line is for the fit, and the dashed line is for the contribution from the e + e − → µ + µ − continuum. Figure 2 (right) shows the corresponding observed cross section, for which both the contributions from the continuum e + e − → µ + µ − and the decay ψ(3686) → µ + µ − are subtracted. Removing the S(4220) from the fit yields a χ 2 change by 23.78, for a reduction of four degrees of freedom, which corresponds to a statistical significance for the structure of 3.9σ.
The systematic uncertainties on the values of the parameters given in Table I originate from three sources: (1) systematic uncertainties on the observed cross sections, (2) uncertainties on the parameters for the ψ(3686), ψ(3770), ψ(4040), ψ(4160), and ψ(4415) states, (3) uncertainties on the c.m. energies. Adding these contributions in quadrature we obtain the total systematic uncertainties for the fit parameters, which are listed as the second uncertainties in Table I. Initial State Radiation distorts the shape of the resonances   Fig. 2 (right)), but also reduce the heights of these peaks, which weakens the significance of the signals. Figure 3 (left) shows the corresponding Born-continuum-dressed-resonance (BC-DR) cross section, which is the observed cross section divided by the ISR correction factor f ISR (s), with f ISR (s) = σ obs µ + µ − (s)/σ D µ + µ − (s), where σ obs µ + µ − (s) is given in Eq. (2) and The BCDR cross section is the sum of the Born continuum cross section of e + e − → µ + µ − and the dressed cross sections for the resonances decaying into µ + µ − . The ISR correction removes the ISR-return events (see cross section around 4.02, 4.20, and 4.36 GeV in Fig.3 (right)) and restores the heights of the signal peaks, making the S(4220) signal to be more pronounced and more clearly seen in the BCDR cross sections. Removing the S(4220) from the fit to the BCDR cross section causes a χ 2 change by 78.20, for a reduction of four degrees of free-dom. This change corresponds to a statistical significance of more than 7σ for the S(4220) structure. Analysis of an ensemble of simulated data sets of e + e − → h c π + π − generated using the Y(4220) and Y(4390) resonance parameters [12] demonstrates that the signal significance of structures seen in the dressed cross section typically exceeds those seen in the observed cross section by about 4 sigma, which is compatible with the increase seen in the data.
Our measured muonic widths for the ψ(4040) and ψ(4415) are consistent within ∼ 1.3 times the uncertainties with theoretical expectations for the electronic widths of these states, which are 1.42 and 0.70 keV, respectively [29].
In summary, we have measured the cross section for e + e − → µ + µ − at c.m. energies from 3.8 to 4.6 GeV. For the first time we have directly measured the muonic widths and branching fractions of ψ(4040), ψ(4160) and ψ(4415), and determined the phases of the decay amplitudes for these three resonances. The relative phases of these three vector states range from (−78 ± 33) to (157 ± 39) degrees.
In addition, we have found evidence for a structure S(4220) lying near to 4.22 GeV/c 2 with a mass of M S(4220) = 4216.7 ± 8.9 ± 4.1 MeV/c 2 , a total width of Γ tot S(4220) = 47.2 ± 22.8 ± 10.5 MeV, and a muonic width of Γ µµ S(4220) = 1.53 ± 1.26 ± 0.54 keV, where the first uncertainties are statistical and the second are systematic. The statistical significance of the S(4220) signal is 3.9σ. The analysis of the BCDR cross section of e + e − → µ + µ − yields a statistical significance of the S(4220) signal of more than 7σ. Although the dimuon branching fractions of the X above DD decays are all at the level of ∼ 10 −5 , the interference of these decays with the e + e − → µ + µ − continuum produces a measurable contribution to the cross section, whose shape is sensitive to new states. Therefore the analysis of the e + e − → µ + µ − cross section in the energy region between 3.73 and 4.8 GeV is a promising way to discover new vector states, complementing the study of the processes e + e − → J/ψX and e + e − → M cc X LH .