First Determination of the Spin and Parity of a Charmed-Strange Baryon, $\Xi_{c}(2970)^+$

We report results from a study of the spin and parity of $\Xi_{c}(2970)^+$ using a $980~\mathrm{fb^{-1}}$ data sample collected by the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. The decay angle distributions in the chain $\Xi_{c}(2970)^+ \to \Xi_c(2645)^{0}\pi^{+} \to \Xi_c^{+}\pi^{-}\pi^{+}$ are analyzed to determine the spin of this charmed-strange baryon. The angular distributions strongly favor the $\Xi_{c}(2970)^+$ spin $J =1/2$ over $3/2$ or $5/2$, under an assumption that the lowest partial wave dominates in the decay. We also measure the ratio of $\Xi_{c}(2970)^+$ decay branching fractions $R={\mathcal{B}[ \Xi_{c}(2970)^+ \to \Xi_c(2645)^{0}\pi^{+} ]} / { \mathcal{B}[ \Xi_{c}(2970)^+ \to \Xi_c^{\prime0}\pi^{+} ]} =1.67 \pm 0.29\mathrm{(stat.)}^{ +0.15}_{ -0.09}\mathrm{(syst.)} \pm 0.25\mathrm{(IS)}$, where the last uncertainty is due to possible isospin-symmetry-breaking effects. This $R$ value favors the spin-parity $J^P=1/2^+$ with the spin of the light-quark degrees of freedom $s_{l}=0$. This is the first determination of the spin and parity of a charmed-strange baryon.


Abstract
We report results from a study of the spin and parity of Ξ c (2970) + using a 980 fb −1 data sample collected by the Belle detector at the KEKB asymmetric-energy e + e − collider. The decay angle distributions in the chain Ξ c (2970) + → Ξ c (2645) 0 π + → Ξ + c π − π + are analyzed to determine the spin of this charmed-strange baryon. The angular distributions strongly favor the Ξ c (2970) + spin J = 1/2 over 3/2 or 5/2, under an assumption that the lowest partial wave dominates in the decay. We also measure the ratio of Ξ c (2970) + decay branching fractions R = B[Ξ c (2970) + → Ξ c (2645) 0 π + ]/B[Ξ c (2970) + → Ξ ′0 c π + ] = 1.67 ± 0.29(stat.) +0.15 −0.09 (syst.) ± 0.25(IS), where the last uncertainty is due to possible isospin-symmetry-breaking effects. This R value favors the spin-parity J P = 1/2 + with the spin of the light-quark degrees of freedom s l = 0. This is the first determination of the spin and parity of a charmed-strange baryon.
Charmed-strange baryons comprise one light (up or down) quark, one strange quark, and a more massive charm quark. They provide an excellent laboratory to test various theoretical models, in which the three constituent quarks are effectively described in terms of a heavy quark plus a light diquark system [1,2]. The ground and excited states of Ξ c baryons have been observed during the last few decades [3]. At present there is no experimental determination of their spins or parities.
Excited Ξ c states with an excitation energy less than 400 MeV can be uniquely identified as particular states predicted by the quark model [4]. However, in the higher excitation region, there are multiple states within the typical mass accuracy of quark-model predictions of around 50 MeV, making a unique identification challenging. In order to identify and understand the nature of excited Ξ c baryons, experimental determination of their spin-parity is indispensable.
In this Letter, we report the first measurement of the spin-parity of a Ξ c baryon. We choose Ξ c (2970), earlier known as Ξ c (2980), an excited state of the lightest charmed-strange baryons, for which a plausible spin-parity assignment is not given by the Particle Data Group [4]. It was first observed in the decay mode Λ + cK π by Belle [5] and later confirmed by BaBar [6] in the same decay mode. It was also observed in the Ξ c (2645)π channel at Belle [7]. Its mass and width have been precisely measured with a larger data sample using the Ξ c (2645)π channel by a recent study [8], which also observed the decay mode Ξ ′ c π for the first time. The high statistics of the Belle data, especially for the Ξ c (2645)π channel, recorded in a clean e + e − environment provides an ideal setting for the experimental determination of the spin and parity of charmed-strange baryons.
Theoretically, there are many possibilities for the spin-parity assignment of Ξ c (2970). For example, a quark-model calculation by Roberts and Pervin [9] listed J P = 1/2 + , 3/2 + , 5/2 + , and 5/2 − as possible candidates. Similarly, most quark-model-based calculations predict the Ξ c (2970) as a 2S state with J P = 1/2 + or 3/2 + [1, 2, 10-12], while some of them find negative parity states in the close vicinity [1,13]. There are even calculations that directly assign negative parity to the Ξ c (2970) [14,15]. The unclear theoretical situation motivates an experimental determination of the spin-parity of the Ξ c (2970) + that will provide important information to test these predictions and help decipher the nature of the state.
In this study, the spin is determined by testing possible spin hypotheses of Ξ c (2970) + with angular analysis of the decay Ξ c (2970) + → Ξ c (2645) 0 π + → Ξ + c π − π + . Similarly, its parity is established from the ratio of branching fractions of the two decays, Ξ c (2970) + → Ξ c (2645) 0 π + and Ξ c (2970) + → Ξ ′0 c π + . We note that recently LHCb observed two new states in the Λ + c K − channel [16] and a narrow third state Ξ c (2965), which is very close in mass to the much wider Ξ c (2970). It is however assumed, because of their significantly different widths and different decay channels in which they are observed, that they are two different states. In this work, it is assumed that the peak structures observed in Ξ c (2645)π and Ξ ′ c π channels come from a single resonance.
The analysis is based on a sample of e + e − annihilation data totaling an integrated luminosity of 980 fb −1 recorded by the Belle detector [17] at the KEKB asymmetric-energy e + e − collider [18]. Belle was a large-solid-angle magnetic spectrometer consisting of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of-flight scintillation counters, and an electromagnetic calorimeter comprised CsI(Tl) crystals, all located inside a superconducting solenoid coil that provided a 1.5 T magnetic field. An iron flux return placed outside of the coil was instrumented to detect K 0 L mesons and muons. Two inner-detector configurations were used: a 2.0 cm radius beampipe and a three-layer SVD were used for the first sample of 156 fb −1 , while a 1.5 cm radius beampipe, a four-layer SVD and a small-cell inner CDC were used to record the remaining 824 fb −1 [19]. Using a GEANT-based Monte Carlo (MC) simulation [20], the detector response and its acceptance are modeled to study the mass resolution of signals and obtain reconstruction efficiencies.
The Ξ c (2970) + is reconstructed in the two decay modes, Ξ c (2645) 0 π + and Ξ ′ 0 c π + with Ξ c (2645) 0 → Ξ + c π − and Ξ ′0 c → Ξ 0 c γ, closely following the earlier analysis by Belle [8]. The only difference is that Ξ + c and Ξ 0 c are reconstructed in the decay modes Ξ + c → Ξ − π + π + and Ξ 0 , which have high statistics with good signal-to-background ratios. The scaled momentum x p = p * c/ s/4 − m 2 c 2 , where p * is the center-of-mass (c.m.) momentum of the Ξ c (2970) + candidate, √ s is the total c.m. energy, and m is the mass of the Ξ c (2970) + candidate, is required to be greater than 0.7. In order to determine the spin of Ξ c (2970) + , two angular distributions of the decay chain Ξ c (2970) + → Ξ c (2645) 0 π + 1 → Ξ + c π − 2 π + 1 are analyzed. The first one is the helicity angle θ h of Ξ c (2970) + , defined as the angle between the direction of the primary pion π + 1 and the opposite of boost direction of the c.m. frame, both calculated in the rest frame of the Ξ c (2970) + . Such an angle was used to determine the spin of Λ c (2880) + [21]. The second one is the helicity angle of Ξ c (2645) 0 , defined as the angle between the direction of the secondary pion π − 2 and the opposite direction of the Ξ c (2970) + , both calculated in the rest frame of the Ξ c (2645) 0 . This angle, referred to as θ c , represents angular correlations of the two pions, because π + 1 and Ξ c (2645) 0 are emitted back to back in the rest frame of Ξ c (2970) + . The angular distributions are obtained by dividing the data into 10 equal bins for cos θ h and cos θ c , each extending for intervals of 0.2. For each cos θ h or cos θ c bin, the yield of Ξ c (2970) + → Ξ c (2645) 0 π + is obtained by fitting the invariant-mass distribution of M(Ξ + c π − π + ) for the Ξ c (2645) 0 signal region and sidebands. These two regions are defined [4]. To consider the nonresonant contribution, which is the direct three-body decay into Ξ + c π − π + , a sideband subtraction is performed. The Ξ c (2970) + signal is parametrized by a Breit-Wigner function convolved with a double-Gaussian resolution function and the background by a first-order polynomial. Parameters for the Breit-Wigner are fixed to the values from the previous Belle measurement [8] while those for the resolution function are determined from an MC simulation. The yields obtained from the fits and efficiencies determined from signal MC events are given in Ref. [22].
The following systematic uncertainties are considered for each cos θ h and cos θ c bin. The resultant systematic uncertainties in the yield of each bin are presented in parentheses. The uncertainty due to the resolution function is checked by changing the width of the core Gaussian component by 10% to consider a possible data-MC difference in resolution (0.2% at most). Also, each resolution parameter is varied within its statistical uncertainty determined from signal MC events (0.1% at most). The statistical uncertainty in the efficiency is negligible. The uncertainty due to the background model is determined by redoing the fit with a second-order polynomial or constant function instead of the first-order polynomial (0.7 -47%). The uncertainty coming from the mass and width of Ξ c (2970) + is determined by changing their values within uncertainties [8] (6.7 -12%). All of these uncertainties are added in quadrature (6.7 -47%).
Yields of the decay Ξ c (2970) + → Ξ c (2645) 0 π + after the Ξ c (2645) 0 sideband subtraction and efficiency correction are shown as a function of cos θ h in Fig. 1. Although the quantum numbers of the Ξ c (2645) have not yet been measured, in the quark model the natural assumption for its spin-parity is J P = 3/2 + . Then the expected decay-angle distributions W J for spin hypotheses of J = 1/2, 3/2, and 5/2 for Ξ c (2970) + are as follows [23]: Here, T = 3 2 ,0)| 2 +|T (p, 1 2 ,0)| 2 and T (p, λ 1 , λ 2 ) is the matrix element of a two-body decay with the momentum p of the daughters in the mother's rest frame and the helicities of daughters being λ 1 for Ξ c (2645) 0 and λ 2 for π + . The parameter ρ ii is the diagonal element of the spin-density matrix of Ξ c (2970) + with helicity i/2. The sum of ρ ii for positive odd integer i is normalized to 1/2.
The fit results are summarized in Table I. Though the best fit is obtained for the spin 1/2 hypothesis, the exclusion level of the spin 3/2 (5/2) hypothesis is as small as 0.8 (0.5) standard deviations. Therefore, the result is inconclusive. In other words, it is consistent with a uniform distribution, which can be exhibited by any spin J if the initial state is unpolarized.
In order to draw a more decisive conclusion, we further analyze the angular correlations of the two pions in the Ξ c (2970) + → Ξ c (2645) 0 π + → Ξ + c π − π + decay. In this case, the expected angular distribution is [23] W (θ c ) = 3 2 ρ * 33 sin 2 θ c + ρ *  where ρ * ii is the diagonal element of the spin-density matrix of Ξ c (2645) 0 with the normalization condition ρ * 11 + ρ * 33 = 1/2. Figure 2 shows the yields of Ξ c (2970) + as a function of cos θ c after the Ξ c (2645) 0 sideband subtraction and efficiency correction. A fit to the expected distribution [Eq. (4)] gives a good χ 2 /n.d.f. = 5.6/8 with ρ * 11 = 0.46 ± 0.04 and ρ * 33 = 0.5 − ρ * 11 = 0.04 ± 0.04, which indicates that the population of helicity 3/2 state is consistent with zero. This result is most consistent with the spin 1/2 hypothesis of Ξ c (2970) + , as only the helicity 1/2 state of Ξ c (2645) 0 can survive due to helicity conservation. Indeed, assuming that the lowest partial wave dominates for the Ξ c (2970) + → Ξ c (2645) 0 π + decay, the expected angular correlations can be calculated as summarized in Table II [24]. Fitting the data to the cases J P = 1/2 ± , 3/2 − , and 5/2 + , we obtain the fit results as summarized in Table III. We find the result to favor the 1/2 ± hypothesis over the 3/2 − (5/2 + ) one at the level of 5.1 (4.0) standard deviations. The exclusion level is even higher for the other hypotheses for which the expected angular distributions are upwardly convex. We note that this result also excludes the Ξ c (2645) spin of 1/2 in which the distribution should be flat, and that the present discussion is still true even if there are two resonances, Ξ c (2970) and Ξ c (2965) [16].
The ratio of branching fractions    [21,25]. In principle, the R value can be determined using the following equation: where N * (N ′ ) is the yield of Ξ c (2970) + in the Ξ c (2645) 0 π + (Ξ ′0 c π + ) decay mode, E * (E ′ i ) is the reconstruction efficiency of Ξ c (2970) + for the decay Ξ c (2645) 0 π + (Ξ ′0 c π + with i = Ξ − π + or Ω − K + mode of Ξ 0 c ), and B + (B 0 i ) is the measured branching fraction of Ξ + c → Ξ − π + π + (Ξ 0 c → i-th subdecay mode) [26][27][28]. In this case, however, the uncertainty will be dominated by the branching fractions of the ground-state Ξ c baryons. Such uncertainties are avoided by calculating the ratio in a different way, with inclusive measurements of Ξ + c and Ξ 0 c and an assumption of isospin symmetry in their inclusive cross sections. We note that this assumption is confirmed within 15% in the Σ ( * ) c case [29]. The branching fraction of Ξ +(0) c in a certain subdecay mode is given as where N(Ξ c +(0) ) i and ǫ are the yield and reconstruction efficiency of the Ξ +(0) c ground states for the i-th subdecay mode, L is the integrated luminosity, and σ Ξc is the inclusive production cross section of Ξ c which is assumed to be the same for Ξ 0 c and Ξ + c . By replacing the ground-state Ξ c branching fractions in Eq. (5) with the values in Eq. (6), R can be rewritten as Here, N * and N ′ are obtained by fitting the Ξ c (2645) 0 π + and Ξ ′0 c π + invariant-mass distributions with all the phase space integrated. For the Ξ c (2645) 0 π + channel, a sideband subtraction is performed. For the Ξ ′0 c π + channel, the fit is performed for the Ξ ′0 c signal region, defined as |M(Ξ 0 [4]. For both decay channels, we perform fits using a Breit-Wigner function convolved with a double Gaussian as signal and a first-order polynomial as background. The invariant-mass distributions together with the fit results are shown in Figs. 3 and 4. Similarly, N(Ξ +/0 c ) are obtained by fitting the invariant-mass distributions of Ξ c candidates. Ground-state Ξ c baryons are reconstructed in a similar way as Ξ c (2970) + ; the only difference being that x p is calculated with the mass of Ξ c and required to be greater than 0.6. The fit is performed with a double-Gaussian function as signal and a first-order polynomial as background.
The following systematic uncertainties are considered for the R measurement. The uncertainty coming from the resolution function is checked by changing the width of the core Gaussian component by 10% to consider possible data-MC difference in resolution (+3.3%/−3.4%). Also, each parameter is varied within its statistical uncertainty determined from signal MC events (0.4%). The statistical uncertainty in the efficiency is negligible. The mass and width of Ξ c (2970) + are changed within their uncertainties [8] (+4.1%/−1.7%). The uncertainty due to the background shape is determined by changing it from a first-order polynomial to a constant function and second-order polynomial (+6.8%/−0.9%). The uncertainty due to the tracking efficiency is 0.35% per track. The systematic uncertainty due to the pion-identification efficiency (1.2%) is obtained using D * + → D 0 π + and D 0 → K − π + decays. Similarly, the uncertainty due to γ reconstruction is obtained from the Σ 0 → Λγ decay and determined to be 3.2%. All of these uncertainties are added in quadrature (+9.2%/−5.2%).
The R value is obtained as 1.67 ± 0.29(stat.) +0.15 −0.09 (syst.) ± 0.25(IS), where the last uncertainty is due to possible isospin-symmetry-breaking effects (15%). As a cross check, we have also calculated the same quantity by using the measured branching fractions of Ξ +/0 c as R = 2.05 ± 0.36(stat.) +0. 18 −0.09 (syst.) +1.75 −0.87 (BF), where the last uncertainty is due to uncertainties in the branching fractions of the ground-state Ξ c baryons. The two values are consistent within uncertainties. We note that the mass spectra of Ξ c (2970) + in this study can be well    c π + invariant-mass distribution for the decay Ξ c (2970) + → Ξ ′0 c π + → Ξ 0 c γπ + . Black points with error bars are data. The fit result (solid blue curve) is also presented along with the background (dashed blue curve).
described by a single resonance with the mass and width from the previous Belle measurement [8].
Heavy-quark spin symmetry (HQSS) predicts R = 1.06 (0.26) for a 1/2 + state with the spin of the light-quark degrees of freedom s l = 0 (1), as calculated using Eq. (3.17) of Ref. [25]. For the case of J P = 1/2 − , we expect R ≪ 1 because the decay to Ξ ′0 c π + is in S wave while that to Ξ c (2645) 0 π + is in D wave. Therefore, our result favors a positive-parity assignment with s l = 0. We note that HQSS predictions could be larger than the quoted value by a factor of ∼ 2 with higher-order terms in (1/m c ) [30], so the result is consistent with the HQSS prediction for J P (s l ) = 1/2 + (0).
The obtained spin-parity assignment is consistent with most quark-model-based calculations [1,2,9,[11][12][13]. However, some of them [1,12] predict J P = 1/2 + with s l = 1 which is inconsistent with our result. We note that J P = 1/2 + are the same as those of the Roper resonance [N(1440)] [31], Λ(1600), and Σ(1660); and interestingly, their excitation energy levels are the same as that of Ξ c (2970) (∼ 500 MeV) even though the quark masses are different. This fact may give a hint at the structure of the Roper resonance. Therefore,

First Determination of the Spin and Parity of a Charmed-Strange
Baryon, Ξ c (2970) +