Evidence of a $J/\psi K_S^0$ structure in $B^0 \to J/\psi \phi K_S^0$ decays

An amplitude analysis of $B^0 \to J/\psi \phi K_S^0$ decays is performed using proton-proton collision data corresponding to an integrated luminosity of 9 ${\rm fb}^{-1}$, collected with the LHCb detector at centre-of-mass energies of 7, 8 and 13 TeV. Evidence with a significance of $4.0$ standard deviations of a structure in the $J/\psi K_S^0$ system, named $T_{\psi s1}^{\theta}(4000)^0$, is seen, with its mass and width measured to be $3991~_{-10}^{+12}~^{+\phantom{0}9}_{-17}$ MeV and $105~_{-25}^{+29}~^{+17}_{-23}$ MeV, respectively, where the first uncertainty is statistical and the second systematic. The $T_{\psi s1}^{\theta}(4000)^0$ state is likely to be the isospin partner of the $T_{\psi s1}^{\theta}(4000)^+$ state, previously observed in the $J/\psi K^+$ system of the $B^+ \to J/\psi \phi K^+$ decay. When isospin symmetry for the charged and neutral $T_{\psi s1}^{\theta}(4000)$ states, is assumed, the signal significance increases to $5.4$ standard deviations.

Hadrons formed by more than three quarks, namely exotic hadrons, can have complex colour and flavour structures, and studies of the structures shed light on the nature of the strong interaction.Since the discovery of the χ c1 (3872) [1], several states compatible with a four or five quark composition have been observed [2], including the fully charmed tetraquark states [3][4][5] and the pentaquark states [6,7].In 2020, the BESIII collaboration reported the observation of the Z cs (3985) + state in the D + s D * 0 and D * + s D 0 mass spectra [8]. 1 Two similar states, T θ ψs1 (4000) + and T ψs1 (4220) + , were observed in B + → J/ψϕK + decays by the LHCb experiment [9]. 2 The observations of the T θ ψs1 (4000) + , T ψs1 (4220) + , and Z cs (3985) + states have stimulated many theoretical studies to interpret their internal structures, such as hadronic molecules [11], compact tetraquarks [12], or threshold effects [13].
Searching for T 0 ψs , the isospin partner of the T + ψs , is an important step in identifying the full SU (3) nonet that involves these states.The quark contents for the T + ψs and T 0 ψs states are ccus and ccds, respectively.Recently, the BESIII experiment reported evidence for the Z cs (3985) 0 state (ccds) in the D + s D * − and D * + s D − mass spectra [14].The B 0 → J/ψϕK 0 S decay [15][16][17] is the ideal candidate to search for such T 0 ψs states.The quark level processes of the B 0 → J/ψϕK 0 S decay are similar to those of the B + → J/ψϕK + decay, as shown in Fig. 1(c,d) of Ref. [16], and the two decays are related by isospin symmetry.
In this Letter, evidence of a J/ψK 0 S structure is reported by making use of an amplitude analysis of B 0 → J/ψϕK 0 S decays.The analysis is based on proton-proton (pp) collision data, collected with the LHCb detector at centre-of-mass energies of 7, 8, and 13 TeV, corresponding to integrated luminosities of 1, 2, and 6 fb −1 , respectively.
The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, described in detail in Refs.[18,19].Simulated samples of the signal decays are used to estimate the effect of reconstruction and event selections.The samples are generated with Pythia [20], EvtGen [21], and the Geant4 toolkit [22] as described in Ref. [23].
For the B 0 → J/ψϕK 0 S decays, the J/ψ, ϕ, and K 0 S candidates are reconstructed by combining two oppositely charged muons, kaons, and pions, respectively, forming vertices detached from any primary vertex (PV).The B 0 candidate is reconstructed by combining the J/ψ, ϕ, and K 0 S candidates, and the resulting decay vertex must be of high quality.In order to improve the resolution of the mass of the B 0 candidates (referred to as m J/ψϕK 0 S ), a kinematic constraint [24] is applied, where the momenta of the final-state particles are recalculated constraining the masses of the J/ψ and K 0 S candidates to their nominal values, and the momentum of B 0 candidate pointing back to its associated PV.The associated PV is the one with the smallest impact parameter χ 2 , defined as the difference in the vertex-fit χ 2 of a given PV with and without the particle under consideration.
The B 0 candidates are selected by a multivariate classifier based on a multilayer perceptron [25,26].The selection is similar to that used in Ref. [9], with the additional requirement on the significance of the flight distance of the K 0 S from the B 0 decay vertex.Only candidates with the mass of the K + K − system within ±15 MeV around the nominal ϕ mass [2] are kept, where the mass resolution is about 0.7 MeV. 3 The multivariate classifier 1 Charge conjugation is implied throughout the Letter unless specified otherwise. 2The new exotic hadron naming convention proposed by the LHCb collaboration [10] is applied.The T θ ψs1 (4000) + , and T ψs1 (4220) + states are named Z cs (4000) + , and Z cs (4220) + in the original papers [8,9], respectively.The superscript θ refers to 1/2 isospin and positive parity. 3Natural units with ℏ = c = 1 are used throughout the Letter.uses eleven variables related to the decay-chain topology, particle transverse momentum, vertex fit quality, and charged particle identification information.The selection criterion on the classifier response is chosen by maximising the figure of merit, S 2 /(S + B) 3/2 [27], where S and B are the yields of signal and background in the signal region, respectively.The signal region in m J/ψϕK 0 S is defined to be ±15 MeV around the nominal B 0 mass [2].An extended maximum-likelihood fit is performed to the reconstructed B 0 mass, as is shown in Fig. 1a.The B 0 signal is described by a Hypatia function [28] and the background is described by an exponential function.The B 0 yield is measured to be 1866 ± 47 in the signal region.The fraction of combinatorial background in the signal region is 6 %.Roughly 4 % of the B 0 yield corresponds to peaking background B 0 → J/ψK + K − K 0 S decays without an intermediate ϕ meson (referred to as non-ϕ).The non-ϕ contribution is neglected in the default amplitude model and is considered as a source of systematic uncertainty.A further kinematic fit is performed to improve the momentum resolution of the final-state particles by constraining the measured B 0 mass to its nominal value.Figure 1b shows the distribution of m 2 J/ψK 0 S versus m 2 J/ψϕ for candidates in the signal region.
An amplitude fit is performed to study the intermediate states.The B 0 → J/ψϕK 0 S sample size is small, while the intermediate contributions are complicated, making the fit difficult to converge.In order to solve this difficulty, a simultaneous fit is performed to the B 0 → J/ψϕK 0 S sample and the B + → J/ψϕK + sample in Ref. [9], where the amplitudes in the two decay modes are related through isospin symmetry.The likelihood for the B 0 decay is given by where ) refer to the probability density functions (PDFs) for the signal and background components, respectively.The superscript i refers to the i-th candidate.The fraction of combinatorial background, β, is fixed to 6 %.The likelihood of the B + decay is similar to Eq. 1, and is described in detail in Ref. [29].The decay kinematics are described by a mass, chosen to be m ϕK 0 S , and five angular variables ⃗ Ω as defined below.The signal PDF is proportional to the incoherent sum of the squared matrix elements for different final-state muon helicities (λ µ + , λ µ − ), and depends on the set of fit parameters, ⃗ ω, which includes masses, widths, and helicity couplings of intermediate states contributing to B 0 → J/ψϕK 0 S decays, The term Φ(m i ϕK 0 S , ⃗ Ω i ) represents the phase-space density, the term ϵ(m i ϕK 0 S , ⃗ Ω i ) represents the efficiency, and the term I(⃗ ω) is a normalisation factor.The efficiency is obtained from simulated samples.Since the B 0 → J/ψϕK 0 S decay is a self-charge-conjugated mode, the data sample contains both B 0 and B 0 decays with unknown b flavour.Therefore, the signal PDF for the neutral B decay averages B 0 and B 0 contributions, where the flavour of B candidates is denoted as f in Eq. 2.
The matrix element ) is constructed based on the helicity formalism [30].Three interfering decay sequences are considered: S , and B 0 → (T ψs1 → J/ψK 0 S )ϕ. 4 The decay sequences can be described by m ϕK 0 S and ⃗ Ω ≡ (θ K * , θ J/ψ , θ ϕ , ∆φ K * J/ψ , ∆φ K * ϕ ), where the θ K * , θ J/ψ , and θ ϕ are the helicity angles of the ϕ, µ, and K + particles in the K * , J/ψ, and ϕ rest frames, respectively.The angles between the K * and J/ψ decay planes, and between the K * and ϕ decay planes in the B rest frame, are denoted as ∆φ K * J/ψ , and ∆φ K * ϕ , respectively.The background PDF P bkg (m i ϕK 0 S , ⃗ Ω i ) is determined from studying the candidates in the sideband of the m J/ψϕK 0 S distribution with the methods outlined in Ref. [29].The default model is taken from Ref.
[9] and is composed of nine K * components, seven X components, and two T ψs1 states (T θ ψs1 (4000) and T ψs1 (4220)), and one J/ψϕ nonresonant component (NR).Relativistic Breit-Wigner (RBW) functions are used to model the lineshapes of these resonances.The amplitudes of the B 0 and the B + decays are formed by intermediate states that are either identical (namely X) or isospin partners (K * or T ψs1 ).Under the assumption of isospin symmetry, the mass, width, and helicity couplings for all the components except for the T θ ψs1 (4000) states are constrained to be identical.In order to test the significance of the T θ ψs1 (4000) 0 signal without prior knowledge of its properties, a fit is performed with the masses, widths, and helicity couplings of T θ ψs1 (4000) 0 and T θ ψs1 (4000) + states allowed to vary independently.In an alternative model, isospin symmetry is imposed for the T θ ψs1 (4000) 0 and T θ ψs1 (4000) + components.The parameters for the T ψs1 (4220) 0 state are always constrained to be identical to the T ψs1 (4220) + state due to the limited size of the B 0 sample.
Figure 2 shows the ϕK, J/ψϕ, and J/ψK mass distributions and the corresponding fit projections of the default model for the two B decay modes.Table 1 summarises measurements of the mass, width, fit fraction of the T θ ψs1 (4000) 0 state, and the mass difference between the T θ ψs1 (4000) 0 and T θ ψs1 (4000) + states, defined as ∆M ≡ M T θ ψs1 (4000) 0 − M T θ ψs1 (4000) + .The fit value of ∆M is zero within uncertainties, consistent with the two states being isospin partners.The fit fraction of each component is defined as the integral of the signal PDF divided by the I(⃗ ω) term.All the fit parameters, including mass, width, and helicity couplings of the intermediate states, of the default model in this analysis are consistent  with the corresponding parameters of the default model for the B + → J/ψϕK + decay in Ref. [9].The estimated systematic uncertainties on the mass, width, fit fraction of the T θ ψs1 (4000) 0 state, and on ∆M are summarised in Table 2.Both the background PDF and efficiency function are described by an expansion with Legendre polynomials and a spherical harmonic function instead of interpolation.The Blatt-Weisskopf hadron size is varied between 1.5 and 4.5 GeV −1 from the default value 3 GeV −1 .The Flatté model [31] is used for the lineshape of the X(4140) state instead of the RBW function to account for the possible additional decay channel D * + s D − s .The J/ψϕ NR shape is changed from constant to be a linear function of m J/ψϕ .The K * states are described by a simplified one-channel K-matrix model [2] instead of a sum of RBW functions.The mass-dependent widths in the RBW function for the K * resonances are calculated using the decay with the largest branching fraction, K * → Kπ or K * → K * (892)π, instead of using the K * → ϕK decay.The spin-parity of the T ψs1 (4220) state is changed from the default 1 + to 1 − to account for its ambiguity between 1 + and 1 − [9].The uncertainty originating from the neglected non-ϕ contribution is evaluated by narrowing the ϕ mass window from ±15 MeV to ±8 MeV.An additional J/ψϕ NR contribution with the spin-parity equal to 1 + or 2 + is included.An extended model with more K * states is also studied, which includes all the K * resonances that are within the allowed phase space, as predicted in Ref. [32].Possible additional X states, with spins ranging from 0 to 2, are checked based on the extended model.The total fit fraction of the default model is 165.2 %, which indicates that the interference between the decay amplitudes is large.An alternative fit constraining the total fit fraction to be smaller than 140 %, corresponding to a reduction by three times its uncertainty, is performed.The final positive (negative) systematic uncertainty is taken as the maximal positive (negative) deviation from the model uncertainties above summed in quadrature with the uncertainty from other sources.These other sources of systematic uncertainty include mis-modelling of the impact parameter χ 2 of B candidates, the finite size of the simulated samples, the fixed masses and widths of the known K * states, the choice of the data sample used to construct the background PDF model, and the uncertainty on the background fraction.The significance of the T θ ψs1 (4000) 0 state is evaluated with a likelihood ratio test.The test statistic is defined as t ≡ −2 ln[L(H 0 )/L(H 1 )], with H 1 and H 0 denoting the default model with and without the T θ ψs1 (4000) 0 state.Here, the term L represents the likelihood of the B 0 decay.Five thousand pseudoexperiments are generated with the H 0 model, where the parameters of the model are fixed to the values determined from data.The sample size of each pseudoexperiment follows a Poisson distribution with its mean being equal to the number of candidates in the data sample.A fit is performed to each pseudoexperiment with the H 0 and H 1 models to determine the test statistic t.The t value from these pseudoexperiments should follows a χ 2 distribution, where the number of degrees-of-freedom is a freely varying parameter.Using the resulting χ 2 distribution, the probability of t > t data is taken as the p-value.The corresponding significance is estimated to be 4.9σ, decreasing to 4.0σ after including systematic uncertainties, which provides evidence for the T θ ψs1 (4000) 0 state.The significance for the T θ ψs1 (4000) 0 state is taken as the smallest significance found when varying the sources of systematic uncertainty.
The significance is also evaluated assuming isospin symmetry for the T θ ψs1 (4000) states, where the test statistic is defined as t ′ ≡ −2 ln[L(H 0 )/L(H ′ 1 )], with L denoting the total likelihood of the B + and B 0 decays.The corresponding fit parameters for the T θ ψs1 (4000) + and T θ ψs1 (4000) 0 states, including mass, width, and helicity couplings, are constrained to be equal between the two states in the H ′ 1 model.A fit is performed to the t ′ distribution from the pseudoexperiments with a Gaussian function.The Gaussian function rather than a χ 2 distribution is used here because the number of degree-of-freedoms of the H 0 and H ′ 1 models are equal.The significance is estimated to be 7.2σ and decreases to 5.4σ after accounting for systematic uncertainties.
In conclusion, an amplitude analysis of the B 0 → J/ψϕK 0 S decay is performed.Evidence of a J/ψK 0 S structure, denoted the T θ ψs1 (4000) 0 state, is obtained with a significance of 4.0σ.The mass and width of this state are measured to be where the first uncertainty is statistical and the second systematic.The mass difference between the T θ ψs1 (4000) 0 and T θ ψs1 (4000) + states is measured to be ∆M which is consistent with the two states being isospin partners.With isospin symmetry imposed for the T θ ψs1 (4000) states, the significance of the T θ ψs1 (4000) 0 structure is measured to be 5.4σ.

Figure 1 : 2 J/ψK 0 S versus m 2 J
Figure 1: (a) Invariant-mass distribution of selected B 0 candidates and corresponding fit result.(b) Distribution of m 2 J/ψK 0 S versus m 2 J/ψϕ for candidates in the ±15 MeV region around the nominal B 0 mass.

Figure 2 :
Figure 2: Distributions of (left) m ϕK , (middle) m J/ψϕ , and (right) m J/ψK , overlaid with the corresponding projections of the default fit model.The upper and lower rows correspond to the B + → J/ψϕK + and B 0 → J/ψϕK 0 S decays, respectively.

Table 1 :
Results for the T θ ψs1 (4000) 0 state from the default model.The first uncertainty is statistical and the second systematic.