Observation of the resonant character of the $Z(4430)^-$ state

Resonant structures in $B^0\to\psi'\pi^-K^+$ decays are analyzed by performing a four-dimensional fit of the decay amplitude, using $pp$ collision data corresponding to $\rm 3 fb^{-1}$ collected with the LHCb detector. The data cannot be described with $K^+\pi^-$ resonances alone, which is confirmed with a model-independent approach. A highly significant $Z(4430)^-\to\psi'\pi^-$ component is required, thus confirming the existence of this state. The observed evolution of the $Z(4430)^-$ amplitude with the $\psi'\pi^-$ mass establishes the resonant nature of this particle. The mass and width measurements are substantially improved. The spin-parity is determined unambiguously to be $1^+$.

In this Letter, we report a 4D model-dependent amplitude fit to a sample of 25 176±174 B 0 → ψ K + π − , ψ → µ + µ − candidates reconstructed with the LHCb detector in pp collision data corresponding to 3 fb −1 collected at √ s = 7 and 8 TeV.The ten-fold increase in signal yield over the previous measurement [27] improves sensitivity to exotic states and allows their resonant nature to be studied in a novel way.We complement the amplitude fit with a model-independent approach [24].
The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, described in detail in Ref. [34].The B 0 candidate selection follows that in Ref. [35] accounting for the different number of final-state pions.It is based on finding (ψ → µ + µ − )K + π − candidates using particle identification information, transverse momentum thresholds and requiring separation of the tracks and of the B 0 vertex from the primary pp interaction points.To improve modeling of the detection efficiency, we exclude regions near the K + π − vs. ψ π − Dalitz plot boundary, which reduces the sample size by 12%.The background fraction is determined from the B 0 candidate invariant mass distribution to be (4.1 ± 0.1)%.The background is dominated by combinations of ψ mesons from B decays with random kaons and pions.
Amplitude models are fit to the data using the unbinned maximum likelihood method.We follow the formalism and notation of Ref. [27] with the 4D amplitude dependent on Φ = (m 2 , where θ ψ is the ψ helicity angle and φ is the angle between the K * and ψ decay planes in the B 0 rest frame.The signal probability density function (PDF), S(Φ), is normalized by summing over simulated events.Since the simulated events are passed through the detector simulation [36], this approach implements 4D efficiency corrections without use of a parameterization.We use B 0 mass sidebands to obtain a parameterization of the background PDF.
As in Ref. [27], our amplitude model includes all known K * 0 → K + π − resonances with nominal mass within or slightly above the kinematic limit (1593 MeV) in B 0 → ψ K + π − decays: K * 0 (800), K * 0 (1430) for J = 0; K * (892), K * (1410) and K * (1680) for J = 1; K * 2 (1430) for J = 2; and K * 3 (1780) for J = 3.We also include a non-resonant (NR) J = 0 term in the fits.We fix the masses and widths of the resonances to the world average values [37], except for the widths of the two dominant contributions, K * (892) and K * 2 (1430), and the poorly known K * 0 (800) mass and width, which are allowed to float in the fit with Gaussian constraints.As an alternative J = 0 model, we use the LASS parameterization [38,39], in which the NR and K * 0 (800) components are replaced with an elastic scattering term (two free parameters) interfering with the K * 0 (1430) resonance.To probe the quality of the likelihood fits, we calculate a binned χ 2 variable using adaptive 4D binning, in which we split the data once in | cos θ ψ |, twice in φ and then repeatedly in m 2 K + π − and m 2 ψ π − preserving any bin content above 20 events, for a total of N bin = 768 bins.Simulations of many pseudoexperiments, each with the same number of signal and background events as in the data sample, show that the p-value of the χ 2 test (p χ 2 ) has an approximately uniform distribution assuming that the number of degrees of freedom (ndf) equals N bin −N par −1, where N par is the number of unconstrained parameters in the fit.Fits with all K * components and either of the two different J = 0 models do not give a satisfactory description of the data; the p χ 2 is below 2 × 10 −6 , equivalent to 4.8σ in the Gaussian distribution.If the K * 3 (1780) component is excluded from the amplitude, the discrepancy increases to 6.3σ.This is supported by an independent study using the model-independent approach developed by the BaBar collaboration [24,25], which does not constrain the analysis to any combination of known K * resonances, but merely restricts their maximal spin.We determine the Legendre polynomial moments of cos θ K * as a function of m K + π − from the sideband-subtracted and efficiency-corrected sample of B 0 → ψ K + π − candidates.Together with the observed m K + π − distribution, the moments corresponding to J ≤ 2 are reflected into the m ψ π − distribution using simulations as described in Ref. [24].As shown in Fig. 1, the K * reflections do not describe the data in the Z(4430) − region.Since a Z(4430) − resonance would contribute to the cos θ K * moments, and also interfere with the K * resonances, it is not possible to determine the Z(4430) − parameters using this approach.The amplitude fit is used instead.
If a Z(4430) − component with is added to the amplitude, the p χ 2 reaches 4% when all the K * → K + π − resonances with a pole mass below the kinematic limit are included.The p χ 2 rises to 12% if the K * (1680) is added (see Fig. 2), but fails to improve when the K * 3 (1780) is also included.Therefore, as in Ref. [27] we choose to estimate the Z − 1 parameters using the model with the K * (1680) as the heaviest K * resonance.In Ref. [27] two independent complex Z − 1 helicity couplings, H Z − λ for λ = 0, +1 (parity conservation requires H Z − −1 = H Z − +1 ), were allowed to float in the fit.The small energy release in the Z − 1 decay suggests neglecting D-wave decays.A likelihood-ratio test is used to discriminate between any pair of amplitude models based on the log-likelihood difference ∆(−2 ln L) [40].The D-wave contribution is found to be insignificant when allowed in the fit, 1.3σ assuming Wilks' theorem2 .Thus, we assume a pure S-wave decay, implying The significance of the Z − 1 is evaluated from the likelihood ratio of the fits without and with the Z − 1 component.Since the condition of the likelihood regularity in Z − 1 mass and width is not satisfied when the no-Z − 1 hypothesis is imposed, use of Wilks' theorem is not justified3 [41].Therefore, pseudoexperiments are used to predict the distribution of ∆(−2 ln L) under the no-Z − 1 hypothesis, which is found to be well described by a χ 2 PDF with ndf = 7.5.Conservatively, we assume ndf = 8, twice the number of free parameters in the Z − 1 amplitude.This yields a Z − 1 significance for the default K * model of 18.7σ.The lowest significance among all the systematic variations to the model discussed below is 13.9σ.
The default fit gives 1430) = (7.0 ± 0.4)% and f K * (1680) = (4.0 ± 1.5)%, which are consistent with the Belle results [27] even without considering systematic uncertainties.Above, the amplitude fraction of any component R is defined as f R = S R (Φ)dΦ/ S(Φ)dΦ, where in S R (Φ) all except the R amplitude terms are set to zero.The sum of all amplitude fractions is not 100% because of interference effects.To assign systematic errors, we: vary the K * models by removing the K * (1680) or adding the K * 3 (1780) in the amplitude (f K * 3 (1780) = (0.5 ± 0.2)%); use the LASS function as an alternative K * S-wave representation; float all K * masses and widths while constraining them to the known values [37]; allow a second Z − component; increase the orbital angular momentum assumed in the B 0 decay; allow a D-wave component in the Z − 1 decay; change the effective hadron size in the Blatt-Weisskopf form factors from the default 1.6 GeV −1 [27] to 3.0 GeV −1 ; let the background fraction float in the fit or neglect the background altogether; tighten the selection criteria probing the efficiency simulation; and use alternative efficiency and background implementations in the fit.We also evaluate the systematic uncertainty from the formulation of the resonant amplitude.In the default fit, we follow the approach of Eq. (2) in Ref. [27] that uses a running mass M R in the (p R /M R ) L R term, where M R is the invariant mass of two daughters of the R resonance; p R is the daughter's momentum in the rest frame of R and L R is the orbital angular momentum of the decay.The more conventional formulation [37,42]  ψ π − increases counterclockwise).The red curve is the prediction from the Breit-Wigner formula with a resonance mass (width) of 4475 (172) MeV and magnitude scaled to intersect the bin with the largest magnitude centered at (4477 MeV) 2 .Units are arbitrary.The phase convention assumes the helicity-zero K * (892) amplitude to be real.
component only.The model-independent analysis has a large statistical uncertainty in the Z − 0 region and shows no deviations of the data from the reflections of the K * degrees of freedom (Fig. 1).Argand diagram studies for the Z − 0 are inconclusive.Therefore, its characterization as a resonance will need confirmation when larger samples become available.
In summary, an amplitude fit to a large sample of B 0 → ψ K + π − decays provides the first independent confirmation of the existence of the Z(4430) − resonance and establishes its spin-parity to be 1 + , both with very high significance.The measured mass, 4475 ± 7 +15 −25 MeV, width, 172±13 +37 −34 MeV, and amplitude fraction, (5.9±0.9 +1.5 −3.3 )%, are consistent with, but more precise than, the Belle results [27].An analysis of the data using the model-independent approach developed by the BaBar collaboration [24] confirms the inconsistencies in the Z(4430) − region between the data and K + π − states with J ≤ 2. The D-wave contribution is found to be insignificant in Z(4430) − decays, as expected for a true state at such mass.The Argand diagram obtained for the Z(4430) − amplitude is consistent with the resonant behavior.For the first time the resonant character is demonstrated in this way among all known candidates for charged four-quark states.

Figure 1 :
Figure 1: Background-subtracted and efficiency-corrected m ψ π − distribution (black data points),superimposed with the reflections of cos θ K * moments up to order four allowing for J(K * ) ≤ 2 (blue line) and their correlated statistical uncertainty (yellow band bounded by blue dashed lines).The distributions have been normalized to unity.

Figure 2 :
Figure 2: Distributions of the fit variables (black data points) together with the projections of the 4D fit.The red solid (brown dashed) histogram represents the total amplitude with (without) the Z − 1 .The other points illustrate various subcomponents of the fit that includes the Z − 1 : the upper (lower) blue points represent the Z − 1 component removed (taken alone).The orange, magenta, cyan, yellow, green, and red points represent the K * (892), total S-wave, K * (1410), K * (1680), K * 2 (1430) and background terms, respectively.

Figure 3 :
Figure 3: Fitted values of the Z − 1 amplitude in six m 2 ψ π − bins, shown in an Argand diagram (connected points with the error bars, m 2ψ π − increases counterclockwise).The red curve is the prediction from the Breit-Wigner formula with a resonance mass (width) of 4475 (172) MeV and magnitude scaled to intersect the bin with the largest magnitude centered at (4477 MeV)2 .Units are arbitrary.The phase convention assumes the helicity-zero K * (892) amplitude to be real.