Model-independent confirmation of the $Z(4430{)}^{-}$ state

The decay $B^0\to \psi (2S)K^{+}{\pi }^{-}$ is analyzed using $3{\mathrm{fb}}^{-1}$ of $pp$ collision data collected with the LHCb detector. A model-independent description of the $\psi (2S)\pi$ mass spectrum is obtained, using as input the $K\pi$ mass spectrum and angular distribution derived directly from data, without requiring a theoretical description of resonance shapes or their interference. The hypothesis that the $\psi (2S)\pi$ mass spectrum can be described in terms of $K\pi$ reflections alone is rejected with more than $8\sigma$ significance. This provides confirmation, in a model-independent way, of the need for an additional resonant component in the mass region of the $Z(4430{)}^{-}$ exotic state.

Figure 1

Spectrum of the $\psi (2S)K\pi$ system invariant mass. Black dots are the data, the continuous (blue) line represents the fit result, and the dashed (red) line represents the background component.

Figure 2

The 2D efficiency is shown in the following planes: (top left) ($m_{K\pi },\cos {\theta }_{K^{*0}}$), (top middle) ($m_{K\pi },\mathrm{\Delta }{\varphi }_{K\pi ,\mu \mu }$), (top right) ($m_{K\pi },\cos {\theta }_{\psi (2S)}$), (bottom left) ($\cos {\theta }_{K^{*0}},\cos {\theta }_{\psi (2S)}$), (bottom middle) ($\cos {\theta }_{K^{*0}},\mathrm{\Delta }{\varphi }_{K\pi ,\mu \mu }$), and (bottom right) ($\cos {\theta }_{\psi (2S)},\mathrm{\Delta }{\varphi }_{K\pi ,\mu \mu }$). Corrections for the efficiency are applied in the 4D space; the 2D plots allow visualization of their behavior.

Figure 3

The $\psi (2S)\pi$ invariant mass resolution as determined from simulated data (red dots). The continuous line is a spline-based interpolation.

Figure 4

Distributions of (left) $m_{K\pi }$ and (right) $\cos {\theta }_{K^{*0}}$, after background subtraction and efficiency correction.

Figure 5

The two-dimensional distributions ($m_{K\pi },\cos {\theta }_{K^{*0}}$) and ($m_{\psi (2S)\pi },\cos {\theta }_{\psi (2S)}$) are shown in the left and the right plots, respectively, after background subtraction and efficiency correction.

Figure 6

Dependence on $m_{K\pi }$ of the first six $K\pi$ moments of the $B^0\to \psi (2S)K^{+}{\pi }^{-}$ decay mode as determined from data.

Figure 7

First six normalized $K\pi$ moments of the $B^0\to \psi (2S)K^{+}{\pi }^{-}$ decay mode as a function of $m_{K\pi }$. The shaded (yellow) bands indicate the $\pm 1\sigma$ variations of the moments.

Figure 8

Background subtracted and efficiency corrected spectrum of $m_{\psi (2S)\pi }$. Black points represent data. Superimposed are the distributions of the Monte Carlo simulation: the dotted (black) line corresponds to the pure phase-space case; in the dash-dotted (red) line, the $m_{K\pi }$ spectrum is weighted to reproduce the experimental distribution; in the continuous (blue) line, the angular structure of the $K\pi$ system is incorporated using Legendre polynomials up to (left) $l_{\max }=4$ and (right) $l_{\max }=6$. The shaded (yellow) bands are related to the uncertainty on normalized moments, which is due to the statistical uncertainty that comes from the data. Therefore, the two uncertainties should not be combined when comparing data and Monte Carlo predictions. See the text for further details.

Figure 9

The experimental spectrum of $m_{\psi (2S)\pi }$ is shown by the black points. Superimposed are the distributions of the Monte Carlo simulation: the dotted (black) line corresponds to the pure phase-space case; in the dash-dotted (red) line, the $m_{K\pi }$ spectrum is weighted to reproduce the experimental distribution; in the continuous (blue) line, the angular structure of the $K\pi$ system is incorporated using Legendre polynomials with index $l_{\max }$ variable according to $m_{K\pi }$ as described in Eq. (4), reaching up to $l_{\max }=4$. The shaded (yellow) bands are related to the uncertainty on normalized moments, which is due to the statistical uncertainty that comes from the data. Therefore, the two uncertainties should not be combined when comparing data and Monte Carlo predictions. See the text for further details.

Figure 10

Black points represent the experimental distributions of $m_{\psi (2S)\pi }$ for the indicated $m_{K\pi }$ intervals. The dash-dotted (red) line is obtained by modifying the $m_{K\pi }$ spectrum of the phase-space simulation according to the $m_{K\pi }$ experimental spectrum. In the continuous (blue) line, the angular structure of the $K\pi$ system is incorporated using Legendre polynomials with variable index $l_{\max }$ according to Eq. (4). The shaded (yellow) bands are related to the uncertainty on normalized moments, which is due to the statistical uncertainty that comes from the data. Therefore, the two uncertainties should not be combined when comparing data and Monte Carlo predictions. See the text for further details.

Figure 11

The experimental spectrum of $m_{\psi (2S)\pi }$ is shown by the black points. Superimposed are the distributions of the Monte Carlo simulation: the dotted (black) line corresponds to the pure phase space case; in the dash-dotted (red) line, the $m_{K\pi }$ spectrum is weighted to reproduce the experimental distribution; in the continuous (blue) line, the angular structure of the $K\pi$ system is incorporated using Legendre polynomials up to $l_{\max }=30$, which implies a full description of the spectrum features even if it corresponds to an unphysical configuration of the $K\pi$ system. The shaded (yellow) bands are related to the uncertainty on normalized moments.

Figure 12

Distributions of $-2\mathrm{\Delta }\mathrm{NLL}$ for the pseudoexperiments (black dots), fitted with a Gaussian function (dashed red line), in three different configurations of the $K\pi$ system angular contributions: (left) $l_{\max }=4$, (middle) $l_{\max }=6$, and (right) $l_{\max }$ variable according to Eq. (4). The black arrow represents the $-2\mathrm{\Delta }\mathrm{NLL}$ value obtained on the data.

Figure 13

Distributions of $-2\mathrm{\Delta }\mathrm{NLL}$ for the pseudoexperiments (black dots), fitted with a Gaussian function (dashed red line), for the region $1000\mathrm{MeV}/c^2<m_{K\pi }<1390\mathrm{MeV}/c^2$ in three different configurations of the $K\pi$ system angular contributions: (left) $l_{\max }=4$, (middle) $l_{\max }=6$, and (right) $l_{\max }$ variable according to Eq. (4). The black arrow represents the $-2\mathrm{\Delta }\mathrm{NLL}$ value obtained on the data.