Experimental constraints on the spin and parity of the $Z\mathbf{(}4430{\mathbf{)}}^{\mathbf{+}}$

We perform a full amplitude analysis of $B^0\to {\psi }^{\prime }{K}^{+}{\pi }^{-}$ decays, with ${\psi }^{\prime }\to {\mu }^{+}{\mu }^{-}$ or $e^{+}{e}^{-}$, to constrain the spin and parity of the $Z(4430{)}^{-}$. The $J^P=1^{+}$ hypothesis is favored over the $0^{-}$, $1^{-}$, $2^{-}$ and $2^{+}$ hypotheses at the levels of $3.4\sigma$, $3.7\sigma$, $4.7\sigma$ and $5.1\sigma$, respectively. The analysis is based on a $711{\mathrm{fb}}^{-1}$ data sample that contains $772\times {10}^6$ $B\overline{B}$ pairs collected at the ${\rm Y}(4S)$ resonance by the Belle detector at the asymmetric-energy $e^{+}{e}^{-}$ collider KEKB.

Figure 1

The $\Delta E$ distribution; the signal and sideband regions are hatched.

Figure 2

The Dalitz plots of (a) the signal region, (b) sidebands, and (c) efficiency.

Figure 3

Efficiency as a function of angular variables.

Figure 4

The results of the fit to background events projected onto the Dalitz variables.

Figure 5

Dalitz plot slices used to present fit results. Vertical divisions are at $(0.796{)}^2{\mathrm{GeV}}^2/c^4$, $(0.996{)}^2{\mathrm{GeV}}^2/c^4$, $(1.332{)}^2{\mathrm{GeV}}^2/c^4$ and $(1.532{)}^2{\mathrm{GeV}}^2/c^4$. Horizontal divisions are at $19.0{\mathrm{GeV}}^2/c^4$ and $20.5{\mathrm{GeV}}^2/c^4$.

Figure 6

The fit results with (solid line) and without (dashed line) $Z^{-}$ ($J^P=1^{+}$) in the default model. The points with error bars are data; the hatched histograms are ${\psi }^{\prime }$ sidebands. Slices are defined in Fig. 5.

Figure 7

Projection of the fit results with the $K^{*}$ veto. The legend is the same as in Fig. 6.

Figure 8

Projections of the fit results with (solid line) and without (dashed line) $Z^{-}$ ($J^P=1^{+}$) onto the angular variables for the entire signal region (no $K^{*}$ veto) in the default model. Points with error bars are data.

Figure 9

Comparison of the $0^{-}$ and $1^{+}$ hypotheses in the default model. The histograms are distributions of $\Delta (-2\ln L)$ in MC pseudoexperiments generated in accordance with the fit results with $0^{-}$ (open histogram) and $1^{+}$ (hatched histogram) signals. The $\Delta (-2\ln L)$ value observed in data is indicated with an arrow.

Figure 10

Definition of the angle between the decay planes for $B^0\to {\psi }^{\prime }(\to {\ell }^{+}{\ell }^{-})K^{*}(\to K^{+}{\pi }^{-})$ decay (in the $B^0$ rest frame).

Figure 11

Definition of the helicity angles for $B^0\to {\psi }^{\prime }(\to {\ell }^{+}{\ell }^{-})K^{*}(\to K^{+}{\pi }^{-})$ decay (in the $K^{*}$ and ${\psi }^{\prime }$ rest frames).

Figure 12

Definition of the $Z(4430{)}^{-}$ helicity angle [in the $Z(4430{)}^{-}$ rest frame].

Figure 13

Definitions of the ${\psi }^{\prime }$ helicity angle and the angle $\tilde{\phi }$ (in the ${\psi }^{\prime }$ rest frame).

Figure 14

Definition of the angle $\alpha$ (in the ${\psi }^{\prime }$ rest frame).