Triangle singularities in (B)over-bar0 → χc1K- π+ relevant to Z1 (4050) and Z2 (4250)

Z(1) (4050) and Z(2) (4250) observed in (B) over bar (0) -> chi(c1) K- pi(+) by the Belle Collaboration are candidates of charged charmoniumlike states that minimally include two quarks and two antiquarks. While Z(1) (4050) and Z(2) (4250) have been interpreted as tetraquark states previously, we propose a completely different scenario based on a kinematical effect called the triangle singularity. We demonstrate that the triangle singularities cause in the chi(c1) pi(+) invariant mass distribution resonance like bumps that fit very well the Belle data. If these bumps are simulated by the Z(1) (4050) and Z(2) (4250) resonance excitations, the spin-parity of them are predicted to be 1(-) for Z(1) (4050) and 1(+) or 1(-) for Z(2) (4250). The bump corresponding to Z(1) (4050) has a highly asymmetric shape, which the Belle data exactly indicate. We show that the asymmetric shape originates from an interplay between the triangle singularity and the opening of the X (3872)pi(+) channel near the triangle-singularity energy. This characteristic lineshape could be used to discriminate different interpretations of Z(1) (4050). An interesting prediction from interpreting Z(1) (4050) and Z(2) (4250) as the triangle singularities is that similar bumps caused by the same mechanisms possibly appear also in (B) over bar (0) -> J/psi K- pi(+) data; the already observed Z(c) (4200) corresponds to Z(2) (4250) of J(P) = 1(+).