$Z_c(4430)$ and $Z_c(4200)$ as triangle singularities

Abstract

$Z_c(4430)$ discovered by the Belle and confirmed by the LHCb in $\bar{B}^0\to\psi(2S)K^-\pi^+$ is generally considered to be a charged charmonium-like state that includes minimally two quarks and two antiquarks. $Z_c(4200)$ found in $\bar{B}^0\to J/\psi K^-\pi^+$ by the Belle is also a good candidate of a charged charmonium-like state. We demonstrate that kinematical singularities in triangle loop diagrams induce a resonance-like behavior that can consistently explain the properties (mass, width, and Argand plot) of $Z_c(4430)$ and $Z_c(4200)$ from the experimental analyses. The triangle diagrams include only experimentally well-established hadrons. Applying this idea to $\Lambda_b^0\to J/\psi p\pi^-$, we also identify triangle singularities that behave like $Z_c(4200)$, but no triangle diagram is available for $Z_c(4430)$. This is consistent with the LHCb's finding that their description of the $\Lambda_b^0\to J/\psi p\pi^-$ data is significantly improved by including a $Z_c(4200)$ contribution while $Z_c(4430)$ seems to hardly contribute. Even though the proposed mechanisms have uncertainty in the absolute strengths which are currently difficult to estimate, they are certainly a compelling alternative to tetraquark-based interpretations of $Z_c(4430)$ and $Z_c(4200)$.