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[Background] $\beta$-decay half-life is sensitive to the shell structure nearthe Fermi levels. Nuclear deformation thus impacts the $\beta$-decayproperties. [Purpose] A first-order shape-phase transition in neutron-rich Zrisotopes is predicted by some models. We investigate the $\beta$-decayhalf-lives of neutron-rich nuclei around $^{110}$Zr, where the shape-phasetransition is predicted to occur, to see if the $\beta$-decay half-life can bean indicator of the shape changes. [Method] The proton-neutron quasiparticlerandom-phase approximation (RPA) is adopted to calculate the Gamow-Tellertransitions. In addition, we apply the quasiparticle phonon-vibrationalcoupling (PVC) to consider the phonon couplings. [Results] The spherical andoblate configurations give similar half-lives but shorter ones than the prolateconfiguration at the RPA level. The PVC effect further reduces the half-livesin general, but the effect is smaller for the deformed configuration than thatfor the spherical one. As a result, it makes the shape change from the oblateconfiguration to the spherical configuration visible. Therefore, a suddenshortening of $\beta$-decay half-lives is always found at the nuclear shapechanges. [Conclusions] $\beta$-decay half-life is an indicator of theshape-phase transition. The shape mixing and the roles of the triaxialdeformation are subject to study in the future.

$\beta$-decay half-lives as an indicator of shape-phase transition in neutron-rich Zr isotopes with particle-vibration coupling effect