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We adapt the proton-neutron finite-amplitude method, which in its originalform is an efficient implementation of the Skyrme quasiparticle random phaseapproximation, to include the coupling of quasiparticles to like-particlephonons. The approach allows us to add beyond-QRPA correlations to computationsof Gamow-Teller strength and $\beta$-decay rates in deformed nuclei for thefirst time. We test the approach in several deformed isotopes for whichmeasured strength distributions are available. The additional correlationsdramatically improve agreement with the data, and will lead to improved global$\beta$-decay rates.

Effects of Quasiparticle-Vibration Coupling on Gamow-Teller Strength and $\beta$ Decay with the Skyrme Proton-Neutron Finite-Amplitude Method