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In this paper, we study the propagations of external fields in Horndeskitheory, including the scalar field, electromagnetic field and Dirac field. Weextensively explore the quasinormal frequencies, time evolution, greybodyfactors and emission rates of those massless perturbing fields by solving thecorresponding master equations in the Horndeski hairy black hole. With the useof both numerical and analytical methods, we disclose thecompetitive/promotional influences of the Horndeski hair, spin and quantummomentum number of the external fields on those phenomenal physics. Our resultsshow that the Horndeski hairy black hole is stable under those perturbations.Moreover, a larger Horndeski hair could enhance the intensity of energyemission rate for Hawking radiation of various particles, indicating thatcomparing to the Schwarzschild black hole, the Horndeski hariy black hole couldhave longer or shorter lifetime depending on the sign of the Horndeski hair.

Perturbations of massless external fields in Horndeski hairy black hole