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Exact black hole solutions in shift symmetric scalar-tensor theories

We derive a variety of exact black hole solutions in a subclass of Horndeski's scalar-tensor theory possessing shift symmetry, $ϕ\toϕ+c$, and reflection symmetry, $ϕ\to-ϕ$. The theory admits two arbitrary functions of $X:=-(\partialϕ)^2/2$, and our solutions are constructed without specifying the concrete form of the two functions, implying that black hole solutions in specific scalar-tensor theories found in the literature can be extended to a more general class of theories with shift symmetry. Our solutions include a black hole in the presence of an effective cosmological constant, the Nariai spacetime, the Lifshitz black hole, and other nontrivial solutions, all of which exhibit nonconstant scalar-field profile.