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Black holes with a nonconstant kinetic term in degenerate higher-order scalar tensor theories

We investigate static and spherically symmetric black hole (BH) solutions in shift-symmetric quadratic-order degenerate higher-order scalar-tensor (DHOST) theories. We allow a nonconstant kinetic term $X=g^{μν} \partial_μϕ\partial_νϕ$ for the scalar field $ϕ$ and assume that $ϕ$ is, like the spacetime, a pure function of the radial coordinate $r$, namely $ϕ=ϕ(r)$. First, we find analytic static and spherically symmetric vacuum solutions in the so-called {\it Class Ia} DHOST theories, which include the quartic Horndeski theories as a subclass. We consider several explicit models in this class and apply our scheme to find the exact vacuum BH solutions. BH solutions obtained in our analysis are neither Schwarzschild or Schwarzschild (anti-) de Sitter. We show that a part of the BH solutions obtained in our analysis are free of ghost and Laplacian instabilities and are also mode stable against the odd-parity perturbations. Finally, we argue the case that the scalar field has a linear time dependence $ϕ=qt+ψ(r)$ and show several simple examples of nontrivial BH solutions with a nonconstant kinetic term obtained analytically and numerically.