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We investigate the tachyonic instability of Kerr-Newman (KN) black hole witha rotation parameter $a$ in the Einstein-Chern-Simons-scalar theory coupledwith a quadratic massive scalar field. This instability analysis corresponds toexploring the onset of spontaneous scalarization for KN black holes. First, wefind no $a$-bound for $\alpha<0$ case by considering (1+1)-dimensionalanalytical method. A direct numerical method is adopted to explore(2+1)-dimensional time evolution of a massive scalar perturbation with positiveand negative $\alpha$ to obtain threshold curves numerically. We obtainthreshold curves $\alpha_{\rm th}(a)$ of tachyonic instability for positive$\alpha$ without any $a$-bounds. We expect to find the same threshold curves$\alpha_{\rm th}(a)$ of tachyonic instability for negative $\alpha$ without any$a$-bound because its linearized scalar theory is invariant under thetransformation of $\alpha\to -\alpha $ and $\theta\to -\theta$. In addition, itis found that the scalar mass term suppresses tachyonic instability of KN blackholes.

Scalarization of Kerr-Newman black holes in the Einstein-Chern-Simons-scalar theory