Application of a coupled-channel Complex Scaling Method with Feshbach projection to the $K^-pp$ system

Kaonic nuclei (nuclear system with anti-kaons) have been an interesting subject in hadron and strange nuclear physics, because the strong attraction between anti-kaon and nucleon might bring exotic properties to that system. In this article, we investigate $K^-pp$ as a prototype of kaonic nuclei. Here, $K^-pp$ is a three-body resonant state in the $\bar{K}NN$-$\pi YN$ coupled channels. ($Y$=$\Lambda$, $\Sigma$) To treat resonant states in a coupled-channel system properly, we propose newly a coupled-channel complex scaling method combined with the Feshbach projection (ccCSM+Feshbach method). In this method, the Feshbach projection is realized with help of so-called the extended closure relation held in the complex scaling method, and a complicated coupled-channel problem is reduced to a simple single-channel problem which one can treat easily. First, we confirm that the ccCSM+Feshbach method completely reproduces results of a full coupled-channel calculation in case of two-body $\bar{K}N$-$\pi Y$ system. We then proceed to study of three-body $\bar{K}NN$-$\pi YN$ system, and successfully find solutions of the $K^-pp$ resonance by imposing self-consistency for the complex $\bar{K}N$ energy. Obtained binding energy of $K^-pp$ is well converged around 27 MeV, with an energy-dependent $\bar{K}N$(-$\pi Y$) potential based on the chiral SU(3) theory, independently of ansatz for the self-consistency. This binding energy is small as ones reported in earlier studies based on chiral models. The decay width of $K^-pp$ strongly depends on the ansatz. We calculate also the correlation density of $NN$ and $\bar{K}N$ pairs by using the obtained complex-scaled wave function of the $K^-pp$ resonance. Effect of the repulsive core of $NN$ potential and survival of $\Lambda^*$ resonance are confirmed.