Complex-Scaling Calculation of Three-Body Resonances Using Complex-Range Gaussian Basis Functions --- Application to 3$\alpha$ resonances in 12C ---

We propose to use the complex-range Gaussian basis functions, {r^l e^{-(1 \pm i\omega)(r/r_n)^2}Y_{lm}(\hat{r}); r_n in a geometric progression}, in the calculation of three-body resonances with the complex-scaling method (CSM) in which use is often made of the real-range Gaussian basis functions, {r^l e^{-(r/r_n)^2}Y_{lm}(\hat{r})}, that are suitable for describing the short-distance structure and the asymptotic decaying behavior of few-body systems. The former basis set is more powerful than the latter when describing the resonant and nonresonant continuum states with highly oscillating amplitude at large scaling angles \theta. We applied the new basis functions to the CSM calculation of the 3\alpha resonances with J=0^+, 2^+ and 4^+ in 12C. The eigenvalue distribution of the complex scaled Hamiltonian becomes more precise and the maximum scaling angle becomes drastically larger (\theta_{max}=16 deg. \arrow 36 deg.) than those given by the use of the real-range Gaussians. Owing to these advantages, we were able to confirm the prediction by Kurokawa and Kato [Phys. Rev. C 71, 021301 (2005)] on the appearance of the new broad 0^+_3 state; we show it as an explicit resonance pole isolated from the 3$\alpha$ continuum.