New broad $0^{+}$ state in ${}^{12}\mathrm{C}$

The $3\alpha$ resonance states in ${}^{12}\mathrm{C}$ are investigated by using the method of analytic continuation in the coupling constant combined with the complex scaling method (ACCC+CSM). We obtain a new $0^{+}$ resonance state with energy $E_r=1.66$ MeV and width $\Gamma =1.48$ MeV, which is of interest as a candidate for the recently observed $0^{+}$ state at $E_x=10.3$ MeV.

Figure 1

Pole distribution for the case of $0^{+}$ state obtained by the CSM. Energies are measured from the $3\alpha$ threshold energy. The crosses are the points of discretized continuum states rotated in the complex energy plane; the circles are the $3\alpha$ resonance states. The scaling angle is ${16}^{ˆ}$.

Figure 2

Trajectories of the new $0^{+}$ state obtained by changing the potential strength parameter on the complex energy and wave number planes. The solid curve is the solution of the ACCC+CSM. The circles are the complex energies obtained by the CSM, which are used for the determination of the coefficients of the Padé approximation in the ACCC+CSM.

Figure 3

Real part of energies of the $0^{+}$ states obtained by changing the potential strength parameter δ. (a) Result with $\mu =0.05$ fm${}^{-2}$. (b) Result with $\mu =0.20$ fm${}^{-2}$. Crosses show the trajectory of the second $0^{+}$ state, and asterisks and open squares are the third and fourth $0^{+}$ states, respectively. For the new $0^{+}$ state, filled circles show the result obtained by the CSM; open circles are that of the ACCC+CSM.