Low-lying 12C continuum states in three-α model

The electromagnetic monopole (E0) transition of the 12C nucleus, the 0 ground state to three α-particles (3α) continuum states, is calculated by applying the Faddeev three-body formalism in coordinate space accommodating the long range Coulomb potential between α particles. Results show that there are two 0 states, C(03 ) and C(04 ) states at excitation energies around 10 MeV, which is consistent with results of recent semi-microscopic calculations and analysis of α-12C inelastic scattering.


Introduction
Low-lying states in 12 C nucleus are interesting subjects to study as a 3α system.For example, it is widely known that the first 0 + resonant state 12 C(0 + 2 ) at the excitation energy E x = 7.65 MeV (so called the Hoyle state) plays an essential role in the synthesis of 12 C from a 3α continuum state (the triple-α process).However, there still have been uncertainties in the 12 C energy level structure at these energies (E x ≈ 10 MeV): While a 0 + state was tentatively listed at E x = 10.3MeV in the compilation of experimental data [1], there is an experimental report that there exist two 0 + states (0 + 3 and 0 + 4 ) [2, 3] near E x = 10 MeV.Theoretically, the existing of two 0 + states in this energy region was predicted by a semi-microscopic model, the orthogonal-condition model (OCM), combined with the complex scaling method [4].
In this paper, I will consider a transition of the 12 C ground state 12 C(0 + 1 ) to 3α 0 + continuum states by the E0 operator and calculate the transition strength as a function of 3α energy E at the center of mass (c.m.) system above the 3α threshold.This is an extension of recent works [5,6], in which 3α continuum states up to E = 0.6 MeV were studied to calculate the reaction rate of the triple-α process at stellar temperature T ∼ 10 9 K, and clarify the decay mode of the Hoyle state.
In Sec. 2, I will give a short description of the formalism to calculate the strength function.In such a treatment, every complicacies arising from nucleon structure of the α-particle are assumed to be incorporated in interaction potentials among the α-particles, which usually consist of two-α potential (2αP) as well as three-α potential (3αP).Such models used in this work and results of the strength function will be presented in Sec. 3. Summary will be given in Sec. 4.

Formalism
We will consider a transition process of the 12 C(0 + 1 ) state |Ψ b to 3α continuum states induced by the E0 operator (see, e.g.[7]), where x is the relative coordinate of α-α pair and y is the relative coordinate of the spectator α-particle with respect to the c.m. of the pair.
With the transition amplitude of the reaction, Ψ (−) qp | Ô(E0)|Ψ b , where |Ψ (−) qp is an eigen-state of 3α Hamiltonian H, and q and p are Jacobi momenta conjugate to the coordinates x and y, respectively, the E0 transition strength function is defined by where E q = 2 m α q 2 and E p = 3 2 4m α p 2 with m α being the α particle mass.Let us define a wave function for the transition process by which has the asymptotic form [5], where ) expresses the effects from the long-range Coulomb interaction, and F (B) ( q, p, E q ) is the breakup amplitude, The wave function Ψ will be obtained by applying the Faddeev three-body formalism to solve integral equations in coordinate space with accommodating the long range Coulomb force effects.(See Refs.[5,6,8] for the details of the calculations.)This procedure provides the amplitude F (B) ( q, p, E q ), and then the E0 strength function is calculated as

Results
In the present work, I used the model D of the Ali-Bodmer potential AB(D) [9] with the (point) Coulomb potential for the α-α interaction.In addition, 3αP of the following form is introduced:  [2,11] renormalized according to Ref. [10] (see the text).

Table 1 .
The range and strength parameters of the 3αP models, Eq. (7), used in this work, and calculated energy of the 12 C ground state with respect to the 3α threshold, E[ 12 C(0 + 1 )].The E0 strength function for a transition from 12 C ground state to 3α 0 + continuum states as a function of the final 3α energy in the c.m. system E. The black, red, green, blue, and aqua lines are calculations by Δ(2.4), Δ(2.6), Δ(3.0), Δ(3.4), and Δ(4.0), respectively.Experimental data are taken from Ref.