Bled Workshops in Physics Vol. 18, No. 1 p. 40
A
Proceedings of the Mini-Workshop
Advances in Hadronic Resonances

Bled, Slovenia, July 2 - 9, 2017
Resonances and strength functions of few-body systems
Y. Suzuki
Department of Physics, Niigata University, Niigata 950-2181, Japan RIKEN Nishina Center, Wako 351-0198, Japan
Abstract. A resonance offers a testing ground for few-body dynamics. Two types of resonances are discussed in detail. One is very narrow Hoyle resonance in 12 C that plays a crucial role in producing that element in stars. The other includes broad high-lying negative-parity resonances in A = 4 nuclei, 4H, 4He, 4Li. The former is dominated by the Coulomb force of three-a particles at large distances, while the latter are by short-ranged nuclear forces. The structure of these resonances is described by different approaches, adiabatic hyperspherical method and correlated Gaussians used for strength function calculations. The localization of the resonance is successfully realized by a complex absorbing potential and a complex scaling method, respectively.
1 Hoyle resonance
The synthesis of 12C is essential to 12C-based life and its process at low temperatures is sequential via a narrow resonance of 8Be:
As predicted by Hoyle, however, an existence of a very narrow resonance at around Ex =7.7 MeV is vital to explain the abundance of 12C element. The resonance is found to be just 0.38 MeV above 3a threshold with its width of 8.5 eV.
Since nobody has ever succeeded in reproducing the Hoyle resonance width, we have undertaken to tackle this problem in the adiabatic hyperspherical method [1,2]. This study has further been motivated by the fact that there exists huge discrepancy in the rate of triple-a reactions, a + a + a —» 12C + y, calculated by several authors [3-5].
In contrast to two-body resonances, the Hoyle resonance is characterized by the followings: (1) 3a particles interact via long-ranged Coulomb force even at large distances. (2) no asymptotic wave function is known. (3) 2a subsystem forms a sharp resonance, which causes successive avoided crossings with three-particle continuum states.
The detail of our approach is given in Refs. [1,2]. The three-body system is completely specified by six coordinates excluding the center-of-mass coordinate. Among six coordinates one is chosen to be the hyperradius of length dimension, and other five coordinates are hyperangles. Among the five angle coordinates
a + a^ 8Be, a + 8Be ^ 12C + y.
(1)
Resonances and strength functions of few-body systems
41
three are Euler angles and two are used to specify the geometry of the three body system. By changing the geometry as much as possible, we can study the adia-batic potential curve of the three-body system as a function of the hyperradius. A resonance can be confined by introducing a complex absorbing potential [6] at large distances of the hyperradius. This method works excellently for quantitatively reproducing the very narrow width of the Hoyle resonance as well as predicting the triple-a reaction rate at low temperatures without relying on any ambiguous ansatz.
2 Resonances in A = 4 nuclei
4He nucleus is doubly magic and its 0+ ground state is strongly bound. The first excited state of 4He is not a negative-party but again 0+. The negative-parity excited states appear above the 3He+p threshold. Seven negative-parity states are known and some of them have very broad widths. There exist isobar resonances in 4H and 4Li that are also very broad. Most of these resonances are identified by R-matrix phenomenology.
These resonances offer typical four-body resonances governed by the nuclear force. The decay channels include not only two-body but three-body systems. To describe the resonance we have employed correlated Gaussians [7,8] that provide us with efficient and accurate performance as few-body basis functions. A general form of the correlated Gaussians is
[0l x Xs]jm exp	aij(ri - rj)2 nTMy,	(2)
where 0L,xS,nT stand for the functions of orbital angular momentum, spin, isospin parts. aij are variational parameters that control the spatial configuration of the system. See also Ref. [9] for recent review on the correlated Gaussians.
The negative-parity resonances may be studied by analyzing strength functions for electromagnetic excitations from the ground state of 4He. Actually we have considered the spin-dipole operator specified by type p and A^ tensor (A=0,1,2)
4
= £[(n - R) x a^T?	(3)
i=1
where the center-of-mass coordinate of A = 4 nucleus, R, is subtracted from the position coordinate ri to make sure excitations of intrinsic motion only and TP distinguishes different types of isospin operators (tx, ty, tz)
( 1	Isoscalar
TP = I 2tz(i)	Isovector	(4)
I tx (i) ± ity (i) Charge — exchange
We calculate the strength function SP(E) corresponding to the response of the 4He ground state ¥0 induced by Op^
SP(E) = |(Yf|OP|JYo}|2S(Ef — Eo — E),	(5)
42 Y. Suzuki
where denotes a sum over all possible final states This strength function can be computed by using the complex scaling method. The important thing for accurate evaluation of Sp(E) is to span possible final configurations as much as possible.
We have studied three negative-parity states with isospin 0 in 4He and four negative-parity states with isospinl in 4He, 4H, 4Li [10-12]. Some of the resonance widths are very broad, and thus it is hard to identify their resonance parameters on the complex plane. However, the strength functions calculated above clearly indicate peaks near the resonance energies. We have confirmed that even the broad resonance can be identified with this calculation.
Acknowledgments
The talk is based on the collaborations with H. Suno and P. Descouvemont for the Hoyle resonance and with W. Horiuchi for A = 4 resonance. The author is grateful to them for many constructive discussions. The author is indebted to the organizers of the workshop for a kind invitation that has led to several in depth communications with the participants.
References
1.	H. Suno, Y. Suzuki, and P. Descouvemont, Phys. Rev. C 91, 014004 (2015).
2.	H. Suno, Y. Suzuki, and P. Descouvemont, Phys. Rev. C 94, 054607 (2016).
3.	K. Ogata, M. Kan, and M. Kamimura, Prog. Theor. Phys. 122,1055 (2009).
4.	N. B. Nguyen, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 87, 054615 (2013).
5.	S. Ishikawa, Phys. Rev. C 87, 055804 (2013).
6.	D. E. Manolopoulos, J. Chem. Phys. 117, 9552 (2002).
7.	K. Varga and Y. Suzuki, Phys. Rev. C 52, 2885 (1995).
8.	Y. Suzuki and K. Varga, Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems, Lecture Notes in Physics, m 54, Springer, Berlin, 1998.
9.	J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Rev. Mod. Phys. 85, 693 (2013).
10.	W. Horiuchi and Y. Suzuki, Phys. Rev. C 78, 034305 (2008).
11.	W. Horiuchi and Y. Suzuki, Phys. Rev. C 85, 054002 (2012).
12.	W. Horiuchi and Y. Suzuki, Phys. Rev. C 87, 034001 (2013).