Applicability of CDCC to the deuteronbreakup at low energies― Is CDCC an alternative to the Faddeev theory? ―RCNP Nuclear Physics Colloquium

Kazuyuki Ogata
Research Center for Nuclear Physics (RCNP), Osaka University
in collaboration withKazuki Yoshida

The Continuum-Discretized Coupled-Channels
method: CDCC (after l-truncation)r (k)
np



 (k)

d R (K)00  0  0   k  k dk0

A

^i

imax

 CDCC   ˆi ˆ ii

cf. M. Kamimura, Yahiro, Iseri, Sakuragi, Kameyama, and Kawai, PTP Suppl. 89, 1 (1986);
N. Austern, Iseri, Kamimura, Kawai, Rawitscher, and Yahiro, Phys. Rep. 154 (1987) 126;
M. Yahiro, Ogata, Matsumoto, and Minomo, PTEP 2012, 01A206 (2012).

Applicability of CDCC to low energy BU process
N. J. Upadhyay et al., PRC 85, 054621 (2012).
12C(d,pn)12C

KO and K. Yoshida, PRC 94, 051603(R) (2016).

gs at Ed = 12 MeV

CDCC

FAGS

CDCC severely overshoots the result ofFAGS,
FAGS. if closed-channels are neglected.

coupling w/ closed channels

Plan of this talkI. Three-body exact theoryL. D. Faddeev, Zh. Eksp. Theor. Fiz. 39, 1459 (1960) [Sov. Phys. JETP 12, 1014 (1961)].

 Three-body scattering problem
 Shortcomings of the Lippmann-Schwinger equation

II. Three-body “exact” theory in a model space
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

 The l-truncation, the center of CDCC
 k-truncation and discretization

III. Numerical verification of the AYK/AKY paper
KO and K. Yoshida, PRC 94, 051603(R) (2016).

 Strong criticism on CDCC for describing deuteron breakup at low energies
 Importance of theclosed channels

IV. Summary

Three-body scattering problem
nnppd

Schroedinger Equation

・Solution 1:

A

Assumptions for simplicity:
・No spins
・No Coulomb
・No absorption (imaginary pot.)
・2-body problem solved
boundary condition (b.c.) not specified

Three-body scattering problem
nnppd

A

Assumptions for simplicity:
・No spins
・No Coulomb
・No absorption (imaginary pot.)
・2-body problem solved

Schroedinger Equation

・Solution 2:(channel Hamiltonian)Lippmann-Schwinger (LS) equation

Problems of the LS equation

1. Absence of the rearrangement channels
2. Divergence problem due to the disconnected diagram
3. Nonuniqueness of the solution

The b.c. of the LS Eq. is not appropriate.
nr
RAppn

R
Ar

The Faddeev theory
L. D. Faddeev, Zh. Eksp. Theor. Fiz. 39, 1459 (1960) [Sov. Phys. JETP 12, 1014 (1961)].

nnpp
dDistorted Faddeev Eqs.

APlan of this talkI. Three-body exact theory
L. D. Faddeev, Zh. Eksp. Theor. Fiz. 39, 1459 (1960) [Sov. Phys. JETP 12, 1014 (1961)].

 Three-body scattering problem
 Shortcomings of the Lippmann-Schwinger equation

II. Three-body “exact” theory in a model space
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

 The l-truncation, the center of CDCC
 k-truncation and discretization

III. Numerical verification of the AYK/AKY paper
KO and K. Yoshida, PRC 94, 051603(R) (2016).

 Strong criticism on CDCC for describing deuteron breakup at low energies Importance of theclosed channels

IV. Summary

Three-body theory in a model space
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

Distorted Faddeev Eqs.

not pair int. but 3-body int.

nR  r/2

rp

RAProblems of the LS equation

1. Absence of the rearrangement channels
2. Divergence problem due to the disconnected diagram✓
3. Nonuniqueness of the solution✓

The b.c. of the LS Eq. is not appropriate.
nrRAppnRAr

l-truncation, the center of CDCC
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

pnRArsmears out ^r w/ the resolution of 1/lmax.
[If lmax → ∞, it means (r’ r).]・We have no rearrangement-like channel in the asymptotic region
because of.
・As lmax increases, the coupling between the 1st Eq. and the othertwo becomes weaker.

Three-body theory in a model space
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

Distorted Faddeev Eqs.

not pair int. but 3-body int.

→0nR  r/2rpR
A

CDCC, as an alternative to the Faddeev theory
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

CDCC solves the following LS eq.:

CDCC gives a proper solution to a three-body scattering problem
if the solution converges w/ respect to l.
・Continuum-Discretization has nothing to do w/ the justification of CDCC.
・l-truncation allows one to truncate also r and k.
・Convergence for other quantities (rmax, kmax, and k, etc.) must be confirmed
to obtain a proper solution to the LS Eq.

The Continuum-Discretized Coupled-Channels
method: CDCC (after l-truncation)r (k)
np



 (k)

d R (K)00  0  0   k  k dk0A

^i

imax

 CDCC   ˆi ˆ ii

cf. M. Kamimura, Yahiro, Iseri, Sakuragi, Kameyama, and Kawai, PTP Suppl. 89, 1 (1986);
N. Austern, Iseri, Kamimura, Kawai, Rawitscher, and Yahiro, Phys. Rep. 154 (1987) 126;
M. Yahiro, Ogata, Matsumoto, and Minomo, PTEP 2012, 01A206 (2012).

Plan of this talkI. Three-body exact theory
L. D. Faddeev, Zh. Eksp. Theor. Fiz. 39, 1459 (1960) [Sov. Phys. JETP 12, 1014 (1961)].

 Three-body scattering problem
 Shortcomings of the Lippmann-Schwinger equation

II. Three-body “exact” theory in a model space
N. Austern, M. Yahiro, and M. Kawai, Phys. Rev. Lett. 63, 2649 (1989);
N. Austern, M. Kawai, and M. Yahiro, Phys. Rev. C 53, 314 (1996).

 The l-truncation, the center of CDCC
 k-truncation and discretization

III. Numerical verification of the AYK/AKY paper
KO and K. Yoshida, PRC 94, 051603(R) (2016).

 Strong criticism on CDCC for describing deuteron breakup at low energies
 Importance of theclosed channels

IV. Summary

Faddeev-AGS vs. CDCC
N. J. Upadhyay, A. Deltuva, F. M. Nunes, PRC 85, 054621 (2012).

Description of deuteron breakup process by CDCC

Incident energyn00p

1

1c

n

22

Ei +  i = E

1
2(Energy Cons.)

0A・Open channels (Ei > 0): directly connected to observables
・Closed channels (Ei < 0): virtual breakup channels
Neglected in the preceding study

Applicability of CDCC to low energy BU process
N. J. Upadhyay et al., PRC 85, 054621 (2012).
12C(d,pn)12C

KO and K. Yoshida, PRC 94, 051603(R) (2016).

gs at Ed = 12 MeV

coupling w/ closed channels

CDCC

FAGS

CDCC severely overshoots the result ofFAGS,FAGS. if closed-channels are neglected.
cf. triple-alpha study

Summary I have recapitulated the three-body scattering theory.
 To use a single LS Eq. is not allowed; we have to rely on the Faddeev theory.
 To use a single LS Eq. w/ the l-truncation is allowed, which gives a proper solution to the three-body scattering problem (for non-rearrangement processes).
 Thus, CDCC is shown to be an alternative to the Faddeev theory.

 We have demonstrated the applicability of CDCC to deuteron breakup
at low energies.
 The failure of CDCC reported by the MSU group is shown to be due to theneglect of the closed-channels; their CDCC model space was not converged.
 The coupling to the closed-channels are crucially important, as suggested by,e.g., Austern et al. in 1987.
 The converged CDCC gives a result that agrees w/ that of FAGS, by which
the theoretical foundation of CDCC has been re-established.