1. Two-body force in three-body system: a case of (d,p) reactions
N.K. Timofeyuk
University of Surrey

2. Do we have a right to solve this equation?
11Be, 19C, 31Ne etc.
Description in terms of three-body model seems to be natural
Target n
core
Three-body Schrodinger equation:
Available methods to solve three-body Schrodinger equation:
Faddeev
CDCC
Hyperspherical harmonics expansion
Integral relations (Kievsky)
Scattering theory on the momentum lattice (Kukulin)
Weinberg basis expansion (Johnson-Tandy)
Lagrange basis expansion (D. Baye)
R-matrix
Eikonal (high energies) …
Do we have a right to solve this equation?

3. Examples: 11Be (neutron binding energy is 0.5 MeV)
Three-body description would be correct if projection of the many- body function on a three-body channel dominated.
Microscopic theories say that a nuclear wave function is a mixture of all possible channels with various core excitations.
Spectroscopic factors in different channels give a good idea about their importance.
Examples: Be (neutron binding energy is 0.5 MeV)
Spectroscopic factors
10Be(0+) Be(2+)
vibrational coupling model
(S. Fortier et al, PLB 461, 22 (1999))
no-core shell model NCSM
(Forssen et al, PRC71, (2009)

4. 11Be(p,d)10Be, E/A = 35 MeV
J.S. Winfield et al, Nucl. Phys. A683, 48 (2001)
n+9Be
Large cross sections for population of weakly-bound states near 6 MeV: are these configurations important?
S~0.8
S~0.2
S=1.40

5. 19C (neutron binding energy is 580 keV) Neutron knockout from 19C:
N.Kobayashi et al, PRC86, (2012)
Inclusive measurements: th = mb exp = 163(12) mb
Core excitations play important role in 19C structure

6. Justification to neglect core excitations: Feshbach formalism
Nucleon scattering:
0 is found from the
two-body equation: A N r
All core excitations are hidden into energy-dependent non-local non-hermitian optical potential.

7. Excluding core-excitations in the A + n + p three-body model
Target n R r p
Ground-state channel function can be found from three-body model
Optical potential for 3-body system has two-body and three-body terms
R.C. Johnson and N.K. Timofeyuk, PRC 89, (2014)

8. Two-body force in a three-body system
R.C. Johnson and N.K. Timofeyuk, PRC 89, (2014)
Target n R r p
Comparison to N-A optical potential:
Two-body force in three-body system differs from two-body optical potential!

9. N-A two-body force in A(d,p)B reactions
The (d,p) amplitude contains projection of the many-body function into the 3-body A+n+p channel.
The most important contribution to the (d,p) reaction comes from small n-p separations.
In the adiabatic approximation the distorted wave function satisfies the equation: p d
rnp n R A B
What is needed for transfer reactions in the N-A force averaged over Vnp

10. Averaging operator
over Vnp and comparing it to two-body optical potential leads to the following conclusion.
Two-body N-A potential for (d,p) reaction is equal to the non-local energy-dependent complex optical N-A potential taken at the effective energy
n-p kinetic energy in deuteron equal approximately 57 MeV
Three-body problem for (d,p) reactions should be solved with energy-independent non-local nucleon potentials taken at effective energy equal to half the deuteron energy plus another 57 MeV.

11. Available optical potentials:
Practically all available phenomenological systematics are local and energy-dependent. Except:
n-A non-local energy-independent potentials, F. Perey and B.Buck, Nucl. Phys. 32, 353 (1962)
Non-local energy independent nucleon potentials for Z=N nuclei, M.M. Gianinni and G. Ricco, Ann. Phys. 102, 458 (1976)
Non-local nucleon potentials with energy-independent real part and energy-dependent imaginary part, M.M. Gianinni, G. Ricco and A. Zucchiatti, Ann. Phys. 124, 208 (1980)
Non-local energy-dependent dispersive optical potential for N+40Ca, M.H. Mahzoon et al, Phys. Rev. Lett. 112, (2014). This potential simultaneously reproduces the data above and below the Fermi energy.
Energy- and mass independent non-local optical potential, Y.Tian, D.-Y. Pang and Z.-Y. Ma, IJMP 24, (2015)

12. Summary
Application of few-body models for complex systems has to be justified.
The three-body model for the A+n+p system contains two-body and three-body contribution. The latter describes interaction of n and p through intermediate recoil excitation of the target.
Two-body n-A and p-A potentials in the three-body A+n+p system differ from those in free n-A and p-A systems.
For A(d,p)B reaction the averaging procedure provides a recepi for n-A and p-B forces. This involves knowing a non-local energy dependent N-A potential at the energy equal to half the deuteron incident energy shifted by the n-p kinetic energy averaged over the short-ranged n-p interaction.