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

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?

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: 11Be (neutron binding energy is 0.5 MeV) Spectroscopic factors 10Be(0+) 10Be(2+) vibrational coupling model 0.84 0.16 (S. Fortier et al, PLB 461, 22 (1999)) no-core shell model NCSM 0.82 0.26 (Forssen et al, PRC71, 044312 (2009)

11Be(p,d)10Be, E/A = 35 MeV J.S. Winfield et al, Nucl. Phys. A683, 48 (2001) n+9Be 6.812 Large cross sections for population of weakly-bound states near 6 MeV: are these configurations important? 2- 6.263 0+ 6.179 1- 5.959 2+ 5.958 2+ 3.368 S~0.8 0+ 0 S~0.2 S=1.40

19C (neutron binding energy is 580 keV) Neutron knockout from 19C: N.Kobayashi et al, PRC86, 054604 (2012) Inclusive measurements: th = 164.8 mb exp = 163(12) mb Core excitations play important role in 19C structure

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.

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, 024605 (2014)

Two-body force in a three-body system R.C. Johnson and N.K. Timofeyuk, PRC 89, 024605 (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!

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

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.

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, 162503 (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, 1550006 (2015)

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.