L.D. Blokhintsev a, A.N. Safronov a, and A.A. Safronov b a Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia b Moscow State Institute of Radioengineering, Electronics and Automation, Moscow, Russia An Analytic Approach to Constructing Effective Local Interactions in Few-Body Systems and Its Application to N 4 He, N 3 H, N 3 He, and 3 He 4 He Scattering

Requirements: 1) Local potential 2) Potential leads to the correct analytic structure of a scattering amplitude S wave scattering: nd, pd, n 3 H, p 3 He, n 4 He, p 4 He, 3 He 4 He А. Coulomb interaction neglected Marchenko equation of the inverse scattering problem:

is a spherical Hankel function. Initially is expressed through the integral of the partial S matrix along the real axis in the complex plane of the relative momentum k and through residues of at the poles corresponding to bound states.

Closing the integration contour as is shown and using the Cauchy theorem, one can express the kernel F l (r,r′) through the discontinuity of S l (k) along the dynamical cut. At that the contribution of poles is cancelled. The sought-for potential is written in the form Fig.1 k plane dynamical cut of S l (k) pole of S l (k) Fig.1

is found by solving the Marchenko or Martin equations. The nearest and the most important singularity is determined by the one-particle transfer. Fig.2

For the pole mechanism the discontinuity of S l (k ) is expressed in terms of the vertex constant (VC) G ABC, which is the matrix element of the virtual A  B+C process in the given partial-wave state. VC G ABC is directly related to the residue in energy of the partial-wave matrix element of elastic BC scattering at the pole corresponding to the bound state A.

G ABC is proportional to the asymptotic normalization coefficient (ANC) с ABC of the wave function (overlap integral) of A nucleus in the B+C channel: G ABC =β с ABC Note. VC is a more fundamental quantity than ANC since it is defined directly in terms of the partial-wave scattering amplitude; on the other hand, the coefficient β contains the model-dependent factor. VC’s and ANC’s are widely used to analyze peripheral nuclear reactions, especially astrophysical nuclear reactions. In particular, the cross section of the radiative capture process B(C,γ)A at astrophysical energies is completely determined by G ABC (or с ABC ).

B. Account of the Coulomb Interaction Coulomb interaction drastically changes the analytic properties of scattering amplitudes. The following Coulomb effects were taken into account Interaction in two-body (A+B) states (by introducing a Coulomb-nuclear scattering amplitude and using the phase function method) Radiative corrections in 3-body intermediate states Radiative corrections in vertex functions

ab Fig.3

The Coulomb-nuclear phase shifts are found by the phase function method, which is convenient to take into account exactly the Coulomb interaction in the two-body initial, final and intermediate states. The equation for the phase function is of the form and are the well-known regular and irregular Coulomb functions, η is the Coulomb parameter. At turns into the usual partial- wave phase shift. The short-range Coulomb-nuclear potential is constructed as described above.

NUMERICAL RESULTS pd scattering ab Fig.4 The contribution of Fig.4b was found to be small.

1 Fig.5. The effective potential for pd scattering in the quartet S state r, fm U (r), MeV

Fig.6. The quartet pd scattering length as a function of the upper integration limit in α. 4 a pd, fm

n 3 H and p 3 He scattering n(p)n(p) n(p)n(p) 3 H( 3 He) Fig.7 d fm

Fig.8 n n n p 3H3H 3H3H p p 3 He n p

Fig.9. The calculated p 3 He scattering phase shift for L=0, J=S=1. Circles, squares and crosses denote the results of the phase-shift analyses. E lab, MeV , degree

Fig.10. The calculated n 3 H scattering phase shift for L=J=S=0. Circles and squares are the results of the four-body calculations (R. Lazauskas et al., 2005) E lab, MeV , degree

Low-energy parameters of NT scattering Processa 1, fmr 1, fma 3, fmr 3, fm n3Hn3H 4.09 a 4.09 b 4.25 c 4.28 d 2.10 a a 3.70 b 3.74 c 3.73 d 1.78 a - p 3 He10.20 a 1.61 a 8.45 a 1.41 a

3 He 4 He SCATTERING 4 He 3 He n Fig.11

Fig.12. The 3 He 4 He S wave phase shift calculated at different values of : 2.8 fm (curve 1), 3.5 fm (curve 2) and 5.0 fm (curve 3) E lab, MeV , degree

Low-energy parameters of 3 He 4 He scattering, fm r, fm (M.Viviani et al., 2005) a, fm

CONCLUSIONS 1. Main characteristics of the processes considered are determined by the most general properties resulting from necessary analyticity requirements. 2. The method suggested can be used to obtain information on vertex constants and asymptotic normalization coefficients.

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