Treatment of Two Nucleons in Three Dimensions

We extend a new treatment proposed for two-nucleon (2N) and three-nucleon (3N) bound states to 2N scattering. This technique takes momentum vectors as variables, thus, avoiding partial wave decomposition, and handles spin operators analytically. We apply the general operator structure of a nucleon-nucleon (NN) potential to the NN T-matrix, which becomes a sum of six terms, each term being scalar products of spin operators and momentum vectors multiplied with scalar functions of vector momenta. Inserting this expansions of the NN force and T-matrix into the Lippmann-Schwinger equation allows to remove the spin dependence by taking traces and yields a set of six coupled equations for the scalar functions found in the expansion of the T-matrix.


Introduction
In Ref. [1] a new formulation for the 2N and 3N bound states in three dimensions has been proposed. In this technique momentum vectors are taken as variables, avoiding a traditional partial-wave decomposition. In addition spin operators occurring as scalar products of spin and momentum vectors -shortly called spin-momentum operators -are evaluated analytically by means of trace operations. In this approach a NN force is employed using its most general operator structure, i.e. as sum of 6 spin-momentum operators multiplied with scalar functions of momenta. A spinmomentum operator representation is used as well for the 2N and 3N bound states, as in Refs. [2] and [3], respectively.
We extend the technique developed in Ref. [1] to NN scattering. This would be an alternative to other threedimensional approach formulated in a momentum-helicity basis [4]. In addition we introduce a new set of spinmomentum operators different from the one used in Ref. [1]. We find one of the spin-momentum operators in Ref. [1] violates time reversal and, therefore, has to be multiplied with a time-reversal violating scalar function. Here we prefer to work with operators, which are also invariant with respect to time reversal. The idea is to apply the general operator structure not only to the NN force but also to the NN T-matrix. The goal is then to find the scalar functions in the expansion of the T-matrix into the spinmomentum operators. First, we insert the spin-momentum operators expansions of the NN interaction and T-matrix into the Lippmann-Schwinger equation. Next by analytical evaluation we remove the spin dependence yielding finally a set of coupled equations for the scalar functions of a e-mail: imamf@fisika.ui.ac.id the T-matrix. Finally we connect the T-matrix to the antisymmetrized scattering amplitude parameterized by the Wolfenstein parameters.

The general operator structure of NN potential
The general operator structure of NN potential reads with V tm t (p ′ , p) being the NN potential projected on the NN total isospin states | tm t as The scalar functions v tm t j (p ′ , p) depend only on the vector momenta. the w j (σ 1 , σ 2 , p ′ , p) are a set of spin-momentum operators, which is time-reversal invariant. As an example a leading order (LO) chiral NN potential is given as [5]

The deuteron
We briefly describe the formulation for the deuteron. The deuteron has total spin 1 and isospin 0. In spin-momentum operator representation the deuteron state is given as [2] where |1m d is the total-spin state with magnetic quantum number m d , φ k (p) scalar functions depending on the magnitude of momenta only, and b k (σ 1 , σ 2 , p) spin-momentum operators given as The scalar functions φ k (p) are connected to the standard partial-wave projected deuteron s-wave ψ 0 (p) and d-wave ψ 2 (p) by [2] ψ 0 (p) = φ 1 (p) Inserting Ψ m d (p) of Eq. (5) and V tm t (p ′ , p) of Eq. (1) into the Schrödinger equation for the deuteron in integral form, To remove the spin dependence from Eq. (9) we project Eq. (9) on 1m d |b i (σ 1 , σ 2 , p) from the left and sum up over m d . We obtain which is a set of two coupled equations for φ k (p), with A d ik (p) and B d i jk ′ (p, p ′ ) being defined as The functions A d ik (p) and B d i jk ′ (p, p ′ ) are scalar functions of the vectors p and p ′ , and need to be calculated only once. As example we have e.g.
The NN scattering observables can be calculated from the anti-symmetrized scattering amplitude M tm t