On the relativistic 3D1 partial-wave contribution to the bound three-nucleon system

The bound state of three nucleons is investigated using the Faddeev equations within the Bethe-Salpeter approach. The relativistic and nonrelativistic nucleon-nucleon interaction is chosen in a multirank separable form. The extension for partial-states with L > 0 is done. Three partial-wave states 1S 0,S 1 and D1 are taken into account. The Gauss quadrature method is used to calculate the integrals and find the triton binding energy by iterations.


Introduction
Three-body calculations in nuclear physics are of great interest used for the description of threenucleon bound states ( 3 He, T ), processes of elastic, inelastic and deep inelastic scattering the leptons off light nuclei and also the hadron-deuteron reactions (for example, pd → pd, pd → ppn).The investigation of the nuclei 3 He and T is also interesting because it allows us to investigate further (in addition to the case of the deuteron) evolution of the bound nucleon thereby contributing to the explanation of so-called EMC-effect.In quantum mechanics the Faddeev equations are commonly used to describe the three-particle systems.The main feature of Faddeev equations is that all particles interact through a pair potential.
However at the high momentum transfer relativistic effects should be taken into account.The Bethe-Salpeter (BS) [1] equation is one of the most consistent approaches to describe the NN interaction.In this formalism, one has to deal with a system of nontrivial integral equations for both the NN scattered states and the bound state -the deuteron.To solve a system of integral equations, it is convenient to use a separable Ansatz [2] for the interaction kernel in the BS equation.In this case, one can transform integral equations into a system of algebraic linear ones which is easy to solve.Parameters of the interaction kernel are found from an analysis of the phase shifts and inelasticity, low-energy parameters and deuteron properties (binding energy, moments, etc.).
The relativistic three-particle systems are described by the Faddeev equations within the BS approach -so called Bethe-Salpeter-Faddeev equations.All nucleons have equal masses and the scalar propagators instead of spinor ones are used for simplicity.The spin-isospin structure of the nucleons is taken into account by using the so-called recoupling-coefficient matrix.The work mainly follows the ideas of the paper [3].In the paper [4] only S partial-states were considered for the one-rank kernel.In this paper the relativistic 3 D 1 partial-wave is added into formalism using the three-rank Graz-II kernel [5].
The paper is organized as follows: in Sec. 2 the two-particle problem is considered, in Sec. 3 three-particle equations and partial-wave decomposition are described.In Sec. 4 the calculations and results are given.The summary is in the Sec. 5.

Two-particle case
Since the formalism of the Faddeev equations is based on the properties of the pair nucleon-nucleon interaction only some conclusions of two-body problem are given here.
The Bethe-Salpeter equation for the relativistic two-particle system is taken in the following form: where T (p, p ; s) is the two-particle T matrix and V(p, p ) -kernel (potential) of the nucleon-nucleon interaction.The free two-particle Green function G(k; s) is expressed, for simplicity, through the scalar propagator of the nucleons To solve equation ( 1) the separable Ansatz for the nucleon-nucleon potential V(p, p ) is used In this case the two-particle T matrix has the following simple form: where

Three-particle case
Neglecting the three-particles interaction we write the equations for three-particle amplitude T = 3 i=1 T (i) in the following form: where G i is the free two-particle ( j and n) Green function (i jn is cyclic permutation of (1,2,3)): The two-particle amplitude T i satisfies the Bethe-Salpeter equation (see previous section).For the system of equal-mass particles the Jacobi momenta can be written in the following form: Using expressions (8) the amplitude T (6) can be rewritten as where P is the total two-body momentum.
For the bound state the total momentum squared s = K 2 is fixed at the mass of the bound state (triton) The equation ( 9) becomes homogeneous for the amplitude Ψ (i) (p i , q i ; s) For the equal-mass case all Ψ (i) functions are equal to each other and we can write To perform the partial-wave decomposition several steps should be made: 1. construct two-body state in its rest frame; 2. boost two-body state into the three-particle rest frame; 3. construct three-body state.
The details of the partial-wave decomposition ideology for the quasi-potential equation can be found in the paper [6].All steps should be made if the particles are considered as spin-one-half with corresponding spinor propagators S (i) .However, as it was said in previous section we simplify the problem and consider the nucleons with scalar propagators (2).In this case the spin-isospin structure of the system can be presented as a simple multiplicative coefficient.
The total orbital angular momentum of the triton then can be written as L = l + λ, where l is the angular momentum corresponding to a nucleon pair with relative momentum p and λ is the angular momentum corresponding to relative momentum q.
To separate the angular dependence the amplitude can be written in the three-particle rest frame in the following form: λL (p 0 , |p|, q 0 , |q|; s)Y (a) λLM ( p, q), ( with the angular part where the two-nucleon state with spin s, angular momentum l and total momentum j (a ≡ 2s+1 l j ) is introduced, C are the Clebsch-Gordan coefficients and Y are the spherical harmonics.If one considers two-nucleon separable interaction (with n i being the separable index) the amplitude Ψ (a)  λL can be written as λL (p 0 , p, q 0 , q; s) = λLn 2 (q 0 , q; s), ( where functions Φ (a) λLi satisfy the following system of integral equations: λλ n 2 n 3 (q 0 , q; q 0 , q ; s) with λλ n 2 n 3 (q 0 , q; q 0 , q ; s) = C (aa ) d cos ϑ qq K (aa ) λλ L (q, q , cos ϑ qq ) and where cos ϑ = ( and is the spin-isospin recoupling-coefficient matrix with (a) In the case of L = 0 we have l = λ = 0 or l = λ = 2 and expressions for non-zero K-functions are: and the K-matrix is: In above formulae the Lorenz transformation from the rest system of two particles to the rest system of three particles is omitted.Since the g-functions depend on invariant p 2 momentum squared and for S partial-wave states the angular part is constant the only variable which should be transformed is the argument of spherical harmonics for 3 D 1 partial-wave state.We believe that such transformation gives small contribution to the binding energy of three-nucleon system.However it should be done for nucleons with spinor propagators.
The system of integral equations (15-17) has several singularities in q 0 complex plain.However in the case of the bound three-particle system ( √ s < 3m N ) all singularities do not cross the path of integration on q 0 and thus do not affect to the Wick-rotation procedure q 0 → iq 4 .
The system (15-17) after the Wick-rotation procedure is well analytically defined and can be solved using various numerical methods.One of them is discussed in the next section.

Solution and results
In the paper the iterations method is used to solve the system of integral equations.The mappings for the variable of integration on q [0, ∞) and q 4 (−∞, ∞) to [−1, 1] interval are introduced.
As it is shown in [7] the binding energy of the three-nucleon system can be found using the following condition for iterated solutions (see details in [7]): where n is a number of iterations.The Graz-II kernels of interaction [5] with three probabilities (p D = 4, 5, 6%) of the 3 D 1 -state are used for calculations.The obtained results are shown in table 1 both for nonrelativistic and relativistic cases and for S -states and (S + D) partial-wave states.The experimental value is 8.48 MeV.As it is seen from the table 1 kernels with different p D can give results for binding energy both bigger and smaller than the experimental value.The relativistic results are systematically higher then nonrelativistic ones to about 3%.Also the 3 D 1 partial-state gives rather small contribution to the three-nucleon binding energy -about −0.5%.

Summary
In the paper three-body system is investigated using Bethe-Salpeter-Faddeev equations.In the calculations the three-rank Graz-II potential is used.The BSF integral equations are solved using the iterations method.The binding energy of the triton and amplitudes of the 1 S 0 , 3 S 1 and 3 D 1 partialwave states of the triton are obtained.
It was shown that kernels with different p D can give results for binding energy both bigger and smaller than the experimental value.The relativistic results are systematically higher then nonrelativistic ones to about 3% and the 3 D 1 partial-state gives rather small contribution to the three-nucleon binding energy -about −0.5%.

Table 1 .
Three-nucleon binding energy calculated with Graz-II separable NN kernel.