On the contribution of the and partialwave states to the binding energy of the triton in the Bethe–Salpeter–Faddeev approach
Аннотация
The influence of the partialwave states with nonzero orbital moment of the nucleon pair on the binding energy of the triton in the relativistic case is considered. The relativistic generalization of the Faddeev equation in the Bethe–Salpeter formalism is applied. Twonucleon matrix is obtained from the Bethe–Salpeter equation with separable kernel of nucleon–nucleon interaction of the rank one. The kernel form factors are the relativistic type of the Yamaguchi functions. The following twonucleon partialwave states are considered: , , , , , . The system of the integral equations are solved by using the iteration method. The binding energy of the triton and threenucleon amplitudes are found. The contribution of the and states to the binding energy of triton is given.
1 Introduction
The study of threenucleon systems has a long history and many works is devoted to the description of such nuclei. One of the most common nonrelativistic descriptions is based on the application of the Faddeev equation with various twoparticle potentials. Among such potentials, there are realistic [1] and separable [2]. Such studies have made it possible to achieve significant progress in the description of static and dynamic properties of threenucleon systems.
In the same time planned experiments on the scattering of electrons by He and H, for instance, Jefferson Lab Experiment E1210103, with the energies of the initial particles up to 12 GeV, require a relativistic description. There are ways of relativizing the nonrelativistic description, and the methods that follow from the first principles of quantum field theory(QFT). Among the latter we single out the quasipotential Gross equation with the exchange kernel of a nucleonnucleon interactions [3], and also a approaches based on the BetheSalpeter formalism with zero range of forces [4], and with a separable kernel of interaction [5,6].
This work develops the ideas represented in the articles [5], where the triton is considered in the state, and [6], where along with the state, the contribution of the state into the twoparticle matrix was considered. To describe a threenucleon bound state, a relativistic generalization of Faddeev equations in the BetheSalpeter formalism – BetheSalpeterFaddeev equation – is used. For simplicity of calculations, we consider nucleons have the same masses and the scalar propagators instead of the spinor ones. The spinisospin structure of the system is described through matrices of recoupling coefficient from one partial state to another.
In previous works [7,8] we considered the case of taking into account the wave not only in the twoparticle matrix, but also its amplitudes in the system of integral equations. In the present paper, the equation is generalized to the case of nonzero values of the angular momentum of a pair of nucleons ( и states). The contributions of the following twoparticle partial states are considered (with a full momentum of twonucleon system ): , , , , , . The resulting system of 12 integral equations for real and imaginary parts of amplitudes is solved by the iteration method and the binding energy of the triton, as well as all threeparticle amplitudes are finding.
The work is organized as follows: after a brief description of the solution BetheSalpeter equations for twonucleon states (sec. 2), the relativistic BetheSalpeterFaddeev equation with scalar propagators is introdused, and the partialwave decomposition is performed (sec. 3). In sec. 4 the results of solving the system of equations and their discussion is represent.
2 Two particles case
Since the kernel of the Faddeev equation, written in integral form, contains a twoparticle matrix we first consider the twobody problem.
The system of two relativistic particles can be described using the BetheSalpeter equation. Written for the twoparticle matrix, it has the next view:
(1) 
where [] – the relative 4momentum of the particles of the system in the initial [final] state, – square of the total 4momentum of the system , – twoparticle matrix, – kernel (potential) of a nucleonnucleon () interaction, – the product of two scalar propagators of nucleons,
(2) 
Considering the equation (1) in the center of mass of system of two particles , it is possible to separate the angular dependence and carry out partialwave decomposition:
(3)  
In the present paper, for solving equation we use the kernel of the interaction in the separable form (rank one):
(4) 
If we substitute in equation (3) the kernel of the interaction as (4), then the twoparticle matrix will also have a separable form:
(5) 
where function :
(6) 
and
(7) 
As form factors of kernel is used relativistic generalization of the Yamaguchitype functions [9,10]
(8) 
(9) 
(10) 
where , , , и – the parameters of the model, which are selected in this way that the calculated values of the observed coincide with the corresponding experimental data for them. As observable quantities in this case can be taken the length and phase of the scattering, the effective radius, and in the case when there is a bound state – deuteron (state), – binding energy. Numerical values of parameters and can be found in [11].
3 Three particles case
The system of three relativistic particles can be described using the Faddeev equations in the BetheSalpeter formalism:
(11) 
where full matrix , – twoparticle Green’s function of particles и ( obeys cyclic permutation):
(12) 
– twoparticle matrix.
For a system of particles with the same masses, Jacobi variables can be introduced:
(13) 
We introduce the amplitude for a bound threeparticle state:
(15) 
where – mass of bound state (triton), – square of the total momentum.
To separation the angular integration and to implement partialwave decomposition it is need to account, that solution for the twoparticle matrix is found in the system of the center of mass of two nucleons but the solution for the threeparticle amplitude is sought in the center of mass system of three nucleons. Since the radial functions depend on the square of the relative 4momentum the Lorentz transformation must be carried out only for arguments of spherical harmonics. In this paper we assume, that the components of the relative 4vectors in the two systems coincide i.e we omit the effects of the Lorentz transformation. In this case, the dependence of the threeparticle amplitudes from two 4vectors и can be divided .
We represent the total orbital angular momentum of a triton in the following form: , where – internal orbital angular momentum of a twoparticle subsystem and – orbital angular momentum of the third particle relative to the twoparticle subsystem.
In order to distinguish the explicit dependence of the amplitude on the angular momentum, we will present it in the following form:
(16)  
where twonucleon states are characterized by  spin, – angular and total momentum. In the equation (16) introduced designation for angular variables of 3vector , – the ClebschGordan coefficients, and – spherical functions.
Using the result of the previous section for the twoparticle matrix (5) and after partialwave decomposition write the amplitude in a separable form:
(17)  
The functions satisfy the following system integral equations:
(18)  
with effective kernels
(19)  
where
(20)  
and
The details of the calculation of the function can be found in [12].
Since we are considering the ground state of a threenucleon system and correspondingly , and the function can be rewritten in the following form:
where correspond to the orbital moments of the partial states .
The accounting of spinisospin structure of the equation kernel can be expressed in terms of matrix of recoupling coefficients from one partial state to another [], which have the following form:
(21) 
4 Numerical calculations and results
In this paper, a homogeneous system of 12 integral equation with a parameter, which is the binding energy of the triton, was solved using the iteration method. A homogeneous system of integral equations has a solution not for all values of parameter, but only for those that satisfy some properties.
To determine the binding energy, the following condition was used (in more detail [13]):
(22) 
where – iteration number.
The procedure for solving the system of integral equations (18)–(20) by the iteration method has good convergence. In numerical calculations of the binding energy of triton and the amplitudes of its states for the Yamaguchi potential, the ratio of the previous iteration to the next did not change with the growth of the iteration number up to the sixth decimal place starting with the 20th iteration.
For the numerical calculation of the integrals, the Gauss method on a twodimensional grid of nodes by dimension was used with mapping . The influence of the number of nodes on the convergence of the result of numerical integration was investigated. For integration on was enough nodes. With further increase quantity of nodes the numerical value of the integral did not change any more. For integration on it was not enough such quantity of nodes. For the study of convergence we increased quantity of nodes to . With further increase quantity of nodes the numerical value of the integral changed only in the fourth decimal place. This accuracy is sufficient and allows us to take into account the contribution of various states to the binding energy.
The table represented the calculated values of the binding energy for different probabilities of the state ().
The values of the binding energy of a triton (MeV)
4  9.221  9.294  9.314  9.287  9.271 

5  8.819  8.909  8.928  8.903  8.889 
6  8.442  8.545  8.562  8.540  8.527 
Experiment 8.48 
The above results show that the main contribution to the binding energy of the triton give an state. Contribution of state is positive and varies from to % depending on the probability of state in deuteron ( = 46 ). Contributions of states have different signs and partially compensate each other, and their total contribution is . So total contribution twoparticle  and partial states with a full angular momenta into the binding energy of a triton is from to %. Comparison of nonrelativistic and relativistic calculations of binding energy was conducted in [5]. The paper shows that relativistic calculation of binding energy in the case of accounting only states more nonrelativistic at MeV.
In Fig. 14 represented graphs of real and imaginary parts of partial amplitudes on variables (at fixed values of ) and (at fixed values of ). As can be seen from the graphs, amplitudes of states dominate wherein other states give a nonzero contribution. However we believe that interference contributions of ,  and states in form factors of the threeparticle system must be taken into account in calculations. Obtained amplitudes will be used to calculate the electromagnetic form factors of the triton using the approximations described in articles [5,6].
5 Conclusion
The solution of the relativistic Bethe  Salpeter  Faddeev equation for a threenucleon system (triton) are considered in article. A relativistic generalization of the partialwave decomposition procedure is carried out, which is spread to the nonzero orbital angular momenta of the interacting pair of nucleons. The case of ,  and partial states of the twoparticle subsystems are considered. The using of partialwave decomposition and potential of interactions in a separable form led to a system of integral equations for the amplitudes of states with different orbital moments of particles in the nucleus. The numerical solution of this system using the iteration method allowed find the binding energy of a triton and amplitudes of its , , , , , states as functions of two variables.
This work was partially supported by the Russian Foundation for Basic Research grants №160200898 and №183200278.
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