A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body t–matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.


Relativistic faddeev equation for three-Body Bound State
The 3B mass operator of three identical particles with mass m and momentum k i , interacting with pairwise interactions, in the relativistic quantum mechanics is defined as The free-body mass operator M 0 in the 3B Hilbert space is given by The 2B boost interactions V V k ij k ≡ embedded in the 3B Hilbert space can be obtained in terms of the relativistic 2B interaction V r by following nonlinear relation^^ω where k = k i + k j is total momentum of the subsystem (ij), p m p ( ) 2 2 2 ω = +ˆ, and p is the relative momentum in the 2B subsystem (ij). The k-dependence arises from this fact that in a 3B system, the 2B subsystems are not at rest. Obviously, for k = 0 the boost interaction reduces to the relativistic interaction, i.e. V k = V r . Since we work in the center of mass frame of the 3B system, k can be considered as the momentum of the spectator particle. The Schrödinger equation for the bound state of three identical particles interacting with pairwise forces is described by Faddeev equation is the 3B mass eigenvalue. The permutation operator is defined as P = P ij P jk + P ik P jk , and t is 2B transition operator. In order to solve the Faddeev Eq. (4) in momentum space, one needs the 3B basis states, which are composed of two relativistic Jacobi momentum vectors pk , defined in details in ref. 44 . The 3B Jacobi momentum variables are defined by boosting the single-particle momenta to the three-body rest frame and then boosting two-body subsystem momenta to the subsystem rest frame. The free-body mass operator M 0 , given in Eq. (2), is diagonal in these basis states where Ω = + k m k ( ) ( ) 2 2 1 2 . As shown in ref. 44 the representation of Faddeev Eq. (4) in the relativistic basis states pk leads to the following three-dimensional integral equation∫ www.nature.com/scientificreports www.nature.com/scientificreports/ , and the Jacobian function N(k, k′) is defined as where N is the square root of the Jacobian which arises from going from individual momenta k and k′ of the subsystem to the relative momentum p and the total 2B momentum k + k′ 44 , In the nonrelativistic limit where the momenta are much smaller than the masses, the Jacobin function N reduces to one, and consequently, the relativistic Jacobi momenta reduce to the nonrelativistic ones. The relativistic shifted momentum arguments are defined as where the permutation coefficients C(k, k′) are defined as 26 2 2 In the nonrelativistic limit, when the momenta are much smaller than the masses, the permutation coefficient C(k, k′) is equal to one. Consequently, the relativistic shifted momenta π ∼ and π reduce to the corresponding nonrelativistic ones. As we mentioned earlier, in this paper we calculate the fully-off-shell boost 2B t-matrices T k (p, p′; ε) by solving the following relativistic Lippmann-Schwinger equation To solve the three-dimensional integral Eq. (6), we choose a coordinate system where k is parallel to Z-axis, p is in the X − Z plane, and the integration vector k′ is free in the space. In a spherical coordinate system the radial distance, the cosine of polar angle, and azimuthal angle for momentum vectors k, p, and k′ are given by , correspondingly. Using this coordinate system, Eq. (6) can be written explicitly as~~'   www.nature.com/scientificreports www.nature.com/scientificreports/ successful application of the iterative scheme proposed by Kamada and Glöckle 52 in the calculation of the relativistic interaction from the nonrelativistic one in a 3D scheme 54 , in this paper we utilize the same method for the calculation of the boost potential. By following 52 , one can obtain the boost interaction from the nonrelativistic interaction by the following quadratic equation^ω 2ˆˆ. Obviously, for k = 0 the Eq. (14) can be reduced to the connection between the relativistic and nonrelativistic interactions as^ω Equations (15) and (14) have same operator forms, where V r and ω p ( ) are replaced by V k and p ( ) k ωˆ. So the procedure of calculation of boost potential V k from the nonrelativistic potential V nr in Eq. (14) is similar to the calculation of the relativistic potential V r from the nonrelativistic potential V nr in Eq. (15). So we can obtain the boost potential matrix elements by taking the nonrelativistic potential and by solving the following three-dimensional integral equation The integral Eq. (16) is solved using an iterative scheme and the details of the calculations are similar to the ones discussed in ref. 54 . We would like to mention that using an averaging scheme on two successive iterations leads to a faster convergence in the calculation of the boost interaction.

numerical Results
In the first step we solve the three-dimensional integral Eq. (16) to calculate the matrix elements of the boost interaction V k (p, p′, x′) from the nonrelativistic interaction V nr (p, p′, x′) by an iterative scheme. The iteration starts with an initial guess and then continued to reach a convergence in the matrix elements of the boost potential with a relative error of 10 −6 at each set of points (p, p′, x′). In Table 1 we show an example of the convergence of the matrix elements of the boost potential V k (p, p′, x′) as a function of the iteration number calculated for three different values of the Jacobi momentum k = 1,5,10 fm −1 for the Malfliet-Tjon-V (MT-V) bare potential in the fixed points (p = 1.05 fm −1 , p′ = 2.60 fm −1 , x′ = 0.50). As one can see for larger values of the Jacobi momentum k the convergence is reached faster. The data of Table 1 are also illustrated in Fig. 1. In Fig. 2 the matrix elements of the nonrelativistic and the boost interactions and also the difference between them are shown. By having the matrix elements of the boost interactions V k (p, p′, x′) we solve the Lippmann-Schwinger integral Eq. (11) to calculate the off-shell boost t-matrices T p p x ( , , ; ) k ε ′ ′ and then symmetrize it on the angle variable to Fig. 3 we show the angular dependencies of the symmetrized 2B boost t-matrices for  www.nature.com/scientificreports www.nature.com/scientificreports/ energies ε = −10, −50, −200 MeV obtained for k = 1 fm −1 and compared with the corresponding nonrelativistic t-matrices. As we expect for higher energies the difference between the boost and the nonrelativistic t-matrices is more visible.
For the calculation of relativistic effects in the 3B binding energy, we solve the three-dimensional integral Eq. (12). To this aim, for discretization of continuous momentum and angle variables, we use Gauss-Legendre quadrature. For angle variables, a linear mapping, and for Jacobi momenta, a hyperbolic-linear mapping is used. The momentum cutoffs used for Jacobi momenta p and k are 60 fm −1 and 20 fm −1 , correspondingly. In Table 2, we present our numerical results for the relativistic 3B binding energy as a function of the number of mesh points for the Jacobi momenta p and k. For discretization of polar and azimuthal angles, we use 40 mesh points. The symmetry property of the Faddeev component on the angle x, i.e. ψ(p, k, x) = ψ(p, k, −x), is employed to save memory and computational time in solving Eq. (12). As one can see, the relativistic 3B binding energy for the MT-V potential converges to −7.5852 MeV.
In Table 3, we present the contribution of different relativistic corrections to the 3B binding energy. While the Faddeev calculations lead to the nonrelativistic 3B binding energy of −7.7380 MeV, the contribution of different relativistic corrections in the 3B binding energy is as So, two of the relativistic corrections, N and 2B t-matrices, decrease the 3B binding energies, and the other two terms, C and G 0 , increase the 3B binding energy. When we apply all the corrections together, we obtain a relativistic 3B energy of −7.5852 MeV, which indicates a reduction of about −0.15 MeV or a percentage difference of 2%.
As we discussed earlier, an alternative approach for the calculation of the boost 2B t-matrices is solving a first resolvent equation, where the boost 2B t-matrices can be obtained directly from the nonrelativistic 2B t-matrices. In this approach, there is no need for the matrix elements of relativistic or boost potentials defined in Eqs. (14) and (15). The implementation of this method to 3B bound state calculations leads to a reduction of about 3.3% in the 3B binding energy for the MT-V potential 44 . As one can see, the calculation of the boost 2B t-matrices using two different methods, i.e. solving the first resolvent equation discussed in ref. 44 7 shows a wide range of decrease between 2 to 6%.
By solving the three-dimensional integral Eq. (12) and obtaining the 3B binding energy and the Faddeev component ψ(p, k, x) one can calculate the 3B wave function Ψ(p, k, x) by adding up the Faddeev components in three different 3B clusters. The details of the calculation are addressed in ref. 44 and here we briefly represent the explicit form of the 3B wave function as   The Jacobian function N is defined in Eq. (8) and the shifted momentum arguments p 2 , k 2 , p 3 , k 3 and the angle variables x 2 , x 3 are defined as       www.nature.com/scientificreports www.nature.com/scientificreports/ The 3B wave function is normalized as i i j j 1 3 To study the contribution of different Faddeev components in the normalization of the 3B wave function, we show the inner product of the Faddeev components, i.e. 〈ψ i |ψ j 〉, in Table 4. The calculated Faddeev components defined in Eq. (22) satisfy the normalization of the 3B wave function given in Eq. (21) with an error of the order of 10 −12 . While three Faddeev components defined in Eq. (18) have completely different momentum dependence, they are all normalized to almost the same value, i.e. 〈ψ i |ψ i 〉 = 0.12658. Moreover, the inner products of the nonidentical Faddeev components leads to the same value of 〈ψ i |ψ j 〉 = 0.10338. Our numerical analysis confirms the normalization condition of the 3B wave function 〈Ψ|Ψ〉 = 3〈ψ|ψ〉 + 3〈ψ|P|ψ〉 = 1 with an accuracy of at least five significant digits.
As a test for the numerical accuracy of our calculations, we calculate the expectation value of the 3B mass operator 〈Ψ|M|Ψ〉 and compare it with the 3B binding energy obtained from the solution of the integral Eq. (12). By considering the definition of the 3B mass operator M in Eq. (1), the expectation value of 〈M〉 can be obtained as where v k (p, p′, x, x′) is the result of the azimuthal angle integration on the matrix elements of the boost potential defined as In Table 5, we present our numerical results for the expectation value of the 3B mass operator compared with the relativistic 3B binding energy. We also show our results for the expectation value of the nonrelativistic 3B Hamiltonian and the nonrelativistic 3B binding energy. As one can see the expectation value of free-body mass operator is 〈M 0 〉 = 3m + 28.3876 MeV, while the expectation value of the boost potential is 〈V k 〉 = −35.9716 MeV which leads to the expectation value of the 3B mass operator of 〈M〉 = 3m − 7.5840 MeV. The difference between the relativistic 3B binding energy, obtained from three-dimensional Faddeev integral Eq. (12), and the expectation value of the 3B mass operator, obtained from Eq. In Table 6 we present the contribution of different Faddeev components in the expectation value of the 3B mass operator 〈ψ i |M|ψ j 〉 − 3m. To this aim the expectation values of the free-body mass operator 〈ψ i |M 0 |ψ j 〉 − 3m and the boost interaction 〈ψ i |V k |ψ j 〉 are calculated for different Faddeev components. As one can see

Summary
The inputs for the kernel of Faddeev and Yakubovsky integral equations, to study the relativistic corrections in three-and four-body bound and scattering states, are the boost 2B t-matrices. A number of theoretical methods are developed to calculate the relativistic and boost 2B t-matrices. The direct solution of the relativistic Lippmann-Schwinger equation using relativistic interactions is one of the methods which has been successfully implemented in traditional partial wave decomposition calculations. One of the novel techniques for the calculation of the relativistic and boost interactions from the nonrelativistic interactions is solving a quadratic equation using an iterative scheme proposed by Kamada and Glöckle. In this paper, we apply the direct calculation of the boost interaction from the nonrelativistic interaction by solving a quadratic equation in a three-dimensional approach. To speed up the convergence in the solution of the quadratic equation, we use an averaging scheme. Then, by solving the Lippmann-Schwinger integral equation with the input boost 2B interaction, the boost 2B t-matrices are obtained as a function of the 2B relative momentum vectors, mainly the magnitude of the initial and final relative momenta and the angle between them. The 2B boost interactions have been then embedded into the kernel of three-dimensional Faddeev integral equations to study the relativistic corrections in the 3B binding energy. For a comprehensive study of the relativistic effects, one must consider four different relativistic corrections: the relativistic free propagator, the relativistic 2B t-matrices, the Jacobian, and the permutation coefficient. The contribution of each relativistic correction is studied in detail. We show that the relativistic corrections lead to a reduction of about 2% in the 3B binding energy. As a test for the numerical accuracy of our calculations, we calculate the expectation value of the 3B mass operator, which is in excellent agreement with the relativistic 3B binding energy. It is shown that in the nonrelativistic limit, where the momenta are much smaller than the masses, the nonrelativistic results are being reproduced from the relativistic ones.

Data availability
The data that support the findings of this study are available from the corresponding author, M.R.H., upon reasonable request.