Derivation of Relativistic Yakubovsky Equations under Poincar\'e Invariance

Relativistic Faddeev-Yakubovsky four-nucleon scattering equations are derived including a 3-body force. We present these equations in the momentum space representation. The quadratic integral equations using the iteration method, in order to obtain boosted potentials and 3-body force, are demonstrated.


Introduction
At high energies one could expect deficiencies in the nonrelativistic Faddeev approach [1,2] in three-nucleon system. We have been constructing a relativistic framework in the form of relativistic Faddeev equations [3][4][5][6][7][8][9][10] according to the Bakamjian-Thomas theory [11]. Not only using the realistic nonrelativistic nucleon-nucleon (NN) potentials but using Kharkov relativistic NN potential [12] we obtained the triton wave function by solving the relativistic Faddeev equation [13][14][15]. However, in the three-body scattering states, the relativistic effects appear to be generally small [16,17] and insufficient to significantly improve the data description. For sensitive observations such as A y puzzles, the relativistic effect has certainly surfaced [18], but the results obtained were in the direction of deterioration.
As the number of particles handled increases, subtle relativistic effects will be accumulated and surface, so here we would like to rewrite the Yakubovsky equations [19], which solve the four-nucleon system exactly, to a relativistic equations as well.
In Section 1 we organize relativistic momenta and its Jacobian. Section 2 deals with rewriting interaction to relativistic potential by Lorentz boost. The boosted potential satisfies the relativistic Lippmann-Schwinger (LS) equation in Sec 3. Section 4 looks back on how the 3body Faddeev equation was relativistically transformed using the boosted potential. In Section 5, we will remodel the four-body Yakubovsky equation and derive the relativistic Yakubovsky equation. In section 6, we derive relativistic equations involving 3-body force. A summary is in section 7.

2 2-body center of mass system
In 2-body systems, these static mass of the particle are given m i (i = 1, 2). The four dimensional intrinsic momenta p i are with By Lorentz transformation L = (L ν µ ) of boosting velocity υ the transformed four-momenta p i are obtained whereυ is the unit vector of υ and light velocity is set to 1. The relativistic total momentum P 12 and the relative momentum k 12 are given [20] as with and where we need pay attention that these equations from Eq.(6) to Eq. (8) are coupled for k 12 .
The momentum k 12 is regarded an instanteneous momentum of center of mass system. We start to enter the center of mass system, which the boosting velocity u is now chosen instead of υ We have and Therefore, we solve them to have k as

Boosted potential
Let us consider two equal mass particles (m = m 1 = m 2 ) which are labeled 1 and 2 in the 2-body center of mass system with interaction v 12 . We have a invariant mass S 12 of the system as where k 12 is the relative momentum between particle 1 and 2.
On the other hand, we leave from the 2-body c. m. system, the total momentum P 12 ≡ p 1 + p 2 is nonzero. The invariant mass S boost 12 is given as Now, one introduce so-called boosted potential V 12 as After quantumization (k 12 →k 12 , v →v and V →V ) the boosted potential operatorV 12 (P 12 ) is still a diagonal operator to the boosting momentum P 12 . We have a boosted Schrödinger equation for the wave function φ 12 4(m 2 +k 2 12 ) + P 2 12 +V 12 (P 12 ) φ 12 = M 2 + P 2 12 φ 12 (19) and a un-boosted one, where M is a eigen value of mass operator Ŝ 12 .

4 Relativistic Faddeev Equations
For the 3-body system, we add to third equal mass particle. There are 3 piarwises (subsystems) denoted not only as (12) but as (23) and (31). For each piarwise (ij) we can define the boosted potential as Eq. (18) by the boosting momentum P i j .
We choose now the boosting momentum P i j = − p k , which means it is 3-body c. m. system (i = k = j); One may naturally have an idea the following 3-body invariant mass S 123 poses a symmetry.
This symmetry helps us to build the relativistic Faddeev equations. After the quantumization we write the relativistic Faddeev equation for bound state as where φ i j is the Faddeev component for the subsystem (ij) with the total wave function Ψ andĜ 0 is the three-body Green's function, andt i j is the t-matrix of subsystem (ij) which is satisfactory with the LS equation; where M 123 is the eigen value of the mass operator Ŝ 123 .
In the case of a system with identical particles, the permutation operators built from transpositions P i j , interchanging particles i and j, are used to express all two-body interaction where we have singled out the (12) pair and denoted V ≡ V 12 , t ≡ t 12 and φ 12 ≡ φ with permutation operator P ≡ P 12 P 23 + P 13 P 23 . We have a simple presentation of Eq. (24).
We choose now the boosting momentum P i j = − p k − p l , which means it is 4-body c. m. system (k = i, j, l, and l = i, j, k); One may naturally have an idea the following 4-body invariant mass S 1234 poses a symmetry.
The most important thing is that during generate the boosting potential V i j the momentum P kl behaves as a parameter. In other word, the boosting potential operatorV i j is diagonal to the momentum P kl . Using the 3-body relative momentum q k between the subsystem (ij) and the third particle, and the [2+2] partition relative momentum s l between the subsystem (ij) and (kl) (see Apppendix A) we rewrite the 4-body invaliant mass S 1234 where q 3 is a relative momentum between subsystem (12) and the third particle, and s 4 is a relative momentum between subsystem (12)
We show the relativistic Yakubovsky equations for bound state.
The relativistic Yakubovsky equation (42) keeps similar form as nonrelativistic one.

Inclusion of 3-body force
Recently [21,22], we obtained the Faddeev equations and Yakubovsky equations including 3-body force. The 3-body force w 123 is naturally decomposed into three parts with w = w

. Instead of Eq. (25) the Faddeev component is defined including a part of 3-body force
Including 3-body force the Faddeev equation for bound state is rewritten as whereτ is defined asτ ≡tP + (1 +tĜ 0 )ŵ(1 + P).
Because the 3-body force are given in 3-body center of mass system, we need not boost the 3-body force.

Conclusion
The relativistic   58) and (59) in Appendices A and B. The quadratic integral equations using the iteration method [7], in order to obtain these boosted potentials and 3-body force, are demonstrated in Appendix C.

B Appendix B
The boosted 3-body force W (3) 123 (p 4 ) is defined as where η can be chosen 1, 2 or 3.

C Appendix C
The boosted potential V 12 ( k, k ; q) ≡ 〈 k|V 12 (q)| k 〉 of Eq. (18) is a solution of the following quadratic integral equation [7].
These equations from (60) to (63) may be solved by the iteration method [7].