Photodisintegration of $^3H$ in a three dimensional Faddeev approach

An interaction of a photon with $^3H$ is invstigated based on a three dimensional Faddeev approach. In this approach the three-nucleon Faddeev equations with two-nucleon interactions are formulated with consideration of the magnitude of the vector Jacobi momenta and the angle between them with the inclusion of the spin-isospin quantum numbers, without employing a partial wave decomposition. In this formulation the two body t-matrices and triton wave function are calculated in the three dimensional approach using AV18 potential. In the first step we use the standard single nucleon current in this article.


Introduction
Since the early days of the study of the nuclear physics so many efforts have been performed on 3N systems considering real or virtual photon interactions [1]- [2]. Also several studies on the behavior of 3N bound states in real or virtual photon absorbtion have been reported [3]- [4]. Before sixties variational approach was used for these calculations and works using this approach are still continuing. After introducing Faddeev formulation for three body systems [5]- [6], new efforts using this scheme were started. As an example one can point out the early calculations of electrodisintegration [7] and photodisintegration [8] with 3 He and 3 H. An improvement in the photodisintegration calculation of the bound and 3N continuum with the same 3N hamiltonian have been performed [9]. There are also other approaches to calculate electromagnetic interactions with light nuclei such as Green-function-Monte-Carlo method [10], hyperspherical harmonic expansions [11], and Lorentz integral transform (LIT) method [12].There is a very good review of Faddeev calculations on the interaction of real or virtual photon with 3 He [13]. In this work like previous calculations the partial wave decomposition has been used. In PW approach one should sum all PW to maximum angular momentum where the calculation is converged. The problem is that in higher energies this maximum angular momentum increases and we should solve more complicated equations. To avoid this complexity one should use vector momentum as basis states [14]. To this aim in the past decade the main steps have been taken by Ohio-Bochum collaboration (Elster, Glöckle et al.) and Bayegan et al. to implement the 3D approach in few-body bound and scattering calculations (see for examples Refs. [15]- [22]). It should be clear that the building blocks to the few-body calculations without angular momentum decoma e-mail: bayegan@khayam.ut.ac.ir position are two-body off-shell t-matrices, which depend on the magnitudes of the initial and final Jacobi momenta and the angle between them. Fachruddin et al. have calculated the NN bound and scattering states in a 3D representation using the Bonn-B and the AV18 potentials [15]- [16]. Recently there has been efforts to do the same calculation using chiral potential [23]. our aim in this work is to formulate photodisintegration of 3 H in a three dimensional Faddeev approach. In the first step we ignore three body forces and we just use the single nucleon current. We will use AV18 potential and triton wave function which has been calculated in our previous work [20].
This manuscript has been organized as follow: in section 2 we explain our basic states and we evaluate all of the matrix elements in these basis. In section 3 we introduce our singularity problem and its solution. We finish in section 4 with a summary and outlook.

Integral equation of nuclear matrix elements without partial wave decomposition
To calculate the photodisintegration observable we first need to calculate nuclear matrix elements in the Faddeev scheme. For more details see Ref. [13].
In above equations t is NN t-operator which obeys Lipmann-Schwinger equations, G 0 is free propagator, P is permutation operator, |ψ is three body bound state and |U is an axillary state. Three body forces have been ignored. |φ 0 is a subsection of the fully antisymmetric free state, |Φ 0 , in which nucleons 2 and 3 are in subsystem.
|φ 0 is also our basic state to solve the integral equation (2) and is antisymmetric under permutation of nucleons 2 and 3.
In equation (4) p and q are jacobi momenta and m's and ν's are the spin and isospin of the individual nucleons respectively.
Orthonormality and completeness relations of these basic states can be considered as bellow: Considering these properties we can rewrite the integral equations (1) and (2) in our basic stats.
The effect of permutation operator on our basic states can be considered as follow: Where: Now with respect to above relation and symmetry considerations we can evaluate equations (7) and (8) as follow: The firs term in the equation (12) can be evaluated as: a pq, m 1 m 2 m 3 ν 1 ν 2 ν 3 |(1 + P)J|ψ Now we concentrate on the elements of these equations i.e. current, two-body t-matrix and triton wave function, more precisely.

current
Considering the symmetry properties we have: Matrix elements of single nucleon current can be evaluated as follow: In above equation Q is the momentum of photon. We need to rewrite the single nucleon current operator in a form which is suitable for our basic states. The current operator which we will use is: Which is summation of convection current and spin current. G E (Q) and G M (Q) are electric and magnetic form factors respectively. For the convection part we have: As we will show we have to choose coordinate system in which the z axis is along the Q vector and we also need tensor component of current so the second and the third terms of the right hand side of equation (17) will vanish.
Thus for the convection current we have: And the tensor component of spin part can also be evaluated as:

two-body t-matrix
Two body t-matrices can be related to the one which calculated in helicity basis: In the above relation z = E − 3q 2 4m is the energy of subsystem. As we know two-body function has a singularity in the energy of deuteron, z = E d . To remove this singularity we should consider t-operator as follow: Two body t-matrix in helicity basis has been calculated before [15]. Where

triton wave function
For evaluating the Triton wave function we need to make a relation between this wave function in our basic states to the one which has been calculated in the following basis [20]: We can relate these states to our free spin and isospin states with Clebsch-Gordan coefficients. Where It is very important to mention that the spin of the nucleons is quantized in the direction of the z axis which in the calculation of wave function it has been chosen to be in the direction of q. But we have to consider the z axis along the direction of incident photon Q. So we should first rotate the spin of the nucleons in our basis to be settled in the direction of q axis. Then we should use Clebsch-Gordan coefficients to obtain the wave function in the calculated basis mentioned in the equation (24): In order to consider the singularity problem we can rewritten the equation (11) an (12) in a unified form ignoring isospin dependent which is similar to spin dependent.