The general configuration-space Faddeev formalism for studying pd scattering

The configuration-space Faddeev equations are derived for p-d scattering taking into account the difference in interaction between the participant particles. Appropriate modifications have been made in the well-known configuration-space equations for n-d scattering. To show the effect of these modifications, the s-wave calculations are performed for bound state and scattering problems. We model the charge symmetry breaking effect for 3H and 3He with a modified Malfliet-Tjon MT I-III potential. Results obtained for elastic n-d and p-d scattering at Elab=14.1 MeV are compared with our prediction (Ref. [1]) and those of the Los-Alamos/Iowa group (Ref. [2]) .


Introduction
The isotopic formalism was developed for the study of neutron-deuteron scattering in the framework of the configuration space Faddeev equations (Ref.[3]).Charge-independence breaking in the threenucleon system was investigated in elastic neutron-deuteron and breakup prosesses (Ref.[4]).Here we study proton-deuteron scattering.Presence of the electromagnetic interaction requires one to consider the neutron and proton to be different particles and precludes literal use of the isotopic formalim of (Ref.[3]).So the FNNM equations used in (Ref.[5]) have to be changed.
Taking the neutron as particle 1 and protons as particles 2 and 3 we have the requirement Ψ(1, 2, 3) = −Ψ(1, 3, 2).To satisfy this condition, we present Ψ in terms of Faddeev components as where it is understood that in Φ(i, k, l) particles are grouped as i + (kl) and Φ 1 is antisymmetric in the last pair of arguments: Φ 1 (1, 2, 3) = −Φ 1 (1,3,2).As function Φ 2 (2, 3, 1) has no definite properties under interchange 3 ↔ 1 we encounter permutations which are not cyclic P 12 (231) = (321), P 13 (123) = (321).In terms of these operators and operators P ± we obtain for the independent components Φ 1 and Φ 2 a system a e-mail: vsuslov@nccu.eduHere v 1 (2, 3) is a sum of the Coulomb and nuclear potentials and v 2 (3, 1) is the pure nuclear potential: The final pair of equations is essentially the same as for the nd case with changes described above.

s-wave approach. Elastic scattering
In the s-wave approach there exists a single equation in the spin-quartet case for quantum state α = {0, 1, 1, 3/2, 0} and our new results for n-d and p-d elastic amplitudes at E lab =14.1 MeV calculated with the Malfliet-Tjon MT-I-III potential do not practically differ from our predictions (Ref.[1]) and those of the Los-Alamos/Iowa group (Ref.[2]) and we do not present them here.However in the spin-doublet case there exist three equations for quantum states α 1 = {0, 0, 0, 1/2, 0} and α 2 = {0, 1, 1, 1/2, 0}, one for Φ 1,α 1 and two for Φ 2,α i , (i=1,2).For the ppn system, a set of equations is writen as: EPJ Web of Conferences 03017-p.2 In the s-wave approach the functions g (±) αα and g (ik)  αα are reduced to functions h 0 0000 (y i /x i , u) = x i y i /(x k y k ).Here (x k , y k ) are the coordinates of the integrand components Φ 1,2 and (k i).
In the deuteron domain (x 2 finite, y 2 → ∞) the asymptotic condition for the component Φ 2 corresponding to elastic channel: In formula (9) F 0 (qy 2 ) and G 0 (qy 2 ) are the regular and irreqular Coulomb functions and ψ l (x 2 ) is the s -wave component of the deuteron wave function (l = 0).Amplitudes Φ 1,α 1 and Φ 2,α 1 have zero initial conditions and zero elastic asymptotics.
In the breakup domain we have for Φ 1,α 1 and Φ 2,α 1,2 the asymptotics: ). (10) In these formulas the Coulomb distorted phases W 1 and W 2 are as following and The two doublet amplitudes are given as follows duh 0 0000 (y 2 /x 2 , u) Equations (6 -8) violate the isospin symmetry because of distinctions between nn, pp and np forces.The charge asymmetry is obtained by allowing the strengths of the central nn and pp forces to be different from np one.We modify MT-I-III potentials to reproduce singlet n-n and p-p scattering data by scaling the pn potential by factors 0.982 and 0.9745 for nn and pp, respectively.While MT-I-III uses for a np , a nn and a pp the value of scattering length is -23.5 fm, we modify the potential to produce their scattering lenghts to agreement with experimental data (Ref.[10]).
The accuracy of this adjustment procedure is checked by calculating binding energies for 3 H and 3 He using MT-I-III potential and its modifications.Our new results and previous ones from Ref. [7] and Ref. [8] obtained in isospin formalism are given in table 1.Our new results calculated applying a new set of three Faddeev equations with modified MT-I-III NN potentials are in a good agreement with those from Ref. [7] and Ref. [8].Results for our Coulomb energy ΔB c and CSB energy ΔB(CBS) are given in table 1.
In the s-wave approach the value of ΔB c is 661 keV slightly different from the result of 693 keV (Ref.[9]).Our result for the charge-symmetry breaking energy ΔB(CSB) is 61 keV close to 71 keV evaluated by Miller et al. [10].Our new results for phase shifts and elasticities for n-d One concludes that CSB is visible in phase shifts and inelasticities for neutron-deuteron and proton-deuteron scattering at E lab =14.1 MeV.However these results have been obtained applying partly artifficial procedure for constructing s-wave singlet nn and pp components of MT-I-III potential.Therefore we are currently extending our studies using the charge dependent AV18 NN potential.

Table 2 .
Spin-doublet case.n-d and p-d elastic shifts and inelasticities at E lab =14.1 MeV.d breakup scattering at E lab =14.1 MeV calculated assuming the neutrons and protons to be distinguishable particles are presented in table 2.