Symmetrical Basis for Faddeev Equations

The angular -momentum reduct ion of the Faddeev equations (1) can be carried out following different approaches. I f the f rame of the S U3 representat ion of the three-part icle states (3) is used, it is known tha t the three-body problem reduces to solving a coupled set of integral equations in only one variable. The analysis of this reduct ion for spinless particles has been carried out by LEE (a). In this orbi ta l -momentum ease an intermediate step through the in t roduct ion of the Omn~s basis is explicitly used for the reduct ion in order to obtain a symmetr ical description. In ref. (3) the way for including the spin and isospin of the particles is also given. The main purpose of this note is to present the generalization of an alternat ive t r e a tm en t of the orbi ta l-angular-momentum analysis, for any value of l, of the th ree-body equations in the S U3 representation, without the abovement ioned in termedia te step. This generalization permits us to take into account

Symmetrical Basis for Faddeev Equations.--It is shown that the Faddeev equations can be treated in the frame of a three-body basis that takes into account the proper symmetries of the three-nucleon states.
The angular-momentum reduction of the Faddeev equations (1) can be carried out following different approaches.If the frame of the S U3 representation of the three-particle states (3) is used, it is known that the three-body problem reduces to solving a coupled set of integral equations in only one variable.The analysis of this reduction for spinless particles has been carried out by LEE (a).In this orbital-momentum ease an intermediate step through the introduction of the Omn~s basis is explicitly used for the reduction in order to obtain a symmetrical description.In ref.
(3) the way for including the spin and isospin of the particles is also given.
The main purpose of this note is to present the generalization of an alternative treatment of the orbital-angular-momentum analysis, for any value of l, of the three-body equations in the S U3 representation, without the abovementioned intermediate step.This generalization permits us to take into account (*) Supported in part by Consejo Nacional de /nvestigaciones Cientificas y T6cnieas de Argentina.
(3) A. DRAGT: Journ. Math. Phys., 6, 533 (1965);Y. SI~o~ov: Soy. Journ. £Vucl. Phys., 3, 461 (!966) where the notation used is self-evident• Our first step is to impose the spin states (1) to be invariant under the action of the symmetric group of three objects, $3.We can build up the elements of this group from the transpositions /)~2 --(12)(3) and the cyclic permutation C = (123).The operator/)13 acting on the basis (1) changes it in a phase factor (--)~+~-~, i.e.P~2 is diagonal in this basis.On the contrary, the matrix elements of the cyclic permutation C are W being a l~acah coefficient.Since C is a unitary operator~ it can be diagonalized through a unitary transformation U. Thus, we take for our spin basis the eigenvectors of matrix (2), i.e. (3) [SM~,; v> = ~ U~]SMs; I> , where v stands for a set of quantum numbers that labels the basis unambiguously.For the most interesting case of three spin-½ particles coupled to S = ½, the symmetrical basis is determined by /o ] (*) Clearly the complete symmetry is obtained when the usual procedure of Young tableaux is applied to the basis.We do not need it explicitly in our calculations.
It will not escape the reader's attention that~ in the case of higher spin, an additional set of operators will be needed in order to break up all the possible degeneracies (~).
It is clear that a similar analysis can be carried out in connection with the isospin.We call the resulting isospin basis I//~; ~>-Having at our disposal the proper spin basis, our next step is to couple it with the orbital SU3 basis obtained from Dragt~s (3) results, to get the total-angular-momentum basis.Formally When projected in momentum space, the three-body wave function is written in terms of p2, the Dalitz-Fabri co-ordinates and the Euler angles.These variables describe the size, shape and orientation of the triangle defined by the tips of the momentum vectors of the particles (henceforth called the momentum triangle).The body-fixed reference frame is chosen with the z-axis being noImal to the momentum triangle plane~ and the x-and y-axis lying along the principal directions of the inertia tensor of the momentum triangle (3).
The transformation brackets from the momentum basis to the S Us basis with spin can be written as follows: We shall need later a rotated fixed-body system, such that its z-axis lies along the momentum p(.2) (which is proportional to q3)-In this case the wave function curt be written as = (~(2), ~/2, 0) being the necessary rotation.Certainly, similar rotations on the other momentum vectors can be performed if needed.
Having obtained the appropriate symmetrical basis for three particles with spin, we shall show how the Fuddeev equations look in terms of the ~ngular-momentum reduction we have just presented.
We shall restrict ourselves to the case of identical particles in order to simplify the algebraic manipulations.The generalization to nonidentical particles follows the same steps in a straighforward manner.In the present case, the Faddeev equations reduce to only one operator equation ( 12) ~(z) = 37(z) + 27(z)Go(z)~r(z), where T(z) is the total three-body scattering operator, Go(z) is the resolvent and t'(z) is the two-body scattering operator.When matrix elements of eq. ( 12) are considered, the main problem is the evaluation of the quantity The angles R and R' and the kets ]S~m,> and [Same} in eq. ( 15) are related by means of a rotation of the body-fixed system through the two-body scattering angle.Introducing now the partial-wave decomposition ( 16) <~(x); S~,m~,[t(z,) ' , ,,, ,. , q('; z').

• <S,~SaS(K~ + m--m')]S,~(Ks
The delta-function that appears in the above equations is eliminated when the integration over the Dalitz-Fabri co-ordinates is performed.In order to show this explicitly and, at the same time, recast eq. ( 17) in a more symmetric way, we introduce Cartesian co-ordinates in the Dalitz plot through x = e cos ~, y = e sin ~v.
Thus the final expression for the matrix element (13) takes the form ,1 Oz,~,'----2J q-1 s,, t,v,j z~,Zs We note that in eq. ( 18) the mentioned &function only implies a restriction on the (x, x')-integration.It is also worth mentioning that the integration involved in the definitions of ~ ,q*~'~'~ '~ S)" (x) and _j, ~x ,, eqs.( 19) and ( 20) respectively, can be performed in many cases analytically.This means a considerable simplification in computing the kernel and the inhomogeneous term of the Faddeev equation, that follows immediately from eq. ( 18).As we have already mentioned, the inclusion of the isospin does not offer any difficulties: we have to consider the direct product of our totally symmetric IPZ2~v; JM> basis with the above-mentioned isospin basis.
The awe-inspiring set of channels that could be present in eq. ( 18) is not a serious computational trouble, at least for the case of identical particles.Indeed, the number of channels is considerably reduced by the requirement of antisymmetry of the three-nucleon states.This last coadition requires which halves the number of physical channels to be considered.Similar restrictions hold for the spin (or isospin)-zero case.
If we are interested in the calculation of three-nucleon low-energy parameters, we shall find that only small values of 2 are important.This means that the rate of convergence found for this symmetrical basis is extremely good (5).
We wish to acknowledge Dr. V. ALESS~t~])RI~I for suggesting this work~ and Dr. C.A. GA]cCIA CA~)m for his helpful discussions and advice.