Three-nucleon equations with separable two-body potentials

Nuclear Physics A159 (1970) 545--554; ( ~ North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

THREE-NUCLEON EQUATIONS W I T H SEPARABLE TWO-BODY POTENTIALS EDWARD H A R M S t

Deportment of Physics and Astronomy, Rensseloer Polytechnic Institute, Troy, N.Y., 12181 t*

Received 31 March 1970 Abstract: Assuming separable two-body potentials, a general derivationis givenof the three-nucleon equations. The derivation allows for the full complexity,in separable form, of the two-nucleon interaction including an arbitrary number of (possibly coupled) partial waves and an arbitrary number of separable terms in each partial wave. All three-body partial waves are considered. Using a convenient notation, the three-body equations and their angular momentum decomposition are easily derived. 1. Introduction

In this paper, we develop the bound state and scattering equations for a system of three nucleons interacting by means of separable two-body potentials. There has been considerable interest in separable potentials due in a large part to the simplifications they bring about in the solution of the three-body problem. Historically their use led to the first 1) "exact" solutions for three-nucleon systems, and up to the present practically all scattering calculations using the Faddeev equations have employed this type of interaction. Separable potentials have been approached from at least two different points of view. Fairly extensive phenomenological fits to the low-J partial waves in the twonucleon system have been given by Tabakin 2) and Mongan 3). Other authors have investigated separable expansions or separable approximations to local potentials. The Weinberg 4) series has been used extensively in three-body calculations 5). A more convenient version of the Weinberg series, the unitary pole expansion 6), has shown excellent convergence properties 7) in three-body bound state calculations for potentials containing the amount of repulsion indicated by two-nucleon scattering data. Many other expansions s) or approximations 9) have been given in the literature. The equations for three-body systems interacting through separable two-body potentials have been given only for particular types of interaction. Lovelace lo) has given equations for systems interacting through central separable potentials containing one separable term in each partial wave. Interactions containing tensor forces have usually been of the Yamaguchi it) type. General three-body equations for such a ) National Science Foundation trainee. Present address: Department of Physics, Fairfield University, Fairfield, Conn. tt Work supported in part by the National Science Foundation. This work is based on part of a Ph.D. thesis submitted to Rensselaer PolytechnicInstitute. 545

546

E.

HARMS

system have been given by Sloan 12). Stagat 13) has shown how rank-N separable potentials of the Yamaguchi form can be incorporated in the three-body bound state equations. In this paper we present a general derivation of the three-body equations assuming separable two-body interactions. Our approach allows for the full complexity, in a separable form, of the two-nucleon potential including an arbitrary number of (possibly coupled) partial waves and an arbitrary number of separable terms in each partial wave. Using a convenient notation, the three-body equations and their angular momentum decomposition are easily derived. All three-body partial waves are considered. The equations derived here should prove useful in the study of polarization in n-d or p-d scattering especially at energies where significant polarization occurs in twonucleon scattering. The equations should also prove useful in the study of the effects of higher two-body partial waves on the triton binding energy and the n-d doublet scattering length. Calculations up to the present 14. t s) have failed to simultaneously reproduce the triton binding energy and latest experimental value for the n-d scattering length, indicating, perhaps, the need for relativistic corrections or many-body forces.

preliminaryr e s u l t s

2. N o t a t i o n a n d

In this section we develop our notation and derive some useful relationships. We use units in which the nucleon mass and h equal 1. We assume that the two-nucleon potential is separable and in momentum space is of the form (or = J, S, T) v(p, q) =

,,.Y',.,.., -

M

g,. ,(v)e/,. L. s( p)

er

,,.

,(q),

with the form factors g~..,(p) and the strengths 2~ being real. Here, P r is the projection operator into the two-particle isospin subspaees (T = 0 or 1) and is given in terms of two-particle isospin states by P r = ~ IT, I>> <
The O2/~,L,s are two-body total angular momentum functions formed from two-body spin and orbital angular momentum states. It is convenient to define the form factor by

(pl g, M, v) = ~ g~,,(p)O'ff~L,S(ap).

(2.1)

L

Using this notation, the potential operator V may be written as

V=

~ a~ M, Vj I

-la, M, v)lT, I>>l<
(2.2)

~'v

or

v = - I z Y X - l(zl,

(2.3)

TWO-BODY

POTENTIALS

547

where we have expressed the sum in a convenient form. A similar notation has been used by Stagat z3). Here, however, we do not distinguish between central or noncentral interactions. Although E contains the full complexity of the two-body interaction, the notation of eq. (2.3) makes it formally equivalent to a one term separable potential. Using this notation the Lippmann-Schwinger equation may be easily.solved. We obtain

T(s) = Iz>tzt(s)
(2.4)

where

-a-'(s)

= ~,+ *,

(2.5)

Go(s) being the free two-particle Green function. More explicitly - bl-'(s)]~.~,.,.,.

~, M. , , . , ,

= 6,.a, 6M.M, 6t, r[k~f,,¢+
=

gL, v(P)gI..¢(P)P s--p 2

P

The sum on L contains two terms if a corresponds to a coupled two-nucleon state, one term otherwise. We will assume that for

= ~ = (1, 1, 0), v equal to one corresponds to the deuteron bound state. The deuteron bound state ket is then given by

_

N

p2+B

,

(2.6)

with N a normalization constant given by

1 _ <¢r~' M, I l G o ( - B ) G o ( - B ) I a B, M, 1>.

N2

For our three-particle momentum states we choose the momentum of particle 0t in the three-body c.m.s, p~ and the relative momentum q~ of the two particles ]~ and other than a

q, = ½(pp- p~).

(2.7)

Only two of the momenta are independent, and we have in particular that t t t -- Z(q,-8,,tsp,) × 6(qp-~a~,a),

(2.8)

/t,, = p,+½rp,

(2.9)

I

548

E. HARMS

e,p =

{ 11 if ( ~ , 3 ) = (1,2)or cyclic, _ if (~t, t ) (2, 1) or cyclic.

(2.10)

The three-particle kinetic energy is given by K

= ~,2 +~}v,2 = p,~+v," p p + v ~ .

(2.11)

We denote by I"1 the potential between particles 2 and 3 and similarly for V2 and Vs , As an operator in the three-body space, V, can be thought of as the direct product between the two-body potential P, and the projection operator 1, into the coordinates of the third particle ct. (We designate two-particle operators by putting hats on them when any misunderstanding may arise.)

1, = lp, ®

Ij. ® It.,

(2.12)

with

lv. 1j. =

=

fo°l

p,)p~

dp,
~. IJ,, l,. m , ) ( j , , l,, re, I. Ju, ma, la

1,. =

,Y__, It,, i,))<
i,I.

if= +~

In the three-body problem it is convenient to modify our notation somewhat from eq. (2.3). After multiplying 17, [eq. (2.2)] by 1, to form an operator in the threeparticle space, we recouple the two-particle and third particle states to form states of conserved three-particle quantum numbers. With the interaction considered here, these quantum numbers will be total angular momentum and its z-projection, total isospin, and its z-projection and parity. For example, denoting the three-particle isospin by z and its z-projection by % we find Pr. ® 1,,, = E I z, "rs(T,,)))<
the sum on z being over ½ and ½ if T= is 1 and over ½ if T, equals 0. We let ~ stand for the three-particle angular momentum, parity, and isospin quantum numbers and r/, stand forj, and 1,. Then defining the states

I~, n,, a,, v,> = I~, ~s(T,)>> x IF, rs, a,, n,, v,>,

(2.13)

with F and F s the three-particle total angular momentum quantum numbers and

It, r s , ~,. n,, v,> = E I~,. M,, v,)lj,, l,, m,)(rrsld, M , j , mD. Ma, ms

we find (2.14) ~, qa, cry, va

Va

TWO-BODY POTENTIAI~

549

The J-j coupling scheme used here is not the only one possible. We might, for example, first couple the third particle spin to the two-body angular momentum states and then couple the resulting states to the third particle orbital angular momentum. The kets I~) contain all of the dependence of V, upon the three-particle angular momentum and isospin variables as well as the dependence upon q~; ~,~-1 is an operator in p.. The two-body T-matrix in the three-particle space, T.(s), is given by.

T,~(s) =

(2.15)

la>*,~"(s)<¢l.

Here zt'(s) is an operator in p. given in terms of A'(s) [see eq. (2.5)] by (p.la

(s)lp.>

-

A"

(2.16)

.

3. Derivation of the three-body equations We now proceed to the development of the three-body equations. Our starting point 12) is the Alt, Grassberger and Sandhas 16) (AGS) formulation of the threebody problem. The three-body scattering amplitudes may be obtained from knowledge of the operators

U,,,#(E) =

(E-K)(1-6,.#)+

~, V~+( E et#y#fl

Vs)G(E)( E V,),

6~¢

(3.1)

y#p

where G(E) is the full three-particleGreen function. All potentials are operators in the three-particle space. If l~b#(~,~#))IP#) represents particle /~ with angular m o m e n t u m quantum numbers r/#incident upon the deuteron bound state of particles a and y and coupled to it to form a three-particlestate with quantum numbers given by ~, then

Ip#>

(3.2)

gives the transition amplitude from this state to the analogous channel state
(3.3) Faddeev-like equations for the

U,,.#(E)have been obtained by AGS. They find that

u,.#(E) = (r-K)(1-~,.#)+ dl~¢ E r,(E)6o(e)u,.#(E).

(3.4)

If we assume that the two-body potential is separable, T6(E) takes the form of eq. (2.15). Using eq. (2.15) in eq. (3.4) and defining the matrix operators

x., #(E) =- <=IGo(E)u., #(E)Go(E)llbh

(3.5)

550

E. HARMS

we find that the X~. B satisfy the following set of operator equations

X,,, p(E) = Z~,,tj(E) + ~ Z,,, ~(E)A~(E)X6,t~(E).

(3.6)

Z,, p(E) = (1-6~p)(~IG0(E)IP)L

(3.7)

6

Here The operators X and Z are operators in the magnitudes of third particle momenta. All other variables have been removed by contracting in eqs. (3.5) and (3.7). This means that when taken into the momentum representation, eq. (3.6) will give a coupled set of one-dimensional integral equations. Following the discussion of Lovelace 10) we now show that on the energy shell defined by eq. (3.3) (aa = (1, 1, 0))

[(P~,IX=.a(E)Ipp)'I {,n'.:

¢,n,

¢a a, v~= 1 ¢pn, v a = 1

1

- N2 (P,l(q~(¢, ~/',)1U,p(E)kb,(~, ~/p))lP~).

(3.8)

We have from eqs. (2.9) and (3.3)

(qJGo(E)l~, r/~, cre, v~ = l>lp~) -

-

1

2

2 ( q , l ~ , r/~, a e, v~ -- 1>

E - ¼p. -- q, ~

B+q~

( q J ¢ , r/~, o ~, v~ = 1>.

Using eq. (2.6) and the definitions (2.13) and (3.5) gives the result (3.8). Thus the knowledge of the X~.p gives the bound state to bound state scattering amplitudes. Since the three nucleons are identical particles, we may xo) further simplify eq. (3.6). In this case the operators Z~,p and ziP(E) have the same structure independent o f ~ and ft. Thus, going into the momentum representation and defining

z(p, p': ~) = , with p = p~ and p' = p~ and

X(p, p': E) -- ~ , ~t

we have

X(p, p': ~) = 2Z(p, p': E)+2

fooZ(p, k: e.)A(e-tk~)X(k,

p': E)k ~ dk.

(3.9)

If we take X on the energy shell, we get the correctly symmetrized scattering amplitude since for identical particles we must sum over all final states in the amplitude (3.2).

TWO-BODY POTENTIALS

551

Using the methods of ref. xo), we may obtain expressions for the break up amplitudes in terms of X(E). The bound state three-body equations are just the homogeneous version of eq. (3.9). (See Fuda 17) for a convenient form for the bound state equations.) In the next section we investigate eqs. (3.9) further, developing an explicit expression for the functions Z(p, p' : E) and investigating the complexity of these equations for different interactions and different three-particle states. 4. Evaluation o f the kernels

The Z(p, p' : E) are matrix functions of the variables p and p' depending parametricaUy upon E. The elements of the matrix are given by

[Z(p, p': E)]~., ~., ,.; ~,. ~,.,, = lp,>(1-6,,)

(4.1)

and are diagonal in ¢. For definiteness we assume at = 1 and fl = 2. Using the definition (2.13) and the fact that Go(E) acts like a unit operator in isospin space, we may write the right hand side of eq. (4.1) as

C(x(TI , T2))lP:>.

(4.2)

The factor C(z(T~, I"2)) is given in terms of Racah coefficients ~8) by

C(x(T,, T2)) = ( - 1)~+r~-'W(½, ½, ~, ½: T2, T~).

(4.3)

The possible values of C(z(TI, T2)) are given in table 1. Normally we will have T = ½. TABLE 1 Spin a n d isospin m a t r i x elements

r,

1"2

C(~(Tx T:))

½ ½

0 1

0 0

-½

½

0

l

½ t

1 1

1 l

½V3 -iV3 -½ l

We now concern ourselves with the matrix element (4.2). From eq. (2.1) we see that the states IF, Fa, a 1, rll,Vl> may contain one or two two-particle orbital angular momentum states

Jr, r3, ~,, ~,, v,> = E IF, r3, ~,, LI

L,,

~/,, Vl>.

(4.4)

552

E. HARMS

The IF,/'3, o"1, LI, qt, vl> are J-j coupling states. We may express them in terms of L-S states (formed by coupling three-particle orbital angular momentum states and spin states) by means of the Wigner 9:/symbols xs). Once in this form the spin matrix elements are easily obtained using methods identical to those for the isospin matrix elements. We obtain the following expression for the function (4.2), (O stands for three-particle spin states and A for three-particle orbital angular momentum states) N(~, A, ~, L t , L2) A, D, L1, L1

x
(4.5)

Here N(~, A, ~2, L1, L2) is given in terms of the C-coefficients (table 1) and 9-j symbols by N(¢, A, I2, L 1 , L2) = C(z(Tt, T2))C(f2(sl, s2)) x ['(2J t + 1)(2/2 + 1)(2jl + 1)(2j2 + 1)]'}(2A + 1)(212+ 1) ×

Sl J ' ~/12

S2

I2 r ) ( A

£2

(4.6)

•

The vectors Igr.,.,,) are the two-body potential form factors and the IA, A3(Lt 11)) are orbital angular momentum states given by 113


= E (AA3IL~ Mlt 113-M)Y~(f]q, )Yza'-u(On~) M

Finally inserting complete sets of momentum states in eq. (4.5) and using eqs. (2.8) through (2.11), we find IP2) (-)1

"(r

" /1' " JJ EaL,.,,,(/~21)Q/1,L,.,,(P2t,P,)*

A3 ~' a'2 " X {~A. L2.12(~11 2, ~2)0L2. v2([212)] E - - p ~

dpt dP2

(4.7)

--p22 -- P l " P2

The angular integrals appearing in eq. (4.7) have been discussed by Balian and Brezin [ref. tg)], Sloan 12) and Harms 6). Putting eq. (4.7) together with eqs. (4.5) and (4.6) gives the desired expression for Z(p, p' : E). The structure of eq. (3.9) is similar to equations derived previously using separable interactions. The elements of the three-body potential function matrix, Z, may be represented diagrammatically as in fig. I. Each element corresponds to a pickup process. The incoming state is formed from a single particle with quantum numbers

T'4~O-BODY POI'~ICFIAI~

553

r/coupled to the two-particle configuration designated by a, v to form a three-body state with quantum numbers ~. The single particle picks up one of the other two particles, producing a configuration a', v' and leaving the other particle in the state r/'. From these considerations we may determine the number of coupled integral equations represented by eq. (3.9). Being diagonal in ~, the equations may be solved independently for each set of three-body quantum numbers. We may couple various third particle states to a given two-body partial wave to form a state w~th the threebody quantum numbers ~. The number of such couplings, N~, is determined by the triangle inequality between j, J and F and by parity considerations. The total number of integral equations is then given by

= E N:N;(O, G

where N~ is the number of different form factors in the partial wave a and N will in general depend upon the three-body state to be studied.

\ \ \ \ \ \ \ b' Fig. 1. Diagrammatic representation of the elements of the three-body Z-matrix.

5. Discussion The solution of the system of eqs. (3.9) poses a difficult numerical problem. Just how difficult depends to a large extent upon how many two-body partial waves must be retained in the interaction. Probably it will be sufficient to solve eq. (3.9) for only a few three-body partial waves. Other partial waves would then be handled approximately. Encouraging results along this line have been reported by Sloan 2o) who uses the two-term separable model of Aaron, Amado and Yam 21). Work with more realistic interactions is needed. The separable fits of Mongan should be quite useful in this context since they provide good fits to the two-body data in a relatively tractable form.

554

e. HARMS

T h e question o f the validity o f s e p a r a b l e p o t e n t i a l s is still quite open. There are i n d i c a t i o n s 6,7) that, due p a r t l y to p o l e d o m i n a n c e a n d p a r t l y to cancellations o f n o n - p o l e terms, a s e p a r a b l e a p p r o x i m a t i o n gives g o o d results for the singlet a n d triplet S-waves o f the t w o - n u c l e o n system. W h e t h e r this is true for o t h e r p a r t i a l waves r e m a i n s to be seen. T h e m e t h o d s used in this p a p : r m a y easily be extended to the p a r t l y s e p a r a b l e a p p r o a c h e s o f R o s e n b e r g 22) a n d Alt, G r a s s b e r g a n d S a n d h a s 16).

References 1) A. N. Mitra, Nucl. Phys. 32 (1962) 529; A. G. Sitenko and V. K. Kharchenko, Nucl. Phys. 49 (1963) 15 2) F. Tabakin, Ann. of Phys. 30 (1964) 51 3) T. R. Mongan, Phys. Rev. 175 (1968) 120; 178 (1969) 1597 4) S. Weinberg, Phys. Rev. 131 (1963) 440 5) J. S. Ball and D. Y. Wong, Phys. Rev. 169 (1968) 1362; A. G. Sitenko, V. F. Karchenko and N. M. Petrov, Phys. Lett. 28B (1968) 308; A. H. Lu, thesis, Rensselaer Polytechnic Institute (1969) unpublished; M. G. Fuda, Phys. Rev. 178 (1969) 1682 6) E. Harms, thesis, Rensselaer Polytechnic Institute, (1969) unpublished; Phys. Rev. CI (1970) 1667 7) E. Harms and J. S. Levinger, Phys. Lett. 30B (1969) 449 8) D, D. Brayshaw, Phys. Rev. 182 (1969) 1658; M. G. Fuda, Phys. Rev. 186 (1969) 1078 9) H. P. Noyes, Phys. Rev. Lett. 15 (1965) 538; K. L. Kowalksi, Phys. Rev. Lett. 15 (1965) 798 10) C. Lovelace, Phys. Rev. 135 (1964) B1225 11) Y. Yamaguchi and Y. Yamaguchi, Phys. Rev. 95 (1954) 1635 12) I. H. Sloan, Nucl. Phys. A139 (1969) 337 13) R. W. Stagat, Nucl. Phys. A125 (1969) 654 14) A. C. Phillips, Nucl. Phys. A107 (1968) 83 15) T. Brady, E. Harms, L. Laroze and J. S. Levinger, Phys. Rev. C2 (1970) 59 16) E. O./kit, P. Grassberger and W. Sandhas, Nucl. Phys. B2 (1967) 167 17) M. G. Fuda, Nucl. Phys. A l l 6 (1968) 83 18) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) 19) R. Balian and E. Brezin, Nuovo Cim. 61B (1969) 403 20) I. H. Sloan, Phys. Rev. 185 (1969) 1361 21) R. Aaron, R. D. Amado and Y. Y. Yam, Phys. Rev. 136 (1964) B650 22) L. Rosenberg, Phys. Rev. 168 (1968) 1756