Bound states in a system of four identical particles

Volume 42B, number 3

PHYSICS LETTERS

11 December 1972

BOUND STATES IN A SYSTEm; OF FOUR IDENTICAL

PARTICLES

V.F. KHARCHENKO and V.E. KUZMICHEV

Institute for Theoretical Physics, Academy of Sciences of the Ukrainian SSR, Kiev, USSR Received 5 October 1972 The energies of the bound S-states of the four identical particles with separable pair interactions are calculated on the basis of the integral equations for the system. It is found, besides the ground states of the four-particle system, a second 0+ bound state. In recent years the integral formulation of the equations of motion for the three-particle system with pair interactions is widely used to investigate the properties of the bound and continuum states of the three-nucleon system. To solve the integral three-particle equations various effective approximate methods have been worked out. In particular, there is a development in the direction based on the separable representation of integral kernels. In this way the dynamic calculations of the quantitative characteristics of a four-particle system seems presently feasible. The technique for removing the singularities, caused by the pair character of interactions in the system, from the kernels of the four-particle integral equations has been given by Faddeev [1, 2] and Yacubovsky [3]. The integral equations determining the wave function of the system of four identical spinless particles have been formulated in ref. [4]. In this case the wave function of the system, being completely symmetric relative to permutations of the particles, is expressed in terms of two functions ~ and ×, (1)

4 = ~ ~(ki/,Pi/,k,qi/k,l )+ ~ x(ki/,kkl, Si/,kt), (ijk,I ) ( ij, kl) which satisfy the following set of the integral equations:

t~(k,p,q) = ~k°(k,p,q) + (~Z- k2 /m - ~ p2 /rn -~3 q2/m) - 1 ×f{Ms(kp;½1,'+kq

'I , k' - ~ qi l ;, ~ - ~ q 2 / m ) ~ ( k ' , q

+1~'3,t ,,ts""~ dk' ok/'

+M(kp;½k' + ~ q'2,k' - ½q'2;~Z- ~ q2/m) x(k', q - ½ q ' , q ' ) ) - -

(2n) 6

'

(2)

x(k,K,s) = x°(k,K,s) + ½(~Z- k2 /m - ~2 /m - s2 /2m) - 1 X f {N 1)(k~;~s+q r , k r ;~Z-s2/2m) ~(k', - s - ~ 2q T ,q F.~s +Ns(2)(kK;½s - q ' , k ' ; 9 " - s 2 / 2 m )

~(k',s

- - 23 .~i, ,tl, , ~.t d J (k- ~' d ) 6q '

.

Here the notations of ref. [4] are used. The summations in (1) are performed over all the different sets of the Jacobi relative momenta. The first and second sums in (1) consist of 12 and 6 terms, respectively. The free terms in (2) are determined by the asymptotic conditions of the problem (see ref. [4] ) , ~ = £ + i0, ~ is the total energy of the four-particle system. The bound states of four particles are described by the homogeneous set of the integral equations corresponding to (2) with qjo = ×o _- 0. 328

Volume 42B, number 3

PHYSICS LETTERS

11 December 1972

Table 1 Binding energies of the ground and excited 0+-states of the two, three and four identical spinless particles in the case of the separable potential (5) - (7). The number of particles

2

3

B.E. (g.s.), MeV (p2/#t) (K2/#t)

2.225

24.55 (0.85)

84.66 (0.75) (0.65)

The number of particles

3

4

B.E. (exc.s.), MeV (p2/#t) (K2/~3t)

2.325 (0.75)

24.87 (0.75) (0.65)

• The integral kernels of the set (2) contain the functions Ms(kP; k , ,p ;Z)and N~si ) (kK; k ,~, ;Z), i = 1,2, characterizing the scattering in subsystems. The function Ms(kP; k'p' ;Z) is the symmetrized three-particle scattering amplitude with the initial and final momenta, in general, off the energy shell. It satisfies the integral equation:

Ms(kP; k'p';Z) = (2rr) 3 ts(k,k'; Z - z} p2/m) 8(p - p ' ) +

ts(k,½p+p;z-~//m) tt

1 [z-(I/m)(p 2 +pp" +p,,2)l-~ M~O~+~p ,p ;Kp ;Z) H

~I

--r

t

(3) (2rr)3 '

where ts(k, k'; Z ) is the symmetrized two-particle scattering amplitude. The vector q'l and q'2in (2) are given by ql, = q , + ~ q ' q2, =q, - ~q" The functions N(1)a s n d Ns(2) are the components of the scattering amplitude of two non-interacting pairs. They satisfy a set of two,coupled integral equations. It is more convenient, however, to consider their sum and difference, N~ +)= N (1) + N (2) a n d N ( - ) = N (1) - Ns(2) , for which the indicated set decomposes into two independent integral equations: N(±)(kK; k'K'; Z ) = (2rr) 3 ts(k,k'; Z - K2/m) [8(K + K') + ~i(K -- K')]

+½ft(k,k";Z-K2/m)[Z-

~ 2 / r n - k " 2 / m ] -1 N~(2)(Kk. .;k . . K ;Z) dk' (21r)3 "

(4)

It must be pointed out that in the derivation of the second equation of the set (57) in re(. [4] the evenness of the wave function under the space reflection of all variables was used implicitly. This equation such as given in re(. [4] is true only for even states. In the general case this equation has the form (2), and the componentsNs (1) and Ns(2) appear separately not being joined together in one term N s = Ns(+). If the pair potential is separable and acts between particles only in the S-state,


(5)

then the set (2) is simplified considerably. In this case the dependences of the matrix elements Ms(k p ;k'p' ;Z) and N(+-)(kK;k'K'; Z) on the variables k and k', and therefore, the dependences of the functions ~ (k, p, q) and ×(k,K,s)on k, are separated out in explicit form [4]. After the partial-wave decomposition the set (2) in the case of the potential (5) is reduced to an infinite set of integral equations for functions of two variables p, q and ~, s. The separable approximation of the integral kernels in the variables p and ~ makes it possible to reduce this set to a set of single-variable integral equations. Let us consider the bound states of the four-particle system with the total orbital momentum ~ = 0. Take the potential function in (5) in the form

g(k) = (2rr) a,= (k 2 +/32)- 1

(6)

To obtain the separable representation of the integral kernels in the variables p and r we apply the Bateman method 329

Volume 42B, number 3

PHYSICS LETTERS

11 December 1972

[5]. Two terms in the Bateman separable representations (IV-- 2) were used in this work. In so doing the Bateman points Pl and K1 were set to zero (Pl = 0, r 1-- 0), and the points P2 and K2 were chosen so that, within the framework of the considered approximation with N = 2, the components of the effective potentials in the single-variable integral equations for four particles might give the strongest attraction. The set of single-variable integral equations was solved numerically. From all the possible values of the oribtal momentum of the relative motion of a particle and the c.m. of two others, k = 0,1,2,..., only the value k = 0 was taken into account in the calculations. The results for the binding energies of two, three and four identical spinless particles in the 0*-states are presented in table 1. They have been obtained with the potential parameters 7 and/3, equal to the parameters of the n-p interaction in the triplet spin state 7t = 0.415 f m - 3 ' fit = 1.450 fm - I

(7)

In the parentheses the values of the Bateman points P2 and K2 are given (Pl = gl = 0). It turns out there are two bound 0+-states of the four-particle system. The binding energy for the excited state of four particles (Q~I = = 24.87 MeV) exceeds only slightly the binding energy for the ground state of three particles (B o = 24.55 MeV). This result suggests that in this excited state the fourth particle is only loosely bound to the complex of the three other particles.* In this respect the four-boson system is similar to the three-boson one. For the latter the excited bound 0+-state is present whenever the two-body subsystem is bound. If we take into account in eq. (2) only one channel, omitting the component X (or that is the same taking Ns(1) = Ns(2) = 0), we find only a single bound four-particle 0+-state ofCBo = 54.40 MeV (X = 0, N = 2, Pl = 0, P2 = 0.75/3) and no excited bound state. The set (2) may be used for the approximate description of the ground state of 4He, if we take the pair potential to be an average of the two-nucleon interactions in the triplet and singlet spin states [6]. Such an approximation is reasonably good in the case of triton [7]. The parameters of the separable potential (5), (6) used in this approach, 7 = ~ (')'t + "Ys)'/3 =/3t = fls' are 7 = 0.353 f m - 3 , / 3 = 1.450 fm -1

(8)

The solution of the corresponding set of single-variable integral equations (X =0, N = 2, Pl = g 1 = 0, P2 = 0.75/3, g2 = 0.65/3) gives in the case of the parameters (8) the following values of the binding energy of c~-particle in the ground state:q3 a = 52.03 MeV. In comparison with the experimental value (c~a(exp.) = 28.3 MeV) we get too much binding in the four-nucleon system. The calculation with the potential (5), (6), (8) overbinds the triton by some 4 MeV [7]. In the case of the parameters (8) there is also the excited four-particle S-state with the binding energy of 12.15 MeV. This value slightly exceeds the binding energy of the triton in the ground state calculated in the same approximation. It should be noted that the treatment of excited states of four nucleons must be performed with due regard for spins and isotopic spins of nucleons, since the admixtures of space functions of mixed permutation symmetry in these states may be considerable. * It is quite possible that the excited 0+-state of 160 considered as consisting of four a-particles is of this nature.

References [1'] L.D. Faddeev, Proc.Problem symp. on nuclear physics, Tbilisi, vol. t (Moscow, 1967) p. 43. [2] L.D. Faddeev, Three body problem in nuclear and particle physics, eds. J.S.C. McKee and P.M. Rolp (North-Holland, Amsterdam, 1970) p. 154. [3] O.A. Jacubovsky, Sov.J.Nucl.Phys. 5 (1967) 1312. [4] V.F. Kharchenko and V.E. Kuzmichev, Nucl.Phys.A183 (1972) 606. [5] V.F. Kharchenko, N.M. Petrov and V.E. Kuzmichev, Phys.Lett. 32B (1970) 19; V.F. Kharchenko, S.A. Storozhenko and V.E. Kuzmichev, Nucl.Phys. A188 (1972) 609. [6] J.M. Blatt and V.F. Weisslopf, Theoretical nuclear physics (John Wiley and Sons, Inc., New York, 1952) ch.5. [7] A.G. Sitenko and V.F. Kharchenko, Nucl.Phys.49 (1963) 15.

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