Some results on the boundary condition model t-matrix

Nuclear Physics A170 (1971) 177-186; Not to be reproduced

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SOME RESULTS ON THE BOUNDARY CONDITION MODEL r-MATRIX 0. P. BAHETHI and M. G. FUDA Department

of Physics

and Astronomy,

Stare University

of New York at Buffalo, Buflalo,

New York 14214

Received 2 March 1971 Abstract: It is shown that the pure boundary condition model (BCM) i-matrix obtained from the Hoenig-Lomon pseudopotential does not satisfy the analyticity requirements suggested by Brayshaw, since the t-matrix possesses unphysical poles. A closed form expression is derived for the t-matrix which arises when the external potential in the BCM is taken to be a square well. A separable expansion for the r-matrix is also derived and compared numerically with the closed form result. The rate of convergence is found to be fast enough to be of practical value in calculations on the three-nucleon system.

1. Introduction Recently several authors have turned their attention to the problem of constructing the f-matrix for the boundary condition model ‘) (BCM), the hope being of course to ultimately carry out calculations on the three-nucleon system with a realistic BCM for the two-nucleon system. By taking the limit of a certain potential model, Kim and Tubis “) have obtained a BCM t-matrix for central as well as tensor forces. One of us (M.G.F.) has shown 3), however, that there is an ambiguity in the fully off-shell BCM t-matrix, since the Hoenig-Lomon “) pseudopotential gives rise to a BCM t-matrix which differs from that of Kim and Tubis. The difference between the two t-matrices vanishes in the two-body problem but remains when more than two bodies are present. Brayshaw “) has demonstrated that it is possible to resolve this ambiguity by imposing reasonable requirements as to the analytic structure and asymptotic behavior of the BCM t-matrix. The Kim-Tubis t-matrix “) satisfies these requirements. We will show in sect. 2 of this paper that the BCM t-matrix derived from the Hoenig-Lomon “) pseudopotential has unphysical poles in it, and is therefore unacceptable if one requiies that Brayshaw’s “) criteria be met. The use of the Kim-Tubis BCM t-matrix in three-body calculations leads to difficulties, since for this t-matrix the Faddeev equations “) do not have a unique solution ‘). A modification of the Faddeev formalism which overcomes this difficulty has been developed by Brayshaw ‘). The modified integral equations have the highly desirable feature that the part of the kernel that arises from the pure BCM is separable. If one has a reasonable separable approximation for the contribution from the external potential then the new equations can, after a partial wave analysis, be reduced to coupled equations in one continuous variable. In this paper we will test a separable 177

expansion of the external potential that was derived previously the external potential we use a square well, since this allows us expression for the exact t-matrix as well as for the terms in the Sect. 2 gives our discussion of’ the Hoenig-Lomon t-matrix.

by one of us “). For to obtain an analytic separable expansion. The exact expression

for the t-matrix arising from an external square-well potential, and the separable expansion of the t-matrix are given in sect. 3. Sect. 4 comparison of one, two and three term separable approximations with the exact results. The convergence is found to be extremely fast of energy and momenta.

2. The Hoenig-Lomon

the formulas for gives a numerical for the f-matrix over a wide range

pseudopotential

The BCM t-matrix arising from the Hoenig-Lomon pseudopotential “) has been derived previously by one of us “). The t-matrix for the Ith partial wave is given by the relations: tl(p, q ; kZ + ic) =

g,(PcLfz)gdqc

Lfl)

+ c(h - b,) w(~c, kc)w,(qc, kc)

2n2kg,(kc;f;)D,(kc)

27c2

g,(kc;h)g,(kc;

dx;_fJ = x$(x)+(l -_fJ.h(x)9

b,) ’

(2.0

(2.2)

D,(x) = xh$+“(x)+(l -jJh;+‘(x),

(2.3)

wix,Y) = ~jl(x>_hG) - xjl(vMx).

(2.4)

Here j,(x) is the usual spherical Bessel function and Al(+)(x) is a spherical HankeI function as normalized in Messiah *), f, is the logarithmic derivative of the radial wave function just outside the core radius c and b, is the logarithmic derivative just inside the core. It is seen from (2.1) and (2.4) that whenp = k or q = k the dependence of the t-matrix on b, vanishes. This is a reflection of the fact that bl is not needed in the pure two-body problem, but only in a system with three or more particles. There is no way of determining b, from two-body data. If one chooses b, equal to f, the second term on the right-hand side of (2.1) vanishes, however this choice leads to unphysical poles in the t-matrix, since g((kc;f,) can vanish. For 1 = 0 the positions of the poles are given by the solutions of the transcendental equation kccotkc

=fO.

(2.5)

As is well known this equation has an infinite number of real solutions, and iffo is greater than one, a purely imaginary solution. Poles that can arise from the vanishing of D, (kc) are not unphysical, since they can be associated with bound states produced by the forces inside the boundary condition radius c [ref. ‘)I. If one chooses b, different from& then the unphysical poles that arise from the vanishing of gL(kc;f,) in the first term on the right-hand side of (2.1) are exactly cancelled by the unphysical poles that arise from the vanishing of gl(kc;f,) in the second term. Unfortunately

BOUNDARY

CONDITION

MODEL t-MATRIX

179

one now has unphysical poles due to the vanishing of g,(kc; b,) in the second term on the right-hand side of (2.1). We therefore conclude that no matter how one chooses the parameter b, there will always be unphysical poles in the BCM t-matrix arising from the Hoenig-Lomon pseudopotential. If one accepts Brayshaw’s “) assumption that the t-matrix should be analytic in the complex energy plane except for the unitarity cut and possible poles for negative energies corresponding to bound states, then one must reject the Hoenig-Lomon BCM t-matrix. We can see no reason for not accepting Brayshaw’s assumptions and therefore conclude that the t-matrix arising from the Hoenig-Lomon pseudopotential is in general unacceptable for calculations in which the full off-shell t-matrix plays a role. 3. Separable approximation for the external potential In this section we investigate the rate of convergence of a separable expansion for the potential outside the core radius. In order to do this we compare the exact t-matrix with approximations to it, obtained by truncating the separable expansion at various stages. As mentioned previously we take a square well for the external potential since this will allow us to obtain most of our results analytically. It was shown in ref. “) that the exact t-matrix for the BCM with an external potential can be obtained in the form T(s) = f(S) + t(‘)(s),

(3.1)

where s is the complex energy, t(s) is the t-matrix for the pure BCM (no external potential), and t(‘)(s) is the contribution from the external potential. The operator t(l)(~) can be obtained from the formula

<~W(‘)(s)ld~)=

s

(PW

C (s)lr)drW(r)(rlT(s)lqlm>.

(3.2)

r2c

Here W(r) is the external potential, (phi

C(s)lr)

= (21-c’)-* [ jl(pr)*

hl+)(kr)] &Z(3), r 2 c, (3.3)

with gr and D[ given by (2.2) and (2.3) respectively, and the quantity (rlr(s)lqlm)

is

the solution of the differential equation [s+ Vz - lV(r)](rlr(s)lqlm)

= (s-q2)(rlqZm),

r > c,

(3.4)

where = (2n2)-‘j,(qr)Y,,(Q

(3.5)

If we let (3.6)

180

0. P. BAHETHI AND M. G. FUDA

then we simply have to solve (3.7) with the boundary conditions dr,(r, 4; s) = 4 Tl(C, q ; s), dr r=c c Tl(r, q; s) N 4-l sin (qr-J&c)+A,k-’ 1’03

(3.8)

exp [i(kr-@n)~],

(3.9)

where A, is independent of r. For the external potential we choose W(r)=

-V,,

=

c
(3.10)

r > d.

0,

In order to construct the operator t(‘)(s), we only need TI in the region c 5 r 5 d. This is given by the relations: Tl(r, q; s) = Bl+‘rhj+‘(Kr)+BI-‘rhj-‘(Kr)+

s-q2 KLq2

rj,(qr),

K2 = V,+s,

c S r 5 d,

(3.11) (3.12)

a,,(k)

= Kchi+“(Kc)-(f,-

a,2(k)

= Kchl-“(Kc)-(f,-l)hI-)(Kc),

(3.14)

a,,(k)

= KdhI”‘(Kd)-G,(k)hl”(Kd),

(3.15)

a22(k) = Kdhi-“(Kd)-G,(k)h!-‘(Kd),

(3.16)

P1(4, k) = - A+$

CqcjXqc) - Cf, -

E2

/%(ak) = G

(k)

Bi-j

Coda

-

hj+‘(kd) _ -

=

, a22

-a12

B2~ll-B1

a1 1 az2 -aI2

(3.13)

(3.17)

(3.18)

(3.19)

’

81a22-P2a12

all

lMv)l,

G(kMq41,

kdhi+“W

=

1

B(+) I

l)hi+‘(Rc),

(3.20)

cl21

alI cl21

(3.21)

BOUNDARY

CONDITION

MODEL r-MATRIX

181

The final expression for t(‘)(s) is obtained by substituting (3.3) and (3.11) into (3.2) The result is (plin(t(‘)(k2 + is)(qlm)

rj,(pr)+

-K2 g,(pc;f,) !?--___

rh(+)

vo (k2 - q2) w rj,(Pr)+ (P’ - 47

Q&c)

(P’-K2)(P2-q2) Vo(k2 - q2)

C

’

(kr), rl(r, 4; s)]

WC)

g,(P”f,)

rhl’)(kr)

,

rj,(qr)

3 I== r=d

(3.22) where 4x44

h(r)1 - x,(r)y;(r)-y,(r)x;(r);

(3.23)

B/+’ and B[-’ are given by (3.20) and (3.21), and g1 and Dl are given by (2.2) and

(2.3). A number of checks were made on this fairly complicated result for t”‘(s). The on-shell limit of (3.22) was checked by adding (3.22) with p = q = k to the pure BCM on-shell t-matrix and seeing if it agreed with the on-shell l-matrix obtained by solving the SchrGdinger equation directly. It did. By lettingf, + co and c --* 0 we were able to verify that the expression for t(i) (s) goes over into the formula for the square-well t-matrix ’ “). We have also checked the hard-core limit (fi -+ co) of (3.22) against an earlier calculation and found agreement. We now turn our attention to the problem of constructing a separable representation for the external potential given by (3.10). It is shown in ref. “) that if one solves the differential equation s;

d2 W+l) dr2

r2

s)=~,~WrMdr; SIT 14dr; r > c,

(3.24)

with the boundary conditions (3.25)

dh(r; s),-*meikr.s

= k2+ie,

E > 0,

(3.26)

or $yl(r; s*)r--,e-“r,

s = k2+ie,

E > 0,

(3.27)

then the external potential W can be represented in the form lV=

2 wl~,l,(s)>(~i,ds*)Iw VT’, * (~~l~(s*)lwI~“l,(s)>

(3.28) ’

0. P. BAHETHI

182

AND M. G. FUDA

where (rp,&)>

= ~~

(3.29)

E;,(i). r

The separable expansion for the operator t(‘)(s) is given by “)

%l(S)= Km.

(3.31)

The factors in (3.30) can be easily worked out by using the relation = <@“&*)lw Et (s*)lPlm)

and eq. (3.3). In order to write out the results for t(‘)(s) when the external potential W is given by (3.10), it is convenient to introduce four quantities clr:, cxY,\,c$, and c&. These are given by eqs. (3.13) - (3.16) if one replaces K everywhere by K,, , where (3.33)

K;t = A,i I/,+k*. The function & is given by &(r;

s) = or:\ rh~+)(RYlr)-a~zl rh{-)(&r),

csrsd,

(3.34)

and the eigenvalues AV1are obtained by solving the equation vz Vl vl vl a1 1 a22 -~12~21

(3.35)

0.

=

The factors in the separable expansion for t(i) (s) are given by
(~)W@vh(~))

v,

w

P2-4

rj,(pr)+

= (27r*)*(K,:-p*)

L v,

dPCkfl)

r hj+‘(kr),h(r;

D,(kc)

r=d

s>r=c

F

(3.36)

II

where the Wronskian IVis defined by (3.23), g1 and D, are given by (2.2) and (2.3), and C#Qis given by (3.34). The normalization factor is given by

(@vlm(~*)JWl@vlm> = - 2+

vl

where Gr is given by (3.19).

BOUNDARY

CONDITION

MODEL t-MATRIX

183

4. Numerical results

We now turn our attention to a numerical comparison of the exact expression for t(‘)(s) given by (3.22) with separable approximations obtained by truncating (3.30). In order to make our comparison as meaningful as possible we have adjusted our potential so as to obtain a reasonable fit to the two-nucleon 3S, and ISo phase shifts. For the core radius c we took a value of 0.71 fm in both the triplet and singlet states since this is approximately the value used in the realistic BCM ‘). This left us with three parameters in each spin state: the logarithmic derivativef,, the depth V,, of the well, and the position d of the outer edge of the well. Two of these parameters were constrained to give the scattering length and effective range and the third was adjusted to give a least-squares fit to the phase shifts in the energy range (lab) from 10 MeV to 350 MeV [ref. “)I. For the scattering lengths and effective ranges we took the values a, = 5.396 fm, r Ot =

1.726 fm,

(3.38)

a, = -23.678 fm, r os = 2.7 fm.

The results are given in table 1. TABLE 1 Potential parameters VO State

% ‘&I

WW

d

Jo

(fm)

22.955

2.1527

0.71

0.4276

8.342

2.8384

0.71

0.5406

We expect the expansion for t(l)(s) [see (3.30)] to converge rapidly when the energy parameter s is near a bound-state or resonance energy, since this corresponds to one of the eigenvalues ryl being close to one. In the 3S, state of the two-nucleon system this occurs when s is close to the deuteron energy (-2.225 MeV); in the ‘S,, state this occurs when s is near the energy of the ‘So virtual state (-0.0663 MeV). As is well known, the pole in the t-matrix corresponding to the virtual state is actually on the unphysical sheet, however since the energy is so small the pole has a strong influence on the low-energy behavior of the ‘S, t-matrix. In order to find the eigenvalues rlvr we solved (3.35) numerically, using an iterative technique. The first three eigenvalues for the 3S, state are shown in fig. 1. We note that at low energies the v = 1 eigenvalue dominates and at high energies all of the eigenvalues approach zero. The eigenvalues for the ‘S, state behave in a similar fashion.

184

0. P. BAHETHI AND M. G. FUDA

In tables 2-4 we show comparisons of the exact values of the S-wave part of t(‘)(s) with separable approximations for t(‘)(s) obtained by keeping one, two or three terms in the S-wave part of (3.30). Table 2 indicates that for small values of the momenta p and q, a one-term separable approximation is quite accurate at low energies. A two-

Fig. 1. The first three eigenvalues for the triplet parameters; s is the two-nucleon MeV divided by 41.47 MeV . fm*.

c.m. energy in

term approximation reduces the error at higher energies to about 5 % and a threeterm approximation is very accurate over the entire energy range. Table 3 shows a similar rate of convergence at higher values of p and q. Table 4 gives some results for the ISo state. A one-term approximation appears to be good only right near zero energy; two terms and three terms give approximations that are as good as the 3S, case. Calculations at other values of the momenta p and q show the same general features: a one-term separable approximation is very good at low energies in the 3S, state; two and three terms give good results in both the 3S1 and ISo state over a wide range of energies. By comparing tables 3 and 4 one sees that for most of the energies the singlet t(l) matrix is smaller in magnitude than the triplet t(l) matrix. This is true in general and is related to the fact that the singlet potential is weaker than the triplet potential. In a calculation on the triton both t-matrices come in with about equal

BOUNDARY

CONDITION

MODEL

185

r-MATRIX

TABLE 2 Values of P)(p,

q; s) for the 3Si state with p = 0 fm-’

and q = 0 fm-’

Separable approximations s(fme2) 0.000 -0.500 -1.000 -1.500 -2.000 -2.500 -3.000 -3.500 -4.000 -4.500 -5.000 -5.500 -6.000 -6.500 -7.000 -7.500 -8.000 -8.500 -9.000

0.32151 -0.21147 -0.15362 -0.13374 -0.12345 -0.11712 -0.11280 -0.10967 -0.10729 -0.10542 -0.10392 -0.10267 -0.10163 -0.10075 -0.09999 -0.09933 -0.09874 -0.09823 -0.09777

-

one term

two terms

three terms

0.32297 -0.20515 -0.14496 -0.12389 -0.11253 -0.10520 -0.09998 -0.09602 -0.09289 -0.09033 -0.08819 -0.08637 -0.08481 -0.08344 -0.08224 -0.08117 -0.08021 -0.07936 -0.07858

0.32165 -0.21071 -0.15230 -0.13249 -0.12210 -0.11557 -0.11101 -0.10761 -0.10497 -0.10284 -0.10108 -0.09961 -0.09835 -0.09726 -0.09631 -0.09547 -0.09473 -0.09406 -0.09347

0.32155 -0.21126 -0.15309 -0.13349 -0.12327 -0.11689 -0.11247 -0.10921 -0.10668 -0.10466 -0.10301 -0.10163 -0.10046 -0.09945 -0.09858 -0.09782 -0.09715 -0.09655 -0.09602

TABLE 3 Values of P(p,

q; s) for the 3S1 state with p =

1 fm-’

and q = 3 fm-i

Separable approximations s(fme2) 0.000 -0.500 -1.000 -1.500 -2.000 -2.500 -3.000 -3.500 -4.000 -4.500 -5.000 -5500 -6.000 -6.500 - 7.000 -7.500 -8.000 -8.500 -9.000

Exact -0.10788 0.03553 0.02089 0.01588 0.01329 0.01170 0.01061 0.00982 0.00922 0.00875 0.00837 0.00806 0.00780 0.00757 0.00738 0.00722 0.00707 0.00694 0.00683

one term

two terms

three terms

- -0.10820 0.03553 0.02113 0.01464 0.01380 0.01185 0.01059 0.00968 0.00899 0.00844 0.00800 0.00763 0.00733 0.00706 0.00684 0.00664 0.00647 0.0063 1 0.00617

-0.10805 0.03554 0.02122 0.01481 0.01405 0.01217 0.01098 0.01013 0.00950 0.00901 0.00861 0.00830 0.00803 0.00781 0.00762 0.00746 0.00732 0.00720 0.00709

-0.10790 0.03554 0.02117 0.01472 0.01394 0.01204 0.01083 0.00997 0.00933 0.00882 0.00842 0.00809 0.00782 0.00759 0.00740 0.00723 0.00709 0.00696 0.00685

0. P. BAHETHI

186

AND

M. G. FUDA

TABLE 4 Values of t(“(p,

9; s) for the ‘So state withp

= 1 fm-’

and q = 3 fm-’

Separable approximations s(fmez) 0.000 -0.500 -1.000 -1.500 -2.000 -2.500 -3.000 -3.500 -4.000 -4.500 -5.000 -5.500 - 6.000 -6.500 -7.000 -7.500 -8.000 -8.500 -9.000

Exact

one term

two terms

three terms

0.21068 0.00756 0.00491 0.00380 0.00317 0.00276 0.00247 0.00225 0.00208 0.00195 0.00184 0.00175 0.00168 0.00161 0.00155 0.00150 0.00146 0.00142 0.00139

0.20849 0.00660 0.00426 0.00331 0.00278 0.00245 0.00222 0.00205 0.00192 0.00182 0.00174 0.00167 0.00162 0.00157 0.00153 0.00150 0.00146 0.00144 0.00141

0.21051 0.00741 0.00478 0.00367 0.00304 0.00264 0.00235 0.00215 0.00199 0.00186 0.00176 0.00167 0.00160 0.00154 0.00149 0.00145 0.00141 0.00138 0.00135

0.21073 0.00759 0.00494 0.00382 0.00318 0.00276 0.00247 0.00225 0.00208 0.00195 0.00184 0.00175 0.00167 0.00161 0.00155 0.00151 0.00146 0.00142 0.00139

weight, therefore one can tolerate a slightly less accurate separable approxima~on in the singlet state. We conclude from the results shown here and others of a similar nature that the separable expansion developed previously “) for the external potential should be of practical value in calculating the three-nucleon binding energy and lowenergy scattering parameters. We intend to carry out calculations on the threenucleon system using this expansion in conjunction with Brayshaw’s ‘) new threepartide formalism. In conclusion we note that the closed form expression for the t(l) matrix given here should be useful in checking numerical methods for obtaining the BCM t-matrix. References 1) G. Breit and W. G. Bouricius, Phys. Rev. 75 (1949) 1029; H. Feshbach and E. L. Lomon, Phys. Rev. 102 (1956) 891; E. L. Lomon and M. Nauenberg, Nucl. Phys. 24 (1961) 474; H. Feshbach, E. L. Lomon and A. Tubis, Phys. Rev. Lett. 6 (1961) 635; E. L. Lomon and H. Feshbach, Rev. Mod. Phys. 39 (1967) 611; Ann. of Phys. 48 (1968) 94 2) Y. E. Kim and A. Tubis, Phys. Rev. Cl (19701 414: Phvs. Rev. C2 (19701 . , 2118 . ’ 3) M. G. Fuda, Phys. Rev.C3 il971) 55 4) M. M. Hoenig and E. L. Lomon, Ann. of Phys. 36 (1966) 363 5) D. D. Brayshaw, Phys. Rev. C3 (1971) 35 6) L. D. Faddeev, ZhETF 39 (1960) 1459; JETP (Sov. Phys.) 12 (1961) 1014 7) D. D. Brayshaw, Phys. Rev. Lett. 26 (1971) 659 8) A. Messiah, Quantum mechanics (Wiley, New York, 1965) 9) H. Feshbach and E. L. Lomon, Ann. of Phys. 29 (1964) 19 10) J. M. J. Van Leeuwen and A. S. Reiner, Physica 27 (1961) 99; J. S. Levinger, A. H. Lu and R. Stagat, Phys. Rev. 179 (1969) 926 11) R. E, Seamon, K. A. Friedman, G. Breit, R. D. Haracz, J. M. Holt and A. Prakash, Phys. Rev. 165 (1968) 1579