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Threshold scattering of the η-meson off light nuclei
5 October
1995
PHYSICS
LETTERS
B
Physics LettersB 359 (1995) 33-38
ELSEVIER
Threshold scattering of the v-meson off light nuclei S.A. Rakityansky a~l, S.A. Sofianosa, W. Sandhasav2, V.B. Belyaev b a Physics Department, University of South Africa, PO. Box 392, Pretoria 0001, South Africa ’ Joint Institute for Nuclear Research, Dubna 141980, Russia Received 3 May 1995; revised manuscript received 18 August Editor: C. Mahaux
1995
Abstract
The scattering lengths of _rl-meson collisions with light nuclei 2H, 3H , ‘He , and 4He are calculated on the basis of few-body equations in coherent approximation. It is found that the v-nucleus scattering length depends strongly on the number of nucleons and the range parameter. By taking into account the off-shell behavior of the qN amplitude, the vly scattering length increases considerably. PACS: 25.80.-e; 21,45.+v; 25.10.+s Keywords: Eta-meson; Scattering-length;
Eta-nucleus
reactions;
Coherent approximation
In q-nucleus collisions at threshold energies, a large Final State Interaction (FSI) is expected due to the N’ ( 1535) &I-resonance. This resonance, which is strongly coupled to the TN and TN channels, lies only N 50 MeV above the qN threshold and has a very broad width of r M 150 MeV [ 11. The v-nucleus dynamics, thus, is of interest from the point of view of both few-body and meson-nucleon physics. In the energy region covering the S1~-resonance, the TN and TN scattering processes are to be considered as a coupled-channel problem [ 2-41. The first of these channels has a long record of theoretical and experimental investigations, while the second one is understood only in a rudimentary way. The rrN and TN channels are connected to each other primarily via the N’ ( 1535) resonance. In fact, the qNN coupling ’ Permanent address: Joint Institute for Nuclear Research, 141980, Russia. 2 Permanent address: Physikalisches Institut, Universitit D-53 115 Bonn, Germany.
Dubna Bonn,
constant was shown to be negligible [5-71 as compared to the one of the qNN* vertex. This latter vertex constant, being hence crucial for determining the TN interaction, is known only with a large uncertainty, so that the TN scattering length inferred from it varies from(0.27+i0.22)fm[2] to(0.55+i0.30)fm[8]. The qN near-threshold interaction, therefore, remains an interesting field of investigation. Of particular relevance in this context is the possibility of v-nucleus bound states [9,10]. Their existence would provide us with an excellent opportunity to answer the abovementioned questions in a reliable manner. Various experimental groups have studied the near-threshold production of v-mesons in photo- and electro-nuclear processes [ 11,121, or in r-nucleus [ 131, N-nucleus [ 14,151, and nucleus-nucleus [ 161 collisions. The expected strong v-nucleus FSI is evident in these measurements and must not be ignored in theoretical treatments. A simple way of taking into account the low-energy
0370.2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlO370-2693(95)01057-2
S.A. Rakityansky et al. / Physics Letters B 359 (1995) 33-38
34
FSI effects in q-production processes is described in Ref. [ 171. It consists in multiplying the reaction amplitude by an energy-dependent factor which contains the final-state scattering length as a parameter. This parameter, hence, is essential in analyzing the increasing number of such production data. To the best of our knowledge, only the scattering-Iength calculations by Wilkin [ 81 for the v collision on 3He, 4He, and ‘Be exist in the literature. There, the lowest-order opticalpotential method was used with a simple TN amplitude having no energy or off-shell dependence. In the present work we calculate the v 2H, q 3H, 7 3He, and r) 4He scattering lengths on the basis of few-body equations with separable 7N amplitudes of the type suggested in Refs. [ 2,3]. The equations employed are obtained within the Finite Rank Approximation (FRA) of the Hamiltonian, proposed in Refs. [ 18,191 as an alternative to the multiple scattering or optical potential theory. Developed originally for the treatment of pion-nucleus scattering, this technique was later on also applied to other questions of fewbody physics, such as the A-nucleus bound state problem [20] or the NN-scattering in a six quark model 1191. The main idea of FRA consists in separating the motion of the projectile and the internal motion of the nucleons inside the nucleus. Correspondingly the total Hamiltonian is split according to H=Ho+V+HA,
(1)
where Ho denotes the kinetic energy operator of the v-meson relative to the center of mass of the nucleus, V = cf_, x the sum of the individual TN-interactions, and HA the Hamiltonian of the nucleus. The scattering amplitude for the transition from the initial state Ik,$e> to the final state Ik’,&>, with Itia> being the ground state of the nucleus, HA~$o>=&o~$o>,
(2)
is given by f(k’,k;z)
=-g
< k’>#olT(z)lk,$o
>,
(3)
where p is the reduced mass of the r]-nucleus system. The T-operator in this amplitude satisfies the Lippmann-Schwinger equation
T(z)=v+vZ_H~_H T(z). A Introducing
(4)
an auxiliary operator Te by
P(z) = v+
v-&w,
(5)
we obtain instead of Eq. (4) T(z)
=7?z)
+~(z)IHA
z - Ho
z
’
- Ho - HA
T(z).
(6)
Our main approximation consists in restricting the spectral decomposition of HA to the ground state I$e >, HA
=
&ol’bo><$ol,
(7)
Physically this so-called coherent approximation [ 21 I, which is widely used in multiple scattering and optical potential approaches, means that during the multiple scattering of the v-meson the nucleus remains unexcited. Eq. (6) then reads T(z)
=p(z)
+Eo~(z)l$o> 1
’ (z -Ho)(z
-HO-&~)
(8)
and, after sandwiching it between the ground state and the relative-momentum states, we obtain for the matrix elements T( k’, k; z ) = < k’, $0 IT( z ) 1k, @o > the integral equation W’,kz) +&I
=
s -
dk”
(2?7)3
x T(k”, k; z).
(z -
$!z)(z
_Eo
-
g,
(9)
Our problem thus is split into two steps. The first consists in evaluating the auxiliary amplitude whichdetermines theinhomogeneity and the kernel of Eq. (9) ; in the second step this equation is to be solved. The auxiliary operator Te describes the scattering of the r]-meson off the nucleons which are fixed in their position within the nucleus. This is easily seen in Eq. (5) which does not contain any operator that acts on the internal nuclear Jacobi coordinates {r}.
S.A. Rakityansky et al. /Physics Letters B 359 (1995) 33-38
All operators in Eq. (5), hence, are diagonal in these variables, so that its momentum representation reads P(k’,
k; r; z) = V(k’, k; r)
dk” V(k’yk”;r)p(k”,k;r;z).
+
(10)
It depends, in other words, only parametrically on the coordinates {r}. After solving this integral equation and determining the ground state wave function t+Ge ( r) by any appropriate bound-state method, the input to Eq. (9) isobtained via
=
s
dr[~o(T)12TO(k’,k;r;Z).
(11)
Since we are interested finally only in the on-shell amplitudes, it suffices to consider the half-on-shell restriction of Eq. (9). That is, we put in all the above relations the parameter z onto the initial energy, z = k’/2p
- IEo( + i0.
(12)
Therefore, the p-matrix enters Eq. (3) off the energy shell, differing thus from the conventional fixed-scatter amplitude for which z = k2/2p. In scattering-length calculations we have k = 0 and hence z = -I&[ + i0, so that Eqs. ( 10) and (9) become nonsingular and easy to handle. From the practical point of view it is convenient to rewrite Eq. (IO) by using the Faddeevtype decomposition
(13)
Introducing
the operators
fi(Z) =K+K
1
pfi(Z>, z - Ho
dk”
(27r)3
ti( k’, k”; r; z ) z-g
p is the canonical
conjugate
to
The amplitude ti describes the scattering of the Qmeson off the i-th nucleon. It is related to the corresponding two-body r,,v-matrix via t;(k’,k;r;z)=f+(k’,k;z)exp[i(k-k’).r;] (16) where ri is the vector from the nuclear center of mass to the i-th nucleon. This vector can be expressed in terms of the Jacobi coordinates {r} shown in Fig. 1. As input information we need these off-shell amplitudes and the ground-state wave functions of the nuclei involved. Due to the dominance of the Sr Iresonance near the threshold energy, we can restrict ourselves to the S-wave TN-interaction. In Ref. [ 91 it was demonstrated that the role played by the higher partial waves is indeed negligible. The ?NN coupling constant is much smaller than the qNN* one [ 5-71. This means that the VN collision goes predominantly via a virtual formation of N* ( 1535), that is, via the VN + N* + TN reaction. Hence, the corresponding amplitude must contain two TN ++ N” vertex functions and an N*-propagator in between. We employ the Yamaguchi-type form
A
?p the fol-
=ti(k’,k;r;z)
J’
+---
systems. The spatial coordinate the momentum k.
(14)
we finally get for the Faddeev components lowing system of integral equations c(k’,k;r;z)
Fig. 1. The Jacobi coordinates {r} for the three and four-nucleon
xTj-)(k”,k;r;z). j#i
(15)
= (k’* + a2)(z
- E. + ir/2)(k2
+ a2) ’
(17)
commonly used in few-body physics, with a simple Breit-Wigner propagator. Two of the four parameters of the t-matrix (17) are immediately fixed, namely the resonance energy EO = 1535 MeV -(MN + M,) and the width l? = 150 MeV [ 11. The parameter A is chosen to provide the correct zero-energy on-shell
36
S.A. Rakityansky et al. / Physics Letters B 359 (1995) 33-38
limit, i.e., to reproduce the known TN scattering length +N,
tgN(o,o,o)
=
-fiL$N.
(18)
71
Finally, in order to lix the parameter (Y, we make use of the results of Refs. [2,3], where the same qN --+ N* vertex function ( k2 + a2) -’ was employed with LY being determined via a two-channel fit to the ?rN + rrN and TN -+ 7N experimental data. Due to experimental uncertainties and differences between the models of the physical processes, there are three different values available for the scattering length in the literature: a@ = (0.27 + i 0.22) fm [ 21, aqN = (0.28+iO.19) fm [2],anda,N= (0.55+iO.30) fm [ 81. For the range parameter LYone also finds three different values: cy = 2.357 fm-’ [2], a = 3.316 fm-’ [ 31, and cy = 7.617 fm-’ [ 21. Since there is no criterion for singling out one of them, we use all 9 combinations of a,,,%,and (Yin our calculation. For the bound states we employed simple Gaussiantype functions
(19)
$,(x,y)
=
(a-)-3/2
xexp[-(:+$)/(2<‘:>)],
&r(X,YI 2) =
xexp[-16:r:>
(20)
b (1671 :r;>)3]3’4 (G+Y’+c)].
(21)
where x,y, and z are the Jacobi vectors depicted in Fig. 1. These functions were constructed to be symmetric with respect to nucleon permutations, and to reproduce the experimental mean square radii: d& [23],
= 1.956 fm [22], ,/G
,/G
= 1.755 fm
= 1.959 fm [23], and d-
=
1.671 fm [ 241. Since the mass radius difference between the 3H and 3He nuclei stems mainly from the mixed symmetric S’ state, which is not included in the above representation Eq. (20)) we also performed
calculations with the average radius J* = I .857 fm. For masses and binding energies of the nuclei we used the experimental values [ 251. The method described above involves only the coherent approximation (7), according to which all excitations of the target nucleus are neglected. This approximation could be shown [ 261, as expected, to be justified at low collision energies and in the case of a wide energy gap between the ground and first excited nuclear state. In our zero-energy calculation the method should, therefore, work particularly well. Among the target nuclei *H, 3H, 3He, and 4He the most accurate results are expected to be those for the 4He nucleus, since its first excited state at 20.21 MeV [ 241 lies comparatively high. The results of our calculations are given in Tables 1 and 2. As can be seen, the v-nucleus scattering length depends strongly on the number of nucleons A, and is sensitive to the range parameter a, i.e. to the off-shell continuation of the TN-amplitude. Prior to this work, Wilkin [ 81 performed calculations of the qA scattering lengths. In these investigations the TN amplitude was simply assumed to be constant and equal to the value of its zero-energy limit, agN = (0.55 + iO.30) fm. It is easily seen that such an assumption is equivalent to using a zero-range, i.e. a S-type interaction. The values of the 773He and r] 4He scattering lengths obtained by Wilkin are a(7 3He) = (-2.31 + i2.57) fm and a(7 4He) = (-2.00 + i0.97) fm. As our results show, the non-zero-range character of the TN interaction, i.e., its off-shell properties change these values significantly. As can be seen in both tables, the real part of the v-nucleus scattering length for the A = 3 case, is sensitive to the choice of the root mean square radius. This sensitivity, however, is significant only when the TN scattering length is large enough generating a negative value for it. More recently [27], Wilkin used the multiplescattering expansion in order to incorporate the nucleon-nucleon correlations. In this approach the ~a and ~;IN scattering lengths are connected via the simple relation a(r],4He)
= (O.l81/a,,v
- 0.281 fm-‘)-I.
(22)
With the three values of a$$ listed in Table 1, this formula gives a(q4He) = (0.99 + i2.67) fm, (1.39 + i2.58) fm, or (-1.38 + i6.95) fm, respectively. As
S.A. Rakityansky et al./Physics
Letters B 359 (1995) 33-38
37
Table 1 The v-nucleus
scattering
length results (in fm) obtained by using the 9 combinations (Y= 2.357 (fm-‘)
of the range parameter
a = 3.316 (fm-‘)
a=
7.617 (fm-‘)
2H “H jHe 4He
0.66 0.91 0.96 0.90
2H
0.75 + i0.73
0.74 + i0.76
0.72 + i0.8 1
YH ?He
1.19+ il.70 1.21 + il.59
4He
1.67+ i3.43
1.11 +i1.81 1.16-ti1.68 1.23 + i3.78
0.93 + il.92 1.04+ il.78 0.53 + i3.94
?H jH ‘He 4He
Table 2 The q-nucleus
1.53 -0.69 0.08 -4.41
scattering
+ + + +
+ + + +
i0.82 il.80 il.72 i3.32
i2.00 i5.13 i5.22 i2.85
0.65 0.81 0.89 0.48
1.38 -1.21 -0.52 -3.73
+ i0.85 + il.88 + il.78 + i3.45
+ + + +
a and the a+, scattering
0.62 0.63 0.76 -0.04
+ i0.89 + il.93 + il.84 + i3.40
1.14+ i2.22 -1.30 + i3.79 -0.79 + i4.2 1 -3.12+ il.85
i2.15 i4.50 i4.83 i2.18
length results (in fm) for the A = 3 nuclei using the average mass mean square radius
0.27 + i0.22
0.28 + i0.19
0.55 + io.30
m
= 1.857 fm
(Y= 3.316 (fm-r)
‘H jHe
0.95 + il.75 0.93 + il.76
0.86 + il.83 0.85 + il.83
0.71 + il.88 0.69 + il.88
0.27 + i0.22
“H ‘He
1.21 + il.63 1.19 + il.65
1.15 + il.74 1.13 + il.75
1.OO + il.84 0.98 + il.85
0.28 + i0. I9
-1.06 + i4.03 - 1.05 + i4.00
0.55 + io.30
-0.29 -0.31
+ i5.24 + i5.18
-0.87 -0.86
+ i4.73 + i4.67
can be seen, the agreement between this simple formula and our microscopic calculation is very poor, especially for large values of the range-parameter LY. If we admit the maximal size of the TN-interaction region found in the literature, which corresponds to cy = 2.357 fm-‘, and use the rather reasonable estimate uV ,v = (0.55 + iO.30) fm by Wilkins, then the corresponding large value of a(v4He) = (-4.41 + i2.86) fm indicates that the condition A 2 12 obtained in Ref. [ 91 for the existence of v-nucleus bound states must be reduced to lower values of A. Since the TN interaction is isospin-independent, the difference between the 7 3H and r] 3He is caused by differences of the sizes and binding energies of these nuclei. Therefore the use of more accurate nuclear wave functions is of utmost importance. Such realistic wave functions can be constructed, for instance, by means of the Integrodifferential Equation Approach (IDEA) [ 28,291, and are in progress.
(fm-‘)
(fm)
(Y= 2.357 (fm-‘)
3H “He
(Y= 7.617
a,N
length
w
(fm)
Financial support by the University of South Africa (UNISA) and the Joint Institute for Nuclear Research, Dubna, is appreciated. Two of us (S.A.R. and W.S.) are grateful to the Physics Department of UNISA for warm hospitality.
References [l] [2] [3] [4] [5] [6] [ 71
Particle Data Group, Phys. Rev. D 50(3) (1994) 1319. R.S. Bhalerao and L.C. Liu, Phys. Rev. L&t. 54 (1985) 865. C. Bennhold and H. Tanabe, Nucl. Phys. A 530 ( 1991) 625. R.J. McLeod and D.J. Ernst, Phys. Rev. C 49 (1994) 1087. W. Grein and E. Oset, Nucl. Phys. A 338 (1980) 332. B.L. Alvaredo and E. Oset, Phys. Lett. B 324 (1994) 125. L. Tiator, C. Bennhold and S.S. Kamalov, preprint MKPHT-94-3, Mainz, 1994. [8] C. Wilkin, Phys. Rev. C 47 (1993) R938. [9] Q. Haider and L.C. Liu, Phys. Lett. B 172 ( 1986) 257. [lo] R.E. Chrien et al., Phys. Rev. Lett. 60 (1988) 2595. [ 1 I] Institut fur Kemphysik. Universit;it Mainz, Jahrcsbericht 1992-1993, p. 167.
38
S.A. Rakityansky et al./Physics Letters B 359 (1995) 33-38
[ 121 M. Wilhelm, Ph.D. thesis, University of Bonn ( 1993). [ 131 C. Peng et al., Phys. Rev. Lett. 63 (1989) 2353. [ 141 J. Berger et al., Phys. Rev. Len. 61 (1988) 919. [ 15) E. Chiavassa et al., Nucl. Phys. A 519 (1990) 413. [ 161 R. Frascaria et al., Phys. Rev. C 50 (1994) R537. [ 171 M. Goldberger and K.M. Watson, Collision Theory (Wiley, New York, 1964) p. 540. [ 181 V.B. Belyaev and J. Wrzecionko, Sov. Journal of Nucl. Phys. 28 (1978) 78. [ 191 V.B. Belyaev, in Lectures on the theory of few-body systems ( Springer-Verlag. Heidelberg, 1990). [ 201 V.B. Belyaev and V.V. Pupyshev, Sov. Journal of Nucl. Phys. 35 (1982) 205. [21] A.K. Kerman, H. MC Manus and R. Thaler, Ann. Phys. (NY) 8 (1959) 551.
[22] S. Klarsfeld, J. Martorell and D.W.L. Sprung, J. Phys. G 10 (1984) 165. [23] A. Amroun et al., Nucl. Phys. A 579 (1994) 596. [ 241 D.R. Tilley, H.R. Weller and GM. Hale, Nucl Phys. A 541 (1992) 1. [ 251 A.H. Wapstm and G. Audi, Nucl. Phys. 432 ( 1985) 1. [26] V.B. Belyaev and V.V. Pupyshev, Sov. Journal of Nucl. Phys. 39 (1984) 594. [27] C. Wilkin, Phys. Lett. B 331 (1994) 267. [28] M. Fabre de la Ripelle, H. Fiedeldey and S.A. Sofianos, Phys. Rev. C 38 (1988) 449. [29] W. Gehm, S.A. Sofianos, H. Fiedeldey and M. Fabre de la Ripelle, Phys. Rev. C 42 (1990) 2322; C 43 (1991) 25; C44 (1991) 81.
1995
PHYSICS
LETTERS
B
Physics LettersB 359 (1995) 33-38
ELSEVIER
Threshold scattering of the v-meson off light nuclei S.A. Rakityansky a~l, S.A. Sofianosa, W. Sandhasav2, V.B. Belyaev b a Physics Department, University of South Africa, PO. Box 392, Pretoria 0001, South Africa ’ Joint Institute for Nuclear Research, Dubna 141980, Russia Received 3 May 1995; revised manuscript received 18 August Editor: C. Mahaux
1995
Abstract
The scattering lengths of _rl-meson collisions with light nuclei 2H, 3H , ‘He , and 4He are calculated on the basis of few-body equations in coherent approximation. It is found that the v-nucleus scattering length depends strongly on the number of nucleons and the range parameter. By taking into account the off-shell behavior of the qN amplitude, the vly scattering length increases considerably. PACS: 25.80.-e; 21,45.+v; 25.10.+s Keywords: Eta-meson; Scattering-length;
Eta-nucleus
reactions;
Coherent approximation
In q-nucleus collisions at threshold energies, a large Final State Interaction (FSI) is expected due to the N’ ( 1535) &I-resonance. This resonance, which is strongly coupled to the TN and TN channels, lies only N 50 MeV above the qN threshold and has a very broad width of r M 150 MeV [ 11. The v-nucleus dynamics, thus, is of interest from the point of view of both few-body and meson-nucleon physics. In the energy region covering the S1~-resonance, the TN and TN scattering processes are to be considered as a coupled-channel problem [ 2-41. The first of these channels has a long record of theoretical and experimental investigations, while the second one is understood only in a rudimentary way. The rrN and TN channels are connected to each other primarily via the N’ ( 1535) resonance. In fact, the qNN coupling ’ Permanent address: Joint Institute for Nuclear Research, 141980, Russia. 2 Permanent address: Physikalisches Institut, Universitit D-53 115 Bonn, Germany.
Dubna Bonn,
constant was shown to be negligible [5-71 as compared to the one of the qNN* vertex. This latter vertex constant, being hence crucial for determining the TN interaction, is known only with a large uncertainty, so that the TN scattering length inferred from it varies from(0.27+i0.22)fm[2] to(0.55+i0.30)fm[8]. The qN near-threshold interaction, therefore, remains an interesting field of investigation. Of particular relevance in this context is the possibility of v-nucleus bound states [9,10]. Their existence would provide us with an excellent opportunity to answer the abovementioned questions in a reliable manner. Various experimental groups have studied the near-threshold production of v-mesons in photo- and electro-nuclear processes [ 11,121, or in r-nucleus [ 131, N-nucleus [ 14,151, and nucleus-nucleus [ 161 collisions. The expected strong v-nucleus FSI is evident in these measurements and must not be ignored in theoretical treatments. A simple way of taking into account the low-energy
0370.2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlO370-2693(95)01057-2
S.A. Rakityansky et al. / Physics Letters B 359 (1995) 33-38
34
FSI effects in q-production processes is described in Ref. [ 171. It consists in multiplying the reaction amplitude by an energy-dependent factor which contains the final-state scattering length as a parameter. This parameter, hence, is essential in analyzing the increasing number of such production data. To the best of our knowledge, only the scattering-Iength calculations by Wilkin [ 81 for the v collision on 3He, 4He, and ‘Be exist in the literature. There, the lowest-order opticalpotential method was used with a simple TN amplitude having no energy or off-shell dependence. In the present work we calculate the v 2H, q 3H, 7 3He, and r) 4He scattering lengths on the basis of few-body equations with separable 7N amplitudes of the type suggested in Refs. [ 2,3]. The equations employed are obtained within the Finite Rank Approximation (FRA) of the Hamiltonian, proposed in Refs. [ 18,191 as an alternative to the multiple scattering or optical potential theory. Developed originally for the treatment of pion-nucleus scattering, this technique was later on also applied to other questions of fewbody physics, such as the A-nucleus bound state problem [20] or the NN-scattering in a six quark model 1191. The main idea of FRA consists in separating the motion of the projectile and the internal motion of the nucleons inside the nucleus. Correspondingly the total Hamiltonian is split according to H=Ho+V+HA,
(1)
where Ho denotes the kinetic energy operator of the v-meson relative to the center of mass of the nucleus, V = cf_, x the sum of the individual TN-interactions, and HA the Hamiltonian of the nucleus. The scattering amplitude for the transition from the initial state Ik,$e> to the final state Ik’,&>, with Itia> being the ground state of the nucleus, HA~$o>=&o~$o>,
(2)
is given by f(k’,k;z)
=-g
< k’>#olT(z)lk,$o
>,
(3)
where p is the reduced mass of the r]-nucleus system. The T-operator in this amplitude satisfies the Lippmann-Schwinger equation
T(z)=v+vZ_H~_H T(z). A Introducing
(4)
an auxiliary operator Te by
P(z) = v+
v-&w,
(5)
we obtain instead of Eq. (4) T(z)
=7?z)
+~(z)IHA
z - Ho
z
’
- Ho - HA
T(z).
(6)
Our main approximation consists in restricting the spectral decomposition of HA to the ground state I$e >, HA
=
&ol’bo><$ol,
(7)
Physically this so-called coherent approximation [ 21 I, which is widely used in multiple scattering and optical potential approaches, means that during the multiple scattering of the v-meson the nucleus remains unexcited. Eq. (6) then reads T(z)
=p(z)
+Eo~(z)l$o> 1
’ (z -Ho)(z
-HO-&~)
(8)
and, after sandwiching it between the ground state and the relative-momentum states, we obtain for the matrix elements T( k’, k; z ) = < k’, $0 IT( z ) 1k, @o > the integral equation W’,kz) +&I
=
s -
dk”
(2?7)3
x T(k”, k; z).
$!z)(z
_Eo
-
g,
(9)
Our problem thus is split into two steps. The first consists in evaluating the auxiliary amplitude
S.A. Rakityansky et al. /Physics Letters B 359 (1995) 33-38
All operators in Eq. (5), hence, are diagonal in these variables, so that its momentum representation reads P(k’,
k; r; z) = V(k’, k; r)
dk” V(k’yk”;r)p(k”,k;r;z).
+
(10)
It depends, in other words, only parametrically on the coordinates {r}. After solving this integral equation and determining the ground state wave function t+Ge ( r) by any appropriate bound-state method, the input
=
s
dr[~o(T)12TO(k’,k;r;Z).
(11)
Since we are interested finally only in the on-shell amplitudes, it suffices to consider the half-on-shell restriction of Eq. (9). That is, we put in all the above relations the parameter z onto the initial energy, z = k’/2p
- IEo( + i0.
(12)
Therefore, the p-matrix enters Eq. (3) off the energy shell, differing thus from the conventional fixed-scatter amplitude for which z = k2/2p. In scattering-length calculations we have k = 0 and hence z = -I&[ + i0, so that Eqs. ( 10) and (9) become nonsingular and easy to handle. From the practical point of view it is convenient to rewrite Eq. (IO) by using the Faddeevtype decomposition
(13)
Introducing
the operators
fi(Z) =K+K
1
pfi(Z>, z - Ho
dk”
(27r)3
ti( k’, k”; r; z ) z-g
p is the canonical
conjugate
to
The amplitude ti describes the scattering of the Qmeson off the i-th nucleon. It is related to the corresponding two-body r,,v-matrix via t;(k’,k;r;z)=f+(k’,k;z)exp[i(k-k’).r;] (16) where ri is the vector from the nuclear center of mass to the i-th nucleon. This vector can be expressed in terms of the Jacobi coordinates {r} shown in Fig. 1. As input information we need these off-shell amplitudes and the ground-state wave functions of the nuclei involved. Due to the dominance of the Sr Iresonance near the threshold energy, we can restrict ourselves to the S-wave TN-interaction. In Ref. [ 91 it was demonstrated that the role played by the higher partial waves is indeed negligible. The ?NN coupling constant is much smaller than the qNN* one [ 5-71. This means that the VN collision goes predominantly via a virtual formation of N* ( 1535), that is, via the VN + N* + TN reaction. Hence, the corresponding amplitude must contain two TN ++ N” vertex functions and an N*-propagator in between. We employ the Yamaguchi-type form
A
?p the fol-
=ti(k’,k;r;z)
J’
+---
systems. The spatial coordinate the momentum k.
(14)
we finally get for the Faddeev components lowing system of integral equations c(k’,k;r;z)
Fig. 1. The Jacobi coordinates {r} for the three and four-nucleon
xTj-)(k”,k;r;z). j#i
(15)
= (k’* + a2)(z
- E. + ir/2)(k2
+ a2) ’
(17)
commonly used in few-body physics, with a simple Breit-Wigner propagator. Two of the four parameters of the t-matrix (17) are immediately fixed, namely the resonance energy EO = 1535 MeV -(MN + M,) and the width l? = 150 MeV [ 11. The parameter A is chosen to provide the correct zero-energy on-shell
36
S.A. Rakityansky et al. / Physics Letters B 359 (1995) 33-38
limit, i.e., to reproduce the known TN scattering length +N,
tgN(o,o,o)
=
-fiL$N.
(18)
71
Finally, in order to lix the parameter (Y, we make use of the results of Refs. [2,3], where the same qN --+ N* vertex function ( k2 + a2) -’ was employed with LY being determined via a two-channel fit to the ?rN + rrN and TN -+ 7N experimental data. Due to experimental uncertainties and differences between the models of the physical processes, there are three different values available for the scattering length in the literature: a@ = (0.27 + i 0.22) fm [ 21, aqN = (0.28+iO.19) fm [2],anda,N= (0.55+iO.30) fm [ 81. For the range parameter LYone also finds three different values: cy = 2.357 fm-’ [2], a = 3.316 fm-’ [ 31, and cy = 7.617 fm-’ [ 21. Since there is no criterion for singling out one of them, we use all 9 combinations of a,,,%,and (Yin our calculation. For the bound states we employed simple Gaussiantype functions
(19)
$,(x,y)
=
(a-
xexp[-(:+$)/(2<‘:>)],
&r(X,YI 2) =
xexp[-16:r:>
(20)
b (1671 :r;>)3]3’4 (G+Y’+c)].
(21)
where x,y, and z are the Jacobi vectors depicted in Fig. 1. These functions were constructed to be symmetric with respect to nucleon permutations, and to reproduce the experimental mean square radii: d& [23],
= 1.956 fm [22], ,/G
,/G
= 1.755 fm
= 1.959 fm [23], and d-
=
1.671 fm [ 241. Since the mass radius difference between the 3H and 3He nuclei stems mainly from the mixed symmetric S’ state, which is not included in the above representation Eq. (20)) we also performed
calculations with the average radius J* = I .857 fm. For masses and binding energies of the nuclei we used the experimental values [ 251. The method described above involves only the coherent approximation (7), according to which all excitations of the target nucleus are neglected. This approximation could be shown [ 261, as expected, to be justified at low collision energies and in the case of a wide energy gap between the ground and first excited nuclear state. In our zero-energy calculation the method should, therefore, work particularly well. Among the target nuclei *H, 3H, 3He, and 4He the most accurate results are expected to be those for the 4He nucleus, since its first excited state at 20.21 MeV [ 241 lies comparatively high. The results of our calculations are given in Tables 1 and 2. As can be seen, the v-nucleus scattering length depends strongly on the number of nucleons A, and is sensitive to the range parameter a, i.e. to the off-shell continuation of the TN-amplitude. Prior to this work, Wilkin [ 81 performed calculations of the qA scattering lengths. In these investigations the TN amplitude was simply assumed to be constant and equal to the value of its zero-energy limit, agN = (0.55 + iO.30) fm. It is easily seen that such an assumption is equivalent to using a zero-range, i.e. a S-type interaction. The values of the 773He and r] 4He scattering lengths obtained by Wilkin are a(7 3He) = (-2.31 + i2.57) fm and a(7 4He) = (-2.00 + i0.97) fm. As our results show, the non-zero-range character of the TN interaction, i.e., its off-shell properties change these values significantly. As can be seen in both tables, the real part of the v-nucleus scattering length for the A = 3 case, is sensitive to the choice of the root mean square radius. This sensitivity, however, is significant only when the TN scattering length is large enough generating a negative value for it. More recently [27], Wilkin used the multiplescattering expansion in order to incorporate the nucleon-nucleon correlations. In this approach the ~a and ~;IN scattering lengths are connected via the simple relation a(r],4He)
= (O.l81/a,,v
- 0.281 fm-‘)-I.
(22)
With the three values of a$$ listed in Table 1, this formula gives a(q4He) = (0.99 + i2.67) fm, (1.39 + i2.58) fm, or (-1.38 + i6.95) fm, respectively. As
S.A. Rakityansky et al./Physics
Letters B 359 (1995) 33-38
37
Table 1 The v-nucleus
scattering
length results (in fm) obtained by using the 9 combinations (Y= 2.357 (fm-‘)
of the range parameter
a = 3.316 (fm-‘)
a=
7.617 (fm-‘)
2H “H jHe 4He
0.66 0.91 0.96 0.90
2H
0.75 + i0.73
0.74 + i0.76
0.72 + i0.8 1
YH ?He
1.19+ il.70 1.21 + il.59
4He
1.67+ i3.43
1.11 +i1.81 1.16-ti1.68 1.23 + i3.78
0.93 + il.92 1.04+ il.78 0.53 + i3.94
?H jH ‘He 4He
Table 2 The q-nucleus
1.53 -0.69 0.08 -4.41
scattering
+ + + +
+ + + +
i0.82 il.80 il.72 i3.32
i2.00 i5.13 i5.22 i2.85
0.65 0.81 0.89 0.48
1.38 -1.21 -0.52 -3.73
+ i0.85 + il.88 + il.78 + i3.45
+ + + +
a and the a+, scattering
0.62 0.63 0.76 -0.04
+ i0.89 + il.93 + il.84 + i3.40
1.14+ i2.22 -1.30 + i3.79 -0.79 + i4.2 1 -3.12+ il.85
i2.15 i4.50 i4.83 i2.18
length results (in fm) for the A = 3 nuclei using the average mass mean square radius
0.27 + i0.22
0.28 + i0.19
0.55 + io.30
m
= 1.857 fm
(Y= 3.316 (fm-r)
‘H jHe
0.95 + il.75 0.93 + il.76
0.86 + il.83 0.85 + il.83
0.71 + il.88 0.69 + il.88
0.27 + i0.22
“H ‘He
1.21 + il.63 1.19 + il.65
1.15 + il.74 1.13 + il.75
1.OO + il.84 0.98 + il.85
0.28 + i0. I9
-1.06 + i4.03 - 1.05 + i4.00
0.55 + io.30
-0.29 -0.31
+ i5.24 + i5.18
-0.87 -0.86
+ i4.73 + i4.67
can be seen, the agreement between this simple formula and our microscopic calculation is very poor, especially for large values of the range-parameter LY. If we admit the maximal size of the TN-interaction region found in the literature, which corresponds to cy = 2.357 fm-‘, and use the rather reasonable estimate uV ,v = (0.55 + iO.30) fm by Wilkins, then the corresponding large value of a(v4He) = (-4.41 + i2.86) fm indicates that the condition A 2 12 obtained in Ref. [ 91 for the existence of v-nucleus bound states must be reduced to lower values of A. Since the TN interaction is isospin-independent, the difference between the 7 3H and r] 3He is caused by differences of the sizes and binding energies of these nuclei. Therefore the use of more accurate nuclear wave functions is of utmost importance. Such realistic wave functions can be constructed, for instance, by means of the Integrodifferential Equation Approach (IDEA) [ 28,291, and are in progress.
(fm-‘)
(fm)
(Y= 2.357 (fm-‘)
3H “He
(Y= 7.617
a,N
length
w
(fm)
Financial support by the University of South Africa (UNISA) and the Joint Institute for Nuclear Research, Dubna, is appreciated. Two of us (S.A.R. and W.S.) are grateful to the Physics Department of UNISA for warm hospitality.
References [l] [2] [3] [4] [5] [6] [ 71
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