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Pion absorption and the pion-deuteron scattering length
Volume 45B, number 5
PHYSICS LETTERS
PION ABSORPTION
AND THE PION-DEUTERON
20 August 1973
SCATTERING
LENGTH*
A.W. THOMAS and I.R. AFNAN School of Physical Sciences, Flinders University of South Australia, Bedford Park, South Australia 5042, Australia
Received 19 June 1973 The solution of the Faddeev equations for the nNN system has been used as a model for studying pion absorption and elastic scattering on the deuteron. Results indicate that absorption is sensitive to the short range behaviour of the deuteron wave function. The contribution of absorption to the ~rd scattering length is found to be small. The pion deuteron system has been of fundamental interest in recent years [1] as the simplest pion nucleus system. We briefly describe a model of this system which enables one, for the first time, to consistently treat both processes: n++d~rr++d
(la)
and Ir++d~p+p
(lb)
while preserving three body unitarity. The results obtained show that the absorption process is very sensitive to the short range behaviour of the N-N interaction, and the d-state probability of the deuteron (PD) [2]. In addition, we show that the contribution of ( I b ) to the real part of the 7r-d scattering length is the same size as its contribution to the imaginary part, thereby resolving a disagreement in the literature [ 3 - 5 ] over the size of this effect. Finally, we show that the Ir-d scattering length is negative (in agreement with experiments on other light nuclei), and that a measurement of this quantity would help to determine the pion nucleon scattering lengths. In our model the rrNN system is treated as a purely three body system, to which we apply the Faddeev equations [6]. One term separable potentials, which have proved useful in many other systems [7], are used for all the two body interactions. Reaction ( l b ) is included by assuming one of the final nucleons is composite [8] - that is, in the P l l channel the ~rN potential has a bound state with binding energy
equal to the pion mass**. We use non-relativistic kinematics, which should be a fair approximation because we are working below break-up threshold. A summary of the two-body interactions used is given in table 1. As well as Phillips potentials [9] for the 3S1-3D 1 N-N channel, we use the U.P.A. of Bhatt et al. [10] to the Reid soft core potential (R.S.C.). Of course, the latter gives the R.S.C. deuteron wave function exactly. For the rr-N system, we find large variation in the $31 data in the literature. In particular, one $31 potential was constructed to fit the 0 - 7 0 0 MeV data of Roper et al. [11] up to 200 MeV quite accurately, but gave much too small a scattering length (a 3 = - 0 . 0 5 # - 1 ) . On the other hand, a fit to Bugg's data [12] in the $31 channel gave a scattering length of a 3 = - 0 . 0 8 6 / I 1, in agreement with Samaranayake and Woolcock [13]. Finally, we observe that the S l l scattering length was correct (a 1 = 0.182/a-1) after just fitting the phase shifts of Roper et al. (0--700 MeV solution), up to 200 MeV. A major aim of this work is to describe the coupling of the absorptive channel ( l b ) to the elastic channel (la). It is clear then, that our model of a composite nucleon must give a good description of the virtual process N' "-~N + lr,** since this is basic to the absorption process. Accordingly, in fig. 1, we compare the N-N phase shifts predicted by the three-body model with the one pion exchange (O.P.E.) phases.
**
* Supported in part by the Australian Research Grants Committee, and the Flinders University Research Budget.
To guarantee the indistinguishability of the simple (N) and composite (N') nucleon, we take a linear combination of Faddeev amplitudes so that the final NN' state is antisymmetric. 437
Volume 45B, number 5
PHYSICS LETTERS
20 August 1973
Table 1 Three-body channels in the Faddeev equation with j~r = 1- (s-wave absorption). Here S is the total channel spin, and L is the spectator particle orbital angular momentum. #
Two-body state
L
S
Two-body potential
1
aS 1-3D 1
0
1
Phillips [9]
IA
3S 1--3D1
0
1
U.P.A. tO R.S.C. [10]
2
1P1
1
1
Mongan Case II [18]
3
S]l
0
1
Fit to 0 - 700 MeV data of Roper et al [ 11], scattering length a 3 = 0.18 lp -1 [ 13 ]
4
Sll
2
1
5
S31
0
1
6
S31
2
1
7
P33
1
1
8
P33
1
2
9
Pll
1
1
Data of Bugg* [ 12], scattering length a 3 = _0.089p-] Fit to (3,3) resonance at 1211-50i MeV [19]
Range same as P33, strength to give B.E.*
* Choice of Pll and P33 was discussed in some detail in ref. [14]. Note that the $31 interaction used there was fitted to the (0-700) MeV data of Roper et al. and had a scattering length of -0.05p -] . I
I
I
i
I
2"O 1.0 ,~
0"0 -I"G
~ -2.o
t z -3"0 z
L
-4"0 -
5"0
I
I
I
I
I
so
loo
1so
2oo
2so
e.~,(MeV) Fig. 1. Comparison of the standard O.P.E. F-wave N-N phase shifts, with those calculated using the present model. The three-body calculation in this case has a two-body 438
interaction only in the P11 channel. The good overall agreement, and the fact that the only ingredient apart from the correct three-body dynamics, is the N' ~ N + ~t vertex, is a strong justification for the present description of the production mechanism. Low energy s-wave pion absorption (or production). by the deuteron, can be described by a single coefficient a [ 1 ], which has been most recently determined as [15] 180 +- 20/ab. We present in table 2 some of our results for a. We observe (i, ii, iii) that a tends to decrease with increasing PD. This is in qualitative agreement with earlier results [2] using the KoltunReitan [16] method, where the amplitude for direct absorption was added to that for one rescattering before absorption. Unfortunately, in that model the second term gives a larger contribution than the first - mainly because of the large contribution from intermediate p-wave pions. This raises doubts about the convergence of the multiple scattering series, which is one reason for the present Faddeev approach. We have tested the sensitivity of a to the presence of all possible two-body channels. We find that the 3P0,1, 2 N-N interactions do not change our results. However, with Phillips deuteron wave function, inclusion o f the 1P 1 N-N channel increases the value of a as seen in table 2 (ii, iv). By omitting various channels we find that this is mainly due to the large rescattering involving the 1P 1 and P33 channels.
Volume 45B, number 5
PHYSICS LETTERS
Table 2 The s-wave pion production cross section and elastic ~rd amplitude. All channels of table 1 are included except those in column five. #
PD (%)
~(#b)
and (fm)
Channels omitted
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)
0 4 7 4 6.58 6.58 6.58 6.58
679 640 607 330 122 220 222 . . . .
-0.029 + 0.019i -0.030+ 0.018i -0.032+ 0.017i -0.034 + 0.092i -0.029 + 0.003i -0.035 + 0.006i -0.036 + 0.006i 0.028
1A 1A 1A 1A, 2 1,2,4,6 1, 2 1 1, 2, 9
To study the importance of the short range N-N interaction in pion production, we have replaced the 3S 1- 3D l potential of Phillips by the UPA representation of the Reid soft core potential, which includes the effects of short range repulsion in the deuteron. Here we find that the large increase in ct due to the 1P 1 N-N interaction disappears (vi, vii), indicating the sensitivity of this process to high Fourier components in the deuteron wave function. Furthermore, when all relevant channels are included, particularly d-wave spectator nucleon with ¢rN pair in S l l and $31 (compare v, vi), our result for c~, of 220/ab, is in good agreement with experiment. Once again we observe qualitative similarities between this result and our earlier calculations [2], where separable potentials with no short range repulsion gave larger values ofc~ than the realistic potentials. Nevertheless the present calculation should be more reliable, since by solving the integral equations, we are summing all terms in the multiple scattering series in a unitary way. In fact, in the present model, the multiple scattering series itself is divergent. We feel it is significant that this calculation agrees with experiment when a realistic deuteron wave function is used, and it is with some confidence that we move to calculate the effect of absorption on the real part of the 7r-d scattering length (Re and ). In table 2 we also give the s-wave rr-d scattering amplitude at - 1 MeV (c.m. energy). We shall refer to this as the scattering length, with an error of a few per cent. Here we find that the multiple scattering series for a~ra converges (three or four terms), provided there is no coupling to the absorptive channel - otherwise the series is divergent. We can write the lrd scattering length as [ 1 ]
I+~/M~ a rd= ~ i ~ / a ~ l a l
20 August 1973 + ~ ~+ h /_a3) a d
(2)
where/a and M are the mass of the pion and nucleon respectively, while a 1 and a 3 are the s-wave lrN scattering lengths. The first term in eq. (2) is due to single scattering of the pion from the ~wo nucleons, while the second term is due to higher order multiple scattering and absorption. Although (at+ 2a3) is not well known experimentally, it is accepted to be small [13], (soft pion theory [17] gives a 1 + 2a 3 = 0).. The uncertainty in a I + 2a 3 arises mainly from the uncertainty in a 3 . We find that Reahd is negative, and relatively insensitive to a 3. For example, with the RSC deuteron, Reahd varies from --0.0455 fm to --0.0365 fm for a 3 in the range ( - 0 . 0 8 6 , - 0 . 0 5 ) / a - t . (For Phillips deuteron wave function the variation is - 0 . 0 4 0 fm to - 0 . 0 3 8 fm.) This is obviously a small effect compared with the change in the first term from +0.010 fm to +0.084 f m - where we have used [13] a 1 = 0 . 1 8 2 / I 1. With the latest value of a 3 [ 13], and the RSC deuteron, we find (vii): Rear d = -0.036 fm. The contribution to ahd from higher order multiple scattering was calculated by Kolybasov and Kudryatsev [4] to be negative ( - 0 . 0 3 7 fm), and in fair agreement with the present calculztion (vii) of - 0 . 0 3 8 fm. However, both the sign and magnitude of the contribution to React d from the absorptive channel is in question. We find (vi, viii) that with the RSC deuteron the contribution to the Reahd from absorption is - 0 . 0 0 7 fm. This is the same size as the contribution to the Imahd (viz. +0.006 fm), and in agreement with the intuitive prediction of Faldt [3] ( - 0 . 0 0 6 8 fm). However, Beder [5] has found a contribution to React d of +0.084 fm, from this process, using an unsubtracted dispersion relation. This is considerably larger than the value we get and opposite in sign. From the theoretical viewpoint we feel these results are encouraging. We have a good description of the rrNN system at low energy, and it should be suitable for extension to higher energies. In particular, one could test the energy dependence of the s-wave pion absorption coefficient c~. Furthermore, the demonstrated sensitivity of our results to the relatively unknown features of the N-N force, like PD, and the 439
Volume 45B, number 5
PHYSICS LETTERS
short range ( < 1 fm) behaviour, suggests that pion absorption should become an important tool in nuclear physics. On the experimental side, we realize the determination of the 7r-d scattering length is quite a difficult experiment. However, in view of our comments on ahd , its direct measurement should give us an alternative method to determine a 1 + 2a 3. We acknowledge many helpful discussions with Professor I.E. McCarthy.
References [1] D.S. Koltun, Advances in Nucl. Phys. 3 (1970) 71. [2] A.W. Thomas and I.R. Afnan, Phys. Rev. Lett. 26 (1971) 906. [3] G. Faldt, Nucl. Phys. B10 (1969) 597. [4] V.M. Kolybasov and A.E. Kudryatsev, NucL Phys. B41 (1972) 510.
440
20 August 1973
[5] D. Beder, Nucl. Phys. B14 (1969) 586. [6] L.D. Faddeev, Soviet. Phys. JETP 12 (1961) 1014. [7] R. Aaron, R.D. Amado andY.Y. Yam, Phys. Rev. 136 (1964) B650; P. Shanley, Phys. Rev. Lett. 21 (1968) 627. [8] C. Lovelace, Phys. Rev. 135 (1964) B1225. [9] A.C. Phillips, Nucl. Phys. A107 (1968) 209. [10] S.C. Bhatt, J.S. Levinger and E. Harms, Phys. Lett. B40 (1972) 23. [ 11 ] L.D. Roper, R.M. Wright and B.T. Feld, Phys. Rev. 138 (1965) B190. [12] B.V. Bugg, Spring School on Pion interaction at low and medium energy, CERN 71-14. [13] V.K. Samaranayake and W.S. Woolcock, Nucl. Phys. B48 (1972) 205. [14] I.R. Afnan and A.W. Thomas, Proc. Int. Conf. on Few particle problems in the nuclear interaction, Los Angeles, 1972 (North-Holland, 1973). [15] C. Ridlard-Serre et al., Nucl. Phys. B20 (1970) 413. [16] D.S. Koltun and A. Reitan, Phys. Rev. 141 (1965) 1413. [17] S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. [18] T.R. Mongan, Phys. Rev. 175 (1968) 1260. [19] J.S. Ball, R. Campbell, P. Lee and G. Shaw, Phys. Rev. Lett. 28 (1972)1143.
PHYSICS LETTERS
PION ABSORPTION
AND THE PION-DEUTERON
20 August 1973
SCATTERING
LENGTH*
A.W. THOMAS and I.R. AFNAN School of Physical Sciences, Flinders University of South Australia, Bedford Park, South Australia 5042, Australia
Received 19 June 1973 The solution of the Faddeev equations for the nNN system has been used as a model for studying pion absorption and elastic scattering on the deuteron. Results indicate that absorption is sensitive to the short range behaviour of the deuteron wave function. The contribution of absorption to the ~rd scattering length is found to be small. The pion deuteron system has been of fundamental interest in recent years [1] as the simplest pion nucleus system. We briefly describe a model of this system which enables one, for the first time, to consistently treat both processes: n++d~rr++d
(la)
and Ir++d~p+p
(lb)
while preserving three body unitarity. The results obtained show that the absorption process is very sensitive to the short range behaviour of the N-N interaction, and the d-state probability of the deuteron (PD) [2]. In addition, we show that the contribution of ( I b ) to the real part of the 7r-d scattering length is the same size as its contribution to the imaginary part, thereby resolving a disagreement in the literature [ 3 - 5 ] over the size of this effect. Finally, we show that the Ir-d scattering length is negative (in agreement with experiments on other light nuclei), and that a measurement of this quantity would help to determine the pion nucleon scattering lengths. In our model the rrNN system is treated as a purely three body system, to which we apply the Faddeev equations [6]. One term separable potentials, which have proved useful in many other systems [7], are used for all the two body interactions. Reaction ( l b ) is included by assuming one of the final nucleons is composite [8] - that is, in the P l l channel the ~rN potential has a bound state with binding energy
equal to the pion mass**. We use non-relativistic kinematics, which should be a fair approximation because we are working below break-up threshold. A summary of the two-body interactions used is given in table 1. As well as Phillips potentials [9] for the 3S1-3D 1 N-N channel, we use the U.P.A. of Bhatt et al. [10] to the Reid soft core potential (R.S.C.). Of course, the latter gives the R.S.C. deuteron wave function exactly. For the rr-N system, we find large variation in the $31 data in the literature. In particular, one $31 potential was constructed to fit the 0 - 7 0 0 MeV data of Roper et al. [11] up to 200 MeV quite accurately, but gave much too small a scattering length (a 3 = - 0 . 0 5 # - 1 ) . On the other hand, a fit to Bugg's data [12] in the $31 channel gave a scattering length of a 3 = - 0 . 0 8 6 / I 1, in agreement with Samaranayake and Woolcock [13]. Finally, we observe that the S l l scattering length was correct (a 1 = 0.182/a-1) after just fitting the phase shifts of Roper et al. (0--700 MeV solution), up to 200 MeV. A major aim of this work is to describe the coupling of the absorptive channel ( l b ) to the elastic channel (la). It is clear then, that our model of a composite nucleon must give a good description of the virtual process N' "-~N + lr,** since this is basic to the absorption process. Accordingly, in fig. 1, we compare the N-N phase shifts predicted by the three-body model with the one pion exchange (O.P.E.) phases.
**
* Supported in part by the Australian Research Grants Committee, and the Flinders University Research Budget.
To guarantee the indistinguishability of the simple (N) and composite (N') nucleon, we take a linear combination of Faddeev amplitudes so that the final NN' state is antisymmetric. 437
Volume 45B, number 5
PHYSICS LETTERS
20 August 1973
Table 1 Three-body channels in the Faddeev equation with j~r = 1- (s-wave absorption). Here S is the total channel spin, and L is the spectator particle orbital angular momentum. #
Two-body state
L
S
Two-body potential
1
aS 1-3D 1
0
1
Phillips [9]
IA
3S 1--3D1
0
1
U.P.A. tO R.S.C. [10]
2
1P1
1
1
Mongan Case II [18]
3
S]l
0
1
Fit to 0 - 700 MeV data of Roper et al [ 11], scattering length a 3 = 0.18 lp -1 [ 13 ]
4
Sll
2
1
5
S31
0
1
6
S31
2
1
7
P33
1
1
8
P33
1
2
9
Pll
1
1
Data of Bugg* [ 12], scattering length a 3 = _0.089p-] Fit to (3,3) resonance at 1211-50i MeV [19]
Range same as P33, strength to give B.E.*
* Choice of Pll and P33 was discussed in some detail in ref. [14]. Note that the $31 interaction used there was fitted to the (0-700) MeV data of Roper et al. and had a scattering length of -0.05p -] . I
I
I
i
I
2"O 1.0 ,~
0"0 -I"G
~ -2.o
t z -3"0 z
L
-4"0 -
5"0
I
I
I
I
I
so
loo
1so
2oo
2so
e.~,(MeV) Fig. 1. Comparison of the standard O.P.E. F-wave N-N phase shifts, with those calculated using the present model. The three-body calculation in this case has a two-body 438
interaction only in the P11 channel. The good overall agreement, and the fact that the only ingredient apart from the correct three-body dynamics, is the N' ~ N + ~t vertex, is a strong justification for the present description of the production mechanism. Low energy s-wave pion absorption (or production). by the deuteron, can be described by a single coefficient a [ 1 ], which has been most recently determined as [15] 180 +- 20/ab. We present in table 2 some of our results for a. We observe (i, ii, iii) that a tends to decrease with increasing PD. This is in qualitative agreement with earlier results [2] using the KoltunReitan [16] method, where the amplitude for direct absorption was added to that for one rescattering before absorption. Unfortunately, in that model the second term gives a larger contribution than the first - mainly because of the large contribution from intermediate p-wave pions. This raises doubts about the convergence of the multiple scattering series, which is one reason for the present Faddeev approach. We have tested the sensitivity of a to the presence of all possible two-body channels. We find that the 3P0,1, 2 N-N interactions do not change our results. However, with Phillips deuteron wave function, inclusion o f the 1P 1 N-N channel increases the value of a as seen in table 2 (ii, iv). By omitting various channels we find that this is mainly due to the large rescattering involving the 1P 1 and P33 channels.
Volume 45B, number 5
PHYSICS LETTERS
Table 2 The s-wave pion production cross section and elastic ~rd amplitude. All channels of table 1 are included except those in column five. #
PD (%)
~(#b)
and (fm)
Channels omitted
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)
0 4 7 4 6.58 6.58 6.58 6.58
679 640 607 330 122 220 222 . . . .
-0.029 + 0.019i -0.030+ 0.018i -0.032+ 0.017i -0.034 + 0.092i -0.029 + 0.003i -0.035 + 0.006i -0.036 + 0.006i 0.028
1A 1A 1A 1A, 2 1,2,4,6 1, 2 1 1, 2, 9
To study the importance of the short range N-N interaction in pion production, we have replaced the 3S 1- 3D l potential of Phillips by the UPA representation of the Reid soft core potential, which includes the effects of short range repulsion in the deuteron. Here we find that the large increase in ct due to the 1P 1 N-N interaction disappears (vi, vii), indicating the sensitivity of this process to high Fourier components in the deuteron wave function. Furthermore, when all relevant channels are included, particularly d-wave spectator nucleon with ¢rN pair in S l l and $31 (compare v, vi), our result for c~, of 220/ab, is in good agreement with experiment. Once again we observe qualitative similarities between this result and our earlier calculations [2], where separable potentials with no short range repulsion gave larger values ofc~ than the realistic potentials. Nevertheless the present calculation should be more reliable, since by solving the integral equations, we are summing all terms in the multiple scattering series in a unitary way. In fact, in the present model, the multiple scattering series itself is divergent. We feel it is significant that this calculation agrees with experiment when a realistic deuteron wave function is used, and it is with some confidence that we move to calculate the effect of absorption on the real part of the 7r-d scattering length (Re and ). In table 2 we also give the s-wave rr-d scattering amplitude at - 1 MeV (c.m. energy). We shall refer to this as the scattering length, with an error of a few per cent. Here we find that the multiple scattering series for a~ra converges (three or four terms), provided there is no coupling to the absorptive channel - otherwise the series is divergent. We can write the lrd scattering length as [ 1 ]
I+~/M~ a rd= ~ i ~ / a ~ l a l
20 August 1973 + ~ ~+ h /_a3) a d
(2)
where/a and M are the mass of the pion and nucleon respectively, while a 1 and a 3 are the s-wave lrN scattering lengths. The first term in eq. (2) is due to single scattering of the pion from the ~wo nucleons, while the second term is due to higher order multiple scattering and absorption. Although (at+ 2a3) is not well known experimentally, it is accepted to be small [13], (soft pion theory [17] gives a 1 + 2a 3 = 0).. The uncertainty in a I + 2a 3 arises mainly from the uncertainty in a 3 . We find that Reahd is negative, and relatively insensitive to a 3. For example, with the RSC deuteron, Reahd varies from --0.0455 fm to --0.0365 fm for a 3 in the range ( - 0 . 0 8 6 , - 0 . 0 5 ) / a - t . (For Phillips deuteron wave function the variation is - 0 . 0 4 0 fm to - 0 . 0 3 8 fm.) This is obviously a small effect compared with the change in the first term from +0.010 fm to +0.084 f m - where we have used [13] a 1 = 0 . 1 8 2 / I 1. With the latest value of a 3 [ 13], and the RSC deuteron, we find (vii): Rear d = -0.036 fm. The contribution to ahd from higher order multiple scattering was calculated by Kolybasov and Kudryatsev [4] to be negative ( - 0 . 0 3 7 fm), and in fair agreement with the present calculztion (vii) of - 0 . 0 3 8 fm. However, both the sign and magnitude of the contribution to React d from the absorptive channel is in question. We find (vi, viii) that with the RSC deuteron the contribution to the Reahd from absorption is - 0 . 0 0 7 fm. This is the same size as the contribution to the Imahd (viz. +0.006 fm), and in agreement with the intuitive prediction of Faldt [3] ( - 0 . 0 0 6 8 fm). However, Beder [5] has found a contribution to React d of +0.084 fm, from this process, using an unsubtracted dispersion relation. This is considerably larger than the value we get and opposite in sign. From the theoretical viewpoint we feel these results are encouraging. We have a good description of the rrNN system at low energy, and it should be suitable for extension to higher energies. In particular, one could test the energy dependence of the s-wave pion absorption coefficient c~. Furthermore, the demonstrated sensitivity of our results to the relatively unknown features of the N-N force, like PD, and the 439
Volume 45B, number 5
PHYSICS LETTERS
short range ( < 1 fm) behaviour, suggests that pion absorption should become an important tool in nuclear physics. On the experimental side, we realize the determination of the 7r-d scattering length is quite a difficult experiment. However, in view of our comments on ahd , its direct measurement should give us an alternative method to determine a 1 + 2a 3. We acknowledge many helpful discussions with Professor I.E. McCarthy.
References [1] D.S. Koltun, Advances in Nucl. Phys. 3 (1970) 71. [2] A.W. Thomas and I.R. Afnan, Phys. Rev. Lett. 26 (1971) 906. [3] G. Faldt, Nucl. Phys. B10 (1969) 597. [4] V.M. Kolybasov and A.E. Kudryatsev, NucL Phys. B41 (1972) 510.
440
20 August 1973
[5] D. Beder, Nucl. Phys. B14 (1969) 586. [6] L.D. Faddeev, Soviet. Phys. JETP 12 (1961) 1014. [7] R. Aaron, R.D. Amado andY.Y. Yam, Phys. Rev. 136 (1964) B650; P. Shanley, Phys. Rev. Lett. 21 (1968) 627. [8] C. Lovelace, Phys. Rev. 135 (1964) B1225. [9] A.C. Phillips, Nucl. Phys. A107 (1968) 209. [10] S.C. Bhatt, J.S. Levinger and E. Harms, Phys. Lett. B40 (1972) 23. [ 11 ] L.D. Roper, R.M. Wright and B.T. Feld, Phys. Rev. 138 (1965) B190. [12] B.V. Bugg, Spring School on Pion interaction at low and medium energy, CERN 71-14. [13] V.K. Samaranayake and W.S. Woolcock, Nucl. Phys. B48 (1972) 205. [14] I.R. Afnan and A.W. Thomas, Proc. Int. Conf. on Few particle problems in the nuclear interaction, Los Angeles, 1972 (North-Holland, 1973). [15] C. Ridlard-Serre et al., Nucl. Phys. B20 (1970) 413. [16] D.S. Koltun and A. Reitan, Phys. Rev. 141 (1965) 1413. [17] S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. [18] T.R. Mongan, Phys. Rev. 175 (1968) 1260. [19] J.S. Ball, R. Campbell, P. Lee and G. Shaw, Phys. Rev. Lett. 28 (1972)1143.