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Recoil and strong formfactor effects in mesonic exchange corrections on the charge distribution of 3He and 3H
12 April 1976
PHYSICS LETTERS
Volume 61B, number 4
RECOIL AND STRONG FORMFACTOR CORRECTIONS
EFFECTS IN MESONIC EXCHANGE
ON THE CHARGE DISTRIBUTION
OF 3 He AND 3 H*
‘W.M. KLOET’ Department of Physics and Astronomy,
University of Maryland, College Park, Maryland 20742,
USA
J.A. TJON Institute for Theoretical Physics, University of Utrecht, Utrecht, The Netherlands
Received 4 December 1975 Effects of meson exchange on the charge distribution of 3He and 3H are determined, including intermediate nucleon-antinucleon states, pion-nucleon resonances and recoil effects. The role of a strong interaction formfactor in these effects is discussed. The calculation is based on a tri-nucleon wavefunction obtained from the Reid soft core potential.
The electric and magnetic properties of a nucleus are conventionally described by the impulse approximation. As a result the electromagnetic operator is a sum of one-body operators, each of which is given by the interaction of a free nucleon with the electromagnetic field. On the other hand it is expected that in certain situations the mesonic degrees of freedom of the nucleus will show up in its electromagnetic properties. Some successful,calculations of these effects have been reported, in particular for the magnetic properties of 3He [l] , slow neutron capture by protons [2] and deuteron electrodesintegration [3]. In all these cases a restriction was made to one-pion exchange. In view of the short range correlations in the nuclear wavefunction, these contributions are the most important ones and effects of exchange of heavy mesons are expected to be much smaller. In recent years it has become evident that it is unlikely to obtain with realistic two-nucleon interactions simultaneous agreement for the binding energy and the charge formfactors of the tri-nucleon system [e.g. 41. In this letter we describe mesonic exchange effects on the charge distribution of 3He and 3H resulting in a considerable improvement of the charge formfactor of 3 He. The exchange corrections included here are shown in fig. 1. In diagram lb the photon interacts with one of the nucleons. Because part of this diagram is already present in the impulse approximation, here only the negative frequency part of the nucleon propagator must be included. In a time ordered picture this part corresponds to the N-N diagram [l-3]. Recently it was shown by Jackson et al. [5] that recoil and renormalization graphs contribute significantly in the deuteron case. The corresponding graphs in the tri-nucleon case are given by figs. le and If. The photon in this case interacts with one of the nucleons while the three-nucleon system is recoiling due to the emission of a pion. It should be noted that some discussion has appeared in the literature in how far these contributions should be included [6]. In an earlier paper [7] the contribution of diagrams la, lb and Id was discussed, but only results for a pure L=O wavefunction were shown. The contribution of each diagram to the charge formfactor is obtained by a method described in ref. [8]. The charge formfactor F&(k) is directly related to the time component of the electromagnetic current by Jo(k) = e F&(k)
.
* Work supported in part by U.S. Energy Research and Development Adm. ’ Present address: Los Alamos Scientific Laboratory, T-5, MS 454, P.O. Box 1663, Los Alamos, NM 87544, USA.
356
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12 April 1976
PHYSICS LETTERS
Volume 61B, number 4
Fig.
Diagrams considered for meson exchange effects on the charge fonnfactors.
For each diagram, starting from the appropriate Feynman amplitude for J,-,(k), the non-relativistic limit is obtained. This leads to a two-nucleon operator (a three nucleon operator for diagram 1f), which must be sandwiched between the non-relativistic three-nucleon wavefunctions and summed over all possible pairs. For the isobar diagram, instead of writing down the Feynman amplitude, a more general procedure was followed. The amplitude for electro pionproduction on nucleons may be used here. The Born part of this amplitude must be excluded since diagram lc only represents excited intermediate states. The relativistic amplitude, as given for instance by Adler [9], is treated in the magnetic dipole approximation, which relates it to p-wave elastic pion-nucleon scattering. Again here the nonrelativistic limit is taken. We include the channels I = l/2, S = l/2, and I= 3/2, S = 3/2. The contributions of the recoil diagrams le and If follow from time-dependent perturbation theory. These two diagrams give a nonzero contribution at k2 = 0, therefore a renormalization of the one-body formfactor must be performed. From now on we will use the term recoilcontribution to indicate the total effect of recoil and corresponding renormalization. The operators for the various exchange corrections are given below. The momenta involved are k for the photon, ki, k;, k; for the three initial nucleons, ki, ki, kk for the three fmal nucleons and we define p, = (ki- kj),, and q,, = (ki-- kj),,. u and r are spin and isospin matrrces respectively. The masses are the nucleon mass M (939 MeV), the pion mass p (139.6 MeV), and the p and w masses mP = 765 MeV and m, = 784 MeV.
g2 q-a(j) oN% _ 1 ii (2n)3 (2M)3 q2+/..?
k-6) [(F,+ G,)@)*r(j) + WV+G&3(j)l ,
n(F, + Gv) q. a(j) {hIi[z(i)X (27r)3 3M2 q2tp2
of*=- 1
r(j)13 [q*k(kj+kitq).U(i)-q*(k~tkitq)k*o(i)] (4)
-4h2T3(j)ik*[qX(k:tki)]t(4h273(j)-hli[r(i)Xr(j)]3)(q-k)‘(k:tki)k’a(i)~
0” ”
=
1
gpNNgpn?g
(27~)~ 4M2mp
oii -mu_
(2~)~
ORecl_ --
ij
1
4M2 m w - g2
(2793 8M2 (q2
,
1
-4’a(j)(j).s(j)[(ltK,){p’k.~(i)-p.kp.a(i)}tip.IkX(k:tki)]], p2tm2 q2tp2
(5)
P
guNNg,,g
1
(3)
1
p2tm$ 1
t p2p2
4’00 r3(j)[(1tKs){p2k*o(i) q2+P2
-p’kp*a(i))
[Fst(j)*t(j)tF,~3(j)14.a(j)q.a(j),
tip*[kX(kitkJ]]
,
(6)
(7)
357
Volume 61B, number 4
PHYSICS LETTERS
,,-b,, 0
4
12 April 1976
5
I 20
!
24
Fig. 2. Charge formfactors for 3He and 3H. Curves a are the one-body formfactor, curves c are the formfactor with exchange corrections including recoil. (A) Curve b does not contain recoil. Experimental points are from ref. [ 171. (B) Curve b is the formfactor including exchange corrections with recoil and a strong nNN formfactor (A = 3.5 cc). Experimental points are from ref. [ 171. (C) Curve b is the formfactor with exchange corrections including recoil and a strong nNN formfactor (A = 3.5 EC).The same form-
factor without recoil but with nNN formfactor practically coincides with curve b.
0@2 = 1 il
g2 __
1
(27r)3 16M2 (q2 + p2)3/2
[Fs+Fv73(k)lr(i).t(i)q.r(i)q.z(j) *
(8)
The couplingconstantsareg=g,+rN= 13.6,g NN=2.56,g,NN=6.82,g,,,=0.39,g,,,= 1.17,&= 3.7,K,=-0.12. The pion electromagnetic formfactor is F,(kf) = mz/( rnz + k2). The k2-dependence of the pry and orry couplings was chosen similar to the form of F,(k2), based on vectormeson dominance, Fv and Fs are the isovector and isoscalar charge formfactors of the nucleon, while G, and G, are its isovector and isoscalar magnetic formfactors. These were obtained from ref. [lo]. h, and h2 are the same combinations of pion-nucleon scattering amplitudes as defined by Chemtob and Rho [ 1, eq. (IV-27)]. The derivation of the above operators is straightforward from Feynman rules, except for the N* contribution. For the latter a detailed derivation of the operator will be given elsewhere. The nuclear wavefunction used was constructed from the Faddeev equations [l l] for the Reid soft core potential [ 121. The components of the wavefunctions taken into account are the totally symmetric S-state and a D-state with 1, = 2 and 1, = 0 [ 111. For this case the operators 0; and Oi* give matrix elements which vanish after integration over over the particle momenta. (We correct hereby the result for the N* contribution as quoted earlier [ 131.) The dominant contributions come from the N-N and recoil diagrams lb and le. In coordinate space the recoil operator probes the nuclear wavefunction at shorter distances than the N-R operator and is therefore strongly dependent on the type of wavefunction employed. Furthermore the S-S and S-D matrix elements for recoil have opposite signs and cancel each other partly. Therefore the recoil effect is strongly dependent on the D-state probability. For the Reid wavefunction, which was used here, the recoil term in 3He is positive for k2 < 17 fme2. The N-R contribution is always negative and falls off less rapidly than the recoil term. Introducing recoil causes the exchange 358
Volume 61B, number 4
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12 April 1976
effect in 3He to be reduced in the intermediate momentum region, but it has little effect on the second maximum. The 3He results are given in fig. 2(A) for the one-body formfactor and the formfactor including exchange corrections with and without recoil. For 3H the recoil contribution is positive for all values of k2 considered and larger than in 3He, while the N-N term is again negative but smaller than in 3He. The resulting formfactor for 3H is shown in fig. 2(B), curve c. We note here that the isoscalar part of the exchange corrections plays a very important role. One of the reasons is that for instance the S-D matrix elements for this case are all isoscalar. This is in contrast with the magnetic formfactor where exchange corrections are mainly isovector. In earlier work on exchange contribution the dependence of the strong nNN coupling constant on the momentum transfer has been ignored. Although this strong formfactor is not precisely known at present, we have used a particular parametrization for the nNN formfactor to study its effect on the exchange charge formfactor. This parametrization is
Fn&q2) = ( A2- p2p2/(h2+q2y2 ,
(9)
with h= 3.5~ obtained from ref. [14]. A strong formfactor tends to cut off large momentum transfer, or in coordinate space it will suppress the short range part of the two-body operators. Its effect therefore is the most significant for the shortest range operators, like those involved in recoil. The results are shown in figs. 2B and C. For 3He the total recoil is so much reduced by the strong formfactor that the final charge formfactor is practically the same with or without recoil. At large momentum transfer (we have considered k2 up to 25 fmM2) the total exchange contribution decreases for 3He as well as for 3H*. We have chosen here a rather low value for the cut-off! parameter X and some authors [ 161 advertise a larger value like X = 7 - 14~. Therefore our example of the effect of a strong formfactor must be viewed as a possible extreme. We have also considered the case where h = 7~ and the final result is between the values for h = 3.5~ and h==. Another widely used parametrization of the strong vertex is the dipole form. Using the dipole form in stead of eq. (9) in this calculation gives no drastic changes in the final results. As expected the dipole form suppresses the short range contribution slightly more than eq. (9) with the same value of X. For example a value of X = 5~ in the dipole form has the same effect on the pair term exchange contribution as h = 3 S/J in eq. (9) in the range of momentum transfer considered here. Comparing the curves from fig. 2 for the one-body formfactor and the corrected formfactor with and without a strong formfactor (X = 3 5 cc), with the experimental data [ 171 for 3 He, one sees that the position of the dip has improved, but the second maximum is still too low. The intermediate region around k2 = 8 fmm2 is slightly lower than experiment. Of course, we are dealing here with a wavefunction corresponding to a binding energy which is too low (& = 6.7 MeV against 8.5 MeV experimentally) and an increase in ET is likely to increase the formfactor at low momentum transfer and consequently improve the fit for 3He. However for the same reason in the case of 3H the disagreement with the datapoint at k2 = 8 fme2 would become worse. In view of this and the difference in behavior for 3H and 3He in the region of the dip and the second maximum, experimental data for the charge formfactor of 3H for 8 < k2< 20 fmv2 would be extremely interesting. * For very.large momentum transfer as in a recent experiment for the deuteron [ 151, the exchange contributions are expected to be suppressed substantially due to this strong formfactor. References [l] M. Chemtob,and M. Rho, Nucl. Phys. Al63 (1971) 1; E.P. Harper, Y.E. Kim, A. Tubis and M. Rho, Phys. Le%t. 40B (1972) 533; E. Hadjimichael and A. Barroso, Phys. Lett. 47B (1973) 103; Nucl. Phys. A238 (1975) 422. [2] D.O. Riska and G.E. Brown, Phys. Lett. 38B (1972) 193. 359
Volume 6lB, number 4 [3] [4] [S] [6]
[7] [8] [9] [lo] [ll] [12] (131 [14] [15] [ 161 [17]
360
PHYSICS LETTERS
12 April 1976
J. Hockert, D.O. Riska, M. Gari and A. Huffman, Nucl. Phys. A217 (1973) 14. Conf. on Few body problems in nuclear and particle physics, Quebec, August 1974. A.D. Jackson, A. Lande and D.O. Riska, Phys. Lett. 55B (1975) 23. R.H. Thompson and L. Heller, Phys. Rev. C7 (1973) 2355; J. Blomqvist, Phys. Lett. 32B (1970) 1; R.M. Woloshyn, Phys. Rev. Cl2 (1975) 901; J.L. Friar, preprint. W.M. Kloet and J.A. Tjon, Phys. Lett. 49B (1974) 419. W.M. Kloet and J.A. Tjon, Nucl. Phys. Al76 (1971) 481. S.L. Adler, Ann. Phys. 50 (1968) 189. F. lachello, A.D. Jackson and A. Lande, Phys. Lett. 43B (1973) 191. R.A. Malfliet and J.A. Tjon, Nucl. Phys. Al27 (1969) 161; Ann. Phys. 61 (1970) 425. R.V. Reid, Ann. Phys. 50 (1968) 411. W.M. Kloet and J.A. Tjon, Conf. on Few body problems in nuclear and particle physics, Quebec, August, 1974, p. 523. K. Bongardt, H. Pilkuhn and H.G. Schlaile, Phys. Rev. Lettt 52B (1974) 271. R.G. Arnold et al., Phys. Rev. Lett. 35 (1975) 776. K. Erkelenz, Phys. Repts. 13C (1974) 208. J.S. McCarthy et al., Phys. Rev. Lett. 25 (1970) 884; H. Collard et al., Phys. Rev. B57 (1965) 138; M. Bernheim et al., Lett. Nuovo Cim. 5 (1972) 431.
PHYSICS LETTERS
Volume 61B, number 4
RECOIL AND STRONG FORMFACTOR CORRECTIONS
EFFECTS IN MESONIC EXCHANGE
ON THE CHARGE DISTRIBUTION
OF 3 He AND 3 H*
‘W.M. KLOET’ Department of Physics and Astronomy,
University of Maryland, College Park, Maryland 20742,
USA
J.A. TJON Institute for Theoretical Physics, University of Utrecht, Utrecht, The Netherlands
Received 4 December 1975 Effects of meson exchange on the charge distribution of 3He and 3H are determined, including intermediate nucleon-antinucleon states, pion-nucleon resonances and recoil effects. The role of a strong interaction formfactor in these effects is discussed. The calculation is based on a tri-nucleon wavefunction obtained from the Reid soft core potential.
The electric and magnetic properties of a nucleus are conventionally described by the impulse approximation. As a result the electromagnetic operator is a sum of one-body operators, each of which is given by the interaction of a free nucleon with the electromagnetic field. On the other hand it is expected that in certain situations the mesonic degrees of freedom of the nucleus will show up in its electromagnetic properties. Some successful,calculations of these effects have been reported, in particular for the magnetic properties of 3He [l] , slow neutron capture by protons [2] and deuteron electrodesintegration [3]. In all these cases a restriction was made to one-pion exchange. In view of the short range correlations in the nuclear wavefunction, these contributions are the most important ones and effects of exchange of heavy mesons are expected to be much smaller. In recent years it has become evident that it is unlikely to obtain with realistic two-nucleon interactions simultaneous agreement for the binding energy and the charge formfactors of the tri-nucleon system [e.g. 41. In this letter we describe mesonic exchange effects on the charge distribution of 3He and 3H resulting in a considerable improvement of the charge formfactor of 3 He. The exchange corrections included here are shown in fig. 1. In diagram lb the photon interacts with one of the nucleons. Because part of this diagram is already present in the impulse approximation, here only the negative frequency part of the nucleon propagator must be included. In a time ordered picture this part corresponds to the N-N diagram [l-3]. Recently it was shown by Jackson et al. [5] that recoil and renormalization graphs contribute significantly in the deuteron case. The corresponding graphs in the tri-nucleon case are given by figs. le and If. The photon in this case interacts with one of the nucleons while the three-nucleon system is recoiling due to the emission of a pion. It should be noted that some discussion has appeared in the literature in how far these contributions should be included [6]. In an earlier paper [7] the contribution of diagrams la, lb and Id was discussed, but only results for a pure L=O wavefunction were shown. The contribution of each diagram to the charge formfactor is obtained by a method described in ref. [8]. The charge formfactor F&(k) is directly related to the time component of the electromagnetic current by Jo(k) = e F&(k)
.
* Work supported in part by U.S. Energy Research and Development Adm. ’ Present address: Los Alamos Scientific Laboratory, T-5, MS 454, P.O. Box 1663, Los Alamos, NM 87544, USA.
356
(1)
12 April 1976
PHYSICS LETTERS
Volume 61B, number 4
Fig.
Diagrams considered for meson exchange effects on the charge fonnfactors.
For each diagram, starting from the appropriate Feynman amplitude for J,-,(k), the non-relativistic limit is obtained. This leads to a two-nucleon operator (a three nucleon operator for diagram 1f), which must be sandwiched between the non-relativistic three-nucleon wavefunctions and summed over all possible pairs. For the isobar diagram, instead of writing down the Feynman amplitude, a more general procedure was followed. The amplitude for electro pionproduction on nucleons may be used here. The Born part of this amplitude must be excluded since diagram lc only represents excited intermediate states. The relativistic amplitude, as given for instance by Adler [9], is treated in the magnetic dipole approximation, which relates it to p-wave elastic pion-nucleon scattering. Again here the nonrelativistic limit is taken. We include the channels I = l/2, S = l/2, and I= 3/2, S = 3/2. The contributions of the recoil diagrams le and If follow from time-dependent perturbation theory. These two diagrams give a nonzero contribution at k2 = 0, therefore a renormalization of the one-body formfactor must be performed. From now on we will use the term recoilcontribution to indicate the total effect of recoil and corresponding renormalization. The operators for the various exchange corrections are given below. The momenta involved are k for the photon, ki, k;, k; for the three initial nucleons, ki, ki, kk for the three fmal nucleons and we define p, = (ki- kj),, and q,, = (ki-- kj),,. u and r are spin and isospin matrrces respectively. The masses are the nucleon mass M (939 MeV), the pion mass p (139.6 MeV), and the p and w masses mP = 765 MeV and m, = 784 MeV.
g2 q-a(j) oN% _ 1 ii (2n)3 (2M)3 q2+/..?
k-6) [(F,+ G,)@)*r(j) + WV+G&3(j)l ,
n(F, + Gv) q. a(j) {hIi[z(i)X (27r)3 3M2 q2tp2
of*=- 1
r(j)13 [q*k(kj+kitq).U(i)-q*(k~tkitq)k*o(i)] (4)
-4h2T3(j)ik*[qX(k:tki)]t(4h273(j)-hli[r(i)Xr(j)]3)(q-k)‘(k:tki)k’a(i)~
0” ”
=
1
gpNNgpn?g
(27~)~ 4M2mp
oii -mu_
(2~)~
ORecl_ --
ij
1
4M2 m w - g2
(2793 8M2 (q2
,
1
-4’a(j)(j).s(j)[(ltK,){p’k.~(i)-p.kp.a(i)}tip.IkX(k:tki)]], p2tm2 q2tp2
(5)
P
guNNg,,g
1
(3)
1
p2tm$ 1
t p2p2
4’00 r3(j)[(1tKs){p2k*o(i) q2+P2
-p’kp*a(i))
[Fst(j)*t(j)tF,~3(j)14.a(j)q.a(j),
tip*[kX(kitkJ]]
,
(6)
(7)
357
Volume 61B, number 4
PHYSICS LETTERS
,,-b,, 0
4
12 April 1976
5
I 20
!
24
Fig. 2. Charge formfactors for 3He and 3H. Curves a are the one-body formfactor, curves c are the formfactor with exchange corrections including recoil. (A) Curve b does not contain recoil. Experimental points are from ref. [ 171. (B) Curve b is the formfactor including exchange corrections with recoil and a strong nNN formfactor (A = 3.5 cc). Experimental points are from ref. [ 171. (C) Curve b is the formfactor with exchange corrections including recoil and a strong nNN formfactor (A = 3.5 EC).The same form-
factor without recoil but with nNN formfactor practically coincides with curve b.
0@2 = 1 il
g2 __
1
(27r)3 16M2 (q2 + p2)3/2
[Fs+Fv73(k)lr(i).t(i)q.r(i)q.z(j) *
(8)
The couplingconstantsareg=g,+rN= 13.6,g NN=2.56,g,NN=6.82,g,,,=0.39,g,,,= 1.17,&= 3.7,K,=-0.12. The pion electromagnetic formfactor is F,(kf) = mz/( rnz + k2). The k2-dependence of the pry and orry couplings was chosen similar to the form of F,(k2), based on vectormeson dominance, Fv and Fs are the isovector and isoscalar charge formfactors of the nucleon, while G, and G, are its isovector and isoscalar magnetic formfactors. These were obtained from ref. [lo]. h, and h2 are the same combinations of pion-nucleon scattering amplitudes as defined by Chemtob and Rho [ 1, eq. (IV-27)]. The derivation of the above operators is straightforward from Feynman rules, except for the N* contribution. For the latter a detailed derivation of the operator will be given elsewhere. The nuclear wavefunction used was constructed from the Faddeev equations [l l] for the Reid soft core potential [ 121. The components of the wavefunctions taken into account are the totally symmetric S-state and a D-state with 1, = 2 and 1, = 0 [ 111. For this case the operators 0; and Oi* give matrix elements which vanish after integration over over the particle momenta. (We correct hereby the result for the N* contribution as quoted earlier [ 131.) The dominant contributions come from the N-N and recoil diagrams lb and le. In coordinate space the recoil operator probes the nuclear wavefunction at shorter distances than the N-R operator and is therefore strongly dependent on the type of wavefunction employed. Furthermore the S-S and S-D matrix elements for recoil have opposite signs and cancel each other partly. Therefore the recoil effect is strongly dependent on the D-state probability. For the Reid wavefunction, which was used here, the recoil term in 3He is positive for k2 < 17 fme2. The N-R contribution is always negative and falls off less rapidly than the recoil term. Introducing recoil causes the exchange 358
Volume 61B, number 4
PHYSICS LETTERS
12 April 1976
effect in 3He to be reduced in the intermediate momentum region, but it has little effect on the second maximum. The 3He results are given in fig. 2(A) for the one-body formfactor and the formfactor including exchange corrections with and without recoil. For 3H the recoil contribution is positive for all values of k2 considered and larger than in 3He, while the N-N term is again negative but smaller than in 3He. The resulting formfactor for 3H is shown in fig. 2(B), curve c. We note here that the isoscalar part of the exchange corrections plays a very important role. One of the reasons is that for instance the S-D matrix elements for this case are all isoscalar. This is in contrast with the magnetic formfactor where exchange corrections are mainly isovector. In earlier work on exchange contribution the dependence of the strong nNN coupling constant on the momentum transfer has been ignored. Although this strong formfactor is not precisely known at present, we have used a particular parametrization for the nNN formfactor to study its effect on the exchange charge formfactor. This parametrization is
Fn&q2) = ( A2- p2p2/(h2+q2y2 ,
(9)
with h= 3.5~ obtained from ref. [14]. A strong formfactor tends to cut off large momentum transfer, or in coordinate space it will suppress the short range part of the two-body operators. Its effect therefore is the most significant for the shortest range operators, like those involved in recoil. The results are shown in figs. 2B and C. For 3He the total recoil is so much reduced by the strong formfactor that the final charge formfactor is practically the same with or without recoil. At large momentum transfer (we have considered k2 up to 25 fmM2) the total exchange contribution decreases for 3He as well as for 3H*. We have chosen here a rather low value for the cut-off! parameter X and some authors [ 161 advertise a larger value like X = 7 - 14~. Therefore our example of the effect of a strong formfactor must be viewed as a possible extreme. We have also considered the case where h = 7~ and the final result is between the values for h = 3.5~ and h==. Another widely used parametrization of the strong vertex is the dipole form. Using the dipole form in stead of eq. (9) in this calculation gives no drastic changes in the final results. As expected the dipole form suppresses the short range contribution slightly more than eq. (9) with the same value of X. For example a value of X = 5~ in the dipole form has the same effect on the pair term exchange contribution as h = 3 S/J in eq. (9) in the range of momentum transfer considered here. Comparing the curves from fig. 2 for the one-body formfactor and the corrected formfactor with and without a strong formfactor (X = 3 5 cc), with the experimental data [ 171 for 3 He, one sees that the position of the dip has improved, but the second maximum is still too low. The intermediate region around k2 = 8 fmm2 is slightly lower than experiment. Of course, we are dealing here with a wavefunction corresponding to a binding energy which is too low (& = 6.7 MeV against 8.5 MeV experimentally) and an increase in ET is likely to increase the formfactor at low momentum transfer and consequently improve the fit for 3He. However for the same reason in the case of 3H the disagreement with the datapoint at k2 = 8 fme2 would become worse. In view of this and the difference in behavior for 3H and 3He in the region of the dip and the second maximum, experimental data for the charge formfactor of 3H for 8 < k2< 20 fmv2 would be extremely interesting. * For very.large momentum transfer as in a recent experiment for the deuteron [ 151, the exchange contributions are expected to be suppressed substantially due to this strong formfactor. References [l] M. Chemtob,and M. Rho, Nucl. Phys. Al63 (1971) 1; E.P. Harper, Y.E. Kim, A. Tubis and M. Rho, Phys. Le%t. 40B (1972) 533; E. Hadjimichael and A. Barroso, Phys. Lett. 47B (1973) 103; Nucl. Phys. A238 (1975) 422. [2] D.O. Riska and G.E. Brown, Phys. Lett. 38B (1972) 193. 359
Volume 6lB, number 4 [3] [4] [S] [6]
[7] [8] [9] [lo] [ll] [12] (131 [14] [15] [ 161 [17]
360
PHYSICS LETTERS
12 April 1976
J. Hockert, D.O. Riska, M. Gari and A. Huffman, Nucl. Phys. A217 (1973) 14. Conf. on Few body problems in nuclear and particle physics, Quebec, August 1974. A.D. Jackson, A. Lande and D.O. Riska, Phys. Lett. 55B (1975) 23. R.H. Thompson and L. Heller, Phys. Rev. C7 (1973) 2355; J. Blomqvist, Phys. Lett. 32B (1970) 1; R.M. Woloshyn, Phys. Rev. Cl2 (1975) 901; J.L. Friar, preprint. W.M. Kloet and J.A. Tjon, Phys. Lett. 49B (1974) 419. W.M. Kloet and J.A. Tjon, Nucl. Phys. Al76 (1971) 481. S.L. Adler, Ann. Phys. 50 (1968) 189. F. lachello, A.D. Jackson and A. Lande, Phys. Lett. 43B (1973) 191. R.A. Malfliet and J.A. Tjon, Nucl. Phys. Al27 (1969) 161; Ann. Phys. 61 (1970) 425. R.V. Reid, Ann. Phys. 50 (1968) 411. W.M. Kloet and J.A. Tjon, Conf. on Few body problems in nuclear and particle physics, Quebec, August, 1974, p. 523. K. Bongardt, H. Pilkuhn and H.G. Schlaile, Phys. Rev. Lettt 52B (1974) 271. R.G. Arnold et al., Phys. Rev. Lett. 35 (1975) 776. K. Erkelenz, Phys. Repts. 13C (1974) 208. J.S. McCarthy et al., Phys. Rev. Lett. 25 (1970) 884; H. Collard et al., Phys. Rev. B57 (1965) 138; M. Bernheim et al., Lett. Nuovo Cim. 5 (1972) 431.