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Isovector magnetic structure of the two- and three-nucleon systems
Volume 244, number 3,4
PHYSICS LETTERS B
26 July 1990
Isovector magnetic structure of the two- and three-nucleon systems R. S c h i a v i l l a Continuous Electron Beam Accelerator Facility, Newport News, VA 23606, USA
and D.O. Riska Department of Physics, University of Helsinki, SF-O0170 Helsinki, Finland
Received 9 March 1990; revised manuscript received 8 May 1990
The backward cross section for electrodisintegration of 2H near threshold and the magnetic form factors of 3H and 3He are found to be in fair agreement with empirical values up to momentum transfers of about l GeV/c, when calculated with the Argonne two-nucleon and Urbana model-VII three-nucleon interactions, exact (A = 2) and variational or 34-channel Faddeev (A---3) wave functions, and a realistic current operator constructed from the Argonne v~4potential model. Below 1 GeV/c the exchange current contribution increases and above 1 GeV/c it strongly decreases the calculated electrodisintegration cross section. Second zeroes are predicted for the magnetic form factors of 3H and 3He at about 1.4 GeV/c.
The b a c k w a r d electrodisintegration cross section for 2H and the magnetic form factors o f 3He and 3H are strongly influenced by the contribution o f isovector exchange current operators [ 1-4 ]. Although the conventional m e t h o d o f constructing these exchange current operators by using phenomenological meson exchange models has led to qualitatively successful predictions o f the empirical values, it is nevertheless unsatisfactory because it does not yield a current operator that satisfies the continuity equation with the potential used to construct the wave functions. To o v e r c o m e this p r o b l e m the exchange current has to be separated into a " m o d e l independent" term, which should be constructed from the potential model so as to satisfy the continuity equation, and a remaining " m o d e l d e p e n d e n t " transverse term that has to be constructed by using meson exchange phenomenology [41. We shall here use the Argonne two-nucleon interaction [ 5 ] to construct the " m o d e l independent" part of the exchange current o p e r a t o r using the m e t h o d s developed in refs. [ 6 - 8 ]. The deuteron and np scattering wave functions are o b t a i n e d by solving the SchriSdinger equation with the same interaction
model, whereas the trinucleon ground states are described by variational wave functions o b t a i n e d for a hamiltonian containing the Argonne two-nucleon and U r b a n a model-VII three nucleon ( U r b a n a - V I I T N I ) interactions [9]. W i t h these wave functions and using a current operator that has a two-body component which satisfies the continuity equation with the Argonne 1314interaction, we have calculated the backward cross section for electrodisintegration o f the deuteron near threshold and the magnetic form factors o f the b o u n d trinucleons. The p r e d i c t e d values for these observables are in fair agreement with the experimental data [ 10-15 ] up to m o m e n t u m transfers o f about 1 G e V / c , with exception o f the region a r o u n d 4 fm -~, where the d a t a are underpredicted. However, it should be p o i n t e d out that until more definite empirical information on the electromagnetic form factors o f the nucleon becomes available, accurate predictions for these (and, in general, electron scattering) observables will not be possible above Q = 1 G e V / c , as the "realistic" semiempirical p a r a m etrizations o f the nucleon form factors differ by large factors in this Q-region. In figs. 1 and 2 we c o m p a r e the calculated cross
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
37 3
Volume 244, number 3,4
''''1
PHYSICS LETTERS B
....
26 July 1990
I .... I .... d (e,e')pn
' A
10-8
I ....
I ....
I ....
d (e,e')pn
i0 -8
>
E
1040
0
ul tfl o tj
, /IA ~~,.... ] ~ ".'
10-12
~E 161° O
~
U
GK
IA+MEC 10-12 b
O3 O3 >
IA+M'EC "' { ~ . " ..... 10-14 -mode[ independent "~ ....
I .... I .... 20 40 Qz (fm-2)
I,l,,
60
80
Fig. 1. Cross section for backward electrodisintegration of 2H near threshold, as a function of the four-momentum transfer, obtained with the IJL [ 16 ] parametrization of the electromagnetic form factors of the nucleon. The results calculated in impulse approximation (IA) are displayed along with those obtained with inclusion of the "model independent" (IA+MEC model independent) and both "model independent" and "model dependent" exchange current contributions. section for backward electrodisintegration o f the deuteron with the empirical values [ 10-12 ]. Both the d a t a and the theoretical results have been averaged over 0 - 3 and 0 - 1 0 MeV internal energy intervals o f the recoiling np pair for the Saclay (Q~< 1 G e V / c , refs. [ 10,11 ] ) a n d SLAC ( Q > 1 G e V / c , ref. [ 12] ) kinematics, respectively. In fig. 1 we display the calculated cross section as o b t a i n e d in impulse approximation, with inclusion o f the " m o d e l i n d e p e n d e n t " exchange currents d e r i v e d from the Argonne v~4 (it should be stressed here that this part o f the exchange current o p e r a t o r contains no free p a r a m e t e r s ) , a n d with in a d d i t i o n the " m o d e l d e p e n d e n t " exchange currents associated with n and p exchange excitation o f an i n t e r m e d i a t e A33 resonance, and the pn7 a n d tony mechanisms. The role o f the exchange current contribution, which at low values o f m o m e n t u m transfer is to increase the calculated cross section, is reversed above l G e V / c , where it serves to reduce the calculated cross section towards the empirical values. In the calculation we have included all partial 374
_
10-I~ ,,,I
....
20
I ....
40 Q2(fm-2 )
I ....
60
80
Fig. 2. Cross section for backward electrodisintegration of 2H near threshold, as a function of the four-momentum transfer, obtained with different parametrizations of the nucleon electromagnetic form factors: D (dipole), GK [17], H [18] and IJL [16]. The theoretical results include the exchange current contributions. waves o f the final np scattering state with full account o f the interaction effects in the relative S-, P- and Dwaves. We have explicitly verified that the numerical i m p o r t a n c e o f the final state interaction in the higher partial waves is negligible [19]. The results displayed in fig. 2 have been o b t a i n e d by using four different p a r a m e t r i z a t i o n s for the electromagnetic form factors o f the nucleon: G K [17], H [18], IJL [16] and finally a dipole form ( D ) , which includes a nonzero neutron electric form factor [20 ]. The exchange current o p e r a t o r used in the present calculations is constructed from the Argonne u~4 interaction with the methods described in ref. [ 8 ], with the exception o f the component that is associated with the s p i n - o r b i t interaction. To construct this part o f the exchange current operator, we assume that the isospin i n d e p e n d e n t central and s p i n - o r b i t components o f the Argonne potential are due to exchanges o f spin zero and spin one bosons (generalized scalar and vector mesons, respectively). We then construct the generalized scalar and vector meson exchange currents by replacing, in the known expressions for the o and to exchange currents, the bare meson prop-
Volume 244, number 3,4
PHYSICS LETTERS B
agators with the corresponding generalized scalar and vector components projected out o f the central and spin-orbit interactions of the Vl4. In the case of the isospin dependent interaction, it is not possible to separate the central and spin-orbit interactions into terms associated with scalar- and vector-like exchanges in a unique way, because of the large isovector tensor coupling to the nucleon. We therefore make in this case the simple assumption that only the exchange current associated with vector-like exchanges (generalized p-meson exchange) is important. The explicit expressions for these exchange current operators are derived in detail in ref. [ 7 ]. Here, we stress that they satisfy the continuity equation with the phenomenological central and spin-orbit interactions, when the single nucleon charge operator includes the relativistic corrections of order M -2 [ 7,21 ]. The "model dependent" part of the exchange current operator includes the n- and p-exchange A33 excitation currents and the pn'/ and (0n? exchange current operators. These have been constructed as in ref. [ 8 ], with the exception that we use the empirical ~,NA transition form factor [22] in the expressions for the n - and p-A33 currents instead of the isovector magnetic form factor of the nucleon. The present calculation is similar in spirit to that in ref. [23], which was based on the parametrized Paris potential [ 24 ]. However, the present results are closer to the empirical values due to the stronger isospin dependent pseudoscalar (n-like) and vector (plike) tensor components of the Argonne u~4 interaction [ 8 ]. Because of the much discussed ambiguity concerning the proper choice o f form factor to be associated with the "model independent" isovector exchange current operator, we emphasize that we (as in ref. [ 8 ] ) use the form factor G v (q2). The sensitivity to the nucleon form factor parametrization in the results in fig. 2 appears to be somewhat smaller than what found in ref. [25 ] in a calculation based on the Paris potential [24], which used the G K [ 17] and dipole form factor parametrizations. The difference is due to the absence of a neutron electric form factor in ref. [25 ]. In figs. 3 and 4 we show the magnetic form factors of 3H and 3He calculated with the same exchange current operator used above and variational wave functions that correspond to the Argonne two-nucleon interaction and Urbana-VII TN1 [9], along
26 July 1990
100
'I ....
I ....
I ....
I ....
3H
10-1 i X IA(FAD) ~X/~IA(VAR)
10-z
"
[}' ~i. "~,,.IA*MEC (VAR)-
I0 -3
i0 -~ -IA*MEC(FAD)YV
~, .. . , '.....'?.;<-- .(,,
I~'~'...... X.
10.5
iyi \ -
10-60
' ......
i
2
4 6 Q(fm-1)
8
10
Fig. 3. Magnetic form factor of 3H, as a function of the four-momentum transfer, obtained with variational (VAR) and 34channel Faddeev (FAD) wave functions. The results obtained without and with inclusion of the exchangecurrent contributions are displayed as IA and IA+MEC, respectively. The H parametrization of the nucleon electromagneticform factors [ 18] is used.
10-1100 ~ ~ I I l, .... [:
\~
I0-2F
.IA (FAD)
10 -z
t
'
I
....
....
~i)~~'k'..IA.MEC(FAD}~ ~,~A.MEC(VARL~
~" 10_31
10-60
1'
~.... I ....
L ,
2
~ ,
,il
f..Z~.~?:: x
I
,
,
I
L, 6 Q (fm 4)
,
,L
,
I ....
8
I
10
Fig. 4. Same as in fig. 3, but of 3He. 375
Volume 244, number 3,4
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with the experimental data. The quality of these wave functions has previously been assessed by quantitatively successful predictions of the binding energies and asymptotic D / S ratios in the d + n and d + p breakup channels of 3H and 3He [9], as well as by direct comparison with results obtained with exact Faddeev [26] and Green's function Monte Carlo [27] wave functions as, for example, those for the two-body correlation functions [28,29 ] and the longitudinal energy-weighted sum rule [ 30 ]. In order to further test their accuracy, we have carried out the calculation o f the magnetic form factors with the 34channel Faddeev wave functions obtained by the Los Alamos group [26,31 ] for the Argonne two-nucleon interaction and a version of the Urbana-VII TNI, in which the strengths of its repulsive and attractive parts are modified so as to reproduce the experimental 3H binding energy. The Monte Carlo method discussed in ref. [8] has been used in both variational and Faddeev calculations. The calculated form factors are in good agreement with the experimental ones over the measured range of m o m e n t u m transfer values, although the uncertainty in the electromagnetic form factors of the nucleon leads to a sizeable uncertainty margin in the predictions (figs. 5 and 6). The form factors calculated with the variational wave functions differ from those of ref. [ 8 ] in that the part of the exchange current operator that is associated with the spin-orbit (and central ) interactions has been constructed so as to satisfy the continuity equation with the relativistic correction in the single nucleon charge operator [7,21 ]. Furthermore, a numerical error in the exchange current contribution due to the excitation o f the A33 resonance, which led to an overestimate of that contribution in ref. [ 8 ], has been corrected. The most prominent features in the calculated magnetic form factors are the predicted zeroes in both form factors around 7 f m - i. For reference, we also list the magnetic m o m e n t values obtained with the Faddeev (variational) wave functions for 3He and 3H, respectively: - 1 . 7 9 3 (-1.783) n.m. and + 2 . 5 9 8 ( + 2 . 5 9 5 ) n.m. in impulse approximation; - 2 . 2 4 1 ( - 2.179) n.m. and + 3.083 ( + 3.071 ) n.m. when the exchange current contribution is included. The latter should be compared with the experimental values - 2 . 1 2 7 n.m. (3He) and + 2 . 9 7 9 n.m. (3H). The present calculations show that a fair descrip376
26 July 1990 10 0
~
1
...
I ....
I ....
I ....
3H
10 "1
10-2
_ _ 10 -~ O ms 10-4
10"5
oj ,,,I,,,,[,
10 4 0
2
,I . . . . 4
6 Q ( f m "1 )
I .... 8
10
Fig. 5. Magnetic form factor of 3H obtained with the variational wave function and inclusion of the exchange current contributions, but using different parametrizations of the nucleon electromagnetic form factors: D (dipole), GK [ 17 ], H [ 18 ] and IJL [16]. 10 o ......
r ....
P.... I'" 3 He
l
10-~
10-=
//GK
10-3
,.=
IJL ~
A
O
~H
10-4
10 5
10 "s
i o
iiJ]llJa
2
,jL,,,,I 4 6 Q ( f m "1 )
....
8
10
Fig. 6. Same as in fig. 5, but of 3He. tion of the three observables considered is obtained for Q~< 1 G e V / c by using the Argonne/)14 interaction to construct, in a consistent way, the exchange current operator and the wave functions o f the two- and
Volume 244, number 3,4
PHYSICS LETTERS B
three-nucleon systems. For Q > 1 G e V / c the prediction of the electrodisintegration cross section is only in good qualitative agreement with the experimental data. It is also evident that the calculated observables are sensitive to the potential models used. Indeed, both the deuteron electrodisintegration cross section [23] and the magnetic form factors of the trinucleons [ 32 ] are clearly underpredicted in comparison with the data at intermediate values of m o m e n t u m transfer, when calculated with the parametrized Paris potential [ 24 ]. The predicted values of the electrodisintegration cross section at large m o m e n t u m transfer could be improved if the strength of the model i n d e p e n d e n t isovector exchange current or, equivalently, the strengths of the isospin dependent pseudoscalar and vector exchange tensor interactions at short range were increased. Above 1 G e V / c one should also in principle include the contributions from exchange current mechanisms that involve excitation of higher n N resonances than the A33. The relative importance of such mechanisms and of explicit relativistic corrections have not so far been systematically explored. Inclusion of explicit resonance configurations in the wave functions may improve the predictions at intermediate values of m o m e n t u m transfer [ 33 ]. Finally, it has to be emphasized that the large uncertainty in the behaviour of the nucleon electromagnetic form factors prevents definitive quantitative predictions above 1 G e V / c [25]. Reducing this uncertainty appears to us as a task of the highest priority. We wish to thank C.R. Chen, J.L. Friar, B.F. Gibson and G.L. Payne for allowing us to use the 3H 34channel Faddeev wave function, V.R. Pandharipande and R.B. Wiringa for interesting conversations. We gratefully acknowledge the support of the US Departm e n t of Energy through the C o n t i n u o u s Electron Beam Accelerator Facility. The calculations were made possible by grants of time on the Cray supercomputer of the National Magnetic Fusion Energy C o m p u t e r Center of the US D e p a r t m e n t of Energy.
26 July 1990
References [ 1] J. Hockert, D.O. Riska, M. Gari and A. Huffman, Nucl. Phys. A 217(1973) 19. 12] A. Barroso and E. Hadjimichael, Nucl. Phys. A 238(1975) 422. [ 3 ] J.F. Mathiot, Phys. Rep. 173 ( 1989) 63. [4] D.O. Riska, Phys. Rep. 181 (1989) 207. [ 5 ] R.B. Wiringa, R.A. Smith and T.A. Ainsworth, Phys. Rev. C29 (1984) 1207. [6] D.O. Riska, Phys. Scr. 31 (1985) 471. [7] J. Carlson, D.O. Riska, R. Schiavilla and R.B. Wiringa, Radiative neutron capture on 3He, preprint CEBAF (1990). [8] R. Schiavilla, V.R. Pandharipande and D.O. Riska, Phys. Rev. C 40 (1989) 2294. [9 ] R. Schiavilla,V.R. Pandharipande and R.B. Wiringa, Nucl. Phys. A 449 (1986) 219. [ 10] M. Bernheim et al., Phys. Rev. Lett. 46 ( 1981 ) 402. [ 11 ] S. Auffret et al., Phys. Rev. Lett. 55 ( 1985 ) 1362. [ 12 ] R.G. Arnold et al., Transverse electrodisintegration of the deuteron in the threshold region at high Q2, preprint SLACPUB-4918 (1989). [ 13] J.M. Cavedon et al., Phys. Rev. Lett. 49 ( 1982) 986. [ 14] P.C. Dunn et al., Phys. Rev. C 27 (1983) 71. [ 15 ] J.P. Juster et al., Phys. Rev. Lett. 55 ( 1985) 2261. [16] F. Iachello, A.D. Jackson and A. Lande, Phys. Len. B 43 (1973) 191. [ 17 ] M. Gari and M. Kriimpelmann,Phys. Lett. B 173 (1986) 10. [ 18] G. H6hler et al., Nucl. Phys. B 114 ( 1976) 505. [ 19] R. Schiavillaand D.O. Riska, in preparation (1990). [20] C. Ciofi degli Atti, Prog. Pan. Nucl. Phys. 3 (1980) 163. [21 ] P.G. Blunden, Nucl. Phys. A 464 (1987) 525. [22] C.E. Carlson, Phys. Rev. D 34 (1986) 2704. [23 ] A. Buchmann,W. Leidemannand H. Arenhrvel, Nucl. Phys. A443 (1985) 726. [24] M. Lacombeet al., Phys. Rev. C 21 (1980) 861. [25] S.K. Singh, W. Leidemann and H. Arenhrvel, Z. Phys. A 331 (1988) 509. [26] C.R. Chen, G.L. Payne, J.L. Friar and B.F. Gibson, Phys. Rev. C 33 (1986) 1740. [27 ] J. Carlson, Phys. Rev. D 36 ( 1987) 2026. [28 ] R. Schiavillaet al., Nucl. Phys. A 473 ( 1987 ) 267. [29 ] J. Carlson, Workshop on Electron-nucleus scattering, eds. A. Fabrocini, S. Fantoni, S. Rosati and M. Viviani (World Scientific,Singapore, 1989). [ 30 ] R. Schiavilla,A. Fabrocini and V.R. Pandharipande, Nucl. Phys. A 473 (1987) 290. [ 31 ] J. Friar, B.F. Gibson and J.L, Payne, private communication (1990); R.B. Wiringa, private communication (1990). [ 32 ] W. Strueve, C. Hajduk, P.U. Sauerand W. Theis,Nucl. Phys. A465 (1987) 651. [33] R. Dymarz and F.C. Khanna, Phys. Rev. C 41 (1990) 828.
377
PHYSICS LETTERS B
26 July 1990
Isovector magnetic structure of the two- and three-nucleon systems R. S c h i a v i l l a Continuous Electron Beam Accelerator Facility, Newport News, VA 23606, USA
and D.O. Riska Department of Physics, University of Helsinki, SF-O0170 Helsinki, Finland
Received 9 March 1990; revised manuscript received 8 May 1990
The backward cross section for electrodisintegration of 2H near threshold and the magnetic form factors of 3H and 3He are found to be in fair agreement with empirical values up to momentum transfers of about l GeV/c, when calculated with the Argonne two-nucleon and Urbana model-VII three-nucleon interactions, exact (A = 2) and variational or 34-channel Faddeev (A---3) wave functions, and a realistic current operator constructed from the Argonne v~4potential model. Below 1 GeV/c the exchange current contribution increases and above 1 GeV/c it strongly decreases the calculated electrodisintegration cross section. Second zeroes are predicted for the magnetic form factors of 3H and 3He at about 1.4 GeV/c.
The b a c k w a r d electrodisintegration cross section for 2H and the magnetic form factors o f 3He and 3H are strongly influenced by the contribution o f isovector exchange current operators [ 1-4 ]. Although the conventional m e t h o d o f constructing these exchange current operators by using phenomenological meson exchange models has led to qualitatively successful predictions o f the empirical values, it is nevertheless unsatisfactory because it does not yield a current operator that satisfies the continuity equation with the potential used to construct the wave functions. To o v e r c o m e this p r o b l e m the exchange current has to be separated into a " m o d e l independent" term, which should be constructed from the potential model so as to satisfy the continuity equation, and a remaining " m o d e l d e p e n d e n t " transverse term that has to be constructed by using meson exchange phenomenology [41. We shall here use the Argonne two-nucleon interaction [ 5 ] to construct the " m o d e l independent" part of the exchange current o p e r a t o r using the m e t h o d s developed in refs. [ 6 - 8 ]. The deuteron and np scattering wave functions are o b t a i n e d by solving the SchriSdinger equation with the same interaction
model, whereas the trinucleon ground states are described by variational wave functions o b t a i n e d for a hamiltonian containing the Argonne two-nucleon and U r b a n a model-VII three nucleon ( U r b a n a - V I I T N I ) interactions [9]. W i t h these wave functions and using a current operator that has a two-body component which satisfies the continuity equation with the Argonne 1314interaction, we have calculated the backward cross section for electrodisintegration o f the deuteron near threshold and the magnetic form factors o f the b o u n d trinucleons. The p r e d i c t e d values for these observables are in fair agreement with the experimental data [ 10-15 ] up to m o m e n t u m transfers o f about 1 G e V / c , with exception o f the region a r o u n d 4 fm -~, where the d a t a are underpredicted. However, it should be p o i n t e d out that until more definite empirical information on the electromagnetic form factors o f the nucleon becomes available, accurate predictions for these (and, in general, electron scattering) observables will not be possible above Q = 1 G e V / c , as the "realistic" semiempirical p a r a m etrizations o f the nucleon form factors differ by large factors in this Q-region. In figs. 1 and 2 we c o m p a r e the calculated cross
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
37 3
Volume 244, number 3,4
''''1
PHYSICS LETTERS B
....
26 July 1990
I .... I .... d (e,e')pn
' A
10-8
I ....
I ....
I ....
d (e,e')pn
i0 -8
>
E
1040
0
ul tfl o tj
, /IA ~~,.... ] ~ ".'
10-12
~E 161° O
~
U
GK
IA+MEC 10-12 b
O3 O3 >
IA+M'EC "' { ~ . " ..... 10-14 -mode[ independent "~ ....
I .... I .... 20 40 Qz (fm-2)
I,l,,
60
80
Fig. 1. Cross section for backward electrodisintegration of 2H near threshold, as a function of the four-momentum transfer, obtained with the IJL [ 16 ] parametrization of the electromagnetic form factors of the nucleon. The results calculated in impulse approximation (IA) are displayed along with those obtained with inclusion of the "model independent" (IA+MEC model independent) and both "model independent" and "model dependent" exchange current contributions. section for backward electrodisintegration o f the deuteron with the empirical values [ 10-12 ]. Both the d a t a and the theoretical results have been averaged over 0 - 3 and 0 - 1 0 MeV internal energy intervals o f the recoiling np pair for the Saclay (Q~< 1 G e V / c , refs. [ 10,11 ] ) a n d SLAC ( Q > 1 G e V / c , ref. [ 12] ) kinematics, respectively. In fig. 1 we display the calculated cross section as o b t a i n e d in impulse approximation, with inclusion o f the " m o d e l i n d e p e n d e n t " exchange currents d e r i v e d from the Argonne v~4 (it should be stressed here that this part o f the exchange current o p e r a t o r contains no free p a r a m e t e r s ) , a n d with in a d d i t i o n the " m o d e l d e p e n d e n t " exchange currents associated with n and p exchange excitation o f an i n t e r m e d i a t e A33 resonance, and the pn7 a n d tony mechanisms. The role o f the exchange current contribution, which at low values o f m o m e n t u m transfer is to increase the calculated cross section, is reversed above l G e V / c , where it serves to reduce the calculated cross section towards the empirical values. In the calculation we have included all partial 374
_
10-I~ ,,,I
....
20
I ....
40 Q2(fm-2 )
I ....
60
80
Fig. 2. Cross section for backward electrodisintegration of 2H near threshold, as a function of the four-momentum transfer, obtained with different parametrizations of the nucleon electromagnetic form factors: D (dipole), GK [17], H [18] and IJL [16]. The theoretical results include the exchange current contributions. waves o f the final np scattering state with full account o f the interaction effects in the relative S-, P- and Dwaves. We have explicitly verified that the numerical i m p o r t a n c e o f the final state interaction in the higher partial waves is negligible [19]. The results displayed in fig. 2 have been o b t a i n e d by using four different p a r a m e t r i z a t i o n s for the electromagnetic form factors o f the nucleon: G K [17], H [18], IJL [16] and finally a dipole form ( D ) , which includes a nonzero neutron electric form factor [20 ]. The exchange current o p e r a t o r used in the present calculations is constructed from the Argonne u~4 interaction with the methods described in ref. [ 8 ], with the exception o f the component that is associated with the s p i n - o r b i t interaction. To construct this part o f the exchange current operator, we assume that the isospin i n d e p e n d e n t central and s p i n - o r b i t components o f the Argonne potential are due to exchanges o f spin zero and spin one bosons (generalized scalar and vector mesons, respectively). We then construct the generalized scalar and vector meson exchange currents by replacing, in the known expressions for the o and to exchange currents, the bare meson prop-
Volume 244, number 3,4
PHYSICS LETTERS B
agators with the corresponding generalized scalar and vector components projected out o f the central and spin-orbit interactions of the Vl4. In the case of the isospin dependent interaction, it is not possible to separate the central and spin-orbit interactions into terms associated with scalar- and vector-like exchanges in a unique way, because of the large isovector tensor coupling to the nucleon. We therefore make in this case the simple assumption that only the exchange current associated with vector-like exchanges (generalized p-meson exchange) is important. The explicit expressions for these exchange current operators are derived in detail in ref. [ 7 ]. Here, we stress that they satisfy the continuity equation with the phenomenological central and spin-orbit interactions, when the single nucleon charge operator includes the relativistic corrections of order M -2 [ 7,21 ]. The "model dependent" part of the exchange current operator includes the n- and p-exchange A33 excitation currents and the pn'/ and (0n? exchange current operators. These have been constructed as in ref. [ 8 ], with the exception that we use the empirical ~,NA transition form factor [22] in the expressions for the n - and p-A33 currents instead of the isovector magnetic form factor of the nucleon. The present calculation is similar in spirit to that in ref. [23], which was based on the parametrized Paris potential [ 24 ]. However, the present results are closer to the empirical values due to the stronger isospin dependent pseudoscalar (n-like) and vector (plike) tensor components of the Argonne u~4 interaction [ 8 ]. Because of the much discussed ambiguity concerning the proper choice o f form factor to be associated with the "model independent" isovector exchange current operator, we emphasize that we (as in ref. [ 8 ] ) use the form factor G v (q2). The sensitivity to the nucleon form factor parametrization in the results in fig. 2 appears to be somewhat smaller than what found in ref. [25 ] in a calculation based on the Paris potential [24], which used the G K [ 17] and dipole form factor parametrizations. The difference is due to the absence of a neutron electric form factor in ref. [25 ]. In figs. 3 and 4 we show the magnetic form factors of 3H and 3He calculated with the same exchange current operator used above and variational wave functions that correspond to the Argonne two-nucleon interaction and Urbana-VII TN1 [9], along
26 July 1990
100
'I ....
I ....
I ....
I ....
3H
10-1 i X IA(FAD) ~X/~IA(VAR)
10-z
"
[}' ~i. "~,,.IA*MEC (VAR)-
I0 -3
i0 -~ -IA*MEC(FAD)YV
~, .. . , '.....'?.;<-- .(,,
I~'~'...... X.
10.5
iyi \ -
10-60
' ......
i
2
4 6 Q(fm-1)
8
10
Fig. 3. Magnetic form factor of 3H, as a function of the four-momentum transfer, obtained with variational (VAR) and 34channel Faddeev (FAD) wave functions. The results obtained without and with inclusion of the exchangecurrent contributions are displayed as IA and IA+MEC, respectively. The H parametrization of the nucleon electromagneticform factors [ 18] is used.
10-1100 ~ ~ I I l, .... [:
\~
I0-2F
.IA (FAD)
10 -z
t
'
I
....
....
~i)~~'k'..IA.MEC(FAD}~ ~,~A.MEC(VARL~
~" 10_31
10-60
1'
~.... I ....
L ,
2
~ ,
,il
f..Z~.~?:: x
I
,
,
I
L, 6 Q (fm 4)
,
,L
,
I ....
8
I
10
Fig. 4. Same as in fig. 3, but of 3He. 375
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with the experimental data. The quality of these wave functions has previously been assessed by quantitatively successful predictions of the binding energies and asymptotic D / S ratios in the d + n and d + p breakup channels of 3H and 3He [9], as well as by direct comparison with results obtained with exact Faddeev [26] and Green's function Monte Carlo [27] wave functions as, for example, those for the two-body correlation functions [28,29 ] and the longitudinal energy-weighted sum rule [ 30 ]. In order to further test their accuracy, we have carried out the calculation o f the magnetic form factors with the 34channel Faddeev wave functions obtained by the Los Alamos group [26,31 ] for the Argonne two-nucleon interaction and a version of the Urbana-VII TNI, in which the strengths of its repulsive and attractive parts are modified so as to reproduce the experimental 3H binding energy. The Monte Carlo method discussed in ref. [8] has been used in both variational and Faddeev calculations. The calculated form factors are in good agreement with the experimental ones over the measured range of m o m e n t u m transfer values, although the uncertainty in the electromagnetic form factors of the nucleon leads to a sizeable uncertainty margin in the predictions (figs. 5 and 6). The form factors calculated with the variational wave functions differ from those of ref. [ 8 ] in that the part of the exchange current operator that is associated with the spin-orbit (and central ) interactions has been constructed so as to satisfy the continuity equation with the relativistic correction in the single nucleon charge operator [7,21 ]. Furthermore, a numerical error in the exchange current contribution due to the excitation o f the A33 resonance, which led to an overestimate of that contribution in ref. [ 8 ], has been corrected. The most prominent features in the calculated magnetic form factors are the predicted zeroes in both form factors around 7 f m - i. For reference, we also list the magnetic m o m e n t values obtained with the Faddeev (variational) wave functions for 3He and 3H, respectively: - 1 . 7 9 3 (-1.783) n.m. and + 2 . 5 9 8 ( + 2 . 5 9 5 ) n.m. in impulse approximation; - 2 . 2 4 1 ( - 2.179) n.m. and + 3.083 ( + 3.071 ) n.m. when the exchange current contribution is included. The latter should be compared with the experimental values - 2 . 1 2 7 n.m. (3He) and + 2 . 9 7 9 n.m. (3H). The present calculations show that a fair descrip376
26 July 1990 10 0
~
1
...
I ....
I ....
I ....
3H
10 "1
10-2
_ _ 10 -~ O ms 10-4
10"5
oj ,,,I,,,,[,
10 4 0
2
,I . . . . 4
6 Q ( f m "1 )
I .... 8
10
Fig. 5. Magnetic form factor of 3H obtained with the variational wave function and inclusion of the exchange current contributions, but using different parametrizations of the nucleon electromagnetic form factors: D (dipole), GK [ 17 ], H [ 18 ] and IJL [16]. 10 o ......
r ....
P.... I'" 3 He
l
10-~
10-=
//GK
10-3
,.=
IJL ~
A
O
~H
10-4
10 5
10 "s
i o
iiJ]llJa
2
,jL,,,,I 4 6 Q ( f m "1 )
....
8
10
Fig. 6. Same as in fig. 5, but of 3He. tion of the three observables considered is obtained for Q~< 1 G e V / c by using the Argonne/)14 interaction to construct, in a consistent way, the exchange current operator and the wave functions o f the two- and
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three-nucleon systems. For Q > 1 G e V / c the prediction of the electrodisintegration cross section is only in good qualitative agreement with the experimental data. It is also evident that the calculated observables are sensitive to the potential models used. Indeed, both the deuteron electrodisintegration cross section [23] and the magnetic form factors of the trinucleons [ 32 ] are clearly underpredicted in comparison with the data at intermediate values of m o m e n t u m transfer, when calculated with the parametrized Paris potential [ 24 ]. The predicted values of the electrodisintegration cross section at large m o m e n t u m transfer could be improved if the strength of the model i n d e p e n d e n t isovector exchange current or, equivalently, the strengths of the isospin dependent pseudoscalar and vector exchange tensor interactions at short range were increased. Above 1 G e V / c one should also in principle include the contributions from exchange current mechanisms that involve excitation of higher n N resonances than the A33. The relative importance of such mechanisms and of explicit relativistic corrections have not so far been systematically explored. Inclusion of explicit resonance configurations in the wave functions may improve the predictions at intermediate values of m o m e n t u m transfer [ 33 ]. Finally, it has to be emphasized that the large uncertainty in the behaviour of the nucleon electromagnetic form factors prevents definitive quantitative predictions above 1 G e V / c [25]. Reducing this uncertainty appears to us as a task of the highest priority. We wish to thank C.R. Chen, J.L. Friar, B.F. Gibson and G.L. Payne for allowing us to use the 3H 34channel Faddeev wave function, V.R. Pandharipande and R.B. Wiringa for interesting conversations. We gratefully acknowledge the support of the US Departm e n t of Energy through the C o n t i n u o u s Electron Beam Accelerator Facility. The calculations were made possible by grants of time on the Cray supercomputer of the National Magnetic Fusion Energy C o m p u t e r Center of the US D e p a r t m e n t of Energy.
26 July 1990
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