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Six-quark bag, exchange currents and trinucleon magnetic moments
Volume 173, number 4
PHYSICS LETTERS B
19 June 1986
SIX-QUARK BAG, EXCHANGE CURRENTS AND TRINUCLEON MAGNETIC MOMENTS R.K. B H A D U R I , M.V.N. M U R T H Y Physics Department, McMaster University, Hamilton, Ontario, Canada L8S 4M1
and E.L. T O M U S I A K Saskatchewan Accelerator Laboratory and Department of Physics, University of Saskatchewan, Saskatoon, Sask., Canada S7N OWO
Received 19 February 1986
The magnetic moments of 3H and 3He are reexamined in the Karl-Miller-Rafelski model of six-quark bag formation. Realistic three-nucleon wavefunctions are taken, and long-range one-pion exchange current corrections are included. It is concluded that the model is compatible with the data.
The ground-state nuclear magnetic moments of 3 He and 3H are known experimentally to very great precision. One could then test a nuclear model where these nuclei consisted only o f three interacting nucleons. The corresponding nonrelativistic wavefunction, using modern two-body potentials, may be numerically obtained to the desired accuracy, and the magnetic moments calculated. Such a calculation, where one assumes that the intrinsic magnetic moments of the bound nucleons remain unaltered from their free values, is referred to as the "impulse approximation". One such recent calculation [ 1 ], using a Reid soft-core potential, yields for the isoscalar part 0.405 n.m., to be compared with the experimental value of 0.4257 n.m. For the isovector part, however, the discrepancy is much larger. The calculation in the impulse approximation yields =2.149 n.m., while experimentally it is - 2 . 5 5 3 2 n.m. These discrepancies (about 16% for the isovector and 5% for the isoscalar moments) are of great interest in nuclear physics, pointing to deviations from the naive nuclear model. In the isoscalar case, the small deviation could be of the same order as relativistic corrections. In the isovector channel, however, meson-exchange currents (MEC) are thought to be very important because o f 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
one-pion exchange [2]. In ref. [1], a calculation o f the contribution o f MEC involving one-pion exchange to the isovector magnetic moment resolved most o f the discrepancy, reducing it from 16% to about 4%. This is shown in more detail #1 in table 1. The conventional nuclear physics viewpoint, with nucleons and pions in the nucleus, seems to be adequate to explain the magnetic moments o f 3He and 3H. Recently, however, much effort is being devoted to unravel quark degrees o f freedom in nuclei [3]. Karl et al. [4] examined the magnetic moments o f the ttuee-nucleon system in this light.They suggested that the deviations in the moments could largely be explain ed if two nucleons, overlapping within a distance r 0 1 fm, form a six-quark bag within the nucleus. Their calculation, however, ignored the noncentral components in the nuclear wavefunctions, and also the possibility of exchange currents for r > r 0 due to pionexchange. Since the six-quark bag formation is expected to take place for small internucleon distances, 41 It has been communicated to us by the authors of ref. [ 1 ] that there was a small error in the published version of the exchange correction term Upair. This has been corrected in table 1. 369
Volume 173, number 4
PHYSICS LETTERS B
19 June 1986
Table 1 lsoscalar and isovector magnetic moments Us and ~v in n.m. No 6-q bag formation a)
With 6-q bag formation
Experiment
r0 = 0 P6q = 0 @ = 0.405 6~ =0 # s Da =~.405 V ~i_= -2.419 ~pair = - 0.283 Ua = 0.095
ro = 1 fm P6q = 12.87% U~,. = 0.351 l J/0 6U~a. = 0.068 us=~.419 v = Ui>ro -1.860 Upair>r0 = -0.218 (~a + ug) ~ (1 - 0.13)(-0.009) = -0.008
Us = 0.4257
~n = 0.086 SU~ag = 0 u v = -2.441
6,~ag = -0.363 u v = -2.449
Uv= -2.5532
a) From ref. [1 ] with corrections, with range parameter A in the pion-nucleon formfactor taken to be 5.8 mrr.
whereas the one-pion-exchange corrections are domin a n t for larger distances, it is reasonable to take account of b o t h , and check if such a calculation is consistent with the data. This we proceed to do. The isoscalar magnetic m o m e n t / a s and the isovector m o m e n t / ~ v are defined as us = 1 [/~(3He ) + # ( 3 H ) ] , Uv = ½ [~t(3 He) - U( 3 H)].
(1)
The results of ref. [1], where the three-nucleon wavef u n c t i o n is obtained from a Faddeev calculation [5] using the Reid soft-core potential, are summarized in the first c o l u m n o f table 1. The trinucleon wavefunction has d o m i n a n t l y 2S1/2 (88.91%) and 4D1/2 (9.34%) c o m p o n e n t s , a small admixture (1.67%) o f the mixed symmetric 2S'1/2_ state, and traces of Pstates. For the calculation o f the one-pion-exchange MEC, three distinct c o n t r i b u t i o n s are included [1]. First is the seagull term, where the p h o t o n is adsorbed at the pion n u c l e o n vertex. Its c o n t r i b u t i o n to the isovector magnetic m o m e n t is d e n o t e d by/-tpair in table 1, and is the most i m p o r t a n t meson correction. Note that a p i o n - n u c l e o n ( m o n o p o l e ) form factor is used in these calculations with a range parameter * 2 ,2 The variation of Upair with different values of the range parameter A is large only for A < 4mrr. If A is increased from 5.8 mrr to 8.6 mrr, the magnitude of ,Upair increases by about 11%. It tends to saturate thereafter, increasing only by a further 5% as A --' ~. 370
A = 5.8 m~r = 4.12 fin - 1 . The other two processes are termed pion and delta-isobar current terms, and their c o n t r i b u t i o n s / a a n d / ~ a to the isovector m o m e n t / a v nearly cancel edch other. The isoscalar m o m e n t / a s remains unaltered, to this order o f the calculation, from its impulse value, as shown in the first c o l u m n o f table 1. To estimate the effect of six-quark bag formation, the corresponding probability P6q is calculated using the three-nucleon w a v e f u n c t i o n t~3N. With the normalization
f
1ff3N(rl,r2,r3)[ 2 d 3 r l d3r2 d3r3 = 1,
(2)
all space the probability P6q is defined as P6q = 3
fl
X 0(r23 -
r2, r3)120(ro -- r12)
r0)0(/'31
--
r0) d3rl
d3r2 d3/.3 .
(3)
Here r12 = [rl2l etc, O(x) is the unit step-function, u n i t y only for x 1> 0 and zero otherwise, and r 0 is a parameter o f the model, explained earlier• The probability P6q for 3H for various values o f r 0 is tabulated [6] in table 2, and is seen to be rapidly changing in the vicinity o f r 0 ~ 1 fro. For calculating the magnetic mom e n t o f the six-quark bag, we use the same model as Karl et al. [4], and take the six quarks to be in a totally symmetric S-state. In the MIT bag model [7], with zero-mass quarks the energy of n quarks in the
Volume 173, number 4
PHYSICS LETTERS B
where/an is the free neutron magnetic moment, and t tau the enhanced moment in the six-quark bag. Similarly,
Table 2 Variation of P6q with the parameter r o a). r o (fm)
P6q (%)
0.8 0.9 1.0 1.1 1.2
4.89 8.46 12.87 17.76 22.71
u(3H)
2
E = 2.04n/R + 47rBR 3 - Z o / R
(n - 3) 2 - Sl2(S12 + 1) (4)
where S12 is the total spin and 112 the/-spin of the n-quark bag. The quark-gluon coupling constant c% is defined as in the bag model [7]. The parameters B, Z 0 and c% are determined by fitting the (n = 3) nucleon mass M N = 940 MeV, the 2x-mass M A = 1232 MeV, and the RMS nucleon radius (r2)~ 2 = 0.88 fm. These are found to b e B 1/4 = 125 MeV, c% = 0.665 and Z 0 = 0.88. For the six-quark bag in theI12 = 0, $12 = 1 channel, we find that R 6 q / R N = 1.28 with these parameters. In the 112 = 1, S12 = 0 channel, the ratio is slightly more, R 6 q / R N = 1.30. Since the magnetic moment of the bag scales with its radius, the magnetic moment of a "nucleon" is enhanced by about 30% when it is part o f a six-quark bag. To estimate the magnetic moments of 3He and 3H, we now perform the three-nucleon calculation for the impulse and exchange correction terms with the condition that every pair o f nucleons is at a distance greater than r 0. In MEC, only the pair term//pair is important, since gTr and/a/, nearly cancel, and is calculated this way. To this is added the simple six-quark bag estimate to obtain the total magnetic moment. Let us denote by/@ro(3He) the three-nucleon part of the magnetic moment o f 3He. In ref. [1], r 0 was zero, and this was the full contribution. Now, following Karl et al. [4], we write u(3He) =/a>r ° (3 He) + P6q (~btPn + 1Ltn) ,
t
1
us=~>r0S + P 6 q [ : ( U p + U ~ ) + : ( ~ p + U n ) ] ,
S-state is
-112(112 + 1)1,
3 + _2 , 1 = / a > r o ( H ) P6q(3/ap +5/ap),
(6)
with/ap the free proton magnetic moment. From eqs. (5), (6), using definition (1), we get
a) Calculated from eq. (3) with Reid soft-core potential [6] for 3H.
- (0.2335c%/R)[9
19 June 1986
(5)
(7)
and _
V
2
F
Ltv-/a>ro-P6q[5(#p
1
/An)+ :(//p - /an) ].
(8)
The result of such a calculation for r 0 = 1 fm is shown r in the second column of table 1. We have taken/a n = 1.3 #n, and similarly for/ap. From table 2, we find that P6q = 3 × 4.29% = 12.87% for r 0 = 1 fm. We denote the bag contribution to themagnetic moS,V ments by ~ ./abag, and these are just the terms with P6q in eqs. (7), (8). Rather than individually calculate /a/, and/a~r with the restriction that each pair distance be greater than r0, we simply multiply the small term (gA +/a~r) by (1 - P 6q)" The impulse contribution to the magnetic moment, lai>ro , and the pair correction term in the second column of table 1 are calculated exactly [6] by integrating for each pair from r 0 onwards. We see from table 1 that both #s and/av for r 0 = 1 fm are in better agreement with the experimental values than the conventional calculation where r 0 = 0. Since our model is rather crude, this should not be taken as evidence for six-quark bag formation. Rather, we take the point of view that aP6q of 10% to 20% in trinucleon systems is compatible with the magnetic moment data, even after the long-range exchange corrections due to pions are included. The point may be raised that pion currents be allowed within the six-quark bag. Note that in such models, pion currents contribute substantially even to the nucleon magnetic moment [8]. Since we are already taking the experimental nucleon moments [see eqs. (7), (8)] in evaluating the bag contribution, we believe such effects are already included to some degree in our model. Finally, in writing equations like (7), we have neglected the nine-quark bag, whose forma tion probability has been found to be negligible for r0 = 1 fm. The authors thank Professor G.A. Miller for discussions. This research was supported by grants from NSERC (Canada). 371
Volume 173, number 4
PHYSICS LETTERS B
References [ 1 ] E.L. Tomusiak, M. Kimura, J.L. Friar, B.F. Gibson, G.L. Payne and J. Dubach, Phys. Rev. C 32 (1985) 2075. [2] J.L. Friar, Lectures on the Three-nucleon problem, in: New vistas in electronic physics, 1985 NATO ASI (Banff, Canada, 1985), eds. E.L. Tomusiak, E. Dressier and H. Kaplan (Plenum, New York), to be published. [31 G.A. Miller, in: Quarks and nuclei, International Review of Nuclear Physics, Vol. I, ed. W. Weise (World Scientific, Singapore, 1985).
372
19 June 1986
[4] G. Karl, G.A. Miller and J. Rafelski, Phys. Lett. B 143 (1984) 326. [5] G.L. Payne, J.L. Friar and B.F. Gibson, Phys. Rev. C 22 (1980) 832. [6 ] E.L. Tomusiak, J.L. Friar, B.F. Gibson and G.L. Payne, private communication. [7] C.W. Wong, UCLA preprint 1985, Rep. Prog. Phys. to be published. [8] A.W. Thomas, Adv. Nucl. Phys. 13 (1983) 1.
PHYSICS LETTERS B
19 June 1986
SIX-QUARK BAG, EXCHANGE CURRENTS AND TRINUCLEON MAGNETIC MOMENTS R.K. B H A D U R I , M.V.N. M U R T H Y Physics Department, McMaster University, Hamilton, Ontario, Canada L8S 4M1
and E.L. T O M U S I A K Saskatchewan Accelerator Laboratory and Department of Physics, University of Saskatchewan, Saskatoon, Sask., Canada S7N OWO
Received 19 February 1986
The magnetic moments of 3H and 3He are reexamined in the Karl-Miller-Rafelski model of six-quark bag formation. Realistic three-nucleon wavefunctions are taken, and long-range one-pion exchange current corrections are included. It is concluded that the model is compatible with the data.
The ground-state nuclear magnetic moments of 3 He and 3H are known experimentally to very great precision. One could then test a nuclear model where these nuclei consisted only o f three interacting nucleons. The corresponding nonrelativistic wavefunction, using modern two-body potentials, may be numerically obtained to the desired accuracy, and the magnetic moments calculated. Such a calculation, where one assumes that the intrinsic magnetic moments of the bound nucleons remain unaltered from their free values, is referred to as the "impulse approximation". One such recent calculation [ 1 ], using a Reid soft-core potential, yields for the isoscalar part 0.405 n.m., to be compared with the experimental value of 0.4257 n.m. For the isovector part, however, the discrepancy is much larger. The calculation in the impulse approximation yields =2.149 n.m., while experimentally it is - 2 . 5 5 3 2 n.m. These discrepancies (about 16% for the isovector and 5% for the isoscalar moments) are of great interest in nuclear physics, pointing to deviations from the naive nuclear model. In the isoscalar case, the small deviation could be of the same order as relativistic corrections. In the isovector channel, however, meson-exchange currents (MEC) are thought to be very important because o f 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
one-pion exchange [2]. In ref. [1], a calculation o f the contribution o f MEC involving one-pion exchange to the isovector magnetic moment resolved most o f the discrepancy, reducing it from 16% to about 4%. This is shown in more detail #1 in table 1. The conventional nuclear physics viewpoint, with nucleons and pions in the nucleus, seems to be adequate to explain the magnetic moments o f 3He and 3H. Recently, however, much effort is being devoted to unravel quark degrees o f freedom in nuclei [3]. Karl et al. [4] examined the magnetic moments o f the ttuee-nucleon system in this light.They suggested that the deviations in the moments could largely be explain ed if two nucleons, overlapping within a distance r 0 1 fm, form a six-quark bag within the nucleus. Their calculation, however, ignored the noncentral components in the nuclear wavefunctions, and also the possibility of exchange currents for r > r 0 due to pionexchange. Since the six-quark bag formation is expected to take place for small internucleon distances, 41 It has been communicated to us by the authors of ref. [ 1 ] that there was a small error in the published version of the exchange correction term Upair. This has been corrected in table 1. 369
Volume 173, number 4
PHYSICS LETTERS B
19 June 1986
Table 1 lsoscalar and isovector magnetic moments Us and ~v in n.m. No 6-q bag formation a)
With 6-q bag formation
Experiment
r0 = 0 P6q = 0 @ = 0.405 6~ =0 # s Da =~.405 V ~i_= -2.419 ~pair = - 0.283 Ua = 0.095
ro = 1 fm P6q = 12.87% U~,. = 0.351 l J/0 6U~a. = 0.068 us=~.419 v = Ui>ro -1.860 Upair>r0 = -0.218 (~a + ug) ~ (1 - 0.13)(-0.009) = -0.008
Us = 0.4257
~n = 0.086 SU~ag = 0 u v = -2.441
6,~ag = -0.363 u v = -2.449
Uv= -2.5532
a) From ref. [1 ] with corrections, with range parameter A in the pion-nucleon formfactor taken to be 5.8 mrr.
whereas the one-pion-exchange corrections are domin a n t for larger distances, it is reasonable to take account of b o t h , and check if such a calculation is consistent with the data. This we proceed to do. The isoscalar magnetic m o m e n t / a s and the isovector m o m e n t / ~ v are defined as us = 1 [/~(3He ) + # ( 3 H ) ] , Uv = ½ [~t(3 He) - U( 3 H)].
(1)
The results of ref. [1], where the three-nucleon wavef u n c t i o n is obtained from a Faddeev calculation [5] using the Reid soft-core potential, are summarized in the first c o l u m n o f table 1. The trinucleon wavefunction has d o m i n a n t l y 2S1/2 (88.91%) and 4D1/2 (9.34%) c o m p o n e n t s , a small admixture (1.67%) o f the mixed symmetric 2S'1/2_ state, and traces of Pstates. For the calculation o f the one-pion-exchange MEC, three distinct c o n t r i b u t i o n s are included [1]. First is the seagull term, where the p h o t o n is adsorbed at the pion n u c l e o n vertex. Its c o n t r i b u t i o n to the isovector magnetic m o m e n t is d e n o t e d by/-tpair in table 1, and is the most i m p o r t a n t meson correction. Note that a p i o n - n u c l e o n ( m o n o p o l e ) form factor is used in these calculations with a range parameter * 2 ,2 The variation of Upair with different values of the range parameter A is large only for A < 4mrr. If A is increased from 5.8 mrr to 8.6 mrr, the magnitude of ,Upair increases by about 11%. It tends to saturate thereafter, increasing only by a further 5% as A --' ~. 370
A = 5.8 m~r = 4.12 fin - 1 . The other two processes are termed pion and delta-isobar current terms, and their c o n t r i b u t i o n s / a a n d / ~ a to the isovector m o m e n t / a v nearly cancel edch other. The isoscalar m o m e n t / a s remains unaltered, to this order o f the calculation, from its impulse value, as shown in the first c o l u m n o f table 1. To estimate the effect of six-quark bag formation, the corresponding probability P6q is calculated using the three-nucleon w a v e f u n c t i o n t~3N. With the normalization
f
1ff3N(rl,r2,r3)[ 2 d 3 r l d3r2 d3r3 = 1,
(2)
all space the probability P6q is defined as P6q = 3
fl
X 0(r23 -
r2, r3)120(ro -- r12)
r0)0(/'31
--
r0) d3rl
d3r2 d3/.3 .
(3)
Here r12 = [rl2l etc, O(x) is the unit step-function, u n i t y only for x 1> 0 and zero otherwise, and r 0 is a parameter o f the model, explained earlier• The probability P6q for 3H for various values o f r 0 is tabulated [6] in table 2, and is seen to be rapidly changing in the vicinity o f r 0 ~ 1 fro. For calculating the magnetic mom e n t o f the six-quark bag, we use the same model as Karl et al. [4], and take the six quarks to be in a totally symmetric S-state. In the MIT bag model [7], with zero-mass quarks the energy of n quarks in the
Volume 173, number 4
PHYSICS LETTERS B
where/an is the free neutron magnetic moment, and t tau the enhanced moment in the six-quark bag. Similarly,
Table 2 Variation of P6q with the parameter r o a). r o (fm)
P6q (%)
0.8 0.9 1.0 1.1 1.2
4.89 8.46 12.87 17.76 22.71
u(3H)
2
E = 2.04n/R + 47rBR 3 - Z o / R
(n - 3) 2 - Sl2(S12 + 1) (4)
where S12 is the total spin and 112 the/-spin of the n-quark bag. The quark-gluon coupling constant c% is defined as in the bag model [7]. The parameters B, Z 0 and c% are determined by fitting the (n = 3) nucleon mass M N = 940 MeV, the 2x-mass M A = 1232 MeV, and the RMS nucleon radius (r2)~ 2 = 0.88 fm. These are found to b e B 1/4 = 125 MeV, c% = 0.665 and Z 0 = 0.88. For the six-quark bag in theI12 = 0, $12 = 1 channel, we find that R 6 q / R N = 1.28 with these parameters. In the 112 = 1, S12 = 0 channel, the ratio is slightly more, R 6 q / R N = 1.30. Since the magnetic moment of the bag scales with its radius, the magnetic moment of a "nucleon" is enhanced by about 30% when it is part o f a six-quark bag. To estimate the magnetic moments of 3He and 3H, we now perform the three-nucleon calculation for the impulse and exchange correction terms with the condition that every pair o f nucleons is at a distance greater than r 0. In MEC, only the pair term//pair is important, since gTr and/a/, nearly cancel, and is calculated this way. To this is added the simple six-quark bag estimate to obtain the total magnetic moment. Let us denote by/@ro(3He) the three-nucleon part of the magnetic moment o f 3He. In ref. [1], r 0 was zero, and this was the full contribution. Now, following Karl et al. [4], we write u(3He) =/a>r ° (3 He) + P6q (~btPn + 1Ltn) ,
t
1
us=~>r0S + P 6 q [ : ( U p + U ~ ) + : ( ~ p + U n ) ] ,
S-state is
-112(112 + 1)1,
3 + _2 , 1 = / a > r o ( H ) P6q(3/ap +5/ap),
(6)
with/ap the free proton magnetic moment. From eqs. (5), (6), using definition (1), we get
a) Calculated from eq. (3) with Reid soft-core potential [6] for 3H.
- (0.2335c%/R)[9
19 June 1986
(5)
(7)
and _
V
2
F
Ltv-/a>ro-P6q[5(#p
1
/An)+ :(//p - /an) ].
(8)
The result of such a calculation for r 0 = 1 fm is shown r in the second column of table 1. We have taken/a n = 1.3 #n, and similarly for/ap. From table 2, we find that P6q = 3 × 4.29% = 12.87% for r 0 = 1 fm. We denote the bag contribution to themagnetic moS,V ments by ~ ./abag, and these are just the terms with P6q in eqs. (7), (8). Rather than individually calculate /a/, and/a~r with the restriction that each pair distance be greater than r0, we simply multiply the small term (gA +/a~r) by (1 - P 6q)" The impulse contribution to the magnetic moment, lai>ro , and the pair correction term in the second column of table 1 are calculated exactly [6] by integrating for each pair from r 0 onwards. We see from table 1 that both #s and/av for r 0 = 1 fm are in better agreement with the experimental values than the conventional calculation where r 0 = 0. Since our model is rather crude, this should not be taken as evidence for six-quark bag formation. Rather, we take the point of view that aP6q of 10% to 20% in trinucleon systems is compatible with the magnetic moment data, even after the long-range exchange corrections due to pions are included. The point may be raised that pion currents be allowed within the six-quark bag. Note that in such models, pion currents contribute substantially even to the nucleon magnetic moment [8]. Since we are already taking the experimental nucleon moments [see eqs. (7), (8)] in evaluating the bag contribution, we believe such effects are already included to some degree in our model. Finally, in writing equations like (7), we have neglected the nine-quark bag, whose forma tion probability has been found to be negligible for r0 = 1 fm. The authors thank Professor G.A. Miller for discussions. This research was supported by grants from NSERC (Canada). 371
Volume 173, number 4
PHYSICS LETTERS B
References [ 1 ] E.L. Tomusiak, M. Kimura, J.L. Friar, B.F. Gibson, G.L. Payne and J. Dubach, Phys. Rev. C 32 (1985) 2075. [2] J.L. Friar, Lectures on the Three-nucleon problem, in: New vistas in electronic physics, 1985 NATO ASI (Banff, Canada, 1985), eds. E.L. Tomusiak, E. Dressier and H. Kaplan (Plenum, New York), to be published. [31 G.A. Miller, in: Quarks and nuclei, International Review of Nuclear Physics, Vol. I, ed. W. Weise (World Scientific, Singapore, 1985).
372
19 June 1986
[4] G. Karl, G.A. Miller and J. Rafelski, Phys. Lett. B 143 (1984) 326. [5] G.L. Payne, J.L. Friar and B.F. Gibson, Phys. Rev. C 22 (1980) 832. [6 ] E.L. Tomusiak, J.L. Friar, B.F. Gibson and G.L. Payne, private communication. [7] C.W. Wong, UCLA preprint 1985, Rep. Prog. Phys. to be published. [8] A.W. Thomas, Adv. Nucl. Phys. 13 (1983) 1.