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Magnetic moments of the nucleon octet in a relativistic light-cone quark model
Volume 200, number 4
PHYSICS LETTERSB
21 January 1988
MAGNETIC M O M E N T S OF THE N U C L E O N OCTET IN A RELATIVISTIC L I G H T - C O N E QUARK M O D E L Zbigniew DZIEMBOWSKI I, Tomasz DZURAK, Adam SZCZEPANIAK Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-O0-681 Warsaw, Poland
and Lech MANKIEWICZ Nicolaus Copernicus Astronomical Centre of Polish Academy of Sciences, Bartycka 18, PL-00-716 Warsaw, Poland
Received 24 September 1987
The magnetic momentsof the baryon octet are calculated in a relativistic constituent quark modelformulated in the light-cone Fock approach. Invokinga natural idea of strangeness-dependent hadron size we find verygoodagreement (to an accuracyof la) for the recent precision hyperonmagnetic-momentdata and reveal 10-15%discrepancyfor the Z - and nucleon. It suggestspions as a missing ingredient in the baryon magnetic moment calculations.
Recently a positive progress has occurred in our understanding of the composition ofhadrons in terms of their quark quanta. Several powerful nonperturbative methods have been developed which allow detailed predictions for the hadronic wavefunctions directly from QCD. Sum rule analysis of Chernyak, Zhitnitsky and Zhitnitsky [ 1 ] and lattice gauge theory calculations [ 2 ] have demonstrated that the nucleon and pion valence-quark distribution amplitudes are highly structured and significantly broader than the nonrelativistic 0-function form. In a recent work [ 3,4] we have attempted to bridge the results ofnonperturbative QCD methods and the quark model approach. Three ideas turn out to be vital for a successful derivation of the basic features of the CZ distribution amplitude, i.e. (a) the use of the lightcone Fock approach, (b) a nonstatic relativistic spin wavefunction, and (c) small transverse size of the valence-quark configuration. The presented model, together with the concept of scale-dependent effective quark mass provides a consistent description of the measured high momentum transfer form factors, This research was supported in part by the Research Program CPBP-0103.
the CZ distribution amplitudes, and some basic lowenergy nucleon and pion properties [5]. The purpose of this paper is to investigate some other predictions of the new relativistic approach. Thus, we present an extension of the nucleon wavefunction to the case of strange panicles and discuss magnetic moments of the nucleon octet. The motivation for the work is provided by new, accurate data [ 6 ] on the magnetic moments of the charged E and F, hyperons, along with substantial discrepancies between the data and the static quark-model predictions. Let us quote as a typical example of the static model results the prediction by Rosner [ 7 ]. Table 1 shows a comparison with the recent experimental data. There are already model-independent analyses of the observed disagreement, due to Franklin [ 8 ] and Lipkin [9]. They come to the conclusion that a good understanding of baryon magnetic moments will require a model with quark-moment contributions which are nonstatic and/or baryon dependent. A role of relativistic effects in the context have also been emphasized by several authors [ 10,11 ]. We use the light-cone formalism [ 12 ], which provides a consistent relativistic framework in momentum space in terms of Fock-state basis defined at
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
539
Volume 200, number 4
PHYSICS LETTERS B
Table 1 Baryon magnetic moments (in nuclear magnetons) in the static constituent quark model (CQM).
21 January 1988
The resultant Lorentz-invariant light-cone wavefunctions are ~,~(i, 2, 3)
Baryon moments
Experiment
/~(p) /t(n) kt(A) /~(Z + ) #(X-) #(.-o) ,u(E )
2.793+0.000 - 1.913+0.000 -0.613+0.004 2.379+0.020 - 1 . 1 4 +0.05 - 1.250+0.014 - 0 . 6 9 +0.04
Static CQM
-- 1/2
2.79 - 1.86 -0.58 2.68 -1.05 - 1.40 -0.47
Their normalizations are given by 2 f [dxd2k±] ~ [~p(1, 2, 3) 12 = 1.
equal-x += t + z, rather than the conventional equalt wavefunctions. With the valence-quark-dominance assumption baryon wavefunctions is taken to be simple generalization of the nonrelativistic constituent quark model one. In view of the relativistic motion of quarks, the momentum distribution is taken to be as a relativistic Gauss,an [ 12 ]
O(x" k ± ' ) = A e x p ( - 6 - ~ B ~
k~,+__m~. x, /
(1)
The baryon states of interest have two identical quarks (except those of the A) which we shall label with i = 1 and 2. The overall symmetry of the wavefunction in momentum, spin, and flavor spaces then implies that the spin-flavor wavefunctions for B = p, n, Z +, Z - , E °, and E - are symmetric under exchange of 1 and 2, but for the A it is ant,symmetric. They have the form ZP(x,, k±,, 2,) = J , ( i , 3, ~) + J , ( 2 , 3, i ) ,
(2a)
with J)(i, ~, 3 ) = aa,(Ma
+P.~,")~',v~2aa3u,,
blxl(MA
+Pv~U)YsVx2U-~3u) ,
(2b)
i, ~ and 3 are collective momentum-helicity indices (xi, k±i, ).~), i= 1, 2, 3. u~ and v~ are the light-cone spinors of ref. [13]. We keep flavor and color implicit. The nonstatic spin wavefunctions (2) are obtained from conventional wavefunctions with jo = ½+ transformed to the light-cone using a Meloshtype rotation of the quark spinors [4]. 540
I's~ = ( B ( P + q , "f) IJ+ ( O)IP + IB(P, ,L) )
=X I[ax d2kl 1~';~"(i', 2', ~')(am/,/~) X y + Qm(Um]x~mm)~B(], 2, ~) ,
(4)
where Qm is the charge of the struck quark with the final momentum k'±m=k±m+( 1 - - X m ) q Z while the spectator quark has final momentum k,, ,= k i i - x , q ± for i # m. Note, that we neglect the quark anomalous magnetic moments. If the light-cone coordinates are chosen [ 15] as pu= (p+, M~/p+, 0±) for the baryon moving along the z-axis and qU= (0, 2P.q/P +, q± ) for the photon, the helicity-flip matrix element of the current j + = j o + j 3 has the simple form
Fs} = - qL Fz( qZ)/Ma ,
and for the A
zA(x,, kzi, ~'i) =
The [dx d2k± ] is the volume element in momentum space. We emphasize the point made earlier that the nucleon wavefunction of the form (3), together with the small transverse size hypothesis provides the essential features of the CZ distribution amplitudes. We start by discussing the magnetic moments. The anomalous magnetic moment of any spin-1/2 system can be identified [ 14] from the spin-flip matrix element of the electromagnetic current f f
where qL=ql--iq 2 is used for the transverse momentum transfer. Hence, the anomalous magnetic moment a = F 2 ( 0 ) becomes
a = - - M B Oq----[ 0 F~f+ q=o
(5)
It was pointed out by several authors [ 16 ] that the spinor rotation of constituent quarks, arising from a Lorentz transformation associated with the boost P" to P" + qU, gives rise to sizable corrections to baryon
Volume 200, number
PHYSICS
4
LETTERS
magnetic moments. Note that the Drell-Yan formula (4) is especially suited to study the effects. It is related to the following advantages obtained by using the light-cone formalism: (i) There is no Wignerlike rotation in (4). (ii) Wavefunctions (3) are invariant under all kinematical Lorentz transformations, which contain the Lorentz boost along the three directions. Thus the simple and exact boost treatment, together with the proper relativistic kinematics of the internal relative motion present in (4), apparently invalidate the ordinary independentquark model additivity assumption. To illustrate numerical importance of the effects we use the basic wavefunctions (3) and formula (5) to calculate the baryon magnetic moments. The parameters entering our expressions for the magnetic moments are the quark masses and the momentum scale (Ye which determines the size of baryon valence wavefunction. We assume that quarks in baryons have typical constituent masses. To be specific we use the values m,,= md= 363 MeV and m,= 538 MeV given by Rosner’s lit to baryon masses. For the momentum scale we have decided to vary it freely in the range 300-500 MeV, in order to show explicitly the dependence on this parameter. The results for the seven measured magnetic moments are given in table 2. Before being compared with experiment, the A moment have to be corrected for A-CO mixing [ 171, which changes p(A) by about -0.04 PN.
General characteristics exhibited are briefly discussed below: Table 2 Batyon magnetic
a) Value corrected
moments
by these results
(in nuclear magnetons)
ffB
P(P)
300 320 340 360 380 400 420 440 460 480 500
2.737 2.718 2.691 2.676 2.653 2.629 2.605 2.580 2.554 2.528 2.502
for the effect of A-Z’
- 1.686 - 1.663 - 1.640 - 1.615 - 1.590 - 1.564 - 1.538 -1.511 - 1.484 - I.457 - 1.429
in the relativistic
-0.638 -0.635 -0.631 -0.621 -0.623 -0.619 -0.614 - 0.609 - 0.604 -0.598 -0.592
mixing of ref.
21 January
B
1988
(a) In the nucleon sector, the theoretical predictions are, for any scale c+, too small compared to the experimental values. If we take the nucleon momentum scale be equal to, say = 320-360 MeV, as in ref. [ 51, then we must consider other effects to account for the missing 1O-l 5% in the observed proton and neutron magnetic moments. (b) In the strange baryon sector a remarkable regularity can be observed. We note that the measured magnetic moments of all hyperons but the C - can be reproduced (to an accuracy of 1a) if one a]]OW an to increase with strangeness. This fit yields the hyperon momentum scale of cr,,= (-Yez 420 MeV and ~~~~440 MeV. For the C- our relativistic calculation gives value discrepant by w 0.1 &,, which is several times the standard deviations of the recent data both from tine-structure splitting in C - exotic atoms [ 181 and beam-polarization-precession technique [ 191. Thus one again has another case which suggests the importance of some other contributions. Let us mention that similar features are observed in the relativistic model calculation of the hyperon axial-vector couplings. The calculation of axial-vector form factors is essentially identical to that of the EM form factor in eq. (4) except of the replacement The prediction on G,/Gv for measured Y++Y0+. transitions [20] are given in table 3. Again the relativistic calculation with a dependence of the hadron size on the number of strange quarks provides very good (to an accuracy of 2~) description of the data. To understand intuitively the observed hierarchy
CQM as functions
2.48 1 2.464 2.446 2.428 2.408 2.389 2.368 2.347 2.325 2.303 2.28 I
_ 1.00 - 1.00 - 1.00 - 1.00 -0.99 -0.99 -0.99 -0.98 -0.98 -0.98 -0.98
of the baryon
_
1.330 1.320 1.309 1.297 1.284 1.272 1.258 1.245 1.231 -1.217 - 1.202
momentum
scale on (in MeV).
-0.60 -0.61 -0.61 -0.62 -0.62 -0.63 -0.64 -0.65 -0.65 -0.65 -0.66
[ 171. 541
Volume 200, number 4
PHYSICS LETTERS B
21January1988
Table 3 Baryon axial-vector couplings in the relativistic CQM. a
Transition E--~A
Z --,A
A~p
Z ~n
300 400 500
0.33 0.32 0.30
0 0 0
0.83 0.71 0.58
-0.28 -0.24 -0.20
experiment [20]
0.25+0.05
0.03+0.08
0.70+0.03
-0.34+0.05
o f the b a r y o n spatial sizes ( w h i c h d e c r e a s e w i t h strangeness) we n o t e t h a t for a C o u l o m b - l i k e p o t e n tial the b o u n d state size is p r o p o r t i o n a l to m - ~, w h e r e m is the r e d u c e d mass. T h u s , in p o t e n t i a l m o d e l s o n e can a n t i c i p a t e a d e c r e a s e o f t h e h a d r o n size w h e n a d d i n g strange q u a r k . In ref. [ 10] Isgur a n d K a r l q u o t e a d e c r e a s e by 4% a n d 13% p e r a d d i t i o n a l strange q u a r k in a h a r m o n i c a n d C o u l o m b p o t e n t i a l , respectively. O u r m a i n c o n c l u s i o n t h e r e f o r e is that the relativistic C Q M , t o g e t h e r w i t h t h e c o n c e p t o f strangenessd e p e n d e n t b a r y o n size, offers a large q u a n t i t a t i v e i m p r o v e m e n t o v e r the n o n s t a t i c d e s c r i p t i o n o f t h e h y p e r o n m a g n e t i c m o m e n t s . W i t h the large b a r y o n dependent non-additive magnetic moment contrib u t i o n s the m o d e l fulfills t h e r e q u i r e m e n t s o f general q u a r k - m o d e l analyses o f refs. [ 8,9 ]. T h e residual d i s a g r e e m e n t j u s t for t h e Z - a n d n u c l e o n suggests the n a t u r e o f the d o m i n a n t m i s s i n g i n g r e d i e n t in the b a r y o n w a v e f u n c t i o n s . It is k n o w n f r o m w o r k o f several a u t h o r s [21 ] that for t h o s e t h r e e b a r y o n s magnetic m o m e n t c o n t r i b u t i o n s f r o m a n o n v a l e n c e q q c o m p o n e n t w i t h p i o n q u a n t u m n u m b e r s are o f numerical importance.
References [ 1 ] V.L. Chernyak and A.R. Zhitnitsky, Phys. Rep. 112 (1979) 173; Nucl. Phys. B 201 (1982) 492; V.L. Chernyak and I.R. Zhitnitsky, Nucl. Phys. B 246 (1984) 52. [2] I.D. King and C.T. Sachrajda, Nucl. Phys. B 279 (1987) 785; G. Martinelli and C.T. Sachrajda, Phys. Lett. B 190 (1987) 151. [3] Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 58 (1987) 2175. [4] Z. Dziembowski, Warsaw University preprint IFT/24/87; IFT/25/87. 542
[ 5] Z. Dziembowski, Warsaw University preprint IFT/26/87; Z. Dziembowski and L. Mankiewicz, in: Proc. Second Intern. Conf. on Hadron spectroscopy, Hadron '87, ed. K. Takamatsu (KEK, Tsukuba, 1987) p.222; J. Bienkowska, Z. Dziembowski and H.J. Weber, Phys. Rev. Lett. 59 (1987) 624. [6] Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Lett. B 170 (1976) 1. [7] J. Rosner, in: High Energy Physics-1980, eds. L. Durand and L.G. Pondrom, AIP Conf. Proc. No. 68 (Am. Institute of Physics, New York, 1981 ). [8] J. Franklin, Phys. Rev. D 20 (1979) 1742; D 29 (1984) 2648. [ 9 ] H. Lipkin, Phys. Rev. D 24 (1981 ) 1437; Nucl. Phys. B 214 (1983) 136. [ 10] N. Isgur and G. Karl, Phys. Rev. D 21 (1980) 3175. [ 11 ] J. Franklin, Phys. Rev. Lett. 45 (1980) 1607; H. Georgi and A. Manohar, Phys. Lett. B 132 (1983) 183; Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 55 (1985) 1839. [ 12] S.J. Brodsky, T. Huang and G.P. Lepage, in: Springer tracks in modern physics, Vol. 100, Quarks and nuclear forces, eds. D. Fries and B. Zeitnitz (Springer, Berlin, 1982); J.M. Namystowski, Progr. Part. Nucl. Phys. 14 (1984) 49. [ 13 ] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22 (1980) 2175. [ 14] S.J. Brodsky and S.D. Drell, Phys. Rev. D 22 (1980) 2236. [ 15 ] S.D. Drell and T.N. Yan, Phys. Rev. Lett. 24 (1970) 181. [ 16] W.-Y.P. Hwang, Z. Phys. C 16 (1983) 327; I. Picek and D. Tadic, Phys. Rev. D 27 (1983) 665. [ 17] R.H. Dalitz and F. yon Hippel, Phys. Lett. 10 (1964) 153; A.J. Macfarlane and E.C.G. Sudarshan, Nuovo Cimento 31 (1964) 1176; J. Franklin, D. Lichtenberg, W. Namgung and D. Carydas, Phys. Rev. D 24 (1981) 2910. [ 18 ] D.W. Hertzog et al., Phys. Rev. Lett. 51 (1983) 1131. [ 19] G. Zapalec et al., Phys. Rev. Lett. 57 (1986) 1526. [20] M. Bourquin et al., Z. Phys. C 21 (1983) 27. [21 ] J.O. Eeg and H. Pilkuhn, Z. Phys. A 287 (1978) 407; G.E. Brown, M. Rho and V. Vento, Phys. Lett. B 97 (1980) 434; F. Mybrer, Phys. Lett. B 125 (1983) 359; G.E. Brown and F. Myhrer, Phys. Lett. B 128 (1983) 229; S. Theberge and A.W. Thomas, Nucl. Phys. A 393 (1983) 252; Z. Dziembowski and L. Mankiewicz, Phys. Rev. D 36 (1987).
PHYSICS LETTERSB
21 January 1988
MAGNETIC M O M E N T S OF THE N U C L E O N OCTET IN A RELATIVISTIC L I G H T - C O N E QUARK M O D E L Zbigniew DZIEMBOWSKI I, Tomasz DZURAK, Adam SZCZEPANIAK Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-O0-681 Warsaw, Poland
and Lech MANKIEWICZ Nicolaus Copernicus Astronomical Centre of Polish Academy of Sciences, Bartycka 18, PL-00-716 Warsaw, Poland
Received 24 September 1987
The magnetic momentsof the baryon octet are calculated in a relativistic constituent quark modelformulated in the light-cone Fock approach. Invokinga natural idea of strangeness-dependent hadron size we find verygoodagreement (to an accuracyof la) for the recent precision hyperonmagnetic-momentdata and reveal 10-15%discrepancyfor the Z - and nucleon. It suggestspions as a missing ingredient in the baryon magnetic moment calculations.
Recently a positive progress has occurred in our understanding of the composition ofhadrons in terms of their quark quanta. Several powerful nonperturbative methods have been developed which allow detailed predictions for the hadronic wavefunctions directly from QCD. Sum rule analysis of Chernyak, Zhitnitsky and Zhitnitsky [ 1 ] and lattice gauge theory calculations [ 2 ] have demonstrated that the nucleon and pion valence-quark distribution amplitudes are highly structured and significantly broader than the nonrelativistic 0-function form. In a recent work [ 3,4] we have attempted to bridge the results ofnonperturbative QCD methods and the quark model approach. Three ideas turn out to be vital for a successful derivation of the basic features of the CZ distribution amplitude, i.e. (a) the use of the lightcone Fock approach, (b) a nonstatic relativistic spin wavefunction, and (c) small transverse size of the valence-quark configuration. The presented model, together with the concept of scale-dependent effective quark mass provides a consistent description of the measured high momentum transfer form factors, This research was supported in part by the Research Program CPBP-0103.
the CZ distribution amplitudes, and some basic lowenergy nucleon and pion properties [5]. The purpose of this paper is to investigate some other predictions of the new relativistic approach. Thus, we present an extension of the nucleon wavefunction to the case of strange panicles and discuss magnetic moments of the nucleon octet. The motivation for the work is provided by new, accurate data [ 6 ] on the magnetic moments of the charged E and F, hyperons, along with substantial discrepancies between the data and the static quark-model predictions. Let us quote as a typical example of the static model results the prediction by Rosner [ 7 ]. Table 1 shows a comparison with the recent experimental data. There are already model-independent analyses of the observed disagreement, due to Franklin [ 8 ] and Lipkin [9]. They come to the conclusion that a good understanding of baryon magnetic moments will require a model with quark-moment contributions which are nonstatic and/or baryon dependent. A role of relativistic effects in the context have also been emphasized by several authors [ 10,11 ]. We use the light-cone formalism [ 12 ], which provides a consistent relativistic framework in momentum space in terms of Fock-state basis defined at
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
539
Volume 200, number 4
PHYSICS LETTERS B
Table 1 Baryon magnetic moments (in nuclear magnetons) in the static constituent quark model (CQM).
21 January 1988
The resultant Lorentz-invariant light-cone wavefunctions are ~,~(i, 2, 3)
Baryon moments
Experiment
/~(p) /t(n) kt(A) /~(Z + ) #(X-) #(.-o) ,u(E )
2.793+0.000 - 1.913+0.000 -0.613+0.004 2.379+0.020 - 1 . 1 4 +0.05 - 1.250+0.014 - 0 . 6 9 +0.04
Static CQM
-- 1/2
2.79 - 1.86 -0.58 2.68 -1.05 - 1.40 -0.47
Their normalizations are given by 2 f [dxd2k±] ~ [~p(1, 2, 3) 12 = 1.
equal-x += t + z, rather than the conventional equalt wavefunctions. With the valence-quark-dominance assumption baryon wavefunctions is taken to be simple generalization of the nonrelativistic constituent quark model one. In view of the relativistic motion of quarks, the momentum distribution is taken to be as a relativistic Gauss,an [ 12 ]
O(x" k ± ' ) = A e x p ( - 6 - ~ B ~
k~,+__m~. x, /
(1)
The baryon states of interest have two identical quarks (except those of the A) which we shall label with i = 1 and 2. The overall symmetry of the wavefunction in momentum, spin, and flavor spaces then implies that the spin-flavor wavefunctions for B = p, n, Z +, Z - , E °, and E - are symmetric under exchange of 1 and 2, but for the A it is ant,symmetric. They have the form ZP(x,, k±,, 2,) = J , ( i , 3, ~) + J , ( 2 , 3, i ) ,
(2a)
with J)(i, ~, 3 ) = aa,(Ma
+P.~,")~',v~2aa3u,,
blxl(MA
+Pv~U)YsVx2U-~3u) ,
(2b)
i, ~ and 3 are collective momentum-helicity indices (xi, k±i, ).~), i= 1, 2, 3. u~ and v~ are the light-cone spinors of ref. [13]. We keep flavor and color implicit. The nonstatic spin wavefunctions (2) are obtained from conventional wavefunctions with jo = ½+ transformed to the light-cone using a Meloshtype rotation of the quark spinors [4]. 540
I's~ = ( B ( P + q , "f) IJ+ ( O)IP + IB(P, ,L) )
=X I[ax d2kl 1~';~"(i', 2', ~')(am/,/~) X y + Qm(Um]x~mm)~B(], 2, ~) ,
(4)
where Qm is the charge of the struck quark with the final momentum k'±m=k±m+( 1 - - X m ) q Z while the spectator quark has final momentum k,, ,= k i i - x , q ± for i # m. Note, that we neglect the quark anomalous magnetic moments. If the light-cone coordinates are chosen [ 15] as pu= (p+, M~/p+, 0±) for the baryon moving along the z-axis and qU= (0, 2P.q/P +, q± ) for the photon, the helicity-flip matrix element of the current j + = j o + j 3 has the simple form
Fs} = - qL Fz( qZ)/Ma ,
and for the A
zA(x,, kzi, ~'i) =
The [dx d2k± ] is the volume element in momentum space. We emphasize the point made earlier that the nucleon wavefunction of the form (3), together with the small transverse size hypothesis provides the essential features of the CZ distribution amplitudes. We start by discussing the magnetic moments. The anomalous magnetic moment of any spin-1/2 system can be identified [ 14] from the spin-flip matrix element of the electromagnetic current f f
where qL=ql--iq 2 is used for the transverse momentum transfer. Hence, the anomalous magnetic moment a = F 2 ( 0 ) becomes
a = - - M B Oq----[ 0 F~f+ q=o
(5)
It was pointed out by several authors [ 16 ] that the spinor rotation of constituent quarks, arising from a Lorentz transformation associated with the boost P" to P" + qU, gives rise to sizable corrections to baryon
Volume 200, number
PHYSICS
4
LETTERS
magnetic moments. Note that the Drell-Yan formula (4) is especially suited to study the effects. It is related to the following advantages obtained by using the light-cone formalism: (i) There is no Wignerlike rotation in (4). (ii) Wavefunctions (3) are invariant under all kinematical Lorentz transformations, which contain the Lorentz boost along the three directions. Thus the simple and exact boost treatment, together with the proper relativistic kinematics of the internal relative motion present in (4), apparently invalidate the ordinary independentquark model additivity assumption. To illustrate numerical importance of the effects we use the basic wavefunctions (3) and formula (5) to calculate the baryon magnetic moments. The parameters entering our expressions for the magnetic moments are the quark masses and the momentum scale (Ye which determines the size of baryon valence wavefunction. We assume that quarks in baryons have typical constituent masses. To be specific we use the values m,,= md= 363 MeV and m,= 538 MeV given by Rosner’s lit to baryon masses. For the momentum scale we have decided to vary it freely in the range 300-500 MeV, in order to show explicitly the dependence on this parameter. The results for the seven measured magnetic moments are given in table 2. Before being compared with experiment, the A moment have to be corrected for A-CO mixing [ 171, which changes p(A) by about -0.04 PN.
General characteristics exhibited are briefly discussed below: Table 2 Batyon magnetic
a) Value corrected
moments
by these results
(in nuclear magnetons)
ffB
P(P)
300 320 340 360 380 400 420 440 460 480 500
2.737 2.718 2.691 2.676 2.653 2.629 2.605 2.580 2.554 2.528 2.502
for the effect of A-Z’
- 1.686 - 1.663 - 1.640 - 1.615 - 1.590 - 1.564 - 1.538 -1.511 - 1.484 - I.457 - 1.429
in the relativistic
-0.638 -0.635 -0.631 -0.621 -0.623 -0.619 -0.614 - 0.609 - 0.604 -0.598 -0.592
mixing of ref.
21 January
B
1988
(a) In the nucleon sector, the theoretical predictions are, for any scale c+, too small compared to the experimental values. If we take the nucleon momentum scale be equal to, say = 320-360 MeV, as in ref. [ 51, then we must consider other effects to account for the missing 1O-l 5% in the observed proton and neutron magnetic moments. (b) In the strange baryon sector a remarkable regularity can be observed. We note that the measured magnetic moments of all hyperons but the C - can be reproduced (to an accuracy of 1a) if one a]]OW an to increase with strangeness. This fit yields the hyperon momentum scale of cr,,= (-Yez 420 MeV and ~~~~440 MeV. For the C- our relativistic calculation gives value discrepant by w 0.1 &,, which is several times the standard deviations of the recent data both from tine-structure splitting in C - exotic atoms [ 181 and beam-polarization-precession technique [ 191. Thus one again has another case which suggests the importance of some other contributions. Let us mention that similar features are observed in the relativistic model calculation of the hyperon axial-vector couplings. The calculation of axial-vector form factors is essentially identical to that of the EM form factor in eq. (4) except of the replacement The prediction on G,/Gv for measured Y++Y0+. transitions [20] are given in table 3. Again the relativistic calculation with a dependence of the hadron size on the number of strange quarks provides very good (to an accuracy of 2~) description of the data. To understand intuitively the observed hierarchy
CQM as functions
2.48 1 2.464 2.446 2.428 2.408 2.389 2.368 2.347 2.325 2.303 2.28 I
_ 1.00 - 1.00 - 1.00 - 1.00 -0.99 -0.99 -0.99 -0.98 -0.98 -0.98 -0.98
of the baryon
_
1.330 1.320 1.309 1.297 1.284 1.272 1.258 1.245 1.231 -1.217 - 1.202
momentum
scale on (in MeV).
-0.60 -0.61 -0.61 -0.62 -0.62 -0.63 -0.64 -0.65 -0.65 -0.65 -0.66
[ 171. 541
Volume 200, number 4
PHYSICS LETTERS B
21January1988
Table 3 Baryon axial-vector couplings in the relativistic CQM. a
Transition E--~A
Z --,A
A~p
Z ~n
300 400 500
0.33 0.32 0.30
0 0 0
0.83 0.71 0.58
-0.28 -0.24 -0.20
experiment [20]
0.25+0.05
0.03+0.08
0.70+0.03
-0.34+0.05
o f the b a r y o n spatial sizes ( w h i c h d e c r e a s e w i t h strangeness) we n o t e t h a t for a C o u l o m b - l i k e p o t e n tial the b o u n d state size is p r o p o r t i o n a l to m - ~, w h e r e m is the r e d u c e d mass. T h u s , in p o t e n t i a l m o d e l s o n e can a n t i c i p a t e a d e c r e a s e o f t h e h a d r o n size w h e n a d d i n g strange q u a r k . In ref. [ 10] Isgur a n d K a r l q u o t e a d e c r e a s e by 4% a n d 13% p e r a d d i t i o n a l strange q u a r k in a h a r m o n i c a n d C o u l o m b p o t e n t i a l , respectively. O u r m a i n c o n c l u s i o n t h e r e f o r e is that the relativistic C Q M , t o g e t h e r w i t h t h e c o n c e p t o f strangenessd e p e n d e n t b a r y o n size, offers a large q u a n t i t a t i v e i m p r o v e m e n t o v e r the n o n s t a t i c d e s c r i p t i o n o f t h e h y p e r o n m a g n e t i c m o m e n t s . W i t h the large b a r y o n dependent non-additive magnetic moment contrib u t i o n s the m o d e l fulfills t h e r e q u i r e m e n t s o f general q u a r k - m o d e l analyses o f refs. [ 8,9 ]. T h e residual d i s a g r e e m e n t j u s t for t h e Z - a n d n u c l e o n suggests the n a t u r e o f the d o m i n a n t m i s s i n g i n g r e d i e n t in the b a r y o n w a v e f u n c t i o n s . It is k n o w n f r o m w o r k o f several a u t h o r s [21 ] that for t h o s e t h r e e b a r y o n s magnetic m o m e n t c o n t r i b u t i o n s f r o m a n o n v a l e n c e q q c o m p o n e n t w i t h p i o n q u a n t u m n u m b e r s are o f numerical importance.
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