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Magnetic moments of hyperons in the bound state approach to the skyrme model
Volume 231, number 4
PHYSICS LETTERS B
16 November 1989
MAGNETIC MOMENTS OF HYPERONS IN T H E B O U N D STATE A P P R O A C H T O T H E S K Y R M E M O D E L J. K U N Z and P.J. M U L D E R S National Institute for Nuclear Physics and High Energy Physics, NIKttEF-K P.O. Box 41882, NL-IO09 DB Amsterdam. The Netherlands
Received 24 July 1989
We construct the electromagnetic current in the bound state approach to the Skyrme model using the Callan-Klebanov ansatz. The current is used to calculate the magnetic moments of the baryons for the case ofmassless pions. The agreement of the magnetic moments in the bound state approach with the experimental magnetic moments is qualitatively satisfying. We disagree with previous work by Nyman and Riska.
In the Skyrme model S U ( 3 ) flavor symmetry is strongly broken due to the mass differences between the nonstrange and the strange mesons, notably the mass difference between pions and kaons, the lightest pseudoscalar mesons. In the simplest version of the Skyrme model [ 1,2 ], based on a derivative expansion with a second order and a fourth order term (the Skyrme term), these are the only mesons taken into account. The other essential part of the model is the Wess-Zumino (WZ) term. The baryons arise as solitons in the Skyrme model in the spirit of the 1/Nc expansion, where Arc is the number of colors in quantum chromodynamics. Callan, Hornbostel and Klebanov [ 3,4 ] showed that a satisfactory description of the mass spectrum of the ground state baryons could be obtained by treating the kaons and pions on a different footing. The classical kaon field vanishes and the kaon fluctuations around the classical hedgehog-like pion field are treated as vibrational modes. Only the SU (2) isospin zero modes are treated via collective variables. The W Z term discriminates between kaons and antikaons. The latter turn out to be bound in certain partial waves. The lowest bound state (with strangeness S = - 1 ) has an energy (o=0.310 MK= 153 MeV. In the background field of the classical hedgehog solution this bosonic state comes with isospin zero and intrinsic spin-parity J P = ½÷. In this paper we follow the procedure of ref. [4], choosing the ansatz for the chiral matrix
(1)
U = ~t;,~,
where ~= x/rET~ contains the pion fields, U~ = exp (i~.lt/,[,,) while E:K= exp (i2~. K,Jf,,) with the Gell-Mann SU ( 3 ) matrices ~.~ (the index a running from 4 to 7 ) describes the kaon fields in terms of the kaon doublets K,
~=4 ,~ _ ;.o.Ko
(:_o.° 0
=x/r}
0
.
(2)
Inserting the above ansatz for the ehiral matrix U in the Skyrme lagrangian and expanding to second order in the kaon gelds [ 3,4 ] leads to
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
335
Volume 231, number 4
PHYSICS LETTERS B
/ F2 /-/'= ~ Tr(a~ U~ a"U=) + ~
16 November 1989
. T r [ a , U~ U~, av U~U~]2+ (D,,K)*D~K-,n~.K*K
1
e21. 2 {2( D;,K)*D~K Tr(,.I;'A ~) + ~ (DvK)*D;'K Tr(c3~Ul i3~U=)-6(D;,K)*[A ~, A "]DvK} i,V,, F~
- ~-2 B;'[KtDuK- (D~K)*K],
(3)
where/-; = 2.[; and A,, = ½(~tB,,~--~0~*),
D;,K= 0~,K+ ½( ~ * 0 ~ + ~ 0 ~ , ~ t ) K .
(4,5)
The last term in eq. (3) originates from the WZ action and is proportional to N¢ and the baryon current 1
where tm 23= I. (We have carefully checked the sign of the WZ term and agree with the signs in refs. [ 3,5 ], we disagree with the signs in refs. [4,6,7]. The above sign renders the antikaons with strangeness S = - 1 bound in the Skyrmion background field ). To construct the baryons we choose the ansatz U~ (r) = exp [ it. PF(r) ] for the classical solution, with the chiral angle F(r) running from F(0 ) = r~ to F(oo ) = 0. The invariance of the background field U~ (r) under combined isospin and spatial rotations enables a partial wave L = I, T = ½, and can be written as
K ( r. t) =e-"°'k(r)r-P:Z,
(7)
where 7. is a two-component isospinor. Because of the WZ term the spectrum is asymmetric for positive and negative energy solutions. Only the negative energy solutions are bound. The explicit solutions for kaon and antikaon modes, k(r))5 (with the index i indicating 1"and J. states respectively), are obtained by solving KleinGordon type of equations as discussed extensively in refs. [4,6,7 ]. The kaon fields are quantized in the familiar way by replacing the mode amplitudes for kaons and antikaons by particle (a;) and antiparticle (b,) creation and annihilation operators. For the quantization of the zeromodes corresponding to the SU (2) isospin symmetry the semiclassical collective coordinate procedure is followed, which leads to the spin and isospin fine structure in the baryon spectrum. One performs time dependent collective isospin rotations/~;= ~A (t)U~A*(t) and K-,A(t)K=e-"°'k(r)A(t)r.PAt(t)A(t)X. The collective spin operator./, of the rotator and the total spin operator J are given by
J~=--if2Tr(,'l*/tr)--CrJK--CKJa,
I=J¢+ L
T=~[at,(lr),jaj-b;(r),jb]]=JK +JK,
(8,9,10)
where the numbers CKand CKare discussed below. The operator T is thus identified as the spin of the kaons. The lowest mode has T = ½. Note further, that the collective spin equals the isospin, g~2 _- 12. The strangeness operator is given by S=a~a,-bib~,. After the semiclassical quantization the mass operator for baryons with Na bound antikaons becomes
M = M d +NK~O+ ~I (l¢ + cr)2=M¢~ +N¢,co+ ~cj 2 +
- - ( 1 c}
~
12+ _ _ c ( c1- ) T'-,
2f2
(11)
where we have omitted the index K on c. The inertia .(2 and the number c are given as integrals involving the chiral angle and the kaon wave function, 336
Volume 231, number4
PHYSICS LETTERSB
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1 2 = -8n ~ I d r r 2 f ~ s i n 2 F [ 1+ e2__~.( ( F ' ) 2 + ~sin2F'~] T-jj, c=l-8n
drk2e)
(12)
~r(r2sinFF')-~sin2Fcos2(lF)
4-ff(r)r2cos2(½F)-
,
(13)
with the antikaon modes normalized to 8gfdrr2k2(r)[f(r)~o+2(r)]=l, where f ( r ) = 1+ [2 sin2(F)/ r2+ (F')21/e2F~ and 2 ( 0 = - (Nc/F~)sin2(F)F '/2~2r 2. Choosing the parameters e and./, to fit the N and A masses, implying Met = 0.865 GeV and 1 / 2 ~ = 98 MeV [ 21, the calculation gives ~o= 153 MeV and c= 0.617. A best fit to the baryon masses with the mass formula of eq. ( l 1 ) yields M~=0.867 GeV, 1/2~2=99.7 MeV, o)=215 MeV and c=0.661. Except for the value of co, which is too small, the fitted quantities are in excellent agreement with the calculated values. Given the lagrangian it is straightforward to obtain the electromagnetic current from Noethers theorem. In the bound state approach the current can be obtained in two ways. The first way starts with the known flavor currents in the SU (3) Skyrme model with quadratic, quartic and WZ term [ 2,81,
Jau = - ~'I.f . Tr( )~.UtOuU) + ( Uc-~Ut) + ~ ei2 Tr( 2a[
[ UtOuU, VtOvVl l ) W ( U~_.V,)
+ 96~n 2N¢ ~,t,o Tr(2~ Ut0%rU*0pUU*0%;) - (U,-*U*).
(14)
The electromagnetic current Ju~r, -J3,u -+ ( l/q/3 )J8 u in the bound state approach is then obtained by substituting the ansatz of eq. ( l ) including the collective variables A (t) into these currents. The second way is to apply Noethers theorem directly to the lagrangian in eq. (3), where the isoscalar baryon current is put in by hand, as in the pure SU(2) case [2]. The two methods yield the same result, providing a check on the quite lengthy calculations. The spatial part of the current has terms parallel and perpendicular to ~. In the magnetic moments only the perpendicular current enters, .Ix = m × r. We obtain for the isoscalar contribution of the electromagnetic current m3=°(r) -
- i Tr(A*Ar3) - s i n 2 ( F ) F ' 2r z 2rP
+ at,(r3)oaj+b,(r3),J b 2] 2 r
k2cosZ(~F) + ~
~4k
cos2(½F) +k2(F ' )2cos2 (~F) + 3kk'F'sin F
,
(15) and for the isovector contribution
m3=t(r)=-Tr(Atr3Ar3){ 2r 2
./'~sinZ(F)
[
4 ( sineF~] 1 + ~e/'. (F')Z+ ---~J_]
+ (a*,a, +b,b*i )k 2 cos2(½F) [ 1 - 4 sin2(½F) ] + ( ata~+b~bt)/ _sin2F 2 -2 ~,4k~c°sZ(½F)[3-8sin2(½F)]+k2(F')2c°s2(~F)[l-18sin2(½F)] e l~,~ + 2(k' )2sin2 (F) + 3kk' F' sin(F)[ 3 - 4 sin2(½F)]) } .
(16)
In order to evaluate the magnetic moments we have to evaluate the operators in eqs. (15) and (16) between 337
Volume 231, number 4
PHYSICS LETTERS B
16 November 1989
baryon states. The operators entering in the isoscalar part are -iTr(A*Jl'r)=(Jc+C.IK+C.IK)/f2 and [a~(r)oaj+b,(r),jb}] = J K - J e ~ , while the operators in the isovector part are T r ( A * r s A r 3 ) = - 7 ( J c , l)J3~l3 with the coefficients Y(Jc, I) given in table 1, and a~a, +b,b t, =NK +N~. Expressed in nuclear magnetons e/2MN the magnetic moment is given by the following expression: lt= ]MNfd3r r2mS(r). For a system consisting of a skyrmion with N~ bound antikaons, we find the magnetic moment operator
I1=al J3 + (a2 +a3Ng )Y(Jo l)j313+aaJ~,, where al =
( 17 )
a~-a4 are functionals of the chiral angle and the kaon wave function,
2MN f 3£2 J 4~r2 dr
-sin2(F)F ' 4n 2
MN (r2)B
-
M2
'
(18) (19)
a2 = ~ ]I//N.Q,
as=~MN f 4ztr2dr[ k2cos2(~F)[l-4sin2(~F)]+
l [ 2~4k s i n 2~Fc2o s
(~F)[3-8sin2(~F)]
e F,~
\q
+kZ(F ' )2cos2 (½F)[I - 1 8 sin2(~F) ] + 2 ( k ' )2sin2 ( F ) + 3kk'F' s i n ( F ) [ 3 - 4 sin2(~F) 1)],
(20)
, f 41rrZdr[ k 2 c o s 2 ( ~ F ) +1~ ( 4k2--cosZ(½F)+k2(F')2cos2(~F)+3kk'F'sinF sin2F )] aa=ca~--~MN e Fn r2 (21) The resulting magnetic moments of the hyperons that consist of a soliton quantized with integer or half integer spin and isospin depending on the number of bound kaons being odd or even respectively, are given in table 2. In table 3 the numerical results arc given. The calculated magnetic moments of the baryons are compared with the experimental magnetic moments. Furthermore the ratios of the magnetic moments to the proton magnetic moment are given. The calculated values are in general too small, by about 30%, just as in the SU(2) Skyrme model. For the ratios, however, the agreement between theory and experiment is remarkably good in view of the fact that they result from a calculation that is parameter-free, except for fitting e and F~ in the nonstrange sector to N and A masses. Notably the calculated ratio p^/pp= -0.221 comes out in perfect agreement with the experimental result p^/pp= - 0 . 2 1 9 + 0.002. In most other cases there is qualitative agreement leaving room for Fable 2 Matrix elements of the magnetic moment operator for baryons ( for the highest spin state ). Table 1
Panicle
Reduction of operator - T r ( A *rsATs) in soliton q u a n t u m states. (1, J¢ )
N .A A
0 ~j~313
~+ I:o
( l, [ )
J313
E-
(~, ~)
8
3
~Jd
-2/x/5
3
~al 4- ~a2
P n
(0, o) (½,~)
(0,0) . ( 1 , 1)
338
- Tr(A ~rjAz3 )
It
A°Z =,, =-
ia4
~,+~a~+ta,-'a, ~a,- ~a~- ~a,- ~a, - ~,~- ~,a~ --
lal
_
2 ~)a2 -- 4~a3 + ~a4
-~a~ +~,~ + ~,~ + ~ ,
Volume 231, number 4
PHYSICS LETTERS B
16 November 1989
Table 3 Numerical results for the magnetic moments of baryons. (Calculated coefficients: a, =0.557, a2=2.39, a3= -0.052, a4= -0.83. Fitted coefficients: a, =0.880, a2= 3.53, a3= - 1.02, a4= - 1.23.) Particle
/~
/l//~p
/l~p
(/~//~,)~,p
,un,
p n N--,A A Z+ 5-o ZA~Z =0 _.Ef~-
1.870 - 1.313 2.251 -0.414 2.066 0.509 - 1.048 -I.557 - 1.153 -0.138 - 1.243
1.000 -0.702 1.204 -0.221 1.105 0.272 -0.560 -0.833 -0.616 -0.074 -0.664
2.793 - 1.9013 ~3 -0.613+0.004 2.42 +_0.05 - 1.157+0.025 -1.61 +_0.08 - 1.250+0.014 -0.69 +-0.04 -
1.000 -0.685 ~ 1.1 -0.219+_0.002 0.87 +_0.02
2.793 - 1.913 2.251 -0.613 2.467 0.791 -0.885 -1.6760 - 1.297 -0.631 - 1.839
-0.414+_0.009 -0.58 _+0.03 -0.448 +0.005 -0.25 +_0.02
improvements. In the last column o f table 3 the result o f a fit of the magnetic m o m e n t s using the operator in eq. ( 1 7 ) is given. In this fit a,, a2, and a4 have been d e t e r m i n e d from the magnetic m o m e n t s o f p, n, and A, while a3 is d e t e r m i n e d from the other magnetic moments. While the fitted values for at, a2, and a4 are about 30% larger, but have the same ratio as the calculated coefficients, the fitted value for a3 is o f the same order o f magnitude as the other coefficients which is considerably ( ~ 20 × ) larger than the calculated coefficient. O u r results differ substantially from those o f N y m a n and Riska [9]. These differences cannot be attributed to the different ansatz, U = x / ~ K U~X/~K with which they start. In their derivation o f the isoscalar contribution to the magnetic moments, N y m a n and Riska have not accounted for the contribution cgJ~t in the collective spin o p e r a t o r (see eq. ( 8 ) ). This contribution affects the coefficient a4 o f J~t, as can be seen from eq. (21). This has also been pointed out in ref. [ 10]. Further, N y m a n and Riska do include one term,/~B, which originates from the W e s s - Z u m i n o term and which is o f higher order in 1/Nc c o m p a r e d to the other contributions. We have o m i t t e d this term in order to be consistent, since in the derivation o f the b o u n d slate lagrangian in eq. ( 3 ) one systematically neglects such higher order contributions [ 3 ]. As expected, however,/IB is small. A final difference is the interaction term in the isovector contribution to the magnetic moments, proportional to n~j313. This term looks like a kaon contribution to the isovector part o f the magnetic moments. We obtain this contribution in both derivations. Its calculated coefficient, a3, however, is quite small c o m p a r e d to the other coefficients. Clearly the calculation o f magnetic m o m e n t s o f hyperons in the bound state approach to the Skyrme model gives reasonable results. Already in the nonstrange sector, the result for the r a t i o / t , / / ~ p = 0 . 7 0 2 [2] turned out to be very close to the experimental result (0.685) and the SU ( 6 ) quark model ratio ( ] ). Also for the hyperons there is a good qualitative agreement with experimental magnetic m o m e n t s and with the results o f the ( b r o k e n ) S U ( 6 ) quark model [ 1 1 ], although the expressions as given in table 2 do not exhibit the same pattern as the ( b r o k e n ) SU ( 6 ) quark model. A reason for the overall reasonable results in the extension from nonstrange to strange baryons is the fact that the construction o f baryons from the soliton core and the antikaons closely follows the construction in the quark model. While in the quark model the nonstrange quarks in the baryons couple to i s o s p i n - s p i n c o m b i n a t i o n s (i, J ) with l=J, the same holds for the soliton core. The role o f the strange quarks is taken over by the antikaons. The difference arising in the symmetry o f the wave functions because the former are fcrmions and the latter are bosons is compensated by the fully a n t i s y m m e t r i c color wave functions o f the quarks in a color singlet hadron. The electromagnetic current can be used to study further properties of baryons such as charge and magnetic current densities, e.g. the electric and magnetic mean square radii. In this case it becomes necessary, however, to consider massive pions since the l/r 2 b e h a v i o r o f F(r) at large distances yields divergent results for the isovector radii [ 2,12 ]. We will present those results elsewhere. 339
Volume 231, number 4
PHYSICS LETTERS B
16 November 1989
T h e a u t h o r s w o u l d like to a c k n o w l e d g e d i s c u s s i o n s with M. C h e m t o b ( S a c l a y ) . T h i s w o r k is s u p p o r t e d by the F o u n d a t i o n for F u n d a m e n t a l R e s e a r c h ( F O M ) a n d the N e t h e r l a n d s O r g a n i z a t i o n for the A d v a n c e m e n t o f Scientific R e s e a r c h ( N W O ) .
References [ 1 ] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 ) 127; Nucl. Phys. 31 (1962) 556. [2] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [3] C.G. Callan and I. Klebanov, Nucl. Phys. B 262 (1985) 365. [4] C.G. Callan, K. Hornborstel and I. Klebanov, Phys. Lett. B 202 (1988) 269. [ 5 ] N. Scoccola, H.Nadeau, M.Nowak and M. Rho, Phys. Left. B 201 ( 1988 ) 425. [6] J. Kunz and P.J. Mulders, Phys. Lett. B 215 (1988) 449. [7] U. Blom, K. Dannbom and D.O. Riska, Nucl. Phys. A 493 (1989) 384. [81 A. Kanazawa, Prog. Theor. Phys. 77 ( 1987 ) 1240. [9] E.M. Nyman and D.O. Riska, University of Helsinki preprint HU-TFT-88-50 (November 1988 ). [ 101 D.P. Min, Y.S. Koh and H.K. Lee, preprint SNU-PHY-NT-8903 (1989). [ 11 ] See e.g.D.H. Perkins, Introduction to high energy physics (Addison-Wesley, Reading, MA, 1989). [121 G.S. Adkins and C.R. Nappi, Nucl. Phys. B 233 (1984) 109.
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PHYSICS LETTERS B
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MAGNETIC MOMENTS OF HYPERONS IN T H E B O U N D STATE A P P R O A C H T O T H E S K Y R M E M O D E L J. K U N Z and P.J. M U L D E R S National Institute for Nuclear Physics and High Energy Physics, NIKttEF-K P.O. Box 41882, NL-IO09 DB Amsterdam. The Netherlands
Received 24 July 1989
We construct the electromagnetic current in the bound state approach to the Skyrme model using the Callan-Klebanov ansatz. The current is used to calculate the magnetic moments of the baryons for the case ofmassless pions. The agreement of the magnetic moments in the bound state approach with the experimental magnetic moments is qualitatively satisfying. We disagree with previous work by Nyman and Riska.
In the Skyrme model S U ( 3 ) flavor symmetry is strongly broken due to the mass differences between the nonstrange and the strange mesons, notably the mass difference between pions and kaons, the lightest pseudoscalar mesons. In the simplest version of the Skyrme model [ 1,2 ], based on a derivative expansion with a second order and a fourth order term (the Skyrme term), these are the only mesons taken into account. The other essential part of the model is the Wess-Zumino (WZ) term. The baryons arise as solitons in the Skyrme model in the spirit of the 1/Nc expansion, where Arc is the number of colors in quantum chromodynamics. Callan, Hornbostel and Klebanov [ 3,4 ] showed that a satisfactory description of the mass spectrum of the ground state baryons could be obtained by treating the kaons and pions on a different footing. The classical kaon field vanishes and the kaon fluctuations around the classical hedgehog-like pion field are treated as vibrational modes. Only the SU (2) isospin zero modes are treated via collective variables. The W Z term discriminates between kaons and antikaons. The latter turn out to be bound in certain partial waves. The lowest bound state (with strangeness S = - 1 ) has an energy (o=0.310 MK= 153 MeV. In the background field of the classical hedgehog solution this bosonic state comes with isospin zero and intrinsic spin-parity J P = ½÷. In this paper we follow the procedure of ref. [4], choosing the ansatz for the chiral matrix
(1)
U = ~t;,~,
where ~= x/rET~ contains the pion fields, U~ = exp (i~.lt/,[,,) while E:K= exp (i2~. K,Jf,,) with the Gell-Mann SU ( 3 ) matrices ~.~ (the index a running from 4 to 7 ) describes the kaon fields in terms of the kaon doublets K,
~=4 ,~ _ ;.o.Ko
(:_o.° 0
=x/r}
0
.
(2)
Inserting the above ansatz for the ehiral matrix U in the Skyrme lagrangian and expanding to second order in the kaon gelds [ 3,4 ] leads to
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
335
Volume 231, number 4
PHYSICS LETTERS B
/ F2 /-/'= ~ Tr(a~ U~ a"U=) + ~
16 November 1989
. T r [ a , U~ U~, av U~U~]2+ (D,,K)*D~K-,n~.K*K
1
e21. 2 {2( D;,K)*D~K Tr(,.I;'A ~) + ~ (DvK)*D;'K Tr(c3~Ul i3~U=)-6(D;,K)*[A ~, A "]DvK} i,V,, F~
- ~-2 B;'[KtDuK- (D~K)*K],
(3)
where/-; = 2.[; and A,, = ½(~tB,,~--~0~*),
D;,K= 0~,K+ ½( ~ * 0 ~ + ~ 0 ~ , ~ t ) K .
(4,5)
The last term in eq. (3) originates from the WZ action and is proportional to N¢ and the baryon current 1
where tm 23= I. (We have carefully checked the sign of the WZ term and agree with the signs in refs. [ 3,5 ], we disagree with the signs in refs. [4,6,7]. The above sign renders the antikaons with strangeness S = - 1 bound in the Skyrmion background field ). To construct the baryons we choose the ansatz U~ (r) = exp [ it. PF(r) ] for the classical solution, with the chiral angle F(r) running from F(0 ) = r~ to F(oo ) = 0. The invariance of the background field U~ (r) under combined isospin and spatial rotations enables a partial wave L = I, T = ½, and can be written as
K ( r. t) =e-"°'k(r)r-P:Z,
(7)
where 7. is a two-component isospinor. Because of the WZ term the spectrum is asymmetric for positive and negative energy solutions. Only the negative energy solutions are bound. The explicit solutions for kaon and antikaon modes, k(r))5 (with the index i indicating 1"and J. states respectively), are obtained by solving KleinGordon type of equations as discussed extensively in refs. [4,6,7 ]. The kaon fields are quantized in the familiar way by replacing the mode amplitudes for kaons and antikaons by particle (a;) and antiparticle (b,) creation and annihilation operators. For the quantization of the zeromodes corresponding to the SU (2) isospin symmetry the semiclassical collective coordinate procedure is followed, which leads to the spin and isospin fine structure in the baryon spectrum. One performs time dependent collective isospin rotations/~;= ~A (t)U~A*(t) and K-,A(t)K=e-"°'k(r)A(t)r.PAt(t)A(t)X. The collective spin operator./, of the rotator and the total spin operator J are given by
J~=--if2Tr(,'l*/tr)--CrJK--CKJa,
I=J¢+ L
T=~[at,(lr),jaj-b;(r),jb]]=JK +JK,
(8,9,10)
where the numbers CKand CKare discussed below. The operator T is thus identified as the spin of the kaons. The lowest mode has T = ½. Note further, that the collective spin equals the isospin, g~2 _- 12. The strangeness operator is given by S=a~a,-bib~,. After the semiclassical quantization the mass operator for baryons with Na bound antikaons becomes
M = M d +NK~O+ ~I (l¢ + cr)2=M¢~ +N¢,co+ ~cj 2 +
- - ( 1 c}
~
12+ _ _ c ( c1- ) T'-,
2f2
(11)
where we have omitted the index K on c. The inertia .(2 and the number c are given as integrals involving the chiral angle and the kaon wave function, 336
Volume 231, number4
PHYSICS LETTERSB
16 November 1989
1 2 = -8n ~ I d r r 2 f ~ s i n 2 F [ 1+ e2__~.( ( F ' ) 2 + ~sin2F'~] T-jj, c=l-8n
drk2e)
(12)
~r(r2sinFF')-~sin2Fcos2(lF)
4-ff(r)r2cos2(½F)-
,
(13)
with the antikaon modes normalized to 8gfdrr2k2(r)[f(r)~o+2(r)]=l, where f ( r ) = 1+ [2 sin2(F)/ r2+ (F')21/e2F~ and 2 ( 0 = - (Nc/F~)sin2(F)F '/2~2r 2. Choosing the parameters e and./, to fit the N and A masses, implying Met = 0.865 GeV and 1 / 2 ~ = 98 MeV [ 21, the calculation gives ~o= 153 MeV and c= 0.617. A best fit to the baryon masses with the mass formula of eq. ( l 1 ) yields M~=0.867 GeV, 1/2~2=99.7 MeV, o)=215 MeV and c=0.661. Except for the value of co, which is too small, the fitted quantities are in excellent agreement with the calculated values. Given the lagrangian it is straightforward to obtain the electromagnetic current from Noethers theorem. In the bound state approach the current can be obtained in two ways. The first way starts with the known flavor currents in the SU (3) Skyrme model with quadratic, quartic and WZ term [ 2,81,
Jau = - ~'I.f . Tr( )~.UtOuU) + ( Uc-~Ut) + ~ ei2 Tr( 2a[
[ UtOuU, VtOvVl l ) W ( U~_.V,)
+ 96~n 2N¢ ~,t,o Tr(2~ Ut0%rU*0pUU*0%;) - (U,-*U*).
(14)
The electromagnetic current Ju~r, -J3,u -+ ( l/q/3 )J8 u in the bound state approach is then obtained by substituting the ansatz of eq. ( l ) including the collective variables A (t) into these currents. The second way is to apply Noethers theorem directly to the lagrangian in eq. (3), where the isoscalar baryon current is put in by hand, as in the pure SU(2) case [2]. The two methods yield the same result, providing a check on the quite lengthy calculations. The spatial part of the current has terms parallel and perpendicular to ~. In the magnetic moments only the perpendicular current enters, .Ix = m × r. We obtain for the isoscalar contribution of the electromagnetic current m3=°(r) -
- i Tr(A*Ar3) - s i n 2 ( F ) F ' 2r z 2rP
+ at,(r3)oaj+b,(r3),J b 2] 2 r
k2cosZ(~F) + ~
~4k
cos2(½F) +k2(F ' )2cos2 (~F) + 3kk'F'sin F
,
(15) and for the isovector contribution
m3=t(r)=-Tr(Atr3Ar3){ 2r 2
./'~sinZ(F)
[
4 ( sineF~] 1 + ~e/'. (F')Z+ ---~J_]
+ (a*,a, +b,b*i )k 2 cos2(½F) [ 1 - 4 sin2(½F) ] + ( ata~+b~bt)/ _sin2F 2 -2 ~,4k~c°sZ(½F)[3-8sin2(½F)]+k2(F')2c°s2(~F)[l-18sin2(½F)] e l~,~ + 2(k' )2sin2 (F) + 3kk' F' sin(F)[ 3 - 4 sin2(½F)]) } .
(16)
In order to evaluate the magnetic moments we have to evaluate the operators in eqs. (15) and (16) between 337
Volume 231, number 4
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16 November 1989
baryon states. The operators entering in the isoscalar part are -iTr(A*Jl'r)=(Jc+C.IK+C.IK)/f2 and [a~(r)oaj+b,(r),jb}] = J K - J e ~ , while the operators in the isovector part are T r ( A * r s A r 3 ) = - 7 ( J c , l)J3~l3 with the coefficients Y(Jc, I) given in table 1, and a~a, +b,b t, =NK +N~. Expressed in nuclear magnetons e/2MN the magnetic moment is given by the following expression: lt= ]MNfd3r r2mS(r). For a system consisting of a skyrmion with N~ bound antikaons, we find the magnetic moment operator
I1=al J3 + (a2 +a3Ng )Y(Jo l)j313+aaJ~,, where al =
( 17 )
a~-a4 are functionals of the chiral angle and the kaon wave function,
2MN f 3£2 J 4~r2 dr
-sin2(F)F ' 4n 2
MN (r2)B
-
M2
'
(18) (19)
a2 = ~ ]I//N.Q,
as=~MN f 4ztr2dr[ k2cos2(~F)[l-4sin2(~F)]+
l [ 2~4k s i n 2~Fc2o s
(~F)[3-8sin2(~F)]
e F,~
\q
+kZ(F ' )2cos2 (½F)[I - 1 8 sin2(~F) ] + 2 ( k ' )2sin2 ( F ) + 3kk'F' s i n ( F ) [ 3 - 4 sin2(~F) 1)],
(20)
, f 41rrZdr[ k 2 c o s 2 ( ~ F ) +1~ ( 4k2--cosZ(½F)+k2(F')2cos2(~F)+3kk'F'sinF sin2F )] aa=ca~--~MN e Fn r2 (21) The resulting magnetic moments of the hyperons that consist of a soliton quantized with integer or half integer spin and isospin depending on the number of bound kaons being odd or even respectively, are given in table 2. In table 3 the numerical results arc given. The calculated magnetic moments of the baryons are compared with the experimental magnetic moments. Furthermore the ratios of the magnetic moments to the proton magnetic moment are given. The calculated values are in general too small, by about 30%, just as in the SU(2) Skyrme model. For the ratios, however, the agreement between theory and experiment is remarkably good in view of the fact that they result from a calculation that is parameter-free, except for fitting e and F~ in the nonstrange sector to N and A masses. Notably the calculated ratio p^/pp= -0.221 comes out in perfect agreement with the experimental result p^/pp= - 0 . 2 1 9 + 0.002. In most other cases there is qualitative agreement leaving room for Fable 2 Matrix elements of the magnetic moment operator for baryons ( for the highest spin state ). Table 1
Panicle
Reduction of operator - T r ( A *rsATs) in soliton q u a n t u m states. (1, J¢ )
N .A A
0 ~j~313
~+ I:o
( l, [ )
J313
E-
(~, ~)
8
3
~Jd
-2/x/5
3
~al 4- ~a2
P n
(0, o) (½,~)
(0,0) . ( 1 , 1)
338
- Tr(A ~rjAz3 )
It
A°Z =,, =-
ia4
~,+~a~+ta,-'a, ~a,- ~a~- ~a,- ~a, - ~,~- ~,a~ --
lal
_
2 ~)a2 -- 4~a3 + ~a4
-~a~ +~,~ + ~,~ + ~ ,
Volume 231, number 4
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16 November 1989
Table 3 Numerical results for the magnetic moments of baryons. (Calculated coefficients: a, =0.557, a2=2.39, a3= -0.052, a4= -0.83. Fitted coefficients: a, =0.880, a2= 3.53, a3= - 1.02, a4= - 1.23.) Particle
/~
/l//~p
/l~p
(/~//~,)~,p
,un,
p n N--,A A Z+ 5-o ZA~Z =0 _.Ef~-
1.870 - 1.313 2.251 -0.414 2.066 0.509 - 1.048 -I.557 - 1.153 -0.138 - 1.243
1.000 -0.702 1.204 -0.221 1.105 0.272 -0.560 -0.833 -0.616 -0.074 -0.664
2.793 - 1.9013 ~3 -0.613+0.004 2.42 +_0.05 - 1.157+0.025 -1.61 +_0.08 - 1.250+0.014 -0.69 +-0.04 -
1.000 -0.685 ~ 1.1 -0.219+_0.002 0.87 +_0.02
2.793 - 1.913 2.251 -0.613 2.467 0.791 -0.885 -1.6760 - 1.297 -0.631 - 1.839
-0.414+_0.009 -0.58 _+0.03 -0.448 +0.005 -0.25 +_0.02
improvements. In the last column o f table 3 the result o f a fit of the magnetic m o m e n t s using the operator in eq. ( 1 7 ) is given. In this fit a,, a2, and a4 have been d e t e r m i n e d from the magnetic m o m e n t s o f p, n, and A, while a3 is d e t e r m i n e d from the other magnetic moments. While the fitted values for at, a2, and a4 are about 30% larger, but have the same ratio as the calculated coefficients, the fitted value for a3 is o f the same order o f magnitude as the other coefficients which is considerably ( ~ 20 × ) larger than the calculated coefficient. O u r results differ substantially from those o f N y m a n and Riska [9]. These differences cannot be attributed to the different ansatz, U = x / ~ K U~X/~K with which they start. In their derivation o f the isoscalar contribution to the magnetic moments, N y m a n and Riska have not accounted for the contribution cgJ~t in the collective spin o p e r a t o r (see eq. ( 8 ) ). This contribution affects the coefficient a4 o f J~t, as can be seen from eq. (21). This has also been pointed out in ref. [ 10]. Further, N y m a n and Riska do include one term,/~B, which originates from the W e s s - Z u m i n o term and which is o f higher order in 1/Nc c o m p a r e d to the other contributions. We have o m i t t e d this term in order to be consistent, since in the derivation o f the b o u n d slate lagrangian in eq. ( 3 ) one systematically neglects such higher order contributions [ 3 ]. As expected, however,/IB is small. A final difference is the interaction term in the isovector contribution to the magnetic moments, proportional to n~j313. This term looks like a kaon contribution to the isovector part o f the magnetic moments. We obtain this contribution in both derivations. Its calculated coefficient, a3, however, is quite small c o m p a r e d to the other coefficients. Clearly the calculation o f magnetic m o m e n t s o f hyperons in the bound state approach to the Skyrme model gives reasonable results. Already in the nonstrange sector, the result for the r a t i o / t , / / ~ p = 0 . 7 0 2 [2] turned out to be very close to the experimental result (0.685) and the SU ( 6 ) quark model ratio ( ] ). Also for the hyperons there is a good qualitative agreement with experimental magnetic m o m e n t s and with the results o f the ( b r o k e n ) S U ( 6 ) quark model [ 1 1 ], although the expressions as given in table 2 do not exhibit the same pattern as the ( b r o k e n ) SU ( 6 ) quark model. A reason for the overall reasonable results in the extension from nonstrange to strange baryons is the fact that the construction o f baryons from the soliton core and the antikaons closely follows the construction in the quark model. While in the quark model the nonstrange quarks in the baryons couple to i s o s p i n - s p i n c o m b i n a t i o n s (i, J ) with l=J, the same holds for the soliton core. The role o f the strange quarks is taken over by the antikaons. The difference arising in the symmetry o f the wave functions because the former are fcrmions and the latter are bosons is compensated by the fully a n t i s y m m e t r i c color wave functions o f the quarks in a color singlet hadron. The electromagnetic current can be used to study further properties of baryons such as charge and magnetic current densities, e.g. the electric and magnetic mean square radii. In this case it becomes necessary, however, to consider massive pions since the l/r 2 b e h a v i o r o f F(r) at large distances yields divergent results for the isovector radii [ 2,12 ]. We will present those results elsewhere. 339
Volume 231, number 4
PHYSICS LETTERS B
16 November 1989
T h e a u t h o r s w o u l d like to a c k n o w l e d g e d i s c u s s i o n s with M. C h e m t o b ( S a c l a y ) . T h i s w o r k is s u p p o r t e d by the F o u n d a t i o n for F u n d a m e n t a l R e s e a r c h ( F O M ) a n d the N e t h e r l a n d s O r g a n i z a t i o n for the A d v a n c e m e n t o f Scientific R e s e a r c h ( N W O ) .
References [ 1 ] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 ) 127; Nucl. Phys. 31 (1962) 556. [2] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [3] C.G. Callan and I. Klebanov, Nucl. Phys. B 262 (1985) 365. [4] C.G. Callan, K. Hornborstel and I. Klebanov, Phys. Lett. B 202 (1988) 269. [ 5 ] N. Scoccola, H.Nadeau, M.Nowak and M. Rho, Phys. Left. B 201 ( 1988 ) 425. [6] J. Kunz and P.J. Mulders, Phys. Lett. B 215 (1988) 449. [7] U. Blom, K. Dannbom and D.O. Riska, Nucl. Phys. A 493 (1989) 384. [81 A. Kanazawa, Prog. Theor. Phys. 77 ( 1987 ) 1240. [9] E.M. Nyman and D.O. Riska, University of Helsinki preprint HU-TFT-88-50 (November 1988 ). [ 101 D.P. Min, Y.S. Koh and H.K. Lee, preprint SNU-PHY-NT-8903 (1989). [ 11 ] See e.g.D.H. Perkins, Introduction to high energy physics (Addison-Wesley, Reading, MA, 1989). [121 G.S. Adkins and C.R. Nappi, Nucl. Phys. B 233 (1984) 109.
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