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Volume 125B, number 5 PIIYSICS LETTERS 9 June 1983
THE QUARK MODEL AND THE STRANGE BARYON MAGNETIC MOMENTS
F. MYHRER Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
Received 16 June 1982 Revised manuscript received 14 September 1982
It is shown that standard quark models have difficulties reproducing the new, precise strange baryon magnetic moments. One possible theoretical resolution is to require that baryons are surrounded by a mesonic cloud in which isoscalar and iso- vector components are enhanced due to strong meson correlations.
The MIT bag model [ 1 ] has been very successful in fitting static hadronic properties. One assumption in ref. [1] is that the baryons have quarks in their low- est spacial state and the sp in- f lavour SU(6) wave func- tions are symmetric. The SU(6) symmetry is broken by requiring the s-quark mass to be heavy. In the chir- al quark bag model [2,3], the magnetic moments of the baryons were recently calculated [4,5]. We will calculate the magnetic moments in the semiclassical chiral quark model [6] , and we will show that there is a crucial contribution missing in present calculations of the baryon magnetic moments, a contribution which was discussed in the work of Brown et al. [7].
The nucleon in the chiral quark bag is an MIT bag surrounded by a pion cloud which must be present in order to ensure the continuity of the axial current. The massless u- and d-quarks satisfy the Dirac equa- tion with a confinement condition on the quark wave function at the bag surface (r = R). Knowing the quark wave function q(r) we can construct an axial current
12,31 1
Au(r ) = -iffl(r)TsT u 5 x q(r), r ~< R ,
Au(r) =/rrD~t t ( r ) , r /> R , (1)
where Du is the covariant derivative [2,3], n the pion field and f,, the pion decay constant (f~r = 95 MeV). Requiring the axial current to be continuous across the bag surface, eq. (1) gives the source for the pion field in terms of the quark wavefunctions on the bag surface.
Tile magnetic moment of the chiral bag model is given by two terms, one where the magnetic field cou- ple to the quarks and the other due to the pion cloud illustrated in fig. 1.
/2 =/dQ + ~z r " (2)
To find the pion field we use the semiclassical chiral bag model when the pion is a small perturbation of the MIT bag. We expand the chiral lagrangian of this model and find that the pion is a free field with a source at the bag surface given by the quark wave function at r = R [3]
V 2 ~ = 0 r>~R , (3a)
i - Y n ( r ) = if~ -1- q 0 ( r ) s x 7 5 q 0 ( r ) ; r = R , (3b)
where q0(r) is the MIT quark wave function [3 ] . In the chiral SU(2) invariant model the u and d quarks, and thereby the pions, are massless and the source of the pion field, eq. (3b) follows from continuity of the axial current. However, to have a more realistic
.1 , / , / , , , , \ \ /
t \ g t
E~ _~'2, E,., r,~ Ea
Fig. 1. The pion cloud contribution to the baryon magnetic moment.
359 0 031-9163/83 /0000-0000/$ 03.00 © 1983 North-Holland
Volume 125B, number 5 PHYSICS LETTERS 9 June 1983
Yukawa-like pion field distribution surrounding the quark core of the baryon, we have to introduce the physical pion mass p 4: 0. We explicitly break chirat SU(2) symmetry by introducing a small quark mass m
10 MeV which results in a pion mass/~ ~ 140 MeV following Gell-Mann et al. [8], see also the discussion by Jaffe [9]. (The exact value of the quark mass m '~ ~ is not critical and we make an error < 1% in our numerical results by using m = 0).
l f f o r the moment we neglect the energy differ- ences in the possible three quark intermediate states we find to lowest order in our perturbation expansion
8~,, . . . . f (u, R)
X ~ (BI(~(i) X *(J))zO:(i) X '~( / ) )31B), (4) i,/
where the sum is over all quarks in the baryon, where ,u = 140 MeV, and
R x2 2 +/.u~
.f(~,R)=(f~R) 2 192u (x ° _ 1)2 [1 +(1 +t.tR)2] 2 '
(S)
where for massless u and d quarks their energy ~ (0 ) = xo/R , (x 0 = 2.043) [1] , and IB) are the MIT baryon states. From eq. (4) we see that the pionic cloud con- tribution is a pure isovector. If we use SU(6) baryon wave functions (all quarks in lowest s-state) then eq. (4) contains a two-quark operator when i 4 : j which is non-zero only for nucleons and for the Y. -+ A magnet-
ic transition [7]. The spin- isospin matrix elements of a baryon B are:
1 ME(B) = -
X ~ (Btl(e(OXe(j))z(~(i)X~(j))31Bt), (6) i,/
which when summed over possible intermediate bary- on states M (neglecting mass differences, i.e., m B = raM) becomes:
ME(B)
= ~ (B3'[ ~.o+(i)z+(i)lM)(Ml ~o_.(j)~'_(j)IBt) M t /
- - (Bt l ~ o__ ( i ) r+( i ) l M)(MI ~ o+(j)r_(/))B~) i /
- (Bt[ ~o+(i)7"__(i)IM)(Ml ~.o_(j) r + ( / ) l B ¢ ) i I
+ (B~I ~ o_(i)r_(i)lM)(MI ~ o + ( ] ' ) % ( j ) l B t ) • i /
For the proton:
ME(p) ~ 2s 4 = ( -O-SM,n-~SM,aO + ~ - 6 M a ÷ + ) . (7a) M
For Y.+ :
ME(X+) ~ 6 8 2 * o ) . (7b) = ( ~ M , A +~6M,'v0 --#6M,Z M
For .-0:
ME(=-O) ~ 1 8 4 = (-~ M , - . - - ~ S M , z , - ) . ( 7 c ) ,*¢1,
The matrix elements for n, Y.-, and - - - equal the cor- responding matrix elements above apart from a minus sign. These matrix elements when summed are:
4 MEt-O) = 1 ME(p) = ~ , ME(Y.+) = ~ , - ~ . (8)
We calculate the pion contribution to the magnetic moment as illustrated in fig. 1 with possible intermedi- ate baryon mass-differences m B - m M 4:0 using the effective Yukawa model of refs. [6,9]. These mass cor- rections 8 will multiply the different matrix elements in eqs. (7). l_x)oking at fig. 1 we see that these correc- tions are proport ional to the integral over intermediate momenta
f dkk dk (°'k2)(°'kl) I = 4u 1~o2 (E B --(F.B EM)I
0 (209)2 -
F o r E a = E M we have I a = f~°Odco k3co -3 where to 0 = 0a 2 + K2)I/2 and we take the cut off• = 500 MeV/c from ref. [6]. This cut-off is provided by the finite source of the pion field ~I/R. Call E B - E M = A and for A 4 :0 we use in our calculations
360
Volume 125B, number 5 PIIYSICS LETTERS 9 June 1983
Table 1 Matrix elements of the isovector contribution, eqs. (4) and (7). The mass-correction factor 6 is calculated from eq. (9). In the last column is given the effective spin-isospin matrix elements eq. (7) when corrected for the mass difference m B - m M ~ 0 due to gluon exchange.
Baryon Intermediate Matrix Mass- Effective baryon M element correction m.e.
6 ME(B)
p n 25/9 1.0 A0 --4/9 0.42 3.15 A+÷ 12/9 0 42
E+ zO 8/9 1.0 E*O --2/9 0.48 1.60 A 6/9 1.22
.--.o _.- 1/9 1.0 -0.10 --*- -4/9 0.48
co o dco k 3
8 = j o z;'. (9)
This means for E B = E M that 8 = 1 in table 1 and eqs. (7). The mass splittings A are due to gluon exchange between quarks only. Part of the physical baryon mass-splitting comes from the pionic cloud as shown [6] and this part should not be included in A in eq. (9). The results of the calculations are given in table 1. The effective ME(B) given in the last column is used to calculate 8/a, o feq . (4) together with f . = 95 MeV and bag radii R found by fitting the baryon mass spec- trum [6]. The resulting pion contributions are given in table 2, column 8ta,,.
The quark contributions come from the quarks in the bag when the photon couples to the "bare" three quark baryon or to an intermediate quark in fig. 1. To calculate this we use the results of the perturbation ex- pansions where the quark wave functions q(r) and en- ergies ~ are perturbed by the coupling of the pion field outside the bag [3,6]. We write
q = q0 +f~Zq2 + ....
= ~20 +fn-2~22 + .... (10)
where q0 is the MIT model quark wave function. The change in quark energy ~2 and the perturbed quark wave functions are calculated in refs. [3 6,10]. The ith quark magnetic moment is
ei / d 2 r r X [q+(r)Qtq(r)]/f d3r q+(r) q(r) (11) I t = ~ -
where e i is the quark charge and at the Dirac matrix. Using expression (10) we find
it= li(MIT) + 8Oz + 8/dN,z~ , (12)
where
it(MIT) e/ f d 3 r r X q~)(r)~q0(r )
= e i ~R (4~20R + 2mR - 3)/[2~20R(~20R - l ) + m R ] , (13)
where m is the mass of quark i: (m u = m d = 0; m s :~ 0). The correction 8/1 z comes from the renormalization of the quark wave functions and is
8it z = -alt(M IT) , (14)
where
a = (f~R) -2 [x0(2x 0 - 3)/2887r(xa - 1) 3 ] Z ' .
Here the spin-isospin matrix element Z' of the opera- tor Zi, ja( i)x( i) .o( j) .e( j ) is summed over all quarks and intermediate baryon states and is calculated in ref. [6].
The contribution from the photon coupling to inter- mediate quark states in fig. I is given by 6/aN, A and is given by
e i fd3r,X (q~ 'q0 * q ~ q 2 ) " ( i ) 811N, A =
= ei ( f .R)-2(R/36rc ) ~ Y/
× [(4x 0 -x02 --3) /(x 0 - l ) 3 ] a ( i ) , (15)
We calculate the baryon magnetic moments using SU(6) spin flavour wave functions. The results are giv- en in table 2 which together with the pionic contribu- tions give the baryon magnetic moments. The numbers are close to what is found by The'berge and Thomas [4,5] in the cloudy bag model without the center of mass corrections.
There are, however, some systematic problems with these results which we will discuss. Our parameters are taken from the fit of baryon masses [6] which give m s = 210 MeV. We will not include CM corrections here
361
Volume 125B, number 5 PHYSICS LETTERS 9 June 1983
Table 2 All magnetic moments are in nucleon magnetons. In calculating the above quantities the values of the quark masses were m u = m d = 0 and m s =- 210 MeV as found in ref. [61. The older experimental values for the strange baryons are uZ- = -1.41 ± 0.25, u.-o = -1.20 ± 0.60, and ta~- = -1.85 ± 0.75 listed by the Particle Data Group [ 12].
Baryon R ~' Quark contributions 6~t n Magnetic moments (fm)
"bare" bag 6~ z 6UN, A sub- model exp. Ref. (MIT) total
P 1.04 48.4 2.00 -0 .84 0.62 1.78 0.51 2.29 2.79 n 1.04 48.4 --1.33 0.55 -0.41 --1.19 -0.51 -1 .70 - 1.91 A 1.10 32.0 -0.56 0 0 -0.56 0 0.56 -0.614 ± 0.005 2;* 1.10 19.0 2.07 -0.27 0.20 2.00 0.24 2.24 2.33 ± 0.13 ~ - 1.10 19.0 -0.75 0.14 -0.10 -0.71 -0 .24 -0.95 -0 .89 ± 0.15 ~-o 1.10 7.2 - 1 . 2 3 0.03 -0.02 -1.22 -0.015 -1.24 -1 .253± 0.014 ---- 1.10 7.2 --0.52 -0.01 0.01 -0.52 0.015 -0.51 -0 .69 ± 0.04
[131 1141 [151
because as discussed by Carlson and Chachkhunashv i l i
[11 ] no t even its sign is well d e t e r m i n e d for the mag-
net ic m o m e n t s .
F r o m table 2 we see tha t the /a A a n d / a ~ resul ts are
close to expe r imen t s , bu t P_.x- is too small. Tile fact
tha t ~u.-- is small is c o m m o n in mos t qua rk model cal-
cu la t ions [1,4,5,1 6 , 1 7 ] , for example De Rfijula et al.
[16] f ind/a A -- - 0 . 6 0 ; / a . - o = - 1 . 3 9 and /a . - - = - 0 . 4 6
nm. Only the a rgumen t s presen ted by Brown et al. [7]
come close to r emedy this. Since the s t range b a r y o n s
A, .-0, and - - now are measured fairly accura te ly , we
will adjus t our pa rame te r s to t ry to fit these three num-
bers. In an SU(6) p ic ture the A magne t ic m o m e n t
equals the s-quark magne t ic m o m e n t , and the next sim-
plest magne t ic m o m e n t in the chiral bag mode l is the
- wi th two s-quarks and a single u or d quark . (As seen
from table 2 the pionic cor rec t ions to the -- magne t i c
m o m e n t are w i th in the small expe r imen ta l er ror-bars
so for the fol lowing a rgumen t we neglect these p ionic
c o n t r i b u t i o n s *~.) In fig. 2 we plot the magne t ic mo-
m e n t s o f the three s t range b a r y o n s as f u n c t i o n s o f the
bag radius R and the s qua rk mass m s. The shaded
areas indicate where the calcula ted magne t ic m o m e n t s
fall w i th in expe r imen ta l errors .
As seen f rom fig. 2 we c a n n o t wi th the two parame-
ters fit the three magne t ic m o m e n t s , no t even wi th a
very small s-quark mass. However , in a chiral bag mod-
:tl A calculation of the K-meson contribution to the ~-o mag- netic moment similar to the pion c',dculation, shows that the kaon contribution is ~ -0.01 nm due in the part to the mass-factor suppression of eq. (5), see also ref. 118].
el one does have a n o t h e r degree of f reedom. In the
chiral bag mode l we have the isovector c o n t r i b u t i o n
f rom unco r r e l a t ed p ions outs ide the bag, eq. (4). This
mode l will the re fore deviate f rom the - 2 ra t io o f neu-
t ron to p r o t o n magne t i c m o m e n t s given by the SU(6)
or the quark model . Thus, as po in ted ou t by Brown et
al. [7] we also need an isoscalar " c l o u d " c o n t r i b u t i o n .
This was a l ready es tabl i shed w i th regard to the nu-
c leon fo rm factor , where no t only the i sovector p ion
c loud c o n t r i b u t i o n needed e n h a n c e m e n t due to at t rac-
tive 7rn forces, bu t an e n h a n c e d isoscalar c o n t r i b u t i o n
was also necessary. This led N a m b u [19] to pos tu la t e
the co-meson which was used by Breit [20] and
Sakura i [21 ] to u n d e r s t a n d par ts o f the n u c l e o n - n u -
c leon repuls ion and L ' S forces. An enhanced isovector
so loo 1so 2o0 MF I M e V I _ - - . . . •
r o ; ; ' / - ,
Fig 2. Magnetic moments of the three strange baryons A, - - and ~-o as functions of the s-quark mass m s and the bag radius R. The magnetic moments are calculated from the quark bag with the very small pionic corrections not included. The shaded area indicates where calculated values fall within exper- imental errors.
362
Volume 125B, number 5 PHYSICS LETTERS 9 June 1983
pionic contribution due to low energy ~rrr scattering, with an added isoscalar contribution (a three-pion state) can reproduce the baryon magnetic moments well [22] when we take the s-quark magnetic moment equal to A's moment. With the numbers in table 2 for ~5/3~ including the correction from nn rescattering [23, 24] which gives 8/aen ff = 1.8 5pTr we [22] find/3p
= 2.70, Pn = -1 .75 ; p2:* = 2.39 and/32:- = -0 .68 , and /3:CA = --1.51 all numbers in nucleon magnetons.
We have here shown that naive quark models have difficulties reproducing the new precise experiments of the strange baryon magnetic moments. In the chiral quark models we have to include bo{h tile strong rrrt attractive isovector forces in the pion cloud as well as introducing an isoscalar coupling, much like what was needed to explain the nucleon form factor, in order to explain the baryon magnetic moments.
This work was initiated after discussions with G.E. Brown and many conversations with him and with A.W. Thomas, S. Th~berge and H. ltCgaasen are acknowledged. This work is partly supported by the NSF EPSCOR grant ISP-80-11451, and NATO grant no. 05782.
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