Jul 7, 1975 - lead to a quark with momenta of order it(l/R) ~ 600 ... will fix the strange quark mass,at least to within ±50. MeV. ... 1 - mR - x/(m R) 2 + x 2.
Volume 57B, number 3
PHYSICS LETTERS
7 July 1975
M A G N E T I C M O M E N T S O F T H E S T R A N G E B A R Y O N S IN T H E B A G M O D E L ¢r E. ALLEN Department of Physics, Massachusettslnsfftute of Technology, Cambridge,Mass. 02139, USA Received 30 May 1975 We break SU(3) symmetry in the bag model by givingthe singlet quark a mass m. The magnetic moment ratios of the strange baryons are calculated and compared with exact SU(6) predictions and with experiment. In the previous work on the bag model of hadrons in the simplest approximation [1, 2], no effort was made to incorporate SU(3) symmetry bereaking el, fects. In fact, the only hadrons considered were the ones which did not include a strange quark. We propose to discuss a simple calculation using the bag model in which SU(3) is broken by giving a mass to the strange quark, leading to an elementary understanding of the megnetic moments of the strange baryons. In this paper we shall not discuss the breaking of the spin degeneracy of the states and hence we shall try to parametrize SU(3) breaking in a way which is in First approximation insensitive to the way in which the spin effects are taken into account. This is important since SU(3) breaking and spin breaking effects are of the same order of magnitude in the low lying hadron states. In the bag model the u and d quarks are massless in first approximation. This is what allowed for a parameterless calculation of the proton magnetic moment (gp = 2.6) and to the excellent value for GA/G v = 1.1 in contrast to the non-relativistic quark model where GA/G V = 5/3 and where gp is fit by an arbitrary adjustment of the quark messes for the u, d quarks [2]. In passing, we also might mention that the assumption of a non-relativistic model for quarks with masses of order 300 MeV seems dubious in view of the fact that the proton size of 10 -13 cm = (1/200) MeV-1 would lead to a quark with momenta of order it(l/R) ~ 600 MeV in the ground state in a non.relativistic confinement model. In this paper we shall continue to take the u, d quarks as massless. We shall give a mess m s to the This work was done in partial fulfillment of the requirements for a desree of Bachelor of Scienceat M.I.T.
strange quark. This will produce a splitting of the baryon spectrum. It will split the proton and (~) by the same amount as the splitting which occurs in the decuplet. Since these mess differences are not the same (Mz - Mp ~ 250 MeV (we choose Z rather than A since the spin arrangements are the same as in the protons) while decuplet spacing = 150 MeV), it is clear that spin effects are also very important in these states. Since these effects may be expected to be opposite in the states (~, p) where the quark spins are parallel, we shall parametrize SU(3) breaking by using the average difference, (150+250)/2 = 200 MeV to determine the strange quark mass. We expect that this will fix the strange quark mass,at least to within ±50 MeV. We f'md below that this determines the strange quark mass to be 300 MeV. We can now compute the magnetic moment of the confined strange quark. Relativistic effects are still very important. A calculation leads to/~s/#d = +0.6. This determines the magnetic moment of the particle (which is the best known from experiment) to be I.tA/~p = - 0 . 2 1 , which is in quite good agreement with experiment. It gives for other magnetic moments the values given in table 1. Table 1 Predictions of ratios of magnetic moments ~//~p) of strange bal'yons,
A° I~+ •"~ •L'~ r
Exact SU(6)
Bag theory
Experimentalvalues [3]
-1/3 1 1/3 -1/3 -2/3 -1/3
-0.21 0.96 O.29 -0.37 -0.49 -0.16
-0.24 ± 0.02 0.92 ± 0.15 -0.57 to +0.28 -0.69 ± 0.27
263
Volume 57B, number 3
PHYSICS LETTERS
The calculations are very elementary and we shall simply outline them since they follow the methods used previously [2]. The bag model in simplest approximation involves quarks moving freely within a cavity of radius R. The radius is determined by the condition that the quark field "pressure" on the surfaces of the cavity be balanced by a universal pressure B which comes by addhag a term -gUVB to the stress energy tensor for hadronic constituents [ 1]. We assume that the u, d quarks are massless and the s quarks have mass m. Both obey the corresponding free Dirac field equations in the lowest baryons, the quarks occupy the lowest mode in the cavity. Hence we may write [2] for a baryon which contains n, u or d quarks and ( 3 - n ) strange quarks all occupying their lowest modes,
7 July 1975
24
22 2O 18
(kR
exp.(kR) 14 12 I0
where x - k R 8
2.04 2
1.5
M = ( 3 - n ) ( 2 - - ~ ) + ( R ) x / ( m R , 2 + x 2 + 3 B R 3.
(1) In (1), we determine R by the balance of the pressure B against the quark field pressure on the surface which gives the equation for R, ~MDR = 0. In ( I ) x = x(mR) is gotten by solving the Dirac equation for a particle of mass m in a spherical cavity with the boundary condition - i ? ' ~ , I , = ~,
(2)
on the surface r = R. Tlus yields the eigenvalue condition for the momentum x in units of 1/R tanx
X =
1 - mR - x/(m R) 2 + x 2
.
(3)
A graph of solutions of (3) is given in fig. 1. We note the limiting values x(0) = 2.04 and x(~°) = lr, which correspond respectively to the massless Dirac equation [2] and the Schr6dinger equation. The SU(3) level spacing is essentially uniform. With B taken from the baryon spectrum and AM = 200 MeV, we fred m = 300 MeV. This corresponds to an mR = 2.5 and x = 2.7. We see that this value for x is in between the corresponding massless limit (2.04) and Schr6dinger equation limit (n). The strange quark is still moving relativistically. We can then compute the magnetic moment of the strange quark from the formula: 264
0
I I
I 2
"1 3
I 4
I 5
I 6
I 7
I 0 8 9 (mR}
Fig. 1.
l l = r- J d 3 r ~ r x q
'tetQg'
(4)
we f'md
Rf. 4,.,R+2mR-3 IUtm,Um(_X/3)
(S)
=-612(t~R)2_2~R +mR J which reduces in the limit m = 0, to the same form as used previously, and as mR ~ ~, to the non-relativistic value, q/2m. We f'md with 300 + 50 MeV for the strange quark mass, #ha(0) = 0.6, where/.tO) is the magnetic moment of the massless quark. This by standard quark model procedures leads to the values given in table 1 for the magnetic moments of the strange baryons. I should like to thank Professor Kenneth Johnson for his suggestion of the problem, and for his patient help in its solution. [1] A. Chodos et al., Phys. Rev. D9 (1974) 3471. [2] A. Chodos et al., Phys. Rev. D10 (1974) 2599. 13] Particle Data Group, Reviewof particle properties (1974).
