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Home , Gluon, Hyperon

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GLUON CONTENT OF THE AND MAGNETIC MOMENTS Friedrich Wagner Max-Planck-Institut ftir Physik und Astrophysik, Munich, Fed.Rep.Germany

ABSTRACT It is shown, that a three component in the octet and decuplet ground state can be quite large, if it is due to a constant color magnetic inside the . In the flavor SU (3) limit it reproduces the usual SU (6) prediction except for a reduction of G /G . Flavor breaking by quark masses improve the agreement of Hyperon magnetic momenA Vts with experiment con­ siderably by including such a gluon component. 410

Introduction I.

The observed hadron spectrum can be explained phenomenologically by a simple model, namely by nonrelativistic mas sive moving in a poten- tial Even details of the baryon spectrum can be accounted for by the Isgur­ 1). 2) Karl model . It is generally believed that such a model can be finally derived from a more fundamental theory where colored light quarks interact via (QCD) mas sless vector fields , the . One proposal to solve this confinement pro­ 3 4 ) blem is made in analogy to superconductivity • where an electric Meissner effect prevents phys ical states other than color singlet states to exist. It seems to me unlikely that all long distance effects result in a nonrela­ QCD tivistic potential for quarks , and gluonic degrees of freedom can be discarded all together. One has already some hints that gluon components in the hadron wave function may be important. The hyperfine splitting of can be under­ S ) stood by a magnetic gluon exchange contribution to the energy . This second order perturbation contribution implies a first order change in the wave­ 6) function . Similarily gluon effects have been advocated to explain the mass splitting and decay rates in the charmonium states . Weak decays of ? , S) D may be affected by gluon effect�. Especially nonleptonic (also K0 ) decays D may proceed via , however the rates are much too smal l, if the quarks are in a . If gluons are present, the quarks can be also in a triplet state, and annihilation is not suppressed. For quantitative s) agreement this component mus t be rather large . In deep inelastic scattering charged constituents carry only half the momentum of the proton . If this effect due to short distance gluons , these may leave some foot prints also at long 1, distances . In this paper we want to investigate the effect. of color magnetic on the hadron wave function . Usually this problem is treated by extrapolation of the short distance perturbation theory . This may be justified for heavy quarks (c ,b, ...) but not for light spectrum quarks (u,d,s) whose compton wave length is comparable to the hadronic size. Rather, we wi ll assume that in this case a slowly varying magnetic field induced by the color-n:� gnetic moments of the quarks leads to an adequate description of the gluonic effects . In analogy to problems in solid state physics one can solve the Hami ltonian for the quarks by a semi­ classical mean field approximation, which is to be discussed in section II. In presence of a color magnetic field the quarks are no longer in a color singlet state. By the Pauli principle this will change also the spin flavor wave-function and one may worry about the successful SU (6) prediction . In section III it will be demonstrated that in the frame work of the additive quark model the observ­ ables essentially do not change by the inclusion of Golor octet 3q configuration 411

in addition to the usual color-s inglet proton or state. We will not assume � any specific potential, but rather fit the free parameters to the observed masses of the octet and decuplet. This fixes the wave-functions , and static properties as G /G or hyperon magnetic moments can be predicted (section IV) . A V

II. Magneto-static Hami ltonian for quarks

We assume that color- electro-static forces lead to the usual nonrelativistic potential Hami ltonian Ho . To Ho we add the magnetic energy H from a constant m color magnetic field B . Its vector potential reads as a

A (r) (1) a

With the conditions

0 (2 )

the homogeneous Maxwell equation are satisfied . In the nonrelativistic approxima­ tion we obtain for H in terms of thE Pauli quark operator m �

a g2 a H A 2 d A A A (3) m _!___m ,,,+"' KB2 a (L + a) + £12 A + abc b c iJ; 2 [ 4 ]

The first term describes the interaction of B with the intrinsic and orbital mag­ a netic moment . Whereas the first two terms are also present in the last term QED , is typical for a SU(n) gauge group with n 3. It describes the coupling of the � quark �+ )ca � to the of the magnetic field. In addition to equs . (1-3) there is the constraint from the inhomogeneous Maxwe ll equation, which we write in analogy to the usual electromagnetic case

B ( ) a 4

where is the susceptibility of the hadronic ma tter. will differ from the X diamagnetic value - due to effects. If electrical con- �2m ) finement is due to magnetic monopoles in the vacuum4 , they will strongly in- fluence will in general depend on B itself. Since we are interested in X . X the effects of a strong B field we assume that is such that B saturates , the X field strength is independent of the strength of the source. Neglecting the L 2 and A terms in H equations (3)-·( ) are nothing else but the Heisenberg ferro- 1 4 ) ) magnet. Also the BCS model9 or the Jona-Lasini0-Nambu modellO have a similar 412

mathematical structure. All these models are solved in the mean field approxima­ tion by replacing the r.h.s. of equ. (4) by its expectation value. For a clas­ sical B field the Hamiltonian (3) can be solved with eigenstate [tf'

B a (5)

the so-called self consistency equation (gap equation). ln our case Ba can b'e used to describe many gluon effects in the hadron wave function. From equ. (4)

we conclude it must transfonn like spin 1 under rotations and like a color octet representation under color transformations .

III. Baryon States with L 0

We want to solve the Hamiltonian (3) for the baryon ground state. These are color singlet states containing three quarks with orbital angular momentum L = 0. The most general state reads as + + + 1\f> - 1 lJ! lJ! lJ! lJ! (r l [o> (6) 13! a 1 a2 a, a,a,a, i a . JIi where the wave-functions 'f (r i) depends on the spin, color and flavor index ai. ai and the quark coordinates ri. The latter requires a specific Ho . We simply assume that this is independent from ai . Therefore the ri dependence can be fac­ tored out and need not to be considered anymore. consists of products of lfa . i color wave-functions Ci and spin flavor wave-functions F. The antisymmetric color singlet wave-function has to be combined with flavor spin F56 (7) lf0 . 1 . SS F or quar k s in a co or octet state C 8 t h e flavor spin part mus t e ba F70, wi'th quark spin S 3/2, 1/2. The quark spin S will be combined with a spin 1 color =k octet object•'ljJa (B) to total spin J ,M state. This leads to the following wave- functions , 2 (note that 1f2 = 0 for the flavor decuplet) (8) rllf S+� Color decuplet can be neglected since in this case H0 is purely repulsive. Occurrence of a F70 wave-function in the baryon groundstate will in general change the successful predictions of the naive SU (6) rr.odel, which are based on the F56 assignment. However these predictions are based on the additive quark model, which tests only the single density ma trix1 1) but not the whole wave-function. For the flavor octet this matrix is characterized by two flavor D. D coupling constants F and If only tpo is present one obtains = 1 and 413

F/D = 2/3. Almost any quark model prediction for the octet (magnetic moments , pseudoscalar coupling constants •••) depend only on this F/D ratio. Only

= = GA/GV F+D 5/3 depends on the absolute size and disagrees with experiment. For the combination lf+ (Lf\ +Lf'2 ) for the octet one obtains the � F/D ratio as for If 0• Therefore �this combination leads to � quark model predictions as the usual tp0• Only the absolute size of D for lf+ is reduced to 2/3. Therefore % · i lf1 _ = (\f1-\f2) GA/GV will be for + alone. Presence of lf will lead to a dif­ ferent F/D ratio and thereby destroy the naive SU (6) pre�dicti ons . The Hami ltonian equ . (3) for the octet (decuplet) will be a 3x3 (2x2) ma trix in the space '-r lf> lf Solving12) the self-consistency equation (5) o ' + _ together with condition (2) we find that lf_ decouples from the octet ground state. Therefore, the admixture of a 70 wave-function induced by the color mag­ netic field does not change the usual SU(6) predictions except for GA/GV. In order to describe the hadron masses we have to introduce the usual flavor break­ ing by assuming a heavier s quark mass. By the same time also coupling to the magnetic color field gets reduced for s quarks � + a A�-+ mu B � a A� B + ms This flavor breaking changes the matrix elements and mixes a 'f_ component and flavor decuplet (singlet) contributions into the octet wave-functions. Since the matrix elements of H will involve integrals over the coordinate r they depend m i ' on the specific Ho . If the B saturates the integrals are the same for the octet and decuplet for the similar radial dependence. Instead of assuming a specific H0 , we rather prefer to fit the seven parameters (integrals of the wave-functions 2 involving Band � terms in equ. (3) , quark mas ses and zero point energy from Ho ) wh ich enter in the matrix of the Hamiltonian) to the observed octet and decuplet masses. The result of the fit is shown in table I. It reproduces the octet and decuplet mas ses up to the level of electromagnetic mass splitting.

Octet Fit Experim. Decuplet Fit Experim.

0.939 0.939 m 1.234 1. � ll 232 ml: 1. 184 1.193 ml: * 1.376 1.383 1. 116 m 1. 112 m_* 1.524 1.532 fl m_ 1.326 1 .318 m'2- 1.674 1 .672

Table 1: Comparison of fitted octet and decuplet masses with 13) experiment

In view of the many parameters this agreement is not too surprising , but it fixes the relative amount of glue 3 quark tft and 3 quark 'f 0 wave-function. It turns out that 'f+ actually dominates . We arrive at the somewhat paradoxial 414

situation , that the proton is 75% of the time a flavor 70 plet and nevertheless we have the same mass spectrum and SU (3) prediction as for the usual 56 as sign­ ment , Knowing the wave-function we can now predict some static properties as G /G and hyperon magnetic moments, A V

III. Predictions

Since dominates the proton wave-function, G /G will be substantially re­ f+ A V duced as compared to the SU (6) value. In fact our value G /G 1.26 is in good A V = 13) 1) agreement with the experimental value 1.254 0.007 . The additive quark model ! reduces all magnetic moments to the quark magnetic moments µ ' µ and µ . Ad us­ u d s j ting µ -2µ and µ to the empirical values of and P moments one can predict u d s A the other moments. In table II the first col umn gives the experimental value, and the second column the predictions of the usual SU (6) model = O) ,

l:+ 2.33 0.13 2.680 2.28 ! - 1.41 0.25 - 1.03 - 0.84 l: ! - 1.253 0.014 1.42 1.24 0 !

- o. 75 0.07 0.50 0.55 !

Table II: Comparison of Hyperon Magnetic Moments Column 3 gives the result with l{J0 only and column 4 the prediction of the wave functions including magnetic gluons . In both cases P and moments are used to A determine the up and strange mass magnetic moment .

They "lgree qualitatively with the data , although there is a 20% deviation in the of I+ and Including our glue component the P/N ratio remains unchanged , case =0 • I+ and are considerably improved . Unfortunately also - is moved away but =0 l: from the SU (6) prediction and is encreased not ,:mough. Since these differ = only from two standard deviations from experiment , we need not to be worried . Besides the ground state there will be other positive states accord­ ing to equ. (8) which have no color singlet component . They all are much higher in mass ( > 1.8 GeV) , They will be mixed with radial excitation occuring in 12) the same energy region. This will be discussed elsewhere Our prediction for r.; /G and the magnetic moments should not be taken too A V seriously. First we used rather drastic assumption about B (constant over hadron a and it strength independent whether it is in a or P) and secondly other effects /J. 15) 16) as relativistic effects , different sizes for proton and the 15' 16) and configuration mixing will affect matrix el.ements of the quark spin 415

operator. We only conclude from our findings , that a large color magnetic field and thereby a dominating quark color octet configuration in the proton is not entirely ruled out by experiment and may improve in the cases of GA/GV and L+ , -o magnetic moments the agreement of theory and experiment. I would like to thank Prof. Gourdin for the hospitality at Paris VI where part of the work has been done and also my theoretical colleagues at Munich for fruitful discussions.

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