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GLUON CONTENT of the PROTON and HYPERON MAGNETIC MOMENTS Friedrich Wagner Max-Planck-Institut Ftir Physik Und Astrophysik, Munich, Fed.Rep.Germany

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GLUON CONTENT of the PROTON and HYPERON MAGNETIC MOMENTS Friedrich Wagner Max-Planck-Institut Ftir Physik Und Astrophysik, Munich, Fed.Rep.Germany 4U9 GLUON CONTENT OF THE PROTON AND HYPERON MAGNETIC MOMENTS Friedrich Wagner Max-Planck-Institut ftir Physik und Astrophysik, Munich, Fed.Rep.Germany ABSTRACT It is shown, that a gluon three quark component in the baryon octet and decuplet ground state can be quite large, if it is due to a constant color magnetic field inside the hadron. In the flavor SU (3) limit it reproduces the usual SU (6) quark model 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 quarks 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 gluons . 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 hadrons 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 mesons ? , S) D may be affected by gluon effect�. Especially nonleptonic (also K0 ) decays D may proceed via annihilation , however the rates are much too smal l, if the quarks are in a spin singlet state. 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 forces 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 charge �+ )ca � to the color charge 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 X diamagnetic value - due to vacuum polarization 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'<Bl . Then a -' ) equ. (4) becomes a nonlinear equation for B 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 he flavor spin part mus te 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 particle 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.
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