PH YS ICAL REVIEW VOLUME 174, NUMBER 5 OCTOB ER 1968 Electromagnetic Properties of BarJJons in a Quark-Diguark Model with Broken SU(6)* JAMEs CARROLLt AND D. B. LICHTENl3ERGf. Physics Department, Tel-Aviv University, Tel-Aviv, Israel AND J. FRANKLIN Physics Department, Temple University, Philadelphia, Pennsylvania 191ZZ (Received 14 June 1968) An investigation of the electromagnetic properties of baryons is made with a quark-diquark model using broken SU(6) wave functions. The magnetic moments are calculated assuming no orbital contributions and equal gyromagnetic ratios for the quarks. The resulting magnetic-moment sum rules deviate slightly from the SU(3) and SU(6) predictions. The electromagnetic mass splittings are assumed to be composed of intrinsic splitting in the quark and diquark, Coulomb terms, and magnetic terms. We obtain fewer sum rules than with the quark model under analogous assumptions about symmetry breaking. I. INTRODUCTION the gyromagnetic ratios of the quark and diquark were taken as two distinct parameters chosen so as to 6t the NUMBER of authors' ' have obtained sum rules experimental value of the proton and neutron magnetic for the electromagnetic mass splittings of hadrons A moments. Recently this quark-diquark model has been and the magnetic moments of baryons on the basis of generalized" so as to be approximately invariant under the quark model. ' In some of these papers, general SU(6). This is done by taking a quark to belong to a properties of two-body quark interactions were as- six-dimensional representation of SU (6) and the diquark sumed' and in others more explicit assumptions were to belong to a 21-dimensional representation of SU(6). made about the details of the quark-quark interaction. ' ' It is the purpose of this paper to calculate the baryon In one specific dynamical model of the quark-quark electromagnetic properties using this generalized version interaction, 7 two quarks form a tightly bound state or a of the quark-diquark model. diquark, which in turn interacts with a third quark to In some earlier papers7 ~ the quark-diquark model has form a baryon. The diquark was assumed in this model been called a fermion-boson model or a triplet-sextet to belong to a six-dimensional representation of SU(3). model. Both of these terms have defects: the erst be- Using this model, the electromagnetic mass splittings of cause the quarks may obey parastatistics, and thus the the hadrons and the baryon magnetic moments have quark not be a fermion and the diquark may not be been calculated. " may a boson. In the second case we used the term "triplet" One defect of the model as originally proposed is that to refer to the quark and "sextet" to refer to the it is not even approximately invariant under SU(6). diquark. However, according to SU(6), the diquark For this reason it was not possible previously to obtain belongs to a 21-dimensional representation of SU (6) and. the striking prediction'e of SU(6) that the ratio of the contains both an SU(3) sextet of spin 1 and an SU(3) proton to the neutron magnetic moment is —~3. Rather, triplet of spin 0. Thus it is more convenient to have the * Work supported in part by the U. S. National Science Founda- terms "sextet" and "triplet" both refer to the diquark tion and in part by the Air Force OfEce of Scientific Research, and to use the term "quark" to refer to the triplet of Ofhce of Aerospace Research, U. S. Air Force, under AFOSR half-integral Grant No. AF EOAR 66-39, through the European Ofhce of spin. Aerospace Research. We shall assume that a diquark has the same quantum t$ Present address: Indiana University, Bloomington, Ind. numbers as a bound state of two quarks with zero 47401. ' H. R. Rubinstein, Phys. Rev. Letters 17, 41 (1966); H. R. orbital angular momentum. Furthermore, we shall use a Rubinstein, F. Scheck, and R. H. Socolow, Phys. Rev. 154, 1608 two-quark model to infer some other properties of the (1967). diquark. Nevertheless, we shall assume the diquark to ' S.Ishida, K. Konno, and H. Shindaira, Nuovo Cimento 46, 194 (1966). be essentially elementary in combining it in an S state s Y. Miyamoto, Progr. Theoret. Phys. (Kyoto) 35, 179 (1966). with a quark to form a baryon. A. D. Dolgov, L. B. Okun, Ya. Pomeranchuk, and V. V. I. In making calculations of electromagnetic properties Solovyev,' Phys. Letters 15, 84 (1956). ' G. Barton and D. Dare, Nuovo Cimento 46, 433 {1966). in the quark-diquark model, we shall use perturbation M. Gell-Mann, Phys. Letters S, 214 (1964); G. Zweig, CERN theory. However, the baryon wave functions will not Report Nos. TH. 401 and TH. 412, 1964 (unpublished}. 'D. B. Lichtenberg and L. J. Tassie, Phys. Rev. 155, 1601 necessarily be SU(6) wave functions but rather wave (1967). functions modified to take into account a syrrunetry s D. B. Lichtenberg, Nuovo Cimento 49, 435 (1967). ' P. D. De Souza and D. B. Lichtenberg, Phys. Rev. 161, 1513 (1967). "D. B.Lichtenberg and L. J. Tassie, Bull. Am. Phys. Soc. 12, ' F. Gursey and L. A. Radicati, Phys. Rev. Letters 13, 173 698 (1967};D. B. Lichtenberg, L. $. Tassie, and P. J. Keleman, (1964); A. Pais, Rev. Mod. Phys. 3S, 215 (1966). Phys. Rev. 167, 1535 (1968}. 1681 1682 J.CARROLL et al. breaking due to the medium-strong interactions. "Thus belonging to the 3 represents, tion of SU(3): we obtain the electromagnetic properties by taking the 21g36+ ~B. expectation values of the electromagnetic operators be- tween baryon states which are almost but not quite The group SU (6) is broken by letting the central mass eigenstates of SU(6). We obtain the SU(6) result by of the sextet diquark be different from the central mass letting certain small parameters which measure the of the triplet diquark. The group SU(3) is also broken. difference from the SU(6) wave functions go to zero. The sextet diquark splits into an isospin triplet of In addition to calculating the baryon electromagnetic hypercharge I'= -'„an isospin doublet with V= —-'„and mass differences and magnetic moments as in Ref. 9, we an isospin singlet with I'= ——,. Similarly, the SU(3) shall also calculate certain transition magnetic moments triplet splits into an isospin singlet of hypercharge —, and and mass differences. In calculating the magnetic mo- an isospin doublet with I'= —3. These isospin multiplets ments of the baryons we shall assume simple additivity are assumed to be split further by the electromagnetic of the magnetic moments of the constituent quark interaction. %e can consider two possibilities: (a) that and diquark. In calculating the electromagnetic mass the electromagnetic splitting is invariant under U-spin splittings of the baryons we shall assume that they arise transformations, and (b) that because of the indirect from two causes: first, from the intrinsic electromagnetic effect of the SU (3)-breaking medium-strong interaction, mass splittings of the quark and diquark multiplets, and there is a small violation of U-spin invariance by the second, from Coulomb and magnetic interactions be- electromagnetic interaction. tween the quark and diquark constituents of the Before discussing the properties of the diquark we baryons. " briefly recapitulate the electromagnetic properties of the We do not treat mesons in this paper for the following quark. The electromagnetic mass-splitting parameter reason: In the model, a meson can be composed of a which splits the quark isospin doublet we call e,. If quark and antiquark or a diquark and antidiquark. U-spin invariance holds, "then the isospin singlet quark (Other combinations, such as quark-antidiquark do not will also have an electromagnetic mass splitting e~. Any have the correct quantum numbers, and must be deviation from U-spin invariance can be absorbed as an assumed to have more energy if, indeed, they are bound electromagnetic correction to the medium-strong mass- at all. ) splitting parameter 5, which separates the isodoublet It is obviously simpler to choose the quark-antiquark. from the isosinglet. model to describe mesons, in which case we have nothing If we assume that the quark magnetic moments p(q;) to add to the usual treatment. ' "Mesons have already are proportional to their charges Q;, we obtain been considered in the SU(3) version of the diquark- u(q') = ~oQ. , antidiquark model, ' but our present attitude is that such states do not correspond to the ground-state 35- where po is a parameter, which will turn out to be the dimensional multiplet of SU(6). If mesons are dis- magnetic moment of the proton, and Q, is the charge of the ith covered which belong to a 27-dimensional representation quark. With this assumption, U-spin invarianre holds, but Eq. is more restrictive than U-spin of SU(3), the diquark-antidiquark model of mesons will (1) invariance. On the other hand, we can assume that merit further consideration, since such a multiplet quarks all have the same gyromagnetic ratio. Then the cannot be achieved in the quark-antiquark model. quark magnetic moments are as follows: The plan of the paper is as follows. In Sec. 2 we dis- = = cuss the electromagnetic properties of the diquark as a p(q~) 3po i p(q2) 3po~cl (~a+ ee) ~ - bound state of two quarks. In Sec. 3 we obtain the ~(q,) = 31.omq/(m—qq Sq+.q), magnetic moment of the baryons including the E*-V where is the quark mass. We define the 5 and Z-A transition magnetic moments and the magnetic m, quantity by moment of the 0 on the basis of the quark-diquark 1 —S=~i,)(m,+S,) .