Volume 60B, number 2 PHYSICS LETTERS 5 January 1976 A UNIVERSAL GAUGE THEORY MODEL BASED ON E6 ~ F. GORSEY and P. RAMOND* Department of Physics, Yale University, New Haven, CT. 06520, USA and P. SIKIVIE Department of Physics and Astronomy, University of Maryland, CollegePark, MD. 20742. USA Received 29 October 1975 A universal gauge theory based on E6 is suggested as a model for the unification of strong, electromagnetic and weak interactions. Left and right components of six quarks (3 light, 3 charmed) and nine leptons (including heavy charged and neutral leptons) are fitted into two 27-dimensional representations of E 6. Two different assignments are proposed. The Weinberg angle and the R ratio are derived and the stability of the proton is discussed. Following the interpretation [1,2] of the quark color group SUC(3) as a subgroup of the automor- phism group G 2 of the octonion algebra and the iden- tification of the exceptional observables of Jordan, von Neumann and Wigner [3] (JNW) with the opera- tors acting on charge space of elementary particles, we are led to consider the Exceptional Lie Groups as gauge groups for Universal Gauge Theories [4]. By universal gauge theory we mean a gauge theory which embeds all known interactions, except gravitation and the CP violating milli-weak interactions, into a simple gauge group. Among the exceptional Lie groups we found E 6 and E 7 to be particularly adequate to play the role of a universal gauge group. In this note we shall treat the case of E 6. E 6 has SU(3) ® SU(3) ® SUC(3) as a maximal compact subgroup. The first two SU(3) are taken to be the SUL(3) and SUR(3 ) act- ing respectively on left-handed and right-handed quark components. The last SU(3) is the quark color group. The fundamental and adjoint representations of E 6 are respectively 27- and 78-dimensional. Their SU(3) ® SU(3) ® SUC(3) decompositions are the following Research (Yale Report COO-3075-118) supported in part by the U.S. Energy Research and Development Agency under contract AT(11-1)-3075. * Present address, Physics Division, California Institute of Technology, Pasadena, CA. 01109, on leave from Yale University. (27) =(3,3,1 c) +(3,1,3 c) +(1,3,3 c) (78) =(8,1,1 c) +(1,8,1 c) +(1,1,8 c) (1) + (3, 3,3 c) + (3,3,3-c). The 27 representation, which is complex, can be re- garded as a 3 × 3 complex octonionic matrix hermit- ean with respect to octonionic conjugation (complex Jordan matrix): cl) j =jT = /3 . (2) 5 We use a T to denote transposition, a bar to denote oc- tonionic conjugation and a star to denote complex conjugation. The 2--7"-can then be represented by J* = j*T. The 78-parameter infinitesimal E 6 transforma- tion of J is given by: 5J = ~-(R, J, S) + iT'S (3) where R, S and T are real traceless hermitean octoionic 3 × 3 matrices (real Jordan matrices), and where we use the notations: T'J=½(TJ +JT), (R,J,S)=(R'J)'S-(S'J)R (4) for the Jordan product and the associator respectively. A E 6 gauge theory is anomaly-free regardless of which representation is chosen for the fermions of the theory. Indeed, the anomalies are proportional [5] to 177