Effects of dimension-5 operators in E₆ grand unified theories

Abstract

The effective dimension-5 operators can be induced by quantum gravity or inspired by string and M theories. They have important impacts on grand unified theories. We investigate their effects for the well known E(6) model. Considering the breaking chains E6 → H = SO(10) × U V ′ (1) → SU (5) × UV (1) × U V ′ (1) → SU (3) × SU (2) × UZ(1) × UV (1) × U V ′ (1) and E6 → H = SO(10) × U V ′ (1) → SU (4) × SUL(2) × SUR(2) × U V ′ (1) → SU (3) × SUL(2) × SUR(2) × US(1) × U V ′ (1), we derive and give all of the Clebsch-Gordan coefficients Φ (r) s,z associated with E6 breaking to the standard model. Physical effects of nonzero vacuum expectations of SM singlets Higgs in E6 grand unified theories are discussed.PACS numbers: 12.10. Dm, 12.10.Kt, Grand unification theories (GUT) are among the most promising models for physics beyond the standard model (SM). Grand unification assumes that the three gauge coupling constants in SM should be unified at a high scale, the unification scale M G , but would be split at low energy due to their different renormalization group evolution from the grand unification scale to low scales. It seems that the assumption is supported by experiments since there is the apparent unification of the measured gauge couplings within the minimal SUSY SM (MSSM) at scale M G ∼ 2 × 10 16GeV [1][2][3][4]. In addition to the GUT scale, one has the Planck scale defined byGeV at which quantum gravity enters [5,6]. It is well-known that the GUT scale M G is smaller than the Planck scale M P l by two orders of magnitude. Therefore, we can investigate unification of particle interactions including effects of gravity in the effective field theory approach. That is, we introduce non-renormalizable higher dimension operators to describe the effects of quantum gravity. They are d ≥ 5 operators which are induced by gravity, enter the Lagrangian scaled by factors Of (M P l ) −(d−4) with order unity coefficients, and are subject only to the constraints of the symmetries (gauge invariance, supersymmetry, etc.) of the low energy theory. The presence of higher-dimensional operators generated at the Planck scale must have impact to GUT and its phenomenology, as has been shown in Refs. [5,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. These operators modify the usual gauge coupling unification condition [7][8][9][10][11][12][13]. They affect analysis of proton decay [5,15,16] and supersymmetric (SUSY) particle spectrum in SUSY GUT and supergravity [17][18][19][20][21][22][23][24]. Therefore, one should consider effects of these higher dimension operators in model building of GUT and SUSY GUT.The effects to the unification of gauge couplings produced by dimension-5 operators which are singlets of the grand unified gauge group G and formed from gauge field strengths G µν and Higgs multiplets H k of Gwhere a,b are group indices and k labels different multiplets, have been examined systematically for G = SU (5), SO(10) in the ref. [25]. In particular, differing from those given before in the literatu...

Gauge coupling unification with extra Higgs doublets

“…Grand unified theories with E 6 gauge group have many theoretical advantages—anomaly free, the largest exceptional Lie group that has chiral representation, a possible unification of quarks, charged leptons, neutrinos and Higgs doublets—and therefore much attention has been paid in the literature .…”

E6GUT through effects of dimension-5 operators

“…. As it is easy to see, from equations (12), (13) and tables in appendix I and in the [55], that for a given specific breaking chain, when two of three numbers…”

“…Effects of higher dimension operators to the unification of gauge couplings have also been investigated in the grand unified gauge group E 6 [20,55]. In the [20], E SU SU SU 6 3 3 3 ´( ) ( ) ( ) has been studied and the corresponding effective contributions to gauge kinetic terms have been given.…”

“…Moreover, the numerical analysis on the unification of gauge couplings and corresponding physical discussions have not been carried out in the paper. To construct E 6 unification models with effects of dimension-5 operators in [55], the author considers only the maximal subgroup H SO U 10 1 =( ) ( )and gives the Clebsch-Gordan coefficients s z r , F ( ) associated with E 6 breaking to the SM for that case. Nevertheless, same as the [20], the numerical analysis on the unification of gauge couplings and corresponding physical discussions have not been carried out.…”

“…Appendix A. Clebsch-Gordan coefficients associated with E 6 breaking to the SM All of the Clebsch-Gordan coefficients s z r , F ( ) associated with E 6 breaking to the SM, in different bases s z , { }, up to a uniform normalization constant for different representations r have been derived and results are given in this appendix. For the subgroup H SO U 10 1 =( ) ( ), the analysis and results have been given in the [55]. We start from H SU SU 6 2 =( ) ( ).…”