Narrow Results

We investigate the SU(3)³ GUT model when signs of the model (precursors), due to low compactification scale, appear before the gauge couplings of the Standard Model get unified. The Kaluza–Klein state contribution seems to lead the gauge couplings to unification through a wide energy scale only in the case when the colour group is augmented to SU(4).

We show that superheavy threshold corrections in the New Minimal Supersymmetric GUT based on the SO(10) Higgs system can comfortably correct the prediction for the value of α₃(M_Z) from the relatively large value predicted by the two-loop RG equations to the central value determined by the current world average. The unification scale is raised above the one-loop value over almost all of the viable parameter space.

Alternative renormalizable minimal non-SUSY SO(10) GUT model is proposed. Instead of a 126-dimensional Higgs field, a 120-dimensional Higgs filed is introduced in addition to a 10-dimensional Higgs field and plays a crucial role to reproduce the realistic charged fermion mass matrices. With contributions of 120 Higgs field, the original Witten’s scenario of inducing the right-handed Majorana neutrino mass through 2-loop diagrams becomes phenomenologically viable. This model inherits the nice features of the conventional renormalizable minimal SO(10) GUT model with $10+\overline{126}$ Higgs fields, while supplemented with a low scale seesaw mechanism due to the 2-loop induced right-handed Majorana neutrino mass.

A string in four dimensions is constructed by supplementing it with 44 Majorana fermions. The later are represented by 11 vectors in the bosonic representation SO(D-1,1). The central charge is 26. The fermions are grouped in such a way that the resulting action is worldsheet supersymmetric. The energy–momentum and current generators satisfy the super-Virasoro algebra. GSO projections are necessary for proving modular invariance. Space–time supersymmetry algebra is deduced and is substantiated for specific modes of zero mass. The symmetry group of the model can descend to the low energy standard model group SU(3)×SU_L(2)×U_Y(1) through the Pati–Salam group.

We study more extensively and completely for global gauge anomalies with some semisimple gauge groups as initiated in Ref. 1. A detailed and complete proof or derivation is provided for the Z₂ global (nonperturbative) gauge anomaly given in Ref. 1 for a gauge theory with the semisimple gauge group SU(2) × SU(2) × SU(2) in D = 4 dimensions and Weyl fermions in the irreducible representation (IR) ω = (2, 2, 2) with 2 denoting the corresponding dimensions. This Z₂ anomaly was used in the discussions related to all the generic SO(10) and supersymmetric SO(10) unification theories¹ for the total generation numbers of fermions and mirror fermions. Our result¹ shows that the global anomaly coefficient formula is given by A(ω) = exp[iπQ₂(□)] = -1 in this case with Q₂(□) being the Dynkin index for SU(8) in the fundamental IR (□) = (8) and that the corresponding gauge transformations need to be topologically nontrivial simultaneously in all the three SU(2) factors for the homotopy group Π₄(SU(2) × SU(2) × SU(2))is also discussed, and as shown by the results¹ the semisimple gauge transformations collectively may have physical consequences which do not correspond to successive simple gauge transformations. The similar result given in Ref. 1 for the Z₂ global gauge anomaly of gauge group SU(2) × SU(2) with Weyl fermions in the IR ω = (2, 2) with 2 denoting the corresponding dimensions is also discussed with proof similar to the case of SU(2) × SU(2) × SU(2). We also give a complete proof for some relevant topological results. We expect that our results and discussions may also be useful in more general studies related to global aspects of gauge theories. Gauge anomalies for the relevant semisimple gauge groups are also briefly discussed in higher dimensions, especially for self-contragredient representations, with discussions involving trace identities relating to Ref. 15. We also relate the discussions to our results and propositions in our previous studies of global gauge anomalies. We also remark the connection of our results and discussions to the total generation numbers in relevant theories.

We study the bridge between the phenomenological mass matrix model and SO(10) GUT. Namely, we consider the four-zero texture model in the framework of the renormalizable SO(10) GUT model. This unification gives more stringent constraints than the case where only either model is considered. However, we can obtain good fitting by expanding the minimal SO(10) GUT to include 120 in addition to 10 and in Yukawa coupling and by considering both type I and type II seesaw mechanisms.

We study the most general renormalizable couplings containing Higgs $H(10)$, $D(120)$, $\bar{\mathrm{\Delta }}(\overline{126})+\mathrm{\Delta }(126)$, $A(45)$, $E(54)$ and $\mathrm{\Phi }(210)$ in the supersymmetric SO(10) models. The Clebsch–Gordan coefficients are calculated using the maximal subgroup $\mathrm{\text{SU}}(5)\times \mathrm{\text{U}}{(1)}_X$.

Recent years have seen increased use of Bayesian model comparison to quantify notions such as naturalness, simplicity, and testability, especially in the area of supersymmetric model building. After demonstrating that Bayesian model comparison can resolve a paradox that has been raised in the literature concerning the naturalness of the proton mass, we apply Bayesian model comparison to GUTs, an area to which it has not been applied before. We find that the GUTs are substantially favored over the nonunifying puzzle model. Of the GUTs we consider, the $B-L$ MSSM GUT is the most favored, but the MSSM GUT is almost equally favored.