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A broad overview of the current status of proton stability in unified models of particle interactions is given which includes non-supersymmetric unification, SUSY and SUGRA unified models, unification based on extra dimensions, and string-M-theory models. The extra dimensional unification includes 5D and 6D and universal extra dimensional (UED) models, and models based on warped geometry. Proton stability in a wide array of string theory and M theory models is reviewed. These include Calabi-Yau models, grand unified models with Kac-Moody levels k > 1, a new class of heterotic string models, models based on intersecting D branes, and string landscape models. The destabilizing effect of quantum gravity on the proton is discussed. The possibility of testing grand unified models, models based on extra dimensions and string-M-theory models via their distinctive modes is investigated. The proposed next generation proton decay experiments, HyperK, UNO, MEMPHYS, ICARUS, LANNDD (DUSEL), and LENA would shed significant light on the nature of unification complementary to the physics at the LHC. Mathematical tools for the computation of proton lifetime are given in the appendices. Prospects for the future are discussed.

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Naturalness bounds on weak scale supersymmetry in the context of radiative breaking of the electro-weak symmetry are analyzed. In the case of minimal supergravity it is found that for low tanβ and for low values of fine tuning Φ, where Φ is defined essentially by the ratio µ 2 /M 2 Z where µ is the Higgs mixing parameter and M Z is the Z boson mass, the allowed values of the universal scalar parameter m 0 , and the universal gaugino mass m 1/2 lie on the surface of an ellipsoid with radii fixed by Φ leading to tightly constrained upper bounds ∼ √ Φ. Thus for tanβ ≤ 2(≤ 5) it is found that the upper limits for the entire set of sparticle masses lie in the range < 700 GeV (< 1.5TeV) for any reasonable range of fine tuning (Φ ≤ 20). However, it is found that there exist regions of the parameter space where the fine tuning does not tightly constrain m 0 and m 1/2 . Effects of non-universalities in the Higgs sector and in the third generation sector on naturalness bounds are also analyzed and it is found that non-universalities can significantly affect the upper bounds. It is also found that achieving the maximum Higgs mass allowed in supergravity unified models requires a high degree of fine tuning. Thus a heavy sparticle spectrum is indicated if the Higgs mass exceeds 120 GeV. The prospect for the discovery of supersymmetry at the Tevatron and at the LHC in view of these results is discussed.

An extension of the standard model of electro-weak interactions by an extra abelian gauge boson is given, in which the extra gauge boson and the hypercharge gauge boson both couple to an axionic scalar in a form that leads to a Stueckelberg mass term. The theory leads to a massive Z ′ whose couplings to fermions are uniquely determined and suppressed by small mixing angles. Such a Z ′ could have a low mass and appear in e + e − collisions as a sharp resonance. The branching ratios into ff species, and the forward-backward asymmetry are found to have distinctive features. The model also predicts a new unit of electric charge e ′ = Q ′ e, where Q ′ is in general irrational, in the coupling of the photon with hidden matter that is neutral under SU(2) L × U(1) Y .The Stueckelberg mechanism [1] gives mass to abelian vector bosons without breaking gauge invariance on the Lagrangian, and thus provides an alternative to the Higgs mechanism [2] to achieve gauge symmetry breaking without spoiling renormalizability. The simplest case is that of one abelian vector boson A µ coupling to one axionic scalar field φ. Here the Lagrangian

A detailed analysis of predictions of SU(5) supergravity grand unification within the framework of radiative breaking of SU(2)xU(l) is given. The masses of the 31 new supersymmetric particles depend on only four parameters. The full parameter space is explored and found to be strongly restricted by proton decay bounds if one requires no extreme fine tuning and the Higgs triplet mass to be < 3MG. The 31 sparticle masses are then strongly correlated, e.g., m& ;S jm^ m& -M? -Zm? , mn^SllO GeV.PACS numbers: 12.10.Gq, 04.65.+e, 13.30.Ce, 14.80.LyThe analyses last year of the high precision 1990 data from the CERN e+e~ collider LEP [1,2] have given the first indication of the validity of the combined ideas of supersymmetry (SUSY) and grand unified theories (GUT's). Thus the LEP data allow an accurate determination of the three coupling constants of the standard model, a\, ai, and «3, at the Z boson mass. Using renormalization-group equations (RGE) one may theoretically check that grand unification does indeed occur at a GUT scale MQ to within 1 standard deviation for the supersymmetric SU(5) GUT theory with one pair of Higgs doublets (the minimal number) due to the assumed presence of the new SUSY particles. With the simplest assumptions that all SUSY particles are degenerate with mass M s , the data give [2] M G = 10 16,±03 GeV, M s = 10 2 -5±1° GeV, and a G _1 =25.7 ± 1.7. Note that the errors (which arise mainly from a 3 =0.113 ±0.005) are anticorrelated, i.e., M G M10 161 GeV)[10 25 / A/s(GeV)] 03 . While the data are consistent with nonminimal SUSY with a larger group which has the same light particle content as the SU(5) theory, it is natural to examine first the SU(5) case.In this paper we consider the predictions of "minimal" SU (5) supergravity grand unification where the supergravity interactions cause the spontaneous breaking of supersymmetry at the Planck scale by a gauge singlet "hidden" sector [3,4]. After the breaking of SU(5) at M G , the theory can be described just below MQ by [3,5] the superpotential W = Wp ) + n 0 H l H 2 +rV (4) /M H , the effective potential 12 K = Z dW dz a + V D + m&\z a \ 2 Uo^3 ) + %o//|//2 + H.c] ,and a gaugino mass term -m\/2k a X a . Here Wy 3) are the cubic Yukawa couplings, V D is the D term, H\,Hi are the two Higgs doublets, {z a } are the spin zero fields, and W (4) are the quartic interactions (which give rise to proton decay) occurring after the elimination of the superheavy color Higgs triplets with mass MH~~MG-At MQ then, aside from the standard model Yukawa coupling constants, minimal SU(5) supergravity GUT models depend on seven parameters: mo, mi/2, AQ, BO, Ho, MG, and ag. This is just two more than in the standard model, and since the LEP data have now determined MQ, CLG, and Mz, there remain effectively four parameters to account for the 31 new SUSY particles. Thus the theory is expected to have considerable predictive power and, as will be seen below, this predictive power is greatly sharpened when the experimental bounds on proton decay are included.Radiative...

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