1. Introduction

Grand unified gauge models (e.g. [1]) typically attempt to combine quarks and leptons (and often also antiquarks and antileptons) as elements of the same irreducible representations of some gauge group (1) G (which must contain the observed low-energy symmetry group ). The gauge bosons (which transform under the adjoint representation of G) can induce transitions between any two members of an irreducible fermion representation. Hence some of them should mediate baryon () and lepton () number violating interactions, in which, for example, quarks decay into leptons and antiquarks (e.g. ). The limit years (e.g. [1]) on the lifetime of the proton suggests, however, that any baryon-violating vector bosons should have masses . Direct evidence for such -violating interactions must presumably come from observation of proton decay. However, if any violation does indeed occur, its suppression at accessible energies due to the large masses of the intermediate bosons, should have been overcome at the extremely high temperatures which presumably existed in the very early universe (e.g. [2]). We shall discuss the constraints on such -violating processes in the standard hot big bang cosmological model necessary to allow the apparent excess of baryons over antibaryons in the present universe. Even if the universe initially (say, at the Planck time) had a non-zero net baryon number (but no other net conserved quantum number), -violating interactions at very early times should relax the asymmetry away, leaving equal numbers of baryons and antibaryons. Then, when the universe cooled to a temperature , the baryons and antibaryons would have annihilated away (2) and the observed baryon number density could not be accounted for. To reconcile the possibility of rapid -violating processes at very high temperatures with the apparent non-zero net baryon number of the universe, it is presumably necessary that a baryon asymmetry should develop from the symmetrical state present after any initial has been erased. (The possibility of the phenomenon was suggested by Sakharov in 1967 [3].) The generation of an asymmetry of the required magnitude places severe constraints on -violating interactions, and therefore on grand unified gauge models. The purpose of this paper is to provide a detailed and systematic description of these constraints. The basic physical phenomena involved in the generation of a baryon excess were discussed in a recent paper by two of us [4] (hereafter referred to as I), where several simple illustrative models were considered. Here, we treat more realistic and complicated gauge models, in which many of the parameters relevant to baryon number generation are determined by the basic structure of the models, rather than being arbitrary, as in the illustrative models of I.

The generation of a baryon excess from a state requires violation, violation and deviations from thermal equilibrium. (Without deviations from equilibrium, no ''direction of time" is distinguished, and invariance renders the , and violations ineffective).

Sect. 2 discusses violation, deriving constraints on its form in grand unified models.

In sect. 3 we discuss the form of violation in grand unified gauge models, and the mechanisms by which it may occur.

Sect. 4 considers the statistical mechanics of baryon number generation. Subsect. 4.2 describes departures from thermal equilibrium for a single massive particle species in an expanding universe. Subsect. 4.3 shows how such a departure from equilibrium may generate asymmetries in quantum number densities associated with light particles. In the realistic models to be treated, many particle species are present: subsect. 4.4 gives the general Boltzmann equations required. Our treatment of statistical mechanics assumes the applicability of the Boltzmann equation. Subsect. 4.5 discusses the limits of validity of this approach, and considers possible extensions. Most of our calculations are performed in the context of the simplest cosmological model in which the early universe is taken homogeneous and isotropic. Subsect. 4.6 discusses the consequences of relaxing this assumption.

In most of the models we consider, the basic process responsible for baryon number generation is the decay of superheavy bosons. -violating effects in these decays must arise from one-loop correction diagrams. Subsect. 5.1 derives the baryon asymmetry generated through such diagrams from the free decay of superheavy bosons. The consequences of these results for -violation parameters in gauge theories are described in subsect. 5.2. Unless supermassive fermions are present, no violation may occur from gauge bosons alone. violation may occur in diagrams with Higgs boson exchange in gauge vector boson decay only when several different representations of Higgs bosons coupling to fermions are present.

The simplest grand unified models are those based on the group SU(5), outlined in subsect. 6.1. Sect. 6 considers in some detail baryon number generation in several simple SU(5) models. In subsect. 6.2, we derive the Boltzmann transport equations for the ''minimal" SU(5) model, involving and Higgs boson multiplets. Subsect. 6.3 demonstrates that violation in this minimal model can occur only at a high order in perturbation theory, and is thus expected to be small. Nevertheless, in subsect. 6.4 we present results on the final baryon number generated in the minimal model. We find that no acceptable choice of parameters yields an adequate baryon asymmetry. Subsect. 6.5 then considers some extensions of the minimal model involving additional Higgs multiplets. With suitable choices of parameters, these models may account for the observed baryon asymmetry.

In sect. 7 we consider grand unified models based on SO(10). Subsect. 7.1 gives a general discussion of these models, emphasizing features not present in SU(5) models. Subsect. 7.2 considers the form of violation, and the possibility of violation not present in SU(5) models. Unbroken SO(10) models exhibit an exact invariance (subsect. 7.3) whose presence would prevent generation of any baryon asymmetry. The consequences of two possible SO(10) symmetry breaking schemes are considered in subsect. 7.4 and 7.5. With suitable choices of parameters, either scheme could be responsible for the observed baryon asymmetry.

SU(5) and SO(10) models represent simple schemes for grand unification. However, it is possible that more complicated models are, in fact, necessary. The cosmological constraints on very large models will be considered further in [5]. A preliminary discussion was given in the preprint version of this paper.

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