III. Baryon-Number Generation

Grand unified models generically involve very massive gauge and Higgs bosons whose couplings violate baryon-number () conservation. Exchanges of such bosons should lead to proton decay. In the very early universe their presence would cause any baryon number introduced as an initial condition to be destroyed (unless this is prevented by other absolute conservation laws), so that the apparent baryon excess in the present universe must have been produced through subsequent -violating -violating processes.

Baryon asymmetry may be generated through the decays of heavy (usually Higgs) bosons, but tends to be destroyed by the corresponding inverse decays and by -violating two-to-two scattering reactions. These processes are represented by the three terms on the right-hand side of the Boltzmann transport equation for the time development of the baryon-number density relative to photon-number density :

Here represents the number density (relative to the photon-number density ) of each species of -violating bosons, with total decay width , -violation parameter , and equilibrium number density . gives the cross section for -violating scattering reactions.

Consider first the simple case (discussed in Sec. 4.4 of Ref. [24]) of identical -violating bosons all with mass and decay width .. If the back reactions represented by the second and third terms in Eq. (3) were absent, the final asymmetry generated in this case would increase linearly with . However, larger also leads to larger back-reaction effects, tending to reduce the final baryon number. The importance of these back reactions depends critically on the magnitude of the scattering cross section. This cross section is obtained as a sum over amplitudes for the exchange of each of the bosons. If all these amplitudes were of the same sign, they would add coherently to give a total cross section ; if they carried random signs, they would add incoherently to give a result . In practice, the amplitudes for exchanges of many bosons in a single irreducible representation of the gauge group are related by Clebsch-Gordan coefficients. The cross section relevant for Eq. (3) is summed over all possible initial and final states, weighted with the square of the baryon-number difference between them.[24] Ignoring this weighting factor (and thus considering the total cross section) one may sum over exchanged bosons and possible initial and final states by standard methods. For the -channel exchange diagram, one obtains a combinatorial factor (e.g., Ref. [25]) , where is related to the quadratic Casimir invariant for the fermion representation, as in Eq. ( 1). The -channel exchange diagram yields the same result, while the rechannel diagram gives , where is the total number of initial and final fermions. Diagrams involving interference between exchanges of bosons in different irreducible representations vanish after summing over all possible initial and final states. These results suggest that the total cross section obtained with bosons grows roughly linearly with . We shall thus assume in Eq. (3) a cross section for . Numerical results using this form were given in Ref. 4. For large , a simple analytical approximation to Eq. (3) yields for the final baryon number the result

where is the total number of particle species (whether -violating or not) contributing to the expansion rate of the universe, as in Eq. (1). and are model-dependent parameters, and typically and [for example, in the SU(5) grand unified model, the relevant Higgs-boson couplings give and , while in the SO(10) grand unified model (with symmetry broken to SU(3)SU(2)U(1)) and Ref. [4]]. Equation (4) manifests the exponential decrease of the final baryon number for large as a consequence of increased back-reaction effects. Notice that the small numerical coefficient in the first term of Eq. (4) typically renders inverse decay unimportant compared to scattering processes. Since present observations imply , Eq. (4) potentially provides an important constraint on . In actual grand unified gauge models, however, one must account for the finite spread in -violating boson masses. Ignoring the resulting modifications for now, one may make some simple estimates based on Eq. (4). Taking and neglecting the term involving , Eq. (4) implies for and for , assuming that the -violation parameter and taking . The largest Higgs-boson couplings (responsible for the quark mass) are expected to give , yielding a significant constraint on .

The derivation of constraints for actual grand unified models requires more detailed consideration of the mechanisms of baryon-number generation. The results of Refs. [3,4], and [24] show that gauge-boson decays do not usually involve sufficient violation to yield significant baryon asymmetry: all baryon asymmetry must therefore come from Higgs-boson decays, with violation arising from one-loop diagrams involving exchange of a -violating Higgs boson or in some cases a gauge bosom The contribution to violation in the decay of a boson from exchange of a heavier boson decreases as . The sign of the contribution depends on the Clebsch-Gordan coefficients involved. If a large number of bosons with approximately equal masses and couplings of random sign are exchanged, then their net contribution decreases like for large . In addition, asymmetry arising from violation generated by -exchange corrections to decays is cancelled by the asymmetry from exchange in decay if and have nearly equal masses and couplings. Net asymmetries can arise only if the bosons have different couplings or are not degenerate.

The results of Ref. [4] indicate that in most cases the final net baryon asymmetry is dominated by the decays of the longest-lived -violating species: effects of shorter-lived species are typically eradicated by back reactions involving the longer-lived species (except when additional conserved or partially conserved quantum numbers exist, as mentioned below). For an adequate final baryon number to survive, it is therefore necessary that back reactions to the decays of the longest-lived species be suitably small. The magnitude of the back reaction depends on the couplings of the decaying species, usually Higgs bosons. In contrast with the case of proton decay, the rate of -violating Higgs-boson interactions in the early universe is determined by the largest Higgs-Yukawa coupling, presumably , rather than by their couplings to the lightest fermion flavor. With such a coupling, back reactions lead to an asymptotically exponential damping of baryon number produced by a single decaying species when its mass is below about .[24] When more than about a hundred -violating bosons with these couplings and all with masses of roughly are present, back reactions reduce the final baryon number by a factor , leaving an inadequate asymmetry. If the boson masses are , only about 20 boson species are permitted if the observed baryon asymmetry is to be accounted for. Because of the exponential dependence of back reactions on the number of contributing species, these bounds are much more stringent than those obtained solely from multiple -violating exchanges in proton decay.

The magnitude of back reactions, and thus the importance of large numbers of -violating species, depends crucially on the couplings of the relevant Higgs bosons. Two effects may increase these couplings over the estimates used above. First, if sufficient Higgs bosons or fermions are present for asymptotic freedom to be lost, as discussed in Sec. 11, then the effective Higgs-Yukawa couplings will increase as described by the appropriate renormalization-group equation. Second, in many models some of the Higgs bosons which couple to fermions do not attain vacuum expectation values, so that the magnitudes of their couplings are not constrained by the values of the light-fermion masses.

While gauge-boson decays rarely generate baryon asymmetry, gauge-boson exchanges contribute back reactions which destroy baryon number. As discussed in Sec. 11, however, the number of gauge bosons (which is determined by the dimensionality of the adjoint representation of the gauge group) is typically much smaller than the number of Higgs bosons. In addition, gauge-boson couplings connect only fermions of the same chirality in a single irreducible representation of the gauge group; Higgs-boson couplings may connect fermions in different irreducible representations and must flip the fermion chirality. [For SU() gauge groups, reducible fermion representations are required for axial anomaly cancellation. With other gauge groups, a single irreducible fermion representation usually suffices.] Thus gauge-boson reactions may neither affect asymmetries between different fermion irreducible representations [e.g., and in SU(5)], nor destroy the total asymmetry in fields of a given chirality in an irreducible representation. One ``zero mode'' exists in the Boltzmann transport equations for each irreducible fermion representation when only vector-boson exchanges are included.[26] Thus gauge-boson exchanges alone are usually insufficient to destroy all asymmetries which have been generated. [4,24]

The discussion above indicates that for an adequate baryon asymmetry to survive, a few -violating Higgs bosons must be significantly lighter than the rest. In general, Higgs-boson masses are determined by minimization of the effective Higgs potential and diagonalization of the resulting mass matrices. Some models exhibit approximate symmetries, broken on scales , which lead to approximately degenerate sets of Higgs bosons corresponding to irreducible representations of the approximate symmetry. Models with large approximate symmetry groups may therefore be unable to account for the observed baryon asymmetry. In the absence of approximate symmetries, one must usually resort to a purely statistical treatment, taking the boson masses as random variables. In the simplest case, one takes the boson masses to be independent random variables, distributed over a definite, say, unit, interval. With 5 particles, the average spacing of the lowest two masses is then ; the spacing decreases to for 10 particles, to with 20, and to with 50. With more than 20 particles, the probability that the lowest two particles are separated by more than 10% in mass falls below 0.1. An alternative and perhaps more realistic approach takes the elements of the boson-mass matrices to be independent Gaussian random variables (with, say, zero mean and unit valance). [Note, however, that the Clebsch-Gordan coefficients on which the mass matrix elements depend are not in fact Gaussian distributed at least in the case of SO(3).[28]] The distribution of eigenvalues for such matrices becomes semicircular when the dimension n of the matrix exceeds about 5.[29] Monte Carlo simulation shows that the mean spacing between the lowest masses, normalized by the total range of masses, falls roughly linearly with n, taking on a value for . For , the probability for a spacing larger than 10% between the lowest masses again falls below 0.1.

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