6. SU(5) Models (13)

6.1. Introduction

SU(5) [42] is the simplest group (and the only one of rank 4) which contains as a subgroup the observed low-energy symmetry group . The gauge vector bosons transform according to the adjoint representation . Fermions come in families, each consisting of 15 left-handed fields, transforming under the reducible representation . Each family has the generic form

where the vector (bold face) indicates transformation as an SU(3) triplet. The superfix denotes charge conjugation (see the appendix).

The SU(5) symmetry is spontaneously broken to by a representation of Higgs bosons. The singlet members of this representation are postulated to attain a large vacuum expectation value , and all surviving members of the representation receive large masses.

Lorentz and SU(5) invariance constrain the possible Higgs representations which may couple to fermions to be those appearing in the decompositions [cf. (5.2.1)]

With the quantum number assignments (6.1.1) only the 5, 15, and 45 representations contain a neutral member which could receive a vacuum expectation value and hence contribute to fermion masses. The decompositions (6.1.2) show that and Higgs representations lead to Dirac masses for the charged fermions. The presence of a would generate a Majorana mass for which cannot ``naturally'' be kept small: the is thus usually excluded. The minimal set of Higgs representations for an SU(5) model is thus together with or . Additional and may be added as required to obtain a suitable fit to the observed fermion masses.

The reducibility of the fermion representation implies that even with a single Higgs representation two independent Yukawa couplings exist: one to , giving D and E masses (equal at unification energies for the case of a single ), and one to , giving the U mass. The complete lagrangian for couplings to fermions may be written for the minimal model in the form

where is the SU(5) gauge coupling constant, and are the two Higgs Yukawa couplings, and and are fermion family indices. The couplings embodied in this lagrangian are illustrated in fig. 14.

[ Figure 14 ] Couplings of fermions (f) to gauge: (V) and Higgs (H) bosons in the minimal SU(5) model.

The SU(5) representations discussed here may be decomposed according to the embedding as

where, in , denotes the SU(3) representation, the SU(2) representation, and the U(1) charge , given by [where is the electric charge and is the diagonal generator of SU(2) normalized so that in the 5 representation of SU(5)].

In subsect. 6.2 we use the quantum number assignments (6.1.4) to determine the minimal set of independent number densities; we then derive the Boltzmann transport equations which these satisfy. In subsect. 6.3 we consider violation in the minimal SU(5) model; subsect. 6.4 gives results for baryon number generation in this model. violation in SU(5) models with additional Higgs representations is analyzed in subsect. 6.5, and results for baryon number generation in some sample models are given.

6.2. Boltzmann Equations for the Minimal SU(5) Model

Given the rates for all reactions in an SU(5) model, one could in principle use the general Boltzmann equations (4.4.6) to determine separately the evolution of the number densities of each of the hundred or so particle species which appear. However, in realistic models where there are conserved or partially conserved global symmetries (see below), or in models where there are cancellations in the contributions to the baryon number from various species as would be the case in symmetric models, it is unnecessary and difficult to calculate numerically the asymmetry in each fermion field and add them together to find the baryon number. For instance in symmetric theories there may be large asymmetries in individual fermion fields that must exactly cancel to give zero baryon number. If the asymmetry in the individual fermion fields were calculated numerically, a numerical accuracy of one part in would be necessary to deduce that the final baryon number was less than . Therefore, we incorporate the conservation laws and calculate directly only combinations of independent asymmetries. Since many of these number densities are, however, related through conservation laws, in the minimal model it is eventually necessary to trace only seven independent combinations of number densities. Note that here, as below, we use the shortened notation for number densities of particle species, and similarly denote the reduced density of a quantum number by the name of that quantum number. For massless fermions, it is sufficient to consider to ; is always close to one, and has no effect on the baryon asymmetry we discuss. Table 4 gives quantum number densities corresponding to asymmetries in each particle species. In this table, asymmetries of colored species have been summed over color. (The asymmetries are, however, not summed over the spin states of the vector bosons.) The quantum number is taken to be 1 for fermions laying in the 5 representation of SU(5), and zero otherwise [43].

[ Table 4 ] Quantum number densities corresponding to asymmetries in particle species of the minimal SU(5) model

At the temperatures of concern in baryon number generation, may be taken as an exact gauged symmetry. In the standard cosmology, the net value of any quantum number associated with a long-range gauge interaction must always vanish. Thus the total electric charge of the universe should be zero. The universe should not only be a color singlet, but also have zero eigenvalues of the commuting generators and of . Similarly, the universe should have . In addition to invariance, the SU(5) model exhibits a further global U(1) invariance corresponding to conservation. There is no necessity for the total of the universe to vanish, but we shall usually take it to do so for simplicity. (In some of the SO(10) models discussed in sect. 7, is violated, and a non-zero net density may develop.) We shall further assume below that and separately are initially zero. The consequences of large initial and are considered in ref. [55].

As mentioned in sect. 4, the rate for exchanges of light bosons (, W, gluons) should be much larger than the rate for -violating reactions induced by exchanges of heavy bosons. Processes such as or serve to maintain all species in equilibrium distributions (kinetic equilibrium), but cannot affect asymmetries between particles and their antiparticles. However, other reactions are relevant in that they quickly share asymmetries generated in one species among a set of species. For example, if an asymmetry developed in ``red'' u quarks, it would immediately be shared among all colors of u quarks by transitions from the ``red'' ones by gluon interactions. The asymmetry in the three color components may thus always be taken equal. Similar effects occur through W interactions, and serve to share asymmetries equally between all weak fermion isodoublets (14) , so that

For quarks, W interactions connect not only members of a single isodoublet, but also, through Cabibbo mixing, different families. An asymmetry generated in one quark species is thus shared with all other quarks, regardless of their weak isospin or family. As discussed below, the rates for reactions depend on Yukawa couplings of Higgs to fermions, which differ between the families. The heaviest family has the largest coupling, so that the fastest changes occur in this family. These changes are nevertheless immediately shared equally among all families. Thus, to a good approximation, one may effectively account for all families by considering only the heaviest one. Finally it is convenient to use any partially conserved quantum number as an independent asymmetry. The partially conserved quantum numbers may be found by finding the zero eigenmodes associated with exchanges of a particular boson [43]. Let be a set of fermion and boson asymmetries, and let be a set of independent quantum number densities , , etc. related to by . The thermalization of a quantum number through reactions of a particular boson is given from eq. (4.3.14) by

where

and represents the change in the value of through the reaction . Boltzmann's theorem requires that the eigenvalues of are all real and non-positive. Any zero eigenvalues reveal additional symmetries; the corresponding eigenvector of number densities is then conserved in reactions [e.g. in vector boson exchanges in SU(5)]. If this eigenvector is conserved in the reactions of all species, then it represents a globally conserved quantum number [e.g. in SU(5)] and results in a further reduction in the number of independent .

Using the constraints discussed above, the asymmetries listed in table 4 may be written in terms of the independent set

as

The time development of number densities in SU(5) models may be obtained by substituting explicit decay branching ratios and scattering cross sections into the coupled Boltzmann equations (4.4.6) and (4.4.7). Table 5 gives the branching ratios for -violating boson decays in SU(5) models, averaged over the boson and antiboson in each case, violation appears in the differences

between boson and antiboson partial decay widths, whose magnitudes are discussed in subsect. 6.3. If they are energetically possible, the decays or proceed at a rate proportional to the SU(5) gauge coupling constant . Their rates are parametrized by

Decays of to fermions are also proportional to , but for S they involve Yukawa couplings, and are . If nearly all X, Y decays are to fermions. If , however, then only a fraction of S decays are to fermions. Taking suggests a branching ratio for S decays to fermions. Since inclusion of modifies the branching ratio to fermions by only 20%, we shall assume for simplicity below that and hence .

[ Table 5 ] -violating boson decays in the minimal SU(5) model

Inserting the branching ratios of table 5 together with relations (6.2.2) into eqs. (4.4.6) and (4.4.7) give the complete Boltzmann transport equations for the evolution of number densities in the minimal SU(5) model:

where

where is the SU(5) gauge coupling constant, and are effective masses for the quarks in the heaviest family (evaluated at an energy scale ). With and , and . In the cross sections given above, is the c.m. energy, taken to be averaged over thermal equilibrium distributions for the incoming particles. The cross sections given ignore the presence of background gas: its effects were discussed in subsect. 4.5, and will be mentioned in subsect. 6.4 below. The and decay widths are also averaged over equilibrium energy distributions.

In the absence of Higgs interactions . Ignoring these interactions, and setting , eq. (6.2.6) simplifies significantly to become

where and are the relevant -violation parameters corresponding to eq. (6.2.6).

6.3. Violation in the Minimal SU(5) Model

In this section, we discuss the magnitude of the -violation parameters appearing in the Boltzmann transport equations (6.2.8) for the minimal SU(5) model. We shall show that violation can occur only in high-order diagrams, and is thus suppressed [24,44,45].

As discussed in subsect. 5.1, -violating decay amplitudes must result from interference of higher order corrections to decays. The lowest order such interference diagrams for the minimal SU(5) model are shown in fig. 15. As discussed in sect. 5, corrections to gauge vector boson decays involving only vector bosons cannot give rise to violation at any order if all fermions are massless. According to the minimal SU(5) model, a single Higgs scalar boson couples to fermions. This representation contains just one -violating component. The result (5.2.5) then shows that no violation may occur in one-loop diagrams of the form illustrated in fig. 11 involving scalar boson exchange in vector boson decay or vector exchange in scalar decay. Application of eq. (5.1.20) shows that no violation may arise to one-loop order from scalar boson exchange in scalar boson decay (as illustrated in fig. 12). Another potential source of violation at lowest order is from diagrams involving three-boson couplings. Since the is a real representation, its vacuum expectation value must be real; in addition, any phase in the vacuum expectation value of may be arranged not to appear in couplings of the -violating component of . Hence, three-boson couplings cannot exhibit violation in the minimal SU(5) model. These results demonstrate that no violation occurs in the one-loop approximation for the minimal SU(5) model.

[ Figure 15 ] First-order corrections to boson decay amplitudes in the minimal SU(5) model. None of the diagrams can give rise to the violation required for baryon number generation.

We now discuss the possibility of violation in higher-order diagrams for the minimal SU(5) model. Since the couplings of gauge vector bosons to fermions and bosons are purely real, addition of further such couplings to the diagrams of fig. 15 cannot yield violation. Similarly, addition of three- or four-boson couplings cannot introduce violation. Any violation must thus occur first in diagrams involving only fermions and Higgs bosons coupling to them. Such violation requires complex phases in the Higgs Yukawa coupling matrices defined in eq. (6.1.3). In terms of these couplings, the violation parameter of (5.1.5) arising from the lowest order diagram fig. 15d may be written as , where the trace is taken over the fermion family indices implicitly carried by the Yukawa couplings . As mentioned above, this quantity is real, so that no violation may result from the diagram of fig. 15d. At the next (three-loop) order, investigation of possible diagrams shows that all yield purely real . and thus cannot introduce violation. For example, the diagram of fig. 16 gives

which is manifestly real. However, in the next order, traces such as

corresponding to the class of diagrams illustrated in fig. 17 need not be real, and may give rise to violation. These diagrams may give rise to small but non-zero values for the -violation parameters appearing in the Boltzmann equations (6.2.11). The result for these parameters may be written as

where is a -violation parameter with . Approximating the Higgs Yukawa couplings by their value for the heaviest fermion family, and taking for the momentum-space factor the volume of available phase space, one obtains the rough estimate

With the usual mechanism for spontaneous symmetry breakdown, so that . In the next section we shall show that unless this inequality is saturated, the baryon asymmetry which may be generated in the minimal SU(5) model is entirely inadequate to explain observational results.

[ Figure 16 ] Example of a second-order correction to scalar boson decay in the minimal SU(5) model. No diagrams at this order can introduce violation in the minimal SU(5) model.

[ Figure 17 ] Examples of a class of third-order corrections to scalar boson decay in the minimal SU(5) model which may potentially give rise to the violation necessary for baryon number production.

6.4. Baryon Number Generation in the Minimal SU(5) Model

In this section, we discuss the generation of a baryon asymmetry in the minimal SU(5) model, using the Boltzmann equations derived in subsect. 6.2, and the -violation parameters discussed in subsect. 6.3. Three basic parameters appear in these calculations: the mass of the gauge vector boson , the mass of the Higgs bosons , and the -violation parameter in the Higgs decays. For the latter parameter, we use the rough estimate (6.3.3) and consider the following cases:

The X boson mass is taken as

in keeping with theoretical and phenomenological estimates. The stability of spontaneous symmetry breakdown in the SU(5) model appears to require

Fig. 18 shows results for the final baryon number density produced in the minimal SU(5) as a function of for the cases (6.4.1a) and (6.4.1b). Explanation of the observed baryon asymmetry would require production of at this epoch: fig. 18 shows that such a result is possible in the minimal SU(5) model only if which would invalidate perturbative methods. (The naive expectation that increases at low energy scales may be invalid if , since the renormalization group equation for then receives both positive and negative contributions of roughly equal size.) Thus one may conclude that the minimal SU(5) model is unable to account for the observed baryon asymmetry. It is nevertheless instructive to consider the origins of the complicated behavior seen in fig. 18.

[ Figure 18 ] Final baryon number generated in the minimal SU(5) model.

We first consider the case . The time development of quantum number densities for this case with and are shown in fig. 19. At high temperatures, S decays generate asymmetries in , and as described by eq. (6.2.8). The primary effect which reduces these asymmetries at lower temperatures is scattering. At high temperatures, the effective magnitude of scattering cross-sections are uncertain (as discussed in subsect. 4.5). The two extreme cases and are shown as solid and dashed lines for in fig. 19a. Although different at high temperatures, they yield identical final results. By virtue of their larger couplings, vector boson exchanges are usually more important than scalar boson exchanges in scattering cross sections. However, in SU(5) models, vector boson exchanges conserve . Hence only Higgs boson exchanges can reduce a non-zero density () generated through S decays at high temperatures. If only vector boson exchanges are considered, then from eq. (6.2.8)

which implies a final equilibrium state with non-zero baryon number

The dotted lines in fig. 19b show results in this case. Changes in require exchange of heavy vector or scalar bosons; however, changes in may occur through light as well as heavy boson exchanges. Typically, vector boson exchanges enforce the relations (6.4.4) during the period when is reduced through heavy Higgs exchanges. At very low temperatures, heavy vector boson exchanges become ineffective, so that remains constant, while light Higgs boson exchanges continue to reduce and . The final value of is essentially determined through the relation (6.4.4) by the value of at the temperature where heavy vector boson exchanges become rare. Note that the larger Higgs Yukawa couplings in the case of fig. 19b cause a much more rapid decrease in than in fig. 19a.

[ Figure 19 ] Time development of quantum number densities in the minimal SU(5) model. The dashed curve in (a) shows results with a modified form for the high-energy scattering cross-section. The dotted curves in (b) show results if light Higgs boson Time development of quantum number densities in the minimal SU(5) model. The dashed curve in (a) shows results with a modified form for the high-energy scattering cross-section. The dotted curves in (b) show results if light Higgs boson exchange reactions are ignored.

The discussion of the previous paragraph applies to the case , in which asymmetries are initially generated in S decays, and subsequently thermalized by X exchanges. If , then the S decays responsible for generation of the asymmetries occur at a sufficiently low temperature that X exchange cannot provide significant thermalization. The presence of the X is thus essentially irrelevant, and the final baryon density behaves as it would with only one particle [I]. The dotted curve in fig. 18 shows the final density if all X exchanges are artificially set to zero. The enhancement around is a transition phenomenon: X exchanges are sufficiently unimportant that is no longer constrained to be proportional to , and thus does not suffer the destruction experienced by as a result of light Higgs exchanges. Since is held fixed, the decrease in for small simply reflects the increasing importance of back reactions when is reduced. The similar conclusions of ref. [46] were based on a calculation which ignored non-thermalizing modes and took the final baryon number to have a simple power law dependence on . Figs. 18 and 19 illustrate the inapplicability of these assumptions.

6.5. Baryon Number Generation in Extended SU(5) Models

The results of sect. 6.4 demonstrate that no viable choice of parameters allows adequate baryon number generation in the minimal SU(5) model. In this section, we consider two simple extensions of the minimal SU(5) model, which can account for the observed baryon asymmetry with suitable choices for parameters.

In the minimal SU(5) model, a single Higgs representation is taken to couple to fermions. This representation contains a single -violating Higgs boson (denoted ) with quantum numbers . We consider two extensions of this minimal model: in model A [12], a second is introduced, and in model B a is added. In both cases we denote the component of the additional Higgs representation by . For the , further B violating bosons occur; we shall however assume that can be arranged to include their effects. The bosons and may in general mix; we denote the resulting mass eigenstate mixtures by and , where . Similarly, we take the light Higgs boson mass eigenstates as and . We write the Yukawa couplings of and (or and ) as and , respectively. These Yukawa couplings satisfy

but are not individually determined. For simplicity we shall, however, take .

The absence of mixing forbids decays of the form , where V is a gauge vector boson. However, in analogy to the case of the minimal SU(5) model discussed in subsect. 6.2, the decays and may occur if they are energetically possible. These decays dilute any violation arising from decays to fermions.

We consider first model A. As discussed in subsect. 5.2, this model allows no violation at first order in gauge vector boson decays; violation may, however, appear in S and decays through and S exchanges respectively. The magnitude of violation in decays is then given in terms of the parameter (such that ) by

Results for S decays are obtained by exchanging S and , and and taking . For one then obtains

which goes to zero as expected when goes to infinity. violation may occur in decays not only through S exchanges, but also through exchanges of light Higgs . As discussed in subsect. 5.1, violation in boson decays may yield an asymmetry in a particular quantum number density only if that quantum number is violated by the exchanged boson in fig. 8. Thus exchanges cannot lead to asymmetries in ; they can, however, contribute to and asymmetries. The magnitude of these contributions is given approximately by

Combining eqs. (6.5.3) and (6.5.4) yields complete quantum number densities generated by free decays

(Results for S decays are obtained by the interchanges , , .) The Boltzmann transport equations for model A are now obtained by replacing every occurrence of S in (6.2.10) with a suitable sum of S and , and inserting the parameters (6.5.5). Equations for analogous to those for must also be added. Finally, the scattering cross-section from S exchange must be supplemented with an analogous term for exchange and with an interference term.

Fig. 20 shows the final baryon number density generated in model A. Note that when our assumption implies that : the ensuing cancellation allows no baryon number generation, as expected from eq. (4.4.4). Fig. 21 shows the time development of quantum number densities in model A for various choices of and .

[ Figure 20 ] Final baryon number generated in an extension of the minimal SU(5) model containing an additional multiplet of Higgs bosons (model A). S and are the two mass eigenstate -violating Higgs bosons.

[ Figure 21 ] Time development of quantum number densities in extended SU(5) model A.

We consider first the region illustrated in fig. 21a. The contributions of various terms in the Boltzmann equation (6.2.8) for this case are shown in fig. 22a. By virtue of eq. (6.5.3), , so that asymmetries generated in S decay are negligible compared to those generated in decay. In addition, no asymmetries may be generated by X decays in model A. decays at high temperatures generate asymmetries in , and . reactions involving X exchange then ``thermalize'' these number densities to the values (6.4.5); S exchange is unimportant since S is both more massive than X and has smaller couplings. The second region of parameters for model A is . The time development of quantum number densities for this region is shown in fig. 21b; contributions to the Boltzmann equation for this case are given in fig. 22b. The initial production of quantum numbers and their thermalization through X exchange reactions here is as in the case discussed above. At low temperatures, however, reactions involving S become important, and determine the final values of the quantum number asymmetries. Notice that the sign of B changes at low temperatures in fig. 21b. This is a consequence of the fact that terms proportional to S in the Boltzmann transport equation involve combinations of , and B in which B may appear with a negative sign. The dominant term governing the time evolution of B for is with similar equations for and . Since , and , this term tends to drive positive. In general there are three linear combinations of , and which decrease as pure exponentials until cut off at temperatures below . is a linear combination of these three exponentials, and its final value depends sensitively on the initial values of , and . For this reason it is not adequate to assume that is produced and damped in successive independent stages as in simple models which treat only one quantum number. For both and inverse decays into S are no longer able to change the sign of the negative produced through decays and hence the final is negative. The results of fig. 21 are for large values of . Most qualitative features remain unchanged when is reduced. However, since decays then occur at lower temperature, back reactions are more important, and the final asymmetries generated tend to be reduced.

[ Figure 22 ] Contributions to in extended SU(5) model A from various terms in the Boltzmann transport equation (6.2.10). Positive contributions are shown as solid lines; negative ones as dashed lines. The terms of sufficient magnitude to appear are: 2: X exchange; 3: ; 6: term; 7: term. The effect of the expansion of the universe included in the left-hand side of the Boltzmann equation is shown as .

Fig. 20 demonstrates that with suitable choices for undetermined parameters, extended SU(5) model A can account for the observed baryon asymmetry. The sign of the final baryon asymmetry cannot, however, be related directly to the sign of the -violation parameter without detailed knowledge of other parameters.

We now consider model B. The couplings of the and to fermions are and , respectively. The difference in Clebsch-Gordan coefficients between these couplings implies that the -violation parameter for scalar-vector diagrams may be non-zero [cf. eq. (5.2.5)]. Hence in this model, asymmetries may be generated through vector boson exchange in scalar boson decay, and in vector boson decay through scalar boson exchange, as well as through scalar boson exchange in scalar boson decay. The relevant -violation parameters in this model are then given by

Terms accounting for light Higgs exchange must be added as before. Inserting these parameters into the Boltzmann transport equations, we obtain the results shown in fig. 23 for the final baryon asymmetry. The possibility of violation in processes involving vector bosons renders these results still more complicated than for model A. In discussing fig. 23, we consider first the region . violation in X decays is suppressed here, since it involves exchange of scalar bosons heavier than X. After asymmetries are initially generated in S and decays, they are thermalized by X exchange reactions to the values (6.4.5). The resulting is then proportional to the original generated. Although the generated by free S and decays would be large since it receives contributions from vector boson exchange, only scalar exchanges can contribute to . Hence the final value of is similar to that obtained in model A. We now consider the region . In this region, the final is dominated by asymmetries generated in S decays; any generated in X or decays is destroyed by processes involving S. generated in S decays is damped by inverse S decay processes; the final obtained varies roughly as : a factor from the -violation parameter (6.5.6), and from the effects of inverse reactions. In the region several sources of baryon number contribute with roughly equal weight, and no simple qualitative explanation of the final results is possible. Fig. 24 gives some examples of the development of the quantum numbers for this model.

[ Figure 23 ] Final baryon number generated in an extension of the minimal SU(5) model involving a as well as a Higgs multiplet (model B). S and are the two mass eigenstate -violating Higgs bosons considered.

[ Figure 24 ] Time development of quantum number densities in some sample cases of extended SU(5) model B.

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