7. SO(10) Models (15)

7.1. Introduction

Although SU(5) grand unified models contain the fewest fundamental fields, they exhibit a number of seemingly undesirable features which may be avoided in models based on larger gauge groups. The first of these features is the assignment of fermions to a reducible representation of SU(5). This has the consequence that particles belong to different irreducible representations from their antiparticles. In addition, axial anomalies cancel only between different irreducible fermion representations, in a seemingly accidental manner. A further feature is the presence of a global conserved quantum number which has no basis in a local gauge invariance. In this section, we consider models based on SO(10) [49], in which these features are removed.

SO(10) symmetry is ultimately broken down to the low-energy symmetry. This breaking may occur through a sequence of stages at different mass scales. Baryon number generation may occur at an intermediate stage in this sequence, when the effective symmetry is larger than . Typical subgroups of SO(10) which may represent intermediate effective symmetries are:

Table 6 lists decompositions of relevant representations of SO(10) in terms of these subgroups. Transformation properties under these decompositions will be denoted as follows. will denote representations of SU(4), , and , respectively. will denote representation of SU(4), , and respectively. is the charge operator , defined to be when acting on an doublet. Finally, as in previous sections, will denote representations of , and [12,50]. The hypercharge is defined so that , where is the electric charge operator.

[ Table 6 ] Decompositions of some representations of SO(10) in terms of the subgroups and

Unbroken SO(10) symmetry exhibits a local gauge invariance associated with (containing the global invariance of SU(5) models as a subgroup). When this invariance is destroyed by spontaneous symmetry breakdown, violating processes may occur. With effective symmetries smaller than SO(10), invariance is preserved whenever a invariance exists. violation may thus occur when the effective symmetry is that of (7.1.1c), but not those of (7.1.1a) or (7.1.1b).

The gauge vector bosons in SO(10) models are taken to transform according to the adjoint representation. This representation contains the -violating bosons of SU(5) models, together with an additional pair of -violating bosons . It also contains a further -conserving multiplet denoted V and the -conserving gauge bosons of , , , and , which we will denote by W, , and respectively.

In SO(10) models each family of left-handed fermions is assigned to the 16-dimensional complex spinor representation, while the corresponding right-handed conjugate fields are assigned to . With this assignment of fermions, no axial anomalies appear (16).

The 16 representation may be decomposed in terms of SU(5) representations as (see table 6)

and thus contains the usual SU(5) fermion fields, together with an additional SU(5) singlet field denoted . The field may be taken as a charge conjugate partner for the . This pairing allows definition of a charge conjugation operator ; no such operator may be defined in SU(5) models because has no suitable partner. Charge conjugation invariance is in general spontaneously broken. It remains unbroken only with the effective symmetry of (7.1.1a), or with the subgroup of (7.1.1a) in which case interchanges and and conjugates representations in SU(4).

The pairing of with provides the possibility of a Dirac mass for the . To avoid this disastrous phenomenon, one must introduce a large () Majorana mass for the , so that the neutral lepton mass matrix takes the form

where is of the order of quark masses [51]. For one eigenvalue of this matrix is and represents the physical N, and the other is and represents a neutrino with small but finite mass. This neutrino mass and the resulting neutrino oscillations are an important prediction of SO(10) models. The Majorana mass term for the has , ; it is thus permitted only if is violated. Hence the must remain massless until the effective symmetry is reduced to (7.1.1c).

The Higgs content of SO(10) models is dictated by the need to break SO(10) down to and by the desire to obtain the observed masses and mixing angles of the fermions. Higgs fields which couple to fermions must appear in the product

where S (A) indicate the symmetric (antisymmetric) product. The 10 contains the usual light doublet , together with a -violating boson denoted S. It also includes the antiparticles and . With a alone, one obtains the tree approximation mass relations

at a mass scale for each family. Attempts to fit the observed mass spectrum more accurately generally lead to models with a rather baroque Higgs sector [52,53]. Inclusion of and will be discussed in subsects. 7.2 and 7.3. Since the Higgs structure of SO(10) models is considerably more complicated than in SU(5) models, we shall below analyze only a few specific cases. Other choices for Higgs structure are expected to yield qualitatively similar results.

As in SU(5) models, Higgs representations which do not couple to fermions must also appear in order to achieve the desired symmetry breakdown. Representations commonly used for this purpose are or a .

In SU(5) models, there are no particles with masses between the scale of breaking and the scale of SU(5) breaking. In SO(10) models, intermediate mass scales are possible, associated with the intermediate symmetries of eq. (7.1.1). If SO(10) breaks first to or to before breaking to , then fits to the weak mixing angle suggest that mass scales as low as may exist. Take and to be the masses of typical -violating vectors and scalars respectively, and and the scales at which and SU(4) break to and , respectively. Then non-observation of proton decay requires that and . Experimental constraints on and are much less stringent: non-observation of muon and electron number violating decays such as gives a lower bound of about for while limits on the strength of right-handed weak currents require only that --300 GeV. Theoretical fits to and the weak mixing angle give a minimum value for of about GeV with typical values being GeV for SO(10) broken first to . If SO(10) breaks first to then typical values are and .

The presence of these intermediate mass scales for spontaneous symmetry breakdown could allow an unbroken gauge symmetry larger to exist even at the temperatures of relevance to production. In the following sections we discuss the production of baryon number for models in which the effective symmetries of (7.1.1a), (7.1.1b) and (7.1.1c) are present.

7.2. and Violation in SO(10) Models

In SO(10) models, the SU(5) singlet fermion is present. If the effective symmetry still contains an unbroken invariance, then the may be assigned a definite conserved , and no new boson couplings are possible. However, when the effective symmetry is that of eq. (7.1.1c), is broken and the may acquire a Majorana mass. The additional classes of boson couplings introduced by the presence of N in this case are listed in table 7. No new assignments for -violating bosons appear in this list. However, comparison of table 7 with tables 2 and 3 shows that several bosons may violate through couplings to N. Two such violating bosons are the S and discussed above. There are in addition violating vector bosons which transform as and and which represent gauge bosons for the SU(4) and subgroups of SO(10). There are also several further violating scalar bosons. These appear in SO(10) representations capable of coupling to fermions as follows:

[ Table 7 ] Quantum numbers for possible vector and scalar pairs of singlet fermions N and quarks q or leptons

We will also consider baryon number generation when the effective symmetry is that of eq. (7.1.1b). Then, in analogy to the discussion of subsect. 6.2, the unbroken symmetry enforces relations between fermions number density asymmetries:

with the number density per color of species . Although symmetry implies we retain the two separately to exhibit the role of the charge conjugation invariance discussed in subsect. 7.3. In terms of the asymmetries (7.2.2), the baryon number density becomes

It is also convenient to introduce a quantum number , defined as the total asymmetry in (left-handed) fermion fields:

Because of their chiral structure, all gauge vector boson interactions conserve . It may nevertheless be violated by Higgs scalar interactions. [In this respect it is analogous to the quantum number introduced for SU(5) models in sect. 6.]

Tables 2, 3 and 7 give possible classes of couplings which respect invariance. If the effective symmetry is larger than , then relations may exist between some of these couplings. It turns out that with an effective symmetry , the only couplings affected in our considerations are those of the component of , which are prevented from exhibiting -violating couplings with this effective symmetry.

7.3. and Violation in SO(10) Models

The generation of a net baryon number from symmetric initial conditions requires the presence of both and violation. In weak interaction models and SU(5) grand unified models no operator may be defined since there is no left-handed antineutrino to form the charge conjugate partner of the lefthanded neutrino. In larger models, such as SO(10) or E(6), each fermion has a potential charge-conjugate partner or is an eigenstate of and a operation may be defined which is a symmetry of the unbroken theory [51]. The production of a -odd quantum number (such as or ) in these models therefore depends on the interplay between the sources of violation and the processes which violate the quantum number under consideration.

The lack of production in a -symmetric theory may be seen by considering the decays of -violating bosons and their antiparticles as well as the decays of their charge conjugate partners and . The produced by the decays of an equal mixture of and into the specific final state and the charge conjugate decays of and into the state is proportional to the quantity [see eq. (5.1.12)]

represents an integral over the intermediate momenta and final state phase space for the decay and is a product of the relevant couplings. The lowest order contributions to and are discussed in sect. 5. and are the corresponding quantities for the charge conjugate reaction. In a -symmetric theory, and , while since is -odd, and , causing to vanish.

As discussed in subsect. 7.1, with the intermediate effective symmetry , or the subgroup of this symmetry, charge conjugation invariance remains unbroken. While this symmetry exists, no baryon number generation may thus occur. Although production requires violation, production of quantum numbers such as , or which are not odd under , may occur even when invariance is unbroken. These asymmetries may then be converted into asymmetries at lower temperatures by and violating reactions. For a final non-zero to result, the second stage must occur at sufficiently high temperatures () that the effects of -violating bosons are still important. Thus if SO(10) symmetry breaking occurs through or , the intermediate symmetry may not persist to temperatures below if an adequate is to be produced [54].

invariance may be broken either by the presence of different masses for the and , or through mass splittings between bosons and their charge conjugate antibosons. A non-zero can occur only with the effective symmetry (7.1.1c). In this case, any of the diagrams in figs. 9--11 may yield - and -violating contributions proportional to .

Under the effective symmetry the adjoint representation of gauge vector bosons has the decomposition given in table 6. The color triplet doublet -violating bosons and and their antiparticles combine to form the representation. With our conventions the have electric charge and the have electric charge . Charge conjugation takes , , , and . production through vector boson reactions therefore requires a mass splitting between the and doublets. This will in general be present if SO(10) is broken to . However, if SO(10) is broken only to , then the splits into and the conjugate state and as a result there is no mass splitting. The -violating vector bosons will therefore be unable to produce a net in their decays or to convert an asymmetry in into an asymmetry in .

The content of the , and Higgs representations coupling to fermions are given in table 6. When is broken to some of these bosons may acquire -violating mass splittings. The usual -violating color triplet S appears along with its antiparticle in , and is thus an eigenstate of . The contains and . However, as discussed in subsect. 7.1, these bosons may violate only when the effective symmetry is rather than . The and appearing in the may nevertheless both violate B and acquire masses which differ between particles and their corresponding antiparticles, and thus violate . After the breaking , these representations may be decomposed as (denoted ), (denoted ) and (denoted ). The requirement of violation therefore requires the presence of a in order for to be produced with effective symmetry . If the effective symmetry is , then since the usual -violating color triplet scalar boson S is transformed into its antiparticle under and hence may not have any -violating mass splitting, production through S decays must be proportional to the -violating mass splitting between and so that

as can be seen by expanding the relevant phase-space integrals in powers of [54]. In the sequel we shall assume that this contribution to production is negligible compared to the contribution from conversion of other asymmetries whose production is not restricted by invariance. Note that more than one family must be present to allow the antisymmetric coupling of the to .

7.4. Generation for SO(10) Models With Effective Symmetry

In this section we describe the calculation of baryon number generation in SO(10) models where is the effective gauge symmetry at temperatures relevant to baryon number production. We shall assume that all production occurs in this phase; the equations derived may nevertheless also be used with suitable initial conditions to describe the development in the phase of a generated at temperatures where a larger effective symmetry exists. If SO(10) breaks first to SU(5) and then to , then fits to the weak mixing angle suggest that but do not constrain the values of , or . Below we shall usually choose the values .

N decays are potentially an important source of and asymmetries in SO(10) models. The N have two distinct types of decay modes. The first are two body decays

with , where is the usual weak doublet. The width for this decay mode is

where is the mass of the relevant charge quark and is the mass of the usual weak boson. The N may also undergo three body decays

mediated by exchange of a supermassive gauge boson coupling to the N (and thus not contained in the of SU(5)). These decays have typical widths given in analogy to decay by

When , two-body decays dominate. These decays violate lepton number, but do not violate baryon number. They may therefore give rise to an asymmetry in but not in . In models where (7.4.3) dominates (as in the unusual case ), N decays may violate baryon number. However, in this case, stringent lower bounds on exist to ensure that N decays should not generate excessive entropy to dilute any produced [35]. We shall not consider N decays below.

We shall consider a specific but presumably typical SO(10) model, in which two couple to fermions. The mass eigenstate -violating Higgs bosons will be denoted by S and . We shall include violation only for exchanges of S in decay and vice-versa. In analogy to the case of SU(5) models discussed in subsect. 6.2, we consider the development of the independent combinations of quantum number densities , , and . Table 8 gives the values of these quantum numbers for the various fields under consideration. Note that since the and form doublets, the asymmetries . Using the decay rates for the X, , W, V, S, and bosons given in table 9 one may derive the following set of Boltzmann equations for the evolution of the independent number densities

The averaged widths and cross sections appearing in these equations are given in subsects. 6.2 and 6.6. The effective -violation parameters, , are given by a sum of the decay modes for each , weighted by the value of created in the decay and multiplied by an overall factor corresponding to the multiplicity of the decaying boson:

We take all the , , , and to be zero. Neglecting compared to , we obtain

so that as expected and vanish in this approximation. Eq. (7.4.7) then yields

with corresponding relations for . In terms of the single -violation parameter these may be written

where as in sect. 6 is the gauge coupling constant, is the mass of the weak gauge boson and is the effective mass of the heaviest family at the unification scale. We take .

[ Table 8 ] Quantum numbers for particles contributing to baryon number production in SO(10) models with effective symmetry

[ Table 9 ] Quantum numbers and partial widths for supermassive boson decay modes in SO(1O) models with effective symmetry

In subsect. 7.3 we showed that the SO(10) model discussed here can generate directly only asymmetries in , and ; asymmetries in may arise only indirectly through conversion of these quantum numbers by inverse decay and scattering processes. Fig. 25 shows the final baryon number and generated in this model, together with the values of and obtained ignoring low temperature light Higgs boson exchanges. The results assume . For , , , and are produced dominantly through the -violating decays of the S with their signs and magnitudes determined by the relations (7.4.8). When the contributions from S decay and decay exactly cancel and no asymmetries are produced. For , decays dominate and since is opposite in sign to the values of the quantum numbers produced differ in sign from the case . For , inverse decays into S tend to damp the asymmetries produced through decay. The final values of the quantum numbers in this case depend sensitively on the values initially produced through decay. A similar phenomenon was noted in subsect. 6.4. For B production in fig. 25 is dominated by inverse decay and scattering processes mediated by . Inspection of the inverse decay terms in (7.4.5) (or of the decay modes given in table 9) reveals that the combination of quantum numbers is conserved in these processes. Hence if only exchange occurred, the final equilibrium values of quantum numbers would be non-zero and given by

where is the initial value of generated through S decays. Since , the processes tend to produce a negative . For , asymmetries are produced through S and decays at temperatures below the mass where reactions are negligible. In this case is dominantly produced through processes involving the X boson. Conservation of in X reactions leads to the equilibrium values [cf. eq. (6.4.5)]

Since the contributions to tend to cancel and the resulting is small. The fact that the X and tend to produce of opposite sign is a consequence of charge conjugation symmetry. As discussed in subsect. 7.3, unbroken invariance would yield and would cause the contributions of X and to production to cancel. For , production is dominated by inverse decays into S. When is sufficiently small, all asymmetries are reduced to zero.

[ Figure 25 ] Final quantum number densities (scaled by the -violation parameter ) generated in an SO(10) model with no intermediate effective symmetry larger than . Results for and , are obtained neglecting effects of light Higgs boson exchange at low temperatures. S and are mass eigenstate Higgs bosons.

If both the S and are sufficiently light then may also be produced directly since in this case the cancellation due to the charge conjugation symmetry is less effective.

The results of fig. 25 demonstrate that for sufficiently light, the model considered in this section can generate sufficient to accord with present observations, even though no is produced directly through -violating decays. The magnitude and sign of the resulting baryon number depend sensitively, however, on the Higgs structure and the masses of the -violating bosons. If is comparable to , then the given in fig. 25 is an underestimate since then production through -violating S decays may not be neglected.

7.5. Generation in SO(10) Models with Effective Symmetry

As described in subsect. 7.3, the production of baryon number in SO(10) models with effective symmetry requires the presence of a with a -violating mass splitting between two of its -violating components. Since the cannot on its own account for observed fermion masses (17) , we include also a . We shall consider only those components in which may attain a -violating mass splitting, and may thus contribute directly to production.

The equations presented here may also be used to track the evolution of asymmetries produced in earlier stages. In particular, with effective symmetry no may be produced due to the unbroken charge conjugation symmetry. This restriction does not apply to asymmetries in . The equation used here may also be used to treat the subsequent conversion of to when is broken. With effective symmetry, may be produced directly through decays of and . symmetry implies that S decays may produce no net (since under ), while the produced through decays must be opposite in sign to that produced in decays. To illustrate the conversion of to we will suppose that no is produced directly through boson decays. This would be the case if asymmetries are produced dominantly through S decays but thermalized by the and bosons.

The quantum number assignments for the various fields are given in table 10. In this table a field stands for the asymmetry per member of an irreducible multiplet of . We will assume that the total charge associated With is zero.

[ Table 10 ] Quantum numbers for particles contributing to baryon number production in SO(10) models with effective symmetry

Using the decay modes for X, , S, , , and listed in table 11, we obtain the following Boltzmann equations for the development of the independent quantum number densities and :

Note that since all bosons are equally likely to decay to states with opposite values of and , the asymmetries in the boson fields do not enter into these equations. The W bosons conserve both and and thus also do not contribute to the development of or .

[ Table 11 ] Quantum numbers and partial widths for supermassive boson decay modes in SO(10) models with effective symmetry

The total widths and cross sections appearing in these equations are given in subsect. 6.5. The effective -violation parameters are

which may be written using the partial widths from table 8 as

Since light Higgs exchange violates , it presumably dominates these -violation parameters. Taking the Yukawa couplings of the and to be equal in magnitude and given by , we then obtain

We will take to be degenerate in mass with in what follows. Since is determined by the mass splitting between and , this choice should have little effect on the final results.

Fig. 26a shows the final baryon number generated in this model, for a variety of values of , and . Figs. 26b, c show the development of and in two characteristic cases, and indicate the dominant processes in each temperature range. An asymmetry in is produced by S, and decays. Asymmetry in must then be generated by conversion of this asymmetry. Only and interactions violate and thus may contribute to .

[ Figure 26 ] Quantum number densities (scaled by the -violation parameter ) generated in an SO(10) model with an intermediate effective symmetry. and are mass eigenstate Higgs bosons occurring in , while S is a Higgs boson from . (a) shows the final baryon number generated for a range of S, and masses. (b) and (c) show the development of the independent quantum number densities and for two characteristic cases.

In fig. 26b, , so that inverse decays first convert positive produced in S decay into a negative . As the temperature falls below the mass, inverse decays into dominate and is driven positive. When is driven negative by and decays the inverse decays drive negative again yielding a negative final baryon number. For , the roles of and are reversed and the final baryon number is positive.

Fig. 26c shows the development of and when . The final produced is positive since . produced in inverse decays is reduced by S inverse decays. For , is produced after the effects of S inverse decays are important and as a result the final is larger than in the previous case.

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