5.1. General Results
In this section we describe the calculation of the parameters which govern the generation of a baryon asymmetry from the basic couplings in a grand unified gauge model.
The basic parameter which enters the Boltzmann transport equations of sect. 4 is the average baryon number produced in the free decays of an equal mixture of particles and their antiparticles :
Here, as in sect. 4, denotes the partial width for decay of to the final state , is the total decay width and is the baryon number of the state f (so that ).
In treating the statistical mechanics of baryon number production it is convenient to choose a basis so that the are mass eigenstates. We assume that the are not eigenstates (which is assured if and have distinct conserved quantum numbers (11) ). Hence the decay process itself must exhibit violation in order for to be non-zero. As discussed below (and proved in general in appendix B of I), this requires interference between the Born amplitude for the decay and a one loop correction with an absorptive part . In addition, the couplings of the particles participating in the decay must be relatively complex.
We consider first the simplest non-trivial case: two massive bosons, X and Y, coupled to four fermion species , , and , through the vertices of fig. 5 and their conjugates (12). In the Born approximation, these vertices lead to the decay processes , , , and the corresponding conjugated processes. We denote the coupling in, for example, the vertex fig. 5a by so that the -conjugate coupling becomes . The quantity here may be considered as a matrix of couplings in the space of possible fermion states . Note that the set of vertices in fig. 5 is invariant under the combined transformations and . This invariance will be used below to obtain results for decays from those for decays. The couplings do not include Lorentz structure which determines, for example, which helicity states of the fermions may contribute.
Born approximations to the X and Y decay rates may be obtained directly from the vertices of fig. 5. For example
Here accounts for the kinematic structure of the process ; it gives the complete result if all couplings are set to one. From eq. (5.1.2) it is evident that , and hence vanishes in this approximation. To obtain a non-zero result for , one must include corrections arising from interference of the one-loop contributions shown in fig. 6 with the Born amplitudes of fig. 5. Consider, for example, the interference of the diagrams of fig. 5b and fig. 6a. The resulting terms in the squared amplitude is shown as fig. 7a. There the dotted line is a ''unitarity cut''; each cut line represents a physical on-mass-shell particle. The amplitude for the diagram fig. 7a is then given by
where the kinematic factor accounts for integration over the final-state phase space of and and over the momenta of the internal and . The complex conjugate diagram, fig. 7b, has the complex conjugate amplitude
Introducing notations for quadratic and quartic combinations of the couplings of fig. 5,
one may write the one loop approximation to the decay rate obtained by adding the results (5.1.2), (5.1.3) and (5.1.4) as
[ Figure 7 ] Squared amplitudes for one-loop corrections to X and Y decays, obtained as interference terms between the diagrams of figs. 5 and 6. The dotted ''unitarity cut'' specifies the physical final state fermions.
In the Born approximation, the kinematic factors are always real. However, the kinematic factors for loop diagrams may have an imaginary part whenever any internal lines have sufficiently small masses that they may propagate on their mass shells in the intermediate state (and thereby sample the piece of the propagator ). In the one-loop diagrams of fig. 7, this occurs when the threshold conditions
are satisfied. With light intermediate fermions, thus always exhibits an imaginary part.
We now consider the -conjugate decay . To obtain the -conjugate amplitude all couplings must be complex conjugated. The kinematic factors are, however, unaffected by the conjugation (this is manifest in the fact that reversal of the direction of fermion lines in a closed loop does not affect the associated Dirac trace). Thus, to one-loop order, the complete result for becomes
The diagrams for the decays and are shown in figs. 7c and d, respectively. The loop diagrams differ from those for the decays and only in that the unitary cut is taken through the and rather than the and lines. In analogy with eqs. (5.1.8) and (5.1.9), we thus obtain
Using the results of eqs. (5.1.7) and (5.1.11) together with eq. (5.1.1) we can compute the average baryon number produced in the free decays of an equal number of X's and 's. The one-loop contribution to this asymmetry from the and final states is given by
The analogous result for the 34 final state is
The kinematic factors and are obtained from diagrams involving two unitarity cuts (as in fig. 8): one through the and lines and the other through the and lines. The resulting quantities are invariant under the combined interchanges and and consequently are equal:
Upon adding the contributions (5.1.12) and (5.1.13) we obtain the final result
The conditions for the kinematic factor to be non-vanishing were given in eqs. (5.1.5) and (5.1.6). A further condition for to be nonvanishing is that both X and Y interactions violate baryon number. If X couplings were -conserving, the two possible final states in X decay would have the same baryon number, so that
and would vanish. Similarly, if Y couplings were -conserving,
and would again vanish. Thus both X and Y couplings must be -violating to obtain a non-vanishing . This is as implied by the general theorem given in appendix B of I. Notice that for (5.1.15) to be nonvanishing, at least two of the must be distinct.
The asymmetry produced in Y and decays may be obtained from (5.1.15) by the transformation , , yielding
It follows that the average baryon number produced in the free decay of an equal number of , Y and is
Even if the and are non-vanishing on their own, for the total to be non-zero the terms in the brace must not cancel. This requires that the particles X and Y be distinct either in mass or in the Lorentz structure of their couplings (e.g. one vector and one scalar) and that . The brace typically vanishes if X and Y are in the same irreducible representation of an unbroken symmetry group.
If more than the minimal set of four fermion species are present, the result (5.1.20) must be summed over all possible contributing . It must also be summed over all possible pairs. Whenever particles have equal masses on the scale of , the corresponding kinematic factors may be factored out of the summation.
The individual baryon asymmetry parameters for X decays enter the complete Boltzmann transport equations discussed in sect. 4. These parameters alone determine the final baryon asymmetry only if back reactions (inverse decays) and scatterings are ignored. The total contribution to the baryon asymmetry from decays of two species X and Y of bosons is thus not in general a simple sum of their corresponding parameters and : if X and Y have different masses, the extent of back reactions is different in the two cases. If, however, X and Y are degenerate in mass, the sum given in eq. (5.1.20) represents their total contribution.
In the derivation of eq. (5.1.15) the particles were assumed to be light fermions of definite baryon number. The result nevertheless remains approximately valid for any particles so long as their masses are much smaller than . Some of the may for example be bosons, which enter through a three-boson coupling vertex, as illustrated in fig. 9. The in eq. (5.1.15) should usually be replaced by the average baryon numbers generated in the decays of the corresponding .
The discussion above concerns the one-loop contributions to baryon asymmetry. In the generic case, an asymmetry occurs at this order if it is to occur at any order. However, in some simple models (such as the minimal SU(5) model treated in subsect. 6.2) the one-loop contribution vanishes, but there are higher loop contributions which are non-zero: in such cases the detailed analysis given above must be suitably generalized by summing over all possible unitarity cuts through the intermediate lines.
5.2. Consequences for Gauge Models
In this section, we give some general results on the value of the -violating parameter defined by eq. (5.1.12) in gauge models.
As demonstrated in sect. 3, the couplings of gauge vector bosons to fermions may always be taken real and diagonal. Couplings of Higgs bosons to fermions and to each other may, however, be complex and induce mixing. After spontaneous symmetry breaking, these couplings may give rise to violation and mixing in the fermion and Higgs boson mass matrices. If fermion masses are neglected, diagrams involving only fermions and gauge vector bosons (fig. 10) can therefore yield no violation. For violation to occur in the decays of superheavy bosons, it is thus necessary for either explicit Higgs bosons or superheavy fermions with complex mixing angles to be present.
Some -violating effects involving Higgs bosons may be investigated before spontaneous symmetry breakdown. If a particular set of Higgs bosons allows violation in the unbroken theory, this violation will remain possible in the broken theory.
Consider first the case of scalar boson (S) exchange in vector boson (V) decay, as illustrated in fig. 11. The diagonal nature of the gauge couplings requires that the fermions and lie in the same irreducible representation f, of the gauge group (and and in ). Scalar bosons contributing to fig. 11 must lie in irreducible representations sit such that
In the absence of mixing between scalar bosons the exchanged S propagator is diagonal. Hence in the notation of subsect. 5.1, the coupling at one end of the exchanged S line is simply the hermitian conjugate of the coupling at the other end: the product of these couplings is thus real, and no violation may occur.
violation may be introduced into fig. 11 through mixing terms in the S propagator arising from mixing which then causes the exchanged mass eigenstate scalar boson S to become in general a linear combination of several components with the same conserved charges. These components may occur within the same irreducible representation of the gauge group, or in different irreducible representations . [Examples of both kinds appear in the illustrative SO(10) models considered in sect. 7.] If a model contains only a single -violating Higgs boson no such mixing is possible, and violation cannot occur at the one-loop level through scalar boson exchange in vector boson decay. This is the case for the minimal SU(5) model discussed in subsect. 6.3. In the general case, we decompose the mass eigenstate field S into its unbroken group eigenstate components according to:
We shall assume for now that just two components are present; the generalization to an arbitrary number will be immediate. In this case,
where we have dropped the real factor corresponding to the gauge boson couplings, and the trace represents a sum over all fermion representations (usually ''families''). Since and , the couplings and are related by a real Clebsch-Gordan coefficient:
Thus, if , vanishes. This effect occurs when all Higgs bosons coupling to fermions have identical group charges, and are distinguished only by a ''family'' index. This is inevitable if all relevant Higgs bosons lie in replications of the same irreducible representation of the gauge group, and if this representation contains only one -violating component. Examples of cases in which are the SU(5) model with a and a (case B in subsect. 6.4) and the SO(10) model with a and a or a . In these models, violation may occur at the one-loop level from scalar boson exchange in vector boson decay. Notice that since in the absence of spontaneous symmetry breakdown, only one of the is non-zero, the result (5.2.5) yields no violation in this case.
The case of vector boson exchange in scalar boson decay (illustrated in fig. 12) is exactly analogous to the case of scalar exchange in vector decay discussed above. When fig. 12 contributes, it is often important by virtue of large value of the vector couplings relative to the scalar ones.
We now consider violation arising from scalar boson () exchange in scalar (S) boson decay, as illustrated in fig. 13. If only one -violating Higgs boson is present, then the decaying and exchanged boson must be identical, and the results of subsect. 5.1 show that fig. 13 can give no violation. This is the case for the minimal SU(5) model. (However, as described in subsect. 6.3, violation may occur in higher order diagrams.) We consider for now the case in which all fermions are effectively massless. Then, in analogy with (5.2.1), the contributing scalar bosons must appear in representations such that
If all the left-handed fermions lie in the same complex irreducible representation, f, (or sequence of such identical representations), then and these constraints become
For low-dimensionality representations, this requires and to be real representations. Hence in SO(10) models where all fermions lie in the 16 representation, only, or may contribute to fig. 13; the which appears in is complex. [For high-dimensional fermion representations, some complex Higgs representations may satisfy (5.2.7): an example is the occurring in the symmetric product of SO(10).] After spontaneous symmetry breakdown, mixing between scalar bosons may occur, and the constraints (5.2.6) are no longer applicable. Thus in both SU(5) models with several Higgs representations coupling to fermions, and in SO(10) models, fig. 13 can yield violation.
The discussion above has assumed that all relevant fermion species are effectively massless. With gauge groups such as SO(10) or E(6), it is common for fermions with SU(2) singlet and thus potentially large mass terms to exist. The effect of such fermions in intermediate states of figs. 10 through 13 is always suppressed by . If only a single massive fermion exists [as in SO(10) models], then it can introduce no -violating effects into fig. 10; a single massive fermion is, however, sufficient to generate violation in figs. 11 and 12 even when (5.2.5) vanishes.