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Articles Cosmology Calculation of Cosmological Baryon Asymmetry in Grand Unified Gauge Models (1982)
Calculation of Cosmological Baryon Asymmetry in Grand Unified Gauge Models (1982) |
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In this section, we discuss the details of 
violation. We consider constraints on the possible forms of -violating couplings, and derive conditions under which and 
are separately violated, but some combination (usually 
) is conserved.
The generic constitution of the three known families of quarks (q) and leptons (
) is summarized in table 1. In considering , violation at high energies, the masses of q, may be neglected, so that the left- and right-handed components of each fermion field may be approximated as independent. Table 1 gives the 
and 
representations under which each field transforms together with the weak hypercharge 
assignment which specifies the final 
transformation properties. We assume, for now, that neutrinos are described by massless Weyl fields. As indicated by present experimental results, we take all 
, 
to transform as doublets under and 
, 
to transform as singlets.
]The quarks in table 1 are assigned baryon number 
; the corresponding antiquarks are assigned 
. The leptons are assigned 
, and antileptons 
. The ``baryon'' and ``lepton'' numbers of other particles are determined solely by their couplings to these quarks and leptons. These couplings are required to satisfy the constraints of 
invariance (at the high energies considered, spontaneous breaking of is insignificant). The couplings conserve and only in so far as can be arranged by assignments or and quantum numbers. If all the quark-lepton systems to which a given particle couples have the same and , then that particle may usefully be assigned a definite and . However, some particles may couple to several systems with different and , in which case no single assignment of 
suffices, and are ``violated" in the interactions of the particles.
Tables 2 and 3 give the quantum numbers for the possible quark and lepton systems to which vector and scalar bosons may couple. Lorentz invariance requires that renormalizable vector couplings have the form 
, and that renormalizable scalar couplings have the form 
, where 
, and 
are vector and scalar fields respectively, and 
are spin 
fields (see the appendix for notation).
[Table 3] Quantum numbers for possible spin 0 (scalar) pairs of
quarks and leptons, to which scalar bosons may couple
The standard Weinberg-Salam model together with QCD involves gauge bosons of , , and . All these bosons are of the type 
defined in table 2. Hence, each gauge boson may be assigned definite and and no or violation may occur. The usual Higgs scalar doublet necessary for spontaneous breaking of 
to 
is of the type 
defined in table 3 and again implies separate and conservation. In grand unified gauge theories, it is common to include both fermion and antifermion fields in the same representation of the gauge group. In these cases, bosons with couplings of types 3, 4 and 5 
may exist. A boson with couplings of type 3 must be a color singlet: it may therefore not participate in couplings 4 and 5, and may thus be assigned a definite . On the other hand, a boson may simultaneously exhibit couplings of types 4 and 5. Such a boson therefore couples to systems with and 
: it may therefore be assigned no definite , and mediates -violating interactions between quarks and leptons. However, although the separate and for cases 4 and 5 differ, the combination is 
in both cases. Thus, invariance and the restriction to the observed fermion fields prevent couplings of bosons to quarks and leptons from violating [7,8,9]. At least for the purposes of these couplings, such bosons may always be assigned a definite . In what follows we will denote the -violating vector bosons with quantum numbers 
by 
and with quantum numbers 
by 
. The possible -violating scalar bosons will be denoted by S (
), (
), and 
(
). Fermi statistics require that and couple only to pairs of fermions in different families.
Another possible scheme for violation involves two bosons [10]: one (say 
) of type 2 and one (say, 
) of type 5. Since and may have the same color and electric charges, -conserving processes such as 
or 
may occur, and give rise indirectly to violation through the different of the systems to which and couple. Similarly, 
mixing may occur between the and states through their interaction with the Higgs condensate. The rate for violation through 
exchanges is then 
: existing limits on the proton lifetime then allow 
as low as 
. Note that 
exchanges conserve 
, and thus violate .
All known fermions carry non-zero color, or electric charges. However, there may exist massive fermions which carry no absolutely-conserved quantum numbers. Such fermions (N) may mix with their antiparticles (charge conjugates) through Majorana mass terms (of the form 
). Clearly, they may not be assigned definite or . If the coupling 
is present, then so may 
be. Thus N does not carry a definite : production and decay of N will lead to violations of conservation. The types of - and -violating bosons allowed in this case are discussed in subsect. 7.2 in the context of SO(10) grand unified models.
