3. Violation

The generation of a baryon excess from an initially symmetric state requires -violating interactions. In this section, we describe the possible forms of -violating couplings between particles appearing in grand unified gauge models, and some mechanisms through which these couplings may occur.

We consider first a complex scalar field . It is necessary to distinguish the field operator from the ``fields'' obtained as the expectation values of this operator in particular states. It is the q-number field operator which appears in the canonical quantization procedure, the c-number field appears in the path integral formalism.

The actions of parity (), charge conjugation () and time reversal () on a complex scalar field are given, up to arbitrary phases, by [11]:

The transformations and are represented by unitary operators, which act on just as on . is an antiunitary operator, which reverses the order of factors in products of field operators. It thus interchanges the bra and ket states in an expectation value, and complex conjugates the field . The combined operator of on yields and is equivalent, as usual, to a generalized Lorentz transformation.

The , and transformations above are modified for particles with spin. Their action on spin fermions is outlined in the appendix. Note that separate and transformations interchange chirality states, while the combined or transformations do not: thus massless particles with only one chirality or helicity state may have definite behavior under . We shall consider the transformations of different chirality states under independently. For spin 1 fields, and transformations reverse, respectively, the space and time components of the polarization vector: they may thus be considered to ``raise" or ``lower" the Lorentz vector index on the vector potential .

The conjugate of a particle which transforms according to a representation r of some internal symmetry group is the corresponding particle in the conjugate representation . Any U(1) factors in the internal symmetry group are associated with charges which are reversed by the action of . If the complete symmetry group is an abelian product of U(1) factors, each field with non-zero charge must be complex. When the symmetry group is non-abelian the U(1) charges are generated by the Cartan subalgebra of the group. In the absence of explicit U(1) factors, the reality or complexity of fields is determined by the representations under which they transform. Three basic classes of representation may be distinguished (e.g. [12]):

Complex: and r are completely inequivalent. The singlet representation appears in the decomposition of , but not in .

Real: A basis exists in which the representation matrices of r are purely real, so that r and are equivalent. The singlet representation appears in the symmetric part of .

Pseudoreal: is unitarily equivalent to r, but there is no basis in which the representation matrices are purely real. The singlet representation appears in the antisymmetric part of . All pseudoreal representations have even dimensionality.

Real and complex representations appear in many models; pseudoreal representations are rare, since they must be used in a ``doubled'' form to allow construction of mass terms for scalar fields. Fermions are usually placed in complex representations; this prevents the possibility of group-invariant fermion mass terms (allowed by chiral symmetries) and avoids unobserved right-handed fermions coupled to the weak current. In most of the discussion below, complex and pseudoreal representations behave similarly: we shall usually mention only complex representations.

The adjoint representation in which gauge vector bosons appear is always real. Hence in a suitable basis, all gauge vector bosons are eigenstates of , but in general have different eigenvalues.

If a set of interactions is to conserve invariance, its lagrangian must be invariant under . The requirement that all invariant lagrangians be hermitian places important constraints on possible violation.

Hermiticity requires each term in a lagrangian to have the form , where is a coupling constant, and is a product of fields. As discussed above, the action of the transformation is , so that . If , violation occurs when . No violation is possible when .

If the lagrangian is not invariant under phase redefinitions of fields, any violation may formally be transferred from one term in the lagrangian to another: its physical effects nevertheless remain unchanged.

Since the gauge vector bosons transform under the real adjoint representation, cubic and quartic couplings between the gauge bosons yield real , and can involve no violation. Similarly, cubic and quartic coupling between Higgs bosons in real representations or between those Higgs bosons and the gauge vector bosons cannot introduce violation. For violation to occur, some Higgs fields must be complex. This may occur because they appear in complex representations of a non-abelian group, or because they exhibit an additional global U(1) invariance. In the latter case, two fields which may individually be real are combined to form a complex field.

In the bare lagrangian, kinetic energy terms for all fields have the form . Couplings of gauge vector bosons obtained by minimal substitution from these terms have the form . Since the gauge vector bosons transform under the real adjoint representation, yielding no violation. However, it is possible that the fermion mass matrix may contain complex entries. Gauge couplings of fermion mass eigenstates may then contain complex mixing angles and exhibit violation. Nevertheless, when all fermions are massless, no such mixing may occur, and violation is again impossible. Even with massive fermions, violation may not be possible in some gauge couplings. For example, if only the left-handed fermion currents which couple to light gauge bosons are considered, unitary rotations may be performed to remove violation unless at least three separate fermion families exist (Kobayashi-Maskawa [13] scheme). violation may nevertheless occur in the right-handed and -violating currents which couple to superheavy bosons even when only one fermion family exists.

violation associated with couplings of Higgs bosons to fermions or to themselves may occur either as a result of explicit complex couplings (``intrinsic violation'') or from a complex vacuum expectation of a Higgs field (``spontaneous violation'' (3) ). For a coupling to exhibit intrinsic violation, hermiticity requires that at least one of the fields involved must be complex. Spontaneous violation requires a -violating vacuum expectation value, which may be associated either with a complex field or with a pseudoscalar real field. Since spontaneous violation requires the presence of a Higgs condensate, it typically disappears at high temperatures (see, however [14]), while intrinsic violation remains unchanged. Symmetry restoration usually occurs at a temperature of the same order as the vacuum expectation value of the Higgs condensate and the mass of the corresponding Higgs boson [15,6]. Spontaneous violation associated with breaking thus cannot survive at the temperatures of relevance to violation. In SU(5) models, only the real representation attains a sufficiently large vacuum expectation value to survive at high temperatures, and thus no spontaneous violation may occur. On the other hand, in SO(10) models, complex or representations as well as real or representations may attain large vacuum expectation values, so that spontaneous violation at high temperatures is possible. Whenever high-temperature spontaneous violation is associated with Higgs bosons which couple to fermions, the large vacuum expectation values necessary must give large masses for some of the fermions. In general, the presence of a large vacuum expectation value for a Higgs field in a complex representation of the gauge group lowers the rank of the effective gauge symmetry by breaking at least one of the U(1) invariances associated with the Cartan subalgebra.

violation in couplings of W bosons to fermions may occur through complex entries in the fermion mass matrix, as mentioned above. These complex entries may arise either from intrinsic complex Yukawa couplings or spontaneously from a complex Higgs vacuum expectation value. It is possible that low-energy violation is a result of -violating Higgs boson exchanges [16] rather than of a small -violating component in W exchange [13]. The best-measured -violating effects in decays do not distinguish between these possibilities. The magnitude of the -violation parameter in decays, or other potential -violating effects in b-quark decays or the neutron electric dipole moment should, however, provide evidence on these possibilities.

Although the QCD lagrangian is invariant, it is possible that instanton effects in the vacuum state may lead to a -violating term in the effective QCD lagrangian (e.g. [17]). The absence of a measured neutron electric dipole moment then requires . Two simple mechanisms which would yield are not viable: the Peccei-Quinn mechanism because a light axion is not observed, and the massless u quark because of conflicts with current algebra results on quark masses (e.g. [17]). terms in the effective QCD lagrangian may also arise from chiral rotations used to render the quark mass matrix real and diagonal: the coefficient of these terms is proportional to the -violating quantity , where is the quark mass matrix. This contribution to receives corrections from higher orders in perturbation theory. If violation in the quark mass matrix occurs through soft terms with dimension 2 or 3 in the Higgs couplings, then the resulting renormalization of is finite. If it occurs by hard terms of dimension 4, may suffer formally infinite renormalization. Any spontaneous violation can contribute only to soft terms: intrinsic violation may yield either hard or soft terms. In requiring to be small, it is perhaps desirable to avoid cases of infinite renormalization, thus favoring violation in soft terms.

We have discussed above violation in models containing fundamental Higgs scalar fields. Models with dynamical symmetry breaking from composite scalar fields may also exhibit violation [18]. Since the fundamental couplings in such models are gauge couplings, all couplings must be CP invariant. However, the minimum of the effective potential for the composite scalar fields may correspond to complex values for the fields, and thus yield spontaneous violation. Another possibility is that the condensate may not be purely scalar, but may contain a -violating pseudoscalar component (c.f. [19]).

At sufficiently high temperatures, any spontaneous violation will usually be restored. When the universe cools below the -violating phase transition, the expectation values of the Higgs fields, and thus the form of the violation, may differ between different domains in the universe. In the simplest case, the expectation value of the Higgs field may be either or , leading to production of baryon asymmetries with opposite signs [20]. At the temperatures at which violation must be present in order for baryon number to be generated, no causal effects may yield correlations in fields over volumes containing more than about particles. Fig. 1 shows schematically a section through a universe consisting of many uncorrelated cells each carrying a positive or negative Higgs field with probabilities . Investigations in percolation theory (e.g. [21]) show that in three dimensions, infinite connected and domains exist with probability one (4). Numerical simulation suggests that all but about 1% of the cells lie in connected domains. Thus, even without correlations between cells, large and domains should exist in the early universe. At the edge of, say, a region, the Higgs field must change sign, and therefore exhibits a non-zero derivative leading to a surface energy density on the boundary. This surface tension tends to collapse the domains. It is, however, possible that the transmission of particles through the domain walls may be sufficiently low that contraction of a domain as a result of surface tension would be opposed by pressure from the enclosed gas. In this case, large domains would tend to become spherical, and survive at least through the period of baryon number generation. Details of the domain walls determine the effect of the expansion of the universe. Constraints on the present energy density of the universe do not allow any domain walls extending over a significant fraction of the universe [22]. However, domains which were stabilized by the pressure of enclosed gas would collapse when the gas recombined. After this point, gravitational clumping and radiation pressure should suffice to hold matter and antimatter domains apart. The viability of such a scheme depends crucially on the amount of energy dissipated in the destruction of the domain walls, which must be determined by detailed calculation.

[ Figure 1 ] Schematic section through a universe consisting of a random mixture of equal numbers of + and - cells in which violation has positive and negative signs. In three dimensions, nearly all the cells are members of infinite connected domains.

Invariance under appears to follow from Lorentz invariance in any lagrangian quantum field theory obeying standard axioms. It is nevertheless conceivable that these axioms are inadequate, and that violations could occur. With invariance, separate violation of and must be accompanied by deviation from thermal equilibrium to provide ``an arrow for time'' and allow baryon asymmetry generation. However, with violation, an asymmetry may be generated even in thermal equilibrium [23]. For example, the mass of a particle and its antiparticle could differ, so that their equilibrium number densities were unequal. Without a specific model for violation, no detailed consequences may be deduced.

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