2. Basic Formalism

2.1. Introduction

Let be the amplitude for a transition from a state to state and let be the state obtained by applying a transformation to Then the theorem (the validity of which is necessary to justify use of quantum field theory) implies that

invariance (and hence, by , invariance), when valid, demands

The requirement of unitarity (that the probabilities for all possible transitions to and from a state should sum to one) yields (1)

But from (2.1.1) (the sum over includes all states and their antistates) (2)

In thermal equilibrium (and in the absence of chemical potentials corresponding to non-zero conserved quantum numbers) all states of a system with a given energy are equally populated (3). Eq. (2.1.4) then shows that transitions from these states (interactions) must produce states and their conjugates in equal numbers. Thus no excess of particles over antiparticles (and hence, for example, a net baryon number) may develop in a system in thermal equilibrium, even if invariance is violated. (A restricted form of this result was given in ref. [5].)

From eqs. (2.1.1) and (2.1.3) one finds

implying that the total cross section for interactions between a set of particles and their conjugates are equal, and that the total decay rate of a particle and its antiparticle must be equal. ( invariance alone implies the equality of the elastic scattering cross sections However, the corresponding result

for specific final states requires invariance (e.g., [10]). Thus if the interaction inducing the decay of a particle (say ) violates invariance, then the decay of a system containing an equal number of and can result in unequal numbers of; say, b and Note that for a system with only two states, unitarity gives so that the result (2.1.6) always holds.

Above, we argued that in thermal equilibrium, no excess of particles over antiparticles may develop. In addition, any pre-existing excess tends to be diminished by interactions: eq. (2.1.5) shows that the total cross sections for the destruction of the states and are equal. Hence, if there exist, say, more than then the rate of destruction is larger than the rate for destruction. Moreover, eq. (2.1.5) implies that in thermal equilibrium and are produced in equal numbers. Thus, interactions tend to destroy any excess of, say, over in thermal equilibrium. According to Boltzmann's theorem (which holds regardless of invariance, as discussed in appendix A) any closed system will evolve on average in the absence of external influences to a state in which all particles not carrying absolutely conserved quantum numbers are distributed equally in phase space. No difference between the densities of any species of particles and their antiparticles may survive (unless they are distinguished by absolutely conserved quantum numbers). However, as discussed in appendix A, the expansion of the universe adds extra terms to the Boltzmann transport equation which invalidate the theorem if some participating particles are massive. The expansion of the universe may therefore prevent the achievement of complete thermal equilibrium and allow a baryon excess to be generated: the relaxation time necessary to destroy the excess often increases faster than the age of the universe (see subsect. 3.2) and hence a net baryon number may persist.

invariance requires hermiticity of the transition matrix and in terms of the matrix the -invariance constraint (2.1.6) becomes

The unitarity requirement [which gave eq. (2.1.3)] is

Thus unitarity constrains possible violations of invariance, and from eq. (2.1.8) one finds that deviations from (2.1.7) must obey

If the rates of transitions are governed by some small parameter, say so that then eq. (2.1.9) shows that any -violating difference must be at least of order (Regardless of perturbation theory, one may show that -violating effects in any scattering process with high c.m. energy are suppressed by [for decays ].) Hence, -violating effects must arise from loop diagram corrections to the processes In appendix B, it is shown that these corrections must also involve -violating interactions [11]. In addition, the intermediate states in the loops must correspond to physical systems (so that the Feynman amplitudes have absorptive parts due to -channel discontinuities), in order to contribute to (2.1.9). Thus even if the intermediate particles have violating complex couplings, they can produce a violation of (2.1.6) only when their masses are sufficiently small to allow them to propagate on their mass-shells in intermediate states. (Note that, as discussed in subsect. 2.4, when absolute conservation laws allow a particle to mix with its own antiparticle [as for -violating mixing may occur without physical intermediate states.)

We assume that the early universe consists primarily of an effective number (see appendix C) of massless particle species (none forming highly degenerate Fermi gases) and we usually take it to be homogeneous and isotropic. Then the Robertson-Walker scale parameter R for the early universe (which corresponds to its radius of curvature) should expand with time (t) according to (e.g., [9]) (4)

where is the energy density of the universe, and is the Planck mass

In keeping with SI conventions, we take eV eV GeV.

Let be the density of a particle species in phase space (i.e., the number of per volume element d d The assumptions of homogeneity and isotropy imply we usually do not display explicitly the dependence of on time. We denote the number of particles per unit volume (of configuration space ) by

where is the number of accessible spin states for the species for for for for particles with small mass, some spin states may be decoupled from interactions). We shall usually make the simplifying approximation that all particles obey Maxwell-Boltzmann statistics and have only one spin state; the small corrections resulting the indistinguishability of the particles are discussed in subsect. 2.4. Then the massless neutral (at least in baryon number) particles (denoted generically as ) which comprise a large fraction of the contents of the early universe, should be Maxwell-Boltzmann distributed at all times due to their interactions, so that

where is their common temperature (referred to as ``the temperature'' of the universe). The expansion of the universe redshifts all like so that and (dots denote time derivatives) (5)

Only for massless particles are the equilibrium self-similar under rescalings of ; for massive particles the mass provides an intrinsic scale and the change their form as the universe expands. The expansion of the universe dilutes the number densities of all types of particles, even in the absence of interactions, at a rate

In keeping with the simple big bang cosmology we shall assume (6) that all species of particles in the universe were initially in thermal equilibrium and spread homogeneously (the gravitational field opposing expansion must, however, remain far from equilibrium). Two effects modify this ``equilibrium'' state. First, regardless of expansion, long-range gravitational forces render a homogeneous state unstable, and lead to clumping. (This formally tends to increase the entropy of the universe, but in fact produces a more ordered state, eventually containing stars, etc.) Second, the expansion of a homogeneous universe can give rise to deviations from equilibrium, some of which may never have time to relax away. The expansion of the universe causes the momenta of all particles to redshift so that So long as the energy density of the universe is dominated by ultrarelativistic particle species, the temperature of the universe will likewise redshift according to The equilibrium for massless particles remain, by construction, unchanged by this expansion (so long as homogeneity is preserved). Since the expansion of the universe is taken to be adiabatic, leading to the energy equation the retention of equilibrium distributions by massless particles shows that the entropy of a universe initially in thermal equilibrium and composed solely of such massless particles (assumed homogeneous and with zero equilibrium chemical potentials) should remain unchanged with time. However, as mentioned above, the equilibrium distributions for massive particles change their form when becomes smaller than Several collision times are then necessary for the actual number distributions of the massive particles to relax into their equilibrium forms; if the rate of expansion is much larger than the rate of interactions, then significant deviations from equilibrium may result. When massive particles are present, therefore, the expansion of the universe is no longer reversible; deviations from thermal equilibrium may occur, and in their relaxation, the entropy of the universe may increase slightly (7). (When equilibrium is destroyed by expansion, the gravitational field opposing the expansion becomes slightly closer to equilibrium; the increase in its entropy compensates the slight decrease in the entropy of the massive particle species.) If the expansion of the universe was slow enough, then any deviations from thermal equilibrium would eventually relax to zero. However, in several cases, the relaxation is always prevented by expansion. One example of this effect is in the survival of massive stable particles from the early universe [13]. At some time after the equilibrium number density of the massive particles starts to decrease rapidly, the rate for annihilation reactions becomes slower than the expansion rate of the universe, and the number of the particles is permanently frozen: the particles are so separated by the expansion that the probability for them to interact becomes negligible. In this way, the expansion of the universe causes the number density to deviate from its equilibrium form, and prevents its relaxation in a finite time. The generation of a baryon asymmetry is an effect of a similar character. That but not violating reactions should occur only too slowly to destroy any baryon asymmetry (deviation from equilibrium) produced placed stringent constraints on models.

2.2. A Very Simple Model

In this subsection, we describe baryon number generation in a very simple model; subsect. 2.3 treats a more complicated and more realistic model, while in subsect. 2.4 we discuss the complications of the general case.

Let b be a nearly massless particle carrying baryon number and its antiparticle, with and let be a massive particle with We allow a small violation of invariance in the rates of scattering processes among the b and and consider the generation of a net baryon number when a system initially symmetric in b and cools, as in the early universe. We take the scattering amplitudes in the simple model to be where is a small coupling constant)

where This parametrization ensures the -invariance constraint (2.1.5) that the total cross sections for and interactions should be equal. invariance would require we shall, however, consider the -violating case so that, for example We assume here that the can interact only through the processes of eq. (2.2.1), but that the also undergo baryon-number conserving interactions (such as or with the other particles in the universe. Such reactions should typically have rates and serve to distribute the b and in phase space in a Maxwell-Boltzmann manner. The time necessary to attain this state of kinetic equilibrium should be much shorter than the time on which the net baryon number changes through the processes of eq. (2.2.1). Hence, at all times

where is a baryon number chemical potential, which is changed only by -violating processes (in the present model, these occur on a time-scale at least longer than the reactions which thermalize the into Maxwell-Boltzmann distributions). The fact that the chemical potentials in and are exactly opposite is a consequence of processes such as which maintain in chemical equilibrium with

The time evolution of the number density and of the total baryon number due to the processes (2.2.1) is described by the Boltzmann transport equations (8)

where the integral operator

represents appropriate integration over initial- and final-state phase space in the scattering processes. (When has only upper or only lower indices, no momentum conservation function is included.) The second term on the left-hand side of eq. (2.2.3) accounts for the dilution of the number densities due to the expansion of the universe, as in eq. (2.1.15). (A proof of its form for Robertson-Walker metrics is given e.g., in ref. [5].) By considering

the explicit expansion term is removed The various terms on the right-hand side of (2.2.3) represent the effects of the processes (2.2.1) (for example, the first term on the right-hand side of (2.2.3a) accounts for the increase in the number of due to ).

To simplify (2.2.3), we first substitute the parametrizations (2.2.1):

Using the 4-momentum conservation function in one may write here

where we have defined

which is the phase-space distribution for a species of particles (perhaps massive) in thermal equilibrium at temperature and with zero chemical potential. The total equilibrium number density is given by (see appendix C)

where is a modified Bessel function (see appendix C) [as and (2.2.9) reverts to the massless result (2.1.13)]. We assume here that and may thus write (2.2.7) in the form

(Non-linear terms in for the model of subsect. 2.3 are discussed in subsect. 2.4.3.) Hence (2.2.6) becomes

The integration over the phase space available for the final momenta and introduces the total scattering cross sections

where is the relative velocity of the incoming particles and which depends only on Eq. (2.2.10) then becomes

where denotes the average of the cross-section over the incoming energy distribution. The first term in eq. (2.2.12a) is simply and is familiar from studies of the survival of stable heavy particles produced in the early universe [13]. The second term in eq. (2.2.12a) contains the two small parameters and and may usually be ignored. The first term in eq. (2.2.12b) is approximately and accounts for the small disparity between b and production in annihilation. The second term in eq. (2.2.12b) is proportional to the total cross section for baryon-number violating interactions, and causes to relax towards zero when the system is in thermal equilibrium ( ). Note that the rate of baryon number generation (2.2.12b) is proportional to the deviation of the number density from its equilibrium value; if then in the present model, the expansion of the universe cannot alter and no net baryon number results [5]. (In subsect. 2.4 we discuss inhomogenities and differential heating effects which may produce non-zero baryon number even if all particles participating directly in -violating processes are massless.)

2.3. A Simple Model (9)

As in the previous model, let b and be nearly massless particles carrying baryon numbers and respectively. Let be some massive boson which mediates baryon-number violating interactions. We take the decay amplitudes for the to be

where now is of order The parametrization (2.3.1) respects the -invariance constraint (2.1.5). invariance would imply but we take Hence a state initially containing an equal number of and will decay, in the absence of back reactions, to a system with a net baryon number ). (Back reactions can be ignored if the are emitted as thermal radiation into an infinite vacuum, or are concentrated into a beam.) conjugation gives the rates for the inverse decay processes

Note that if and decay preferentially produce b (i.e., then according to invariance, inverse decay processes must preferentially destroy Thus, if only decay and inverse decay are considered, a system even in thermal equilibrium cannot fail to generate a net baryon number. (This rather relevant point has also been noted in ref. [14], but appears to have been neglected elsewhere.) However, according to eq. (2.1.4), which follows purely from invariance and unitarity, no excess of b over can develop in thermal equilibrium. We take the total rate for decay to be Then eq. (2.1.9) shows that -violating effects in these decays must be at least (hence Eq. (2.1.4) applies only when summed over all possible initial states which can produce to a given order in Decay and inverse decay are, however, not the only possible interactions between and to scattering processes, such as mediated by -channel exchange, also occur. We show below that after including these processes, eq. (2.1.4) is respected, and no baryon excess develops in thermal equilibrium.

Although the and in (2.3.1) are taken to have identical decay modes, we shall, for simplicity ignore any mixing between them (until subsect. 2.4.4). This may be enforced by considering two species of b (each with ), and taking The formulae below are unaffected by these distinctions. Note that, as discussed in subsect. 2.4, the model of this section may be slightly simplified by taking and to be indistinguishable, so that and is an eigenstate of Then invariance requires or This case is exemplified by semileptonic decay, where violation is revealed in

2.3.1. The Number Density.

In calculating the time evolution of the number density, we work to at which only the decay and inverse decay processes of equations (2.3.1) and (2.3.2) contribute. In addition to these baryon number violating interactions, the may also undergo baryon number conserving interactions (such as or ) with other particles in the universe. Typically, these processes will be and therefore occur on a longer time-scale than the decays we consider. However, it is possible that the typical coupling constants involved are larger than for the -violating decays, or that the number of light particle species so that may undergo several -conserving scatterings before decay. In this case, as with the b in the model of subsect. 2.2, the will be brought into kinetic equilibrium before they decay, and assume a Maxwell-Boltzmann distribution in phase space, so that

Processes such as would lead to The chemical potentials in eq. (2.3.3) are in any case determined by the processes (2.3.1) and (2.3.2) in which single are created or destroyed.

To the number density evolves with time according to the equation [analogous to (2.2.3a)]

The first two terms on the right-hand side account for decays, while the second two represent inverse decay processes. In addition to the reactions (2.3.1) and (2.3.2) in which a net baryon number is created or destroyed the also undergo baryon-conserving interactions with other particles in the universe. These -conserving processes (such as occur at a rate at temperatures they are much faster than any -violating interactions mediated by exchange. The relative rates of the various processes at high temperatures are discussed in sect. 4, and it seems likely that in most cases, -conserving reactions occur with larger rates than do -violating ones. Hence the should be Maxwell-Boltzmann distributed in phase space as in eq. (2.2.2), with their chemical potential determined by -violating processes. Using the momentum conservation function in one may then write (assuming )

where, as above, is the distribution of in thermal equilibrium at temperature and with zero chemical potential. On inserting eq. (2.3.5) into eq. (2.3.4), the and integrations are weighted only by the matrix element and the available phase space; they therefore yield simply the total decay rates (10)

where is the rate measured in the rest frame of the decaying Eq. (2.3.4) may then be written in the form

Performing the final integration, and using the parametrization (2.3.2) then gives

where denotes the total decay rate averaged over the time-dilation factors for the decaying particles. [In writing eq. (2.3.8) we have made the approximation that the actual momentum distribution does not differ from the equilibrium form sufficiently to affect the time-dilation factor. This is certainly the case if thermalizing reactions occur sufficiently fast to produce an distribution of the form (2.3.3). Note that we assume all decaying to be exactly on-shell; in practice they should have a distribution of invariant masses peaked at and with a width (11): we make the narrow resonance approximation ] Charge conjugation and so that by invariance) gives the corresponding equation for the number density

It is convenient to write eqs. (2.3.8) and (2.3.9) in terms of

and

2.3.2. The Number Density.

The Boltzmann equation for the b number density in the model of equations (2.3.1) and (2.3.2) is

The first term in eq. (2.3.11) accounts for the decay and inverse decay processes , , and The second term in (2.3.11) accounts for those scattering processes that are not already included as successive inverse decay and decay processes, (as would with a real intermediate ) The amplitude for due to -channel exchange of a single contains two terms: a part corresponding to the propagation of an on-shell intermediate (which is important only when the incoming energies lie within the resonance curve), and, as usual, a part accounting for off-shell exchange. [The - and -channel exchange diagrams at lowest order receive no contributions from physical intermediate states. Note that processes such as are energetically forbidden in (2.3.11).] We write

where denotes the contribution from physical intermediate states, already included in the first term of the Boltzmann equation (2.3.11) as successive lower-order and processes.

Subtracting from eq. (2.3.11) the charge-conjugated equation for the number density, we obtain an equation for the evolution of the total baryon number density

Notice that, as mentioned above, even when the are in thermal equilibrium, so that the two terms on the right-hand side of this equation do not individually vanish even when the scattering processes must conspire with decay and inverse decay processes to maintain thermal equilibrium.

In the model of this section, the only -violating reactions which may occur to are and But the unitarity requirement (2.1.3) then demands for the total matrix elements of these processes. However, in the which actually enter the Boltzmann equation (2.3.13), the part which arises from real intermediate exchanges already accounted for by the first term of eq. (2.3.13), has been subtracted out. Unlike the total (and hence ) may differ at between and In the narrow-width approximation (which has already been made in (2.3.13) by assigning the decaying and definite mass ) the contributions of a real intermediate to the process and become

Because of the factor arising from the integral under the resonance curve, these terms are of order rather than as expected for scatterings. Using the fact that in our model for the total amplitudes [in general these terms may have a -violating difference ], we may write the difference appearing in the second term on the right-hand side of eq. (2.3.13) as

Then the complete second term in eq. (2.3.13) may be written in the form

where the mass-shell delta function has allowed us to replace by [The extra terms which may in general appear in (2.3.15) from -violating loop corrections to genuine scattering processes may contain parts not proportional to but these cannot be retained consistently in view of other approximations.] To simplify this we apply the results

and (from appendix C)

so that eq. (2.3.15) becomes just

and thus elegantly cancels the first term of (2.3.13) in thermal equilibrium, as required by eq. (2.1.4). (This seemingly miraculous result may formally be obtained by considering the sum of double cuts in vacuum diagrams with finite temperature propagators, without treating separately one- and two-body initial states as is done here.)

The last term of eq. (2.3.13) may be written using eq. (2.2.11) in the form

where in the contribution from real intermediate exchange in the -channel has been subtracted out by replacing the full propagator by its principal part. In addition, since is approximated by [No factor, as in eq. (2.3.18), appears here, since the incoming c.m.s. energy is no longer constrained to be and at low temperatures will be much smaller.]

Finally, therefore, eq. (2.3.13) for the time development of the baryon number density may be written in the simple form

as expected from the discussion of subsect. 2.1, the rate of baryon generation vanishes when the system is in thermal equilibrium while any pre-existing baryon number is destroyed at a rate governed by the total rate for B-violating processes. Note that any possible violation in the reactions would be ineffective at producing a baryon excess, since the masslessness of the prevents deviations from thermal equilibrium [5].

In sect. 3 we discuss the solution of eq. (2.3.10) and (2.3.20). First, however, we consider some possible complications.

2.4. Complications

2.4.1. More Particles and More Decay Modes.

The evolution of the number density of a massive particle species due to decay and inverse decay processes is given in direct analogy with eq. (2.3.4) by

But, so long as all excesses of particles over antiparticles are small,

and hence

where denotes the number of particles in the state This equation holds even if several massive species are present. (If some may mix, further complications may occur, as discussed below.) By charge conjugation and subtraction, we obtain from (2.4.3) equations analogous to (2.3.10) (and with corresponding approximations):

In analogy with eq. (2.3.13), the density of a quantum number violated in decays and scatterings involving particles evolves according to [sums on run over both particles and antiparticles and for simplicity we assume that any particle for which carries baryon number]

Using CPT invariance, the second term may immediately be rewritten as

which is equal to the first term when while the fourth term may be simplified to

which has exactly the form necessary to negate the second term, as required by the theorem (2.1.4). The second and fifth terms in eq. (2.4.5) may be written in the manifestly negative forms

and

respectively, exhibiting the fact that -violating processes in thermal equilibrium must always act to destroy any initial net baryon number. Performing the remaining phase space integrations, eq. (2.4.5) may finally be written as

In many supposedly more realistic models, it is necessary to generalize these equations to describe the development of several approximately-conserved quantum numbers (e.g., Often some combination of the quantum numbers (e.g., ) may be absolutely conserved (typically in order to conserve fermion number). Note that if heavy unstable fermions are present, they may be treated in this analysis as

2.4.2. Spin and Statistics.

In the discussion above, we have assumed that all particles have only one spin state, and obey Maxwell-Boltzmann statistics. Accounting for more spin states changes no formulae: inclusion of appropriate Fermi-Dirac or Bose-Einstein statistics complicates the proof of the theorems discussed in subsect. 2.1 (see appendix A) and yields some small corrections.

For a particle with accessible spin states, we define to be the phase-space density for each single spin state, but take to be the total number density of summed over spins, and thus write The matrix elements are taken to be summed over the possible spin states of initial and final particles, so that the total rates for reactions are given by products of the form without further spin factors. To write these rates in terms of the total initial particle number densities rather than would require division by the requisite factors. However, since we define (as usual) cross sections and widths to be averaged, rather than summed, over initial spin states, the rates written in terms of and these cross-sections require no explicit spin multiplicity factors.

When the density of a species of particles in phase space becomes close to one, so that cells at least in some region of phase space have a high probability to be occupied, the rates for reactions in which are produced must be modified to account for quantum statistics effects. If is a fermion, then these rates contain a factor for each produced, thereby implementing the exclusion principle that no more than one (with a given spin direction) may occupy a single cell in phase space. If instead obeys Bose-Einstein statistics, then each produced introduces a factor to account for stimulated emission, as discussed in appendix A. These factors appear not only for the final particles produced in a process, but also for each intermediate virtual particle and hence modify the unitarity relation (2.1.3) [to (A.22)]. This will guarantee the cancellation between the two-body processes, and decay-inverse decay as discussed in subsect. 2.3. Taking to be fermions, and a boson, eq. (2.3.4) becomes

We again assume that are in kinetic equilibrium. If is Fermi-Dirac (Bose-Einstein) distributed, then

so that the product of in eq. (2.4.10) may be written as

where the second equality follows from the energy conservation function We now assume that the fermion chemical potential is small, and take so that (2.4.10) becomes

where we have defined

Quantum statistics corrections should be small so long as

The correction to the decay rate of particle with mass decaying at rest to two massless fermions in the presence of a gas of in thermal equilibrium at a temperature is given simply by

As the density of the c gas goes to zero, and If, as in (2.4.13), the are fermions, then when of the final-state phase space is excluded by Pauli's principle ( climbs slowly up to 1 as decreases; for ). When the are bosons, the presence of an ambient gas causes stimulated emission, and increases the decay rate. In fact, as reflecting the approach to Bose condensation in the (As decreases, falls steadily to 1; for ) The correction (2.4.16) is for decay at rest: for high-energy A, tends to one. In the region which dominates baryon production is close to one. Suppression of decays by Pauli exclusion merely causes to be generated at slightly lower temperatures: its final value is unaffected, except in as far as the processes which destroy are less effective. The correction is reversed if the decay products are bosons rather than fermions.

If we assume that the actual distribution is not far from its equilibrium form (but perhaps at a different temperature) in most regions of phase space, then the approximation (2.4.15b) is good. However, for a Bose gas, when the phase-space density may become large. The approximation (2.4.15b) should nevertheless remain valid for two reasons. First, the equilibrium distribution will also become large, tending to cancel the growth in Also the region or phase space where is expected to be large is for small and small. Since most baryon production occurs at only in the small region is However, in this region the distribution function is multiplied by in calculating the number density, which lessens the contribution of the region where

2.4.3. Large Baryon Excesses.

In 2.3, we always assumed that and made the linear approximation

However, in most cases, the formalism of subsect. 2.3 does not require this approximation. Retaining the full non-linear form

the final equations (2.3.10) and (2.3.20) become

where we have used the fact that

In the limit but ignoring Pauli exclusion effects, these reduce to

For very large the b or should form a degenerate Fermi gas; while exclusion effects render the or elastic scattering cross section much suppressed, they do not affect the baryon number destruction processes of eq. (2.4.20) since the phase space available to the products of these reactions is unrestricted by the presence of the Fermi gas. As shown in eq. (4.8), even if degenerate massless particles dominate the energy density and hence expansion rate of the universe, the expansion term absorbed on the left-hand side of eqs. (2.4.20) remains unchanged. For the usual hot universes considered in sect. 3, is sufficiently small that the non-linear terms in eq. (2.4.19) are entirely irrelevant. [Note that, as shown in sect. 3, if the initial then the final generated is always less than hence, for example, the last term in eq. (2.4.19) cannot dominate even if is very small.] Notice that chemical potentials associated with quantum numbers which are genuinely conserved in the processes considered exactly cancel out in all equations [e.g., (2.2.7)].

2.4.4. Mixing.

As mentioned in subsect. 2.3, when it is not forbidden by absolute conservation laws (as exhibited for example by their ability to decay into the same final state), and should mix (12); the mixed states and diagonalize the hamiltonian and have definite masses and decay widths (typically, the and will be split by an amount -violating effects in decays may then arise in two ways: either because the eigenstates consist of a combination of which is not a eigenstate, or because the final decays of the exhibit violation (in the manner described in subsect. 2.3). The observed violation in system appears to be dominantly of the former type.

We first consider the case in which are the eigenstates:

Then invariance requires

The unitarity and -invariance constraint (2.1.5) is impotent in this case; the result (2.1.9) still applies, however, so that -violating differences must again be in perturbation theory. For example, if for some state

then (we take the mixing to occur through an intermediate state with in these cases none of the decay final states need be identical)

which may differ by (If s was a eigenstate, so that then When are eigenstates, their number densities individually satisfy equations analogous to those derived in subsect. 2.3 (on setting and identifying and there). Notice that with the choice (2.4.23) of matrix elements, the state may yield only and only Because of the mixing, it is the rather than states which exhibit the characteristic time dependence; the number of or oscillates at the beat frequency when the are at rest). [The case of the system is slightly more complicated than that treated here. In practical experiments, the are produced by strong interaction processes for which the matrix elements are arranged somewhat analogously to (2.4.23), in that strangeness +1 initial states give only and only (assuming the recoil final particle has ). The weak interactions responsible for mixing and decay are not involved in their production.]

The second type of violation arises when and are no longer eigenstates as in (2.4.21), but are rather of the form (in the system, these combinations are conventionally denoted and (e.g., [15]))

where measures the impurity of the states. -- mixing occurs when the matrix elements are non-zero (if they involve as intermediate states shared decay modes, they are typically ). In the eigenvectors (2.4.25) of the -- propagation matrix, the parameter measures the -violating difference of the ratio from one. Whereas unitarity places severe constraints (2.1.9) on violation in these constraints do not apply to the unsquared amplitudes: to obtain it is sufficient for interactions generating to have complex coupling constants (which appear conjugated in ); no further restrictions apply. (In the system, may well arise from relatively complex couplings of and quarks in box diagrams with virtual WW intermediate states (e.g., [12]). We shall assume here that the decays of the states are conserving (as appears to be the case for ); all -violating effects arise from the -- mixing which causes the eigenstates of the hamiltonian not to be eigenstates. We take the decay amplitudes

so that the decay rates for the states (2.4.25) become (for simplicity taking real)

(If is a eigenstate with then in (2.4.26), this is the case for the final state in decays. In semileptonic decays, the rule implies ) (The value of in the simple model (2.4.27) is determined from the total and decay rates by a quadratic equation. In general, satisfies the unitarity constraint [15].)

The violation in the rates (2.4.27) can generate an excess of over in a system which is initially symmetrical in and For example, the free decay (without back reactions) of and produced in equal numbers, and with wave functions of random phase, as when they are emitted in thermal radiation, generates an excess of over given by

2.4.5. Multiple Temperatures and Annihilation Heating.

Eq. (2.1.4) shows that if all particles are in thermal equilibrium with zero chemical potential, then no baryon number can be generated. Once a massless particle species has been brought into kinetic equilibrium, equation (2.1.13) shows that the expansion of the universe alone cannot destroy its equilibrium distribution in phase space. Thus other influences are necessary to modify the distributions of massless particles so as to allow reactions between them to generate baryon number. One possible such influence might be the presence or growth of inhomogeneities in the universe. Another possible mechanism would be differential heating of various massless particle species by the annihilation products of other, massive, particles decoupling from thermal equilibrium. As mentioned in subsect. 2.1, the expansion of a homogeneous universe (containing weakly interacting particles) should approximately conserve the entropy

where is the effective number of particle species at temperature for each boson spin state with and for non-degenerate ultrarelativistic fermion spin states), as described in appendix is the pressure (13): for an ultrarelativistic ideal gas in equilibrium while for dust (For any ideal equilibrium gas, where Systems whose components interact strongly may have pressures up to Such pressure should probably occur if the universe undergoes a phase transition.) As falls below the mass of a particular species, the contribution of that species to the energy density of the universe (and hence to ) drops rapidly to zero (unless the particles carry a non-vanishing chemical potential). The energy density originally carried by the disappearing species is transferred to its lighter annihilation products; their interactions with the rest of the universe raise the temperatures of other particle species in such a way as to conserve (2.4.29), so that constant. However, the rate at which the energy is shared among the species depends on their cross sections for interaction with the annihilation products. If the cross section for a particular species is too small, it may never receive its full share of the energy, and remain at a lower temperature. (This behavior is exhibited by light neutrinos in the present universe; below MeV, the rate for reactions becomes very small, and when the annihilate at MeV, all their energy goes into photons. Consequently, the temperature of photons in the present universe should be about times higher than that of the neutrinos. Similarly, heavy quark species should dominantly annihilate into gluons, which heat the lighter quarks, but not leptons, in the universe.) In eq. (2.2.12), for example, the rate of baryon number generation is proportional to the difference of the actual number density from its equilibrium value with and at the temperature of the Even if so that the phase-space distribution remains of the form (2.2.13), its temperature may differ from that of the (because of weaker interactions with annihilation products), and baryon number generation may occur until the and are brought to the same temperature. Thus baryon-number conserving annihilation of massive species can indirectly result in baryon number generation by -violating reactions between massless particles.

In models such as those of subsects 2.2 and 2.3, the baryon number of the universe remains constant after the temperature has fallen below the masses of the particles mediating -violating interactions. When lighter species of particles annihilate, they increase the temperature, and thus number, of photons, but leave the baryon number unchanged. These effects reduce the original produced by a factor

In typical grand unified models, this factor is (14).

The approximate conservation of entropy for the universe may of course be drastically violated if its contents undergo a first order phase transition (with specific latent heat ). Such a transition would reduce by a factor As mentioned above, most phase transitions, regardless of order, would result in a temporary increase in the pressure of the universe. The phase transitions associated with the spontaneous breaking of gauge symmetries (mentioned below) are expected to be second-order, or first-order with very small latent heats. The phase transition at in asymptotically free theories (either or a higher ``technicolor'' group) to confinement and chiral symmetry breaking is also probably second order.

2.4.6. Phase Transitions and Spontaneous Symmetry Restoration.

The only known method for breaking local gauge symmetries and providing masses for gauge bosons without destroying renormalizability is by the introduction of Higgs fields with non-zero vacuum expectation values With this mechanism, the mass of a gauge boson is given by

where is the gauge coupling constant, and is determined by minimizing the effective potential

which gives the energy density of the ``vacuum'' as a function of the strength of a uniform classical Higgs field The masses of the quantized fluctuations in this condensate (Higgs particles H) are given by

The vacuum energy density implied by minimizing then has the absurdly large value

If this energy density is present, it presumably has no gravitational effects; to accord with observation it must otherwise be delicately cancelled by a cosmological term in the Einstein field equations. (When the symmetry is restored, the Higgs condensate energy density disappears, leaving uncancelled the cosmological term; however, at the relevant temperatures, its effects on the expansion rate of the universe are typically overwhelmed by the radiation energy density present.) The energy density (2.4.34) perhaps discredits any cosmological considerations of the Higgs mechanism.

At zero temperature, will always be given by minimizing the ``vacuum'' energy (2.4.32). However, at high temperatures, thermal fluctuations in become much larger than and the ``vacuum'' state is no longer concentrated at the minimum of The mean Higgs condensate strength at high temperatures is typically given by (15)

(In the region close to the phase transition the perturbative methods used to derive (2.4.35) fail, so that the precise nature of the transition cannot be determined.) For therefore, the gauge bosons and Higgs particles will be effectively massless; when the universe cools below their masses grow slowly to the values. Note that the constraint necessary for spontaneous symmetry breakdown to occur at low temperatures implies

Above this critical temperature, all particles should be effectively massless, and therefore exist in their equilibrium number densities, so that

When the universe cools below the particles become massive and may decay. The smaller the ratio is, the smaller will be the back reactions to these decays, and thus the larger the final baryon number generated. However, the bound (2.3.36) suggests that the values of for which back reactions will not destroy the baryon number produced are not in fact much extended by these considerations of symmetry restoration.

In addition to the masses of the decaying particles, spontaneous symmetry breakdown may also determine the strength of violation (e.g., [11]). In this case, -violating effects should disappear above a critical temperature If then most decays will be conserving, and so no baryon asymmetry may be generated.

An intriguing (but probably irrelevant) possibility is that domains with different ``order parameters'', (typically ) signalling different symmetry breaking, may have formed in the early universe just below the critical temperature. If, as in the phase transition leading to different values of imply different vacuum energies, then the ``true vacuum'' in which is at its global minimum should quickly overwhelm the regions of false vacuum. However, if there are many possible as characterized, for example, by a phase angle, which give the same vacuum energy, then domains may survive. Thus it is possible that the sign and perhaps magnitude of violation may initially have differed from one region (domain) in the universe to another. However, it is probable that insufficient surface tension would exist to prevent the domains from mixing freely. Moreover, the maximum size of a domain is presumably governed by the distance over which a light signal could have propagated by the time of the phase transition: larger regions could not yet be in ``causal contact'' and therefore could not act collectively. At the temperatures GeV probably relevant for -violating processes, the maximum number of particles in a domain is The possibility that domains in which baryons were generated should have collected together and repelled antibaryons seems extremely implausible. (Note that the ``Leidenfrost effect'' by which radiation pressure from annihilation may hold matter and antimatter regions apart is entirely impotent at since it relies on the conversion of rest energy into photon momentum in annihilation.)

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