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There are observational and theoretical indications that the local preponderance of baryons over antibaryons extends throughout the universe (at least since the time when the temperature MeV) with an average ratio of baryon to photon densities [1] If baryon number were absolutely conserved in all processes, this small baryon excess must have been present since the beginning of the universe. However, many grand unified gauge models [2] require superheavy particles (typically with masses ) which mediate baryon- and lepton-number violating interactions. Any direct evidence for these must presumably come from an observation of proton decay. In the standard hot big bang model [1], the temperature T (of light particle species) in the early universe fell with time according to (taking units such that where

and is the Planck mass, while gives the effective number of particle species in equilibrium for each ultrarelativistic boson (nondegenerate fermion) spin state). At temperatures -violating interactions should have been important, and they should probably have destroyed or at least much diminished any initial baryon excess. (This occurs even when, for example, is absolutely conserved, since then an initial baryon excess would presumably be accompanied by a lepton excess, so as to maintain the accurate charge neutrality of the universe.) It is interesting (and in some models necessary) to postulate that -violating interactions in the very early universe could give rise to a calculable baryon excess even from an initially symmetrical state. For this to be possible, the rates for reactions producing baryons and antibaryons must differ, and hence the interactions responsible must violate and invariance. We describe here a simple but general method for calculating generation in any specific model. We clarify and extend previous estimates [3]. A detailed account of our work is given in ref. [4].

Let be the amplitude for transitions from the state to and let be the conjugate of Then invariance demands while invariance would require Unitarity (transitions to and from must occur with total probability 1) demands (1) (e.g., ref. [5]) ; combining this with the constraint of invariance yields (the sum over includes all states and their antistates)

In thermal equilibrium (and in the absence of chemical potentials representing nonzero conserved quantum numbers) all states of a system with a given energy are equally populated. Then the last equality in eq. (1) shows that transitions from these states (interactions) must produce and in equal numbers; thus no excess of particles over antiparticles may develop in a system in thermal equilibrium, even if is violated. In addition, the first equality in eq. (1) shows that the total cross sections for destroying particles and antiparticles must be equal. Since in thermal equilibrium no excess of over may develop, this implies that any initial excess must be destroyed.

The phase space distribution (number per unit cell (2) for a species develops with time (on average) according to a Boltzmann transport equation. A closed system with no external influences obeys Boltzmann's H-theorem [which holds regardless of (i.e., ) invariance (e.g., ref. [4])], so that from any initial state the evolve (on average) to their equilibrium forms for which and no baryon excess may survive.

However, in an expanding universe, extra terms must be added to the Boltzmann equations, and if some participating particles are massive (3) , a baryon excess may be generated; the relaxation time necessary to destroy the excess often increases faster than the age of the universe (4).

Eq. (1) requires that the total rates for processes with particle and antiparticle initial states be equal. violation allows the rates for specific conjugate reactions to differ; unitarity nevertheless requires (5):

Hence the fractional difference between conjugate rates must be at least where is some coupling constant (6). Moreover, the loop diagrams giving violation must allow physical intermediate states . (These loop corrections must be usually also -violating to give a difference in rates when summed over all final states with a given [4,6].)

Let be an ``(anti)baryon'' with For simplicity we assume here that all particles (including photons) obey Maxwell-Boltzmann statistics and have only one spin state. In our first (very simple) model, we consider violating reactions involving and a heavy neutral particle ; we take their rates to be (this parametrization ensures unitarity and invariance)

where measures the magnitude of (and ) violation. The number of a species per unit volume decreases with time even without collisions in an expanding universe according to ( is the Robertson--Walker scale factor; dots denote time derivatives)

The are also changed by collisions; the (average) time development of the and baryon number densities is given by the Boltzmann equations where is a massless particle;

where the operator represents suitable integration over initial and final state momenta. We assume that the undergo baryon-conserving collisions with a frequency much higher than the rate on which changes (as is presumably the case in realistic models). They are therefore always in kinetic equilibrium with the rest of the universe, and hence Maxwell--Boltzmann distributed in phase space:

is a baryon number chemical potential, which is changed only by -violating processes, and would vanish if chemical equilibrium prevailed. Assuming , one may use energy conservation in eq. (5) to write

where is the equilibrium distribution of at temperature The equilibrium number density

where is a modified Bessel function [7] [as as Then substituting the parametrization (3) and performing phase space integrations, eq. (5) becomes

where is the cross section corresponding to averaged over a flux incoming particles in equilibrium energy distributions. Eq. (7b) exhibits the necessity of deviation from equilibrium for generation, and the destruction of in equilibrium. It also demonstrates that if and will always be zero. This simple model demonstrates all the conditions necessary for baryon generation.

We now turn to a slightly more realistic but more complicated model in which massive particles decay to with rates

Note that if decays preferentially produce then invariance implies that are preferentially destroyed in inverse processes; thus decays and inverse decays (DID) alone would generate a net even if all particles were in thermal equilibrium, in contravention of the theorem (1) (7). However, the violation parameter is and hence changes in from DID are of the same order as scattering processes, such as It will turn out that -channel exchange of nearly on-shell in cancels the DID contribution to so as to recover in thermal equilibrium. In direct analogy with eqs. (5) and (7), and using the assumption (6), the equation for the evolution of the number density becomes

the corresponding equation for is obtained by charge conjugation The in eq. (9) is the total decay width multiplied by the time dilation factor and averaged over the equilibrium X energy distribution (8). The baryon concentration evolves according to

where the first term is from DID (and does not separately vanish when ) while the second two terms arise from scatterings. The DID term accounts for sequential inverse decay and decay processes involving real : these are therefore subtracted from the true scattering terms by writing where is the amplitude for due to on-shell -channel exchange. In the narrow width approximation,

the presence of the denominator renders it According to the theorem (1), the violating difference of total rates Hence and the second term in eq. (10) becomes thereby elegantly cancelling the first term in thermal equilibrium. Finally, therefore

[ Figure 1 ] The development of baryon number density (solid curves) as a function of inverse temperature in the model of eq. (ll) for various choices of parameters (unless otherwise indicated, and ). The dashed and dotted lines give and respectively. In all cases we have taken the -violation parameter (even when is changed). (Results depend only on through the dimensionless combination here we take in the definition of Note that inhomogeneities in the early universe may be manifest in different expansion rates and hence different effective for different regions. The final produced could vary considerably between the regions.)

The differential eqs. (9) and (11) must now be solved with the initial condition and possibly an initial baryon density Fig. 1 shows the solutions with guesses for parameters based on the SU(5) model [2] and ; (vector decays), or (scalar decays)]. If all initially in thermal equilibrium decayed with no back reaction, the generated would be simply For small or large this upper limit is approached. (At small series solution of eqs. (10) and (11) gives , , , where For baryon number is destroyed by reactions with roughly like (9) , so that constant as but if is small, the final is much diminished from its value at higher The generated is always roughly linearly proportional to but is a sensitive function of and ; for realistic values of these parameters, a numerical solution is probably essential. Previous treatments of baryon number generation [3] have assumed that or Fig. 1 demonstrates that intermediate results are probable.

According to eq. (11), any baryon excess existing at the Planck time should be diminished by inverse decays at so that any initial should be reduced by a factor before violating processes can generate at -violating scatterings at temperatures should reduce an initial by a factor . One might expect that at high energies due to -channel vector exchange; however, the effective presumably relevant for the Boltzmann equation is rather where the Debye screening length In this approximation and higher multiplicity collisions are probably no more effective at destroying an initial than are inverse decays.

We conclude therefore that -violating reactions in the very early universe might well destroy any initial baryon number existing around the Planck time requiring subsequent - and -violating interactions to generate the observed baryon asymmetry. The methods described here allow a calculation of the resulting baryon excess in any specific model; the simple examples considered suggest that the observed should place stringent constraints on parameters of the model (10) [8].

We are grateful to many people for discussions, including A.D. Dolgov, S. Frautschi, William A. Fowler, G.C. Fox, S.E. Koonin, D.L. Tubbs and R.V. Wagoner. We thank T. Goldman for collaboration in the early stages of this work.

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