Conditions for baryogenesis
95
the first condition is that there are baryon-number non-conserving interactions
in nature. Aside from the baryon-number asymmetry itself, there is no direct
experimental evidence of such interactions and any theory which contains them
is constrained by the current lower bound on the lifetime ('Cp ) of the proton [7]
'Cp ~ 10 31 _10 33 yr.
(4.28)
The generation of a non-zero baryon number for the universe, or
'baryogenesis'. also requires that there are C- and CP-violating interactions in
nature. To see this, suppose that a process i _ f. with initial state i and final state
f. violates baryon-number conservation, so Bi - B f #- O. If charge conjugation
C were an exact symmetry. then the process I-i. where I is obtained from i by
replacing all particles by their antiparticles and similarly for i. would occur at the
same rate as the fonner. Since Bi = - Bi and B i = - B f' the net baryon number
produced by the two processes Bi - B f + Bi + B i is zero. A similar argument
applies if CP-invariance were exact; parity reversal P reverses the momenta of all
participating particles but when these are integrated over the (identical) allowed
phase space the net baryon number produced by the two processes is again zero.
The TCP-invariance of any particle physics model (T is time reversal) ensures that
if there is CP-violation, then there is also T-violation and it is easy to see that if
T-invariance were an exact symmetry. then baryon number would be conserved.
Of course. we have long known that C-invariance is violated by weak interactions
and that CP-violation occurs at the milliweak level in kaon decays [8]. There is
also limited evidence for T-violation in kaon decays [9]. Thus there is no a priori
need for new physics from this condition.
The final condition is that the baryon-number non-conserving processes
occur when the universe is not in thermal equilibrium. To see this, consider a
particle X with non-zero baryon number in thennal equilibrium at a temperature
T «mx. The number density nx of X particles is given by
nx ~ gx(mxT)3/2 e (-m x+Jl.xlT
(4.29)
where JLX is the chemical potential. Likewise, in thermal equilibrium the number
density n i of the antiparticles X is
ni ~ gx(mxT)3/2e(-mx+Jl.x lT •
(4.30)
If X. X participate in baryon-number non-conserving processes, as required by
the first condition, then the process
XX-XX
(4.31 )
is allowed and, in equilibrium, this requires
2JLX = 2JLi'
(4.32)
95
the first condition is that there are baryon-number non-conserving interactions
in nature. Aside from the baryon-number asymmetry itself, there is no direct
experimental evidence of such interactions and any theory which contains them
is constrained by the current lower bound on the lifetime ('Cp ) of the proton [7]
'Cp ~ 10 31 _10 33 yr.
(4.28)
The generation of a non-zero baryon number for the universe, or
'baryogenesis'. also requires that there are C- and CP-violating interactions in
nature. To see this, suppose that a process i _ f. with initial state i and final state
f. violates baryon-number conservation, so Bi - B f #- O. If charge conjugation
C were an exact symmetry. then the process I-i. where I is obtained from i by
replacing all particles by their antiparticles and similarly for i. would occur at the
same rate as the fonner. Since Bi = - Bi and B i = - B f' the net baryon number
produced by the two processes Bi - B f + Bi + B i is zero. A similar argument
applies if CP-invariance were exact; parity reversal P reverses the momenta of all
participating particles but when these are integrated over the (identical) allowed
phase space the net baryon number produced by the two processes is again zero.
The TCP-invariance of any particle physics model (T is time reversal) ensures that
if there is CP-violation, then there is also T-violation and it is easy to see that if
T-invariance were an exact symmetry. then baryon number would be conserved.
Of course. we have long known that C-invariance is violated by weak interactions
and that CP-violation occurs at the milliweak level in kaon decays [8]. There is
also limited evidence for T-violation in kaon decays [9]. Thus there is no a priori
need for new physics from this condition.
The final condition is that the baryon-number non-conserving processes
occur when the universe is not in thermal equilibrium. To see this, consider a
particle X with non-zero baryon number in thennal equilibrium at a temperature
T «mx. The number density nx of X particles is given by
nx ~ gx(mxT)3/2 e (-m x+Jl.xlT
(4.29)
where JLX is the chemical potential. Likewise, in thermal equilibrium the number
density n i of the antiparticles X is
ni ~ gx(mxT)3/2e(-mx+Jl.x lT •
(4.30)
If X. X participate in baryon-number non-conserving processes, as required by
the first condition, then the process
XX-XX
(4.31 )
is allowed and, in equilibrium, this requires
2JLX = 2JLi'
(4.32)