Out-of-equilibrium decay of heavy particles
97
X particle, which controls the numbers of X and X particles. Unless this is
sufficiently rapid, thermal equilibrium densities cannot be maintained as T falls.
If at some temperature T ~ m x the X bosons cannot decay in the time scale H- 1
associated with the expansion of the universe, then they decouple from the thermal
bath while they are still relativistic and their densities satisfy (4.35). Thus at a
lower temperature T ~ m x, their abundance is much larger than the equilibrium
densities satisfying (4.36). The condition for this to happen is that
rx ~ HIT=mx.
(4.37)
Using (4.21), this requires
1/2
2
3 ( 5 )
-1/2
mx ~ 21r -;
N.
rxmp
(4.38)
assuming that all (relativistic) particle species are at the same temperature. so
g •. T = N •. The overabundance which occurs when this condition is satisfied
allows the possibility ofbaryogenesis. Whether or not it is satisfied depends upon
the particular GUT model in which the X particles arise.
If X is a superheavy gauge boson. for example,
rx '" ClGmx
(4.39)
where ClG = g~/41r is the GUT 'fine structure constant'. Then (4.38) gives
> -1/2
mx '" N.
ClGmp.
(4.40)
For the non-supersyrrunetric SU(5) GUT (which. incidentally, does not satisfy
the constraint (4.28) on the proton lifetime),
I
15
427
ClG '" ;J1
mx'" 10 GeV
N·=T
(4.41)
provided that the colour triplet, electroweak Higgs particles are superheavy, the
constraint (4.40) is not obviously satisfied. However. for the supersyrrunetric
SU(5) GUT,
I
ClG '" B
mx '" 2 x 10 16 GeV
915
N·=T
(4.42)
the larger values of N. and mx outweigh the larger value of ClG and the nonequilibrium condition (4.38) is marginally satisfied.
However. if X is a superbeavy Higgs particle. its decay width
rx '" (=~ Y
(4.43)
ClGmX
can be much smaller than that of the superheavy gauge boson, because the Yukawa
coupling is suppressed (unless f is a top quark) by a factor m f / m w relative to the