3.4 Lepton era 91established from observations and is of orderB 10 −^10 − 10 −^9 .This means that
the entropy per one baryon or, equivalently, the number of photons
(
nγ∼s∼T^3)
per baryon, is very large,∼ 109 –10^10 .The lepton-to-entropy ratiosLiare not so
well established. The most severe limits onLiare indirect. We will see in the next
chapter that the total fermion number is not conserved at temperatures higher than
100 GeV and, as a result, the combination
B+a(
Le+Lμ+Lτ)
,
wherea∼O( 1 ),vanishes. Hence, if there are no special cancellations between the
lepton numbers, their absolute values cannot significantly exceed the baryon num-
ber, that is,|Li|< 10 −^9. Limits from more direct observations are much weaker.
If the temperature is higher than the mass of a particular particle, the particle
is relativistic and many particle–antiparticle pairs are created from the vacuum,
so that the number density of pairs is of order the number density of photons,
nγT^3 .As the temperature drops below the mass, most of these pairs annihilate
and finally only the particle excess survives. Let us determine when the numbers of
particle–antiparticle pairs become negligible. The particle excess is characterized
by a constant number
β=n−n ̄
s, (3.74)
which can be either a baryon number, a lepton number or electric charge. Solving
this together with the equation
nn ̄
s^2∼
(m
T) 3
exp(
−
2 m
T)
, (3.75)
which follows from (3.61), we obtain
n
sβ
2+
√
β^2
4+
(m
T) 3
exp(
−
2 m
T)
;
n ̄
s−
β
2+
√
β^2
4+
(m
T) 3
exp(
−
2 m
T)
.
(3.76)
It is clear that the number density of particle–antiparticle pairs becomes negligible
compared to the particle excess when the second term under the square root becomes
smaller than the first. Forβ1 this occurs at
m
T>ln(
2
β)
+
3
2
ln(
ln(
2
β)
+···
)
. (3.77)
For example, ifβ 10 −^9 ,the particle–antiparticle pairs can be neglected when
the temperature drops by a factor of 25 below the mass. Thus, the number of