Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 197

Fig. 4.17.

The situation is very similar in non-Abelian gauge theories, where the divergences
of the left- and right-handed currents are given by

∂μJLμ=−cL

g^2

16 π^2

tr

(

FF ̃

)

,∂μJRμ=cR

g^2

16 π^2

tr

(

FF ̃

)

, (4.193)

respectively. In a theory where the right- and left-handed fermions are coupled to
the gauge field with the same strength, as for instance in quantum chromodynamics,
cL=cR=1 for every flavor, and the total current and as a result the fermion number
are conserved. On the other hand, the difference between the numbers of right- and
left-handed fermions,

Q 5 ≡

∫(

JR^0 −JL^0

)

d^3 x=NR−NL, (4.194)

changes in instanton transitions. Using Gauss’s theorem and taking into account
(4.183), we obtain from (4.193)

Q 5 =Q 5 f−Qin 5 =

g^2

8 π^2

∫

tr

(

FF ̃

)

d^4 x= 2 , (4.195)

that is, the corresponding fermion flips its helicity in the instanton.

Violation of fermion numberThe situation is more interesting inchiraltheories,
where the right- and left-handed particles are coupled to the gauge fields differ-
ently. Let us consider electroweak theory at temperaturesT>100 GeV,where the
symmetry is restored and the rate of topological transitions is very high. TheSU( 2 )
gauge fields interact only with left-handed fermions and have the same strength for
each doublet; thereforecL= 1 ,cR=0 and

∂μ(f)JLμ=−

g^2

16 π^2

tr

(

FF ̃

)

, (4.196)

wheref indicates the corresponding fermion doublet and runs from 1 to 12.For
instance,f=1 for the first lepton family,

( 1 )Jμ

L =e ̄Lγ

μeL+ν ̄eγμνe, (4.197)