Barron-number non-conservation in the Standard Model
119
where Ncs is the Chem-Simons number. It derives from the existence of a 3form K(3) which arises because the 4-form tr(W(2) 1\ W(2» is exact; W(2) ==
WILV dx lL 1\ dx v is the field strength 2-form. Thus, in our (vacuum) case, Ncs
is just (minus) the winding number n.
Evidently electroweak theory, unlike QED, has an infinity of topologically
distinct vacua, which may be labelled with their Chem-Simons number. These
vacua are not physically distinct [36]: they are gauge transformations of each
other, as we have emphasized. Now consider the total baryon number
B == / d 3 x jO(B)
(4.155)
and let the gauge and Higgs fields be general and have time dependence. Then
the baryon number B will also be time-dependent. In a time interval (Ij, If), the
change is
ll.B = 1" dl aoB = / d 4 x alLjlL(B)
(4.156)
assuming that j(B) vanishes at spatial infinity. Note that using (4.130) this ;s
gauge invariant. Using (4. 134),(4.136) and (4.143), we see that
2/
ll.B = - N Gg2 d 4 x a KIL
3211'2
IL
NGg~
(4.157)
= - 3211'2 ll.K.
Now suppose that in this time interval the gauge and Higgs fields traverse a
non-contractible loop in the field configuration space, starting and finishing in a
vacuum configuration. Then the change ll.K is given just by the vacuum formula
(4.153), which gives
ll.B = -NGll.n = NGll.Ncs.
(4.158)
In other words, in the standard model, baryon-number non-conservation arises
in integer multiples of NG whenever the initial and final vacuum states are
topologically distinct.
It is instructive to consider what is happening using Dirac's picture of the
vacuum as the state with all negative-energy fermion levels filled [37]. We start
(at t;) and finish (at t f) in this state. In the presence of non-zero field strengths
and Higgs fields, we expect the energy levels to be displaced at intermediate
times. In the (trivial) case of QED, for example. although the levels are perturbed
from their values at tit they each return to their original level at t/, because the
electromagnetic field interacts with the left- and right-chiral fermion components
with equal strength. However, the baryogenesis with which we are concerned
derives from the chiral nature of electroweak theory: only left-chiral fermions
interact with the SU(2) gauge bosons. Depending on the field configurations. this
119
where Ncs is the Chem-Simons number. It derives from the existence of a 3form K(3) which arises because the 4-form tr(W(2) 1\ W(2» is exact; W(2) ==
WILV dx lL 1\ dx v is the field strength 2-form. Thus, in our (vacuum) case, Ncs
is just (minus) the winding number n.
Evidently electroweak theory, unlike QED, has an infinity of topologically
distinct vacua, which may be labelled with their Chem-Simons number. These
vacua are not physically distinct [36]: they are gauge transformations of each
other, as we have emphasized. Now consider the total baryon number
B == / d 3 x jO(B)
(4.155)
and let the gauge and Higgs fields be general and have time dependence. Then
the baryon number B will also be time-dependent. In a time interval (Ij, If), the
change is
ll.B = 1" dl aoB = / d 4 x alLjlL(B)
(4.156)
assuming that j(B) vanishes at spatial infinity. Note that using (4.130) this ;s
gauge invariant. Using (4. 134),(4.136) and (4.143), we see that
2/
ll.B = - N Gg2 d 4 x a KIL
3211'2
IL
NGg~
(4.157)
= - 3211'2 ll.K.
Now suppose that in this time interval the gauge and Higgs fields traverse a
non-contractible loop in the field configuration space, starting and finishing in a
vacuum configuration. Then the change ll.K is given just by the vacuum formula
(4.153), which gives
ll.B = -NGll.n = NGll.Ncs.
(4.158)
In other words, in the standard model, baryon-number non-conservation arises
in integer multiples of NG whenever the initial and final vacuum states are
topologically distinct.
It is instructive to consider what is happening using Dirac's picture of the
vacuum as the state with all negative-energy fermion levels filled [37]. We start
(at t;) and finish (at t f) in this state. In the presence of non-zero field strengths
and Higgs fields, we expect the energy levels to be displaced at intermediate
times. In the (trivial) case of QED, for example. although the levels are perturbed
from their values at tit they each return to their original level at t/, because the
electromagnetic field interacts with the left- and right-chiral fermion components
with equal strength. However, the baryogenesis with which we are concerned
derives from the chiral nature of electroweak theory: only left-chiral fermions
interact with the SU(2) gauge bosons. Depending on the field configurations. this