Baryon-number non-conservation in the Standard Model
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background vacuum fields in the SU(2)L x U(l)y theory. The standard solution
is to take all fields to be zero except for the Higgs doublet for which
t/>vac =
(4.138)
_1
..ti (0)_ v = t/>o
where v is constant. Both KJ.I. and kJ.l. are then zero. of course. Because of the
SU(2)L x U(l)y gauge invariance of the Lagrangian. we could as well choose a
gauge transformation of this solution. Then.
t/>vac(x) = U(X)UI (x)t/>o
(4.139)
where UI (x) is a general element of U(l)y and U(x) is likewise a general element
of SU(2)L. so we may write
U(x) = a(x)h + iT . b(x)
(4.140)
with
det U(x) = a(x)2 + b(x) . b(x) = I.
(4.141)
Thus. U (x) can be regarded as a mapping from spacetime into the three-sphere
(S3) which is the group space of SU(2). In this gauge. the U(l) vector potential
is non-zero:
B;ac = i.(CJJ.l.UI)U I
I
(4.142)
gl
but, of course. the (gauge-invariant) field strength B;~c is still zero and kJ.l. remains
zero:
k vac = o.
(4.143)
J.I.
Similarly. in this gauge. the S U (2) gauge field becomes
.
1
W vac = ~(CJ U)U- I == -T . W
(4.144)
J.I.
g2 J.I.
2
J.I.
and. with the parametrization (4.140). this gives
2
W~ac = --(aCJJ.l.b - bCJJ.l.a + CJJ.l.b x b)
(4.145)
g2
For future reference, we note that, using (4.141),
W~ac . W~ac = 42 [(CJJ.l.a)(CJva) + (CJJ.l. b ) . (CJ)./b)]
g2
4
(4.146)
=2YJ.l .v
g2
where YJ.l.V is the metric on S3 for the spacetime coordinates. The vacuum state
described by (4.144) may be taken to be time-independent and such that U ~ 12