Baryon·number non·conservation in the Standard Model
115
= d 4 x (Ja",(1;Fy'" B",1/I)
(4.122)
where B", is the baryon number of 1/1. Since the remainder of the Lagrangian
density is invariant under the (global and therefore the local) transformation, the
(Noether) current
j!B) == L ~y",B",1/I
(4.123)
associated with baryon number, is divergenceless
' "
a'" j~B) = 0
(4.124)
and B is conserved at every order of perturbation theory. Similarly, for the lepton
currents,
j~Lt) == L ~y",Llt1/l
(4.125)
t
However, classical symmetries such as these are generally not preserved at the
quantum level [32-34). In particular, in a chiral theory. in which the left· and
right·chiral fermion states are coupled differently, as occurs in electroweak theory,
there are chiral anomalies resulting from the non· invariance of the field measure
V1/IV1;F in the functional integral determining the generating function [34J. This
non·invariance is equivalent to a further change "S' in the action
8S' = 32~Z f d4x(J(x)[tr[{F~~), P",,,(L)}BJ - tr[{F~!), P",,,(R)}BIl (4.126)
where F~~) == t a ~~L) is the gauge field strength coupled to the left-chiral
component 1/IL == !(l - Ys)1/I of 1/1; t a is the generally matrix-valued coupling
of the representation of the gauge group to which the left·chiral component 1/IL
of the fermion belongs, P~~) == !E",,,pa Fpa(L) is the dual field strength and {, }
denotes anticommutator. The trace is over all fermions 1/1. Similarly for F~~).
Thus, combining this with (4.122), we see that quantum effects require that
alolj,(B) = __ 1_ tr[{F(L) P",,,(L)}B _ (F(R) P",,,(R)}B]
(4.127)
'"
321rz
101" '
101" '
•
For the standard model, we note that there is no contribution to the divergence
from the SU(3)c group, because, for each quark flavour,
F(L) - F(R)
(4.128)
"''' -
IoIIJ'
The left-chiral quark states Q L of the first generation are coupled both to the
SU(2)L field strength W: IJ (a = 1,2,3) with t a = gZ!T" (T" are the 2 x 2 Pauli
matrices) and to the U(I)y field strength B",IJ with strength g) YL = g) A. The
f