Physical Foundations of Cosmology

164 The very early universe

W− W+ W+

νR νR νL

CP

e⎯L eL+ e+R

∼

Fig. 4.9.

The term describing the charged weak interactions for quarks can be written as
(4.87):

g

√

2

∑

i

((

u ̄iLγμdL′i

)

Wμ++

( ̄

d′LiγμuiL

)

Wμ−

)

, (4.89)

or, after rewriting it in terms of quark flavors, as

g

√

2

(

Vji

(

u ̄iLγμdLj

)

Wμ++

(

Vij

)∗(

d ̄LjγμuiL

)

Wμ−

)

. (4.90)

UnderCPtransformation, the first term becomes

Vij

(

u ̄iLγμdLj

)

Wμ+→Vji

(

d ̄LjγμuiL

)

Wμ−, (4.91)

and coincides with the second term only ifVij=

(

Vji

)∗

.In other words, (4.90) is

CP-invariant only if the Kobayashi–Maskawa matrix is real-valued; otherwiseCP
is violated. An arbitrary 3×3 unitary matrix

Vji=rijexp

(

iθij

)

is characterized by three independent real numbersrand by six independent phases
θ.The quark Lagrangian is invariant with respect to global quark rotations:qi→
exp(iαi)qi.Using the six independent parameters for six quarks, we can eliminate
fiveθphases as having no physical meaning. One phase is left over, however,
because bilinear quark combinations are insensitive to the overall phase of the
quark rotation. Because of this one remaining phase factor the Kobayashi–Maskawa
matrix will generally have complex elements and therefore one can expectCP
violation. ThisCPviolation is due to the complex-valued coupling constant in the
charged weak interaction term.

Problem 4.15Could we expectCPviolation in a model with only two quark
generations, where the quark mixing is entirely characterized by Cabibbo angle?

The violation ofCPsymmetry was first observed in 1964 in kaonK^0 (d ̄s)decay
and then, in 2001, in theB^0 (db ̄) meson system. There is strong evidence that the
Kobayashi–Maskawa mechanism is responsible for thisCPviolation.