320
20. Introduction to the Phenomenology of Weak Interactions
However, it would also be desirable to have a measure of CP-violation
that was independent of quark rephasing. Consider one of the off-diagonal
unitarity conditions,
∗
∗
∗
V ud V ub + V cd V cb + V td V tb = 0.
(20.167)
(Note that the complex conjugate of this equation gives another, independent,
∗
condition.) The best-measured of these products is V cd V ; dividing by this
cb
quantity, (20.167) can be written as
1 + z 1 + z 2 = 0,
(20.168)
where
∗
∗
V td V tb
V ud V ub
z 1 =
, z 2 =
.
(20.169)
∗
∗
V cd V cb
V cd V cb
When viewed in the complex plane, relation (20.168) is the statement that the
vectors (1,0), z 1 and z 2 close to form a triangle as shown in figure 20.10, one
of 6 such unitarity triangles that can be formed. The area Δ of this triangle
is
(
)
∗
∗
1
1
V ud V V V tb
∗
ub td
Δ = Im(z 2 z 1 ) = Im
.
(20.170)
2
2
|V cd | 2 |V cb | 2
Recalling that a rephasing multiplies V ij by expi(α i − α j ), we see that Δ is
rephasing invariant; in particular, so is the numerator J where
∗
∗
J ≡ Im(V ud V tb V
(20.171)
ub V td )
is a Jarlskog invariant (Jarlskog 1985). J may be thought of as follows: (i)
strike out the ‘c’ row and ‘s’ column of V; (ii) take the complex conjugate of
the off-diagonal elements in the 2 × 2 matrix that remains; (iii) multiply the
four elements and take the imaginary part. There is nothing special about
this particular row and column: there are nine different ways of choosing to
pair one row with one column, but all such Js are equal up to a sign, because
of the unitarity of V. In the parametrization (20.166), J takes the form
2
J = c 12 s 12 c 23 s 23 c 13 s 13 sin δ,
(20.172)
which vanishes if any θ ij = 0, or π/2, or if δ = 0 or π.
The CKM matrix is an integral part of the Standard Model, and testing
its validity is an important experimental goal. Various tests are possible.
Consider first the magnitudes of the CKM elements. These must satisfy six
relations following from the unitarity of V: namely, the sum of the squares
of the absolute values of the elements of each row, and of each column, must
add up to unity.
The magnitudes of the six elements of the first two rows have been determined from measurements of semileptonic decay rates: for example, the
amplitude for the tree-level process d → u + e
− + ¯
ν e is proportional to V ud .
20. Introduction to the Phenomenology of Weak Interactions
However, it would also be desirable to have a measure of CP-violation
that was independent of quark rephasing. Consider one of the off-diagonal
unitarity conditions,
∗
∗
∗
V ud V ub + V cd V cb + V td V tb = 0.
(20.167)
(Note that the complex conjugate of this equation gives another, independent,
∗
condition.) The best-measured of these products is V cd V ; dividing by this
cb
quantity, (20.167) can be written as
1 + z 1 + z 2 = 0,
(20.168)
where
∗
∗
V td V tb
V ud V ub
z 1 =
, z 2 =
.
(20.169)
∗
∗
V cd V cb
V cd V cb
When viewed in the complex plane, relation (20.168) is the statement that the
vectors (1,0), z 1 and z 2 close to form a triangle as shown in figure 20.10, one
of 6 such unitarity triangles that can be formed. The area Δ of this triangle
is
(
)
∗
∗
1
1
V ud V V V tb
∗
ub td
Δ = Im(z 2 z 1 ) = Im
.
(20.170)
2
2
|V cd | 2 |V cb | 2
Recalling that a rephasing multiplies V ij by expi(α i − α j ), we see that Δ is
rephasing invariant; in particular, so is the numerator J where
∗
∗
J ≡ Im(V ud V tb V
(20.171)
ub V td )
is a Jarlskog invariant (Jarlskog 1985). J may be thought of as follows: (i)
strike out the ‘c’ row and ‘s’ column of V; (ii) take the complex conjugate of
the off-diagonal elements in the 2 × 2 matrix that remains; (iii) multiply the
four elements and take the imaginary part. There is nothing special about
this particular row and column: there are nine different ways of choosing to
pair one row with one column, but all such Js are equal up to a sign, because
of the unitarity of V. In the parametrization (20.166), J takes the form
2
J = c 12 s 12 c 23 s 23 c 13 s 13 sin δ,
(20.172)
which vanishes if any θ ij = 0, or π/2, or if δ = 0 or π.
The CKM matrix is an integral part of the Standard Model, and testing
its validity is an important experimental goal. Various tests are possible.
Consider first the magnitudes of the CKM elements. These must satisfy six
relations following from the unitarity of V: namely, the sum of the squares
of the absolute values of the elements of each row, and of each column, must
add up to unity.
The magnitudes of the six elements of the first two rows have been determined from measurements of semileptonic decay rates: for example, the
amplitude for the tree-level process d → u + e
− + ¯
ν e is proportional to V ud .