Stars as Laboratories for Fundamental Physics - MPP Theory Group

262 Chapter 7
Including the third generation, the mixing is induced by the threedimensional
Cabbibo-Kobayashi-Maskawa (CKM) matrix V . After removing
all unphysical phases by an appropriate redefinition of the quark
fields this unitary matrix is given in terms of four significant parameters.
The Particle Data Group (1994) recommends a standard parametrization
in terms of three two-family mixing angles θ ij < π/2 and one phase
0 ≤ δ < 2π. With C ij ≡ cos θ ij and S ij ≡ sin θ ij this standard form is
(Fritzsch and Plankl 1987)
⎛
⎞ ⎛
1 0 0 C 13 0 S 13 e −iδ ⎞ ⎛
⎞
C 12 S 12 0
⎜
⎟ ⎜
⎟ ⎜
⎟
V = ⎝ 0 C 23 S 23 ⎠ ⎝ 0 1 0 ⎠ ⎝ −S 12 C 12 0 ⎠
0 −S 23 C 23 −S 13 e iδ 0 C 13 0 0 1
⎛
C 12 C 13 S 12 C 13 S 13 e −iδ ⎞
⎜
= ⎝ −C 23 S 12 − C 12 S 23 S 13 e iδ C 12 C 23 − S 12 S 23 S 13 e iδ ⎟
C 13 S 23 ⎠
S 12 S 23 − C 12 C 23 S 13 e iδ −C 12 S 23 − C 23 S 12 S 13 e iδ C 13 C 23
≈
⎛
⎜
⎝
1 S 12 S 13 e −iδ
−S 12 1 S 23
S 12 S 23 − S 13 e iδ −S 23 1
⎞
⎟
⎠ . (7.6)
Experimentally one has the 90% CL ranges (Particle Data Group 1994)
0.218 < S 12 < 0.224,
0.032 < S 23 < 0.048,
0.002 < S 13 < 0.005. (7.7)
The approximation in Eq. (7.6) is justified by the small mixing angles.
They and the CP-violating phase δ = 3.3×10 −3 (Wolfenstein 1986) are
measured parameters of the standard model which, like the fermion
masses, are not theoretically accounted for at the present time.
If neutrinos have masses one naturally expects that they follow a
similar scheme and so the “weak interaction eigenstates” ν l (l = e, µ, τ)
are thought to be given as linear superpositions of the mass eigenstates
ν i by virtue of
3∑
ν l = U li ν i , (7.8)
i=1
where the unitary matrix U plays the role of the CKM matrix. Unless
otherwise stated ν 1 will always refer to the dominant mass admixture
of ν e and so forth. It seems plausible that m 1 < m 2 < m 3 , a hierarchy
that is often assumed.