Quark flavor mixing

Mixing between different generations arises from the explicit breaking of cus-todial SU(2) symmetry through the Yukawa couplings of the quarks. To illustrate, the Standard Model Lagrangian can be divided into 3 parts, ℒ_SM = ℒ_kinetic + ℒ_Higgs+ ℒ_Yukawa, and the quark Yukawa interaction is given by

−ℒ^quarks_Yukawa= 𝑌_𝑖𝑗^d𝑄^I_L𝑖𝜑𝐷^I_R𝑗+ 𝑌_𝑖𝑗^u𝑄^I_L𝑖𝜀𝜑^∗𝑈_R𝑗^I + h.c.,

where 𝑖, 𝑗 = 1, 2, 3 are generation labels, 𝑌^uand 𝑌^dare 3×3 complex matrices, φ is the Higgs field, and ε is the rank-2 antisymmetric tensor. 𝑄^I_Lare left-handed quark doublets, and 𝐷^I_R(𝑈_R^I) are right-handed down(up)-type quark singlets, all in the weak eigenstates. When φ acquires a vacuum expectation value, 𝜑 = (0, 𝑣/√

2),

.. the Yukawa interactions give rise to quark mass terms

−ℒ^𝑞_M = (𝑀_𝑑)_𝑖𝑗𝐷^I_L𝑖𝐷^I_R𝑗+ (𝑀_𝑢)_𝑖𝑗𝑈_L𝑖^I 𝑈_R𝑗^I + h.c.,

with the 3 × 3 mass matrices

𝑀_𝑑 = 𝑣

√2𝑌^𝑑, 𝑀_𝑢 = 𝑣

√2𝑌^𝑢

and 𝑈_L𝑖^I , 𝐷^I_L𝑖being parts of the same SU(2)_Ldoublet, 𝑄^I_L𝑖. One can use unitary ma-trices 𝑉_L^𝑢(𝑑)and 𝑉_R^𝑢(𝑑)to change the mass matrices from the basis of flavor eigen-states to that of mass eigeneigen-states

𝑉_L^𝑢(𝑑)𝑀^𝑢(𝑑)𝑉_L^{𝑢(𝑑)†} = diag (𝑚_u(d), 𝑚_c(s), 𝑚_t(b)) ,

where the mass 𝑚_𝑞 are real. Then, the doublet in the interaction basis (with su-perscript I) are expressed in terms of the mass basis (no susu-perscript) as

𝑄^I_L=⎛⎜⎜

By convention, (𝑈_L^𝑢†)_𝑖𝑗 is pulled out, so that the transformation only acts on the down-type quarks. Hence, the charged-current weak interaction in ℒ_kineticis mod-ified by the product of the diagonalizing matrices, or the Cabbibo-Kobayashi-Maskawa (CKM) matrix [42]

It is the misalignment between these two bases that leads to the quark mixing.

Being the product of unitary matrices, the CKM matrix is itself unitary (𝑉 𝑉^†= 𝐼). Out of the free parameters of 3 real numbers and 6 complex phases, 5 phases

.. can be rotated away without making any observable effect¹³, leaving 3 real Eu-ler angles 𝜃₁₂, 𝜃₁₃, 𝜃₂₃ and 1 irreducible complex phase δ. This corresponds to 3 rotations in real, 3-dimensional space [45]

𝑈₁₂ =

and another unitary matrix with the 𝒞𝒫-violating phase

𝑈_𝛿=

This complex phase is the only source of 𝒞𝒫 violation in the Standard Model (ne-glecting the θ-term of the strong interaction). The canonical way to parametrize the CKM matrix is [46]

𝑉_CKM = 𝑈₂₃𝑈_𝛿^†𝑈₁₃𝑈_𝛿𝑈₁₂

These 4 parameters are not predicted by the Standard Model, and thus have to be

13We are free to transform the quark fields as 𝑑𝑗→ 𝑒^{𝑖𝜑𝑑𝑗}𝑑_𝑗, 𝑢_𝑗 → 𝑒^{𝑖𝜑𝑢𝑗}𝑢_𝑗. This has no observ-able effect (up to redefining the Yukawa coupling constants), except that the CKM matrix elements are now

𝑉_𝑗𝑘𝑒^{𝑖(𝜑𝑗𝑑−𝜑𝑘𝑢)}. (1.5)

There are 5 independent phase differences in these expressions. Thus, up to 5 complex phases in the CKM matrix elements 𝑉𝑗𝑘can be eliminated by choosing the appropriate phases 𝜑^𝑑𝑗 and 𝜑^𝑢𝑘

[43,44].

.. determined by various experimental measurements. A recent fitting result is [31]

0.97434^+0.00011_−0.00012 0.22506 ± 0.00050 0.00357 ± 0.00015 0.22492 ± 0.00050 0.97351 ± 0.00013 0.00411 ± 0.0013

0.00875^+0.00032_−0.00033 0.0403 ± 0.0013 0.99915 ± 0.00005

⎞⎟ are only slightly mixed. This leads to an alternative parametrization using the ex-pansion of a small parameter 𝜆 = |𝑉_us| ≈ 0.23 [47], so that the hierarchy becomes more visible

where all the rest parameters 𝐴, 𝜌 and η are of order 1. All the calculation in section 1.3adopts this parametrization, in which case all effects of 𝒞𝒫 violation in the Standard Model is proportional to 𝜂. Of course, the physics prediction would be the same in any other convention.

As physical quantities are independent of phase convention, the magnitude of 𝒞𝒫 violation can be defined as the Jarlskog parameter [48]

ℐ𝑚 (𝑉_𝑖𝑗𝑉_𝑘𝑙𝑉_𝑖𝑙^∗𝑉_𝑘𝑗^∗) = 𝐽

3

∑

𝑚,𝑛=1

𝜀_𝑖𝑘𝑚𝜀_𝑗𝑙𝑛,

which is invariant under the phase transformation in Eq. (1.5). In terms of the above parametrization,

𝐽 = 𝑐₁₂𝑐₂₃𝑐²₁₃𝑠₁₂𝑠₂₃𝑠²₁₃sin 𝛿 ≃ 𝜆⁶𝐴²𝜂 = (3.04^+0.21_−0.20× 10⁻⁵) .

In addition, if the d, s, b quarks were degenerate in mass, we could redefine the states so that each quark only couples to the same generation. Therefore, a

basis-.. invariant measure of the 𝒞𝒫 violation is

𝑑_𝒞𝒫 = sin(𝜃₁₂) sin(𝜃₂₃) sin(𝜃₁₃) sin 𝛿_𝒞𝒫 (1.6)

⋅(𝑚²_t − 𝑚²_c)(𝑚²_t − 𝑚²_u)(𝑚²_c − 𝑚²_u)(𝑚²_b− 𝑚²_s)(𝑚²_b− 𝑚²_d)(𝑚²_s − 𝑚²_d)

The above describes the mechanism to produce baryon asymmetry in the Stan-dard Model, but to explain the degree of the asymmetry, its magnitude also needs to match the scale of 𝜂 in Eq. (1.1). A widely accepted argument [49,50,51] states that the only natural energy scale at which the baryon asymmetry is generated is the temperature of electroweak phase transition 𝑇_EW ≈ 100 GeV. Therefore, a dimensionless number made by 𝑑_𝒞𝒫 and 𝑇_EW should be greater than the baryon asymmetry

𝜂 ≲ 𝑑_𝒞𝒫

𝒩_eff𝑇_EW¹² ≈ 10⁻²⁰.

Since this number falls short of 𝜂 by more than 10 orders of magnitude, it is impos-sible for the Standard Model to explain the observed asymmetry. This argument has been questioned in Ref. [52]. Nevertheless, there is little dispute over the third Sakharov condition. In order for the generated asymmetry not to be washed out, the electroweak phase transition must be a first order phase transition, which put a limit of ~90 GeV on the Higgs mass [53]. The discovery of a 125 GeV Higgs ex-cludes this possibility. Therefore, the baryon asymmetry strongly suggests that there are new physics beyond the Standard Model.

Theories of baryogenesis [54, 55] extends the Standard Model to provide dy-namical mechanisms that can account for the observed baryon asymmetry, based on the 3 Sakharov conditions. For instance, theories of leptogenesis seek new source of 𝒞𝒫 violation through 𝐵 + 𝐿 violating but 𝐵 − 𝐿 conserving processes in the lepton sector. The two Higgs doublet models extend the Higgs field to con-trol flavor violation. Many theories provide falsifiable prediction at the current level of experimental precision. Since the Standard Model 𝒞𝒫 violation has been established to be the dominant source for observations in the B-meson system,

.. the focus of the experiments has shifted from changing the overall picture of the Standard Model to seek small deviation from its prediction that may hint new physics. Although 𝜂 strongly motivated the prosper of new theories, as a mere number, it tells nothing about the detailed mechanism of baryogenesis. Conse-quently, it bears little to no discriminating power to kill new theories. In contrast, it is the flavor physics measurements that can test the theoretical predictions in detail, and ultimately nail the coffin of unfortunate theories.