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1.4 $\mathit{C}\mathit{P}$ Violation in the Standard Model

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1.4 $\mathit{C}\mathit{P}$ Violation in the Standard Model

1.4.1 The CKM Picture of $\mathit{C}\mathit{P}$ Violation

In the Standard Model (SM) [24] of $\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{3}{\mathrm{\right)}}_{\mathit{C}}\mathrm{×}\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{2}{\mathrm{\right)}}_{\mathit{L}}\mathrm{×}\mathit{U}\mathrm{\left(}\mathrm{1}{\mathrm{\right)}}_{\mathit{Y}}$ gauge symmetry with three fermion generations, $\mathit{C}\mathit{P}$ violation arises fr om a single phase in the mixing matrix for quarks [3]. Each quark generation consists of three multiplets:

${\mathit{Q}}_{\mathit{L}}^{\mathit{I}}\mathrm{=}\left(\begin{array}{c}{\mathit{U}}_{\mathit{L}}^{\mathit{I}}\\ {\mathit{D}}_{\mathit{L}}^{\mathit{I}}\end{array}\right)\mathrm{=}\mathrm{\left(}\mathrm{3}\mathrm{,}\mathrm{2}{\mathrm{\right)}}_{\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{6}}\mathrm{,}\mathrm{ }{\mathit{u}}_{\mathit{R}}^{\mathit{I}}\mathrm{=}\mathrm{\left(}\mathrm{3}\mathrm{,}\mathrm{1}{\mathrm{\right)}}_{\mathrm{+}\mathrm{2}\mathrm{/}\mathrm{3}}\mathrm{,}\mathrm{ }{\mathit{d}}_{\mathit{R}}^{\mathit{I}}\mathrm{=}\mathrm{\left(}\mathrm{3}\mathrm{,}\mathrm{1}{\mathrm{\right)}}_{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{3}}$, (1.62)

where $\mathrm{\left(}$3, $\mathrm{2}{\mathrm{\right)}}_{\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{6}}$ denotes a triplet of $\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{3}{\mathrm{\right)}}_{\mathit{C}}$, doublet of $\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{2}{\mathrm{\right)}}_{\mathit{L}}$ with hypercharge $\mathit{Y}\mathrm{=}\mathit{Q}\mathrm{⇔}{\mathit{T}}_{\mathrm{3}}\mathrm{=}$ $\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{6}$, and similarly for the other representations. The interactions of quarks with the $\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{2}{\mathrm{\right)}}_{\mathit{L}}$ gauge bosons are given by

${{L}}_{\mathit{W}}\mathrm{=}{\mathrm{\infty }}_{\mathrm{2}}^{\mathrm{1}}\mathit{g}\overline{{\mathit{Q}}_{\mathit{L}\mathit{i}}^{\mathit{I}}}{\mathit{\gamma }}^{\mathit{\mu }}{\mathit{\tau }}^{\mathit{a}}{\mathrm{1}}_{\mathit{i}\mathit{j}}{\mathit{Q}}_{\mathit{L}\mathit{j}}^{\mathit{I}}{\mathit{W}}_{\mathit{\mu }}^{\mathit{a}}$, (1.63)

where ${\mathit{\gamma }}^{\mathit{\mu }}$ operates in Lorentz space, ${\mathit{\tau }}^{\mathit{a}}$ operates in $\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{2}{\mathrm{\right)}}_{\mathit{L}}$ space and 1 is the unit matrix operating in generation (flavor) space. This unit matrix is written explicitly to make the transformation to mass eigenbasis clearer. The interactions ofquarks with the single Higgs scalar doublet $\mathit{\phi }\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{2}{\mathrm{\right)}}_{\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{2}}$ ofthe Standard Model are given by

${{L}}_{\mathit{Y}}\mathrm{=}\mathrm{a}{\mathrm{e}}_{\mathit{i}\mathit{j}}\overline{{\mathit{Q}}_{\mathit{L}\mathit{i}}^{\mathit{I}}}\mathit{\phi }{\mathit{d}}_{\mathit{R}\mathit{j}}^{\mathit{I}}\mathrm{⇔}{\mathit{F}}_{\mathit{i}\mathit{j}}\overline{{\mathit{Q}}_{\mathit{L}\mathit{i}}^{\mathit{I}}}\stackrel{\mathrm{˜}}{\mathrm{ũ}}{\mathit{u}}_{\mathit{R}\mathit{j}}^{\mathit{I}}\mathrm{+}$ Hermitian conjugate; (1.64)

where $\mathit{G}$ and $\mathit{F}$ are general complex $\mathrm{3}\mathrm{×}\mathrm{3}$ matrices. Their complex nature is the source of $\mathit{C}\mathit{P}$ violation in the Standard Model. With the spontaneous symmetry breaking, $\mathit{S}\mathit{U}\mathrm{\left(}\mathrm{2}{\mathrm{\right)}}_{\mathit{L}}\mathrm{×}\mathit{U}\mathrm{\left(}\mathrm{1}{\mathrm{\right)}}_{\mathit{Y}}\mathrm{\to }$ $\mathit{U}\mathrm{\left(}\mathrm{1}{\mathrm{\right)}}_{\mathrm{E}\mathrm{M}}$ due to $\mathrm{⟨}\mathit{\phi }\mathrm{⟩}\mathrm{\ne }\mathrm{0}$, the two components of the quark doublet become distinguishable, as are the three members ofthe ${\mathit{W}}^{\mathit{\mu }}$ triplet. The charged current interaction in (1.63) is given by

(1.65)

gu

ij

Li

Lj

The mass terms that arise fr om the replacement ${R}\mathit{e}\mathrm{\left(}{\mathit{\phi }}^{\mathrm{0}}\mathrm{\right)}\mathrm{\to }\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{\left(}\mathit{v}\mathrm{+}{\mathit{H}}^{\mathrm{0}}\mathrm{\right)}$ in (1.64) are given by

(1.66)

$\mathrm{v}\mathrm{G}\mathrm{d}$

$\mathrm{v}\mathrm{F}\mathrm{u}$

$\mathrm{R}\mathrm{j}$ $\mathrm{+}$ Hermitian conjugate;

Li

Li

Rj

namely

${\mathit{M}}_{\mathit{d}}\mathrm{=}\mathit{G}\mathit{v}\mathrm{/}\sqrt{\mathrm{2}}\mathrm{,}\mathrm{ }{\mathit{M}}_{\mathit{u}}\mathrm{=}\mathit{F}\mathit{v}\mathrm{/}\sqrt{\mathrm{2}}$. (1.67)

The phase information is now contained in these mass matrices. To transform to the mass eigen‐ basis, one defines four unitary matrices such that

${\mathit{V}}_{\mathit{d}\mathit{L}}{\mathit{M}}_{\mathit{d}}{\mathit{V}}_{\mathit{d}\mathit{R}}^{\mathrm{†}}\mathrm{=}{\mathit{M}}_{\mathit{d}}^{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}\mathrm{,}\mathrm{ }{\mathit{V}}_{\mathit{u}\mathit{L}}{\mathit{M}}_{\mathit{u}}{\mathit{V}}_{\mathit{u}\mathit{R}}^{\mathrm{†}}\mathrm{=}{\mathit{M}}_{\mathit{u}}^{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$, (1.68)

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