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1.3 The Three Types of $\mathit{C}\mathit{P}$ Violation in $\mathit{B}$ Decays 15

In terms $\mathrm{o}\mathrm{f}\mathrm{|}\mathit{q}\mathrm{/}\mathit{p}\mathrm{|}\mathrm{,}$

${\mathit{a}}_{\mathrm{s}\mathrm{1}}\mathrm{=}\frac{\mathrm{1}\mathrm{⇔}\mathrm{|}\mathit{q}\mathrm{/}\mathit{p}{\mathrm{|}}^{\mathrm{4}}}{\mathrm{1}\mathrm{+}\mathrm{|}\mathit{q}\mathrm{/}\mathit{p}{\mathrm{|}}^{\mathrm{4}}}$, (1.54)

which follows from

$\mathrm{⟨}{\mathrm{\ell }}^{\mathrm{-}}\mathit{\nu }\mathit{X}\mathrm{|}\mathit{H}\mathrm{|}{\mathit{B}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{⟩}\mathrm{=}\mathrm{\left(}\mathit{q}\mathrm{/}\mathit{p}\mathrm{\right)}{\mathit{g}}_{\mathrm{-}}\mathrm{\left(}\mathit{t}\mathrm{\right)}{\mathit{A}}^{\mathrm{*}}\mathrm{,}\mathrm{ }\mathrm{⟨}{\mathrm{\ell }}^{\mathrm{+}}\mathit{\nu }\mathit{X}\mathrm{|}\mathit{H}\mathrm{|}{\overline{\mathit{B}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{⟩}\mathrm{=}\mathrm{\left(}\mathit{p}\mathrm{/}\mathit{q}\mathrm{\right)}{\mathit{g}}_{\mathrm{-}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathit{A}$. (1.55)

As can be seen from the discussion in Section 1.2.3, eff ects of $\mathit{C}\mathit{P}$ violation in mixing in neutral ${\mathit{B}}_{\mathit{d}}$ decays, such as the asymmetries in semileptonic decays, are expected to be small, ${O}\mathrm{\left(}\mathrm{1}{\mathrm{0}}^{\mathrm{-}\mathrm{2}}\mathrm{\right)}$ . Moreover, to calculate the deviation of $\mathit{q}\mathrm{/}\mathit{p}$ fr om a pure phase, one needs to calculate ${\mathrm{I}}_{\mathrm{1}\mathrm{2}}^{\mathrm{⌝}}$ and ${\mathit{M}}_{\mathrm{1}\mathrm{2}}\mathrm{.}$ This involves large hadronic uncertainties, in particular in the hadronization models for ${\mathrm{I}}_{\mathrm{1}\mathrm{2}}^{\mathrm{⌝}}$. The overall uncertainty is easily a factor of 23 in $\mathrm{|}\mathit{q}\mathrm{/}\mathit{p}\mathrm{|}\mathrm{⇔}\mathrm{1}\mathrm{\left[}\mathrm{1}\mathrm{0}\mathrm{\right]}$. Thus even if such asymmetries are observed, it will be difficult to relate their rates to fundamental CKM parameters.

1.3.3 $\mathit{C}\mathit{P}$ Violation in the Interference Between Decays With and Without Mixing

Finally, consider neutral $\mathit{B}$ decays into final $\mathit{C}\mathit{P}$ eigenstates, ${\mathit{f}}_{\mathit{C}\mathit{P}}\mathrm{\left[}\mathrm{1}\mathrm{4}$, 15, 16$\mathrm{\right]}$. Such states are accessible in both ${\mathit{B}}^{\mathrm{0}}$ and $\mathrm{-}\mathrm{◃}\mathit{B}$ decays. The quantity of interest here that is independent of phase conventions and physically meaningful is $\mathit{\lambda }$ ofEq. (1.42), $\mathit{\lambda }\mathrm{=}{\mathit{\eta }}_{{\mathit{f}}_{\mathit{C}\mathit{P}}}\frac{\mathit{q}}{\mathit{p}}\frac{{\overline{\mathit{A}}}_{{\overline{\mathit{f}}}_{\mathit{C}\mathit{P}}}}{{\mathit{A}}_{{\mathit{f}}_{\mathit{C}\mathit{P}}}}$. When $\mathit{C}\mathit{P}$ is conserved, $\mathrm{|}\mathit{q}\mathrm{/}\mathit{p}\mathrm{|}\mathrm{=}\mathrm{1}\mathrm{,}$ $\mathrm{|}{\overline{\mathit{A}}}_{{\overline{\mathit{f}}}_{\mathit{C}\mathit{P}}}\mathrm{/}{\mathit{A}}_{{\mathit{f}}_{\mathit{C}\mathit{P}}}\mathrm{|}\mathrm{=}\mathrm{1}$, and furthermore, the relative phase between $\mathrm{\left(}\mathit{q}\mathrm{/}\mathit{p}\mathrm{\right)}$ and $\mathrm{\left(}{\overline{\mathit{A}}}_{{\overline{\mathit{f}}}_{\mathit{C}\mathit{P}}}\mathrm{/}{\mathit{A}}_{{\mathit{f}}_{\mathit{C}\mathit{P}}}\mathrm{\right)}$ vanishes. Therefore, Eq. (1.42) implies

$\mathit{\lambda }\mathrm{\ne }\mathrm{±}\mathrm{1}$ $\mathrm{⇒}$ $\mathit{C}\mathit{P}$ violation: (1.56)

Note that both $\mathit{C}\mathit{P}$ violation in decay (1.47) and $\mathit{C}\mathit{P}$ violation in mixing (1.52) lead to (1.56) through jj $\mathrm{\ne }\mathrm{1}$. However, it is possible that, to a good approximation, $\mathrm{|}\mathit{q}\mathrm{/}\mathit{p}\mathrm{|}\mathrm{=}\mathrm{1}$ and $\mathrm{|}\overline{\mathit{A}}\mathrm{/}\mathit{A}\mathrm{|}\mathrm{=}\mathrm{1}\mathrm{,}$ yet there is $\mathit{C}\mathit{P}$ violation:

$\mathrm{|}\mathit{\lambda }\mathrm{|}\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{ }{I}\mathit{m}\mathit{\lambda }\mathrm{\ne }\mathrm{0}$. (1.57)

This type of $\mathit{C}\mathit{P}$ violation is called $\mathit{C}\mathit{P}$ violation in the intererence betw een decays with and without mixing here; sometimes this is abbreviated as ""interference between mixing and decay.“ As explained in Section 1.6, this type of $\mathit{C}\mathit{P}$ violation has also been observed in the neutral kaon system.

For the neutral $\mathit{B}$ system, $\mathit{C}\mathit{P}$ violation in the interference between decays with and without mixing can be observed by comparing decays into final $\mathit{C}\mathit{P}$ eigenstates of a time‐evolving neutral $\mathit{B}$ state that begins at time zero as ${\mathit{B}}^{\mathrm{0}}$ to those ofthe state that begins as a ${\overline{\mathit{B}}}^{\mathrm{0}}$:

$\mathit{a}$ ${\mathit{f}}_{\mathit{C}\mathit{P}}\mathrm{=}\frac{{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\mathit{B}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\to }{\mathit{f}}_{\mathit{C}\mathit{P}}\mathrm{\right)}\mathrm{⇔}{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\overline{\mathit{B}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\to }{\mathit{f}}_{\mathit{C}\mathit{P}}\mathrm{\right)}}{{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\mathit{B}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\to }{\mathit{f}}_{\mathit{C}\mathit{P}}\mathrm{\right)}\mathrm{+}{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\overline{\mathit{B}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\to }{\mathit{f}}_{\mathit{C}\mathit{P}}\mathrm{\right)}}$. (1.58)

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