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1.3 The Three Types of $\mathit{C}\mathit{P}$ Violation in $\mathit{B}$ Decays 13

A second type of phase can appear in scattering or decay amplitudes even when the Lagrangian is real. Such phases do not violate $\mathit{C}\mathit{P}$, since they appear in ${\mathit{A}}_{\mathit{f}}$ and ${\overline{\mathit{A}}}_{\overline{\mathit{f}}}$ with the same sign. Their origin is the possible contribution fr om intermediate on‐shell states in the decay process, that is an absorptive part of an amplitude that has contributions from coupled channels. Usually the dominant rescattering is due to strong interactions, hence the designation strong phases“ for the phase shiftts so induced. Again only the relative strong phases of diff erent terms in a scattering amplitude have physical content, an overall phase rotation ofthe entire amplitude has no physical consequences.

Thus it is useful to write each contribution to $\mathit{A}$ in three parts: its magnitude ${\mathit{A}}_{\mathit{i}}$, its weak‐phase term ${\mathit{e}}^{\mathit{i}{\mathit{\phi }}_{\mathit{i}}}$, and its strong phase term ${\mathit{e}}^{\mathit{i}{\mathit{\delta }}_{\mathit{i}}}$. Then, if several amplitudes contribute to ${\mathit{B}}^{\mathrm{0}}\mathrm{\to }\mathit{f}$, the amplitude ${\mathit{A}}_{\mathit{f}}$ (see (1.20)) and the $\mathit{C}\mathit{P}$ conjugate amplitude ${\overline{\mathit{A}}}_{\overline{\mathit{f}}}$ (see (1.21)) are given by:

${\mathit{A}}_{\mathit{f}}\mathrm{=}\sum _{\mathit{i}}{\mathit{A}}_{\mathit{i}}{\mathit{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\delta }}_{\mathit{i}}\mathrm{+}{\mathit{\phi }}_{\mathit{i}}\mathrm{\right)}}\mathrm{,}\mathrm{ }{\overline{\mathit{A}}}_{\overline{\mathit{f}}}\mathrm{=}{\mathit{e}}^{\mathrm{2}\mathit{i}\mathrm{\left(}{\mathit{\xi }}_{\mathit{f}}\mathrm{-}{\mathit{\xi }}_{\mathit{B}}\mathrm{\right)}}\sum _{\mathit{i}}{\mathit{A}}_{\mathit{i}}{\mathit{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\delta }}_{\mathit{i}}\mathrm{-}{\mathit{\phi }}_{\mathit{i}}\mathrm{\right)}}$, (1.45)

where ${\mathit{\xi }}_{\mathit{f}}$ and ${\mathit{\xi }}_{\mathit{B}}$ are defined in 1.2.2. (If $\mathit{f}$ is a $\mathit{C}\mathit{P}$ eigenstate then ${\mathit{e}}^{\mathrm{2}\mathit{i}{\mathit{\xi }}_{\mathit{f}}}\mathrm{=}\mathrm{±}\mathrm{1}$ is its $\mathit{C}\mathit{P}$ eigenvalue.) The convention‐independent quantity is then

$\mathrm{|}\frac{{\overline{\mathit{A}}}_{\overline{\mathit{f}}}}{{\mathit{A}}_{\mathit{f}}}\mathrm{|}\mathrm{=}\mathrm{|}\frac{{\mathrm{\Sigma }}_{\mathit{i}}{\mathit{A}}_{\mathit{i}}{\mathit{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\delta }}_{\mathit{i}}\mathrm{-}{\mathit{\phi }}_{\mathit{i}}\mathrm{\right)}}}{{\mathrm{\Sigma }}_{\mathit{i}}{\mathit{A}}_{\mathit{i}}{\mathit{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\delta }}_{\mathit{i}}\mathrm{+}{\mathit{\phi }}_{\mathit{i}}\mathrm{\right)}}}\mathrm{|}$. (1.46)

When $\mathit{C}\mathit{P}$ is conserved, the weak phases ${\mathit{\phi }}_{\mathit{i}}$ are all equal. Therefore, fr om Eq. (1.46) one sees that

$\mathrm{|}{\overline{\mathit{A}}}_{\overline{\mathit{f}}}\mathrm{/}{\mathit{A}}_{\mathit{f}}\mathrm{|}\mathrm{\ne }\mathrm{1}$ $\mathrm{⇒}$ $\mathit{C}\mathit{P}$ violation. (1.47)

This type of $\mathit{C}\mathit{P}$ violation is here called $\mathit{C}\mathit{P}$ violation in decay. It is oft en also called direct $\mathit{C}\mathit{P}$ violation. It results fr om the $\mathit{C}\mathit{P}$‐violating interference among various terms in the decay amplitude. From Eq. (1.46) it can be seen that a $\mathit{C}\mathit{P}$ violation of this type will not occur unless at least two terms that have diff erent weak phases acquire diff erent strong phases, since:

$\mathrm{|}\mathit{A}{\mathrm{|}}^{\mathrm{2}}\mathrm{⇔}\mathrm{|}\overline{\mathit{A}}{\mathrm{|}}^{\mathrm{2}}\mathrm{=}\mathrm{⇔}\mathrm{2}\sum _{\mathit{i}\mathrm{,}\mathit{j}}{\mathit{A}}_{\mathit{i}}{\mathit{A}}_{\mathit{j}}\mathrm{sin}\mathrm{\left(}{\mathit{\phi }}_{\mathit{i}}\mathrm{⇔}{\mathit{\phi }}_{\mathit{j}}\mathrm{\right)}\mathrm{sin}\mathrm{\left(}{\mathit{\delta }}_{\mathit{i}}\mathrm{⇔}{\mathit{\delta }}_{\mathit{j}}\mathrm{\right)}$ . (1.48)

Any $\mathit{C}\mathit{P}$ asymmetries in charged $\mathit{B}$ decays,

${\mathit{a}}_{\mathit{f}}\mathrm{=}\frac{{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\mathit{B}}^{\mathrm{+}}\mathrm{\to }\mathit{f}\mathrm{\right)}\mathrm{⇔}{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\mathit{B}}^{\mathrm{-}}\mathrm{\to }\overline{\mathit{f}}\mathrm{\right)}}{{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\mathit{B}}^{\mathrm{+}}\mathrm{\to }\mathit{f}\mathrm{\right)}\mathrm{+}{\mathrm{I}}^{\mathrm{⌝}}\mathrm{\left(}{\mathit{B}}^{\mathrm{-}}\mathrm{\to }\overline{\mathit{f}}\mathrm{\right)}}$, (1.49)

are fr om $\mathit{C}\mathit{P}$ violation in decay. In terms ofthe decay amplitudes

${\mathit{a}}_{\mathit{f}}\mathrm{=}\frac{\mathrm{1}\mathrm{⇔}\mathrm{|}\overline{\mathit{A}}\mathrm{/}\mathit{A}{\mathrm{|}}^{\mathrm{2}}}{\mathrm{1}\mathrm{+}\mathrm{|}\overline{\mathit{A}}\mathrm{/}\mathit{A}{\mathrm{|}}^{\mathrm{2}}}$. (1.50)

$\mathit{C}\mathit{P}$ violation in decays can also occur for neutral meson decays, where it competes with the other two types of $\mathit{C}\mathit{P}$ violation eff ects described below. There is as yet no unambiguous experimental

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