Chapter

33 CKM Phases and Interference CP Violation

https://doi.org/10.1093/oso/9780198503996.003.0033

Pages

398–408

Published:

July 1999

Abstract

The parameter relevant for interference CP violation in the transitions B A,IA1 in that model. In this chapter we assume that qs /PB,, is given by a phase, but allow this phase to differ from the standard-model (SM) one-cf. eqn (30.33). As for the decay amplitudes, we assume that they are dominate by SM diagrams, but allow non-unitarity of the CKM matrix to alter the corresponding phases from their allowed values in the SM. We start by assuming that the decay amplitudes are dominated by a single weak phase. study the effect of subdominant contributions to the decay, still within the SM.Let us assume that the amplitudes are dominated by the tree-level diagrams of the SM.

33.1 Parameters $λ_{f}$ at tree level

The parameter relevant for interference CP violation in the transitions $B_{q}^{0} \to f$ and $\bar{B_{q}^{0}} \to f$ is

λ_{f} = \frac{q_{B_{q}}}{p_{B_{q}}} \frac{{\bar{A}}_{f}}{A_{f}} .

(33.1)

In order to find its value in a given model one must compute both $q_{B_{q}} / p_{B_{q}}$ and ${\bar{A}}_{f} / A_{f}$ in that model. In this chapter we assume that $q_{B_{q}} / p_{B_{q}}$ is given by a phase, but allow this phase to differ from the standard-model (SM) one—cf. eqn (30.33).

As for the decay amplitudes, we assume that they are dominated by SM diagrams, but allow non-unitarity of the CKM matrix to alter the corresponding phases from their allowed values in the SM. We start by assuming that the decay amplitudes are dominated by a single weak phase. In that case, ${\bar{A}}_{f} / A_{f}$ reduces to a phase and, with our assumption that $∣ q / p ∣ = 1$ ⁠, so does $λ_{f}$ ⁠. Later, we shall study the effect of subdominant contributions to the decay, still within the SM.

33.1.1 $B_{d}^{0} \to π^{+} π^{-}$

At quark level, the decay $B_{d}^{0} \to π^{+} π^{-}$ is $\bar{b} \to \bar{u} u \bar{d}$ ⁠, while $\bar{B_{d}^{0}} \to π^{+} π^{-}$ is $b \to \bar{u} ud$ ⁠. Let us assume that the amplitudes are dominated by the tree-level diagrams of the SM. Then, their ratio is

\frac{{\bar{A}}_{π^{+} π^{-}}}{A_{π^{+} π^{-}}} = \frac{V_{ub} V_{ud}^{*} 〈 π^{+} π^{-} ∣ {(\bar{u} b)}_{V - A} {(\bar{d} u)}_{V - A} ∣ \bar{B_{d}^{0}} 〉}{V_{ub}^{*} V_{ud} 〈 π^{+} π^{-} ∣ {(\bar{b} u)}_{V - A} {(\bar{u} d)}_{V - A} ∣ B_{d}^{0} 〉} .

(33.2)

We use CP symmetry of the strong interactions to relate the two matrix elements to each other. As in the decays of the neutral kaons to two pions, the state $π^{+} π^{-}$ is CP-even. Therefore,

\frac{{\bar{A}}_{π^{+} π^{-}}}{A_{π^{+} π^{-}}} = e^{i (ξ_{b} - ξ_{d} - ξ_{B_{d}})} \frac{V_{ub} V_{ud}^{*}}{V_{ub}^{*} V_{ud}} .

(33.3)

Then, using eqn (30.33),

\begin{array}{c} λ_{B_{d}^{0} \to π^{+} π^{-}} = - sign (B_{B_{d}}) \frac{V_{tb}^{*} V_{td}}{V_{tb} V_{td}^{*}} \frac{V_{ub} V_{ud}^{*}}{V_{ub}^{*} V_{ud}} \\ = - sign (B_{B_{d}}) e^{2 iα} . \end{array}

(33.4)

(The spurious phases brought about by CP transformations plague the unphysical $q_{B_{d}} / p_{B_{d}}$ and ${\bar{A}}_{π^{+} π^{-}} / A_{π^{+} π^{-}}$ but drop out in their product—as they should, because $λ_{B_{d}^{0} \to π^{+} π^{-}}$ is a physical quantity.) Thus, if there were no penguin diagrams, the rate asymmetries would measure the phase $α$ ⁠.

33.1.2 $B_{d}^{0} \to J / ψ K_{S}$

Let us now consider the $B_{d}^{0} \to J / ψ K_{S}$ decays. $J / ψ$ has spin one and is CP-even, while $K_{S}$ has spin zero and is almost CP-even. Since $B_{d}^{0}$ has spin zero, the $J / ψ$ and the $K_{S}$ of the final state must have a relative $l = 1$ angular momentum. As a result, the $J / ψ K_{S}$ final state must be CP-odd.

The final state $J / ψ K_{S}$ is common to $B_{d}^{0}$ and $\bar{B_{d}^{0}}$ ⁠. But, in the spectator-quark approximation, $B_{d}^{0} = \bar{b} d$ only decays to $K^{0} = \bar{s} d$ and not to $\bar{K^{0}} = s \bar{d}$ ⁠; conversely, $\bar{B_{d}^{0}}$ decays to $\bar{K^{0}}$ but not to $K^{0}$ ⁠. Using the reciprocal basis of § 6.8,

〈 {\tilde{K}}_{S} ∣ = \frac{1}{2} (〈 K^{0} ∣ p_{K}^{- 1} - 〈 \bar{K^{0}} ∣ q_{K}^{- 1}) .

(33.5)

Therefore,

\begin{array}{c} 〈 J / ψ K_{S} ∣ T ∣ B_{d}^{0} 〉 = \frac{1}{2 p_{K}} 〈 J / ψ K^{0} ∣ T ∣ B_{d}^{0} 〉, \\ 〈 J / ψ K_{S} ∣ T ∣ \bar{B_{d}^{0}} 〉 = - \frac{1}{2 q_{K}} 〈 J / ψ \bar{K^{0}} ∣ T ∣ \bar{B_{d}^{0}} 〉 . \end{array}

(33.6)

Thus,

\frac{{\bar{A}}_{J / ψ K_{S}}}{A_{J / ψ K_{S}}} = - \frac{p_{K}}{q_{K}} \frac{〈 J / ψ \bar{K^{0}} ∣ T ∣ \bar{B_{d}^{0}} 〉}{〈 J / ψ K^{0} ∣ T ∣ B_{d}^{0} 〉} .

(33.7)

Whenever $B_{d}^{0}$ decays into a $K_{S}$ one must include a factor $- p_{K} / q_{K}$ in $λ$ ⁠. Similarly, a $K_{L}$ in the final state requires a factor $+ p_{K} / q_{K}$ ⁠. The reason is simple: $A_{f}$ refers to the amplitude of $B^{0}$ into a given final state f, and not only to the diagram from which it originates. For decays with $K_{S}$ or $K_{L}$ in the final state, the overall amplitude will involve the amplitudes into the flavour eigenstates, and the transformation from the flavour into the mass eigenstates.

We now assume that the decay is dominated by the tree-level amplitude. Then,

\frac{{\bar{A}}_{J / ψ K_{s}}}{A_{J / ψ K_{S}}} = - \frac{p_{K}}{q_{K}} \frac{V_{cb} V_{cs}^{*} 〈 J / ψ \bar{K^{0}} ∣ {(\bar{c} b)}_{V - A} {(\bar{s} c)}_{V - A} ∣ \bar{B_{d}^{0}} 〉}{V_{cb}^{*} V_{cs} 〈 J / ψ K^{0} ∣ {(\bar{b} c)}_{V - A} {(\bar{c} s)}_{V - A} ∣ B_{d}^{0} 〉} .

(33.8)

Using CP symmetry to relate the matrix elements, one obtains

\frac{{\bar{A}}_{J / ψ K_{S}}}{A_{J / ψ K_{S}}} = \frac{p_{K}}{q_{K}} e^{i (ξ_{K} - ξ_{B_{d}} + ξ_{b} - ξ_{s})} \frac{V_{cb} V_{cs}^{*}}{V_{cb}^{*} V_{cs}},

(33.9)

where a minus sign has been included to take into account the fact that $J / ψ$ and $K_{S}$ are in a relative $l = 1$ state. Using the q/p factors for the $K^{0} - \bar{K^{0}}$ and $B_{d}^{0} - \bar{B_{d}^{0}}$ systems in eqns (30.18) and (30.33), respectively, we arrive at the final result:

\begin{matrix} λ_{B_{d}^{0} \to J / ψ K_{S}} = sign (B_{B_{d}}) \frac{V_{cs} V_{cd}^{*}}{V_{cs}^{*} V_{cd}} e^{2 i ϵ^{'}} \frac{V_{tb}^{*} V_{td}}{V_{tb} V_{td}^{*}} \frac{V_{cb} V_{cs}^{*}}{V_{cb}^{*} V_{cs}} \\ = sign (B_{B_{d}}) e^{- 2 i (β - ϵ^{'})} . \end{matrix}

(33.10)

The analysis is the same for a final state $J / ψ K_{L}$ ⁠, except that the extra factor due to $K^{0} - \bar{K^{0}}$ mixing is $+ p_{K} / q_{K}$ instead of $- p_{K} / q_{K}$ ⁠. Hence,

λ_{B_{d}^{0} \to J / ψ K_{L}} = - λ_{B_{d}^{0} \to J / ψ K_{S}} = - sign (B_{B_{d}}) e^{- 2 i (β - ϵ^{'})} .

(33.11)

The difference in sign might be expected from the fact that $J / ψ K_{L}$ is CP-even, while $J / ψ K_{S}$ is CP-odd. (In both cases, $J / ψ$ and the kaon are in a p wave.)

33.1.3 $B_{s}^{0} \to ρ K_{S}$

The computation of $λ_{B_{s}^{0} \to ρ K_{S}}$ follows a similar route. The meson $ρ$ has spin one and is CP-even. In the spectator approximation $B_{s}^{0}$ decays into $\bar{K^{0}}$ but not into $K^{0}$ ⁠. As a result,

\begin{array}{c} 〈 ρ K_{S} ∣ T ∣ B_{s}^{0} 〉 = - \frac{1}{2 q_{K}} 〈 ρ \bar{K^{0}} ∣ T ∣ B_{s}^{0} 〉, \\ 〈 ρ K_{S} ∣ T ∣ \bar{B_{s}^{0}} 〉 = \frac{1}{2 p_{K}} 〈 ρ K^{0} ∣ T ∣ \bar{B_{s}^{0}} 〉, \end{array}

(33.12)

and

λ_{B_{s}^{0} \to ρ K_{S}} = - \frac{q_{K}}{p_{K}} \frac{q_{B_{s}}}{p_{B_{s}}} \frac{〈 ρ K^{0} ∣ T ∣ \bar{B_{s}^{0}} 〉}{〈 ρ \bar{K^{0}} ∣ T ∣ B_{s}^{0} 〉} .

(33.13)

Therefore, when $B_{s}^{0}$ decays into a $K_{S}$ ⁠, or $K_{L}$ ⁠, we introduce an extra $- q_{K} / p_{K}$ ⁠, or $+ q_{K} / p_{K}$ ⁠, respectively. Assuming that the decay is dominated by the current–current operators, we find

\begin{matrix} \frac{〈 ρ K^{0} | T | \bar{B_{s}^{0}} 〉}{〈 ρ \bar{K^{0}} | T | B_{s}^{0} 〉} = \frac{V_{u b} V_{u d}^{*} 〈 ρ K^{0} | {(\bar{u} b)}_{V - A} {(\bar{d} u)}_{V - A} | \bar{B_{s}^{0}} 〉}{V_{u b}^{*} V_{u d} 〈 ρ \bar{K^{0}} | {(\bar{b} u)}_{V - A} {(\bar{u} d)}_{V - A} | B_{s}^{0} 〉} \\ = - e^{i (ξ_{b} - ξ_{d} - ξ_{B_{s}} - ξ_{K})} \frac{V_{u b} V_{u d}^{*}}{V_{u b}^{*} V_{u d}} . \end{matrix}

(33.14)

We thus arrive at

\begin{array}{l} λ_{B_{s}^{0} \to ρ K_{S}} = sign (B_{B_{s}}) \frac{V_{cs}^{*} V_{cd}}{V_{cs} V_{cd}^{*}} e^{- 2 i ϵ^{'}} \frac{V_{tb}^{*} V_{ts}}{V_{tb} V_{ts}^{*}} \frac{V_{ub} V_{ud}^{*}}{V_{ub}^{*} V_{ud}} \\ = sign (B_{B_{s}}) e^{- 2 i (γ - ϵ + ϵ^{'})} . \end{array}

(33.15)

As $ϵ$ and $ϵ^{'}$ are very small, the CP-violating rate asymmetry in $B_{s}^{0} \to ρ K_{S}$ would measure the angle $γ$ ⁠, if the decay were dominated by the tree-level diagram. Similarly,

λ_{B_{s}^{0} \to ρ K_{L}} = - λ_{B_{s}^{0} \to ρ K_{S}} = - sign (B_{B_{s}}) e^{- 2 i (γ - ϵ + ϵ^{'})} .

(33.16)

The examples above illustrate two important features of the determination of CP-violating asymmetries in a given model. Firstly, when the calculations are careful and consistent, the spurious phases in the CP transformation cancel out amongst the various factors. Secondly, the value of $λ_{f}$ can, in some cases at least, be predicted. Both its magnitude and its sign are determined by the model.⁸⁸¹

33.1.4 Consequences of single-phase dominance

Let us assume an idealized situation in which the decays $\bar{b} \to \bar{α} α \bar{s}$ and $\bar{b} \to \bar{α} α \bar{d}$ (⁠ $α$ is an up-type quark) are dominated by tree-level diagrams, while the decays $\bar{b} \to \bar{s} s \bar{k}$ (k is a down-type quark) are dominated by the gluonic penguin with intermediate top quark.⁸⁹

Following Soares (unpublished thesis, 1993), we define $\tilde{a}$ by

η_{f} \tilde{a} = a^{int} = \frac{2 Im λ_{f}}{1 + {∣ λ_{f} ∣}^{2}} .

(33.17)

We may table the predictions of the SM for $\tilde{a}$ without specifying the CP eigen-value· $η_{f}$ ⁠. Since the CP-transformation phases cancel out in $λ_{f}$ ⁠, we may drop them altogether. We find that the correct results may be reproduced with⁹⁰

\tilde{a} = sin 2 (ϕ_{diagram} - ϕ_{K} - ϕ_{M}),

(33.18)

where

ϕ_{diagram} = {\begin{array}{l} \arg (V_{α b}^{*} V_{α k}) & for decays \bar{b} \to \bar{α} α \bar{k}, \\ \arg (V_{t b}^{*} V_{t k}) & for decays \bar{b} \to \bar{k} s \bar{s}; \end{array}

(33.19)

ϕ_{K} = {\begin{array}{l} \arg (V_{c s} V_{c d}^{*}) + ϵ^{'} & for an unpaired K_{S, L} in B_{d}^{0} decays, \\ \arg (V_{c s}^{*} V_{c d}) - ϵ^{'} & for an unpaired K_{S, L} in B_{s}^{0} decays, \\ 0 & otherwise; \end{array}

(33.20)

ϕ_{M} = {\begin{array}{l} \arg (V_{t b}^{*} V_{t d}) & for B_{d}^{0} decays, \\ \arg (V_{t b}^{*} V_{t s}) & for B_{s}^{0} decays . \end{array}

(33.21)

Equation (33.18) should be compared with eqn (28.14), where $ϕ_{A} = ϕ_{diagram} - ϕ_{K}$ ⁠. Notice that, from now on, we assume the ‘bag parameters’ $B_{B_{q}}$ to be positive. As we have seen in this section, all $λ_{f}$ change sign when the bag parameters change sign, and therefore the SM predictions for the CP asymmetries also change sign in that case (Grossman et al. 1997b).

33.2 New physics in the mixing and the parameters $λ_{f}$

The four phases $β, γ, ϵ$ ⁠, and $ϵ^{'}$ determine the phase structure of the $3 \times 3$ submatrix of an extended CKM matrix. Assuming that the decays are dominated by SM diagrams, all new-physics effects appear in the mixing of neutral mesons, and in deviations of those four angles from their allowed ranges in the SM. New-physics effects in the mixing will in general involve new phases. We have introduced phases $θ_{d}$ and $θ_{s}$ —see eqn (30.32)—measuring the deviations of the phases of $q_{B_{d}} / p_{B_{d}}$ and $q_{B_{s}} / p_{B_{s}}$ ⁠, respectively, from their SM values. As a consequence, eqn (33.21) gets substituted by

ϕ_{M} = {\begin{array}{l} \arg (V_{t b}^{*} V_{t d}) + θ_{d} & for B_{d}^{0} decays, \\ \arg (V_{t b}^{*} V_{t s}) + θ_{s} & for B_{s}^{0} decays . \end{array}

(33.22)

The resulting asymmetries in $B_{d}^{0}$ decays are listed in Table 33.1. Analogously, the $\tilde{a}$ parameters for $B_{s}^{0}$ decays are given in Table 33.2 (Silva and Wolfenstein 1997).

Table 33.1

Open in new tab

CP-violating asymmetries in

B_{d}^{0}

decays. It is assumed that the decay is dominated by a single weak phase and that the only manifestation of new physics is in

B_{d}^{0} - \bar{B_{d}^{0}}

mixing.

Quark process	Sample decay mode	$\tilde{a}$
$\bar{b} \to \bar{c} c \bar{s}$	$J / ψ K_{S}$	$sin 2 (β - ϵ^{'} - θ_{d})$
$\bar{b} \to \bar{c} c \bar{d}$	$D^{+} D^{-}$	$sin 2 (β - θ_{d})$
$\bar{b} \to \bar{u} u \bar{d}$	$π^{+} π^{-}$	$- sin 2 (α + θ_{d})$
$\bar{b} \to \bar{s} s \bar{s}$	$ϕ K_{S}$	$sin 2 (β + ϵ - ϵ^{'} - θ_{d})$

Quark process	Sample decay mode	$\tilde{a}$
$\bar{b} \to \bar{c} c \bar{s}$	$J / ψ K_{S}$	$sin 2 (β - ϵ^{'} - θ_{d})$
$\bar{b} \to \bar{c} c \bar{d}$	$D^{+} D^{-}$	$sin 2 (β - θ_{d})$
$\bar{b} \to \bar{u} u \bar{d}$	$π^{+} π^{-}$	$- sin 2 (α + θ_{d})$
$\bar{b} \to \bar{s} s \bar{s}$	$ϕ K_{S}$	$sin 2 (β + ϵ - ϵ^{'} - θ_{d})$

Table 33.2

Open in new tab

The same as Table 33.1, but for

B_{s}^{0}

decays.

Quark process	Sample decay mode	$\tilde{a}$
$\bar{b} \to \bar{c} c \bar{s}$	$D_{s}^{+} D_{s}^{-}$	$- sin 2 (ϵ + θ_{s})$
$\bar{b} \to \bar{c} c \bar{d}$	$J / ψ K_{S}$	$- sin 2 (ϵ - ϵ^{'} + θ_{s})$
$\bar{b} \to \bar{u} u \bar{d}$	$ρ K_{S}$	$sin 2 (γ - ϵ + ϵ^{'} - θ_{s})$
$\bar{b} \to \bar{s} s \bar{s}$	$η^{'} η^{'}$	$- sin 2 θ_{s}$

Quark process	Sample decay mode	$\tilde{a}$
$\bar{b} \to \bar{c} c \bar{s}$	$D_{s}^{+} D_{s}^{-}$	$- sin 2 (ϵ + θ_{s})$
$\bar{b} \to \bar{c} c \bar{d}$	$J / ψ K_{S}$	$- sin 2 (ϵ - ϵ^{'} + θ_{s})$
$\bar{b} \to \bar{u} u \bar{d}$	$ρ K_{S}$	$sin 2 (γ - ϵ + ϵ^{'} - θ_{s})$
$\bar{b} \to \bar{s} s \bar{s}$	$η^{'} η^{'}$	$- sin 2 θ_{s}$

Recalling that $α = π - β - γ$ ⁠, we see that $θ_{d}$ always appears in the combination $β - θ_{d}$ ⁠. Thus, we need some further input to disentangle $β$ from $θ_{d}$ ⁠. Clearly, $ϵ$ can be extracted independently of $β$ and $γ$ ⁠. Now, using the unitarity of the $3 \times 3$ CKM matrix, Aleksan et al. (1994) have proved that

sin ϵ \approx {∣ \frac{V_{us}}{V_{ud}} ∣}^{2} \frac{sin β sin γ}{sin (β + γ)} .

(33.23)

The power of this relation lies in the fact that the ratio $∣ V_{us} / V_{ud} ∣$ is known to high precision. If there is no violation of unitarity, we may use this to disentangle $β$ from $θ_{d}$ ⁠. However, it is rather difficult to distinguish $θ_{d}$ from non-unitarity of the CKM matrix in a completely model-independent fashion (Silva and Wolfenstein 1997).

Notice that the phase $ϵ^{'}$ always appears in connection with the formula for $q_{K} / p_{K}$ (Cohen et al. 1997). But, as discussed in § 28.5.2, its inclusion is irrelevant for most realistic models.

We have not included the decays $\bar{b} \to \bar{s} s \bar{d}$ in our tables because, as we know from Chapter 31, these are likely to be affected by penguin diagrams with virtual charm and up quarks. The decays $\bar{b} \to \bar{s} u \bar{u}$ are also not shown. These have a large penguin contribution and might even be dominated by penguin amplitudes—in which case, they will effectively measure the weak phase due to the gluonic penguin diagrams with intermediate top quark. On the other hand, we did include in the tables the decays $\bar{b} \to \bar{c} c \bar{d}$ and $\bar{b} \to \bar{u} u \bar{d}$ ⁠, and used only the phase from the dominant amplitude. These decays are expected to be affected by gluonic penguins but, to the extent that $p_{t} / t$ is suppressed, the penguin amplitude will not be larger than the tree-level one. The case is worse for those decays originating in colour-suppressed tree-level diagrams, such as $B_{s}^{0} \to ρ K_{S}$ ⁠. The effect of a second phase, coming from subdominant diagrams, will be analysed in detail in § 33.4.

33.3 $α + β + γ = π$

In our framework the relation $α + β + γ = π (mod 2 π)$ is true by definition. Some authors refer to ‘tests’ of this relation, because they adopt a different definition for $α, β$ ⁠, and $γ$ ⁠. Their definition is the following: $α$ is what one measures in $B_{d}^{0} \to π^{+} π^{-}, β$ is what one measures in $B_{d}^{0} \to J / ψ K_{S},$ and $γ$ is what one measures in $B_{s}^{0} \to ρ K_{S} \cdot$ ⁹¹ In our language,

\begin{array}{l} α_{B_{d}^{0} \to π^{+} π^{-}} = α + θ_{d}, \\ β_{B_{d}^{0} \to J / ψ K_{S}} = β - ϵ^{'} - θ_{d}, \\ γ_{B_{s}^{0} \to ρ K_{S}} = γ - ϵ + ϵ^{'} - θ_{s} . \end{array}

(33.24)

Those authors want to test whether the sum

Σ \equiv α_{B_{d}^{0} \to π^{+} π^{-}} + β_{B_{d}^{0} \to J / ψ K_{S}} + γ_{B_{s}^{0} \to ρ K_{S}}

(33.25)

is equal to $π$ ⁠.

As one gathers from eqns (33.24), in models with new physics in the mixing one has

\begin{array}{l} Σ = α + β + γ - ϵ - θ_{s} \\ = π - (ϵ + θ_{s}) . \end{array}

(33.26)

Nir and Silverman (1990) have pointed out that a much simpler and equivalent test would be to measure directly the phase $ϵ_{B_{s}^{0} \to D_{s}^{+} D_{s}^{-}}$ in the decay asymmetry $B_{s}^{0} \to D_{s}^{+} D_{s}^{-}$ which is

ϵ_{B_{s}^{0} \to D_{s}^{+} D_{s}^{-}} = ϵ + θ_{s},

(33.27)

as seen in Table 33.2.

In the SM, $θ_{s} = 0$ and $ϵ \sim λ^{2}$ ⁠. Therefore, $Σ = π$ should be satisfied to excelent accuracy. If one finds that $Σ$ differs from $π$ by a large amount, then there is new physics in the mixing $(θ_{s} \neq 0)$ and/or $ϵ$ must be larger than in the SM, thus signalling non-unitarity of the CKM matrix.

33.4 Corrections induced by the subleading amplitudes

33.4.1 General discussion

We now use the diagrammatic analysis of Chapter 31 to try and get an estimate of the corrections imposed by subleading diagrams. The purpose of this section is to find out whether the results in Tables 33.1 and 33.2 are really meaningful, or whether the subleading amplitudes are likely to distort the picture conveyed in those tables.

When one goes beyond the approximation of single-phase dominance there is some arbitrariness in the choice of the second phase. For decays which have a tree-level contribution, it is customary to write the decay amplitude in terms of the phase of the tree-level diagram and of the phase of the penguin diagram with a virtual top quark running in the loop, as we have done in § 31.6. However, due to the unitarity of the CKM matrix, the decay amplitudes may be written in terms of any pair of parameters $λ_{α}$ ⁠. This arbitrariness may lead into erroneous estimates of the corrections introduced by the subleading diagram.

As an example, take the decay $\bar{b} \to \bar{s} c \bar{c}$ ⁠. Its amplitude may be written as

A_{\bar{b} \to \bar{s} c \bar{c}} = V_{cb}^{*} V_{cs} t + V_{tb}^{*} V_{ts} p_{t},

(33.28)

where t and $p_{t}$ are essentially the tree-level amplitude and the top-quark gluonic-penguin amplitude,⁹² respectively, with the CKM factors explicitly factored out. Since $∣ V_{cb} V_{cs} ∣ \approx ∣ V_{tb} V_{ts} ∣$ ⁠, we might be tempted to say that the correction introduced by the gluonic penguin is $\sim ∣ p_{t} / t ∣$ ⁠. On the other hand, we may rewrite eqn (33.28) as

A_{\bar{b} \to \bar{s} c \bar{c}} = V_{cb}^{*} V_{cs} (t - p_{t}) - V_{ub}^{*} V_{us} p_{t} .

(33.29)

In this case, we would estimate the correction to be of order

∣ \frac{V_{ub} V_{us} p_{t}}{V_{cb} V_{cs} (t - p_{t})} ∣ \sim λ^{2} ∣ \frac{p_{t}}{t} ∣ .

(33.30)

One thus obtains two very different estimates for the same quantity. Which of them is correct?

One must recall that the quantity that one wants to compute is $\tilde{a}$ ⁠. In the presence of only one weak phase $ϕ_{1}$ one has $\tilde{a} = sin 2 ϕ_{1}$ ⁠. But, in the presence of two contributions with different weak phases, one gets from eqn (28.18),

\tilde{a} \approx sin 2 ϕ_{1} - 2 r sin (ϕ_{1} - ϕ_{2}) cos 2 ϕ_{1} cos Δ,

(33.31)

when the ratio of amplitudes, r, is small. Thus, the important quantity is not r itself, but rather the product $r sin (ϕ_{1} - ϕ_{2})$ ⁠. For instance, in eqn (33.28) both terms have the same phase but for a correction $ϵ$ ⁠, and therefore $sin (ϕ_{1} - ϕ_{2}) = sin ϵ \sim λ^{2}$ ⁠. This should be multiplied by $r \sim ∣ p_{t} / t ∣$ ⁠. Similarly, in eqn (33.29) the two terms have phase difference $γ$ ⁠, which is large and does not introduce any further suppression. But, this appears multiplied by $r \sim λ^{2} ∣ p_{t} / t ∣$ ⁠. If one takes this into account, evaluates the matrix elements involved, and does not do any undue approximations, the results obtained using eqn (33.28) or using eqn (33.29) are the same, as they must be.

It should be stressed that the same $r sin (ϕ_{1} - ϕ_{2})$ combination shows up in the direct-CP-violating term $a^{dir}$ of eqn (28.17). Indeed,

a^{dir} \approx - 2 r sin (ϕ_{1} - ϕ_{2}) sin Δ .

(33.32)

It will be convenient to define the deviation that the $a^{int}$ suffers due to the presence of a second weak phase:

\begin{array}{l} Δ \tilde{a} \equiv \tilde{a} - sin 2 ϕ_{1} \\ \approx - 2 r sin (ϕ_{1} - ϕ_{2}) cos 2 ϕ_{1} cos Δ . \end{array}

(33.33)

We have listed in Tables 31.3–31.7 the various types of decays into CP eigenstates. We shall now go through those tables, use the phase convention for the CKM matrix elements introduced in eqn (28.24), and list the values of $ϕ_{1}, ϕ_{2}$ ⁠, and r for some decays. We shall build upon work done by Grossman and Worah (1997).

33.4.2 Decays with a tree-level contribution

•

$\bar{b} \to \bar{c} c \bar{s}$ ⁠. Examples: $B_{s}^{0} \to D_{s}^{+} D_{s}^{-}$ and $B_{s}^{0} \overset{c}{\to} J / ψϕ$ (both with $ϕ_{1} = - ϵ + π),$ ⁹³ and $B_{d}^{0} \overset{c}{\to} J / ψ K_{S}$ (with $ϕ_{1} = β - ϵ^{'} + π$ ⁠). The symbol $\overset{c}{\to}$ means that these decays proceed at tree level through colour-suppressed diagrams, if we neglect final-state-interaction rescattering effects. Since the leading contribution has CKM factor $V_{cb}^{*} V_{cs}$ and the main gluonic-penguin amplitude has weak phase $V_{tb}^{*} V_{ts}$ ⁠, the correction $Δ \tilde{a} \propto r sin (ϕ_{1} - ϕ_{2})$ is of order

∣ \frac{p}{t} ∣ sin ϵ or ∣ \frac{p}{c} ∣ sin ϵ .

(33.34)

The second value holds for decays proceeding at tree level through colour-suppressed diagrams. As $ϵ \sim λ^{2}$ ⁠, the corrections are rather small, and therefore these decays should be dominated by a single weak phase.

•

$\bar{b} \to \bar{c} c \bar{d}$ ⁠. Examples: $B_{d}^{0} \to D^{+} D^{-}$ and $B_{d}^{0} \overset{c}{\to} J / ψ π^{0}$ (both with $ϕ_{1} = β + π$ ⁠), and $B_{s}^{0} \overset{c}{\to} J / ψ K_{S}$ (with $ϕ_{1} = - ϵ + ϵ^{'} + π$ ⁠). In this case, $Δ \tilde{a}$ is estimated to be

∣ \frac{p}{t} ∣ R_{t} sin β or ∣ \frac{p}{c} ∣ R_{t} sin β .

(33.35)

Since $β$ is unsuppressed, the gluonic-penguin contribution is not negligible. If one trusts the usual $λ^{2}$ estimate for $∣ p / t ∣$ ⁠, the effect in $B_{d}^{0} \to D^{+} D^{-}$ is smaller than 5%, but it may well turn out to be larger, should $∣ p / t ∣$ be of order $λ$ ⁠. The situation is worse in colour-suppressed decays, since there the tree-level diagrams are smaller.

•

$\bar{b} \to \bar{u} u \bar{d}$ ⁠. Examples: $B_{d} \to π^{+} π^{-}$ and $B_{d} \overset{c}{\to} π^{0} π^{0}$ (both with $ϕ_{1} = - α + π$ ⁠), and $B_{s}^{0} \overset{c}{\to} ρ K_{S}$ (with $ϕ_{1} = γ - ϵ + ϵ^{'}$ ⁠). Now $Δ \tilde{a}$ is proportional to

∣ \frac{p}{t} ∣ \frac{R_{t}}{R_{b}} sin α or ∣ \frac{p}{c} ∣ \frac{R_{t}}{R_{b}} sin α .

(33.36)

This is worse than the previous case because $R_{b}$ is smaller than unity. As a consequence, the measurement of $α$ in the process $B_{d} \to π^{+} π^{-}$ suffers from large uncertainties. The extraction of $γ$ from the decay $B_{s}^{0} \overset{c}{\to} ρ K_{S}$ is even worse, due to the colour suppression of the tree-level amplitude.

•

$\bar{b} \to \bar{u} u \bar{s}$ ⁠. Examples: $B_{s}^{0} \to K^{+} K^{-}$ ⁠, with $ϕ_{1} = γ - ϵ + ϵ^{'} + π$ ⁠, and $B_{d}^{0} \overset{c}{\to} K_{S} π^{0}$ ⁠, with $ϕ_{1} = - α$ ⁠. Here $Δ \tilde{a}$ should be of order

∣ \frac{p}{t} ∣ \frac{1}{λ^{2} R_{b}} sin γ or ∣ \frac{p}{c} ∣ \frac{1}{λ^{2} R_{b}} sin γ .

(33.37)

Here, the suppression of the tree-level diagrams is so effective, that it is probably better to think of these decays as measuring the weak phase of the top-mediated gluonic penguins, with the tree-level diagrams acting as the pollutant (Ciuchini et al. 1997b). Then, the CP-violating phase in the decay $B_{s}^{0} \to K^{+} K^{-}$ would be close to $ϕ_{2} = 0$ ⁠, while $B_{d}^{0} \overset{c}{\to} K_{S} π^{0}$ would be dominated by the phase $ϕ_{2} = β + ϵ - ϵ^{'}$ ⁠.

There is one exception to this analysis of decays $\bar{b} \to \bar{s} u \bar{u}$ ⁠. The transition $B_{s}^{0} \overset{c}{\to} {ϕπ}^{0}$ only has a colour-suppressed tree-level contribution, with angle $ϕ_{1} = γ - ϵ + ϵ^{'} + π$ ⁠, since it is not affected by gluonic penguins. As explained before, this is due to the fact that, in the gluonic penguin, the $q \bar{q}$ pair arises out of the gluon in an $I = 0$ state, while $π^{0}$ belongs to an isospin triplet. However, this decay is affected by the electroweak penguins, which are even expected to be dominant (Buras and Fleischer 1997).

33.4.3 Pure penguin decays

Naively, one would think that the phase of a pure-penguin decay amplitude would be equal that of the penguin diagram with intermediate top quark. However, the subleading penguin diagrams with intermediate up and charm quarks may be relevant. The importance of the subleading penguin effects in pure penguin decays was first pointed out by Gérard and Hou (1991a,b) and by Simma etal. (1991), in the context of direct CP violation. These subleading effects also generate a correction of the interference CP violation term.

•

$\bar{b} \to \bar{s} d \bar{d}$ ⁠. Example: $B_{s}^{0} \to K^{0} \bar{K^{0}}$ ⁠. This decay cannot proceed through a tree-level diagram. For an intermediate top quark in the gluonic penguin we find $ϕ_{1} = 0$ ⁠. This gets a correction proportional to $∣ p_{c} / p_{t} ∣ sin ϵ$ from the charm-quark gluonic penguin. Since this is very small, the decay $B_{s}^{0} \to K^{0} \bar{K^{0}}$ is often proposed as ideal to look for new physics.

•

$\bar{b} \to \bar{s} s \bar{s}$ ⁠. Examples: $B_{d}^{0} \to ϕ K_{S}$ ⁠, with $ϕ_{1} = β + ϵ - ϵ^{'}$ ⁠, and $B_{s}^{0} \to η^{'} η^{'}$ ⁠, with $ϕ_{1} = 0$ ⁠. The gluonic penguin with an intermediate top quark has CKM factor $V_{tb}^{*} V_{ts}$ ⁠; the gluonic penguin with an intermediate charm quark gives a correction of order $∣ p_{c} / p_{t} ∣ sin ϵ$ ⁠. For $B_{d} \to ϕ K_{S}$ ⁠, this correction must be added to a second one which is due to the small $u \bar{u}$ and $d \bar{d}$ components of $ϕ$ ⁠. As a consequence, that decay may also proceed through the tree-level diagrams $\bar{b} \to \bar{u} u \bar{s}$ ⁠. Grossman and Worah (1997) have estimated the second uncertainty to be of order 1%, leading to a combined correction of order 4%.

•

$\bar{b} \to \bar{d} s \bar{s}$ ⁠: Examples: $B_{d}^{0} \to K^{0} \bar{K^{0}}$ and $B_{d}^{0} \to {ϕπ}^{0}$ (both with $ϕ_{1} = 0$ ⁠), and $B_{s}^{0} \to ϕ K_{S}$ (with $ϕ_{1} = - β - ϵ + ϵ^{'}$ ⁠). Here, the correction is proportional to $∣ p_{c} / p_{t} ∣ sin β / R_{t}$ ⁠. Fleischer (1994c) has estimated that the corrections could lead to a CP-violating asymmetry as large as 50%. Particularly interesting is the decay $B_{d}^{0} \to {ϕπ}^{0}$ ⁠, which proceeds via a singlet penguin. This gets a small contribution from the $\bar{b} \to \bar{u} u \bar{d}$ tree-level diagram, due to the small $u \bar{u}$ component of $ϕ$ ⁠. However, this decay is likely to be affected by rescattering effects.

33.5 $B_{d}^{0} \to J / ψ K_{S}$ is the gold-plated decay

We have concentrated our efforts in trying to identify decay channels which are dominated, in the SM, by a single weak phase. In such cases one can extract that CKM phase, unless there are new-physics contributions either to the mixing or to the decay amplitudes. Unfortunately, only a few cases satisfy these conditions. We have come in this chapter to the following conclusions (Grossman and Worah 1997). The quark decay $\bar{b} \to \bar{c} c \bar{s}$ should provide clean measurements of CKM phases. The decay $\bar{b} \to \bar{s} s \bar{s}$ should have only small uncertainties from other SM contributions. We will show in § 34.5, in which we discuss the decays with $Δ C \neq 0 \neq Δ U$ ⁠, that this is also the case for the decay $\bar{b} \to \bar{c} u \bar{d}$ ⁠. Most other decays are likely to suffer from pollution due to a subdominant amplitude with a different weak phase.

Therefore, it appears that experimentalists should devote special attention to decays which at quark level are $\bar{b} \to \bar{c} c \bar{s}$ ⁠. These include $B_{d}^{0} \to J / ψ K_{S}$ ⁠, which measures $sin 2 (β - ϵ^{'} - θ_{d})$ ⁠, and $B_{s}^{0} \to D_{s}^{+} D_{s}^{-}$ ⁠, which measures $sin 2 (ϵ + θ_{d})$ ⁠. In the SM, $θ_{d} = 0$ and $sin 2 β$ is positive, see eqn (18.30). On the other hand, $ϵ \sim λ^{2}$ is very small, and therefore the CP asymmetry in $B_{s}^{0} \to D_{s}^{+} D_{s}^{-}$ is predicted to be small.

The decay $B_{d}^{0} \to J / ψ K_{S}$ has other advantages. Machines working at the $ϒ (4 S)$ provide a very clean source of mesons $B_{d}^{0}$ and $\bar{B_{d}^{0}}$ ⁠; this is to be contrasted with the situation for mesons $B_{s}^{0}$ and $\bar{B_{s}^{0}}$ which, in the near future, will only be produced at hadron machines, where the background constitutes a severe challenge. Moreover, both the $J / ψ$ and the $K_{S}$ are easy to detect, the $J / ψ$ through its decay into two muons, with a branching ratio $\sim 6 %$ (the decay into two electrons has the same branching ratio), and the $K_{S}$ through its decay into $π^{+} π^{-}$ ⁠; the final state is then composed entirely of charged particles which are easy to detect. For these reasons, $B_{d}^{0} \to J / ψ K_{S}$ has been termed the gold-plated decay: it should provide a very clean measurement of $sin 2 β$ in the SM. This determines $β$ up to a fourfold ambiguity.

The existence of two different amplitudes with different weak phases brings into play hadronic uncertainties, due both to uncancelled operator matrix elements and to the presence of unknown final-state-interaction CP-even phases. In Chapters 35, 36, and 37 we shall discuss methods that can be applied to some of these more challenging cases. The general idea is that one may be able to relate several decay channels in order to circumvent the hadronic uncertainties and extract the weak phases.

Notes

⁸⁸

Strictly speaking, it is only after we have chosen by convention $Δ m > 0$ that the sign of $λ_{f}$ acquires physical significance. The convention $Δ m > 0$ is implicit in our computation of q/p for all neutral-meson systems.

⁸⁹

CP-eigenstate final states arising from the $Δ U \neq 0 \neq Δ C$ decays of the type $\bar{b} \to \bar{α} β \bar{k}$ with $α \neq β$ must involve a state $D_{f_{cp}}$ in the final state and can be reached by two distinct tree-level diagrams. We shall treat these channels after we discuss cascade decays in the next chapter.

⁹⁰

This is equivalent to writing $η_{f} λ_{f} = - exp [- 2 i (ϕ_{diagram} - ϕ_{K} - ϕ_{M})]$ ⁠. But notice that, depending on the phase convention used for the CKM matrix elements, the $ϕ_{K}$ and $ϕ_{M}$ defined in eqns (33.20) and (33.21) may differ by $π$ from the similar parameters defined in § 28.3 through $q / p = - e^{2 i ϕ_{M}}$ ⁠. Naturally, working consistently with either definition leads to the same results.

⁹¹

It is assumed that the penguin pollution in $B_{d}^{0} \to π^{+} π^{-}$ and in $B_{s}^{0} \to ρ K_{S}$ has been removed by some method.

⁹²

In all rigour, we should use $t + p_{c} - p_{u}$ instead of t, and $p_{t} - p_{u}$ instead of $p_{t}$ ⁠. We implicitly assume that t dominates the first term, while the $m_{t}$ -enhanced portion of the gluonic penguin dominates the second term.

⁹³

We include here phases of $π$ ⁠, which are relevant for the caculation of $ϕ_{1} - ϕ_{2}$ ⁠.

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CP Violation. G. C. Branco, L. Lavoura, and J. P. Silva, Oxford University Press. © G. C. Branco, L. Lavoura, and J. P. Silva 2024. DOI: 10.1093/oso/9780198503996.003.0033

This is an open access publication, available online and distributed under the terms of a Creative Commons Attribution-Non Commercial-No Derivatives 4.0 International licence (CC BY-NC-ND 4.0), a copy of which is available at https://creativecommons.org/licenses/by-nc-nd/4.0/. Subject to this license, all rights are reserved.

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Contents

33 CKM Phases and Interference CP Violation

Abstract

33.1 Parameters $λ_{f}$ at tree level

33.1.1 $B_{d}^{0} \to π^{+} π^{-}$

33.1.2 $B_{d}^{0} \to J / ψ K_{S}$

33.1.3 $B_{s}^{0} \to ρ K_{S}$

33.1.4 Consequences of single-phase dominance

33.2 New physics in the mixing and the parameters $λ_{f}$

33.3 $α + β + γ = π$

33.4 Corrections induced by the subleading amplitudes

33.4.1 General discussion

33.4.2 Decays with a tree-level contribution

33.4.3 Pure penguin decays

33.5 $B_{d}^{0} \to J / ψ K_{S}$ is the gold-plated decay

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Contents

33 CKM Phases and Interference CP Violation

Abstract

33.1 Parameters λf at tree level

33.1.1 Bd0→π+π−

33.1.2 Bd0→J/ψKS

33.1.3 Bs0→ρKS

33.1.4 Consequences of single-phase dominance

33.2 New physics in the mixing and the parameters λf

33.3 α+β+γ=π

33.4 Corrections induced by the subleading amplitudes

33.4.1 General discussion

33.4.2 Decays with a tree-level contribution

33.4.3 Pure penguin decays

33.5 Bd0→J/ψKS is the gold-plated decay

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33.1 Parameters $λ_{f}$ at tree level

33.1.1 $B_{d}^{0} \to π^{+} π^{-}$

33.1.2 $B_{d}^{0} \to J / ψ K_{S}$

33.1.3 $B_{s}^{0} \to ρ K_{S}$

33.2 New physics in the mixing and the parameters $λ_{f}$

33.3 $α + β + γ = π$

33.5 $B_{d}^{0} \to J / ψ K_{S}$ is the gold-plated decay