34 Cascade Decays

In this chapter we discuss decay chains with a neutral-meson system in an intermediate state. These are known as cascade decays. We want to study cases where the decays of B mesons proceed through intermediate states containing neutral kaons or neutral D mesons. Examples are $Bd→J/ψK→J/ψfK$ and $Bd→π0D→π0fD$⁠. In this notation, $fK$ designates a set of particles which one identifies experimentally, in particular through their invariant mass, as coming from the decay of a neutral kaon; and, similarly, $fD$ is a set of particles originating in the decay of either $D0$ or $D0¯$⁠.

A correct understanding of cascade decays is important because the CP violation present in the intermediate-meson system will in general show up in the calculation of CP-violating observables for the overall decay chain. In particular, in a cascade decay in which a neutral-meson system decays into a lighter neutral-meson system, it may be possible to use the CP properties of the latter as an analyser for the CP properties of the former. This is now fully appreciated, following work by Azimov (1989, 1990) and by Kayser and Stodolsky (1996).

We shall discuss decay chains with intermediate neutral kaons and neutral D-mesons separately. We start with a pedagogical discussion of the $Bd→J/ψK→J/ψfK$ decay chain. This case is simplified by the fact that the decays $Bd0→K0¯$ and $Bd0¯→K0$ are forbidden (Azimov 1989, 1990; Dass and Sarma 1992; Kayser and Stodolsky 1996; Azimov et al. 1997; Kayser 1997). A similar situation occurs with the $Bs→J/ψK→J/ψfK$ decays considered by Azimov and Dunietz (1997). Next, we discuss the $B→XD→XfD$ decay chains, under the assumption that the mixing in the $D0−D0¯$ system is negligible. Finally, we include an elementary introduction to those decays in which the initial state can decay into both flavour eigenstates of the neutral-meson system in the intermediate state. These cases have been introduced by Meca and Silva (1998) and by Amorim et al. (1999). Examples include $D+→π+K→π+fK$⁠, and also, if we allow for sizeable new-physics contributions to $D0−D0¯$ mixing, $B+→K+D→K+fD$⁠. In these cases, there is CP violation in the interference between the decays from the initial state and the mixing in the intermediate neutral-meson system. This involves new CP-violating parameters, beyond the ones discussed thus far in this book (Meca and Silva 1998; Amorim et al. 1999).

When discussing cascade decays it is important to take into account the specific experimental conditions. We illustrate this point by considering the process $Bd→J/ψK→J/ψ(π+π−)K$⁠, where $(π+π−)K$ refers to two pions with an invariant mass equal to the one of the neutral kaon. Our discussion follows closely the presentation in recent reviews by Kayser—see for example Kayser (1997).⁹⁴

The two kaon masses, $mL$ and $mS$⁠, differ very little and are not resolved in this experiment. Thus, the mass eigenstates of the intermediate neutral kaon may only be resolved through the lifetimes $(τS≪τL)$⁠.

There are two decay times involved. There is the time $tB$ between the production of $Bd$ and its decay into $J/ψK$⁠, and the time $tK$ between this event and the decay of the neutral kaon into two pions. The time interval $tB$ is measured in the rest frame of $Bd$⁠, while $tK$ is measured in the rest frame of the neutral kaons.

This decay is represented in Fig. 34.1. There are four paths leading from $Bd0$ to the final state. By means of a suitable choice of the two decay times, the interference among these four paths can be studied in order to determine different combinations of CKM phases (Azimov 1989, 1990; Kayser and Stodolsky 1996).

The $π+π−$ pair is CP-even. If CP was conserved, then the mass eigenstates would coincide with the CP eigenstates in both neutral-meson systems. We would then have $CP(BH)=CP(J/ψKS)=CP(J/ψπ+π−)=−1$⁠, and $CP(BL)=CP(J/ψKL)=+1.$⁹⁵ Then, the decays represented by solid lines conserve CP, while the decays represented by dashed lines violate CP.

Most analyses in the literature neglect CP violation in the kaon decays. Then, $KL$ cannot decay into two pions and there are only two paths; $BH→J/ψKS→J/ψ(π+π−)K$ and $BL→J/ψKS→J/ψ(π+π−)K$⁠, which interfere. Both $Bd0$ and $Bd0¯$ have $BH$ and $BL$ components, so both contain interference terms. The interference involves the parameter $λBd0→J/ψKS$ computed in § 33.1.2. Neglecting $ϵ′$ and using eqns (28.6), we then have

These equations display a term proportional to $sin2β$⁠, which arises from the interference between the two competing paths.

However, this cannot be the whole picture for the decay chain $Bd0→J/ψK→J/ψ(π+π−)K$⁠. In fact, if $tK$ is much larger than the lifetime of $KS$⁠, the $KS$ component will have completely decayed away, and the two pions must necessarily have come from a $KL$⁠. (Of course, the rate will be exceedingly small, but this is a useful thought experiment.) For this case, we combine eqns (28.6) and (33.11) to get

Again, there is an interference term proportional to $sin2β$⁠, but with the opposite sign. The reason is that $λBd0→J/ψKL=−λBd0→J/ψKS$⁠.

It is true that $A(KS→π+π−)$ is three orders of magnitude larger than $A(KL→π+π−)$⁠. But, as this thought experiment illustrates, when discussing observables concerning cascade decays, one must take into account the specific experimental conditions. The experiments on $B→J/ψK→J/ψ(π+π−)K$ will be looking at small times $tK$⁠. Therefore, the $J/ψ(π+π−)K$ events detected are over-whelmingly due to $J/ψKS$⁠, due both to the huge ratio $A(KS→π+π−)/(KL→π+π−)$ and to the $tK$ interval probed.

In this section we have considered two limiting cases of the decay chain $Bd0→J/ψK→J/ψ(π+π−−)K$⁠: the case of times $tK∼τS$⁠, in which all decays go through $KS$⁠, and the case of times $tK≫τS$⁠, in which all decays go through $KL$⁠. The fact that both these decays can occur is a consequence of CP violation in the kaon sector (which allows both $KS$ and $KL$ to decay into the same CP eigenstate $π+π−$⁠). In the next section we shall develop formulas valid for all times $tK$⁠.

Let us consider a meson $Bd0$ produced at $t=0$⁠. At a later time $tB$⁠, it will be a linear combination of $Bd0$ and $Bd0¯$ given by eqn (9.1):

where

We shall neglect the width difference in the $Bd0−Bd0¯$ system and write

To lowest order in the weak interaction, the decays $Bd0→K0¯$ and $Bd0¯→K0$ are forbidden. Therefore, the decay of $Bd0(tB)$ into $J/ψK$ produces a kaon state given by (Azimov 1989, 1990)

Using eqns (8.6) and the fact that

we find

When $ϵ′$ is neglected one has $λBd0→J/ψKS=exp(−2iβ)$ in the SM. Thus (Kayser 1997)

A similar analysis leads to

By collecting events with a particular $tB$ we may tune the composition of the kaon state, much as we do with a regenerator (Azimov 1989, 1990; Kayser 1997).

One may use our knowledge of the kaon system to learn about the B systems by using cascade decays. Azimov (1989, 1990) has pointed out that, in particular, we may determine whether the heaviest $Bd$ mass eigenstate in mostly CP-even or mostly CP-odd, as well as the sign of $ΔΓ$ for the $Bd0−Bd0¯$ system, because we know this information for the kaon system. This strategy has also been explored by Dass and Sarma (1992), Kayser and Stodolsky (1996), Azimov et al. (1997), and Kayser (1997). A similar technique may be used to measure $Δm$ in the $Bs0−Bs0¯$ system through the cascade decay $Bs0→J/ψK→J/ψfK$⁠, even when $xs≫1$ (Azimov and Dunietz 1997).

Once created, the states $∣KfromBdo(tB)〉$ and $∣KfromBd0¯(tB)〉$ will evolve in time, and decay into the final state $fK$ at time $tK$⁠. After a tedious but straightforward calculation, we find

where $ΓS(ΓL)$ is the width of $KS(KL)$⁠, and we have defined $Γ≡(ΓS+ΓL)/2$ and

as in eqn (8.25).

The behaviour of the expression in eqn (34.11) varies according to whether $∣ηf∣=∣AKL→f/AKS→f∣$ is larger or smaller than 1. If $∣ηf∣≫1$ the decay is dominated by $KL$⁠, for all times $tK$⁠. If $∣ηf∣≪1$ the structure of the decay changes with time. For $tK∼τS$ that expression is dominated by the decays of $KS$ in the first line, and we reproduce eqn (34.1). For $tK≫τS$ any event is due to the decays of $KL$ in the second line, and we reproduce eqn (34.2). At intermediate times one is sensitive to the interference between the decays of $KS$ and of $KL$⁠. Instead of depending on $sin2β$⁠, as do eqns (34.1) and (34.2), the interference term probes $cos2β$⁠. This is interesting, because it may allow us to determine a different trigonometric function of the CP-violating phase in $Bd0→J/ψK$ (Kayser 1997).

Let us consider a CP-allowed decay of $KS$⁠, such as $π+π−$⁠. The interference term becomes relevant when

i.e., for

For $f=π+π−$ we have $∣η+−∣∼2.3×10−3$ and the interference term is important for $tK∼11τS$—see Fig. 8.2. Unfortunately, by then the decay rate is suppressed by a factor $e−11∼10−5$ compared to what it was at $tK=0$⁠. Therefore, the interference should be easiest to detect for decays with $∣ηf∣≈1$⁠, such as the semileptonic decays—see § 34.3.4.

Let us integrate the rate in eqn (34.11) over $tK$⁠. We obtain

where $xK=ΔmK/Γ≈1$⁠. The interference term has a piece proportional to $cos(ΔmBtB)$⁠, which fakes a (very small) direct CP violation in $Bd0→J/ψKS$⁠. The other piece is proportional to $sin(ΔmBtB)$ and to $cos2β$⁠. Its coefficient, $xKcosϕf−sinϕf$⁠, almost vanishes if f is a two-pion state, because $ϕ+−$ and $ϕ00$ are very close to $45°$⁠. The effect of the interference terms becomes even smaller once a cut in $tK$ is imposed in the integration (keeping $tK$ smaller than a few times $τS$⁠) because that helps in identifying the $KS$⁠.

We see from eqns (34.11) and (34.15) that the cascade decay $Bd0→J/ψK→J/ψfK$ may be used to determine $cos2β$ (Kayser 1997). In order to do this we must maximize the interference term. This happens when the decay amplitudes of $KL$ and $KS$ into the final state f are of similar magnitude, so that all terms are comparable at small times, when the exponentials are close to one.

Explicitly, Kayser (1997) has proposed to use the decays $Bd0→J/ψK→J/ψ(πlνl)K$⁠. In this case, $∣AKL→f∣≈∣AKS→f∣$ and the amplitudes factor out. Moreover, this cascade decay has a rate comparable to that of $Bd0→J/ψK→J/ψ(π+π−)K$⁠, because

see eqns (8.8) and (8.9). However, while the $KS$ term dominates in the cascade decay $Bd0→J/ψK→J/ψ(π+π−)K$⁠, the interference term only accounts for approximately

of the cascade decay $Bd0→J/ψK→J/ψ(πlνl)K$ (Kayser 1997).⁹⁶ Therefore, measuring $cos2β$ will require a few hundred times more events $Bd0→J/πK$ than measuring $sin2β$⁠.

Actually, the need for large data samples may not be as acute as it seems. Indeed, once $sin2β$ is measured one only needs to determine the sign of $cos2β$⁠. This measurement will reduce the fourfold ambiguity in $β$ to the twofold ambiguity $β→β+π$ (see Chapter 38).

Consider a decay chain of the type $B→XD→XfD$⁠. In most cases of interest, the state f may be reached from both $D0$ and $D0¯$⁠, and both contributions must be included. A simple algorithm to calculate such decay amplitudes arises from the observation that the linear combination of $D0$ and $D0¯$ that decays into f is

Here, $∣cf∣2+∣c¯f∣2=1$ and

Indeed, this state is orthogonal to

which clearly cannot decay into f.

We frequently use the linear combination of D mesons which decays into a CP eigenstate $fcp,$ designated by $Dfcp$⁠:

In general, the states $Dfcp$ do not coincide with the CP eigenstates $D±$⁠. If the decay of $D0$ into $fcp$ is dominated by a single weak phase, then CP relates the numerator and the denominator in eqn (34.19), there is no direct CP violation in the decay, and $∣cf∣=∣c¯f∣=1/2$⁠.

We shall also be interested in the decays of D mesons into flavour-specific final states. In this case, either $cf$ or $c¯f$ vanish, implying that the decaying meson was a flavour eigenstate, either $D0$ or $D0¯$⁠. Clearly, by suitably choosing the final state f we may pick up different combinations of $D0$ and $D0¯$ in the decay chain $B→XD→XfD$⁠.

The crucial physical input in the usual analysis of cascade decays involving an intermediate neutral-D meson is the following: the mixing parameters x and y for the $D0−D0¯$ system are very small. (In Appendix. E we give both the experimental results which justify this assertion, and the theoretical expectations for those mixing parameters, in the context of the SM.) Because of the smallness of x and y the D mesons suffer almost no oscillation before they decay. This means that the linear combination of $∣D0〉$ and $∣D0¯〉$ created by the decay of the B meson is identical to the linear combination of $∣D0〉$ and $∣D0¯〉$ that later decays. This implies that we do not have to compute the full decay chain, expressed in terms of a time $tB$ and a time $tD$⁠, as we did in the previous section; the $tD$-dependence just factors out as $exp(−ΓDtD)$⁠, where $ΓD$ is the (mean) decay width of $D0$ and $D0¯$⁠.

There are in the literature a number of proposals to determine CKM phases by comparing the decays $B→XD0,B→XD0¯$⁠, and $B→XfD$⁠. Such methods exploit the existence of a linear equation among the decay amplitudes,

Such triangular relations lie at the heart of several methods to determine the angle $γ$⁠. The simplest methods of this type use a CP eigenstate, $f=fcp$⁠. That is, they compare $B→XD0$ and $B→XD0¯$ with $B→XDfcp$⁠. Gronau and London (1991) studied $Bd0→DKS$ and $Bs0→Dϕ$⁠, while Gronau and Wyler (1991) proposed $B±→DK±$⁠. As a result, such proposals are sometimes known collectively as ‘the Gronau–London–Wyler method’.

So far, we have not yet calculated the parameter $λf$ for any $ΔC≠0≠ΔU$ decay. Clearly, all the $ΔC≠0≠ΔU$ decays into CP eigenstates must contain $Dfcp$ in the final state. This final state is identified through its subsequent decay into a CP eigenstate $fcp$ such as $ππ,K+K−$⁠, or $π0KS$⁠. Therefore, that decay enters into the correct calculation of any CP-violation parameter for the overall decay chain. As seen in the previous section, this requires a knowledge of $cf/c¯f$⁠.

We shall assume that the decays of D mesons are dominated by the tree-level diagrams of the SM. If $fcp=π+π−$⁠, then the auxiliary quantity required is

where $Γμ≡γμ(1−γ5)$⁠, and $CP∣D0〉=eiξD∣D0¯〉$⁠. Similarly, if $fcp=K+K−$⁠, we need

Theoretically, the practical difference between using $fcp=π+π−$ and using $fcp=K+K−$ is very small because

and $ϵ′$ is of order $λ4$ in the SM. Many authors neglect $ϵ′$ and use $π+π−$ or $K+K−$ indifferently.

The distinctive feature of these $ΔC≠0≠ΔU$ decays, which we have listed in Table 31.4, is that they cannot proceed through gluonic penguins. Two tree-level diagrams, $b¯→c¯us¯$ and $b¯→u¯cs¯$⁠, contribute almost equally to the decays $Bd0→DfcpKS$ and $Bs0→Dfcpϕ$⁠. These diagrams have comparable magnitudes and different weak phases and, therefore, $λf$ is not a pure phase and we do not measure a CKM phase directly. Fortunately, one may still compare these decays with the corresponding decays in which $Dfcp$ is substituted by either $D0$ or $D0¯$⁠. We will show in § 36.3 how this can be used to extract the CKM phase $γ$⁠.

The situation with the decays $Bd0→Dfcpπ0$ and $Bs0→DfcpKS$ is rather different. These decays get their main contribution from the tree-level diagram $b¯→c¯ud¯$⁠. The diagram $b¯→u¯cd¯$ also contributes, but it is suppressed by $∣Vub∗Vcd∣/∣Vcb∗Vud∣$⁠, which is of order $λ2Rb$⁠. In the factorization approximation, the hadronic matrix elements are the same and we find

Therefore, the rate asymmetry for these decays is approximately given by a single weak phase. To find it, we neglect the doubly Cabbibo-suppressed decays $B0→D0$ and $B0¯→D0¯$⁠. Using eqn (34.22) we get

Now,

where we have used eqn (32.21) on the last step. Combining eqn (34.28) with eqns (30.33) and (34.23), we get

Analogously,