37 Extracting CKM Phases with Bs0 Decays

In this chapter we shall discuss some methods available to extract CKM phases with the help of $Bs0$ decays. In the near future, $Bs0−Bs0¯$ pairs will only be produced at DESY, Fermilab, and LHC. In all these cases, the $Bs0−Bs0¯$ pairs are created in uncorrelated initial states. Moreover, they are produced together with a large number of other states originating in $bb¯$ quark pairs, such as $Bs0B−X+,Bd0B−X+,Bs0Bd0¯Xs¯d0,$ etc. This means that one can study both $Bd0$ and $Bs0$ at these facilities, but that such studies must face a daunting background.

As happens with $Bd0$ studies at the $ϒ(4S)$⁠, full reconstruction of the events is extremely inefficient, and one usually looks for the semileptonic decay of one b-hadron (the tagging decay), only tracing the time dependence of the other b-hadron. For exclusive decays, such as $Bs0→fcp$ or $Bd0→fcp$⁠, this procedure still requires the reconstruction of the exclusive channel of interest. There is, however, a very big difference with respect to experiments performed at the $ϒ(4S)$⁠. There, the anti-correlated nature of the initial state guarantees that the semileptonic decay in one side of the detector does tag the flavour of the meson on the opposite side at the same instant. For uncorrelated initial states of neutral B mesons there is no such effect, and one must take the mistags into account through a dilution factor $DM=1−2χ0=(1−y2)/(1+x2)$⁠, cf. § 9.10.2. Since the semileptonic decay may come from any of the b-hadrons on the tagging side, what is of interest is the dilution factor averaged over all the b-hadron species that can decay into the channels used for tagging.

Besides these differences, that have to do with the initial states produced in the experiments, there are a number of important differences between the $Bd0−Bd0¯$ and $Bs0−Bs0¯$ systems. In the latter system

These characteristics will be exploited in the following sections in order to determine CP-violating phases in the CKM matrix.

We have seen that the decays $b¯→c¯cs¯$ are the only ones where there is virtually no trace of a second CKM phase in the decay amplitude. This led naturally to the gold-plated decay $Bd0→J/ψKS$ in § 33.5, which measures $sin2(β−ϵ′−θd)$⁠. Its $Bs0$ counterpart is the decay $Bs0→Ds+Ds−$⁠, which measures $sin2(ϵ+θs)$⁠. In the SM, $ϵ∼λ2$ and this CP-violating asymmetry is down by an order of magnitude with respect to the one in $Bd0→J/ψKS$⁠. Moreover, in the near future $Bs0$ will only be produced in the harsh hadronic environment, and $Ds±$ mesons may be hard to detect. Therefore, although this is dominated by a single weak phase, the measurement of $ϵ$ will not be immediate. The asymmetry in $Bs0→J/ψϕ$also measures $sin2(ϵ+θd)$⁠, but one must study the angular distributions to disentangle the CP-even and CP-odd components of the final state.

However, precisely because the SM predicts that this asymmetry should be small, this is a perfect channel to look for new physics in $Bs0−Bs0¯$ mixing. In particular, there could be a significant new phase $θs$ in that mixing (Nir and Silverman 1990).

Aleksan et al. (1992) proposed a method to measure the angle $γ$ using $Bs0$ decays into non-CP eigenstates such as $Ds±K∓,Ds±K∗∓$⁠, and $Ds∗±K∓$⁠. For example, one would look for the time-dependent decays

These final states can be reached by colour-allowed tree-level (T) diagrams $b¯→c¯us¯$ and $b¯→u¯cs¯$⁠, but also by W-exchange diagrams (E). The latter, however, have the same CKM phase as the T diagrams. Moreover, there are no penguin amplitudes contributing to these decays, and rescattering effects cannot bring in a new weak phase. The rates for these decays are estimated to be $∼10−4$⁠, while the decay rate for $Bs0→ρ0KS$ is estimated to be $∼10−7$ (Aleksan et al. 1992).

Thus, extracting $γ$ with $Bs0→D±K∓$ has two advantages over $Bs0→ρ0KS$⁠: the decay rates are much larger and there is only one weak phase at play. The disadvantage of working with non-CP eigenstates is that one must trace the time-dependence of four decay rates in order to extract $λf$ and $λ¯f¯$ and recover the weak phase from them.

The four amplitudes relevant for the $Bs0→Ds±K∓$ analysis are

with the operators defined in eqns (32.3). Therefore,

Using the CP transformation of the operators in eqns (32.13) it is easy to prove that

The big difference with respect to decays into CP eigenstates is that, here, the CP transformation does not relate $A¯f$⁠, in the numerator of $λf$⁠, with the amplitude in the denominator, $Af$⁠, but rather with $Af¯$⁠. Therefore, one cannot proceed with the calculation of $λf$ without either making some phase choice, or else using some symmetry to relate the numerator with the denominator. Still, we may assert that

is a measurable quantity, as it should be. If we measure the real and imaginary parts of $λf$ and $λ¯f¯,$ we can determine $γ+ϵ′−2ϵ−2θs$⁠. However, if $∣ΔΓ∣$ turns out to be small in the $Bs0−Bs0¯$ system, we can only determine the imaginary parts of $λf$ and $λ¯f¯$⁠, as well as their magnitudes. Then, the sine of the relevant phase may be recovered up to a fourfold ambiguity from

where

This result is easier to understand if we choose a convention in which the CP transformation includes no phases. Then, we may write $q/p=−e2iϕM$⁠, and

Therefore,

where $ϕ=ϕa+ϕb−2ϕM$ and $Δ=δb−δa$ are measurable weak and strong phases, respectively. In our example, $ϕ=γ+ϵ′−2ϵ−2θs$⁠. We see that $∣λf∣,Imλf,$ and $Reλf$ measure B/A, $B/Asin(ϕ−Δ)$⁠, and $−B/Acos(ϕ−Δ)$⁠, respectively. Similarly, $∣λ¯f¯∣,Imλ¯f¯$⁠, and $Reλ¯f¯$ measure B/A, $−B/Asin(ϕ+Δ)$⁠, and $−B/Acos(ϕ+Δ)$⁠, respectively. We can recover $ϕ$ from these measurements.

Let us look back at eqn (37.6). One of the signs on the RHS of eqn (37.6) yields the true value of $sin2ϕ$⁠; the other sign gives $cos2Δ$ instead. We stress that this trigonometric exercise is only needed if $∣ΔΓ∣$ is small, in which case we can only measure $Imλf,Imλ¯f¯,∣λf∣$⁠, and $∣λ¯f¯∣$⁠. The ambiguity in $sin2ϕ$ is removed if $∣ΔΓ∣$ is large enough to allow for the additional measurement of the real parts, $Reλf$ and $Reλ¯f¯$⁠.

The fact that we are probing $sin2γ$ and that we do so through eqn (37.6) has three interesting consequences. Firstly, if one searches for $γ$ with CP-violating asymmetries alone, one is really probing $sin2γ$⁠. As pointed out before, present constraints allow this to be zero, while $sin2γ$ is guaranteed to be nonzero. Indeed, we have seen in § 18.5 that $0.33≤sin2γ≤1$ in the SM. Secondly, since $BKsinγ$ is positive, if we assume that $BK$ is positive, then eqn (37.6) only retains a twofold ambiguity. Thirdly, Aleksan et al. (1992) argue that $Δ$ is likely to be very small. If this turns out to be the case, then the angle $γ$ may be extracted from eqn (37.6) without any ambiguity at all.

This method is analogous to the Gronau–London $Bd0→DKS$ method discussed in § 36.6, only with the spectator quark d being substituted by s. Both methods were suggested in the same article by Gronau and London (1991).

Here, one looks for the tagged decays

The idea is the same as in § 36.3 and 36.6.1, and one also measures the sine of the angle in eqn (36.34), $γ±ϵ′$⁠, up to a fourfold discrete ambiguity. Again, the hierarchy of the amplitudes makes this method difficult to implement in practice.

There is a subtle difference between this method and those other methods related to it by a change in the spectator quark: here we must follow the time dependence of the decays. In § 36.3 that was not needed because there is no mixing in charged decays. In § 36.6.1 that was also not needed, although there is $Bd0−Bd0¯$ mixing. The point there was that, at the $ϒ(4S)$⁠, the terms proportional to $Imλf$ drop out from the tagged, time-integrated decay rates $Γ[B0→f]$ and $Γ[B0¯→f]$⁠. Therefore, from these two observables we could extract the two unknowns $∣Af∣$ and $∣A¯f∣$⁠. Here the situation is different because the $Bs0−Bs0¯$ pairs to be detected in experiments will not come from an odd-parity correlated state.

We have seen in § 36.6.2 that the interference terms in the time-dependent decays $Bd0→{D0,D0¯}KS$ can be used to measure $2β+γ$⁠, in the SM. Analogously, we may use the $Bs0→{D0,D0¯}ϕ$ decays to find $−2ϵ+γ$⁠. The difference is that, in this case, we use the mixing in the $Bs−Bs0¯$ system, rather than the mixing in the $Bd−Bd0¯$ and $K0−K0¯$ systems. This is just the colour-suppressed version of the same quark processes $b¯→c¯us¯$ and $b¯→u¯cs¯$ involved in the Aleksan–Dunietz–Kayser method. Therefore, it also measures the sine of the angle $γ+ϵ′−2ϵ−2θs,$ up to a fourfold discrete ambiguity.

A large $∣ΔΓ∣/Γ$ opens up the possibility for a new class of experiments to determine weak phases: those performed with untagged data samples (Dunietz 1995). Untagged data samples are interesting because they are readily produced at $e+e−$ and $pp¯$ machines; there is no need to tag and no extra cost in statistics (Dunietz 1995).

We have already introduced untagged decays in § 9.6. We recall that the untagged decay rates may be written as in eqn (9.50):

where H and I are given by eqns (9.15):

The function H depends on exponentials and on $ΔΓ$⁠, while I is an oscillatory function of $Δmt$⁠. Clearly, if $δ$ is small, then the oscillatory function I drops out of the untagged decay rates. This is an advantage of untagged decays because, if $Δm$ is very large, the time oscillations in I are too fast to be observed anyway. However, one must keep in mind that the corrections due to $δ$ should be included whenever untagged decays are being used to look for very small effects in H.

Henceforth we take $δ=0$⁠. Then,

A difference between these two untagged decays violates CP. If the decays are dominated by a single weak phase, then there is no direct CP violation, $∣Af∣=∣A¯f¯∣,∣λf∣=∣λ¯f¯∣$⁠, and the only CP violation arises from $Reλf≠Reλ¯f¯$⁠.

If f is a CP eigenstate, then H is CP-conserving, and I is CP-violating. In this case, $Γf(t)=Γf¯(t)$⁠. If there is no direct or mixing CP violation, then $∣λf∣=1$⁠, and H only measures CP violation to the extent that $∣Reλf∣$ differs from unity.

We are now in a position to appreciate the systematic analysis performed by Dunietz (1995). Firstly we consider decays into CP eigenstates such as $Bs→Ds+Ds−$⁠. In § 37.2 we have seen that tagged decays in this channel measure $sin2(ϵ+θs)$⁠. Here, we notice that untagged decays allow us to measure the cosine of the same angle, $cos2(ϵ+θs)$⁠. Instead of $Ds+Ds−$ we may use $J/ψϕ$ or $Ds∗+Ds∗−$⁠. These are combinations of CP-even and (small) CP-odd components, which can be resolved by means of an analysis of the angular distributions (Dunietz et al. 1991; Dighe et al. 1996c), even in the case of untagged data samples (Dunietz and Fleischer 1997). In the SM, $θs=0,ϵ∼λ2$⁠, and this experiment looks extremely difficult, since we will be looking at the difference $cos2ϵ−1∼λ4$⁠. However, in some models beyond the SM there may be a large $θs$⁠. Untagged decays can be used to look for it.

Secondly, Dunietz (1995) has proposed a version of the Aleksan–Dunietz–Kayser method of § 37.3 in which one uses instead untagged decays. This requires the value of $∣Af∣$ as an additional input. In fact, only if we know this magnitude from theory can we extract $Reλf$ and $Reλ¯f¯$ from the fit to the time dependences in eqns (37.13). To see how to extract the weak phase from these observables, we use eqn (37.9), from which

where $ϕ=γ+ϵ′−2ϵ−2θs$⁠. We are now left with the trivial trigonometric exercise in eqn (36.7). The usual remarks about the fourfold ambiguity in $sinγ$ apply. Notice that the CP-violating quantity $c+−c−=−2sinϕsinΔ$ is only different from zero if there is a non-negligible final-state phase difference $Δ$⁠. Dunietz (1995) pointed out that $Δ$ is likely to be very small for colour-allowed decays, in which case $c+=c−$ would measure $cosϕ$ directly.

Thirdly, one could perform a $Bs0→{D0,D0¯,Dfcp}ϕ$ analysis with untagged decays (Dunietz 1995). The idea is that one can extract from the time dependences of the untagged decays the observables

with $f=D0ϕ,f¯=D0¯ϕ$⁠, and $fcp=Dfcpϕ$⁠. This allows the extraction of the difference of weak phases, the difference of strong phases, and the common magnitude $∣λf∣=∣λ¯f¯∣$⁠. Contrary to the previous case, here there is no need for a theoretical input concerning amplitudes.