28 Introduction

In this part of the book we apply the observables described in Part I to the study of CP violation in the $Bd0−Bd0¯$ and $Bs0−Bs0¯$ systems. These mesons, together with $B+=b¯u$⁠, can be found in Table 28.1. There is also experimental evidence for $Bc+=b¯c$⁠, but at present little is known about its properties. We shall sometimes use the notation $B0$ to refer to either $Bd0$ or $Bs0$⁠. When specifying the flavour of the light quark clarifies a particular expression, we shall use instead $Bq0$⁠, where q may be either d or s.

The resonance $ϒ(4S)=b¯b$⁠, with mass $10580.0±3.5MeV$⁠, width $21±4MeV$⁠, and parity and C-parity –1, is experimentally very important. It is copiously produced at the $e+e−$ colliders called B factories, like those at SLAC and KEK. The branching ratios for the decays of $ϒ(4S)$ into $B+−B−$ pairs and into $Bd0−Bd0¯$ pairs are close to 50% each.

The most compelling reason to study the decays of B mesons is to learn more about the mechanism behind CP violation. In particular, we would like to test the Standard Model with three generations of quarks and leptons (SM). In the SM, CP violation is accommodated through a single irremovable complex phase present in the Cabibbo–Kobayashi–Maskawa (CKM) matrix V. This may be represented in a rephasing-invariant manner by $J≡Im(VusVcbVub∗Vcs∗)$ (Jarlskog 1985; Dunietz et al. 1985)—see eqn (13.23). The fact that there is only one independent CP-violating quantity in the SM means that this is a very predictive model. Thus far, the SM picture of CP violation has only been tested to the extent that it is consistent with the experimental value of the CP-violating parameter $ϵK$ in the neutral-kaon system. Experiments with B decays will enable us to learn more about the CKM matrix.

Unfortunately, this is not a straightforward endeavour. First, there are experimental errors and uncertainties. Besides, the SM and other theoretical models are written in terms of quarks, while experiments are performed with hadrons. The relation between quarks and hadrons involves low-energy strong interactions and, as a consequence, the extraction of CKM parameters from experiment is generally plagued by theoretical uncertainties. Furthermore, $∣J∣∼3×10−5$ is very small. As a result, large CP-violating asymmetries should only be found in channels with small branching ratios; conversely, channels with large branching ratios are likely to display small CP-violating asymmetries. This fact drives the need for large statistics and, therefore, for experiments producing large numbers of B mesons—the B factories.

As a consequence of both the theoretical and the experimental uncertainties, one should try to determine the CKM matrix elements in various different ways. Our final aim should be to overconstrain the SM and, thus, test it.

In B physics in general, and in particular when discussing the $Bd0−Bd0¯$ system, an important role is played by the unitarity triangle discussed in § 13.5. This triangle represents the unitarity equation

in the complex plane—see Fig. 13.1. The angles between the sides of the unitarity triangle are

In § 18.5 we have summarized the constraints on the unitarity triangle. Those constraints follow from the available experimental information on the moduli of the CKM matrix elements, together with the SM fits of the mass difference in the $Bd0−Bd0¯$ system and of $ϵK$ in the $K0−K0¯$ system. This results in⁷²

The main goal of experiments at B factories will be to measure the phases in eqns (28.2), thereby testing the consistency of the SM. Towards this end, a number of experiments will be performed looking for CP-violating asymmetries in the decays of neutral B mesons into CP eigenstates.

We have defined in Chapter 7 the parameters $λf$ for the decays of two mixing neutral mesons $P0$ and $P0¯$ into a final state f:

where $A¯f=A(P0¯→f)$ and $Af=A(P0→f)$ are the decay amplitudes of $P0¯$ and $P0$⁠, respectively. The parameters q and p describe the transformation from the flavour basis into the basis of the eigenstates of evolution:

In B decays one generally assumes, based on both experimental and theoretical arguments, that there is no CP violation in the mixing: $∣q/p∣=1$⁠. There are then two possible forms of CP violation: direct CP violation, and CP violation in the interference between the mixing and the decays. For decays into a CP eigenstate f, these two forms of CP violation are measured by $∣A¯f∣−∣Af∣$ and by $Imλf$⁠, respectively. Most of the study of CP violation in B decays hinges on measuring these quantities, and interpreting the meaning of the measured values, for various CP-eigenstate final states $fcp$⁠.

A good gauge to evaluate our progress in the study of CP violation is to see whether we are able to disprove the superweak theory of Wolfenstein (1964). As described in § 7.3, the superweak theory asserts that there is no CP violation in the decay amplitudes. This has two consequences. Firstly, there is no direct CP violation: $∣A¯f∣=∣Af∣$ when f is a CP eigenstate. Secondly, all parameters $λf$ are equal up to their sign. Thus, if f and g are two CP eigenstates with CP-parities $ηf=±1$ and $ηg=±1$⁠, respectively, then the superweak theory predicts $λf=ηfηgλg$⁠.

We have derived in Chapter 9 the basic formulae for the decays of tagged mesons $P0$ and $P0¯$ into final states f which are common to both mesons. In this part of the book we shall usually work under the approximations $ΔΓ=0$ and $∣q/p∣=1$⁠.⁷³ Using these approximations we get

For the time-integrated decay rates one has

We can see from these expressions that the parameters $λf$ are observables. They are crucial in the study of CP violation in neutral-meson systems.

Most articles in the literature refer to the CP-violating asymmetry built from tagged decays into CP eigenstates f. Using the decay rates in eqns (28.6) we find

where we define

Similarly, from eqns (28.7) one derives

The asymmetries $ACP(t)$ and $ACP$ violate CP when the final state f is an eigenstate of CP, which we denote $fcp$⁠: indeed, $aint∝Imλf$ measures interference CP violation, and $adir∝1−∣λf∣2∝∣Af∣2−∣A¯f∣2$ (remember that we are assuming $∣q/p∣=1$⁠) measures direct CP violation.

Let us consider the decays of $B0$ and $B0¯$ into a CP eigenstate f, and assume that they are dominated by a single weak phase $ϕA$⁠:

We have used the CP symmetry of the strong interactions to go from the first to the second equation; in this chapter we neglect spurious phases in the CP transformation, which will however be carefully taken into account in the following chapters. The weak phase $ϕA$ can usually be determined directly from the Lagrangian. In the SM, it is the phase of a certain combination of CKM matrix elements. On the other hand, the computation of the modulus A and of the strong phase $δ$ of the decay amplitudes is usually plagued by uncertain or unknown hadronic matrix elements.

Since $∣q/p∣=1$⁠, we may write $q/p=−e2iϕM$⁠. As a result,

We conclude that, under these conditions, $λf$ is a pure phase. This occurs because we may relate the numerator $A¯f$ and the denominator $Af$ via the CP symmetry. Due to the presence of a single weak phase $ϕA$⁠, both the magnitude A and the strong phase $δ$ of the decay amplitudes cancel out in the ratio. This is welcome, because these quantities suffer from hadronic uncertainties. In $λf$ only the weak phase $2(ϕA−ϕM)$ remains. Notice that $ϕA$ and $ϕM$ are not separately rephasing-invariant; however, their difference $ϕA−ϕM$ is rephasing-invariant, and it can be measured, as shown by eqns (28.6) and (28.13).

When eqn (28.13) holds, there is no direct CP violation, because $∣λf∣=1$ and then $adir=0$⁠. Moreover, the interference CP violation is given by $Imλf=ηfsin2(ϕA−ϕM)$⁠. As a result, the tagged, time-dependent CP asymmetry in eqn (28.8) reduces to

This is the Holy Grail of CP-violation measurements in B decays: when the decays into a CP eigenstate are dominated by a single weak phase, the CP asymmetry measures directly a weak phase in the Lagrangian. For certain decays in the SM, that phase is related to $α,β$⁠, and $γ$⁠.

It is important to notice the oscillatory dependence on the sine of $Δmt$ of the CP asymmetry in eqn (28.14). As seen in eqn (28.8), this happens because direct CP violation, represented by $adir,$ vanishes when the decays are dominated by a single weak phase. In the presence of direct CP violation there is another oscillatory term, but now involving the cosine of $Δmt$⁠.

It is also important to stress the advantages of working with final states which are CP eigenstates. If $f¯≠f$⁠, we must compare the decays into f and into $f¯$ in order to study CP violation, and the experimental task becomes more demanding.

Thus far, CP violation has only been detected in the quark sector, and we are unavoidably confronted with strong interactions. They bring with them several difficulties to be faced:

We now discuss the impact that these effects have on the measurement of weak phases with CP asymmetries.

In § 28.3 we have shown that, if there is only one weak phase contributing to the decay amplitudes, then the CP asymmetry measures that weak phase. In general, the presence of another diagram with a different weak phase destroys that simple result. In most cases, this is due to the presence of penguin diagrams, in which case this effect is referred to as penguin pollution.

When there are two weak phases contributing to the decay amplitudes, the latter may be written as⁷⁴

The real numbers $A1$ and $A2$ are the moduli of the interfering amplitudes. The weak phases $ϕA1,ϕA2,$ and $ϕM$ are not rephasing-invariant. On the other hand, the differences $ϕ1≡ϕA1−ϕM,ϕ2≡ϕA2−ϕM$⁠, and $Δ≡δ2−δ1$ can be measured.

Let us see how the observables in eqns (28.9) and (28.10) are related to the weak and strong phases. In the approximation where $r=A2/A1$ is small, we find

Then,

When the decays are dominated by a single weak phase, i.e., when either $r≈0$ or $ϕ2≈ϕ1$⁠, there is no direct CP violation and $aint$ measures a single weak phase $ϕ1$⁠. This is the situation discussed in § 28.3. However, it is a very particular situation; in general, $aint$ does not measure a single weak phase.

This problem is not rooted in the presence of direct CP violation. Indeed, if we assume the FSI to be vanishingly small, we have $Δ=0$ and there is no direct CP violation—$adir=0$⁠. The parameter $λf$ is then a pure phase but, still, $aint$ does not measure a single weak phase (Gronau 1993). Indeed, when $Δ=0$ we can write

where $δϕ1$ is defined by

Therefore,

The presence of a second amplitude with a different weak phase, $ϕ2≠ϕ1$⁠, may spoil the measurement of $sin2ϕ1$⁠, even if the second amplitude has the same strong phase as the first one. This occurs even for moderate values of r (Gronau 1993).

We shall allow for new phases to be present in $B0−B0¯$ mixing, see § 24.8.2 and 30.4.2. On the other hand, we shall in general assume that the decay amplitudes are given by SM diagrams. For decays that occur through unsuppressed SM tree-level diagrams, it is likely that no new physics can occur at a competing level, since that new physics should have been detected elsewhere. The situation is of course different for decays that are strongly suppressed in the SM.

We shall also allow for a possible non-unitarity of the CKM matrix. If the CKM matrix is not unitary, eqn (28.1) does not hold; the usual relations between the angles $α,β$⁠, and $γ$ and the sides of the unitarity triangle are destroyed, and the bounds in eqn (28.3) cease to be valid. Nevertheless, if the decay amplitudes are dominated by SM diagrams, as we assume, then the weak phases in those decay amplitudes will still be controlled by the phases in the CKM matrix.

The three phases introduced in eqn (28.2) are not independent; they satisfy by definition

It is important to stress that using these three phases is redundant, because they are linearly dependent. In general, we shall take $β$ and $γ$ to be the fundamental phases, and we shall consider $α$ as just a linear combination of $β$ and $γ$⁠.

In any model, as in the SM, in which eqn (28.1) holds, the phases $α,β,$ and $γ$ may be geometrically pictured as the angles between the sides of the unitarity triangle. Then, $α+β+γ$ may be either $π$ or $5π$⁠, but not $3π$ (Grossman et al. 1997a). Indeed, if the angles are interior to the triangle, then they lie in the range $[0,π]$ and add up to $π$⁠; if the angles are exterior to the triangle then they lie in the range $[π,2π]$ and add up to $5π$⁠. This is a restriction on eqn (28.22).

Aleksan et al. (1994) have introduced two further phases, $ϵ$ and $ϵ′$⁠:⁷⁵

They have shown—see § 16.3.3—that, in the SM, one can parametrize the $3×3$ unitary CKM matrix V, moduli and phases alike, with only four phases $β,γ,ϵ$⁠, and $ϵ′$ (Aleksan et al. 1994). These four phases are useful even in the presence of new physics. Indeed, it is easy to see that $β,γ,ϵ$, and  $ϵ′$  parametrize all the phases in the usual  $3×3$  submatrix of the generalized CKM matrix. This happens because the $3×3$ submatrix of the generalized CKM matrix has nine phases. By rephasing the six quarks it is possible to redefine away five phases. Therefore, there are only four physical phases in the submatrix. We may choose a phase convention such that its phase structure is given by

The choice of phases in eqn (28.24) is useful in order to identify rapidly the phase of any rephasing-invariant combination of CKM matrix elements.

We have seen in § 18.5 that, in the SM, the angle $β$ is quite well determined: $10°<β<30°$⁠. The angle $γ$ should lie between $35°$ and $135°$⁠. Finally, $α$ is expected to lie between $35°$ and $120°$⁠. These values correspond to the bounds in eqns (28.3).

Thus, $α,β$⁠, and $γ$ are in principle large angles. On the other hand, as Aleksan et al. (1994) pointed out, within the three-generation SM $ϵ$ and $ϵ′$ must be very small: $ϵ≲λ2∼0.05$ and $ϵ′≲λ4∼0.0025$⁠. This can easily be checked by considering the Wolfenstein parametrization of the CKM matrix, and in particular eqns (16.27) and (16.29). One sees that

Notice that $η=0.33±0.10$—see § 18.5—entails a further suppression of the values of $ϵ$ and $ϵ′$⁠, beyond the one given by the powers of $λ=0.22$⁠.

We conclude that, in the SM, the CKM matrix contains only two independent large phases. Once $β$ is measured, only one large phase—say, $γ$—remains to be determined. From our point of view, measuring $α$ is just a different way of measuring $γ$⁠.

When there is new physics the bounds on $α,β$⁠, and $γ$ given in § 28.5.1 are relaxed. Also, $ϵ$ can in principle be large. On the other hand, in most models of new physics the $3×3$ CKM matrix is a submatrix of a larger $n×n$ unitary matrix $V̂$⁠; this holds, in particular, in the SM with more than three generations, as well as in a model with vector-like isosinglet or isodoublet quarks, or with mirror quarks. In those models, one may use the fact that, experimentally, $∣Vud∣$and $∣Vcs∣$ are very close to one, together with the orthonormality of the first and second columns of $V̂$⁠, to prove that $ϵ′$ will still be rather small. One derives

As a consequence,

in the last step we have used the lower bounds from Chapter 15,

Thus, $∣ϵ′∣<0.2$ (Kurimoto and Tomita 1997). This is a much poorer bound than in the SM, where $∣ϵ′∣$ is two orders of magnitude smaller, but it holds in most models of new physics. Only rather contrived models, which change the normalization of the CKM matrix, might avoid this conclusion; it is difficult to imagine such models which pass all the experimental constraints. In particular, a large $ϵ′$ may lead to problems with the CP-violating parameter $ϵK$ of the neutral-kaon system: this is because $Γ21∝λu2$⁠, while $M21$ has pieces proportional to $λu2,λuλc,$ and $λc2$⁠, where $λα≡Vus∗Vud$⁠. If $ϵ′$ is large then the phase difference between $λu$ and $λc$ is large, and the phase of $M21$ may turn out too different from the one of $Γ21$⁠, destroying the fit of $ϵK$ in the SM.

Several important issues arise when one considers the precise conditions to be faced in a real experiment. Extracting information on $λf$ is not as simple as the discussions in Chapter 9 might lead us to believe. In general, there are several meson configurations in the initial state; tagging the initial flavour of a decaying meson may not be trivial; and the ability to trace the time dependence of the decays is limited by the capabilities of vertex reconstruction. Some of these problems are discussed briefly in Chapter 29.

In order to compute the rephasing-invariant parameters $λf$ one needs to compute the phase-convention-dependent quantities q/p and $A¯f/Af$ separately. Each of these two quantities depends on the CP-transformation phases for the quark fields and for the hadron state vectors; in their product, however, those spurious phases cancel out. The computation of q/p for the neutral-meson systems $K0−K0¯,Bd0−Bd0¯$⁠, and $Bs0−Bs0¯$ is the subject of Chapter 30. We show that, in all these cases, $∣q/p∣$ is very close to unity.⁷⁶ This is done using only phenomenological arguments and the known experimental values. We then proceed to find out the phase of q/p in the SM; in other models, we parametrize the phase of q/p in the $Bq0−Bq0¯$ system by means of an extra phase $θq$⁠.

The decay amplitudes are the subject of Chapters 31 and 32. We discuss the decay amplitudes in the SM from a diagrammatic point of view in Chapter 31. A more formal presentation, using effective Hamiltonians, is left to Chapter 32, which however is inessential for most of the rest of the book. The elementary quark-decay-diagram analysis of Chapter 31 is most convenient in order to identify the weak phases present in each amplitude.

Throughout Part IV, we assume that the strong-interaction rescatterings among the various final states of the weak decays are negligible or, at least, they do not affect the result of a given calculation. This is a central assumption in most treatments of CP violation in the $B0−B0¯$ systems; indeed, no one really knows how to treat the FSI, both because of their non-perturbative nature and because they may mix many different final states. Another restriction in this book is that we concentrate on decays into two-meson final states. These are the easiest final states to describe, and the ones most commonly treated in the literature. We shall concentrate on decays into final states that are either flavour-specific or CP eigenstates. We also consider some decays into non-CP eigenstates. We mention only briefly other possibilities like inclusive decays, semi-inclusive decays, and decays into states where an angular decomposition permits the distinction between CP-even and CP-odd components. These have been summarized by Dunietz (1994), where references to the original literature can be found.

We emphasize that, for the methods analysed in this book, we assume that the strong-interaction final-state rescatterings do not mix decays which occur through different quark processes. This allows us to estimate the relative contributions of the different quark-level diagrams, as is done by most authors. Although useful, such estimates should not be taken too seriously. Rescattering effects may alter them—see, for example, Blok et al. (1997), Ciuchini et al. (1997a), Gérard and Weyers (1997), and Neubert (1997).

Chapters 29–32 contain detailed discussions which may be skipped in a cursory reading. Readers eager to get a quick acquaintance with CP violation in B decays may want to proceeed to Chapter 33 immediately, and refer to earlier chapters as the need arises. Chapter 33 is the crucial one of Part IV. In it, we put together the results for q/p and $A¯f/Af$ and compute the parameters $λf.$ In particular, in § 33.1 we assume that only the tree-level SM amplitudes contribute to the decays, and show that the CP asymmetries in $Bd0→π+π−,Bd0→J/ψKS,$ and $Bs0→ρ0KS$ measure $sin2α,sin2β,$ and $sin2γ$⁠, respectively. Unfortunately, the penguin pollution spoils the theoretical interpretation of the first and third asymmetries as $sin2α$ and $sin2γ$⁠, respectively. On the other hand, the CP asymmetry in the decay $Bd0→J/ψKS$ measures $sin2β$ almost without hadronic uncertainties.

In Chapter 34 we study decay chains that involve an intermediate neutral-meson system. As shown in Appendix E, the mixing in the $D0−D0¯$ system is very small in the SM. Therefore, we assume that this mixing vanishes when we consider decay chains which include $D0$ and/or $D0¯$ at an intermediate stage. This restriction will be lifted in § 34.6, where we discuss the most general cascade decay chain. That decay chain requires the introduction of new CP-violating parameters, beyond the ones used in most of this book.

Several methods have been devised to overcome the problem of penguin pollution. Decays of the type $B→D→f$ have been used by some authors to gain access to the phase $γ$⁠. Other possibilities to determine weak phases involve the use of isospin or SU(3)-flavour symmetries in order to relate different decay channels. Methods using the decays of the $Bd0−Bd0¯$ system are treated in Chapters 35 and 36; those using the decays of the $Bs0−Bs0¯$ system are presented in Chapter 37.

All these methods give access to trigonometric functions of the weak phases, rather than to the phases themselves. The extraction of the values of phases from the values of trigonometric functions thereof suffers from discrete ambiguities. In Chapter 38 we discuss ways to resolve these ambiguities.

Chapters 35–38 include mostly relatively recent contributions by a variety of authors. Many proposals must still be tested in order to check their feasibility. As a result, the subjects covered in those chapters are likely to evolve more rapidly than those in the rest of the book.