B Physics and Cp Violation
Abstract
After an introduction to the StandardModel description of violation and a brief look at the present status of this phenomenon in the kaon system, a classification of nonleptonic decays is given and the formalism of – mixing is discussed. We then turn to the factory benchmark modes, violation in charged decays, and the meson system. Finally, we focus both on decays, which play an important role to probe the CKM angle , and on the , system, which allows an interesting determination of and .
DESY 00–170 
hep–ph/0011323 
November 2000 
(Invited lecture at NATO ASI 2000, Cascais, Portugal, 26 June – 7 July, 2000)
1 Introduction
The violation of the symmetry, where and denote the chargeconjugation and paritytransformation operators, respectively, is one of the fundamental phenomena in particle physics. Although weak interactions are neither invariant under , nor invariant under , it was originally believed that the product was preserved. Consider, for instance, the process
(1) 
Here the lefthanded state is not observed in nature; only after performing an additional parity transformation we obtain the righthanded electron antineutrino. In 1964, it was then found experimentally through the observation of decays that weak interactions are not invariant under transformations [1]. So far, violation has only been observed in the kaon system, and we still have few experimental insights into this phenomenon. However, the measurement of asymmetries should also be “around the corner” in decays [2], which are currently explored in great detail at the factories. For a collection of basic references on violation and physics, the reader is referred to Refs. [3, 4].
Studies of violating effects are very exciting, since physics beyond the Standard Model is usually associated with new sources for violation. Important examples are nonminimal supersymmetry, leftrightsymmetric models, models with extended Higgs sectors, and many other scenarios for “new” physics [5]. In this context, it is also interesting to note that the evidence for neutrino masses we got during the recent years points towards physics beyond the Standard Model [6], raising the question of violation in the neutrino sector [7], which may be studied in the more distant future at factories. In cosmology, violation plays also a crucial role: one of the necessary conditions to generate the matter–antimatter asymmetry of our Universe is – in addition to baryon number violation and deviations from thermal equilibrium – that the elementary interactions have to violate (and ) [8, 9]. Recent model calculations indicate, however, that the violation present in the Standard Model is too small to generate the observed matter–antimatter asymmetry of [10].
Concerning quantitative tests of the StandardModel description of violation, the meson system is particularly promising. In the search for new physics, it is crucial to have violating decay processes available that can be analysed reliably within the framework of the Standard Model, which will be the major topic of these lectures. Presently, we are at the beginning of the factory era in particle physics, and in the summer of 2000, the BaBar (SLAC) and Belle (KEK) collaborations have already reported their first results. Moreover, HERAB (DESY) has seen first events, CLEOIII (Cornell) has started taking data, and run II of the Tevatron (Fermilab) will follow next spring. A lot of interesting physics will also be left for “secondgeneration” experiments at hadron machines, LHCb (CERN) and BTeV (Fermilab). Detailed studies of the physics potentials of BaBar, run II of the Tevatron, and the LHC can be found in Ref. [11].
The outline of these lectures is as follows: in Section 2, we discuss the StandardModel description of violation. After a brief look at the present status of violation in the kaon system in Section 3, we turn to the system in Section 4 by giving a classification of nonleptonic decays and introducing lowenergy effective Hamiltonians. A key element for violation in the system – the formalism of – mixing – is presented in Section 5, and is applied to important factory benchmark modes in Section 6. We then turn to violation in charged decays in Section 7, and discuss the meson system – the “El Dorado” for hadron machines – in Section 8. The remainder of these lectures is devoted to two more recent developments: the phenomenology of decays, which is the topic of Section 9, and the , system, which is discussed Section 10. Before concluding in Section 12, we make a few comments on other interesting rare decays in Section 11.
2 The StandardModel Description of CP Violation
Within the Standard Model of electroweak interactions [12], violation is closely related to the Cabibbo–Kobayashi–Maskawa (CKM) matrix [13, 14], connecting the electroweak eigenstates of the down, strange and bottom quarks with their mass eigenstates through the following unitary transformation:
(2) 
The elements of the CKM matrix describe chargedcurrent couplings, as can be seen easily by expressing the nonleptonic chargedcurrent interaction Lagrangian in terms of the mass eigenstates appearing in (2):
(3) 
where the gauge coupling is related to the gauge group , and the field corresponds to the charged bosons.
2.1 Parametrizations of the CKM Matrix
The phase structure of the CKM matrix is not unique, as we may perform the following phase transformations:
(4) 
which are related to redefinitions of the up and downtype quark fields:
(5) 
Using these transformations, it can be shown that the general generation quarkmixingmatrix is described by parameters, which consist of Eulertype angles, and complex phases. In the twogeneration case [13], we arrive therefore at
(6) 
where can be determined from decays.
In the case of three generations, three Eulertype angles and a single complex phase are needed to parametrize the CKM matrix. This complex phase allows us to accommodate violation in the Standard Model, as was pointed out by Kobayashi and Maskawa in 1973 [14]. In the “standard parametrization”, the threegeneration CKM matrix takes the form
(7) 
where and . Another interesting parametrization of the CKM matrix was proposed by Fritzsch and Xing [15], which is based on the hierarchical structure of the quark mass spectrum.
In Fig. 1, the hierarchy of the strengths of the quark transitions mediated through chargedcurrent interactions is illustrated. In the standard parametrization (7), it is reflected by
(8) 
If we introduce new parameters , , and by imposing the relations
(9) 
and go back to the standard parametrization (7), we arrive at
(10) 
This is the “Wolfenstein parametrization” of the CKM matrix [16]. It corresponds to an expansion in powers of the small quantity , and is very useful for phenomenological applications. A detailed discussion of the nexttoleading order terms in can be found in Ref. [17].
2.2 Further Requirements for CP Violation
As we have just seen, three generations are necessary to accommodate violation in the Standard Model. However, still more conditions have to be satisfied. They can be summarized as follows:
(11)  
where
(12) 
The “Jarlskog Parameter” represents a measure of the “strength” of violation within the Standard Model [18]. Using the Wolfenstein parametrization, we obtain
(13) 
Consequently, violation is a small effect in the Standard Model. However, typically new complex couplings are present in scenarios for new physics, yielding additional sources for violation.
2.3 The Unitarity Triangles of the CKM Matrix
Concerning tests of the CKM picture of violation, the central targets are the unitarity triangles of the CKM matrix. The unitarity of the CKM matrix, which is described by
(14) 
leads to a set of 12 equations, consisting of 6 normalization relations and 6 orthogonality relations. The latter can be represented as 6 triangles in the complex plane, all having the same area, [19]. However, in only two of them, all three sides are of comparable magnitude , while in the remaining ones, one side is suppressed relative to the others by or . The orthogonality relations describing the nonsquashed triangles are given by
(15)  
(16) 
At leading order in , these relations agree with each other, and yield
(17) 
Consequently, they describe the same triangle in the – plane^{1}^{1}1Usually, the triangle relation (17) is divided by the overall normalization ., which is usually referred to as “the” unitarity triangle of the CKM matrix [20]. However, in the era of secondgeneration experiments, the experimental accuracy will be so tremendous that we will also have to take into account the nexttoleading order terms of the Wolfenstein expansion, and will have to distinguish between the unitarity triangles described by (15) and (16). They are illustrated in Fig. 2, where and are related to the Wolfenstein parameters and through [17]
(18) 
Note that . The sides and of the unitarity triangle shown in Fig. 2 (a) are given as follows:
(19)  
(20) 
and will show up at several places throughout these lectures.
2.4 Towards an Allowed Range in the – Plane
The parameter introduced in (19), i.e. the ratio , can be determined through semileptonic and decays. It fixes a circle in the – plane around with radius . The second side of the unitarity triangle shown in Fig. 2 (a) can be determined through – mixing. It fixes another circle in the – plane, which is centered at and has radius . Finally, using experimental information on an observable , which describes “indirect” violation in the neutral kaon system and will be discussed in the next section, a hyperbola in the – plane can be fixed. These contours are sketched in Fig. 3; their intersection gives the apex of the unitarity triangle shown in Fig. 2 (a). The contours that are implied by – mixing and depend on , the topquark mass, QCD corrections, and nonperturbative parameters (for a review, see [4]). This feature leads to strong correlations between theoretical and experimental uncertainties. A detailed recent analysis was performed by Ali and London [21], who find the following ranges:
(21) 
We shall come back to this issue in Subsection 8.1, where we emphasize that the present experimental lower bound on – mixing has already a very important impact on the allowed range in the – plane (see Fig. 16).
3 A Brief Look at CP Violation in the Kaon System
Although the discovery of violation goes back to 1964 [1], so far this phenomenon could only be observed in the meson system. Here it is described by two complex quantities, called and , which are defined by the following ratios of decay amplitudes:
(22) 
While parametrizes “indirect” violation, originating from the fact that the mass eigenstates of the neutral kaon system are not eigenstates, the quantity Re measures “direct” violation in transitions. The violating observable plays an important role to constrain the unitarity triangle [4, 21] and implies – using reasonable assumptions about certain hadronic parameters – in particular a positive value of the Wolfenstein parameter . In 1999, new measurements of Re have demonstrated that this observable is nonzero, thereby excluding “superweak” models of violation [22]:
(23) 
Unfortunately, the calculations of Re are very involved and suffer at present from large hadronic uncertainties [25]. Consequently, this observable does not allow a powerful test of the violating sector of the Standard Model, unless the hadronic matrix elements of the relevant operators can be brought under better control.
In order to test the StandardModel description of violation, the rare decays and are more promising, and may allow a determination of with respectable accuracy [26]. Yet it is clear that the kaon system by itself cannot provide the whole picture of violation, and therefore it is essential to study violation outside this system. In this respect, meson decays appear to be most promising. There are of course also other interesting probes to explore violation, for example, the neutral meson system or electric dipole moments, which will, however, not be addressed further in these lectures.
4 Nonleptonic B Decays
With respect to testing the StandardModel description of violation, the major role is played by nonleptonic decays, which are mediated by quarklevel transitions ().
4.1 Classification
There are two kinds of topologies contributing to nonleptonic decays: treediagramlike and “penguin” topologies. The latter consist of gluonic (QCD) and electroweak (EW) penguins. In Figs. 4–6, the corresponding leadingorder Feynman diagrams are shown. Depending on the flavour content of their final states, we may classify decays as follows:

: only tree diagrams contribute.

: tree and penguin diagrams contribute.

: only penguin diagrams contribute.