The analysis of CP asymmetries in K and B decays – and likewise for charm – is clearly hampered by our failure to accurately evaluate hadronic matrix elements, since those are shaped by non-perturbative dynamics. There are, however, fermionic systems that are not subject to non-perturbative dynamics thus making our calculational tools more powerful. These are leptons – electrons, muons, τ leptons and neutrinos – and top quarks. The electron's EDM has been discussed in Section 3.6 and CP violation in neutrino oscillations in Section 16; the decays of charged leptons will be addressed here. As pointed out in Section 10.10.3 the aforementioned gain in calculational control comes with a price – namely at best small CP asymmetries. We will see that final state distributions rather than partial widths probably have the best chance to reveal CP violation.

Production and decay of top quarks

The existence of all members of three quark families has been established with the top quark being discovered last. Even before that time it had been realized that the top quark, once it becomes sufficiently massive, will decay (semi-)weakly – t → bW – before it can hadronize; i.e. top states decay as quarks rather than hadrons. This transition occurs around the 110–130 GeV region (for |V (tb)| ∼ 1), i.e. well below the mass value now observed. Non-perturbative dynamics thus plays hardly any role in top decays, and the strong forces can be treated perturbatively. While this is certainly good news for our ability to calculate observables, it carries also a negative message concerning the observability of CP asymmetries.

The sciences in general and physics in particular are full of fascinating phenomena; this is why they have attracted intense human interest early on and have kept it ever since. Yet even so we feel that the question to which degree nature is invariant under time reversal and CP transformations is so fundamental that it richly deserves its own comprehensive monograph. Two lines of reasoning – different, though not unrelated to each other – lead us to this conclusion. The first relies on multi-layered considerations, the second is based on a property inferred for the whole universe.

• The first line of reasoning centres on the important role symmetries have always played in physics. It has been recognized only last century, though, how central and crucial this role actually is, and this insight forms one of the lasting legacies of modern physics to human perception of nature and thus to human culture. The connection between continuous symmetries – like translational and rotational invariance – and conserved quantities – momentum and angular momentum for these examples – has been formulated through Noether's theorems. The pioneering work of Wigner and others revealed how atomic and nuclear spectra that appeared at first sight to be quite complicated could be understood through an analysis of underlying symmetry groups, even when they hold only in an approximate sense. This line of reasoning was successfully applied to nuclear and elementary particle physics through the introduction of isospin symmetry SU(2), which was later generalized to SU(3) symmetry in particle physics.

Some discoveries in the sciences profoundly change how we view nature. The discovery of parity violation in the weak interactions in 1956 certainly falls into this illustrious category. Yet it just started the shift to a new perspective; it was the discovery of CP violation in 1964 by Christenson, Cronin, Fitch and Turlay at Brookhaven National Lab – completely unexpected to almost all despite the experience of 8 years earlier – that established the new paradigm that even in the microscopic regime symmetries should not be assumed to hold a priori, but have to be subjected to determined experimental scrutiny.

It would seem that after the initial period of discoveries little progress has been achieved, since despite dedicated efforts CP violation has not been observed outside the decays of KL mesons, nor can we claim to have come to a real understanding of this fundamental phenomenon.

We have, however, ample reason to expect imminent dramatic changes. Firstly, direct CP violation has been observed in KL decays. Secondly, our phenomenological and theoretical descriptions have been refined to the point that we can predict with confidence that the known forces of nature will generate huge CP asymmetries, which could even be close to 100%, in the decays of so-called beauty mesons. Dedicated experiments are being set up to start taking data that would reveal such effects before the turn of the millennium. What they observe – or do not observe – will shape our knowledge of nature's fundamental forces.

We consider it thus an opportune time to take stock, to represent CP invariance and its limitations in its full multi-layered complexity.

A host of mostly theoretical arguments points to the presence of New Physics (beyond the SM Higgs state) at the 1TeV scale. This led to the expectation that experiments at the LHC will find direct evidence for new degrees of freedom. This confidence has been moderated recently for some people by the fact that neither precision measurements of electroweak observables (like the masses of the weak bosons) nor the detailed data on B decays have so far shown any evidence for such New Physics entering there through quantum corrections. One potential conclusion is that the mass scale of the New Physics states – like SUSY quanta – is significantly higher than the 1 TeV scale and might be beyond the reach of the LHC for direct production. In any case, we view it as a challenge to increase our efforts in B studies.

Alternatively one can search for a principle to suppress sufficiently the impact of New Physics on B and K decays, even when the New Physics quanta enter with masses below 1 TeV. In the SUSY framework described in Chapter 19, that amounts to requiring the mass insertion parameters Δij to satisfy the bounds listed in Table 19.2 (and potentially tighter ones in the future) while retaining squark and gluino masses below 1TeV or so. This might be feasible along the lines sketched in Section 19.4.4.

On minimal flavour violation

As discussed in the preceding Chapter minimal SUSY extensions of the SM can be constructed in such a way that no additional sources of CP violation arise.

The discovery that the weak forces break previously unquestioned discrete symmetries – first parity P and charge conjugation C, then CP and T – had a revolutionizing impact on our perception of nature and how we analyse the elements of its grand design. We realized that symmetries should not be taken for granted; some even began questioning that sacrosanct fruit of quantum field theory, CPT invariance. We learnt from the violation of CP symmetry – not from that of P and C separately – that left and right or positive and negative charge are dynamically distinct rather than being mere labels based on a convention; furthermore that nature distinguishes between past and future even on the microscopic level. From 1964 to 2001 CP violation had been observed only in a single system – the decays of KL mesons – as a seemingly unobtrusive phenomenon. Yet even so we had come to understand that it represents not only a profound intellectual insight, but has also many and far-reaching concrete consequences.

• The huge predominance of matter over antimatter apparently observed in our universe requires CP violation if it is to be understood as dynamically generated rather than merely reflecting the initial conditions.

• Once the dynamics are sufficiently complex to support CP violation, the latter can manifest itself in numerous different ways; we can even say the floodgates open.

• The three-family SM can implement CP violation through the KM mechanism without requiring so-far unobserved degrees of freedom. It is already highly non-trivial that it can accommodate the data on ∈K and ∈k and ∈′ within the uncertainties.

• […]

In this chapter we discuss a particular class of N = 2 supersymmetric gauge theories in which non-Abelian strings were found. One can pose the question: what is so special about these models that makes an Abelian ZN string become non-Abelian? Models we will dwell on below have both gauge and flavor symmetries broken by the condensation of scalar fields. The common feature of these models is that some global diagonal combination of color and flavor groups survive the breaking. We consider the case when this diagonal group is SU(N)C+F, where the subscript C + F means a combination of global color and flavor groups. The presence of this unbroken subgroup is responsible for the occurrence of the orientational zero modes of the string which entail its non-Abelian nature.

Clearly, the presence of supersymmetry is not important for the construction of non-Abelian strings. In particular, while here we focus on the BPS non-Abelian strings in N = 2 supersymmetric gauge theories, in Chapter 5 we review non-Abelian strings in N = 1 supersymmetric theories and in Chapter 6 in nonsupersymmetric theories.

Basic model: N = 2 SQCD

The model we will deal with derives from N = 2 SQCD with the gauge group SU(N + 1) and Nf = N flavors of the fundamental matter hypermultiplets which we will call quarks [3]. At a generic point on the Coulomb branch of this theory, the gauge group is broken down to U(1)N.

Ever since 't Hooft [124] and Mandelstam [125] put forward the hypothesis of the dual Meissner effect to explain color confinement in non-Abelian gauge theories, people were trying to find a controllable approximation in which one could reliably demonstrate the occurrence of the dual Meissner effect in these theories. A breakthrough achievement was the Seiberg–Witten solution [2] of N = 2 supersymmetric Yang–Mills theory. They found massless monopoles and, adding a small (N = 2)-breaking deformation, proved that they condense creating strings carrying a chromoelectric flux. It was a great success in qualitative understanding of color confinement.

A more careful examination shows, however, that details of the Seiberg–Witten confinement are quite different from those we expect in QCD-like theories. Indeed, a crucial aspect of Ref. [2] is that the SU(N) gauge symmetry is first broken, at a high scale, down to U(1)N-1, which is then completely broken at a much lower scale where condensation of magnetic monopoles occurs. Correspondingly, the strings in the Seiberg–Witten solution are, in fact, Abelian strings [36] of the Abrikosov–Nielsen–Olesen (ANO) type which results, in turn, in confinement whose structure does not resemble at all that of QCD. In particular, the “hadronic” spectrum is much richer than that in QCD [126, 127, 128, 35, 129].