This book takes a fresh approach to the teaching of discrete symmetries which are central to fundamental physics: mirror symmetry, matter/anti-matter symmetry, and time reversal. It is self-contained and includes detailed discussions of relevant experiments - conveying some of the fascination and intellectual challenges of experimental physics.

This lecture note provides a tutorial review of non-Abelian discrete groups and presents applications to particle physics where discrete symmetries constitute an important principle for model building. While Abelian discrete symmetries are often imposed in order to control couplings for particle physics—particularly model building beyond the standard model—non-Abelian discrete symmetries have been applied particularly to understand the three-generation flavor structure. The non-Abelian discrete symmetries are indeed considered to be the most attractive choice for a flavor sector: Model builders have tried to derive experimental values of quark and lepton masses, mixing angles and CP phases on the assumption of non-Abelian discrete flavor symmetries of quarks and leptons, yet lepton mixing has already been intensively discussed in this context as well. Possible origins of the non-Abelian discrete symmetry for flavors are another topic of interest, as they can arise from an underlying theory, e.g., the string theory or compactification via orbifolding as geometrical symmetries such as modular symmetries, thereby providing a possible bridge between the underlying theory and corresponding low-energy sector of particle physics. The book offers explicit introduction to the group theoretical aspects of many concrete groups, and readers learn how to derive conjugacy classes, characters, representations, tensor products, and automorphisms for these groups (with a finite number) when algebraic relations are given, thereby enabling readers to apply this to other groups of interest. Further, CP symmetry and modular symmetry are also presented.

Symmetries and their conservation laws play a fundamental role in Physics. Among them, the discrete symmetries corresponding to charge (C), parity (P), and time (T) transformations are extensively used in the theory of the elementary particles and their interactions (so called Standard Model (SM)) to give the basis of the fundamental physical description of nature. Eventual discoveries of violations of these symmetries become a crucial test for our understanding of the nature. It was assumed that the three discrete symmetries were not violated until 1956 when it was found that P is violated in the weak interaction. Soon it was understood that also the C is violated in the weak interaction. At that time these two violated symmetries were replaced by their combination, CP, which was considered a new fundamental symmetry. In 1964 also the CP was found violated in the case of the neutral K meson system. Since that year there were many achievements in theories and experiments in order to explain this symmetry violation. In the last five years the main contribution comes from the discovery of the CP violation in B meson system. In this note we will describe briefly how the CP violation is described in the SM and the main experimental results obtained in the B mesons system.

Discrete symmetries as parity P, charge conjugation C and time reversal T, as well as their combinations such as CP and CPT play a fundamental role in particle physics. The CP symmetry violation was discovered unexpectedly in 1964 during the regeneration studies of the neutral K mesons, and result in a far-reaching consequences. In the framework of the Standard Model it implies the existence of the third generation of quarks. Moreover, it is a very important mechanism which could have an essential contribution to the asymmetry between matter and antimatter in the Universe. Parameters describing CP violation in neutral kaon decays were measured with a good precision and at present the main experimental effort is focused on studies of the neutral B and D meson systems. However, there are still several interesting open issues in the kaon physics, like rare Ks meson decays. This work is focused on the measurement of the rare CP violating Ks 3 0 decay branching ratio based on the data sample gathered with the KLOE detector operating at the -factory DA NE in the Italian National Center for Nuclear Physics in Frascati.

Electric dipole moments (EDMs) have interested physicists since 1950, when it was first suggested that there was no experimental evidence that nuclear forces are symmetric under parity (P) transformation. This question was regarded as speculative because the existence of an EDM, in addition to P violation, requires a violation of time-reversal (T) symmetry. In 1964 it was discovered that the invariance under CP transformation, which combines charge conjugation (C) with parity, is violated in K-meson decays. This provided a new incentive for EDM searches. Since the combined operations of CPT are expected to leave a system invariant, breakdown of CP invariance should be accompanied by a violation of time-reversal symmetry. Thus there is a reason to expect that EDMs should exist at some level. The original neutron EDM experiments were later supplemented with checks of T invariance in atoms and molecules. These investigations are pursued now by many groups. Over the years, the upper limit on the neutron EDM has been improved by seven orders of magnitude, and the upper limit on the electron EDM obtained in atomic experiments is even more strict.