Finally, we will be able to present a grand summary of weak decays, list open questions, andpresent a shoping list for the coming years.energy observables that could be useful for the search for NP, in particular beyond the reach of the LHC. Subsequently, with the goal to distinguish various NP models we will discuss DNA charts proposed already in a number of papers.

Next we will discussflavor-changing neutral current (FCNC) processes. In the SM such processes can only occur first at the one-loop level due to Glashow–liopoulos–Maiani (GIM) mechanism, a very important mechanism in the field of weak decays. We will first present the general structure of these processes, which will involve a set of basic master one-loop functions. We will calculate most ofthese functions explicitly without including first QCD corrections. Subsequently we will show how these functions enter the operator product expansion. Nextwe will include QCD and electroweak corrections to a number of FCNC processes. This will result in a number of effective Hamiltonians for most important processes thatinvolve new operators not encountered in previous chapter: the so-called penguin operators. In this step we will study several properties of these Hamiltonians but we will postpone detailed phenomenology of these processesto later chapters.

We will next attempt a grand summary of NP models discussed by us comparing the different patterns of flavor violation present in these models. While they have been discussed already previously in some detail, it will be instructive to compare how different models face various anomalies listed in Chapter 12. This chapter is really a general view on the literature related to various anomalies and after being completed will hopefully motivate the readers to study numerous papers containing many ideas with the goal to construct their own models.

In order to describe properly the physics above the electroweak scale, wewill discuss next the so-called gauge invariant Standard Model Effective Theory (SMEFT). This theory became very popular in view of the fact that the LHC until now did not discover directly any new particles. It offers a model independent formulation of NP beyond the SM in terms of operators that are invariant under the full SM gauge group. This is a very important framework, and many techniques developed in this decade in the context of SMEFT are crucial for a serious phenomenological applications in order to obtain model independent results. Moreover, SMEFT technology plays an important role in the derivation of predictions in concrete models as well.We will see this in this chapter and subsequent chapters of our book. But already in this chapter we will discuss implications of SMEFT in a number of general NP scenarios in which only a subset of operators is relevant. Moreover, we will present a few general formulas valid for any NP model, in particular the one for the ratio epsilon'/epsilon.

In order to illustrate what happens in specific models we will discuss three classes of models: the so-called 331 models, models with heavy vector-like quarks and models with leptoquarks, either of spin-0 or spin-1. We will concentrate on models with leptoquarks having sufficiently low masses so that they could be in principle discovered by the LHC. But even if the masses of new particles in the models considered will turn out to be beyond the reach of the LHC, their presence can still be identifiedin the rare processes discussed by us. These three classes of models are sufficiently simple so that we can present them here in some details. The new aspect of these models will be larger involvement of leptons than in previous sections. Yet, in thischapter we will confine the phenomenology to quark flavour observables postponing the discussion of lepton flavor violation to the next chapter. There are several more complicated models that we will only briefly mention providing list of references to papers investigatingweak decays in these models.

This chapter deals with basic aspects of flavor physics in the charm sector. It isnot as detailed as other ones. Yet, it contains many fundamentals related to CP violation and mixing in the charm sector.

In this chapter we present a detailed discussion of the particle-antiparticle mixing in the SM and perform the classification of various types of CP violation. We will discuss the determination of the CKM parameters in tree-level decays and also the determination of the so-called unitarity triangle from particle-antipartcle mixing both from Band K meson systems and tree-level decays in question. In this context we will present various methods for the determination of the angles in this triangle from certain leading decays. The considered decays are subject to only small contributions from NP so that the CKM parameters can be extracted without knowledge of what happens beyond the SM. This is useful as then the results of tree-level determinations of CKM parameterscan be used to find SM predictions forparticle-antiparticle mixing observables considered already in this chapter and for rare processes considered in the next two chapters. Both particle-antiparticle mixing and rare processes are subject to potential NP contributions and could help us to identify NP through the pattern of deviations of the data for themfrom SM predictions.

In this chapter we will discuss flavor physics of leptons, in particular charged lepton decays mediated by flavor changing neutral currents. While very strongly suppressed within the SM, lepton flavor violation (LFV)can be by many orders of magnitude larger in NP models. Therefore finding such transitions would be a clear signal of NP.The nice feature of most of these decays is the absence of QCD corrections, making the theoretical analysis much simpler than was the case of meson decays. Another important topic are electric dipole moments (EDMs) of various atoms, molecules, nuclei, and nucleons that all are very strongly suppressed in the SM. They test CP violation without flavor violation. Similar to LFV, a measurement of any nonvanishing EDM would be a clear signal of NP at work. But in contrast to LFV there are large nonperturbative uncertainties in this case. In this part we will also discuss anomalous magnetic moments of charged leptons. We will present a compendium of formulas for many observables. These formulasare quite general so that one canuse them for various models. We will illustrate them on a few examples.

Chapter 14 will be rather general, and it will be strategically useful to look next at specific simple examples of NP. Having already all relevant SM formulas for flavor observables at hand, it will be of interest to see how they generalize beyond the SM. In this step we will first discuss the concept of minimal flavor violation (MFV), that is the most modest modification of the flavor-violating effects found in the SM. In these models a number of strict correlations between flavor observablesare found. Next we will go go beyond MFV models considering the so-called simplified models in which NP contributions to flavor-violating processes are dominated by tree-level exchanges of neutral bosons, in particular the SM Z-boson, a heavy neutral gauge boson Z', and a heavy scalaror pseudoscalar. In these models FCNC processes appear already at tree-level modifying in a striking manner the pattern of flavor violation found in the SM and in MFV models. Moreover, right-handed currents will enter the scene. We will see that some of thesesimplified models will give us some insight into the origin of anomalies found in previous chapters.

We will next move to discuss the Standard Model (SM) of electroweak and strong interactions. We will present the basic SM Lagrangian, andwe will discuss various properties of this simplest theory, but we will not do any phenomenology at this stage. There are many excellent books on the dynamics of the SM, and we will in most cases skip derivations of well-known formulae to save space and in particular energy for the later steps that are not covered in detail in textbooks. The most important result of this step will be collection of the Feynman rules for those interactions present in the SM that we will need for explicit calculations as we proceed. For readers' convenience we will collect these rules in an Appendix.

The phenomenological applications of theformulae collected in Chapter 6 require input from lattice QCD (LQCD) and other nonperturbative approches. In particular one needsthe values of hadronic matrix elements of quark currents and those of four-quark operators. In this chapter we will collect all results for hadronic matrix elements, weak decay constants, and other nonperturbative objects in one place so that they can be looked up quickly whenever necessary. But before presenting the results obtained by purely numerical LQCD calculations, we will discuss the application of the so-calleddual QCD approach to weak decays, which was developed by Bardeen, Gerard, and myself over many years. While not as accurate as the lattice QCD method, it offers a better insight in the dominant QCD dynamics at long-distance scales than LQCD itself. The application of LQCD to nonleptonic decays of B mesons is very difficult, and therefore we will devote a section to the QCD factorization approach, stressing its successes and limitations.

In this chapter most important rare $K$ and B decaysand related observables are discussed. The presentation is based on the information gained in previous chapters. In particular wecollect many formulas that are necessary to perform the phenomenology of weak decays of mesons within the SM. This chapter is rather long but a very important one for the subsequent chapters. The general expressions for various observablesremain often unchanged in many NP models, and only the master functions of Chapter 6 will be modified. But in order to be prepared for the study of the physics beyond the SM, we will already here present rather general formulas that contain new operators that are absent in the SM or are very strongly suppressed.