This short chapter touches on the limitations of the SM. The SM does not include gravity, and it does not explain the major components of the mass–energy budget of the universe, dark matter and dark energy, the latter being probably the cosmological constant. CP violation in the quark sector is too small to explain the matter–antimatter asymmetry of the Universe, but, if confirmed, the non-SM CP violation in the neutrino sector might be large enough. The ‘strong CP violation’ problem might be solved with the existence of a very light particle, the axion; experiments are reaching the requested sensitivity. Supersymmetric particles present in some extensions of the SM have been searched for, but not found so far.

The SM contains too many free parameters: the masses of the fermions and of the bosons, and the mixing angles. The masses of the fermions, from neutrinos to the top quark, span 13 orders of magnitude. Why such big difference? Why is mixing small in the quark sector, and large in the neutrino sector? Why do the proton and the electron have exactly equal (and opposite) charges? Why are there just three families? Are there any spatial dimensions beyond the three we know? And so on.

In classical electrodynamics (CED), the most important quantities are the electric and magnetic fields, which directly determine the forces. In quantum electrodynamics (QED), the potentials are the most important quantities; they determine the energy and momentum exchange between the EM field and matter. Gauge invariance, which in CED is just a mathematical curiosity, becomes fundamental in QED, ruling the gauge symmetry that determines the interaction itself.

The Lamb experiment that opened the way is discussed in detail.

Feynman diagrams are graphic representations of mathematical expressions of scattering or decay amplitudes. Without going into the mathematics, we use them to visually suggest the underlying physics. We show how the propagator describes virtual particles, and how uncertainty and relativity principles, joined, imply the existence of antimatter.

The fine-structure constant, which is the dimensionless expression of the electromagnetic charge, depends on the momentum transfer between the probe and the target charge in the scattering experiment we are performing. The ‘running’ of the coupling constants is a property of all the interactions.

The highest precision measurements and theoretical predictions of the magnetic moments of the electron and of the muon. The precision frontier to search for new physics.

The principle of relativity requires that no interaction can propagate instantly. Gravitational waves (GW) must exist, propagating with the same speed as light. The specific characters of GW are predicted by Einstein’s general relativity (GR). After decades of efforts to develop detectors, on 11 February 2016, the LIGO and Virgo Collaboration published the discovery of a GW.

The elements of GR relevant for GW production, propagation and detection. How the GR field, which is the dimensionless metric tensor, differs from the other fundamental fields, which have physical dimensions. The instruments and the discovery. After the first observation, dozens of gravitational signals have been detected, the vast majority from merging black holes and one, on 17 August 2017, from the merger of neutron stars. In this case, electromagnetic signals are expected, and have been detected, providing unique information to astrophysics and to fundamental physics as well. The measurement of the speed of the GW and the establishment of a bound on the mass of the graviton.

Oscillations between members of flavoured, electrically neutral meson pairs and the CP violation are phenomena strictly connected with the mixing. However, CP is more general, having been observed also in the decay of charged mesons.

CP violation was first observed in the neutral K system. We see the states of definite strangeness, those of definite CP and those with definite mass and lifetime. The oscillation between the former states, the mathematical expressions and the experimental evidence.

The oscillations and CP violation in the B0 system, and the beautiful experimental results obtained at dedicated high-luminosity electron–positron colliders, the ‘beauty factories’. Beauty physics at the dedicated experiment LHCb at LHC, in particular for the B0, that is not accessible to beauty factories. Examples of CP violation in B0. The recent discovery of CP violation in the charm sector.

How the many different measurements can be put together to test the SM with the unitary triangle.

Neutrinos are the most difficult particle to study, because they interact only via weak interactions. However, they have given revolutionary surprises, and it is with neutrinos that physics beyond the SM has been discovered. In the SM, neutrino masses are rigorously zero, but experiments show that they do have a mass. In the SM, neutrino flavour eigenstates are mass eigenstates; experiments show that they are mixtures of them. Two discoveries proved this. One is neutrino oscillations, discovered in atmospheric neutrinos, the other is the adiabatic flavour conversion in matter, discovered in solar neutrinos.

These were with natural neutrinos. Several experiments have been, and are being, performed with artificial neutrinos from reactors or accelerators to measure with increasing accuracy the neutrino mixing matrix and the mass spectrum. We found that the neutrino mixing is much larger than that of the quarks. Nobody knows why.

The SM assumes neutrinos to be different from antineutrinos, but no experimental proof of it exists. Neutrinos and antineutrinos may well be the same particle, a Majorana spinor. We see how this is searched for by looking for the extremely rare double beta decay.

The weak interaction was proposed by Fermi in 1933, to interpret the beta decay. The interaction Lagrangian is the product of two charged currents (CC) – one of the nucleons, one of the leptons. It was later discovered that parity and charge conjugation are not conserved and that the structure of the charged currents is a combination of vector and axial currents, V–A. The beautiful Goldhaber experiment on the helicity of the neutrino.

The coupling of all leptons is universal, but not that of the quarks. To obtain universality, Cabibbo introduced the concept of mixing of the hadronic currents, namely of quarks. Then the Glashow–Iliopoulos–Maiani mechanism solved a problem introducing the hypothesis that a fourth quark would exist, the charm, completing a doublet with the strange one. With the discovery of two more quarks, the quark mixing matrix contains a phase factor that is the origin of CP violation in the Standard Model.

The weak neutral currents were discovered with the Gargamelle bubble chamber at CERN in 1973. This showed a close similarity between weak and electromagnetic interactions and opened the way to their unification.

If the mass of a hadron is large enough, decays into final states that can be reached by strong interaction, that is, without violating any selection rule, are possible. The lifetime is then extremely short, of the order of a yoctosecond (10−24 s). These hadrons decay practically where they were born. We show how they are observed as ‘resonances’.

Hadrons, both baryons and mesons, were discovered in rapidly increasing numbers in the 1950s and early 1960s. How their quantum numbers, spin, parity and isospin were measured. Gradually it became clear that hadrons with the same spin and parity could be grouped in multiplets of the SU(3) symmetry. Proposal of the quark model and experimental verifications of its predictions. With increasing accelerator energies, more surprises were to come. The quarks are not only the three originally known, u, d and s, but three more exist, c, b and t. And more leptons were found, in total three ‘families’ of fundamental fermions, each with two quarks, a charged lepton and its neutrino.

In modern physics, symmetries are a powerful tool to constrain the form of equations, namely the Lagrangian that describes the system. Equations are assumed to be invariant under the transformation of a given group, which may be discrete or a continuous Lie group. Classification of the various types of symmetry. The concept of spontaneous symmetry breaking. It will evolve into the Higgs mechanism, which gives origin to the masses of the vector bosons that mediate the weak interactions, of the quarks and of the charged leptons.

The discrete symmetries, in particular the parity and the particle–antiparticle conjugation operations and the corresponding quantum numbers.

An important dynamical symmetry of the hadrons, the invariance of the Lagrangian under rigid rotations in an ‘internal’ space, the isospin space. The unitary group is SU(2).