The effective Lagrangian method was developed by Weinberg [463] and independently by Wilczek and Zee [464]. It can be seen as a general, powerful method which allows us to quantitatively describe the effects of physics beyond the SM. The idea is that the SM is very effective at describing with high precision all experimental observations up to the tera-electronvolt scale, i.e., at “low energy.”

The equation, developed by Dirac as the union of quantum mechanics and relativity, historically led to the prediction of the existence of a new form of matter – the antimatter – previously unsuspected and unobserved and which was experimentally confirmed several years later with the discovery of the positron. The equation also entailed the explanation of spin. Altogether it represented one of the great triumphs of theoretical physics. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1/2 particles. In the Standard Model all fundamental building blocks of matter – the quarks and leptons – are represented with such Dirac fields.

During the 1940s and 1950s, the studies continued on the 𝛽 decays. It was found that not all 𝛽 decays occur between nuclear states with identical angular momenta, so the Fermi allowed transitions defined in Section 21.3, which represent a 𝛥𝐽 = 0 operator (see Eq. (21.31)), could not be a complete description.

The successful development of QED represented a great achievement: the theory was very useful, it handled matter and antimatter (electrons and positrons), it introduced the technique of renormalization, and it proved to be extremely useful and precise (for example in computing the anomalous magnetic moments). Nonetheless, QED could not simply explain even the existence of the nucleus of atoms! Indeed, what holds the nucleus together?

Lorentz symmetry is at the core of modern physics: the kinematical laws of special relativity and Maxwell’s field equations in the theory of electromagnetism respect it. The direct relativistic extension of the Schrödinger equation leads to the Klein–Gordon equation, which will be interpreted in the context of the second quantization, to describe bosons. In the Standard Model all interactions are induced by intermediate vector gauge boson fields and the Higgs boson is represented by a scalar boson field.

Hadrons are copiously produced at high-energy electron–positron or hadron–hadron colliders and provide a well-suited environment to study QCD. Electron–positron colliders are particularly well suited, since the tree-level process is the 𝑠-channel annihilation into a virtual photon and a quark–antiquark pair in the final state.

When we first observe the Universe, it might appear to us as a very complex object. One of the primary goals of the philosophy of Nature (or simply Physics) is to “reduce” (“simplify”) this picture in order to find out what the most fundamental constituents of matter (i.e., the atoms from the Greek word indivisible) are and to understand the basic forces by which they interact in the otherwise void space, along the line of thinking of Demokritos who wrote “Nothing exists except atoms and empty space.”

The weak force acts among all elementary fermions, that is, all leptons (𝑒, 𝜇, 𝜏, 𝜈) and all quarks (𝑢, 𝑑, 𝑠, …).

Particle accelerators are devices that produce different kinds of energetic, high-intensity beams of stable particles (𝑒, 𝑝, …). They possess many fields of application: nuclear and particle physics, material science, chemistry, biology, medicine, isotope production, medical imaging, medical treatments – just to name a few. Beams of accelerated primary particles can be used to produce beams of secondary metastable particles, such as 𝜇’s, 𝜋’s, 𝐾’s, etc

The Poincaré group and its Lorentz subgroup are of great importance because invariance under the Poincaré group is a fundamental symmetry in particle physics. For example, a relativistic quantum field theory must have a Poincaré-invariant Lagrangian. This means that its fields must transform under representations of the Poincaré group and Poincaré invariance must be implemented. Here we will discuss some properties of the Lorentz and Poincaré groups.

In 1961 M. Gell-Mann published a report entitled The Eightfold way: a theory of strong interactions symmetry. He wrote: “It has seemed likely for many years that the strongly interacting particles, grouped as they are into isotopic multiplets, would show traces of a higher symmetry that is somehow broken. Under the higher symmetry, the eight familiar baryons would be degenerate and form a super-multiplet.

We have now collected all the building blocks to write down the most up-to-date theory of particle physics. As we have seen in the previous chapters, it took several decades to develop it, basically in several stages during the second half of the 20th century. Its current form was finalized during the 1970s. The Standard Model (SM) describes the electromagnetic, weak, and strong interactions (but not gravity) among fundamental fermions of Nature.

The SM is without doubt a great success of particle physics. It describes a very wide range of precise experimental measurements remarkably well. Nonetheless, it is still considered as a model based on perturbative quantum field theory constructed on fundamental gauge symmetries that are spontaneously broken.