Couplings and asymptotic freedom

Each of the four fundamental interactions can be characterized by a dimensionless parameter expressed in terms of universal constants, the coupling constant. In the lagrangian of strong interactions, the coupling constant is parameterized by the gauge coupling parameter,, giving in natural units . However, in a general field theory, the effective coupling constant is not a constant, … Continue reading Couplings and asymptotic freedom

Beyond the Standard Model

After the discovery of a Higgs boson, all the particles included in the SM have been experimentally observed. Despite its extraordinary phenomenological success, the SM is considered an incomplete theory. It manifests some theoretical underlying problems which suggest the need for a "deeper" theory of which the Standard Model could be just an effectual theory. Furthermore, … Continue reading Beyond the Standard Model

The Higgs mechanism

The simplest way to implement the SSB in the electroweak theory is achieved by adding the Higgs field, a complex scalar isospin doublet with four degrees of freedom, Without affecting the gauge invariance, it is possible to add to the Lagrangian of the electroweak interaction (*) the term which indicates the SSB, with  and in the potential explicited … Continue reading The Higgs mechanism

Spontaneous symmetry breaking

One of the most important missing aspects of the SM theory has been for a long the origin of mass of fundamental particles. The Lagrangian of the full Standard Model (*) well-describes the interactions of matter and radiation, but it does not include any mass term for the introduced bosons, while fermion masses are put by hand via … Continue reading Spontaneous symmetry breaking


Fields with physical significance are those that verify the Euler--Lagrange equations, or that satisfy the Hamilton's principle. For such systems, a general and systematic procedure is available to establish conservation theorems and constants of motion, as a consequence of invariance properties. Thus, conservation laws and selection rules observed in Nature may be imposed as symmetries of … Continue reading Symmetries

So, what about the Standard Model?

It is now possible to state that the Standard Model is a gauge theory invariant under  symmetry group given by the direct product of the symmetry group of the strong interaction, , with that of electroweak interaction, which also contains the symmetry group of electromagnetic interaction . The general form involving these fields is where: indicate leptons … Continue reading So, what about the Standard Model?

Electromagnetic interactions

The electromagnetic interaction is described by a gauge theory headed by the symmetry group whose density of the Lagrangian can be written as: where is the electromagnetic field tensor and is the gauge covariant derivative, with indicating the covariant four-potential of the electromagnetic field, generated by the charge ; is the external field due to … Continue reading Electromagnetic interactions