The Standard Model (SM) was motivated by the weak interaction observed in many decay processes starting from β decay, inverse β decay, muon decay, and many other fundamental processes in nuclear physics. The first attempt to understand the nature of weak interaction in neutron β decay was done by Fermi by taking analogous current-current interaction in Quantum Electrodynamics (QED). The neutron β decay can be explained quite well at energy scale around few GeV by using a particular form of the Fermi Theory in the picture of current-current interaction Hamiltonian density as
HI = GF
√2Jµ†Jµ, (2.1)
where the current is written in terms of hadronic and leptonic currents of the form
Jµ = Jµhad+ Jleptµ (2.2)
= pγ¯ µ(1− γ5)n + ¯ν γµ(1− γ5)e , (2.3) where p, n, ν, and e stand for fermion fields of the proton, neutron, neutrino, and electron.
The constant GF that appears in the above equation is called the Fermi coupling constant and
its value has been determined from low energy experiment as [11]
GF = 1.1663787× 10−5GeV−2. (2.4)
From the magnitude of Fermi constant above, the weak interaction indeed quite weak, and extremely short ranged. All particles that exist in nature (hadrons and leptons) were known to interact weakly. Furthermore, their interactions are represented by SU (2)L multiplets of weak isospin group. As a consequence, the currents in Eq. (2.3) encodes the weak isospin structure within it, and the corresponding currents are called weak isospin currents. The V− A (vector minus axial vector) structure of the weak currents stems from the fact that the parity is violated in weak processes. Moreover, this also implies that only left-handed component of the fermion fields contribute in the weak interaction. The electric charge of any particles involved in weak interaction can be realized by assigning additive quantum number known as weak hypercharge U (1)Y such that
Q= T3+Y , (2.5)
where Q, T3 and Y correspond to the electric charge (in unit of e), the third component of weak isospin generator, and the weak hypercharge of the particle. This assignment was result from the experimental fact that both weak isospin and weak hypercharge quantum number are violated in weak interaction but leave the electric charge intact.
The shortcoming of the four fermion interaction described above is that they are non-renormalizable. This can easily be seen from the existence of inverse square dimension of the Fermi coupling constant GF. At the same era, it was shown that the gauge theory can be used to explain physical interaction and it is compatible with renormalizability. Thus, it is natural to incorporate the gauge theory to describe the weak interaction such that it takes the form of V− A structure in the low energy limit. The gauge theory dictates the need of the mediators in every interaction. This so called gauge boson acts as a messenger that brings the information being exchanged during the interaction. Due to the short ranged nature of the weak interaction, one expects that the gauge boson to be massive. This is where the Higgs mechanism enters the game. Utilizing the Higgs mechanism will cause the gauge symmetry to be spontaneously broken, but as an advantage, one obtains massive gauge bosons. This is exactly what we want, since the weak interaction violates both of weak isospin and weak hypercharge quantum number, one can employ the Higgs mechanism to spontaneously break the SU (2)L and U (1)Y symmetry to get the corresponding massive gauge bosons while preserving the electromagnetic gauge symmetry. Diagrammatically, this
Fig. 2.1The fundamental particles in the Standard Model. (Source: wikipedia.org)
can be written in the following
SSB : SU (2)L×U(1)Y → U(1)EM, (2.6)
where SSB denotes spontaneously symmetry breaking. With this, one can construct the gauge theory of the electroweak interaction or known as the Standard Model (SM).
The Standard Model (SM) was developed to explain the weak and electromagnetic interaction of all particles observed in nature. All fundamental particles known so far are listed in Fig. 2.1. One notices that instead of lepton and hadron that appear in the V− A current, the SM matter fields consist of leptons and quarks which appear in three families.
This is due to the fact that hadron is the bound state of quarks. Quarks and leptons are spin 1/2 elementary particles that respect Fermi-Dirac statistics [12] widely known as fermions. The main difference between those two species is that quarks are involved in strong interaction which is responsible to hold protons and neutrons inside the nucleus, while leptons are inert to it.
According to the weak isospin quantum number (flavor), there are 6 types of quarks and leptons, six flavors for quarks: up-quark (u), down-quark (d), charm-quark (c), strange-quark (s), top-quark (t) and bottom-quark (b), and six flavors for leptons: electron (e), electron neutrino (νe), muon (µ), muon neutrino (νµ), tau (τ) and tau neutrino (ντ). They are arranged into three groups with similar quantum number assignments even though their masses are not the same. Each family consists of two quark and two lepton flavors. As an example, the second family has c, s quarks and νµ, µ leptons. In unit of e > 0, up-type quarks (u, c,t) have similar electric charges of +23, and down-type quarks (d, s, b) have electric charges
−13. For leptonic groups, e, µ, τ have identical electric charges of−1 and the neutrinos are electrically neutral. Moreover, for each fermion listed on the Fig. 2.1, there is a corresponding antiparticle which has the same properties as the original fermion except its opposite electric
charge. The antiparticle of u quark is anti u quark ( ¯u), the antiparticle of electron (e−) is the positron (e+) etc. It is known that leptons exist as free particles in nature while quarks always appear in form of hadronic bound states due to the nature of strong interaction. Furthermore, hadron that is formed by odd number of quarks is identified as baryon, while meson consists of quark and antiquark pair.
Moving forward to the messenger of the SM interaction, there are gauge bosons that follow the Bose-Einstein statistics [13]. The bosons are characterized by their integral spin quantum number. The gauge bosons are spin 1 particles that mediate every interaction described by the gauge interaction. The weak interaction is mediated by three massive gauge bosons W+, W− and Z. The electromagnetic interaction occurs by exchanging massless spin one photon (γ). Furthermore, these two interactions belong to the unified framework of the electroweak interaction. Finally, the strong interaction which is responsible for the nuclear forces requires massless colored gluons as their messenger.
Lastly, the SM requires a special boson to generate masses for all the fundamental particles listed above. This boson is called the Higgs boson which is a spin zero scalar particle responsible for the spontaneously symmetry breaking in SM. This boson was discovered in 2012 [7, 8] at the large hadron collider (LHC). It was shown that this particle has a mass around 125 GeV [14] while its other properties are still under careful investigation.