II.2 Symmetry and Super Symmetry in the Standard Model

The Standard Model of Elementary Particles was discovered in the 1960s when physicists realized that Quantum Electrodynamics broke down at extremely high energies. This inconsistency led to extended Quantum Field Theory, which unified the electromagnetic and weak interactions into the electroweak theory. The Standard Model also contains a high energy unification of the electroweak with the strong force, described by Quantum Chromodynamics. However, although there are some postulates about gravity as another gauge theory, but this is not yet fully understood.

In 1964, Peter Higgs, and other physicists, proposed the Higgs mechanism that suggested the existence of a particle that was confirmed in 2013. This prediction of the Higgs boson, that explained how inertial mass is formed, established the Standard model as the most comprehensive description of particle physics, but it can t be considered complete without explaining gravity. It explains three of the four known fundamental interactions, and classifies all the known elementary particles.

The Standard Model was developed by the collective work of many scientists from around the World, and it finalized after the experimental confirmation of the existence of quarks in 1968. The Standard Model predicted various properties of weak neutral currents and the  and  bosons, with great accuracy. Other predicted particles were later confirmed, including the top quark in 1995, the tau neutrino in 2000, and the Higgs boson in 2013.

However, some other related phenomena are still unexplained, such as gravity and baryon asymmetry, in addition to its huge discrepancy in calculating the vacuum energy density which is part of the cosmological constant problems. Additionally, the Standard Model does not incorporate neutrino oscillations and their non-zero masses, also known as Yang-Mills mass gap problem.

The Standard Model explains matter and energy in terms of the kinematics and interactions of elementary particles. It have succeeded in describing the behavior and interaction, of all known forms of matter and energy, using a small set of fundamental laws and theories. However, the major goal of physics is to unite all of these theories, in addition to gravity, into one integrated Theory of Everything, which should be able to explain the behavior of all matter and energy, and allow the derivation of all the other known laws, that would be some of its special cases.

The Quantum Field Theory provides the mathematical framework for the Standard Model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time.

The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.

The global Poincar symmetry is postulated for all relativistic Quantum Field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance, central to the theory of Special Relativity.

The Standard Model is a non-Abelian gauge theory quantized through path-integrals. The local  gauge symmetry is an internal symmetry that essentially defines the Standard Model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions.

The fields fall into different representations of the various symmetry groups of the Standard Model: ,  and . Upon writing the most general Lagrangian, one finds that the dynamics depends on 19 parameters, whose numerical values are arbitrary and unrelated, but they are established by experiment.

Figure II.3: The Standard Model of Elementary Particles, in addition to the hypothesized Graviton.

As summarized in Figure II.3, the Standard Model classifies the elementary particles as: fermions, gauge bosons, and the Higgs boson, which in turn can be distinguished by other characteristics, such as spin, charge, color charge, generation, and mass. There are 12 elementary particles of spin-half which are the fermions. According to the spin-statistics theorem, fermions respect the Pauli exclusion principle. Each fermion also has a corresponding antiparticle, that is equal in mass but having opposite charge.

The  fermions are the six quarks: up, down, charm, strange, top, bottom, and the six leptons: electron, electron neutrino, muon, muon neutrino, tau, tau neutrino. The defining property of the quarks is that they carry color charge, and hence, interact via the strong interaction. The quarks are strongly bound to one another, forming color-neutral composite particles (hadrons) containing either a quark and an anti-quark (mesons) or three quarks (baryons). There are no free quarks in nature, which is called color confinement. This phenomena is explained in the Duality of Time Theory as a result of their being confined in lower dimensions, which also explains why the gluons have mass, despite being bosons.

Quarks also carry electric charge and weak iso-spin. Hence, they interact with other fermions both electromagnetically and via the weak interaction. The familiar proton and neutron are the two baryons having the smallest mass, while the remaining six fermions do not carry color charge, and they are called leptons. The three neutrinos also do not carry electric charge either, so their motion is directly influenced only by the weak nuclear force, which makes them notoriously difficult to detect.

There are three generations, as shown in Figure II.3. Each member of a higher generation has greater mass than the corresponding particles of lower generations. The first generation charged particles do not decay; hence all ordinary (baryonic) matter is made of such particles. Specifically, all atoms consist of electrons orbiting around atomic nuclei, ultimately constituted of up and down quarks.

Second and third generation charged particles, on the other hand, decay with very short half lives, and are observed only in very high-energy environments, while neutrinos of all generations do not decay, but rarely interact with baryonic matter.

Gauge bosons are the force carriers that mediate the strong, weak, and electromagnetic fundamental interactions. At a macroscopic level, electromagnetism allows particles to interact with one another via electric and magnetic fields, just as gravitation allows particles with mass to attract one another. The Standard Model explains such forces as resulting from matter particles exchanging gauge bosons which are known as force mediating particles . All gauge boson have spin one, so they do not follow the Pauli exclusion principle that constrains fermions. That is why photons, for example, do not have a theoretical limit on their spatial density.

Photons mediate the electromagnetic force between electrically charged particles, and they are massless. The , , and  gauge bosons mediate the weak interactions between particles of different flavors, i.e. all quarks and leptons. These gauge bosons, however, have considerable mass, as in Figure II.3. The  bosons also carry an electric charge, so they couple to the electromagnetic interaction, and the weak interactions involving these charged bosons exclusively act on left-handed particles and right-handed antiparticles. On the other hand, he electrically neutral  boson interacts with both left-handed particles and antiparticles.

Gluons are massless and they mediate the strong interactions between color charged particles which are the quarks only. There are eight gluons, labeled by a combination of color charge and anti-color charge, e.g. red-anti-green. Because the gluons have an effective color charge, they can also interact among themselves. The gluons and their interactions are described by the theory of Quantum Chromodynamics.

As we mentioned in section II.1.5 above, the Higgs boson was predicted in 1964 and confirmed by the Large Hadron Collider in 2013. The Higgs boson explains why the other elementary particles, except the photon and gluon, are massive, or actually why the photon has no mass, while the  and  bosons are very heavy. The Higgs boson generates the masses of the leptons and quarks, and it is very massive it decays almost immediately when created, so that only a very high-energy particle accelerator can observe and record it.

However, there are still many unsolved problems with the Standard Model, mainly because it is actually anad hocmodel which depends on 19 parameters that have to be fixed experimentally, it does not explain why particle have their measured masses and other coupling constants. Another problem is the asymmetry of matter and antimatter in the visible Universe, which is composed almost entirely of ordinary matter.

There is also no clue as to why there are three generations of particles, as we notice in Figure II.3. Additionally, neutrino masses, Dark Matter and gravity are not explained at all in this model. Most of these problems are easily explained in the Duality of Time, as we shall discuss further in section III.3.2 below and also in Chapter II.

As we shall discuss also in section II.2.6, the problem of the Yang-Mills existence and mass gap, is also one of the major unsolved problems in physics and mathematics, and it is one of the seven Millennium Prize Problems that were stated by the Clay Mathematics Institute in 2000. This problem is related to the Standard Model because it is formulated as a non-Abelian gauge theory.

Therefore, there had been various theoretical and experimental attempts to extend the Standard Model into a Unified field theory, or a Theory of Everything, to tackle these problems and include gravity. There are yet no accepted or verified concepts in this regard. We shall discuss some of these extensions in the Chapter II when we tackle the problem of super symmetry.