Grand Unified Theory

A Grand Unified Theory is any model in particle physics that merges the electromagnetic, weak, and strong forces into a single force at high energies. Although this unified force has not been directly observed, many GUT models theorize its existence. If the unification of these three interactions is possible, it raises the possibility that there was a grand unification epoch in the very early universe in which these three fundamental interactions were not yet distinct.
Experiments have confirmed that at high energy, the electromagnetic interaction and weak interaction unify into a single combined electroweak interaction. GUT models predict that at even higher energy, the strong and electroweak interactions will unify into one electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers, but one unified coupling constant. Unifying gravity with the electronuclear interaction would provide a more comprehensive theory of everything rather than a Grand Unified Theory. Thus, GUTs are often seen as an intermediate step towards a TOE.
The novel particles predicted by GUT models are expected to have extremely high masses—around the GUT scale of —and so are well beyond the reach of any foreseen particle hadron collider experiments. Therefore, the particles predicted by GUT models will be unable to be observed directly, and instead the effects of grand unification might be detected through indirect observations of the following:

proton decay,
electric dipole moments of elementary particles,
or the properties of neutrinos.

Some GUTs, such as the Pati–Salam model, predict the existence of magnetic monopoles.
While GUTs might be expected to offer simplicity over the complications present in the Standard Model, realistic models remain complicated because they need to introduce additional fields and interactions, or even additional dimensions of space, in order to reproduce observed fermion masses and mixing angles. This difficulty, in turn, may be related to the existence of family symmetries beyond the conventional GUT models. Due to this and the lack of any observed effect of grand unification so far, there is no generally accepted GUT model.
Models that do not unify the three interactions using one simple group as the gauge symmetry but do so using semisimple groups can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.

History

Historically, the first true GUT, which was based on the simple Lie group, was proposed by Howard Georgi and Sheldon Glashow in 1974. The Georgi–Glashow model was preceded by the semisimple Lie algebra Pati–Salam model by Abdus Salam and Jogesh Pati also in 1974, who pioneered the idea to unify gauge interactions.
The acronym GUT was first coined in 1978 by CERN researchers John Ellis, Andrzej Buras, Mary K. Gaillard, and Dimitri Nanopoulos, however in the final version of their paper they opted for the less anatomical GUM. Nanopoulos later that year was the first to use the acronym in a paper.

Motivation

The fact that the electric charges of electrons and protons seem to cancel each other exactly to extreme precision is essential for the existence of the macroscopic world as we know it, but this important property of elementary particles is not explained in the Standard Model of particle physics. While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups and which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry which in principle allows for arbitrary charge assignments. The observed charge quantization, namely the postulation that all known elementary particles carry electric charges which are exact multiples of one-third of the "elementary" charge, has led to the idea that hypercharge interactions and possibly the strong and weak interactions might be embedded in one Grand Unified interaction described by a single, larger simple symmetry group containing the Standard Model. This would automatically predict the quantized nature and values of all elementary particle charges. Since this also results in a prediction for the relative strengths of the fundamental interactions which we observe, in particular, the weak mixing angle, grand unification ideally reduces the number of independent input parameters but is also constrained by observations.
Grand unification is reminiscent of the unification of electric and magnetic forces by Maxwell's field theory of electromagnetism in the 19th century, but its physical implications and mathematical structure are qualitatively different.

Unification of matter particles

SU(5)

is the simplest GUT. The smallest simple Lie group which contains the Standard Model, and upon which the first Grand Unified Theory was based, is
Such group symmetries allow the reinterpretation of several known particles, including the photon, W and Z bosons, and gluon, as different states of a single particle field. However, it is not obvious that the simplest possible choices for the extended "Grand Unified" symmetry should yield the correct inventory of elementary particles. The fact that all currently known matter particles fit perfectly into three copies of the smallest group representations of and immediately carry the correct observed charges, is one of the first and most important reasons why people believe that a Grand Unified Theory might actually be realized in nature.
The two smallest irreducible representations of are and. In the standard assignment, the contains the charge conjugates of the right-handed down-type quark color triplet and a left-handed lepton isospin doublet, while the contains the six up-type quark components, the left-handed down-type quark color triplet, and the right-handed electron. This scheme has to be replicated for each of the three known generations of matter. It is notable that the theory is anomaly free with this matter content.
The hypothetical right-handed neutrinos are a singlet of, which means its mass is not forbidden by any symmetry; it doesn't need a spontaneous electroweak symmetry breaking which explains why its mass would be heavy.

SO(10)

The next simple Lie group which contains the Standard Model is
Here, the unification of matter is even more complete, since the irreducible spinor representation contains both the and of and a right-handed neutrino, and thus the complete particle content of one generation of the extended Standard Model with neutrino masses. This is already the largest simple group that achieves the unification of matter in a scheme involving only the already known matter particles.
Since different Standard Model fermions are grouped together in larger representations, GUTs specifically predict relations among the fermion masses, such as between the electron and the down quark, the muon and the strange quark, and the tau lepton and the bottom quark for and. Some of these mass relations hold approximately, but most don't.
The boson matrix for is found by taking the matrix from the representation of and adding an extra row and column for the right-handed neutrino. The bosons are found by adding a partner to each of the 20 charged bosons and adding an extra heavy neutral Z-boson to make 5 neutral bosons in total. The boson matrix will have a boson or its new partner in each row and column. These pairs combine to create the familiar 16D Dirac spinor matrices of.

E₆

In some forms of string theory, including E₈ × E₈ heterotic string theory, the resultant four-dimensional theory after spontaneous compactification on a six-dimensional Calabi–Yau manifold resembles a GUT based on the group E₆. Notably E₆ is the only exceptional simple Lie group to have any complex representations, a requirement for a theory to contain chiral fermions. Hence the other four can't be the gauge group of a GUT.

Extended Grand Unified Theories

Non-chiral extensions of the Standard Model with vectorlike split-multiplet particle spectra which naturally appear in the higher SU GUTs considerably modify the desert physics and lead to the realistic grand unification for conventional three quark-lepton families even without using supersymmetry. On the other hand, due to a new missing VEV mechanism emerging in the supersymmetric SU GUT the simultaneous solution to the gauge hierarchy problem and problem of unification of flavor can be argued.
GUTs with four families / generations, SU: Assuming 4 generations of fermions instead of 3 makes a total of types of particles. These can be put into representations of. This can be divided into which is the theory together with some heavy bosons which act on the generation number.
GUTs with four families / generations, O: Again assuming 4 generations of fermions, the 128 particles and anti-particles can be put into a single spinor representation of.

Symplectic groups and quaternion representations

Symplectic gauge groups could also be considered. For example, has a representation in terms of quaternion unitary matrices which has a dimensional real representation and so might be considered as a candidate for a gauge group. has 32 charged bosons and 4 neutral bosons. Its subgroups include so can at least contain the gluons and photon of. Although it's probably not possible to have weak bosons acting on chiral fermions in this representation. A quaternion representation of the fermions might be:
A further complication with quaternion representations of fermions is that there are two types of multiplication: left multiplication and right multiplication which must be taken into account. It turns out that including left and right-handed quaternion matrices is equivalent to including a single right-multiplication by a unit quaternion which adds an extra SU and so has an extra neutral boson and two more charged bosons. Thus the group of left- and right-handed quaternion matrices is which does include the Standard Model bosons:
If is a quaternion valued spinor, is quaternion hermitian matrix coming from and is a pure vector quaternion then the interaction term is: