Editing Grand Unified Theory (section)

== Unification of matter particles ==
{{For|an elementary introduction to how Lie algebras are related to particle physics|Particle physics and representation theory}}
[[File:SU(5) representation of fermions.png|thumb|Schematic representation of fermions and bosons in {{math|SU(5)}} GUT showing {{math|'''5''' + '''10'''}} split in the multiplets. Neutral bosons (photon, Z-boson, and neutral gluons) are not shown but occupy the diagonal entries of the matrix in complex superpositions.]]

=== SU(5) ===
{{main|Georgi–Glashow model}}

[[File:Georgi-Glashow charges.svg|200px|right|thumb|The pattern of [[weak isospin]]s, weak hypercharges, and strong charges for particles in the [[Georgi–Glashow model|SU(5) model]], rotated by the predicted [[weak mixing angle]], showing electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes twelve colored X bosons, responsible for proton decay.]]

{{math|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
: <math> \rm SU(5) \supset SU(3)\times SU(2)\times U(1) .</math>

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 representation]]s of {{math|SU(5)}} 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 [[group representation#Reducibility|irreducible representations]] of {{math|SU(5)}} are {{math|'''5'''}} (the defining representation) and {{math|'''10'''}}. (These bold numbers indicate the dimension of the representation.) In the standard assignment, the {{math|'''5'''}} contains the [[Charge Conjugation|charge conjugates]] of the right-handed [[Quark|down-type quark]] [[Quantum chromodynamics|color]] [[Triplet state|triplet]] and a left-handed [[lepton]] [[isospin]] [[Doublet state|doublet]], while the {{math|'''10'''}} contains the six [[Quark|up-type quark]] components, the left-handed down-type quark [[Quantum chromodynamics|color]] triplet, and the right-handed [[electron]]. This scheme has to be replicated for each of the three known [[Generation (particle physics)|generations of matter]]. It is notable that the theory is [[Gauge anomaly|anomaly free]] with this matter content.

The hypothetical [[neutrino#Chirality|right-handed neutrinos]] are a singlet of {{math|SU(5)}}, 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{{clarify|date=March 2016}} (see [[seesaw mechanism]]).

=== SO(10) ===
{{main|SO(10)}}

[[File:E6GUT.svg|200px|right|thumb|The pattern of [[weak isospin]], W, weaker isospin, W′, strong g3 and g8, and baryon minus lepton, B, charges for particles in the [[SO(10)]] Grand Unified Theory, rotated to show the embedding in [[E6 (mathematics)|E<sub>6</sub>]].]]

The next simple Lie group which contains the standard model is
: <math>\rm SO(10)\supset SU(5)\supset SU(3)\times SU(2)\times U(1) .</math>

Here, the unification of matter is even more complete, since the [[group representation|irreducible]] [[spinor]] [[group representation|representation]] {{math|'''16'''}} contains both the {{math|{{overline|'''5'''}}}} and {{math|'''10'''}} of {{math|SU(5)}} and a right-handed neutrino, and thus the complete particle content of one generation of the extended [[standard model]] with [[Neutrino|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 (apart from the [[Higgs mechanism|Higgs sector]]).

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 {{math|SU(5)}} and {{math|SO(10)}}. Some of these mass relations hold approximately, but most don't (see [[Georgi-Jarlskog mass relation]]).

The boson matrix for {{math|SO(10)}} is found by taking the {{math|15 × 15}} matrix from the {{math|'''10''' + '''5'''}} representation of {{math|SU(5)}} 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 (2 right-handed W bosons, 6 massive charged gluons and 12 X/Y type 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 {{math|SO(10)}}.

=== E<sub>6</sub> ===
{{main|E6 (mathematics)}}
In some forms of [[string theory]], including E<sub>8</sub>&nbsp;&times;&nbsp;E<sub>8</sub> [[heterotic string theory]], the resultant four-dimensional theory after spontaneous [[Compactification (physics)|compactification]] on a six-dimensional [[Calabi–Yau manifold]] resembles a GUT based on the group [[E6 (mathematics)|E<sub>6</sub>]]. Notably E<sub>6</sub> is the only [[exceptional simple Lie group]] to have any [[complex representation]]s, a requirement for a theory to contain chiral fermions (namely all weakly-interacting fermions). Hence the other four ([[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]]) can't be the gauge group of a GUT.{{Citation needed|date=June 2022}}

=== Extended Grand Unified Theories ===
Non-chiral extensions of the Standard Model with vectorlike split-multiplet particle spectra which naturally appear in the higher SU(N) GUTs considerably modify the desert physics and lead to the realistic (string-scale) grand unification for conventional three quark-lepton families even without using [[supersymmetry]] (see below). On the other hand, due to a new missing VEV mechanism emerging in the supersymmetric SU(8) GUT the simultaneous solution to the gauge hierarchy (doublet-triplet splitting) problem and problem of unification of flavor can be argued.<ref>J.L.Chkareuli, SU(N) SUSY GUTS WITH STRING REMNANTS: MINIMAL SU(5) AND BEYOND, Invited Talk given at 29th International Conference on High-Energy Physics (ICHEP 98), Vancouver, 23–29 July 1998. In *Vancouver 1998, High energy physics, vol. 2 1669–73</ref>

'''GUTs with four families / generations, SU(8)''': Assuming 4 generations of fermions instead of 3 makes a total of {{math|'''64'''}} types of particles. These can be put into {{math|'''64''' {{=}} '''8''' + '''56'''}} representations of {{math|SU(8)}}. This can be divided into {{math|SU(5) × SU(3)<sub>F</sub> × U(1)}} which is the {{math|SU(5)}} theory together with some heavy bosons which act on the generation number.

'''GUTs with four families / generations, O(16)''': Again assuming 4 generations of fermions, the '''128''' particles and anti-particles can be put into a single spinor representation of {{math|O(16)}}.

=== Symplectic groups and quaternion representations ===
Symplectic gauge groups could also be considered. For example, {{math|Sp(8)}} (which is called {{math|Sp(4)}} in the article [[symplectic group]]) has a representation in terms of {{math|4 × 4}} quaternion unitary matrices which has a {{math|'''16'''}} dimensional real representation and so might be considered as a candidate for a gauge group. {{math|Sp(8)}} has 32 charged bosons and 4 neutral bosons. Its subgroups include {{math|SU(4)}} so can at least contain the gluons and photon of {{math|SU(3) × U(1)}}. 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:
: <math>
\begin{bmatrix}
e   + i\ \overline{e}            + j\ v             + k\ \overline{v} \\
u_r + i\ \overline{u}_\mathrm{\overline r} + j\ d_\mathrm{r} + k\ \overline{d}_\mathrm{\overline r} \\
u_g + i\ \overline{u}_\mathrm{\overline g} + j\ d_\mathrm{g} + k\ \overline{d}_\mathrm{\overline g} \\
u_b + i\ \overline{u}_\mathrm{\overline b} + j\ d_\mathrm{b} + k\ \overline{d}_\mathrm{\overline b} \\
\end{bmatrix}_\mathrm{L}
</math>

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 {{math|4 × 4}} quaternion matrices is equivalent to including a single right-multiplication by a unit quaternion which adds an extra SU(2) and so has an extra neutral boson and two more charged bosons. Thus the group of left- and right-handed {{math|4 × 4}} quaternion matrices is {{math|Sp(8) × SU(2)}} which does include the standard model bosons:
: <math> \mathrm{ SU(4,\mathbb{H})_L\times \mathbb{H}_R = Sp(8)\times SU(2) \supset SU(4)\times SU(2) \supset SU(3)\times SU(2)\times U(1) }</math>

If <math>\psi</math> is a quaternion valued spinor, <math>A^{ab}_\mu</math> is quaternion hermitian {{math|4 × 4}} matrix coming from {{math|Sp(8)}} and <math>B_\mu</math> is a pure vector quaternion (both of which are 4-vector bosons) then the interaction term is:
: <math>\ \overline{\psi^{a}} \gamma_\mu\left( A^{ab}_\mu\psi^b + \psi^a B_\mu \right)\ </math>

=== Octonion representations ===
It can be noted that a generation of 16&nbsp;fermions can be put into the form of an [[octonion]] with each element of the octonion being an 8-vector. If the 3&nbsp;generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional (Grassmann) [[Jordan algebra]], which has the symmetry group of one of the exceptional Lie groups ({{math|F}}{{sub|4}}, {{math|E}}{{sub|6}}, {{math|E}}{{sub|7}}, or {{math|E}}{{sub|8}}) depending on the details.
: <math>
\psi=
\begin{bmatrix}
a & e & \mu \\
\overline{e} & b & \tau \\
\overline{\mu} & \overline{\tau} & c
\end{bmatrix}
</math>
: <math>\ [\psi_A,\psi_B] \subset \mathrm{J}_3(\mathbb{O})\ </math>

Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that {{math|E}}{{sub|6}} has subgroup {{math|O(10)}} and so is big enough to include the Standard Model. An {{math|E}}{{sub|8}} gauge group, for example, would have 8&nbsp;neutral bosons, 120&nbsp;charged bosons and 120&nbsp;charged anti-bosons. To account for the 248&nbsp;fermions in the lowest multiplet of {{math|E}}{{sub|8}}, these would either have to include anti-particles (and so have [[baryogenesis]]), have new undiscovered particles, or have gravity-like ([[spin connection]]) bosons affecting elements of the particles spin direction. Each of these possesses theoretical problems.

=== Beyond Lie groups ===
Other structures have been suggested including [[Lie 3-algebra]]s and [[Lie superalgebra]]s. Neither of these fit with [[Yang–Mills theory]]. In particular Lie superalgebras would introduce bosons with incorrect{{clarify|date=January 2016}} statistics. [[Supersymmetry]], however, does fit with Yang–Mills.