View source for Grand Unified Theory

{{Short description|Comprehensive physical model}}
{{redirect|Grand Unification|the albums|Grand Unification (Fightstar album)|and|Grand Unification (Milford Graves album)}}
{{Not to be confused with|Unified field theory}}
{{Beyond the Standard Model|expanded=Theories}}

A '''Grand Unified Theory''' ('''GUT''') is any [[Mathematical model|model]] in [[particle physics]] that merges the [[electromagnetism|electromagnetic]], [[weak interaction|weak]], and [[strong interaction|strong]] [[fundamental interaction|forces]] (the three [[gauge theory|gauge interactions]] of the [[Standard Model]]) into a single force at high [[energy|energies]]. Although this [[Unification (physics)|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 [[Chronology of the universe#Very early universe|very early universe]] in which these three [[fundamental interaction]]s were not yet distinct.

Experiments have confirmed that at high energy, the electromagnetic interaction and weak interaction unify into a single combined [[electroweak interaction]].<ref>{{Cite arXiv |last=Altareli |first=Guido |title=The Standard Electroweak Theory and Beyond |eprint=hep-ph/9811456 |date=1998-11-24}}</ref> GUT models predict that at even [[grand unification energy|higher energy]], the strong and electroweak interactions will unify into one electronuclear interaction. This interaction is characterized by one larger [[gauge theory|gauge symmetry]] and thus several [[force carrier]]s, but one unified [[coupling constant]]. Unifying [[gravity]] with the electronuclear interaction would provide a more comprehensive [[theory of everything]] (TOE) 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 <math> 10^{16} </math> GeV (just three orders of magnitude below the [[Planck units#Planck scale|Planck scale]] of <math> 10^{19} </math> GeV)—and so are well beyond the reach of any foreseen [[collider|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 moment]]s of [[elementary particle]]s, 
* or the properties of [[neutrino]]s.<ref name="Ross">{{cite book |last=Ross |first=G. |date=1984 |title=Grand Unified Theories |publisher=[[Westview Press]] |isbn=978-0-8053-6968-7}}</ref>

Some GUTs, such as the [[Pati–Salam model]], predict the existence of [[magnetic monopole]]s.

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{{clarify|empirical, suspected, or theoretical existence?|date=June 2020}} 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 [[reductive group|semisimple groups]] can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.

{{unsolved|physics|Are the three forces of the Standard Model unified at high energies? By which symmetry is this unification governed? Can the Grand Unification Theory explain the number of fermion generations and their masses?}}

==History==
Historically, the first true GUT, which was based on the [[simple Lie group]] {{math|[[SU(5)]]}}, was proposed by [[Howard Georgi]] and [[Sheldon Glashow]] in 1974.<ref>{{cite journal |last1=Georgi |first1=H. |last2=Glashow |first2=S.L.
 |s2cid=9063239 |date=1974 |title=Unity of All Elementary Particle Forces |journal=[[Physical Review Letters]] |volume=32 |issue=8 |pages=438–41 |doi=10.1103/PhysRevLett.32.438|bibcode=1974PhRvL..32..438G}}</ref> The [[Georgi–Glashow model]] was preceded by the [[semisimple Lie algebra]] Pati–Salam model by [[Abdus Salam]] and [[Jogesh Pati]] also in 1974,<ref>{{cite journal |last1=Pati |first1=J. |last2=Salam |first2=A. |date=1974 |title=Lepton Number as the Fourth Color |journal=[[Physical Review D]] |volume=10 |issue=1 |pages=275–89 |doi=10.1103/PhysRevD.10.275 |bibcode = 1974PhRvD..10..275P }}</ref> who pioneered the idea to unify gauge interactions.

The acronym GUT was first coined in 1978 by CERN researchers [[John Ellis (physicist, born 1946)|John Ellis]], [[Andrzej Buras]], [[Mary K. Gaillard]], and [[Dimitri Nanopoulos]], however in the final version of their paper<ref>{{cite journal |last1=Buras |first1=A.J.
 |last2=Ellis |first2=J. |last3=Gaillard |first3=M.K. |last4=Nanopoulos |first4=D.V. |date=1978 |title=Aspects of the grand unification of strong, weak and electromagnetic interactions |journal=[[Nuclear Physics B]] |volume=135 |issue=1 |pages=66–92
 |doi=10.1016/0550-3213(78)90214-6 |url=http://cdsweb.cern.ch/record/132734/files/197712054.pdf |archive-url=https://web.archive.org/web/20141229062727/http://cdsweb.cern.ch/record/132734/files/197712054.pdf |archive-date=2014-12-29 |url-status=live |access-date=2011-03-21|bibcode = 1978NuPhB.135...66B }}</ref> they opted for the less anatomical GUM (Grand Unification Mass). Nanopoulos later that year was the first to use<ref>{{cite journal |last=Nanopoulos |first=D.V. |date=1979 |title=Protons Are Not Forever |url=http://www-spires.fnal.gov/spires/find/hep/www?rawcmd=r+HUTP-78-A062 |journal=[[Orbis Scientiae]] |volume=1 |page=91 |id=Harvard Preprint HUTP-78/A062}}</ref> the acronym in a paper.<ref>{{cite journal |last=Ellis |first=J. |title=Physics gets physical |journal=[[Nature (journal)|Nature]] |volume=415 |page=957 |date=2002 |doi=10.1038/415957b |issue=6875 |bibcode=2002Natur.415..957E |pmid=11875539|doi-access=free }}</ref>

==Motivation==
The fact that the [[electric charge]]s of [[electron]]s and [[proton]]s 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 interaction|strong]] and weak interactions within the Standard Model is based on gauge symmetries governed by the [[simple Lie group|simple symmetry groups]] {{math|[[SU(3)]]}} and {{math|[[SU(2)]]}} which allow only discrete charges, the remaining component, the [[weak hypercharge]] interaction is described by an [[Abelian group|abelian symmetry]] {{math|[[U(1)]]}} which in principle allows for arbitrary charge assignments.<ref group=note>There are however certain constraints on the choice of particle charges from theoretical consistency, in particular [[gauge anomaly|anomaly cancellation]].</ref> The observed [[charge quantization]], namely the postulation that all known [[elementary particle]]s carry electric charges which are exact multiples of one-third of the [[elementary charge|"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 equations|Maxwell's field theory of electromagnetism]] in the 19th century, but its physical implications and mathematical structure are qualitatively different.

==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.

==Unification of forces and the role of supersymmetry==
The unification of forces is possible due to the energy scale dependence of force [[coupling constant|coupling parameters]] in [[quantum field theory]] called [[renormalization group|renormalization group "running"]], which allows parameters with vastly different values at usual energies to converge to a single value at a much higher energy scale.<ref name="Ross" />

The [[renormalization group]] running of the three gauge couplings in the Standard Model has been found to nearly, but not quite, meet at the same point if the [[hypercharge]] is normalized so that it is consistent with {{math|SU(5)}} or {{math|SO(10)}} GUTs, which are precisely the GUT groups which lead to a simple fermion unification. This is a significant result, as other Lie groups lead to different normalizations. However, if the supersymmetric extension [[Minimal Supersymmetric Standard Model|MSSM]] is used instead of the Standard Model, the match becomes much more accurate. In this case, the coupling constants of the strong and electroweak interactions meet at the [[grand unification energy]], also known as the GUT scale:

:<math>\Lambda_{\text{GUT}} \approx 10^{16}\,\text{GeV}</math>.

It is commonly believed that this matching is unlikely to be a coincidence, and is often quoted as one of the main motivations to further investigate supersymmetric theories despite the fact that no supersymmetric partner particles have been experimentally observed. Also, most model builders simply assume supersymmetry because it solves the [[hierarchy problem]]—i.e., it stabilizes the electroweak [[MSSM Higgs mass|Higgs mass]] against [[Renormalization|radiative correction]]s.<ref>{{Cite journal |arxiv = hep-ph/9702371|doi = 10.1142/S0217751X9800038X|title = The Future of Particle Physics as a Natural Science|journal = International Journal of Modern Physics A|volume = 13|issue = 6|pages = 863–886|year = 1998|last1 = Wilczek|first1 = Frank|s2cid = 14354139|bibcode = 1998IJMPA..13..863W}}</ref>

==Neutrino masses==
Since [[Majorana fermion|Majorana]] masses of the right-handed neutrino are forbidden by {{math|SO(10)}} symmetry, {{math|SO(10)}} GUTs predict the Majorana masses of right-handed neutrinos to be close to the [[GUT scale]] where the symmetry is [[Spontaneous symmetry breaking|spontaneously broken]] in those models. In supersymmetric GUTs, this scale tends to be larger than would be desirable to obtain realistic masses of the light, mostly left-handed neutrinos (see [[neutrino oscillation]]) via the [[seesaw mechanism]]. These predictions are independent of the [[Georgi–Jarlskog mass relation]]s, wherein some GUTs predict other fermion mass ratios.

==Proposed theories==
Several theories have been proposed, but none is currently universally accepted. An even more ambitious theory that includes ''all'' fundamental forces, including [[gravitation]], is termed a theory of everything. Some common mainstream GUT models are:

* [[Pati–Salam model]] — {{math|SU(4) × SU(2) × SU(2)}}
* [[Georgi–Glashow model]] — {{math|SU(5)}}; and [[Flipped SU(5)|Flipped {{math|SU(5)}}]] — {{math|SU(5) × U(1)}}
* [[SO(10) (physics)|{{math|SO(10)}} model]]; and [[flipped SO(10)|Flipped {{math|SO(10)}}]] — {{math|SO(10) × U(1)}}
* [[E6 (mathematics)#Importance in physics|{{math|E{{sub|6}}}} model]]; and [[Trinification]] — {{math|SU(3) × SU(3) × SU(3)}}
* minimal [[left-right model]] — {{math|SU(3){{sub|C}} × SU(2){{sub|L}} × SU(2){{sub|R}} × U(1){{sub|B − L}} }}
* [[331 model]] — {{math|SU(3){{sub|C}} × SU(3){{sub|L}} × U(1){{sub|X}} }}
* [[chiral color]]

Not quite GUTs:
{{div col begin|colwidth=10em}}
* [[Technicolor (physics)|Technicolor models]]
* [[Little Higgs]]
* [[String theory]]
* [[Causal fermion system]]s
* [[M-theory]]
* [[Preon]]s
* [[Loop quantum gravity]]
* [[Causal dynamical triangulation| Causal dynamical triangulation theory]]
{{div col end}}
''Note'': These models refer to [[Lie algebra]]s not to [[Lie group]]s. The Lie group could be <math>[\text{SU}(4) \times \text{SU}(2) \times \text{SU}(2)] / \mathbb{Z}_2,</math> just to take a random example.

The most promising candidate is {{math|[[SO(10)]]}}.<ref>
{{cite book
 |last=Grumiller |first=Daniel
 |date=2010
 |title=Fundamental Interactions: A memorial volume for Wolfgang Kummer
 |publisher=World Scientific
 |isbn=978-981-4277-83-9
 |page=351
 |url=https://books.google.com/books?id=kmxkDQAAQBAJ&pg=PA351
 |via=Google Books
}}
</ref><ref>
{{cite conference
 |last=Tsybychev  |first=Dmitri
 |year=2004
 |title=Status of searches for Higgs and physics beyond the standard model at CDF
 |editor1-last=Nath      |editor1-first=Pran  
 |editor2-last=Vaughn    |editor2-first=Michael T. 
 |editor3-last=Alverson  |editor3-first=George
 |editor4-last=Barberis  |editor4-first=Emanuela 
 |publication-date=19 August 2005 
 |conference=Tenth International Symposium on Particles, Strings and Cosmology – PaSCos 2004 
 |book-title=Proceedings of the Tenth International Symposium on Particles, Strings and Cosmology – The Pran Nath Festschrift – Pascos 2004
 |place=Boston, MA
 |publisher=[[Northeastern University]] / World Scientific
 |isbn=978-981-4479-96-7
 |lang=en
 |url=https://books.google.com/books?id=M6_KCgAAQBAJ&pg=RA1-PA265
 |via=Google Books
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 |quote=Part&nbsp;I: Particles, Strings, and Cosmology; Part&nbsp;II: Themes in Unification.
}}
</ref>
(Minimal) {{math|SO(10)}} does not contain any [[exotic fermion]]s (i.e. additional fermions besides the Standard Model fermions and the right-handed neutrino), and it unifies each generation into a single [[irreducible representation]]. A number of other GUT models are based upon subgroups of {{math|SO(10)}}. They are the minimal [[left-right model]], {{math|[[SU(5)]]}}, [[flipped SU(5)|flipped {{math|SU(5)}}]] and the Pati–Salam model. The GUT group {{math|E{{sub|6}}}} contains {{math|SO(10)}}, but models based upon it are significantly more complicated. The primary reason for studying {{math|E{{sub|6}}}} models comes from {{math|E{{sub|8}} × E{{sub|8}}}} [[heterotic string theory]].

GUT models generically predict the existence of [[topological defect]]s such as [[magnetic monopoles|monopoles]], [[cosmic strings]], [[Domain wall (string theory)|domain wall]]s, and others. But none have been observed. Their absence is known as the [[monopole problem]] in [[physical cosmology|cosmology]]. Many GUT models also predict [[proton decay]], although not the Pati–Salam model. As of now, proton decay has never been experimentally observed. The minimal experimental limit on the proton's lifetime pretty much rules out minimal {{math|SU(5)}} and heavily constrains the other models. The lack of detected supersymmetry to date also constrains many models.

<gallery caption="Proton Decay. These graphics refer to the [[X and Y bosons|{{math|X}}&nbsp;boson]] and [[Higgs boson]] families." widths="250px" heights="300px" perrow="3">

File:Proton_decay2.svg|Dimension 6&nbsp;proton decay mediated by the {{math|X}}&nbsp;boson <math>\ (3,2)_{-\tfrac{5}{6}}\ </math> in {{math|SU(5)}} GUT

File:proton decay3.svg|Dimension 6&nbsp;proton decay mediated by the {{math|X}}&nbsp;boson <math>\ (3,2)_{\tfrac{1}{6}}\ </math> in flipped {{math|SU(5)}} GUT

File:proton decay4.svg|Dimension 6&nbsp;proton decay mediated by the triplet Higgs <math>\ T(3,1)_{-\tfrac{1}{3}}</math> and the anti-triplet Higgs <math>\ \bar{T} (\bar{3},1)_{\tfrac{1}{3}}</math> in {{math|SU(5)}} GUT\ </gallery>

Some GUT theories like {{math|SU(5)}} and {{math|SO(10)}} suffer from what is called the [[doublet-triplet problem]]. These theories predict that for each electroweak Higgs doublet, there is a corresponding [[color charge|colored]] Higgs triplet field with a very small mass (many orders of magnitude smaller than the GUT scale here). In theory, unifying quarks with [[lepton]]s, the Higgs doublet would also be unified with a Higgs triplet. Such triplets have not been observed. They would also cause extremely rapid proton decay (far below current experimental limits) and prevent the gauge coupling strengths from running together in the renormalization group.

Most GUT models require a threefold replication of the matter fields. As such, they do not explain why there are three generations of fermions. Most GUT models also fail to explain the [[little hierarchy]] between the fermion masses for different generations.

==Ingredients==
A GUT model consists of a [[gauge group]] which is a [[compact Lie group]], a [[connection form]] for that Lie group, a [[Yang–Mills action]] for that connection given by an [[Invariant (mathematics)|invariant]] [[symmetric bilinear form]] over its Lie algebra (which is specified by a coupling constant for each factor), a [[Higgs sector]] consisting of a number of scalar fields taking on values within real/complex [[representations of Lie groups|representations]] of the Lie group and chiral [[Weyl fermion]]s taking on values within a complex rep of the Lie group. The Lie group contains the [[Standard Model group]] and the Higgs fields acquire [[VEV]]s leading to a [[spontaneous symmetry breaking]] to the Standard Model. The Weyl fermions represent matter.

==Current evidence==
The discovery of [[neutrino oscillation]]s indicates that the Standard Model is incomplete, but there is currently no clear evidence that nature is described by any Grand Unified Theory. Neutrino oscillations have led to renewed interest toward certain GUT such as {{math|SO(10)}}.

One of the few possible experimental tests of certain GUT is proton decay and also fermion masses. There are a few more special tests for supersymmetric GUT. However, minimum proton lifetimes from research (at or exceeding the {{10^|34}}~{{10^|35}}&nbsp;year range) have ruled out simpler GUTs and most non-SUSY models.
<ref>
{{cite journal
 |last1 = Mine           |first1 = Shunichi
 |year = 2023
 |title = Nucleon decay: theory and experimental overview
 |journal = Zenodo
 |doi = 10.5281/zenodo.10493165
}}
</ref>
The maximum upper limit on proton lifetime (if unstable), is calculated at 6×{{10^|39}}&nbsp;years for SUSY models and 1.4×{{10^|36}}&nbsp;years for minimal non-SUSY GUTs.<ref>
{{cite journal
 |last1 = Nath           |first1 = Pran
 |last2 = Fileviez Pérez |first2 = Pavel
 |year = 2007
 |title = Proton stability in grand unified theories, in strings, and in branes
 |journal = Physics Reports
 |volume = 441  |issue = 5–6  |pages = 191–317
 |s2cid = 119542637  |bibcode = 2007PhR...441..191N
 |arxiv = hep-ph/0601023  |doi = 10.1016/j.physrep.2007.02.010
}}
</ref>

The [[gauge coupling]] strengths of QCD, the [[weak interaction]] and [[hypercharge]] seem to meet at a common length scale called the [[Grand unification energy|GUT scale]] and equal approximately to {{10^|16}}&nbsp;GeV (slightly less than the [[Planck energy]] of {{10^|19}}&nbsp;GeV), which is somewhat suggestive. This interesting numerical observation is called the ''gauge coupling unification'', and it works particularly well if one assumes the existence of [[superpartner]]s of the Standard Model particles. Still, it is possible to achieve the same by postulating, for instance, that ordinary (non supersymmetric) {{math|SO(10)}} models break with an intermediate gauge scale, such as the one of Pati–Salam group.

==See also==
*[[B − L]] quantum number
*[[Classical unified field theories]]
*[[Paradigm shift]]
*[[Physics beyond the Standard Model]]
*[[Theory of everything]]
*[[X and Y bosons]]

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

==Further reading==
*[[Stephen Hawking]], [[A Brief History of Time]], includes a brief popular overview.
*{{cite journal |doi=10.4249/scholarpedia.11419|title=Grand unification|journal=Scholarpedia|volume=7|issue=10|pages=11419|year=2012|last1=Langacker|first1=Paul|bibcode=2012SchpJ...711419L|doi-access=free}}

==External links==
*[https://web.archive.org/web/20180829120432/http://math.ucr.edu/~huerta/oral.pdf The Algebra of Grand Unified Theories]

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