View source for SO(10)

[[File:SO10.svg|300px|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) model, rotated to show the embedding of the [[Georgi–Glashow model]] and [[Standard Model]], with electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes 30 colored X bosons, responsible for [[proton decay]], and two W' bosons.]]

[[File:E6GUT.svg|300px|right|thumb|The pattern of charges for particles in the SO(10) model, rotated to show the embedding in [[E6 (mathematics)|E6]].]]

In [[particle physics]], '''SO(10)''' refers to a [[grand unified theory]] (GUT) based on the [[spin group]] Spin(10).  The shortened name SO(10) is conventional<ref>{{Cite journal |last=Baez |first=John |last2=Huerta |first2=John |date=2010-03-11 |title=The algebra of grand unified theories |url=http://www.ams.org/journal-getitem?pii=S0273-0979-10-01294-2 |journal=Bulletin of the American Mathematical Society |language=en |volume=47 |issue=3 |pages=483–552 |doi=10.1090/S0273-0979-10-01294-2 |issn=0273-0979|doi-access=free |arxiv=0904.1556 }}</ref> among physicists, and derives from the [[Lie algebra]] or less precisely the [[Lie group]] of SO(10), which is a [[special orthogonal group]] that is [[Double covering group|double covered]] by Spin(10).

SO(10) subsumes the [[Georgi–Glashow model|Georgi–Glashow]] and [[Pati–Salam model]]s, and unifies all [[fermion]]s in a [[Generation (particle physics)|generation]] into a single field.  This requires 12 new [[gauge boson]]s, in addition to the 12 of [[Georgi–Glashow model|SU(5)]] and 9 of [[Pati–Salam model|SU(4)×SU(2)×SU(2)]].

==History==

Before the [[SU(5)]] theory behind the [[Georgi–Glashow model]],<ref name="GG">{{cite journal |first1=Howard |last1=Georgi |first2=Sheldon |last2=Glashow |s2cid=9063239 |title=Unity of All Elementary-Particle Forces |journal=[[Physical Review Letters]] |volume=32 |issue=8 |year=1974 |pages=438 |doi=10.1103/PhysRevLett.32.438 |bibcode = 1974PhRvL..32..438G }}</ref> [[Harald Fritzsch]] and [[Peter Minkowski]], and independently [[Howard Georgi]], found that all the matter contents are incorporated into a single representation, [[spinorial]] 16 of SO(10).<ref name="Minkowski1975">{{cite journal|author=Fritzsch, H.|author2=Minkowski, P.|title=Unified interactions of leptons and hadrons|journal=Annals of Physics|volume=93|pages=193–266|year=1975|issue=1–2|doi=10.1016/0003-4916(75)90211-0|bibcode=1975AnPhy..93..193F}}</ref> However, Georgi found the SO(10) theory just a few hours before finding SU(5) at the end of 1973.<ref>{{Cite web |last=Physics |first=American Institute of |date=2021-09-24 |title=Howard Georgi |url=https://www.aip.org/history-programs/niels-bohr-library/oral-histories/44877 |access-date=2022-12-12 |website=www.aip.org |language=en}}</ref>

==Important subgroups==

It has the [[branching rule]]s to [SU(5)&times;U(1)<sub>χ</sub>]/'''Z'''<sub>5</sub>.
:<math> 45 \rightarrow 24_0 \oplus 10_{-4} \oplus \overline{10}_4 \oplus 1_0</math>
:<math> 16 \rightarrow 10_1 \oplus \bar{5}_{-3} \oplus 1_5</math>
:<math> 10 \rightarrow 5_{-2} \oplus \bar{5}_2.</math>
If the [[hypercharge]] is contained within SU(5), this is the conventional [[Georgi–Glashow model]], with the 16 as the matter fields, the 10 as the electroweak Higgs field and the 24 within the 45 as the GUT Higgs field. The [[superpotential]] may then include [[renormalizable]] terms of the form ''Tr''(45 ⋅ 45); ''Tr''(45 ⋅ 45 ⋅ 45); 10 ⋅ 45 ⋅ 10, 10 ⋅ 16* ⋅ 16 and 16* ⋅ 16. The first three are responsible to the [[gauge symmetry]] breaking at low energies and give the [[Peter Higgs|Higgs]] mass, and the latter two give the matter particles masses and their [[Yukawa coupling]]s to the Higgs.

There is another possible branching, under which the hypercharge is a linear combination of an SU(5) generator and χ. This is known as [[flipped SU(5)]].

Another important subgroup is either [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/'''Z'''<sub>2</sub> or '''Z'''<sub>2</sub> ⋊ [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/'''Z'''<sub>2</sub> depending upon whether or not the [[left-right symmetry]] is broken, yielding the [[Pati–Salam model]], whose branching rule is
:<math> 45 \rightarrow (15,1,1) \oplus (6,2,2) \oplus (1,3,1) \oplus (1,1,3)</math>
:<math> 16 \rightarrow (4,2,1)\oplus (\bar 4,1,2). </math>

==Spontaneous symmetry breaking==
The symmetry breaking of SO(10) is usually done with a combination of (( a 45<sub>H</sub> OR a 54<sub>H</sub>) AND ((a 16<sub>H</sub> AND a <math>\overline{16}_H</math>) OR (a 126<sub>H</sub> AND a <math>\overline{126}_H</math>)) ).

Let's say we choose a 54<sub>H</sub>. When this Higgs field acquires a GUT scale [[VEV]], we have a symmetry breaking to '''Z'''<sub>2</sub> ⋊ [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/'''Z'''<sub>2</sub>, i.e. the [[Pati–Salam model]] with a '''Z'''<sub>2</sub> [[left-right symmetry]].

If we have a 45<sub>H</sub> instead, this Higgs field can acquire any VEV in a two dimensional subspace without breaking the standard model. Depending on the direction of this linear combination, we can break the symmetry to SU(5)&times;U(1), the [[Georgi–Glashow model]] with a U(1) (diag(1,1,1,1,1,-1,-1,-1,-1,-1)), [[flipped SU(5)]] (diag(1,1,1,-1,-1,-1,-1,-1,1,1)), SU(4)&times;SU(2)&times;U(1) (diag(0,0,0,1,1,0,0,0,-1,-1)), the minimal [[left-right model]] (diag(1,1,1,0,0,-1,-1,-1,0,0)) or SU(3)&times;SU(2)&times;U(1)&times;U(1) for any other nonzero VEV.

The choice diag(1,1,1,0,0,-1,-1,-1,0,0) is called the [[Dimopoulos-Wilczek mechanism]] aka the "missing VEV mechanism" and it is proportional to [[B−L]].

The choice of a 16<sub>H</sub> and a <math>\overline{16}_H</math> breaks the gauge group down to the Georgi–Glashow SU(5). The same comment applies to the choice of a 126<sub>H</sub> and a <math>\overline{126}_H</math>.

It is the combination of BOTH a 45/54 and a 16/<math>\overline{16}</math> or 126/<math>\overline{126}</math> which breaks SO(10) down to the [[Standard Model]].

==The electroweak Higgs and the doublet-triplet splitting problem==
The electroweak Higgs doublets come from an SO(10) 10<sub>H</sub>. Unfortunately, this same 10 also contains triplets. The masses of the doublets have to be stabilized at the electroweak scale, which is many orders of magnitude smaller than the GUT scale whereas the triplets have to be really heavy in order to prevent triplet-mediated [[proton decay]]s. See [[doublet-triplet splitting problem]].

Among the solutions for it is the Dimopoulos-Wilczek mechanism, or the choice of diag(1,1,1,0,0,-1,-1,-1,0,0) of <45>. Unfortunately, this is not stable once the 16/<math>\overline{16}</math> or 126/<math>\overline{126}</math> sector interacts with the 45 sector.<ref>*{{cite journal |author=[[John Baez|J.C. Baez]], J. Huerta |arxiv=0904.1556 |title=The Algebra of Grand Unified Theories |journal=Bull. Am. Math. Soc. |year=2010 |volume=47 |issue=3 |pages=483–552 |doi=10.1090/S0273-0979-10-01294-2 |s2cid=2941843 }}</ref>

==Content==
===Matter===

[[File:SO(10) - 16 Weight Diagram.svg|right|400px]]

The matter representations come in three copies (generations) of the 16 representation. The [[Yukawa coupling]] is 10<sub>H</sub> 16<sub>f</sub> 16<sub>f</sub>. This includes a right-handed neutrino.  One may either include three copies of [[Singlet state|singlet]] representations {{mvar|φ}} and a Yukawa coupling <math><\overline{16}_H>16_f \phi</math> (the "double seesaw mechanism");  or else, add the Yukawa interaction <math><\overline{126}_H> 16_f 16_f</math> or add the [[nonrenormalizable]] coupling <math><\overline{16}_H><\overline{16}_H>16_f 16_f</math>. See [[seesaw mechanism]].

The 16<sub>f</sub> field branches to [SU(5)&times;U(1)<sub>χ</sub>]/'''Z'''<sub>5</sub> and SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub> as

:<math> 16 \rightarrow 10_1 \oplus \bar{5}_{-3} \oplus 1_5</math>
:<math> 16 \rightarrow (4,2,1)\oplus (\bar 4,1,2).</math>

=== Gauge fields ===

The 45 field branches to [SU(5)&times;U(1)<sub>χ</sub>]/'''Z'''<sub>5</sub> and SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub> as

:<math> 45 \rightarrow 24_0 \oplus 10_{-4} \oplus \overline{10}_4 \oplus 1_0</math>
:<math> 45 \rightarrow (15,1,1) \oplus (6,2,2) \oplus (1,3,1) \oplus (1,1,3)</math>

and to the standard model [SU(3)<sub>C</sub> × SU(2)<sub>L</sub> × U(1)<sub>Y</sub>]/'''Z'''<sub>6</sub> as

:<math>\begin{align} 45 \rightarrow & (8,1)_0 \oplus (1,3)_0 \oplus (1,1)_0 \oplus\\
&(3,2)_{-\frac{5}{6}} \oplus (\bar{3},2)_{\frac{5}{6}} \oplus\\
&(3,1)_{\frac{2}{3}} \oplus (\bar{3},1)_{-\frac{2}{3}} \oplus (1,1)_1 \oplus (1,1)_{-1} \oplus (1,1)_0\oplus\\
&(3,2)_{\frac{1}{6}} \oplus (\bar{3},2)_{-\frac{1}{6}}.\\
\end{align}</math>

The four lines are the SU(3)<sub>C</sub>, SU(2)<sub>L</sub>, and U(1)<sub>B−L</sub> bosons; the [[Georgi–Glashow model|SU(5)]] leptoquarks which don't mutate [[X (charge)|X charge]]; the [[Pati–Salam model|Pati-Salam]] leptoquarks and SU(2)<sub>R</sub> bosons; and the new SO(10) leptoquarks. (The standard [[electroweak]] U(1)<sub>Y</sub> is a linear combination of the {{math|(1,1)<sub>0</sub>}} bosons.)

==Proton decay==
<gallery caption="These graphics refer to the [[X and Y bosons|X bosons]] and [[Higgs boson]]s." widths="125px" heights="150px" perrow="3">

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

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

</gallery>

Note that SO(10) contains both the Georgi–Glashow SU(5) and flipped SU(5).

== Anomaly free from local and global anomalies ==

It has been long known that the SO(10) model is free from all perturbative local anomalies, computable by Feynman diagrams. However, it only became clear in 2018 that the SO(10) model is also free from all [[Anomaly_(physics)#Witten_anomaly_and_Wang-Wen-Witten_anomaly|nonperturbative global anomalies]] on non-spin manifolds --- an important rule for confirming the consistency of SO(10) grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on [[Spin_structure|non-spin manifolds]].<ref name="1809.11171">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | title=Nonperturbative definition of the standard models | journal=Physical Review Research  | volume=2 | issue=2 | date=1 June 2020 | issn=2469-9896 | doi=10.1103/PhysRevResearch.2.023356 | page=023356 |arxiv=1809.11171| bibcode= 2018arXiv180911171W| s2cid=53346597 }}</ref><ref name="1810.00844">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | last3=Witten | first3=Edward | title=A New SU(2) Anomaly | journal=Journal of Mathematical Physics | volume=60 | issue=5 | date= May 2019 | issn= 1089-7658 | doi=10.1063/1.5082852 | page=052301 |arxiv=1810.00844| bibcode=2019JMP....60e2301W | s2cid=85543591 }}</ref>

== See also ==
* [[Flipped SO(10)]]

==Notes==
{{reflist}}

[[Category:Grand Unified Theory]]