Editing Flipped SU(5)

{{short description|Grand unified theory}}
The '''Flipped SU(5) model''' is a [[grand unified theory]] (GUT) first contemplated by [[Stephen Barr]] in 1982,<ref>{{cite journal | last=Barr | first=S.M. | title=A new symmetry breaking pattern for SO(10) and proton decay | journal=Physics Letters B | volume=112 | issue=3 | year=1982 | doi=10.1016/0370-2693(82)90966-2 | pages=219–222}}</ref> and by [[Dimitri Nanopoulos]] and others in 1984.<ref>{{cite journal | last1=Derendinger | first1=J.-P. | last2=Kim | first2=Jihn E. | last3=Nanopoulos | first3=D.V. | title=Anti-Su(5) | journal=Physics Letters B | volume=139 | issue=3 | year=1984 | doi=10.1016/0370-2693(84)91238-3 | pages=170–176| url=https://cds.cern.ch/record/149877 }}</ref><ref>Stenger, Victor J., ''Quantum Gods: Creation, Chaos and the Search for Cosmic Consciousness'', Prometheus Books, 2009, 61. {{ISBN|978-1-59102-713-3}}</ref> [[Ignatios Antoniadis]], [[John Ellis (physicist, born 1946)|John Ellis]], [[John Hagelin]], and [[Dimitri Nanopoulos]] developed the supersymmetric flipped SU(5), derived from the deeper-level superstring.<ref>{{cite journal | last1=Antoniadis | first1=I. | last2=Ellis | first2=John | last3=Hagelin | first3=J.S. | last4=Nanopoulos | first4=D.V. | title=GUT model-building with fermionic four-dimensional strings | journal=Physics Letters B | volume=205 | issue=4 | year=1988 | doi=10.1016/0370-2693(88)90978-1 | pages=459–465| osti=1448495 | url=https://cds.cern.ch/record/184491 }}</ref><ref>Freedman, D. H. "The new theory of everything", ''Discover'', 1991, 54–61.</ref>

In 2010, efforts to explain the theoretical underpinnings for observed neutrino masses were being developed in the context of supersymmetric flipped {{math|SU(5)}}.<ref>{{cite journal | last1=Rizos | first1=J. | last2=Tamvakis | first2=K. | title=Hierarchical neutrino masses and mixing in flipped-SU(5) | journal=Physics Letters B | volume=685 | issue=1 | year=2010 | issn=0370-2693 | doi=10.1016/j.physletb.2010.01.038 | pages=67–71|arxiv=0912.3997| s2cid=119210871 }}</ref>

Flipped {{math|SU(5)}} is not a fully unified model, because the {{math|U(1)<sub>Y</sub>}} factor of the [[Standard Model]] gauge group is within the {{math|U(1)}} factor of the GUT group. The addition of states below ''M''x in this model, while solving certain threshold correction issues in [[string theory]], makes the model merely descriptive, rather than predictive.<ref>[https://books.google.com/books?id=p8hs0QHxk_oC&pg=PA194 Barcow, Timothy ''et al.'', ''Electroweak symmetry breaking and new physics at the TeV scale''] World Scientific, 1996, 194. {{ISBN|978-981-02-2631-2}}</ref>

==The model==
The flipped {{math|SU(5)}} model states that the [[gauge group]] is:

:{{math|([[Special unitary group|SU(5)]] × [[Unitary group|U(1)]]<sub>''χ''</sub>)/'''Z'''<sub>5</sub>}}

Fermions form three families, each consisting of the [[Representations of Lie groups/algebras|representations]]

:{{math|{{overline|'''5'''}}<sub>−3</sub>}} for the lepton doublet, L, and the up quarks {{mvar|u<sup>c</sup>}};
:{{math|'''10'''<sub>1</sub>}} for the quark doublet, Q, the down quark, {{mvar|d<sup>c</sup>}} and the right-handed neutrino, {{math|N}};
:{{math|'''1'''<sub>5</sub>}} for the charged leptons, {{mvar|e<sup>c</sup>}}.

This assignment includes three right-handed neutrinos, which have never been observed, but are often postulated to explain the lightness of the observed neutrinos and [[neutrino oscillation]]s. There is also a {{math|[[Representations of Lie groups/algebras|'''10'''<sub>1</sub>]]}} and/or {{math|{{overline|'''10'''}}<sub>−1</sub>}} called the Higgs fields which acquire a [[VEV]], yielding the [[spontaneous symmetry breaking]]

:{{math|(SU(5) × U(1)<sub>''χ''</sub>)/'''Z'''<sub>5</sub> → (SU(3) × SU(2) × U(1)<sub>Y</sub>)/'''Z'''<sub>6</sub>}}

The {{math|SU(5)}} representations [[restricted representation|transform under this subgroup]] as the reducible representation as follows:

:<math>\bar{5}_{-3}\to (\bar{3},1)_{-\frac{2}{3}}\oplus (1,2)_{-\frac{1}{2}}</math> (u<sup>c</sup> and l)
:<math>10_{1}\to (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{\frac{1}{3}}\oplus (1,1)_0</math> (q, d<sup>c</sup> and ν<sup>c</sup>)
:<math>1_{5}\to (1,1)_1</math> (e<sup>c</sup>)
:<math>24_0\to (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{\frac{1}{6}}\oplus (\bar{3},2)_{-\frac{1}{6}}</math>.

==Comparison with the standard SU(5)==
The name "flipped" {{math|SU(5)}} arose in comparison to the "standard" {{math|SU(5)}} [[Georgi–Glashow model]], in which {{mvar|u<sup>c</sup>}} and {{mvar|d<sup>c</sup>}} quark are respectively assigned to the {{math|'''10'''}} and {{math|'''5'''}} representation. In comparison with the standard {{math|SU(5)}}, the flipped {{math|SU(5)}} can accomplish the spontaneous symmetry breaking using Higgs fields of dimension 10, while the standard {{math|SU(5)}} typically requires a 24-dimensional Higgs.<ref>L.~F.~Li,
``[https://inspirehep.net/files/74d321cb7f023581a5311642aacb7711 Group Theory of the Spontaneously Broken Gauge Symmetries],''
Phys. Rev. D 9, 1723-1739 (1974)
[https://doi.org/10.1103/PhysRevD.9.1723 doi:10.1103/PhysRevD.9.1723]</ref>

The [[sign convention]] for {{math|U(1)<sub>''χ''</sub>}} varies from article/book to article.

The hypercharge Y/2 is a linear combination (sum) of the following:

:<math>\begin{pmatrix}{1 \over 15}&0&0&0&0\\0&{1 \over 15}&0&0&0\\0&0&{1 \over 15}&0&0\\0&0&0&-{1 \over 10}&0\\0&0&0&0&-{1 \over 10}\end{pmatrix}\in \text{SU}(5), \qquad \chi/5.</math>

There are also the additional fields {{math|'''5'''<sub>−2</sub>}} and {{math|{{overline|'''5'''}}<sub>2</sub>}} containing the [[electroweak]] [[Higgs doublet]]s.

Calling the [[Representations of Lie groups/algebras|representations]] for example, {{math|{{overline|'''5'''}}<sub>−3</sub>}} and {{math|'''24'''<sub>0</sub>}} is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by [[Young tableau]]x or [[Dynkin diagram]]s with numbers on their vertices, and is a standard used by GUT
theorists.

Since the [[homotopy group]]

:<math>\pi_2\left(\frac{[SU(5)\times U(1)_\chi]/\mathbf{Z}_5}{[SU(3)\times SU(2)\times U(1)_Y]/\mathbf{Z}_6}\right)=0</math>

this model does not predict [[Magnetic monopole|monopoles]]. See [['t Hooft–Polyakov monopole]].

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

==Minimal supersymmetric flipped SU(5)==

===Spacetime===
The {{math|''N'' {{=}} 1}} superspace extension of {{math|3 + 1}} Minkowski spacetime

===Spatial symmetry===
{{math|''N'' {{=}} 1}} SUSY over {{math|3 + 1}} Minkowski spacetime with [[R-symmetry]]

===Gauge symmetry group===
{{math|(SU(5) × U(1)<sub>χ</sub>)/'''Z'''<sub>5</sub>}}

===Global internal symmetry===
{{math|'''Z'''<sub>2</sub>}} (matter parity) not related to {{math|U(1)<sub>R</sub>}} in any way for this particular model

===Vector superfields===
Those associated with the {{math|SU(5) × U(1)<sub>''χ''</sub>}} gauge symmetry

===Chiral superfields===
As complex representations:

{| class="wikitable sortable"
!label||description||multiplicity||{{math|SU(5) × U(1)<sub>''χ''</sub>}} rep||{{math|'''Z'''<sub>2</sub>}} rep||{{math|U(1)<sub>R</sub>}}
|- align=center
|{{math|'''10'''<sub>H</sub>}}||GUT Higgs field||{{math|1}}||{{math|'''10'''<sub>1</sub>}}||+||{{math|0}}
|- align=center
|{{math|{{overline|'''10'''}}<sub>H</sub>}}||GUT Higgs field||{{math|1}}||{{math|{{overline|'''10'''}}<sub>−1</sub>}}||+||{{math|0}}
|- align=center
|{{math|H<sub>u</sub>}}||electroweak Higgs field||{{math|1}}||{{math|{{overline|'''5'''}}<sub>2</sub>}}||+||{{math|2}}
|- align=center
|{{math|H<sub>d</sub>}}||electroweak Higgs field||{{math|1}}||{{math|'''5'''<sub>−2</sub>}}||+||{{math|2}}
|- align=center
|{{math|{{overline|'''5'''}}}}||matter fields||{{math|3}}||{{math|{{overline|'''5'''}}<sub>−3</sub>}}||-||{{math|0}}
|- align=center
|{{math|'''10'''}}||matter fields||{{math|3}}||{{math|'''10'''<sub>1</sub>}}||-||{{math|0}}
|- align=center
|{{math|'''1'''}}||left-handed positron||{{math|3}}||{{math|'''1'''<sub>5</sub>}}||-||{{math|0}}
|- align=center
|{{mvar|φ}}||sterile neutrino (optional)||{{math|3}}||{{math|'''1'''<sub>0</sub>}}||-||{{math|2}}
|- align=center
|{{mvar|S}}||singlet||{{math|1}}||{{math|'''1'''<sub>0</sub>}}||+||{{math|2}}
|}

===Superpotential===
A generic invariant renormalizable superpotential is a (complex) {{math|SU(5) × U(1)<sub>''χ''</sub> × '''Z'''<sub>2</sub>}} invariant cubic polynomial in the superfields which has an {{math|R}}-charge of 2. It is a linear combination of the following terms:

<math>\begin{matrix}
S&S\\
S 10_H \overline{10}_H & S 10_H^{\alpha\beta} \overline{10}_{H\alpha\beta}\\
10_H 10_H H_d&\epsilon_{\alpha\beta\gamma\delta\epsilon}10_H^{\alpha\beta}10_H^{\gamma\delta} H_d^{\epsilon}\\
\overline{10}_H\overline{10}_H H_u&\epsilon^{\alpha\beta\gamma\delta\epsilon}\overline{10}_{H\alpha\beta}\overline{10}_{H\gamma\delta}H_{u\epsilon}\\
H_d 10 10&\epsilon_{\alpha\beta\gamma\delta\epsilon}H_d^{\alpha}10_i^{\beta\gamma}10_j^{\delta\epsilon}\\
H_d \bar{5} 1 &H_d^\alpha \bar{5}_{i\alpha} 1_j\\
H_u 10 \bar{5}&H_{u\alpha} 10_i^{\alpha\beta} \bar{5}_{j\beta}\\
\overline{10}_H 10 \phi&\overline{10}_{H\alpha\beta} 10_i^{\alpha\beta} \phi_j\\
\end{matrix} </math>

The second column expands each term in index notation (neglecting the proper normalization coefficient). {{mvar|i}} and {{mvar|j}} are the generation indices. The coupling {{math|H<sub>d</sub> '''10'''<sub>''i''</sub> '''10'''<sub>''j''</sub>}} has coefficients which are symmetric in {{mvar|i}} and {{mvar|j}}.

In those models without the optional {{mvar|φ}} sterile neutrinos, we add the [[nonrenormalizable]] couplings instead.

<math>\begin{matrix}
(\overline{10}_H 10)(\overline{10}_H 10)&\overline{10}_{H\alpha\beta}10^{\alpha\beta}_i \overline{10}_{H\gamma\delta} 10^{\gamma\delta}_j\\
\overline{10}_H 10 \overline{10}_H 10&\overline{10}_{H\alpha\beta}10^{\beta\gamma}_i\overline{10}_{H\gamma\delta}10^{\delta\alpha}_j
\end{matrix}</math>

These couplings do break the R-symmetry.

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

== References ==
{{Reflist}}

{{DEFAULTSORT:Flipped Su(5)}}
[[Category:Grand Unified Theory]]