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The Flipped SU(5) model is a GUT theory which states that the gauge group is:

/${\displaystyle \mathbb {Z} _{5}}$

Fermions form three families, each consisting of the representations

${\displaystyle {\bar {5}}_{-3}}$ for the lepton doublet, L, and the up quarks ${\displaystyle u^{c}}$;

${\displaystyle 10_{1}}$ for the quark doublet,Q ,the down quark, ${\displaystyle d^{c}}$ and the right-handed neutrino, N;

${\displaystyle 1_{5}}$ for the charged leptons,${\displaystyle e^{c}}$.

It is noticeable that this assignment includes three right-handed neutrinos, which are never been observed, but are often postulated to explain the lightness of the observed neutrinos and neutrino oscillations. There is also a ${\displaystyle 10_{1}}$ and/or ${\displaystyle {\bar {10}}_{-1}}$ called the Higgs fields which acquire a VEV, yielding the spontaneous symmetry breaking

${\displaystyle /\mathbb {Z} _{5}}$ to ${\displaystyle /\mathbb {Z} _{6}}$

The SU(5) representations transform under this subgroup as the reducible representatio as follows:

${\displaystyle {\bar {5}}_{-3}\rightarrow ({\bar {3}},1)_{-{\frac {2}{3}}}\oplus (1,2)_{-{\frac {1}{2}}}}$ (u and l)

${\displaystyle 10_{1}\rightarrow (3,2)_{\frac {1}{6}}\oplus ({\bar {3}},1)_{\frac {1}{3}}\oplus (1,1)_{0}}$ (q, d and ν)

${\displaystyle 1_{5}\rightarrow (1,1)_{1}}$ (e)

${\displaystyle 24_{0}\rightarrow (8,1)_{0}\oplus (1,3)_{0}\oplus (1,1)_{0}\oplus (3,2)_{\frac {1}{6}}\oplus ({\bar {3}},2)_{-{\frac {1}{6}}}}$.

Comparison with the standard SU(5)

The name "flipped" SU(5) arose in comparison with the "standard" SU(5) model of Georgi-Glashow, in which ${\displaystyle u^{c}}$ and ${\displaystyle d^{c}}$ quark are respectively assigned to the 10 and 5 representation. In comparison with the standard SU(5), the flipped SU(5) can accomplish the spontaneous symmetry breaking using Higgs fields of dimension 10, while the standard SU(5) need both a 5- and 45-dimensional Higgs.

The sign convention for U(1)_χ varies from article/book to article.

The hypercharge Y/2 is a linear combination (sum) of the ${\displaystyle {\begin{pmatrix}{1 \over 15}&0&0&0&0\\0&{1 \over 15}&0&0&0\\0&0&{1 \over 15}&0&0\\0&0&0&-{1 \over 10}&0\\0&0&0&0&-{1 \over 10}\end{pmatrix}}}$ of SU(5) and χ/5.

There are also the additional fields 5_-2 and ${\displaystyle {\bar {5}}_{2}}$ containing the electroweak Higgs doublets.

Of course, calling the representations things like ${\displaystyle {\bar {5}}_{-3}}$ and 24₀ is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, it is standard among GUT theorists.

Since the homotopy group

${\displaystyle \pi _{2}\left({\frac {/\mathbb {Z} _{5}}{/\mathbb {Z} _{6}}}\right)=0}$

this model does not predicts monopoles. See Hooft-Polyakov monopole.

This theory was invented by Dimitri Nanopoulos, with some collaboration by John Hagelin and John Ellis.

Minimal supersymmetric flipped SU(5)

spacetime

The N=1 superspace extension of 3+1 Minkowski spacetime

spatial symmetry

N=1 SUSY over 3+1 Minkowski spacetime with R-symmetry

gauge symmetry group

/Z₅

global internal symmetry

Z₂ (matter parity) not related to U(1)_R in any way for this particular model

vector superfields

Those associated with the SU(5)× U(1)_χ gauge symmetry

chiral superfields

As complex representations:

label	description	multiplicity	SU(5)× U(1)χ rep	${\displaystyle \mathbb {Z} _{2}}$ rep	U(1)R
10H	GUT Higgs field	1	101	+	0
${\displaystyle {\bar {10}}_{H}}$	GUT Higgs field	1	${\displaystyle {\overline {10}}_{-1}}$	+	0
Hu	electroweak Higgs field	1	${\displaystyle {\bar {5}}_{2}}$	+	2
Hd	electroweak Higgs field	1	${\displaystyle 5_{-2}}$	+	2
${\displaystyle {\bar {5}}}$	matter fields	3	${\displaystyle {\bar {5}}_{-3}}$	-	0
10	matter fields	3	101	-	0
1	left-handed positron	3	15	-	0
φ	sterile neutrino (optional)	3	10	-	2
S	singlet	1	10	+	2

Superpotential

A generic invariant renormalizable superpotential is a (complex) ${\displaystyle SU(5)\times U(1)_{\chi }\times \mathbb {Z} _{2}}$ invariant cubic polynomial in the superfields which has an R-charge of 2. It is a linear combination of the following terms: ${\displaystyle {\begin{matrix}S&S\\S10_{H}{\overline {10}}_{H}&S10_{H}^{\alpha \beta }{\overline {10}}_{H\alpha \beta }\\10_{H}10_{H}H_{d}&\epsilon _{\alpha \beta \gamma \delta \epsilon }10_{H}^{\alpha \beta }10_{H}^{\gamma \delta }H_{d}^{\epsilon }\\{\overline {10}}_{H}{\overline {10}}_{H}H_{u}&\epsilon ^{\alpha \beta \gamma \delta \epsilon }{\overline {10}}_{H\alpha \beta }{\overline {10}}_{H\gamma \delta }H_{u\epsilon }\\H_{d}1010&\epsilon _{\alpha \beta \gamma \delta \epsilon }H_{d}^{\alpha }10_{i}^{\beta \gamma }10_{j}^{\delta \epsilon }\\H_{d}{\bar {5}}1&H_{d}^{\alpha }{\bar {5}}_{i\alpha }1_{j}\\H_{u}10{\bar {5}}&H_{u\alpha }10_{i}^{\alpha \beta }{\bar {5}}_{j\beta }\\{\overline {10}}_{H}10\phi &{\overline {10}}_{H\alpha \beta }10_{i}^{\alpha \beta }\phi _{j}\\\end{matrix}}}$

The second column expands each term in index notation (neglecting the proper normalization coefficient). i and j are the generation indices. The coupling H_d 10_i 10_j has coefficients which are symmetric in i and j.

In those models without the optional φ sterile neutrinos, we add the nonrenormalizable couplings

${\displaystyle {\begin{matrix}({\overline {10}}_{H}10)({\overline {10}}_{H}10)&{\overline {10}}_{H\alpha \beta }10_{i}^{\alpha \beta }{\overline {10}}_{H\gamma \delta }10_{j}^{\gamma \delta }\\{\overline {10}}_{H}10{\overline {10}}_{H}10&{\overline {10}}_{H\alpha \beta }10_{i}^{\beta \gamma }{\overline {10}}_{H\gamma \delta }10_{j}^{\delta \alpha }\end{matrix}}}$

instead. These couplings do break the R-symmetry, though.

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