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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 have 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 representation 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.

Supersymmetric flipped SU(5)

Firstly, as to the fundamental significance of the theory, supersymmetric flipped SU(5) is the most successful Grand Unified Theory, because it:

1. Is fully consistent with all experimental data (unlike minimal SU(5), for example);
2. Is free from a severe technical problem, known variously as the “doublet-triplet splitting problem,” the “fine-tuning problem,” or the “gauge hierarchy problem” that plagues all other GUTs;
3. Makes successful predictions for particle masses, including nearly massless neutrinos and the correct bottom quark to taon mass ratio, in the context of string-derived flipped SU(5);
4. Is the only GUT that appears to be consistent with superstring superunification.

Flipped SU(5) had been contemplated by Stephen Barr and by Dmitri Nanopoulos, as well as others. But the enduring “gauge hierarchy problem,” among other issues, had prevented its being given serious consideration. In a major advancement of the theory, John Hagelin developed the supersymmetric flipped SU(5), derived from the deeper-level superstring. Hagelin, with the help of John Ellis in working out some of the details, originated a simple set of formulas that appeared to solve all the major problems plaguing earlier grand unified theories, including the recalcitrant gauge hierarchy problem. Hagelin faxed the results to their collaborator, Dmitri Nanopoulos, and enthusiastically scrawled on the top of the fax, “Isn’t this the prettiest GUT you’ve ever seen?” This derivation was first presented in (Antoniadis, Ellis, Hagelin and Nanopoulos, 1987) and in more detail in (Campbell, Ellis, Hagelin, Nanopoulos, and Ticciatti, 1987), and its implications worked out in numerous subsequent papers with these and other collaborators. Over the next several years, the four principle collaborators, John Hagelin, John Ellis, Dmitri Nanopoulos and Ignatios Antoniadis, met together for several intensive conferences to more fully develop the theory.

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.

1. Antoniadis I, Ellis J, Hagelin J, and Nanopoulos D. "Supersymmetric Flipped SU(5) Revitalized", Phys Lett 1987; 194B: 231.
2. Campbell B, Ellis J, Hagelin J, Nanopoulos D, and Ticciatti R. "Flipped SU(5) from Manifold Compactification of the 10-Dimensional Heterotic String", Phys Lett 1987; 198B: 200.
3. Dickie N. "John Hagelin and the Constitution of the Universe", The Iowa Source. February 1992; pp. 1, 10-13.
4. Freedman DH. "The new theory of everything", Discover, 1991, pp 54–61.

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