Flipped SU(5)

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

[ SU(5) × U(1)_χ ] /$mathbb\{Z\}\_5$
Fermions form three families, each consisting of the representations
for the lepton doublet, L, and the up quarks $u^c$;

$10\_1$ for the quark doublet,Q ,the down quark, $d^c$ and the right-handed neutrino, N;

$1\_5$ for the charged leptons,$e^c$.
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 $10\_1$ and/or called the Higgs fields which acquire a VEV, yielding the spontaneous symmetry breaking

$[SU(5)\; imes\; U(1)\_chi]\; /mathbb\{Z\}\_5$ to $[SU(3)\; imes\; SU(2)\; imes\; U(1)\_Y]\; /mathbb\{Z\}\_6$
The SU(5) representations transform under this subgroup as the reducible representatio as follows:: (u^c and l): (q, d^c and ν^c): $1\_\{5\}\; ightarrow\; (1,1)\_1$ (e^c)

:.

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 $u^c$ and $d^c$ quark are respectively assigned to the 10 and 5 representation. In comparison with the standard SU(5), the flipped SU(5) can accomplish the sopontaneous 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 of SU(5) and χ/5.

There are also the additional fields 5_-2 and containing the electroweak Higgs doublets.

Of course, calling the representations things like 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 :$pi\_2left(frac\{\; [SU(5)\; imes\; U(1)\_chi]\; /mathbb\{Z\}\_5\}\{\; [SU(3)\; imes\; SU(2)\; imes\; U(1)\_Y]\; /mathbb\{Z\}\_6\}\; ight)=0$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)

pacetime

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

patial symmetry

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

gauge symmetry group

[SU(5)× U(1)_χ] /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:

uperpotential

A generic invariant renormalizable superpotential is a (complex) $SU(5)\; imes\; U(1)\_chi\; imesmathbb\{Z\}\_2$ invariant cubic polynomial in the superfields which has an R-charge of 2. It is a linear combination of the following terms:

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

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