Flipped SU(5)

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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

$\bar{5}_{-3}$ for the lepton doublet, L, and the up quarks

u c

;

101

for the quark doublet,Q ,the down quark,

d c

and the right-handed neutrino, N;

15

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 $101$ and/or $\bar{10}_{-1}$ called the Higgs fields which acquire a VEV, yielding the spontaneous symmetry breaking

$[SU(5)\times U(1)_\chi]/\mathbb{Z}_5$ to $[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6$

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

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

$10_{1}\rightarrow (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{\frac{1}{3}}\oplus (1,1)_0$ (q, d^c and ν^c)

$1_{5}\rightarrow (1,1)_1$ (e^c)

$24_0\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{\frac{1}{6}}\oplus (\bar{3},2)_{-\frac{1}{6}}$ .

1 Comparison with the standard SU(5)
2 Minimal supersymmetric flipped SU(5)
3 spacetime
4 spatial symmetry
5 gauge symmetry group
6 global internal symmetry
7 vector superfields
8 chiral superfields
9 Superpotential

[edit] 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 $\begin{pmatrix}{1 \over 15}&0&0&0&0\\0&{1 \over 15}&0&0&0\\0&0&{1 \over 15}&0&0\\0&0&0&-{1 \over 10}&0\\0&0&0&0&-{1 \over 10}\end{pmatrix}$ of SU(5) and χ/5.

There are also the additional fields 5_-2 and $\bar{5}_2$ containing the electroweak Higgs doublets.

Of course, calling the representations things like $\bar{5}_{-3}$ 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

$\pi_2\left(\frac{[SU(5)\times U(1)_\chi]/\mathbb{Z}_5}{[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6}\right)=0$

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.

[edit] Minimal supersymmetric flipped SU(5)

[edit] spacetime

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

[edit] spatial symmetry

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

[edit] gauge symmetry group

[SU(5)× U(1)_χ]/Z₅

[edit] global internal symmetry

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

[edit] vector superfields

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

[edit] chiral superfields

As complex representations:

label	description	multiplicity	SU(5)× U(1)_χ rep	$\mathbb{Z}_2$ rep	U(1)_R
10_H	GUT Higgs field	1	10₁	+	0
$\bar{10}_H$	GUT Higgs field	1	$\overline{10}_{-1}$	+	0
H_u	electroweak Higgs field	1	$\bar{5}_2$	+	2
H_d	electroweak Higgs field	1	$5 - 2$	+	2
$\bar{5}$	matter fields	3	$\bar{5}_{-3}$	-	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

[edit] Superpotential

A generic invariant renormalizable superpotential is a (complex) $SU(5)\times U(1)_\chi\times\mathbb{Z}_2$ invariant cubic polynomial in the superfields which has an R-charge of 2. It is a linear combination of the following terms: $\begin{matrix} S&S\\ S 10_H \overline{10}_H&S 10_H^{\alpha\beta} \overline{10}_{H\alpha\beta}\\ 10_H 10_H H_d&\epsilon_{\alpha\beta\gamma\delta\epsilon}10_H^{\alpha\beta}10_H^{\gamma\delta} H_d^{\epsilon}\\ \overline{10}_H\overline{10}_H H_u&\epsilon^{\alpha\beta\gamma\delta\epsilon}\overline{10}_{H\alpha\beta}\overline{10}_{H\gamma\delta}H_{u\epsilon}\\ H_d 10 10&\epsilon_{\alpha\beta\gamma\delta\epsilon}H_d^{\alpha}10_i^{\beta\gamma}10_j^{\delta\epsilon}\\ H_d \bar{5} 1 &H_d^\alpha \bar{5}_{i\alpha} 1_j\\ H_u 10 \bar{5}&H_{u\alpha} 10_i^{\alpha\beta} \bar{5}_{j\beta}\\ \overline{10}_H 10 \phi&\overline{10}_{H\alpha\beta} 10_i^{\alpha\beta} \phi_j\\ \end{matrix}$

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

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