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 ]/<math>\mathbb{Z}_5</math> and the fermions form three families, each consisting of the representations <math>\bar{5}_{-3}</math>, 10₁ and 1₅. This includes three right-handed neutrinos, which is consistent with the observed neutrino oscillations. There is also a 10₁ and/or <math>\bar{10}_{-1}</math> called the Higgs field which acquires a VEV. This results in a spontaneous symmetry breaking from

<math>[SU(5)\times U(1)_\chi]/\mathbb{Z}_5</math>

to

<math>[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6</math>

and also,

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

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

<math>1_{5}\rightarrow (1,1)_1</math> (e^c)

Compare to the Georgi-Glashow model. The left-handed antifermions are flipped, hence the name flipped SU(5).

<math>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}}</math>. See restricted representation.

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

The hypercharge Y/2 is a linear combination (sum) of the <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}</math> of SU(5) and χ/5.

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

Of course, calling the representations things like <math>\bar{5}_{-3}</math> 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

<math>\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</math>

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.

Image:Proton decay3.png

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

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

[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:

label	description	multiplicity	SU(5)× U(1)χ rep	<math>\mathbb{Z}_2</math> rep	U(1)R
10H	GUT Higgs field	1	101	+	0
<math>\bar{10}_H</math>	GUT Higgs field	1	<math>\overline{10}_{-1}</math>	+	0
Hu	electroweak Higgs field	1	<math>\bar{5}_2</math>	+	2
Hd	electroweak Higgs field	1	<math>5_{-2}</math>	+	2
<math>\bar{5}</math>	matter fields	3	<math>\bar{5}_{-3}</math>	-	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) <math>SU(5)\times U(1)_\chi\times\mathbb{Z}_2</math> invariant cubic polynomial in the superfields which has an 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). i and j are the generation indices. The coupling H_d 10_{i 10j has coefficients which are symmetric in i and j.}

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

<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>

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