Flipped SU(5)

Template:Short description The Flipped SU(5) model is a grand unified theory (GUT) first contemplated by Stephen Barr in 1982,<ref>Template:Cite journal</ref> and by Dimitri Nanopoulos and others in 1984.<ref>Template:Cite journal</ref><ref>Stenger, Victor J., Quantum Gods: Creation, Chaos and the Search for Cosmic Consciousness, Prometheus Books, 2009, 61. Template:ISBN</ref> Ignatios Antoniadis, John Ellis, John Hagelin, and Dimitri Nanopoulos developed the supersymmetric flipped SU(5), derived from the deeper-level superstring.<ref>Template:Cite journal</ref><ref>Freedman, D. H. "The new theory of everything", Discover, 1991, 54–61.</ref>

In 2010, efforts to explain the theoretical underpinnings for observed neutrino masses were being developed in the context of supersymmetric flipped Template:Math.<ref>Template:Cite journal</ref>

Flipped Template:Math is not a fully unified model, because the Template:Math factor of the Standard Model gauge group is within the Template:Math factor of the GUT group. The addition of states below Mx in this model, while solving certain threshold correction issues in string theory, makes the model merely descriptive, rather than predictive.<ref>Barcow, Timothy et al., Electroweak symmetry breaking and new physics at the TeV scale World Scientific, 1996, 194. Template:ISBN</ref>

The model

The flipped Template:Math model states that the gauge group is:

Template:Math

Fermions form three families, each consisting of the representations

Template:Math for the lepton doublet, L, and the up quarks Template:Mvar;

Template:Math for the quark doublet, Q, the down quark, Template:Mvar and the right-handed neutrino, Template:Math;

Template:Math for the charged leptons, Template:Mvar.

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 Template:Math and/or Template:Math called the Higgs fields which acquire a VEV, yielding the spontaneous symmetry breaking

Template:Math

The Template:Math representations transform under this subgroup as the reducible representation as follows:

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

<math>10_{1}\to (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}\to (1,1)_1</math> (e^c)

<math>24_0\to (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>.

Comparison with the standard SU(5)

The name "flipped" Template:Math arose in comparison to the "standard" Template:Math Georgi–Glashow model, in which Template:Mvar and Template:Mvar quark are respectively assigned to the Template:Math and Template:Math representation. In comparison with the standard Template:Math, the flipped Template:Math can accomplish the spontaneous symmetry breaking using Higgs fields of dimension 10, while the standard Template:Math typically requires a 24-dimensional Higgs.<ref>L.~F.~Li, ``Group Theory of the Spontaneously Broken Gauge Symmetries, Phys. Rev. D 9, 1723-1739 (1974) doi:10.1103/PhysRevD.9.1723</ref>

The sign convention for Template:Math varies from article/book to article.

The hypercharge Y/2 is a linear combination (sum) of the following:

<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}\in \text{SU}(5), \qquad \chi/5.</math>

There are also the additional fields Template:Math and Template:Math containing the electroweak Higgs doublets.

Calling the representations for example, Template:Math and Template:Math 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, and is a standard used by GUT theorists.

Since the homotopy group

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

this model does not predict monopoles. See 't Hooft–Polyakov monopole.

File:Proton decay3.svg

Dimension 6 proton decay mediated by the Template:Mvar boson <math>(3,2)_{\frac{1}{6}}</math> in flipped Template:Math GUT

Minimal supersymmetric flipped SU(5)

Spacetime

The Template:Math superspace extension of Template:Math Minkowski spacetime

Spatial symmetry

Template:Math SUSY over Template:Math Minkowski spacetime with R-symmetry

Global internal symmetry

Template:Math (matter parity) not related to Template:Math in any way for this particular model

Vector superfields

Those associated with the Template:Math gauge symmetry

Chiral superfields

As complex representations:

label	description	multiplicity	Template:Math rep	Template:Math rep	Template:Math
Template:Math	GUT Higgs field	Template:Math	Template:Math	+	Template:Math
Template:Math	GUT Higgs field	Template:Math	Template:Math	+	Template:Math
Template:Math	electroweak Higgs field	Template:Math	Template:Math	+	Template:Math
Template:Math	electroweak Higgs field	Template:Math	Template:Math	+	Template:Math
Template:Math	matter fields	Template:Math	Template:Math	-	Template:Math
Template:Math	matter fields	Template:Math	Template:Math	-	Template:Math
Template:Math	left-handed positron	Template:Math	Template:Math	-	Template:Math
Template:Mvar	sterile neutrino (optional)	Template:Math	Template:Math	-	Template:Math
Template:Mvar	singlet	Template:Math	Template:Math	+	Template:Math

Superpotential

A generic invariant renormalizable superpotential is a (complex) Template:Math invariant cubic polynomial in the superfields which has an Template:Math-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). Template:Mvar and Template:Mvar are the generation indices. The coupling Template:Math has coefficients which are symmetric in Template:Mvar and Template:Mvar.

In those models without the optional Template:Mvar sterile neutrinos, we add the nonrenormalizable couplings instead.

<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>

These couplings do break the R-symmetry.

See also

• Flipped SO(10)