Flipped SU(5)

The Flipped SU(5) model is a GUT theory which states that the gauge group is [ SU(5) × U(1)_χ ]/$\mathbb {Z} _{5}$ and the fermions form three families, each consisting of the representations ${\bar {5}}_{-3}$, 10₁ and 1₅. This includes three right-handed neutrinos, which is consistent with the observed neutrino oscillations. There is also a 10₁ and/or ${\bar {10}}_{-1}$ called the Higgs field which acquires a VEV. This results in a spontaneous symmetry breaking from

$[SU(5)\times U(1)_{\chi }]/\mathbb {Z} _{5}$

to

$[SU(3)\times SU(2)\times U(1)_{Y}]/\mathbb {Z} _{6}$

and also,

${\bar {5}}_{-3}\rightarrow ({\bar {3}},1)_{-{\frac {2}{3}}}\oplus (1,2)_{-{\frac {1}{2}}}$ (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)

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

$24_{0}\rightarrow (8,1)_{0}\oplus (1,3)_{0}\oplus (1,1)_{0}\oplus (3,2)_{\frac {1}{6}}\oplus ({\bar {3}},2)_{-{\frac {1}{6}}}$. 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 ${\begin{pmatrix}{1 \over 15}&0&0&0&0\\0&{1 \over 15}&0&0&0\\0&0&{1 \over 15}&0&0\\0&0&0&-{1 \over 10}&0\\0&0&0&0&-{1 \over 10}\end{pmatrix}}$ 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.

To do:

This theory was invented by ???.

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	$\mathbb {Z} _{2}$ rep	U(1)R
10H	GUT Higgs field	1	101	+	0
${\bar {10}}_{H}$	GUT Higgs field	1	${\overline {10}}_{-1}$	+	0
Hu	electroweak Higgs field	1	${\bar {5}}_{2}$	+	2
Hd	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	3	10	-	2
S	singlet	1	10	+	2

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: ${\begin{matrix}S&S\\S10_{H}{\overline {10}}_{H}&S10_{H}^{\alpha \beta }{\overline {10}}_{H\alpha \beta }\\10_{H}10_{H}H_{d}&\epsilon _{\alpha \beta \gamma \delta \epsilon }10_{H}^{\alpha \beta }10_{H}^{\gamma \delta }H_{d}^{\epsilon }\\{\overline {10}}_{H}{\overline {10}}_{H}H_{u}&\epsilon ^{\alpha \beta \gamma \delta \epsilon }{\overline {10}}_{H\alpha \beta }{\overline {10}}_{H\gamma \delta }H_{u\epsilon }\\H_{d}1010&\epsilon _{\alpha \beta \gamma \delta \epsilon }H_{d}^{\alpha }10_{i}^{\beta \gamma }10_{j}^{\delta \epsilon }\\H_{d}{\bar {5}}1&H_{d}^{\alpha }{\bar {5}}_{i\alpha }1_{j}\\H_{u}10{\bar {5}}&H_{u\alpha }10_{i}^{\alpha \beta }{\bar {5}}_{j\beta }\\{\overline {10}}_{H}10\phi &{\overline {10}}_{H\alpha \beta }10_{i}^{\alpha \beta }\phi _{j}\\\end{matrix}}$

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 10j has coefficients which are symmetric in i and j.}