Pati–Salam model

is purely a physicist's convention(source?), 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.

The weak hypercharge, Y, is the sum of the two matrices:

{\begin{pmatrix}{\frac {1}{3}}&0&0&0\\0&{\frac {1}{3}}&0&0\\0&0&{\frac {1}{3}}&0\\0&0&0&-1\end{pmatrix}}\in {\text{SU}}(4),\qquad {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\in {\text{SU}}(2)_{\text{R}}

It is possible to extend the Pati–Salam group so that it has two connected components. The relevant group is now the semidirect product $\left([SU(4)\times SU(2)_{L}\times SU(2)_{R}]/\mathbf {Z} _{2}\right)\rtimes \mathbf {Z} _{2}$ . The last $Z 2$ also needs explaining. It corresponds to an

B−L

.

Since the homotopy group

\pi _{2}\left({\frac {SU(4)\times SU(2)}{[SU(3)\times U(1)]/\mathbf {Z} _{3}}}\right)=\mathbf {Z} ,

this model predicts monopoles. See 't Hooft–Polyakov monopole.

This model was invented by Jogesh Pati and Abdus Salam.

This model doesn't predict gauge mediated proton decay (unless it is embedded within an even larger GUT group).

Differences from the SU(5) unification

As mentioned above, both the Pati–Salam and

SO(10)

unification. The difference between the two models then lies in the way that the

SO(10)

symmetry is broken, generating different particles that may or may not be important at low scales and accessible by current experiments. If we look at the individual models, the most important difference is in the origin of the weak hypercharge

. In the

SU(5)

model by itself there is no left-right symmetry (although there could be one in a larger unification in which the model is embedded), and the weak hypercharge is treated separately from the color charge. In the Pati–Salam model, part of the weak hypercharge (often called

U(1) B-L

) starts being unified with the color charge in the

SU(4) C

group, while the other part of the weak hypercharge is in the

SU(2) R

. When those two groups break then the two parts together eventually unify into the usual weak hypercharge

U(1) Y

.

Minimal supersymmetric Pati–Salam

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

$(SU(4) \times SU(2) L \times SU(2) R)/ Z 2$

Global internal symmetry

$U(1) A$

Vector superfields

Those associated with the $SU(4) \times SU(2) L \times SU(2) R$ gauge symmetry

Chiral superfields

As complex representations:

label	description	multiplicity	$SU(4) \times SU(2) L \times SU(2) R$ rep	R	A
$(4, 1, 2) H$	GUT Higgs field	$1$	$(4, 1, 2)$	$0$	$0$
$(4, 1, 2) H$	GUT Higgs field	$1$	$(4, 1, 2)$	$0$	$0$
$S$	singlet	$1$	$(1, 1, 1)$	$2$	$0$
$(1, 2, 2) H$	electroweak Higgs field	$1$	$(1, 2, 2)$	$0$	$0$
$(6, 1, 1) H$	no name	$1$	$(6, 1, 1)$	$2$	$0$
$(4, 2, 1)$	left handed matter field	$3$	$(4, 2, 1)$	$1$	$1$
$(4, 1, 2)$	right handed matter field including right handed (sterile or heavy) neutrinos	$3$	$(4, 1, 2)$	$1$	$-1$

Superpotential

A generic invariant renormalizable superpotential is a (complex) $SU(4) \times SU(2) L \times SU(2) R$ and $U(1) R$ invariant cubic polynomial in the superfields. It is a linear combination of the following terms: