Pati–Salam model

Template:Short description Template:No footnotes In physics, the Pati–Salam model is a Grand Unified Theory (GUT) proposed in 1974 by Abdus Salam and Jogesh Pati. Like other GUTs, its goal is to explain the seeming arbitrariness and complexity of the Standard Model in terms of a simpler, more fundamental theory that unifies what are in the Standard Model disparate particles and forces. The Pati–Salam unification is based on there being four quark color charges, dubbed red, green, blue and violet (or originally lilac), instead of the conventional three, with the new "violet" quark being identified with the leptons. The model also has left–right symmetry and predicts the existence of a high energy right handed weak interaction with heavy W' and Z' bosons and right-handed neutrinos.

Originally the fourth color was labelled "lilac" to alliterate with "lepton".Template:Sfn Pati–Salam is an alternative to the [[Georgi–Glashow model|Georgi–Glashow Template:Math unification]] also proposed in 1974. Both can be embedded within an [[SO(10) (physics)|Template:Math unification model]].

Core theory

The Pati–Salam model states that the gauge group is either Template:Math or Template:Math and the fermions form three families, each consisting of the [[Representations of Lie groups/algebras|representations Template:Math]] and Template:Math. This needs some explanation. The center of Template:Math is Template:Math. The Template:Math in the quotient refers to the two element subgroup generated by the element of the center corresponding to the two element of Template:Math and the 1 elements of Template:Math and Template:Math. This includes the right-handed neutrino. See neutrino oscillations. There is also a Template:Math and/or a Template:Math scalar field called the Higgs field which acquires a non-zero VEV. This results in a spontaneous symmetry breaking from Template:Math to Template:Math or from Template:Math to Template:Math and also,

Template:Math

Template:Math

Template:Math

Template:Math

Template:Math

See restricted representation. Of course, calling the representations things like Template:Math and Template:Math 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:

<math>\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}}</math>

It is possible to extend the Pati–Salam group so that it has two connected components. The relevant group is now the semidirect product <math>\left ([SU(4)\times SU(2)_L\times SU(2)_R]/\mathbf{Z}_2\right )\rtimes\mathbf{Z}_2</math>. The last Template:Math also needs explaining. It corresponds to an automorphism of the (unextended) Pati–Salam group which is the composition of an involutive outer automorphism of Template:Math which isn't an inner automorphism with interchanging the left and right copies of Template:Math. This explains the name left and right and is one of the main motivations for originally studying this model. This extra "left-right symmetry" restores the concept of parity which had been shown not to hold at low energy scales for the weak interaction. In this extended model, Template:Math is an irrep and so is Template:Math. This is the simplest extension of the minimal left-right model unifying QCD with B−L.

Since the homotopy group

<math>\pi_2\left(\frac{SU(4)\times SU(2)}{[SU(3)\times U(1)]/\mathbf{Z}_3}\right)=\mathbf{Z},</math>

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 [[Georgi–Glashow model|Georgi–Glashow Template:Math]] unification models can be embedded in a [[SO(10) (physics)|Template:Math unification]]. The difference between the two models then lies in the way that the Template:Math 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 Template:Math 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 Template:Math) starts being unified with the color charge in the Template:Math group, while the other part of the weak hypercharge is in the Template:Math. When those two groups break then the two parts together eventually unify into the usual weak hypercharge Template:Math.

Minimal supersymmetric Pati–Salam

Spacetime

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

Spatial symmetry

N=1 SUSY over Template:Math Minkowski spacetime with R-symmetry

Vector superfields

Those associated with the Template:Math gauge symmetry

Chiral superfields

As complex representations:

label	description	multiplicity	Template:Math rep	R	A
Template:Math	GUT Higgs field	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	GUT Higgs field	Template:Math	Template:Math	Template:Math	Template:Math
Template:Mvar	singlet	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	electroweak Higgs field	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	no name	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	left handed matter field	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	right handed matter field including right handed (sterile or heavy) neutrinos	Template:Math	Template:Math	Template:Math	Template:Math

Superpotential

A generic invariant renormalizable superpotential is a (complex) Template:Math and Template:Math invariant cubic polynomial in the superfields. It is a linear combination of the following terms:

<math>\begin{matrix}

S \\ S(4,1,2)_H (\bar{4},1,2)_H\\ S(1,2,2)_H (1,2,2)_H \\ (6,1,1)_H (4,1,2)_H (4,1,2)_H\\ (6,1,1)_H (\bar{4},1,2)_H (\bar{4},1,2)_H\\ (1,2,2)_H (4,2,1)_i (\bar{4},1,2)_j\\ (4,1,2)_H (\bar{4},1,2)_i \phi_j\\ \end{matrix} </math>

<math>i</math> and <math>j</math> are the generation indices.

Left-right extension

We can extend this model to include left-right symmetry. For that, we need the additional chiral multiplets Template:Math and Template:Math.

Sources

• Graham G. Ross, Grand Unified Theories, Benjamin/Cummings, 1985, Template:ISBN
• Anthony Zee, Quantum Field Theory in a Nutshell, Princeton U. Press, Princeton, 2003, Template:ISBN

References

• Template:Cite journal
• Template:Cite journal

External links

• Template:Cite journal – Fusion of all three quarks is the only decay mechanism mediated by the Higgs particle, not the gauge bosons, in the Pati–Salam model
• The Algebra of Grand Unified Theories John Huerta. Slide show: contains an overview of Pati–Salam
• the Pati-Salam model Motivation for the Pati–Salam model