In physics, the Pati-Salam model is a Grand Unification Theory (GUT) which states that the gauge group is either SU(4) × SU(2)_L× SU(2)_R or ( SU(4) × SU(2)_L× SU(2)_R ) / Z₂ and the fermions form three families, each consisting of the representations (4,2,1) and (\bar 4,1,2). This needs some explanation. The center of SU(4)× SU(2)_L× SU(2)_R is Z₄× Z_2L× Z_2R. The Z₂ in the quotient refers to the two element subgroup generated by the element of the center corresponding to the 2 element of Z₄ and the 1 elements of Z_2L and Z_2R. This includes the right-handed neutrino, which is now likely believed to exist. See neutrino oscillations. There is also a (4,1,2) and/or a (\bar 4,1,2) scalar field called the Higgs field which acquires a VEV. This results in a spontaneous symmetry breaking from SU(4)\times SU(2)_L\times SU(2)_R to [SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_3 or from [SU(4)\times SU(2)_L\times SU(2)_R]/\mathbb{Z}_2 to [SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6 and also,

(4,2,1)\rightarrow (3,2)_{\frac{1}{6}}\oplus (1,2)_{-\frac{1}{2}} (q and l),

(\bar{4},1,2)\rightarrow (\bar{3},1)_{\frac{1}{3}}\oplus (\bar{3},1)_{-\frac{2}{3}}\oplus (1,1)_1\oplus (1,1)_0 (d^c, u^c, e^c and ν^c),

(6,1,1)\rightarrow (3,1)_{-\frac{1}{3}}\oplus (\bar{3},1)_{\frac{1}{3}}, (1,3,1)\rightarrow (1,3)_0 and (1,1,3)\rightarrow (1,1)_1\oplus (1,1)_0\oplus (1,1)_{-1}. See restricted representation. Of course, calling the representations things like (\bar{4},1,2) and (6,1,1) 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.

The hypercharge, Y/2 is the sum of \begin{pmatrix}\frac{1}{6}&0&0&0\\0&\frac{1}{6}&0&0\\0&0&\frac{1}{6}&0\\0&0&0&-\frac{1}{2}\end{pmatrix} of SU(4) and \begin{pmatrix}\frac{1}{2}&0\\0&-\frac{1}{2}\end{pmatrix} of SU(2)_R

Actually, it is possible to extend the Pati-Salam group so that it has two connected components. The relevant group is now the semidirect product \{[SU(4)\times SU(2)_L\times SU(2)_R]/\mathbb{Z}_2\}\rtimes\mathbb{Z}_2. The last Z₂ also needs explaining. It corresponds to an automorphism of the (unextended) Pati-Salam group which is the composition of an involutive outer automorphism of SU(4) which isn't an inner automorphism with interchanging the left and right copies of SU(2). 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, (4,2,1)\oplus(\bar{4},1,2) is an irrep and so is (4,1,2)\oplus(\bar{4},2,1). This is the simplest extension of the minimal left-right model unifying QCD with B?L.

Since the homotopy group \pi_2\left(\frac{SU(4)\times SU(2)}{[SU(3)\times U(1)]/\mathbb{Z}_3}\right)=\mathbb{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, of course).

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)× SU(2)_L × SU(2)_R]/Z₂

Global internal symmetry

U(1)_A

Vector superfields

Those associated with the SU(4)× SU(2)_L × SU(2)_R gauge symmetry

Chiral superfields

As complex representations:

!label!!description!!multiplicity!!SU(4)×SU(2)_L× SU(2)_R rep!!R!!A |- |(4,1,2)_H||GUT Higgs field||1||(4,1,2)||0||0 |- |(\bar{4},1,2)_H||GUT Higgs field||1||(\bar{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)||matter field||3||(4,2,1)||1||1 |- |(\bar{4},1,2)||matter field||3||(\bar{4},1,2)||1||-1 |- |\phi||sterile neutrino||3||(1,1,1)||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: \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}

i and j 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 (4,2,1)_H and (\bar{4},2,1)_H

References

J. Pati and A. Salam, Phys. Rev. D10 (1974), 275.

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