Editing Pati–Salam model

{{Short description|A Grand Unification Theory proposed in 1974 by nobel laureate Abdus Salam and Jogesh Pati}}
{{no footnotes|date=April 2021}}
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 charge]]s, dubbed red, green, blue and violet (or originally lilac), instead of the conventional three, with the new "violet" quark being identified with the [[lepton]]s. 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 [[neutrino]]s.

Originally the fourth color was labelled "'''l'''ilac" to alliterate with "'''l'''epton".{{sfn|Pati|Salam|1974}} Pati–Salam is an alternative to the [[Georgi–Glashow model|Georgi–Glashow {{math|SU(5)}} unification]] also proposed in 1974. Both can be embedded within an [[SO(10) (physics)|{{math|SO(10)}} unification model]].

== Core theory ==
The Pati–Salam model states that the [[gauge group]] is either {{math|[[Special unitary group|SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]]}} or {{math|[[Special unitary group|(SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>)/'''Z'''<sub>2</sub>]]}} and the fermions form three families, each consisting of the [[Representations of Lie groups/algebras|representations {{math|('''4''', '''2''', '''1''')}}]] and {{math|({{overline|'''4'''}}, '''1''', '''2''')}}. This needs some explanation. The [[Center (group theory)|center]] of {{math|SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} is {{math|'''Z'''<sub>4</sub> × '''Z'''<sub>2L</sub> × '''Z'''<sub>2R</sub>}}. The {{math|'''Z'''<sub>2</sub>}} in the quotient refers to the two element subgroup generated by the element of the center corresponding to the two element of {{math|'''Z'''<sub>4</sub>}} and the 1 elements of {{math|'''Z'''<sub>2L</sub>}} and {{math|'''Z'''<sub>2R</sub>}}. This includes the right-handed neutrino. See [[neutrino oscillation]]s. There is also a {{math|('''4''', '''1''', '''2''')}} and/or a {{math|({{overline|'''4'''}}, '''1''', '''2''')}} [[scalar field]] called the [[Higgs field]] which acquires a non-zero [[Vacuum expectation value|VEV]]. This results in a [[spontaneous symmetry breaking]] from {{math|SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} to {{math|(SU(3) × SU(2) × U(1)<sub>Y</sub>)/'''Z'''<sub>3</sub>}} or from {{math|(SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>)/'''Z'''<sub>2</sub>}} to {{math|(SU(3) × SU(2) × U(1)<sub>Y</sub>)/'''Z'''<sub>6</sub>}} and also,

:{{math|('''4''', '''2''', '''1''') → ('''3''', '''2''')<sub>{{sfrac|1|6}}</sub> ⊕ ('''1''', '''2''')<sub>− {{sfrac|1|2}}</sub>&nbsp;&nbsp;&nbsp;&nbsp;(''q'' & ''l'')}}
:{{math|({{overline|'''4'''}}, '''1''', '''2''') → ({{overline|'''3'''}}, '''1''')<sub>{{sfrac|1|3}}</sub> ⊕ ({{overline|'''3'''}}, '''1''')<sub>− {{sfrac|2|3}}</sub> ⊕ ('''1''', '''1''')<sub>1</sub> ⊕ ('''1''', '''1''')<sub>0</sub>&nbsp;&nbsp;&nbsp;&nbsp;(''d&thinsp;<sup>c</sup>'', ''u<sup>c</sup>'', ''e<sup>c</sup>'' & ''ν<sup>c</sup>'')}}
:{{math|('''6''', '''1''', '''1''') → ('''3''', '''1''')<sub>− {{sfrac|1|3}}</sub> ⊕ ({{overline|'''3'''}}, '''1''')<sub>{{sfrac|1|3}}</sub>}}
:{{math|('''1''', '''3''', '''1''') → ('''1''', '''3''')<sub>0</sub>}}
:{{math|('''1''', '''1''', '''3''') → ('''1''', '''1''')<sub>1</sub> ⊕ ('''1''', '''1''')<sub>0</sub> ⊕ ('''1''', '''1''')<sub>−1</sub>}}

See [[restricted representation]]. Of course, calling the [[Representations of Lie groups/algebras|representations]] things like {{math|({{overline|'''4'''}}, '''1''', '''2''')}} and {{math|('''6''', '''1''', '''1''')}} is purely a physicist's convention(source?), not a mathematician's convention, where representations are either labelled by [[Young tableau]]x or [[Dynkin diagram]]s 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 space|connected component]]s. 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 {{math|'''Z'''<sub>2</sub>}} also needs explaining. It corresponds to an [[automorphism]] of the (unextended) Pati–Salam group which is the [[Function composition|composition]] of an [[Involution (mathematics)|involutive]] [[outer automorphism]] of {{math|SU(4)}} which isn't an [[inner automorphism]] with interchanging the left and right copies of {{math|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 (physics)|parity]] which had been shown not to hold at low energy scales for the [[weak interaction]]. In this extended model, {{math|('''4''', '''2''', '''1''') ⊕ ({{overline|'''4'''}}, '''1''', '''2''')}} is an [[irreducible representation|irrep]] and so is {{math|('''4''', '''1''', '''2''') ⊕ ({{overline|'''4'''}}, '''2''', '''1''')}}. This is the simplest extension of the minimal [[left-right model]] unifying [[Quantum chromodynamics|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 [[Magnetic monopole|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 {{math|SU(5)}}]] unification models can be embedded in a [[SO(10) (physics)|{{math|SO(10)}} unification]]. The difference between the two models then lies in the way that the {{math|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 {{math|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 {{math|U(1)<sub>B-L</sub>}}) starts being unified with the color charge in the {{math|SU(4)<sub>C</sub>}} group, while the other part of the weak hypercharge is in the {{math|SU(2)<sub>R</sub>}}. When those two groups break then the two parts together eventually unify into the usual weak hypercharge {{math|U(1)<sub>Y</sub>}}.

==Minimal supersymmetric Pati–Salam==

===Spacetime===
The {{math|''N'' {{=}} 1}} superspace extension of {{math|3 + 1}} Minkowski spacetime

===Spatial symmetry===
N=1 SUSY over {{math|3 + 1}} Minkowski spacetime with [[R-symmetry]]

===Gauge symmetry group===
{{math|(SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>)/'''Z'''<sub>2</sub>}}

===Global internal symmetry===
{{math|U(1)<sub>A</sub>}}

===Vector superfields===
Those associated with the {{math|SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} gauge symmetry

===Chiral superfields===
As complex representations:

{| class="wikitable"
!label!!description!!multiplicity!!{{math|SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} rep!!R!!A
|-
|{{math|({{overline|'''4'''}}, '''1''', '''2''')<sub>H</sub>}}||GUT Higgs field||{{math|1}}||{{math|('''4''', '''1''', '''2''')}}||{{math|0}}||{{math|0}}
|-
|{{math|({{overline|'''4'''}}, '''1''', '''2''')<sub>H</sub>}}||GUT Higgs field||{{math|1}}||{{math|({{overline|'''4'''}}, '''1''', '''2''')}}||{{math|0}}||{{math|0}}
|-
|{{mvar|S}}||singlet||{{math|1}}||{{math|('''1''', '''1''', '''1''')}}||{{math|2}}||{{math|0}}
|-
|{{math|('''1''', '''2''', '''2''')<sub>H</sub>}}||electroweak Higgs field||{{math|1}}||{{math|('''1''', '''2''', '''2''')}}||{{math|0}}||{{math|0}}
|-
|{{math|('''6''', '''1''', '''1''')<sub>H</sub>}}||no name||{{math|1}}||{{math|('''6''', '''1''', '''1''')}}||{{math|2}}||{{math|0}}
|-
|{{math|('''4''', '''2''', '''1''')}}||left handed matter field||{{math|3}}||{{math|('''4''', '''2''', '''1''')}}||{{math|1}}||{{math|1}}
|-
|{{math|({{overline|'''4'''}}, '''1''', '''2''')}}||right handed matter field including right handed (sterile or heavy) neutrinos||{{math|3}}||{{math|({{overline|'''4'''}}, '''1''', '''2''')}}||{{math|1}}||{{math|−1}}
|-
|}

===Superpotential===
A generic invariant renormalizable superpotential is a (complex) {{math|SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} and {{math|U(1)<sub>R</sub>}} 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 {{math|('''4''', '''2''', '''1''')<sub>H</sub>}} and {{math|({{overline|'''4'''}}, '''2''', '''1''')<sub>H</sub>}}.

==Sources==
* Graham G. Ross, ''Grand Unified Theories'', Benjamin/Cummings, 1985, {{ISBN|0-8053-6968-6}}
* Anthony Zee, ''Quantum Field Theory in a Nutshell'', Princeton U. Press, Princeton, 2003, {{ISBN|0-691-01019-6}}

==References==
{{reflist|25em}}

* {{cite journal | last1=Pati | first1=Jogesh C. | last2=Salam | first2=Abdus | title=Lepton number as the fourth "color" | journal=Physical Review D | volume=10 | issue=1 | date=1 June 1974 | issn=0556-2821 | doi=10.1103/physrevd.10.275 | pages=275–289| bibcode=1974PhRvD..10..275P }}
* {{Cite journal|author-link=John Baez |last=Baez |first=John C. |last2=Huerta |first2=J. |arxiv=0904.1556 |title=The Algebra of Grand Unified Theories |journal= Bulletin of the American Mathematical Society|year=2010 |volume=47 |issue=3 |pages=483–552 |doi=10.1090/S0273-0979-10-01294-2 |s2cid=2941843 }}

==External links==
* {{cite journal |bibcode=1985ZPhyC..27..321W |doi=10.1007/BF01556623 |title=Proton decay, annihilation or fusion? |year=1985 |last1=Wu |first1=Dan-di |last2=Li |first2=Tie-Zhong |journal=Zeitschrift für Physik C |volume=27 |issue=2 |pages=321–323 |s2cid=121868029 }} – Fusion of all three quarks is the only decay mechanism mediated by the [[Higgs particle]], not the [[gauge bosons]], in the Pati–Salam model
*[http://math.ucr.edu/~huerta/oral.pdf The Algebra of Grand Unified Theories] John Huerta. Slide show: contains an overview of Pati–Salam
*[http://math.ucr.edu/~huerta/guts/node18.html the Pati-Salam model] Motivation for the Pati–Salam model

{{DEFAULTSORT:Pati-Salam model}}
[[Category:Grand Unified Theory]]
[[Category:Abdus Salam]]