ABSTRACT (original article): To explain quark and lepton masses and mixing angles, one has to extend the standard model, and the usual practice is to put the quarks and leptons into irreducible representations of discrete groups. We argue that discrete flavor symmetries (and their concomitant problems) can be avoided if we extend the gauge group. In the framework of SU(12) we give explicit examples of models having varying degrees of predictability obtained by scanning over groups and representations and identifying cases with operators contributing to mass and mixing matrices that need little fine-tuning of prefactors. Fitting with quark and lepton masses run to the GUT scale and known mixing angles allows us to make predictions for the neutrino masses and hierarchy, the octant of the atmospheric mixing angle, leptonic CP violation, Majorana phases, and the effective mass observed in neutrinoless double beta decay. CITATION (original article): C.H. Albright, R. Feger, T.W. Kephart (2016), Phys. Rev. D 93, 075032. https://doi.org/10.1103/PhysRevD.93.075032 [arXiv:1601.07523]

Preamble

This notebook requires the Wolfram paclet LieART, which can be installed via
In[]:=
PacletInstall["LieART/LieART"]
Out[]=
PacletObject
Name: LieART/LieART
Version: 2.1.1

Please move the accompanying package files (GrandUnifiedTheories.m, LieARTExtension.m, ModelFit.m, and init.m) into a subdirectory named GrandUnifiedTheories. This subdirectory can be installed in standard package locations or placed in the same directory as this notebook. It is loaded by:
In[]:=
SetDirectory[NotebookDirectory[]];​​<<GrandUnifiedTheories`

SU(12) Unification Models

We found economical sets of models for SU(12) as unification group:
In[]:=
algebra=SU12
Out[]=
SU(12)
The group has eleven antisymmetric irreducible representations (irreps), excluding the singlet, also known as basic irreps:
In[]:=
BasicIrreps[algebra]
Out[]=
{12,66,220,495,792,924,
792
,
495
,
220
,
66
,
12
}
of which ten are complex:
In[]:=
complexBasicIrreps=Select[BasicIrreps[algebra],ComplexIrrepQ]
Out[]=
{12,66,220,495,792,
792
,
495
,
220
,
66
,
12
}
and one is real, and hence cannot be used for chiral fermion assignments:
In[]:=
Select[BasicIrreps[algebra],RealIrrepQ]
Out[]=
{924}
The irreps can be represented by single-column Young tableaux:
In[]:=
DisplayYoungTableaux[BasicIrreps[algebra]]
Out[]=
12
66
220
495
792
924
792
495
220
66
12
​
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Fermion Assignments

To avoid the violation of gauge-symmetry currents, the anomalies of fermion irreps need to cancel out, leading to an anomaly-free model. The anomalies of SU(12) basic irreps are:
In[]:=
TableForm[Transpose[{#,Anomaly[#]}&/@BasicIrreps[algebra]],TableHeadings->{{"Irrep","Anomaly"},None}]
Out[]//TableForm=
Irrep
12
66
220
495
792
924
792
495
220
66
12
Anomaly
-1
-8
-27
-48
-42
0
42
48
27
8
1
Anomaly-free sets containing three families of SU(5) fermions with at most twenty irreps are:
In[]:=
su12AnomalyFreeSetCoefficients=FindSuitableRepresentationVar[algebra,3,NoPairs->True,NumberOfFamilies3,WithRestgroupFalse,MaxNumberOfIrreps20];Column[su12AnomalyFreeSets=IrrepSum/@SortBy[su12AnomalyFreeSetCoefficients,Total[#]&]]
Out[]=
66+495+2(
220
)+2(
12
)
220+3(
66
)+3(
12
)
3(12)+220+792+
495
+3(
66
)
66+2(495)+
792
+2(
220
)+8(
12
)
220+495+
792
+3(
66
)+9(
12
)
3(66)+3(792)+3(
495
)+6(
12
)
3(66)+2(792)+2(
495
)+12(
12
)
Two of the sets are of special interest and further analysed in the following:
In[]:=
(setsOfSpecialInterest=su12AnomalyFreeSets[[{1,4}]])//TableForm
Out[]//TableForm=
66+495+2(
220
)+2(
12
)
66+2(495)+
792
+2(
220
)+8(
12
)
Using the SU(12)SU(5) branching rules for the contained irreps
In[]:=
irrepsInSetsOfInterest=Reverse[Union[Join@@(IrrepList/@setsOfSpecialInterest)]];​​IrrepRule[#,DecomposeIrrep[#,SU5]]&/@irrepsInSetsOfInterest//TableForm
Out[]//TableForm=
66→21(1)+7(5)+10
495→35(1)+35(5)+
5
+21(10)+7(
10
)
792
→22(1)+7(5)+35(
5
)+21(10)+35(
10
)
220
→35(1)+21(
5
)+10+7(
10
)
12
→7(1)+
5
we obtain the following sets of SU(5) irreps, with three chiral families 3(
10
+
5
+
1
) and conjugate paired irreps that become massive at the SU(12) scale:
In[]:=
IrrepRule[#,IrrepPlus@@DecomposeIrrep[IrrepList[#],SU5]]&/@setsOfSpecialInterest//TableForm
Out[]//TableForm=
66+495+2(
220
)+2(
12
)→140(1)+42(5)+45(
5
)+24(10)+21(
10
)
66+2(495)+
792
+2(
220
)+8(
12
)→239(1)+84(5)+87(
5
)+66(10)+63(
10
)
The three SU(5) families can be selected from the following possible embeddings of the
10
of SU(5)
In[]:=
embedded10Irreps=EmbeddedIrrep[Irrep[SU5][10],#]&/@Select[irrepsInSetsOfInterest,Contains10s]
Out[]=
{(10)66,(10)495,(10)
792
,(10)
220
}
and
5
of SU(5):
In[]:=
embedded5barIrreps=EmbeddedIrrep[Irrep[SU5][Bar[5]],#]&/@Select[irrepsInSetsOfInterest,Contains5Bars]
Out[]=
{(
5
)495,(
5
)
792
,(
5
)
220
,(
5
)
12
}
The regular embeddings of SU(5) in SU(12) irreps displayed as boxes filled with grey on the top of the SU(12) Young tableaux:
In[]:=
DisplayYoungTableaux[Join[embedded10Irreps,embedded5barIrreps]]
Out[]=
(10)66
(10)495
(10)
792
(10)
220
(
5
)495
(
5
)
792
(
5
)
220
(
5
)
12

Higgs Assignments

We construct singlet-Higgs conjugate pairs from the basic irreps of SU(12) analogously:
In[]:=
embeddedSinglets=EmbeddedIrrep[Irrep[SU5][1],#]&/@Select[complexBasicIrreps,Contains1s]
Out[]=
{(1)12,(1)66,(1)220,(1)495,(1)792,(1)
792
,(1)
495
,(1)
220
,(1)
66
,(1)
12
}
and display them as embedded Young tableaux:
In[]:=
DisplayYoungTableaux[embeddedSinglets]
Out[]=
(1)12
(1)66
(1)220
(1)495
(1)792
(1)
792
(1)
495
(1)
220
(1)
66
(1)
12
An SU(5) adjoint-Higgs, the
24
, must be present to break the SU(5) GUT scale down to the Standard Model. It is best to include it in the complex pair
5148
and
5148
, with the following decomposition to SU(5):
In[]:=
IrrepRule[Irrep[SU12][5148],IrrepPlus@@DecomposeIrrep[Irrep[SU12][5148],SU5]]
Out[]=
5148→224(1)+245(5)+7(
5
)+147(10)+49(
10
)+35(15)+24+21(
40
)+7(
45
)
hence the adjoint-Higgs contribution is
In[]:=
adjointHiggs={EmbeddedIrrep[Irrep[SU5][24],Irrep[SU12][5148]],EmbeddedIrrep[Irrep[SU5][24],Irrep[SU12][Bar[5148]]]}
Out[]=
{(24)5148,(24)
5148
}
which can be represented as Young Tableaux:
In[]:=
DisplayYoungTableaux[adjointHiggs]
Out[]=
(24)5148
(24)
5148
​
​
​
​
​
​

Model Construction

Renormalizable dim-4 operators with two fermions and a Higgs are SU(5) and SU(12) singlets, i.e., the five SU(5) boxes in the Young tableau are aligned on top of the remaining seven SU(12) boxes (only tableaux leading to the singlet are shown and singlet columns are not removed):
Effective higher dimensional operators can be constructed by insertions of massive fermion pairs and Higgs in tree diagrams. Massive fermion pairs at the SU(12) scale are formed from complex irreps and can be selected from

Construction of the Mass Matrix Elements

The 33 component of the up-quark Yukawa matrix (U33) represents the largest source of the top quark mass and hence must be a dim-4 renormalizable contribution. With the possible fermion and Higgs assignments discussed in the first section, it turn out that the only possible 3-point vertex singlet is
All other matrix elements must be of dimension five or higher, which involves massive fermion pair and singlet or adjoint Higgs fields.
Dim-4 contributions may also arise for elements of the the down-quark Yukawa matrix:
which depends on the choice of Higgs assignments:
The elements of the heavy right-handed Majorana mass matrix with only singlet fermions reads:

Illustrated Model

We illustrate the derivation and fit of the mass matrices for a model with interesting mixing results. We enter the set of fermions, massive fermions and Higgs as lists:
A more readable form of the family structure, massive fermions and Higgs is as follows:

Mass Matrices

Including the coefficients of the used (4th) anomaly-free set for the fermion assignment and the mass matrices in the model, we can display it with all mass term diagrams arranged by their dimensions, as given in Appendix B in the original article:

Quark and Lepton Sector Fits

The mass matrices, more specifically the prefactors, ϵ and κ as well as the vacuum expectation values are fit to phenomenological data at the SU(5) unification scale. These are the masses or mass differences for neutrinos and mixing angles and phases. The fit is performed in two steps with the first for the quark sector only and a second including the lepton sector, but keeping the initial values from the quark sector fit.

Quark Sector Fit

Initial values of prefactors are set randomly near one and fit parameters and assumptions for numerical values of parameters, such as scales and phases, are set explicitly:
We display the quark model and fit results, including the prefactors in the mass matrices and show the phenomenological values at the SU(5) unification scale used in the fit for comparison. Note that the fit prefactors deviate from those given in the article, as several minima exist, that lead to the desired phenomenology:

Lepton Sector Fit

The full and final fit includes the lepton sector, especially the neutrino phenomenology, as far as known at the time of the publication of the original article in 2016. The results of the quark-sector fit are used as initial values of the quark-sector fit parameters:
We display the full quark- and lepton-model fit results, including mass matrices with fit prefactors an the phenomenology used in the fit for comparison:

Model Scan

Scan Strategy

The successful SU(12) models listed in the original article were found by a model scan, which is organized in enclosing loops. First, a list of anomaly-free sets of basic irreps of SU(12) that yield three families on the SU(5) level is constructed:
The scan loops over all anomaly-free set of irreps and for each generates all possible up-type quark assignments. Here we select the anomaly-free set discussed above:
For each up-type quark assignment yielding admissible mass-matrix dimension, all down-type quark assignments are generated and looped over:
In a third loop the scan constructs subsets of electroweak Higgs doublets, SU(5) Higgs singlets and massive-fermion pairs, up to a given maximal number. Only those sets of Higgs are kept that yield a dim-4 mass term for the U33 element:
After a successful fit of the quark sector and charged leptons the scan performs another loop over Majorana neutrino assignments, which are numerous:
A final fit of the complete quark- and lepton-sector model is performed with quark-sector initial values from the respective fit. A successful model is displayed as shown for the model of interest as panel of the fit results and a list of mass-term diagrams and collected for the final output.

Scan Execution

Fit parameters and assumptions for numerical values of parameters:
We limit the scan to the selected anomaly-free set of irreps and up-type assignments discussed above as well as the maximum number of singlet-Higgs and massive fermion pairs to keep the scan as short as possible. Remove the anomaly-free set and up-type assignments selection and increase the maximum number of massive fermion pairs up to five to obtain the models listed in the original article. Note that models might not appear, if the fit cannot reach the desired goodness within the limited number of tries. For the same reason, new models not appearing in the article may appear. If a model is found the fit parameters can differ from those in the article as several minima are possible due to the large number of parameters.

Model Summary

Table

Plots

CITE THIS NOTEBOOK

Unification of gauge, family, and flavor symmetries illustrated in gauged SU(12) models​
by by Carl H. Albright, Robert P. Feger, and Thomas W. Kephart
Wolfram Community, STAFF PICKS, January 21, 2026
​https://community.wolfram.com/groups/-/m/t/3619584