# Systematic study of the $SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}$ local
gauge symmetry

Richard H. Benavides  Instituto Tecnológico Metropolitano, ITM, Facultad de
Ciencias Exactas y Aplicadas, calle 73#76A-354, vía el volador, Medellín,
Colombia.  Yithsbey Giraldo  Departamento de Física, Universidad de Nariño,
A.A. 1175, San Juan de Pasto, Colombia.  Luis Muñoz  Instituto Tecnológico
Metropolitano, ITM, Facultad de Ciencias Exactas y Aplicadas, calle
73#76A-354, vía el volador, Medellín, Colombia.  William A. Ponce  Instituto
de Física, Universidad de Antioquia, A.A. 1226, Medellín, Colombia.  Eduardo
Rojas  Departamento de Física, Universidad de Nariño, A.A. 1175, San Juan de
Pasto, Colombia.

###### Abstract

We review in a systematic way how anomaly free $SU(3)_{c}\otimes
SU(3)_{L}\otimes U(1)_{x}$ models without exotic electric charges can be
constructed, using as basis closed sets of fermions which includes each one
the particles and antiparticles of all the electrically charged fields. Our
analysis reproduce not only the known models in the literature, but also shows
the existence of several more independent models for one and three families
not considered so far. A phenomenological analysis of the new models is done,
where the lowest limits at a 95 % CL on the gauge boson masses are presented.

###### pacs:

12.60.-i, 11.15.-q

##  I Introduction

The impressive success of the Standard Model (SM) based on the local gauge
group $SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}$ with the color sector
$SU(3)_{c}$ confined and the flavor sector $SU(2)_{L}\otimes U(1)_{Y}$ hidden
and broken spontaneously by the minimal Higgs mechanism Donoghue:1992dd , has
not been able enough to provide explanation for several fundamental issues,
among them: the hierarchical masses and the mixing angles for both, the quark
and the lepton sectors CarcamoHernandez:2015hbk ; Csaki:2015xpj ;
Novichkov:2021evw ; Rodejohann:2019izm ; Giraldo:2015cpp , charge quantization
deSousaPires:1998jc ; Dong:2005ebq ; RomeroAbad:2020uvo ; Pisano:1996ht ;
Doff:1998we ; deSousaPires:1999ca ; Dong:2005ebq , the strong CP violation
Pal:1994ba ; Neves:2021usp ; Dias:2003zt ; Dias:2003iq , the small neutrino
masses and their oscillations Boucenna:2014dia ; Binh:2021yfy , and last but
not least, the presence of dark matter and dark energy in the universe
Kelso:2014qka ; Mizukoshi:2010ky ; Ruiz-Alvarez:2012nvg ; Profumo:2013sca ;
Filippi:2005mt . Because of this, many physicists believe that the SM does not
stand for the final theory, representing only an effective model originated
from a more fundamental one.

Minimal extensions of the SM arise either by adding new fields, or by
enlarging the local gauge group (adding a right handed neutrino field
constitute its simples extension, something that ameliorate, but not solve
some of the problems mentioned above).

Simple extensions of the local gauge group consider an electroweak sector with
an extra abelian symmetry $SU(2)_{L}\otimes U(1)_{x}\otimes U(1)_{z}$
Ponce:1987wb ; Benavides:2018rgh , or either the so called left right
symmetric model $SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{(B-L)}$
Mohapatra:1974gc ; Mohapatra:1979ia ; Mohapatra:1980qe , and also
$SU(3)_{L}\otimes U(1)_{X}$, being the last one we are going to consider in
this study Valle:1983dk ; Pisano:1992bxx ; Frampton:1992wt ; Ponce:2001jn ;
Ponce:2002sg .

##  II 3-3-1 Models

In what follows we assume that the electroweak gauge group is
$SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}$ (3-3-1 for short) in which the
electroweak sector of the standard model $SU(2)_{L}\otimes U(1)_{Y}$ is
extended to $SU(3)_{L}\otimes U(1)_{X}$. We also assume that, as in the SM,
the color group $SU(3)_{c}$ is vector-like (free of anomalies) and that the
left-handed quarks (color triplets) and left-handed leptons (color singlets)
transform only under the two fundamental representations of $SU(3)_{L}$ (the 3
and $3^{*}$).

Two classes of models will show up: universal one family models where the
anomalies cancel in each family as in the SM, and family models where the
anomalies cancel by an interplay between the several families.

For the 3-3-1 models, the most general electric charge operator in the
extended electroweak sector is

$Q=a\lambda_{3}+\frac{1}{\sqrt{3}}b\lambda_{8}+XI_{3},$ (1)

where $\lambda_{\alpha},\;\alpha=1,2,\dots,8$ are the Gell-Mann matrices for
$SU(3)_{L}$ normalized as
Tr$(\lambda_{\alpha}\lambda_{\beta})=2\delta_{\alpha\beta}$ and
$I_{3}=Dg(1,1,1)$ is the diagonal $3\times 3$ unit matrix. If one assumes
$a=1/2$, the isospin $SU(2)_{L}$ of the SM is entirely embedded in
$SU(3)_{L}$; $b$ is a free parameter which fixes the model and the $X$ values
are obtained by anomaly cancellation. The 8 gauge fields $A_{\mu}^{\alpha}$ of
$SU(3)_{L}$ may be written as Ponce:2001jn ; Ponce:2002sg

$\sum_{\alpha}\lambda_{\alpha}A^{\alpha}_{\mu}=\sqrt{2}\left(\begin{array}[]{ccc}D^{0}_{1\mu}&W^{+}_{\mu}&K_{\mu}^{(b+1/2)}\\\
W^{-}_{\mu}&D^{0}_{2\mu}&K_{\mu}^{(b-1/2)}\\\
K_{\mu}^{-(b+1/2)}&K_{\mu}^{-(b-1/2)}&D^{0}_{3\mu}\\\ \end{array}\right),$ (2)

where
$D^{0}_{1\mu}=A_{\mu}^{3}/\sqrt{2}+A_{\mu}^{8}/\sqrt{6},\;D^{0}_{2\mu}=-A_{\mu}^{3}/\sqrt{2}+A_{\mu}^{8}/\sqrt{6},$
and $D^{0}_{3\mu}=-2A_{\mu}^{8}/\sqrt{6}.$ The upper indices on the gauge
bosons in Eq. (2) stand for the electric charge of the particles, some of them
being functions of the $b$ parameter Byakti:2020ipa .

##  III The Minimal Model

In Ref. Valle:1983dk ; Frampton:1992wt  it has been shown that, for $b=3/2$,
the following fermion structure is free of all the gauge anomalies:
$\psi_{lL}^{T}=(\nu_{l}^{0},l^{-},l^{+})_{L}\sim(1,3,0),\;Q_{iL}^{T}=(d_{i},u_{i},X_{i})_{L}\sim(3,3^{*},-1/3),\;Q_{3L}^{T}=(u_{3},d_{3},Y)\sim(3,3,2/3)$,
where $l=e,\mu,\tau$ is a family lepton index, $i=1,2$ for the first two quark
families, and the numbers after the similarity sign means 3-3-1
representations. The right handed fields are
$u_{aL}^{c}\sim(3^{*},1,-2/3),\;d_{aL}^{c}\sim(3^{*},1,1/3),\;X_{iL}^{c}\sim(3^{*},1,4/3)$
and $Y_{L}^{c}\sim(3^{*},1,-5/3)$, where $a=1,2,3$ is the quark family index,
and there are two exotic quarks with electric charge $-4/3\;(X_{i})$ and other
with electric charge 5/3 $(Y)$. This version is called minimal in the
literature, because it does not make use of exotic leptons, including possible
right-handed neutrinos.

##  IV  3-3-1 Models Without Exotic Electric Charges

If one wishes to avoid exotic electric charges in the fermion and boson
sectors as the ones present in the minimal (3-3-1) model, one must choose
$b=1/2$ in Eq. (1) as shown in Ref Ponce:2001jn ; Ponce:2002sg .

To begin with our systematic analysis, let us start with closed fermion
structures consisting of only one left handed $SU(3)_{L}$ triplet and right
handed singlets, where for “closed” we mean structures containing the
antiparticles of all the electric charged particles. Following the notation in
Ref. Ponce:2001jn ; Ponce:2002sg , there are only six of such structures
containing at least one of the fermion fields in one family of the SM, or in
its minimal extension with right-handed neutrino fields:

there are only six such structures that contain at least one of the fermion
fields in a family of SMthere are only six such structures that contain at
least one of the fermion fields in a family of SM

  * •

$S_{1}=[(\nu^{0}_{e},e^{-},E_{1}^{-})\oplus e^{+}\oplus E_{1}^{+}]_{L}$ with
quantum numbers $(1,3,-2/3);(1,1,1)$ and $(1,1,1)$ respectively.

  * •

$S_{2}=[(e^{-},\nu_{e}^{0},N_{1}^{0})\oplus e^{+}]_{L}$ with quantum numbers
$(1,3^{*},-1/3)$ and $(1,1,1)$ respectively.

  * •

$S_{3}=[(d,u,U)\oplus u^{c}\oplus d^{c}\oplus U^{c}]_{L}$ with quantum numbers
$(3,3^{*},1/3)\;$; $(3^{*},1,-2/3)\;$; $(3^{*},1,1/3)$ and $(3^{*},1,-2/3)$
respectively.

  * •

$S_{4}=[(u,d,D)\oplus u^{c}\oplus d^{c}\oplus D^{c}]_{L}$ with quantum numbers
$(3,3,0)\;$; $(3^{*},1,-2/3)\;$; $(3^{*},1,1/3)$ and $(3^{*},1,1/3)$
respectively.

  * •

$S_{5}=[(N_{2}^{0},E_{2}^{+},e^{+})\oplus E^{-}_{2}\oplus e^{-}]_{L}$ with
quantum numbers $(1,3^{*},2/3)\;\;(1,1,-1)$, and $(1,1,-1)$ respectively.

  * •

$S_{6}=[(E^{+}_{3},N_{3}^{0},N_{4}^{0})\oplus E_{3}^{-}]_{L}$ with quantum
numbers $(1,3,1/3)$ and $(1,1,-1)$ respectively,

where for phenomenological reasons we allow for the precense of several exotic
leptons (charged and neutral), but only one exotic quark of each type (down-
type or up-type). In the former sets, $N^{0}_{1}$ and $N_{4}^{0}$ can play the
role of the right handed neutrino field $\nu_{e}^{0c}$ in an SO(10) basis.

Notice that the value $b=1/2$ in Eq. (1) implies that the electric charge of
the last two components of a 3 or a $3^{*}$ of $SU(3)_{L}$ are of the same
value. At this point, our approach is different to the one presented in Ref.
Ponce:2001jn ; Ponce:2002sg , the difference being that only one fundamental
representation of $SU(3)_{L}$ (triplet or anti-triplet) is used in each set,
instead of the composite ones present in the original reference Valle:1983dk ;
Pisano:1992bxx ; Frampton:1992wt  (such composite lepton structures will
appear anon in our systematic analysis).

Anomalies | $S_{1}$ | $S_{2}$ | $S_{3}$ | $S_{4}$ | $S_{5}$ | $S_{6}$
---|---|---|---|---|---|---
$[SU(3)_{C}]^{2}U(1)_{X}$ | 0 | 0 | 0 | 0 | 0 | 0
$[SU(3)_{L}]^{2}U(1)_{X}$ | $-2/3$ | $-1/3$ | 1 | 0 | 2/3 | 1/3
$[Grav]^{2}U(1)_{X}$ | 0 | 0 | 0 | 0 | 0 | 0
$[U(1)_{X}]^{3}$ | 10/9 | 8/9 | $-4/3$ | $-2/3$ | -10/9 | -8/9
$[SU(3)_{L}]^{3}$ | 1 | $-1$ | $-3$ | 3 | $-1$ | 1
Table 1: Anomalies for some 3-3-1 fermion fields structures

The several gauge anomalies calculated for these six sets are shown in Table
1; where notice that the anomaly values for $S_{1},\;S_{2},\;S_{3}$ and
$S_{4}$ coincide with the ones presented in Ref Ponce:2001jn ; Ponce:2002sg ,
being the values for $S_{5}$ and $S_{6}$ new results.

Now, if we want to consider only one family of quarks, either the sets $S_{3}$
or $S_{4}$ are enough, but for 3 quark families, one of the following
combinations must be used: $3S_{3},\;3S_{4},(2S_{3}+S_{4})$ and
$(S_{3}+2S_{4})$, where the first two ones are associated with universal
models in the quark sector.

Right from Table 1 it is simple to read the following sets free of anomalies:

  * •

Model I: $2S_{2}+S_{4}+S_{5}$,

  * •

Model J: $2S_{1}+S_{3}+S_{6}$,

  * •

Model A: $3S_{2}+S_{3}+2S_{4}$,

  * •

Model B: $3S_{1}+2S_{3}+S_{4}$,

where the structures I and J acad  contain only one family of quarks, and A
and B are three family quark models (here we are following the notation in
Ref. acad ). But, can we view I and J as one family (“universal”) anomaly free
models? the answer is yes if we allow models with exotic electrons and new
electric neutral particles. As a matter of fact, we can writte the particle
content for Model I as acad :

$\displaystyle[(e^{-},\nu^{0}_{e},N^{0}_{1})$ $\displaystyle\oplus$
$\displaystyle e^{+}\oplus(E^{-}_{1},N^{0}_{2},N^{0}_{3})\oplus
E^{+}_{1}\oplus$ $\displaystyle(N^{0}_{4},E^{+}_{2},E^{+}_{3})$
$\displaystyle\oplus$ $\displaystyle E_{2}^{-}\oplus
E^{-}_{3}\oplus(u,d,D)\oplus u^{c}\oplus d^{c}\oplus D^{c}]_{L}.$

Ugly as it may be, due to the precense of several exotic leptons, some with
the same quantun numbers of the ordinary ones, we may said that this model is
not yet excluded from present phenomenology.

In a similar way, the particle content of the structure J can be written as
acad :

$\displaystyle[(\nu^{0}_{e},e^{-},E_{1}^{-})$ $\displaystyle\oplus$
$\displaystyle e^{+}\oplus E^{+}_{1}\oplus(N_{1},E^{-}_{2},E^{-}_{3})\oplus
E^{+}_{2}\oplus E^{+}_{3}\oplus$
$\displaystyle(E^{+}_{4},N_{2}^{0},N_{3}^{0})$ $\displaystyle\oplus$
$\displaystyle E_{4}^{-}\oplus(d,u,U)\oplus u^{c}\oplus d^{c}\oplus
U^{c}]_{L}.$

The other two structures A and B correspond to two well known non universal
models already present in the literature; A being named as a “3-3-1 model with
right-handed neutrinos” Montero:1992jk ; Foot:1994ym ; Benavides:2009cn  and B
named as a “3-3-1 model with exotic charged leptons” Ozer:1995xi ;
Ponce:2006au ; Salazar:2007ym .

An unrealistic two family anomaly free structure is for example
$S_{1}+S_{2}+S_{3}+S_{4}$ (unrealistic because there is strong evidence for at
least three families in nature Donoghue:1992dd ).

The next strategy is to use the lepton sets $S_{1},\;S_{2},\;S_{5}$ and
$S_{6}$ to build non vector-like new sets of leptons (vector-like sets are
free of anomalies by definition Donoghue:1992dd , and quark sector anomalies
must cancel out with those of the lepton sector). Notice that vector-like sets
as for example $S_{1}+S_{5}$ and $S_{2}+S_{6}$ are free of anomalies and not
suitable for constructing realistic models due to the non zero anomalies in
the quark sector. For the same reason we exclude from our analysis vector like
structures as
$(\nu^{0}_{e},e^{-},E^{-})\oplus(N^{0},e^{+},E^{+})\sim(1,3,-2/3)\oplus(1,3^{*},2/3)$
and
$(e^{-},\nu^{0}_{e},N^{0}_{1})\oplus(E^{+},N^{0}_{2},N^{0}_{3})\sim(1,3^{*},-1/3)\oplus(1,3,1/3)$.

To be systematic, let us start first with sets of leptons containing only two
$SU(3)_{L}$ triplets or anti-triplets:

  * •

$S_{7}=[(e^{-},\nu^{0}_{e},N_{1}^{0})\oplus(N_{2}^{0},E^{+},e^{+})\oplus
E^{-}]_{L}$ with quantum numbers $(1,3^{*},-1/3)\;$; $(1,3^{*},2/3)$ and
$(1,1,-1)$ respectively.

  * •

$S_{8}=[(\nu^{0}_{e},e^{-},E^{-})\oplus(E^{+},N_{1}^{0},N_{2}^{0})\oplus
e^{+}]_{L}$ with quantum numbers $(1,3,-2/3)$, $(1,3,1/3)$ and $(1,1,1)$
respectively.

Notice again $S_{7}+S_{8}$ is vector-like and in consequence is free of
anomalies and unsuitable for our building process.

The next step is to include lepton sets with three $SU(3)_{L}$ triplets or
anti-triplets:

  * •

$S_{9}=[(e^{-},\nu_{e},N_{1}^{0})\oplus(E^{-},N_{2}^{0},N_{3}^{0})\oplus(N_{4}^{0},E^{+},e^{+})]_{L}$
with quantum numbers $(1,3^{*},-1/3)$;$(1,3^{*},-1/3)$ and $(1,3^{*},2/3)$
respectively.

  * •

$S_{10}=[(\nu_{e},e^{-},E_{1}^{-})\oplus(E^{+}_{2},N_{1}^{0},N_{2}^{0})\oplus
e^{+}\oplus(N_{3}^{0},E^{-}_{2},E_{3}^{-})\oplus E_{1}^{+}\oplus
E_{3}^{+}]_{L}$ with quantum numbers
$(1,3,-2/3);\;\;(1,3,1/3);\;(1,1,1);\;(1,3,-2/3)$; $(1,1,1)$, and $(1,1,1)$
respectively.

  * •

$S_{11}=[(e^{-},\nu_{e},N_{1}^{0})\oplus(N_{2}^{0},E^{+}_{1},e^{+})\oplus(N_{3}^{0},E^{+}_{2},E^{+}_{3})\oplus
E^{-}_{1}\oplus E^{-}_{2}\oplus E^{-}_{3}]_{L}$ with quantum numbers
$(1,3^{*},-1/3);\;\;(1,3^{*},2/3);\;\;(1,3^{*},2/3);\;\;(1,1,-1);\;\;$
$(1,1,-1)$, and $(1,1,-1)$ respectively.

  * •

$S_{12}=[(\nu_{e}^{0},e^{-},E_{1}^{-})\oplus(E^{+}_{1},N_{1}^{0},N_{2}^{0})\oplus(E^{+}_{2},N^{0}_{3},N^{0}_{4})\oplus
e^{+}\oplus E_{2}^{-}]_{L}$ with quantum numbers $(1,3,-2/3)$;
$(1,3,1/3);\;(1,3,1/3);\;\;(1,1,1)$, and $(1,1,-1)$; respectively.

The anomalies for these new lepton sets are given in Table 2.

At this step we can stop combining new sets, and analyze the rich structure we
have gotten so far with the anomaly values presented in the former two Tables:

Anomalies | $S_{7}$ | $S_{8}$ | $S_{9}$ | $S_{10}$ | $S_{11}$ | $S_{12}$
---|---|---|---|---|---|---
$[SU(3)_{C}]^{2}U(1)_{X}$ | 0 | 0 | 0 | 0 | 0 | 0
$[SU(3)_{L}]^{2}U(1)_{X}$ | 1/3 | -1/3 | 0 | -1 | 1 | 0
$[Grav]^{2}U(1)_{X}$ | 0 | 0 | 0 | 0 | 0 | 0
$[U(1)_{X}]^{3}$ | $-2/9$ | $2/9$ | 2/3 | 4/3 | -4/3 | -2/3
$[SU(3)_{L}]^{3}$ | $-2$ | 2 | $-3$ | 3 | $-3$ | 3
Table 2: Anomalies for the 3-3-1 non Vector-Like lepton fields structures

A simple computer program allow us to construct Table 3, which by the way is
our main result of the first part of this paper. Let us see:

##  V Irreducible Anomaly free sets

Table 3 lists all the basic and irreducible sets of multiplets of quarks and
leptons which are free of anomalies (hereafter, Irreducible Anomaly-Free-Set
(IAFS)), classified according to their quark content. These sets can be
combined in several different ways in order to construct anomaly free three
family models. In the first column in Table 3 the index $i$ lists the IAFSs,
the second column shows various possible irreducible lepton sets free of
anomalies. A closer look to the sets of the second column shows that all of
them are vector-like structures, not suitable to build simple models, but
useful to build complex anomaly free models, as we will see ahead. Column
three contains one of the most important results of our analysis. As a matter
of fact, it lists 20 universal models (one family models), where only four of
them are reported in the literature so far. Let us see:

$Q^{I}_{1}=S_{4}+S_{9}$ called carbon copy one or “Model G” in Ref.
Ponce:2001jn ; Ponce:2002sg . This structure can be embedded in the
unification group E(6) according to Ref. Sanchez:2001ua , with some
phenomenology of this structure already presented in the same reference.

$Q^{I}_{2}=S_{3}+S_{10}$ called carbon copy two or “Model H” in Ref.
Ponce:2001jn ; Ponce:2002sg . It can be embedded in the unification group
$SU(6)\otimes U(1)$ according to Ref. Martinez:2001mu , with some
phenomenology of this structure already presented in the same reference.

$Q^{I}_{5}=2S_{1}+S_{3}+S_{6}$ and $Q^{I}_{6}=2S_{2}+S_{4}+S_{5}$ were
introduced in Re. acad  where they were named as “Model J” and “Model I”,
respectively.

The other 16 structures correspond to totally new, one family (universal)
models, which will be analyzed in the second part of this paper. But there is
much more in the third column of Table 3: we can construct three family models
using sets in column three not only by iterating three times every structure,
but also combining them by taking two equal and one diferent (producing in
this way 380 three family models), or by combining three different sets
(producing in this way 1140 three family models). Let us see some of them:
$2Q^{I}_{1}+Q^{I}_{2}=2S_{4}+2S_{9}+S_{3}+S_{10}$ which is denoted as Model E
in Ref. Ponce:2001jn ; Ponce:2002sg , called hybrid model one in that paper.
$Q^{I}_{1}+2Q^{I}_{2}=S_{4}+S_{9}+2S_{3}+2S_{10}$ which is denoted as Model F
in Ref. Ponce:2001jn ; Ponce:2002sg , called hybrid model two in that paper.

Column four in Table 3 shows an amusing result; aparently it is related to
unphysical structures (only two families of quarks). But the point is that
each one of the seven entries in this column can be combined with each one of
the 20 entries in the third column, in order to produce a three family quark
structure, for a total of 140 anomaly free three family models, most of them
new ones. Let us see some examples:
$Q^{I}_{1}+Q^{II}_{1}=S_{1}+S_{2}+S_{3}+2S_{4}+S_{9}$ which is denoted as
Model C in Ref. Ponce:2001jn ; Ponce:2002sg , called a model with unique
lepton generation one in such paper.
$Q^{I}_{2}+Q^{II}_{1}=S_{1}+S_{2}+2S_{3}+S_{4}+S_{10}$ which is denoted as
Model D in Ref. Ponce:2001jn ; Ponce:2002sg , called a model with unique
lepton generation two in the same paper.

The last column in Table 3 shows the simplest and most economical three family
structures available for 3-3-1 models without exotic electric charges. As a
matter of fact, $Q^{III}_{1}$ is the well known 3-3-1 model with right handed
neutrinos Montero:1992jk ; Foot:1994ym ; Benavides:2009cn , and $Q^{III}_{2}$
is the 3-3-1 model with exotic electrons Ozer:1995xi ; Ponce:2006au ;
Salazar:2007ym . All of these are well known models and they are summarized in
Table 4.

At this point we can use the second column in Table 3, by adding one or more
entries in the Table to any anomaly free three family structure constructed
from the other three columns in the way we have explained it above. In this
way we end up with an immeasurable number of anomaly free fermion structures,
constructed out of the 12 fermion structures (10 lepton and two quark)
introduced in Sec. IV.

Model count summary: leaving aside the IAFSs of the second column in Table 3,
we can make a very illuminating analysis to exemplify the model building
process. Each one of the IAFSs in the third column ($Q^{I}$) in Table 3
contains, at least, one family of SM leptons and quarks, so that, if we
threefold this structure, one IAFS for each family, it is possible to have a
universal model. Following this procedure, we can obtain 20 models. Another
possibility is to choose the same IAFSs for two families and a different one
for the third family, in this case we have $19\times 20=380$ possibilities.
Taking a different IAFS for each family we will have $\binom{20}{3}=1140$
different models. Adding these results with the 2 models coming from the fifth
column in Table 3, and the $7\times 20=140$ combinations between the IAFSs in
the columns $Q^{I}$ and $Q^{II}$, we obtain a total of 1682 models with a
minimum content of exotic quarks. It is clear that from the arbitrary union of
IAFSs it is possible to obtain an infinite number of models, but in general,
these models will have too many exotic particles.

Irreducible anomaly free sets
---
$i$ | Vector-like lepton sets ($L_{i}$) | One quark set ($Q^{I}_{i}$) | Two quark sets ($Q^{I\\!I}_{i}$) | Three quark sets ($Q^{I\\!I\\!I}_{i}$)
1 | $S_{1}+S_{5}$ | $S_{4}+S_{9}$ | $S_{1}+S_{2}+S_{3}+S_{4}$ | $3S_{2}+S_{3}+2S_{4}$
2 | $S_{2}+S_{6}$ | $S_{3}+S_{10}$ | $2S_{1}+S_{3}+S_{4}+S_{7}$ | $3S_{1}+2S_{3}+S_{4}$
3 | $S_{7}+S_{8}$ | $S_{2}+S_{4}+S_{7}$ | $2S_{2}+S_{3}+S_{4}+S_{8}$ |
4 | $S_{10}+S_{11}$ | $S_{1}+S_{3}+S_{8}$ | $3S_{2}+S_{3}+S_{4}+S_{12}$ |
5 | $S_{9}+S_{12}$ | $2S_{1}+S_{3}+S_{6}$ | $3S_{1}+2S_{3}+S_{12}$ |
6 | $S_{1}+S_{6}+S_{7}$ | $2S_{2}+S_{4}+S_{5}$ | $3S_{2}+2S_{4}+S_{11}$ |
7 | $S_{6}+S_{8}+S_{9}$ | $S_{1}+S_{4}+2S_{7}$ | $3S_{1}+S_{3}+S_{4}+S_{11}$ |
8 | $S_{2}+S_{5}+S_{8}$ | $S_{2}+S_{3}+2S_{8}$ |  |
9 | $S_{5}+S_{7}+S_{10}$ | $S_{1}+S_{2}+S_{3}+S_{12}$ |  |
10 | $S_{2}+S_{7}+S_{12}$ | $S_{1}+S_{2}+S_{4}+S_{11}$ |  |
11 | $S_{1}+S_{8}+S_{11}$ | $S_{4}+3S_{7}+S_{10}$ |  |
12 | $S_{1}+2S_{6}+S_{9}$ | $S_{3}+3S_{8}+S_{9}$ |  |
13 | $S_{6}+2S_{7}+S_{10}$ | $2S_{1}+S_{3}+S_{7}+S_{12}$ |  |
14 | $S_{5}+2S_{8}+S_{9}$ | $2S_{1}+S_{4}+S_{7}+S_{11}$ |  |
15 | $S_{5}+S_{6}+S_{9}+S_{10}$ | $2S_{2}+S_{3}+S_{8}+S_{12}$ |  |
16 | $S_{2}+2S_{5}+S_{10}$ | $2S_{2}+S_{4}+S_{8}+S_{11}$ |  |
17 | $S_{1}+2S_{7}+S_{12}$ | $3S_{2}+S_{3}+2S_{12}$ |  |
18 | $S_{1}+S_{2}+S_{11}+S_{12}$ | $3S_{2}+S_{4}+S_{11}+S_{12}$ |  |
19 | $S_{2}+2S_{8}+S_{11}$ | $3S_{1}+S_{3}+S_{11}+S_{12}$ |  |
20 | $2S_{1}+S_{6}+S_{11}$ | $3S_{1}+S_{4}+2S_{11}$ |  |
21 | $2S_{2}+S_{5}+S_{12}$ |  |  |
Table 3:  IAFSs. Any general Anomaly Free-Set (AFS) containing quarks, must be a combination of IAFSs (i.e., $L_{i}$, $Q^{I}$, $Q^{II}$ and $Q^{III}$) even for more than three families. For leptons, the second column (L) is not exhaustive and it was not possible to account for all the possibilities. Name |  Model |  AFS
---|---|---
Model A | $Q^{I\\!I\\!I}_{1}$ | $3S_{2}+S_{3}+2S_{4}$
Model B | $Q^{I\\!I\\!I}_{2}$ | $3S_{1}+2S_{3}+S_{4}$
Model C | $Q^{I}_{1}+Q^{I\\!I}_{1}$ | $S_{1}+S_{2}+S_{3}+2S_{4}+S_{9}$
Model D | $Q^{I}_{2}+Q^{I\\!I}_{1}$ | $S_{1}+S_{2}+2S_{3}+S_{4}+S_{10}$
Model E | $Q^{I}_{2}+2\>Q^{I}_{1}$ | $2S_{4}+2S_{9}+S_{3}+S_{10}$
Model F | $2\>Q^{I}_{2}+Q^{I}_{1}$ | $S_{4}+S_{9}+2S_{3}+2S_{10}$
Model G | $3\>Q^{I}_{1}$ | $3(S_{4}+S_{9})$
Model H | $3\>Q^{I}_{2}$ | $3(S_{3}+S_{10})$
Model I | $3\>Q^{I}_{6}$ | $3(2S_{2}+S_{4}+S_{5})$
Model J | $3\>Q^{I}_{5}$ | $3(2S_{1}+S_{3}+S_{6})$
Table 4: The 3-3-1 models already reported in the literature. Particular
embeddings are assumed for the AFSs, which are well known in the literature
Ponce:2001jn ; Ponce:2002sg .

##  VI Embeddings of the SM fermions

From the previous section, we have several anomaly-free representations for
the 3-3-1 gauge group. From now on we will make a distinction between an
Anomaly Free-Set (AFS) and a particular embedding corresponding to a
distinguishable phenomenological model. In these representations, there are
several ways to assign the SM particles within the available multiplets. For
the quark sector, this identification is easy since there are only two sets of
multiplets of quarks, ${S}_{3}$ and ${S}_{4}$, depending on whether the SM
quark doublet is within a $SU(3)_{L}$ triplet or an anti-triplet. For the
lepton sector, there are several sets of multiplets $S_{i}$ by allowing the
right-handed charged lepton to be the third component of a $SU(3)_{L}$ triplet
or a singlet. Once the $S_{i}$ are chosen for quarks and leptons, there are
still several choices for the SM particles within the multiplets.

In order to illustrate the possible embeddings, we will consider the AFSs
associated with the models A, E, G, $Q^{I}_{3}$, I and $Q^{I}_{7}$ . For model
A, its particle content is indicated in Table 4, and there is only one
possible identification for the SM particles in the lepton sector
($3S_{2}^{\ell+e^{+}}$), in this case we have three identical copies of the
set of multiplets $S_{2}$. In this case the SM left-handed lepton doublet
($\ell$) is embedded in the $SU(3)_{L}$ triplet of $S_{2}$ and the right-
handed charged lepton ($e^{+}$) in the singlet. Additionally, the model
contains three exotic neutral particles, $N^{0}$. In Tables 6 and 7 the SM
particle content is shown in the superscript of the fermion sets $S_{i}$,
avoiding any mention of exotic particles.

Another interesting example is the model E which has a rich content of
particles and therefore its corresponding AFS has many possible embeddings, as
shown in Table 4 and 6. For example, we can choose the following embedding for
the SM particles:

$\begin{split}2S_{9}+S_{10}&=2\left[\underbrace{(e^{-},\nu_{e}^{0},N_{1}^{0})}_{\textrm{SM}}\oplus(N_{4}^{0},E^{+},\underbrace{e^{+}}_{\text{SM}})\oplus(E^{-},N_{2}^{0},N_{3}^{0})\right]_{L}\\\
&+\left[\underbrace{(\nu_{e},e^{-},E_{1}^{-})\oplus
e^{+}}_{\textrm{SM}}\oplus(E^{+}_{2},N_{1}^{0},N_{2}^{0})\oplus(N_{3}^{0},E_{2}^{-},E_{3}^{-})\oplus
E_{1}^{+}\oplus E_{3}^{+}\right]_{L}.\end{split}$

In this embedding, two SM lepton families are put in $2S_{9}$ and one SM
family in $S_{10}$. In $S_{9}$ the right-handed lepton is in the third
component of an anti-triplet. The third family is embedded into $S_{10}$ where
the right-handed charged lepton is a singlet. The sets, $S_{9}$ and $S_{10}$,
have different particle content and quantum numbers. This particular embedding
is usually known as the model E, which is not universal in the lepton sector
and it corresponds to the model $E^{1}$ in Table 6.

In the two sets of particles, $2S_{9}$, there are 4 anti-triplets of
$SU(3)_{L}$ with identical quantum numbers, by embedding the three left-handed
lepton doublets of the SM in these anti-triplets and the three SM right-handed
leptons into the $S_{10}$ singlets, we obtain a new embedding which
corresponds to the model $E^{5}$ in Table 6, as follows:

$\begin{split}&\left[\underbrace{(e^{-},\nu_{e}^{0},N_{1}^{0})\oplus(E^{-},N_{2}^{0},N_{3}^{0})\oplus(e^{-},\nu_{e}^{0},N_{1}^{0})}_{\textrm{
SM}}\oplus(E^{-},N_{2}^{0},N_{3}^{0})\oplus(N_{4}^{0},E^{+},e^{+})\oplus(N_{4}^{0},E^{+},e^{+})\right]_{L}\\\
&+\left[\underbrace{e^{+}\oplus E_{1}^{+}\oplus
E_{3}^{+}}_{\textrm{SM}}\oplus(\nu_{e},e^{-},E_{1}^{-})\oplus(E^{+}_{2},N_{1}^{0},N_{2}^{0})\oplus(N_{3}^{0},E_{2}^{-},E_{3}^{-})\right]_{L}\,.\end{split}$

At a phenomenological level, the last embedding, which is equivalent to the
model A, exceeds it by the exotic vector-likes:
$(E^{-},N_{2}^{0},N_{3}^{0})\oplus(E^{+}_{2},N_{1}^{0},N_{2}^{0})$,
$(N_{4}^{0},E^{+},e^{+})\oplus(\nu_{e},e^{-},E_{1}^{-})$ and
$(N_{4}^{0},E^{+},e^{+})\oplus(N_{3}^{0},E_{2}^{-},E_{3}^{-})$. This result
implies that adding vector-like lepton content to an anomaly-free set of
fermions could result in a different non-trivial model by taking a different
embedding. The embeddings for the models $I^{j}$ and $Q_{7}^{Ij}$ are given in
the respective Tables 6 and 7. These models are always universal in the quark
sector, which is very convenient to avoid FCNC. There are four $I^{j}$
embeddings, two universal and two non-universal as shown in Table 6. The
$I^{1}$ model has the same SM particle content as model A, with additional
exotic leptons. There are 20 possible $Q^{Ij}_{7}$ embeddings, with only 4 of
them universal, as listed in Table 7. Interesting examples of AFSs that do not
contain the vector-like structures listed in column 2 in Table 3 are: model G
and the models $Q_{3}^{Ij}$. Model G is universal and it has only one
embedding, while the $Q_{3}^{Ij}$ models are universal in the quark sector and
for some of its embeddings in the lepton sector.

##  VII Collider Constraints

In general, each one of the possible embeddings has a different phenomenology.
For collider constraints only the $Z^{\prime}$ couplings to the SM particles
matter Langacker:2008yv ; Cao:2016uur ; Fonseca:2016tbn . In the lepton
sector, these couplings depend on whether left-handed lepton doublet $\ell$ is
in a $SU(3)_{L}$ triplet or anti-triplet. In the same way, the $Z^{\prime}$
coupling for the right-handed charged lepton $e^{+}$ depends on whether is a
$SU(3)_{L}$ singlet or it is the third component of a $SU(3)_{L}$ anti-
triplet. Similar caveats hold for the SM quark doublets $q$ and the right-
handed quarks as shown in the Table 5.

We obtain the lower limit on the $Z^{\prime}$ mass in Table 5, from the
intersection of the 95% CL upper limit on the cross-section from searches of
high-mass dilepton resonances at the ATLAS experiment ATLAS:2019erb  with the
theoretical cross-section reported in Erler:2011ud . In the reference
ATLAS:2019erb  the upper limits on the cross-section go up to 6 TeV, however,
we extrapolate up to 7 TeV in order to obtain the restrictions for the
simplified models $C_{4}+\bar{q}$ and $C_{4}+q$ in Table 5. The cross-section
depends on the $Z^{\prime}$ charges as they are given the Appendix A. The
ATLAS data was obtained from proton-proton collisions at a center-of-mass
energy of $\sqrt{s}=13$ TeV during Run 2 of the Large Hadron Collider and
correspond to an integrated luminosity of 139 fb-1. Further details are shown
in references Erler:2011ud ; Salazar:2015gxa ; Benavides:2018fzm . We obtain
the constraints in Tables 6 and 7 from those shown in Table 5. We only report
lower limits for embeddings for which it is possible to choose the same
$Z^{\prime}$ charges for the first two families. Under these assumptions, for
the models A, $C^{j}$, E , G, I, $Q_{3}^{I_{j}}$ and $Q_{7}^{I_{j}}$, in
Tables 6 and 7, the left-handed quark doublet $q$ is part of a $SU(3)_{L}$
triplet. For the remaining models, in the mentioned tables, $q$ is part of a
$SU(3)_{L}$ anti-triplet.

Model | SM Particle | Short | LHC-Lower limit
---|---|---|---
| embedding | Notation | in TeV
$C_{1}+{\bar{q}}$ |  $\ell\subset 3$, $e^{+}\subset 1$, $q\subset 3^{*}$ |  $\ell$+$e^{+}$+$\bar{q}$ | 5.53
$C_{1}+{q}$ |  $\ell\subset 3$, $e^{+}\subset 1$, $q\subset 3$ |  $\ell$+$e^{+}$+$q$ | 5.33
$C_{2}+{\bar{q}}$ |  $\ell\subset 3^{*}$, $e^{+}\subset 1$, $q\subset 3^{*}$ |  $\bar{\ell}$+$e^{+}$+$\bar{q}$ | 4.98
$C_{2}+{q}$ |  $\ell\subset 3^{*}$, $e^{+}\subset 1$, $q\subset 3$ |  $\bar{\ell}$\+ $e^{+}$+$q$ | 4.87
$C_{3}+{\bar{q}}$ |  $\ell\subset 3^{*}$, $e^{+}\subset 3^{*}$, $q\subset 3^{*}$ |  $\bar{\ell}$\+ $e^{\prime+}$+$\bar{q}$ | 5.75
$C_{3}+{q}$ |  $\ell\subset 3^{*}$, $e^{+}\subset 3^{*}$, $q\subset 3$ |  $\bar{\ell}$\+ $e^{\prime+}$+$q$ | 5.53
$C_{4}+{\bar{q}}$ |  $\ell\subset 3$, $e^{+}\subset 3^{*}$, $q\subset 3^{*}$ |  $\ell$\+ $e^{\prime+}$+$\bar{q}$ | 7.00
$C_{4}+{q}$ |  $\ell\subset 3$, $e^{+}\subset 3^{*}$, $q\subset 3$ |  $\ell$\+ $e^{\prime+}$+$q$ | 6.52
Table 5:  Simplified 3-3-1 models. The collider constraints depend on whether the SM left-handed doublets $\ell$ or $q$ are contained in a $SU(3)_{L}$ triplet or anti-triplet. In short notation $q$, $\bar{q}$ or $\ell$, $\bar{\ell}$. The notation $e^{+\prime}$ is used if the right-handed lepton is the third component of an anti-triplet, otherwise $e^{+}$. In the first column, $C_{1}$ corresponds to the embedding $\ell\subset 3$, $e^{+}\subset 1$, in a similar way are defined the remaining $C_{i}$. The $Z^{\prime}$ charges for the $C_{i}$, $q$ and $\bar{q}$ are given in Tables 8, 9, 10, 11, 12 and 13, respectively. In the last column the 95% CL lower limits on the $Z^{\prime}$ mass, coming from the Drell-Yan process $\bar{p}p\rightarrow Z^{\prime}_{\mu}$. To obtain the constraints, we assumed that the first two families have the same $Z^{\prime}$ charges.  Model | $j$ | SM Lepton Embeddings | Universal — | $2+1$ | — Lepton Configuration | — LHC-Lower limit (TeV)
---|---|---|---|---|---|---
A | - | $3S_{2}^{\bar{\ell}+e^{+}}$ | ✓ | $\times$ | $3C_{2}$ | 4.87
B | - | $3S_{1}^{\ell+e^{+}}$ | ✓ | $\times$ | $3C_{1}$ | 5.53
Cj | 1 | $S_{1}^{\ell+e^{+}}+S_{2}^{\bar{\ell}+e^{+}}+S_{9}^{\bar{\ell}+e^{\prime+}}$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{3}$ |
2 | $(S_{1}^{\ell}+S_{9}^{e^{\prime+}})+S_{2}^{\bar{\ell}+e^{+}}+(S_{9}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | ✓ | $2C_{2}+C_{4}$ | 4.87
Dj | 1 | $S_{1}^{\ell+e^{+}}+S_{2}^{\bar{\ell}+e^{+}}+S_{10}^{\ell+e^{+}}$ | $\times$ | ✓ | $2C_{1}+C_{2}$ | 5.53
2 | $S_{1}^{\ell+e^{+}}+S_{10}^{2\ell+2e^{+}}$ | ✓ | $\times$ | $3C_{1}$ | 5.53
Ej | 1 | $2S_{9}^{\bar{\ell}+e^{\prime+}}+S_{10}^{\ell+e^{+}}$ | $\times$ | ✓ | $C_{1}+2C_{3}$ | 5.75
2 | $S_{9}^{2\bar{\ell}}+S_{10}^{\ell+3e^{+}}$ | $\times$ | ✓ | $C_{1}+2C_{2}$ | 4.98
3 | $S_{9}^{2\bar{\ell}+e^{\prime+}}+S_{9}^{\bar{\ell}+e^{\prime+}}+S_{10}^{e^{+}}$ | $\times$ | ✓ | $C_{2}+2C_{3}$ | 5.75
4 | $S_{9}^{2\bar{\ell}+e^{\prime+}}+S_{9}^{\bar{\ell}}+S_{10}^{2e^{+}}$ | $\times$ | ✓ | $2C_{2}+C_{3}$ | 4.98
5 | $S_{9}^{2\bar{\ell}}+S_{9}^{\bar{\ell}}+S_{10}^{3e^{+}}$ | ✓ | $\times$ | $3C_{2}$ | 4.98
6 | $S_{9}^{2\bar{\ell}+e^{\prime+}}+S_{10}^{\ell+2e^{+}}$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{3}$ |
7 | $(S_{9}^{2\ell+e^{\prime+}}+S_{10}^{e^{+}})+(S_{10}^{\ell}+S_{9}^{e^{\prime+}})$ | $\times$ | $\times$ | $C_{2}+C_{3}+C_{4}$ |
8 | $(S_{9}^{2\ell}+S_{10}^{2e^{+}})+(S_{10}^{\ell}+S_{9}^{e^{\prime+}})$ | $\times$ | ✓ | $2C_{2}+C_{4}$ | 4.98
| 9 | $S_{9}^{\bar{\ell}+e^{\prime+}}+S_{10}^{2\ell+2e^{+}}$ | $\times$ | ✓ | $2C_{1}+C_{3}$ | 5.53
| 10 | $S_{9}^{\ell+e^{\prime+}}+(S_{10}^{2\ell+e^{+}}+S_{9}^{e^{\prime+}})$ | $\times$ | $\times$ | $C_{1}+C_{3}+C_{4}$ |
| 11 | $S_{9}^{\bar{\ell}}+S_{10}^{2\ell+3e^{+}}$ | $\times$ | ✓ | $2C_{1}+C_{2}$ | 5.53
| 12 | $(S_{9}^{\bar{\ell}}+S_{10}^{e^{+}})+(S_{10}^{2\ell+e^{+}}+S_{9}^{e^{\prime+}})$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{4}$ |
| 13 | $(S_{9}^{\bar{\ell}}+S_{10}^{e^{+}})+(S_{10}^{2\ell}+2S_{9}^{e^{\prime+}})$ | $\times$ | ✓ | $C_{2}+2C_{4}$ | 7.0
Fj | 1 | $S_{9}^{\bar{\ell}+e^{\prime+}}+S_{10}^{2\ell+2e^{+}}$ | $\times$ | ✓ | $2C_{1}+C_{3}$ | 5.53
2 | $(S_{9}^{\bar{\ell}}+S_{10}^{e^{+}})+S_{10}^{2\ell+2e^{+}}$ | $\times$ | ✓ | $2C_{1}+C_{2}$ | 5.53
3 | $(S_{9}^{\bar{\ell}}+S_{10}^{e^{+}})+(S_{10}^{2\ell+e^{+}}+S_{9}^{e^{\prime+}})$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{4}$ |
4 | $(S_{9}^{2\bar{\ell}+e^{\prime+}}+S_{10}^{e^{+}})+S_{10}^{\ell+e^{+}}$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{3}$ |
5 | $(S_{9}^{2\bar{\ell}}+S_{10}^{2e^{+}})+S_{10}^{\ell+e^{+}}$ | $\times$ | ✓ | $C_{1}+2C_{2}$ | 4.98
6 | $(S_{9}^{2\bar{\ell}}+S_{10}^{2e^{+}})+(S_{10}^{\ell}+S_{9}^{e^{\prime+}})$ | $\times$ | ✓ | $2C_{2}+C_{4}$ | 4.98
| 7 | $S_{10}^{2\ell+2e^{+}}+S_{10}^{\ell+e^{+}}$ | ✓ | $\times$ | $3C_{1}$ | 5.53
| 8 | $S_{10}^{2\ell+2e^{+}}+S_{10}^{\ell}+S_{9}^{e^{\prime+}}$ | $\times$ | ✓ | $2C_{1}+C_{4}$ | 5.53
G | - | $3S_{9}^{\bar{\ell}+e^{\prime+}}$ | ✓ | $\times$ | $3C_{3}$ | 5.53
H | - | $3S_{10}^{\ell+e^{+}}$ | ✓ | $\times$ | $3C_{1}$ | 5.53
Ij | 1 | $3S_{2}^{\bar{\ell}+e^{+}}$ | ✓ | $\times$ | $3C_{2}$ | 4.87
2 | $2S_{2}^{\bar{\ell}+e^{+}}+S_{2}^{\bar{\ell}}+S_{5}^{e^{\prime+}}$ | $\times$ | ✓ | $2C_{2}+C_{3}$ | 4.87
3 | $S_{2}^{\bar{\ell}+e^{+}}+2S_{2}^{\bar{\ell}}+2S_{5}^{e^{\prime+}}$ | $\times$ | ✓ | $C_{2}+2C_{3}$ | 5.53
| 4 | $3S_{2}^{\bar{\ell}}+3S_{5}^{e^{\prime+}}$ | ✓ | $\times$ | $3C_{3}$ | 5.53
J | - | $3S_{1}^{\ell+e^{+}}$ | ✓ | $\times$ | $3C_{1}$ | 5.53
Table 6: Alternative embeddings for the classical AFSs. The superscripts
correspond to the particle content of the SM, where $\ell$ ($\bar{\ell}$)
stands for a left-handed lepton doublet embedded in a $SU(3)_{L}$ triplet
(anti-triplet), and $e^{\prime+}$ ($e^{+}$) is the right-handed charged lepton
embedded in a $SU(3)_{L}$ triplet (singlet). The lepton content of the model
$C_{i}$ was defined in Table 5. The check mark $\checkmark$ means that at
least two families (2+1) or three families (universal) have the same charges
under the gauge symmetry, the cross $\times$ stands for the opposite. LHC
constraints are obtained from Table 5 for embeddings for which we can choose
the same $Z^{\prime}$ charges for the first two families, otherwise we leave
the space blank.

Model | j | SM Lepton Embeddings | Universal — | $2+1$ | — Lepton Configuration | — LHC-Lower limit (TeV)
---|---|---|---|---|---|---
$Q^{Ij}_{3}$ | 1 | $3S_{2}^{\bar{\ell}+e^{+}}$ | ✓ | $\times$ | $3C_{2}$ | 4.87
2 | $2S_{2}^{\bar{\ell}+e^{+}}+(S_{2}^{\bar{\ell}}+S_{7}^{e^{\prime+}})$ | $\times$ | ✓ | $2C_{2}+C_{3}$ | 4.87
3 | $S_{2}^{\bar{\ell}+e^{+}}+(2S_{2}^{\bar{\ell}}+2S_{7}^{e^{\prime+}})$ | $\times$ | ✓ | $C_{2}+2C_{3}$ | 5.53
4 | $3S_{2}^{\bar{\ell}}+3S_{7}^{e^{\prime+}}$ | ✓ | $\times$ | $3C_{3}$ | 5.53
$Q^{Ij}_{7}$ | 1 | $3S_{1}^{\ell+e^{+}}$ | ✓ | $\times$ | $3C_{1}$ | 5.33
2 | $2S_{1}^{\ell+e^{+}}+(S_{1}^{\ell}+S_{7}^{e^{\prime+}})$ | $\times$ | ✓ | $2C_{1}+C_{4}$ | 5.33
3 | $S_{1}^{\ell+e^{+}}+(2S_{1}^{\ell}+2S_{7}^{e^{\prime+}})$ | $\times$ | ✓ | $C_{1}+2C_{4}$ | 6.52
4 | $3S_{1}^{\ell}+3S_{7}^{e^{\prime+}}$ | ✓ | $\times$ | $3C_{4}$ | 6.52
5 | $2S_{1}^{\ell+e^{+}}+S_{7}^{\bar{\ell}+e^{\prime+}}$ | $\times$ | ✓ | $2C_{1}+C_{3}$ | 5.33
6 | $2S_{1}^{\ell+e^{+}}+(S_{7}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | ✓ | $2C_{1}+C_{2}$ | 5.33
7 | $S_{1}^{\ell+e^{+}}+(S_{1}^{\ell}+S_{7}^{e^{\prime+}})+S_{7}^{\bar{\ell}+e^{\prime+}}$ | $\times$ | $\times$ | $C_{1}+C_{3}+C_{4}$ |
8 | $S_{1}^{\ell+e^{+}}+(S_{1}^{\ell}+S_{7}^{e^{\prime+}})+(S_{7}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{4}$ |
9 | $(2S_{1}^{\ell}+2S_{7}^{e^{\prime+}})+S_{7}^{\bar{\ell}+e^{\prime+}}$ | $\times$ | ✓ | $C_{3}+2C_{4}$ | 6.52
10 | $(2S_{1}^{\ell}+2S_{7}^{e^{\prime+}})+(S_{7}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | ✓ | $C_{2}+2C_{4}$ | 6.52
11 | $S_{1}^{\ell+e^{+}}+2S_{7}^{\bar{\ell}+e^{\prime+}}$ | $\times$ | ✓ | $C_{1}+2C_{3}$ | 5.53
12 | $S_{1}^{\ell+e^{+}}+S_{7}^{\bar{\ell}+e^{\prime+}}+(S_{7}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | $\times$ | $C_{1}+C_{2}+C_{3}$ |
13 | $S_{1}^{\ell+e^{+}}+(2S_{7}^{\bar{\ell}}+2S_{1}^{e^{+}})$ | $\times$ | ✓ | $C_{1}+2C_{2}$ | 4.87
14 | $(S_{1}^{\ell}+S_{7}^{e^{\prime+}})+2S_{7}^{\bar{\ell}+e^{\prime+}}$ | $\times$ | ✓ | $2C_{3}+C_{4}$ | 5.53
15 | $(S_{1}^{\ell}+S_{7}^{e^{\prime+}})+S_{7}^{\bar{\ell}+e^{\prime+}}+(S_{7}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | $\times$ | $C_{2}+C_{3}+C_{4}$ |
16 | $(S_{1}^{\ell}+S_{7}^{e^{\prime+}})+(2S_{7}^{\bar{\ell}}+2S_{1}^{e^{+}})$ | $\times$ | ✓ | $2C_{2}+C_{4}$ | 4.87
17 | $3S_{7}^{\bar{\ell}+e^{\prime+}}$ | ✓ | $\times$ | $3C_{3}$ | 5.53
18 | $2S_{7}^{\bar{\ell}+e^{\prime+}}+(S_{7}^{\bar{\ell}}+S_{1}^{e^{+}})$ | $\times$ | ✓ | $C_{2}+2C_{3}$ | 5.53
19 | $S_{7}^{\bar{\ell}+e^{\prime+}}+(2S_{7}^{\bar{\ell}}+2S_{1}^{e^{+}})$ | $\times$ | ✓ | $2C_{2}+C_{3}$ | 4.87
| 20 | $3S_{7}^{\bar{\ell}}+3S_{1}^{e^{+}}$ | ✓ | $\times$ | $3C_{2}$ | 4.87
Table 7: Alternative embeddings for new anomaly-free sets. The superscripts
correspond to the particle content of the SM, where $\ell$ ($\bar{\ell}$)
stands for a left-handed lepton doublet embedded in a $SU(3)_{L}$ triplet
(anti-triplet), and $e^{\prime+}$ ($e^{+}$) is the right-handed charged lepton
embedded in a $SU(3)_{L}$ triplet (singlet). The lepton content of the model
$C_{i}$ was defined in Table 5. The check mark $\checkmark$ means that at
least two families (2+1) or three families (universal) have the same charges
under the gauge symmetry, the cross $\times$ stands for the opposite. LHC
constraints are obtained from Table 5 for embeddings for which we can choose
the same $Z^{\prime}$ charges for the first two families, otherwise we leave
the space blank.

##  VIII Conclusions

Restricting ourselves to models without exotic electric charges, we have built
12 sets of particles $S_{i}$ from triplets, antitriplets and singlets of
$SU(3)_{L}\otimes U(1)_{X}$. These sets are constructed in such a way that
they contain the charged particles and their respective antiparticles,
following a similar procedure to that in references Ponce:2001jn ;
Ponce:2002sg . With these sets, we built the IAFSs $L_{i}$, $Q_{i}^{I}$,
$Q_{i}^{II}$ and $Q_{i}^{III}$ depending on their quark content, as it is
shown in Table III. From the IAFSs it is possible to systematically build
3-3-1 models. It is important to realize that if we restrict the AFSs to a
minimum content of vector-like structures (i.e, $L_{i}$), having a lepton and
quark sector consistent with the SM, our analysis is reduced to the AFSs that
containt the classical 3-3-1 models reported in Ponce:2001jn ; Ponce:2002sg .
However, if we allow alternative embeddings for SM particles within $S_{i}$,
we get new phenomenological distinguishable models. Table 6 lists all the
possible embeddings for the sets of fermions that originate the models
reported in references Ponce:2001jn ; Ponce:2002sg . In Tables 6 and 7,
$C^{1}$, $D^{1}$, $E^{1}$, $F^{1}$ and $I^{1}$ correspond to the models C, D,
E, F  and I in references Ponce:2001jn ; Ponce:2002sg .

By combining the IAFSs from Table 3, it is possible to find a large number of
models. By restricting to models with a minimal content of exotic fermions, we
found 1682 models which could be of phenomenological interest. To exemplify
the new realistic AFSs that can be formed, we reported some of them, with
their corresponding embeddings, in Table 7. We also report LHC constraints for
models with the first two families having SM fermions with identical charges,
including some of the classical 3-3-1 models, as reported in Table 5. From
this Table we can see that, independent of the model, the mass value of the
new neutral gauge boson for all the 3-3-1 models considered in this paper, is
above 4.87 TeV.

##  IX Acknowledgments

R. H. B. and L. M. thank the “Dirección de investigaciones ITM” project number
P20246. We thank Financial support from “Patrimonio Autónomo Fondo Nacional de
Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José
de Caldas”, and “Sostenibilidad-UDEA”. This research was partly supported by
the “Vicerrectoría de Investigaciones e Interacción Social VIIS de la
Universidad de Nariño”, project numbers 1928 and 2172.

##  Appendix A $Z^{\prime}$ Charges

$C_{1}$, $\ell\subset 3$, $e\subset 1$ (as in $S_{1}$)
---
Fields | Vectorial | Axial
$\nu_{e}$ | $-\frac{1}{2}\left(\frac{\cos\theta}{\delta}+\sin\theta\right)$ | $-\frac{1}{2}\left(\frac{\cos\theta}{\delta}+\sin\theta\right)$
$e$ | $\frac{1}{2}\left[\sin\theta(1-4\sin^{2}\theta_{W})-\frac{\cos\theta}{\delta}(1+2\sin^{2}\theta_{W})\right]$ | $\frac{1}{2}\left[\sin\theta-\frac{\cos\theta}{\delta}(1-2\sin^{2}\theta_{W})\right]$
Table 8: $\bar{f}f\rightarrow Z^{\prime}_{\mu}$ $C_{2}$, $\ell\subset 3^{*}$,
$e\subset 1$ (as in $S_{2}$)
---
Fields | Vectorial | Axial
$\nu_{e}$ | $\frac{\cos\theta}{\delta}\left(\frac{1}{2}-\sin^{2}\theta_{W}\right)-\frac{\sin\theta}{2}$ | $\frac{\cos\theta}{\delta}\left(\frac{1}{2}-\sin^{2}\theta_{W}\right)-\frac{\sin\theta}{2}$
$e$ | $\sin\theta\left(\frac{1}{2}-2\sin\theta_{W}^{2}\right)+\frac{\cos\theta}{\delta}\left(\frac{1}{2}-2\sin\theta_{W}^{2}\right)$ | $\frac{1}{2}\left(\frac{\cos\theta}{\delta}+\sin\theta\right)$
Table 9: $\bar{f}f\rightarrow Z^{\prime}_{\mu}$ $C_{3}$, $\ell\subset 3^{*}$,
$e\subset 3^{*}$ (as in $S_{7}$)
---
Fields | Vectorial | Axial
$\nu_{e}$ | $\frac{1}{2}\left[\frac{\cos\theta}{\delta}(1-2\sin^{2}\theta_{W})-\sin\theta\right]$ | $\frac{1}{2}\left[\frac{\cos\theta}{\delta}(1-2\sin^{2}\theta_{W})-\sin\theta\right]$
$e$ | $\sin\theta\left(\frac{1}{2}-2\sin^{2}\theta_{W}\right)+\frac{3\cos\theta}{2\delta}(1-2\sin^{2}\theta_{W})$ | $\frac{1}{2}\left[\sin\theta-\frac{\cos\theta}{\delta}(1-2\sin^{2}\theta_{W})\right]$
Table 10: $\bar{f}f\rightarrow Z^{\prime}_{\mu}$ $C_{4}$, $\ell\subset 3$,
$e\subset 3^{*}$
---
Fields | Vectorial | Axial
$\nu_{e}$ | $\frac{1}{2}\left(\sin\theta+\frac{\cos\theta}{\delta}\right)$ | $\frac{1}{2}\left(\sin\theta+\frac{\cos\theta}{\delta}\right)$
$e$ | $\frac{1}{2}\left[\sin\theta(1-4\sin^{2}\theta_{W})+\frac{\cos\theta}{\delta}(1-2\sin^{2}\theta_{W})\right]$ | $\frac{1}{2}\left[\sin\theta(1-2\sin^{2}\theta_{W})-\frac{3\cos\theta}{\delta}\right]$
Table 11: $\bar{f}f\rightarrow Z^{\prime}_{\mu}$ $\bar{q}$, $q\subset 3^{*}$
(as in $S_{3}$)
---
Fields | Vectorial | Axial
$u$ | $\frac{1}{6}\left[\sin\theta\left(-3+8\sin^{2}\theta_{W}\right)+\frac{\cos\theta}{\delta}\left(3+2\sin^{2}\theta_{W}\right)\right]$ | $\frac{1}{2}\left[\frac{\cos\theta}{\delta}\left(1-2\sin^{2}\theta_{W}\right)-\sin\theta\right]$
$d$ | $\left[\frac{1}{2}-\frac{2}{3}\sin^{2}\theta_{W}\right]\left[\sin\theta+\frac{\cos\theta}{\delta}\right]$ | $\frac{1}{2}\left[\sin\theta+\frac{\cos\theta}{\delta}\right]$
Table 12: $\bar{f}f\rightarrow Z^{\prime}_{\mu}$ $q$, $q\subset 3$ (as in
$S_{4}$)
---
Fields | Vectorial | Axial
$u$ | $\frac{1}{6}\left[\sin\theta\left(-3+8\sin^{2}\theta_{W}\right)+\frac{\cos\theta}{\delta}\left(5-8\cos^{2}\theta_{W}\right)\right]$ | $-\frac{1}{2}\left[\frac{\cos\theta}{\delta}+\sin\theta\right]$
$d$ | $\frac{1}{6}\left[\sin\theta\left(3-4\sin^{2}\theta_{W}\right)-\frac{\cos\theta}{\delta}\left(3-2\sin^{2}\theta_{W}\right)\right]$ | $\frac{1}{2}\left[\sin\theta-\frac{\cos\theta}{\delta}\left(1-2\sin^{2}\theta_{W}\right)\right]$
Table 13: $\bar{f}f\rightarrow Z^{\prime}_{\mu}$

where $\theta$ is an angle mixing between $Z$ and $Z^{\prime}$ bosons,
$\theta_{W}$ is the Weinberg angle and $\delta=\sqrt{4\cos^{2}\theta_{W}-1}$.

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the state space explosion problem and hence makes our approach efficient and
terminating.

To inspect a given specified actor of $M$ against the generated assumption, we
have manually check its compliance and then exploited mCRL2 to generate
$\rho_{c_{3}}(T_{M})$. To this end, $A_{M}$ was specified as a process with
two subprocesses as before. The subprocess which implements the input-
enableness of the actor, results in $A_{M}$ receiving any of its messages
which implements the functionality of $A_{M^{c}}$ and there is no need to
implement $o_{i}^{d}$’s as in mCRL2 we can implement a process such that send
actions manifest in the resulting LTS without the need for the recipient of
messages to exist. We used CADP666http://cadp.inria.fr/ to compare the traces
of $\rho_{c_{3}}(T_{M})$ and $T_{A}$ due to its support for systems with a
huge number of states and transitions.

To validate our results, we verified the system containing open system actors
and the given $M$ against the given property. To this aim, the composition of
the closed system $A_{M}\interleave O$ and $P_{\it err}$ was generated using
the mCRL2 toolset again (similar to the above). The property is satisfied by
the system if the $\pi$ state is not reachable in the resulting LTS. As we
have implemented the $\pi$ state with a $\pi$ action terminating in a deadlock
state, the $\pi$ state is reachable if the action $\pi$ exists and is
reachable from the initial state in the LTS $A_{M}\interleave O$ and $P_{\it
err}$. This is checked by the mCRL2 model checker tool.

Table 4: Evaluation results

Implemented Approach System under Analysis Property LTS Generated Assumption
States Transitions Time(s) States Transitions Asynchronous AG EFTS 127 186 43
42 417 Quadri. - Single Init 10 10 1 0 0 Quadri. - Multiple Init 182 343 17 8
22 Mutex 15 21 0 14 68 Tailored Direct AG EFTS 224328$>$ 823293$>$ 3600 - -
Quadri. - Single Init 1510 4830 135 20 59 Quadri. - Multiple Init 688 2001 96
9 25 Mutex 22 57 1 21 85

####  4.3.1 Exploiting a More Precise $\mathit{Info}$

If $\mathit{Info}$ is more accurate, in the sense of being stateful, our
approach - with minor changes due to extending $\mathit{Info}$ and generating
$\mathit{InfM}$\- still works. However the set of possible $M$s which comply
to $\mathit{Info}$ will become more limited. We have conducted a set of
experiments on our running example to illustrate how using a more accurate
$\mathit{Info}$ affects the performance of our assumption generation
algorithm. As a more precise $\mathit{Info}$ leads to a more precise
$\mathit{InfM}$, we have directly used a more precise $\mathit{InfM}$ in our
experiments to observe its impact on the resulting assumption. Such precise
specifications can be derived by adding the variable $\it taken$ to
$\mathit{InfM}$, specified by Figure 8, and defining its behavior based on
this variable while processing $\it reqL$, $\it reqR$, and $\it release$.
Conversely, they can be characterized based on the lines removed from $M$,
specified by Figure 11a. All resulting specifications definitely comply with
$\mathit{Info}$. The experiment results, shown in the Table 5, indicate that a
more precise $\mathit{InfM}$ may not affect the performance as extra precision
may result in larger property LTSs and assumptions for different values of a
variable. However, all experiments result smaller LTSs compared to the direct
approach. For the cases that the $\pi$ state is not reachable in their
property LTSs, the algorithm converges faster and it terminates immediately.

Table 5: Effect of a more precise $\mathit{InfM}$ for Mutex system Lines Omitted from Figure 11a | Property LTS | Generated Assumption
---|---|---
States | Transitions | Time | States | Transitions | Time
| 3,5,7-10,14,16,
---
17-19,23:($\mathit{Info}$)
15 | 21 | 3518576 | 14 | 68 | 3087559909
7,16,23 | 15 | 21 | 3546567 | 14 | 68 | 3081696082
16,23 | 16 | 21 | 3565696 | 15 | 72 | 3079037789
23 | 14 | 18 | 3527906 | - | - | 3527906
\- :($M$) | 13 | 18 | 3407075 | - | - | 3407075
7,23 | 16 | 21 | 3561963 | 15 | 72 | 3073369906
7 | 18 | 24 | 3617947 | 11 | 53 | 3079523449
7,16 | 15 | 21 | 3551699 | 14 | 68 | 3081989530
16 | 18 | 24 | 3989771 | 11 | 53 | 3082167745

##  5 Related Work

Compositional verification has been proposed as a way to address the state
space explosion problem in model checking clarke1999model . It uses divide-
and-conquer approach to verify a system against a property by verifying each
of its components individually against a local property
clarke1989compositional ; grumberg1994model , a technique that is often called
quotienting. In the context of process algebra as the language for describing
reactive systems and Henessy-Milner logic as the logical specification
formalism for describing properties, decompositional reasoning has been
originally studies in larsen1986context ; larsen1991compositionality  and has
been further developed in, e.g., bloom2004precongruence ;
fokkink2006compositionality ; raclet2008residual . A related line of research
is the one devoted to compositional proof systems, e.g., AndersenSW94  which
determines satisfaction between algebraic processes and assertions of the
modal $\mu$-calculus. This technique provides a set of rules guided by the
structure of the processes. The idea of compositional verification can be
formulated as finding a proof for a term with a set of meta-variables,
distinct from recursive variables. These meta-variables denote undefined
processes. Thus, proofs carried out with such variables appearing in the
terms, are valid for all instantiations of the variable. Such a parameterized
proof results in an assertion that should be satisfied by the meta-variable.
This assertion expresses possible instantiations for the meta-variable and
hence, plays the role of an assumption in our case. A given $M$ guarantees the
whole system satisfies $P$ if a proof can be found for the assertion. Although
this approach is also applicable to non-safety properties, finding such proofs
and assertions cannot be automated and needs an exhaustive user involvement.

To successfully verify an isolated component, it is often needed to have some
knowledge about the environment interacting with the component, and to this
end assume-guarantee reasoning approaches have been proposed to address the
issue jones1983specification ; pnueli1985transition . These approaches
guarantee a system $S$ composed of components $M_{1}$ and $M_{2}$ satisfies a
property $P$ if $M_{1}$ under an assumption $A$ satisfies $P$ and also $M_{2}$
satisfies $A$ in the context of all possible environments. As we explained
earlier, our problem is reduced to the assume-guarantee reasoning problem
where $M_{2}$ maps to the unspecified component and $M_{1}$ plays the role of
the open system. To the best of our knowledge the assume-guarantee reasoning
has not been specifically associated with asynchronous environments.

In recent work such as alur2005symbolic ; nam2008automatic ;
puasuareanu2008learning ; chaki2007optimized , the assumption has been
generated through learning-based approaches by using the $L^{*}$ algorithm,
first proposed by Angulin angluin1987learning . Given a set of positive and
negative traces, an initial assumption is generated which will be iteratively
refined by posing questions to an oracle. These approaches are not time-
efficient due to additional overhead to learn assumptions cobleigh2008breaking
. Also the generated assumption may become very large and also the
verification may fail in some cases nam2006learning ; alur2005symbolic . These
approaches are useful when the computations for the generation of the weakest
assumption run out of memory. The other fact that should be considered is that
in the $L^{*}$-based approaches in order to produce the assumption, the
component $M_{2}$ is incorporated in the iterative solution which is in
contrast to our original goal of reusing the produced assumption in different
settings and specifically with different $M_{2}$ components. In other words,
despite the solution of alur2005symbolic ; nam2008automatic ;
puasuareanu2008learning ; chaki2007optimized , our solution is independent
from $M_{2}$ and there is no need to generate a new assumption for each
different $M_{2}$ component. In the future, we plan to investigate more on how
to incorporate learning techniques into our asynchronous setting.

Counter example abstraction refinement technique has been associated with
assume-guarantee reasoning in bobaru2008automated  by iteratively computing
assumptions as conservative abstractions of the interface behavior of $M_{2}$
which concerns interactions with $M_{1}$. In each iteration, the composition
of $M_{1}$ and $A$ is checked against $P$ and based on the achieved
counterexample, $A$ is refined accordingly which makes the approach
computationally expensive. Furthermore the aforementioned issue of reusability
due to the iterative incorporation of $M_{2}$ in computations is also observed
in this solution. We also take advantage of an abstract interface of $M_{2}$
in our approach using interaction information of $M_{1}$, however there is no
need for iterative checkings.

The direct approach toward the generation of the weakest assumption has been
applied in work such as giann ; cheung1999checking . In particular in giann ,
the weakest assumption for the environment of a component is generated such
that the component satisfies a required property. But the generated assumption
may become too large and the computation may run out of memory
cobleigh2003learning . We have extended the work in giann  to asynchronous
systems and also made some improvements on handling internal actions. By
assuming some information on the interactions of $O$, our approach produces an
assumption which is noticeably smaller than the generated assumption in giann
.

In feng2007relationship ; gupta2007automated ; grinchtein2006inferring ;
pena1999new , the problem of finding the separating DFA of two disjoint
regular languages, so called concurrent separation logic, has been mapped to
the problem of assume-guarantee reasoning. To this aim, the separating
automaton $A$ is generated which accepts the language of $M_{2}$ but rejects
the language of $M_{1}^{\prime}$, where $M_{1}^{\prime}$ accepts the language
of $M_{1}$ and does not satisfy $P$. The work in gupta2007automated ;
grinchtein2006inferring  exploits an iterative method using boolean
satisfiability solvers. Generally all of these approaches are computationally
expensive while gupta2007automated  is only suited for hardware verification.

The theory of supervisory control and controller synthesis also tackles a
similar problem but in a very different context. The goal is to synthesis a
controller called the supervisor, such that the behaviors of the system called
plant, are controlled so that the requirements of the plant are satisfied
ramadge1987supervisory ; cassandras2009introduction . The theory mainly
focuses on hardware systems, therefore assuming that the controller reacts
sufficiently fast on machine input, the supervisor and the plant are
originally modeled as synchronizing processes ramadge1987supervisory ;
cassandras2009introduction  and the supervisor allows the occurrence of events
of the plan by synchronizing with them and prevents them by refusing to
synchronize. This also prevails in modern state-of-the art approaches
heymann1998discrete ; fabian1996non ; overkamp1997supervisory . The notion of
asynchrony in this context is mostly referred to as events occurring without
any reference to a clock ramadge1984supervisory ; cortadella1997petrify ;
cortadella2012logic  which is very different from the notion of asynchrony in
our context. The problem of controller synthesis is mapped to our context such
that the supervisor is mapped to the unspecified component, system
requirements play the role of the system properties and the plant is the open
system. However, the fundamental difference is that system requirements in a
controller synthesis are limited to the behaviors of the plant and the
generated supervisor describes a “well-defined” plant markovski2016process ;
van2015maximal . However in our case, the behaviors of the unspecified
component are mentioned in the property and the generated specification
describes the unspecified component. In other words a synthesized controller
is a dependent entity controlling the behaviors of the plant, but in our
context we synthesis a component which is a part of the system and has its own
independent functionality.

##  6 Conclusion and Future Work

We developed a framework for asynchronous systems to successfully verify
safety properties of such systems in the presence of a modified or an added
component. This is done by generating the weakest assumption for the component
to be checked against. By applying this framework, we ignore model checking of
the whole system with each modification made to a component of the system. We
adapted the direct approach of the synchronous settings to make it terminating
and more efficient in time and computations. To this aim, we generate an over
approximated component instead of the missing component to make the system of
asynchronous components closed and used the special composition to build the
property LTS to eliminate the error traces simultaneously. We have illustrated
the applicability of our work on two real world case studies and shown that
the generated assumptions are significantly smaller. We have also relaxed the
condition which restricts $M$ to the actions of $P$ by considering the
internal actions of the unspecified component in our computations. Therefore,
our approach eliminates false positive results as a consequence of such a
condition (In giann  internal actions are led to the sink state after which
observable actions may violate the property). Such improvements are not
limited to the asynchronous settings and therefore can be adopted.

As future work we focus on improving the third step of our algorithm: instead
of completing the generated LTS and exploiting conformance checking to prevent
false positive, we find conditions to steer the completion phase. Another
improvement is to imply some reductions on the generated LTS of a single actor
(regarding its interactions) in order to reduce the time and resources for
checking an actor against the generated assumption. Furthermore, providing an
approach to validate a rejected $M$ by our compliance checking is another
research direction. Extending our framework with data can also be considered.
We also plan to investigate on how to incorporate grammar inference techniques
into our asynchronous setting.

## Acknowledgements

We would like to thank Mohammad Reza Mousavi for his helpful discussion on the
paper.

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##  Appendix A Actors of the EFT System

⬇

1 actor PoS(2){

3 int isCanceled;

4 int flag1;

5 //1: ”Insufficient Credit” has

6 // happened while ”cancelPurchase”

7 // has been sent

8 int flag2;

9 //1: ”Authentication error” has

10 // happened while ”cancelPurchase”

11 // has been int sent

12 int flag3;

13 //1: ”cancelPurchase” has been sent

15 insertCard{

16 user!insertPassword;

17 }

18 passwordIs{

19 user!chooseTransaction;

20 }

21 balanceRequest{

22 eftSwitchCore!balanceRequest;

23 }

24 purchaseRequest{

25 eftSwitchCore!purchaseRequest;

26 }

27 cancelPurchase{

28 isCanceled=1;

29 eftSwitchCore!cancelPurchase;

30 }

32 balanceResponse{

33 user!balanceResponse;

34 }

35 purchaseSuccessful{

36 if(isCanceled==1){

37 flag3=1;}

38 else{

39 user!purchaseSuccessful;}

40 }

41 authenticationError{

42 if(isCanceled==1){

43 flag2=1;}

44 else{

45 user!authenticationError;

46 isCanceled=0;}

47 }

49 insufficientCredit{

50 if(isCanceled==1){

51 flag1=1;}

52 else{

53 user!insufficientCredit;

54 isCanceled=0;}

55 }

57 purchaseCanceled{

58 if(flag1==1){

59 user!insufficientCredit;

60 flag1=0;}

61 else{ if(flag2==1){

62 user!authenticationError;

63 flag2=0;}

64 else {if(flag3==1){

65 user!purchaseCanceled;

66 flag3=0;}

67 else{}}}

68 isCanceled=0;

69 }

70 }

Figure 21: Actors of the Open System of EFT System

⬇

1 actor user(2){

3 int scenario;

4 start{

5 PoS!insertCard;

6 }

7 insertPassword{

8 PoS!passwordIs;

9 }

10 chooseTransaction{

11 scenario=?(0,1,2);

12 if(scenario==0){

13 PoS!balanceRequest;}

14 else{ if(scenario==1){

15 PoS!purchaseRequest;}

16 else{ if(scenario==2){

17 PoS!purchaseRequest;

18 PoS!cancelPurchase;}

19 else{}}}

20 cenario=0;

21 }

22 balanceResponse{

23 self!start;

24 }

25 purchaseSuccessful{

26 self!start;

27 }

28 authenticationError{

29 self!start;

30 }

31 insufficientCredit{

32 self!start;

33 }

34 purchaseCanceled{

35 self!start;

36 }

37 }

39 actor EFTSwitchCore(2){

41 int wrongPassword;

42 balanceRequest{

43 wrongPassword=?(0,1);

44 if(wrongPassword==0){

45 balanceTransaction!start;}

46 else{

47 PoS!authenticationError;}

48 wrongPassword=0;

49 }

50 purchaseRequest{

51 wrongPassword=?(0,1);

52 if(wrongPassword==1){

53 purchaseTransaction!start;}

54 else{

55 PoS!authenticationError;}

56 wrongPassword=0;

57 }

58 cancelPurchase{

59 purchaseTransaction!cancelPurchase;

60 }

61 }

63 actor balanceTransaction(2){

65 start{

66 core!balanceRequest;

67 }

68 balanceResponse{

69 PoS!balanceResponse;

70 }

71 }

73 actor core(2){

75 int noCredit;

76 balanceRequest{

77 balanceTransaction!balanceResponse;

78 }

79 purchaseRequest{

80 noCredit=?(0,1);

81 if(noCredit==0){

82 purchaseTransaction!purchaseSuccessful;}

83 else{

84 purchaseTransaction!insufficientCredit;}

85 noCredit=0;

86 }

87 cancelPurchase{

88 purchaseTransaction!purchaseCanceled;

89 }

90 }

92 main {

93 user!start;

94 }

Figure 22: Actors of the Open System of EFT System

⬇

1actor purchaseTransaction(2){

3 int l;

4 start{

5 core!purchaseRequest;

6 }

7 purchaseSuccessful{

8 PoS!purchaseSuccessful;

9 }

10 insufficientCredit{

11 PoS!insufficientCredit;

12 }

13 cancelPurchase{

14 int l=?(0,1);

15 if(l==1){

16 core!cancelPurchase;

17 }

18 else{

19 PoS!purchaseCanceled;

20 }

21 }

22 purchaseCanceled{

23 PoS!purchaseCanceled;

24 }

25}

Figure 23: Actor $A_{\mathit{InfM}}$ of EFT System