Table Of ContentA supersymmetric 3-4-1 model
M. C. Rodriguez
Fundac¸˜ao Universidade Federal do Rio Grande-FURG
Departamento de F´ısica
Av. It´alia, km 8, Campus Carreiros
7
0 96201-900, Rio Grande, RS
0
Brazil
2
n Abstract
a
J
We build the complete supersymmetric version of a 3-4-1 gauge
1
model using the superfield formalism. We point out that a discrete
1
symmetry, similar to the R-symmetry in the minimal supersymmetric
1 standard model, is possible to be defined in this model. Hence we
v
have both R-conserving and R-violating possibilities. We also discuss
8
8 some phenomenological results coming from this model.
0
1 PACS numbers: 12.60.-i, 12.60.Jv
0
7
0
1 Introduction
/
h
p
- The full symmetry of the so called Standard Model (SM) is the gauge group
p
e SU(3) SU(2) U(1) . Nevertheless, the SM is not considered as the
c L Y
h ⊗ ⊗
ultimate theory since neither the fundamental parameters, masses and cou-
:
v
plings, nor the symmetry pattern are predicted. Even though many aspects
i
X of the SM are experimentally supported to a very accuracy, the embedding
r of the model into a more general framework is to be expected.
a
Some of these possibilities is that, at energies of a few TeVs, the gauge
symmetry may be SU(3) SU(3) U(1) (3-3-1 for shortness) [1, 2,
c L N
⊗ ⊗
3]. Recently, the supersymmetric version of these model have alreday benn
constructed in [4, 5]. These 3-3-1 models can be embedded in a model with
3-4-1, its mean SU(3) SU(4) U(1) gauge symmetry [6].
c L N
⊗ ⊗
In SU(4) U(1) , the most general expression for the electric charge
L N
⊗
generator is a linear combination of the four diagonal generators of the gauge
group
1 b c
Q = aλ + λ + λ +NI
3 8 15 4 4
2 √3 √6 ! ×
1 b c 1 b c 1 2b c c
= diag a+ + +N, a+ + +N, − + +N, − +N ,
"2 3 6! 2 − 3 6! 2 3 6! 4 #
(1)
where λ , being the Gell-Mann matrices for SU(4) , see [7, 8], normalized as
i L
Tr(λ λ ) = 2δ , I = diag(1,1,1,1) is the diagonal 4 4 unit matrix, and
i j ij 4 4
× ×
a, b and c are free parameters to be fixed next. Therefore, there is an infinite
number of models can, in principle, be constructed.
A model with the SU(4) U(1) symmetry in the lepton sector, quarks
⊗
were not considered on this work, was suggested some years ago in Ref. [9],
in wchich the magnetic moment of neutrinos arises as the result of charged
scalars that belong to an SU(4) sextet, and the mass of neutrino arises at
two-loop level as the result of electroweak radiative correction.
The 3-4-1 model in Ref. [6] contain exotic electric charges only in the
quark sector, while leptons have ordinary electric charges and gauge bosons
haveintegerelectriccharges. Thebestfeatureofthismodelisthatitprovides
us with an alternative to the problem of the number N of fermion families.
f
These sort of models are anomaly free only if there are equal number of
quadriplet and anti-quadriplet (considering the color degrees of freedom),
and furthermore requiring the sum of all fermion charges to vanish. Two of
the three quark generations transform identically and one generation, it does
not matter which one, transforms in a different representation of SU(4)
L
⊗
U(1) . This means that in these models as in the SU(3) SU(3) U(1)
N c L N
⊗ ⊗
ones [1], in order to cancel anomalies, the number of families (N ) must be
f
divisible by the number of color degrees of freedom (n). This fact, together
with asymptotic freedom in QCD, the model predicts that the number of
generations must be three and only three.
Ontheotherhand,atlowenergiesthesemodelsareindistinguishablefrom
the SM. There is a very nice review about this kind of model see [10, 11].
This make 3-4-1 model interesting by their own. In this article we construct
the supersymmetric version of the model in Ref [6].
The outline of the paper is as follows. In Sec. 2 we present the represen-
tation content of the supersymmetric 3-4-1 model. We build the lagrangian
in Sec. 3. While in Sec. 4, we discuss the double charged charginos inthis
model, while in the last section we present our conclusion.
2 The model
In this section (Sec. 2.1) we review the non-supersymmetric 3-4-1 model of
Refs. [6]and add thesuperpartners (Sec. 2.2) of the usual particles of the non
supersimmetric model. The superfields, useful to construct the supersimmet-
ric lagrangian of the model, associated with the particles of this model are
introduced in section (Sec. 2.3).
2.1 The representation content
In the model of Ref. [6], the free parameters for the eletric charge generators
are
a = 1, b = 1, c = 4, (2)
− −
and Eq.(1) can be rewritten as
1 1 4
Q = λ λ λ +NI ,
2 3 − √3 8 − √6 15! 4×4
= diag(N,N 1,N,N +1). (3)
−
However, let usfirstconsider theparticlecontent ofthemodelwithout su-
persymmetry. We have the leptons transforming in the lowest representation
of SU(4) the quartet1 in the following way
L
ν
a
l
L = a (1,4,0), a = 1,2,3. (4)
aL νc ∼
a
lc
a L
In parenthesis it appears the transformations properties under the respective
factors (SU(3) ,SU(4) ,U(1) ).
C L N
In the quark sector, one quark family is also put in the quartet represen-
tation
u
1
d 2
Q = 1 3,4, , (5)
1L u′ ∼ (cid:18) 3(cid:19)
J
L
1In the same way as proposed by Voloshin [9] in order to understand the existence of
neutrinos with large magnetic moment and small mass.
and the respective singlets are given by
2 1
uc 3 ,1, , dc 3 ,1, ,
1L ∼ ∗ −3 1L ∼ ∗ 3
(cid:18) (cid:19) (cid:18) (cid:19)
2 5
uc 3 ,1, , Jc 3 ,1, , (6)
′L ∼ ∗ −3 L ∼ ∗ −3
(cid:18) (cid:19) (cid:18) (cid:19)
writing all the fields as left-handed; u and J are new quarks with charge
′
+2/3 and +5/3 respectively.
The others two quark generations, as we have explained in the introduc-
tion, we put in the anti-quartet representation
d d
2 3
u u
Q = 2 3,4 , 1 , Q = 3 3,4 , 1 , (7)
2L d′1 ∼ ∗ −3 3L d′2 ∼ ∗ −3
(cid:16) (cid:17) (cid:16) (cid:17)
j j
1 L 2 L
and also with the respective singlets,
2 1
uc 3 ,1, , dc 3 ,1, ,
αL ∼ ∗ −3 αL ∼ ∗ 3
(cid:18) (cid:19) (cid:18) (cid:19)
1 4
dc 3 ,1, , jc 3 ,1, , (8)
′βL ∼ ∗ 3 βL ∼ ∗ 3
(cid:18) (cid:19) (cid:18) (cid:19)
j and d , β = 1,2 are new quarks with charge 4/3 and 1/3 respec-
β ′β − −
tively, while α = 2,3 is the familly index for the quarks. We remind that in
Eqs. (4,5,6,7,8) all fields are still symmetry eigenstates.
Ontheotherhand, thescalars, inquartet,whicharenecessary togenerate
the quark masses are
η0
1
η
η = 1− (1,4,0)
η0 ∼
2
η+
2
ρ+
1
ρ0
ρ = (1,4,1)
ρ+ ∼
2
ρ++
χ
−1
χ
χ = −− (1,4, 1). (9)
χ−2 ∼ −
χ0
In order to avoid mixing among primed and unprimed quarks, we have to
introduce an extra scalar transforming like η but with different vacuum ex-
pectation value (VEV)
φ0
1
φ
φ = −1 (1,4,0) . (10)
φ0 ∼
2
φ+
2
In order to obtain massive charged leptons it is necessary to introduce the
following symmetric anti-decuplet
H0 H+ H0 H
1 1 2 2−
H+ H++ H+ H0
H = 1 1 3 3 (1,10 ,0). (11)
H20 H3+ H40 H4− ∼ ∗
H2− H30 H4− H2−−
then the charged leptons get a mass but neutrinos remain massless, at least
at tree level.
2.2 Supersymmetric partners
Now, we introduce the minimal set of particles in order to implement the
supersymmetry [12]. We have the sleptons corresponding to the leptons
in Eq. (4); squarks related to the quarks in Eqs.(6)-(8); and the Higgsinos
related to the scalars given in Eqs. (9) and (11). Then, we have to introduce
the following additional particles
u˜ d˜
1 α
d˜ 2 u˜ 1
Q˜ = 1 3,4, , Q˜ = α 3,4 , ,
1L u˜′ ∼ (cid:18) 3(cid:19) αL d˜′β ∼ (cid:18) ∗ −3(cid:19)
J˜ L ˜jβ L
ν˜
a
˜
l
L˜ = a (1,4,0),
aL ν˜c ∼
a
˜lac L
2 1
uc 3 ,1, , dc 3 ,1, ,
iL ∼ ∗ −3 iL ∼ ∗ 3
(cid:18) (cid:19) (cid:18) (cid:19)
2 5
uc 3 ,1, , Jc 3 ,1, ,
′L ∼ ∗ −3 L ∼ ∗ −3
(cid:18) (cid:19) (cid:18) (cid:19)
1 4
dc 3 ,1, , jc 3 ,1, . (12)
′βL ∼ ∗ 3 βL ∼ ∗ 3
(cid:18) (cid:19) (cid:18) (cid:19)
Where α = 2,3 and β = 1,2. The higgsinos of these model are given by
η˜0 φ˜0
1 1
˜
η˜ φ
η˜ = 1− (1,4,0), φ˜= −1 (1,4,0),
η˜0 ∼ φ˜0 ∼
η˜2+2 φ˜+22
ρ˜+ χ˜
1 −1
ρ˜0 χ˜
ρ˜ = (1,4,1), χ˜ = −− (1,4, 1),
ρ˜+2 ∼ χ˜−2 ∼ −
ρ˜++ χ˜0
(13)
H˜0 H˜+ H˜0 H˜
1 1 2 2−
H˜+ H˜++ H˜+ H˜0
H˜ = 1 1 3 3 (1,10 ,0). (14)
H˜0 H˜+ H˜0 H˜ ∗
2 3 4 4−
H˜2− H˜30 H˜4− H˜2−−
Besides, inordertotocancelchiralanomaliesgeneratedbythesuperpart-
ners of the scalars, we have to add the following higgsinos in the respective
anti-quartet representation,
η0 φ0
1′ ′1
η+ φ+
η = 1′ (1,4 ,0), φ = ′1 (1,4 ,0)
′ η2′0 ∼ ∗ ′ φ′20 ∼ ∗
η2′− φ′2−
ρ χ+
′1− ′1
ρ0 χ++
ρ = ′ (1,4 , 1), χ = ′ (1,4 ,1), (15)
′ ρ′2− ∼ ∗ − ′ χ′2+ ∼ ∗
ρ χ0
′−− ′
and the decuplet
H 0 H H 0 H +
1′ 1′− 2′ 2′
H H H H 0
H = 1′− 1′−− 3′− 3′ (1,10,0). (16)
′ H2′0 H3′− H4′0 H4′+ ∼
H2′+ H3′0 H4′+ H2′++
Their superpartners, higgsinos, are
η˜0 φ˜0
1′ ′1
η˜+ φ˜+
η˜ = 1′ (1,4 ,0), φ˜ = ′1 (1,4 ,0),
′ η˜η˜2′2′−0 ∼ ∗ ′ φφ˜˜′2′2−0 ∼ ∗
ρ˜ χ˜+
′1− ′1
ρ˜0 χ˜++
ρ˜ = ′ (1,4 , 1), χ˜ = ′ (1,4 ,1), (17)
′ ρ˜′2− ∼ ∗ − ′ χ˜′2+ ∼ ∗
ρ˜ χ˜0
′−− ′
H˜ 0 H˜ H˜ 0 H˜ +
1′ 1′− 2′ 2′
H˜ H˜ H˜ H˜ 0
H˜ = 1′− 1′−− 3′− 3′ (1,10,0). (18)
′ H˜ 0 H˜ H˜ 0 H˜ + ∼
2′ 3′− 4′ 4′
H˜2′+ H˜3′0 H˜4′+ H˜2′++
The vev of our scalars are given by
v u z w
η = ,0,0,0 , ρ = 0, ,0,0 , φ = 0,0, ,0 , χ = 0,0,0, ,
h i √2 ! h i √2 ! h i √2 ! h i √2!
x
H0 = , H0 = H0 = H0 = 0,
h 3i √2 h 1i h 2i h 4i
v u z w
′ ′ ′ ′
η = ,0,0,0 , ρ = 0, ,0,0 , φ = 0,0, ,0 , χ = 0,0,0, ,
′ ′ ′
h i 2 ! h i √2 ! h i √2 ! h i √2!
x
H 0 = ′ , H 0 = H 0 = H 0 = 0.
h 3′ i √2 h 1′ i h 2′ i h 4′ i
(19)
Concerning the gauge bosons and their superpartners, if we denote the
gluons by gb the respective superparticles, the gluinos, are denoted by λb ,
C
with b = 1,...,8; and in the electroweak sector we have Va, with a =
1,...,15; the gauge boson of SU(4) , and their gauginos partners λb ; finally
L A
we have the gauge boson of U(1) , denoted by V , and its supersymmetric
N ′
partner λ .
B
2.3 Superfields
The superfields formalism is useful in writing the Lagrangian which is mani-
festly invariant under the supersymmetric transformations [13] with fermions
and scalars put in chiral superfields while the gauge bosons in vector super-
ˆ
fields. As usual the superfield of a field φ will be denoted by φ [12]. The
chiral superfield of a multiplet φ is denoted by
1
φˆ φˆ(x,θ,θ¯) = φ˜(x)+i θσmθ¯∂ φ˜(x)+ θθ θ¯θ¯2φ˜(x)
m
≡ 4
i
+√2 θφ(x)+ θθ θ¯σ¯m∂ φ(x)
m
√2
+θθ F (x), (20)
φ
while the vector superfield is given by
Vˆ(x,θ,θ¯) = θσmθ¯V (x)+iθθθ¯V˜(x) iθ¯θ¯θV˜(x)
m
− −
1
¯¯
+ θθθθD(x). (21)
2
The fields F and D are auxiliary fields which are needed to close the su-
persymmetric algebra and eventually will be eliminated using their motion
equations.
Summaryzing, we have in the 3-4-1 supersymmetric model the following
superfields: Lˆ , Qˆ , ηˆ, ρˆ, χˆ, φˆ, Hˆ; ηˆ, ρˆ, χˆ, φˆ, Hˆ ; uˆc , dˆc , u, d ,
1,2,3 1,2,3 ′ ′ ′ ′ ′ 1,2,3 1,2,3 ′ ′1,2
Jˆ and ˆj , i.e., 28 chiral superfields, and 24 vector superfields: Vˆa, Vˆα and
1,2
Vˆ .
′
3 The Lagrangian
With the superfields introduced in the last section we can built a supersym-
metric invariant lagrangian. It has the following form
= + . (22)
341 SUSY soft
L L L
Here is the supersymmetric piece, while explicitly breaks SUSY.
SUSY soft
L L
Below we will write each of these lagrangians in terms of the respective
superfields.
3.1 The Supersymmetric Term.
The supersymmetric term can be divided as follows
= + + + , (23)
SUSY Lepton Quarks Gauge Scalar
L L L L L
where each term is given by
= d4θ Lˆ¯e2gVˆLˆ , (24)
L´epton
L
Z h i
= d4θ Qˆ¯ e[2gsVˆC+2gVˆ+g′(23)Vˆ′]Qˆ + Qˆ¯ e 2gsVˆC+2gVˆ¯+g′(−31)Vˆ′ Qˆ
LQuarks ( 1 1 α h i α
Z
+ uˆ¯ce 2gsVˆ¯C+g′(−23)Vˆ′ uˆc +dˆ¯ce 2gsVˆ¯C+g′(31)Vˆ′ dˆc +Jˆ¯ce 2gsVˆ¯C+g′(−53)Vˆ′ Jˆc
i i i i
h i h i h i
+ ˆ¯jcβe 2gsVˆ¯C+g′(43)Vˆ′ ˆjβc +uˆ¯′ce[2gsVˆ¯C+g′(−32)Vˆ′]uˆ′c +dˆ¯′βce[2gsVˆ¯C+g′(13)Vˆ′]dˆ′βc ,
h i )
(25)
where α = 2,3 and β = 1,2, while the third term is
1 1 1
= d2θ Tr[W W ]+ d2θ Tr[W W ]+ d2θW W
Gauge C C L L ′ ′
L 4 4 4
Z Z Z
1 1 1
+ d2θ¯Tr[W¯ W¯ ]+ d2θ¯Tr[W¯ W¯ ]+ d2θ¯W¯ W¯
C C L L ′ ′
4 4 4
Z Z Z
(26)
where Vˆ = TaVˆa, Vˆ = TiVˆi and Ta = λa/2 are the generators of SU(3) i.e.,
c c
a = 1, ,8, and Ti = λi/2 are the generators of SU(4) i.e., i = 1, ,15,
··· ···
and g , g and g are the gauge coupling of SU(3) , SU(4) and U(1) . Wa,
s ′ C L N c
Wi and W are the strength fields, and they are given by
′
1
Wa = D¯D¯e 2gsVˆcD e 2gsVˆc
αc −8g − α −
s
1
Wa = D¯D¯e 2gVˆD e 2gVˆ
α −8g − α −
1
W = D¯D¯D Vˆ . (27)
α′ −4 α ′
Finally
= d4θ ηˆ¯e2gVˆηˆ+ρˆ¯e(2gVˆ+g′Vˆ′)ρˆ+χˆ¯e(2gVˆ g′Vˆ′)χˆ+φˆ¯e2gVˆφˆ+Hˆ¯e2gVˆHˆ
Escalar −
L
Z (cid:20)
+ ηˆ¯′e2gVˆ¯ηˆ′ +ρˆ¯′e 2gVˆ¯−g′Vˆ′ ρˆ′ +χˆ¯′e 2gVˆ¯+g′Vˆ′ χˆ′ +φˆ¯′e2gVˆφˆ′ +Hˆ¯′e2gVˆ¯Hˆ′
(cid:16) (cid:17) (cid:16) (cid:17) #
+ d2θW + d2θ¯W (28)
Z Z
where W is the superpotential, which we discuss in the next subsection.
3.2 Superpotential.
The superpotential of our model is given by
W W
2 3
W = + , (29)
2 3
with W having only two chiral superfields and the terms permitted by our
2
symmetry are
3 3
W = µ Lˆ ηˆ+ µ Lˆ φˆ+µ ηˆηˆ+µ φˆφˆ+µ ηˆφˆ+µ φˆηˆ+µ ρˆρˆ+µ χˆχˆ+µ HˆHˆ ,
2 0a aL ′ 1a aL ′ η ′ φ ′ 2 ′ 3 ′ ρ ′ χ ′ H ′
a=1 a=1
X X
(30)
and in the case of three chiral superfields the terms are
3 3 3 3 3 3 3
W = λ ǫLˆ Lˆ Lˆ + λ ǫLˆ Lˆ ηˆ+ λ ǫLˆ Lˆ φˆ
3 1abc aL bL cL 2ab aL bL 3ab aL bL
a=1b=1c=1 a=1b=1 a=1b=1
XXX XX XX
