We investigate the vacuum stability conditions of theSupersymmetric reduced minimal 331 model (SUSYRM331) that createimportant  consequences on  Higgs mass spectrum as well assoft-parameters of the model. We prove that if this condition issatisfied then all Higgses are massive. Furthermore,soft-parameters should be in order of \(SU(3)_L\) scale. Based onthis, we investigate in detail masses of CP-even neutral  anddoubly charged Higgses in the model. The  neutral Higgs sectorincludes one light Higgs with mass at tree level \(m_{H^0_1}\simeq m_Z|\cos 2\gamma |  92.0\) GeV and three other heavy Higgses. Fordoubly charged Higgses, there may exist a light Higgs which can beobserved by recent colliders such as LHC

Hue, Le Tho

Phuong, Ly Thi Mai

Thao, Nguyen Huy

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 23, No. 3 (2013), pp. 203-209
VACUUM STABILITY IN SUPERSYMMETRIC REDUCED
MINIMAL 3-3-1 MODEL
NGUYEN HUY THAO, LY THI MAI PHUONG
Department of Physics,
Hanoi University of Education No. 2, Vinh Phuc, Vietnam
LE THO HUE
Institute of Physics,
Vietnam Academy of Science and Technology, Hanoi, Vietnam
Email: nhthao@grad.iop.vast.ac.vn
Received 01 September 2013;
Accepted for publication 17 September 2013 ;
Abstract. We investigate the vacuum stability conditions of the supersymmetric reduced minimal
331 model (SUSYRM331) that create important consequences on Higgs mass spectrum as well as
soft-parameters of the model. We prove that if this condition is satised then all Higgses are
massive. Furthermore, soft-parameters should be in order of SU(3)L scale. Based on this, we
investigate in detail masses of CP-even neutral and doubly charged Higgses in the model. The
neutral Higgs sector includes one light Higgs with mass at tree level mH01
' mZ j cos 2j < 92:0
GeV and three other heavy Higgses. For doubly charged Higgses, there may exist a light Higgs
which can be observed by recent colliders such as LHC.
I. INTRODUCTION
The discovery of a new scalar particle with mass around 126:5 GeV by the Large
Hadron Collider (LHC) [1,2] is a milestone of particle physics at this time. Although this
new particle inherits many properties of the Standard Model (SM) Higgs, both theoretical
and experimental physics need more time and data to conrm whether it is really the SM
Higgs or a Higgs of some model beyond SM. The fact now is that a new model beyond
SM must contain at least one Higgs satisfying all properties of the new Higgs mentioned
above. In this work, we show that the SUSYRM331 in the presence of B=-type term
always contains a Higgs discovered by LHC and therefore it is a realistic model. We
emphasize that Higgs sector was investigated in [3] without B=-type terms.
Another aim of this work is reinvestigating the Higgs spectrum of the SUSYRE331
[3] in the presence of b-type terms (b
0 + b0 +H:c). This will guarantee the vacuum
stability of the model. But terms contributing to Higgs masses now become more com-
plicating and the mass spectrums of dierent charged Higgs will directly depend on the
scale of both b and b. We only concentrate on the real neutral and double charged Higgs
masses because they are not calculated exactly but we can nd them with high accuracy.
204 NGUYEN HUY THAO, LY THI MAI PHUONG, AND LE THO HUE
II. HIGGS POTENTIAL
The SUSYRM331 model was built in [3]. In this work, we only investigate the
minimum of the scalar potiential and the Higgs sector of the model. First, the scalar
potential of the model is
VSUSYRM331 = VD + VF + Vsoft; (1)
Vsoft =  LSMT = m2+m2 +m20 00 +m20 00  
 
b
0 + b0 + h:c:

;
VD =
g02
12
(  00   + 00)2 + g
2
8
X
i;j
 
i
a
ijj + i
a
ijj   0iaij 0j   0iaij 0j
2
;
VF =  LF =
X
F
FF =
X
i

2
0i
2 + 
2
0i
2 + 
2
i
2 + 
2
i
2 ;
where m; m; m0 and m0 have the mass dimension; b as well as b are assumed to be
real and positive to make sure the non-zero and real values of VEVs. Note that Vsoft in
this work is revised by adding the B=-type terms.
In the Higgs sector, all of the four neutral scalar components 0; 0; 00; 00 gain
non-zero vacuum expectation values (VEVs). The expansions of the neutral scalars around
their VEVs are dened as
<  > =
1p
2
 
0 v +H + iF 0
T
; < 0 >=
1p
2
 
0 v0 +H0 + iF0 0
T
;
<  > =
1p
2
 
0 0 v +H + iF
T
; < 0 >=
1p
2
 
0 0 v0 +H0 + iF0
T
:
The minimum of the Higgs potential (1) appears when all linear Higgs terms in this
potential vanish, namely
m2 +
1
4
2 =
b
t
  2g
2 + g21
12
v02(t2   1) +
g2 + g21
12
w02(t2   1);
m2 +
1
4
2 =
b
t
+
g2 + g21
12
v02(t2   1) 
2g2 + g21
12
w02(t2   1);
m2 +m
2
0 +
1
2
2 = b

t +
1
t

;m2 +m
2
0 +
1
2
2 = b

t +
1
t

; (2)
where two new notations are used, t = tan  =
v
v0 ; t = tan =
w
w0 : Comparing with the
gauge boson masses calculated in [3], we remind that
m2W =
g2
4
(v2 + v02) =
g2
4
v02(t2 + 1); m
2
V =
g2
4
(w2 + w02) =
g2
4
w02(t2 + 1): (3)
VACUUM STABILITY IN SUSYRM 331 MODEL 205
Masses of these two gauge bosons will be used as independent parameters in our calcula-
tion. Four equations (2) now can be rewritten in forms of
m2 +
1
4
2 =
b
t
  1 + t
2
3
m2V cos 2 +
t2 + 2
3
m2W cos 2; (4)
m2 +
2
4
=
b
t
  2 + t
2
3
m2V cos 2 +
1 + t2
3
m2W cos 2; (5)
s2  sin 2 = 2b
m2 +m
2
0 +
1
2
2

; s2  sin 2 = 2b
m2 +m
2
0 +
1
2
2

: (6)
From two equations (4) and (5) we have
c2  cos 2 =
 (m2 + 
2

4   bt )(1 + 2s2W ) + 2(m2 +
2
4   bt )c2W
m2W
;
c2  cos 2 =
(m2 +
2
4   bt )(1 + 2s2W )  2(m2 +
2
4   bt )c2W
m2V
: (7)
Because of the property of the general SUSY331 models, m2W  m2V and in addition
jc2 j; jc2j  1, it can be deduced that two quantities (m2 + 
2

4   bt ) and (m2 +
2
4   bt )
should be in the same order of O(m2W ) or O(m2V ). Similar to the case of the SUSYE331
model [6], to avoid the very light Higgses contained in the model, we only consider the
case of two quantities having order of O(m2V ), i.e the TeV scale.
III. HIGGS SPECTRUM
Before investigating the mass spectrum of real neutral and doubly charged Higgses,
we note that the SUSYRM331 model also contain pseudo-scalar neutral and singly charged
Higgses. It is easy to show the mass eigenvalues and eigenstates can be found exactly,
for example see [7]. We will use masses of two pseudo-scalar Higgses as independent
parameters characterized for soft and B=-terms. They are dened as follows
m2A1 =
2b
s2
= m2 +m
2
0 +
1
2
2; m
2
A2 =
2b
s2
= m2 +m
2
0 +
1
2
2: (8)
III.1. Real scalar neutral Higgs
The squared mass matrix of these Higges can be written in the form of
LrH =
1
2
(H; H0 ; H; H0) M2p(H; H0 ; H; H0)T (9)
with
M24S =
0BB@
m2S11 m
2
S12 m
2
S13 m
2
S14
m2S22 m
2
S23 m
2
S24
m2S33 m
2
S34
m2S44
1CCA ; (10)
206 NGUYEN HUY THAO, LY THI MAI PHUONG, AND LE THO HUE
where formulas of elements are
m2S11 =
b
t
+
1
6
g2(2 + t2)t2v
02; m2S12 =  b  
1
6
g2(2 + t2)tv
02;
m2S13 =  
1
6
g2(1 + t2)ttv
0w0; m2S14 =
1
6
g2(1 + t2)tv
0w0;
m2S22 = bt +
1
6
g2(2 + t2)v02; m2S23 =
1
6
g2(1 + t2)tv
0w0;
m2S24 =  
1
6
g2(1 + t2)v0w0; m2S33 =
b
t
+
1
6
g2(2 + t2)t2w
02;
m2S34 =  b  
1
6
g2(2 + t2)tw
02; m2S44 = bt +
1
6
g2(2 + t2)w02:
Each eigenvalues of this matrix  = m2H0 is one physical mass of CP-even neutral Higgs. It
must be satised the equation det(M24S I4) = 0, or equivalent f() = 0, where f() can
be reduced to the more simple form by dening new variable and parameters as follows:
 = Xm2V ;m
2
W = m2V ; m2A1 = k1m2V ; m2A2 = k2m2V (11)
This equation now can be written in the form
f(X) = X4 + a1X
3 + a2X
2 + a3X + a4 = 0; (12)
where
a1 =  4
3
  k1   k2 + 4s
2
W
 1 + 4s2W
+
4c2W
 3 + 12s2W
 ;
a2 =
 4c22k2c2W + k1( 4  3k2 + 4(1 + 3k2)s2W )
 3 + 12s2W
  4(1 + c
2
2k1c
2
W + k2c
2
W )
 3 + 12s2W
 ;
a3 =  
4c22k1k2c
2
W
3  12s2W
  4(c
2
2k2 + c
2
2k1(1 + k2   k2s2W ))
3  12s2W
 ;
a4 =
4c22c
2
2k1k2
3  12s2W
 : (13)
For light neutral Higgses, we denote its mass as X1 = X
0
1   where X 01  O(1). Inserting
this value into (12) then force the factor of lowest order of  to be zero, we obtain
X 01 '
c22
c2W
) m2H01 'M
2
Zc
2
2 : (14)
It is emphasized that although this value of light Higgs mass, MZ jc2 j < 92: GeV, is much
lighter than the value of 126:5 GeV discovered by LHC, the loop corrections will enhance
this mass to the consistent value. This conclusion is the same as the case of the well-known
Minimal Supersymmetric Standard Model (MSSM).
For heavy neutral Higgses, we denote their masses as Xi = X
0
i +X
00
i   where both
X 0i; X
00
i  O(1) (i = 2; 3; 4). Then these masses can be written in the from
m2H0i
= X 0im
2
V +X
00
i m2W +O()m2W : (15)
VACUUM STABILITY IN SUSYRM 331 MODEL 207
where main contributions are
m2H02
' X 02m2V =m2A1 ; (16)
m2H03;4
' X 03;4 m2V =
1
6
0@ 4c2W
1  4s2W
+ 3m2A2 
s
 48c
2
2m
2
A2
m2V
1  4s2W
+

3m2A2 +
4c2Wm
2
V
1  4s2W
21A :
(17)
III.2. Doubly charged Higgs boson
The mass terms of the doubly charged boson are
LH =
 
++ 0++ ++ 0++
M2H      0      0   T ; (18)
M2H =
g2
4
0BBB@
m2++    4bg2   v02t v0w0tt  v0w0t
m20++0   v
0w0t v0w0
m2++    4bg2   w02t2
m20++0  
1CCCA (19)
m2++   =
4b
g2t
+ v02 + w02(t2   1); m20++0   = v02t +
4bt
g2
+ w02(1  t2);
m2++   = v
02(t2   1) +
4b
g2t
+ w02; m20++0   = v
02(1  t2) + w02t2 +
4bt
g2
:
Finding the mass eigenvalues of doubly charged is similar to the case of the real scalar
neutral Higgs shown above. The eigenvalues of this matrix  = m2H must be satised
the equation det(M2H I4) = 0. Using parameters in (11), this equation can be written
in the form of
g(X) = X3 +AX2 +BX + C = 0 (20)
A =  (1 + k1 + k2 + );
B =  c22   c222 + k1 + (+ k1)k2 + c2c2(k1 + (2 + k2));
C = (1 + )(c2   c2(+ k1))(c2   c2+ c2k2): (21)
Masses of these Higgses are presented as
m2H = X
0m2V +X
00 m2W +O()m2W : (22)
By analogy, it can be shown that main contributions to masses of doubly charged are
m2
H1
' X 01 m2V = m2V +m2A2
m2
H2;3
' X 02;3 m2V = m2A1 +
q
4c22m
4
V   4c2c2m2Vm2A1 +m4A1
X"i = A=B; (23)
A = c22k1 + c
2
2(1 + k2)  c2c2(k1(1 + k2)  (2 + k2)( 1 +X 0i)) + (k2  X 0i)X 0i;
B = c22   c2c2k1   k1(1 + k2   2X 0i) + (2 + 2k2   3X 0i)X 0i:
208 NGUYEN HUY THAO, LY THI MAI PHUONG, AND LE THO HUE
1000 1500 2000 2500 3000 3500
1
10
100
1000
mA2 @ GeVD
m
H
0
2
@
G
eV
2 D
(a)
1000 1500 2000 2500 3000 3500
-2´106
0
2´106
4´106
6´106
8´106
mA1 @ GeVD
m
H
±
±
2
@
G
eV
2 D
(b)
Fig. 1. Masses of Higgses as functions of mA2 (a) or mA1 (b), where mV = 2:0
TeV, mW = 80:4 GeV and s
2
W = 0:231. The left panel presents masses of neutral
Higgses withmA1 = 1:0 TeV, tan  = 50, and tan = 30. The right panel presents
squared masses of doubly charged Higgses with mV = 2:5 TeV, mA2 = 1:0 TeV,
tan  = 10 and tan = 30.
For an illustration of ours result, masses of Higgses are shown in Fig. 1 with some
xed values of parameters. This choice of the parameters satises the scale of SU(3)L
being TeV scale. The large values of tan  and tan make sure the loop corrections to
the lightest neutral Higgs mass being consistent with present values of 126:5 GeV. In the
left panel of the Fig.1, we draw four values of CP-even neutral Higgs masses, where the
green line corresponds to the lightest mass and three blue curves present masses of three
heavy masses of TeV scale. In the right panel, we see that in order to cancel doubly
charged Higgs Tachyon, i.e m2H must be positive, mA1 is bounded from below, 2:0 TeV
for example in this case. In addition, if values of mA1 is very close to 2.0 TeV, there
VACUUM STABILITY IN SUSYRM 331 MODEL 209
exists a very light doubly charged Higgs. This is very important because this light doubly
charged Higgs can be detected by recent colliders.
IV. CONCLUSIONS
To guarantee the stability of VEVs in the SUSYRM331, it is needed to add some
b-type terms into Vsoft. As a consequence, soft parameters are in order of SU(3)L scale.
Further more, all Higgses of the model are massive. There are only light neutral Higgs
with mass at tree level is, the same mass of the light Higgs concerned in MSSM. The model
contains three other heavy real neutral Higgses, two pseudo neutral, two singly charged,
and three doubly charged Higgses. The model may contains one light doubly charged
Higgs if mA1 get a suitable values. This will be an important signal to distinguish the
SUSYRM331 model with many other models without doubly charged Higgses.
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 103.01-2011.63.
REFERENCES
[1] The ATLAS Collaboration, Phys. Lett. B 716 (2012) 1, arXiv:1207.7214.
[2] The CMS Collaboration, G. Aad et al., Phys. Lett. B 716 (2012) 30, arXiv:1207.7235.
[3] D. T. Huong, L. T. Hue, M. C. Rodriguez, H. N. Long, Nucl.Phys. B 870 (2013) 293.
[4] M. C. Rodriguez, Journ. Mod. Phys. 2 (2011) 1193 .
[5] L. Girardello and M. T. Grisaru, Nucl. Phys. B 194 (1982) 65 .
[6] D. T. Binh, L. T. Hue, D. T. Huong, H. N. Long, arXiv: hep-ph/1308.3085.
[7] J. G. Ferreira, C. A. de S. Pires, P. S. Rodrigues da Silva, A. Sampieri, arXiv: hep-ph/1308.0575.


Vacuum Stability in Supersymmetric Reduced Minimal 3-3-1 Model

http://vjs.ac.vn/index.php/cip/article/view/3237/pdf

Vacuum Stability in Supersymmetric Reduced Minimal 3-3-1 Model

Abstract

Similar works

Full text

Available Versions

Vietnam Academy of Science and Technology: Journals Online