Page 1
arXiv:hep-ph/9904388v1 19 Apr 1999
HIP-1999-16/TH
Phenomenology of light Higgs bosons in
supersymmetric left-right models
K. Huitua, P. N. Panditaa,b,cand K. Puolam¨ akia
aHelsinki Institute of Physics, P.O.Box 9, FIN-00014 University of Helsinki,
Finland
bDeutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22603 Hamburg,
Germany
cDepartment of Physics, North Eastern Hill University, Shillong 793022, India1
Abstract
We carry out a detailed analysis of the light Higgs bosons in supersymmetric left-
right models (SLRM). This includes models with minimal particle content and those
with additional Higgs superfields. We also consider models with non-renormalizable
higher-dimensional terms. We obtain an upper bound on the mass of the lightest
CP-even neutral Higgs boson in these models. The upper bound depends only on
the gauge couplings, and the vacuum expectation values of those neutral Higgs fields
which control the spontaneous breakdown of the SU(2)L×U(1)Y gauge symmetry.
We calculate the one-loop radiative corrections to this upper bound, and evaluate
it numerically in the minimal version of the supersymmetric left-right model. We
consider the couplings of this lightest CP-even Higgs boson to the fermions, and
show that in a phenomenologically viable model the branching ratios are similar to
the corresponding branching ratios in the minimal supersymmetric standard model
(MSSM). We then study the most promising particle for distinguishing the SLRM
from other models, namely the doubly charged Higgs boson. We obtain the mass
of this doubly charged Higgs boson in different types of supersymmetric left-right
models, and discuss its phenomenology.
Key words: Supersymmetry; left-right symmetry; Higgs boson; doubly charged
Higgs boson
PACS: 12.60.Jv; 11.30.Fs; 14.80.Cp
1Permanent address
Preprint submitted to Elsevier Preprint 1 February 2008
Page 2
1Introduction
One of the central problems of particle physics is to understand how the electroweak
scale associated with the mass of the W boson is generated, and why it is so small as
compared to the Planck scale associated with the Newton’s constant. In the Standard
Model (SM) the electroweak scale is generated through the vacuum expectation value
(VEV) of the neutral component of a Higgs doublet [1]. Apart from the fact that this
VEV is an arbitrary parameter in the SM, the mass parameter of the Higgs field suffers
from quadratic divergences, making the weak scale unstable under radiative corrections.
Supersymmetry is at present the only known framework in which the weak scale is stable
under radiative corrections [2], although it does not explain how such a small scale arises in
the first place. As such, considerable importance attaches to the study of supersymmetric
models, especially the Minimal Supersymmetric Standard Model (MSSM), based on the
gauge group SU(2)L× U(1)Y, with two Higgs doublet superfields. It is well known that,
because of underlying gauge invariance and supersymmetry (SUSY), the lightest Higgs
boson in MSSM has a tree level upper bound of MZ(the mass of Z boson) on its mass [3].
Although radiative corrections [4] to the tree level result can be appreciable, these depend
only logarithmically on the SUSY breaking scale, and are, therefore, under control. This
results in an upper bound of about 125 − 135 GeV on the one-loop radiatively corrected
mass [5] of the lightest Higgs boson in MSSM2. Because of the presence of the additional
trilinear Yukawa couplings, such a tight constraint on the mass of the lightest Higgs boson
need not a priori hold in extensions of MSSM based on the gauge group SU(2)L×U(1)Y
with an extended Higgs sector. Nevertheless, it has been shown that the upper bound
on the lightest Higgs boson mass in these models depends only on the weak scale, and
dimensionless coupling constants (and only logarithmically on the SUSY breaking scale),
and is calculable if all the couplings remain perturbative below some scale Λ [7–13]. This
upper bound can vary between 150 GeV to 200 GeV depending on the Higgs structure of
the underlying supersymmetric model. Thus, nonobservation of such a light Higgs boson
below this upper bound will rule out an entire class of supersymmetric models based on
the gauge group SU(2)L× U(1)Y.
The existence of the upper bound on the lightest Higgs boson mass in MSSM (with
arbitrary Higgs sectors) has been investigated in a situation where the underlying super-
symmetric model respects baryon (B) and lepton (L) number conservation. However, it
is well known that gauge invariance, supersymmetry and renormalizibility allow B and L
violating terms in the superpotentioal of the MSSM [14]. The strength of these lepton and
baryon number violating terms is, however, severely limited by phenomenological [15,16],
and cosmological [17] constraints. Indeed, unless the strength of baryon-number violating
term is less than 10−13, it will lead to contradiction with the present lower limits on the
lifetime of the proton. The usual strategy to prevent the appearance of B and L violating
2The two-loop corrections to the Higgs boson mass matrix in the MSSM are significant, and
can reduce the lightest Higgs boson mass by up to ∼ 20 GeV as compared to its one-loop value
[6].
2
Page 3
couplings in MSSM is to invoke a discrete Z2symmetry [18] known as matter parity, or
R-parity. The matter parity of each superfield may be defined as
(matter parity) ≡ (−1)3(B−L). (1)
The multiplicative conservation of matter parity forbids all the renormalizable B and L
violating terms in the superpotential of MSSM. Equivalently, the R-parity of any com-
ponent field is defined by Rp = (−1)3(B−L)+2S, where S is the spin of the field. Since
(−1)2Sis conserved in any Lorentz-invariant interaction, matter parity conservation and
R-parity conservation are equivalent. Conservation of R-parity then immediately implies
that superpartners can be produced only in pairs, and that the lightest supersymmetric
particle (LSP) is absolutely stable.
Although the Minimal Supersymmetric Standard Model with R-parity conservation can
provide a description of nature which is consistent with all known observations, the as-
sumption of Rpconservation appears to be ad hoc, since it is not required for the internal
consistency of MSSM. Furthermore, all global symmetries, discrete or continuous, could
be violated by the Planck scale physics effects [19]. The problem becomes acute for low
energy supersymmetric models, because B and L are no longer automatic symmetries
of the Lagrangian, as they are in the Standard Model. It is, therefore, more appealing
to have a supersymmetric theory where R-parity is related to a gauge symmetry, and
its conservation is automatic because of the invariance of the underlying theory under
this gauge symmetry. Fortunately, there is a compelling scenario which does provide for
exact R-parity conservation due to a deeper principle. Indeed, Rp conservation follows
automatically in certain theories with gauged (B − L), as is suggested by the fact that
matter parity is simply a Z2subgroup of (B − L). It has been noted by several authors
[20,21] that if the gauge symmetry of MSSM is extended to SU(2)L×U(1)I3R×U(1)B−L,
or SU(2)L× SU(2)R× U(1)B−L, the theory becomes automatically R-parity conserving.
Such a left-right supersymmetric theory (SLRM) solves the problems of explicit B and
L violation of MSSM, and has received much attention recently [22–28]. Of course left-
right symmetric theories are also interesting in their own right, for among other appealing
features, they offer a simple and natural explanation for the smallness of neutrino mass
through the so called see-saw mechanism [29].
Such a naturally R-parity conserving theory necessarily involves the extension of the
Standard Model gauge group, and since the extended gauge symmetry has to be broken,
it involves a “new scale”, the scale of left-right symmetry breaking, beyond the SUSY
and SU(2)L× U(1)Y breaking scales of MSSM. In [30] we showed that in the SLRM
with minimal particle content the upper bound on the mass of the lightest neutral Higgs
boson depends only on gauge couplings and those VEVs which break the SU(2)L×U(1)Y
symmetry. Here we will present a detailed analysis of the Higgs sector of the left-right
supersymmetric models, and consider some of the distinguishing features of the lightest
Higgs boson in these models. In the SLRMs there are typically also other light Higgs
particles. The light doubly charged Higgs boson, which we will consider in detail as well,
should provide a clear signal in experiments.
3
Page 4
The plan of the paper is as follows. In section 2, we review the various left-right supersym-
metric models that we consider in this paper. In section 3, we obtain the tree level upper
bound on the mass of the lightest CP-even Higgs boson for the various SLRMs considered
in section 2. We use a general procedure to obtain this (tree-level) upper bound on the
mass of the lightest CP-even Higgs boson in models with extended gauge groups, such
as SLRMs. We show that the upper bound so obtained in the renormalizable models is
independent of the supersymmetry breaking scale, as well as the left-right breaking scale.
In the case of models containing non-renormalizable terms, although the upper bound de-
pends on the left-right breaking scale, the dependence is extremely weak, being suppressed
by powers of Planck mass.
In section 4 we calculate the radiative corrections to the upper bound on the mass of the
lightest Higgs boson, and show that the most important radiative corrections arising from
quark-squark loops are of the same type as in the MSSM based on SU(2)L×U(1)Y. The
radiatively corrected upper bound so obtained is numerically considerably larger than the
corresponding bound in the MSSM, but for most of the parameter space is below 200 GeV.
In section 5 we consider the branching ratios of the lightest neutral Higgs boson, and find
that its couplings are similar to the corresponding Higgs couplings in the Standard Model
in the decoupling limit.
In section 6 we discuss the mass of the lightest doubly charged Higgs boson and the
possibility of its detection at colliders. In section 7, we present our conclusions. The
full scalar potential of the minimal supersymmetric left-right model is presented in the
Appendix A.
2The Higgs sector of the left-right supersymmetric models
In this section we briefly review the minimal supersymmetric left-right model, and then
discuss models which have an extended Higgs sector, and finally discuss models with
non-renormalizable interaction terms in the superpotential.
The minimal SLRM is based on the gauge group SU(3)C×SU(2)L×SU(2)R×U(1)B−L.
The matter fields of this model consist of the three families of quark and lepton chiral
superfields with the following transformation properties under the gauge group:
Q =
U
D
∼ (3,2,1,1
3), Qc=
Dc
Ec
Uc
∼ (3∗,1,2,−1
3),
L =
ν
E
∼ (1,2,1,−1), Lc=
νc
∼ (1,1,2,1), (2)
4
Page 5
where the numbers in the brackets denote the quantum numbers under SU(3)C×SU(2)L×
SU(2)R×U(1)B−L. The Higgs sector consists of the bidoublet and triplet Higgs superfields:
Φ =
Φ0
∆0
1Φ+
1
Φ−
2 Φ0
2
∼ (1,2,2,0), χ =
χ0
1χ+
1
χ−
2 χ0
2
∼ (1,2,2,0),
δ++
R
δ++
L
∆R=
1
√2∆−
∆−−
R
RR
−1
√2∆−
R
∼ (1,1,3,−2), δR=
1
√2δ+
δ0
R
R
−1
√2δ+
R
∼ (1,1,3,2),
∼ (1,3,1,2).
∆L=
1
√2∆−
∆−−
L
L
∆0
L
−1
√2∆−
L
∼ (1,3,1,−2), δL=
1
√2δ+
δ0
L
L
−1
√2δ+
L
(3)
There are two bidoublet superfields in order to implement the SU(2)L×U(1)Y breaking,
and to generate a nontrivial Kobayashi-Maskawa matrix. Furthermore, two SU(2)RHiggs
triplet superfields ∆Rand δRwith opposite (B − L) are necessary to break the left-right
symmetry spontaneously, and to cancel triangle gauge anomalies due to the fermionic
superpartners. The gauge symmetry is supplemented by a discrete left-right symmetry
under which the fields can be chosen to transform as
Q ↔ Qc, L ↔ Lc, Φ ↔ −τ2ΦTτ2, χ ↔ −τ2χTτ2, ∆R↔ δL, δR↔ ∆L.(4)
Thus, the SU(2)Ltriplets ∆Land δLare needed in order to make the Lagrangian fully
symmetric under L ↔ R transformation, although these are not needed phenomenologi-
cally for symmetry breaking, or the see-saw mechanism.
The most general gauge invariant superpotential involving these superfields can be written
as (generation indices suppressed)
Wmin=hΦQQTiτ2ΦQc+ hχQQTiτ2χQc+ hΦLLTiτ2ΦLc+ hχLLTiτ2χLc
+hδLLTiτ2δLL + h∆RLcTiτ2∆RLc+ µ1Tr(iτ2ΦTiτ2χ) + µ′
+µ′′
1Tr(iτ2ΦTiτ2Φ)
1Tr(iτ2χTiτ2χ) + Tr(µ2L∆LδL+ µ2R∆RδR).(5)
The scalar potential can be calculated from
V = VF+ VD+ Vsoft,(6)
where VF, VD, and Vsoftrepresent the contribution of F-terms, the D-terms, and the soft
SUSY breaking terms, respectively. The full scalar potential can be found in Appendix
A. The general form of the vacuum expectation values of the various scalar fields which
preserve the U(1)emgauge invariance can be written as
5
Page 6
?Φ? =
κ1
0
0 eiφ1κ′
1
, ?χ? =
eiφ2κ′
20
0κ2
, ?∆R? =
0 v∆R
0 0
, ?δR? =
0 0
vδR0
,
?∆L? =
0 v∆L
0 0
, ?δL? =
0 0
vδL0
, ?L? =
σL
0
, ?Lc? =
0
σR
.(7)
We note that the triplet vacuum expectation values v∆Rand vδRrepresent the scale of
SU(2)Rbreaking and are, according to the lower bounds [31] on heavy W- and Z-boson
masses, in the range v∆R,vδR>∼1 TeV. These represent a new scale, the right-handed
breaking scale. We note that κ′
bosons and to the flavour changing neutral currents, and are usually assumed to vanish.
Furthermore, since the electroweak ρ parameter is close to unity, ρ = 0.9998±0.0008 [31],
the triplet vacuum expectation values v∆Land vδLmust be small.
1and κ′
2contribute to the mixing of the charged gauge
The Yukawa coupling hχL is proportional to the neutrino Dirac mass mD. The light
neutrino mass in the see-saw mechanism is given by ∼ m2
the Majorana mass. The magnitude of hχLis not accurately determined given the present
upper limit on the light neutrino masses. On the other hand the Yukawa coupling hΦLis
proportional to the electron mass and is, thus, small.
D/mM, where mM= h∆Rv∆Ris
In the minimal model described above, parity cannot be spontaneously broken at the
renormalizable level without spontaneous breaking of R-parity. This may be cured by
adding more fields to the theory. In [24,32] it was suggested that a parity-odd singlet,
coupled appropriately to triplet fields, be introduced so as to ensure proper symmetry
breaking. This leads to a set of degenerate minima connected by a flat direction, all of
them breaking parity. When soft SUSY breaking terms are switched on, the degeneracy
is lifted, but the global minimum that results breaks U(1)em. Because of the flat direction
connecting the minima, there is no hope that the fields remain in the phenomenologically
acceptable vacuum, which rolls down to global minimum after SUSY is softly broken. The
only option that is left is to have a relatively low SU(2)Rbreaking scale, with sponta-
neously broken R-parity (?νc? ≡ σRis non-zero). We note that present experiments allow
for a low SU(2)Rbreaking scale.
There is an alternative to the minimal left-right supersymmetric model which involves the
addition of a couple of triplet fields, ΩL(1,3,1,0) and ΩR(1,1,3,0), instead of a singlet
Higgs superfield, to the minimal model [28]. In these extended models the breaking of
SU(2)Ris achieved in two stages. In the first stage the gauge group SU(2)L× SU(2)R×
U(1)B−Lis broken to an intermediate symmetry group SU(2)L×U(1)R×U(1)B−L, and at
the second stage U(1)R×U(1)B−Lis broken to U(1)Y at a lower scale. In this theory there
is only one parity-breaking minimum, in contrast to the minimal model, which respects
the electromagnetic gauge invariance. The superpotential for this class of models obtains
additional terms involving the triplet fields ΩLand ΩR:
6
Page 7
WΩ=Wmin+1
2µΩLTrΩ2
?
?
L+1
2µΩRTrΩ2
R+ aLTr∆LΩLδL+ aRTr∆RΩRδR
+TrΩL
αLΦiτ2χTiτ2+ αL′Φiτ2ΦTiτ2+ αL′′χiτ2χTiτ2
?
?
+TrΩR
αRiτ2ΦTiτ2χ + αR′iτ2ΦTiτ2Φ + αR′′iτ2χTiτ2χ,(8)
where Wminis the superpotential (5) of the minimal left-right model. In these models the
see-saw mechanism takes its canonical form with mν≃ m2
trino Dirac mass. In this case the low-energy effective theory is the MSSM with unbroken
R-parity, and contains besides the usual MSSM states, a triplet of Higgs scalars much
lighter than the B − L breaking scale.
D/MBL, where mDis the neu-
A second option is to add non-renormalizable terms to the Lagrangian of the minimal
left-right supersymmetric model, while retaining the minimal Higgs content [21,33,34]
The superpotential for this class of models can be written as
WNR=Wmin+aL
2M(Tr∆LδL)2+aR
L+bR
2M(Tr∆RδR)2+
c
MTr∆LδLTr∆RδR
+bL
2MTr∆2
+λijkl
M
+αijR
MTr∆RδRiτ2ΦT
+ηij
MTrΦi∆Riτ2ΦT
+kql
LTrδ2
2MTr∆2
RTrδ2
R+
1
M
?
d1Tr∆2
LTrδ2
R+ d2Trδ2
LTr∆2
R
?
Triτ2ΦT
iiτ2ΦjTriτ2ΦT
kiτ2Φl+αijL
MTr∆LδLΦiiτ2ΦT
jiτ2
iiτ2Φj+
1
MTrτ2ΦT
iτ2Φj[βijLTr∆LδL+ βijRTr∆RδR]
jiτ2δL+ηij
MTrΦiδRiτ2ΦT
MQTiτ2QQcTiτ2Qc+kll
jiτ2∆L
MQTiτ2LQcTiτ2Lc+kqq
+1
M[jLQTiτ2QQTiτ2L + jRQcTiτ2QcQcTiτ2Lc].
MLTiτ2LLcTiτ2Lc
(9)
It has been shown that the addition of non-renormalizable terms suppressed by a high scale
such as Planck mass, M ∼ MPlanck∼ 1019GeV, with the minimal field content ensures the
correct pattern of symmetry breaking in the supersymmetric left-right model. In particular
the scale of parity breakdown is predicted to be in the intermediate region MR>∼1010−
1011GeV, and R-parity remains exact. This theory contains singly charged and doubly
charged Higgs scalars with a mass of order M2
accessible. However, what is different is the nature of see-saw mechanism. Whereas in
the renormalizable version the see-saw mechanism takes its canonical form, in the non-
renormalizable case it takes a form similar to what occurs in the non-supersymmetric
left-right models, with the neutrino mass depending on the unknown parameters of the
Higgs potential. This in general leads to different neutrino mass spectra, which can be
experimentally distinguished.
R/MPlanck, which may be experimentally
7
Page 8
3The tree-level upper bound on the lightest Higgs mass
Given the fact that the Higgs sector of SLRM models contain a large number of Higgs
multiplets, and the VEVs of some of the Higgs fields involve possibly large mass scales com-
pared to the electroweak and SUSY breaking scales, it is important to ask what is the mass
of the lightest Higgs boson in these models. The upper bound on the lightest Higgs boson
mass in the minimal model was derived in [30] in the limit when κ′
using the fact that for any Hermitean matrix the smallest eigenvalue must be smaller than
that of its upper left corner 2 × 2 submatrix. In the basis in which the first two indices
correspond to (Φ0
1,κ′
2,σL,v∆L,vδL→ 0,
1,χ0
2), we find for matrix elements m2
11,m2
22,m2
12(see Appendix A)
m2
11=−m2
Φχ
κ2
κ1
κ1
κ2
+1
2(g2
L+ g2
R)κ2
1,
m2
22=−m2
Φχ
+1
2(g2
L+ g2
R)κ2
2,
m2
12=m2
Φχ−1
2(g2
L+ g2
R)κ1κ2.(10)
It follows that the upper bound on the lightest Higgs boson mass in the minimal super-
symmetric left-right model can be written as [30]:
m2
h≤1
2(g2
L+ g2
R)(κ2
1+ κ2
2)cos22β =
?
1 +g2
R
g2
L
?
m2
WLcos22β, (11)
where tanβ = κ2/κ1. The upper bound (11) is not only independent of the supersymmetry
breaking parameters (as in the case of supersymmetric models based on SU(2)L×U(1)Y),
but it is also independent of the SU(2)Rbreaking scale, which, a priori, can be large. The
upper bound is controlled by the weak scale vacuum expectation value, κ2
dimensionless gauge couplings (gLand gR) only. Since the former is essentially fixed by
the electroweak scale, the gauge couplings gLand gRdetermine the bound.
1+ κ2
2, and the
We will see below that even when κ′
the lightest Higgs mass at the tree level does not depend on either the right-handed
breaking scale or the SUSY breaking scale. A general method to find an upper limit
for the lightest Higgs mass in models based on SU(2)L× U(1)Y and maximally quartic
potentials was presented in [35]. We will apply this method to the case of SLRMs, with
possible nonrenormalizable terms, in an appropriate manner.
1,κ′
2,σL,v∆L,vδLare non-zero, the upper bound on
Consider a set of scalar fields Φjtransforming under SU(2)L, and define a discrete trans-
formation P : Φj→ (−1)2TjΦjof these fields, where 2Tj+ 1 is the dimension of SU(2)L
representation3. By setting the P-even fields to their VEVs, a normalized field can be
3One could as well choose any other spontaneously broken U(1) [35], but P provides the best
limit in the case of SLRM.
8
Page 9
defined in the direction of SU(2)Lbreaking, φ =
the fields orthogonal to φ have zero VEVs. Since the original lagrangian is invariant un-
der the transformatiom P, the potential must be invariant under P, so that the potential
contains only even powers of φ,
1
v0
?
oddviΦi, where v2
0=?
iv2
i, and all
V (φ) = V (0) −1
2m2φ2+1
8λφφ4+1
6Aφ6+ ....(12)
We take into account only the leading nonrenormalizable terms and thus φ6is the largest
power in the potential V (φ). If φ were the mass eigenstate, then by using the minimization
condition the Higgs mass would be λφv2
provides an upper bound on the mass of the lightest Higgs boson,
0+ 4Av4
0. In the general case, this expression
m2
h≤ λφv2
0+ 4Av4
0.(13)
Since only the SU(2)Ldoublet fields are relevant for obtaining this bound, adding extra
singlets or triplets in the model has no effect on the bound. Thus, there are three separate
cases to be considered for deriving an upper bound on the mass of the lightest Higgs boson
in the models discussed in the last section, namely: (A) R-parity is broken (sneutrinos get
VEVs), (B) R-parity is conserved because there are additional triplets, and (C) R-parity
is conserved because there are nonrenormalizable terms.
Let us first consider the situation in the minimal model, case (A). We define a new neutral
scalar field in the direction of the breaking in the SU(2)Ldoublet space:
Φ0=1
v
?
κ1ReΦ0
1+ κ′
1ReΦ0
2+ κ′
2Reχ0
1+ κ2Reχ0
2+ σLRe ˜ ν
?
,(14)
where
v2= κ2
1+ κ′2
1+ κ2
2+ κ′2
2+ σ2
L.(15)
All the SU(2)L doublet fields which are orthogonal to Φ0have vanishing VEVs. We
calculate next the quartic term (Φ0)4in the potential. It has contributions from both the
F-terms and the D-terms, and we find that the upper bound on the mass of the lightest
Higgs boson in the minimal SLRM is
m2
h≤
1
2v2
?
g2
L(ω2
κ+ σ2
L)2+ g2
Rω4
κ+ g2
B−Lσ4
L+ 8(hΦLκ′
1+ hχLκ2)2σ2
L+ 8h2
∆Lσ4
L
?
, (16)
where
ω2
κ= κ2
1− κ2
2− κ′2
1+ κ′2
2.(17)
9
Page 10
As is evident, the upper bound (16) is independent of SUSY and right-handed breaking
scales, and depends only on the dimensionless gauge and Yukawa couplings, and vacuum
expectation values which are determined by the weak scale:
m2
WL=1
2g2
L
?
κ2
1+ κ2
2+ κ′2
1+ κ′2
2+ σ2
L+ 2v2
∆L+ 2v2
δL
?
+ O
?κ
′2m2
m2
WR
WL
?
. (18)
The triplet VEVs v∆Land vδLmust be small in order to maintain ρ ≃ 1. In the limit
when κ′
1,κ′
2,σL,v∆L,vδL→ 0, the bound (16) reduces to the upper bound (11).
It is obvious that the addition of extra triplets does not change this bound. Thus, the
bound for the case (B), the SLRM with additional triplets to ensure that R-parity is not
spontaneously broken, can be obtained from (16) by taking the limit σL→ 0.
The total number of nonrenormalizable terms in case (C) is rather large. All the coefficients
in nonrenormalizable terms are proportional to inverse powers of a large scale. Thus the
largest contribution comes from those terms which have the smallest number of large
scales and the largest number of potentially large VEVs from SU(2)Rtriplets. We recall
that the terms of the form (Φ0)4and (Φ0)6in the potential are needed to determine
the contributions to the mass bound. The nonrenormalizable terms have no effect on the
contribution from D-terms. The leading terms in the superpotential which can give a
(Φ0)4and (Φ0)6type F-term contribution are of the type
WNR= ATr(iτ2ΦTiτ2χ)Tr(∆RδR) + BTr(iτ2ΦTiτ2χ)2, (19)
i.e. one term with two and another with four bidoublet fields. Here A, B ∼ 1/MPlanck.
With SU(2)Lsinglets fixed to their VEVs, the corresponding F-terms are
VNR= O(ABv∆RvδR)(Φ0)4+ O(B2)(Φ0)6.
The contribution to the Higgs mass bound from these nonrenormalizable terms is
(20)
O(v2
R/M2
Planck)?Φ0?2+ O(1/M2
Planck)?Φ0?4. (21)
If the VEV vR∼ 1010GeV in these models, the contribution is numerically negligible.
Therefore the upper bound for this class of models is essentially the same as in the case
(B).
4Radiative corrections
Since it is known that the radiative corrections to the lightest Higgs mass are significant
in the MSSM, as well as its extensions based on the SU(2)L× U(1)Y, it is important
10
Page 11
to consider the radiative corrections to the upper bound on the lightest Higgs boson
mass obtained in the previous section. In this section we discuss the one-loop radiative
corrections to the upper bound on the lightest Higgs mass in the minimal supersymmetric
left-right model, which was obtained in last section. We shall use the method of one-loop
effective potential [36] for the calculation of radiative corrections, where the effective
potential may be expressed as the sum of the tree-level potential plus a correction coming
from the sum of one-loop diagrams with external lines having zero momenta,
V1−loop= Vtree+ ∆V1, (22)
where Vtreeis the tree level potential (6) evaluated at the appropriate running scale Q,
and ∆V1is the one loop correction given by
∆V1=
1
64π2
?
i
(−1)2Ji(2Ji+ 1)m4
i
?
lnm2
i
Q2−3
2
?
, (23)
where miis the field dependent mass eigenvalue of the ith particle of spin Ji. The dominant
contribution to (23) comes from top-stop (t−˜t) system. However, under certain conditions
the contribution of bottom-sbottom (b−˜b) can be nonneglible. We shall include both these
contributions in our calculations of the radiative corrections.
In order to evaluate the contributions of top-stop and bottom-sbottom to (23), we need
the stop and sbottom mass matrices for the SLRM. From (A.1), (A.2) and (A.3), it is
straightforward to calculate squark mass matrices [27]. Ignoring the interfamily mixing,
the part of the potential containing the stop and sbottom mass terms can be written as
Vsquark=
?
U∗
LU∗
R
?
˜ MU
UL
UR
+
?
D∗
LD∗
R
?
˜ MD
DL
DR
,(24)
where the mass matrix elements for the stop are
(˜ MU)U∗
LUL= ˜ m2
Q+ m2
u+1
4g2
Lω2
κ−1
6g2
B−Lω2
R,
(˜ MU)U∗
RUL=hΦQAΦQκ′
+(hφLhφQ+ hχLhχQ)σLσR
?
(˜ MU)U∗
1+ hχQAχQκ2− µ1(hφQκ′
2+ hχQκ1) − 2hφQµ′
1κ1− 2hχQµ′′
1κ′
2
= (˜ MU)U∗
LUR
u+1
?∗,
RUR=m2
Qc + m2
4g2
R(ω2
κ− 2ω2
R) +1
6g2
B−Lω2
R,(25)
while for the sbottom these are
11
Page 12
(˜ MD)D∗
LDL=m2
Q+ m2
d−1
4g2
Lω2
κ−1
6g2
B−Lω2
2+ µ1(hφQκ2+ hχQκ′
R,
(˜ MD)D∗
RDL=−hΦQAΦQκ1− hχQAχQκ′
= (˜ MD)D∗
LDR
d−1
1) + 2hφQµ′
1κ′
1+ 2hχQµ′′
1κ2
?
?∗,
(˜ MD)D∗
RDR=m2
Qc + m2
4g2
R(ω2
κ− 2ω2
R) +1
6g2
B−Lω2
R, (26)
where top and sbottom squared masses are given by (hχQκ2)2and (hΦQκ1)2, respectively,
m2
Q, m2
Qc, AΦQand AχQare soft supersymmetry breaking parameters (see eq. (A.3)), and
ω2
R= v2
∆R− v2
δR−1
2σ2
R, ω2
κ= κ2
1+ κ′
2
2− κ2
2− κ′
1
2. (27)
In order that SU(3)C×U(1)emis unbroken, none of the physical squared masses of squarks
can be negative. Necessarily then all the diagonal elements of the squark mass matrices
should be non-negative. Combining the diagonal elements of the stop and sbottom mass
matrices leads to the inequality
m2
Q+ m2
Qc ≥ |1
2g2
Rω2
R| =1
2g2
R|v2
∆R− v2
δR−1
2σ2
R|.(28)
where we have ignored terms which are of the order of the weak scale or less.
The eigenvalues m2
(m2
˜t1,2,m2
˜b2)
˜b1,2of the stop and sbottom mass squared matrices are given by
˜t1> m2
˜t2,m2
˜b1> m2
m2
m2
˜t1,2= m2
˜b1,2= m2
˜t± ∆2
˜b± ∆2
˜t,
˜b,(29)
where
m2
˜t=1
2
?
??
m2
Q+ m2
Qc + 2m2
t+1
4(g2
L+ g2
R)ω2
κ−1
2g2
R−1
Rω2
R
?
,
∆2
˜t=1
2
+4[hχQAtκ2− hχQ˜ µtκ1]2?1
m2
Q− m2
Qc +1
4(g2
L− g2
R)ω2
κ+1
2g2
Rω2
3g2
B−Lω2
R
?2
2,(30)
and
m2
˜b=1
2
?
m2
Q+ m2
Qc + 2m2
b−1
4(g2
L+ g2
R)ω2
κ+1
2g2
Rω2
R
?
,
12
Page 13
∆2
˜b=1
2
+4[hΦQAbκ1− hΦQ˜ µbκ2]2?1
Here we have defined
??
m2
Q− m2
Qc −1
4(g2
L− g2
R)ω2
κ−1
2g2
Rω2
R−1
3g2
B−Lω2
R
?2
2. (31)
˜ µt≡ µ1+ 2µ′
1
mb
mttanβ, ˜ µb≡ µ1+ 2µ′′
1
mt
mbcotβ, At≡ AχQ, Ab≡ AφQ. (32)
Using eqs. (29), (30) and (31) in (23), we have calculated the radiatively-corrected ex-
pressions for the matrix elements of the upper left corner 2 × 2 submatrix of the 10 × 10
CP-even Higgs mass matrix. After imposing the appropriate one-loop minimization con-
ditions, we find the following form for the radiatively corrected upper left corner 2 × 2
submatrix of CP-even Higgs mass matrix:
1
2(g2
L+ g2
R)κ2
1
−1
2(g2
L+ g2
R)κ1κ2
−1
2(g2
L+ g2
R)κ1κ2
1
2(g2
L+ g2
R)κ2
2
+
tanβ −1
−1 cotβ
?∆
2
?
+
3g2
L
16π2m2
WL
∆11∆12
∆12∆22
(33)
,
where
∆=
?
−2m2
Φχ+
3
32π2
?
g2
L
sin2β
m2
m2
t
W
f(m2
?
At˜ µt
f(m2
˜t1) − f(m2
m2
˜t2)
˜t1− m2
˜b2)
,
˜t2
+
3
32π2
?
g2
L
cos2β
m2
m2
b
W
?
Ab˜ µb
˜b1) − f(m2
m2
˜b1− m2
˜b2
(34)
f(x2) = 2x2(ln(x2/Q2) − 1),(35)
and
∆11=
m4
b
cos2β
ln
m2
A2
˜b1m2
m4
b
˜b2
+2Ab(Ab− ˜ µbtanβ)
b(Ab− ˜ µbtanβ)2
?
m2
˜b1− m2
˜b2
ln
m2
(At− ˜ µtcotβ)2˜ µ2
(m2
˜b1
m2
˜b2
+
m4
b
cos2β
m2
˜b1− m2
˜b2
?2
g(m2
˜b1,m2
˜b2) +
m4
sin2β
t
t
˜t1− m2
˜t2)2
g(m2
˜t1,m2
˜t2),
(36)
13
Page 14
∆22=
m4
sin2β
t
?
ln
?m2
˜t1m2
m4
t
˜t2
?
+2At(At− ˜ µtcotβ)
m2
˜t1− m2
˜t2
ln
?m2
m2
(Ab− ˜ µbtanβ)2˜ µ2
?
˜t1
˜t2
??
+
m4
sin2β
t
A2
t(At− ˜ µtcotβ)2
?
m2
˜t1− m2
˜t2
?2
g(m2
˜t1,m2
˜t2) +
m4
b
cos2β
b
m2
˜b1− m2
˜b2
?2
g(m2
˜b1,m2
˜b2),
(37)
∆12=
m4
sin2β
t
(At− ˜ µtcotβ)(−˜ µt)
m2
˜t1− m2
˜t2
×
?
ln
?m2
m2
˜t1
˜t2
?
+At(At− ˜ µtcotβ)
m2
˜t1− m2
˜t2
g(m2
˜t1,m2
˜t2)
?
+
m4
b
cos2β
(Ab− ˜ µbtanβ)(−˜ µb)
m2
˜b1− m2
˜b2
×
ln
m2
˜b1
m2
˜b2
+Ab(Ab− ˜ µbtanβ)
m2
˜b1− m2
˜b2
g(m2
˜b1,m2
˜b2)
,
(38)
with
g(m2
1,m2
2) = 2 −m2
1+ m2
m2
2
1− m2
2
ln
?m2
m2
1
2
?
.(39)
We have neglected D-terms in the squark masses, because these are small, and, since we
are including only the quark-squark contributions to ∆V1, in order to gain approximate
independence of the renormalization scale Q (see also the inequality (28)).
Using eqs. (33) and (34), we obtain the one-loop radiatively corrected upper bound on
the lightest Higgs boson mass in the SLRM:
m2
h≤1
2
?
(g2
L+ g2
R)
?
κ2
1+ κ2
2
?
cos22β +
3g2
L
8π2m2
WL
?
∆11cos2β + ∆22sin2β + ∆12sin2β
??
(40)
For tanβ<∼20, one can neglect the b-quark contribution in the radiative corrections.
Then, in the approximation [37,38]
|m2
˜t1− m2
˜t2| ≪ |m2
˜t1+ m2
˜t2|,(41)
the upper bound (40) on the lightest Higgs mass reduces to
m2
h≤1
2
+3g2
8π2m2
?
(g2
L+ g2
R)
?
ln(m2
κ2
1+ κ2
2
?
˜t2
cos22β
Lm4
t
WL
?
˜t1m2
m4
t
) + 2
˜A2
M2
t
s
(1 −
˜A2
t
12M2
s
) − 8µ′′4
1
3M4
s
??
(42)
14
Page 15
Fig. 1. The upper bound on the radiatively corrected mass of the lightest neutral Higgs boson
for two different values of tanβ. The right-handed scale MRis indicated in the Figure. The bi-
and trilinear soft supersymmetry breaking parameters are 1 TeV (solid line) and 10 TeV (dashed
line). Supersymmetric Higgs mixing parameters are assumed to vanish, and mtop= 175 GeV.
where˜At= At−µ1cotβ, and Msis the supersymmetry breaking scale (2M2
In this limit the upper bound eq. (42) on the lightest Higgs mass in the supersymmetric
left-right model differs in form from the corresponding MSSM upper bound only because
of µ′′
|˜At| = (√6)Ms
s= m2
˜t1+m2
˜t2).
1being nonzero. The upper bound is maximised by
(43)
for a given value of µ′′
1.
The radiatively corrected upper bound (40) on the mass of the lightest Higgs boson is
plotted in Fig.1 as a function of the large scale Λ up to which the supersymmetric left-right
model remains perturbative. The upper bound comes from the requirement that all the
gauge couplings of the SLRM remain perturbative below the scale Λ. We have taken into
account the dominant radiative corrections coming from the quark and squark loops in our
calculations. In Fig.1 we have taken two values of tanβ = 2 and tanβ = 20. In the figure
the upper bound is shown for two different values of the SU(2)Rbreaking scale, MR= 10
TeV and MR= 1010GeV, respectively, and for two values of soft supersymmetry breaking
mass parameter, Ms= 1 TeV and Ms= 10 TeV. It is seen from this figure that if the
difference between the SU(2)Rbreaking scale and the large scale Λ is more than two orders
of magnitude, the radiatively corrected upper bound on the mass of the lightest Higgs
boson remains below 250 GeV. For large values of Λ the upper bound is below 200 GeV.
The upper bound increases with increasing MRand with increasing soft supersymmetry
breaking parameters. It is considerably larger than the corresponding upper bound in the
MSSM.
15
Download full-text