Charting generalized supersoft supersymmetry
Revised: May
Charting generalized supersoft supersymmetry
Sabyasachi Chakraborty 0 1 4
Adam Martin 0 1 2
Tuhin S. Roy 0 1 3 4
0 Los Alamos , NM 87545 , U.S.A
1 Mumbai 400005 , India
2 Department of Physics, University of Notre Dame
3 Theory Division T2, Los Alamos National laboratory
4 Department of Theoretical Physics, Tata Institute of Fundamental Research
Without any shred of evidence for new physics from LHC, the last hiding spots of natural electroweak supersymmetry seem to lie either in compressed spectra or in spectra where scalars are suppressed with respect to the gauginos. While in the MSSM (or in any theory where supersymmetry is broken by the F vev of a chiral spurion), a hierarchy between scalar and gaugino masses requires special constructions, it is automatic in scenarios where supersymmetry is broken by Dvev of a real spurion. In the latter framework, gaugino mediated contributions to scalar soft masses are nite (loop suppressed but not logenhanced), a feature often referred to as supersoftness". Though phenomenologically attractive, pure supersoft models su er from the problem, potential colorbreaking minima, large T parameter, etc. These problems can be overcome without sacri cing the model's virtues by departing from pure supersoftness and including type e ective operators at the messenger scale, that use the same Dvev, a framework known as generalized supersoft supersymmetry. The main purpose of this paper is to point out that the new operators also solve the last remaining issue associated with supersoft spectra, namely that a right handed (RH) slepton is predicted to be the lightest superpartner, rendering the setup cosmologically unfeasible. In particular, we show that the operators in generalized supersoft generate a new source for scalar masses, which can raise the RHslepton mass above bino due to corrections from renormalisation group evolutions (RGEs). In fact, a mild tuning can open up the binoRH slepton coannihilation regime for a thermal dark matter. We derive the full set of RGEs required to determine the spectrum at low energies. Beginning with input conditions at a high scale, we show that completely viable spectra
can be achieved

2
3
1 Introduction
The framework 4
Solutions
4.1
Analytical solutions
Renormalization group equations
3.1
A toy model
3.1.1
3.1.2
Detour: consistency check
Towards the full Lagrangian
3.2
Renormalization group equations in the full model
4.1.1
4.1.2
4.1.3
4.1.4
First and second generation sfermions
Colored sector
Electroweak sector
Electroweak symmetry breaking
5
Numerical results
5.1
5.2
5.3
5.4
5.5
5.6
Righthanded slepton masses
Dark matter constraints
Higgsino masses
Dark matter direct detection
Higgs mass and new superpotential terms
Benchmark points
6
Summary and conclusion
A Neutralino and chargino mass matrices
B Switching on the Yukawa coupling C
Higgs mass matrix
D RGEs with
S, T
E Tadpole issue
{ 1 {
Introduction
Supersymmetry at the electroweak scale remains one of the most celebrated solutions to the
hierarchy problem of the Standard Model (SM) to date. However, the LHC experiments
have not yet found any signi cant excesses in their pursuit of superpartners. After the
most recent run, containing an integrated luminosity of nearly 36 fb 1 at a center of mass
energy of 13 TeV, the constraints on the superpartner masses are becoming quite stringent.
For example, the lack of events in the jets plus missing energy search has excluded
degenerate squark/gluino scenarios up to masses of 2 TeV, and squarks (including the stop)
are now ruled out up to a TeV or so provided they can decay to comparatively lighter
neutralinos [1{3]. However, it should be emphasized that these experimental exclusions
are drawn assuming simpli ed scenarios within the paradigm of the Minimal
Supersymmetric Standard Model (MSSM), and are subject to change with more involved topologies.
Within the MSSM, the increased top squark and gluino masses have profound implications
for naturalness, as both scales feed into the soft mass of the Higgs via renormalization,
dragging it upwards and requiring
netuned cancellations among parameters.1 In short,
the promise of a natural MSSM explanation for the weak scale is fading, even if we neglect
other generic MSSM problems such as rapid proton decay from dimension
ve operators,
excessive avor changing neutral currents, and additional CP violating phases.
While the MSSM is the most well studied framework of weak scale supersymmetry,
it does not provide all aspects of a general model of electroweak scale supersymmetry.
Speci cally, the MSSM is an infrared e ective supersymmetric framework with
superpartner masses sourced by the supersymmetrybreaking vacuum expectation value (vev) of
the F component of a chiral spurion. Alternative theories, where supersymmetry
breaking is sourced from Dvev of a real spurion, are qualitatively distinct and bring several
new niceties:
The primary operators sourced by the Dvev generate Dirac masses for gauginos.
This requires one to extend the theory to include new chiral super elds in the adjoint
representations, as the Dvev mass terms pair up the gauginos with the fermion
components of the appropriate adjoint super eld, e.g. gluino with color octet fermion,
wino with SU(2) triplet fermion, etc. In the absence of any other mass terms,
gauginos are purely Dirac. This can be contrasted to the MSSM, where gauginos are
purely Majorana.
It turns out that one can not write similar Dvev sourced operators that can
generate scalar masses at the messenger scale. Further, because of the speci c structure
of gaugino masses, the gauginomediated contribution to scaler masses do not get
any logenhancement and remain
nite. This property that the gaugino generated
sfermion masses are insensitive to even logarithmic dependence of the ultraviolet
(UV) scale, is often referred to as `supersoft' [15]  as opposed to the MSSM, where
1Exceptions exist within MSSM where scalar masses can somewhat decouple from gaugino masses.
Examples include models with double protection [4{10], scalar sequestering [11{13], twin SUSY [14] etc.
{ 2 {
a logarithmic dependence remains (`soft'). As a result, the scalar masses at the weak
scale are loop suppressed with respect to the gaugino masses.
The resultant mild split in the squarkgluinos masses are of utmost importance,
especially when one calculates the bound in the jets + missing energy (MET) channel.
Heavier gluinos imply suppressed gluino pair and squarkgluino pair production.
Additionally, pair production of samechirality squarks (pp ! q~Lq~L) is not possible
if gluinos are pure Dirac, as it requires a chirality
ipping Majorana mass
insertion. The reduced crosssections relax the bound on the rst two generation squark
masses [16, 17].
As the gaugino are Dirac particles, the theory (minus the Higgs sector) respects a
continuous U(1)R symmetry. This symmetry suppresses supersymmetric
contributions to avor changing neutral currents and electric dipole moments [15, 18]. More
model building can elevate the U(1)R symmetry to be a symmetry of the full
Lagrangian, which ameliorates many of the avor and CP di culties of the prototypical
MSSM [19, 20].
While the above features are attractive, the supersoft framework does have its share
of issues:
The ` ' problem is severe for supersoft models. A nonzero
term is essential in
a supersymmetry framework as it controls the mass of the charginos. The LEP
experiments have placed model independent bounds on light charged states, e.g.,
m ~+ > 97 GeV at 95% C.L [21], therefore a small
1
elegant solution  at least in the framework of the MSSM  is given by the
GuidiceMasiero mechanism [22], however this mechanism requires an F type spurion and
thus will not work for supersoft models. Another solution was proposed in ref. [15],
where the conformal compensator eld generates the mass for the higgsinos. While
viable, this mechanism relies on a conspiracy among the supersymmetry breaking
scale and the Planck scale. A third solution is to include a gauge singlet S in the
theory and the superpotential interaction SHuHd (e.g., NMSSM [23]) which becomes
an e ective term once the scalar component of S acquires a vev. Supersoft models
automatically include the gauge singlet required for this approach as the Dirac partner
of the bino. Gauge singlets do require care, however, and may lead to power law UV
term is catastrophic. The most
sensitivity [24].
The infamous `lemontwist' operators [15, 25, 26] can break color or generate a too
large a T parameter. Upon replacing the real spurion by its Dvev, one nds mass
squared of the adjoint scalars to be linear in the coupling constants of these operator.
In a large part of the parameter space, the imaginary components of these adjoints
may acquire vevs leading to dangerous charge and color breaking vacuum. Previous
solutions to this problem require deviation from the supersoftness, resulting in a low
energy theory with Majorana gauginos or extended messenger sectors [26, 27]. A
viable solution of this problem was put forward in terms of Goldstone gauginos [28],
{ 3 {
HJEP05(218)76
where the right handed gaugino is a pseudoGoldstone eld originating from the
spontaneous breaking of an anomalous avor symmetry.
In the framework of pure supersoft supersymmetry breaking, D atness is a natural
direction, leading to vanishing D terms in the potential [15]. In the generic MSSM,
the Dterm contribution leads to a tree level mass of the Higgs boson as MZ cos 2 .
Given the discovery of a Higgs with mass close to 125 GeV, a vanishing Dterm is
not a good starting point, as one needs to depend on large quartic corrections at
the tree level (NMSSM like) or at the oneloop level to achieve the correct value.
Most Rsymmetric models with Dirac gauginos therefore include both F  and
Dtype breaking to enhance the Higgs quartic term, which consequently increases the
Higgs mass. A partial list of such frameworks can be found in the literature [29{79].
Another problematic issue of the supersoft scenario is dark matter (DM). If the Dvev
spurion remains the only source of superpartner masses and the mediation scale is
high, then the lightest supersymmetric particle (LSP) is a right handed (RH) slepton,
leading to contradictions with cosmological observations. Lowering the mediation
scale, the gravitino becomes the LSP, but this brings its own issues: i.) a lower
mediation scale means the conformal compensator mechanism for generating a
fails, and ii.) the gravitino LSP scenario is highly constrained because of the long
lived charged particle searches [
80
] or  if the gravitino mass is less than a few keV
 constraints arise from prompt multijet or multilepton nal states [81, 82].
A recently proposed framework, `generalized supersoft supersymmetry' (GSS) [83] is
free of many of these issues yet maintains key phenomenological features of pure supersoft
models, such as nite gaugino mediated contribution to scalar soft masses. Crucially, GSS
ameliorates the above issues without introducing additional matter or additional sources
of supersymmetry breaking. Instead, in GSS one adds a set of new operators formed
from matter super elds and the same Dtype spurion used to give gauginos mass. When
written with chiral adjoints, the new operators give supersymmetric masses to the adjoints
and allow one to avoid colorbreaking and T parameter issues. When written using Higgses,
one nds two parameters (namely, u and d) are generated, both of which scale with
the supersymmetry breaking Dvev. If u =
d, these operators can be replaced by the
supersymmetric term with
=
u =
d, which solves the
problem in the same spirit
as that of GuidiceMasiero mechanism.
However when
d, supersoftness is lost
as the hyperchage Dterm gets a vev and log sensitivity (i.e., soft) creeps back into the
scalar masses.
The goal of this paper is to esh out the spectrum of generalized supersoft theories
and to determine the parameter regions where they can be viable. While at rst glance
the loss of supersoftness when
u 6= d appears to be a aw in the idea, we will show that
it turns out to be a feature that can be utilized to resolve the DM issue that plagues pure
supersoft setups. It turns out that this theory with
usual supersoft picture with a supersymmetric ` ' term
term with DY /
, and nonholomorphic trilinears (such as
2
u
d 6= 0 is equivalent to the
= u+ d , a hypercharge
D
hyuq~u~, etc.). In particular
{ 4 {
the nonzero Dterms provide a boost to all scalar soft masses, while Dterm operators
involving the adjoint super elds allow more exibility in the gaugino masses. These e ects
combine to open up a swath of parameter space where the theory satis es the observed
Higgs mass and achieves the correct thermal relic dark matter abundance through
binoslepton coannihilation, all in addition to the usual supersoft phenomenological bene ts.
Additionally, whenever
d, nonstandard supersymmetry breaking operators arise
which give subtle e ects in the running of soft parameters. To map out the e ects of
the GSS UV inputs, we derive the full set of renormalization group equations (RGEs)
and provide full numerical results for important outputs such as the Higgs mass and relic
abundance. It turns out that RGEs with these unconventional operators are quite subtle
and previous attempts [84, 85] miss several important e ects. To give a more intuitive
understandings of electroweak parameters in terms of high scale inputs, we also derive
analytical solutions to the RGE under simplifying assumptions.
The rest of this paper is organized as follows: in section 2 we begin by laying out
the elds, scales, and operators we will consider. Next, in section 3 we examine how the
radiative generation of soft masses in our theory di ers from the conventional MSSM and
pure supersoft setups, using a toy model to illustrate the key features. In section 4 and 5,
we turn to the IR spectra. Many of the gross features of the spectrum can be understood
using simpli ed renormalization group equations that admit analytic solutions, however
we will resort to numerics when calculating quantities like the Higgs mass and dark matter
constraints. This section concludes with a few benchmark points that pass all tests. Finally,
a summary and discussion of our results is given in section 6 .
2
The framework
The skeleton of any weak scale model of supersymmetry typically consists of two sectors 
one containing the MSSM
elds, and the other, known as the hidden sector, responsible for
supersymmetry breaking, linked by a \messenger" sector.2 The messenger sector provides
a scale (namely,
mess) that characterizes all contact operators among the hidden and
visible elds. The supersymmetry breaking scale (or rather the supersymmetry breaking
vev, namely
int) is generated in the hidden sector. Note that the observables at the
electroweak scale are various superpartner masses, which are functions of both
int and
mess, as well as dimensionless numbers, that represent details of messenger mechanisms,
and renormalization e ects. In fact, renormalization can be quite tricky especially if the
mess is taken as the input scale. As shown in refs. [11, 87], the superpartner masses are
renormalized due to interactions of hidden sectors from the scale
mess to int, in addition to
the usual e ects from visible sector interactions. The lack of knowledge of the exact nature
of hidden sector dynamics can actually lead to order one uncertainty in the superpartner
masses as calculated using massmarket softwares such as SOFTSUSY [88], SPheno [89, 90],
SuSpect [91] etc, which completely neglect these e ects. Therefore, out the two natural
choices for setting the UV input scale of the theory ( mess or int) we choose int; below
this scale the hidden sector decouples and the superpartner masses renormalize only due to
2Examples of single site model do exist, see ref. [86].
{ 5 {
visible sector interactions. In other words, we use the superpartner masses and couplings
at the scale of renormalization r =
int to be the input conditions in order to evaluate
the spectrum at the electroweak scale. Directly using
int as the input scale naturally
raises questions regarding the nature of hidden sector interactions and the details of the
messengers that result in the used boundary conditions. We leave this discussion (and
further UV completion) for future work.
We reiterate that int is the scale at which contact operators are turned into masses
for visible sector
elds. In order to simplify, we begin with a hidden sector where the
supersymmetry breaking is captured in the vev of a single real eld R, with the
following conditions:
(2.1)
HJEP05(218)76
(2.2)
(2.3)
R
y = R
and
D
8
Here, D's are the usual chiral covariant derivative and D is the gauge auxiliary
Further, we rede ne R ! R= dmess, where d is the engineering dimension of the operator
elds.
at
mess. This rede nition sets the engineering dimension of R to be 0.
The visible sector in generalized supersoft supersymmetry extended beyond the MSSM
content to include three extra chiral super elds in the adjoint representation of the three SM
gauge groups, i.e. we include a color octet
3, a weak triplet
2, and a chargeneutral 1
.
Given this eld content, we will work with the minimal set of contact operators (between the
hidden and visible sector elds) capable of generating a viable IR spectrum. First, consider
the conventional supersoft operator [15] that gives rise to Dirac type gaugino masses.
2
1 Z d2 !i( int)
1
mess
D2D RW ;i i
!
MDi ( int) i i ;
where
MDi ( int) = !i( int)
D
mess
;
where Wi is the eldstrength chiral super elds for the ith SM gauge group (i is not
summed over, but spinor Lorentz indices
are summed) and !i are dimensionless coupling
constants. In eq. (2.2) we have suppressed any gauge group indices. By design, the
supersymmetry breaking vev of R picks out gauginos ( i) from Wi and the fermionic component
of i elds (denoted by i). Another operator often used in this context is known as the
`lemontwist' operators [15]:
1 Z d2 !qq0 2mess
1
2
where
D2DR
2
QQ0
!
bqq0 ( int) q 0q ;
bqq0 ( int) = !qq0 ( int) D2mess
;
2
where Q and Q0 represent visible sector chiral super elds with scalar components q; 0q,
and the coupling constant !qq0 is nonzero only if QQ0 is a gauge singlet. Examples of
such operators include
i2, and HuHd. These operators are problematic because they give
opposite sign mass to the real and imaginary parts of the scalars and thus can drive elds
tachyonic [15].
{ 6 {
In addition to eq. (2.2) and (2.3), we use the operators proposed in ref. [83] to generate
the
term.
4
1 Z d2 !Dqq0
1
mess
D2 (D RD Q) Q0
!
Dqq0 ( int)
2 q q0 + FQ Q0 ;
where
Dqq0 ( int) = !Dqq0 ( int)
D
mess
1
;
where FQ represents the auxiliary component of the eld Q. As in eq. (2.3), the coupling
constant !Dqq0 are nonzero only if the chiral operator QQ0 is a gauge singlet. The potential
generated after eliminating the auxiliary elds is given by
crucially, if Q and Q0 represent di erent elds, the equation above only gives mass for
Q0scalars. For nonzero @@WQ , eq. (2.4) also gives rise to trilinear scalar mass terms. As
an explicit example, substituting Q ! Hu and Q0 ! Hd, eq. (2.4) generates masses for
higgsinos, the scalar hd, and a nonholomorphic scalar trilinear operator hy q~u~. If we ip
d
Hu and Hd (namely, Q ! Hd and Q0 ! Hu), eq. (2.4) instead gives rise to masses for
the higgsinos and hu scalar, and operator hyuq~d~. Including both possibilities and denoting
the mass scales of the two operators by
d and
u, we nd the following terms in the
Lagrangian: 1 2
L
( d + d) H~uH~d
a result identical to what we would get from superpotential
HuHd. To make this
supersymmetric limit more apparent we can rewrite eq. (2.6) in terms of a new superpotential
term (namely, the term) and soft supersymmetry breaking mass terms:
W
Lsoft
HuHd
way, the supersymmetric limit corresponds to Lsoft ! 0. In addition, supersoft models are
well motivated as the avor and CP violating e ects are suppressed. In our scenario, we
are generating nonstandard trilinear scalar terms proportional to Yu and Yd. As a result,
minimal avor violation (MFV) would ensure that avor issues are under control.
As pointed out in [83], the operators in eq. (2.4) can also be used to solve potential
color breaking or a large T parameter issues in supersoft models by preventing scalars from
elds from becoming tachyonic. In detail, one needs to include the operators in eq. (2.4),
with QQ0 replaced by Tr( 2a). Repeating the same steps as shown after eq. (2.4), this
results in a mass for both the fermion and scalar components of
a. The Majorana mass
for a upsets the cancellation of the SU(2) and U(1) Dterms that occurs in pure supersoft
{ 7 {
(2.4)
(2.5)
scenarios, regenerating the treelevel Higgs quartic. Additionally, the scalar mass squared
from eq. (2.4) is positive for both the real and imaginary parts of the adjoint scalar, thus the
issue of tachyonic masses can be avoided provided contributions from eq. (2.4) are larger
than any `lemon twist' adjoint masses.
At this point, let us reiterate that we are still describing the theory at the scale of
renormalization r =
IR. In order to calculate the spectrum at the electroweak scale, these
operators must be renormalized below
int. Renormalization gives rise to new
counterterms and changes the couplings of various operators. In particular, one
nds that the
nice relationship between ; ~u;d; m~ hu;d , are modi ed and need to be treated independently.
We therefore, collect all the terms described above, include new operators to account for
generated counterterms as follows:
W = Yu QHuU + Yd QHdD + Y LHdE +
HuHd +
stands for q~i; u~i; d~i; ~li; e~i, and i denotes the family. Coe cients of many of these
operators are either zero or related to each other at the input scale int, as discussed before.
Before we proceed any further, let us completely specify the coe cients at the input
scale as well as establish our notation. Our UV inputs are:
The Dirac gaugino masses and supersymmetric masses for the adjoints (
elds), as
well as the supersymmetric
parameter:
MDa ( int) = M D0a ;
M a ( int) = M 0a ;
( int) =
0 ;
(2.9)
where a runs over the three gauge groups in the SM.
Soft masses for scalars
m~ 2 ( int) = 0;
m~ 2hu ( int) =
Another important quantity is the Hypercharge Dterm de ned as S
where
runs over all particles, and q represent corresponding hyperchages.
Therefore, we nd the boundary value of S to be
P
:
(2.10)
q m~ 2 ,
(2.11)
(2.12)
The initial conditions for the soft trilinear ~ operators are completely speci ed in
terms of the Higgs sector parameters:
S ( int) =
~u ( int) =
~
d= ( int) =
We devote the next section for deriving the renormalization group equations for the
coupling constants given in eq. (2.8).
{ 8 {
help us to disregard complexities due to the gaugino masses.
3
Renormalization group equations
With the exception of the nonholomorphic trilinear terms (namely the operators) in
eq. (2.8), RGEs of all other operators are well known and widely used. The e ect of the
operators, on the other hand, are extremely nontrivial and subtle. Early e orts in deriving
these missed several e ects [84, 85], and the RGEs of these operators and their e ects in
RGEs of other operators get more complicated in the presence of various Yukawa terms,
gauge couplings, and, in particular, the hypercharge Dterm. In order to get things correct,
we begin with a simple toy model consisting of:
HJEP05(218)76
1. A minimal set of degrees of freedoms;
2. Only a single Yukawa coupling;
3. Only global symmetries: this assumption allows us to disregard complexities due to
gaugino masses. Towards the end we will gauge one of the global U(1) in the model,
and study the e ects of operators, in the presence of Dterm.
RGEs of the full model of eq. (2.8) is given in section 3.1.2. Readers interested in seeing
the nal form can simply jump to section 3.1.2.
3.1
A toy model
1
2
In order to understand the nontrivial e ects of the operator in eq. (2.4) we construct a
toy model of the visible sector described by ve multiplets (namely, Q; U; D; Hu and Hd)
charged under the global groups G1
G2
U(1)X
U(1)R as described in table 1, along with
a single superpotential term (namely, Y QHuU ). The nongeneric nature of the holomorphic
superpotential allows us to get away with not writing any other marginal interaction.
With this particle content, the only invariant, holomorphic bilinear we can form is
HuHd. Consequently, there can be two operators of the form eq. (2.4) (namely, one term
with derivative acting on Hu, and the second with derivative on Hd). At the input scale int,
this give rise to masses for higgsinos, scalar Higgses as well as nonholomorphic scalar
trilinear.
r = int :
L
( u + d) H~uH~d
u 2 jhuj2
d 2 jhdj2
Y dhydq~u~ + h.c.: (3.1)
{ 9 {
soft terms:
with
In eq. (3.1) we have explicitly written down r = int in order to indicate the fact that the
relationships exhibited amongst masses of Higgs scalars, higgsinos, as well as the couplings
is only valid at the scale
int. In order to renormalize the theory below
int, additional
counterterms are needed, and this is where the full symmetry structure in table 1 is helpful
since it restricts the number of operators we need to consider considerably. Further, the
D multiplet does not have any interaction at all, and as a result no new counterterm
involving D needs to be written down (another way to see it is the fact that with the
current interactions, there is an additional U(1) symmetry under which only D is charged,
and this restricts more counterterms). Including all of the counterterms we need to take
into account (and that are invariant under all global symmetries mentioned above), the
Lagrangian in this toy model at any scale r
int, is given by:
W = Y QHuU ;
L
H~uH~d
Terms corresponding to traditional aterms, such as huq~u~ (hyuq~u~), or the b term break
U(1)R (U(1)X ) and therefore will not be generated via loops. Note that the massessquared
parameters in eq. (3.2), such as m2hu , refer to the full masses of the scalar elds. Following
the logic of eq. (2.7), this Lagrangian can be reexpressed as a superpotential piece plus
W = Y QHuU +
HuHd ;
Lsoft
m~q2~;u~;d~ = mq2~;u~;d~ : (3.4)
supersymmetric limit corresponds to: m2hu;d ! j j2 ;
!
; m2
q~;u~;d~ ! 0.
The supersymmetric limit of eq. (3.3) is obvious, namely Lsoft ! 0, while for (3.2), the
It is instructive to derive the RGEs both in term of massparameters from eq. (3.2) and
in terms of softmass parameters in eq. (3.3), then match the two apprroaches using the
substitutions stated above for a consistency check. However, to save space in this writeup
we show only one derivation, the functions of the operators in eq. (3.2); the functions
for the soft parameters can be derived using the substitutions in eq. (3.4)
At one loop order, the functions can be evaluated diagrammatically. In addition to
the standard diagrams one encounters in MSSM calculations, we need to take into account
new diagrams due to the
operators (e.g., gure 1). Starting from eq. (3.2), the full
functions for the Yukawa coupling Y , the higgsino mass parameter , the nonholomorphic
q˜, u˜, hd
q˜, u˜, hd
⇠ u⇤Y
⇠ uY ⇤
q˜, u˜, hd
qe, ue
˜
Hu
q, u
Y
Y ⇤
qe, ue
scalar trilinear parameter u, and various scalar masssquared parameters are given below.
mq2~ + m2u~ + m2hu + j uj
2
2
2 j j
2 j j
2
2
;
;
3.1.1
Detour: consistency check
(namely, j j2) at all scales,
As a consistency check for the RGEs derived above due to the unconventional
operator,
consider taking the supersymmetric limit. In particular, we look at the mass parameter
m2hd  in the supersymmetric limit, m2hd should be equal to the higgsino mass parameter
d
lim
! dt
m2hd
j j
2
= 0 :
Staring at eq. (3.6), (3.12), we see our RGE pass this check.
3.1.2
Towards the full Lagrangian
To formulate the RGEs of the full theory as given in eq. (2.8), one needs to incorporate
important complexicites:
If the superpotential in eq. (3.2) is expanded to include new marginal interactions
involving the D super eld, such as Y QHdD, the accidental global U(1) symmetry
with D is lost and new counterterms are needed. For example, in the presence of
both Y and Y , the operator hydq~u~ shown in gure 2 is permitted.
Y ⇠ u⇤
q˜
Y¯ 2
hd
q˜
If one gauges U(1)X , the e ect of its Dterms need to be taken into account, even if we
refrain from adding mass term for the gaugino (so that the U(1)R remains unbroken,
and we can keep using the symmetry arguments in order to restrict counterterms).
The impact of gauging shows up in two places. First, the anomalous dimensions of
all super elds charged under U(1)X changes because of the gauge
elds. Second,
new additive contributions to the RGEs arise because of the U(1)X Dterm. The
contribution can be summarized in terms of the parameter SX .
HJEP05(218)76
SX
m~2
X q m~ 2 ;
!
m~ 2
gX2 q SX ;
(SX ) = gX2 SX
12 jY j2
u
+ ~u
+ ~u
+ 12 Y 2
d
+ ~d
+ ~d
(3.14)
(3.15)
;
(3.16)
where
runs over all scalars of the model, q represents 's charge under U(1)X , and
gX represents the gauge coupling constant.
Importantly, these RGE hold as long as U(1)R remains unbroken. Therefore, even
in the presence of U(1)R preserving gaugino mass terms (i.e., Dirac gaugino masses)
the above oneloop results prevail.
if one adds a b term to the toy model, it is multiplicatively renormalized and does
not enter the RGE for any other parameters. Both e ects stem from the fact that
b is the only U(1)R breaking term and has the wrong mass dimension to radiatively
generate aterms.
The experience with the toy model (with or without added complications) paves way
for us to write down the RGEs of the full model, as shown in the next section.
3.2
Renormalization group equations in the full model
We will take the approximation that only the third generation of the Yukawa couplings are
nonzero. This reduces Yukawa coupling matrices to
0 0 y
(3.17)
+ ~u
+ ~d
m~ 2u~3
m~ 2~
d3
m~ 2~
`3
m~e2~3
= 4 jytj2 m~ q2~ + m~2u~ + m~2hu +4 jytj2
= 4 jybj2
= 2 jy j2
= 4 jy j2
m~ q2~ + m~2~ + m~2hd
d
+4 jybj2
m~ 2L~ + m~e2~ + m~2hd
m~ 2L~ + m~e2~ + m~2hd
+2 jy j2
+4 jy j2
= 6 jytj2 m~ q2~ + m~2u~ + m~2hu
+ 6 jybj2
+ 2 jy j2
+ 6 jytj2
+ ~u
+ ~
+ ~u
+
51 g12S ;
u
d
+ ~u
+ ~d
+ ~d
+
53 g12S ;
53 g12S :
When expressing the RGE in the full model, we will work in the more familiar language of
running supersymmetric and supersymmetry breaking parameters, in the spirit of eq. (3.3).
To split RGE for full masses into supersymmetric and soft pieces, one needs to express the
soft mass parameters in terms of the mass parameters in eq. (3.2) using eq. (3.4), then
apply equations eqs. (3.5){(3.12).
Written in this form and using the reduced Yukawa
matrices, the functions of the soft masses are:
16 2
= 2 jytj2 m~ q2~ + m~2u~ + m~2hu
+ 2 jybj2
+ ~u
+ ~d
+ ~
+ ~
+ ~d
(3.18)
+
54 g12S ; (3.19)
52 g12S ; (3.20)
53 g12S ; (3.21)
56 g12S ; (3.22)
+
The soft mass RGE must be complemented by the RGE for the Higgs sector
parameters, ~u; ~d; ~ ; and b :
= 3 jytj2 + 3 jybj2 + jy j2 ~
u
2 jybj2 ~u + ~d + ~u 3g22 +
= 3 jytj2 + 3 jybj2 + jy j2 ~
d
2 jytj2 ~u + ~d
+ 2 jy j2 ~
~
d + ~d 3g22 +
= 3 jytj2 + 3 jybj2 + jy j2 ~
+ 6 jybj2 ~
d
~
+ ~
3g22 +
The notation for left and right handed sleptons are `~ and e~ respectively. The RGEs of Dirac
gauginos and Majorana adjoint fermions can be found in [56]. Examining these RGE, there
are several features worth mentioning. First, the rst and second generation squarks and
sleptons can be found by setting the Yukawa couplings in eq. (3.18){(3.22) to zero. As
the traditional gaugino mediated contribution to the soft mass RGEs is absent because of
the Dirac nature of our gauginos, the rst and second generation squarks/sleptons only
renormalize due to hypercharge Dterm. Written compactly,
is a rst or second generations sfermion with hypercharge Y . A second feature of
these RGE is the unusual piece proportional to u
seen in e.g. eq. (3.18). As shown in
gure 1, this piece can be traced to insertions of trilinear scalar term from
operator and
The nal ingredient needed to complete the RGE for this theory is the running of S:
u
d
+ ~d
+ ~u
+ ~d
+ ~u
+ 4y2
d
+ ~d
+ ~d
:
(3.31)
In the limit the ~ parameters are taken to zero, eq. (3.31) reduces to its conventional
Our objective in this section is to determine the spectra at IR and other parameters after
solving equations listed in eqs. (3.18){(3.31) using all the initial conditions speci ed at the
boundary int (eq. (3.1)). Before we proceed any further, however, we give the schematics of
the scales we target. We think that having a prior understanding of the scales (i.e., sizes of
various terms) allows one to visualize and consequently appreciate the solutions, especially
the analytical part, better. We have plotted a schematic representation of superpartner
mass spectrum in
gure 3. Here is a simpli ed summary of the mass scales.
The key spectral features are:
As expected, the lowest lying scale represents the mass of the LSP. For the relic
abundance to work out we rely on coannihilation of the LSP (mostly bino) with RH
sleptons. This scale, therefore, also represents masses of the RH sleptons. We take
this opportunity to reiterate that gaugino mediated contribution (loop suppressed
and
nite) from bino can not generate this massscale. In this work, we generate this
scale by the hypercharge Dterm S
gives the size of the Sterm.
. The mass of dark matter therefore also directly
Decays of left handed sleptons to LSP give rise to hard leptons and consequently
these need to be signi cantly heavier than the LSP mass scale (or the scale of RH
⇠ 1 TeV
⇠ 500 GeV
⇠ 100 GeV
Gluinos
Weak adjoint fermions
Weak gauginos
Squarks
LH slepton masses
RH slepton masses
bino mass
Directly set by initial conditions
Dominant contribution comes from
threshold corrections from gluinos
Dominant contribution comes from
threshold corrections from weak gauginos
gives the size of S
Dominant contribution comes from S
Seesaw in neutralino mass matrix
resulting in a mostly Majorana bino LSP
HJEP05(218)76
sleptons). Setting the LH sleptons above the LHC bound, therefore, gives rise to
a second scale in the spectrum. Masses for the LH sleptons are generated by nite
corrections from Wino masses. This allows us, inturn, to set the scale for wino mass.
Finally, LHC bounds on colored particles imply that squarks need to be signi
cantly heavier. The primary source for squark masses are
nite corrections from
gluinos. Setting the squark masses to be around the TeV scale, one then nds masses
for gluinos.
The nal piece is the mass scale of the higgsinos. This can be determined either
by dark matter direct detection constraints which limits the higgsino fraction in the
lightest binolike neutralino or by direct LHC searches. At LHC, heavy higgsinos can
decay to the LSP associated with Higgs or Zboson. The nonobservation of such
events puts an upper bound on the higgsino masses.
4.1
Analytical solutions
It is clear that, even at one loop order, we need to solve the RGE eq. (3.18){(3.31)
numerically. However, in order to develop some intuition for the gross features of the spectrum,
we start by making some simplifying assumptions which allow us to solve the RGE
analytically. As we show later in this subsection, most of the phenomenological aspects of
this work, such as nding a viable candidate for dark matter or the spectrum of colored
particles, can be understood within this simpli ed picture. Calculating the Higgs mass,
however, requires more careful considerations and will only be discussed in the context of
full numerical solutions.
To simplify the RGE, in the following subsections we will ignore all Yukawa couplings
except for the top Yukawa yt. Even though we use nonzero S0, we will use ~0 = 0, which
ensures that none of the
operators will play a role. While this approximation may seem
unjusti ed given our initial conditions, we nd that the full, numerical solution derived
later is well approximated by the results we derive with ~0 = 0.
First and second generation sfermions
As mentioned earlier, the rst and second generation sfermions only run because of the
hypercharge Dterm S
. With the Yukawa couplings zeroed, the running of S is easy to
work out:
S( IR) = S0 g1( int)
;
where b1 is the beta function for hypercharge and we have imposed the boundary conditions
from eq. (2.10). Plugging eq. (4.1) into the sfermion RGE (eq. (3.30)), we nd
m~ 2 ( IR) =
6
5 qY S0
4
1( IR) log
int
IR
;
where qY is the hypercharge of the sfermion. No matter what sign we choose for S0, this
(radiative) mass squared will be negative for some matter
elds (q~i; u~i; d~i; ~li; e~i), simply
because they don't all have the same sign hypercharge. If eq. (4.2) were the only
contribution to the sfermion masses, this result would be fatal as charge/color breaking minima
would occur. Fortunately, as illustrated in gure 3, the LH sfermion masses receive positive
de nite and
nite contributions from loops of gauginos [15]. As eq. (4.2) is proportional
handed sleptons. For example, if int = 1011 GeV we need jS0j
to the hypercharge coupling and only logarithmically sensitive to
int, it is entirely
possible for the nite contribution proportional to
2( IR); 3( IR) to dominate over eq. (4.2).
This logic suggests that we should choose S0 < 0, so that eq. (4.1) is positive for the right
(700 GeV)2 in order
to achieve 100 GeV right handed slepton masses. Such a value of S can be obtained by
properly choosing UV inputs
. This choice for S0 renders the contribution of
eq. (4.2) to m~q2~ and m~2~ also positive, while m~2~ and m~2u~ receive a negative contribution. As
d `
we will show, negative m~2~
`
; m~ 2u~ must be o set by the
requirement that m~2~( IR) > 0; m~ 2u~( IR) > 0 can be used to restrict the input wino/gluino
`
mass parameters. In order to impose this restriction, we rst need to know how the IR
nite wino and gluino loops, and the
gaugino masses depend on the int inputs.
4.1.2
Colored sector
The gluino sector (gluino + adjoint partner) masses at IR are straightforward to calculate.
Using the boundary conditions set in section 4, the renormalized masses are.
s ( int) =
s ( IR) ;
( IR) =
s
s ( IR)
s ( int)
s
Z 3
Z 3
( int) M 3
( IR)
3 s
Z 3
Z 3
;
( IR)
( int) MD3 ( int) = MD3 ( int)
3 s
2
;
( int) = M 3
;
(4.1)
(4.2)
HJEP05(218)76
(4.3)
(4.4)
(4.5)
(4.6)
˜
R
q˜
g˜
elds. The purely scalar loop cancels the
logarithmic divergence which appears in the prototypical gaugino mediated correction to squark
masses.
where Z 3 is the eld strength renormalization of the color adjoint and eq. (4.3) contains
the wellknown result that the QCD gauge coupling does not run with the supersoft eld
content. Using
3 to denote the fermion within
3, the color adjoint fermion masses can
be written in matrix form as3
Depending on the relative strength of the Dirac mass of gluino (MD3 ) and the Majorana
mass of 3 (M 3 ), three qualitatively distinct spectrum at the scale IR emerge. i.)
primarily Majorana gluinos, ii.) primarily Dirac gluinos, and iii.) mixed MajoranaDirac gluinos.
For the coupling structure and some collider implications of the di erent possibilities, see
ref. [92].
The gluino mass matrix can acquire all the three types of textures even if M 3 and MD3
start out being equal at the messenger scale (depending on whether
3 self interactions
are present or not). It is instructive to properly diagonalize the mass matrix and write the
e ective theory in terms of mass eigenstates
g~ !
l
g~h
=
cos2 g =
0
1
cos g sin g
sin g cos g
!
A ;
In the above, g~l and g~h represent the light and heavy gluino spinors with masses Mgl and
Mgh respectively.
L
Mgl =
1
2
1
2
Mgl g~lg~l +
Mgh g~hg~h ;
1
2
q
and
Mgh =
1
2
M 3 + q
M 23 + 4M D23
:
and sin g = cos g = 1=p2.
This decomposition is valid even if gluinos are purely Dirac; there, M 3 = 0, Mgl =
Mgh ,
One of the important features of the generalized supersoft spectrum is that 
regardless of the Majorana/Dirac composition of the gluino in all these three cases  squark
3The gluino g~ does not acquire a Majorana mass.
(4.7)
(4.8)
(4.9)
(4.10)
masses remain supersoft, i.e. the gluino mediated squarks masses do not pick up any log int
sensitivity. Below, we verify this statement using the diagrams in gure 4:
m~q2 =
s C2 (r)
Z 4
d k
cos2 g
k2
Mg2l +
k2
sin2 g
Mg2h
+
s C2 (r)
Z 4
d k
M D23
k2(k2
m2R )
s C2 (r) cos2 gMg2l + sin2 gMg2h log i2nt +
s C2 (r) M D23 log i2nt : (4.11)
Here, m~q2 is the nite correction squark mass generated at IR and C2(r) is the quadratic
casimir. Examining eq. (4.11), the rst integral is due to gl and gh running in the loop,
and the second integral is due to the real part of the scalar octet of mass m R
. The log i2nt
term is cancelled between the two terms, as
Cleaning up eq. (4.11), the gluinoinduced contributions to the soft masses (squared) of
, m~2~) can be written in terms of the mass eigenvalues and the gluino
d
m~ q2~ =
s C2 (r)
M D23 log
M D23
m23 + M D23 cos 2 g log tan2 g
:
(4.13)
where m 3 is the mass of the scalar adjoint component in
3
.
At tree level,
m 3 = 2MD3 , though running and the existence of Majorana masses will change the
relationship somewhat.
If we assume that nite gluino contribution represents the squark masses, we can
compare eq. (4.13) to the LHC bounds on colored sparticles to get an idea for the allowed
ranges of UV inputs M D03 ; M 03 . The IR masses for the squarks and lightest gluino sector
are shown below in
gure 5 as a function of the input masses. As the squark masses are
radiatively generated and
nite, the squarks are signi cantly lighter than the gluinos for
a given set of inputs. The shape of the squark mass curves can be understood from the
fact that the Majorana gluino sector mass runs signi cantly faster than the Dirac mass.
For M 03
M D03 , M 3
( IR)
MD3 ( IR), e ectively, the gluino is Majorana like with
eigenvalue M D23 =M 3 . In such a scenario, the log emerges in the gluino mediated
correction to the scalar masses but with a cut o
M 3 , i.e., Log (M 3 =MD3 ). In assuming the
squark masses are set by eq. (4.13), we have ignored: i.)
nite (positive) contributions
from loops of SU(2) or U(1) gauginos, ii.) the logenhanced hypercharge Dterm
contribution proportional to S0. While necessary for getting the exact spectrum of the theory,
these contributions are both subdominant to eq. (4.13); the SU(2) and U(1) gaugino loops
are suppressed by the smaller EW couplings, and the S0 contribution is small because,
as explained in section 4.1.1, it sets the mass of the lightest sfermions. Therefore, our
assumption that eq. (4.13) sets the squark mass is justi ed, and we can use gure 5 to rule
out M D03 . 2 TeV. These bounds are rough, as the detailed phenomenology will depend
on how `Diraclike' vs. `Majoranalike' the lightest gluino eigenstate is; see refs. [16, 92].
0 ⌃
(G1000
1500
1000
500
0
1000
Dirac mass M D03 and adjoint Majorana mass M 03 . Right: mass of the lightest eigenvalue of the
gluino sector as a function of the same UV inputs. For both plots, we used the tree level relation
m 3 = 2MD3 for simplicity and took IR = 103 GeV,
int = 1011 GeV and
3( IR) =
3( int) =
0:118 as inputs. The starred point (M D03 ; M 03 ) = (2:48 TeV; 0 TeV) for BP1 and (M D03 ; M 03 ) =
(2:65 TeV; 940 GeV) for BP2, yields m~q~ ' 1:8 TeV which satisfy the present LHC bounds [1, 2], and
a gluino mass of 7 TeV and 4:6 TeV respectively.
4.1.3
Electroweak sector
Following the same procedure as above, we can calculate the electroweak gaugino masses
(both Dirac and Majorana pieces). One di erence in the electroweak case is that the gauge
couplings do run in a supersoft theory, with beta function coe cients b1 = 33=5 and b2 = 3.
Working in the yt ! 0 limit, the masses at the IR scale are:
1
i ( int)
=
1
i ( IR)
2
ba log
Z a ( IR) = Z a ( int)
MDa ( IR) = M D0a
M a ( IR) = M 0i
i ( R)
i ( int)
a ( IR) 2
i ( int)
i ( IR)
i ( int)
1 C2(Adj)
ba
2C2(Adj)
ba
int
IR
2C2(Adj)
ba
;
;
;
;
(4.14)
(4.15)
(4.16)
(4.17)
where a = 1; 2 and C2(Adj) is the quadratic Casimir of the adjoint representation.
Focusing rst on the SU(2) sector, loops of winos and SU(2) adjoints generate a nite
correction to the mass of all SU(2) charged sfermions. The contribution has the same form
as eqs. (4.7){(4.13), but with MD2 replacing MD3 , 2 replacing 3, and 2 replacing g:
m~ `2~ =
2 C2 (r)
M D22 log
M D22
m22 + M D22 cos 2 2 log tan2 2
:
(4.18)
This e ect is particularly important for the lefthanded sleptons, as the contribution to
their masses from the S0 piece is negative (see discussion following eq. (4.2)) and they
0.025
˜
m
1
2 e˜
˜
m
3
2 e˜
˜
m
(5.4)
(5.5)
0103 104 105 106 107 108 109 1010 1011
by S0 as a function of the intermediate scale for di erent tan
values of 2.5 (redsolid), 20
(bluedashed) and 40 (blackdotted). In the left panel we present the mass splitting between the rst and
third generation of right handed sleptons as a function of tan . The tan
dependence manifests
itself in the RGEs.
where  as discussed earlier  the largest regions parameter space will have the bino (LSP)
and lightest right handed selectron nearly degenerate. As such, a mass splitting between
slepton generations means the heavier sleptons are slightly heavier than the LSP and can
decay `~ ! ` ~01. For sleptons in the 100 GeV range, bounds from lepton plus missing energy
searches are quite stringent [100, 101]. To avoid these bounds without raising the overall
mass scale, we need to quench the splitting by restricting parameters to low to moderate
tan . As we shall see in the next section, the small tan
region also well motivated from
the perspective of Higgs mass.
To obtain a rough estimate on the size of S0, we present two cases with di erent values
of tan : 2.5 and 40. To obtain a right slepton IR mass of around 100 GeV for these tan
values, one would require
p
jS0j =
m~ e~
= 2:5, the e ect of the tau Yukawa coupling in the RGEs can be neglected to a
very good approximation. Hence, typical values such as 0d = 800 GeV and
0u = 500 GeV
satis es the left hand side of eq. (5.4). Furthermore, requiring the rightsleptons to be
nearly degenerate results in
m~ e2~
j m~e~j2 < 1% =) tan
. 10:
From eq. (5.5) and for m~e~
100 GeV, the degeneracy between the slepton generations turn
out to be around 10 GeV. In the next section we look at the DM constraints which gives
a range of the LSPright slepton masses where relic density can be satis ed.
5.2
In the absence of coannihilation, the relic density of a binolike neutralino can be
approximated as [96]
other states, say r < 0:8, the present value of the observed relic density 0.1199 [102] readily
limits m~e~ . 100 GeV. Such slepton masses are very tightly constrained from the LEP
data [21]. As a result, in most of the allowed parameter space of slepton masses, the bino
su ers from overabundance.
In our setup, the bino is nearly massdegenerate with right handed sleptons, and
consequently coannihilation becomes important. Roughly, whenever m
then these slightly heavier degrees of freedom are thermally accessible and are therefore
m~ e~
m~01
Tf ,
nearly abundant as the relic species. Since Tf
degeneracy needed for coannihilation is
m~01=25 [96], we nd the degree of
The higgsino mass is essentially tied with the right handed slepton masses through their
initial conditions. Assuming again
0u = 500 GeV and 0d = 800 GeV, the right panel of the
gure 8 shows that the higgsino mass at IR is driven by
( IR) tan =2:5
1:025
0d + 0u
2
660 GeV:
(5.9)
Another way to make higgsinos heavy would rely on the modi cation of the messenger scale.
diately sets d( int)
again u( int)
Higgsino masses
m~ e~
m~01 =
m~01
m
m~01
5%:
Once coannihilation becomes important, the slepton self interactions and interactions with
the bino [103] set the relic abundance. These additional interactions relax the bound on
the slepton masses and overabundance issue can be avoided if
m~ e~
me01 . 500GeV;
which is well above the current LHC bound on sleptons nearly degenerate with the
LSP [94]. For a dedicated and more robust analysis, we implemented the e ective
model in SARAH4.11.0 [104, 105] and generated the spectrum using SPheno3.3.3 [89, 90].
We have varied the masses of the lightest neutralino state and the slepton masses
(assumed to be degenerate). If the LSP is neutralino then only the spectrum
le is fed
to micrOMEGAsv3 [106] for computing relic abundance and dark matter direct detection
rates. We nd that coannihilation works e ciently for m~01
me~
400 GeV. This
imme2 TeV and ( IR)
600 GeV. This limit is obtained by considering
(5.6)
(5.7)
(5.8)
IR
μ (μIR) 1.03
10
5
μ (μIR)
m˜e˜ (μIR)
1103 104 105 106 107 108 109 1010 1011 0104 105 106 107 108 109 1010 1011
gure we show the running of the higgsino mass scaled with
its initial condition for a
xed value of tan
= 2:5. On the right panel we elucidate how the low
energy constraints can be translated to a bound on the intermediate scale. These constraints can
come from either DM direct detection results or collider experiments. The yellow shaded region is
excluded from the direct collider searches for right slepton/LSP masses close to 100 GeV for the
same initial conditions as discussed before.
In the right panel of gure 8 we show how low energy constraints can signi cantly
constrain the messenger scale in our scenario. The constraints are two folds, rst from direct
detection experiments. The limit from direct detection experiment constrains the higgsino
fraction in the lightest neutralino. As a result, higgsino should be heavier compared to
the LSP or the lightest slepton. In our case, the lightest neutralino is a predominant
mixture of the bino and singlino gauge eigenstates and the higgsino admixture is negligible.
Therefore, the stringent constraint from the DM direct detection can be avoided easily.
However, to push the Higgs mass to the observed value, as elaborated in the next section,
we require introducing new superpotential terms. For larger couplings, this increases the
higgsino component in the lightest neutralino state. Secondly, collider experiment can also
provide stringent constraint on the ratio of the higgsino and slepton masses. For example,
our spectrum has the following hierarchical structure where NLSPs are the right handed
slepton and LSP is the binosinglino admixture neutralino. Higgsinos are heavier than the
sleptons. Such a higgsino after electroweak production can decay to a Z ~01 or h ~01 [93, 107].
However, all these modes are phase space suppressed for ~02
the dominant decay mode would be `~`. The limits on this particular
~
0
3
robust [101]. In our case, we took a conservative approach and used
200 GeV. In such cases
nal state is very
( IR)=m~ `~R ( IR)
For our choice of parameters, we observe from
gure 8 that the UV scale should be less than
1011 GeV or so. Changing the initial values would modify the results as the dependence of
these two parameters are di erent for higgsino and slepton mass runnings.
5.4
Dark matter direct detection
Our framework also needs to be consistent with the null results in DM direct detection.
When the squarks are heavy, the spinindependent interaction between DM and nuclei
comes from Higgs exchange and thus it depends crucially on the Higgs coupling to the
lightest neutralino. The Higgsneutralino coupling, in turn, depends on the higgsino mass
parameter. For a given LSP bino mass we can translate limits from direct detection
directly into limits on the higgsino mass. The spinindependent crosssection can be well
approximated by the following relation [108]
SI '
where GF ; MZ ; mred are the Fermi coupling constant, Z boson mass, and DMnucleon
reduced mass, and Fh, Ih are coupling and kinematic factors. In the limit when the wino
and heavy Higgs are heavy and e ectively decoupled (in addition to the squarks),
HJEP05(218)76
Fh =
N11N14 sin W ;
Ih =
X kqhmqhN jqqjN i:
q
h
Here, N11; N14 are the bino and higgsino fraction in the lightest neutralino, kuptype =
h
cos = sin , kdowntype =
sin = cos , and we have already assumed cos
! 1 and
sin
! 0 by decoupling the heavy Higgs. Because of the Dirac nature of the gaugino
masses and the extra interactions involving the adjoint (e.g., SHuHd coupling in the
superpotential), the neutralino mass matrix no longer has the MSSM form. Taking the ratio
of the DM direct detection in GSS to the prototypical MSSM, the only factors that don't
drop out are the N11; N14,
(5.10)
(5.11)
(5.12)
GSS
SI
SI
MSSM '
N1G1SS N1G4SS 2
jN11N14j2
:
For typical values
= 300 GeV, M ~01
section in GSS turns out to be an order of magnitude less than in the MSSM, as the
binosinglino mixing in GSS dominates over binohiggsino mixing. This implies lower higgsino
150 GeV, tan
= 4, the direct detection
crossmasses are possible in our framework.
The constraints on the right handed slepton masses and the higgsino mass can be
the contours of right handed slepton mass (red) and higgsino mass (blue) in the 0u
e ectively translated to constrain the two input parameters 0u and 0d. In gure 9 we show
 0d plane.
As already stated, one needs to have 0d >
0u in order to get positive de nite right handed
slepton mass. As a result, the grey shaded region is not viable. The masses of right handed
sleptons and the LSP should be nearly degenerate in order to satisfy the relic abundance.
Moreover, DM direct detection experiments sets a limit on the binohiggsino mixing and
e ectively on the higgsino mass parameter, . Therefore, given the values of m`~R and
one can readily constrain the input parameters
gure in the left panel. In such cases, the lightest neutralino is a predominant admixture
of the bino and the singlino instead of binohiggsino. As a result, the dark matter direct
detection constraints are irrelevant. However, collider constraints are certainly applicable
and we discuss those issues in detail later. The gure in the right panel is obtained by xing
S = 0:8. A nonzero
S is important to satisfy the Higgs mass constraints as discussed in
section 5.5. It is obvious, that order one value of S increases the higgsino fraction in the
lightest neutralino and hence direct detection constraints become important. The yellow
T
0μu 0.8
0 u
μ
5
0
0
Direct detection
constraint
μ0d (TeV)
u 0d plane. To obtain positive de nite mass for the right handed sleptons one needs to have
0
0
u, therefore, ruling out the shaded region. For de nite values of right handed slepton and
higgsino masses, designated by the crossing of the contours, the two input parameters
can be readily obtained. We have
xed
int = 1011 GeV and tan
= 2:5. We have
xed
0u and 0d
S = 0,
for the gure in the left panel. In such a scenario, the direct detection constraints are not relevant.
However, for the gure in the right panel, we have
xed S = 0:8 and show the excluded region
after taking into account constraints from direct detection experiments
shaded region in the right panel plot of gure 9 shows the excluded parameter space when
direct detection results (LUX WS 201416) [97] are taken into consideration. We reiterate
that the LEP results exclude parameter space where slepton masses are below 100 GeV.
mass and degree of etL
5.5
Higgs mass and new superpotential terms
In previous sections we have shown how the inputs 0u
; 0d, the Dirac and Majorana gaugino
masses and, to some extent, tan
are constrained by collider physics and dark matter.
What remains to be done is to see what Higgs masses are possible in the `surviving' regions.
As already mentioned in section 4.1.4, the traditional MSSM Higgs quartic terms get
depleted due to the presence of Dirac gauginos. However, new superpotential couplings such
as S generates additional quartic terms which are NMSSM like. Under some simpli ed
assumptions such as i.) integrating out the adjoint scalar elds, ii.) assuming T = 0, for
simplicity, the Higgs mass can be obtained by diagonalizing the scalar mass matrix in the
basis (hu; hd) given in appendix C.
To fully answer the question of the Higgs mass, we need to go beyond treelevel. The
largest looplevel contribution comes from the top squarks and is governed by their overall
etR mixing. In GSS, the top squark mass matrix, neglecting Dterm
mt2~ =
m2Q3 + mt2
~uytvd
~uytvd
m2u3 + mt2
!
;
(5.13)
where the terms ~u is the supersymmetry breaking trilinear interaction originating from
eq. (2.8). It is well known that the top squark contribution to the Higgs mass is largest
when there is substantial etL
etR mixing [
98
]. In GSS, the mixing angle in the stopsector,
tan 2 =
m2Q3
2 ~uymtvd2u3 ;
is proportional to vd
cos , while the mixing angle in the MSSM
sin . The di erence
can be traced to the unusual structure of the scalar trilinears in GSS and, since one usually
wants to take tan
large to maximize the tree level Higgs quartic, leads to suppressed stop
sector mixing. Suppressed stop sector mixing can be overcome by taking large ~u, though
this is an unattractive option as it will increase the traditional ne tuning measure of the
setup. Moreover, even if we set aside tuning concerns for the moment, we nd that even
very large values of ~u are unable to push the Higgs mass to more than 100 GeV (recent
works, studying the phenomenological implications of such nonstandard soft
supersymmetry breaking terms in MSSM can be found in [109, 110]). In addition to the tree level
terms, we have also included two loop corrections from the stop sector [111] (for three loop
corrections in MSSM see [112]) and considered mt~1
are possible if we consider heavier stops, though at the expense of increased tuning.
1:8 TeV.5 Higher Higgs masses
Hence, the most natural way to increase the Higgs mass is to extend the theory with
additional F terms, as in the NMSSM. In GSS, this extension requires no new matter as
the theory already contains a SU(2) singlet and triplet super eld. Including interactions
between these super eld and the Higgses, the GSS superpotential is modi ed to
WGSS = W + SS Hu Hd + T Hu T Hd;
(5.15)
with W given in eq. (2.8). The modi ed superpotential generates a new tree level quartic
for the Higgs. Speci cally, taking S 6= 0; T = 0 for simplicity, two new Higgs potential
terms are generated,
V2 = j Sj
V3 =
2 "
4
p2g0 S
4
In the limit MD
4M D21
MD1
M 21 + 4M D21
(5.14)
HJEP05(218)76
M 21 +4M D21 h2uh2d +
M 21 + 4M D21 huhd h2u +h2d
M 1
2
M 21 +4M D21
h2u + h2d
M 1 huhd
h
2
d
h
2u :
h2u + h2d 2 ;
(5.16)
M ; , the quartic no longer vanishes but instead takes on a NMSSM
form, e.g., V = j sj2 h2uh2d and can generate a large tree level Higgs mass.6
4
In the left panel of
gure 10 we show the Higgs mass with new tree level quartic
contributions after taking into account two loop corrections from the stop sector ( m~t~
5In the presence of additional superpotential interactions (eq. (5.15)), the electroweak adjoints also
contribute to the Higgs mass and can in principle be included. At one loop, these contributions are /
and are logarithmically sensitive to the di erence between the adjoint fermion and scalar masses. However,
unlike in the topstop sector, the ratio of adjoint fermion to scalar masses is O(1), thereby suppressing
the adjoint contribution to the Higgs mass. We have therefore neglected this (positive de nite) piece for
4
S
; 4T
6The dependent pieces do not vanish in the D at limit so they will alter the stabilization conditions,
as well as the relation between the b and the other inputs.
4.5
3.5
2.5
b
24G
eV
1
26 G
eV
BP1
2
1
HJEP05(218)76
0.6
loop corrections from the stop sector ( m~t~
1:8 TeV). The MSSM tree level contribution is depleted
due to our choice of M
and MD's (same as BP1). However, additional F term contributions occur
from the superpotential coupling
S. Right: running of superpotential coupling
S (shown by
the redsolid curve on the left panel) and top Yukawa coupling (bluesolid curve). We have xed
( IR) = 0:8 and tan
= 3:6 (same as BP1) to satisfy Higgs mass constraints.
1:8 TeV). The MSSM tree level contribution gets reduced due to the presence of M i and
MDi , however additional F term contributions from the superpotential coupling
S helps.
The values shown in
gure 10 are the IR values, so one might worry that the relatively
large
values needed for mh = 125 GeV grow even larger in the UV and lead to a violation
of perturbativity. In addition to their own running, the
couplings modify the anomalous
dimensions of the Higgs elds and thus contribute to the running of the top and bottom
Yukawa couplings:
We reiterate, the RGEs of the scalar masses are shown to run from
therefore integrating out any hidden sector e ects. However, 's are superpotential coupling
and should remain perturbative upto the GUT scale. The running of the singlet coupling to
the Higgs elds is shown in eq. (5.17) and depicted in the right panel of gure 10. We have
chosen ( IR) = 0:8 and tan
= 3:6 (same as BP1) to satisfy Higgs mass constraints. The
coupling remains perturbative up to the GUT scale. However, due to larger group theory
factors in the anomalous dimension the triplet coupling diverges more rapidly, becoming
nonperturbative at an energy close to 1010 GeV unless additional structure is added to
the theory.
The new superpotential couplings generate new trilinear interactions in the scalar
potential and modify the running of the Higgs soft masses m~hu , m~hd and couplings ~u; ~d.
:
int
(5.17)
1011 GeV,
The rest remains una ected. The additional couplings in the Lagrangian
L ! L
S d jhuj2 S
S u jhdj2 S + triplet trilinear terms
(5.18)
The modi ed RGE are presented in appendix D. One important thing to note that the
presence of the singlet eld and nonstandard soft terms can give rise to dangerous
tadpole diagrams which might destabilize the hierarchy [113, 114]. We discuss such issues in
appendix E.
Finally, we provide two benchmark points to show that the framework is consistent with all
the phenomenological observations including the Higgs mass, DM relic density, DM direct
detection and collider results. The UV and IR parameters for these benchmarks, along
with mh;
h2 and the DMnucleon spinindependent cross section are shown in table 2.
As discussed in section 5.5, a viable Higgs mass is most naturally reached in GSS when
one admits superpotential interactions between the SU(2); U(1) adjoint super eld and the
Higgses. While either
S, T or both can be used, however, T = 0 is the safer choice,
in the sense that it requires fewer assumptions about the UV. Therefore, both of the
benchmark setups have S 6= 0; T = 0.
Inspecting the benchmarks, both points share the feature that q~; u~; d~; `~; e~ are massless at
int and have 0d;u split in a way that yields positive RH slepton masses, 0d
0
100 GeV.
To augment the Higgs mass, both benchmarks have S 6= 0; O(1). Order one values of S
enhance the higgsino fraction of the lightest bino sector
eld [115]. Hence, to overcome
stringent constraints from the DM direct detection,7 the higgsino mass  set by
needs to be around a TeV. We note in passing that such a problem does not arise when one
makes the triplet coupling
T large to x the Higgs mass. As this coupling only increases
the higgsino fraction in the winotripletino like neutralino, considered to be heavy in our
case. For both benchmarks, the b ( int) values have been determined numerically following
the logic established in section 4.1.4 with re nements due to S 6= 0 and are similar in size
0
u;d 
to the other UV inputs.
The largest di erence between the points is the mass of the triplet and octet fermions.
In BP1, M 2 = M 3 = 0 in the UV, which, since the running of these masses is
multiplicative, implies M 2 = M 3 = 0 in the IR as well.8 We cannot further simplify things
by choosing M 1 = 0 without destroying the seesaw mechanism in the bino sector. A
second di erence between the benchmarks is the LSP mass, set to be 100 GeV in BP1 and
150 GeV in BP2. The fact that the LSP mass is higher in BP2 while the higgsino mass
remains the same as in BP1 results in slightly enhanced direct detection limit.
Finally, one set of bounds we have yet to address are the LHC limits on electroweakinos.
The bounds on chargino/heavier neutralinos ( ~02; ~0) production can be quite stringent,
3
1 TeV if the chargino/neutralinos decay primarily to sleptons [116, 117]. Fortunately,
7We have used micrOMEGAsv3 for DM direct detection crosssection. For
100 GeV neutralino, the
crosssection should be less than 9
8There is no phenomenological disaster associated with M 2 = M 3 = 0 as the 2;3 elds will receive a
nite, looplevel contribution from gaugino loops, see section 4.1.2 and 4.1.3
m~ q~;u~;d~;`~;e~
Parameters ( IR)
int
MD3
MD2
MD1
b
0
d
0
MD3
MD2
MD1
M 3
M 2
M 1
b
S
tan
Outputs
m~ g~1;g~2
m~ q~;u~;d~
m~ t~1;2
m~ ~2;3
m~ ~
`
m~ e~
m~ ~01
mh;H
h
2
SI
02 ! h ~01
03 ! Z ~01
03 ! h ~01
1011 GeV
2:48 TeV
4:72 TeV
556 GeV
0
0
0
0
0
1.97 TeV
1.39 TeV
1.15 GeV
0:7392 TeV2
BP1
7:00 TeV
5:00 TeV
515 GeV
2.50 TeV
1.026 TeV
0:7492 TeV2
1011 GeV
2.65 TeV
567 GeV
940 GeV
3.45 TeV
1.30 TeV
0
1.475 TeV
1.0 TeV
0:8582 TeV2
BP2
7.50 TeV
5.50 TeV
525 GeV
7.50 TeV
5.50 TeV
1.65 TeV
1.076 TeV
0:8672 TeV2
the Higgs mass and DM relic density and direct detection constraints. The DM direct detection
limit for a 100 GeV WIMP is around 9
on left handed sleptons push the mass of the winos into the multiTeV range, while DM
and Higgs constraints prefer higgsinos at 1 TeV. As a result, ~ ; ~02; ~03 are predominantly
higgsino and thus have Yukawasuppressed couplings to SM fermions (this is exacerbated by
the small tan
in BP1 and BP2). With sfermionfermion decays suppressed, the charginos
and neutralinos decay preferentially to gauge bosons and Higgses, modes with much looser
constrains. The dominance of the gauge/Higgs boson branching ratios of ~ ; ~02; ~03 in BP1
and BP2 can be seen in the bottom rows of table 2.
These two benchmarks have been chosen with particles sitting just outside the existing
bounds. In the near future, both points would be exposed through jets plus missing energy
searches (sensitive to q~; u~; d~) or leptons plus missing energy (sensitive to `~). However,
the squarks and left handed sleptons can easily been taken heavier without ruining the
main features of GSS, namely the near degeneracy of the right handed sleptons with the
LSP. Compressed spectra, especially among weakly interacting states, are hard to probe at
the LHC, though studies with displaced vertices [118, 119], softtracks [120, 121], or initial
state radiation [122{126] can be a useful tool to look for such regions. Should a compressed
sleptonLSP sector be discovered at the LHC, the next step towards singling out GSS as
the underlying framework would be to verify the hierarchical structure of the spectrum, for
example, a right handed slepton signal without any sign of squarks, left handed sleptons,
or gauginos. As the absence of other states is a rather unsatisfactory discriminator among
models, a more concrete signal is the presence of SU(2), SU(3) adjoint scalars. The masses
of these states is more model dependent (e.g. compare BP1 and BP2) and it is possible
that they are light enough to yield a signal at the LHC [127], however it is likely that a
future, higher energy machine is required to explore the spectrum fully.
6
Summary and conclusion
The recent run of the LHC has put stringent constraints on the superpartner masses.
In fact, the strongest limit is drawn when the gluinos/squark masses are well separated
from the LSP mass, reaching almost 2 TeV. A heavy gluino raises the soft mass of Higgs
elds through renormalization group evolutions. As these parameters are a measure of
the
ne tuning, the nonobservation of the superpartners has resulted into
nely tuned
regions of parameter space for most supersymmetry models. Frameworks with supersoft
supersymmetry with Dirac gauginos are well motivated in this light. The supersoft nature
of the gluinos ensure that the gaugino mediated correction to the squark masses are nite
and not log enhanced. Therefore, Dirac gluinos can be naturally heavy. Consequently,
the pair production of the gluinos goes down signi cantly due to kinematic suppression.
Moreover, the production of same chirality squarks are also less as this requires chirality
ipping Majorana gaugino masses in the propagator. The reduction in the production
crosssection of the squarks weakens the constraints on the squark masses signi cantly.
Hence, supersoft models are often coined as `supersafe' in the literature. An additional
virtue of this framework is avor and CP violating e ects are well under control.
However, supersoft frameworks also su er from a few drawbacks. First, the
GuidiceMasiero mechanism is unavailable so viable
values require a conspiracy between the
supersymmetry breaking scale and the Planck scale. Further, the natural D at direction
of the Higgs potential sets the tree level quartic term to zero, making it very di cult to t
the observed Higgs mass of 125 GeV. Finally, supersoft models contain additional scalars
in the adjoint representation of the SM gauge group that often acquire negative squared
masses, resulting in a color breaking vacuum. An interesting way to resolve these three
issues is to supplement the theory with additional, potentially nonsupersoft operators
involving the same Dterm vev used to generate gaugino masses, the socalled generalized
supersoft framework. The additional operators in GSS generate
terms proportional to
the supersymmetry breaking vev and positive de nite masses (squared) for the scalar
components of the adjoint chiral super elds. While economically solving the
and adjoint
masses issues is a step forward, dark matter remains an issue in GSS; right handed
sleptons only receive mass through the
nite correction from the bino and therefore seem to
be destined to be the LSP unless the mediation scale is low.
In this work we mapped out the parameter space of GSS, paying particular attention to
the DM problem. We have shown that bino LSP can be avoided while maintaining a high
mediation scale and without new
elds in parameter regions where the two GSS
terms
are unequal. If
u 6=
d, supersoftness is lost and loop suppressed, log divergent pieces
from the hypercharge Dterm contribute to the running of the scalar masses. For the right
choice of inputs, these hypercharge contributions can lift up the RH slepton mass above the
bino. Focusing on the region where the bino is the LSP, we explored how well the setup can
satisfy additional constraints such as the 125 GeV Higgs mass, correct relic abundance 
achieved whenever the bino and RH slepton masses are similar and coannihilation becomes
important  and compatibility with the latest LHC results.
The RH slepton masses are controlled by the di erence of the UV
parameters while
the higgsino mass is set by the sum. If we insist that the LSP bino is a valid thermal
relic DM that escapes all direct detection bounds, the two constraints become tightly
correlated, since the slepton mass controls the degree of coannhiliation while the higgsino
mass controls the strength of the Higgsexchange DMnucleon interaction.
Within this
parameter region, we
nd squark and slepton collider bounds can be satis ed, but the
Higgs mass is generically too low unless one resorts to large loop corrections. Rather than
resort to heavy stops, we showed how additional, NMSSMtype interactions can be used to
lift the Higgs mass. These NMSSMtype interactions require no additional eld content,
as the bino's Dirac partner is a gauge singlet. The interpolation between IR constraints
and UV inputs requires some care, as nonstandard supersymmetry breaking interactions
are generated whenever
u 6=
d and enter nontrivially into the RGE.
Even though we supplement this work with full numerical solutions, we focus rather
on calculating general features of the spectrum, which we derive using analytical solutions
whenever we can after making various simplifying assumptions. In fact, most of the
features of the electroweak spectra get captured even after these simpli cations. Instead of
scanning the full parameter space numerically for allowed regions, this approach allows us
to generate intuitions about how to convert various experimental bounds into bounds in
results of this paper came from taking the masses and parameters of the theory at the
supersymmetry breaking scale (
D) to be inputs. The lack of knowledge in the exact nature
of supersymmetry breaking dynamics does not allow us directly to consider the parameters
at the messenger scale as inputs. A derivation of renormalization group equations of these
new operators  including the operators proposed in eq. (2.4)  in the presence of
arbitrary hidden sector dynamics (in the fashion of [87]) would allow one to relate these masses
and parameters of the messenger scale and the supersymmetry breaking scale. To further
UV complete this framework, more detailed messenger sector model building is required.
Acknowledgments
No. PHY1520966.
The work of AM was partially supported by the National Science Foundation under Grant
A
Neutralino and chargino mass matrices
The fermion sector in our case is di erent from the usual MSSM structure. The neutralino
mass matrix in the basis (B~; S~; W~ 0; T~0; H~u0; H~d0) looks like
HJEP05(218)76
16 2 dS =
dt
16 2 d ~u
16 2 dyt
dt ' 3 jytj2 ~u;
dt ' yt 6 jytj2
~ 2
u
+ ~u
+ ~u
136 g32 :
M ~0 = B
B
B
BB MD1
BB g0vu
MD1
M 1
0
0
MD2
gvu
2
2
2
pSvu gvd
2
MD2
M 2
T vd
T vu
2
2
g0vu
2
pSvd
2
gvu
T vd
2
2
0
g0vd 1
2
pSvu C
gvd2 CC
C
2T vu CC :
M ~+ = BMD2
0 0
p
2
MD2
M 2
2
gvu 1
p
2
pTv2d CA :
Also the chargino mass matrix written in the basis (W~ ; T~ ; H~d ) and (W~ +; T~+; H~u+) takes
the following shape
B
Switching on the Yukawa coupling
We now switch on the Yukawa couplings and show how the right handed slepton masses
get generated through RGEs. For this the following equations are required to be solved
through some approximate means. Such as
(A.1)
(A.2)
(B.1)
(B.2)
(B.3)
Eq. (B.3) can be simpli ed to obtain the following form
Z
d log jytj2 =
4 2
dt jytj2
2g32 Z
3 2
dt:
This can be further simpli ed to
and used for the solution of ~u, which we nd
4 2
dt jytj2 = log
jyt( )j
jyt( )j
2 !
2
8 S=3 #
;
(B.5)
;
16 2 dS
dt
66 2
5 g1S '
12 jytj2 ~ 2
u :
The nal part is the computation of S which directly goes into the RGEs of the scalar
masses. We simplify by considering
(B.4)
(B.6)
(B.7)
(B.9)
(B.10)
S(t) =
2
g1(t) 2
2 ~u(t3)
2
6
4 jg1(t3)j2 +
b1 Z
8 2
dt3 ~u(t3) 7 :
2
3t
5
t0
Although, the nal result is not really transparent from eq. (B.10), however it is conspicuous
that a nonzero S requires a nonzero ~u=d at int.
One can treat the above equation as the most general linear rst order ordinary di erential
equations. The term in the right hand can be regarded as a source or a driving term for
the inhomogeneous ordinary di erential equation. The solution is straightforward which
involves a integrating factor. The nal solution can be written in the closed form as
Z t
t0
g1(t) 2 Z t
2
d ~u(t3)
dt3
3
27 :
5
Z t
2
Using integration by parts one can further simply this to obtain
2
Z t3
d log g1(t2)
; (B.8)
The scalar mass matrix elements written in the basis (hu; hd) after integrating out the
adjoint scalars and assuming T = 0, turns out to be
M121 = b cot
1
g2M 22
4 4M D22 + M 22
4M D21 + M 21
v2 sin2
2
2
M222 = b tan
g0 s MD1 M 1
v
+ p
2 2 4M D21 + M 21 ( 2 + cos 2 ) cot
p
+2 2 sg0
2
MD1 v
4M D21 + M 21 sin2
1
g2M 22
4 4M D22 + M 22
4M D21 + M 21
2 2
v
2 s2 4M D21 + M 21 sin2 ;
v2 cos2
M 1 v
2
g0 s MD1 M 1
v
+ p
2 2 4M D21 + M 21 (2 + cos 2 ) tan
2 4M D21 + M 21 sin 3 sec
p
+2 2 sg0
2
MD1 v
4M D21 + M 21 cos2
2 s2 4M D21 + M 21 cos2 ;
M122 =
b
1
g2M 22
4 4M D22 + M 22
v2 sin cos
2
s
2
s
2 2
v
M 1 v
2
2 4M D21 + M 21 cos 3 csc
2p2 4M D21 + M 21 cos 2 +
2
2v2 3M 1
s
2
+ 4M D21
4M D21 + M 21
2 2 sin 2
:
(C.1)
Furthermore, integrating out the Dirac adjoint scalars and adding the two minimization
equations for hu and hd we nd
2b
sin 2
= j uj2 + j dj2 + 2 s2v2
M D21
2
4M D21 + M 21
g0 sv2
p
MD1 M 1
2 4M D21 + M 21 cot 2
(C.2)
p2g0 sv
2
MD1
4M D21 + M 21 cos 2
M 1
4 4M D21 + M 21 ( 7 + 5 cos 2 ) cot :
v
For a xed rightslepton and higgsino masses, u and
d is completely xed. Therefore,
given the values of gaugino masses and tan , one completely
xes b . The size of b also
controls the heavy and charged Higgs masses. Hence, the whole spectrum gets determined.
D
RGEs with
S,
T
The inclusion of the superpotential and nonstandard soft supersymmetry breaking terms
proportional to
S and
T modi es the anomalous dimensions of the Higgs elds. This
in turn modi es the RGEs of the soft supersymmetry breaking Higgs mass parameters
and obviously the term and nonstandard supersymmetry breaking terms proportional
S
S
˜
Hu
˜
Hd
μ
gauge singlet chiral super eld.
Tadpole issue
h ~ i
d + 2 j Sj2 + 6 j T j
2 ~ ;
d
u + 2 j Sj2 + 6 j T j
2 ~u:
+ 2 j Sj2 + 6 j T j
+ 2 j Sj2 + 6 j T j
2 hm~ 2hu + m~2hd + 2 nj ~dj2 + ~d
2 hm~ 2hu + m~2hd + 2 nj ~uj2 + ~u
+ ~d
+ ~u
oi ;
oi ;
Another important aspect of our case is the e ect of nonstandard soft terms in the
presence of the singlet. These terms have been traditionally neglected in models with gauge
singlets as they could give rise to dangerous tadpole diagrams which might destabilize the
hierarchy [113, 114]. It is important to see the e ect of such terms in GSS. These diagrams
can be evaluated in a straight forward manner and the terms which gives rise to hard
breaking are
S =
S
2
16 2 int
Since we have chosen ( int) = ( 0u + 0d)=2, therefore these tadpole diagrams do not give
rise to hard breaking at int.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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