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arXiv:1206.3910v1 [hep-ph] 18 Jun 2012
Yukawa coupling unification in SO(10) with positive µ and a
heavier gluino
Anjan S. Joshipura∗and Ketan M. Patel†
Physical Research Laboratory, Navarangpura, Ahmedabad-380 009, India.
Abstract
The t − b − τ unification with positive Higgs mass parameter µ in the minimal supersymmetric
standard model prefers “just so” Higgs splitting and a light gluino ? 500 GeV which appears to be
ruled out by the recent LHC searches. We reanalyze constraints on soft supersymmetry breaking
parameters in this scenario allowing independent splittings among squarks and Higgs doublets at
the grand unification scale and show that it is possible to obtain t − b − τ unification and satisfy
experimental constraints on gluino mass without raising supersymmetry breaking scale to very
high value ∼ 20 TeV. We discuss the origin of independent squark and Higgs splittings in realistic
SO(10) models. Just so Higgs splitting can be induced without significantly affecting the t−b−τ
unification in SO(10) models containing Higgs fields transforming as 10 + 126 + 126 + 210. This
splitting arises in the presence of non-universal boundary conditions from mixing between 10 and
other Higgs fields. Similarly, if additional matter fields are introduced then their mixing with the
matter multiplet 16 is shown to generate the squark splitting required to raise the gluino mass
within the t − b − τ unified models with positive µ.
∗anjan@prl.res.in
†kmpatel@prl.res.in
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I. INTRODUCTION
Grand unified theories (GUTs) based on SO(10) group not only unify the gauge inter-
actions but also lead to a unified framework for the Yukawa couplings and hence fermion
masses. In particular, SO(10) model with a 10-plet of higgs coupling dominantly to the
third generation implies an equality yt= yb= yτ of the t − b − τ Yukawa couplings at the
GUT scale. Quite independently, the renormalization group (RG) running of the Yukawa
couplings in a softly broken minimal supersymmetric standard model (MSSM) can lead [1–6]
to the t−b−τ unification at the GUT scale making the supersymmetric SO(10) broken to
the MSSM at the GUT scale an attractive theory of unification.
t−b−τ unification at the GUT scale is however not the most generic property of the MSSM
but follows only for a restricted set of boundary conditions for the soft supersymmetric
breaking terms. These restrictions mainly arise due to the need of significant threshold
corrections [2, 3] to the b quark mass required for the t−b−τ unification and difficulties in
achieving the radiative electroweak symmetry breaking (REWSB) in the presence of large b
and τ Yukawa couplings [3, 4]. Both of these depend on the soft breaking sector. It is realized
[5, 6] that t − b − τ unification generally requires departure from the universal boundary
conditions assumed within the minimal supergravity (mSUGRA) framework. Universality of
the gaugino masses is enforced by the SO(10) invariance if it is assumed that supersymmetry
(SUSY) is not broken at the GUT scale by a non-trivial representation contained in the
symmetric product of two adjoints of SO(10). In contrast, the soft masses m16, m10 for
sfermions belonging to 16Mand the Higgs scalars belonging to the 10Hrepresentations are
allowed to be different and are also required to be so to obtain t − b − τ unification. In
addition to this SO(10) preserving non-universality, one also needs to introduce explicit
SO(10) breaking non-universality. Such non-universality can be induced spontaneously by
a non-zero D-term (DT) which introduces splitting within 16M of squarks and 10H of the
Higgs simultaneously [6, 7]. In several situations, one also needs to assume that only the
MSSM Higgs fields Hu, Hdsplit at the GUT scale. This is termed as “just so” Higgs splitting
(HS) [8].
Restrictions placed on soft parameters by the t − b − τ unification, the LEP and LHC
bounds on the masses of the SUSY particles and other flavor violating observables have been
worked out in detail in number of papers [8–13]. Two viable scenarios and their properties
have been identified (see [14] for details and references therein). These depend on the sign of
the µ parameter of the MSSM. For example, one can achieve an exact t−b−τ unification in
mSUGRA itself for negative µ. But this needs very heavy SUSY spectrum with m0∼ 5−12
TeV and m1/2∼ (1.5 − 2)m0[10]. Also, perfect t − b − τ unification with relatively light
SUSY spectrum (∼ 2 TeV) can be obtained with the introduction of DT or purely Higgs
splitting. This appears to be the best and testable scenario as far as the t−b−τ unification is
concerned. But the supersymmetric contribution to the muon (g−2) is negative in this case.
This adds to the existing discrepancy between theory [15] and experiments [16]. Scenario
with positive µ proves better and allows the theoretical prediction for (g − 2) to agree with
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experiments within 3σ. In this case, achieving t−b−τ unification becomes considerably more
difficult. The mSUGRA in this case at best allows unification at 65% level [10]. Even with
non-universal boundary conditions, one needs specific relations between the soft parameters,
m10∼ 1.2m16, A0∼ −2m16and m1/2≪ m16together with tanβ ∼ 50 in order to achieve
t − b − τ unification [9]. The DT splitting in this case, allows unification at most 90% level
but this requires m16 ? 10 TeV and a gluino mass < 500 GeV. Just so HS works much
better than DT splitting and leads to the perfect Yukawa unification for m16∼ 10 TeV but
gluino is still light. In both these scenarios, the sparticle mass spectrum is characterized by
lighter gluino which is within the reach of current LHC searches at√s = 7 TeV, whereas all
scalar sparticles have masses beyond the TeV scale. The light gluino mainly decays through
a three body channel ˜ g → b¯b˜ χ0leading to multijets plus missing transverse energy. The final
states may also contain dileptons if˜t is lighter than˜b. Recently, the ATLAS experiment
with 2.05 fb−1data collected at√s = 7 TeV has excluded the light gluino masses below 620
GeV in SO(10)+HS model [17]. As we will show later, this experimental limit on gluino
mass rules out t − b − τ unification better than 90% for m16∼ 10 TeV. The strong bound
on the gluino mass follows from the need of suitable threshold corrections in bottom quark
mass to achieve t−b−τ unification which requires the hierarchy m1/2≪ m16. One can thus
raise the value of m1/2and hence the bound on the gluino mass by raising m16. The gluino
mass can be pushed in this case beyond the present experimental limit but at the cost of
choosing m16? 20 TeV [18].
The best viable scenario with µ > 0 corresponds to just so Higgs splitting and large
m16and m10. Theoretically, both these features are unsatisfactory. A large SUSY scale is
unnatural and just so HS breaks SO(10) explicitly. Just so HS can be indirectly introduced
through the right handed neutrinos at the intermediate scale. Apart from causing problem
with the gauge coupling unification, this case also does not do as well as the arbitrarily
introduced just so HS, see [12]. We wish to discuss here possible ways to improve on both
these aspects. Specifically, we show that just so HS arises naturally in realistic SO(10)
models containing additional Higgs fields, e.g. the one transforming as 126,126, and 210
representations under SO(10). Realistic fermion masses can be obtained if these fields are
present and the SU(2)Ldoublets contained in them mix with each other. Moreover, all the
angles involved in such mixing need not be small. We show that significant Higgs doublet
mixing can generate just so HS in the presence of non-universal but SO(10) preserving
boundary conditions at the GUT scale without inducing any splitting between squarks or
without significantly upsetting the t − b − τ Yukawa unification.
SO(10) models also allow an interesting possibility of matter fields mixing among them-
selves [19]. This leads to just so squark/slepton splitting similar to the just so HS occurring
due to Higgs mixing. Impact of such mixing was used earlier [19] to obtain departure from
the b-τ unification that follows in SU(5) or SO(10) models. Here we discuss phenomeno-
logical implications of such mixing in the context of the t − b − τ unification. We discuss
explicit example leading to independent squark and Higgs splitting and show that the pres-
ence of such splittings helps in raising the gluino mass prediction without raising the SUSY
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parameters m16to high values around 20 TeV.
This paper is organized as follows. In the next section, we present a short review of the
basic features of the t−b−τ unification and discuss the existing phenomenological results.
We also update the existing results incorporating the recent limit on Bs→ µ+µ−from LHCb
[20]. The viability of t−b−τ unified solutions in the presence of independent Higgs splitting
and squark splitting (SS) are discussed in Section III. In Section IV, we discuss how HS and
SS can arise in the realistic versions of SO(10) models. The study is summarized in the last
section.
II.
t − b − τ UNIFICATION IN MSSM
In this section, we review numerical and analytic results presented in the literature [8–13]
in the context of the t − b − τ unification. We have re derived several existing numerical
results in a way which optimizes the rate of the Bs→ µ+µ−to make it consistent with the
recent more stringent experimental [20] bound without pushing the SUSY scale to a higher
value. Before discussing this, we summarize aspects of the t−b−τ unification which allows
understanding of the salient key features.
The hypothesis of the t − b − τ unification assumes that at the GUT scale
mb
vcosβ= yτ≡
yb≡
mτ
vcosβ= yt≡
mt
v sinβ,
(1)
where v ≈ 174 GeV. This equation is motivated by a simple SO(10) model containing only
a single 10-plet Higgs. The appropriate choice of the free parameter tanβ in Eq. (1) can
always allow equality of ytwith ybor yτat the GUT scale. However it is well known [13] that
yband yτ and hence all three of them derived from Eq. (1) using the experimental values
for fermion masses extrapolated to the GUT scale do not unify for any value of tanβ. The
degree of unification of three Yukawas or lack of it is usually measured by the parameter
Rtbτ≡ R =Max.(yt,yb,yτ)
Min.(yt,yb,yτ). (2)
The parameter Rtbτ is defined at the GUT scale. Variation of Rtbτ with tanβ obtained
by using the tree level Yukawa couplings and fermion masses at MZ is shown in Fig. (1).
For comparison we also show similar ratios Rbτ and Rtτ defined using only two of the
Yukawa couplings. The extrapolation from MZto the GUT scale is done using the 1-loop
RG equations which are independent of the details of the soft SUSY breaking. Fig. (1)
explicitly shows that the three Yukawas do not meet for any tanβ and the reason is that b
and τ Yukawa couplings never meet. The best value of Rtbτ seen from Fig. (1) is around
1.2. Thus any scheme which tries to achieve t−b−τ unification should do better than this
tree level value.
It is known that the tree level Yukawa couplings, particularly that of the b quark receive [2,
3] significantly large radiative corrections once the supersymmetry is broken. The corrected
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102030
tan Β
40 5060
1.0
1.2
1.4
1.6
1.8
2.0
R
Rb Τ
Rtb Τ
Rt Τ
FIG. 1. Tree level Yukawa unification as a function of tanβ. Rijdefined in Eq. (2) measures the
closeness of yiand yjat the GUT scale.
ybcan be written as
yb= ytree
b
cosβ(1 + ∆y˜ g
b+ ∆y˜ χ
b+ ...) , (3)
where (...) contains the electroweak suppressed SUSY corrections, the standard model (SM)
electroweak corrections and logarithmic corrections which are sub dominant. The dominant
correction ∆y˜ g
[8]
b≈2g2
b(∆y˜ χ
b) induced by the gluino (chargino) exchange is approximately given by
∆y˜ g
3
3πµtanβm˜ g
m2
˜b2
, (4)
∆y˜ χ±
b
≈
y2
t
16π2µtanβAt
m2
˜t2
, (5)
where m˜b2(m˜t2) is mass of the heaviest sbottom (stop).
The presence of tanβ makes the radiative corrections significant. The corrections to top
Yukawa is inversely proportional to tanβ while corrections to tau Yukawa is proportional to
tanβ but electroweak suppressed. In order to achieve unification, one needs to reduce Rbτ
and hence ybcompared to its tree level value by about 10-20%. This requires that ∆y˜ g
in Eq. (3) should be negative. Since the gluino induced contribution dominates over most of
the parameter space, one can make the radiative corrections negative by choosing a negative
µ. As a result, models with negative µ achieve t−b−τ unification more easily. For positive
µ, the chargino contribution has to dominate over gluino and it should be negative. This
can be satisfied with a negative A0and light gluino with |A0|, m16≫ m1/2. As a result, all
the scenarios of t − b − τ unification with positive µ lead to a light gluino and very heavy
SUSY spectrum as borne out by the detailed numerical analysis [8–12].
The requirement of a light gluino as argued above directly conflicts with the requirement
of the REWSB unless an explicit HS is introduced. This can be seen as follows. In large
b+∆y˜ χ
b
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tanβ limit, the REWSB can be achieved if
− m2
∆m2
Hu> M2
H≡ m2
Z/2 ,
Hd− m2
Hu> M2
Z, (6)
where mHu,dare soft scalar masses evaluated at the weak scale. Starting with a positive value
at MGUT, m2
satisfied. But the large yb, yτas required in the t − b − τ unification drives m2
negative and conflicts with the second requirement. In addition to the Yukawas, the gaugino
and scalar mass terms also contribute to the ∆m2
the latter is negative, see the semi analytic solution of the 1-loop RG equations presented
for example in [4, 5]. In Fig. (2), we show the running ∆m2
Hugets driven to large negative values by a large ytand the first equation gets
Hdeven more
H. The former contribution is positive while
Hfor different values of m1/2.
2468 1012 14
?0.1
0.0
0.1
0.2
0.3
log ?Μ?GeV?
?mH
2?m0
2
m1? 2?0
m1? 2?m0
m1? 2?1.5m0
m1? 2?0
FIG. 2. The solution of 1-loop RGE equation for ∆m2
H= m2
Hd− m2
Huin mSUGRA for m0(≡
m10= m16) = 1 TeV, A0= 0 and tanβ=50.
As can be seen from the figure, second of Eq. (6) can be satisfied by choosing m1/2> m0
such that the gaugino induced contribution in ∆m2
m1/2and hence light gluino around m˜ g≤ 500 GeV is required if significant corrections to yb
is to be obtained in case of µ > 0. This corresponds to m1/2≤ 200 GeV for which REWSB
cannot be achieved unless one introduces splitting between Higgs fields at the GUT scale
itself. This is clearly seen from Fig. (2). Moreover, the case in which only Higgs splitting is
considered is more favorable than the D-term splitting. This follows [8] qualitatively from
Eqs. (4, 5) which implies
|∆y˜ g
|∆y˜ χ±
One finds m˜t2∼ m˜b2for mSUGRA as well for just so HS. The D-term splitting introduces
sfermion splitting together with Higgs splitting and leads to m˜t2> m˜b2. As a result, one
needs to choose even a lighter gluino or a larger |At| to suppress the gluino induced correc-
tions in yb. As we will show later in this paper, the additional squark splitting can instead
Hdominates. On the other hand, small
b|
b|
≈ 11πm˜ g
|At|
m2
m2
˜t2
˜b2
. (7)
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reduce the ratio m˜t2/m˜b2and make it less than one. This allows significantly higher gluino
mass.
The above qualitative features are borne out by several numerical studies presented in a
number of papers [9–12]. A list of different scenarios proposed to achieve t−b−τ unification
for µ > 0 is given in Table (1) in Ref. [12]. Among all the proposals, the SO(10) model
with just so HS is found as the best scenario which leads to an exact t − b − τ unification
corresponding to R = 1. It is shown [10] that HS works particularly well for large m16
and the Yukawa unification R ? 1.02 can be achieved if m16 ? 10 TeV. We update this
analysis for the following reasons. In [10], the t − b − τ unified solutions were obtained
considering the experimental constraint BF(Bs→ µ+µ−)(exp)< 2.6×10−6. The recent data
collected by LHCb [20] experiment has improved this bound significantly. The current limit
BF(Bs→ µ+µ−)(exp)< 4.5×10−9is three order of magnitude stronger than old bound. As a
result, all the solutions obtained in [10] are found inconsistent with new limit on Bs→ µ+µ−
(see, Tabel (1) in [10]). We repeat the old analysis considering the new limits on Bs→ µ+µ−
and b → sγ. In addition, we also consider the present constraints on B → τντ [21] which
was not considered in the old analysis.
We use the ISASUGRA subroutine of ISAJET 7.82 [22] in our numerical analysis. For
given boundary conditions (soft SUSY parameters at the GUT scale and mt, tanβ at the
weak scale), ISASUGRA solves full 2-loop MSSM RG equations and incorporates 1-loop
SUSY threshold corrections in all the MSSM sparticles and in the masses of third generation
fermions. Moreover, it checks for (a) non-techyonic solutions and (b) consistent REWSB
using the minimization of one-loop corrected effective MSSM Higgs potential. Once these
conditions are satisfied, we calculate R using Eq. (2). Then using CERN’s subroutine
MINUIT, we minimize R. Finally, we calculate branching factor for b → sγ, Bs→ µ+µ−
using the IsaTools package [22] and B → τντ using the expressions given in [23]. For
completeness, we also estimate the relic abundance of neutralino dark matter ΩCDMh2and
the SUSY contribution to anomalous magnetic moment of muon ∆aµ(where aµ= (g−2)/2)
using IsaTools. On the acquired solutions, we apply the following constraints obtained from
experimental data:
BF(Bs→ µ+µ−) < 4.5 × 10−9
BF(b → sγ)
BF(B → τντ)
[20] (8)
2.78 × 10−4≤
0.62 × 10−4≤
≤ 4.32 × 10−4(3σ)
≤ 2.66 × 10−4(3σ)
[21](9)
[24](10)
Further, we impose the mass bounds given in PDG [25] on all sparticles including the LEP
[26] bound on the mass of lightest Higgs (mh> 114.4 GeV). We use mt= 172.9 GeV in our
analysis. The recent LHC limit on gluino mass is not considered here. We will discuss it in
detail in the next section.
The results of our numerical analysis are displayed in Table (I). We study three different
cases:
1. In case I, we do not impose constraint (8) and minimize R for fixed m16= 10 TeV. The
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Parameter Case ICase IICase III Case IV
m16
m1/2
A0/m16
m10/m16
tanβ
∆m2
10000 100001500020000
34.05 43.0 51.07 62.65
−2.29
1.08
−2.26
1.11
−2.29
1.08
−2.45
0.94
51.39 51.6251.3954.81
H/m2
R
10
0.250.280.250.39
1.011.041.031.02
m˜ g
m˜ χ0
m˜ χ0
m˜ χ+
345.0 367.9 487.7 634.9
1,2
49.6, 118.9 53.2, 127.1 73.6, 177.9 101.4, 241.7
3,4
6658.5, 6658.6 6460.2, 6460.3 9966.9, 9966.9 17102, 17103
1,2
119.6, 6650.6 126.5, 6452.6 183.7, 9953.2 242.9, 17097
m˜ uL,R
m˜dL,R
m˜ νe,τ
m˜ eL,R
m˜t1,2
m˜b1,2
m˜ τ1,2
mh
mH
mA
mH+
9984.3, 9891.6 9988.4, 9874.7 14988, 14850 20010, 19782
9984.7, 10043 9988.8, 1005214988, 1507520010, 20126
9926.0, 7374.4 9912.4, 7274.3 14893, 1119319841, 14636
9924.7, 10134 9911.2, 10159 14890, 15199 19838, 20312
2554.9, 3181.62506.2, 3110.1 3957.5, 4859.7 5915.9, 6807.6
2997.6, 3341.7 2902.4, 3226.2 4724.0, 5097.9 6538.7, 6984.1
3904.9, 7365.2 3664.9, 7278.2 6329.0, 11192 7549.7, 14617
126.0 125.8 127.8 125.0
3463.6 4353.4 5664.2 8683.1
3441.4 4325.4 5627.4 8626.5
3465.5
3.09 × 10−4
0.79 × 10−4
4.55 × 10−9
0.024 × 10−10
2737
4354.9
3.07 × 10−4
0.79 × 10−4
4.22 × 10−9
0.026 × 10−10
719
5665.3
3.06 × 10−4
0.79 × 10−4
4.15 × 10−9
0.008 × 10−10
39866
8683.8
3.06 × 10−4
0.79 × 10−4
3.96 × 10−9
0.008 × 10−11
14115
BF(b → sγ)
BF(B → τντ)
BF(Bs→ µ+µ−)
∆aµ
ΩCDMh2
TABLE I. The benchmark solutions obtained for t−b−τ Yukawa unification in SO(10)+HS model
for positive µ. Different columns correspond to different cases discussed in the text. All masses
are in GeV units.
best unification found corresponding to R = 1.01. The solution predicts BF(Bs→
µ+µ−) = 4.55 × 10−9which is slightly above the upper bound (8).
2. The dominant contribution to Bs→ µ+µ−in MSSM is proportional to m−4
mAis the mass of pseudo-scalar Higgs. Using this fact, we simultaneously maximize
mAand minimize R in case II for m16= 10 TeV. As a result, we get a lower value of
BF(Bs→ µ+µ−) which is consistent with experimental limit (8). However one gets a
A[27] where
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slight declination in Yukawa unification in this case.
3. Case III and IV correspond to an obvious way of decreasing SUSY contribution to
flavor violation namely, uplifting the SUSY scale. We take m16= 15 and 20 TeV which
increase the masses of all SUSY spectrum including mA. The unification achieved in
these cases is 97-98%.
It is clear from the results of our analysis that a very good t − b − τ Yukawa unification
can still be achieved with “low” m16 in SO(10)+HS model without violating the present
experimental constraint on Bs→ µ+µ−. The calculated values of BF(b → sγ), BF(B → τντ)
and ∆aµshown in Table (I) are almost similar to their standard model values. The SUSY
contributions to these processes are negligible due to the heavy sparticle spectrum one
typically gets in t− b− τ unified solutions for µ > 0. Among the other well known features
of t − b − τ unification with positive µ are relatively heavier Higgs mh∼ 125 − 130 GeV
arising due to large m16and the condition A0∼ −2m16[9, 28] and pure bino like lightest
neutralino which leads to the over abundance of the neutralino dark matter [11]. One would
need additional mechanism, e.g. tiny R parity violation [29] to reduce this abundance.
III.
t − b − τ UNIFICATION AND HEAVIER GLUINO
As noted earlier, the t−b−τ unified solutions for positive µ generally require very light
gluino mass ? 500 GeV. The direct SUSY searches at the LHC has now excluded m˜ g? 620
GeV in SO(10)+HS model [17]. As a result, the solutions displayed in first three columns
in Table (I) are ruled out. It is recently shown that consistent t − b − τ unification with
heavier gluino can be obtained by increasing m16[18]. In fact one needs m16? 20 TeV to
evade the present LHC bound on gluino mass e.g. case IV in Table (I). We propose here
an alternate way to obtain heavier gluino in t − b − τ unified solution without increasing
m16. As mentioned in Section II, the ratio |∆y˜ g
suppression if m˜t2< m˜b2. This allows t−b−τ unification with heavier gluinos. The required
mass hierarchy m˜t2< m˜b2can be obtained if the appropriate squark splitting is introduced
at the GUT scale. For example, consider SU(5) invariant boundary conditions:
b|/|∆y˜ χ±
b| in Eq. (7) can get an additional
m2
m2
˜Q= m2
˜L= m2
˜U= m2
˜ D≡ m2
˜E≡ m2
16+ ∆m2
16,
S. (11)
The origin of such splitting in an SO(10) model will be discussed in the next section. As we
will show later in Eq. (31, 32), ∆m2
choice ∆m2
its value obtained with universal squark masses at the GUT scale. This leads to m˜t2< m˜b2
at weak scale which is the case of our interest.
To quantify the effect of squark splitting, we study the above case through detailed
numerical analysis. Fixing m16= 10 TeV we perform a random scan over all the remaining
Sis allowed to take any values greater than −m2
16. The
S> 0 raises the mass of one eigenstate of sbottom squarks, i.e. m˜b2, compared to
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soft parameters and tanβ. The analysis is performed for two scenarios: (1) only Higgs
splitting, i.e.
for ∆m2
∆m2
and experimental constraints discussed in Section II. The results of numerical analysis are
shown in Fig. (3).
S= 0 and (2) Higgs splitting + Squark splitting (HS+SS) with
S> 0. We employ the same numerical technique and apply all the theoretical
H,∆m2
0.5 1.01.5 2.02.5
1.00
1.05
1.10
1.15
1.20
mg??TeV?
Rtb Τ
0.5 1.01.52.0 2.5
1.00
1.05
1.10
1.15
1.20
mg??TeV?
Rtb Τ
FIG. 3. Solutions of t − b − τ Yukawa unification as a function of gluino mass for m16= 10 TeV.
The figure in the left (right) panel shows the solutions obtained with only HS (HS+SS) at the GUT
scale in SO(10). The vertical line corresponds to the lower bound on gluino mass in SO(10)+HS
model derived from the recent ATLAS data [17]. All the points shown are consistent with various
phenomenological constraints discussed in the text.
As can be seen from Fig. (3), the lower bound on gluino mass in SO(10)+HS model
rules out the Yukawa unification R < 1.08. In the presence of additional squark splitting,
the unification up to 99% can be achieved without violating the present LHC limit on
gluino mass. Also, a relatively heavier gluino up to 1.5 TeV is obtained assuming at most
10% deviation in Yukawa unification without uplifting m16. It can be also seen that the
number of valid solutions obtained with HS+SS are more compared to those obtained with
only HS. In Fig. (4), we show the ratio m˜t2/m˜b2in HS and HS+SS models. The ratio
substantially decreases in HS+SS model compared to its value without squark splitting. As
we mentioned earlier in this section, this allows a heavier gluino in the spectrum without
uplifting the SUSY braking scale. Note that with m16 = 10 TeV, m˜ g > 1.5 TeV cannot
be obtained for R ? 1.1 even in the HS+SS model. This range of gluino mass is still in
the reach of LHC and its future operations at√s = 14 TeV can significantly constraint the
parameter space of HS+SS model if not rule it out completely.
IV.
t − b − τ UNIFICATION AND REALISTIC SO(10)
We consider two categories of SO(10) models. One in which Higgs sector is extended
to obtain realistic fermion masses and mixing and the other in which one introduces also
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0.40.60.81.0
1.00
1.05
1.10
1.15
1.20
mt??mb
?
Rtb Τ
FIG. 4. Squark mass ratio m˜t2/m˜b2obtained in HS (blue points) and HS+SS (red points) models.
additional matter multiplet at MGUT. The former class of models lead to just so HS and
the latter also to an independent squark splitting. We discuss them in turn.
A. Just so HS in realistic SO(10)
SO(10) model containing a 10-plet of Higgs field 10H allows the following term in the
superpotential:
Y1016F16F10H,(12)
where 16F refers to the matter multiplet and Y10 to the Yukawa coupling matrix in the
generation space. The t − b − τ unification follows under two assumptions:
(A) Third generation of fermions obtain their masses only from Eq. (12).
(B) The MSSM fields Huand Hdreside solely in 10H.
One however needs to violate both these assumptions in order to obtain correct masses for all
fermions and non-trivial mixing among them. We estimate the effects of these violations on
the t−b−τ unification. We will take as an example a popular minimal renormalizable SO(10)
model [30] which is found to explain fermion masses and mixing in a number of situations,
for instance, see [31] and references therein. The model contains a 126H field to generate
neutrino mass and a 126Hto preserve supersymmetry at the GUT scale. In addition, it has
a Higgs transforming as 210Hrepresentation of SO(10) which breaks SO(10) to MSSM.
Due to the presence of additional Higgs fields particularly 210H, SU(2)Ldoublets resid-
ing in various Higgs fields mix with each other. This mixing plays an important role in
generating the right type of the second generation masses [30]. But as we show below, this
mixing also generates just so Higgs splitting required to obtain t−b−τ unification if the soft
masses of the 10H, 126Hand 210Hfields are non-universal. On the negative side, the pres-
ence of 126Hand the Higgs mixing also lead to departures from the exact t−b−τ unification.
11
Page 12
The 10H, 126H, 126H, 210Hfields each contain four down-type and four up-type standard
model doublets. They mix and produce four mass eigenstates denoted as hu,d
a:
hu,d
b
= Ou,d
abφu,d
a
,(13)
where φu,d
through fine tuning only Hu≡ hu
terms of various SO(10) Higgs prior to the SO(10) breaking:
a
are components of bi-doublets in 10H, 126H, 126H, 210H. One assumes that
1and Hd≡ hd
1remain light. Consider now the soft mass
Vsoft∋ m2
1616†
F16F+ m2
1010†
H10H+ m2
126(126†
H126H+ 126†
H126H) + m2
210210†
H210H. (14)
Here we have assumed equal masses for the 126Hand 126Hfields to avoid non-zero D-term.
The masses of the other Higgs multiplets are taken non-universal. Substitution of Eq. (13)
in Eq. (14) leads to
Vsoft∋ m2
+ (m2
16(˜Q†˜Q +˜U†˜U +˜D†˜D +˜L†˜L +˜E†˜E)
10|Ou
+ (u → d).
11|2+ m2
126(|Ou
21|2+ |Ou
31|2) + m2
210|Ou
41|2)H†
uHu
(15)
The above equation leads to the following boundary conditions at the GUT scale
m2
˜ Q= m2
Hu,d= m2
˜U= m2
126+ |Ou,d
˜ D= m2
11|2(m2
˜L= m2
10− m2
˜ E= m2
126) + |Ou,d
16,
m2
41|2(m2
210− m2
126) .(16)
It is seen that the Higgs mixing generated through Eq. (13) has produced the desirable
splitting only among Hu,dmasses without splitting squarks from each other unlike in case
of the popular D-term splitting.
Let us now look at the impact of Higgs mixing on the t−b−τ unification. The presence
of 126Hfield modifies Eq. (12) to
Y1016F16F10H+ Y12616F16F126H,(17)
where Y126additional Yukawa coupling matrix. By substituting Eq. (13) in Eq. (17) one
arrives at the charged fermion mass matrices:
Md= υd(Y10Od
Ml= υd(Y10Od
Mu= υu(Y10Ou
11+ Y126Od
11− 3Y126Od
11+ Y126Ou
21),(18)
21),(19)
21),(20)
where υu,ddenote the vacuum expectation values of the neutral component of Hu,d. We can
go to a basis with Y10diagonal. Neglecting the contribution of 126Hfor the time being, one
has
yb= yτ=Od
11
Ou
11
yt.(21)
12
Page 13
Thus one source of the departure from the t − b − τ unification is the ratio Od
parametric form of the matrices Ou,dis worked out [32, 33] in the model under consideration
and we closely follow the notation in [33]. This is based on the Higgs superpotential
11/Ou
11. The
WH= M10102
+ λ 2103
H+ M2102102
H+ M126126H126H
H+ η 210H126H126H+ 210H10H(α126H+ α126H).(22)
The Higgs mass matrices and hence Ou,dfollow from the above superpotential after the
SO(10) breaking and in the most general situation with SO(10) breaking to standard model
one obtains (see, Eqs. (C18, C19) in [33])
√6α
p3φd
?
Hd= Nd
?
2p5
x − 1φd
2p5
x − 1φu
1−
η(2x − 1)(x + 1)p5
√6¯ α
2−
√6¯ α
η(3x − 1)(x3+ 5x − 1)φd
√6α
η(3x − 1)(x3+ 5x − 1)φu
3+ ¯ ασ
mφp′
3φd
4
?
?
,
Hu= Nu
1−
η(2x − 1)(x + 1)p5
p3φu
3−
2− ασ
mφp′
3φu
4
,
(23)
where Nu,dare overall normalization constants. x is an arbitrary parameter and p3,p′
polynomial in x. The expressions of elements of Ou,dcan be read from the above equation.
The ratio Ou
nearly one in a number of situations. Obvious case is the limit Nu= Nd≈ 1 corresponding
to the situation in which extra Higgs fields’ φu,d
In general, various φu
10H, 126H, 126H, 210Hrepresentations. They are thus distinguished by the SU(2)Rgroup.
One therefore automatically has Ou= Odas long as SU(2)Ris unbroken. This happens
[33] for x = 0. One can then show from Eq. (23) that Ou= Odin this case. Another
interesting limit corresponds to choosing α = ¯ α in Eq. (23). In this limit, Ou?= Odbut
still Ou
SU(2)Ris broken. This case also corresponds to Ou
this limit as follows from Eq. (16). But mild deviation from the limit α = ¯ α can generate
sizable HS and approximate t−b−τ unification for large ranges in other parameter. This is
illustrated in Fig. (5) where we showNu
of ǫ = 1/2(¯ α − α) = 0.1, 0.2. The remaining unknown parameters appearing in Eq. (23)
are equated to 1. It is seen that for most values of the unknown parameter x one obtains
almost exact t − b − τ unification, i.e.
Another threat to the t−b−τ unification comes due to the presence of the Y126Yukawa
couplings in Eq. (18). This effect is somewhat model-dependent and we estimate it by
specializing to the case of the second and third generations. We can write the charged
fermion mass matrices as
?
xh3+ x3
?
sxh3+ sx3
3,p5are
11/Od
11which measures deviation from t − b − τ unification can be exactly or
2,3,4contribution to Hu,dare sub-dominant.
aare components of SU(2)L× SU(2)Rbi-doublets residing in
aand φd
11= Od
11thus one obtains exact yb = ytin Eq. (21) even in the situations where
41= −Od
41and thus HS also vanish in
Ndand |Od
41
41| as a function of x for two specific values
Ou
Nd
Nu= 1 and non-zero HS [34] as given in Eq. (16).
Md= υd
h2+ x2
x
?
;Ml= υd
?
h2− 3x2
−3x
−3x
h3− 3x3
?
;
Mu= υuOu
11
Od
11
h2+ sx2
sx
?
,(24)
13
Page 14
0.6 0.81.0 1.21.4 1.61.82.0
1.0
1.1
1.2
1.3
1.4
x
Nu
Nd
&
O41
d
O41
u
Nu? Nd
?O41
d???O41
u?
?O41
d???O41
u?
FIG. 5. The Higgs mixing parametersNu
Nd(continuous lines) and |Od
41
41| (dashed lines) as a function
Ou
of x in the minimal SUSY SO(10) model. The blue (red) line corresponds to parameter ǫ = 0.1
(0.2).
where s ≡Ou
symmetric Od
Approximate t − b − τ unification is obtained with the hierarchy x,x2,x3≪ h3. Assuming
h2≪ x2then leads to the desirable mass relation 3ms= mµ. Given this hierarchy one finds
for real parameters
x2∼ms
21
Od
21
21Y126. Several of these can be determined from the known masses and mixing.
Od
Ou
11
11
. Here h2,3refer to elements of the diagonal Y10Od
11and x2,x,x3to that of
mb;x ∼ mbVcb(1 − s);s ∼mb
ms
mc
mt
The ratio x2/x3remains undermined. If type II seesaw dominates and Y126is to provide the
neutrino masses and mixing then x2∼ x3[35]. In this case, one finds
yb≈ h3
?
?
11
1 +ms
mb
??1 + O(V2
??1 + O(V2
1 +mc
mt
cb)?,
cb)?,
yτ≈ h3
yt≈Ou
1 − 3ms
?
mb
Od
11
h3
?
, (25)
where we have used Eq. (1). Using value for the mass ratios at the GUT scale for tanβ = 50
[36],
The ms,mcdependent terms in Eq. (25) get suppressed if one assumes x3≪ x2. In this
case, one can still obtain b − τ unification and reproduce the second generation masses and
Cabibbo mixing. The Higgs mixing factor can be nearly one as argued before. It is thus
quite plausible that one can obtain almost exact Yukawa unification not just in simplified
but also in more realistic GUT based on SO(10).
ms
mb≈ 0.016,mc
mt≈ 0.0023 already implies about 6% departure from b − τ unification.
14
Page 15
B.Squark splitting and t − b − τ unification
The squark splitting can be induced by adding new matter fields having the same quan-
tum numbers as some of the squarks and letting them mix with the normal squarks. The
minimal possibility at the SO(10) level is introduction of a 10M field. This contains fields
transforming under an SU(5) subgroup of SO(10) as 5′
with 5M contained in the matter multiplet 16M of SO(10). This mixing can generate the
squark splitting. Let us discuss details within a specific model which has been studied ex-
tensively [37, 38] for several different reasons. The model contains three generations of 16M
and three copies of 10M. We will explicitly consider only the third generation and a 10Min
the following. The Higgs sector consists of the usual 10Hsupplemented by 16H+ 16Hand
45H+ 54H. The 16His introduced to break the B − L gauge symmetry and 45H+ 54Hare
needed to complete the breakdown of SO(10) to SM. This model serves as a good example
in which (a) independent squark and Higgs splitting can be generated and (b) there exit
ranges of parameters for which t − b − τ unification is approximately maintained. We shall
discuss only a part of the superpotential relevant to describe Higgs and squark mixing, see
[38] for a general discussion of the model. The superpotential of the model can be divided
in two parts one describing matter-Higgs interaction and the other describing Higgs-Higgs
interactions:
M+ 5′
M. Of these, the 5′
Mcan mix
WM= Y 16M16M10H+ F16M10M16H+ M10M10M,
WH= M1616H16H+ M1010H10H+ H16H16H10H+ H′16H16H10H.(26)
The above superpotential is designed to respect the matter parity under which all the matter
(Higgs) fields are odd (even). This symmetry is essential for preventing renormalizable
baryon and lepton number violating terms. Scalar components of none of the matter fields
acquire vacuum expectation value (vev) and thus matter parity remains unbroken. Only
fields appearing in the above superpotential and acquiring the GUT scale vev are 1H+ 1H
contained in 16H+ 16Hof SO(10). Thus after the GUT scale breaking, above superpotial
maintains invariance under the SU(5) subgroup of SO(10). As a result, the mixing between
the following SU(5) components is allowed and arise from Eq. (26):
?
5l
5h
M
M
?
= R(θ)
?
5M
5′
M
?
;
?
5l
5h
H
H
?
= R(γ)
?
5H
5′
H
?
;
?
5l
5h
H
H
?
= R(δ)
?
5H
5′
H
?
,(27)
where
R(j) =
?
cosj −sinj
sinj cosj
?
.
The fields with (without) prime are component of the original 10 (16+ 16) of SO(10). It is
assumed that fields labeled with superscript l are kept light by fine tuning and those with
the superscript h pick up masses at the GUT scale. The mixing angles appearing above are
15
Page 16
related to parameters in Eq. (26) and are explicitly given as
tanθ =Fv16
M
,
tan2γ =
2v16(HM10+ H′M16)
(M2
2v16(H′M10+ HM16)
(M2
16− M2
10) + v2
16(H2− H′2),
16(H2− H′2),
tan2δ =
16− M2
10) − v2
(28)
where v16is a vev of SU(5) singlets residing in 16H+16H. The light fields transform under
SU(5) as 5H+ 5H of Higgs and 10M+ 1M+ 5M which together makes 16M and 10H of
SO(10). The effective Yukawa couplings of these fields can be worked out from Eqs. (26,
27) in a straightforward way. One finds
Weff= Y sδ10M10M5l
H+ (Y sγcθ+ Fsθcγ)10M5l
M5l
H, (29)
where sj= sinj and cj= cosj.
The Higgs mixing as given in Eq. (27) leads to both the squark and Higgs splitting
through SO(10) invariant non-universal boundary conditions. Consider the following soft
terms:
Vsoft∋ m2
= m2
+ m′2
1616†
16(10†
16(10†
M16M+ m2
M10M+ 5†
H10H+ 5†
1010†
M5M+ 1†
H5H+ 1†
M10M+ m′2
M1M) + m2
H1H) + m′2
1616†
H16H+ m′2
10(5′†
10(5′†
1010†
H10H
M5′
H5′
M5′
H5′
M+ 5′†
H+ 5′†
M)
H).(30)
Substitution of Eq. (27) in Eq. (30) leads to the following boundary conditions for the soft
mass parameters of squarks and Higgs:
m2
m2
m2
m2
˜ Q= m2
˜L= m2
Hd= m′2
Hu= m′2
˜U= m2
˜D= m2
16c2
16c2
˜ E= m2
16+ s2
γ+ m′2
δ+ m′2
16,
θ(m2
10− m2
16),
10s2
10s2
γ,
δ
(31)
resulting in the following squarks and Higgs splitting
∆m2
∆m2
S≡ m2
H≡ m2
˜ Q− m2
Hd− m2
˜ D= s2
Hu= (s2
θ(m2
10− m2
γ− s2
16),
δ)(m′2
10− m′2
16).(32)
Thus model under consideration simultaneously generates independent mixing among
squarks and Higgses that result into squark splitting and Higgs splitting respectively. The
mixing angles which generate these splitting also lead to departure from the exact t−b−τ
unification as before. But the exact b-τ unification follows for arbitrary values of these
mixing angles. Even t − b − τ unification also holds approximately in limiting cases, e.g.
Fsθ≪ 1 and H = H′. Another interesting limit corresponds to sδ≈ sγcθand Fsθ≪ 1. In
this limit one gets yb= yτ≈ ytand simultaneously non-zero splittings, ∆m2
m10= m′
required for the REWSB when ∆m2
H≈ s2
γ∆m2
H> 0 as
Sfor
10and m16= m′
16in Eq. (32). This relation automatically implies ∆m2
Sis chosen positive to raise the gluino mass limit.
16
Page 17
V.SUMMARY
We have addressed two important issues in this paper in the context of the t − b − τ
unification in SO(10) broken to MSSM with a positive µ parameter. The existing analysis
[9–12] have either assumed only HS with degenerate squarks at the GUT scale or a D-term
splitting in which case squark splitting is correlated to the HS. This scenario appears to be
ruled out save for very high SUSY scale m16around 20 TeV [18]. Detailed phenomenological
analysis presented here shows that independent and positive squark splitting ∆m2
can change the allowed gluino mass and scenario can be compatible with t − b − τ unifica-
tion R ∼ 1.01 and the recent ATLAS bound m˜ g> 620 GeV. Moreover, R < 1.1 requires
m˜ g< 1.5 TeV. Thus viability of the t−b−τ unification can be tested in future at LHC with
√s = 14 TeV. The other issue addressed here concerns the origin of just so HS. While just
so HS is introduced as a phenomenological parameter in many works [9–12], its origin is not
justified in most of the existing analysis, see however [6, 39]. We have taken here a concrete
realistic model [30] used to understand fermion mass spectrum and shown within it that
just so HS is an automatic consequence of the non-universal boundary conditions and Higgs
mixing. It is also shown that one can obtain sizable just so HS and retain almost exact
t − b − τ unification in this realistic model. Independent squark splitting required to relax
the gluino mass bound is also shown to follow in an extended model in which squarks mix
with additional matter multiplet. One may conclude based on the analysis presented here
that t−b−τ unification with positive µ is still phenomenologically viable and theoretically
well-founded.
S, Eq. (11)
ACKNOWLEDGMENTS
ASJ thanks the Department of Science and Technology, Government of India for support
under the J. C. Bose National Fellowship programme, grant no. SR/S2/JCB-31/2010. KMP
would like to thank Partha Konar for useful discussions on ISAJET package.
[1] B. Ananthanarayan, G. Lazarides and Q. Shafi, Phys. Rev. D 44, 1613 (1991);
B. Ananthanarayan, G. Lazarides and Q. Shafi, Phys. Lett. B 300, 245 (1993);
G. W. Anderson, S. Raby, S. Dimopoulos and L. J. Hall, Phys. Rev. D 47, 3702 (1993)
[hep-ph/9209250].
[2] R. Hempfling, Phys. Rev. D 49, 6168 (1994);
L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D 50, 7048 (1994) [hep-ph/9306309];
T. Blazek, S. Raby and S. Pokorski, Phys. Rev. D 52, 4151 (1995) [hep-ph/9504364].
17
Page 18
[3] M. S. Carena, M. Olechowski, S. Pokorski and C. E. M. Wagner, Nucl. Phys. B 426, 269
(1994) [hep-ph/9402253].
[4] M. Olechowski and S. Pokorski, Phys. Lett. B 344, 201 (1995) [hep-ph/9407404].
[5] H. Murayama, M. Olechowski and S. Pokorski, Phys. Lett. B 371, 57 (1996) [hep-ph/9510327].
[6] R. Rattazzi and U. Sarid, Phys. Rev. D 53, 1553 (1996) [hep-ph/9505428].
[7] M. Drees, Phys. Lett. B 181, 279 (1986);
C. F. Kolda and S. P. Martin, Phys. Rev. D 53, 3871 (1996) [hep-ph/9503445].
[8] T. Blazek, R. Dermisek and S. Raby, Phys. Rev. Lett. 88, 111804 (2002) [hep-ph/0107097];
T. Blazek, R. Dermisek and S. Raby, Phys. Rev. D 65, 115004 (2002) [hep-ph/0201081].
[9] H. Baer and J. Ferrandis, Phys. Rev. Lett. 87, 211803 (2001) [hep-ph/0106352].
[10] D. Auto, H. Baer, C. Balazs, A. Belyaev, J. Ferrandis and X. Tata, JHEP 0306, 023 (2003)
[hep-ph/0302155].
[11] H. Baer, S. Kraml, S. Sekmen and H. Summy, JHEP 0803, 056 (2008) [arXiv:0801.1831
[hep-ph]];
H. Baer, M. Haider, S. Kraml, S. Sekmen and H. Summy, JCAP 0902, 002 (2009)
[arXiv:0812.2693 [hep-ph]].
[12] H. Baer, S. Kraml and S. Sekmen, JHEP 0909, 005 (2009) [arXiv:0908.0134 [hep-ph]].
[13] I. Gogoladze, R. Khalid, S. Raza and Q. Shafi, JHEP 1012, 055 (2010) [arXiv:1008.2765
[hep-ph]];
I. Gogoladze, Q. Shafi and C. S. Un, Phys. Lett. B 704, 201 (2011) [arXiv:1107.1228 [hep-ph]];
M. Badziak, M. Olechowski and S. Pokorski, JHEP 1108, 147 (2011) [arXiv:1107.2764 [hep-
ph]].
[14] M. Badziak, arXiv:1205.6232 [hep-ph].
[15] M. Passera, W. J. Marciano and A. Sirlin, Phys. Rev. D 78, 013009 (2008) [arXiv:0804.1142
[hep-ph]];
M. Passera, W. J. Marciano and A. Sirlin, AIP Conf. Proc. 1078, 378 (2009) [arXiv:0809.4062
[hep-ph]].
[16] G. W. Bennett et al. [Muon G-2 Collaboration], Phys. Rev. D 73, 072003 (2006)
[hep-ex/0602035].
[17] G. Aad et al. [ATLAS Collaboration], [arXiv:1203.6193 [hep-ex]].
[18] H. Baer, S. Raza and Q. Shafi, arXiv:1201.5668 [hep-ph].
[19] S. Dimopoulos and A. Pomarol, Phys. Lett. B 353, 222 (1995) [hep-ph/9502397].
[20] R. Aaij et al. [LHCb Collaboration], arXiv:1203.4493 [hep-ex].
[21] D. Asner et al. [Heavy Flavor Averaging Group Collaboration], arXiv:1010.1589 [hep-ex].
[22] H. Baer, F. E. Paige, S. D. Protopopescu and X. Tata, hep-ph/0312045.
[23] D. Eriksson, F. Mahmoudi and O. Stal, JHEP 0811, 035 (2008) [arXiv:0808.3551 [hep-ph]].
[24] R. Barlow [BABAR Collaboration], arXiv:1102.1267 [hep-ex].
[25] Particle Data Group Collaboration, K. Nakamura et. al., J.Phys.G G37 (2010) 075021.
[26] S. Schael et. al., Eur.Phys.J. C47 (2006) 547–587,
18
Page 19
[27] G. L. Kane, C. Kolda and J. E. Lennon, hep-ph/0310042.
[28] H. Baer, V. Barger and A. Mustafayev, Phys. Rev. D 85, 075010 (2012) [arXiv:1112.3017
[hep-ph]].
[29] V. Berezinsky, A. S. Joshipura and J. W. F. Valle, Phys. Rev. D 57, 147 (1998)
[hep-ph/9608307].
[30] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 70, 2845 (1993) [hep-ph/9209215].
[31] A. S. Joshipura and K. M. Patel, Phys. Rev. D 83, 095002 (2011) [arXiv:1102.5148 [hep-ph]].
[32] C. S. Aulakh and A. Girdhar, Int. J. Mod. Phys. A 20, 865 (2005) [hep-ph/0204097];
T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac and N. Okada, J. Math. Phys. 46, 033505
(2005) [arXiv:hep-ph/0405300].
[33] B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, Phys. Rev. D 70, 035007 (2004)
[hep-ph/0402122].
[34] This remains true for most values of x which are not very near to zeros of the polynomial p3
or p′
3.
[35] B.Bajc, G.Senjanovic and F. Vissani,Phys.Rev. Lett.
90, 051802(2003)
[arXiv:hep-ph/0210207].
[36] G. Ross and M. Serna, Phys. Lett. B 664, 97 (2008) [arXiv:0704.1248 [hep-ph]].
[37] S. M. Barr, Phys. Rev. D 21, 1424 (1980);
Y. Nomura and T. Yanagida, Phys. Rev. D 59, 017303 (1999) [hep-ph/9807325];
M. Frigerio, P. Hosteins, S. Lavignac and A. Romanino, Nucl. Phys. B 806, 84 (2009)
[arXiv:0804.0801 [hep-ph]];
S. M. Barr, Phys. Rev. D 76, 105024 (2007) [arXiv:0706.1490 [hep-ph]].
[38] M. Malinsky, Phys. Rev. D 77, 055016 (2008) [arXiv:0710.0581 [hep-ph]];
M. Heinze and M. Malinsky, Phys. Rev. D 83, 035018 (2011) [arXiv:1008.4813 [hep-ph]].
[39] S. Dimopoulos and L. J. Hall, Phys. Lett. B 344, 185 (1995) [hep-ph/9411273].
19
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