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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 10 12 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 I Case IICase IIICase IV

m16

m1/2

A0/m16

m10/m16

tanβ

∆m2

10000 1000015000 20000

34.05 43.0 51.0762.65

−2.29

1.08

−2.26

1.11

−2.29

1.08

−2.45

0.94

51.39 51.62 51.3954.81

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, 6460.39966.9, 9966.9 17102, 17103

1,2

119.6, 6650.6126.5, 6452.6 183.7, 9953.2242.9, 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, 9891.69988.4, 9874.7 14988, 1485020010, 19782

9984.7, 100439988.8, 1005214988, 1507520010, 20126

9926.0, 7374.49912.4, 7274.314893, 1119319841, 14636

9924.7, 101349911.2, 1015914890, 1519919838, 20312

2554.9, 3181.62506.2, 3110.13957.5, 4859.75915.9, 6807.6

2997.6, 3341.72902.4, 3226.24724.0, 5097.96538.7, 6984.1

3904.9, 7365.23664.9, 7278.26329.0, 111927549.7, 14617

126.0125.8127.8 125.0

3463.64353.45664.28683.1

3441.44325.45627.48626.5

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

0.51.01.52.02.5

1.00

1.05

1.10

1.15

1.20

mg??TeV?

Rtb Τ

0.51.0 1.52.0 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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0.40.60.8 1.0

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

0.60.81.0 1.21.41.61.8 2.0

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