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arXiv:0910.2732v2 [hep-ph] 13 Nov 2009

Extra vector-like matter and the lightest Higgs scalar boson mass

in low-energy supersymmetry

Stephen P. Martin

Department of Physics, Northern Illinois University, DeKalb IL 60115, and

Fermi National Accelerator Laboratory, P.O. Box 500, Batavia IL 60510.

The lightest Higgs scalar boson mass in supersymmetry can be raised significantly

by extra vector-like quark and lepton supermultiplets with large Yukawa couplings

but dominantly electroweak-singlet masses. I consider models of this type that main-

tain perturbative gauge coupling unification. The impact of the new particles on

precision electroweak observables is found to be moderate, with the fit to Z-pole

data as good or better than that of the Standard Model even if the new Yukawa cou-

plings are as large as their fixed-point values and the extra vector-like quark masses

are as light as 400 GeV. I study the size of corrections to the lightest Higgs boson

mass, taking into account the fixed-point behavior of the scalar trilinear couplings.

I also discuss the decay branchings ratios of the lightest new quarks and leptons and

general features of the resulting collider signatures.

Contents

I. Introduction

2

II. Supersymmetric models with new vector-like fields

A. Field and particle content

B. Renormalization group running

C. Fine-tuning considerations

4

4

6

11

III. Corrections to the lightest Higgs scalar boson mass

14

IV. Precision electroweak effects

17

V. Collider phenomenology of the extra fermions

A. The LND model

B. The QUE model

C. The QDEE model

20

22

24

26

VI. Outlook

28

Appendix A: Contributions to precision electroweak parameters

29

Appendix B: Formulas for decay widths of new quarks and leptons

35

References

39

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2

I.INTRODUCTION

The Minimal Supersymmetric Standard Model [1] (MSSM) predicts that the lightest neutral

Higgs boson, h0, has a mass that can only exceed that of the Z0boson by virtue of radiative correc-

tions. If the superpartners are not too heavy, then it becomes a challenge to evade the constraints

on h0set by CERN LEPII e+e−collider searches. On the other hand, larger superpartner masses

tend to require some tuning in order to accommodate the electroweak symmetry breaking scale.

In recent years this has motivated an exploration of models that extend the MSSM and can raise

the prediction for mh0.

In the MSSM, the largest radiative corrections to mh0 come from loop diagrams involving top

quarks and squarks, and are proportional to the fourth power of the top Yukawa coupling. This

suggests that one can further raise the Higgs mass by introducing new heavy supermultiplets

with associated large Yukawa couplings. In recent years there has been renewed interest [2–20]

in the possibility of a fourth family of quarks and leptons, which can be reconciled with precision

electroweak constraints with or without supersymmetry. However, within the context of super-

symmetry, if the new heavy supermultiplets are chiral (e.g. a sequential fourth family), then in

order to evade discovery at the Fermilab Tevatron p¯ p collider the Yukawa couplings would have to

be so large that perturbation theory would break down not far above the electroweak scale. This

would negate the success of apparent gauge coupling unification in the MSSM. Furthermore, the

corrections to precision electroweak physics would rule out such models without some fine tuning.

These problems can be avoided if the extra supermultiplets are instead vector-like, as proposed

in [21–24]. If the scalar members of the new supermultiplets are heavier than the fermions, then

there is a positive correction to mh0. As I will show below, the corrections to precision electroweak

parameters decouple fast enough to render them benign.

To illustrate the general structure of such models, suppose that the new left-handed chiral

supermultiplets include an SU(2)Ldoublet Φ with weak hypercharge Y and an SU(2)Lsinglet φ

with weak hypercharge −Y − 1/2, and Φ and φ with the opposite gauge quantum numbers. The

fields Φ and φ transform as the same representation of SU(3)C(either a singlet, a fundamental, or

an anti-fundamental), and Φ and φ transform appropriately as the opposite. The superpotential

allows the terms:

W = MΦΦΦ + Mφφφ + kHuΦφ − hHdΦφ,(1.1)

where MΦand Mφare vector-like (gauge-singlet) masses, and k and h are Yukawa couplings to the

weak hypercharge +1/2 and −1/2 MSSM Higgs fields Huand Hd, respectively. In the following,

I will consistently use the letter k for Yukawa couplings of new fields to Hu, and h for couplings

to Hd. Products of weak isospin doublet fields implicitly have their SU(2)L indices contracted

with an antisymmetric tensor ǫ12= −ǫ21= 1, with the first component of every doublet having

weak isospin T3= 1/2 and the second T3= −1/2. So, for example, ΦΦ = Φ1Φ2− Φ2Φ1, with the

components Φ1, Φ2, Φ1, and Φ2having electric charges Y +1/2, Y −1/2, −Y +1/2, and −Y −1/2

respectively.

The scalar members of the new chiral supermultiplets participate in soft supersymmetry break-

ing Lagrangian terms:

−Lsoft=

?

bΦΦΦ + bφφφ + akHuΦφ − ahHdΦφ

?

+ c.c. + m2

Φ|Φ|2+ m2

φ|φ|2,(1.2)

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where I use the same name for each chiral superfield and its scalar component.

The fermion content of this model consists of two Dirac fermion-anti-fermion pairs with electric

charges ±(Y +1/2) and one Dirac fermion-anti-fermion pair with electric charges ±(Y −1/2). The

doubly degenerate squared-mass eigenvalues of the fermions with charge ±(Y + 1/2) are obtained

at tree-level by diagonalizing the matrix

m2

F=

MFM†

F

0

0M†

FMF

(1.3)

with

MF=

MΦ kvu

hvd Mφ

,(1.4)

which is assumed to be dominated by the MΦand Mφentries on the diagonal. Here vu= vsinβ

and vd= v cosβ are the vacuum expectation values (VEVs) of the MSSM Higgs fields Huand Hd,

in a normalization where v ≈ 175 GeV. The scalar partners of these have a squared-mass matrix

given by, in the basis (Φ,φ,Φ∗,φ∗):

m2

S= m2

F+

m2

Φ+ ∆ 1

2,Y +1

2

0b∗

Φ

a∗

kvu− kµvd

b∗

φ

0m2

φ+ ∆0,Y +1

2

a∗

hvd− hµvu

Φ+ ∆−1

0

bΦ

ahvd− hµ∗vu m2

bφ

2,−Y −1

2

0

akvu− kµ∗vd

m2

φ+ ∆0,−Y −1

2

(1.5)

where the ∆T3,q= [T3− q sin2θW]cos(2β)m2

isospin and electric charge. The scalar particle squared-mass eigenvalues of eq. (1.5) are presumably

larger than those of their fermionic partners because of the effects of m2

a significant positive one-loop correction to m2

are largest if the k-type Yukawa coupling is as large as possible, i.e. near its infrared quasi-fixed

point.

The fermions of charge ±(Y − 1/2) have squared mass M2

squared-mass matrix

Zare electroweak D-terms, with T3and q the weak

Φ, m2

φ, m2

Φand m2

φ, inducing

h0. If tanβ is not too small, the corrections to m2

h0

Φ, and their scalar partners have a

|MΦ|2+ m2

Φ+ ∆−1

2,Y −1

2

−b∗

Φ+ ∆ 1

Φ

−bΦ

|MΦ|2+ m2

2,−Y +1

2

.(1.6)

These particles do not contribute to m2

do not have Yukawa couplings to the neutral Higgs boson. Since that contribution is therefore

parametrically suppressed, it will be neglected in the following.

With the phases of Hu and Hdchosen so that their vacuum expectation values (VEVs) are

real, then in complete generality only three of the new parameters MΦ, Mφ, k and h can be

h0except through the small electroweak D-terms, since they

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simultaneously chosen real and positive by convention. Nevertheless, I will take all four to be real

and positive below. (I will usually be assuming that the magnitude of at least one of the new

Yukawa couplings is small, so that the potential CP-violating effects are negligible anyway.)

In the MSSM, the running gauge couplings extrapolated to very high mass scales appear to

approximately unify near Q = Munif = 2.4 × 1016GeV. In order to maintain this success, it is

necessary to include additional chiral supermultiplets, besides the ones just mentioned. These

other fields again do not have Yukawa couplings to the Higgs boson, so their contribution to ∆m2

will be neglected below.

I will be assuming that the superpotential vector-like mass terms are not much larger than

the TeV scale. This can be accomplished by whatever mechanism also generates the µ term in

the MSSM. For example, it may be that the terms MΦand Mφare forbidden at tree-level in the

renormalizable Lagrangian, and arise from non-renormalizable terms in the superpotential of the

form:

h0

W =

λ

MPlSSΦΦ +

λ′

MPlSSφφ,

(1.7)

after the scalar components of singlet supermultiplets S and S obtain vacuum expectation values of

order the geometric mean of the Planck and soft supersymmetry-breaking scales. Then MΦ,Mφ∼

TeV can be natural, just as for µ in the MSSM.

In the remainder of this paper, I will discuss aspects of the phenomenology of models of this type,

concentrating on the particle content and renormalization group running (section 2), corrections to

mh0 (section 3), precision electroweak corrections (section 4), and branching ratios and signatures

for the lightest of the new fermions in each model (section 5).

< 1

II.SUPERSYMMETRIC MODELS WITH NEW VECTOR-LIKE FIELDS

A.Field and particle content

To construct and describe models, consider the following possible fields defined by their trans-

formation properties under SU(3)C× SU(2)L× U(1)Y:

Q = (3,2,1/6),Q = (3,2,−1/6),

D = (3,1,1/3),

U = (3,1,2/3),U = (3,1,−2/3),

L = (1,2,1/2),D = (3,1,−1/3),

E = (1,1,−1),

L = (1,2,−1/2),

N = (1,1,0),E = (1,1,1),N = (1,1,0).(2.1)

Restricting the new supermultiplets to this list assures that small mixings with the MSSM fields

can eliminate stable exotic particles which could be disastrous relics from the early universe. In

this paper, I will reserve the above capital letters for new extra chiral supermultiplets, and use

lowercase letters for the MSSM quark and lepton supermultiplets:

qi= (3,2,1/6),ui= (3,1,−2/3),

ei= (1,1,1),

di= (3,1,1/3),

ℓi= (1,2,−1/2),

Hu= (1,2,1/2),Hd= (1,2,−1/2).(2.2)

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with i = 1,2,3 denoting the three families. So the MSSM superpotential, in the approximation

that only third-family Yukawa couplings are included, is:

W = µHuHd+ ytHuq3u3− ybHdq3d3− yτHdℓ3e3.(2.3)

It is well-known that gauge coupling unification is maintained if the new fields taken together

transform as complete SU(5) multiplets. However, this is not a necessary condition. There are

three types of models that can successfully maintain perturbative gauge coupling unification with

the masses of new extra chiral supermultiplets at the TeV scale.

First, there is a model to be called the “LND model” in this paper, consisting of chiral super-

multiplets L,L,N,N,D,D, with a superpotential

W = MLLL + MNNN + MDDD + kNHuLN − hNHdLN.(2.4)

Here L,L play the role of Φ,Φ and N,N the role of φ,φ in eqs. (1.1)-(1.6). In most of the following,

I will consider only the case that the multiplicity of each of these fields is 1, although 1, 2, or 3 copies

of each would be consistent with perturbative gauge coupling unification. These fields consist of a

5+5 of SU(5), plus a pair†of singlet fields. The non-MSSM mass eigenstate fermions consist of a

charged lepton τ′, a pair of neutral fermions ν′

are complex scalars ˜ τ′

1,2. The primes are used to distinguish these states from

those of the usual MSSM that have the same charges.

Second, one has a model consisting of a 10+10 of SU(5), to be called the “QUE model” below,

consisting of fields Q,Q,U,U,E,E with a superpotential

1,2, and a charge −1/3 quark b′. Their superpartners

1,2, ˜ ν′

1,2,3,4, and˜b′

W = MQQQ + MUUU + MEEE + kUHuQU − hUHdQU.(2.5)

The non-MSSM particles in this case consist of charge +2/3 quarks t′

and a charged lepton τ′, and their scalar partners˜t′

Third, one has a “QDEE model” consisting of fields Q,Q,D,D,Ei,Ei(i = 1,2) with a super-

potential

1,2, a charge −1/3 quark b′,

1,2.

1,2,3,4,˜b′

1,2and ˜ τ′

W = MQQQ + MUDD + MEiEiEi+ kDHuQD − hDHdQD.(2.6)

Although this particle content does not happen to contain complete multiplets of SU(5), it still

gives perturbative gauge coupling unification. The non-MSSM particles in this model consist of

charge −1/3 quarks b′

partners˜b′

1,2,3,4.

The field and particle content of these three models is summarized in Table I.

In reference [24], it is suggested that a model with extra chiral supermultiplets in 5+5+10+10

of SU(5), or equivalently (if a pair of singlets is added) 16 + 16 of SO(10), will also result in

1,2, a charge +2/3 quark t′, and two charged leptons τ′

1,2and ˜ τ′

1,2, and their scalar

1,2,3,4,˜t′

†Here I choose the minimal model of this type that includes Yukawa couplings of the kind mentioned in the

Introduction while not violating lepton number. It is also possible to identify the fields N and N, since they are

gauge singlets, or to eliminate them (and their Yukawa couplings) entirely.

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ModelNew supermultipletsNew particles

Φ,Φ φ,φothersspin 1/2 spin 0

LNDL,LN,N D,Dν′

t′

b′

1,2τ′b′

1,2b′τ′

1,2t′τ′

˜ ν′

˜t′

˜b′

1,2,3,4˜ τ′

1,2,3,4˜b′

1,2,3,4˜t′

1,2˜b′

1,2˜ τ′

1,2˜ τ′

1,2

QUEQ,QU,U E,E

1,2

QDEEQ,QD,DE1,2,E1,2

1,2

1,2,3,4

TABLE I: The new chiral supermultiplets and the new particle content of the models discussed in this paper.

The notation for Φ,Φ,φ,φ follows that of the Introduction.

gauge coupling unification. However, the multi-loop running of gauge couplings actually renders

them non-perturbative below the putative unification scale, unless the new particles have masses

well above the 1 TeV scale. For example, working to three-loop order, if one requires that the

unified coupling (defined to be the common value of α1and α2at their meeting point) satisfies

the perturbativity condition αunif < 0.35, then the average threshold of the new particles must

exceed 5 TeV if the MSSM particles are treated as having a common threshold at or below 1 TeV

as suggested by naturalness and the little hierarchy problem. In that case, the new particles will

certainly decouple from LHC phenomenology. Even if one allows the MSSM soft mass scale to

be as heavy as the new particles, treating all non-Standard Model particles as having a common

threshold, I find that this threshold must be at least 2.8 TeV if the new Yukawa couplings vanish

and at least 2.1 TeV if the new Yukawa couplings are as large as their fixed-point values. While

such heavy mass spectra are possible, they go directly against the motivation provided by the little

hierarchy problem. Furthermore, at the scale of apparent unification of α1and α2in such models,

the value of α3is considerably smaller, rendering the apparent unification of gauge couplings at best

completely accidental, dependent on the whim of out-of-control high-scale threshold corrections. I

will therefore not consider that model further here, although it could be viable if one accepts the

loss of perturbative unification and control at high scales. The collider phenomenology should be

qualitatively similar to that of the LND and QUE models, since the particle content is just the

union of them.

B.Renormalization group running

The unification of running gauge couplings in the MSSM, LND, and QUE models is shown

in Figure 1. In this graph, 3-loop beta functions are used for the MSSM gauge couplings, and

mt= 173.1 GeV and tanβ = 10, and all non-Standard-Model particles are taken to decouple at

Q = 600 GeV. (The Yukawa couplings kNand hN in the LND model and kUand hUin the QUE

model are set to 0 here for simplicity; they do not have a dramatic effect on the results as long as

they are at or below their fixed-point trajectories.) The running for the QDEE model is not shown,

because it is very similar to that for the QUE model. Indeed, it will turn out that many features

of the QUE and QDEE models are similar, insofar as the U + U fields can be interchanged with

the D+D+E+E fields. This similarity does not extend, however, to the collider phenomenology

as discussed in section 5. Note that the unification scale, defined as the renormalization scale Q at

which α1= α2, is somewhat higher with the extra chiral supermultiplets in place; in the MSSM,

Munif≈ 2.4 × 1016GeV, but Munif≈ 2.65 × 1016GeV in the LND model, and Munif≈ 8.3 × 1016

GeV in the QUE and QDEE models. The strong coupling α3misses the unified α1and α2, but by

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FIG. 1: Gauge coupling unification in the MSSM,

LND and QUE models. The running is performed

with 3-loop beta functions, with all particles be-

yond the Standard Model taken to decouple at Q =

600 GeV, and mt= 173.1 GeV with tanβ = 10.

2468 1012 1416

Log10(Q/GeV)

0

10

20

30

40

50

60

α-1

MSSM

MSSM + 5 + 5

MSSM + 10 + 10

U(1)

SU(2)

SU(3)

__

2468101214 16

Log10(Q/GeV)

0

1

2

3

4

kN

2468 10121416

Log10(Q/GeV)

0

1

2

3

4

kU

FIG. 2: Renormalization group trajectories near the fixed point for kNin the LND model (left panel) and kU

in the QUE model (right panel), showing the infrared-stable quasi-fixed point behaviors. Here mt= 173.1

GeV and tanβ = 10 are assumed.

a small amount that can be reasonably ascribed to threshold corrections of whatever new physics

occurs at Munif.

The largest corrections to mh0 are obtained when the new Yukawa couplings of the type kN,

kU, or kDare as large as possible in the LND, QUE, and QDEE models respectively. These new

Yukawa couplings have infrared quasi-fixed point behavior, which limits how large they can be at

the TeV scale while staying consistent with perturbative unification. This is illustrated in Figure

2, which shows the renormalization group running‡of the kNcoupling in the LND model and kU

in the QUE model. The running of kDin the QDEE model is very similar to the latter (and so

‡In this paper, I use 3-loop beta functions for the gauge couplings and gaugino masses, and 2-loop beta functions

for the Yukawa couplings, soft scalar trilinear couplings, and soft scalar squared masses. These can be obtained

quite straightforwardly from the general results listed in [25–27], and so are not given explicitly here.

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FIG. 3: The contours represent the infrared-stable

quasi-fixed points of the 2-loop renormalization

group equations in the plane of Yukawa couplings

(ki,hi) evaluated at Q = 500 GeV. The allowed

perturbative regions (defined by ki,hi < 3 at

Q = Munif) are to the left and below the contours.

The long dashed (blue) line corresponds to kN,hN

in the LND model. The solid (black) line corre-

sponds to kU,hUin the QUE model, and the nearly

overlapping short dashed (red) line corresponds to

kD,hDin the QDEE model. Here mt= 173.1 GeV

and tanβ = 10 are assumed. The very nearly rect-

angular shape of these contours reflects the absence

of direct coupling between the Yukawa couplings in

the one-loop β functions.

00.20.4 0.60.8

1

1.2

k

0.0

0.2

0.4

0.6

0.8

1.0

1.2

h

is not shown). In this paper, I will somewhat arbitrarily define the fixed-point trajectories to be

those for which the extreme Yukawa couplings are equal to§3 at the scale Munifwhere α1and α2

unify. Then, assuming that only one of the new Yukawa couplings is turned on at a time, and that

tanβ = 10 with mt= 173.1 GeV and with all new particle thresholds taken to be at Q = 600 GeV,

the fixed point values also evaluated at Q = 600 GeV are

LND model:kN= 0.765orhN= 0.905,(2.7)

QUE model:kU= 1.050orhU= 1.203,(2.8)

QDEE model:kD= 1.043orhD= 1.196. (2.9)

Turning on both Yukawa couplings at the same time in each model hardly affects the results at all,

because ki,hidecouple from each other’s beta functions at one loop order for each of i = N,U,D.

This is illustrated by the very nearly rectangular shape of the fixed-line contours in Figure 3.

The phenomenology of supersymmetric models is crucially dependent on the ratios of gaugino

masses. In the MSSM, it is will known that if the gaugino masses unify at Munif, then working

to one-loop order they obey M1/α1= M2/α2= M3/α3= m1/2/αunif, and this relation has only

moderate corrections from higher-loop contributions to the beta functions. The presence of extra

matter particles strongly affects this prediction, however. In Table II, the predictions for M1,

M2, and M3at Q = 1 TeV are given for the MSSM, the LND model, the QUE model, and the

QDEE model. In the latter two cases, I distinguish between the cases of vanishing extra Yukawa

couplings and the fixed-point trajectories with (kU,hU) = (3,0) and (kD,hD) = (3,0), respectively,

at Q = Munif. As before, I have used tanβ = 10 and mt= 173.1 GeV, and taken all new particle

thresholds to be at Q = 600 GeV, and assumed for simplicity that the new scalar trilinear couplings

vanish at Q = Munif. The results will change slightly if these assumptions are modified, but there

§Formally, it turns out that the 2-loop and 3-loop beta functions for these Yukawa couplings have ultraviolet-stable

fixed points, although these occur at such large values (> 5) that they cannot be trusted to reflect the true

behavior. Simply requiring the high-scale value of the Yukawa couplings to be somewhat smaller avoids this issue.

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M1/m1/2

M2/m1/2

M3/m1/2

M2/M1

M3/M1

MSSM 0.410.77 2.281.88 5.53

LND0.320.591.75 1.86 5.52

QUE (kU= 0)

(kU(Munif) = 3)

QDEE (kD= 0)

(kD(Munif) = 3)

0.0970.1470.5711.525.90

0.1090.1760.6171.61 5.66

0.094 0.153 0.572 1.62 6.08

0.107 0.1780.615 1.665.72

TABLE II: Gaugino masses Ma/m1/2for (a = 1,2,3) and ratios of gaugino masses M2/M1and M3/M1,

evaluated at Q = 1 TeV in the models described in the text, assuming unified gaugino masses m1/2at Munif.

FIG. 4: Running of gaugino masses in the QUE

model, assuming a unified value m1/2 at Munif.

The gluino mass parameter M3is evolved accord-

ing to the 1, 2, and 3 loop beta functions in the top

three lines. The 1 and 3 loop beta functions are

shown for the parameters M1 (bottom two lines)

and M2, for which the 2-loop and 3-loop results

are not visually distinguishable. Note the signifi-

cant running of M3due to multi-loop effects.

24681012 14 16

Log10(Q/GeV)

0

0.2

0.4

0.6

0.8

1

1.2

Ma/m1/2

1 loop

2 loop

3 loop

M3

M2

M1

are a couple of striking and robust features to be pointed out about the gaugino masses in these

models. First, because the unified gauge couplings are so much larger in the models with extra

matter than in the MSSM, the gaugino masses at the TeV scale are suppressed relative to m1/2

by a significant amount compared to the MSSM. Secondly, the one-loop prediction for the ratios

of gaugino masses is very strongly violated by two-loop effects¶in the extended models, which

were evidently neglected in [24]. For example, in both the QUE and QDEE models, the one-loop

prediction is that M3= m1/2, independent of Q, since the one-loop beta function for M3happens

to vanish. However, the correct result is that M3does run significantly, with M3/m1/2reduced by

some 40% from unity, depending on the Yukawa coupling value. This reflects, in part, the accidental

vanishing of the one-loop beta function; in contrast, the three-loop contribution to the running is

quite small compared to the two-loop one. This is illustrated for the QUE model in Figure 4, which

shows the renormalization-scale dependence of the running gaugino mass parameters M1, M2, and

M3in the QUE model [with (kU,hU) = (3,0) at Q = Munif], evolved according to the 1, 2, and

3-loop beta functions.

Another notable feature of the extended models is that they permit gaugino mass domination

¶Similar effects have been noted long ago in the context of “semi-perturbative unification” [28].

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m˜ q

m˜ ¯ u

m˜ ¯d

2.00

m˜ℓ

0.67

m˜ ¯ e

MSSM2.082.01 0.37

LND 1.891.821.810.63 0.35

QUE (kU= 0)

(kU(Munif) = 3)

QDEE (kD= 0)

(kD(Munif) = 3) 1.30

1.24 1.201.190.45 0.28

1.29 1.24 1.240.47 0.30

1.241.20 1.200.450.28

1.25 1.240.470.30

TABLE III: Ratios of first- and second-family MSSM squark and slepton mass parameters to m1/2, evaluated

at Q = 1 TeV, assuming unified gaugino mass dominance at Q = Munif(m2

0= 0 and A0= 0).

for the soft supersymmetry breaking terms at the unification scale, according to which all soft scalar

masses and scalar trilinear couplings are assumed negligible compared to the gaugino masses, or

A0= 0, m2

0= 0 in the usual mSUGRA language. In the MSSM, this “no-scale” boundary condition

is problematic if applied strictly, because it predicts that the lightest supersymmetric particle (LSP)

is not a neutralino. However, in the QUE and QDEE models, the increased size of the gauge

couplings at high scales gives extra gaugino-mediated renormalization group contributions to the

scalar squared masses, so that they are safely heavier than the bino-like LSP. For the squarks and

sleptons of the first two families, this is illustrated in Table III (for the same models as in Table

II), by giving the ratios of the running masses to the unified gaugino mass parameter m1/2. The

contributions of the gaugino masses to the new extra squarks and sleptons in the LND, QUE, and

QDEE models are listed below:

LND:(mD,mD,mL,mL,mN,mN) = (1.80,1.80,0.63,0.63,0,0),

(mQ,mQ,mU,mU,mE,mE) = (1.17,1.29,1.25,0.94,0.267,0.299),

(mQ,mQ,mD,mD,mE,mE) = (1.30,1.18,0.94,1.24,0.266,0.304).

(2.10)

QUE:(2.11)

QDEE:(2.12)

Here I have chosen to display the results for boundary conditions at Munifof kN = hN = 0 and

for kU = 3,hU = 0 and for kD= 3,hD = 0, respectively. It should be noted that these are all

running mass parameters, and the physical mass parameters will be different. Also, if there are

non-zero contributions to the running scalar squared masses and scalar trilinear couplings at Munif,

the results will of course change. For example, including a non-zero common m2

will raise all of the scalar squared masses, yielding a more degenerate scalar mass spectrum.

For the QUE and QDEE models, we see from Tables II and III and eqs. (2.11) and (2.12)

that the bino mass parameter is well over a factor of 2 smaller than the lightest slepton mass, for

unified, dominant gaugino masses. Since neutralino mixing only decreases the LSP mass compared

to the bino mass parameter, the LSP will be a neutralino. In contrast, for the LND models, the

gaugino mass dominance boundary condition would predict that the scalar component of N or N

(a non-MSSM sneutrino) should be the LSP, and should be nearly massless. In fact, including a

non-zero Yukawa coupling hNor kNwould give the corresponding scalar a negative squared mass.

If there is an additional positive contribution to that sneutrino mass, then it can be the LSP, and

it might be interesting to consider it as a possible dark matter candidate.

The corrections to the lightest Higgs squared mass considered in the next section depend on

the scalar trilinear coupling akN, akU, or akDof the type appearing in eq. (1.2). It is therefore

0, as in mSUGRA,

Page 11

11

2468 1012 1416

Log10(Q/GeV)

-3

-2

-1

0

1

2

3

Ak/m1/2

246810 121416

Log10(Q/GeV)

-3

-2

-1

0

1

2

3

Ak/m1/2

FIG. 5: Renormalization group running of scalar trilinear couplings AkNin the LND model (left panel)

and AkUin the QUE model (right panel), normalized to m1/2, the common gaugino mass parameter at

the unification scale Munif. The different lines correspond to different boundary conditions at Munif. The

corresponding Yukawa couplings kNand kUare taken to be near their fixed-point trajectories, with kN= 3

and kU= 3 at Munif. The running of AkDin the QDEE model is very similar to that shown here for AkU.

useful to note that these couplings have a strongly attractive fixed-point behavior in the infrared

when the corresponding superpotential couplings kN, kU and kDare near their fixed points. To

illustrate this, consider the quantities

AkN≡ akN/kN,AkU≡ akU/kU,AkD≡ akD/kD,(2.13)

for the LND, QUE, and QDEE models, respectively. The renormalization group runnings of AkN

and AkU(each normalized to m1/2) are shown in Figure 5, for various input values at the unification

scale. The running of AkNin the LND model is seen to have a mild focusing behavior, leading

to values at the weak scale of −0.1∼< AkN/m1/2 ∼< 0.6 for input values at Munif in the range

−3∼

behavior leading to −0.5∼< AkU/m1/2∼< −0.3 at the weak scale. The running of AkDin the QDEE

model is very similar (and so is not shown). It is useful to note that in the cases of AkUin the QUE

model and AkDin the QDEE model, most of the contribution to the running comes from the gluino

mass parameter. This will still be true if one does not assume gaugino mass unification, provided

only that the gluino mass parameter M3is not very small compared to the bino and wino mass

parameters M1and M2. Therefore, the previous results concerning the fixed-point behavior of AkD

and AkUremain approximately valid if m1/2is replaced by the value of M3at the unification scale.

< AkN/m1/2∼

< 3. In the case of AkUin the QUE model, one finds an even stronger focusing

C.Fine-tuning considerations

One of the primary model-building motivations in recent years is the supersymmetric little

hierarchy problem, which concerns the tuning required to obtain the electroweak scale, given the

large supersymmetry breaking effects needed to avoid a light Higgs boson that should have been

Page 12

12

seen at LEP and to evade direct searches for superpartners at LEP and the Tevatron. One way to

express this problem is to note that the Z boson mass is related to the parameters |µ| and m2

near the weak scale by:

Hu

−1

2m2

Z= |µ|2+ m2

Hu+

1

2vu

∂

∂vu∆V + O(1/tan2β),(2.14)

where ∆V is the radiative part of the effective potential. Although there can be no such thing

as an objective measure of fine tuning in parameter space, the cancellation needed between |µ|2

and m2

Hucan be taken as an indication of how “difficult” it is to achieve the observed weak scale.

Large values of −m2

In the MSSM, with the gauge and Yukawa couplings taken to be the values of the infamous

benchmark point SPS1a′[29] for a concrete example, one finds

Hurequire more tuning in this sense.

−m2

Hu= 1.82ˆ

M2

3− 0.212ˆ

t− 0.64ˆ m2

M2

2+ 0.156ˆ

Hu+ 0.36 ˆ m2

M3ˆ

˜ q3+ 0.28 ˆ m2

M2+ 0.023ˆM1ˆ

M3− 0.32ˆ Atˆ

˜ u3+ ....

M3− 0.07ˆ Atˆ

M2

+0.11ˆ A2

(2.15)

Here m2

presumably not too large. The non-MSSM particle thresholds are also taken to be at the same

scale. The hats on the parameters on the right side denote that they are inputs at the apparent

unification scale Munif= 2.4 × 1016GeV. They consist of gaugino masses ˆ

masses ˆ m2

small coefficients. Note that the gaugino masses and scalar squared masses are not assumed to be

unified here. The essence of the supersymmetric little hierarchy problem is that after constraints

from non-observation of the lightest Higgs boson, the charged supersymmetric particles, and from

the relic abundance of dark matter are taken into account, the remaining parameter space tends

to yield −m2

noted long ago in ref. [30] that the gluino mass parameter M3is actually mostly responsible for

the tuning needed in m2

Hu, because of its large coefficient as seen in eq. (2.15), and this problem

can be ameliorated significantly by taking |ˆ

achieved in non-mSUGRA models. For example, taking |ˆ

between theˆM2

M2

Now let us compare to the corresponding formulas in the LND, QUE and QDEE models under

study here. For the QUE model, I find near the fixed point kU= 1.05 with hU= 0 that the most

significant contributions are approximately:

Huon the left side is evaluated at the scale Q = 600 GeV, where corrections to ∆V are

M1,2,3, scalar squared

Hu, ˆ m2

˜ q3and ˆ m2

˜ u3, andˆAt≡ at/yt. I have neglected to write other contributions with

Hu≫ m2

Z/2, so that some fine adjustment is needed between −m2

Huand |µ|2. It was

M3/ˆ

M2| smaller than unity at Munif. This can easily be

M3/ˆ

M2| ∼ 1/3 produces near cancellation

2terms with opposite signs in eq. (2.15), yielding a smaller value for m2

3andˆ

Hu.

−m2

Hu= 2.10ˆM2

3+ 0.035ˆ

M3+ 0.014ˆ AkUˆ

−0.17ˆ m2

M2

2+ 0.019ˆM2

M2+ 0.057ˆ A2

˜ u3+ 0.47m2

1− 0.014ˆM3ˆ

t− 0.015ˆ AtˆAkU+ 0.25ˆ A2

Q+ 0.40m2

M2− 0.075ˆ Atˆ

M3− 0.016ˆ Atˆ

M2

+0.022ˆ AkUˆ

kU

Hu+ 0.34 ˆ m2

˜ q3+ 0.27 ˆ m2

U+ .... (2.16)

Again the hats on parameters on the right side denote their status as input values at Munif, and

m2

Huon the left side is evaluated at Q = 600 GeV, which is also where the new particle thresholds

are placed, and tanβ = 10. The result of eq. (2.16) seems to reflect a worsening of the little

hierarchy problem, since the contribution to −m2

MSSM case, while the physical MSSM superpartner masses are actually lower for fixed values of

Huproportional toˆ M2

3is even larger than in the

Page 13

13

the input soft parameters, as can be seen from Tables II and III. This implies that for a given scale

of physical superpartner masses, including notably the top squarks that contribute strongly to m2

one will need larger −m2

two. Note also that since the contribution proportional toˆ

cannot be a cancellation as in the MSSM for large |ˆ

there is the fact that there are large positive corrections to m2

in the following section, so that the top squark masses need not be so large.

It is interesting to compare with the corresponding result when the new Yukawa coupling kUis

instead taken to vanish:

h0,

Hu, and thus larger |µ|2, and so a more delicate cancellation between the

M2

M2/ˆ

M3|. Counteracting these considerations,

h0from the new particles, as discussed

2is positive (and quite small), there

−m2

Hu= 1.14ˆM2

3− 0.107ˆ

t− 0.70ˆ m2

M2

2+ 0.153ˆ

Hu+ 0.30 ˆ m2

M3ˆ

M2+ 0.022ˆ

˜ q3+ 0.21 ˆ m2

M1ˆ

M3− 0.436ˆ Atˆ

˜ u3+ ...

M3− 0.090ˆ Atˆ M2

+0.125ˆ A2

(2.17)

for kU= 0. Here the impact on fine-tuning is less because the coefficient ofˆ

is no large positive contribution from the new scalar soft masses, and the possibility of significant

cancellation between the gluino and wino mass contributions (if |ˆ

counteracting this, there is no large positive contribution to m2

when kU= 0.

Results for the QDEE model are quite similar. At the fixed point with kD= 1.043, I find

M2

3is reduced, there

M2/ˆ

M3| > 1) is restored. But,

h0from the extra vector-like sector

−m2

Hu= 2.12ˆM2

3+ 0.034ˆ

M3+ 0.014ˆ AkDˆ

−0.22ˆ m2

M2

2+ 0.006ˆM2

M2+ 0.054ˆ A2

˜ u3+ 0.37m2

1− 0.013ˆM3ˆ

t− 0.027ˆ AtˆAkD+ 0.12ˆ A2

Q+ 0.39m2

M2− 0.085ˆ Atˆ

M3− 0.017ˆ Atˆ

M2

+0.029ˆ AkDˆ

kD

Hu+ 0.33 ˆ m2

˜ q3+ 0.26 ˆ m2

D+ ...,(2.18)

and for kD= 0,

−m2

Hu= 1.15ˆM2

3− 0.106ˆ

t− 0.70ˆ m2

M2

2+ 0.154ˆ

Hu+ 0.30 ˆ m2

M3ˆ

M2+ 0.024ˆ

˜ q3+ 0.21 ˆ m2

M1ˆ

M3− 0.439ˆ Atˆ

˜ u3+ ....

M3− 0.090ˆ Atˆ M2

+0.125ˆ A2

(2.19)

The same general comments therefore apply for the QDEE model as for the QUE model.

Treating the LND model in the same way, I find for kN= 0.765:

−m2

Hu= 1.74ˆ M2

3− 0.166ˆ

M3+ 0.11ˆ A2

−0.62ˆ m2

M2

2+ 0.131ˆ

t− 0.04ˆ AtˆAkN+ 0.05ˆ A2

Hu+ 0.38 ˆ m2

M3ˆ

M2+ 0.020ˆM3ˆ

M1− 0.33ˆ Atˆ

kN

M3− 0.06ˆ Atˆ

M2

+0.07ˆ AkNˆ

˜ q3+ 0.30ˆ m2

˜ u3+ ....(2.20)

The dependence on the soft parameters in the sector of new extra particles is very slight, due to the

fact that the fixed-point Yukawa coupling is not too large. Here we see that even at its fixed point

the LND model is qualitatively quite similar to the MSSM, in that a ratio of |ˆ

1 at the unification scale can reduce −m2

with |µ|2. For comparison, the result with kN= 0 is

M2/ˆ

M3| larger than

Huand therefore mitigate the amount of tuning required

−m2

Hu= 1.68ˆM2

3− 0.178ˆM2

2+ 0.164ˆM3ˆ

M2+ 0.020ˆM3ˆ

M1− 0.36ˆ Atˆ

M3− 0.08ˆ Atˆ

M2

Page 14

14

+0.12ˆ A2

t− 0.66ˆ m2

Hu+ 0.34 ˆ m2

˜ q3+ 0.26 ˆ m2

˜ u3+ ...,(2.21)

which shows quite similar characteristics.

Summarizing the preceding discussion, there are two general counteracting effects on the little

hierarchy problem from introducing vector-like supermultiplets with large Yukawa couplings. The

impact of contributions to −m2

rection to m2

h0 discussed in the next section works to mitigate the problem. (Ref. [24] obtained

qualitatively similar results, but with quite different numerical details, presumably due to neglect

of higher-loop contributions to the running of gaugino masses, as noted above.) I will make no

attempt to further quantify the competition between these two competing and opposite impacts

on the little hierarchy problem, because there is simply no such thing as an objective measure on

parameter space, and because there is great latitude in choosing the remaining parameters anyway.

Hugenerally tends to worsen the problem, but the additional cor-

III. CORRECTIONS TO THE LIGHTEST HIGGS SCALAR BOSON MASS

The contributions of the new supermultiplets to the lightest Higgs scalar boson mass can be

computed using the effective potential approximation, which amounts to neglecting non-zero exter-

nal momentum effects in h0self-energy diagrams. Since m2

particle masses, this approximation is quite good for these contributions. The one-loop contribution

to the effective potential due to the supermultiplets in eqs. (1.1)-(1.5) is:

h0is much smaller than any of the new

∆V = 2Nc

4

?

i=1

[F(M2

Si) − F(M2

Fi)], (3.1)

where Nc is the number of colors of Φ, and M2

eqs. (1.3) and (1.5), and F(x) = x2[ln(x/Q2) − 3/2]/64π2. Here Q is the renormalization scale. I

will assume the decoupling approximation that the neutral Higgs mixing angle is α ≈ β − π/2,

which is valid if m2

Siand M2

Fiare the squared-mass eigenvalues of

A0≫ m2

h0. Then the correction to m2

h0is

∆m2

h0 =

?sin2β

2

?∂2

∂v2

u

−1

vu

∂

∂vu

?

+cos2β

2

?∂2

∂v2

d

−1

vd

∂

∂vd

?

+ sinβ cosβ

∂2

∂vu∂vd

?

∆V.(3.2)

Before presenting some numerical results, it is useful to note a relatively simple analytical

result that can be obtained if the superpotential vector-like fermion masses are taken to be equal

(MΦ= Mφ≡ MF) and the soft supersymmetry-breaking non-holomorphic masses are equal (m2

m2

bΦand bφare neglected. Then, writing

Φ=

Φ= m2

φ= m2

φ≡ m2), and the small electroweak D-terms and the holomorphic soft mass terms

M2

x ≡ M2

¯k ≡ k sinβ,

Xk≡ Ak− µcotβ,

S= M2

F+ m2= average scalar mass

S/M2

F

¯h ≡ hcosβ,

(3.3)

(3.4)

(3.5)

Xh≡ Ah− µtanβ,(3.6)

Page 15

15

FIG. 6: The functions f(x) and fmax(x) described in the

text, graphed as a function of√x = the average ratio of

scalar to fermion masses.

12345

x1/2

0

1

2

3

4

5

f(x)

fmax(x)

and expanding to leading order in the normalized Yukawa couplings¯k and¯h, one obtains:

∆m2

h0 =

Ncv2

4π2

?

?

¯k4

?

f(x) +

X2

xM2(1 −

x)(1 −1

k

1

3x) −

X4

k

12x2M4

?

+¯k3¯h−2

?

?

3(2 −1

−(1 −1

−2

x) − Xk(2Xk+ Xh)/(3x2M2)

?

+¯k2¯h2

x)2− (Xk+ Xh)2/(3x2M2)

3(2 −1

X2

h

xM2(1 −

?

+¯k¯h3

x)(1 −1

x) − Xh(2Xh+ Xk)/(3x2M2)

1

3x) −

12x2M4

?

+¯h4

?

f(x) +

X4

h

??

.(3.7)

where

f(x) ≡ ln(x) −1

6(5 −1

x)(1 −1

x).(3.8)

It is often a good approximation to keep only the contribution proportional to¯k4, corresponding

to the case where ktanβ ≫ h. In that limit, eq. (3.7) agrees with the result given in [24], which

can be rewritten as simply:

∆m2

h0 =

Nc

4π2k4v2sin4β

?

f(x) +

X2

xM2(1 −

k

1

3x) −

X4

k

12x2M4

?

.(3.9)

Note that x is, to first approximation, the ratio of the mean squared masses of the scalars to the

fermions. A key feature of the result for ∆m2

h0is that the contribution of the vector-like particles

does not decouple with the overall extra particle mass scale, provided that there is a hierarchy x

maintained between the scalar and fermion squared masses. To get an idea of the impact of this

hierarchy, the function f(x) is depicted in Figure 6. In the limit of unbroken supersymmetry, f(1) =

0, and f(x) monotonically increases for scalars heavier than fermions (x > 1). The other significant

feature that could lead to enhanced ∆m2

h0 is the mixing parameterized by Xk. The maximum

possible value of the Xkcontribution in eq. (3.9) is obtained when X2

k= 2M2(3x − 1), leading

Page 16

16

FIG. 7: Estimates for the corrections to the Higgs mass

as a function of√x, where x = (M2+m2)/M2is the ra-

tio of the mean scalar squared mass to the mean fermion

squared mass, in the simplified model framework used in

eq. (3.9) of the text, using Nc= 3 and k4v2sin4β = (190

GeV)2, corresponding roughly to the QUE or QDEE

model near the fixed point with reasonably large tanβ.

The lower line is the no-mixing case Xk= 0, and the up-

per line is the maximal mixing case Xk= 2M2(3x − 1).

The Higgs mass before the correction is taken to be 110

GeV.

12345

x1/2

0

10

20

30

40

50

60

∆mh [GeV]

No mixing

Maximal mixing

FIG. 8: Corrections to mh0 in the QUE model with

kU= 1.05, for varying m1/2with other parameters

described in the text. Here MF = 400, 600, and

800 GeV is the vector-like superpotential fermion

mass term, and MS is the geometric mean of the

new up-type scalar masses. The upper and lower

lines in each case correspond to Ak = −0.5m1/2

and Ak= −0.3m1/2, respectively, at the TeV scale.

The value of mh0 before these corrections is as-

sumed to be 110 GeV.

6008001000

MS [GeV]

120014001600

0

5

10

15

20

25

∆mh [GeV]

MF = 400 GeV

MF = 600 GeV

MF = 800 GeV

to a “maximal mixing” result ∆m2

This function is also graphed in Figure 6 to show the maximal effects of mixing from the new

fermion sector. In Figure 7, I show an estimate of the corresponding corrections to ∆mh0, taking

Nc= 3 and k4v2sin4β = (190 GeV)2(corresponding roughly to the QUE or QDEE model near

the fixed point with reasonably large tanβ) and assuming that the predicted Higgs mass before

?

The previous depiction may be too simplistic, since the superpotential and soft supersymmetry-

breaking masses need not have the simple degeneracies that were assumed. Also, as found in

the previous section, the scalar trilinear coupling has a fixed point behavior that implies that

the mixing is neither maximal nor zero (but closer to the latter).

is therefore as depicted in Figure 8. Here, I take scalar masses inspired by the renormaliza-

tion group solutions of the previous section for the QUE model.

cases for the vector-like superpotential masses at the TeV scale, MQ= MU ≡ MF = 400, 600,

and 800 GeV. The Yukawa couplings are taken to be at the fixed-point values kU = 1.05 and

hU = 0.The soft supersymmetry-breaking terms are parameterized by (mQ,mQ,mU,mU) =

(1.17,1.29,1.25,0.94)m1/2and (bQ,bU) = −(MQ,MU)m1/2, and Ak= −0.3m1/2and −0.5m1/2,

h0=

Nc

4π2k4v2sin4βfmax(x) where fmax= f(x) + (3 − 1/x)2/3.

the correction is 110 GeV, so that ∆mh0 =

(110 GeV)2+ ∆m2

h0− 110 GeV.

A more realistic estimate

In particular, I take three

Page 17

17

all at a renormalization scale of 1 TeV. The results turn out to be not very sensitive to bQ or

bU, or to the MSSM supersymmetric Higgs mass parameter µ (taken to be 800 GeV here), or to

tanβ as long as it is not too small (tanβ = 10 was used here). Figure 8 shows the results for

?

masses. Quite similar results obtain for the QDEE model at the fixed point with kD = 1.043,

hD= 0.

Figure 8 illustrates that the contribution of the new extra particles to mh0 is probably much less

than the “maximal mixing” scenario, if one assumes that the TeV-scale parameters (particularly the

scalar trilinear coupling ak) can be obtained by renormalization group running from the unification

scale. Note that the models illustrated in Figure 8 represent gaugino-mass dominated examples.

If one assumes that the soft scalar squared masses at Munif actually have significant positive

values (from, for example, running between Munifand MPlanck), then the low-scale model will be

even closer to the no-mixing scenario, since the diagonal entries in the scalar mass matrix will be

enhanced, while the mixing terms are still subject to the strong focusing behavior seen in Figure

5.

In the case of the minimal LND model, one expects the maximum contributions to ∆m2

to be suppressed by a factor of roughly (kN/kU)4/Nc ≈ (0.765/1.05)4/3 ≈ 0.094. This leads

to corrections that are typically not large compared to the inherent uncertainties in the total

prediction. This counts against the minimal LND model as a way of significantly increasing the

Higgs mass. One can also consider n > 1 copies of the LND model, with each kNYukawa coupling

near a common fixed point to maximize ∆m2

h0. However, then the common fixed point value is even

smaller, with kN= 0.695 for n = 2, and kN= 0.650 for n = 3. (A much more significant correction

to m2

h0 can occur if one enhances the model with several copies of the extra fields connected by

the “lateral” gauge group idea of [23].)

∆mh0 ≡

(110 GeV)2+ ∆m2

h0− 110 GeV, as a function of MS, the geometric mean of the scalar

h0

IV. PRECISION ELECTROWEAK EFFECTS

Because the Yukawa couplings responsible for large effects on m2

try of the Higgs sector, it is necessary to consider the possibility of constraints due to precision

electroweak observables arising from virtual corrections to electroweak vector boson self-energies.

In this section, I will show that these corrections are actually benign (and much smaller than

previously estimated), at least if one uses only Mt, MW, and Z-peak observables as in the LEP

Electroweak Working Group analyses [31, 32] rather than including also low-energy observables as

in [33]. The essential reason for this is that the corrections decouple with larger vector-like masses,

even if the Yukawa couplings are large and soft supersymmetry breaking effects produce a large

scalar-fermion hierarchy. Indeed, they decouple even when the corrections to m2

The most important new physics contributions to precision electroweak observables can be

summarized in terms of the Peskin-Takeuchi S and T parameters [34]. For the measurements of

Standard Model observables, I use the updated values:

h0break the custodial symme-

h0do not.

s2

eff= 0.23153 ± 0.00016

MW = 80.399 ± 0.025 GeV

Γℓ = 83.985 ± 0.086 MeV

h(MZ) = 0.02758 ± 0.00035

ref. [31] (4.1)

ref. [32, 35](4.2)

ref. [31](4.3)

∆α(5)

ref. [31](4.4)

Page 18

18

Mt = 173.1 ± 1.3 GeV

αs(MZ) = 0.1187 ± 0.0020

ref. [36](4.5)

ref. [33](4.6)

with MZ= 91.1875 GeV held fixed. For the Standard Model predictions for s2

terms of the other parameters, I use refs. [37], [38], and [39], respectively. These values are then

used to determine the best experimental fit values and the 68% and 95% confidence level (CL)

ellipses for S and T, relative to a Standard Model template with Mt= 173.1 GeV and Mh= 115

GeV, using

eff, MW, and Γℓin

s2

eff

(s2

M2

(M2

eff)SM

= 1 +

α

4s2

2c2WS +αc2

Wc2WS −αc2

α

W

c2W

T, (4.7)

W

W)SM

Γℓ

(Γℓ)SM

= 1 −

W

c2W

T, (4.8)

= 1 − αdWS + α(1 + s2

2WdW)T, (4.9)

where sW = sinθW, cW = cosθW, s2W = sin(2θW), c2W = cos(2θW), and dW = (1 − 4s2

4s2

W)c2W]. The best fit turns out to be S = 0.057 and T = 0.080.

The new physics contributions to S and T are given in terms of one-loop corrections to the

electroweak vector boson self-energies ΠWW, ΠZZ, ΠZγ and Πγγ, which are computed for each

of the LND, QUE and QDEE models in Appendix A. They are dominated by the contributions

from the fermions when the soft supersymmetry-breaking scalar masses are large. It is useful and

instructive to consider the simplified example that occurs when, in the notation of the Introduction,

MΦ= Mφ= MF with an expansion in small mu≡ kvu, md≡ hvd, and MW. Then one finds for

the new fermion contributions:

W)/[(1 −

W+ 8s4

∆T =

Nc

WM2

480πs2

Nc

30πM2

WM2

F

?13(m4

u+ m2

u+ m4

d) + 2(m3

umd+ m3

dmu) + 18m2

um2

d

?

(4.10)

∆S =

F

?4(m2

d) + mumd(3 + 20YΦ)?, (4.11)

where YΦis the weak hypercharge of the left-handed fermion doublet, denoted Φ in the Introduction,

that has a Yukawa coupling to Hu(so that YΦ= −1/2, 1/6, and −1/6 for the LND, QUE, and

QDEE models respectively). Equations (4.10), (4.11) agree†with the results found in [40, 41].

An important feature of this is that the corrections decouple quadratically with increasing MF,

regardless of the soft supersymmetry breaking terms. This is in contrast to the contributions

to ∆m2

h0, which do not decouple as long as there is a hierarchy between the scalar and fermion

masses within a heavy supermultiplet. It also contrasts with the situation for chiral fermions (as

in a sequential fourth family), which yields much larger ∆S, ∆T.

†However, note that the result for ∆T quoted in ref. [24] actually corresponds to the improbable case hvd = kvu,

rather than h = 0. So, for small h, the actual correction to ∆T is almost a factor of 4 smaller than their estimate.

As a result, much smaller values for MF are admissible than would be indicated by ref. [24].

Page 19

19

FIG. 9: Corrections to electroweak precision observ-

ables S,T from the LND model at the fixed point

(kN,hN) = (0.765,0), for varying ML = MN =

mτ′ > 100 GeV, in the limit of heavy scalar super-

partners. The seven dots on the line segment corre-

spond to mτ′ = 100,120,150,200,250,400 GeV and

∞, from top to bottom. The experimental best fit is

shown as the × at (∆S,∆T) = (0.057,0.080). Also

shown are the 68% and 95% CL ellipses, obtained

as described in the text. The point ∆S = ∆T = 0

is defined to be the Standard Model prediction for

mt= 173.1 GeV and mh0 = 115 GeV.

-0.2-0.100.10.20.3

∆S

-0.2

-0.1

0

0.1

0.2

0.3

∆T

95%

68%

If h = 0, then the results of eqs. (4.10), (4.11) from the fermions become, numerically:

∆T = 0.54Nck4sin4(β)

?100 GeV

?100 GeV

MF

?2

?2

,(4.12)

∆S = 0.13Nck2sin2(β)

MF

.(4.13)

These rough formulas show that it is not too hard to obtain agreement with the precision elec-

troweak data, provided that MF is not too small, but it should be noted that especially for light

new fermions with mass of order 100 GeV, the expansion in large MF is not very accurate, with

eqs. (4.12) and (4.13) overestimating the actual corrections.

A more precise evaluation, using the formulas of Appendix A, is shown in figures 9 and 10,

which compares the experimental best fit and 68% and 95% CL ellipses to the predictions from

the models. Note that in these figures I do not include the contributions from the ordinary MSSM

superpartners, which are typically not very large and which become small quadratically with large

soft supersymmetry breaking masses. Figure 9 shows the corrections for the LND model at the

Yukawa coupling fixed point (kN,hN) = (0.765,0), for varying MN= ML= mτ′ > 100 GeV as a

line segment with dots at mτ′ = 100,120,150,200,250,400 GeV and ∞. These contributions are

due to the fermions ν′

Note that in the LND model b′and˜b′

1,2do not contribute to S,T as defined above, since they

do not have Yukawa couplings to the Higgs sector. Figure 9 shows that even for mτ′ as small as

100 GeV, the S and T parameters remain within the 68% CL ellipse, and can even give a slightly

better fit to the experimental results provided that mτ′∼> 120 GeV. If the Yukawa coupling kNis

less than the fixed point value, or if ML< MN, then the corrections to S and T are smaller, for a

given mτ′.

Figure 10 shows the corrections for the QUE model at the Yukawa coupling fixed point

(kU,hU) = (1.050,0), for varying MU = MQ = mb′ as a line segment with dots at mt′

275,300,350,400,500,700,1000 GeV and ∞. [For a comparison to the approximate formulas (4.12)

and (4.13), the appropriate values are MF = mb′ ≈ 355, 381, 432, 483, 584, 786, 1088 GeV and

∞, respectively.] Here, I have included the contributions from the scalar states˜t1,2,3,4and˜b1,2,

1,2,τ′, with their scalar superpartners assumed heavy enough to decouple.

1=

Page 20

20

FIG. 10: Corrections to electroweak precision observ-

ables S,T from the QUE model at the fixed point

(kU,hU) = (1.050,0), for varying MQ= MU = mb′,

with m1/2 = 600 GeV and AkU= −0.4m1/2and

m0= 0 and bQ= bU= −m1/2MQ, using eqs. (2.11)

and (2.13). The eight dots on the line segment corre-

spond to mt′

and ∞, from top to bottom. The experimental best

fit is shown as the × at (∆S,∆T) = (0.057,0.080).

Also shown are the 68% and 95% CL ellipses, ob-

tained as described in the text. The point ∆S =

∆T = 0 is defined to be the Standard Model pre-

diction for mt = 173.1 GeV and mh0 = 115 GeV.

Results for the QDEE model are very similar.

1= 275,300,350,400,500,700,1000 GeV

-0.2-0.100.10.2 0.3

∆S

-0.2

-0.1

0

0.1

0.2

0.3

∆T

95%

68%

obtained for m1/2= 600 GeV and AkU= −0.4m1/2and m0= 0 and bQ= bU= −m1/2MQ, using

eqs. (2.11) and (2.13). Smaller values of m1/2would imply a chargino lighter than the LEP2 bound;

see Table II. From Figure 10 we see that a slightly better fit than the Standard Model can be

obtained for mt′

the 95% CL ellipse. The corrections to S and T for a given mt′

is even better) if any of the following conditions apply: the Yukawa coupling kUis below its fixed

point value, m1/2or m0is larger so that the new squarks are heavier, or MQ?= MU. For t′

less than about 400 GeV, the fit also improves slightly if mh0 is larger than 115 GeV.

I have also looked at the QDEE model at its fixed point (kD,hD) = (1.043,0), with scalar

squared-mass soft terms given by eq. (2.12) with m1/2= 600 GeV. The results are nearly identical

to those found in the QUE model in figure 10 with the values for mt′

not depicted.

If one also included lower energy data as used in [33], the fits to S and T would be somewhat

worse, so it is important to keep in mind that the results above are sensitive to the choice of

following the LEP Electroweak Working Group [31, 32] in using fits based on the Z-pole data and

mtand mW. With this caveat, one may conclude that the models considered here fit at least as

well as the Standard Model, provided that the new quarks with large Yukawa couplings are heavier

than roughly 400 GeV, and can do even better if the new squarks are heavy enough to decouple.

1 ∼> 400 GeV, but even for mt′

1as light as 275 GeV, the corrections remain within

1are even smaller (and so the fit

1masses

1replaced by mb′

1, and so are

V. COLLIDER PHENOMENOLOGY OF THE EXTRA FERMIONS

The extra particles in the models discussed above will add considerable richness to the already

complicated LHC phenomenology of the MSSM. A full discussion of the different signals, and how

to disentangle them, is beyond the scope of the present paper, but it is likely that the most impor-

tant distinguishing collider signals will arise from production of the new fermions, especially the

new quarks. This is simply because of the relatively large production cross-section compared to

the scalars, which are presumably much heavier due to the effects of soft supersymmetry break-

ing masses. One can therefore expect signals from direction pair production of the lightest new

quark, and possibly also from cascade decays of somewhat heavier fermions down to them. For

Page 21

21

concreteness, I will concentrate on only the final states from decays of the lightest new quark in

each model. In general, the lightest new quark and the lightest new lepton would be stable, were

it not for mixing with the Standard Model fermions. At least some small such mixing is necessary

to avoid a cosmological disaster from unwanted heavy relics. If the mixing is very small, then the

new fermions could be quasi-stable, with decay lengths on the scale of collider detectors. Then the

collider signatures will involve particles that leave highly-ionizing and slow tracks in the detectors,

or feature macroscopic decay kinks or charge-changing tracks. These can be either the new charged

leptons or hadronic bound states of the new quarks. Such signals have been discussed before in a

variety of different model-building contexts; for some reviews, see refs. [49–52].

In the following, I will assume that the mixing of the new fermions with Standard Model fermions

is large enough to provide for prompt decays. Mixing of the new fermions with the first and second

family Standard Model fermions is highly constrained by flavor-changing neutral currents, since the

vector-like gauge quantum number assignments eliminate the GIM-type suppression. Therefore, I

will assume that the mixing is with the third Standard Model family, for which the constraints are

much easier to satisfy. Then the final states of the decays will always involve a single third-family

quark or lepton, together with a W, Z, or h0boson. Below, I will discuss the possibilities for the

branching ratios of the new quarks and leptons, and their dependence on the type of mixing.

There are existing limits on the extra quarks coming from Tevatron, although these have mostly

been found with assumed 100% branching ratios for particular decay modes (which as we will see

below is not necessarily likely). The current limits are, for prompt decays:

• mt′ > 311 GeV for BR(t′→ Wq) = 1, based on 2.8 fb−1[42]

• mb′ > 325 GeV for BR(b′→ Wt) = 1, based on 2.7 fb−1[43]

• mb′ > 268 GeV for BR(b′→ Zb) = 1, based on 1.06 fb−1[44]

• mb′ > 295 GeV for BR(b′→ Wt,Zb,h0b) = 0.5,0.25,0.25, based on 1.2 fb−1[45]

and for quasi-stable quarks:

• mt′ > 220 GeV, based on dE/dx for 90 pb−1at√s = 1.8 TeV [46]

• mb′ > 190 GeV, based on dE/dx for 90 pb−1at√s = 1.8 TeV [46]

• mb′ > 170 GeV for 3mm < cτb′ < 20mm, based on 163 pb−1[47]

Also, if the cross-section upper bound found from time-of-flight measurements with 1.0 fb−1in

ref. [48] for stable top squarks also applies to stable t′quarks with no change in efficiency, then I

estimate a bound mt′∼> 360 GeV should be obtainable, with a somewhat weaker bound for stable

b′due to a lower detector efficiency.

At hadron colliders, the production cross-section of the new quarks is due to gg and qq initial

states and is mediated by the strong interactions, and so is nearly model-independent when ex-

pressed as a function of the mass. The leading order cross-section is shown in figure 11 for the

Tevatron pp collider at√s = 1.96 TeV and for the LHC pp collider with√s = 7,10,12, and 14

TeV. Note the Tevatron will probably be unable to strengthen the existing constraints very signif-

icantly, at least for promptly decaying new quarks, due to the rather steep fall of the production

cross-section with mass. At the LHC pair production of mostly vector-like quarks should provide

a robust signal; see for example studies (in diverse other model contexts) in refs. [6, 54–58]. (Note

Page 22

22

FIG. 11: Production cross-section for new

quarks as a function of the mass, for the Teva-

tron p¯ p collisions at√s = 1.96 TeV, and for

the LHC pp collisions with√s = 7,10,12,

and 14 TeV. The graph was made at lead-

ing order using CTEQ5LO parton distribu-

tion functions [53] with Q = mq′ and apply-

ing a K factor of 1.5 for LHC and 1.25 for

Tevatron.

200

400

600800 1000 1200 1400

Mq’ [GeV]

0.001

0.01

0.1

1

10

100

1000

Cross section [pb]

14 TeV

12 TeV

10 TeV

7 TeV

1.96 TeV

that in the models under study here, there is no reason that the flavor-violating charged current

couplings should be large enough to enable a viable signal from single q′production in association

with a Standard Model fermion through t-channel W exchange, unlike in other model contexts as

studied in refs. [59–65].) The branching ratios and possible signals for the LND, QUE, and QDEE

models are examined below.

A.The LND model

In the LND model, the fermions consist of a b′, τ′, and two neutral fermions ν′

is always lighter than the τ′. The fermions b′and ν′

with the Standard Model fermions from the superpotential

1and ν′

2. The ν′

1

1can therefore decay only through their mixing

W = −ǫDHdq3D + ǫNHuℓ3N − ǫEHdLe3, (5.1)

where ǫD, ǫN, and ǫE are new Yukawa couplings that are assumed here to be small enough to

provide mass mixings that can be treated as perturbations compared to the other entries in the

mass matrices.

First consider the decays of b′. The mass matrix for the down-type quarks resulting from

eqs. (2.3), (2.4), and (5.1) is:

Md=

MD

0

ǫDvd ybvd

, (5.2)

with eigenstates b and b′. The b′decay can take place only through the ǫDcoupling, to final states

Wt, Zb, and h0b. Formulas for these decay widths are given in Appendix B. To leading order, the

branching ratios only depend on the mass of the b′, and the results are graphed in Figure 12. Note

Page 23

23

200400600 8001000

mb’[GeV]

0

0.2

0.4

0.6

0.8

1

b’ Branching Ratios

hb

Wt

Zb

100 200

mν’[GeV]

300400

0

0.2

0.4

0.6

0.8

1

ν’ Branching Ratios

hν

Wτ

Zν

FIG. 12: The branching ratios of the lightest new quark b′(left panel) and the lightest new lepton ν′

panel) in the LND model. The ν′

(not shown).

1(right

1results assume that ǫN≫ ǫE; if instead ǫE≫ ǫNthen BR(ν′

1→ Wτ) = 1

that in the limit of large mb′, the branching ratios are “democratic” between charged and neutral

currents, approaching 0.5, 0.25, and 0.25 for Wt, Zb, and h0b respectively, in accord with the

Goldstone boson equivalence theorem. However, for smaller masses, kinematic suppression reduces

the Wt branching ratio, so that, for example, the three final states have comparable branching

ratios for mb′ in the vicinity of 300 to 400 GeV.

The LHC signals include pp → b′

W’s decay leptonically and the other two W’s decay hadronically, this leads to a same-charge

dilepton plus multi-jets (including two b jets) plus missing transverse energy signal, with a total

branching ratio as high as 25%. This signal is also the basis for the current Tevatron bound

mb′ > 325 GeV, but this assumes BR(b′→ Wt) = 100%; since the actual branching ratio predicted

by the LND model for that mass range is more than a factor of 3 smaller, the model prediction

for the signal in the channel that was searched is more than an order of magnitude smaller, and

decreases sharply for lower mb′

or more b jets, coming mostly from events with h0b → bb¯b decays but also from Zb → bb¯b. The

Tevatron limit [44] of mb′ > 268 GeV from assuming BR(b′→ Zb) = 100% is in a mass range where

the actual branching ratio is about 0.55, so the actual predicted signal from the LND model is

more than a factor of 3 smaller. The limit of mb′ > 295 GeV from [45], a search which is motivated

in part by [66, 67], is based on the idealized large mass limit “democratic” branching, but in the

relevant mass range the model prediction has BR(b′→ Wt) more than a factor of 2 smaller, and

decreasing very rapidly for smaller mb′, due to the kinematic suppression. The neutral current

decays, including Z → ℓ+ℓ−, could also play an important role at the LHC, see for example [58]

for a similar case.

The decay of ν′

1in the LND model is dependent on two different mixing Yukawa couplings

ǫN and ǫE. The mass matrix for the neutral leptons in the (L,N,ℓ3,L,N) basis resulting from

1¯b′

1→ W+W−t¯t → W+W−W+W−b¯b. When two same-charge

1. In over half of the other b′

1¯b′

1production events, there will be four

Page 24

24

eqs. (2.3), (2.4) and (5.1) is

0Mν

MT

ν

0

,whereMT

ν=

ML

hNvd

0

kNvu MN

ǫNvU

, (5.3)

with the masses of the Standard Model neutrinos neglected. The corresponding mass eigenstates

are a Standard Model neutrino ν and two extra massive neutrino states ν′

matrix for the charged leptons is

1and ν′

2. The mass

Me=

−ML ǫEvd

0yτvd

.(5.4)

Formulas for the resulting decay widths for ν′

If one assumes that ǫE≫ ǫN, then the decay ν′

opposite limit applies, ǫN ≫ ǫE, then the branching ratios as a function of mν′

the right panel of Figure 12. Note that in the limit of large mν′

Zν, and h0ν asymptote to 0.5, 0.25, 0.25 respectively when ǫN dominates, again in accordance

with Goldstone boson equivalence with equal charged and neutral currents. So, depending on

which Yukawa coupling dominates, one could have interesting hadron collider signatures from

ν′

and Zh0+ Emiss

T

. So far, there are no published limits specifically on mν′ based on collider pair

production with these final states. If ML∼

than ν′

production, followed by τ′→ W(∗)ν′. It should also be noted that production of ν′

well be dominated by cascade decays from heavier strongly interacting superpartners.

1→ Wτ and Zν and h0ν are given in Appendix B.

1→ Wτ has a nearly 100% branching ratio. If the

1are as shown in

1, the branching ratios for Wτ,

1¯ ν′

1production, such as W+W−τ+τ−, and h0h0+ Emiss

T

, and ZZ + Emiss

T

, and Wh0+ Emiss

T

< MN in this model, then τ′will be not much heavier

1, and so there will be additional contributions to the signal from τ′ν′

1production and τ′+τ′−

1,2and τ′

1might

B.The QUE model

In the QUE model, the lightest of the new quarks is always the charge 2/3 quark t′

pair-produced at hadron colliders, it can decay due to mixing with the Standard Model fermions

through the superpotential

1. After being

W = ǫUHuq3U + ǫ′

UHuQu3− ǫDHdQd3,(5.5)

where ǫU, ǫ′

as perturbations compared to other entries in the mass matrices. The resulting mass matrices for

U, and ǫDare new Yukawa couplings that are assumed here to be small enough to treat

Page 25

25

300400 500600700800

mt’[GeV]

0

0.2

0.4

0.6

0.8

1

t’ Branching Ratios

ht

Wb

Zt

300400500600700 800

mt’[GeV]

0

0.2

0.4

0.6

0.8

1

t’ Branching Ratios

ht

Wb

Zt

FIG. 13: Branching ratios for the lightest extra quark, t′

Wb, Zt, and h0t, as a function of m˜ t1. The left panel shows the “democratic” case that arises when ǫU

dominates (with equal charged and neutral currents), and the right panel shows the “W-phobic” (mostly

neutral current) case that arises when ǫ′

then BR(t′→ Wb) = 1 (not shown).

1, in the QUE model with MQ= MU, to final states

Udominates. In the “W-philic” case that arises when ǫDdominates,

the up-type quarks and down-type quarks are

Mu=

MQ kUvu ǫ′

Uvu

hUvd MU

0

0ǫUvu ytvu

,Md=

−MQ ǫDvd

0ybvd

,(5.6)

with mass eigenstates t,t′

to Wb, Zt, and h0t are presented in Appendix B. I will concentrate on the three cases where one

of the mixing Yukawa couplings in eq. (5.5) dominates over the other two. The branching ratios

depend on the mass of t′

1and on the type of mixing. If ǫD provides the dominant effect, then

the decays are dominantly charged-current, or “W-philic”, with BR(t′

scenario for which the Tevatron limit is now mt′ > 311 GeV [42]. If instead ǫ′

the decays are dominantly neutral-current, or “W-phobic”; in the limit of large mt′, the branching

ratios asymptote to BR(t′

dominates, the the decays are “democratic”, with branching ratios for Wb, Zt, and h0approaching

0.5, 0.25, and 0.25 respectively in the large mt′

as a function of mt′

(The results are only mildly sensitive to the last two assumptions.) By taking the different mixing

Yukawa couplings ǫU, ǫ′

the branching ratios, but it seems reasonable to assume that one of the individual mixing Yukawa

couplings dominates in the absence of some organizing principle. So the possible signatures will

include W+W−b¯b, (similar to the Standard Model t¯t signature, but with larger invariant masses;

1,t′

2and b,b′respectively. Formulas for the resulting decay widths for t′

1

1→ Wb) = 1. This is the

Udominates, then

1→ Wb) = 0 and BR(t′

1→ Zt) = BR(t′

1→ h0t) = 0.5. Finally, if ǫU

1limit. Numerical results are shown in Figure 13

1, for the case that kU is at its fixed point value, and hU= 0, and MQ= MU.

U, and ǫDto be comparable, one can get essentially any result one wants for

Page 26

26

FIG. 14: Branching ratios for τ′decays to Wν,

Zτ, and h0τ in the QUE and QDEE models, as a

function of mτ′.

100200300400500600

mτ’[GeV]

0

0.2

0.4

0.6

0.8

1

τ’ Branching Ratios

hτ

Wν

Zτ

see [6, 54, 55, 57, 58] for recent studies of comparable signals), and ZZt¯t and h0h0t¯t, etc. If

MQ∼

an additional component of the signal from b′¯t′and b′¯b′production, followed by b′→ W(∗)t′.

The τ′in the QUE model mixes with the Standard Model τ lepton through a superpotential

term:

< MU in this model, then the b′will be not much heavier than the t′, and one should expect

W = −ǫEHdℓ3E.(5.7)

The mass matrix for the charged leptons resulting from this and eqs. (2.3) and (2.5) is:

Me=

ME

0

ǫEvd yτvd

,(5.8)

with mass eigenstates τ and τ′. It follows that τ′can decay to Wν, Zτ, and h0τ, with decay

widths that are computed in Appendix B. Because there is only one relevant Yukawa mixing term,

the branching ratios depend only on mτ′. They are shown in Figure 14, assuming mh0 = 115 GeV.

The largest branching ratio for τ′is always to Wν, and in the large mτ′ limit, Goldstone boson

equivalence provides that the Wν, Zτ, and h0τ branching ratios approach 0.5, 0.25, and 0.25,

respectively. The most immediately relevant searches at hadron colliders will be in the mass range

of mτ′ just above 100 GeV, where the electroweak pair-production cross-section can be sufficiently

large, and limits do not presently exist. However, note that the appearance of τ′

dominated by cascade decays from heavier strongly interacting superpartners.

1could easily be

C.The QDEE model

In the QDEE model, the new fermions consist of a b′

lighter charge −1/3 quark b′

1, b′

2, t′, and τ′

1, τ′

2. In this model, the

1in the QDEE model

1is always lighter than the t′. The decays of b′

Page 27

27

are brought about by superpotential mixing terms with third-family quarks:

W = −ǫDHdq3D − ǫ′

DHdQd3+ ǫUHuQu3.(5.9)

In the gauge eigenstate basis, the resulting mass matrices for the down-type quarks and up-type

quarks are

Md=

MQ kDvu

0

hDvd MD ǫDvd

ǫ′

Dvd

0ybvd

,Mu=

−MQ

0

ǫUvu ytvu

.(5.10)

with mass eigenstates b,b′

to Wt, Zb, and h0b are given in Appendix B. As in the case of the QUE model, I will consider

the three cases where one of the mixing Yukawa couplings in eq. (5.9) dominates over the other

two. Then the branching ratios depend on the mass of b′

provides the dominant effect, then the decays are dominantly charged-current, or “W-philic”, with

BR(b′

1→ Wt) = 1 provided that it is kinematically allowed. The resulting signal at hadron colliders

will be b′

is presently mb′ > 325 GeV [43], based on the same-charge dilepton plus b-jets signal already

mentioned above for the LND model. If instead ǫ′

neutral-current, or “W-phobic”, with BR(b′

1→ Wt) = 0; in the limit of large mb′, the branching

ratios slowly approach BR(b′

Finally, if ǫDis dominant, the the decays are “democratic”, with branching ratios for Wt, Zb, and

h0b approaching 0.5, 0.25, and 0.25 respectively in the large mb′

ratios are shown in Figure 15 as a function of mb′

fixed point value, and hD= 0 and MQ= MD. (However, it should be noted that, unlike in the

QUE model case, the results shown are somewhat sensitive to the last of these assumptions.) Note

that in the “democratic” case, the branching ratios are similar to what one obtains for the b′of

the LND model. The CDF limit mb′ > 295 GeV was obtained in the idealized case of branching

ratios obtained in the high mass limit, but for finite mb′, the actual BR(b′→ Wt) is much smaller

and BR(b′→ h0b) is larger. In contrast, the same-charge dilepton signal from b′

turned off in the “W-phobic” case, where the largest overall branching ratio is typically to six b

quarks, yielding the interesting possible signal b′

Z bosons are unfortunately suppressed by both small BR(Z → ℓ+ℓ−) and small BR(b′

in this case. If MQ∼< MD in this model, then the t′will be not much heavier than the b′, and

one should expect an additional component of the signal from t′¯b′and t′¯t′production, followed by

t′→ W(∗)b′.

For τ′

1in the QDEE model, the branching ratio situation is essentially the same as for the QUE

model as discussed above.

1,b′

2and t,t′respectively. Formulas for the resulting decay widths for b′

1

1and on the type of mixing. If ǫU

1¯b′

1→ W+W−t¯t → W+W+W−W−b¯b. This is the scenario for which the Tevatron limit

Dis dominant, then the decays are dominantly

1→ Zb) = BR(b′

1→ h0b) = 0.5, but with h0b larger for finite masses.

1limit. The predicted branching

1for the latter two cases, assumings kDis at its

1b′

1→ W+W−t¯t is

1b′

1→ h0h0b¯b → bbb¯b¯b¯b. Decays to leptons through

1→ Zb)

Page 28

28

300400

mb’[GeV]

500600700 800

0

0.2

0.4

0.6

0.8

1

b’ Branching Ratios

hb

Wt

Zb

300 400

mb’[GeV]

500600 700800

0

0.2

0.4

0.6

0.8

1

b’ Branching Ratios

hb

Zb

FIG. 15: Branching ratios for the lightest extra quark, b′

states Wt, Zb, and h0b. The left panel shows the “democratic” case that ǫDdominates, and the right panel

shows the “W-phobic” case that ǫ′

case that ǫUdominates leading to mostly charged-current decays, then BR(b′

1, in the QDEE model with MQ= MD, to final

Ddominates, leading to mostly neutral-current decays. In the “W-philic”

1→ Wt) = 1 (not shown).

VI.OUTLOOK

In this paper, I have studied supersymmetric models that have vector-like fermions that are

consistent with perturbative gauge coupling unification and have large Yukawa couplings that can

significantly raise the Higgs mass in supersymmetry. Some of the more important features found

for these models are:

• There are three types of models consistent with perturbative gauge coupling unification and

all new particles near the TeV scale. The first type (LND) contains up to three copies of

the 5 + 5 of SU(5). The second type (QUE) contains a 10 + 10 of SU(5). The third type

(QDEE) is not classifiable in terms of complete representations of SU(5), but consists of the

fields Q,D,E,E and their conjugates.

• A complete vector-like family (i.e. a 16 + 16 of SO(10)) could also be entertained, but

was not considered here because a multi-loop renormalization group analysis shows that this

would forfeit perturbative unification and high-scale control unless (at least some of) the

new particles are much heavier than 1 TeV.

• The constraints imposed by oblique corrections to electroweak observables are rather mild,

especially in comparison to the corresponding constraints on a chiral fourth family, and are

easily accommodated by present data as long as the new quarks with Yukawa couplings are

heavier than about 400 GeV, and perhaps considerably lower.

• The model framework is consistent with the hypothesis that gaugino masses dominate soft

supersymmetry breaking near the unification scale, without problems from sleptons being

too light as is the case in so-called mSUGRA models.

Page 29

29

• The lightest Higgs mass can be substantially raised in the QUE and QDEE models if the

Yukawa couplings are near their fixed points. However, the extent of this is limited if one

takes seriously the prediction for the fixed point behavior of the scalar trilinear couplings,

which limits the mixing in the new squark sector.

at MF = 400 GeV, and their scalar partners have an average mass of MS = 1000 GeV,

then one finds an increase in mh0 of up to about 15 GeV (see Figure 8). For larger MS,

this contribution increases, but at the expense of apparently more severe fine-tuning of the

electroweak scale.

For example, if the new quarks are

• Despite the sizable positive contribution to the lightest Higgs, the contributions to the µ

parameter are also raised, so it is difficult to make any unambiguous claim for an improvement

in the supersymmetric little hierarchy problem.

• The new fermions can decay through any mixture of neutral and charged currents to third-

family fermions and W,Z,h0weak bosons, but with different combinations correlated to the

possible superpotential couplings that mix the new fermions with the Standard Model ones.

• Existing bounds from direct searches at the Tevatron do not significantly constrain the pa-

rameter space of these models after precision electroweak constraints are taken into account.

The collider phenomenology of the MSSM augmented by the new particles in these models should

be both rich and confusing, leading to a difficult challenge at the LHC and beyond in deciphering

the new discoveries.

Appendix A: Contributions to precision electroweak parameters

This Appendix gives formulas for the contributions of the new chiral supermultiplets to the

Peskin-Takeuchi precision electroweak parameters [34]. For convenience I will follow the notations

and conventions of [68]. The oblique parameters S and T are defined in terms of electroweak vector

boson self-energies by

αS

Wc2

4s2

W

αT = ΠWW(0)/M2

=

?

ΠZZ(M2

Z) − ΠZZ(0) −

W− ΠZZ(0)/M2

c2W

cWsWΠZγ(M2

Z) − Πγγ(M2

Z)

?

/M2

Z,(A.1)

Z.(A.2)

In the following, the one-loop integral functions G(x), H(x,y), B(x,y), and F(x,y) are as defined

in ref. [68], and particle names should be understood to stand for the squared mass when used as

an argument of one of these functions, which also have an implicit argument s which is identified

with the invariant mass of the self-energy function in which they appear.

1. Corrections to electroweak vector boson self-energies in the LND model

For the LND model, define the gauge eigenstate new neutral lepton mass matrix by

Mν=

ML

kNvu

hNvd MN

,(A.3)

Page 30

30

and unitary mixing matrices L and R by

L∗MνR†= diag(mν′

1,mν′

2),(A.4)

and note mτ′ = ML. Then the (ν′

self-energies are:

1,ν′

2,τ′) fermion contributions to the electroweak vector boson

∆Πγγ = −Nc

∆ΠZγ = −Nc

∆ΠZZ = −Nc

16π22g2s2

We2

eG(τ′), (A.5)

16π2gsWee(gZ

?

2

?

?

τ′τ′†− gZ

¯ τ′¯ τ′†)G(τ′),(A.6)

16π2

(|gZ

τ′τ′†|2+ |gZ

¯ τ′¯ τ′†|2)G(τ′)

+

i,j=1

(|gZ

ν′

iν′†

j|2+ |gZ

¯ ν′

i¯ ν′†

j|2)H(ν′

i,ν′

j) − 4Re(gZ

ν′

iν′†

jgZ

¯ ν′

i¯ ν′†

j)mν′

imν′

jB(ν′

i,ν′

j)

?

, (A.7)

∆ΠWW = −Nc

16π2

i=1,2

?

(|gW

ν′

iτ′†|2+ |gW

¯ ν′

i¯ τ′†|2)H(τ′,ν′

i) − 4Re(gW

ν′

iτ′†gW

¯ ν′

i¯ τ′†)mτ′mν′

iB(τ′,ν′

i)

?

, (A.8)

where Nc= 1 and ee= −1 and the massive vector boson couplings with the new leptons are:

g

2cWL∗

g

cW

gW

ν′

gZ

ν′

iν′†

j

=

i1Lj1,gZ

¯ ν′

i¯ ν′†

j

= −

?

g

2cWR∗

i1Rj1,(A.9)

gZ

τ′τ′† = −gZ

¯ τ′¯ τ′†=

i1/√2,

?

−1

2− ees2

gW

¯ ν′

W

,(A.10)

iτ′† = gL∗

i¯ τ′†= −gR∗

i1/√2.(A.11)

To obtain the (˜ ν′

1,2,3,4, ˜ τ′

1,2) scalar contribution, consider the new sneutrino squared-mass matrix:

M2

˜ ν= M2

ν+

m2

L+ ∆ 1

2,0

0b∗

L

a∗

kNvu− µkNvd

b∗

N

0m2

N

a∗

hNvd− µhNvu

m2

L+ ∆−1

bL

ahNvd− µ∗hNvu

2,0

0

akNvu− µ∗kNvd

bN

0m2

N

, (A.12)

where the supersymmetric part (also equal to the fermion squared-mass matrix) is:

M2

ν=

MνM†

ν

0

0M†

νMν

.(A.13)

Page 31

31

Also, the new charged slepton squared-mass matrix is given by:

M2

˜ e=

M2

L+ m2

L+ ∆−1

2,−1

−b∗

L

−bL

M2

L+ m2

L+ ∆ 1

2,1

.(A.14)

Now define unitary scalar mixing matrices U and V by:

UM2

˜ νU†= diag(m2

˜ ν′

1,m2

˜ ν′

2,m2

˜ ν′

3,m2

˜ ν′

4), V M2

˜ eV†= diag(m2

˜ τ′

1,m2

˜ τ′

2).(A.15)

Then the scalar contributions to the vector boson self-energies are:

∆Πγγ =

Nc

16π2g2s2

We2

e

2

?

2

?

i=1

F(˜ τ′

i, ˜ τ′

i), (A.16)

∆ΠZγ =

Nc

16π2gsWee

i=1

gZ

˜ τ′

i˜ τ′∗

iF(˜ τ′

i, ˜ τ′

i), (A.17)

∆ΠZZ =

Nc

16π2

?

4

?

2

i,j=1

|gZ

˜ ν′

i˜ ν′∗

j|2F(˜ ν′

i, ˜ ν′

j) +

2

?

i,j=1

|gZ

˜ τ′

i˜ τ′∗

j|2F(˜ τ′

i, ˜ τ′

j)

, (A.18)

∆ΠWW =

Nc

16π2

i=1

4

?

j=1

|gW

˜ τ′

i˜ ν′∗

j|2F(˜ τ′

i, ˜ ν′

j),(A.19)

where the vector boson couplings with the new sleptons are:

gZ

˜ ν′

i˜ ν′∗

j

=

g

2cW(U∗

= g(V∗

i1Uj1+ U∗

i3Uj3),gZ

˜ τ′

i˜ τ′∗

j

=

g

cW

?

−1

2− ees2

W

?

δij,(A.20)

gW

˜ τ′

i˜ ν′∗

j

i1Uj1− V∗

i2Uj3)/√2.(A.21)

2. Corrections to electroweak vector boson self-energies in the QUE model

For the QUE model, the gauge eigenstate new up-type quark mass matrix is:

Mu=

MQ kUvu

hUvd MU

, (A.22)

with unitary mixing matrices L and R defined by

L∗MuR†= diag(mt′

1,mt′

2), (A.23)

Page 32

32

and mb′ = MQ. Then the (t′

energies are:

1,t′

2,b′) fermion contributions to the electroweak vector boson self-

∆Πγγ = −Nc

16π22g2s2

W

?

e2

u

?

(gZ

i=1,2

G(t′

i) + e2

dG(b′)

?

,(A.24)

∆ΠZγ = −Nc

16π2gsW

?

eu

?

i=1,2

t′

it′†

i

− gZ

¯t′

i¯t′†

i)G(t′

i) + ed(gZ

b′b′†− gZ

¯b′¯b′†)G(b′)

?

,(A.25)

∆ΠZZ = −Nc

16π2

?

(|gZ

b′b′†|2+ |gZ

¯b′¯b′†|2)G(b′)

+

2

?

i,j=1

(|gZ

t′

it′†

j|2+ |gZ

¯t′

i¯t′†

j|2)H(t′

i,t′

j) − 4Re(gZ

t′

it′†

jgZ

¯t′

i¯t′†

j)mt′

imt′

jB(t′

i,t′

j)

?

,(A.26)

∆ΠWW = −Nc

16π2

?

i=1,2

?

(|gW

t′

ib′†|2+ |gW

¯t′

i¯b′†|2)H(b′,t′

i) − 4Re(gW

t′

ib′†gW

¯t′

i¯b′†)mb′mt′

iB(b′,t′

i)

?

, (A.27)

where Nc= 3 and eu= 2/3 and ed= −1/3 and the massive vector boson couplings with the new

quarks are:

gZ

t′

it′†

j

=

g

cW

?1

2L∗

i1Lj1− eus2

g

cW

Wδij

?

,gZ

¯t′

i¯t′†

j

=

g

cW

?

−1

2R∗

i1Rj1+ eus2

Wδij

?

, (A.28)

gZ

b′b′† = −gZ

gW

t′

¯b′¯b′†=

i1/√2,

?

−1

2− eds2

gW

¯t′

W

?

,(A.29)

ib′† = gL∗

i¯b′†= −gR∗

i1/√2. (A.30)

To obtain the (˜t′

trix:

1,2,3,4,˜b′

1,2) scalar contribution, consider the up-type squark squared-mass ma-

M2

˜ u= M2

u+

m2

Q+ ∆ 1

2,2

3

0b∗

Q

a∗

kUvu− µkUvd

b∗

U

0m2

U+ ∆0,2

3

a∗

hUvd− µhUvu

bQ

ahUvd− µ∗hUvu m2

Q+ ∆−1

2,−2

3

0

akUvu− µ∗kUvd

bU

0m2

U+ ∆0,−2

3

,(A.31)

where the supersymmetric part (also equal to the fermion squared-mass matrix) is:

M2

u=

MuM†

u

0

0M†

uMu

. (A.32)

Page 33

33

Also, the down-type squark mass matrix is

M2

˜d=

M2

Q+ m2

Q+ ∆−1

2,−1

3

−b∗

Q

−bQ

M2

Q+ m2

Q+ ∆ 1

2,1

3

. (A.33)

Now define unitary scalar mixing matrices U and V by

UM2

˜ uU†= diag(m2

˜t′

1,m2

˜t′

2,m2

˜t′

3,m2

˜t′

4),V M2

˜dV†= diag(m2

˜b′

1,m2

˜b′

2). (A.34)

Then the scalar contributions to the vector boson self-energies are:

∆Πγγ =

Nc

16π2g2s2

W

?

e2

u

4

?

4

?

i=1

F(˜t′

i,˜t′

i) + e2

d

2

?

i=1

F(˜b′

i,˜b′

i)

?

, (A.35)

∆ΠZγ =

Nc

16π2gsW

?

eu

i=1

gZ

˜t′

i˜t′∗

iF(˜t′

i,˜t′

i) + ed

2

?

i=1

gZ

˜b′

i˜b′∗

iF(˜b′

i,˜b′

i)

?

,(A.36)

∆ΠZZ =

Nc

16π2

?

4

?

2

i,j=1

|gZ

˜t′

i˜t′∗

j|2F(˜t′

i,˜t′

j) +

2

?

i,j=1

|gZ

˜b′

i˜b′∗

j|2F(˜b′

i,˜b′

j)

,(A.37)

∆ΠWW =

Nc

16π2

i=1

4

?

j=1

|gW

˜b′

i˜t′∗

j|2F(˜b′

i,˜t′

j),(A.38)

where the vector boson couplings with the new squarks are:

gZ

˜t′

i˜t′∗

j

=

g

cW

?1

i1Uj1− V∗

2(U∗

i1Uj1+ U∗

i2Uj3)/√2.

i3Uj3) − eus2

Wδij

?

,gZ

˜b′

i˜b′∗

j

=

g

cW

?

−1

2− eds2

W

?

δij, (A.39)

gW

˜b′

i˜t′∗

j

= g(V∗

(A.40)

3. Corrections to electroweak vector boson self-energies in the QDEE model

For the QDEE model, define the gauge eigenstate new down-type quark mass matrix by:

Md=

MQ kDvu

hDvd MD

(A.41)

and unitary mixing matrices L and R by:

R∗MdL†= diag(mb′

1,mb′

2), (A.42)

Page 34

34

and note mt′ = MQ. Then the (b′

self-energies are:

1,b′

2,t′) fermion contributions to the electroweak vector boson

∆Πγγ = −Nc

16π22g2s2

W

?

e2

d

?

(gZ

i=1,2

G(b′

i) + e2

uG(t′)

?

,(A.43)

∆ΠZγ = −Nc

16π2gsW

?

ed

?

i=1,2

b′

ib′†

i

− gZ

¯b′

i¯b′†

i)G(b′

i) + eu(gZ

t′t′†− gZ

¯t′¯t′†)G(t′)

?

,(A.44)

∆ΠZZ = −Nc

16π2

?

(|gZ

t′t′†|2+ |gZ

¯t′¯t′†|2)G(t′)

+

2

?

i,j=1

(|gZ

b′

ib′†

j|2+ |gZ

¯b′

i¯b′†

j|2)H(b′

i,b′

j) − 4Re(gZ

b′

ib′†

jgZ

¯b′

i¯b′†

j)mb′

imb′

jB(b′

i,b′

j)

?

,(A.45)

∆ΠWW = −Nc

16π2

?

i=1,2

?

(|gW

b′

it′†|2+ |gW

¯b′

i¯t′†|2)H(t′,b′

i) − 4Re(gW

b′

it′†gW

¯b′

i¯t′†)mt′mb′

iB(t′,b′

i)

?

, (A.46)

where Nc= 3 and eu= 2/3 and ed= −1/3 and the massive vector boson couplings with the new

quarks are:

gZ

b′

ib′†

j

=

g

cW

?

−1

2L∗

i1Lj1− eds2

g

cW

i1/√2,

Wδij

?

,

,gZ

¯b′

i¯b′†

j

=

g

cW

?1

2R∗

i1Rj1+ eds2

Wδij

?

, (A.47)

gZ

t′t′† = −gZ

gW

b′

¯t′¯t′†=

?1

2− eus2

W

?

(A.48)

it′† = −gL∗

gW

¯b′

i¯t′†= gR∗

i1/√2. (A.49)

To obtain the (˜b′

matrix:

1,2,3,4,˜t′

1,2) scalar contribution, start with the down-type squark squared-mass

M2

˜d= M2

d+

m2

Q+ ∆ 1

2,1

3

0b∗

Q

a∗

kDvu− µkDvd

b∗

D

0m2

D+ ∆0,1

3

a∗

hDvd− µhDvu

bQ

ahDvd− µ∗hDvu m2

Q+ ∆−1

2,−1

3

0

akDvu− µ∗kDvd

bD

0m2

D+ ∆0,−1

3

, (A.50)

where the supersymmetric part (also equal to the fermion squared-mass matrix) is:

M2

d=

MdM†

d

0

0M†

dMd

. (A.51)

Page 35

35

Also, the up-type squark squared-mass matrix is given by:

M2

˜ u=

M2

Q+ m2

Q+ ∆−1

2,−2

3

−b∗

Q

−bQ

M2

Q+ m2

Q+ ∆ 1

2,2

3

.(A.52)

Now define unitary scalar mixing matrices U and V by:

UM2

˜dU†= diag(m2

˜b′

1,m2

˜b′

2,m2

˜b′

3,m2

˜b′

4),V M2

˜ uV†= diag(m2

˜t′

1,m2

˜t′

2).(A.53)

Then the scalar contributions to the vector boson self-energies are:

∆Πγγ =

Nc

16π2g2s2

W

?

e2

d

4

?

?

i=1

F(˜b′

i,˜b′

i) + e2

u

2

?

i=1

F(˜t′

i,˜t′

i)

?

, (A.54)

∆ΠZγ =

Nc

16π2gsW

?

−ed

4

i=1

gZ

˜b′∗

i˜b′

iF(˜b′

i,˜b′

i) − eu

2

?

i=1

gZ

˜t′∗

i˜t′

iF(˜t′

i,˜t′

i)

?

, (A.55)

∆ΠZZ =

Nc

16π2

?

4

?

2

i,j=1

|gZ

˜b′∗

i˜b′

j|2F(˜b′

i,˜b′

j) +

2

?

i,j=1

|gZ

˜t′∗

i˜t′

j|2F(˜t′

i,˜t′

j)

,(A.56)

∆ΠWW =

Nc

16π2

i=1

4

?

j=1

|gW

˜t′∗

i˜b′

j|2F(˜t′

i,˜b′

j),(A.57)

where the vector boson couplings with the new squarks are

gZ

˜b′∗

i˜b′

j

=

g

cW

?1

i1Uj1− V∗

2(U∗

i1Uj1+ U∗

i2Uj3)/√2.

i3Uj3) + eds2

Wδij

?

,gZ

˜t′∗

i˜t′

j=

g

cW

?

−1

2+ eus2

W

?

δij, (A.58)

gW

˜t′∗

i˜b′

j

= g(V∗

(A.59)

Appendix B: Formulas for decay widths of new quarks and leptons

This Appendix gives formulas for the decay widths of the lightest of the new quarks and leptons

to Standard Model states. These decays are assumed to be mediated by Yukawa couplings that

provide small mass mixings that can be treated as perturbations compared to the other entries in

the mass matrices. In the following, λ(x,y,z) = x2+ y2+ z2− 2xy − 2xz − 2yz.

1. Decays of b′in the LND model

In the LND model, the lightest quark b′can decay to Standard Model states because of the

mixing Yukawa parameter ǫD in eq. (5.1). In terms of the mass matrix Mdin eq. (5.2), define

unitary mixing matrices L and R by:

L∗MdR†= diag(mb,mb′).(B.1)

Page 36

36

The relevant couplings of b′to Standard Model particles are

gW

b′t† = gL∗

b′¯b= −sin(α)(ybR12+ ǫDR11)L22/√2,

yh0

22/√2,gZ

b′b†= −

g

2cWL∗

22L12,(B.2)

yh0

¯b′b= −sin(α)(ybR22+ ǫDR21)L12/√2.

(B.3)

(B.4)

It follows that the decay widths of b′are:

Γ(b′→ Wt) =

Γ(b′→ Zb) =

Γ(b′→ h0b) =

mb′

32π|gW

mb′

32π|gZ

mb′

32π

b′t†|2λ1/2(1,rW,rt)(1 + rt− 2rW+ (1 − rt)2/rW),

b′b†|2(1 − rZ)2(2 + 1/rZ),

?

i/m2

(B.5)

(B.6)

|yh0

b′¯b|2+ |yh0

¯b′b|2?

(1 − rh0)2, (B.7)

where mbis neglected for kinematic purposes and ri= m2

b′ for i = Z,W,h0.

2. Decays of ν′

1in the LND model

Consider the decays of ν′

the superpotential mixing terms ǫNand ǫEin eq. (5.1). Define unitary mixing matrices L (3 × 3)

and R (2 × 2) in terms of the neutral lepton mass matrix in eq. (5.3) by:

1, the lighter new neutral lepton in the LND model, brought about by

R∗MT

νL†=

0 mν′

1

0

00mν′

2

(B.8)

where we are neglecting the tau neutrino mass. Also define unitary matrices L′and R′in terms of

the charged lepton mass matrix in eq. (5.4) by:

L′∗MeR′†= diag(mτ,mτ′).(B.9)

Then the relevant couplings of ν′

1to Standard Model particles are:

gW

ν′

1τ† = g(L∗

21L′

11+ L∗

23L′

12)/√2gW

¯ ν′

1¯ τ†= gR∗

11R′

11/√2(B.10)

gZ

ν′

1ν† =

g

2cW(L∗

cosα

√2

21L11+ L∗

23L13)(B.11)

yh0

¯ ν′

1ν=

(ǫNL13+ kNL11)R12−sinα

√2

hNL12R11.(B.12)

It follows that the decay widths of ν′

1are:

Γ(ν′

1→ Wτ) =

Γ(ν′

1→ Zντ) =

mν′

32π(1 − rW)2(2 + 1/rW)(|gW

mν′

1

32π(1 − rZ)2(2 + 1/rZ)|gZ

1

ν′

1τ†|2+ |gW

¯ ν′

1¯ τ†|2), (B.13)

ν′

1ν†|2,(B.14)

Page 37

37

Γ(ν′

1→ h0ντ) =

mν′

32π(1 − rh0)2|yh0

1

¯ ν′

1ν|2,(B.15)

where mτand mντare neglected for kinematic purposes and ri= m2

i/m2

ν′

1for i = Z,W,h0.

3. Decays of t′

1in the QUE model

Consider the decays of t′

superpotential mixing terms in eq. (5.5). Define unitary mixing matrices L, R, L′, R′in terms of

the mass matrices in eq. (5.6) by:

1, the lightest new quark in the QUE model, brought about by the

L∗MuR†= diag(mt,mt′

1,mt′

2),L′∗MdR′†= diag(mb,mb′). (B.16)

Then the relevant couplings of t′

1to Standard Model particles are:

gW

t′

1b† = g(L∗

21L′

11+ L∗

23L′

12)/√2,gW

¯t′

1¯b†= gR∗

21R′

g

2cWR∗

11/√2, (B.17)

gZ

t′

1t† =

g

2cW(L∗

cosα

√2

cosα

√2

21L11+ L∗

23L13),gZ

¯t′

1¯t†= −

21R11,(B.18)

yh0

t′

1¯t=

?ǫUL23R12+ ǫ′

?ǫUL13R22+ ǫ′

UL21R13+ kUL21R12+ ytL23R13

?−sinα

?−sinα

√2

hUL22R11, (B.19)

yh0

¯t′

1t=

UL11R23+ kUL11R22+ ytL13R23

√2

hUL12R21.(B.20)

It follows that the decay widths of t′

1are:

Γ(t′

1→ Wb) =

Γ(t′

1→ Zt) =

mt′

32π(1 − rW)2(2 + 1/rW)(|gW

mt′

1

32πλ1/2(1,rZ,rt)

+12√rtRe(gZ

t′

mt′

1

32πλ1/2(1,rh0,rt)

1

t′

1b†|2+ |gW

¯t′

1¯b†|2), (B.21)

?

1¯t†)

(1 + rt− 2rZ+ (1 − rt)2/rZ)(|gZ

?

?

t′

1t†|2+ |gZ

¯t′

1¯t†|2)

1t†gZ

¯t′

, (B.22)

Γ(t′

1→ h0t) =

(1 + rt− rh0)(|yh0

t′

1¯t|2+ |yh0

¯t′

1t|2) + 4√rtRe(yh0

¯t′

1tyh0

t′

1¯t)

?

,(B.23)

where the bottom quark is treated as massless for purposes of kinematics and ri= m2

i = t,Z,W,h0.

i/m2

t′

1for

4. Decays of b′

1in the QDEE model

Consider the decays of b′

superpotential mixing terms in eq. (5.9). Define unitary mixing matrices R, L, R′, L′in terms of

the mass matrices in eq. (5.10) by:

1, the lightest new quark in the QDEE model, brought about by the

R∗MdL†= diag(mb,mb′

1,mb′

2),R′∗MuL′†= diag(mt,mt′). (B.24)

Then the relevant couplings of b′

1to Standard Model particles are:

gW

b′

1t† = g(L∗

21L′

11+ L∗

23L′

12)/√2,gW

¯b′

1¯t†= gR∗

21R′

11/√2,(B.25)

Page 38

38

gZ

b′

1b† = −

1¯b= −sinα

1b= −sinα

g

2cW(L∗

21L11+ L∗

23L13),gZ

¯b′

1¯b†=

g

2cWR∗

21R11,(B.26)

yh0

b′

√2

?ǫDL23R12+ ǫ′

?ǫDL13R22+ ǫ′

DL21R13+ hDL21R12+ ybL23R13

?+cosα

?+cosα

√2

kDL22R11,(B.27)

yh0

¯b′

√2

DL11R23+ hDL11R22+ ybL13R23

√2

kDL12R21. (B.28)

It follows that the decay widths of b′

1are:

Γ(b′

1→ Wt) =

mb′

32πλ1/2(1,rW,rt)

+12√rtRe(gW

mb′

1

32π(1 − rZ)2(2 + 1/rZ)(|gZ

mb′

1

32π(1 − rh0)2(|yh0

1

?

(1 + rt− 2rW+ (1 − rt)2/rW)(|gW

?

b′

¯b′

b′

1t†|2+ |gW

¯b′

1¯t†|2)

b′

1t†gW

¯b′

1¯t†),(B.29)

Γ(b′

1→ Zb) =

1b†|2+ |gZ

1¯b†|2),(B.30)

Γ(b′

1→ h0b) =

b′

1¯b|2+ |yh0

¯b′

1b|2),(B.31)

where the bottom quark is treated as massless for purposes of kinematics and ri= m2

i = t,Z,W,h0.

i/m2

b′

1for

5. Decays of τ′in the QUE and QDEE models

Consider the decays of τ′in the QUE model, brought about by the superpotential mixing term

ǫEin eq. (5.7). In terms of the mass matrix eq. (5.8), define unitary mixing matrices L and R by:

L∗MeR†= diag(mτ,mτ′).(B.32)

Then the relevant couplings of τ′to Standard Model particles are:

gW

τ′ν† = gL∗

τ′¯ τ= −sin(α)L22(yτR12+ ǫER11)/√2,

yh0

22/√2,gZ

τ′τ†= −

g

2cWL∗

22L12,(B.33)

yh0

¯ τ′τ= −sin(α)L12(yτR22+ ǫER21)/√2.

(B.34)

(B.35)

It follows that the decay widths of τ′are:

Γ(τ′→ Wν) =

Γ(τ′→ Zτ) =

Γ(τ′→ h0τ) =

mτ′

32π(1 − rW)2(2 + 1/rW)|gW

mτ′

32π(1 − rZ)2(2 + 1/rZ)|gZ

mτ′

32π(1 − rh0)2(|yh0

τ′ν†|2,

τ′τ†|2,

¯ τ′τ|2),

(B.36)

(B.37)

τ′¯ τ|2+ |yh0

(B.38)

where ri= m2

model, the same calculation holds, provided that ME is replaced by ME1corresponding to the

lighter mass eigenstate mτ′.

i/m2

τ′ for i = Z,W,h0, and mτ is neglected for kinematic purposes. In the QDEE

Acknowledgments: I am indebted to James Wells for useful comments. This work was supported

in part by the National Science Foundation grant number PHY-0757325.

Page 39

39

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