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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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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).

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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.