Page 1
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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3
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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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)
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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)
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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
Note added: shortly after the present paper, one with some related subject matter appeared [69].
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