Testing Little Higgs Mechanism at Future Colliders

0.23 MB·English

by  Keisuke Harigaya

IPMU-11-0110
September 23, 2011
Testing Little Higgs Mechanism at Future Colliders
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p Keisuke Harigaya(a), Shigeki Matsumoto(a),
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S Mihoko M. Nojiri(b,a) and Kohsaku Tobioka(a)

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h (a)IPMU, TODIAS, University of Tokyo, Kashiwa, 277-8583, Japan
p
(b)Theory Group, KEK, Tsukuba, 305-0801, Japan
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1
v
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4 Abstract
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4
. In the framework of the little higgs scenario, coupling constants of several
9
0 interactions are related to each other to guarantee the stability of the higgs
1
1 bosonmassatone-looplevel. Thisrelationiscalledthelittlehiggsmechanism.
:
v We discuss how accurately the relation can be tested at future e+e− colliders,
i
X with especially focusing on the top sector of the scenario using a method of
r
a effective lagrangian. In order to test the mechanism at the top sector, it is
important to measure the Yukawa coupling of the top partner. We consider
higgs associated production and threshold production of the top partner, and
findthatthemechanismcan betested precisely usingtheassociate production
whenthecenter ofmassenergyislargeenough. Thethresholdproductionalso
allows us to test it even if the center mass energy is not so large.
1 Introduction
It is well known that the standard model (SM) has a serious problem called the little
hierarchy problem [1], which is essentially from quadratically divergent corrections
to the higgs mass term. The little higgs scenario [2] has been proposed to solve
the problem. In the scenario, the higgs boson is regarded as a pseudo Nambu-
Goldstone boson associated with a spontaneous breaking of a global symmetry at
the energy scale of (10) TeV. Explicit breaking terms of the symmetry are also
O
specially arranged to cancel the quadratically divergent corrections at one-loop level.
The mechanism of this cancellation is sometimes called the little higgs mechanism,
which is commonly equipped in all models of the little higgs scenario.
The little higgs mechanism predicts the existence of new particles at the scale
of (1) TeV, which are called little higgs partners. The mechanism also predicts
O
some relations between coupling constants of SM interactions and those of the new
particles. Among the partners, the top partner is the most important one, because
it is responsible for the cancelation of the largest quadratically divergent correction
to the higgs mass term. The top partner has a color charge and could be produced
in near future [3] at the large hadron collider (LHC) [4]. The discovery of the top
partner, however, doesnotmeantheconfirmationofthelittlehiggsscenario, because
new particles which are similar to the top partner are also predicted in various new
physics extensions of the SM. In order to test the little higgs scenario, we have to
verify the relation between interactions predicted by the little higgs mechanism.
This verification requires us to measure the Yukawa coupling of the top part-
ner. Future linear colliders such as the international linear collider (ILC) [5] and the
compact linear collider (CLIC) [6] give a good opportunity for coupling measure-
ments [7]. Following four processes are considered in this article; higgs associated
productions (e+e− TT¯h, tT¯h + Tt¯h) and threshold productions (e+e− TT¯,
→ →
Tt¯+ tT¯), where t, T, and h are top quark, top partner, and higgs boson, respec-
tively. We found that the coupling can be measured precisely using the associate
production e+e− TT¯h when the center of mass energy is large enough. The
→
threshold production e+e− TT¯ also allows us to measure it with the same preci-
→
sion. Interestingly, with smaller center of mass energy, it is even possible to measure
the coupling using the threshold production e+e− t¯T +Tt¯.
→
This article is organized as follows. In the next section, we introduce the effective
lagrangian to describe the top sector of the little higgs scenario. The little higgs
1
mechanism is quantitatively described using the lagrangian. In sections 3 and 4,
higgs associated and threshold productions of the top partner are discussed with
particularly focusing on how the cross sections of these processes are sensitive to the
Yukawa coupling of the top partner. In section 5, we consider how accurately the
Yukawa coupling can be measured using the processes discussed in previous sections,
and discuss the capability of future linear colliders to test the little higgs mechanism.
Section 6 is devoted to summary of our discussions.
2 Top sector of the little higgs scenario
The method using an effective lagrangian is adopted to investigate the top sector of
the little higgs scenario. In the following, we introduce the lagrangian and discuss
what kinds of interactions are predicted from it. We finally mention our strategy to
test the little higgs mechanism at future linear colliders such as the ILC.
2.1 Effective lagrangian
In the little higgs scenario, the vector-like quark called the top partner is necessar-
ily introduced, which is responsible for the cancellation of quadratically divergent
corrections to higgs mass term from the top quark [2]. Interactions between higgs
boson, top quark, and top partner are described by the effective lagrangian,1
= m U¯ U y Q¯ Hcu y Q¯ HcU
eff U L R 3 3L 3R U 3L R
L − − −
(λ/m )U¯ u H 2 (λ′/m )U¯ U H 2 +h.c., (1)
U L 3R U L R
− | | − | |
where Q = (u ,b )T and u are third generation left- and right-handed quarks,
3L 3L 3L 3R
while U and U are left- and right-handed top partners. Higgs boson is denoted
L R
by Hc, where the superscript ’c’ denotes charge conjugation. Quantum numbers of
these fields are shown in Table 1. Here, we postulate that top partners couple only
to third generation quarks to avoid flavor changing processes. Model parameters
m , y , y and λ are taken to be real by appropriate redefinitions of the fields. On
U 3 U
the other hand, the parameter λ′ can be complex in general. We take, however, this
parameter to also be real because of the little higgs mechanism discussed below.
1
For simplicity, we have omitted to write gauge interactions of third generationquarks and top
partners in the effective lagrangian. See appendix A for the derivation of the lagrangian.
2
Q u U U H
3L 3R L R
SU(3) 3 3 3 3 1
c
SU(2) 2 1 1 1 2
L
U(1) 1/6 2/3 2/3 2/3 1/2
Y
Table 1: Quantum numbers of Q , u , U , U and H.
3L 3R L R
The little higgs mechanism at the top sector can be quantitatively defended by
using the effective lagrangian in eq.(1). Since quadratically divergent corrections to
the higgs mass term should be cancelled with each other at 1-looplevel, the following
relation between the coupling constants (y , y , and λ′) is required,
3 U
2λ′ = y2+y2, (2)
− 3 U
which is nothing but the little higgs mechanism at the top sector. It is thus very
important to confirm the relation experimentally. The purpose of this article is to
clarify what kind of observation is the most efficient for this confirmation.
Once H acquires the vacuum expectation value Hc = (v/√2,0) with being
h i
v 246 GeV, the electroweak symmetry (SU(2) U(1) ) is broken, and third
L Y
≃ ×
generation quarks are mixed with top partners. Mass matrix of these particles is
A B u m 0 t
u¯ U¯ 3R = t¯ T¯ t R , (3)
3L L L L
C D! U ! 0 m ! T !
R T R
(cid:16) (cid:17) (cid:16) (cid:17)
where A, B, C, and D are defined by A y v/√2, B y v/√2, C λv2/(2m ),
3 U U
≡ ≡ ≡
and D m + λ′v2/(2m ), respectively. We call a Dirac fermion t composed of
U U
≡
t and t the top quark in following discussions. We also call T the top partner
L R
which is defined by T and T in the same manner as t. Mixing angles for left- and
L R
right-handed quarks to diagonarize the mass matrix are then defined by
t cosθ sinθ u t cosθ sinθ u
L L L 3L R R R 3R
= − , = − . (4)
T ! sinθ cosθ ! U ! T ! sinθ cosθ ! U !
L L L L R R R R
Using model parameters (m , y , y , λ, λ′), which are defining the effective la-
U 3 U
grangian, mass eigenvalues (m , m ) and mixing angles (tanθ , tanθ ) are
t T L R
m = (A2 +B2 +C2 +D2 ∆)/2 y v/√2, (5)
t 3
− ≃
m = p(A2 +B2 +C2 +D2 +∆)/2 m , (6)
T U
≃
tanθ = (p∆+A2 +B2 C2 D2)/(2AC +2BD) y v/(√2m ), (7)
L U U
− − ≃
tanθ = (∆+A2 B2 +C2 D2)/(2AB +2CD) (y y +λ)v2/(2m2),(8)
R − − ≃ 3 U U
3
where ∆ in above expressions is defined by ∆ (A2 +B2 +C2 +D2)2 4(AD
≡ { − −
BC)2 1/2. Last term in each expression is the leading approximation of (v/m ).
U
} O
2.2 Interactions
Here, we discuss interactions predicted by the effective lagrangian in eq.(1) (and
eq.(23)). Though the effective lagrangian in eq.(1) is originally defined by the model
parameters (m , y , y , λ, λ′), we use following five parameters (m , m , sinθ , λ,
U 3 U t T L
λ′) as fundamental ones defining the lagrangian in following discussions. Parameters
(m , y , y , tanθ ) are therefore given as functions of the fundamental parameters
U 3 U R
(m , m , sinθ , λ, λ′), which are obtained numerically by solving eqs.(5)-(8). Gauge
t T L
and Yukawa interactions including t, T and b (bottom quark) are then given by
2e 2e gc s
= g t¯G/t g T¯G/T t¯A/t T¯A/T L L(T¯Z/P t+h.c.)
int s s L
L − − − 3 − 3 − 2c
W
g 2s2 c2 g 2s2 s2
t¯Z/ W + LP t T¯Z/ W + LP T
L L
−c − 3 2 − c − 3 2
W (cid:18) (cid:19) W (cid:18) (cid:19)
gc gs
L(¯bW/ P t+h.c.) L(¯bW/ P T +h.c.)
L L
−√2 − √2
y t¯th y T¯Th (T¯[y P +y P ]th+h.c.), (9)
t T L L R R
− − −
where G/ = Ga(λa/2)γµ, W/ = W γµ, Z/ = Z γµ, and A/ = A γµ are gluon, W
µ µ µ µ
boson, Z boson, and photon fields with λa and γµ being Gell-Mann and gamma
matrices, while h denotes higgs field. Coupling constants associated with SU(3) ,
c
SU(2) , and U(1) gauge interactions are denoted by g , g, and e = gs with
L EM s W
being s (c ) = sinθ (cosθ ), where θ is the Weinberg angle. We have also used
W W W W W
the notation, s (c ) = sinθ (cosθ ). Coupling constants associated with Yukawa
L L L L
interactions (y , y , y , y ) have complicated forms, and these are given by
t T L R
c s v m
y = L (c y s y ) L (c λ s λ′) t, (10)
t R 3 R U R R
√2 − − m − ≃ v
U
s c v m v
y = L (c y +s y )+ L (c λ′ +s λ) Ts2 + c λ′, (11)
T √2 R U R 3 m R R ≃ v L m L
U T
c s v s m
y = L (c y +s y ) L (c λ′ +s λ) L T, (12)
L R U R 3 R R
√2 − m ≃ v
U
s c v s m λv
y = L (c y s y )+ L (c λ s λ′) L t + , (13)
R R 3 R U R R
√2 − m − ≃ v m
U T
where it should be emphasized again that the parameters (c , s , y , y , m ) are
R R 3 U U
obtained as functions of the fundamental parameters (m , m , sinθ , λ, λ′). It can
t T L
be seen in eq.(9) that weak gauge interactions of the top partner are governed by
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the mixing angle s , because these interactions are originally from those of SU(2)
L L
doublet field Q through the mixing between Q and U . On the other hand,
3L 3L L
Yukawa interactions depend not only on s but also on other parameters.
L
The little higgs mechanism at the top sector is quantitatively described by the
relation between coupling constants y , y , and λ′, as shown in eq.(2). It can be seen
3 U
in eqs.(5)-(8) that first two parameters y and y arealmost determined by masses of
3 U
top quark and top partner (m and m ), and the mixing angle s . These parameters
t T L
are not difficult to be measured precisely when the top partner is discovered. On
the other hand, in order to determine the last parameter λ′, we have to measure the
Yukawa coupling of the top partner y , as can be seen in eqs.(10)-(13).
T
2.3 Representative point
Wefirstpostulatethatseveral physical quantities(m , s , andbranchingfractionsof
T L
the T decay) have already been measured precisely. This is actually possible because
m and the branching fractions can be measured accurately by observing the pair
T
production of top partners, while s can be determined by the single production of
L
the top partner [3]. As a representative point of these observables, we take
m = 400GeV, sinθ = 0.2, Br(T th)/Br(T bW) = 0.98, (14)
T L
→ →
where the value of third observable, namely, the ratio of branching fractions, corre-
sponds to the one which is obtained by assuming λ = 0 with keeping the relation in
eq.(2). Higgs mass is fixed to be m = 120 GeV. It is worth notifying that the point
h
satisfies all phenomenological constraints and is also attractive from the viewpoint
of naturalness on the little hierarchy problem. For more details, see appendix B.
Since the mass of top quark m has already been measured precisely [8], there are
t
four free parameters in the effective lagrangian of eq.(1). It is therefore possible to
test the little higgs mechanism by measuring one more observable. This observable
should be sensitive to λ′, as pointed out in previous subsection. Since gauge interac-
tions of the top partner depend only on s , we should focus on Yukawa interactions.
L
In Fig. 1, we have shown coupling constants of Yukawa interactions (y , y , y , y )
t T L R
as functions of λ′ in unit of λ′ (y2+y2)/2, so that λ′ = λ′ corresponds to the
cr ≡ − 3 U cr
prediction of the mechanism. Other model parameters are fixed to be those satisfy-
ing the conditions in eq.(14). It can be seen that the Yukawa coupling between top
partners y is the most sensitive against the change of λ′ as expected. As a result,
T
we should focus on physical quantities involving this Yukawa coupling.
5
1.0
y
t
y
0.5 L
y
R
0.0
y
T
−0.5
Yukawa couplings
−1.0
−1 0 1 2 3
λ’ (in unit of λ ’ )
cr
Figure 1: Yukawa couplings (y , y , y , y ) as a function of λ′ (in unit of λ′ ).
t T L R cr
3 Associate Productions
It can be easily imagined that top partner productions associated with a higgs boson
enable us to explore the little higgs mechanism. It is actually possible to measure
the Yukawa coupling between top partners (y ) simply by measuring cross sections
T
of the processes with an appropriate center of mass energy. There are two higgs
associated processes. One is the higgs production associating with a top quark and
atoppartnerproduction(e+e− tT¯h&Tt¯h), andanotheristheprocessassociating
→
with two top partners (e+e− TT¯h). At the former process, the center of mass
→
energy √s m + m + m is, at least, required, which corresponds to about 700
t T h
≥
GeV in our representative point. On the other hand, at the latter process, the center
of mass energy should be higher than √s 2m + m , which corresponds to 920
T h
≥
GeV in the representative point. In this section, we consider how the cross sections
of these processes are sensitive against the change of the parameter λ′, and discuss
which process is suitable for the confirmation of the little higgs mechanism.
3.1 The e+e tT¯h + Tt¯h process
−
→
We first consider the higgs production process associating with a top quark and a
top partner. The sum of the cross sections, σ(e+e− tT¯h) +σ(e+e− Tt¯h), are
→ →
shown in Fig.2 (upper part of the left panel) as a function of center of mass energy
with several choices of λ′. Other model parameters to depict the figure are fixed
according to the representative point in eq.(14). In order to see the sensitivity of the
cross section against the change of λ′, we also plot the deviation of the cross section
6
1000 10
− − −
t T h + T t h production T T h production
σn (fb) 100 σn (fb) 1
o 10 o
cti cti
e e
s λ’ = 3λ’ s
Cross 1 λλ’’ == 2 λλ’’cccrrr Cross 0.1 λ’ = λ’cr
0.1 λ’ = 0λ’cr λ’ = −λ’cr λ’ = 2λ’cr
λ’ = −λ’cr λ’ = 0λ’cr λ’ = 3λ’cr
8
0.2
σ/σ σ/σ 4
δ 0 δ
0
−0.2
700 800 900 1000 1100 1200 1000 1200 1400 1600
Center of mass energy (GeV) Center of mass energy (GeV)
Figure 2: Cross sections for higgs associated productions e+e− tT¯h+Tt¯h and e+e−
→ →
TT¯h as a function of center of mass energy. Results for several values of λ′ are shown.
from the one predicted by the little higgs mechanism, namely, δσ/σ [σ(λ′)
≡ −
σ(λ′ )]/σ(λ′ ) (lower part of the left panel). It can be seen from the figure that the
cr cr
deviation becomes almost zero when the center of mass energy exceeds 800 GeV.
This is because, with a center of mass energy higher than this value, two on-shell
top partners can be produced, whose production cross section and branching ratio
T thareindependent oftheparameterλ′, andthisprocessdominatestheassociate
→
production. On the other hand, the center of mass energy below 800 GeV, we can
expect about 10% deviation. The cross section is, however, quite small below 800
GeV, which is about 0.1 fb. After all, the measurement of the Yukawa coupling λ′
turns out to be difficult in this associate production.
3.2 The e+e TT¯h process
−
→
Next, we consider the higgs production process associating with two top partners.
As in the case of previous associate production, we show its production cross section
σ(e+e− TT¯h) in Fig.2 as a function of center of mass energy with various λ′
→
(upper part of the right panel). The deviation of the cross section from the little
higgs prediction is also shown (lower part of the right panel). It can be seen from
the figure that the cross section depends strongly on the value of λ′, which enable
us to explore the little higgs mechanism accurately using this association process,
7
though the production cross section itself is not so large.
4 Threshold productions
We next consider top partner productions, e+e− TT¯ and e+e− tT¯ + Tt¯, at
→ →
the region of their threshold energies. Since the cross sections of these processes are
significantly affected by the exchange of virtual higgs bosons due to the threshold
singularity [9], the Yukawa coupling y is expected to be measured precisely. In this
T
section, we consider how the cross sections are sensitive to the parameter λ′.
4.1 Cross section formula
The cross section of top quark pair production at the e+e− collider (e+e− tt¯)
→
is known to have threshold singularities due to exchanges of soft gluons and higgs
bosons between top quarks [9]. Productions of top partner are, in the same man-
ner, expected to have the same singularities, which are quantitatively obtained by
using a method of non-relativistic field theory [10]. Derivations of non-relativistic
lagrangians for top partner productions and resultant cross section formulae are dis-
cussed in appendix C. For the case of top partner pair production (e+e− TT¯),
→
the cross section at their threshold energy √s 2m is obtained to be
T
≃
16Q2Q2 12Q Q v v 6v2(v2 +a2)
σ = T e + T e T e + T e e Im G (√s 2m ;0,0) , (15)
TT s2 s(s m2) (s m )2 TT − T
(cid:20) − Z − Z (cid:21)
(cid:2) (cid:3)
where Q = 2e/3, v = (g/c )( 2s2 /3+s2/4), Q = e, v = (g/c )( 1/4+s2 ),
T T W − W L e − e W − W
and a = g/(4c ). The mass of Z boson is denoted by m . The green function
e W Z
G (E;r,r′) in above formula satisfies the following Schro¨dinger equation,
TT
2 Γ
∇r +V (r) E i T G (E,r,r′) = δ3(r r′), (16)
TT TT
−m − − 2 −
(cid:20) T (cid:21)
with appropriate boundary conditions [9]. Here, Γ is the total decay width of the
T
top partner. All information of soft gluon and higgs exchanges between top partners
are involved in the potential term V (r), which is explicitly given by
TT
1 4α
V (r) = s +α e−mh|r| , (17)
TT − r 3 T
| | (cid:18) (cid:19)
whereα = g2/(4π)andα = y2/(4π). InadditiontoQCDandYukawainteractions,
s s T T
the electroweak interaction (the exchange of Z bosons between top partners) also
8
contributes to the potential. This effect is however negligible and not included in
our calculations. One of the purposes of this article is to clarify the effectiveness
of threshold productions to measure the Yukawa coupling y , we only consider the
T
potentialattheleadingordercalculation. Inordertocomparetheoreticalpredictions
withexperimental resultsveryprecisely, weshouldincludehigherordercontributions
especially from QCD interactions [10]. Since those calculations are beyond the scope
of this article and our conclusion is expected not to be altered even if we include
those effects, we omit the higher order contributions in our calculation.
On the other hand, for the case of top quark and top partner production, the
cross section at the threshold energy region (√s m +m ) turns out to be
t T
≃
12v2 (v2 +a2)
σ = tT e e Im G (√s m m ,0,0) , (18)
tT (s m2)2 tT − t − T
− Z
(cid:2) (cid:3)
where v = gc s /(4s ). The green function G (E;r,r′) satisfies the equation,
tT L L W tT
2 Γ
∇r +V (r) E i tT G (E,r,r′) = δ3(r r′), (19)
tT tT
−2µ − − 2 −
(cid:20) tT (cid:21)
where µ is the inertial mass of top quark and top partner, µ = m m /(m +m ),
tT tT t T t T
while Γ is the averaged decay width of these particles, Γ = (Γ +Γ )/2 with Γ
tT tT t T t
being the total decay width of the top quark. The potential V (r) is given by
tT
1 4α y y
V (r) = s + t Te−mh|r| . (20)
tT − r 3 4π
| | (cid:18) (cid:19)
Finally, we would like to add a comment on the scale of the strong coupling α
s
which appears in the potential terms V (r) and V (r). The scale (µ) is taken to
TT TT
be the solution of the following self-consistency equation of the coupling,
m 4
µ = α(run)(µ), (21)
2 × 3 × s
(run)
where α (x) is the running coupling of the strong interaction at the scale x, and
s
m is the inertial mass of a two-body system, namely, m = m /2 for top partner
T
pair production and m = m m /(m +m ) for the production of top quark and top
t T t T
partner. This prescription is known to make higher order QCD corrections to the
Coulomb potential small at the Bohr radius [11].
4.2 The e+e TT¯ process
−
→
We first consider the threshold production of top partner pair. The resultant cross
section, which is obtained using the formula in eq.(15), is shown in the left panel of
9