Dark Matter Implication and Collider Phenomenologyshufang/talk/IHDM_DM_collider_talk.pdf · C.P....

S. Su

Inert Higgs Doublet Model:

Dark Matter Implication and Collider Phenomenology

In collaboration with E. Dolle, X. Miao and B. Thomas

Shufang Su • U. of Arizona

S. Su 2

Outline -

Introduction

Inert Higgs Doublet Model

Experimental and Theoretical ConstraintsDark Matter Relic Density

Collider Phenomenology

S. Su 3

Dark matter candidate-

WIMP: natural candidate for dark matter

WMAP: Ωh2=0.112 ± 0.009

•Spin 1/2 DM: neutralino in MSSM•Spin 1 DM: LKP photon in UED•Spin 0 DM

-SM + extra scalar S

-SM + extra SU(2)L scalar doublet: WIMP

V. Silveira and A. Zee, PLB 161, 136 (1985)J. McDonald, PRD 50, 3637 (1994)C.P. Burgess, M. Pospelov and T. ter Veldhuis, NPB 619, 709 (2001)

G. Servant and I.M.P.Tait, NPB 650, 391 (2003)

This paper is organized as follows. In the next section we identify the three pa-

rameters which describe the model, and determine the general conditions which lead

to acceptable masses and to su!ciently stable dark matter. In section 3 we calculate

the annihilation cross section of S-particles and give the resulting cosmic abundance as

a function of masses and couplings. This calculation is similar to the analysis of Ref.

[10]. We perform the numerical analysis for the most interesting part of the parameter

space, with 100 GeV ! mh ! 200 GeV and 10 GeV ! mS ! 100 GeV. In section 4

we obtain the cross section for elastic scattering with ordinary matter and apply the

constraints, imposed by direct and indirect searches. Section 5 computes the cross

sections for the missing energy events which are predicted for colliders due to the pair

production of S particles. It also contains a prediction for the degradation of the Higgs

boson signal at hadronic colliders, when the Higgs boson is allowed to decay into a pair

of S particles. Our conclusions are reserved for section 6.

2 The Model’s Lagrangian

The lagrangian which describes our model has the following simple form:

L = LSM +1

2!µS !µS "

m20

2S2 "

"S

4S4 " " S2 H†H, (2.1)

where H and LSM respectively denote the Standard Model Higgs doublet and la-

grangian, and S is a real scalar field which does not transform under the Standard

Model gauge group. (Lagrangians similar to this have been considered as models for

strongly-interacting dark matter [14] and as potential complications for Higgs searches

[15]. The same number of free parameters appears in the simplest Q-ball models [16].)

We assume S to be the only new degree of freedom relevant at the electroweak scale,

permitting the neglect of nonrenormalizable couplings in eq. (2.1), which contains all

possible renormalizable interactions consistent with the field content and the symmetry

S # "S.

Within this framework the properties of the field S are described by three param-

eters. Two of these, "S and m0 are internal to the S sector, characterizing the S mass

and the strength of its self-interactions. Of these, "S is largely unconstrained and can

be chosen arbitrarily. We need only assume it to be small enough to permit the pertur-

4

Viable DM region: 100 GeV ≤ mh ≤ 200 GeV, 10 ≤ mS ≤ 100 GeVfine tuned !

S. Su 3



WMAP: Ωh2=0.112 ± 0.009





















L = LSM +1

2!µS !µS "

m20

2S2 "

"S

4S4 " " S2 H†H, (2.1)









S # "S.





4


20 40 60 80 100

-4

-3

-2

-1

20 40 60 80 100

-4

-3

-2

-1

20 40 60 80 100

-4

-3

-2

-1

20 40 60 80 100

-4

-3

-2

-1

mS [GeV]

mS [GeV]

mS [GeV]

mS [GeV]

mh = 100GeV

mh = 140GeV

mh = 120 GeV

mh = 200 GeV

log

!lo

g!

log

!lo

g!

Figure 2: Four samples of the log !–mS relationship between ! and mS, which gives

the correct cosmic abundance of S scalars. For these plots the Higgs mass is chosen to

be 100, 120, 140, and 200 GeV. The abundance is chosen to be !sh2 = 0.3

These expressions may be used with standard results for the Standard Model Higgs

decay widths to predict how the primordial S-particle abundance depends on the pa-

rameters mS and !. Standard procedures [18] give the present density of S particles

to be

!sh2 =

(1.07 · 109) xf

g1/2! MPl GeV !"vrel"

. (3.5)

Here g!, as usual, counts the degrees of freedom in equilibrium at annihilation, xf =

mS/Tf is the inverse freeze-out temperature in units of mS, which can be determined

by solving the equation xf # ln!

0.038(g!xf )"1/2MPl mS!"v""

. Finally, !· · ·" denotes

the relevant thermal average.

The abundance constraint is obtained by requiring !sh2 to be in the cosmologically

preferred range, which imposes a relation between ! and mS. For our numerical results

we use !sh2 = 0.3, which corresponds to a large value for !s = 0.6, perhaps the largest

possible value consistent with observations. We choose such a large !s in order to

be conservative in our later predictions for the signals to be expected in dark matter

9

C.P. Burgess, M. Pospelov and T. ter Veldhuis, NPB 619, 709 (2001)

S. Su 3



WMAP: Ωh2=0.112 ± 0.009





















L = LSM +1

2!µS !µS "

m20

2S2 "

"S

4S4 " " S2 H†H, (2.1)









S # "S.





4


S. Su 4

Inert Higgs Doublet Model-

➡ Introduce another Higgs field that only couples to gauge sector

impose Z2 parity: SM particles + , extra Higgs: -

(DµH2)†(DµH2) Q†LH2uR

Inert Higgs Doublet Model (IHDM)• first proposed in late 70’s, neutrino mass

• naturalness problem in SM• appears in left-right Twin Higgs models

• majorana neutrino mass• electroweak symmetry breaking• grand unification• leptogenesis

N.G.Deshpande and E. Ma, PRD 18, 2574 (1978)

R. Barbieri, L.J. Hall and V.S. Rychkov, PRD 74, 015007 (2006)

Z. Chacko, H.S. Goh and R. Harnik, JHEP 0601, 108 (2006)

E. Ma, PRD 73, 077301 (2006)

T. Hambye and M.H.G. Tytgat, PLB 659, 651 (2008)

M. Lisanti and J.G. Wacker, 0704.2816

E. Ma, MPLA 21, 1777 (2006), T. Hambye, K. Kannike, E. Ma and M. Raidal, PRD75, 095003 (2007)

S. Su

Dark matter studies-

• relic density• neutrino signature

• gamma ray spectrum

• positron and antiproton • LEP II limit • collider SA with A→Sll • direct detection

L. Honorez, E. Nezri, J. Oliver, M. Tytgat, JCAP 0702, 028 (2007)

P. Agrawall, E.M. Dolle and C.A. Krenke, PRD 79, 015015 (2009)S. Andreas, M.H.G. Tygat and Q. Swillens, 0901.1750

L. Honorez, E. Nezri, J. Oliver, M. Tytgat, JCAP 0702, 028 (2007)M. Gustafsson, E. Lundstrom, L. Bergstrom and J. Edsjo, PRL99, 041301 (2007)

E. Nezri, M.H.G. Tytgat and G. Vertongen, 0901.2556

E. Lundstrom, M. Gustafsson and J. Edsjo, 0810.3924

Q.H. Cao, E. Ma and G. Rajasekaran, PRD 76, 095011 (2007)

R. Barbieri, L.J. Hall and V.S. Rychkov, PRD 74, 015007 (2006)L. Honorez, E. Nezri, J. Oliver, M. Tytgat, JCAP 0702, 028 (2007)D. Majumdar and A. Ghosal, MPLA23, 2011 (2008)

H1 = HSM

relic density could be obtained for the dark matter mass around 40 GeV ! 80 GeV or largerthan 600 GeV. Ref. [14, 15] studied the neutrino signatures from dark matter annihilationin the IHDM. Continuous gamma ray spectrum from fragmentation and monochromaticgamma ray lines are studied in Ref. [13] and [16] respectively. Positron and antiprotonsignatures are studied in Ref. [17]. There are also collider analysis on the LEP II limit forthe IHDM [18] as well as collider signatures of SA associated production with A " Sl+l!

at the LHC [19]. Direct detection of the IHDM dark matter has been studied in [7, 13, 20].In this work, we performed a complete analyzed the dark matter relic in the IHDM

over the whole parameter space, taken into account various theoretical and experimentalconstraints on the IHDM. The latest results of the collider constraints based on !0

1!02 search

at the LEP are imposed. Unlike in Ref. [13], in which only a low SM Higgs mass mh = 100GeV and 200 GeV are considered, we also considered a high Higgs mass mh = 500 GeV.In Ref. [13], the mass splitting between A, H± and the dark matter candidate S is fixed tobe 10 (5) GeV and 50 (10) GeV respectively for low (high) mass region. We studied thecases when the mass splittings between A, H± and the dark matter candidate S are small,in which the coannihilation plays an important role, as well as the cases when the masssplittings are large. In regions that overlap with those analyzed in Ref. [13], our resultsagree with the literature. We identified additional regions of parameter space, in whichthe dark matter relic density is also consistent with the WMAP result but was overlookedbefore. We also present our results in the parameter spaces of physical Higgs masses andHiggs couplings, which can easily be used for the purpose of collider study and dark matterdetections.

The rest of the paper is organized as follows: Sec. II briefly present the IHDM. Wediscussed the theoretical and experimental constraints on the model parameter space inSec. III. Sec. IV presented our results on the relic density analysis. We concluded in Sec. V.

II. THE INERT HIGGS DOUBLET MODEL

The IHDM is an extension of the Higgs sector of the SM. Besides the usual Higgs doubletH1, additional Higgs doublet H2 is introduced:

H2 =

!

H+

(S + iA)/#

2

"

, (2)

which is charged under SU(2)L $ U(1)Y as (2, 1/2). Unlike the SM Higgs boson, whichcouples to both the gauge bosons and matter fermions, the extra Higgs doublet couplesto the gauge sector only. Such couplings can be guaranteed by imposing a Z2 symmetry(sometimes also called as matter parity) where all the particles except H2 are even under theZ2. While H1 obtains a vacuum expectation value (VEV) v/

#2 = 174 GeV as in the SM,

H2 does not obtain a VEV: %H2& = 0. The Z2 symmetry is, therefore, not spontaneouslybroken. The lightest particle in H2 is stable and could be a good dark matter candidate.

3

lightest one: DM candicate

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Z2

H1 H2

Z2

H1 ! H1 H2 ! "H2.

Z2

Z2

H2

#H2$ ! 0

V = µ21|H1|2 + µ2

2|H2|2 + !1|H1|4 + !2|H2|4 + !3|H1|2|H2|2 + !4|H†1H2|2 +

!5

2

!

(H†1H2)

2 + h.c."

.

U(1) !5 = 0

SU(2) % U(1) H1

#H1$ =v&2

v = "µ21/!1 = 248 µ2

2 > 0

#H2$ = 0.

The most general potential in the Higgs sector can be written as

V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.

(3)Notice that the usual mixing term µ2

12H†1H2 is forbidden by the Z2 symmetry. After

electroweak symmetry breaking, three degree of freedom in H1 are eaten by massive gaugebosons W±, Z. We are left with one physical Higgs boson h, which resembles the SM Higgsboson. The mass of h is related to !1 via

m2h = !2µ2

1 = 2!1v2. (4)

The masses of the scalars in H2 doublet are related to the parameters in the Higgs potentialas

m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)

We define the mass di!erence "1 and "2 as

"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)

It is obvious that !4 controls the mass splitting between the charged states and the neutralstates, while !5 controls the mass splitting between the CP odd state and CP even states.In our analysis below, we assume that the CP even scalar S is LIP, therefore the darkmatter candidate: "1 > 0, "2 > 0. The numerical results obtained below are similar if A isthe LIP.

The Higgs potential in Eq. (3) has seven parameters:

(µ21, µ

22, !1, !2, !3, !4, !5). (10)

They could be replaced by the Higgs VEV v, physical Higgs masses, mass splittings, andsum of quartic couplings !L = !3 + !4 + !5 as

(v, mh, mS, "1, "2, !2, !L). (11)

In particular, !L shows up in the couplings SSh and SShh, which is relevant for the darkmatter annihilation process. It is therefore convenient to pick !L as a model parameter.Quartic coupling !2 only shows up in self-couplings involving S, A and H±. It does notplay an important role for the dark matter analysis that we present below.

4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

[

(H†1H2)

2 + h.c.]

.




m2h = −2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 − !5)v2/2. (7)

(8)

We define the mass difference "1 and "2 as

"1 = mH± − mS = −(!4 + !5)v2

2(mH± + mS), "2 = mA − mS = −

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+λ1|H1|4+λ2|H2|4+λ3|H1|2|H2|2+λ4|H†1H2|2+

λ5

2

[

(H†1H2)

2 + h.c.]

.



electroweak symmetry breaking, three degree of freedom in H1 are eaten by massive gaugebosons W±, Z. We are left with one physical Higgs boson h, which resembles the SM Higgsboson. The mass of h is related to λ1 via

m2h = −2µ2

1 = 2λ1v2. (4)


m2H± = µ2

2 + λ3v2/2, (5)

m2S = µ2

2 + (λ3 + λ4 + λ5)v2/2, (6)

m2A = µ2

2 + (λ3 + λ4 − λ5)v2/2. (7)

(8)

We define the mass di!erence δ1 and δ2 as

δ1 = mH± − mS = −(λ4 + λ5)v2

2(mH± + mS), δ2 = mA − mS = −

λ5v2

(mA + mS). (9)

It is obvious that λ4 controls the mass splitting between the charged states and the neutralstates, while λ5 controls the mass splitting between the CP odd state and CP even states.In our analysis below, we assume that the CP even scalar S is LIP, therefore the darkmatter candidate: δ1 > 0, δ2 > 0. The numerical results obtained below are similar if A isthe LIP.


(µ21, µ

22, λ1, λ2, λ3, λ4, λ5). (10)

They could be replaced by the Higgs VEV v, physical Higgs masses, mass splittings, andsum of quartic couplings λL = λ3 + λ4 + λ5 as

(v, mh, mS, δ1, δ2, λ2, λL). (11)

In particular, λL shows up in the couplings SSh and SShh, which is relevant for the darkmatter annihilation process. It is therefore convenient to pick λL as a model parameter.Quartic coupling λ2 only shows up in self-couplings involving S, A and H±. It does notplay an important role for the dark matter analysis that we present below.

4

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Z2

H1 H2

Z2

H1 ! H1 H2 ! "H2.

Z2

Z2

H2

#H2$ ! 0

V = µ21|H1|2 + µ2

2|H2|2 + !1|H1|4 + !2|H2|4 + !3|H1|2|H2|2 + !4|H†1H2|2 +

!5

2

!

(H†1H2)

2 + h.c."

.

U(1) !5 = 0

SU(2) % U(1) H1

#H1$ =v&2

v = "µ21/!1 = 248 µ2

2 > 0

#H2$ = 0.


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

[

(H†1H2)

2 + h.c.]

.




m2h = −2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 − !5)v2/2. (7)

(8)


"1 = mH± − mS = −(!4 + !5)v2

2(mH± + mS), "2 = mA − mS = −

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+λ1|H1|4+λ2|H2|4+λ3|H1|2|H2|2+λ4|H†1H2|2+

λ5

2

[

(H†1H2)

2 + h.c.]

.




m2h = −2µ2

1 = 2λ1v2. (4)


m2H± = µ2

2 + λ3v2/2, (5)

m2S = µ2

2 + (λ3 + λ4 + λ5)v2/2, (6)

m2A = µ2

2 + (λ3 + λ4 − λ5)v2/2. (7)

(8)


δ1 = mH± − mS = −(λ4 + λ5)v2

2(mH± + mS), δ2 = mA − mS = −

λ5v2

(mA + mS). (9)



(µ21, µ

22, λ1, λ2, λ3, λ4, λ5). (10)


(v, mh, mS, δ1, δ2, λ2, λL). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4

SSh coupling

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Z2

H1 H2

Z2

H1 ! H1 H2 ! "H2.

Z2

Z2

H2

#H2$ ! 0

V = µ21|H1|2 + µ2

2|H2|2 + !1|H1|4 + !2|H2|4 + !3|H1|2|H2|2 + !4|H†1H2|2 +

!5

2

!

(H†1H2)

2 + h.c."

.

U(1) !5 = 0

SU(2) % U(1) H1

#H1$ =v&2

v = "µ21/!1 = 248 µ2

2 > 0

#H2$ = 0.


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

[

(H†1H2)

2 + h.c.]

.




m2h = −2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 − !5)v2/2. (7)

(8)


"1 = mH± − mS = −(!4 + !5)v2

2(mH± + mS), "2 = mA − mS = −

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4


V = µ21|H1|2+µ2

2|H2|2+λ1|H1|4+λ2|H2|4+λ3|H1|2|H2|2+λ4|H†1H2|2+

λ5

2

[

(H†1H2)

2 + h.c.]

.




m2h = −2µ2

1 = 2λ1v2. (4)


m2H± = µ2

2 + λ3v2/2, (5)

m2S = µ2

2 + (λ3 + λ4 + λ5)v2/2, (6)

m2A = µ2

2 + (λ3 + λ4 − λ5)v2/2. (7)

(8)


δ1 = mH± − mS = −(λ4 + λ5)v2

2(mH± + mS), δ2 = mA − mS = −

λ5v2

(mA + mS). (9)



(µ21, µ

22, λ1, λ2, λ3, λ4, λ5). (10)


(v, mh, mS, δ1, δ2, λ2, λL). (11)


4

SAH± couplingsnot important for DM analysis


V = µ21|H1|2+µ2

2|H2|2+!1|H1|4+!2|H2|4+!3|H1|2|H2|2+!4|H†1H2|2+

!5

2

!

(H†1H2)

2 + h.c."

.




m2h = !2µ2

1 = 2!1v2. (4)


m2H± = µ2

2 + !3v2/2, (5)

m2S = µ2

2 + (!3 + !4 + !5)v2/2, (6)

m2A = µ2

2 + (!3 + !4 ! !5)v2/2. (7)

(8)


"1 = mH± ! mS = !(!4 + !5)v2

2(mH± + mS), "2 = mA ! mS = !

!5v2

(mA + mS). (9)



(µ21, µ

22, !1, !2, !3, !4, !5). (10)


(v, mh, mS, "1, "2, !2, !L). (11)


4

SSh coupling

S. Su 7

Constraints-

✦ W and Z decay width

W→S/A + H±, Z→S+A, H+H-

III. THEORETICAL AND EXPERIMENTAL CONSTRAINTS

There are various experimental constraints on the IHDM, both from direct searches andindirect electroweak precision test.

• W and Z decay widthFor light H±, S and A, it could lead to the deviation of Z and W decay widths fromthe SM value. On the other hand, !W,Z has been measured precisely at the LEPand the Tevatron[21], which agrees well with the SM prediction. Therefore regions inwhich decay processes W± ! S/A+H±, Z ! H+H!, SA are kinematically allowedhave already been excluded. These translate into constraints in the mass regions

2mS + !1 > mW , 2mS + !1 + !2 > mW ,

2mS + !2 > mZ , 2mS + 2!1 > mZ . (12)

• Direct collider searchesLight neutral and charged Higgs has been searched at the LEP and the Tevatron.Limits from conventional searches for H±, S and A, however, does not apply for theHiggses in the IHDM since those searches reply on the Higgs decaying to fermionpairs. Neutral and charged Higgses in the IHDM, on the other hand, does not coupleto fermions. In particular, the charged Higgs has been searched for at the LEPand the Tevatron [22, 23]. A lower mass bound of 74 " 79 GeV at 95% C.L. isobtained at the LEP [22] considering H+ ! cs, H+ ! "+#, or H+ ! W "A. A morerecent search at CDF[23] studied the charged Higgs produced in the top quark decayt ! H+b, with H+ further decaying into a pair of quarks, leptons, or W+$. All ofthose searches, however, rely on the couplings of the charged Higgs with fermions,which is absent for H± in the IHDM. Therefore, the collider direct search limit onthe charged Higgs mass does not apply.

Experimental signatures for S, A, H±, however, is similar to that of the neutralinoand charginos in the Minimal Supersymmetric Standard Model (MSSM), as Sappears as missing energy at colliders, similar to the lightest supersymmetricparticle(LSP) in the MSSM. In particular, searches of e+e! ! %0

1%02 can be

interpreted as searches for e+e! ! SA. Therefore, the null result for the neutralinoand chargino searches at the LEP can be used to set limits on m±

H and mA in theIHDM. A recent analysis [18] argued that a direct application of the upper limit onthe LEP II %0

1%02 cross section to the IHDM is oversimplified due to the di"erence

between the MSSM process %01%

02 and IHDM process SA. Based on DELPHI analyses

of e+e! ! %01%

02 with %0

2 ! %01qq, µ

+µ!, e+e!, Ref. [18] determined the e#cienciesfor the corresponding MSSM and IHDM processes after cuts using Monte-Carlosimulations. The ratio of the e#ciencies is then used to rescale the MSSM crosssection upper limit and applied to the IHDM. It is shown that regions satisfyingmS < 80 GeV, mA < 100 GeV and !2 > 8 GeV are excluded by LEP II MSSMsearches. This limit is stronger than previous estimations by the direct applicationof the %0

1%02 cross section upper limit to the IHDM [7, 24]. For !2 < 8 GeV, however,

5

✦ LEP II constraints

Neutral and charged Higgs searches at LEP and Tevatron: does not applyrely on VVh coupling and the couplings of Higgses to fermions

Charged Higgs searches: mH± > 74 −79 GeV at 95% C.L.LEP: CDF: t→H+b, with H+ decays to qq’, lν, W+ϕ

III. THEORETICAL AND EXPERIMENTAL CONSTRAINTS

There are various experimental constraints on the IHDM, both from direct searches andindirect electroweak precision test.

• W and Z decay widthFor light H±, S and A, it could lead to the deviation of Z and W decay widths fromthe SM value. On the other hand, !W,Z has been measured precisely at the LEPand the Tevatron[21], which agrees well with the SM prediction. Therefore regions inwhich decay processes W± ! S/A+H±, Z ! H+H!, SA are kinematically allowedhave already been excluded. These translate into constraints in the mass regions

2mS + !1 > mW , 2mS + !1 + !2 > mW ,

2mS + !2 > mZ , 2mS + 2!1 > mZ . (12)

• Direct collider searchesLight neutral and charged Higgs has been searched at the LEP and the Tevatron.Limits from conventional searches for H±, S and A, however, does not apply for theHiggses in the IHDM since those searches reply on the Higgs decaying to fermionpairs. Neutral and charged Higgses in the IHDM, on the other hand, does not coupleto fermions. In particular, the charged Higgs has been searched for at the LEPand the Tevatron [22, 23]. A lower mass bound of 74 " 79 GeV at 95% C.L. isobtained at the LEP [22] considering H+ ! cs, H+ ! "+#, or H+ ! W "A. A morerecent search at CDF[23] studied the charged Higgs produced in the top quark decayt ! H+b, with H+ further decaying into a pair of quarks, leptons, or W+$. All ofthose searches, however, rely on the couplings of the charged Higgs with fermions,which is absent for H± in the IHDM. Therefore, the collider direct search limit onthe charged Higgs mass does not apply.

Experimental signatures for S, A, H±, however, is similar to that of the neutralinoand charginos in the Minimal Supersymmetric Standard Model (MSSM), as Sappears as missing energy at colliders, similar to the lightest supersymmetricparticle(LSP) in the MSSM. In particular, searches of e+e! ! %0

1%02 can be

interpreted as searches for e+e! ! SA. Therefore, the null result for the neutralinoand chargino searches at the LEP can be used to set limits on m±

H and mA in theIHDM. A recent analysis [18] argued that a direct application of the upper limit onthe LEP II %0

1%02 cross section to the IHDM is oversimplified due to the di"erence

between the MSSM process %01%

02 and IHDM process SA. Based on DELPHI analyses

of e+e! ! %01%

02 with %0

2 ! %01qq, µ

+µ!, e+e!, Ref. [18] determined the e#cienciesfor the corresponding MSSM and IHDM processes after cuts using Monte-Carlosimulations. The ratio of the e#ciencies is then used to rescale the MSSM crosssection upper limit and applied to the IHDM. It is shown that regions satisfyingmS < 80 GeV, mA < 100 GeV and !2 > 8 GeV are excluded by LEP II MSSMsearches. This limit is stronger than previous estimations by the direct applicationof the %0

1%02 cross section upper limit to the IHDM [7, 24]. For !2 < 8 GeV, however,

5

S. Su 8

Constraints-

MSSM searches: e+e−→ χ10 χ20 with χ20 → χ10 qq/µµ/eesimilar to e+e− → SA with A → Sqq/µµ/ee

8

mass A0 [GeV]

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sH

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xcluded

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FIG. 6: Production cross section upper limits as extractedfrom Fig. 13(d) in [16]. For models inside the (red) solid[(green) dashed] contour the limits are rescaled by a factor0.9 (1.1) before being applied to H0A0 production. The solid(dark blue) contour lines indicate the e+e! ! H0A0 crosssection. The (red) dotted-shaded region, where mH0+mA0 <mZ , is excluded by LEP I data on the Z boson width. Theupper right dashed line shows the LEP II kinematical limit.

modifications might be needed (this calculation is donewith MadGraph/MadEvent).

Finally, we calculate the H0A0 production cross sec-tion as a function of mH0 and mA0 , and compare it withour derived cross section upper limits in order to con-strain the IDM parameter space.

IV. RESULTS

Under our imposed cuts the resulting IDM and MSSMe!ciencies turn out to be quite similar, an appealing,although not at all trivial, result.

The e!ciencies are first determined for each individ-ual channel (qq, µ+µ!, e+e!), after which those are com-bined into an e!ciency representing the actual branchingratio. This combination is done by weighting the chan-nels in accordance with the decay branching ratios ofthe Z boson (i.e. the qq e!ciency is given the highestweight).

In general we observe that the ratio between our de-rived IDM and MSSM e!ciencies is quite insensitive tothe very details of the imposed cuts, and we estimate oursensitivity in determining this ratio to be of the order of10 %.

We find that whenever mH0 ! 80 GeV the IDM e!-ciencies typically are a few percent higher than those ofthe corresponding MSSM models. An important obser-vation is that we find no mass combinations in this regionwhere the MSSM gives a higher e!ciency than the IDM,and it is therefore appropriate to apply at least as hard

0 50 100 150 2000

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20

30

40

50

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100

110

mass A0 [GeV]

mas

sH

0[G

eV]

LE

P I e

xcluded

LE

P II e

nerg

y reach

LEP excluded

FIG. 7: LEP exclusion plot. The (red) dotted-shaded regionindicates the region of the (mH0 ,mA0) plane excluded by LEPdata. The lower left triangle, where mH0 + mA0 < mZ , isexcluded by LEP I data on the Z boson width. The remainingpart of the shaded region is excluded by our LEP II analysis.Shown is also the LEP II kinematical limit. Since we areassuming mH0 <mA0 the upper left region is not accessible.

production cross section upper limits on the inert scalarsas those put on the neutralinos in [16].

In the specific region defined by 8 GeV< "m <15 GeVand mH0 ! 85 GeV, the IDM e!ciencies are found to beabout a factor 1.15-1.20 higher than those of the MSSM.On noting that the models with the lowest "m have aslightly higher branching into neutrinos compared to or-dinary Z boson decay, we in this region adopt a conser-vative factor of 0.9 with which we rescale the neutralinoproduction limits given in Fig. 13(d) in [16]. This regionis encircled with a (green) dashed line in Fig. 6.

Among the remaining mH0 " 80 GeV models we findsome for which the ratio between the IDM and MSSMe!ciencies drops down to 0.9. We therefore use a factorof 1.1 for the rescaling here, and this region is encircledwith a red solid line in Fig. 6.

Except for in the low "m and high mH0 regions men-tioned above we find it appropriate to apply the sameproduction limits as for the neutralinos. While this mightbe argued to be too conservative, the points where harderlimits could possibly be imposed are anyway far from ex-cluding any IDM model.

By utilizing the limits on the !01!

02 production from

Fig. 13(d) in [16] we find, after rescaling, upper limits onthe H0A0 production cross section as a function of mH0

and mA0 . The cross section limits, and the regions wherewe impose rescaling, are found in Fig. 6. Comparingthese with the calculated e+e! ! H0A0 cross sections,which also are shown in Fig. 6, finally tells us which IDMmodels are excluded.

The resulting exclusion plot is shown in Fig. 7.Roughly speaking, our LEP II analysis exclude models



S. Su 8

Constraints-


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IV. RESULTS





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xcluded

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nerg

y reach

LEP excluded







02 production from






LEP ImS +mA < mZ

excluded

S. Su 8

Constraints-


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IV. RESULTS





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xcluded

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nerg

y reach

LEP excluded







02 production from






LEP ImS +mA < mZ

excluded

LEP IIδ2 > 8 GeV mS < 80 GeVmA < 100 GeVexcluded

S. Su 8

Constraints-


8

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IV. RESULTS





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02 production from






δ2 < 8 GeV mS +mA > mZ allowed

LEP ImS +mA < mZ

excluded

LEP IIδ2 > 8 GeV mS < 80 GeVmA < 100 GeVexcluded

S. Su 9

Electroweak Precision Test-

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

T

68 % CL

U=0

mt

mh

mt= 172.7 ± 2.9 GeV

mh= 114...1000 GeV

Figure 1: (Adapted from [8].) Dependence of the S, T parameters on the Higgs mass. The thickblack band marks mh = 400 ! 600 GeV.

3 The Inert Doublet Model

In this section we will present what seems to us the most attractive realization of the improvednaturalness idea. Some alternatives are described in Section 4.

3.1 The Model

We consider the most general two-Higgs doublet model that possesses the parity

H2 " !H2 (8)

with all other fields invariant. This parity imposes natural flavor conservation in the Higgssector[9]3, implying that only H1 couples to matter. The scalar potential is

V = µ21|H1|2 + µ2

2|H2|2 + !1|H1|4 + !2|H2|4 + !3|H1|2|H2|2

+ !4|H†1H2|2 +

!5

2[(H†

1H2)2 + h.c.]. (9)

We assume that the parameters of this potential yield an asymmetric phase: H1 acquires a vevbut H2 does not4 This is not the well-studied standard phase of the theory that has both vevs

3In standard nomenclature this would be called Type I 2HDM, except that we reverse the usual roles of H1 andH2.

4This phase of the unbroken parity was considered recently in [10] motivated by neutrino physics. We thankE. Ma for bringing this to our attention.

5

mh=90 +36 - 27 GeVmh< 163 GeV @ 95% C.L.

✦ Electroweak precision test

http://lepewwg.web.cern.ch/LEPEWWG/



S. Su 10

Electroweak Precision Test-

mS=75 GeV!

1 [GeV]

!2 [G

eV

]

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!1 [GeV]

!2 [G

eV

]

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

FIG. 1: Plot of the allowed region in mass splittings !1 ! !2 given the constraints on S ! T from

the precision electroweak measurements. The value for the SM Higgs mass is set to be 120 GeVfor the left plot and 500 GeV for the right plot. mS is taken to be 75 GeV.

between the dark matter and the scattering nuclei in such scattering processes. Spin-independent DM!nuclei scattering cross section via h-exchange is many order ofmagnitude smaller than the sensitivity of the current and near future dark matterdirect detection experiments. Therefore we do not consider the dark matter directdetection constraints as long as !2 is not extremely small.

The bounds from indirect dark matter detection (gamma ray, for example) is veryweak on the IHDM parameter spaces. Moreover, those bounds are subject to thelarge astrophysical uncertainties involved in those observations. Therefore, we don’tconsider the constraints from indirect dark matter detection as well.

In addition, we impose the following theoretical constraints:

• Vacuum stabilityWe require the vacuum stability of the Higgs potential at tree level, which leads to

"1,2 > 0 (13)

"3, "3 + "4 ! |"5| > !2!

"1"2. (14)

• PerturbativityWe require that the correction to the beta function of "1 from non-SM quarticcouplings is less than the 50% of the SM term 24"2

1 [7]. This amounts to theconstraint:

"23 + ("3 + "4)

2 + "25 < 12"2

1, (15)

The evolution of the remaining quartic couplings does not lead to extra constraints.In addition, we require the quartic coupling "2 to be in the perturbativity region:

"2 < 1 (16)

7

mh=120 GeV mh=500 GeV

✦ Electroweak precision test

relic density could be obtained for the dark matter mass around 40 GeV ! 80 GeV or largerthan 600 GeV. Ref. [14, 15] studied the neutrino signatures from dark matter annihilationin the IHDM. Continuous gamma ray spectrum from fragmentation and monochromaticgamma ray lines are studied in Ref. [13] and [16] respectively. Positron and antiprotonsignatures are studied in Ref. [17]. There are also collider analysis on the LEP II limit forthe IHDM [18] as well as collider signatures of SA associated production with A " Sl+l!

at the LHC [19]. Direct detection of the IHDM dark matter has been studied in [7, 13, 20].In this work, we performed a complete analyzed the dark matter relic in the IHDM

over the whole parameter space, taken into account various theoretical and experimentalconstraints on the IHDM. The latest results of the collider constraints based on !0

1!02 search

at the LEP are imposed. Unlike in Ref. [13], in which only a low SM Higgs mass mh = 100GeV and 200 GeV are considered, we also considered a high Higgs mass mh = 500 GeV.In Ref. [13], the mass splitting between A, H± and the dark matter candidate S is fixed tobe 10 (5) GeV and 50 (10) GeV respectively for low (high) mass region. We studied thecases when the mass splittings between A, H± and the dark matter candidate S are small,in which the coannihilation plays an important role, as well as the cases when the masssplittings are large. In regions that overlap with those analyzed in Ref. [13], our resultsagree with the literature. We identified additional regions of parameter space, in whichthe dark matter relic density is also consistent with the WMAP result but was overlookedbefore. We also present our results in the parameter spaces of physical Higgs masses andHiggs couplings, which can easily be used for the purpose of collider study and dark matterdetections.

The rest of the paper is organized as follows: Sec. II briefly present the IHDM. Wediscussed the theoretical and experimental constraints on the model parameter space inSec. III. Sec. IV presented our results on the relic density analysis. We concluded in Sec. V.

II. THE INERT HIGGS DOUBLET MODEL

The IHDM is an extension of the Higgs sector of the SM. Besides the usual Higgs doubletH1, additional Higgs doublet H2 is introduced:

H2 =

!

H+

(S + iA)/#

2

"

, (2)

which is charged under SU(2)L $ U(1)Y as (2, 1/2). Unlike the SM Higgs boson, whichcouples to both the gauge bosons and matter fermions, the extra Higgs doublet couplesto the gauge sector only. Such couplings can be guaranteed by imposing a Z2 symmetry(sometimes also called as matter parity) where all the particles except H2 are even under theZ2. While H1 obtains a vacuum expectation value (VEV) v/

#2 = 174 GeV as in the SM,

H2 does not obtain a VEV: %H2& = 0. The Z2 symmetry is, therefore, not spontaneouslybroken. The lightest particle in H2 is stable and could be a good dark matter candidate.

3

S. Su 11

Dark matter direct detection-

Current CDMS limit: 10-42 cm2

S A

Z

q q

Avoid such constraints if |mS-mA|> 0.1 GeV

!SI(Z) =G2

F m2N

2"(N ! (1! s2

W )Z)2 " 10!31cm2

✦ Dark matter direct detection

S. Su 12

Theoretical constraints-

✦ Vacuum stability

!1 [GeV]

!2 [

Ge

V]

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"1,2 > 0 (13)

"3, "3 + "4 ! |"5| > !2!

"1"2. (14)



"23 + ("3 + "4)

2 + "25 < 12"2

1, (15)


"2 < 1 (16)

7

✦ Perturbativity: RG running of λ1 from non-SM terms < 50% of SM term

!1 [GeV]

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eV

]

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"1,2 > 0 (13)

"3, "3 + "4 ! |"5| > !2!

"1"2. (14)



"23 + ("3 + "4)

2 + "25 < 12"2

1, (15)


"2 < 1 (16)

7

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Ge

V]

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"23 + ("3 + "4)

2 + "25 < 12"2

1, (15)


"2 < 1 (16)

7✦ Naturalness: loop correction < tree level for cut off up to 1.5 TeV

• NaturalnessWe also impose naturalness constraints, which is to require that the loop correctionto the Higgs mass parameters not to exceed the tree level value for a cut o! up to1.5 TeV. This leads to the constraints that [7]

|2!3 + !4| <! 9 (17)

µ2>! (1, 2.5

!

!2,!

|2!3 + !4|)120 GeV (18)

did you take into account these two constraints?

IV. RELIC DENSITY ANALYSIS

We analyzed the relic density using the program MicrOMEGAs[28]. This program solvesthe Boltzmann equation numerically, using the program CalcHEP[29] to calculate all therelevant cross sections. When the mass splittings between the dark matter candidate andother particles are small, coannihilation e!ects are also included.

In our analyses, we fixed the SM Higgs mass to be 120 GeV (preferred by the EWPT inthe SM) or 500 GeV (preferred by naturalness, and could be consistent with EWPT withthe radiative contributions from H±, S and A.). We fixed !2 = xxx since it does not enterthe dark matter relic density analysis. We studied both the cases of small mass splittingsand large mass splittings. When "1 ("2) is small, coannihilation between S and H±(A) isimportant. Part of our study overlap with analysis in Ref. [13], which, only consider smallHiggs mass mh = 120 GeV and 200 GeV with fixed mass splittings ("1, "2) = (50, 10) GeVfor low mass region and ("1, "2) = (10, 5) GeV for high mass region. Results in Ref. [13]were presented in µ2 "mS plane. We present our results on !L "mS plane in stead, whichis more transparent since mS and !L are the two relevant parameters for the dark matteranalyses, which also enters the analyses of dark matter direct and indirect detection. Inparticular, the value of !L is closely related to the neutrino flux and gamma ray flux fromdark matter annihilation. Larger !L typically leads to enhanced flux, and have a betterchance to be detected at future neutrino or gamma ray telescopes.

We discuss below in detail two mass regions of mS that could provide the amount ofdark matter relic density consistent with the WMAP result at the 3 # level: (A) low massregion where mS < 100 GeV, and (B) high mass region where 400 GeV < mS < a fewTeV.

A. Low mass region

To illustrate the dependence of the relic density "h2 on mS, !L and mass splittings"1,2, we presented in Fig. 2 various curves of "h2 vs. mS for varying "2 (left plot) andvarying !L (right plot). The horizontal band indicates the 3 # region that is consistentwith relic density measurement from WMAP: 0.085 < "h2 < 0.139. A light SM Higgsmass mh = 120 GeV is used in both plots.

8

S. Su 13

Dark matter relic density-

• coannihilation of S, A

• coannihilation of S, H±

• Use MicrOMEGA /CalCHEP to calculate the relic density

➡ low mass region: mS < 100 GeV

➡ high mass region: mS > 400 GeV

δ1=mH±-mS

δ2=mA-mS

A

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• δ2=mA-mS = 10 GeV

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0.05

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mS

! h

2FIG. 2: The dependence of the relic density on mS for mh=120 GeV and !1 = 50 GeV. In theleft plot, !2 is chosen to be 10 (solid line), 7 (dashed line) and 5 (dash-dotted line) while "L is

fixed to be 0.01. In the right plot, "L is chosen to be 0.01 (solid line), 0.05 (dashed line) and 0.1(dash-dotted line) while !2 is fixed to be 10 GeV. The horizontal band indicates the 3 # regionthat is consistent with relic density measurement from WMAP: 0.085 < !h2 < 0.139. larger

fonts.

The red curve in the left plot corresponds to (!1, !2) = (50, 10) GeV. Therefore,coannihilation between S and A is e!ective while S and h± coannihilation is not. Forsmall mS, SS ! bb via SM h exchange dominates, with cross section proportional to"2

L. The cross section is typically small due to the small bottom Yukawa coupling, whichleads to relic density too big that overclose the Universe. When mS gets larger, SA ! qqcoannihilation via Z exchange becomes more and more important. The relic density entersthe WMAP 3# region for mS around 35 GeV. The coannihilation cross section maximizes atZ-pole: mS +mA " mZ , corresponding to the dip around mS " 40 GeV. Annihilation crosssection decreases when mS increases to be away from the Z-pole region, which makes therelic density falls back to the allowed region. As mS gets larger, SS ! WW ! annihilationvia h starts to dominates and the relic density enters the 3# region again. For mS " mh/2,SS annihilation hits h-pole, corresponding to the second dip around mS " 60 GeV. Theannihilation cross section gets smaller once mS leaves the mh pole region. When mS

>" mW ,annihilation into WW opens. The annihilation cross section quickly increases and the relicdensity drops below the WMAP observed value.

The dahsed and dot-dashed curves in the left plot of Fig. 2 shows the relic densitydependence for !2=7, 5 GeV respectively. The coannihilation e!ects gets stronger forsmaller mass splittings. Therefore, for most of the mS region between 40 # 80 GeV, theannihilation cross section is too large and the relic density is too small.

Three curves in the right plot of Fig. 2 correspond to "L=0.01 (solid), 0.05 (dashed),0.1 (dash-dotted), respectively, while (!1, !2) is fixed to be (50, 10) GeV. Similar features ofZ-pole and h-pole appear. The relic density is smaller for larger "L, since SS annihilation

9

mh=120

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Low mass region: vary λL

-

δ1=mH±-mS = 50 GeV, δ2=mA-mS = 10 GeV,

10 20 30 40 50 60 70 80 90 1000

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FIG. 2: The dependence of the relic density on mS for mh=120 GeV and !1 = 50 GeV. In theleft plot, !2 is chosen to be 10 (solid line), 7 (dashed line) and 5 (dash-dotted line) while "L is


fonts.






9

λL=

mh=120

S. Su 16

Low mass region: low mh-

120_50_10via h-exchange is increased due to the increased SSh coupling.

mS [GeV]

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FIG. 3: WMAP 3! allowed region (enclosed by blue curves) in mS ! "L plane for mh=120

GeV. The mass splittings are chosen to be (#1, #2) = (50, 10) GeV (left plot), (50,8) GeV(right plot). Shaded regions are excluded either by LEP I+II searches (yellow, light shade),

electroweak precision constraints (organge, medium shade), vacuum stability (red, dark shade),and perturbativity (hatched region).

Fig. 3 shows the relic density in mS ! !L plane for mS < 100 GeV with mh = 120GeV for ("1, "2) =(50, 10) GeV (left plot), (50, 8) GeV (right plot). Regions enclosed bythe two blue curves are those consistent with the 3 # region of relic density from WMAP.Shaded regions are excluded either by LEP I+II searches (yellow, light shade), electroweakprecision constraints (organge, medium shade), vacuum stability (red, dark shade), andperturbativity (hatched region). how about naturalness?

For ("1, "2) =(50, 10) GeV (left plot), the gap around mS " 40 GeV corresponds to theZ-pole. The gap around mS " 60 GeV correspinds to the h-pole. The LEP constraints(yellow region) is very strong due to the stronger constraints on mA with not so small masssplitting: mA

>" 100 GeV. this bound is stronger than the one that you indicatedin the plot. The precision electroweak constraints (orange region) is weak since "1 > "2

is slightly preferred by the fit to the S ! T contour. Given all the constraints, only a tinyregion around mS " 80 GeV survives. The value for !L for the allowed region, however,could be as large as -0.2. This large value of !L would be important for generating a largesignal for the indirect detection of dark matter.

The LEP constraints on mS and mA, however, are weakened for small mass splitting"2

<" 8 GeV. mS as low as 40 GeV is still allowed. update the plot with LEP ImS + mA > mZ constraints.. In the right plot of Fig. 3, the allowed parameter region isgiven for ("1, "2) =(50, 8) GeV. In most of the mS region between 40 GeV and 60 GeV,the relic density for the dark matter is too small due to the large coannihilation SA crosssection. However, there are viable region for 60 GeV < mS < 80 GeV with !L as large as±0.1 is allowed.

10

120_50_8

Z h

weak LEP II constraints

update 120_50_8 plot with LEP I

S. Su 17

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120_10_50

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FIG. 4: WMAP 3! allowed region (enclosed by blue curves) in mS ! "L plane for mh=120 GeV.The mass splittings are chosen to be (#1, #2) = (10, 50) GeV (left plot), (10,10) GeV (right plot).

Shaded regions are excluded by various theoretical and experimental constraints; see caption ofFig. 3 for explanation. make a plot for (10,8).

The left plot of Fig. 4 shows the allowed relic density region for (!1, !2) =(10, 50) GeVand mh = 120 GeV. The dark matter annihilation behaves similar to (50,10) case, onlythat the previous SA coannihilation via Z is replaced by SH± coannihilation via W±. TheLEP constraints (yellow region) becomes weakerer comparing to (!1, !2) = (50, 10) GeVcase, due to the weaker constraints on mA. The precision electroweak constraints (orange)region, however, becomes stronger. Given all the theoretical and experimental constraints,60 GeV < mS < 80 GeV with !0.1 < "L < 0.1 survives.

When !1 and !2 are both small, the co-annihilations between S, A and H± are relevantand the allowed 3 # region shrinks due to the enhanced annihilation cross section. for(!1, !2)=(10, 10) GeV, there is no allowed region survives given all the theoretical andexperimental constraints, which is shown in the right plot of Fig. 4. try (10,8) withweaker LEP II constraints, see if there is any region survives.

Fig. 5 shows the allowed relic density region for mh = 120 GeV when both !1 and !2 arelarge: (!1, !2) = (50, 50) GeV (left plot) and (!1, !2) = (100, 100) GeV (right plot). The gapin the low mass region corresponding to the Z-pole or W -pole disappear, since the masssplitting is too large for the coannihilation process to be important. The h-pole, however,still survives due to the low value of mh = 120 GeV that we pick. The LEP I+II constraints(yellow region) are weaker because of the large value of !2. The perturbativity constraints(hatched region), however, are stronger, since larger mass splittings !1,2 correspond to largervalues for "4 and "5. Region with 50 GeV < mS < 90 GeV with a negative !0.2 < "L < 0is consistent with the WMAP 3 # region. how about the LEP II constraints thatmS > 80 GeV?

For (!1, !2) = (100, 100) GeV, the perturbativity constrain is so strong that only a small

11

120_10_10

need 10_8

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Low mass region: low mh

h

S. Su 18

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The mass splittings are chosen to be (#1, #2) = (50, 50) GeV (left plot), (100,100) GeV (right plot).Shaded regions are excluded by various theoretical and experimental constraints; see caption ofFig. 3 for explanation.

window around mS " 25 GeV survives. Again, how about LEP II constraints?

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Shaded regions are excluded by various theoretical and experimental constraints; see caption ofFig. 3 for explanation.

Fig. 6 shows the allowed relic density region for a high value of the SM Higgs massmh = 500 GeV. Electroweak precision constraints requires a large value for δ1 (as shownin Fig. 1) in order to compensate for the ∆S > 0, ∆T < 0 contribute from a heavy SM

12

120_100_100

strong pert. constraints

Low mass region: low mh

h

where is 15 GeV constraints on mS comeing from?

S. Su 19

Low mass region: high mh-

500_250_110

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Fig. 6 shows the allowed relic density region for a high value of the SM Higgs massmh = 500 GeV. Electroweak precision constraints requires a large value for !1 (as shownin Fig. 1) in order to compensate for the !S > 0, !T < 0 contribute from a heavy SM

12

500_180_8

Z

update 500_120_8 plot with LEP Iunderstand the horizontal band

S. Su 20

High mass region: low mh-

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The mass splittings are chosen to be (#1, #2) = (1, 1) GeV (left plot), (1, 10) GeV (right plot). Redregion are excluded by vacuum stability while he hatched region are excluded by perturbativityconstraints.

(hatched region), however, shifts to the left for larger !1,2. Therefore, no allowed regionleft if at least one of !1,2

>" xxx GeV. increase one of the delta to see at which deltathere is no allowed region.

For a large SM Higgs mass mh = 500 GeV, large mass splitting !1>" 150 GeV is needed

to satisfy the precision electroweak constraints. There is no region in mS !"L survive afterall the experimental and theoretical constraints are taken into account. does the relicdensity region change for mh? cross section gets smaller, could not compensatefor large !1.

V. CONCLUSION

We studied the simple extension of the SM Higgs sector when an extra inert Higgsdoublet is introduced that couples to the gauge sector only. The lighter of the neutralcomponents could be a good dark matter candidate. We explored the parameter regionsof the IHDM, taken into account the relic density constraints from WMAP and varioustheoretical and experimental constraints. We showed that there are five distinctive regionsthat could provide the right amount of cold dark matter in the Universe while satisfy allthe constraints.

• (I) Low mh, mS " 20 GeV, "L " !0.2 for large !1 and !2.

• (II) Low mh, 60 GeV < mS < 80 GeV, !0.2 < "L < 0.2 when at least one of !1, !2

is large.

14

120_1_10

why perturbativity constraints strongger?

S. Su 21

Viable region for relic density-

DM SM h mS δ1, δ2 λL

(I) low mS low mh ~20 GeV both large ~ -0.2

(II) 60 − 80 GeV at least one is large

-0.2 − 0.2

(III) high mh 50 − 75 GeV large δ1δ2 < 8 GeV

-1 − 3

(IV) ~ 75 GeV large δ1, δ2 -1 − 3

(V) high mS low mh 500 − 1000 GeV small δ1, δ2 -0.2 − 0.2

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!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5

mS [GeV]

!L

10 20 30 40 50 60 70 80 90 100!0.5

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5




mS [GeV]

!L

10 20 30 40 50 60 70 80 90 100!10

!8

!6

!4

!2

0

2

4

6

8

10

mS [GeV]

!L

10 20 30 40 50 60 70 80 90 100!10

!8

!6

!4

!2

0

2

4

6

8

10




12

mS [GeV]

!L

10 20 30 40 50 60 70 80 90 100!0.5

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5

mS [GeV]

!L

10 20 30 40 50 60 70 80 90 100!1

!0.75

!0.5

!0.25

0

0.25

0.5

0.75


Shaded regions are excluded by various theoretical and experimental constraints; see caption ofFig. 3 for explanation. make a plot for (10,8).

The left plot of Fig. 4 shows the allowed relic density region for (!1, !2) =(10, 50) GeVand mh = 120 GeV. The dark matter annihilation behaves similar to (50,10) case, onlythat the previous SA coannihilation via Z is replaced by SH± coannihilation via W±. TheLEP constraints (yellow region) becomes weakerer comparing to (!1, !2) = (50, 10) GeVcase, due to the weaker constraints on mA. The precision electroweak constraints (orange)region, however, becomes stronger. Given all the theoretical and experimental constraints,60 GeV < mS < 80 GeV with !0.1 < "L < 0.1 survives.

When !1 and !2 are both small, the co-annihilations between S, A and H± are relevantand the allowed 3 # region shrinks due to the enhanced annihilation cross section. for(!1, !2)=(10, 10) GeV, there is no allowed region survives given all the theoretical andexperimental constraints, which is shown in the right plot of Fig. 4. try (10,8) withweaker LEP II constraints, see if there is any region survives.

Fig. 5 shows the allowed relic density region for mh = 120 GeV when both !1 and !2 arelarge: (!1, !2) = (50, 50) GeV (left plot) and (!1, !2) = (100, 100) GeV (right plot). The gapin the low mass region corresponding to the Z-pole or W -pole disappear, since the masssplitting is too large for the coannihilation process to be important. The h-pole, however,still survives due to the low value of mh = 120 GeV that we pick. The LEP I+II constraints(yellow region) are weaker because of the large value of !2. The perturbativity constraints(hatched region), however, are stronger, since larger mass splittings !1,2 correspond to largervalues for "4 and "5. Region with 50 GeV < mS < 90 GeV with a negative !0.2 < "L < 0is consistent with the WMAP 3 # region. how about the LEP II constraints thatmS > 80 GeV?

For (!1, !2) = (100, 100) GeV, the perturbativity constrain is so strong that only a small

11

mS [GeV]

!L

400 500 600 700 800 900 1000 1100 1200!0.5

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5

mS [GeV]

!L

400 500 600 700 800 900 1000 1100 1200!0.5

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5


The mass splittings are chosen to be (#1, #2) = (1, 1) GeV (left plot), (1, 10) GeV (right plot). Redregion are excluded by vacuum stability while he hatched region are excluded by perturbativityconstraints.

(hatched region), however, shifts to the left for larger !1,2. Therefore, no allowed regionleft if at least one of !1,2

>" xxx GeV. increase one of the delta to see at which deltathere is no allowed region.

For a large SM Higgs mass mh = 500 GeV, large mass splitting !1>" 150 GeV is needed

to satisfy the precision electroweak constraints. There is no region in mS !"L survive afterall the experimental and theoretical constraints are taken into account. does the relicdensity region change for mh? cross section gets smaller, could not compensatefor large !1.

V. CONCLUSION

We studied the simple extension of the SM Higgs sector when an extra inert Higgsdoublet is introduced that couples to the gauge sector only. The lighter of the neutralcomponents could be a good dark matter candidate. We explored the parameter regionsof the IHDM, taken into account the relic density constraints from WMAP and varioustheoretical and experimental constraints. We showed that there are five distinctive regionsthat could provide the right amount of cold dark matter in the Universe while satisfy allthe constraints.

• (I) Low mh, mS " 20 GeV, "L " !0.2 for large !1 and !2.

• (II) Low mh, 60 GeV < mS < 80 GeV, !0.2 < "L < 0.2 when at least one of !1, !2

is large.

14

(I) (II) (III) (IV) (V)

update plots

S. Su 22

Collider signatures-

+W (*)

+ Z (*)

+W(*)

pp! AH± ! SSZ(!)W (!), SSZ(!)Z(!)W (!), SSW (!)W (!)W (!)

pp! SH± ! SSW (!), SSZ(!)W (!)

pp! SA! SSZ(!), SSW (!)W (!)

pp! H+H! ! SSW (")W ("), SSW (")W (")Z("), SSW (")W (")Z(")Z(").

H±

A

S

Signatures: jets + leptons + missing ET

jets and leptons could be soft for small splittings

S. Su 23

Leptonic signals-

Focus on purely leptonic signals• single lepton: SH±

• dilepton: SA, H+H-

• trilepton: AH±

• ...

Dominant background

• WW, ZZ, WZ

Benchmark points

mS (GeV) (δ1, δ2) (GeV) λL

(I) 25 (100,100) − 0.2

(II) 79 (50,10) − 0.18

(III) 73 (10,50) 0

(IV) 82 (50,50) − 0.2

S. Su 24

Sig. vs. Bg. cross sections-

σ (I) (II) (III) (IV)

S(pb)

pp→SA 0.543 0.475 0.251 0.181

pp→H+H- 0.103 0.093 0.438 0.0852

pp→SH± 0.920 0.356 1.076 0.319

pp→AH± 0.189 0.303 0.372 0.157

B(pb)

pp→WW 71.8

pp→ZZ 10.0

pp→WZ 27

S. Su 25

Dilepton signal from Z*-

Signal: pp→SA→SSZ(*)→SSl+l-

Background:

• from IHDM pp→H+H-→SSW(*)W(*)→SSl+l-νν• from SM

- pp→WW→l+l-νν- pp→ZZ→l+l-νν

Dilepton signals:

• pp→SA→SSZ(*)→SSl+l- • pp→H+H-→SSW(*)W(*)→SSl+l-νν

Two isolated opposite charge e or µ with missing ET, no hard jets.

S. Su 26

Dilepton signal: point I-

•Benchmark I: mS = 25 GeV, (δ1, δ2)=(100,100) GeV

Signal: pp→SA→SSZ→SSl+l-, l=e,µ


! "!! #!! $!! %!! &!!"!

!$

"!!#

"!!"

"!!

'(

")!*+!)+'(

*

*

,,

--

./

01

! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

S. Su 26





! "!! #!! $!! %!! &!!"!

!$

"!!#

"!!"

"!!

'(

")!*+!)+'(

*

*

,,

--

./

01

! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

S. Su 26





! "!! #!! $!! %!! &!!"!

!$

"!!#

"!!"

"!!

'(

")!*+!)+'(

*

*

,,

--

./

01

! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

S. Su 27



Cuts• HT >150 GeV• MET >15 GeV• PTl >15 GeV, |ηl|<2.5 • ΔR(ll)>0.4• PTZ >50 GeV• 80 GeV < Mll < 100 GeV• no jets with PTi>20 GeV, |ηj|<2.5

before cuts after cuts

S (fb) SA 25.92 5.768

B (fb) H+H- 3.57 0.048

WW 1836 1.156

WZ 174.6 3.762L=100 fb-1 S/B 0.117

s/√(B) 8.22

S. Su 28

-

•Benchmark II: mS = 79 GeV, (δ1, δ2)=(50,10) GeV

Signal: pp→SA→SSZ*→SSl+l-, l=e,µ

Soft leptons. Difficult.

Dilepton signal: point II

S. Su 29

Low mass region: point III-

•Benchmark III: mS = 73 GeV, (δ1, δ2)=(10,50) GeV


Two isolated opposite charge e or µ with large missing ET, no hard jets.

S. Su 30

Low mass region: point III-

•Benchmark III: mS = 73 GeV, (δ1, δ2)=(10,50) GeV

Cuts (change)• HT >150 GeV• MET >15 GeV• PTl >15 GeV, |ηl|<2.5 • ΔR(ll)>0.4• PTZ >50 GeV• 70 GeV < Mll < 110 GeV• no jets with PTi>20 GeV, |ηj|<2.5


S (fb) SA

B (fb) H+H-

WW

WZL=100 fb-1 S/B

s/√(B)

S. Su 31

-

Low mass region: point IV

•Benchmark IV: mS = 82 GeV, (δ1, δ2)=(50,50) GeV



! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

S. Su 31

-





! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

S. Su 31

-





! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

! "! #!! #"! $!! $"!#!

!%

#!!$

#!!#

#!!

&'((')*+,

#-!./!-/&+,

.

.

00

11

2&

34

S. Su 31

-





! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

! "! #!! #"! $!! $"!#!

!%

#!!$

#!!#

#!!

&'((')*+,

#-!./!-/&+,

.

.

00

11

2&

34

S. Su 31

-





! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

! "! #!! #"! $!! $"!#!

!%

#!!$

#!!#

#!!

&'((')*+,

#-!./!-/&+,

.

.

00

11

2&

34

!! !"#$ " "#$ !!"

!%

!"!&

!"!!

!""

'()*!"++,

!-#./#-/'()*!"++,

.

.

00

11

23

45

S. Su 31

-





! "! #!! #"! $!! $"! %!!#!

!%

#!!$

#!!#

#!!

&''

#(!)*!(*&''

)

)

++

,,

-&

./

! "! #!! #"! $!! $"!#!

!%

#!!$

#!!#

#!!

&'((')*+,

#-!./!-/&+,

.

.

00

11

2&

34

!! !"#$ " "#$ !!"

!%

!"!&

!"!!

!""

'()*!"++,

!-#./#-/'()*!"++,

.

.

00

11

23

45

S. Su 32

-



Cuts • HT >200 GeV• MET >100 GeV• PTl >15 GeV, |ηl|<2.5 • cos(ϕll)>0.8• ΔR(ll)>0.4• PTZ >100 GeV• Mll < 40 GeV• no jets with PTi>20 GeV, |ηj|<2.5


S (fb) SA 5.22 0.132

B (fb) H+H- 1.08 0

WW 1836 0.022

WZ 174.6 0.0032L=100 fb-1 S/B 5.24

s/√(B) 8.33

S. Su 33

Collider reach @ LHC-

mS (δ1, δ2) S B S/B S/√(B)

GeV GeV fb fb L=100 fb-1

(I) 25 (100,100) 5.768 4.966 0.117 8.22

(II) 75 (50,10) difficult(III) 75 (10,50)

(IV) 75 (50,50) 0.132 0.025 5.24 8.33

pp→SA→SSZ(*)→SSl+l-

S. Su 34

Conclusions

IHDM: provide a scalar WIMP dark matter candidate

Viable regions of parameter spaces provide correct relic density

Rich collider phenomenology

dilepton signal from SA production observable for large δ2

-

DM SM h mS δ1, δ2 λL

(I) low mS low mh ~20 GeV both large ~ -0.2

(II) 60 − 80 GeV at least one is large

-0.2 − 0.2

(III) high mh 50 − 75 GeV large δ1δ2 < 8 GeV

-1 − 3

(IV) ~ 75 GeV large δ1, δ2 -1 − 3

(V) high mS low mh 500 − 1000 GeV small δ1, δ2 -0.2 − 0.2

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