Higgs portal vector dark matter: revisited

Abstract

We revisit the Higgs portal vector dark matter model including a hidden sector Higgs field that generates the mass of the vector dark matter. The model becomes renormalizable and has two scalar bosons, the mixtures of the standard model (SM) Higgs and the hidden sector Higgs bosons. The strong bound from direct detection such as XENON100 is evaded due to the cancellation mechanism between the contributions from two scalar bosons. As a result, the model becomes still viable in large range of dark matter mass, contrary to some claims in the literature. The Higgs properties are also affected, the signal strengths for the Higgs boson search being universally suppressed relative to the SM value, which could be tested at the LHC in the future.

Published for SISSA by Springer

Received: December 14, 2012 Revised: March 18, 2013 Accepted: April 13, 2013

Published: May 7, 2013

Higgs portal vector dark matter: revisited

Seungwon Baek, P. Ko, Wan-Il Park and Eibun SenahaSchool of Physics, KIAS, Seoul 130-722, Korea

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected]

Abstract: We revisit the Higgs portal vector dark matter model including a hidden sector Higgs eld that generates the mass of the vector dark matter. The model becomes renormalizable and has two scalar bosons, the mixtures of the standard model (SM) Higgs and the hidden sector Higgs bosons. The strong bound from direct detection such as XENON100 is evaded due to the cancellation mechanism between the contributions from two scalar bosons. As a result, the model becomes still viable in large range of dark matter mass, contrary to some claims in the literature. The Higgs properties are also a ected, the signal strengths for the Higgs boson search being universally suppressed relative to the SM value, which could be tested at the LHC in the future.

Keywords: Higgs Physics, Beyond Standard Model, Cosmology of Theories beyond the SM

ArXiv ePrint: 1212.2131

c

JHEP05(2013)036

[circlecopyrt] SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP05(2013)036

Web End =10.1007/JHEP05(2013)036

Contents

1 Introduction 1

2 The model Lagrangian for vector dark matter 2

3 Phenomenology 33.1 Dark matter phenomenology 33.2 Collider phenomenology 63.3 The EFT as a limit of the full theory for m2 ! 1 9

4 Vacuum stability and perturbativity of Higgs quartic couplings 12

5 Conclusions 14

A One-loop functions of Higgs quartic couplings 15

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1 Introduction

The so-called Higgs portal cold dark matter (CDM) model is an interesting possibility for the nonbaryonic dark matter of the universe. The dark matter elds are assumed to be the standard model (SM) gauge singlets, and could be a scalar (S), a singlet fermion ( ) or a vector boson (X) depending on their spin. The Lagrangian of these CDMs are usually taken as [14]

Lscalar = 12@S@S

1

2m2SS2

2 HHS2

4 S4 (1.1)

Lfermion = [i [notdef] @ m ]

HS

S

H

HH (1.2)

Lvector =

1 4X X +

1

2m2XXX +

1 4 X(XX)2 +

1

2 HXHHXX. (1.3)

Dark matter elds (S, , X) are assumed to be odd under some discrete Z2 symmetry: (S, , X) ! (S, , X) in order to guarantee the stability of CDM. This symmetry

removes the kinetic mixing between the X and the U(1)Y gauge eld B , making X stable.

The scalar CDM model (1.1) is satisfactory both theoretically and phenomenologically, as long as Z2 symmetry is unbroken. The model is renormalizable and can be considered to high energy scale as long as the Landau pole is not hit. A large region of parameter space is still allowed by the relic density and direct detection experiments [3]. On the other hand, the other two cases have problems.

Let us rst consider the fermionic CDM model (1.2). This model is nonrenormalizable, and has to be UV completed. The simplest way to achieve the UV completion of (1.2) is to introduce a real singlet scalar eld as proposed in ref. [5, 6] by some of us. We observed that

1

there are two Higgs-like scalar bosons which interfere destructively in the spin-independent cross section of the singlet fermion CDM on nucleon. The strong constraint from direct detection experiments such as XENON100 [7] or CDMS [8] can be relaxed signicantly. On the other hand, the e ective eld theory (EFT) based on the Lagrangian (1.2) is strongly constrained for DM masses below about 2 TeV [13], although the EFT with pseudo-scalar Higgs portal suggested in [4] can be still consistent with the current direct search bound even for light DM masses. The decoupling of the 2nd scalar boson occurs rather slowly, since the mass mixing between the SM Higgs boson and the new singlet scalar is due to the dim-2 operator [6]. Also the mixing between two scalar bosons makes the signal strength of two physical Higgs-like bosons less than one, and make it di cult to detect both of them at the LHC. Since there is now an evidence for a new boson at 125 GeV at the LHC [9, 10], the 2nd scalar boson in the singlet fermion DM model is very di cult to be observed at the LHC because its signal strength is much less than 1 [6, 11]. Also an extra singlet scalar solves the vacuum instability problem for mH = 125 GeV in the SM [1113], making the electroweak (EW) vacuum stable up to Planck scale for mt = 173.2 GeV. These phenomena would be very generic in general hidden sector DM models [14]. In short, it is very important to consider a renormalizable model when one considers the phenomenology of a singlet fermion CDM.

Now let us turn to the Higgs portal vector dark matter described by (1.3) [13]. This model is very simple, compact and seemingly renormalizable since it has only dim-2 and dim-4 operators. However, it is not really renormalizable and violates unitarity, just like the intermediate vector boson model for massive weak gauge bosons before Higgs mechanism was developed. The Higgs portal VDM model based on (1.3) is a sort of an e ective Lagrangian which has to be UV completed. It lacks the dark Higgs eld, (x), that would generate the dark gauge eld mass and will mix with the SM Higgs eld, H(x), after U(1)X symmetry breaking. Therefore the model (1.3) does not capture dark matter or

Higgs boson phenomenology correctly. It is the purpose of this work to propose a simple UV completion of the model (1.3) with hidden sector U(1)X gauge symmetry (see also ref. [15] for a similar approach), and deduce the correct phenomenology of vector CDM and two Higgs-like scalar bosons. Vector dark matter models in extended gauge symmetries can be found in [1622]. Qualitative aspects of our model are similar to those presented in refs. [6, 11], although there are some quantitative di erences due to the vector nature of the CDM.

This work is organized as follows. In section 2, we dene the model by including the hidden sector Higgs eld that generates the vector dark matter mass by the usual Higgs mechanism. Then we present dark matter and collider phenomenology in the following section. We also compare the full theory with the EFT, and discuss the region in which the EFT approach is valid. The vacuum structure and the vacuum stability issues are discussed in section 4, and the results are summarized in section 5.

2 The model Lagrangian for vector dark matter

Let us consider a vector boson dark matter, X, which is assumed to be a gauge boson associated with Abelian dark gauge symmetry U(1)X. The simplest model will be without

2

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any matter elds charged under U(1)X except for a complex scalar, , whose VEV will generate the mass for X (see also ref. [15]):

LV DM =

14X X + (D )(D ) [parenleftbigg]

v2 2

2

HH v2H2[parenrightbigg] [parenleftbigg] v2 2[parenrightbigg], (2.1)

in addition to the SM Lagrangian which includes the Higgs potential term

LSM = H HH v2H 2

2. (2.2)

H

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The covariant derivative is dened as

D = (@ + igXQ X) ,

where Q QX( ) is the U(1)X charge of and we will take Q = 1 throughout the paper.

Assuming that the U(1)X-charged complex scalar develops a nonzero VEV, v , and thus breaks U(1)X spontaneously,

= 1

p2 (v + '(x)) .

Therefore the Abelian vector boson X gets mass MX = gX[notdef]Q [notdef]v , and the hidden sector

Higgs eld (or dark Higgs eld) '(x) will mix with the SM Higgs eld h(x) through Higgs portal of the H term. The mixing matrix O between the two scalar elds is dened as

h '

= O H1

H2

[parenrightBigg] [parenleftBigg][parenleftBigg]

c s

s c

[parenrightBigg][parenleftBigg][parenleftBigg]

H1 H2

!, (2.3)

where s (c ) sin (cos ), h, ' are the interaction eigenstates and Hi(i = 1, 2) are the

mass eigenstates with masses mi. The mass matrix in the basis (h, ') can be written in terms either of Lagrangian parameters or of the physical parameters as follows:

2 Hv2H H vHv H vHv 2 v2

[parenrightBigg]

= m21c2 + m22s2 (m22 m21)s c

(m22 m21)s c m21s2 + m22c2 [parenrightBigg]

. (2.4)

3 Phenomenology

3.1 Dark matter phenomenology

The observed present cold dark matter density, CDMh2 [similarequal] 0.1123 [notdef] 0.0035 [24], is approximately related to the thermally averaged annihilation cross section at freeze-out temperature, [angbracketleft]v[angbracketright]fz, as

CDMh2 = 3 [notdef] 1027cm3/s

hv[angbracketright]fz

. (3.1)

3

m1 [Equal] 125[LParen1] GeV[RParen1] ,m 2 [Equal] 150[LParen1] GeV[RParen1] ,a [Equal] p 4

104

100

1

0.01

10[Minus] 4

[CapOmega] Xh2

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50 100 150 200

Figure 1. The thermal relic density Xh2 of the vector dark matter as of function of the dark matter mass, MX. For this plot we xed m1 = 125 GeV , m2 = 150 GeV , = /4 and the purple (blue) line corresponds to gX = 0.05 (0.5). The horizontal line is the central value of the current relic density Xh2 = 0.1123 [24].

So we require [angbracketleft]v[angbracketright]fz 3 [notdef] 1026cm3/s to obtain the correct relic density. We have used

the micrOmegas v.2.4.5 [25] to calculate thermal relic density and direct detection cross section of the VDM in our model.

In gure 1 we show the thermal relic density as a function of the dark matter mass, MX. For this plot we xed m1 = 125 GeV , m2 = 150 GeV , = /4 and the purple (blue) line corresponds to gX = 0.05 (0.5). We can see two resonance dips at MX = mi/2 (i = 1, 2). The VDMs can annihilate into the SM particles in the S-wave state, which is di erent from the singlet fermionic dark matter case studied in [6] where the annihilation occurs in the P-wave state. As a result the annihilation cross section for the vector dark matter is generally O(10100) larger than that of the SFDM. And the current relic density

can be explained more easily even at non-resonance region. (See the blue line in gure 1.) The di erence between the two curves becomes larger for MX > 125 GeV. This is because the channels XX ! HiHj (i, j = 1, 2) which begin to open for MX > 125 GeV are sensitive

to gX and they give larger annihilation cross sections as the coupling gX increases.

One important e ect when considering the full theory, which we found in ref. [6], is that a generic cancellation occurs in the dark matter and nucleon scattering amplitude, which can not be observed in the e ective Lagrangian approach.1 This is because the transformation matrix between the interaction eigenstates and the mass eigenstates in the scalar sector is an orthogonal matrix. The dark matter and nucleon elastic scattering cross

1In general the cancellation mechanism can also work in the annihilation process for the relic density. However, the di erent decay widths for the H1 and H2 and/or other processes such as annihilations into scalar particle pairs makes it less e ective than in the direct detection process. As a result, the annihilation process and the direct detection process are not strictly proportional to each other in our scenario.

4

M X [LParen1] GeV[RParen1]

m1=125[LParen1]GeV[RParen1]

m2=135[LParen1]GeV[RParen1]

0.5

m2=150[LParen1]GeV[RParen1]

m2=200[LParen1]GeV[RParen1]

m2=300[LParen1]GeV[RParen1]

m2=700[LParen1]GeV[RParen1]

0.0 0.2 0.4 0.6 0.8 1.0

Figure 2. The excluded region in the (gX, )-plane. Each colored region is excluded by XENON100 direct detection experiment for the m2 value given in the plot. We xed MX =70 GeV , m1 = 125 GeV. The black solid (dashed, long-dashed, dotted, dot-dashed) curve corresponds to Xh2 = 0.1123 for m2 = 135(150, 200, 300, 700) GeV. Therefore, the VDM as light as MX = 70 GeV is allowed by both the relic density and the XENON100 constraints either by the cancellation mechanism for m2 = 135 GeV or by the resonant annihilation for m2 = 150 GeV. The entire region is also allowed by the S, T, U-parameters at 99% condence level except that only the range (0.63, 0.63) and (0.42, 0.42) of are allowed for m2 = 300 GeV and m2 = 700 GeV,

respectively [6].

section is proportional to the following factor:

p /

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

, (3.2)

where q is the momentum transfer of the dark matter. When m1 m2 or [notdef]q2[notdef] m2i,

we have p 0 due to the orthogonality of the mixing matrix O. This cancellation phe

nomenon is quite similar to the GIM-mechanism [26] in the quark (or lepton) avor violating neutral current processes. In gure 2, we show the excluded region in the (gX, )-plane by the non-observation of dark matter by the XENON100 which currently gives the strongest bound on the dark matter direct detection cross section [7]. Each colored region is excluded by XENON100 direct detection experiment for the m2 value given in the plot. We xed

MX = 70 GeV , m1 = 125 GeV for the plot. The black solid (dashed, long-dashed, dotted, dot-dashed) curve corresponds to Xh2 = 0.1123 for m2 = 135 (150, 200, 300, 700) GeV.

5

0.0

-0.5

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gX

Xi=1,2

OhiO'i q2 m2i [vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2

0.4

0.3

0.2

0.1

T

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[SolidSquare]

[SolidSquare] [SolidSquare] [SolidSquare]

[SolidCircle] [SolidCircle][SolidCircle][SolidCircle]

[Minus] 0.3 [Minus] 0.2 [Minus] 0.1 0.0 0.1 0.2 0.3

S

Figure 3. The predictions of (S, T )-parameters in our model for (m1, m2) = (25, 125), (50, 125), (75, 125), (100, 125), (125, 125), (125, 250), (125, 500), (125, 750) GeV from above. The green (red) dots are for = 45 (20 ). The thick black line is the prediction of the SM with the mH in the range [125, 720] GeV. The ellipses represent 68, 95, 99% CL experimental lines from inside out.

The case m2 = 150 GeV is close to the resonance (m2 = 2MX) and shows quite di erent behavior from the other cases. So the VDM as light as MX = 70 GeV, even if it is o the resonance region, can be consistent with both the relic density and the XENON100 experiment by the cancellation mechanism when H2 is light. This can be compared with the EFT approach based on the Lagrangian (1.3) where MX [lessorsimilar] 300 GeV is already excluded by the direct search limit [2] (See also the blue line in gure 6 (a)). The entire region is also allowed by the electroweak precision S, T, U-parameters at 99% condence level except that only the range (0.63, 0.63) and (0.42, 0.42) of are allowed for m2 = 300 GeV and

m2 = 700 GeV, respectively [6].

The predictions of our model on the S, T parameters assuming U = 0 are shown in gure 3 for the choices (m1, m2) = (25, 125), (50, 125), (75, 125), (100, 125), (125, 125), (125, 250), (125, 500), (125, 750) GeV from above. The green (red) dots are for = 45 (20 ). The thick black line is the prediction of the SM with the mH in the range [125, 720] GeV. The ellipses represent 68, 95, 99% CL experimental lines from inside out.

3.2 Collider phenomenology

Since the scalar sector is extended, the Higgs phenomenology is di erent from that of the SM. In this subsection we study the possibility that the second Higgs which our model predicts could be discovered at the LHC. We will also see that the combination of the

6

0.0

[SolidCircle]

[SolidSquare][SolidSquare][SolidSquare][SolidSquare]

[SolidCircle]

[SolidCircle]

[SolidCircle]

[Minus] 0.1

[Minus] 0.2

collider signatures and the DM direct searches is robust enough to exclude or conrm our model in the on-going LHC and the next generation DM direct detection experiments.

The signal strength of a scalar boson Hi=1,2 dened as

ri

(pp ! Hi)B(Hi ! fSM) [(pp ! Hi)B(Hi ! fSM)]SM

(3.3)

can be measured at the LHC. Here i = 1, 2 and fSM is a specic SM nal state which the scalar boson Hi can decay into. In our model it can be written in terms of tot,SMi (i = 1, 2)

which is the total decay width of Hi in the SM assuming Hi is a pure SM Higgs and toti which is the total decay width of Hi in our model [6, 11]:

ri = O4hi tot,SMi

tot i

, (3.4)

where Oh1 = c , Oh2 = s . The total decay widths can be decomposed as

tot1 = c2 tot,SM1 + s2 tot,hid1,

tot2 = s2 tot,SM2 + c2 tot,hid2 + (H2 ! H1H1), (3.5)

where tot,hidi is the total decay width of Hi into the hidden sector assuming Hi is a pure SM-singlet scalar. The channel H2 ! H1H1 opens when m2 > 2m1. From the eqs. (3.4)

and (3.5) it is obvious that ri < 1 in our model. Therefore if the excess of the signal strength in some channels like H ! above the SM prediction at the LHC remains

in the future data, our model will either be excluded or need to be extended (two Higgs doublet portal to a hidden sector dark matter, for example). From r1 + r2 < 1 [6, 11] we

obtain r2 < 0.3 for the second Higgs boson, when we identify the observed new boson at 125 GeV (whose signal strength is greater than 0.7 at 2 level [27] 2) as one of the two Higgs-like scalar bosons in our model.

The correlation between r1 and r2 can be seen in gure 4 where we show only the region r1 > 0.7. For this plot we scanned the parameters gX, MX, , m2 in the range, 0 < gX < 1,10 GeV < MX < 1000GeV, /2 < < /2, m1(= 125 GeV) < m2 < 2000 GeV for the

panel (a), and 10 GeV < m1 < m2(= 125 GeV) for the panel (b). All the points pass the constraints: Xh2 < 0.1228 (the 3 upper bound of the relic density), the upper bound on the XENON100 direct detection cross section, and the bound on the S, T -parameters at 99% CL. The big (small) points (do not) satisfy the WMAP relic density constraint within 3 , while the red-(blue-)colored points can (cannot) be probed at the planned XENON1T direct detection experiment [29]. In both plots, the big red points on the straight line, r1 + r2 = 1, are those with Hi ! XX and H2 ! H1H1 suppressed. In the panel (a),

the sizable contribution from the H2 ! H1H1 channel allows the big red points below the

r1 + r2 = 1 line.

In gure 5, we show the allowed mixing angle as a function of the second Higgs mass. We xed the SM-like Higgs mass to be 125 GeV. Color scheme is the same as

2We used only the ATLAS value since there is no combined result. The corresponding value for the CMS can be found in [28].

7

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[LParen1] a[RParen1] m1 [LParen1] [Equal]125GeV[RParen1] [Less]m 2

0.30

0.25

0.20

0.15

0.10

0.05

0.00 0.70 0.75 0.80 0.85 0.90 0.95 1.00

0.30

0.25

0.20

0.15

0.10

0.05

0.00 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Figure 4. The scatter plot in (a) (r1, r2) for m1(= 125 GeV) < m2 and (b) (r2, r1) for m1 < m2(= 125 GeV). The big (small) points (do not) satisfy the WMAP relic density constraint within 3 , while the red-(blue-)colored points can (cannot) be probed at the planned XENON1T direct detection experiment.

gure 4 except that black points are excluded by the LHC Higgs search, i.e. r < 0.7. We can see the maximal mixing angle = /4 (black points near m2 125 GeV) is excluded

by the LHC Higgs search. Also the light scalar with mass less than 125 GeV, if exists, should be singlet-like.

In gure 6, we show a scatter plot of p as a function of MX. The big (small) points (do not) satisfy the WMAP relic density constraint within 3 , while the red-(black-)colored points gives r1 > 0.7(r1 < 0.7). The Gray region is excluded by the XENON100 experiment. The dashed line denotes the sensitivity of the next XENON experiment, XENON1T. We note that many points are still allowed by the WMAP relic density constraint, the

8

r 2

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r1

[LParen1] b[RParen1] m1 [Less]m 2 [LParen1] [Equal]125GeV[RParen1]

r 1

r2

1.5

1.0

0.5

0.0

[Minus] 0.5

[Minus] 1.0

[Minus] 1.5

a

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10 20 50 100 200 500 1000 2000

Figure 5. The allowed mixing angle as a function of the second Higgs mass. We xed the SM-like Higgs mass to be 125 GeV. The big (small) points (do not) satisfy the WMAP relic density constraint within 3, while the red-(blue-)colored points can (cannot) be probed at the planned XENON1T experiment. The black points are excluded by the LHC Higgs search, i.e. r < 0.7.

XENON100 direct detection experiment, and also by the constraint r1 > 0.7 which is in the ball park of the LHC Higgs search bound. On the other hand, the e ective eld theory approach considered in refs. [1] strongly constrains the vector dark matter scenario. We can also see that there is no point below about MX 50 GeV in gure 6 (a). It is because

the Higgs exchanged dark matter annihilation channel does not allow the resonance and the relic density is larger than the WMAP measurement. Most of the big red points are within the reach of the XENON1T sensitivity, and our model can be tested in the next generation dark matter detection experiment.

3.3 The EFT as a limit of the full theory for m2 ! 1In this subsection we consider the EFT in (1.3) as a limit of the full theory in (2.1) when m2 ! 1. We keep nite the full theory parameters: H, , H and MX(= gXv ), while

taking v ! 1. We trade H for the experimentally measured m1 using the relation

2 Hv2H = m21 + ( H vHv )2

2 v2 m21

. (3.6)

For large v , m2 ( ) is proportional to v (1/v ). The light dark matter (MX m2) is

possible when g2X . In other words, we should note that the EFT is valid only when

m2 ! 1, ! 0, g2X and it is a very restricted region. The term HHXX can be

generated both at tree- and loop-level [2]. Considering the tree-level diagram only, we get

HX =

where

e 112 112c /v and 112 is the H1 H1 H2 coupling constant given by 112 = H

h(c3 2c s2 )v + (s3 2s c2 )vH[bracketrightBig]+ 6 Hs c2 vH + 6 c s2 v . (3.8)

9

The 2nd Higgs mass [LParen1] GeV[RParen1]

2M2X

e 112m22, (3.7)

[LParen1]a[RParen1] m1[LParen1]=125GeV[RParen1]<m2

10-40

10-42

2 [RParen1]

10-44

p[LParen1]cm

10-46

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10-48

-50

10 10 20 50 100 200 500 1000

MX[LParen1]GeV[RParen1]

[LParen1] b[RParen1] m1 [Less] m 2 [LParen1] [Equal] 125GeV[RParen1]

10[Minus] 40

10[Minus] 42

2 [RParen1]

10[Minus] 44

s p[LParen1]cm

10[Minus] 46

10[Minus] 48

[Minus] 50

10 10 20 50 100 200 500 1000

M X [LParen1] GeV[RParen1]

Figure 6. The scatter plot of p as a function of MX. The big (small) points (do not) satisfy the WMAP relic density constraint within 3 , while the red-(black-)colored points gives r1 > 0.7(r1 <

0.7). The gray region is excluded by the XENON100 experiment. The dashed line denotes the sensitivity of the next XENON experiment, XENON1T. The solid blue line in panel (a) represents the prediction of the EFT approach in (1.3).

The elastic cross section p of the VDM X scattering o the proton in the full theory is obtained as

fullp = 42X

gXs c mp 2vH

2[parenleftbigg]1m21 1 m22

2f2p, (3.9)

Pq=u,d,s fpq + 2/9(1 Pq=u,d,s fpq) 0.468 [23].

The EFT predicts the corresponding cross section to be

EFTp = 42X

where X = MXmp/(MX + mp) and fp =

2 1m4hf2p. (3.10)

Using the relations (2.4), (3.7), (3.8) and identifying m1 and mh with the observed Higgs

10

HXmp 4MX

1.0

0.9

0.8

EFT

full s p

0.7

s p

0.6

0.5

0.4

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1000 2000 3000

1500

m2[LParen1]GeV[RParen1]

Figure 7. The ratio fullp/EFTp as a function of m2 for several values of H : H = 0.5, 0.3, 0.1 (blue, purple, green). We x MX = 300 GeV, m1 = 125 GeV = 0.175.

10

1

[LParen1][CapOmega]h2 [RParen1]full [LParen1][CapOmega]h2 [RParen1]EFT

0.1

0.01

0.001

10[Minus]4

10[Minus]5

1000 2000 3000

1500

m2[LParen1]GeV[RParen1]

Figure 8. The ratio ( h2)full/( h2)EFT as a function of m2 for several values of MX: MX = 150, 300, 500 GeV (blue, purple, green). We x m1 = 125 GeV, = 0.175 and H = 0.1.

mass (125 GeV) in their respective theories, we obtainfullp

EFTp = [parenleftbigg]

H

e 112

2. (3.11)

This ratio approaches to one as v ! 1. In gure 7, we show the ratio fullp/EFTp as a

function of m2 to see how quickly the full theory prediction approaches that of the EFT. We choose three di erent values for H , H = 0.5, 0.3, 0.1 (blue, purple, green), and x other parameters: MX = 300 GeV, m1 = 125 GeV, = 0.175. We can see that the EFT predictions agree well with those of the full theory within a few percent when m2 [greaterorsimilar] 2000 GeV.

Figure 8 shows the ratio of relic density predictions in the full theory and the EFT, ( h2)full/( h2)EFT , as a function of m2. Since the dependence on the coupling H is not

11

appreciable,3 we take several values of MX instead: MX = 150, 300, 500 GeV (blue, purple, green), although MX = 150 GeV is already excluded by the direct search limit as can be seen in gure 6 (a). We x m1 = 125 GeV, = 0.175 and H = 0.1. There is a sharp increase in the green line at m2 [similarequal] MX. This is because the dominant process XX ! H2H2

is kinematically closed at the point and the annihilation cross section decreases abruptly in the full theory. We can also see the resonance e ects of the full theory. Both e ects are absent in the EFT. We can see that the lighter the DM is, the faster the full theory approaches the EFT.

4 Vacuum stability and perturbativity of Higgs quartic couplings

In this section, we analyze vacuum stability and perturbativity of Higgs quartic couplings. To make the Higgs potential be bounded-from-below, we require

H > 0, > 0, 2

p H < H , (4.1)

where the last condition applies for H < 0. We also require

det M2Higgs = det 2 Hv2H H vHv

H vHv 2 v2

[parenrightBigg]

= (4 H 2H )v2Hv2 > 0. (4.2)

Since there is additional direction of , the Higgs potential can have minima other than our EW vacuum. In the following, we investigate whether the EW vacuum is global or not. We closely follow the analysis done in ref. [11].

The tree-level e ective potential takes the U(1)X symmetric form

V0('H, ' ) = H4 ('4H 2v2H'2H) +

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4 ('4 2v2 '2 ) +

4 ('2H'2 '2Hv2 v2H'2 ),

H

(4.3)

where 'H and ' are spacetime-independent classical elds. Following the refs. [30, 31], we dene the various vacua as follows:

EW : vH = 246 GeV, v [negationslash]= 0, (4.4)

SYM : vH = v = 0, (4.5)

I : vH = 0, v [negationslash]= 0, (4.6)

II : vH [negationslash]= 0, v = 0. (4.7)

Unlike the general Higgs potential, only nontrivial phase may be the I-phase. Such a minimum is given by

v = [notdef]

rv2 + H 2 v2H. (4.8)

3This is partly because the XX ! H2 ! H1H1 process is important and the amplitudes of which are

exactly the same in both the full theory and the EFT.

12

Allowed Region

Allowed Region

Figure 9. The vacuum stability and perturbativity constraints in the -m2 plane. We take m1 = 125 GeV, gX = 0.05, MX = m2/2 and v = MX/(gXQ ).

The di erences of vacuum energies of the I and the EW phases is

V (I)0(0, v ) V (EW)0(vH, v ) =

= 1

16 (4 H 2H )v4H, (4.9)

where we have used eq. (4.8) in the second line. Therefore, as long as eqs. (4.1) and (4.2) are satised, the EW vacuum is always the global minimum. Note that this is not the case for the generic Higgs potential [11].

Although the EW vacuum is stable at the EW scale, its stability up to Planck scale (MPl [similarequal] 1.22[notdef]1019 GeV) is nontrivial question since a renormalization group (RG) e ect of

the top quark can drive H negative at certain high-energy scale, leading to an unbounded-from-below Higgs potential or a minimum that may be deeper than the EW vacuum. We will work out this question by solving RG equations with respect to the Higgs quartic couplings and the U(1)X gauge coupling. The one-loop functions of those couplings are listed in appendix A. In addition to the vacuum stability, we also take account of the perturbativity of the couplings. To be specic, we impose i(Q) < 4 (i = H, H , ) and

g2X(Q) < 4 up to Q = MPl.

Figure 9 shows the vacuum stability and the perturbativity constraints in the -m2 plane. We take m1 = 125 GeV, gX = 0.05, MX = m2/2 and v = MX/(gXQ ). The vacuum stability constraint is denoted by red line; i.e., the region above the red line is allowed for > 0, and it is the other way around for < 0. The perturbativity requirement is represented by blue line; i.e., the region below the blue line is allowed for > 0, and

13

JHEP05(2013)036

4 v4H +

4 v2Hv2

H

H

4 (v4 v4 )

JHEP05(2013)036

Figure 10. The vacuum stability and perturbativity constraints in the MX-m2 plane. We set gX = 0.1 (Left Panel) and 0.5 (Right Panel) with being = 0.1.

it is the other way around for < 0. For < 0, the region above the dotted black line is excluded by eq. (4.1). Putting all together, for > 0 the region between the red and blue lines is allowed while for < 0 the region between the dotted black and blue lines is allowed. It should be noted that since the coe cient of H in H is doubled in comparison with the real singlet case, the improvement of the vacuum stability by the increase of H or, equivalently , is more e ective. However, unlike the general Higgs potential involving explicit U(1)X breaking terms, the EW vacuum cannot be stable up to Planck scale if is exactly zero.

In gure 10, we show the vacuum stability and perturbativity constraints in the MX-

m2 plane. We x = 0.1 varying gX, i.e., gX = 0.1 (Left Panel) and 0.5 (Right Panel). Once gX is xed, the small MX is realized by a small v . In such a case, the large m2 is possible only by a large since m2 [similarequal] p2 v for a small . This explains the regions

excluded by (Q) > 4 in both plots. Indeed, the gX = 0.5 case yields the severer constraints. As for the vacuum stability constraint, the change of gX has little e ect on it, which can be understood from the expression of H, eq. (A.2).

5 Conclusions

In this paper, we revisited the Higgs portal vector dark matter including the hidden sector Higgs eld that provides the vector dark matter mass. Including the hidden sector Higgs eld makes the model renormalizable and unitary. The constraint from direct detection cross section (XENON100) still allows a large parameter space in this model. On the contrary to some claims that the Higgs portal dark matter model is strongly constrained by XENON100 data, we showed that the model is still viable. It is crucial to work with a model that is renormalizable, and not with e ective lagrangian, as in the Higgs portal fermion DM model in ref. [6, 11] Including the hidden sector Higgs eld also improves the

14

vacuum stability of the model for mH = 125 GeV upto the Planck scale as in ref. [11]. Our model can be tested at colliders by searching for the 2nd Higgs boson and/or the signal strength of the 125 GeV Higgs boson. It would take long in order to observe the 2nd Higgs boson since its signal strength is smaller than 0.3. In our model, ri is universally suppressed relative to the SM case for all channels. This could be a useful criterion when the signal strengths of 125 GeV Higgs boson are measured with smaller uncertainties. If ri is not universally suppressed or larger than one, then our model shall be excluded.

Acknowledgments

We are grateful to Yasaman Farzan for bringing ref. [15] to our attention. This work is supported in part by NRF Research Grant 2012R1A2A1A01006053 (PK and SB), and by SRC program of NRF Grant No. 20120001176 funded by MEST through Korea Neutrino Research Center at Seoul National University (PK). WIP is supported in part by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2012-0003102).

A One-loop functions of Higgs quartic couplings

The renormalization group equation and the functions are given by

d (t)d log(Q) = , (A.1)

where

H = 1

162

24 2H + 2H 6y4t +3 8

n2g42 + (g22 + g21)2[bracerightBig] H

JHEP05(2013)036

n3(3g22 + g21) 12y2t[bracerightBig][bracketrightbigg],

(A.2)

H = 1

2 H (6 H + 4 + 2 H ) H

32(3g22 + g21) 6y2t + 6g2XQ2 [bracketrightbigg], (A.3)

= 1

162

162

h2( 2H + 10 2 + 3g4XQ4 ) 12 g2XQ2 [bracketrightBig], (A.4)

gX = 1

13g3XQ2 . (A.5)

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