Study of WIMP annihilations into a pair of on-shell scalar mediators

# Study of WIMP annihilations into a pair of on-shell scalar mediators

Lian-Bao Jia School of Science, Southwest University of Science and Technology, Mianyang 621010, P. R. China
###### Abstract

In this article, we focus on a new scalar mediated scalar/vectorial WIMPs (weakly interacting massive particles) with ’s mass slightly below the WIMP mass. To explain the Galactic center 1 - 3 GeV gamma-ray excess, here we consider the case that a WIMP pair predominantly annihilates into an on-shell pair with mainly decaying to . The masses of WIMPs are in a range about 14 - 22 GeV, and the annihilations of WIMPs are phase space suppressed today. In this annihilation scheme, the couplings of the - standard model (SM) particles are almost arbitrary small, and the WIMP-nucleus spin-independent scattering can be tolerant by the present dark matter (DM) direct detections. A scalar mediator-Higgs field mixing is introduced, which is small and available. The lower limit on the couplings of the -SM particles set by the thermal equilibrium in the early universe is derived, and this constraint is above the neutrino background for scalar DM in direct detections. The WIMPs may be detectable at the upgraded DM direct detection experiment in the next few years, and the exotic decay , the production of may be observable at future high-luminosity collider.

## I Introduction

The weakly interacting massive particle (WIMP) type dark matter (DM) attracts much attention in DM direct detections, and the cold DM relic density can be derived from thermally freeze-out WIMPs. Today, the compatible confident events are still absent in DM direct detection experiments, and the recent search results of CRESST-II Angloher:2015ewa (), CDMSlite Agnese:2015nto (), LUX Akerib:2015rjg () and XENON1T Aprile:2015uzo () set stringent constraints on the WIMP-nucleus spin-independent (SI) scattering. Even with these rigorous constraints, the case of the SI interaction being dominant in WIMP-nucleus scattering can still be allowed by the present direct detections, and a feasible scenario will be investigated in this work with the possible DM signatures from indirect detections.

The cosmic ray observations, such as -rays, neutrinos, positrons, and antiprotons from DM dense regions, may indirectly reveal properties of WIMPs. The recent 1-3 GeV gamma-ray excess from the Galactic center may be due to WIMP annihilations, for WIMPs in a mass range about 35-50 GeV annihilating into with corresponding annihilation cross section cms Goodenough:2009gk (); Hooper:2011ti (); Abazajian:2014fta (); Daylan:2014rsa (); Calore:2014xka (); Alves:2014yha (); Zhou:2014lva (), or WIMPs in a mass range about 7 -11 GeV annihilating into with the annihilation cross section cms (20% to also allowed) Hooper:2010mq (); Hooper:2011ti (); Abazajian:2014fta (); Calore:2014xka (); Daylan:2014rsa (). In this work, we focus on the latter case, that is, the main WIMP annihilation products in SM sector are pairs (see Refs. Lacroix:2014eea (); Yu:2014mfa (); Ibarra:2015fqa (); Kim:2015fpa () for more discussions). Moreover, with a small number of visible matter in dwarf satellite galaxies, gamma rays from the DM-dominant dwarf galaxies provide significant information about WIMPs. The mode galactic center GeV gamma-ray excess can be compatible with the recent results from the new dwarf spheroidal galaxy observations Geringer-Sameth:2015lua (); Drlica-Wagner:2015xua (); Ackermann:2015zua (); Li:2015kag ().

New physics beyond the standard model (SM) is needed to yield the main product in SM sector in WIMP annihilations. The leptophilic WIMPs were discussed in the literature Baltz:2002we (); Chen:2008dh (); Pospelov:2008jd (); Cholis:2008qq (); Fox:2008kb (); Cao:2009yy (); Bi:2009uj (); Ibarra:2009bm (); Kopp:2009et (); Cohen:2009fz (). Here we consider that a new scalar mediates the interactions between the SM charged leptons and scalar/vectorial WIMPs (the annihilation of fermionic WIMPs is p-wave suppressed today), and the new couplings of the mediator to leptons are proportional to the lepton masses. If the scalar mediator is lighter than the WIMP mass, the way of a WIMP pair annihilating into an on-shell mediator pair is allowed (see e.g. Refs. Martin:2014sxa (); Abdullah:2014lla (); Rajaraman:2015xka (); Cline:2015qha () for more). In this case, the scalar mediator’s couplings to SM particles can be almost arbitrarily small.111There is a lower bound about the couplings, which is from the thermal equilibrium in the early universe. To fit the GeV gamma-ray excess and meanwhile evade present constraints from DM direct detections and collider experiments, we focus on the case that the mediator is lighter than the WIMP mass and the mediator-tau lepton coupling is much smaller than the mediator-WIMP coupling. Thus, the dominant annihilation mechanism of WIMPs is that a WIMP pair annihilates into an on-shell mediator pair which mainly decays to the heaviest leptons . The case of mode dominant is naturally compatible with the antiproton spectrum observations from PAMELA Adriani:2010rc (), and is tolerant by the smooth positron spectrum of AMS-02 Bergstrom:2013jra (); Hooper:2012gq (); Ibarra:2013zia ().

A small scalar mediator-Higgs field mixing is discussed, and the mixing is small enough to keep dominant in the scalar mediator decays. With a small mixing introduced, one prospect is that the WIMP-target nucleus SI scattering may be detectable at the upgraded DM direct detection experiment in the next few years, and another prospect is that the scalar mediator may be observable in the future high-luminosity experiment. In fact, the small mediator-Higgs mixing can play an important role to the thermal equilibrium between DM and SM sectors in the early universe. The reaction rates of SM particles WIMPs should be larger than the expansion rate of the universe for some time in the early universe, and this sets a lower bound about the couplings of the scalar mediator to SM particles. The lower bound of the coupling gives a lower limit on the cross section of WIMP-target nucleus SI scattering. These will be explored in this paper.

This article is organized as follows. After this introduction, the form of the interactions in new sector and the annihilation cross section of scalar/vectorial WIMPs are given in section II. Next we give a detailed analysis about scalar WIMPs in section III, including the constraints and the test at future experiment. In section IV, we give a brief discussion about the test of vectorial WIMPs. The conclusions and some discussions are given in the last section.

## Ii Interactions between WIMPs and SM

In this article, we focus on scalar/vectorial WIMPs, with a new scalar field mediating the interactions between WIMPs and SM particles.

### ii.1 The new sector interactions

Consider that a real scalar field mediates the interactions between scalar/vectorial WIMPs and SM particles, with favoring SM leptons. Let us formulate the corresponding interactions. Following the forms in Refs Pospelov:2007mp (); Batell:2012mj (); Liu:2014cma (); Abdallah:2015ter (), the effective interactions of to scalar/vectorial WIMPs, charged lepton (, , ), Higgs field and self-interactions are taken as

 LiS = −λ2Φ2S∗S−μΦS∗S−μ33!Φ3−λ44!Φ4−λlΦ¯ll (1) −λ′S∗S(H†H−v22)−λhΦ2(H†H−v22)−μhΦ(H†H−v22),
 LiV = λ2Φ2V∗μVμ+μΦV∗μVμ−μ33!Φ3−λ44!Φ4−λlΦ¯ll (2) +λ′V∗μVμ(H†H−v22)−λhΦ2(H†H−v22)−μhΦ(H†H−v22),

where is a scalar WIMP field, is a vectorial WIMP field, and a symmetry is introduced to let WIMPs stable. The Yukawa type coupling parameter is proportional to the charged lepton mass. is the vacuum expectation value with 246 GeV, and is chosen for no vacuum expectation value obtained Pospelov:2007mp (); Batell:2012mj (). The parameter can be rewritten as , , with a dimensionless parameter and , the scalar, vectorial WIMP mass respectively. The self-interaction terms of are included, and the contribution from the cubic term may need be considered in WIMP annihilations in some cases. The term is the DM-Higgs field interaction, which is also included for completeness.

The scalar component of Higgs field and the scalar field can mix after the electroweak symmetry breaking, giving the mass eigenstates , in the form

 (hϕ)=[cosθsinθ−sinθcosθ](h′Φ). (3)

Here is the mixing angle, and one has

 tan2θ=2vμhm2h′−m2Φ. (4)

For the Higgs sector being affected by interactions as small as possible, here we suppose that the interactions are relatively small, i.e. in the case of and . For the mass eigenstates, one then has , . Thus, the value of can be very small, i.e. , , and this is necessary to be compatible with experimental constraints. The value should be small enough to keep dominant in ’s decay, and this is essential to explain the Galactic center gamma ray excess.

Here we give a brief discussion about the term in Eqs. (1), (2). In the case of being not much smaller than , , the SM-like Higgs boson may have an appreciable contribution to the WIMP annihilations. Due to the definite Higgs-nucleon coupling, would have a significant contribution to the WIMP-nucleus scattering, while the Higgs portal DM is rigorously constrained by the direct detections. The circumstance may also occur that the annihilation of WIMPs is mainly via the interactions mediated by , while the WIMP-nucleus scattering is mainly via the interactions mediated by . As we focus on the new scalar portal DM, i.e. the case of , , and a further request of to let the WIMP-nucleus scattering dominantly mediated by . The contribution from term is neglected in this paper.

### ii.2 Annihilations of WIMPs

The case the scalar mediator is lighter than the WIMP mass , , is of our concern in this paper. When the coupling , , a WIMP pair predominantly annihilates into an on-shell pair. The particle mainly decays to , and the gamma rays from mode can reveal some properties of WIMPs. The differential gamma-ray flux from DM annihilation is

 E2γdΦγdEγ=⟨σannvr⟩0J8πm2DM∑iBRiE2γdNiγdEγ, (5)

where is the thermally averaged DM annihilation cross section today, and is the annihilation factor. To fit the galactic center gamma-ray excess via mode as mentioned by the introduction (a WIMP pair annihilates into a pair, with WIMP mass about 7 -11 GeV and the annihilation cross section cms ), an alternative scheme is via the process of a WIMP pair , with the WIMP mass being twice GeV and the mass close to the WIMP mass . In this case, the thermally averaged annihilation cross section today is cms (nearly of that at the thermally freeze-out temperature). The scheme above is of our concern.

#### ii.2.1 Scalar WIMPs

Let us consider the scalar WIMPs first. The process is dominant in the WIMP annihilation, as shown in Fig. 1. The WIMP annihilation cross section in one particle rest frame is

 σannvr≃12βf32π(s−2m2S)(λ+2k2m2Sm2ϕ−2m2S+kk3mSmϕ4m2S−m2ϕ)2. (6)

Here the factor arises from the required type in annihilations, is the relative velocity of two WIMP particles, and is the total invariant mass squared. The parameter in Eq. (1) is rewritten as . is a kinematic factor, with

 βf=√1−4m2ϕs. (7)

Due to the factor, when the mediator mass is slightly below the WIMP mass , i.e. being close to the threshold of , the thermally averaged annihilation cross section today (in the limit) is more suppressed in phase space compared with the cross section at the freeze-out temperature . For this thermally freeze-out case, the key factor is failed to be expanded in Taylor series of .

The present DM relic density and the parameter (with ) can be approximately written as Kolb:1990vq (); Griest:1990kh ()

 ΩDh2≃1.07×109GeV−1Jann√g∗mPl, (8)
 xf≃ln0.038c(c+2)gmPlmS⟨σannvr⟩f√g∗xf, (9)

with

 Jann=∫∞xf⟨σannvr⟩x2d\emphx. (10)

Here is the Hubble constant (in units of 100 km/(Mpc)), and is the number of the relativistic degrees of freedom with masses less than the temperature . is the Planck mass with the value GeV, and is the degrees of freedom of DM. The parameter is of order one, and is taken here. The thermally averaged annihilation cross section is Gondolo:1990dk (); Cannoni:2013bza ()

 ⟨σannvr⟩ = 2xK22(x)∫∞0dε√ε(1+2ε) (11) ×K1(2x√1+ε)σannvr,

with . is the th order modified Bessel function.

#### ii.2.2 Vectorial WIMPs

Now let us turn to the vectorial WIMPs. The process is dominant in the vectorial WIMP annihilation. When is slightly below , the annihilation cross section is

 σannvr≃12βf96π(s−2m2V)(λ+2k2m2Vm2ϕ−2m2V+kk3mVmϕ4m2V−m2ϕ)2. (12)

For vectorial WIMPs, the thermally averaged annihilation cross section, the relic density are similar to the scalar case, with the corresponding parameter inputs in calculations.

## Iii Analysis of Scalar WIMPs

Here we give a detailed analysis about the scalar WIMPs, and the case of vectorial WIMPs is similar.

### iii.1 The constraints of WIMP annihilations

The process of WIMP pair is dominant in WIMP annihilations. For the mode in WIMP annihilations, the present thermally averaged cross section set by the Galactic center gamma-ray excess is cms, with the mass of WIMPs in the range 14-22 GeV. The cold DM relic density today is Ade:2015xua (). These constraints are taken to restrict the parameter spaces.

Define , with and close to 1. The factor plays a key role in fixing the ratio /, and the value of can be set by and . The numerical results of the thermally averaged annihilation cross sections are shown in Fig. 2, for WIMP masses in the range 14-22 GeV. When changes in the range , approximately varies from cms to cms. The derived annihilation cross section range of together with the WIMP mass range of concern can give an interpretation about the Galactic center GeV gamma-ray excess.

Since the value of is obtained by the constraints, the coupling between WIMPs and the mediator is also determined. Taking , we have

 |λ+kk3/3−2k2|∼2.0×10−3mS(GeV), (13)

with in units of GeV. The coupling in the WIMP- trilinear term plays an important role in the WIMP-target nucleus scattering in DM direct detections. In the case of the WIMP- trilinear term dominating the WIMP annihilations, we have

 |k|=|μmS|∼3.16×10−2√mS(GeV). (14)

If the contribution from , () terms are significant in WIMP annihilations, the relation

 λ∼2k2−kk3/3±2.0×10−3mS (15)

needs to be taken care of. In the case of a large value together with a comparable large and a mall value, or a large value together with a comparable large and a mall value, the s-channel annihilation of WIMP directly annihilating into SM particles can be enhanced. As we focus on the case of such direct annihilation being suppressed in this paper, e.g. the value of being order of , and here a range of

 k2≲1.4×10−3mS(GeV), (16)

is considered in calculations.

### iii.2 The constraints of ϕ

Now we give a brief discussion about the coupling of to SM particles, that is, the ’s value and the mixing angle . Some parameters are inputted as follows, GeV, GeV, GeV, GeV, with the results from PDG Agashe:2014kda ().

#### iii.2.1 The λτ value

As discussed above, the case of s-channel suppressed in WIMP annihilations is of our concern, i.e. . Here the value in the case of the term dominant is taken to restrict the ’s value, that is

 λτ≪3.16×10−2√mS(GeV). (17)

The decay width of is

 Γϕ≃mϕ8π[λ2τ(1−4m2τm2ϕ)3/2+3m2bv2sin2θ(1−4m2bm2ϕ)3/2], (18)

with term dominant. Taking the limit of in Eq. (17), we have . Thus, the ’s range of concern is feasible.

The particle contributes to the muon , and the one-loop result is Hektor:2015zba ()

 aϕμ≃λ2μ8π2m2μm2ϕ(lnm2ϕm2μ−76). (19)

The difference between experiment and theory is Agashe:2014kda ()

 Δaμ=aexpμ−aSMμ=288(63)(49)×10−11. (20)

Taking the replacement and in Eq. (19), we can find that the upper limit of in Eq. (17) is tolerant by the muon result.

#### iii.2.2 The mixing angle θ

For the WIMP mass range of concern, according to Eq. (18), if the channel is not larger than 20%, the value should satisfy the relation (when is compatible with 0.01, this constraint is relaxed). The Higgs hunt results at LEP Barate:2003sz () set an upper limit on ,

 sin2θ≲0.1 (ϕ→τ¯τ),sin2θBϕ→b¯b≲2×10−2 . (21)

We can see that the constraints from LEP are mild. The ATLAS ATLAS:2011cea () and CMS Chatrchyan:2012am () search constraints about light Higgs-like particles can be approximately written as Clarke:2013aya (); Haisch:2016hzu ()

 sin2θBϕ→μ+μ−≲BSMh→μ+μ− . (22)

With these constraints, we obtain an upper limit of

 sin2θ≲6×10−2 ,and|sinθ|≲20λτ. (23)

The constrains of the DM direct detection and Higgs boson decay will be discussed in the following.

### iii.3 Thermal equilibrium constraints

In the early universe, the WIMPs and SM particles are in thermal equilibrium. The reaction rates of WIMP pairs SM particles exceed the expansion rate of the universe for some time,

 ⟨σannvr⟩neq≳1.66√g∗ T2mPl, (24)

where is the corresponding equilibrium number density, with for fermions in the relativistic limit. For SM particles WIMP pairs, the annihilation cross section of each fermion specie is

 σannvr=λ2SMk2√1−4m2S/s32π(s−2m2f)m2S(s−4m2f)(s−m2ϕ)2, (25)

with for charged leptons, quarks, respectively. The reaction rate can set a lower bound about the ’s coupling to SM particles (see e.g. Refs. Chu:2011be (); Dolan:2014ska () for more). If the mixing angle is tiny, with the contribution mainly from the annihilation, the reaction rate can give a lower bound on . However, in this case, the WIMPs are insensitive in target nucleus scattering detections, and traces of are difficult to be observed at collider experiment. Here we focus on the interesting case of contribution dominating the SM particle reaction rate at . By the calculation, we can obtain that the contribution is dominant when is some times larger than (this value corresponding to the nearly equal contributions of and ), e.g. . Moreover, an appreciable value is available in interpreting the Galactic center gamma ray excess. Now, we have a range of ,

 10λτ≲|sinθ|≲20λτ. (26)

For the case of the contribution dominating the SM reaction rate, according to Eq. (24), we obtain that the constraint can be written as

 sin2θk2m2S(GeV)≳2.2×108π3√g∗ mtζ(3)mPl≈8.5×10−7. (27)

This constraint is taken as the lower bound for the mixing angle and the parameter . In fact, the constraint is valid for the WIMPs in a general mass range of .

### iii.4 DM direct detection

Here we turn to the direct detection of WIMPs. The WIMP-target nucleus scattering is mainly mediated by . The effective coupling between and nucleon can be set by , and is the Higgs-nucleon coupling, with He:2008qm () adopted here.222There is an uncertainty about the value of the Higgs-nucleon coupling. See e.g. Refs Ellis:2000ds (); Gondolo:2004sc (); Alarcon:2011zs (); Cheng:2012qr () for more. The cross section of the WIMP-nucleon SI elastic scattering is

 σel≃sin2θk2m2Sg2hNNm2N4π(mS+mN)2m4ϕ, (28)

where is the nucleon mass.

For WIMPs in the mass range of concern, the recent DM searching results of LUX Akerib:2015rjg () and XENON1T Aprile:2015uzo () set stringent upper limits on the mixing angle and the parameter . In addition, the thermal equilibrium condition requirement of Eq. (27) gives a lower bound on the parameters. Taking , and considering the neutrino background Billard:2013qya () in detections, the tolerant hunting region of the cross section is depicted in Fig. 3, for WIMPs in an ordinary mass range 10 - 30 GeV. The filled region is for the potential mass range 14 - 22 GeV of concern, which is indicated by the galactic center gamma ray excess, and the allowed region of the cross section is . The parameter spaces are set by the thermal equilibrium limit and the recent XENON1T results. For 14 - 22 GeV, the upper limit of XENON1T is fitted in the form

 a×(mS)b(GeV)×g2hNNm2N4π(mS+mN)2m4S, (29)

with the fitting values , . Thus, in the WIMP mass range of concern, the constraints of and can be expressed as

 8.5×10−7≲sin2θk2m2S(GeV)≲2.61×10−5(mS20)3.71. (30)

This is the parameter space allowed, and it is detectable in DM direct detections in the future.

### iii.5 New sector search at collider

#### iii.5.1 New channels for Higgs decays

After the discovery of the SM-like Higgs boson at LHC Aad:2012tfa (); Chatrchyan:2012xdj (), the exploration of the Higgs portal new physics attracts much attention in recent years. In our scheme, the Higgs boson can decay into a WIMP pair, and the decay width is

 Γh→S∗S=sin2θk2m2S16πmh ⎷1−4m2Sm2h. (31)

Taking 125 GeV Aad:2015zhl (), the total width of SM Higgs is GeV Denner:2011mq (); Agashe:2014kda (). With the constraints of the thermal equilibrium limit and the recent XENON1T results, i.e. Eq. (30), the branching ratio of Higgs boson decaying into a scalar WIMP pair is

 3.3×10−8≲Bh→S∗S≲1.0×10−6(mS20)3.71. (32)

This invisible branching ratio is very small and difficult to investigate at present and in the future collider experiment.

According to Eq. (1), the channel of Higgs boson decaying into a on-shell pair is allowed for the mass of concern. For , , the decay width of can be approximately written as

 Γh→ϕϕ≈λ2hv2cos6θ32πmh ⎷1−4m2ϕm2h, (33)

with the terms neglected. This channel should be small compared with the SM leading channel , i.e. . As a rough estimate, an upper limit is taken in discussions (i.e. the decay width GeV).

The decay channel may be detectable at the future precise Higgs decay measurement via the process . According to Eqs. (18), (26), the branching ratios of the two main decay channels of are

 6.4 %≲Bϕ→b¯b≲21 %,Bϕ→τ¯τ≃1−Bϕ→b¯b, (34)

with GeV adopted as input. Thus, the upper limits of the exotic branching ratios in decay are as follows:

 Bh→ϕϕ→(τ¯τ)(τ¯τ)≃Γh→ϕϕ B2ϕ→τ¯τΓh(SM)+Γh→ϕϕ≲(3.4−4.7)×10−3, (35) Bh→ϕϕ→(τ¯τ)(b¯b)≃2Γh→ϕϕ Bϕ→τ¯τBϕ→b¯bΓh(SM)+Γh→ϕϕ≲(0.64−1.8)×10−3, (36) Bh→ϕϕ→(b¯b)(b¯b)≃Γh→ϕϕ B2ϕ→b¯bΓh(SM)+Γh→ϕϕ≲(0.22−2.4)×10−4. (37)

Due to the missing neutrino(s) in decay, the resolution of is poor (about 15%) Agashe:2014kda (). Thus, the above three channels are comparable in the Higgs decay search. As the collider has a more clean environment compared with the hadron collider, here we focus on the precise tests of Higgs decay channels at the future collider. At the center of mass energy 250 - 350 GeV, the dominant Higgs production mechanism is via the Higgs-strahlung process , and this can be employed for the precise measurement of the Higgs decays. The cross sections of Higgs-strahlung mode in collisions at GeV are 211, 134 fb Dawson:2013bba (), respectively. For GeV, there are about Higgs events produced at a high integrated luminosity of 500 fb 5 ab. In this case, if the decay width of is near the upper limit, there will be tens hundreds tagging events of via the three decay modes discussed above. Thus, the decay can be investigated at the future Higgs factory, or the corresponding limit is set by the experiment.

#### iii.5.2 Production of ϕ at collider

Now we turn to the production at collider. Due to the messy background at the hadron collider, the constraints from ATLAS ATLAS:2011cea () and CMS Chatrchyan:2012am () are mild on the teens/tens GeV of concern, as discussed above. Here we focus on the search of at high energy collider, and this clean environment machine is good for high precise studies. The dominant production processes of are the strahlung, the fusion, and the fusion, as depicted in Fig. 4.

The strahlung process is similar to the case of Higgs boson production, and the corresponding cross section can be written as

 σe+e−→Zϕ=sin2θG2Fm4Z96πs(v2e+a2e)ββ2+12m2Z/s(1−m2Z/s)2, (38)

where Agashe:2014kda () is the Fermi coupling constant, and , are the vector, axial-vector current parameters, respectively. is the phase space factor, with

 β=√(1−m2ϕs−m2Zs)2−4m2ϕm2Zs2. (39)

The cross section of the vector boson (, ) fusion process can be written in the form Zerwas:desy (); Kilian:1995tr (); Djouadi:1996uj ()

 σ=sin2θG3FM4v64√2π3∫1κϕdx∫1xdy[1+(y−x)/κv]2[(^v2+^a2)2f(x,y)+4^v2^a2g(x,y)], (40) f(x,y)=(2xy3−1+2xy2+2+x2y−12)[z1+z−log(1+z)]+xy3z2(1−y)1+z g(x,y)=(−xy2+2+x2y−12)[z1+z−log(1+z)]

with for the , boson respectively, , , and . , are the electron couplings to the vector bosons, with for the boson, and , for the boson.

Let us give a further discussion about the range of before evaluating the production cross section of . From Eq. (16) and Eq. (30), we can derive a lower bound , with

 6.1×10−4m3S≲sin2θl≲1.9×10−2m3S(mS20)3.71. (41)

In addition, should be much smaller than 1. As a rough estimate, an alteration of order about the Higgs production and decay is tolerant by the present experiment. Here, an upper limit is taken. In SM particle scattering processes, the contribution from Higgs boson keeps dominant among the ’s contributions.

With in this paper, we consider the production of in the range 14 - 22 GeV. Fixing 20 GeV, the dependence of the strahlung, fusion, and fusion cross sections with the center of mass energy is depicted in Fig. (5), for varying in a range 150 - 500 GeV. The upper limit, lower limit of the production cross sections are corresponding to , , respectively. It can be seen that, for below 400 GeV, the main production process of is the strahlung mechanism. At a given center of mass energy 250 GeV (the potential Higgs production energy), the production cross sections of for in the range 14 - 22 GeV is shown in Fig. (6), with the same upper limit, lower limit in the processes of strahlung, fusion, and fusion as that of Fig. (5). At the same value, the cross section changes slowly with . Thus, the upper limit results of production cross section given in Fig. (5) are roughly the cross section of with the mass of 14 - 22 GeV.

If the value is near the upper limit of the parameter space, the signature of may appear at the future high luminosity collider. Considering below 400 GeV, the main production mechanism of is via the strahlung. In the case , there is about a hundred events produced at 250 GeV with a integrated luminosity of 200 fb. The dominant final state of is , and the second branching fraction of final state is about 6.4% 21%. The particle can be searched by the final states ( )( ), ( )( ). In fact, as shown in Fig. (7), it is better to test the non-standard model -like particle at a low center of mass energy collider with a high luminosity, e.g. 120 - 150 GeV with the energy above the production threshold. For , there are about 800 (150 GeV) - 2000 (120 GeV) production events with a integrated luminosity of 200 fb. Thus, the new particle with near the upper limit of the parameter space can leave traces at the future high luminosity collider, or the upper limit of is reduced by the search result.

## Iv Analysis of vectorial WIMPs

The vectorial WIMPs is similar to the case of scalar WIMPs. To satisfy the corresponding constraints, the value of is approximately in the same range as the scalar WIMPs. The cross section of the vectorial WIMP-nucleon SI elastic scattering is

 σel≃sin2θk2m2Vg2hNNm2N4π(mV+mN)2m4ϕ. (42)

In the following, we just focus on the significant differences for vectorial WIMPs, and give a brief discussion about them.

For vectorial WIMPs, with 0.994, we have

 |λ+kk3/3−2k2|∼√3×2.0×10−3mV(GeV). (43)

In the thermal equilibrium era of the early universe, the reaction rates of SM particles WIMP pairs exceed the expansion rate of the universe. The cross section of each SM fermion specie annihilating into a vectorial WIMP pair is

 σannvr=λ2SMk2√1−4m2V/s32π(s−2m2f)m2V(s−4m2f)(s−m2ϕ)2[2+(s−2m2V)24m4V]. (44)

At , the reaction rates of SM particles are significant enhanced by the longitudinal polarization of vectorial WIMPs, e.g. the enhancement over for . Consider the contribution dominating the SM particle reaction rate, and we have a range of ,

 2λτ≲|sinθ|≲20λτ. (45)

The constraint of the thermal equilibrium can be approximately written as

 sin2θk2m2V(GeV)≳5.8×102π3√g∗ mtζ(3)mPl(mV20)4≈2.3×10−12(mV20)4. (46)

For the vectorial WIMPs of concern, this thermal equilibrium constraint is below the neutrino background in DM direct detections, and the lower bound of the production cross section at collider is reduced by an order factor compared with the case of scalar WIMPs.

## V Conclusion and discussion

The scalar and vectorial WIMPs have been studied in this article, with a new scalar as the mediator and the mass of being slightly below the WIMP mass. The dominant annihilation products of WIMPs are on-shell pairs with mainly decaying into , and the WIMP annihilations are phase space suppressed today. For masses of WIMPs in a range about 14 - 22 GeV, the annihilation cross section cms today can be obtained to meet the Galactic center GeV gamma-ray excess. Due to the nearly arbitrary small couplings between and SM particles, the WIMP-target nucleus SI scattering can be tolerant by the present stringent constraints of DM direct detections.

The upper limit of the ’s coupling to lepton is discussed with the constraints of WIMP annihilations, and the limit is tolerant by the muon result. The scalar mediator-Higgs mixing angle should be small enough to keep dominant in the scalar ’s decay, and the upper limit of from collider experiment is mild. The thermal equilibrium in the early universe sets an lower bound on the reaction rates of SM particles. Considering the contribution is dominant in SM particle reaction rates, we have derived an lower limit about the mixing angle and the coupling of the WIMP- trilinear term. For vectorial WIMPs, the reaction rate of WIMP pair is dramatically enhanced by the longitudinal polarization of vectorial WIMPs.

For the scalar WIMP-nucleon SI elastic scattering of concern, we obtain that the bound from the thermal equilibrium sets a minimum scattering cross section above the neutrino background. The present parameter spaces are set by the XENON1T result and the bound from thermal equilibrium, and the allowed region of the elastic scattering cross section is derived, with . The range of the scattering cross section can be examined at the future DM ultimate direct detection experiments, such as LUX-ZEPLIN (LZ) Akerib:2015cja (), XENONnT Aprile:2015uzo () and DARWIN Aalbers:2016jon (). Thus, for WIMPs of concern, the future DM direct detections can give an answer about whether the scalar WIMP candidates exist or not. For vectorial WIMPs, the bound from the thermal equilibrium is below the neutrino background in direct detections, and this type WIMPs cannot be ruled out by the future DM ultimate direct detections.

The tests of the new sector at collider are as follows: i). The Higgs boson can decay into a WIMP pair, while this invisible decay is tiny and difficult to be explored at collider. ii). The decay channel may leave traces at the future collider with the precise Higgs decay measurement, e.g. via the Higgs-strahlung process at GeV, and a high luminosity of 500 fb 5 ab is needed. iii). For 400 GeV and above the threshold, the production of is mainly via the strahlung mechanism at collider. The signature of with the value near the upper limit may appear at the future collider, and it is better to test the non-standard model -like particle at a low center of mass energy collider with a high luminosity, e.g. 120 - 150 GeV with the energy above the production threshold and the luminosity up to about 200 fb.

The future collider, such as the Circular Electron Positron Collider (CEPC) CEPC-Report (), the International Linear Collider (ILC) Baer:2013cma (), and FCC-ee(TLEP) Gomez-Ceballos:2013zzn (), may do the job to investigate the decay and the production of . We look forward to the future tests of the WIMPs of concern via DM indirect detections, DM direct detections and the hunt at collider.

###### Acknowledgements.
Thank Xuewen Liu for useful discussions. This work was supported by the National Natural Science Foundation of China under Contract No. 11505144, and the Research Fund for the Doctoral Program of the Southwest University of Science and Technology under Contract No. 15zx7102.

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