1 Introduction
Abstract

We promote the idea of multi-component Dark Matter (DM) to explain results from both direct and indirect detection experiments. In these models as contribution of each DM candidate to relic abundance is summed up to meet WMAP/Planck measurements of , these candidates have larger annihilation cross-sections compared to the single-component DM models. This results in larger -ray flux in indirect detection experiments of DM. We illustrate this fact by introducing an extra scalar to the popular single real scalar DM model. We also present detailed calculations for the vacuum stability bounds, perturbative unitarity and triviality constraints on this model. As direct detection experimental results still show some conflict, we kept our options open, discussing different scenarios with different DM mass zones. In the framework of our model we make an interesting observation: The existing direct detection experiments like CDMS II, CoGeNT, CRESST II, XENON 100 or LUX together with the observation of excess low energy -ray from Galactic Centre and Fermi Bubble by FGST already have the capability to distinguish between different DM halo profiles.

A Possible Explanation of Low Energy -ray Excess from Galactic Centre and Fermi Bubble by a Dark Matter Model with Two Real Scalars

Kamakshya Prasad Modak111kamakshya.modak@saha.ac.in, Debasish Majumdar222debasish.majumdar@saha.ac.in, Subhendu Rakshit333rakshit@iiti.ac.in

Astroparticle Physics and Cosmology Division, Saha Institute of Nuclear Physics,

1/AF Bidhannagar, Kolkata 700064, India.

Discipline of Physics, Indian Institute of Technology Indore,

IET-DAVV Campus, Indore 452017, India.

1 Introduction

The overwhelming cosmological and astrophysical evidences have now established the existence of an unknown non-luminous matter present in the universe in enormous amount, namely the dark matter (DM). Experiments like Wilkinson Microwave Anisotropy Probe (WMAP) [1], BOSS [2] or more recently Planck [3] measure the baryonic fraction precisely to consolidate the fact that this non-baryonic DM constitutes around 26.5% of the content of the universe. The particle nature of DM candidate is still unknown. The relic density of dark matter deduced from cosmological observations mentioned above tends to suggest that most of the DM could be made of weakly interacting massive particles (WIMPs) [4, 5, 6, 7] and they are non-relativistic or cold in nature. This calls for an extension of the standard model (SM) of particle physics. Many such extensions have been suggested in the literature in the framework of supersymmetry, extra dimensions, axion etc. Models such as Kaluza Klein [8], inert triplet [9] or supersymmetry breaking models like mAMSB [10] predict very massive DM whereas models like SMSSM [11], axion [12] predict DM of lower mass. Phenomenology of simpler extensions of SM like fermionic DM model [13] or inert doublet model [14] has been elaborately studied. Amongst all such options, extending the scalar sector is particularly interesting because of its simplicity.

The minimal extension with a single gauge singlet real scalar stabilised by a symmetry in the context of dark matter was proposed by Silveira and Zee in Ref. [15] and then it was extensively studied in the literature [16] - [37]. In Ref. [38] the singlet scalar DM model has been discussed with a global U(1) symmetry.

Amongst the non-minimal extensions, a DM model where SM is extended by a complex singlet scalar has been considered in Refs. [39, 40, 41]. A DM model with two real scalars has been discussed in Refs. [42, 43], where one scalar is protected by a symmetry, but the symmetry protecting the other one spontaneously breaks. In all these non-minimally extended models there is, however, only one DM candidate.

A pertinent question to ask at this stage is that, if our visible sector is enriched with so many particles, why the DM should be composed of only one component? We therefore intend to discuss in this paper a model with two DM candidates. In some earlier works [44] the idea of multicomponent dark matter has been discussed in details. We extend the SM with two gauge singlet real scalars protected by unbroken symmetries. As mentioned in the abstract, the advantage of such a multi-component DM model is that the DM annihilation can be enhanced so that one can expect spectacular signals in the indirect detection experiments. Because of this simple fact, the thermal averaged annihilation cross-sections in this model can enjoy enhancement upto two orders of magnitude compared to the models with one real scalar. As a result, in the present model with two real scalars we can make an attempt to explain both direct and indirect detection DM experimental observations, which was not possible with the model with a single real scalar.

Direct detection DM experiments can detect DM by measuring the recoil energy of a target nucleon of detecting material in case a DM particle happens to scatter off such nucleons. Experiments like CDMS [45, 46], DAMA [47], CoGeNT [48] or CRESST [49] present their results indicating allowed zones in the scattering cross-section – DM mass plane. These experiments seem to prefer low dark matter masses 10 GeV. Some earlier works on 10 GeV DM mass have been done [50, 51]. XENON 100 [52, 53], however, did not observe any potential DM event contradicting claims of the earlier experiments and has presented an upper bound on DM-nucleon scattering cross-section for various DM masses. Recent findings by LUX [54] have fortified claims by XENON 100 collaboration.

The indirect detection of DM involves detecting the particles and their subsequent decay products, produced due to DM annihilations. Huge concentration of DM are expected at the centre of gravitating bodies such as the Sun or the galactic centre (GC) as they can capture DM particles over time.

The region in and around the GC are looked for detecting the dark matter annihilation products such as , etc. Fermi Gamma Space Telescope (FGST), operated from mid of ’08, have been looking for the gamma ray from the GC [55]. The low energy gamma ray from GC shows some bumpy structures around a few GeV which cannot be properly explained by known astrophysics. A plausible explanation of such a non-power law spectrum is provided by DM annihilations [56, 57].

The emission of gamma rays from Fermi Bubble may also be partially caused by DM annihilations. The Fermi Bubble is a lobular structure of gamma ray emission zone both upward and downward from the galactic plane has been discovered recently by Fermi’s Large Area Telescope [58]. The lobes spread up to a few kpc above and below the galactic plane and emit gamma ray with energy extending from a few GeV to about a hundred GeV. The gamma emission is supposed to be produced from the inverse Compton scattering (ICS) of cosmic ray electrons. But more involved study of this emission reveals that while the spectra from the high galactic latitude region can be explained by ICS taking into consideration cosmic electron distribution, it cannot satisfactorily explain the emission from the lower latitudes. The -ray flux from possible DM annihilation in the galactic halo may help explain this apparent anomaly [59, 60, 61].

As mentioned earlier, in this work, the proposed model with two gauge singlet real scalars protected by a symmetry is confronted with the experimental findings of both direct and indirect DM experiments. As direct DM experimental results are contradictory, we keep an open mind in considering them. We find constraints on the parameter space from these results and then try to constrain them further imposing the requirement of producing the relic density of the DM candidates consistent with Planck observations. We then choose benchmark points from this constrained parameter space to explain results from indirect detection experiments from DM annihilation taking into consideration different DM halo profiles.

The paper is organised as follows. In Sec. 2 we discuss the theoretical framework of our proposed model. The theoretical constraints from vacuum stability, perturbative unitarity, triviality and experimental constraints from the invisible branching ratio of the Higgs boson have been discussed in Sec. 3. The next section contains the relevant relic density calculations. The model is confronted with direct detection experiments and Planck observations in Sec. 5. Explanation of the observed excess of -ray from GC and Fermi bubble by our model is studied in Sec. 6. We conclude in Sec. 7.

2 Theoretical Framework

We propose a model where two real scalar singlets ( and ) are added to the standard model.

The general form of the renormalisable scalar potential is then given by,

 V(H,S,S′) = m22H†H+λ4(H†H)2 (1) +δ12H†HS+δ22H†HS2+δ1m22λS+k22S2+k33S3+k44S4 +δ′12H†HS′+δ′22H†HS′2+δ′1m22λS′+k′22S′2+k′33S′3+k′44S′4 +δ′′22H†HS′S+k′′22SS′+13(ka3SSS′+kb3SS′S′) +14(ka4SSS′S′+kb4SSSS′+kc4SS′S′S′),

where is the ordinary (SM) Higgs doublet. In the above ’s denote the couplings between the singlets and the Higgs and ’s are the couplings between these singlets themselves.

The stability of DM particles is achieved by imposing a discreet symmetry onto the Lagrangian. Depending on whether and are odd under the same or not, we discuss two scenarios for completeness.

2.1 Lagrangian Invariant under Z2×Z2

If only and are odd under the same , and the rest of the particles are even,

 (SS′) Z2×Z2−−−−→ (−S−S′), (2)

some parameters of the potential vanish:

 δ1=k3=δ′1=k′3=ka3=kb3=0, (3)

so that the scalar potential (1) reduces to the following,

 V(H,S,S′) = m22H†H+λ4(H†H)2 (4) +δ22H†HS2+k22S2+k44S4 +δ′22H†HS′2+k′22S′2+k′44S′4 +δ′′22H†HS′S+k′′22SS′ +14(ka4SSS′S′+kb4SSSS′+kc4SS′S′S′).

After the spontaneous symmetry breaking the mass matrix for and is given by

 MSS′=(k2+δ2v2/2δ′′2v2/4+k′′2/2δ′′2v2/4+k′′2/2k′2+δ′2v2/2)≡(M11M12M12M22).

denotes the vacuum expectation value of the Higgs. After diagonalisation the masses of the physical eigenstates and are given by

 M2S1 = cos2θM11+sin2θM22+2cosθsinθM12 (5) M2S2 = cos2θM22+sin2θM11−2cosθsinθM12, (6)

where

 tan2θ=2M12M11−M22. (7)

2.2 Lagrangian Invariant under Z2×Z′2

If and are stabilised by different discrete symmetries,

 SZ2−→−S and S′Z′2−→−S′, (8)

, so that the scalar potential (4) further reduces to

 V(H,S,S′) = m22H†H+λ4(H†H)2 (9) +δ22H†HS2+k22S2+k44S4 +δ′22H†HS′2+k′22S′2+k′44S′4 +14ka4SSS′S′.

After spontaneous symmetry breaking the respective masses of and are given by

 M2S = k2+δ2v22 (10) M2S′ = k′2+δ′2v22. (11)

The four beyond SM parameters determining the masses of the scalars are , , and .

In both and cases if scattering processes can be avoided, the model can give rise to a two-component DM scenario. However, as the later case has fewer number of beyond SM parameters, in the following we will restrict ourselves only to the invariant Lagrangian.

3 Constraints on Model Parameters

The extra scalars present in the model modifies the scalar potential. Hence it is prudent to revisit constraints emanating from vacuum stability conditions and triviality of the Higgs potential. Perturbative unitarity can also get affected by these scalars. Limits on the invisible decay width of Higgs from LHC severely restricts such models. In the following we elaborate on these constraints.

3.1 Vacuum Stability Conditions

Calculating the exact vacuum stability conditions for any model is generally difficult. However, for many dark matter models the quartic part of the scalar potential can be expressed as quadratic form () with the squares of real fields as single entity. Lagrangian respecting symmetry which ensures the stability of scalar dark matter have the terms which can be expressed like that. The scalar potential of our proposed model can also be expressed in a similar form as above because of preservation of symmetry. The criteria for copositivity allow one to derive properly the analytic vacuum stability conditions for the such matrix from which sufficient conditions for vacuum stability can be obtained.444 Derivation of the necessary and sufficient conditions for the model is much simpler with copositivity than with the other used formalisms.

The necessary conditions for a symmetric matrix of order 3 to be copositive are given by [62, 63, 64, 65],

 a11⩾0,a22⩾0,a33⩾0,¯a12=a12+√a11a22⩾0,¯a13=a13+√a11a33⩾0,¯a23=a23+√a22a33⩾0, (12)

and

 √a11a22a33+a12√a33+a13√a22+a23√a11+√2¯a12¯a13¯a23⩾0. (13)

The last criterion given in Eq. (13) is a simplified form of the two conditions (Eqs. (14) and (15)) below

 √a11a22a33+a12√a33+a13√a22+a23√a11 ⩾0, (14) detA=a11a22a33−(a212a33+a213a22+a11a223)+2a12a13a23 ⩾0, (15)

where one or the other inequality has to be satisfied [63]555The criterion, is a part of well known Sylvester’s criterion for positive semidefiniteness.. The conditions Eq. (12) impose that the three principal submatrices of are copositive.

The matrix of quartic couplings in the basis for the potential Eq. (9) is given by

 4Λ=⎛⎜ ⎜ ⎜ ⎜⎝λδ2δ′2δ2k4ka42δ′2ka42k′4⎞⎟ ⎟ ⎟ ⎟⎠. (16)

Copositivity criteria of Eqs. (12) and (13) yield the necessary and sufficient vacuum stability conditions,

 λ⩾ 0,k4⩾0,k′4⩾0,δ2+√λk4⩾0,δ′2+√λk′4⩾0,ka4+√k4k′4⩾0, (17)

and

 √λk4k′4+δ2√k′4+δ′2√k4+2k′4√λ+√(δ2+√λk4)(δ′2+√λk′4)(ka4+√k4k′4)⩾0. (18)

The conditions of Eqs. (17) and (18) simply determine the vacuum stability bounds on our model. We restrict the parameter space by these conditions for later calculation.

3.2 Perturbative Unitarity Bounds

The potential of the model is bounded from below if Eq. (17) and Eq. (18) are simultaneously satisfied. Then, and or

 δ22

The Higgs mechanism generates a mass of for the Higgs and also contributes to the mass of the particle

 M2S = k2+δ2v22 (20) M2S′ = k′2+δ′2v22, (21)

For and , be a local minimum we should have and are required. This is also a global minimum as long as and [17]. The potential of the scalar sector after electroweak symmetry breaking in the unitary gauge can be written as,

 VSS′H = λ4H4+m24H2+m2v2H+vλH3+3v2λ2H2+v3λH (22) +δ22H2S2+vδ2HS2+v2δ22S2+k22S2+k44S4+δ′22H2S′2 +vδ′2HS′2+v2δ′22S′2+k′22S2+k′44S′4+ka44S2S′2.

After that tree-level perturbative unitarity [66] to scalar elastic scattering processes have been applied in this model (Eq. (22)). The zeroth partial wave amplitude,

 a0=132π√4pCMfpCMis∫+1−1T2→2dcosθ (23)

must satisfy the condition [67]. In the above, is the centre of mass (CM) energy, are the initial and final momenta in CM system and denotes the matrix element for processes with being the incident angle between two incoming particles.

The possible two particle states are and the scattering processes include many possible diagrams such as , , , , , , , , , , , . The matrix elements () for the above processes are calculated from the tree level Feynman diagrams for corresponding scattering and given by,

 THH→HH = 3M2Hv2(1+3M2H(1s−M2H+1t−M2H+1u−M2H)), (24) TSS→SS = 6k4+δ2(δ2v2s−M2H+δ2v2t−M2H+δ2v2u−M2H), (25) TSS→HH = δ2(1+3M2H1s−M2H+δ2v2(1t−M2S+1u−M2S)), (26) THS→HS = δ2(1+v2(δ2s−M2S+3λt−M2H+δ2u−M2S)), (27) TS′S′→HH = δ′2(1+3M2H1s−M2H+δ′2v2(1t−M2S+1u−M2S)), (28) TS′S′→SS = ka4+(δ2δ′2v2s−M2H), (29) THS′→HS′ = δ′2(1+v2(δ′2s−M2S′+3λt−M2H+δ′2u−M2S′)), (30) TS′S→S′S = ka4+(δ2δ′2v2t−M2H), (31) TS′S′→S′S′ = 6k′4+δ′2(δ′2v2s−M2H+δ′2v2t−M2H+δ2v2u−M2H). (32)

Now using Eq. (23), we have calculated the partial wave amplitude for each of the scattering processes and the coupled amplitude can be written as a matrix form,

 M=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝aHH→HH0aHH→SS0aHH→SH0aHH→S′S′0aHH→S′S0aHH→S′H0aSS→HH0aSS→SS0aSS→SH0aSS→S′S′0aSS→S′S0aSS→S′H0aSH→HH0aSH→SS0aSH→SH0aSH→S′S′0aSH→S′S0aSH→S′H0aS′S′→HH0aS′S′→SS0aS′S′→SH0aS′S′→S′S′0aS′S′→S′S0aS′S′→S′H0aS′S→HH0aS′S→SS0aS′S→SH0aS′S→S′S′0aS′S→S′S0aS′S→S′H0aS′H→HH0aS′H→SS0aS′H→SH0aS′H→S′S′0aS′H→S′S0aS′H→S′H0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⟶s≫M2H,M2S,M2S′116π⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝3λδ20δ′200δ26k40ka400002δ2000δ′2ka406k′4000000ka40000002δ′2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (33)

Requiring for each individual process above we obtain

 forHH→HH MH≤√8π3v, (34) forHS→HS and HH→SS |δ2|≤8π, (35) forSS→SS k4≤86π, (36) forS′S′→S′S′ k′4≤86π, (37) forS′S′→SS and S′S→S′S |ka4|≤8π, (38) forHS′→HS′ and HH→S′S′ |δ′2|≤8π. (39)

3.3 Triviality Bound

The requirement for ‘triviality bound’ on any model is guaranteed by one of the conditions that the renormalization group evolution should not push the quartic coupling constant of such models (say, ) to infinite value up to the ultraviolet cut-off scale of the model. This requires that Landau pole of the Higgs boson should be in higher scale than .

Therefore to check the triviality in our model, namely the two scalar singlet model with symmetry, we have to solve the renormalization group (RG) evolution equations for all the running parameters of this model. We have chosen only one-loop contribution in determining the beta functions for our model. The RG equations for the couplings in the model, namely, , , , , , , , are thus obtained at one-loop level as

 16π2dδ2dt = 4δ22+δ′2ka4+δ2(2γh+6k4+3λ), 16π2dδ′2dt = 4δ′22+δ2ka4+δ′2(2γh+6k′4+3λ), 16π2dλdt = 6λ2+4λγh−24y4t (42) +32(g41+2g21g22+3g42)+2δ22+2δ′22, 16π2dk2dt = 2m2δ2+6k2k4+ka4δ′2, 16π2dk′2dt = 2m2δ′2+6k′2k′4+ka4δ2, 16π2dk4dt = 18k24+12ka42+2δ22, (45)
 16π2dk′4dt = 18k′24+12ka42+2δ′22, 16π2dka4dt = 4ka42+6ka4(k4+k′4)+4δ2δ′2, (47)

where . In Eq. (3.3) Eq. (47) denotes the renormalization scale and is an arbitrary scale. Here and , , are , gauge couplings and top Yukawa coupling, respectively. In our calculation, the RG equations for gauge and top Yukawa couplings are also taken into account. We have taken the initial condition, for running of top Yukawa coupling, where and is strong coupling at the scale of [68]

We have solved all the RG equations given above and checked the consistency of all the quartic couplings within the suitably chosen scale of the theory. For the initialisation, we have taken corresponding to the recent value of Higgs mass, 126 GeV. The other initial values of parameters in our model should have been chosen from the allowed region of parameter space.

In Fig. 1 the variation of different parameters () with scale used in this model is shown. The benchmark point 4A of Table 4 has been chosen for assigning initial values in the evaluation of the running of various couplings. The variation of mass of each scalar (or ) with energy scale can be obtained from the plots as it is determined by couplings and (or and ). Although we have solved the RG equation for and , the influence of and on is very small as we can see from Eq. (42) that the RG equation of is deviated from the SM RG equation only by the almost smooth term . But the allowed region for ‘triviality bound’ for a given Higgs mass shrinks as the term, starts growing.

3.4 Constraints from Invisible Higgs Decay Width

If kinematically allowed, Higgs boson can decay to or . Such invisible decay channels are severely restricted by the present data from Large Hadron Collider (LHC). The branching fraction

 B(H→inv)=ΓinvΓSM+Γinv (48)

is bounded at 95% CL to be less than 19% by the global fits to the Higgs data keeping Higgs to fermion couplings fixed to their SM values. If such a restriction is lifted and additional particles are allowed in the loops the bound get relaxed to  [69]. denotes the SM Higgs decay width and is the invisible Higgs decay width, which in our model is given by [36],

 Γinv=v232πMH⎛⎜⎝δ22 ⎷1−4M2SM2H+δ′22 ⎷1−4M2S′M2H⎞⎟⎠. (49)

The benchmark point 4A in Table 4, consistent with the XENON 100 direct detection results, gives which at present is allowed at 95% CL [70, 71, 72, 73, 74]. However as we intend to interpret the low mass regions of dark matter claimed to be probed by several other dark matter direct search experiments (CDMS II, CRESST, CoGeNT etc.) along with indirect searches (low energy -ray from Fermi Bubble and Galactic Centre), in some cases and the Higgs boson decays invisibly to or , with a disfavoured by the LHC observations. This is a well known problem with all such models, where DM annihilation is mediated by the SM Higgs. It can be circumvented by adding extra degrees of freedom lighter than DM particles [75]. In that case it is possible to delineate the dominant Higgs-mediated annihilation channels from the diagrams contributing to the invisible Higgs decay. Such an elaborate model building issue is out of the scope of the present work as we feel given the non-observation of low mass DM events by XENON 100 and LUX, it is still premature to rely on CDMS II and other experiments betting on the low mass DM. For quantitative estimations we have chosen the Higgs to be the  GeV SM Higgs as a benchmark.

4 Calculation of Relic Abundance

In order to calculate the relic abundance for the dark matter candidates in the present formalism, we need to solve the relevant Boltzmann equations.

In presence of one singlet scalar , the Boltzmann eqn. is given by

 dnSdt+3HnS=−⟨σv⟩(n2S−n2Seq), (50)

where and are the number density and equilibrium number density of the singlet scalar, . is the thermal averaged annihilation cross-section of dark matter annihilating to SM particles.

Defining dimensionless quantities and , where is the total entropy density, Eq. (50) can be written in the form,

 dYdx=−(45πG)−1/2g1/2∗MSx2⟨σv⟩(Y2−Y2eq), (51)

where is the degrees of freedom. The relic density (value of at ) is obtained by integrating Eq. (51) from initial value, to final value, , where and are the present photon temperature (2.726 K) and freeze-out temperature respectively.

The relic density of a dark matter candidate, , in the units of critical energy density, , can be expressed as

 ΩS=MSnρcr=MSe0Y0ρcr, (52)

where is the present entropy density evaluated at . It follows that knowing , we can compute the relic density of the dark matter candidate from the relation [76],

 ΩSh2 = 2.755×108MS(in GeV)GeVY0. (53)

In Eq. (53) is the Hubble constant. The Planck survey provides the constraints on the dark matter density from precision measurements of anisotropy of cosmic microwave background radiation as

 0.1165<ΩDMh2<0.1227, (54)

consistent with the previous WMAP measurement .

In our model with two real scalars, both and contribute to the relic density. Their individual contributions can be obtained by solving the following coupled Boltzmann equations,

 dnSdt+3HnS = −⟨σv⟩SS→XX(n2S−n2Seq)−⟨σv⟩SS→