Journal of Physics G: Nuclear and Particle Physics PAPER Scalar dark matter candidates in a two inert Higgs doublet model To cite this article: E C F S Fortes et al 2015 J. Phys. G: Nucl. Part. Phys. 42 105003   View the article online for updates and enhancements. Related content Prediction of h Z from h at the LHC for the IMDS3 model E C F S Fortes, A C B Machado, J Montaño et al. - Two component dark matter: a possible explanation of 130 GeV -ray line from the galactic centre Anirban Biswas, Debasish Majumdar, Arunansu Sil et al. - Novel gamma-ray spectral features in the inert doublet model Camilo Garcia-Cely and Alejandro Ibarra - Recent citations Spontaneous symmetry breaking in three- Higgs-doublet S 3-symmetric models D Emmanuel-Costa et al - High-scale validity of a model with three Higgs doublets Nabarun Chakrabarty - Spontaneous symmetry breaking in the S 3-symmetric scalar sector D. 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Scalar dark matter candidates in a two inert Higgs doublet model E C F S Fortes, A C B Machado1, J Montaño and V Pleitez Instituto de Física Teórica–Universidade Estadual Paulista R. Dr. Bento Teobaldo Ferraz 271, Barra Funda São Paulo-SP, 01140–070, Brazil E-mail: elaine@ift.unesp.br, ana@ift.unesp.br, montano@ift.unesp.br and vicente@ift. unesp.br Received 24 February 2015, revised 21 July 2015 Accepted for publication 5 August 2015 Published 9 September 2015 Abstract We study a two scalar inert doublet model (IDMS3) which is stabilized by an S3 symmetry considering two scenarios: (a) two of the scalars in each charged sector are mass degenerated due to a residual 2 symmetry, (b) there is no mass degeneracy because of the introduction of soft terms that break the 2 symmetry. A major difference between this model and the one with only one inert Higgs doublet (IDM) is that it has twice the number of possible dark matter (DM) candidates. In particular we show that for the case (a) it is possible to have two CP even DM candidates. Besides, in case (b) the upper limit to the intermediate mass region for the one IDM in our model is larger, once we obtain a DM candidate with mass equals to 257 GeV. Keywords: dark matter, inert doublets, multi Higgs model (Some figures may appear in colour only in the online journal) 1. Introduction The existence of dark matter (DM) has been well established since the early astronomical [1] and cosmological observations [2–5]. For more recent data see [6]. It accounts for approx- imatively 23% of the composition of the Universe. Moreover, these observational evidence justify the experimental searches trying to find events that can be interpreted as direct manifestations of DM. Some of them are astronomical observations [7–11], and others like DAMA [12], CoGeNT [13], CDMS [14], XENON [15, 16], and LUX [17] are experiments trying to measure the recoil energy of nuclei if it scatters with the DM. Journal of Physics G: Nuclear and Particle Physics J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 (11pp) doi:10.1088/0954-3899/42/10/105003 1 Author to whom all correspondence should be addressed. 0954-3899/15/105003+11$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1 mailto:elaine@ift.unesp.br mailto:ana@ift.unesp.br mailto:montano@ift.unesp.br mailto:vicente@ift.unesp.br mailto:vicente@ift.unesp.br http://dx.doi.org/10.1088/0954-3899/42/10/105003 http://crossmark.crossref.org/dialog/?doi=10.1088/0954-3899/42/10/105003&domain=pdf&date_stamp=2015-09-09 http://crossmark.crossref.org/dialog/?doi=10.1088/0954-3899/42/10/105003&domain=pdf&date_stamp=2015-09-09 Models which contain DM candidates have to explain among other aspects, the DM density, which is h h 0.1196 0.00312 2 cr rW = =  where h is the scale factor for Hubble expansion [18], H G3 8c 0 2 ( )r p= is the critical density of the Universe, and H0 is the current value of the Hubble constant [6]. Much effort has been employed in order to discover or interpret DM signals. It is possible that it consists of one or more elementary particles which interact very weakly with ordinary matter. One of the most common scenarios are supersymmetric models [19]. In fact, in this kind of model, the lightest supersymmetric particle (neutralino) is prevented of interacting by the R parity with the known particles. The neutralino is an example of cold dark matter (CDM), i.e. a kind of DM which is not relativistic at the time of freeze out. Of course, there are other possibilities, for instance, Kaluza–Klein states in models with universal [20, 21] or warped [22] extra dimensions, stable states in little Higgs theories [23] and a number of models with extra heavy neutrinos. Some other alternative scenarios for DM consider self-interacting DM [24] and warm DM [25]. Other ambitious scenarios consider asymmetric DM models. They have their motivation based on the simi- larity of mass densities of the DM DM( )r and that of the visible matter B( )r observed 5BDMr r » and try to explain this rate. Consequently, most of these models are based on the hypothesis that the present abundance of DM and visible matter have the same origin [26, 27]. An additional and interesting scenario which contains DM candidates is the inert doublet model (IDM) [28–32]. It is a minimal extension of the standard model (SM) which contains a second Higgs doublet (H2) with no direct couplings to quarks and leptons and not contribute to the spontaneous symmetry braken. The first time that the phenomenology of an inert doublet was considered was in the context of neutrino physics [33] and also in the context of the problem of naturalness [34]. In all cases, the inert doublet was possible due to a 2 symmetry under which H H2 2 - and all the other fields are even. In particular, this discrete symmetry forbids interactions like H H H H1 2 2 2( )( )† † , being H1 the SM Higgs doublet, and prohibits fully the interactions with fermions. Nowadays astronomical observations suggest that the DM spectrum may be composed of several particles [35–37]. In this work, we study the three Higgs doublet model with an S3 symmetry, proposed in [38] in which, besides the SM like doublet there are two additional inert doublets here denoted by H2 and H3. It is this S3 symmetry and an appropriate vacuum alignment that allows us to obtain a model with two inert doublets. Although this model can accommodate four DM candidates, it is out of the scope of this paper to realize a complete study of all the possibilities for the DM scenarios. We will only analyze only two scenarios which already show the difference from the IDM model: scenario 1 in which the extra scalars are mass degenerated and the other, scenario 2, in which soft terms breaking a residual 2 symmetry, are added, resulting in non degenerated masses for this extra scalars. In scenario 1, we have two CP even DM components, each one corresponding to 50% of the DM relic density. Moreover, in scenario 2 we obtain a solution for the DM candidate with a mass which is out the intermediate region obtained for the IDM. The paper is organized as follows. In section 2 we briefly present the model. In section 3 we briefly describe the theoretical framework for the calculations for DM abundance and in the section 4 we show the parameter choices suitable for the DM candidate and the numerical results. Finally in the last section section, section 5, we summarize our conclusions. J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 E C F S Fortes et al 2 2. The model In the context of the SM the number of scalar doublets can be arbitrary. An interesting case occurs when the number of these fields is the same as the number of the fermion families, i.e. just three. In this case the S3 symmetry is, probably, the most interesting one because it is the minimal non-abelian discrete symmetry with one doublet and one singlet irreducible representations. The model that we will consider here has the three Higgs doublets transforming as 2, 1( )+ under SU U2 1L Y( ) ( )Ä and under S3 as S H D D D H H 1, , , 2. 1 1 1 2 2 3) ( )( ( ) = ~ = @ ~ The necessary conditions under which the vacuum alignment v v1 SM= (vSM is the SM VEV ∼246 GeV) and v v 02 3= = , allow a scalar potential bounded from below and a stable minimum as has been shown in [38]. With this vacuum alignment and, since the quarks and leptons are singlet of S3, the two Higgs doublet D D,1 2 do not couple to fermions and do not contribute to the spontaneous symmetry breakdown, i.e they are inert. They couple only to the gauge bosons and this vacuum alignment also implies in a residual 2 symmetry in which the two inert doublets are mass degenerate in each charged sector. In this case the mass spectra is m v m m v m m v m m v , 1 2 , 1 2 , 1 4 2 , 2 h H H d A A d h h d 2 4 SM 2 2 2 2 SM 2 2 2 2 SM 2 2 2 2 5 SM 2 2 0 3 0 2 3 2 3 ( ) ( ) l m l m l m l = = = + ¢ = = +  = = ++ + where mh 2 is the SM-like Higgs boson (here denoted by h), 0d 2m > came from the term D Dd 2 1[ ]†m in the scalar potential, where 25 6 7l l l l¢ = + + and 25 6 7l l l l = + - , with 4,5,6,7l are quartic coupling constants in the scalar potential. We call this scenario 1. If the residual 2 symmetry is softly broken by adding non-diagonal quadratic terms in the inert sector, the mass degeneracy is broken and the mass spectra becomes m m m m m m m m m m m m , , , , , , 3 H H H H A A A A h h h h 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2 0 3 0 3 0 2 2 3 3 2 2 2 2 ( ) n n n n n n = - = + = - = + = - = ++ + + + and we call this scenario 2. In the case of mass degenerate scalars, the lightest scalars can be DM candidates and we will choose the CP even ones. In the case of no mass degeneracy it is possible that the lightest one is the DM candidate or we can have up to four DM components. For the scenario 1, our parameter choice enables us to establish the follow order for the mass of the scalars: m m mA h H2,3 2,3 2,3 0> >+ . Since H2,3 0 are the lightest neutral scalars, their decays are kinematically forbidden. With the rearrangement of the parameters, instead of choosing H2,3 0 , we could choose the CP odd scalars A2,3 as the DM candidates, if they were the lightest ones, and the same conclusions would remain valid for this scenario. In scenario 2, H2 0 accounts for all the hDM 2W contribution. For each scenario, with the couplings shown in table 1, we choose two set of parameters for scenario 1 and five sets of parameters for scenario 2, as is shown in table 2. J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 E C F S Fortes et al 3 3. DM abundance Preliminary analysis showing that this model can accommodate at least one DM candidate was done in [38]. Here, this will be confirmed by a more detailed analysis. In order to calculate the DM abundance we have used the MicrOMEGAs package to solve numerically the Boltzmann equation after implementing all the interactions of the model in the CalcHEP package [39]. The Feynman diagrams for the annihilations process for this model are given in figure 1. Let us consider for instance, the model of inert doublets with non degenerated mass (scenario 2). In this case, as we already said in section 2, H2 0 is our DM candidate. The evolution of the numerical density n of H2 0, at the temperature T in the early Universe, is given by the Boltzmann equation, which is written in simplified form as follows [40]: Y y g G m y v Y Y d d 45 , 4 H 2 ann 2 eq 22 0 ( ) ( )* p s= - - here Y n s= , s is the entropy per unity of volume, Yeq is the Y value in the thermal equilibrium, y m TH2 0= . The parameter G in equation (4) is the Newton gravitational constant, anns is the cross section for annihilation of the particle H2 0 and v is the relative velocity, and the symbol represents thermal average. Finally, g* is a parameter that measures the effective number of degrees of freedom at freeze-out, which is expressed as g g T T g T T 7 8 , 5 i i i i i i bosons 4 fermions 4 ( ) * å å= + = = ⎜ ⎟ ⎜ ⎟⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ Table 1. Interactions and the corresponding couplings Interaction Coupling H H h2 0 2 0 , H H h3 0 3 0 vSMl¢ H H hh2 0 2 0 , H H hh3 0 3 0 l¢ H H W W2 0 2 0 + -, H H W W3 0 3 0 + - g 2 2 H H ZZ2 0 2 0 , H H ZZ3 0 3 0 cos 1 tang 2 2 2 W W 2 ( )+q q H H h h2 0 2 0 2 2 + -, H H h h2 0 2 0 3 3 + - − 1 3( )l l+ H H h h3 0 3 0 2 2 + -, H H h h3 0 3 0 3 3 + - − 1 3( )l l+ H H h2 0 2 2 + - 0 H H h2 0 3 3 + - 0 H A Z2 0 2 0 , H A Z3 0 3 0 g cos 1 tan 2W W- q + q H h W2 0 2 + - i g 2 H h W3 0 3 + - i g 2 J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 E C F S Fortes et al 4 Table 2. Parameters choice for scenario 1 and 2 with m 125h = GeV. The other masses units are in GeV, vs is in units of10 cm s26 3 1- - and the units for SIs are in cm .2 The parameters 0.34l = and 0.45l = for scenarios 1a,1b, 2a, 2b, 0.35l = - for scenario 2c, 0.25l = for 2d, finally for 2e we used 1.35l = and 0.2l = . Parameter Scenario 1a Scenario 1b Scenario 2a Scenario 2b Scenario 2c Scenario 2d Scenario 2e mH2 0 54.12 79.77 63.46 59.01 161.80 130.20 256.7 mH3 0 54.12 79.77 76.20 72.52 167.21 142.94 263.67 mA2 0 112.44 127.98 117.19 117.22 190.89 163.30 263.64 mA3 0 112.44 127.98 131.22 131.22 199.79 173.62 270.47 mh2 + 85.02 95.38 83.11 83.12 87.56 101.71 224.69 mh3 + 85.02 95.38 101.91 101.91 105.56 117.57 232.66 dm 48.52 78.05 72.05 72.05 167 134.6 255.5 ν − − 41.7 41.7 41.7 41.7 42.7 l¢ 0.019 0.009 0.019 0.001 0.001 0.019 0.08 W 0.12 0.12 0.118 0.119 0.12 0.11 0.11 vs 0.0831 0.003 5 0.0012 0.99 0.841 0.317 proton SIs 7.33 10 46´ - 7.46 10 47´ - 5.30 10 46´ - 1.7 10 48´ - 2.18 10 49´ - 1.23 10 46´ - 5.28 10 46´ - neutron SIs 8.38 10 46´ - 8.55 10 47´ - 6.06 10 46´ - 1.9 10 48´ - 2.50 10 49´ - 1.41 10 46´ - 6.07 10 46´ - J. P hys. G : N ucl. P art. P hys. 42 (2015) 105003 E C F S F ortes et al 5 where the sums runs over only those species with mass m TH2 0  [41]. The model studied here has, besides the SM particles, eight extra scalars (A A H H h h, , , , ,2 0 3 0 2 0 3 0 2 3  ). So, considering, for instance T 300 GeV we obtain g 114.75 * » . To find Y0, the present value of Y, equation (4) must be integrated between y = 0 and y m TH0 02 0= . Once this value is found, the contribution of H2 0 to DM density is m s Y . 6h H 0 0 c 2 2 0 ( ) r W = The same calculations hold for the scenario 1, however in this scenario we consider that the DM has 50% of each component H H,2 0 3 0( ). 4. Results and comments The main numerical results for this model using the parameters presented in table 2. For more detail about the interactions of the model see [42]. For both scenarios (1 and 2), we have considered some set of parameters, so we call these scenarios respectively scenario 1a, 1b and scenario 2a, 2b, 2c, 2d and 2e. For scenario 1a, the dominant annihilation channels are: 39% relative to H H bb3 0 3 0  , 39% to H H bb2 0 2 0  , 5% to H H GG3 0 3 0  , 5% to H H GG2 0 2 0  , 4% to H H3 0 3 0 t t + -, 4% to H H2 0 2 0 t t + -, 2% to H H cc3 0 3 0  and 2% due to H H cc2 0 2 0  . The contributions of the two candidates (H H,3 0 2 0) to the Higgs invisible decay is 34.8%. The Higgs invisible decay depends strongly on the parameter l¢. In scenario 1, another choice of parameters which Figure 1. Feynman diagrams for lowest order of two Hi which annihilate into a pair of fermions and anti-fermions, W W+ -, ZZ and Higgs. J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 E C F S Fortes et al 6 brings null contributions to this invisible decay is reached with the numbers presented in scenario 1b. The dominant annihilation channels are in this case 50% relative to H H W W3 0 3 0  + - and 50% relative to H H W W2 0 2 0  + -. Next we consider scenario 2, in which H2 0 is the DM candidate. In scenario 2a, the dominant annihilation channels are respectively 77% relative to H H bb2 0 2 0  , 11% to H H GG2 0 2 0  , 8% to H H2 0 2 0 t t + - and 3% due to H H cc2 0 2 0  . In this scenario, H2 0 does not contribute to the Higgs invisible decay. In scenario 2b, the dominant annihilation channels are 78% relative to H H bb2 0 2 0  , 10% relative to H H GG2 0 2 0  , 8% relative to H H2 0 2 0 t t + - and 3% relative to H H cc2 0 2 0  . In scenario 2c, the dominant annihilation channel is 100% due to H H W W2 0 2 0  + -. In scenario 2d, the dominant annihilation channels are respectively 97% relative to W W+ - and 3% relative to hh, finally, in scenario 2e, the dominant annihilation channels are 81% em a a,2 2, 13% em tt e 6% em W W+ -. For all the scenarios discussed above, the cold DM-nucleons amplitudes are in agreement with CoGent, DAMA, LUX, XENON100. The scenario 2b and 2c are in agreement with the predictions of XENON1T for SIs . In this scenario, a negative 5l favors mainly the Higgs decay into two neutral gauge bosons [42]. Due to the smallness of 0.001l¢ = , in scenario 2b the branching h H H2 0 2 0 is negligible ( 5 10 4» ´ - ), since this Higgs decay is very sensible to this parameter. Figure 2 shows the data presented in table 2 compared to the experimental results for SIs considered in the experiments CoGent, DAMA, LUX, XENON100 and XENON1T. In order to adjust the DM solutions it is necessary to fix some parameters. Oncel¢,l and 5l are already set, we can determine the values for 6l and 7l using the relations: 25 6 7l l l l¢ = + + and 25 6 7l l l l = + - . The numerical results for each scenario are given in table 3. The figure 3 shows the allowed region for the scalar potential couplings 1l and 3l . In order to obtain this region we have considered the relations between the scalar potential coupling constants, which ensures that the scalar potential is bounded from below, for more detail see [38], namely: 0, 0, 0. 7 1 3 5 4 1 3 4 1 3 ( ) ( ) ( )    l l l l l l l l l l + + +  + + Figure 2. Limits for SIs according to the experiments CoGent, DAMA, XENON100, XENON1T and LUX. The points X X X, ,1a 1b 2a , X X X, ,2b 2c 2d and X2e are the ones which refer to scenarios 1a, 1b, 2a, 2b, 2c, 2d and 2e given in Table 2. J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 E C F S Fortes et al 7 Table 3. Used parameters of table 2 considering 2 and 46 5 7l l l l l l l= ¢ +  - = ¢ - . Parameter Scenario 1a Scenario 1b Scenario 2a Scenario 2b Scenario 2c Scenario 2d Scenario 2e 6l −0.223 −0.223 −0.223 0.471 0.041 −0.02 −1.16 7l −0.080 −0.083 −0.083 −0.085 −0.085 −0.080 −0.03 J. P hys. G : N ucl. P art. P hys. 42 (2015) 105003 E C F S F ortes et al 8 We have considered the parameter choice of table 2 and the above restrictions. The allowed regions are given in figure 3(a) for scenarios 1a, 1b, 2a and 2b, and in figure 3(b) for scenario 2c. The remaining free parameters now are 2l and 8l . We should mention that 2l is the quartic term coupling in interactions between inert scalars. 5. Conclusion Here we have considered a two IDM with an S3 symmetry. The model has, besides the SM particles, eight scalars bosons which are inert, i.e. they do not contribute to the spontaneous electroweak symmetry breaking. They interact only among themselves and with the gauge bosons through trilinear and quartic interactions, here only the latter one is important. We have considered here the simplest cases. On one hand, in the case of degenerated masses Figure 3. Using equation (7) and the values in table 2 we obtain the allowed region for the scalar potencial couplings and1 3l l . In (a) for scenarios 1a, 1b, 2a and 2b. In (b) for scenarios 2c, in (c) for 2d and in (d) for 2e. J. Phys. G: Nucl. Part. Phys. 42 (2015) 105003 E C F S Fortes et al 9 (scenario 1), the two real neutral scalars play the role of DM candidates each one with a 50% contribution to the relic density. On the other hand, in the case of non-degenerated masses (scenario 2), one of the neutral scalars is the DM candidate. It is well known that in the IDM there are three allowed regions of masses that are compatible with observed value of hDM 2W : (i) 1 GeV mH 0  10 GeV; (ii) 40–160 GeV, and (iii) 500 GeV mH 0  10 TeV [43, 44]. However, we have obtain a solution with mass equal to 257 GeV, see point 2e in figure 2. Moreover, from table 2 and figure 2 we can see that the spin-independent elastic cross section, SIs , is in good agreement with the results of experiments LUX and XENON100 for the mass range of DM considered here. 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Phys. 42 (2015) 105003 E C F S Fortes et al 11 http://dx.doi.org/10.1142/S0217751X13300287 http://dx.doi.org/10.1088/1126-6708/2009/07/090 http://dx.doi.org/10.1088/1126-6708/2009/07/090 http://dx.doi.org/10.1007/JHEP05(2010)066 http://dx.doi.org/10.1007/JHEP05(2010)066 http://dx.doi.org/10.1007/JHEP09(2010)046 http://dx.doi.org/10.1103/PhysRevD.86.075019 http://dx.doi.org/10.1103/PhysRevD.73.077301 http://dx.doi.org/10.1103/PhysRevD.74.015007 http://dx.doi.org/10.1143/PTP.126.855 http://dx.doi.org/10.1088/1475-7516/2013/12/049 http://dx.doi.org/10.1088/1475-7516/2013/12/049 http://arXiv.org/abs/1205.0995 http://dx.doi.org/10.1016/j.cpc.2013.01.014 http://dx.doi.org/10.1007/JHEP09(2013)055 http://dx.doi.org/10.1007/JHEP09(2013)055 http://arXiv.org/abs/1106.1719 1. Introduction 2. The model 3. DM abundance 4. Results and comments 5. Conclusion Acknowledgments References