Relic density in nonuniversal gaugino mass models with SO(10) GUT symmetry
aa r X i v : . [ h e p - ph ] A p r HIP-2008-41/THDO-TH-08/09
Relic density in nonuniversal gaugino mass models with SO(10) GUT symmetry
Katri Huitu ∗ Department of Physics, and Helsinki Institute of Physics,P.O. Box 64, FIN-00014 University of Helsinki, Finland
Jari Laamanen † Institut f¨ur Physik, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany (Dated: November 8, 2018)Nonuniversal boundary conditions in grand unified theories can lead to nonuniversal gauginomasses at the unification scale. In R -parity preserving theories the lightest supersymmetric particleis a natural candidate for the dark matter. The composition of the lightest neutralino and theidentity of the next-to-lightest supersymmetric particle are studied, when nonuniversal gauginomasses come from representations of SO(10). In these cases, the thermal relic density compatiblewith the Wilkinson Microwave Anisotropy Probe observations is found. Relic densities are comparedwith the universal case. Mass spectra in the studied cases are discussed. PACS numbers: 12.60.Jv, 95.35.+dKeywords: Gaugino masses, relic density, dark matter
I. INTRODUCTION
The phenomenology of supersymmetric models de-pends crucially on the compositions of neutralinos andcharginos, if the lightest neutralino is the lightest super-symmetric particle (LSP). In addition to the laboratorystudies, relevant input is obtained from the dark mattersearches, where the the Wilkinson Microwave AnisotropyProbe (
WMAP ) satellite has put precise limits on therelic density. Supersymmetric theories which preserve R -parity contain a natural candidate for the cold darkmatter particle. Neutralino LSP can provide the appro-priate relic density.In many supergravity type models the lightest neu-tralino is binolike, which often leads to too large thermalrelic density, as compared to the limits provided by the WMAP experiment [1]. When the gaugino masses arenot universal at the grand unification scale, the resultingneutralino composition changes from the case of univer-sal gaugino masses [2]. In this paper, the thermal relicdensity of the neutralino LSP is studied, when gauginomasses are due to nonuniversal representations of SO(10)grand unified theory (GUT) [3, 4]. Dark matter in aparticular gauge symmetry breaking chain of the SO(10)GUT in the case of universal gaugino masses has beenrecently studied in [5]. Some phenomenological aspectsof SO(10) GUTs with nonuniversal gaugino masses havebeen considered in [6, 7].SO(10) has many attractive features among the GUTmodels. One of the most appealing properties is thatone family of matter fermions can be put into a sin-gle 16-dimensional irreducible spinor representation ofSO(10), including the right-handed neutrino [8, 9]. In ∗ Electronic address: katri.huitu@helsinki.fi † Electronic address: [email protected] addition, SO(10) allows possibility for the Yukawa cou-pling unification and representations are anomaly free.Conservation of R -parity, which forbids the unwanteddimension-five operators leading to rapid proton decay,may result from the SO(10) symmetry breaking. Thedoublet-triplet splitting could be achieved using, e.g. ,the so-called Dimopoulos-Wilczek mechanism [10]. Be-cause the SO(10) gauge symmetry breaks down to thestandard model (SM) gauge symmetry through some in-termediate group, the SO(10) GUT offers several possi-bilities for the model building. For example, it can con-tain as a subgroup the Pati-Salam SU(4) × SU(2) × SU(2)model.Gaugino masses originate from the non-renormalizableterms in the N = 1 supergravity Lagrangian involvingthe gauge kinetic function f ab (Φ) [11]. The gauge part ofthe Lagrangian contains the gauge kinetic function cou-pling with two field strength superfields W a . The La-grangian for the coupling can be written as L gk = Z d θf ab (Φ) W a W b + H.C. , (1)where a and b are gauge group indices (for example, a, b = 1 , , ...,
45 for SO(10)), and repeated indices aresummed over. The function f ab (Φ) is an analytic func-tion of the chiral superfields Φ in the theory. The chiralsuperfields Φ consist of a set of gauge singlet superfieldsΦ s and gauge nonsinglet superfields Φ n under the grandunified group. The gauge kinetic function f ab (Φ) can beexpanded, f ab (Φ) = f (Φ s ) δ ab + X n f n (Φ s ) Φ nab M P + · · · , (2)where Φ s and Φ n are the singlet and nonsinglet chiral su-perfields, respectively. Here f (Φ s ) and f n (Φ s ) are func-tions of gauge singlet superfields Φ s , and M P is somelarge scale. In order to generate a mass term for the gaug-inos, the gauge kinetic function must be non-minimal,i.e., it must not be a constant [12]. When F Φ gets avacuum expectation value ( vev ) h F Φ i , the interaction (1)gives rise to gaugino masses: L gk ⊃ h F Φ i ab M P λ a λ b + H.C. , (3)where λ a,b are gaugino fields. The nonuniversal gauginomasses are generated by the nonsinglet chiral superfieldΦ n that appears linearly in the gauge kinetic function f ab (Φ) in Eq. (2).Gauginos belong to the adjoint representation of thegauge group, which in the case of SO(10) is the di-mensional representation. Because Eq. (3) must be gaugeinvariant, Φ and F Φ must belong to some of the fol-lowing representations appearing in the symmetric prod-uct of the two dimensional representations of SO(10)[13, 14]:( ⊗ ) Symm = ⊕ ⊕ ⊕ . (4)The representations , and may lead tononuniversal gaugino masses, while the dimensionalrepresentation gives manifestly the universal gauginomasses. The relations between the gaugino masses aredetermined by the representation invariants, and are spe-cific for each of the representations. Because the gaugekinetic function in Eq. (2) can get contributions fromseveral different Φ’s, a linear combination of any of therepresentations is also possible. In that case the gaug-ino mass terms are not uniquely determined anymore, incontrast to the contribution purely from one representa-tion. Here we assume that the dominant component ofthe gaugino masses comes from only one representation.This gives us a clear understanding of the role of differentrepresentations. II. DARK MATTER IN SO(10)REPRESENTATIONSA. Breaking Chains: SO(10) → H → SM The GUT group SO(10) breaks down to the standardmodel gauge group SU(3) × SU(2) × U(1) via some inter-mediate gauge group H . Therefore the gaugino massrelations depend also on the gauge group breaking chain,in addition to the representation invariants coming fromthe gauge kinetic function. Moreover, the intermediatebreaking scale affects also the generated gaugino massesvia heavy gauge supermultiplets that correspond to thebroken generators. However, if the gauge breaking fromthe GUT group to the SM group takes place at the GUTscale, these loop-induced messenger contributions [15]can be neglected in comparison to the tree-level contri-butions. Some fits to the experimental data in SO(10)GUT indicate that the two breaking scales are very closeto each other, see [16, 17], although realistic models ex-ist also with large splitting of the scales [18]. In this work we assume that the breaking from SO(10) to theSM gauge group happens at the GUT scale, and that theGUT breaking does not affect the gauge coupling unifi-cation.We will study the representations and in theright-hand side of Eq. (4). The interesting breakingchains of and are included also in the breakingchains of . Table I shows possible SO(10) breakingchains [14, 19, 20], which include the standard modelgauge group, for the two chosen representations. Someof the subgroups lead to universal gaugino masses, orto massless gauginos [14], and we do not consider them.We will limit ourselves to the intermediate gauge groupsSU(4) × SU(2) × SU(2), SU(2) × SO(7) and SU(5) × U(1).TABLE I: Breaking chains of SO(10) representations and which include the SM gauge group. F Φ H Subgroup description SU(4) × SU(2) × SU(2)
Pati-Salam
SU(2) × SO(7)SO(9)
Universal gauginos
SU(4) × SU(2) × SU(2)
Massless gluino
SU(3) × SU(2) × SU(2) × U(1)
Massless
SU(2) L gauginos SU(3) × SU(2) × U(1) × U(1)SU(5) × U(1) “Flipped”
SU(5)
Table II displays the ratios of resulting gaugino massesat the tree level as they arise when F Φ belongs to theabove-mentioned representations of SO(10) or singlet[14]. The resulting 1-loop relations at the electroweakscale are also displayed. These values and the resultingTABLE II: Ratios of the gaugino masses at the GUTscale in the normalization M ( GU T ) = 1, and at theelectroweak scale in the normalization M ( EW ) = 1. F Φ H M G1 M G2 M G3 M EW M EW M EW .
14 0 .
29 1 SU(4) × SU(2) × SU(2) -1 -1.5 1 − . − .
44 1 SU(2) × SO(7) . − .
68 1
SU(5) × U(1) -96/25 1 1 − .
56 0 .
29 1 relic densities can be compared with the universal andnonuniversal representations resulting in the SU(5) GUTmodel [21, 22]. Since we assume breaking at one scale,the universal model with which we should compare inthe SO(10) GUT is similar to the universal model in theSU(5) GUT. In the nonuniversal representations, the re-lations between gaugino masses change. Thus, e.g. , the -dimensional Pati-Salam model of SO(10) may seemat first glance rather similar to of SU(5), but we willsee that the twice as large bino component has a largeeffect to the relic density. The bino and wino mass pa-rameters affect directly the lightest neutralino mass andproperties. They also affect the value of the µ -parameterthrough the renormalization group equations (RGE) andthe radiative electroweak symmetry breaking (rEWSB),therefore controlling also the Higgsino component in thelightest neutralino. Since the lightest neutralino masslimit can be deduced from the chargino mass limit, thenonuniversal gaugino masses change the lower limit forthe neutralino mass: for the neutralino mass limit issmaller than in the universal case, while for the masslimit is close to the chargino mass limit. B. Calculation of Dark Matter Relic Density
We calculate the
SUSY spectrum for each model withthe program S OFTSUSY (version 2.0.11) [23], and theresulting relic density with the program m icrOMEGAs(version 2.0.7) [24, 25, 26]. For the relic density, we usehere the WMAP combined 3 yr limits [1]Ω
CDM h = 0 . +0 . − . (2 σ ) . (5)In all the figures that we show below, the filling denotedby wmap is the WMAP -preferred region. For the b → sγ experimental branching fraction, we have used the twosigma world average [27], BR ( b → sγ ) = (355 ± +9 − ± × − . (6)The areas enclosed by the bsg contour are disallowed bythe b → sγ constraint. For the particle masses, the fol-lowing limits are applied [26]: m ˜ e R > . m ˜ µ R >
95 GeV, m ˜ τ > . m ˜ ν i >
43 GeV, and m ˜ χ ± > . lep shows an area where the experimentalmass limits are not met, rge shows an area where thereis no radiative EWSB , and lsp the area where neutralinois not the LSP. The curve m h = 114 GeV is depictedin the figures (dash-dotted line denoted by h ). For theshown parameter regions, when otherwise experimentallyallowed, Higgs is always heavier than 91 GeV, which isthe Higgs mass limit in MSSM for tan β ≥
10 assumingmaximal top mixing [28].
1. Representation 54
The area of preferred thermal relic density for the twochains of the dimensional representation are shown forsets of parameters in Figs. 1, 2, and 3. In each set of threefigures, the first figure (a) represents the neutralino relicdensity for given parameters with collider constraints de-picted in the plot, the second figure (b) shows, for thesame parameters, the identity of the next-to-lightest su-persymmetric particle, and the third figure (c) shows thelightest neutralino composition in RGB-color encoding,( i.e. , colors, or hues of black and white, indicate the par-ticle as shown in the figure; therefore the mixture of the colors, or hues of black and white, describes the nature ofthe e χ -composition). In each of the figures, the WMAP -preferred relic density filling is also superimposed to thegraph.As can be seen from the Table II, the lightest neu-tralino is expected to be bino rather than wino. Thelarge bino component tends to suppress the neutralinoannihilation cross section, since bino lacks the s-channel Z -boson annihilation mode. A substantial Higgsino com-ponent is usually needed to help to increase the annihi-lation rate, unless there happens to be coannihilation or, e.g. , an open Higgs s-channel annihilation mode avail-able.In Fig. 1 relic density, the next-to-lightest supersym-metric particle (NLSP) and LSP composition in thebreaking chain SU(2) × SO(7) are shown. Because e χ ismostly bino, the spectrum with preferred relic density isquite light and conflicts with collider constraints in someparts of the parameter space. With increasing gauginomasses also the Higgsino component in the neutralinoLSP increases, and at the point where the change to dom-inantly Higgsino LSP occurs, also the relic density drops.The overall relic density is not very high, thus allowing awider WMAP -preferred region than, e.g. , in the singlet, i.e. , mSUGRA case [22]. For a given M , the correspond-ing M is smaller than in the singlet case, which resultsin a smaller µ value at the EW-scale. This has an effectof an increasing the Higgsino component in the lightestneutralino thus boosting the annihilation. The NLSPis chargino, and with increasing Higgsino component iteventually becomes the LSP.In the allowed region, M is less than 740 GeV, whichrestricts the lightest chargino mass to values less than ∼
150 GeV. The lower limit for the chargino mass is theLEP limit. The partners of the SM fermions are heavierthan 300–500 GeV. Thus, assuming that the neutralinois responsible for the dark matter, this breaking chainof the gauge symmetry has as a robust prediction forthe upper limit of the chargino mass, and furthermoreit is lighter than the squarks and sleptons. For M ∼
350 GeV the whole neutralino and chargino spectrum,and even the gluino, is lighter than the sfermions for theLEP allowed and
WMAP -preferred region. The same istrue also for M ∼
630 GeV and m > ∼
940 GeV for the
WMAP -preferred region.The b → sγ constraint cuts away a considerable areafrom the otherwise allowed region. Including a 10% errorin theoretical calculations of the decay, the constraintsfrom b → sγ loosen considerably, and all of the other-wise allowed WMAP area becomes available. The lightestHiggs is always lighter than 114 GeV, but heavier than91 GeV.The effect of varying the universal trilinear coupling A is shown in Fig. 2. In contrast to the previous figure, herethe sign of the µ -parameter is chosen to be negative. Ingeneral, giving the A -parameter a nonzero value tendsto increase the relic density. Increasing | A | will helpthe m H u to run larger negative values during the RG-
200 400 600 800 1000 1200 1400 250 500 750 1000 1250 1500lepwmaplsprgeh 0 0.2 0.4 0.6
PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (a) Relic density
200 400 600 800 1000 1200 1400 250 500 750 1000 1250 1500wmap
PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (b) NLSP map s 250 500 750 1000 1250 1500 200 400 600 800 1000 1200 1400 wmap PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (c) Neutralino composition FIG. 1: Relic density Ω χ h in the representation with H =SU(2) × SO(7) in the ( M , m ) plane fortan β = 10 , sgn( µ ) = +1 , A = 0. In 1a the dark shaded areas represent the larger relic density. The filling denotedby wmap is the WMAP preferred region, lep shows an area, where the experimental mass limits are not met, rge shows an area where there is no radiative
EWSB , and lsp the area where neutralino is not the LSP. h encloses thearea with m h <
114 GeV, and in the following figures bsg the area disallowed by b → sγ limits. In 1b the NLSPwithin the same region is plotted, and in 1c the e χ composition.evolution, and therefore to increase the actual (absolute)value for the µ -parameter via the rEWSB. This in turnfavors the bino component in the e χ composition in theSU(2) × SO(7) chain. The effect can be seen in the LSP-composition Fig. 2c. The preferred
WMAP region stillfollows the transition zone of the e χ from bino to Hig-gsino. The effect of negative µ is most visible in the factthat the NSLP in this case is e χ , since negative µ tends toincrease the lightest chargino mass (at least in the limitof | M | < | µ | ). The b → sγ constrains an area with neg-ative A values at M < e χ compositionare plotted for the SU(4) × SU(2) × SU(2) breaking chain.In this breaking chain the | M | and | M | values are closerto each other at the EW-scale than in the SU(2) × SO(7)chain, but still wider spread than in the singlet. Thathas an effect of increasing the µ value, therefore result-ing in a smaller Higgsino component for the equal M values for the two chains of the dimensional represen-tation. Again the WMAP allowed narrow region followsthe transition from the e χ from bino to Higgsino. The b → sγ and Higgs boson 114 GeV limits cut pieces fromnear the low M , m -values. The NLSP is mostly thelightest chargino, but an interesting region exists withsmall m near the area, where e χ is no longer the LSP;along the line of transition from the stau NLSP to thesmuon NLSP there is a narrow region, where e χ , stau andsmuon masses are very close to each other, and the coan-nihilations may reduce the relic density to an acceptablelevel. For example, for M = 1400 GeV, the LSP mass is around 410 GeV, and for M = 1000 GeV, the LSP massis 290 GeV for the WMAP -preferred area. However, thisarea is highly prone to numerical subtleties, and the or-dering of the LSP identity changes in the preferred relicdensity range, when comparing the output of differentspectrum calculators. Reducing the m parameter fur-ther makes the stau become the LSP. From the colliderpoint of view, such regions may be especially interesting,as they would lead to quasistable smuons or staus.
2. Representation 210
In the representation we inspected the break-ing chain through the intermediate gauge groupSU(5) × U(1), called flipped SU(5) [29, 30, 31, 32]. InFig. 4a the area of preferred thermal relic density in therepresentation is plotted for a set of (GUT scale)parameters. For the chosen parameters, rather large
WMAP -preferred regions are found for large values of M and/or m parameters. When not Higgsino, the lightestneutralino is expected to be wino, rather than bino (seeTable II), and therefore the neutralino relic density tobe very small. In general, due to the wino being thesmallest of the two electroweak gaugino parameters, itcharacterizes the lightest neutralino. Since the lightestchargino is characterized also by this parameter, for alarge part of the parameter space, the masses of the e χ and e χ ± are very close to each other, which boosts therapid neutralino relic density annihilation. The situationresembles the one arising in the anomaly mediated SUSY breaking scenario, where also both the lightest neutralinoand chargino are characterized by the wino mass param- -6000-4000-2000 0 2000 4000 6000 500 1000 1500 2000 2500 3000lepwmaplsprgebsgh 0 10 20 30 40
PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (a) Relic density -6000-4000-2000 0 2000 4000 6000 100 500 1000 1500 2000 2500 3000wmap PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (b) NLSP map s 100 500 1000 1500 2000 2500 3000-6000-4000-2000 0 2000 4000 6000 wmap PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (c) Neutralino composition FIG. 2: Representation : H =SU(2) × SO(7), tan β = 10 , sgn( µ ) = − , m = 1 TeV. Otherwise as in Fig. 1. PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (a) Relic density PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (b) NLSP map s 100 1000 2000 3000 4000 5000 6000 100 1000 2000 3000 4000 5000 6000 wmap PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (c) Neutralino composition FIG. 3: Representation with H =SU(4) × SU(2) × SU(2) in the ( M , m ) plane fortan β = 10 , sgn( µ ) = +1 , A = 0. Otherwise as in Fig. 1.eter.Since both the wino and Higgsino have a large anni-hilation cross section, the WMAP -preferred relic densityregion does not have to follow the transition zone of e χ from wino to Higgsino. The increase of the relic den-sity to the observed level is mainly due to the increase ofthe mass parameters M and m . An interesting changein the pattern can be seen on the diagonal of the fig-ures, where the e χ and e χ ± , and also at some point e χ ,masses are very close to each other leading to enhancedcoannihilation through the processes e χ e χ ± → q u q d and e χ e χ , e χ ± e χ ± → qq, ℓℓ, W + W − , which allows the accept-able parameter space to extend to very large values of M and m .The WMAP -preferred region is very wide as comparedto, e.g. , the universal model or other models in SU(5)[22, 33]. The spectrum is relatively heavy for the
WMAP -preferred region, around a couple of TeV. This leads to a wide range in the parameter space, but from the pointof view of the coming Large Hadron Collider such a massspectrum may be problematic. However, the represen-tation produces naturally a neutralino with massaround a TeV, which seems to be favorable in the viewof the recent PAMELA [34] and ATIC [35] results of theexcess positron and positron+electron flux. Under cer-tain circumstances, a nearby clump of 600-1000 GeV neu-tralino LSP could fit into these observations [36].In Fig. 5 the area of preferred thermal relic densityin the representation is plotted for the same set of(GUT scale) parameters as in Fig. 4, except that nowthe trilinear A -parameter is varied along the y-axis, andthe m is set to 1 TeV. The interesting feature in thisfigure is the existence of the pseudoscalar Higgs anni-hilation channel through the M -values. This reducesgreatly the relic density in the parameter space where theLSP mass equals or is less than half of the A -Higgs mass. PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (a) Relic density
100 1000 2000 3000 4000 5000 6000 100 1000 2000 3000 4000 5000wmap
PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (b) NLSP map s 100 1000 2000 3000 4000 5000 100 1000 2000 3000 4000 5000 6000 wmap PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H (c) Neutralino composition FIG. 4: Representation : tan β = 10 , sgn( µ ) = +1 , A = 0. Otherwise as in Fig. 1. -6000-5000-4000-3000-2000-1000 0 2000 2200 2400 2600 2800 3000wmap 0 0.1 0.2 0.3 PSfrag replacements m M A ˜ t ˜ τ ˜ µ e χ ± e χ e χ rge e B f W e H FIG. 5: Relic density in
Rep 210 :tan β = 10 , sgn( µ ) = +1 , m = 1 TeV. The colliderconstraints are fulfilled.This has the effect of pushing the WMAP -preferred relicdensity region to heavier neutralino masses, and there-fore to larger M values. The width of the WMAP re-gion is naturally the same as before, since the top ofthe figure, where A = 0, coincides with Fig. 4 with m = 1000 GeV. With these parameters, the NLSP isalways the lighter chargino, and the lightest neutralino isdominantly a wino, which can also be read from Fig. 4c. III. DISCUSSION AND SUMMARY
We studied the dark matter allowed regions in theSO(10) GUT representations, of which all but the singletmay lead to nonuniversal gaugino masses. The
WMAP -preferred relic density regions are quite distinct for differ-ent representations, thus leading to quite different parti-cle spectra for each representation. In the representation , the lightest neutralino ispredominantly a bino, leading to the narrow areas ofthe WMAP -favored region. The excessive relic densityis diluted either by increasing the Higgsino componentor by coannihilation with other particles. The break-ing chain SU(2) × SO(7) predicts an upper limit for thelighter chargino mass for the chosen parameters. Forpart of the
WMAP allowed region, the whole neutralinoand chargino spectrum is lighter than the spectrum ofsfermions. For the SU(4) × SU(2) × SU(2) breaking chain,the relic density area is narrow in the parameter space.Interestingly, there may exist a region, where the stau,smuon and the lightest neutralino masses are in a veryclose range to each other. This can lead to long-livedstaus and smuons, which may be stable in the collidertime scale.In the dimensional representation the lightest neu-tralino is either wino or Higgsino, which leads to a lowthermal relic density. In addition, the lightest charginoand the lightest neutralino tend to be close in mass,thus providing a coannihilation channel. The preferredrelic density area is quite large. The sparticle spectrumis heavy, as compared to the universal mSUGRA case.Only in a small part of the
WMAP -preferred parameterregion are a few
SUSY particles expected to be within thekinematic reach of the
LHC .In this work we have studied each representation sepa-rately. It is obvious that if several representations affectsimultaneously the composition of neutralinos, the possi-ble
WMAP -preferred region in the parameter space maybe relaxed. However, if the neutralinos and charginos arefound with a certain mass pattern, it helps to understandthe relation of the lightest neutralino with dark matter,if characteristics of each representation are known.
IV. ACKNOWLEDGMENTS
The work of K.H. is supported by the Academy of Fin-land (Project No. 115032). The work of J.L. is supported by the Bundesministerium f¨ur Bildung und Forschung,Berlin-Bonn. [1] D. N. Spergel et al. (WMAP), Astrophys. J. Suppl. ,377 (2007), astro-ph/0603449.[2] K. Huitu, J. Laamanen, P. N. Pandita, and S. Roy, Phys.Rev.
D72 , 055013 (2005), hep-ph/0502100.[3] H. Fritzsch and P. Minkowski, Ann. Phys. , 193 (1975).[4] Z.-Y. Zhao, J. Phys. G8 , 1019 (1982).[5] M. Drees and J. M. Kim (2008), 0810.1875.[6] S. Bhattacharya, A. Datta, and B. Mukhopadhyaya,JHEP , 080 (2007), 0708.2427.[7] B. Ananthanarayan and P. N. Pandita, Int. J. Mod. Phys. A22 , 3229 (2007), 0706.2560.[8] R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. , 912 (1980).[9] B. Bajc, G. Senjanovic, and F. Vissani, in *Bu-dapest 2001, High energy physics* hep2001/198 (2001),hep-ph/0110310.[10] S. Dimopoulos and F. Wilczek, in Proceedings of the 19thCourse of the International School of Subnuclear Physics,Erice, Italy, 1981 , edited by A. Zichichi (Plenum Press,New York, 1983).[11] E. Cremmer, S. Ferrara, L. Girardello, and A. VanProeyen, Phys. Lett.
B116 , 231 (1982).[12] S. Ferrara, L. Girardello, and H. P. Nilles, Phys. Lett.
B125 , 457 (1983).[13] R. Slansky, Phys. Rept. , 1 (1981).[14] N. Chamoun, C.-S. Huang, C. Liu, and X.-H. Wu, Nucl.Phys. B624 , 81 (2002), hep-ph/0110332.[15] G. F. Giudice and R. Rattazzi, Nucl. Phys.
B511 , 25(1998), hep-ph/9706540.[16] C. S. Aulakh and S. K. Garg (2008), 0807.0917.[17] C. S. Aulakh and S. K. Garg, Nucl. Phys.
B757 , 47(2006), hep-ph/0512224.[18] B. Bajc, I. Dorsner, and M. Nemevsek, JHEP , 007(2008), 0809.1069.[19] C. S. Aulakh and R. N. Mohapatra, Phys. Rev. D28 , 217 (1983).[20] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac,and N. Okada, J. Math. Phys. , 033505 (2005),hep-ph/0405300.[21] S. F. King, J. P. Roberts, and D. P. Roy (2007),arXiv:0705.4219.[22] K. Huitu et al., Eur. Phys. J. C58 , 591 (2008), 0808.3094.[23] B. C. Allanach, Comput. Phys. Commun. , 305(2002), hep-ph/0104145.[24] G. Belanger, F. Boudjema, A. Pukhov, and A. Se-menov, Comput. Phys. Commun. , 103 (2002),hep-ph/0112278.[25] G. Belanger, F. Boudjema, A. Pukhov, and A. Se-menov, Comput. Phys. Commun. , 577 (2006),hep-ph/0405253.[26] G. Belanger, F. Boudjema, A. Pukhov, and A. Se-menov, Comput. Phys. Commun. , 367 (2007),hep-ph/0607059.[27] E. Barberio et al. (Heavy Flavor Averaging Group(HFAG)) (2007), 0704.3575.[28]
Search for charged Higgs bosons: Preliminary combinedresults using LEP data collected at energies up to 209-GeV (2001), hep-ex/0107031, hep-ex/0107031.[29] A. De Rujula, H. Georgi, and S. L. Glashow, Phys. Rev.Lett. , 413 (1980).[30] S. M. Barr, Phys. Lett. B112 , 219 (1982).[31] J. P. Derendinger, J. E. Kim, and D. V. Nanopoulos,Phys. Lett.
B139 , 170 (1984).[32] I. Antoniadis, J. R. Ellis, J. S. Hagelin, and D. V.Nanopoulos, Phys. Lett.
B194 , 231 (1987).[33] K. Huitu, J. Laamanen, and S. Roy (2007), 0710.2058.[34] O. Adriani et al. (2008), 0810.4994.[35] J. Chang et al., Nature456