We investigate perturbative unitarity constraints in a model for a singlet
scalar dark matter candidate. Considering elastic two particle scattering
processes of the Higgs particle and the dark matter candidate, a real
Klein-Gordon scalar field, perturbative unitarity constrains the self-couplings
of the scalar fields.Comment: 6 pages, 4 eps figure

Cynolter, G.

Lendvai, E.

Pocsik, G.

English

arXiv

Cynolter, G

Lendvai, E

Pócsik, G

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0410102v1  7 Oct 2004ITP-Budapest 614hep-ph/0410102Note on Unitarity Constraints in a Model for aSinglet Salar Dark Matter CandidateG. Cynolter∗, E. Lendvai∗and G. Pósik†∗Theoretial Physis Researh Group of Hungarian Aademy of Sienes,Eötvös University, Budapest, Hungary†Institute for Theoretial Physis, Eötvös Lorand University, Budapest,HungaryAbstratWe investigate perturbative unitarity onstraints in a model for a sin-glet salar dark matter andidate. Considering elasti two partile sat-tering proesses of the Higgs partile and the dark matter andidate, areal Klein-Gordon salar eld, perturbative unitarity onstrains the self-ouplings of the salar elds.Despite the unique suess of the Standard Model (SM) in the partile physisaelerator experiments, there have been strong indiations of fundamentalphysis related to partile physis beyond the SM in the past few years, suh asneutrino osillations and masses, the baryon asymmetry of the universe annotbe explained by the SM, reent ts to the osmologial parameters [1℄ need darkenergy. About 23% of the total energy density of the universe is made up ofsome dark matter. Assuming that gravity does not hange signiantly at dis-tanes larger than a few kpc the dark matter must be non-baryoni to maintainthe suess of big bang nuleosynthesis.These problems were addressed among others in [2℄ where a possible minimalextension of the SM was proposed. They added the minimal number (6) of newdegrees of freedom purely to answer the empirial hallenges. The non-baryonidark matter andidate is assumed to be a Z2 symmetri gauge singlet salar eld,S. It an aount for the observed dark matter abundane and is onsistent withthe limit from CDMS-II experiment [3℄. [2℄ onsidered a few onsequenes of1the model suh as triviality and stability of the Higgs potential, Higgs deaysinto new partiles.In this note we apply perturbative unitarity [4℄ to a singlet dark mattereld S oupled to the Higgs eld H and itself in the SM ompleted by termsdesribing S,H interations. The onstraints are valid also for [2℄ beause theinaton eld of [2℄ is too heavy to partiipate in the sattering. We get variousupper bounds for the relevant salar ouplings.Start with the minimal renormalizable extension of the Standard Modelproviding a salar non-baryoni dark matter andidate S. S is required to beodd under a Z2 symmetry in order to be stable. The odd singlet salar anhave renormalizable interation only with the standard Higgs and not with theordinary fermions and gauge bosons. This model was proposed earlier in theliterature by Silveira and Zee [5℄, a omplex salar eld ase was studied in [6℄.The parameters of the model were rst onstrained by Burgess et al. [7℄, laterin [2℄.The Lagrangian of the salar setor isLSH = |DµH |2−λ2∣∣∣∣H†H − v22∣∣∣∣+12∂µS∂µS− 12m20S2− k2H†H S2− λS4!S4. (1)Beside the usual Higgs potential parameters λ,v = 254 GeV, there are three newparameters m0, k, λS determining the properties of S. The potential of the H-Ssetor is bounded from below if λ, λS > 0 and k > 0 or3k2 < λSλ for k < 0. (2)The Higgs eld gets a vauum expetation value v while 〈S〉 = 0 in order torespet the Z2 symmetry. The Higgs mehanism generates a mass of m2H = λv2for the Higgs and also ontributes to the mass of the S partilem2S = m20 +12kv2. (3)m2s > 0 is required for 〈H〉 = (0, v/√2) and 〈S〉 = 0 be a loal minimum. Thisis also a global minimum as long as m20 > − 12v2√13λλS [7℄. After eletroweaksymmetry breaking in the unitarity gauge the potential of the salar setorbeomesVSH =λ8H4+λ2H3v+λ2H2v2+12(m20 +12kv2)S2+k2vHS2+k4H2S2+λS4!S4.(4)Next we apply tree-level perturbative unitarity [4℄ to salar elasti satteringproesses in the model (4). The zeroth partial wave amplitude,a0 =132π√4pCMf pCMis∫ +1−1T2→2d cosθ (5)2must satisfy |Rea0| ≤ 12 . s is the entre of mass (CM) energy and pCMi,f are theinital and nal momenta in CM system.There are three possible two partile states HH,HS, SS and four satter-ing proesses. Inlusion of the gauge bosons does not signiantly alter ouronsideration sine they do not interat with the new singlet salar S at treelevel.1.) HH → HH . The tree-graphs ontributing to this proess are drawn inFig. 1.HHHHHHHHHHHHHHHHHHHFigure 1: Tree level Feynman diagrams for HH→HH in the SMWe getTHH→HH = 3m2Hv2(1 + 3m2H(1s−m2H+1t−m2H+1u−m2H)). (6)2.) SS → SS. The ontat graph and the H-exhange graphs an be seenin Fig. 2.SSSSSSHSSSSHSSSSHSSFigure 2: Tree level Feynman diagrams for SS→SSTSS→SS = λS + k(kv2s−m2H+kv2t−m2H+kv2u−m2H). (7)3.) SS → HH .SSHHSSHHHSSSHHSSSHHFigure 3: Tree level Feynman diagrams for SS→HH3The T matrix (Fig. 3) isTSS→HH = k(1 + 3m2H1s−m2H+ kv2(1t−m2S+1u−m2S)). (8)4.) HS → HS. The T matrix (Fig. 4.) isTHS→HS = k(1 + v2(ks−m2S+3λt−m2H+ku−m2S)). (9)SHSH HSSSHSHHSHSHSSHFigure 4: Tree level Feynman diagrams for SH→SHFrom (5) we have the following partial wave projetion of the oupled systemin the J=0 hannel:a0(HH → HH) =3m2H16πv2√1− 4m2Hs(1 +3m2Hs−m2H− 6m2Hs− 4m2Hln(sm2H− 3)),a0( SS → SS ) =116π√1− 4m2Ss(λS +k2v2s−m2H− 2k2v2s− 4m2Sln(s− 4m2Sm2H+ 1)),a0(SS → HH) =k16πκSS→HH(1 + 3m2H(1s−m2H)− 2kv2√s− 4m2S√s− 4m2Hln(1 +2√s− 4m2S√s− 4m2Hs−√s− 4m2S√s− 4m2H − 2m2h)), (10)a0(SH → SH) =k16πκSH→SH(1 +kv2s−m2H− 3sAm2H lns− 3m2Hs (2m2S −m2H)− (m2H −m2S)2−− sAkv2 ln(s− 3m2S)s+(m2H −m2S)2s (2m2H −m2S)),whereA = s2−2s(m2H +m2S)+(m2H −m2s)2, κSS−>HH =(1− 4m2Ss)1/4 (1− 4m2Hs)1/4and κSH−>SH =√(1− (mH+mS)2s)(1− (mH−mS)2s)are kinematial fators.Perturbative unitarity will be imposed in the high energy limit s ≫ m2H ,m2S .The oupled amplitudes areaHH→HH0 aHH→SS0 aHH→SH0aSS→HH0 aSS→SS0 aSS→SH0aSH→HH0 aSH→SS0 aSH→SH0 −→s≫m2H,m2S116π3λ k 0k λS 00 0 k .4Requiring |Rea0| ≤ 12 for eah individual proess above we obtainHH → HH mH ≤√8π3v. (11)HS → HS and HH → SS |k| ≤ 8π. (12)SS → SS λS ≤ 8π. (13)While (11) is the well known bound in [4℄, (12) shows that the maximumontribution of the Higgs mehanism to mS is 900 GeV.(12) an be improved for k < 0 by using (11,13) in the positivity relation−k ≤ 8π3∼ 4.2 and 12kv2 & −(520 GeV)2. (14)For mH = 150GeV, λ = 0.38 this goes into12kv2 & −(230 GeV)2.These bounds an be rened onsidering the partial wave unitarity for thethree oupled hannel system HH, SS, SH and onstraining the eigenhannelwith the highest eigenvalue. Atually SH deouples from HH and SS and theremaining two eigenvalues are µ1,2 =3λ+λS2 ± 12√(3λ+ λS)2+ 4k2, providingthe onstraintλS +k28π≤ 8π − 3λ. (15)The onstraint (15) ontains all the previous bounds (11, 12, 13).After the new measurement of the top mass [8℄ the upper bound of the Higgsmass from radiative orretions is 251 GeV at 95 % C.L. and the diret lowerbound is 114.5 GeV from LEP2 implying the range 0.21 − 0.97 for λ. We see,however, that for λ = 0.21 − 0.97 (mH = 114.5GeV-251 GeV) the right handside of (15) hanges very small, 24.5-22.1, and λS +k28π . 8π. Only a heavyHiggs would provide a stronger upper bound.In onlusion we have onsidered a simple non-baryoni dark matter andi-date model added to the Standard Model. We have alulated the J=0 partialwave amplitudes for the two partile elasti sattering proesses and found thatthe salar ouplings of the model are restrited.In general our results did not restrit the S mass, however, assuming mSomes from the Higgs mehanism we get mS < 900GeV. The model an aountfor the dark matter in the universe and is onsistent with the limits from CDMS-II. Perturbative unitarity onstraints allow also higher λS , k than those obtainedfrom stability and triviality desribed in [2℄.5Referenes[1℄ D.N. Spergel et al. [WMAP Collaboration℄, Astrophys. J. Suppl. 148 (2003)175.[2℄ H.Davoudiasl, R.Kitano, T.Li and H.Murayama, arXiv:hep-ph/0405097.[3℄ D.S.Akerib et al. [CDMS Collaboration℄, arXiv:astro-ph/0405033[4℄ B.W. Lee, C. Quigg and H.B.Thaker, Phys. Rev. D 16 (1977) 1519.[5℄ V.Silveira and A.Zee, Phys. Lett. B 161 (1985) 136.[6℄ J.MDonald, Phys. Rev. D 50 (1994) 3637.[7℄ C.P.Burgess, M.Pospelov and T.ter Veldhuis, Nul. Phys. B619 (2001) 709[arXiv:hep-ph/0011335℄.[8℄ DØ Collaboration, Nature 429, 638 (2004)6

Note on Unitarity Constraints in a Model for a Singlet Scalar Dark Matter Candidate

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Note on Unitarity Constraints in a Model for a Singlet Scalar Dark Matter Candidate

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