NOTE ON UNITARITY CONSTRAINTS IN A MODEL FOR A SINGLET SCALAR DARK MATTER CANDIDATE

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1 Vol ) ACTA PYICA POLONICA B No 3 NOTE ON UNITARITY CONTRAINT IN A MODEL FOR A INGLET CALAR DARK MATTER CANDIDATE G. Cynolter, E. Lendvai Theoretical Physics Research Group of ungarian Academy of ciences Eötvös University, Budapest, ungary and G. Pócsik Institute for Theoretical Physics Eötvös Lorand University, Budapest, ungary Received October, 004) We investigate perturbative unitarity constraints in a model for a singlet scalar dark matter candidate. Considering elastic two particle scattering processes of the iggs particle and the dark matter candidate, a real Klein Gordon scalar field, perturbative unitarity constrains the self-couplings of the scalar fields. PAC numbers:.60.fr, j, d Despite the unique success of the tandard Model M) in the particle physics accelerator experiments, there have been strong indications of fundamental physics related to particle physics beyond the M in the past few years, such as neutrino oscillations and masses, the baryon asymmetry of the Universe cannot be explained by the M, recent fits to the cosmological parameters [] need dark energy. About 3% of the total energy density of the Universe is made up of some dark matter. Assuming that gravity does not change significantly at distances larger than a few kpc the dark matter must be non-baryonic to maintain the success of big bang nucleosynthesis. These problems were addressed among others in [] where a possible minimal extension of the M was proposed. They added the minimal number 6) of new degrees of freedom purely to answer the empirical challenges. The non-baryonic dark matter candidate is assumed to be a Z symmetric gauge singlet scalar field,. It can account for the observed dark matter abundance and is consistent with the limit from CDM-II experiment [3]. 87)

2 88 G. Cynolter, E. Lendvai, G. Pócsik Davoudiasl et al., [] considered a few consequences of the model such as triviality and stability of the iggs potential, iggs decays into new particles. In this note we apply perturbative unitarity [4] to a singlet dark matter field coupled to the iggs field and itself in the M completed by terms describing, interactions. The constraints are valid also for [] because the inflaton field of [] is too heavy to participate in the scattering. We get various upper bounds for the relevant scalar couplings. tart with the minimal renormalizable extension of the tandard Model providing a scalar non-baryonic dark matter candidate. is required to be odd under a Z symmetry in order to be stable. The odd singlet scalar can have renormalizable interaction only with the standard iggs and not with the ordinary fermions and gauge bosons. This model was proposed earlier in the literature by ilveira and Zee [5], a complex scalar field case was studied in [6]. The parameters of the model were first constrained by Burgess et al. [7,8], later in []. The Lagrangian of the scalar sector is L = D µ λ v µ µ m 0 k λ 4! 4. ) Beside the usual iggs potential parameters λ, v = 54 GeV, there are three new parameters m 0,k,λ determining the properties of. The potential of the sector is bounded from below if λ,λ > 0 and k > 0 or 3k < λ λ, for k < 0. ) The iggs field gets a vacuum expectation value v while = 0 in order to respect the Z symmetry. The iggs mechanism generates a mass of m = λv for the iggs and also contributes to the mass of the particle m = m 0 kv. 3) m s > 0 is required for = 0,v/ ) and = 0 be a local minimum. This is also a global minimum as long as m 0 > v 3 λλ [8]. After electroweak symmetry breaking in the unitarity gauge the potential of the scalar sector becomes V = λ 8 4 λ 3 v λ v m 0 ) kv k v k 4 λ 4! 4. 4)

3 Note on Unitarity Constraints in a Model for Next we apply tree-level perturbative unitarity [4] to scalar elastic scattering processes in the model 4). The zeroth partial wave amplitude a 0 = 4p CM f p CM i T dcosθ 5) 3π s must satisfy Re a 0. s is the centre of mass CM) energy and pcm i,f are the initial and final momenta in CM system. There are three possible two particle states,, and four scattering processes. Inclusion of the gauge bosons does not significantly alter our consideration since they do not interact with the new singlet scalar at tree level. ). The tree-graphs contributing to this process are drawn in Fig.. We get Fig.. Tree level Feynman diagrams for in the M. T = 3 m v 3m s m t m u m )). 6) ). The contact graph and the -exchange graphs can be seen in Fig.. Fig.. Tree level Feynman diagrams for. kv T = λ k s m kv t m ) kv u m. 7)

4 830 G. Cynolter, E. Lendvai, G. Pócsik 3). The T matrix Fig. 3) is Fig. 3. Tree level Feynman diagrams for. T = k 3m s m kv t m u m )). 8) 4). The T matrix Fig. 4.) is k T = k v s m 3λ t m k u m )). 9) Fig. 4. Tree level Feynman diagrams for. From 5) we have the following partial wave projection of the coupled system in the J = 0 channel a 0 ) = 3m 6πv 4m s 3m s m 6m s 4m )) s ln m 3, a 0 ) = 4m λ k v 6π s s m k v )) s 4m ln s 4m m,

5 a 0 ) = a 0 ) = where and Note on Unitarity Constraints in a Model for k 6π κ 3m s m ) kv s 4m s 4m )) s 4m ln s 4m s, s 4m s 4m m h k 6π κ kv s m 3s A m ln s 3m s m m ) m m ) s ) s 3m A kv ln s m m) ) s m m ), 0) A = s s m m ) m m s), ) /4 κ > = 4m 4m s s κ > = m m ) s ) /4 ) m ) m ) s are kinematical factors. Perturbative unitarity will be imposed in the high energy limit s m,. The coupled amplitudes are a 0 a 0 a 0 a 0 a 0 a 0 3λ k 0 k λ 0. a0 a0 a0 6π s m,m 0 0 k m Requiring Re a 0 for each individual process above we obtain 8π m 3 v, ) and k 8π, ) λ 8π. 3) While ) is the well known bound in [4], ) shows that the maximum contribution of the iggs mechanism to m is 900 GeV. Eq. ) can be improved for k < 0 by using ), 3) in the positivity relation k 8π 3 4. and kv 50 GeV). 4) For m = 50GeV, λ = 0.38 this goes into kv 30GeV).

6 83 G. Cynolter, E. Lendvai, G. Pócsik These bounds can be refined considering the partial wave unitarity for the three coupled channel system,, and constraining the eigenchannel with the highest eigenvalue. Actually decouples from and and the remaining two eigenvalues are providing the constraint µ, = 3λ λ ± λ k 8π 3λ λ ) 4k, 8π 3λ. 5) The constraint 5) contains all the previous bounds ) 3). After the new measurement of the top mass [9] the upper bound of the iggs mass from radiative corrections is 5GeV at 95 % C.L. and the direct lower bound is 4.5GeV from LEP implying the range for λ. We see, however, that for λ = m = 4.5GeV 5GeV) the right hand side of 5) changes very small, 4.5., and λ k /8π) 8π. Only a heavy iggs would provide a stronger upper bound. In conclusion we have considered a simple non-baryonic dark matter candidate model added to the tandard Model. We have calculated the J = 0 partial wave amplitudes for the two particle elastic scattering processes and found that the scalar couplings of the model are restricted. In general our results did not restrict the mass, however, assuming m comes from the iggs mechanism we get m < 900GeV. The model can account for the dark matter in the Universe and is consistent with the limits from CDM-II. Perturbative unitarity constraints allow also higher λ,k than those obtained from stability and triviality described in []. REFERENCE [] D.N. pergel et al., WMAP Collaboration, Astrophys. J. uppl. 48, ). []. Davoudiasl, R. Kitano, T. Li,. Murayama, hep-ph/ [3] D.. Akerib et al., CDM Collaboration, astro-ph/ [4] B.W. Lee, C. Quigg,.B. Thacker, Phys. Rev. D6, ). [5] V. ilveira, A. Zee, Phys. Lett. B6, ). [6] J. McDonald, Phys. Rev. D50, ). [7] M.C. Bento, O. Bertolami, R. Rosenfeld, Phys. Lett. B58, 76 00). [8] C.P. Burgess, M. Pospelov, T. ter Veldhuis, Nucl. Phys. B69, ). [9] DZero Collaboration, Nature 49, ).

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