Neutralino Relic Density in a Supersymmetric U(1)' Model

We study properties of the lightest neutralino (\chi) and calculate its cosmological relic density in a supersymmetric U(1)' model with a secluded U(1)' breaking sector (the S-model). The lightest neutralino mass is smaller than in the minimal supersymmetric standard model; for instance, m_\chi<100 GeV in the limit that the U(1)' gaugino mass is large compared to the electroweak scale. We find that the Z-\chi-\chi coupling can be enhanced due to the singlino components in the extended neutralino sector. Neutralino annihilation through the Z-resonance then reproduces the measured cold dark matter density over broad regions of the model parameter space.


I. INTRODUCTION
Recent mapping of the Cosmic Microwave Background (CMB) anisotropy [1] has provided precision information on the densities of matter and dark energy in the Universe. The major part of the matter is non-relativistic and non-baryonic (cold and dark). When the Sloan Digital Sky Survey (SDSS) data on large scale structure are analyzed in combination with the Wilkinson Microwave Anisotropy Probe (WMAP) data, a cold dark matter (CDM) relic density Ω CDM h 2 = 0.12 ± 0.01 (SDSS + WMAP) (1) is found [2]. This very restrictive range for Ω CDM h 2 has significant impact on the allowed masses and couplings of cold dark matter particles. A neutral, stable, massive particle that interacts weakly is a natural candidate for CDM. The Standard Model (SM) does not have a particle with these properties but a supersymmetric model with R-parity conservation, one of the best motivated new physics possibilities, does. In the Minimal Supersymmetric Standard Model (MSSM) the lightest neutralino is the favored lightest supersymmetric particle (LSP).
The lightest neutralinos existed abundantly in the early Universe in thermal equilibrium with other particles, where their pair annihilations were balanced by pair creation. As the Universe cooled, the neutralino density became Boltzmann-suppressed. Deviation from thermal equilibrium began when the temperature reached the freeze-out temperature T f ≃ m χ 0 /20. After the temperature dropped to ∼ 1 5 T f , the annihilation rate became equal to the expansion rate, and the neutralino relic density was n χ 0 = H/ σv , where H is the Hubble expansion rate at that temperature. Here σv is the thermally averaged cross section times neutralino velocity [3]. The remaining CDM density relative to the critical density is Ω χ 0 = n χ 0 m χ 0 /ρ c = Hm χ 0 / ( σv ρ c ) . ( A viable CDM model must reproduce the recent precise measurement of Ω CDM , preferably without fine-tuning of the model parameters. The full test of neutralino dark matter can be accomplished with the direct detection of the LSP in collider experiments and/or in elastic scattering experiments at underground detectors. The parameters of the model must quantitatively explain Ω χ 0 h 2 ≃ 0.12 and the direct detection rates. There have been extensive theoretical studies of the relic density [4,5,6]. In the minimal supergravity model (mSUGRA) annihilations of stable supersymmetric particles can reproduce the right order of magnitude. However, the allowed regions of mSUGRA parameter space are becoming increasingly constrained [7] by the combination of the recent relic density and the LEP data and now allow only a few regions of parameter space including (i) pair annihilation of neutralinos with dominant bino composition through A 0 , H 0 -resonances at high tan β (the so-called A-funnel) [8], (ii) neutralino-stau coannihilation [9], and (iii) annihilation of neutralinos with mixed gaugino-Higgsino components in the focus point region [10]. A nearly pure bino LSP state can give the right size of the relic density [11] and also satisfy the required radiative electroweak symmetry breaking (EWSB) in unified models [12]. In SO(10) grand unified supergravity models with Yukawa unification, it is more difficult to obtain the required relic density, but it is still possible with specific scalar mass patterns [13].
The MSSM suffers from the µ-problem [14] and the lack of a sufficiently strong first order phase transition for electroweak baryogenesis (EWBG) over most of the parameter space 1 . The missing ingredients of possible TeV-scale new physics that would cure these problems may modify the properties of the CDM candidate. The Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) [16] avoids the problems by introducing an additional SM singlet Higgs and a discrete symmetry Z 3 , but it then suffers from the cosmological domain wall problem [17].
A natural extension of the MSSM that avoids the above difficulties is a supersymmetric U(1) ′ model, with Higgs singlets to break the additional Abelian symmetry spontaneously at the TeV scale 2 . Additional U(1) gauge symmetries are predicted in many types of new theories, including superstrings [20], grand unification, extra dimensions [21], dynamical symmetry breaking [22], and the little Higgs model [23]. In this letter we investigate the properties of the lightest neutralino and evaluate its relic density 3 in an extended model of the MSSM with an extra U(1) gauge symmetry and four extra Higgs singlets (S, S 1 , S 2 , and S 3 ) [25]. The superpotential is 4 where h s and λ s are dimensionless parameters. We call this the S-model. The S-model solves the µ-problem by generating an effective µ parameter The model is free from the domain wall problem since there is no discrete symmetry [25,26,27]. The S-model can also provide a sufficiently strong first order phase transition and new sources of CP violation for EWBG [28]. The Z ′ has a large mass 5 generated by singlet Higgs fields, S 1,2,3 , which acquire large (TeV scale) VEVs for small λ s because of an almost F and D flat direction. These multiple singlets allow µ eff to be of the electroweak scale while keeping the Z ′ heavier than the experimental limit. Electroweak symmetry breaking is driven by electroweak scale trilinear terms. This leads to tan β ≡ v 2 /v 1 ≃ 1, while solutions without unwanted global minima at H 0 i = 0 typically have S < ∼ 1.5 H 0 i [25]. However, both of these conditions can be relaxed somewhat [31]. All dimensionful supersymmetry-breaking parameters are close to the electroweak scale. The squark and slepton masses are similar to those of the MSSM, and the soft supersymmetry breaking Higgs and singlet masses are smaller than the Higgs doublet mass in the MSSM.
In our analysis, we consider the limit with M ′ 1 ≫ M 1,2 , where M ′ 1 , M 1 , and M 2 are the gaugino masses for U(1) ′ , U(1) Y , and SU(2) L , respectively [25,31]. This limit suppresses the effects of the U(1) ′ Higgs charges and greatly reduces the number of free parameters 1 To have a strong first-order phase transition in the MSSM, the light Higgs mass should be only slightly above the LEP experimental bound and the light stop should be lighter than the top [15]. 2 For another extension of the MSSM using a discrete symmetry but free of domain wall problem, see the nMSSM model [18,19]. 3 For an earlier study of the relic density in a U (1) ′ model with one singlet, in a different approach and framework, see Ref. [24]. 4 For a U (1) ′ model with only one singlet, see Ref. [26]. It is more difficult to obtain M Z ′ ≫ M Z in such models. 5 The CDF limit is M Z ′ > ∼ 500 − 800 GeV, depending on the model [29,30].
in the neutralino mass matrix. The allowed mass range of the lightest neutralino then is limited (m χ 0 < ∼ 100 GeV) compared to that of the MSSM, due to the large component of the singlino in the neutralino, which we assume is the LSP. While the small mass of the LSP makes the heavy fermion and gauge boson channels of the MSSM irrelevant, it opens up new relevant channels, particularly the Z-resonance and the light Higgs resonance channels. The lighter Higgs scalar (h 0 ) in this model is a mixture of the Higgs doublets and singlets, and it can be much lighter than the LEP bound of m h 0 > 115 GeV that applies to the SM-like Higgs bosons [25,31].
In the MSSM, Z-resonance annihilation is not likely to be a relevant channel because the Zχ 0 χ 0 coupling is small when tan β ∼ 1. However, when the singlino component is introduced into the neutralino sector, the Zχ 0 χ 0 coupling can be enhanced to give sufficiently large χ 0 χ 0 annihilation. This Z-resonance annihilation alone, with suitable parameter values, can reproduce the acceptable cold dark matter relic density in most of the allowed χ 0 mass range in the S-model.

II. MASS AND COUPLING OF THE LIGHTEST NEUTRALINO
We consider a scenario with a massive Z ′ and VEVs of S i (i = 1, 2, 3) that are large compared to other electroweak scale parameters. There is an approximate decoupling of the neutralinos associated with the Z ′ and the S i , and the effective neutralino mass matrix for the remaining neutralinos in the basis of B , where e = g 1 cos θ W = g 2 sin θ W . The VEVs of the Higgs doublets are H 0 GeV and the VEV of the Higgs singlet is S 0 ≡ s √ 2 . The mass eigenstates are ordered as m χ 0 = m χ 0 1 < m χ 0 2 < · · · < m χ 0 5 . This mass matrix leads to a kind of see-saw mechanism [19,25,33] that makes the lightest neutralino mass very small. In our analysis we scanned over the parameters tan β = 1.03, 1.5, 2.0, 2.5 h s = 0.1 to 0.75 (in steps of 0.01) M 2 = −500 GeV to 500 GeV (in steps of 1 GeV) s = 100 GeV to 500 GeV (in steps of 1 GeV) We use the gaugino mass unification relations (M 1 = 5 The U(1) ′ charge dependence vanishes from the effective mass matrix for large M ′ 1 . 6 In this limit, the neutralino matrix is basically the same as that of the one-singlet models and shares many of the same properties [19,32].
The MSSM mass matrix 7 corresponds to dropping the last row and column in the matrix of Eq. (5) and taking µ = h s s/ √ 2. The MSSM upper bound on m χ 0 is very sensitive to the value of M 2 , while that of the S-model depends sensitively on the value of h s . In both models, the upper bound on the lightest neutralino mass has its largest value when tan β is 1.03 (or 1) 8 . When tan β ∼ 1, the m χ 0 bound in the S-model increases almost linearly with h s . We choose our upper bound of h s to be 0.75 as advocated in Ref. [25] to keep the running of h s finite up to Plank scale [19,25].
The small LSP mass makes the Z-resonance contribution to the relic density relevant in the S-model. For Zχ 0 i χ 0 j and Z ′ χ 0 i χ 0 j interactions, the general Lagrangian is given by [33] which, in terms of neutralino mass eigenstates, can be written as Here the couplings are defined as where the matrix N relates the weak and mass bases.
is composed of the two doublet Higgsino components coupling to the Z boson. A massive 7 The MSSM parameter range considered here (especially for tan β) is not experimentally allowed (e.g. by the lightest Higgs mass), but is chosen to demonstrate the effects of adding the U (1) ′ . 8 Since the exact value tan β = 1 would make the coupling of Zχ 0 χ 0 vanish in both the MSSM and the S-model, we choose tan β = 1.03 instead of 1. 9 The upper bound on m χ 0 in the different limits are: We scanned 100 to 1000 GeV for M 2 and s i and assumed h s < ∼ 0.75 with λ s < ∼ 0.2. For the U (1) ′ couplings, various E 6 model charge assignments (χ, ψ, η) and the GUT motivated coupling constant g Z ′ = 5 3 g 1 were assumed. For a full 9 × 9 neutralino mass matrix, see Ref. [25,31]. (TeV-scale) Z ′ would not provide sufficiently large annihilation for our light neutralinos, and thus in our analysis we use only the Z-resonance annihilation.
The large difference in |N 13 | 2 and |N 14 | 2 for the S-model is remarkable in view of the fact that tan β = 1 makes theH 0 1 andH 0 2 parts in the mass matrix of Eq. (5) the same up to the sign, which leads one to expect |N 13 | 2 −|N 14 | 2 ∼ 0, as is the case in the MSSM (Figure 1 (b)). Thus, the addition of the singlino component plays a critical role in enhancing the difference of |N 13 | 2 and |N 14 | 2 , which is particularly important for the χ 0 in the S-model to generate sufficient annihilation mediated by the Z boson. The suppressed coupling (by tan β ∼ 1) can still be large enough to give an acceptable relic density. Figure 1 also shows that the gaugino components |N 11 | 2 (bino) and |N 12 | 2 (wino), especially the bino, are dominant for relatively small values of |M 2 | in both models.
We present |O ′′ 11 | ≡ |O ′′L 11 | = |O ′′R 11 | for the S-model and the MSSM in . For tan β ∼ 1 (Figure 2 (a)), the Zχ 0 χ 0 coupling in the S-model is at its maximum for M 2 ≃ 300 GeV and there is a relatively small but still noticeable peak in the MSSM at M 2 ≃ −250 GeV. The χ 0 relic density depends on not only the coupling but also whether m χ 0 is near M Z /2.   Figures 1 and 2 (a) are used. The LSP mass in the S-model is almost constant for M 2 < ∼ −200 GeV and M 2 > ∼ 300 GeV, and it is smaller than in the MSSM. The lightest neutralino masses in both models have practically identical dependence on M 2 before they reach the flat curves in our parameter choice (tan β ≃ 1). Enhancements of couplings in the S-model and the MSSM are found around m χ 0 ∼ 80 GeV and 130 GeV, respectively, that is, for the flat parts of the mass curves in Figure 3 (a).

III. NEUTRALINO ANNIHILATION MEDIATED BY THE Z BOSON
The relic density of the lightest neutralino is inversely proportional to the annihilation cross section of χ 0 pairs. When kinematically allowed, the neutralino pairs annihilate into pairs of fermions, gauge bosons, and Higgs bosons through s, t, and u channel diagrams. In general, the annihilation cross section is greatly enhanced by the Z boson or the Higgs boson (φ 0 ) poles, and the relic density is correspondingly suppressed when 2m The Higgs masses and couplings in the S-model depend on several free parameters. Moreover, the light Higgs width is very narrow and a fine-tuning of parameters would be required for neutralino annihilation through φ 0 to generate the appropriate relic density, and the other Higgs bosons would often be too heavy to have resonance effects because the LSP is light [31]. We are therefore interested in the case that the Zχ 0 χ 0 coupling is sizable and Z mediated annihilation alone can lead to an acceptable relic density (Ω CDM h 2 ). A more complete investigation is in progress to include all Higgs resonance effects. We require a lower bound on m χ 0 of M Z /2 so that Z → χ 0 χ 0 does not affect the Z width significantly 10 . Then the lightest neutralino mass in the S-model is constrained to the range with the upper bound from Eq. (6). We evaluate the annihilation cross section including interferences among all diagrams by calculating the amplitude of each diagram with the helicity amplitude formalism, then numerically evaluate the full matrix element squared. The neutralino relic density is calculated with relativistic Boltzmann averaging, neutralino annihilation threshold effects and Breit-Wigner poles [5,6,8].
We show the neutralino relic density (Ω χ 0 h 2 ) versus m χ 0 in Figure 4 for both the Smodel 11 and the MSSM for the tan β choices (a) tan β = 1.03, (b) tan β = 2.0, and (c) tan β = 2.5. We included all possible channels for the MSSM calculation; for the S-model we included only the kinematically relevant s-channel processes with the Zχ 0 χ 0 coupling, a SM fermion). We scanned over −500 GeV < M 2 < 500 GeV for h s = 0.75, and s = 250 GeV to calculate m χ 0 and evaluate the Zχ 0 χ 0 coupling. There are several interesting aspects to note. 10 Since ∆Γ inv < 2.0 MeV (95% C.L.) [34], smaller m χ 0 is possible [19]. For reasonably smaller masses, our results are unchanged. 11 For similar relic density results in the nMSSM, see Ref. [19]. For the NMSSM relic density calculations, see Ref. [35]. • When m χ 0 ≃ M Z /2, there is a dip in the curve of relic density versus m χ 0 since there is a peak in the annihilation cross section enhanced by the Z-pole.
• In the S-model, a positive value of M 2 gives two acceptable CDM density regions (around the Z pole and the enhanced coupling region such as m χ 0 ∼ 80 GeV in the tan β = 1.03 case) while a negative M 2 leads to Ω χ 0 h 2 > ∼ 1 except for the singlinodominated region.
• The MSSM can also give an acceptable CDM density, but the model is excluded for tan β ∼ 1 by the LEP Higgs bound, owing to its small Higgs mass. The S-model Higgs contains a singlet component and the light Higgs can be compatible with the LEP constraint [31].
• The small isolated region for relatively large m χ 0 corresponds to M 2 < 0 with the lightest neutralino being singlino-dominated in the S-model 12 . As observed in Figure  1, a sudden sizable deviation of |N 13 | 2 from |N 14 | 2 in a singlino dominated region (M 2 < ∼ −150 GeV for tan β = 1.03) provides a sudden drop of Ω χ 0 h 2 from the rest of the M 2 < 0 curve. Also shown is the region excluded by the LEP 2 chargino mass bound with m χ + 1 < 104 GeV [36] The parameter points at which the light chargino is the LSP were omitted. The tree level mass for the lighter chargino (m χ + 1 < m χ + 2 ) is evaluated with the chargino mass matrix There are three separate regions that have an acceptable CDM density (filled square): (P1) Near the Z pole, (P2) enhanced coupling region (tan β ≈ 1 case), and (P3) isolated singlino region. In general, the Zχ 0 χ 0 coupling is enhanced by the singlino component. There appears a sudden peak of the enhancement (P2) when tan β ≈ 1 as we can see from Figure 2 (a). In the enhanced coupling region (P2), even when the χ 0 mass is significantly distant from the Z pole, an acceptable relic density can be obtained. The isolated singlino region (P3) is singlino-dominated and happens for M 2 < 0.
For tan β = 1.03, there is a small region in the M 2 -s plane with M 2 > 0 that satisfies the relic density and LEP chargino mass constraints. The solution in this region is due to the enhanced Zχ 0 χ 0 coupling. For tan β = 2.0, there is a large acceptable region with M 2 < 0. With M 2 > 0 most of the parameter regions that give the relic density are excluded by the LEP 2 chargino search. For tan β = 2.5, there is a large region with M 2 > 0 that reproduces the observable relic density and is consistent with the chargino mass limits.
We numerically checked, with suitable parameter values including different h s , that the S-model can reproduce the observed relic density for most of the theoretically allowed m χ 0 range without violating the LEP constraints. However, for a relatively large neutralino mass, i.e., m χ 0 ≈ 80 to 100 GeV, it becomes hard to satisfy both the relic density and the LEP constraints since those m χ 0 values occur only with tan β close to 1 and h s close to its maximum (0.75), but then the chargino constraint becomes severe, as can be seen in Figure  5 (a).

IV. CONCLUSIONS
We have studied the properties of the lightest neutralino in a supersymmetric U(1) ′ model with a secluded U(1) ′ breaking sector [25]. In this model, tan β ∼ 1 is required for natural electroweak symmetry breaking. In general, the model allows a lightest neutralino mass up to about 300 GeV (depending on the charge assignments and input mass limits) with the gaugino unification assumption M 1 ′ = M 1 = 5 3 g 2 1 g 2 2 M 2 , but only up to m χ 0 < ∼ 100 GeV in the limit with M 1 ′ ≫ M 1 = 5 3 g 2 1 g 2 2 M 2 . We quantitatively studied the S-model in the limit where M 1 ′ , s i are much larger than the electroweak scale. In this limit the Zχ 0 χ 0 coupling is enhanced compared to the MSSM due to the singlino component. This allows the χ 0 pair annihilation rate via the Z resonance channel alone to reproduce the observed relic density of the cold dark matter. In addition, the doublet-singlet mixed nature of the Higgs bosons allows the lightest Higgs bosons to have a small mass without violating the LEP constraint on SM-like Higgs mass. The S-model explains the relic density over a considerable fraction of the parameter space.