Impact on the Light Higgsino-LSP Scenario from Physics beyond the Minimal Supersymmetric Standard Model

The modest addition of the dimension-5 term lambda(H_u.H_d)^2/M to the superpotential of the minimal supersymmetric standard model (MSSM) originated from physics beyond the MSSM (BMSSM) has a significant impact on the scenario of the Higgsino-dominated neutralino state being the lightest supersymmetric particle (LSP). It increases the mass difference between the LSP and the lighter chargino as well as that between the LSP and the second-lightest neutralino. This enhances the LHC discovery potential of the chargino and neutralino decays, producing more energetic charged leptons or pions than the decays without the BMSSM corrections. Furthermore, the coannihilation between the lighter chargino or second-lightest neutralino and the LSP is reduced substantially such that the LSP mass does not have to be very heavy. Consequently, an almost pure Higgsino LSP with its mass ~100 GeV in the BMSSM can account for all the relic density of cold dark matter in the Universe unless tan(beta) is too large.

Introduction -The presence of cold dark matter (CDM) in our Universe is now well established by the very precise measurement of the cosmic microwave background radiation in the Wilkinson Microwave Anisotropy Probe (WMAP) experiment [1]. A nominal 3σ range of the CDM relic density is where h is the Hubble constant in units of 100 km/Mpc/s.
One of the most appealing and natural CDM particle candidates is provided by supersymmetric models with R-parity conservation [2]. This R-parity conservation ensures the stability of the lightest supersymmetric particle (LSP) so that the LSP can be CDM. The LSP is in general the lightest neutralino, a linear combination of neutral electroweak (EW) gauginos and Higgsinos. Since the LSP nature depends on its compositions, its detection can vary a lot.
An interesting scenario is the Higgsino-like LSP, which can arise from a number of supersymmetry breaking models, e.g., focus-point supersymmetry models [3] or whenever the µ parameter is much smaller than the Bino and Wino masses [4]. In the minimal supersymmetric standard model (MSSM) [5], the Higgsino-LSP scenario implies nearly-degenerate Higgsino states: Coannihilation is too efficient so that the observed CDM relic density requires a rather heavy Higgsino state with mass around 1-1.2 TeV [6]. Moreover, the mass degeneracies between the LSP ( χ 0 1 ) and the lighter chargino/second-lightest neutralino ( χ ± 1 / χ 0 2 ) generate too soft decay products for detecting the states χ ± 1 and χ 0 2 at the LHC. Thus, the Higgsino-LSP in the MSSM posts a difficult scenario at the LHC.
In this Letter we show that the modest addition of a dimension-5 term λ ( H u · H d ) 2 /M to the MSSM superpotential, a scenario beyond MSSM (BMSSM) [7], alleviates the difficulties of the Higgsino-LSP scenario. It has been discussed that non-renormalizable and highdimensional operators in new physics models can yield important consequences in low energy phenomenology [8,9]. As shall be demonstrated in the following, this dimension-5 term, λ ( H u · H d ) 2 /M, lifts up the degeneracy between the states χ 0 1 and χ ± 1 / χ 0 2 [10], thus enhancing the discovery potential of the chargino and neutralino decays with more energetic charged leptons or pions than the decays in the MSSM. In addition, the coannihilation of the LSP with χ ± 1 or χ 0 2 is reduced substantially such that a light Higgsino-LSP with a mass around 100 GeV can accommodate the WMAP data on the CDM relic density.
BMSSM -Albeit many virtues of low energy supersymmetry (SUSY), fine-tuning in the lightest Higgs boson mass m h motivates additional degrees of freedom to the MSSM [10]. New interactions beyond the MSSM at the TeV scale M may be encoded in higherdimensional operators. Recently, Dine et al. [7] have shown that the most general dimension-5 superpotential term for the MSSM Higgs sector is with the SU (2) for the up-type and down-type Higgs doublet superfields, H u and H d , respectively. This dimension-5 operator has been shown to raise easily the lightest Higgs boson mass above the LEP bound without loss of naturalness [11].
Another dimension-5 operator, which breaks SUSY and affects the Higgs spectrum, is where Z = θ 2 m SUSY is the spurion field with the SUSY breaking scale m SUSY [7]. If m SUSY ≃ |µ|, the correction to m h comes dominantly from the supersymmetric operator in Eq. (2) rather than that in Eq. (3). The correction is given to leading order in the dimensionless parameter ε ≡ λ µ/M by where v ≈ 246 GeV is the Higgs vacuum expectation value and the second expression holds for large tan β. For simplicity we take the CP-conserving framework and decoupling limit of m A ≫ m Z in the following. Note that the BMSSM correction is inversely proportional to tan β in contrast to the conventional radiative corrections. The correction δm h normalized by m h around the LEP bound is roughly δm h /m h ≃ 20 ε/ tan β for m A ≫ m Z . For ε = 0.05 the correction can be as large as 50% (10%) for tan β = 2 (10). Therefore, the LEP bound is easily satisfied by a positive ε ∼ 0.1.
The interaction terms that involve only the Higgsino fields ( H u,d ) and Higgs fields (H u,d ) are given by After EW symmetry and SUSY breaking, the modified neutralino mass matrix M N in the and the modified chargino mass matrix M C in the { W − , H − } basis reads: where s β = sin β, s W = sin θ W , etc. To leading order in ε, the BMSSM effects on the masses of χ 0 1 , χ 0 2 , and χ ± 1 are, in the light Higgsino case (M 1,2 ≫ m Z , µ), [12] m e where in the first equation the upper (lower) sign is for χ 0 1 ( χ 0 2 ) mass. With increasing ε both the χ 0 1 and χ ± 1 masses decrease; the χ 0 2 mass increases for large tan β 4 but decreases for small tan β < ∼ 4, if the m W /M 1,2 correction is ignored. However, since the LSP mass drops faster than the χ ± 1 mass with increasing ε, sizable mass differences between χ ± 1 and χ 0 1 as well as between χ 0 2 and χ 0 1 are developed. Due to the decreasing mass of χ ± 1 with ε, the lower mass bound of m e χ ± 1 > ∼ 94 GeV [13] can constrain the BMSSM light Higgsino-LSP scenario. If µ is negative, the second term in the expressions of m e χ ± 1 in Eq. (8) slows down the lighter chargino mass, which leads to larger mass difference between χ 0 1 and χ ± 1 . Therefore, the negative µ case accommodates larger parameter space to explain all the WMAP data by the Higgsino-LSP. However we note that the combined analysis for the anomalous magnetic moment of the muon as well as b → sγ prefers a positive µ [14]. In what follows, therefore, we take the case of positive µ.
Collider detection -It is well known [12] that in the Higgsino-LSP scenario the mass degeneracy among χ 0 1 , χ 0 2 , and χ ± 1 renders their detection at colliders extremely difficult, even though radiative corrections can increase the mass difference by a few GeV [15] 1 . The decay products of χ ± 1 or χ 0 2 are too soft for their detection. In the BMSSM, the new contribution from the dimension-5 term in Eq. (2)   TeV, µ = 120 GeV and tan β = 3. In this parameter set, the LSP mass is 70 − 110 GeV.
For ε = 0.05 -0.1, the mass difference between χ ± 1 and χ 0 1 is about 15 − 30 GeV and the mass difference between χ 0 2 and χ 0 1 is about 15-40 GeV. These mass splittings are much larger than those due to radiative corrections, which are typically a few GeV. Such sizable mass differences can help us detect the lighter chargino or second-lightest neutralino by tagging more energetic charged leptons or jets in the decay products. Consequently, the phenomenological impact of the BMSSM corrections on the SUSY search at the LHC is expected significant. Nevertheless, we do not perform any full-fledged analysis in the present work, expecting that such a comprehensive analysis will lead to almost the same physical conclusions as those described above.
Dark Matter -In the MSSM Higgsino-LSP scenario, the strong coannihilation due to mass degeneracy pushes the Higgsino mass rather high, about 1-1.2 TeV, to account for the CDM relic density in Eq. (1). We note, in passing, that the MSSM radiative corrections affect the relic density rather mildly [17]. In the BMSSM, however, the large mass differences of ∆m 0 and ∆m ± can suppress the coannihilation effectively; a much lighter Higgsino-LSP can account for the CDM relic density.
Another compelling feature in the BMSSM arises from the modified χ 0 1 -χ 0 1 -Z coupling which is proportional to (|N 13 | 2 − |N 14 | 2 ). The ε dependence can be easily seen from the first row of the matrix to leading order in the BMSSM corrections, as where ε h = εv 2 c 2β /(4µ 2 ), and we ignore small terms of O(m W /M 1 ) as well as an overall phase [18]. In the MSSM (ε = 0), the light Higgsino-LSP scenario implies an almost vanishing In the BMSSM, the modified neutralino mixing matrix N in Eq. (9) leads to a sizable χ 0 1 -χ 0 1 -Z vertex, linearly proportional to ε. Therefore, the annihilation process χ 0 1 χ 0 1 → Z → ff can be enhanced by the BMSSM corrections, giving a profound effect on the relic density of the LSP.
For a simple quantitative estimate we use a useful formula for the CDM relic density [2] We include all the 2 → 2 self-annihilation and coannihilation processes in calculating the effective annihilation cross section σ eff . Since the mass difference ∆m 0 is substantially larger than ∆m ± for ε ∼ 0.1 (see Fig. 1), we ignore the χ 0 1 χ 0 2 coannihilation in estimating σ eff v and use the following formula for a crude estimate of the thermally-averaged effective annihilation cross section, taking into account the coannihilation from χ ± 1 : We take the freeze-out temperature T f = m e χ 0 1 /25 and the relative velocity v ij = 0.3 during the freeze-out. For the self-annihilation cross section (σ e χ 0 1 e χ 0 1 ) we consider the processes 19]. In the following numerical analysis we fix m e Q L = m e q R = m e l = −A = m A = 1 TeV and include all the radiative corrections to the masses of neutralinos, charginos, and the Higgs bosons.
In Fig. 2, we show the tan β-and M 1 -dependence of the relic density Ω e H h 2 versus ε with the fixed µ = 120 GeV. In Fig. 2(a), we fix tan β = 3 and take three typical values of M 1 = 300, 500, 700 GeV. As expected, the relic density increases with ε due to the suppressed coannihilation. In addition, we observe that decreasing M 1 reduces the LSP relic density. This is because the effective annihilation cross section σ eff , dominated by the process χ 0 1 χ 0 1 → ff for sizable ε, is enhanced by the stronger Bino-Higgsino mixing for smaller M 1 . Nevertheless the M 1 -dependence of the relic density is rather mild.
On the other hand, the tan β-dependence of Ω e H h 2 is strong, as can be seen from Fig. 2(b). We take tan β = 2, 3, 4, 5, 10 for the fixed µ = 120 GeV and M 1 = 500 GeV. The tan β = 2 case has the curve terminated at a large ε as m e χ ± 1 gets below the experimental bound, m e χ ± 1 > 100 GeV. Up to ε ≈ 0.08, the relic density is increasing with ε, because of more suppressed coannihilation. And Ω e H is larger for smaller tan β, which can be attributed to the tan β-dependence of the masses in Eq. (8): Small tan β ≃ 1 maximally reduces the LSP mass, and thus enhances the mass difference ∆m ± , suppressing the coannihilation.
The slope of increasing Ω e H around ε ≃ 0.08 is very steep, due to the kinematic closure of  We set µ = 120 GeV and other SUSY mass parameters to 1 TeV. as ε increases, the χ 0 1 -χ 0 1 -Z vertex becomes stronger and thus the self-annihilation process χ 0 1 χ 0 1 → Z → ff is enhanced. Finally we observed that the WMAP data of Ω e H h 2 ≃ 0.1 with positive µ strongly prefer small tan β. If tan β > ∼ 4, the WMAP data can be only partially explained. Of course, if µ is negative, a much larger parameter space is allowed for the WMAP data.
In order to understand the behavior of the relic density against ε (especially for the  self-annihilation while the W ± Z/W ± γ modes from the χ 0 1 χ ± 1 coannihilation. bumpy shape), we calculate the contribution of each channel for small tan β = 2 case. e −∆m ± /T f . For the χ 0 1 χ 0 1 self-annihilation, the W + W − and ff modes are the dominant channels. As ε increases, the LSP mass m e χ 0 1 decreases. For ε 0.072 the W + W − mode is kinematically closed. The self-annihilation process χ 0 1 χ 0 1 → ff is enhanced due to the χ 0 1 -χ 0 1 -Z vertex being stronger with increasing ε. On the other hand, the coannihilation channels, dominant for small ε, become suppressed with larger ε due to larger mass splittings. As a whole, we have the bump-shaped distribution of the relic density as a function of ε in Fig.2(b). One technical issue can arise when we calculate the thermally averaged annihilation cross section near W + W − thresholds. Since we fix the relative velocity, there is some discrepancy in the cross section which is sensitive to the actual relative velocities near the threshold. A more detailed analysis based on the exact Boltzman equation will be required for more accurate estimates of the cross sections, which is, however, beyond the scope of the present short report.
Another important experimental test for the Higgsino-LSP scenario is the spin- independent scattering cross section σ SI χp of the LSP with nucleons. We have calculated the scattering cross section σ SI χp based on the input parameters given in Ref. [20]. As shown in Fig. 4, if ε is larger than 0.02, the elastic scattering cross section σ SI χp is reduced far below the current limits, mainly because the lightest Higgs boson mass m h increases with the BMSSM corrections. Here we briefly comment on the spin-dependent scattering cross sections. The spin-dependent scattering process, which is experimentally more difficult to extract because of the lack of coherent enhancement unlike the spin-independent process, is mainly mediated by the Z boson. It is enhanced by the stronger χ 0 1 -χ 0 1 -Z coupling with increasing ε. , the χ ± 1 mass (all masses in units of 100 GeV), the Higgsino fraction P e H of the LSP, and the spin-independent scattering cross section σ SI χp in units of 10 −44 cm 2 . As ε increases, the Higgs mass m h increases but both m e χ 0 1 and m e χ ± 1 decrease. However, the mass difference ∆m ± increases while the Higgsino fraction P e H of the LSP stays high ∼ 98%. We see a dramatic enhancement of the Higgsino-LSP relic density Ω e H h 2 near ε = 0.08 or larger, where χ 0 1 χ 0 1 → W + W − channel is kinematically closed. For our parameter choice, the light Higgsino-LSP of its mass about 82 GeV, when ε ≈ 0.085, can explain all the observed CDM relic density in the Universe. For other choices of parameters we can still accommodate a light Higgsino-LSP as a primary candidate for the CDM relic density observed by the WMAP unless tan β is large.
Conclusions -The fine-tuning of the lightest Higgs boson mass m h in the MSSM motivates additional interactions beyond the MSSM around the TeV scale. We have checked that, in the effective Lagrangian approach, the least-suppressed dimension-5 operator λ ( H u · H d ) 2 /M added to the MSSM superpotential can usually increase m h sufficiently for a moderate tan β and it can significantly affect the light Higgsino-LSP scenario. It lifts up the mass degeneracy between the LSP and the lighter chargino and that between the LSP and the second-lightest neutralino as much as a few tens of GeV. As a result, it is expected to improve significantly the chance of detecting the decay products of charginos and neutralinos at the LHC. Another important impact is on the Higgsino-LSP dark matter. Since the mass splittings suppress the coannihilation processes, the WMAP narrow band for the CDM relic density can accommodate a light Higgsino-LSP particle of its mass around 100 GeV for rather small tan β.