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Dark matter constraints in the minimal and nonminimal SUSY standard model

We determine the allowed parameter space and the particle spectra of the minimal SUSY standard model (MSSM) and nonminimal SUSY standard model (NMSSM) imposing correct electroweak gauge-symmetry breaking and recent experimental constraints. The parameters of the models are evolved with the SUSY RGEs assuming universality at the GUT scale. Applying the new unbounded from below (UFB) constraints we can exclude the LSP singlinos and light scalar and pseudoscalar Higgs singlets of the NMSSM. This exclusion removes the experimental possibility to distinguish between the MSSM and NMSSM via the recently proposed search for an additional cascade produced in the decay of the bino into the LSP singlino. Furthermore the effects of the dark matter condition for the MSSM and NMSSM are investigated and the differences concerning the parameter space, the SUSY particle and Higgs sector are discussed.


Introduction
If the standard model (SM) is embedded in a grand unified theory (GUT), finetuning is needed in order to have a Higgs mass of the order of the electroweak scale and not the GUT scale. The quadratic divergences in the radiative corrections to the Higgs mass are the reason for this naturalness problem in the SM. SUSY models have the advantage of being free of quadratic divergences due to cancellations in Feynman graphs with particles and their supersymmetric partners. The simplest SUSY extension of the SM is the minimal SUSY standard model (MSSM) [1,2] with two Higgs doublets H 1 and H 2 . However the MSSM has also a naturalness problem why the µ-parameter of the term µ H 1 H 2 in the superpotential should be much smaller than the Planck mass. The nonminimal SUSY standard model (NMSSM) [3] is a minimal extension of the MSSM. In this model a Higgs singlet N is added to the Higgs sector of the MSSM. Moreover the µ-parameter of the MSSM can be dynamically generated via µ = λx with the term λ H 1 H 2 N in the superpotential of the NMSSM, if the Higgs singlet acquires a vaccum expectation value (VEV) N = x. SUSY models with Higgs singlets can be derived from superstring inspired E 6 or SU(5) × U(1) GUT models and they offer the possibility of spontaneous breaking of the CP symmetry. The discrete Z 3 symmetry of the NMSSM causes a domain wall problem. In the early universe the Z 3 symmetry is spontaneously broken during the electoweak phase transition. Domains of different degenerate vacua develop which are separated by domain walls. Since these domain walls would dominate the energy density of the universe and disturb primordial nucleosynthesis, they have to disappear before the onset of nucleosynthesis. Possible solutions to the domain wall problem and the hierarchy problem reintroduce the terms µ H 1 H 2 and µ ′ N 2 in the superpotential of the NMSSM and possess a gauged R-symmetry or a target space duality symmetry at the Planck scale [4]. In these more general models the NMSSM is approximated for very small values of the parameters µ and µ ′ .
The addition of a Higgs singlet superfield leads to two extra Higgses and one extra neutralino in the NMSSM compared to the particle spectrum of the MSSM. Since the Higgs singlets and the singlinoÑ can mix with the other Higgses and neutralinos, the phe-nomenology of the NMSSM may be modified compared to the MSSM. The phenomenology of the NMSSM has been investigated in several papers [5] - [15]. In SUSY models with R-parity conservation the lightest SUSY particle (LSP) cannot decay and is therefore stable. Especially a neutral LSP like the lightest neutralino is a good candidate for cold dark matter. In the MSSM the dark matter neutralino is mostly a bino. In the NMSSM there exists the alternative of a dark matter neutralino with a larger singlino portion.
The cosmology of these dark matter singlinos has been discussed in [16] at a given fixed low energy scale. The experimentally and cosmologically allowed parameter space of the NMSSM and the dark matter neutralinos have been investigated in [17] using SUSY RG evolutions.
In the present paper we compare the allowed parameter space and the particle spectra of the MSSM and NMSSM. Assuming universality at the GUT scale the parameters of the models are evolved to the electroweak scale with the SUSY RGEs (given in [18] and Appendix A2 for the NMSSM). The Higgs potential is minimized and additional theoretical constraints and recent experimental bounds are imposed. Besides charge and color breaking (CCB) constraints we include here new unbounded from below (UFB) constraints [19].
Furthermore the consequences of the dark matter condition for the MSSM and NMSSM are investigated. In the present paper we give a more detailed analysis of the two models and the dark matter condition than in our previous paper [20].
In section 2 we introduce the NMSSM. The additional theoretical and experimental constraints are discussed in section 3. In section 4 we describe the dark matter constraints.
The procedure of finding solutions of the NMSSM is explained in section 5. In section 6 the parameter space and the particle spectra of the MSSM and NMSSM are investigated without and with the dark matter condition. In Appendix A1 we give the complete formulas for the neutralino annihilation cross section in the NMSSM which are needed to calculate the relic neutralino density. In Appendix A2 the SUSY RGEs [18] for the NMSSM are listed.

The NMSSM
The Higgs sector of the MSSM consists of two Higgs doublets H 1 and H 2 . In the NMSSM [3] an additional Higgs singlet N is introduced. The superpotential of the NMSSM contains only Yukawa terms with dimensionless couplings Q is a squark doublet.Ũ andD are up-type and down-type squark singlets. ǫ is the antisymmetric tensor with ǫ 12 = 1. The SUSY breaking is parametrized by different types of SUSY soft breaking terms in the lagrangian: In SUSY models the tree-level scalar potential is given by the formula: where A i are scalar fields and T a ij are the gauge group generators. The part of the scalar potential with only Higgs fields corresponds to the Higgs potential. The neutral part of the tree-level Higgs potential is given by: In order to calculate the Higgs VEVs reliably we use the effective 1 loop Higgs potential consisting of the tree-level Higgs potential and the radiative corrections The radiative corrections [6] to the effective Higgs potential arising from all particles and antiparticles with field-dependent mass m i , spin S i and color degrees of freedom C i are given by Q is the renormalisation scale which we take to be equal to the electroweak scale M weak ≃ 100 GeV . Here we only consider the important contributions of the top quark and stops t 1,2 . The electroweak gauge-symmetry SU(2) I × U(1) Y is spontaneously broken to the electromagnetic gauge-symmetry U(1) em by the Higgs VEVs H 0 i = v i with i = 1, 2 and N = x. The local minimum of the effective 1 loop Higgs potential is determined by the three minimum conditions for the VEVs: where v = v 2 1 + v 2 2 = 174 GeV , tan β = v 2 /v 1 and Σ i = ∂V Higgs rad /∂v 2 i . The neutralinos are Majorana particles and mixed states of the gauginos and Higgsinos.
The mass and composition of the neutralinos is defined by the following part of the In this basis the symmetric mass matrix M of the neutralinos has the form: With µ = λ x the first 4 × 4 submatrix recovers the mass matrix of the MSSM [2]. The mass of the neutralinos is here obtained by diagonalising the mass matrix M with the orthogonal matrix N. (Then some mass eigenvalues may be negative [21].) The neutralinosχ 0 i (i = 1 − 5) are ordered with increasing mass |m i |, thusχ 0 1 is the LSP neutralino. The matrix elements N ij (i, j = 1 − 5) describe the composition of the neutralinoχ 0 i in the basis Ψ j (Eq. (13)). For example the bino portion of the LSP neutralino is given by N 2 11 and the singlino portion of the LSP neutralino by N 2 15 . The diagonalising of the mass matrix M leads to a polynomial equation of degree 5, which cannot be solved analytically in general. Therefore we calculate the eigenvalues and eigenvectors of the symmetric mass matrix M numerically by using a subroutine in the NAG Fortran Library.
The possibility of gauge coupling unification is an important motivation for SUSY GUTs.
In contrast to non-supersymmetric GUTs [22], the gauge couplings in SUSY GUTs are allowed to unify g a (M X ) = g 5 ≃ 0.72 at a scale M X ≃ 1.6 × 10 16 GeV . Within a minimal N = 1 supergravity framework universality naturally arises for the SUSY soft breaking parameters: The Yukawa couplings at the GUT scale take the values λ(M X ) = λ 0 , k(M X ) = k 0  (7)).

Theoretical and experimental constraints
The solutions of the SUSY RGEs ( [18], Appendix A2) have to fulfill several theoretical and experimental constraints. The minimum of the effective 1 loop Higgs potential V Higgs 1loop (Eq. (7)) should be not only a local minimum but also a stable minimum. A local minimum is stable against tunnelling into a lower minimum, if its lifetime is large enough (of the order of the age of the universe). A global minimum is certainly stable, but it may be difficult to prove, that it is really a global minimum. Here we only investigate, whether the physical minimum is lower than the unphysical minima of V Higgs 1loop (v 1 , v 2 , x) with at least one vanishing VEV [5].
The conservation of charge and color would be broken by slepton and squark VEVs. To avoid a scalar potential with such a minimum, the trilinear couplings have to fulfill the following conditions [18]: In [19] improved charge and color breaking (CCB) constraints and unbounded from below (UFB) constraints are analysed for the MSSM. The by far strongest of these new constraints is the UFB-3 constraint (Eq. (33) in the first publication of Ref. [19]), in contrast to the remaining ones. To apply the UFB-3 constraint to the NMSSM we have to modify the formulas in [19]: In the NMSSM a dangerous field direction of the tree-level scalar potential V tree (Eq.(3)) occurs for VEVs of the neutral Higgs H 0 2 , the electron sneutrinõ ν e denoted by L 1 =L 0 1 and the stausτ − L andτ + R denoted byL − 3 andẼ 3 . The VEVs of the staus are introduced in order to cancel the F-term of H 0 1 : For the VEVs of the staus we choose the relation e =L − 3 =Ẽ 3 . In analogy to the MSSM we have replaced in Eq. (20) µ by λ x where the Higgs singlet VEV N = x is determined by the physical minimum of the NMSSM Higgs potential. Since we consider here the scalar potential and not only the Higgs potential, the Higgs singlet VEV x may be optimized further. But here we don't investigate this more general case. The NMSSM tree-level scalar potential along the described field direction is given by: Now we can determine the dangerous field direction of V tree (Eq.(21)), which is given by: provided that For smaller values of |H 0 2 | we obtain |L 1 | = 0. In the field direction of Eq. (22) and (23) the tree-level scalar potential of the NMSSM reads: For |L 1 | = 0 the tree-level scalar potential is given by: Since the tree-level scalar potential V U F B−3 should be larger than the physical minimum of the tree-level Higgs potential V Higgs tree (Eq.(6)), the UFB-3 constraint can be written as The tree-level Higgs potential V Higgs Because we assume h t ≫ h b , h τ , we restrict | tan β | to be smaller than 20 [7]. For the Higgs singlet VEV x we choose the range |x| < 80 T eV [7,8,13]. In the present paper we choose the parameters m 0 and |m 1/2 | to be smaller than 1500 GeV . This restriction is motivated by requiring the absence of too much fine-tuning [13], which disfavors solutions with enormous sensitivity of m Z to the GUT scale parameters h t0 or λ 0 and k 0 . The dark matter constraints further restrict the allowed parameter region of m 0 and |m 1/2 | as shown in Fig.1.
For the top quark pole mass we take the range 169 GeV < m t < 181 GeV corresponding to the recent world average measurement [23]. Since charged or colored massive stable particles are cosmologically disfavored LSPs, we assume the lightest neutralino to be the LSP and impose the constraints mχ0 1 < mτ 1 and mχ0 1 < mt 1 . The SUSY particles in the NMSSM have to fulfill the following experimental conditions: mν ≥ 41.8 GeV [24], mẽ R ≥ 70GeV [25], mq ≥ 176GeV (ifmg < 300GeV ) [24] and mq ≥ 70GeV (for all mg) [26], [30] or the coupling of the lightest scalar Higgs S 1 to the Z boson is reduced compared to the SM [31] assuming visible decays with branching ratios like the SM Higgs

Dark matter constraints
Inflationary cosmological models are proposed to solve the flatness, horizon and magnetic monopole problem. In these models the universe is flat with a density equal to the critical density ρ crit = 1.054×10 −5 h 2 0 GeV /cm 3 implying a cosmological density parameter of Ω = ρ/ρ crit = 1. The Hubble constant H 0 = 100 h 0 km s −1 Mpc −1 is restricted by observations to the range 0.4 < h 0 < 1 of the scaled Hubble constant h 0 . To be consistent with the observed abundances of the isotopes D, 3 He, 4 He and 7 Li the big bang nucleosynthesis model restricts the baryon density to Ω baryons ≤ 0.1. The discrepancy between the cosmological density Ω = ρ/ρ crit = 1 and the baryon density requires a huge amount of dark matter in the universe. One of the favored theories to explain the structure formation of the universe is the cold + hot dark matter model (CHDM) [33].
In this model the dark matter consists of hot dark matter (massiv neutrinos) and cold dark matter (neutralinos) with Ω hot ≃ 0.3 and Ω cold ≃ 0.65. With the allowed range of the scaled Hubble constant h 0 the condition for the relic density of the neutralinos in the CHDM model reads The relic neutralino density can be calculated by solving the Boltzmann equation for the early universe. The Boltzmann equation describes the particle densities in the cosmic plasma governed by the annihilation cross section. The annihilation cross section of the neutralinos can be expanded in powers of the relative velocity v of the non-relativistic neutralinos: The thermally averaged annihilation cross section [34] - [36] is then given by < σ ann v > = x 1.
During the expansion of the universe its temperature falls and reaches the freeze-out temperature of the neutralinos T f r , at which the neutralinos drop out of the thermal equilibrium with the cosmic plasma. The freeze-out temperature of the neutralinos T f r follows from the equation [34] - [37] x f r = ln 0.0764 |mχ0 where M P = 1.221×10 19 GeV is the Planck mass, g * ≈ 81 is the effective number of degrees [36] the relic neutralino density [34] - [37] is given by The expansion of the annihilation cross section σ ann v = a+bv 2 is not a good approximation in the vicinity of poles or thresholds [38]. In the present work the case of coannihilation is not important [17,39], since the LSP neutralino in the MSSM and NMSSM is mostly bino-like or a mixed state with small mass (see Fig.4). Furthermore the LSP singlinos [9,10,13] of the NMSSM are now excluded by the UFB-3 constraint (Eq. (27)) as discussed in section 6.
Compared to the MSSM the annihilation cross section of the neutralinos changes in the NMSSM because of more particles in this model, which can mix, and modified vertices. In our numerical analysis we consider all possible decay channels. The relevant formulas for the NMSSM are given in the Appendix A1. They are obtained by substituting the MSSM couplings in [34] by the corresponding NMSSM couplings and by slightly modifying the partial wave amplitudes in [34].

Procedure of finding solutions
The SUSY soft breaking parameters m i and A i and the Yukawa couplings λ, k and h t are evolved with the SUSY RGEs ( [18], Appendix A2) from the GUT scale M X to the electroweak scale M weak . Since we assume universality at the GUT scale M X (Eq. (16)) we have only 6 GUT scale parameters m 0 , m 1/2 , A 0 , λ 0 , k 0 and h t0 . Together with the low energy parameters tan β and x there are 8 parameters in the NMSSM with only 5 of them being independent. In our procedure we take λ 0 , k 0 , h t0 , tan β and x as independent input parameters. The remaining 3 parameters (m 0 , m 1/2 , A 0 ) are then determined by the three minimum conditions (9)(10)(11) and are numerically calculated by the method described in [7,20]: The three minimum conditions (9-11) depend only on the low energy parameters Since λ 0 , k 0 and h t0 are (randomly choosen) fixed input parameters, all low energy parameters p k (M weak ) are functions f k (which can be constant) of the GUT scale parameters m 0 , m 1/2 and A 0 only: To obtain the functions f k we numerically solve the SUSY RGEs ( [18], Appendix A2) . A solution to a system of first-order differential equations can be found with the method of Runge, Kutta and Merson. A subroutine for this numerical method is available in the CERN Program Library. An additional subroutine has to be written which calculates the derivatives dp k /dt (k = 1−n) depending on the current values of t and p 1 , ..., p n . In the three minimum conditions (31) we then substitute the low energy parameters p k (M weak ) by the functions f k and obtain the generic The SUSY breaking parameters m 0 , m 1/2 and A 0 are then determined by the zero points of . In order to find a solution of Eq. (33) we first minimize 3 i=1 G i (m 0 , m 1/2 , A 0 ) 2 and then we check, whether the minimum is also a zero point. The minimization is performed by using the simplex method of Nelder and Mead. A corresponding minimization routine SIMPLEX is available in the MINUIT package of the CERN Program Library. The numerical procedure needs starting values for the parameters m 0 , m 1/2 and A 0 , which are randomly generated using a logarithmic measure within their ranges.
To obtain a representative sample of solutions of the NMSSM, we randomly generate  7)) has a local minimum for the input parameters tan β and x, we impose the additional theoretical and experimental constraints described in section 3.
The very strong UFB-3 constraint (Eq. (27)) excludes about 80% of the solutions which fulfill all the other constraints of section 3. For every solution we calculate the masses of all SUSY particles [2,40] and Higgses [3,6,12,13]. Altogether we obtain about 6600 solutions which fulfill all requirements. The ranges of the parameters, for which we find solutions, are given by m 0 ∈ [ 60 GeV , 1500 GeV ], |m 1/2 | ∈ [ 120 GeV , 1500 GeV ], plane. The parameter region with small m 0 and large |m 1/2 | is excluded by the UFB-3 constraint [19]. Furthermore in the region |m 1/2 | ≫ m 0 the stau or stop would be the LSP.
But a charged or colored LSP like the stau or stop is cosmologically disfavored [41]. In the upper right picture the cosmologically allowed solutions are shown. The dark matter condition leads to an upper bound of 600 GeV for |m 1/2 | and leaves m 0 unconstrained.
Values of m 0 larger than about 300GeV are cosmologically allowed due to Z-pole or Higgspole enhanced neutralino pair annihilation. Comparable scatter-plots for the constrained MSSM are discussed in [42]. In that paper upper bounds of 940GeV for |m 1/2 | and 500GeV for m 0 are deduced for m t = 170GeV . The lower left picture shows the NMSSM solutions.
The effect of the UFB-3 constraint (Eq. (27)) in the NMSSM is comparable to the MSSM.
In the NMSSM this constraint also excludes the parameter region with small m 0 and large |m 1/2 |. The reason, why the case m 0 ≫ |m 1/2 | is suppressed in the NMSSM, is discussed in [9]. It follows from the condition, that the minimum of the scalar potential does not break the conservation of charge and color, and from the condition that the physical minimum is lower than the symmetric vacuum, 9 m GeV for the lighter selectron as discussed in [20] and below. This can be understood from the fact [34], that the LSP neutralino in the NMSSM is mostly bino-like (see Fig.4) and the annihilation cross section is dominated by sfermion exchange.

Conclusions
In the present paper we have made a comparison between the MSSM and NMSSM concerning the parameter space and the particle spectra. In contrast to the MSSM, the case m 0 ≫ |m 1/2 | is disfavored in the NMSSM as discussed in [9] and the squark masses are nearly proportional to the gluino mass (cf. Figs. (6) and (7)). Furthermore the gluino and  Fig.4 and Fig.9 and discussed in section 6. In this way the UFB-3 constraint strongly reduces the differences between the NMSSM and the MSSM. for the cosmologically allowed pseudoscalar Higgs P 1 is somewhat lower in the NMSSM m P 1 < 900 GeV (Fig.9).
Despite of the discussed differences, it may be difficult to distinguish experimentally between the MSSM and NMSSM. One possibility is the measurement of smaller couplings of the neutral Higgses in the NMSSM, which result from the mixing of the ordinary Higgs states with the Higgs singlets as discussed in [8,9] and [11] - [13]. Another possibility suggested in [9,10] is the search for an additional cascade in the NMSSM, which is produced in the decay of the bino-like second lightest neutralino into the LSP singlino. Since the LSP singlinos are excluded by the UFB-3 constraint this additional cascade is not a way to discriminate between the NMSSM and MSSM any more.

Appendix A1: Annihilation cross section
To calculate the relic neutralino density (Eq. (30)) in the NMSSM we need the annihilation cross section of the neutralinos into the different decay products X and Y .
where λ X and λ Y are the helicities of particle X and Y . Within the partial wave formalism the annihilation cross section is obtained [34] as: s ≈ 4m 2 χ is the center of mass energy squared. S is a symmetry factor, which is 2 if X = Y . Summation over final helicities is implicit in Eq. (34). The partial wave amplitude A( 2S+1 L J ) describes annihilation from an initial state with spin S, orbital angular momentum L and total angular momentum J. The partial wave amplitudes A( 2S+1 L J ) for the NMSSM are obtained by slightly modifying the partial wave amplitudes of the MSSM in [34] and by using the corresponding NMSSM couplings. Thereby we have corrected some misprints [44] in [34]. One modification of the neutralino annihilation is the exchange of 5 neutralinos in the NMSSM instead of 4 neutralinos in the MSSM. Other modifications concern the exchange of the scalar Higgses S 1 , S 2 and S 3 in the NMSSM instead of H 1 and H 2 in the MSSM and the exchange of the pseudoscalar Higgses P 1 and P 2 in the NMSSM instead of P in the MSSM. In addition the NMSSM coupling g α−+ (Eq.(98)) has to be inserted in Eq.(86). In the MSSM there is only one pseudoscalar Higgs P and the coupling g α−+ turns out to be 1. In the following the partial wave amplitudes A( 2S+1 L J ) of the NMSSM for all decay channels are given in leading order in the relative velocity v of the non-relativistic neutralinos. Previously we give some NMSSM couplings and definitions. The couplings of the W boson to the charginoχ + j and the LSP neutralino, which is denoted with index 0 in [34] instead of 1, have the same form in the MSSM and NMSSM, but a different matrix N (Eq.(15)) enters: The mass matrix of the charginos is diagonalized by the orthogonal matrices U and V , which are explicitly given in [40]. In the NMSSM the couplings of the charged Higgs to the charginoχ + j and the LSP neutralino has to be modified compared to the MSSM: In the MSSM and NMSSM the coupling of the Z boson to the neutralinosχ 0 i andχ 0 j is given by: In the NMSSM the coupling of the scalar Higgs S a to the neutralinosχ 0 i andχ 0 j changes compared to the MSSM and can be written as: The 3 × 3 mass matrix of the scalar Higgses S i [3] in the basis (H 0 is diagonalised by the matrix U S . The scalar Higgses S i (i = 1 − 3) are ordered with increasing mass. The coupling of the pseudoscalar Higgs P α to the neutralinos is also different in the MSSM and NMSSM. In the NMSSM this coupling has the following form: The 3 × 3 mass matrix of the pseudoscalar Higgses P α and the neutral Goldstone boson [3] in the basis (H 0 is here diagonalised by the 3 × 3 matrix U P . The pseudoscalar Higgses P α (i = 1, 2) are ordered with increasing mass. In [34] the following definitions are used: The W bosons are produced by Z boson and scalar Higgs S i (i = 1, 2, 3) exchange in the s-channel and charginoχ + j (j = 1, 2) exchange in the t-and u-channel.
are the rescaled widths of the scalar Higgses S i . The coupling of the scalar Higgs S i to the LSP neutralinos and to the W bosons is combined to F i . In the NMSSM F i is given by: The Z bosons are produced by scalar Higgs S i (i = 1, 2, 3) exchange in the s-channel and neutralinoχ 0 j (j = 1 − 5) exchange in the t-and u-channel. (67) The Z boson and the scalar Higgs S a are produced by Z boson and pseudoscalar Higgs P α (α = 1, 2) exchange in the s-channel and neutralinoχ 0 j (j = 1 − 5) exchange in the tand u-channel.
In [34] the following definition is used: G Pα = Γ Pα m Pα /m 2 χ are the rescaled widths of the pseudoscalar Higgses P α . The couplings appearing in the neutralinoχ 0 j exchange are combined to J j : L α is the product of the coupling of the pseudoscalar Higgs P α to the LSP neutralinos and to the Z boson and the scalar Higgs S a . In the NMSSM L α reads: O consists of the couplings appearing in the Z boson exchange. In the NMSSM O is given by: The Z boson and the pseudoscalar Higgs P α are produced by scalar Higgs S i (i = 1, 2, 3) exchange in the s-channel and neutralinoχ 0 j (j = 1 − 5) exchange in the t-and u-channel.
In [34] the following definition is used: The couplings appearing in the neutralinoχ 0 j exchange are combined to K j : L i is the product of the coupling of the scalar Higgs S i to the LSP neutralinos and to the Z boson and the pseudoscalar Higgs P α . In the NMSSM L i has the following form: The W boson and charged Higgs are produced by scalar Higgs S i (i = 1, 2, 3) and pseudoscalar Higgs P α (α = 1, 2) exchange in the s-channel and charginoχ + j (j = 1, 2) exchange in the t-and u-channel.
In [34] the following definition is used: The couplings appearing in the charginoχ + j exchange are combined to: L i is the product of the coupling of the scalar Higgs S i to the LSP neutralinos and to the W boson and the charged Higgs. In the NMSSM L i is given by: In the NMSSM the coupling g α−+ of the pseudoscalar Higgs P α to the W boson and charged Higgs reads: In the MSSM with only one pseudoscalar Higgs P the coupling g α−+ turns out to be 1 [34].
The scalar Higgses S a and S b are produced by scalar Higgs S i (i = 1, 2, 3) exchange in the s-channel and neutralinoχ 0 j (j = 1 − 5) exchange in the t-and u-channel.
A( 3 P 0 ) : The pseudoscalar Higgses P α and P β are also produced by scalar Higgs S i (i = 1, 2, 3) exchange and neutralinoχ 0 j (j = 1 − 5) exchange. The corresponding amplitudes for P α and P β production can be obtained by replacing a, b by α, β or S a , S b by P α , P β and R j by −R j . In the NMSSM the trilinear scalar Higgs coupling g aaa is given by: The trilinear scalar Higgs coupling g aab (or g aba ) has the following form: Finally the trilinear scalar Higgs coupling g 123 (or g 132 or g 231 ) reads: The trilinear scalar and pseudoscalar Higgs coupling g ααa (or g aαα ) is given by: The trilinear scalar and pseudoscalar Higgs coupling g αβa (or g aαβ or g aβα ) can be written as: The scalar Higgs S a and the pseudoscalar Higgs P β are produced by Z boson and pseudoscalar Higgs P α (α = 1, 2) exchange in the s-channel and neutralinoχ 0 j (j = 1 − 5) exchange in the t-and u-channel.
G Z = Γ Z m Z / m 2 χ is the rescaled widths of the Z boson. The couplings appearing in the Z boson exchange are combined to g aβZ . In the NMSSM g aβZ has the following form: The charged Higgses are produced by Z boson and scalar Higgs S i (i = 1, 2, 3) exchange in the s-channel and charginoχ + j (j = 1, 2) exchange in the t-and u-channel.
A( 3 P 0 ) : A( 3 P 2 ) : g +−i is the product of the coupling of the scalar Higgs S i to the LSP neutralinos and to the charged Higgses. In the NMSSM g +−i is given by: The fermions are produced by Z boson, scalar Higgs S i (i = 1, 2, 3) and pseudoscalar Higgs P α (α = 1, 2) exchange in the s-channel and sfermionf 1,2 exchange in the t-and u-channel.
The sfermion mass eigenstatesf 1,2 are defined bỹ The couplings of the mixed sfermionf 1,2 to the fermion and LSP neutralino can be written as: where the couplings of the unmixed sfermionf L,R to the fermion and LSP neutralino are given by: Y a0 = √ 2 g 2 tan θ W e fa N 01 (126) In the NMSSM the couplings of the Higgses to the fermions have to be modified compared to the MSSM: