CERN-PH-TH/2008-219

Neutralino Dark Matter in the USSM

[8mm] J. Kalinowski, S.F. King and J.P. Roberts

[3mm] Physics Department, University of Warsaw, 00-681 Warsaw, Poland

[3mm] Theory Division, CERN, CH-1211 Geneva 23, Switzerland

[3mm] School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, U.K.

[3mm] Center for Cosmology and Particle Physics, New York University,

New York, NY 10003, USA

[1mm]

This paper provides a comprehensive discussion of neutralino dark matter within classes of extended supersymmetric models referred to as the USSM containing one additional SM singlet Higgs plus an extra , together with their superpartners the singlino and bino’. These extra states of the USSM can significantly modify the nature and properties of neutralino dark matter relative to that of the minimal (or even next-to-minimal) supersymmetric standard models. We derive the Feynman rules for the USSM and calculate the dark matter relic abundance and direct detection rates for elastic scattering in the USSM for interesting regions of parameter space where the largest differences are expected.

## 1 Introduction

One of the benefits of weak scale supersymmetry (SUSY) with conserved R parity is that the lightest supersymmetric particle (LSP) is absolutely stable, and provides a weakly interacting massive particle (WIMP) candidate [1, 2] capable of accounting for the observed cold dark matter (CDM) relic density [3]. In particular, the lightest neutralino in SUSY models is an excellent candidate, providing its mass, composition and interactions are suitably tuned to result in the correct value of . The minimal supersymmetric standard model (MSSM) has become a widely studied paradigm [4]. However the stringent upper bound on the Higgs boson mass in the MSSM combined with its experimental lower bound from LEP has led to some tension in the electroweak symmetry breaking sector, roughly characterized by a fine-tuning of parameters at the percent level [5]. While the experimental elusiveness of the Higgs boson may cast some doubt on the MSSM, there are a host of non-minimal SUSY models which predict a heavier and/or more weakly coupled Higgs boson [4].

A further reason to move beyond the minimal case is the so-called mu problem of the MSSM [6]. The MSSM contains a bilinear mass term that couples the two Higgs doublets with a dimensionful coupling . This term is SUSY preserving, and as such only has two natural values, and (unless special forms of the Kahler metric are assumed). However experimental data and the stability of the Higgs mass requires that be of the order of the SUSY breaking scale. In non-minimal SUSY models the mu problem is solved by setting and including an additional superfield , a singlet under the Standard Model (SM) gauge group, which couples to the Higgs doublet superfields according to , where is a dimensionless coupling constant. We shall refer to such models generically as singlet SUSY models. Such a coupling replaces the SUSY Higgs/Higgsino mass term of the MSSM. The singlet vacuum expectation value (VeV) then dynamically generates a SUSY Higgs/Higgsino mass near the weak scale as required. This results in an increased Higgs boson mass upper bound depending on the value of , and hence a welcome reduction in electroweak fine tuning in addition to solving the problem of the MSSM [7].

However, although an extra singlet superfield seems like a minor modification to the MSSM, which does no harm to either gauge coupling unification or neutralino dark matter, there are further costs involved in this scenario since the introduction of the singlet superfield leads to an additional accidental global (Peccei-Quinn (PQ) [8]) symmetry which will result in a weak scale massless axion when it is spontaneously broken by [9]. Since such an axion has not been observed experimentally, it must be removed somehow. This can be done in several ways resulting in different non-minimal SUSY models, each involving additional fields and/or parameters. For example, the classic solution to this problem is to introduce a singlet term , as in the next-to-minimal supersymmetric standard model (NMSSM) [10], which reduces the PQ symmetry to the discrete symmetry . The subsequent breaking of a discrete symmetry at the weak scale can lead to cosmological domain walls which would overclose the Universe. This can be avoided by breaking the symmetry explicitly without upsetting the hierarchy problem by non-renormalizable operators that obey a -symmetry [11], or by removing the term altogether [12].

Another solution to the axion problem of singlet models, which we follow, is to promote the PQ symmetry to an Abelian gauge symmetry [13]. The idea is that the extra gauge boson will eat the troublesome axion via the Higgs mechanism resulting in a massive at the TeV scale. The essential additional elements of such a scenario then consist of two extra superfields relative to those of the MSSM, namely the singlet superfield and the gauge superfield . The scenario involving only the MSSM superfields plus these two additional superfields, may be considered as a phenomenological model in its own right which has been referred to as the USSM. In the USSM, then, the MSSM particle spectrum is extended by a new CP-even Higgs boson , a gauge boson and two neutral –inos: a singlino and a bino’ while other sectors are not enlarged. The presence of new singlino and bino’ states greatly modifies the phenomenology of the neutralino sector both at colliders and in cosmology-related processes. The collider phenomenology and cosmology of the USSM has been studied in [14, 15, 16, 17, 18, 19, 20], which we briefly review as follows.

The collider phenomenology of the USSM has recently been considered in [18]. The neutralino production cross sections and their decay branching fractions depend crucially on their masses and composition with respect to the MSSM case. If the new -ino states are heavy, their influence on the MSSM neutralinos is small. In contrast, if the singlino mass scale is low, the production rates can be quite different and, since there are more neutralinos, the decay chains of sparticles can be longer. Moreover, if the mass gaps between the MSSM and new -inos are very small, the standard decay modes are almost shut and radiative transitions between neutralino states with a soft photon may be dominant. In such a case the decay chains can be apparently shorter, a feature which is of relevance for the LHC experiments.

The dark matter phenomenology of the USSM was first studied in [15, 16], and more recently in [19]. In [15], the analysis was performed in a scenario with a very light and considered the case in which the LSP was a very light singlino. This allowed the authors to consider the annihilation of dark matter in the early universe to be dominated by s-channel processes, allowing an analytic solution to the dark matter relic density to be obtained. In the full parameter space of the USSM, this is just one possibility, and indeed such a light is heavily disfavored by current data. In [19] the recoil detection of the dark matter candidate in the USSM (and other non-minimal SUSY models) was considered.

In this paper, we provide an up to date and comprehensive analysis of neutralino dark matter in the USSM111The recent observation of a positron excess by the PAMELA collaboration [21] have caused a flurry of speculation that the high energy positrons are produced by annihilating dark matter in the galactic halo[22]. An alternative explanation is that astrophysical sources could account for the positron excess - in particular nearby pulsars [23]. It is unclear as yet which of these explanations is correct and as a result we do not address the PAMELA results further in this work.. We provide a complete discussion of the extended gauge, neutralino, Higgs squark and slepton sectors in the USSM, and using the LanHEP [24] package, derive all the new Feynman rules involving these extended sectors. We first provide a complete qualitative discussion of the new annihilation channels relevant for the calculation of the cold dark matter relic density for the neutralino LSP in the USSM. We also discuss the elastic scattering cross section for the neutralino LSP in the USSM, including both spin-independent and spin-dependent parts of the cross sections, relevant for the direct dark matter search experiments. We then survey the parameter space of the USSM, and discuss quantitatively how the nature and composition of the neutralino LSP can be significantly altered compared to that in the MSSM due to the extra singlino and bino’ states, for different ranges of parameters. The Feynman rules are then implemented into the micrOMEGAs [25] package in order to calculate the relic density for the corresponding regions of parameter space. This provides a full calculation of the annihilation channels including co-annihilation and careful treatment of resonances as well as accurately calculating the relic density for an arbitrary admixture of states. In this way we extend the analysis of USSM neutralino dark matter annihilation beyond the specific cases previously studied in the literature. We also perform an equally general calculation of the direct detection cross-sections for USSM dark matter for elastic neutralino–nuclei scattering.

It is worth emphasizing that the USSM is not a complete model, since from its definition it does not include the additional superfields at the TeV scale, charged under the gauged Abelian symmetry, which are necessarily present in order to cancel the fermionic gauge anomalies involving the gauge symmetry. For example, a well motivated and elegant solution to the problem of anomaly cancelation is to identify the Abelian gauge group as a subgroup of and then cancel the anomalies by assuming complete 27 dimensional representations of matter down to the TeV scale. With the further requirement that the right-handed neutrino carries zero charge under the Abelian gauge group (in order to have a high scale see-saw mechanism) this then specifies the theory uniquely as the ESSM [26, 27]. However our working assumption is that the additional matter superfields required to cancel anomalies are heavy compared to the mass. The USSM considered in this paper may thus be regarded as a low-energy truncation of the ESSM model, with other ESSM fields assumed heavy, and the charge assignments under the extra as given in [27] and summarized in Section 2.

Despite that fact that the USSM must be regarded as a truncation of a complete model, it makes sense to study the physics and cosmology in the USSM since it provides a simplified setting to learn about crucial features which will be relevant to any complete model involving an additional gauge group and a singlet. For example, as already mentioned the neutralino LSP in the USSM may have components of the extra gaugino and singlino , in addition to the usual MSSM neutralino states. Naively we might expect that the dark matter phenomenology of such regions would be similar to that of singlino dark matter in the NMSSM. However this is not the case. The inclusion of the bino’ state, as well as the lack of a cubic interaction term , results in a significant change in the phenomenology. Also the neutralino mass spectrum in the USSM is very different from that of the NMSSM as the singlino mass is determined indirectly by a mini-see-saw mechanism involving the bino’ soft mass parameter rather than through a diagonal mass term arising from the cubic . The lack of a cubic interaction term also restricts the annihilation modes of the singlino, making it dominantly reliant on annihilations involving non-singlet Higgs bosons and higgsinos. As the USSM has a different Higgs spectrum to the NMSSM, notably in the pseudoscalar Higgs sector, the Higgs dominated annihilation channels of the USSM singlino are significantly modified with respect to the NMSSM singlino. As Higgs exchange diagrams dominate the direct detection phenomenology, the difference in the Higgs spectrum and the singlino interactions results in significant differences in the direct detection predictions as well.

The remainder of the paper is organized as follows. In section 2 we shall define the Lagrangian of the USSM and discuss the Higgs, , neutralino and sfermion sectors. In sections 3 and 4 we shall give an overview of the important features of the relic density calculation and the direct detection calculation, respectively, highlighting the main differences to the MSSM. In section 5 we present the results of the full numerical calculations for both the relic density and the direct detection cross-section. It will be performed in two physically interesting scenarios: (A) with the MSSM higgsino and gaugino mass parameters fixed, while the mass of the extra U(1) gaugino taken free (to complement the collider phenomenology discussed in Ref. [18]); (B) with GUT-unified gaugino masses. Section 6 summarizes and concludes the paper. The mass matrix structure of the extended Higgs scalar sector, and a discussion of the Feynman rules involving the extended neutralino sector in the USSM are given in a pair of appendices.

## 2 The USSM model

Including the extra U(1) symmetry, the gauge group of the model is with the couplings , respectively. In addition to the MSSM superfields, the model includes a new vector superfield and a new iso–singlet Higgs superfield . The usual MSSM Yukawa terms of the MSSM superpotential (i.e. without the term) are augmented by an additional term that couples the iso–singlet to the two iso–doublet Higgs fields:

 ^W=^WY+λ^S(^Hu^Hd). (1)

The coupling is dimensionless. Gauge invariance of the superpotential under U(1) requires the U(1) charges to satisfy and corresponding relations between the U(1) charges of Higgs and matter fields. In the following we adopt the U(1) charges as in the ESSM model [27], see Table 1. (For notational convenience we will also use , and .) The effective parameter is generated by the vacuum expectation value of the scalar –field.

The USSM particle content, in addition to the MSSM particles, includes a single extra scalar state, a new Abelian gauge boson and an additional neutral higgsino and gaugino state. The chargino sector remains unaltered, while the sfermion scalar potential receives additional D-terms.

### 2.1 The Abelian gauge sector

With two Abelian gauge factors, U(1) and U(1), the two sectors can mix through the coupling of the kinetic parts [28],

 Lgauge=132∫d2θ{^WY^WY+^WX^WX+2sinχ^WY^WX}, (2)

where and are the corresponding chiral superfields. The gauge/gaugino part of the Lagrangian can be converted back to the canonical form by the GL(2,) transformation from the original superfield basis to the new one  [28, 29]:

 (^WY^WX)=(1−tanχ01/cosχ)(^WB^WB′). (3)

This transformation alters the U(1)U(1) part of the covariant derivative to

 Dμ = ∂μ+igYYiBμ+i(−gYYitanχ+gXcosχQXi)B′μ (4) = ∂μ+ig1YiBμ+ig′1Q′iB′μ, (5)

where we introduced the notation , . We will also use for the low-energy (non-GUT normalized) hypercharge gauge coupling.

With the above mixing matrix the hypercharge sector of the Standard Model is left unaltered, while the effective U(1) charge is shifted from its original value to

 Q′i=QXi−g1g′1Yitanχ. (6)

As a result of the kinetic mixing, new interactions among the gauge bosons and matter fields are generated even for matter fields with zero U(1) charge.

In the ESSM the two U(1) gauge groups are automatically orthogonal at the GUT scale and the RG running effects give acceptable small mixing at the low scale [27, 30] providing in a natural way the general agreement between SM analyses and precision data [31]. Therefore in the reminder of the paper we will simply write instead of .

After breaking the electroweak and U(1) symmetries spontaneously due to non–zero vacuum expectation values of the iso–doublet and the iso–singlet Higgs fields,

 (7)

the mass matrix takes the form

 M2ZZ′=(M2ZΔ2Δ2M2Z′), (8)

where

 M2Z = g′2+g224v2 Δ2 = g′1√g′2+g222v2(Q1cos2β−Q2sin2β) M2Z′ = g′21v2(Q21cos2β+Q22sin2β)+g′21Q2Sv2S (9)

We then diagonalise the mass matrix to give the mass eigenstates:

 (Z1Z2)=Dij(ZZ′)=(cosαZZ′sinαZZ′−sinαZZ′cosαZZ′)(ZZ′) (10)

where the resultant masses and mixing angle are given by

 M2Z1,Z2=12(M2Z+M2Z′∓√(M2Z−M2Z′)2+4Δ4) (11) αZZ′=12arctan(2Δ2M2Z′−M2Z) (12)

### 2.2 The Higgs sector

In the charged sector it is convenient to introduce the basis as:

 G−=H−dcosβ−H+∗usinβ (13) H+=H−∗dsinβ+H+ucosβ

After the gauge symmetry breaking, two Goldstone modes from the original and doublets are eaten by fields leaving two physical charged Higgs bosons , with the mass

 m2H±=√2λAλsin2βvS−λ22v2+g222v2+Δ±, (14)

where the trilinear coupling is the soft-SUSY breaking counterpart of , and the one-loop corrections are the same as in the MSSM [32] with the effective parameter given by

 μ≡λvS√2.

In the CP-conserving model the CP-even and CP-odd scalar Higgs component fields do not mix. The CP-even sector involves , and fields. The mass matrix of the CP-even Higgs scalars has been calculated to one-loop in Refs. [27, 33] in the field space basis . This basis222Note that are not the MSSM-like eigenstates. is rotated by an angle with respect to the interaction basis,

 √2Re⎛⎜⎝H0dH0uS⎞⎟⎠=⎛⎜⎝cosβ−sinβ0sinβcosβ0001⎞⎟⎠⎛⎜⎝hHN⎞⎟⎠+⎛⎜⎝vcosβvsinβvS⎞⎟⎠ (15)

The explicit form of is given in Appendix A. It can be diagonalized by a orthogonal mixing matrix , i.e.

 M2diagH=OTM2evenO (16)

by going to the mass eigenstates basis

 (H1,H2,H3)=(h,H,N)O (17)

in which, by convention, mass eigenstates are ordered by mass, .

It will be convenient to introduce a mixing matrix ,

 O′=⎛⎜⎝cosβ−sinβ0sinβcosβ0001⎞⎟⎠O, (18)

that enters the Feynman rules. It is a superposition of two rotations in eqs. (15) and (17) and links the interaction eigenstates directly to the CP-even mass eigenstates .

The imaginary parts of the neutral components of the Higgs doublets and Higgs singlet compose the CP-odd sector of the model. In the field basis defined by

 √2ImH0d = Gcosβ+(Acosϕ−G′sinϕ)sinβ √2ImH0u = −Gsinβ+(Acosϕ−G′sinϕ)cosβ √2ImS = Asinϕ+G′cosϕ (19)

the massless pseudoscalar fields are absorbed to after the electroweak gauge symmetry breaking. The physical CP-odd Higgs boson acquires mass

 m2A=√2λAλsin2ϕv+ΔEA (20)

where and the one-loop correction is given in Appendix A.

Note that the Higgs sector of this model involves only one physical CP-odd pseudoscalar as in the MSSM, since, unlike the NMSSM, the extra CP-odd state arising from the singlet is eaten by the . However, there are three CP-even scalars, one more than in the MSSM, where the extra singlet state arises from the extra singlet as in the NMSSM. The characteristic Higgs mass spectrum in this model is governed by the value of . For small values of , say , the Higgs spectrum resembles that of the MSSM, with the heaviest CP-even Higgs scalar being predominantly composed of the singlet scalar state, and being approximately degenerate with the CP-odd pseudoscalar and the charged Higgs states when their masses exceed about 500 GeV. In this regime the lightest CP-even Higgs scalar is Standard Model like, and respects the MSSM mass bound. On the other hand, for large values of , say , a viable Higgs mass spectrum only occurs for a very large CP-odd Higgs mass, say TeV, with the heaviest CP-even Higgs scalar being non-singlet and degenerate with the the CP-odd and charged Higgs states. The second heaviest Higgs scalar is comprised mainly of the singlet state and is thus unobservable, while the lightest CP-even Higgs scalar is Standard Model like but may significantly exceed the MSSM bound. For more details concerning the Higgs sector see [27].

### 2.3 The neutralino sector

The Lagrangian of the neutralino system follows from the superpotential in Eq.(1), complemented by the gaugino SU(2), U(1) and U(1) mass terms of the soft–supersymmetry breaking electroweak Lagrangian:

 Lgauginomass = −12M2~Wa~Wa−12MY~Y~Y−12MX~X~X−MYX~Y~X+h.c. (21)

where the (), and are the (two–component) SU(2), U(1) and U(1) gaugino fields, and () are the corresponding soft-SUSY breaking mass parameters. After performing the transformation of gauge superfields to the gauge boson eigenstate basis, Eq.(3), the Lagrangian takes the form

 Lgauginomass = −12M2~Wa~Wa−12M1~B~B−12M′1~B′~B′−MK~B~B′+h.c., (22)

where

 M′1≡MXcos2χ−2sinχcos2χMYX+MYtan2χ,MK≡MYXcosχ−MYtanχ, (23)

and we introduce the conventional notation for the U(1) bino mass . In parallel to the gauge kinetic mixing discussed in Sect.2.1, the Abelian gaugino mixing mass parameter is assumed small compared with the mass scales of the gaugino and higgsino fields.

Notice that the gauge kinetic term mixing (and the corresponding soft-SUSY breaking mass) can be a source of mass splitting between the and gauginos in models with universal gaugino masses . Since the mixing angle must be small, as required by data [31], the splitting is very small. The splitting could be enhanced if additional U(1) gauge factors in the hidden sector were present that mix via the kinetic term with the visible sector.333 Since the fields in the hidden sector are generally considered to be heavy enough and the hidden-visible mixing is expected to be small, their effect on the visible gauge sector can be negligible. Nevertheless, the mass of the Abelian gaugino in the visible sector can obtain a substantial contribution, as advocated in [34]. In our phenomenological analyses, therefore, we will consider two scenarios: (A) with taken as a free parameter, independent from ; and (B) with tight to and by a unification of gaugino masses at the GUT scale.

After breaking the electroweak and U(1) symmetries spontaneously the doublet higgsino mass and the doublet higgsino–singlet higgsino mixing parameters are generated

 μ≡λvS√2andμλ≡λv√2. (24)

The USSM neutral gaugino–higgsino mass matrix in a basis of two–component spinor fields can be written in the following block matrix form

 M~χ0=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝M10−MZcβsWMZsβsW0MK0M2MZcβcW−MZsβcW00−MZcβsWMZcβcW0−μ−μλsβQ1g′1vcβMZsβsW−MZsβcW−μ0−μλcβQ2g′1vsβ\cline1−600−μλsβ−μλcβ0QSg′1vSMK0Q1g′1vcβQ2g′1vsβQSg′1vSM′1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (25)

where the upper-left is the neutral gaugino–higgsino mass matrix of the MSSM, the lower-right corresponds to the new sector containing the singlet higgsino (singlino) and the new U(1)–gaugino that is orthogonal to the bino , and off-diagonal describes the coupling of the two sectors via the neutralino mass matrix (, , and are the sine and cosine of the electroweak mixing angle ). Notice the see-saw type structure of the new sector due to the absence of a diagonal mass parameter for the singlino , which is in direct contrast to the NMSSM in which the cubic self-interaction generates a singlet mass term[10]. For the same reason, in the USSM the lightest neutralino can never be bino’-dominated.

In general, the neutralino mass matrix is a complex symmetric matrix. To transform this matrix to the diagonal form, we introduce a unitary matrix such that

 ~χ0k=Nkℓ(~B,~W3,~Hd,~Hu,~S,~B′)ℓ, (26)

where the physical neutralino states are ordered according to ascending absolute mass values. The eigenvalues of the above matrix can be of both signs; the negative signs are incorporated to the mixing matrix . Mathematically, this procedure of transforming a general complex symmetric matrix to the diagonal form with non-negative diagonal elements is called the Takagi diagonalization, or the singular value decomposition [18, 35]. Physically, the unitary matrix determines the couplings of the mass–eigenstates to other particles.

Although the complexity of neutralino sector increases dramatically by this extension as compared to the MSSM (which can be solved analytically), the structure remains transparent since, in fact, the original MSSM and the new degrees of freedom are coupled weakly. must be small by the requirement that the mixing of the and sectors satisfy experimental limits. The remaining off-diagonal terms are suppressed with respect to the corresponding block diagonal terms by a factor of . Since sets the mass of the , this results in being roughly an order of magnitude greater than . Therefore in physically interesting case of weak couplings of both the MSSM higgsino doublets to the singlet higgsino and to the U(1) gaugino, and the coupling of the U(1) and U(1) gaugino singlets, the remaining terms in the off-diagonal submatrix in Eq.(25) are small. Then, an approximate analytical solution can be found following a two-step diagonalization procedure given in Ref. [18]. In the first step the MSSM submatrix and the new singlino–U(1) gaugino submatrix are separately diagonalised. In the second step a block–diagonalization removes the non–zero off–diagonal blocks while leaving the diagonal blocks approximately diagonal up to second order, due to the weak coupling of the two subsystems.

### 2.4 The sfermion sector

As explained in the Introduction, we assume the exotic squarks to be substantially heavier than the MSSM fields. However the structure of the MSSM squarks gets modified by the presence of extra U(1). Both the squarks and sleptons are important to our analysis and so we briefly describe the new ingredients in the sfermion mass matrix (neglecting the possibility of flavor and CP violation)

 M2~f = ⎛⎜⎝m2~F+m2f+Δ~fmf(Af−μ(tanβ)−2I3f)mf(Af−μ(tanβ)−2I3f)m2~f+m2f+Δ~f∗⎞⎟⎠. (27)

where , are the sfermion soft-supersymmetry breaking parameters for the quark and lepton doublets and singlets and is the trilinear coupling, while is the corresponding fermion mass and the -terms receive additional U(1) terms

 Δ~f=M2Zcos2β(I3f−efs2W)+12g′21Q~f[v2(Q1cos2β+Q2sin2β)+QSv2S] (28)

where and are the weak isospin and electric charge and the U(1) charges are for the left fields. Explicitly, we have for squarks

 Δ~u = M2Zcos2β(12−23s2W)+180g′21[−v2(2sin2β+3cos2β)+5v2S] Δ~u∗ = M2Zcos2β23s2W+180g′21[−v2(2sin2β+3cos2β)+5v2S] Δ~d = M2Zcos2β(−12+13% s2W)+180g′21[−v2(2sin2β+3cos2β)+5v2S] Δ~d∗ = M2Zcos2β(−13s2W)+280g′21[−v2(2sin2β+3cos2β)+5v2S] (29)

and for sleptons

 Δ~ν = M2Zcos2β(12)+280g′21[−v2(2sin2β+3cos2β)+5v2S] Δ~e = M2Zcos2β(−12+s2W)+280g′21[−v2(2sin2β+3cos2β)+5v2S] Δ~e∗ = M2Zcos2β(−s2W)+180g′21[−v2(2sin2β+3cos2β)+5v2S] (30)

Note that here is the GUT normalized gauge coupling analogous to the GUT normalized hypercharge gauge coupling in the MSSM.

The diagonal form of the sfermion mass matrix is obtained, as usual, by a 2x2 rotation in the plane

 M2diag~f=UT~fM2~fU~f (31)

and the mass eigenstates are defined according to

 (~f1~f2)=U†~f(~fL~fR) (32)

with the convention that .

## 3 Calculating the relic density

The calculation of the neutralino LSP relic density in the MSSM is well known [1, 2] and has been widely studied in the general MSSM [36] and the constrained MSSM [37]. The calculation of the relic density in the NMSSM has also been extensively studied [38]. The differences between the MSSM relic density calculation and the USSM calculation arise through the extension of the particle spectrum and through the new interactions that are introduced. We have implemented all new interactions into the micrOMEGAs[25] code using LanHep[24] to generate the feynman rules. MicrOMEGAs takes full account of all annihilation and coannihilation processes and calculates their effect whenever they are relevant. Nevetheless, from the form of these alterations we would like to make some general observations before we go on to consider the details of the calculations.

The USSM extends the neutralino sector by adding two new states to the spectrum: the bino’ and singlino components. This results in two extra neutralinos. However for the relic density calculation we are only interested in the lightest neutralinos, so the primary effect will be through the magnitude of the singlino and bino’ components in the lightest neutralino. In what follows we will be interested in the scenarios in which the lightest neutralino has a significant singlino component and a small but non-zero bino’ component. Therefore it is informative to consider the general form of the interactions that arise from the singlino and bino’ components of the lightest neutralino before considering specific diagrams.

The bino’ component is always subdominant to the singlino component due to the see-saw structure of the extra sector of the neutralino mass matrix in Eq. 25. The form of the interactions that arise from the inclusion of the bino’ component closely mirror those of the bino component, except for the different coupling constant and charges under the new .

The singlino component is another matter. It gives rise to a new type of neutralino interaction from the term in the superpotential that will be seen to dominate the annihilation processes of neutralinos with significant singlino components. This term means that if the lightest neutralino has significant singlino and higgsino components then it will couple strongly to Higgs bosons with a significant or component, usually the lighter Higgs bosons, in the spectrum. Moreover, the absence of the singlet cubic term , in contrast the the NMSSM, implies that the singlino-dominated LSP needs an admixture of MSSM higgsinos to annihilate to Higgs bosons.

On the other hand, the singlino component does not interact with the or gauginos. Therefore a significant singlino component in the lightest neutralino will suppress annihilations to or bosons.

Finally, there is no coupling of the singlino component to fermions. Thus a significant singlino component in the lightest neutralino will also suppress annihilation to fermions.

Having noted these general features we will now consider the specific behavior of the different annihilation diagrams.

### 3.1 t-channel diagrams

• Gauge boson final states

Fig. 1 shows the t-channel diagrams available for annihilation of neutralinos to gauge bosons. The vertex is given in Eq. (B.1). Note that the coupling of neutralinos to the component of the state is precisely that of the MSSM coupling. As the component dominates the state, a singlino dominated LSP will not annihilate strongly to bosons.

In contrast there is a strong coupling from the MSSM-higgsino components as well as the singlino component to the component of the state. Notice also that the MSSM-higgsino components of the LSP enter with the same sign in the coupling to the , unlike in the coupling to the , where they tend to cancel each other. As the boson is dominantly any LSP with a non-zero higgsino or singlino fraction will annihilate to bosons when such a final state is kinematically allowed. Unfortunately the is required to be heavy by experimental limits, so annihilation of the lightest neutralinos to final states involving one is hard to achieve and annihilation to two bosons is impossible.

The second diagram of Fig. 1 shows the t-channel annihilation to final states. Eqs. (B.2) and (B.3) give the relevant coupling and show that the singlino and bino’ components do not couple to the wino component of charginos or to the bosons. This means that a large singlino or bino’ component in the LSP will suppress annihilation to bosons in the final state.

• Higgs boson final states

Fig. 2 shows the available t-channel processes for the annihilation of neutralinos to final state Higgs bosons. Due to the term in the superpotential and the D-terms there are significant differences between these diagrams in the USSM and the MSSM. The vertex given in Eq. (B.6) is the relevant vertex in this first diagram.

First note that the bino’ component of one neutralino couples with the higgsino component of the other and the component of the final state Higgs boson in the same way as the equivalent coupling of the bino or wino components. In addition there is an extra term which couples the bino’ component of one neutralino to the singlino component of the other and to the singlet component of the Higgs boson in the final state. This means that if the lightest neutralino is dominantly singlino, it will annihilate to final state Higgs bosons with a significant singlet component through the exchange of a neutralino with a significant bino’ component in the t-channel. Unfortunately these processes are disfavored for the same reason as annihilation to final states containing a . The Higgs boson with a significant singlet component will have a mass comparable to the boson and thus a final state with two such Higgs bosons will be impossible and even one will often be kinematically ruled out.

Of more interest is the term in this vertex that couples a singlino component of one neutralino to a higgsino component of the other neutralino and the components of the Higgs boson with a strength . If the lightest neutralino is dominantly singlino then two LSPs can exchange a dominantly higgsino neutralino in the t-channel to produce two Higgs bosons in the final state. This is a channel that is always present if the lightest neutralinos are heavy enough to produce two light Higgs bosons in the final state. Obviously, if both and are lighter than the lightest neutralino then there will be more available channels. As the singlino couples predominantly to Higgs states, this channel provides the strongest annihilation mechanism for a neutralino with a large singlino component. This amplitude will be maximised for three degenerate mixed state neutralinos with strong higgsino and singlino components that are heavier in mass than the lightest two Higgs states. The addition of this vertex also allows for a new annihilation process for a dominantly higgsino neutralino through the exchange of a t-channel neutralino with a substantial singlino component.

The middle diagram of Fig. 2 shows the annihilation to final state pseudoscalar Higgs bosons. The relevant vertex is given in Eq. (B.10). The first line gives the familiar MSSM vertex for the coupling of a or component of a neutralino to a higgsino component and a pseudoscalar Higgs. This is modified by an overall factor of which determines the magnitude of the MSSM-like components of the pseudoscalar Higgs over the singlet contribution. As , the suppression from is small. This is the same as saying that the pseudoscalar Higgs generally only has a very small singlet component. The analogue of the interaction terms appears for the bino’. The bino’ component also couples to the singlino component of the second neutralino and the singlet component of the final state pseudoscalar Higgs. This term is suppressed due to the small singlet component of the bosons in the final state. These interactions determine the strength of the annihilation of a dominantly gaugino LSP to pseudoscalar Higgs bosons through the exchange of a dominantly higgsino (or singlino) neutralino.

More interesting contributions come from the term in the superpotential. These provide a coupling, albeit suppressed by a factor of . Such a coupling does not appear in the MSSM. There is also a term that couples with no suppression. Once again this produces a strong annihilation channel for a neutralino with a substantial singlino component through t-channel neutralino exchange where the neutralino exchanged in the t-channel must have a significant higgsino component. This is the analogue of the process we discussed in some detail for the scalar Higgs final states and will, kinematics allowing, give a strong annihilation channel for a dominantly singlino neutralino as long as there is a light neutralino in the spectrum with a substantial higgsino component to be exchanged in the t-channel.

The final diagram of Fig. 2 shows annihilation to charged Higgs boson final states. The relevant vertex is given in Eq. (B.14). The vertex includes a interaction that parallels the familiar and interactions to the higgsino component of the chargino and a charged Higgs boson. There is also a term that arises from the superpotential term. This allows for a neutralino with a substantial singlino component to annihilate to charged Higgs bosons via t-channel chargino exchange as long as there are light charginos with a significant higgsino component and the final state charged Higgs bosons are kinematically allowed. In contrast to the previous two diagrams, this one does not add an extra annihilation channel for a dominantly higgsino neutralino. In the first two diagrams there is the new possibility in which a dominantly singlino neutralino is exchanged in the t-channel. In the third diagram there is no such process as there is no singlino component in the charginos.

From an analysis of the processes with Higgs bosons in the final state we see that there will be a strong annihilation cross-section for a neutralino with a large singlino component to light Higgs bosons if there is a light neutralino with a substantial higgsino component in the spectrum and the Higgs boson final states are kinematically allowed. We also note that the allows for new couplings between neutralinos and Higgs bosons that will alter the annihilation of dominantly higgsino neutralinos with respect to their behavior in the MSSM.

• Mixed boson final states

It is quite possible to have an unmatched pair of bosons in the final state of a t-channel annihilation diagram. We do not need to go through the details of all possible diagrams. Instead we just note that a neutralino with a significant singlino component will dominantly annihilate to final states made up of Higgs bosons. The strength of such channels will depend upon the size of the singlino component in the lightest neutralino, the mass of the neutralinos with substantial higgsino components that will be exchanged in the t-channel, and the mass of the final state Higgs bosons.

• Fermion final states

Finally we consider the t-channel annihilation diagram to final state fermions through the diagram given in Fig. 3. The squark vertices are given in Eqs. (B.18) and (B.19). The couplings of the bino and wino components of the neutralino are the same as in the MSSM. Note that there is an extra coupling of the bino’ component of the neutralino to the squark-quark pair that is of the same order of magnitude as for the . As the bino’ is only ever a subdominant component of the lightest neutralino, and as the annihilation to fermions is relatively weak in the first place, we can expect that interactions of this form will have little impact on the annihilation cross-section. However, if the lightest neutralino is too light to annihilate to final state Higgs bosons, this channel will remain open and can dominate though it will give a relic density well in excess of that measured by WMAP.

### 3.2 s-channel diagrams

Fig. 4 shows the possible s-channel processes available for the annihilation of a pair of neutralinos. The first diagram shows the annihilation through and intermediate gauge boson. The relevant coupling of two neutralinos to a is given in Eq. (B.1). As before we note that the singlino component of the neutralino only couples to the component of the gauge boson. This means that if the lightest neutralino has a significant singlino component, then annihilations through an s-channel will be suppressed as the component of the is required to be very small. On the other hand, the has a large component. Therefore a lightest neutralino with a substantial singlino component will annihilate through an s-channel .

The second diagram shows the annihilation of neutralinos through an s-channel scalar or pseudoscalar Higgs boson. The relevant couplings are given in Eq. (B.6) and Eq. (B.10) respectively. Consider first the case in which the s-channel Higgs boson is dominantly composed of the singlet Higgs. Since the bino or wino components of the lightest neutralino only couple to the non-singlet Higgs components of the s-channel Higgs a light neutralino that is dominantly wino or bino will not annihilate strongly through a dominantly singlet Higgs. However there is a coupling of the singlet component of the Higgs boson to the higgsino components of the lightest neutralino. This provides a strong channel when on-resonance for annihilation of a light neutralino with a large higgsino component. There is also a strong coupling if the lightest neutralino has significant bino’ and singlino components. Thus we expect a light neutralino with strong mixing between higgsino, singlino and bino’ terms to annihilate strongly through s-channel heavy Higgs exchange where the heavy higgs has a large singlet Higgs component.

If the s-channel Higgs boson does not have a large singlet Higgs component then the story is somewhat different. In this case the light neutralino needs to have a significant higgsino fraction along with a substantial contribution from one of the other non-higgsino states. This situation is mirrored in the case of the pseudoscalar Higgs.

From this we see that we have a new annihilation channel for neutralinos with a significant higgsino fraction through a dominantly singlet Higgs in the s-channel. We also see that a light neutralino with a substantial singlino-higgsino mixture will annihilate strongly through the whole range of s-channel Higgs exchange processes.

### 3.3 Coannihilation

As well as the annihilation of two identical neutralinos, it is often the case that coannhilation between the LSP and the NLSP (and sometimes even heavier states) can be important. This process is normally important for a dominantly MSSM-higgsino or wino neutralino LSP. In these situations there is an automatic near degeneracy in the mass of the lightest neutralino with the mass of the lightest chargino and, in the case of the higgsinos, also with the next-to-lightest neutralino. A dominantly singlino LSP does not have an automatic degeneracy with other states. However, it is possible for a singlino neutralino to be exactly degenerate with other states - something that does not happen in the MSSM due to the signs of the terms in the neutralino mixing matrix. In these cases we would expect the effect of coannihilation to be important.

Therefore we expect coannihilation processes to only be significant in regions of the parameter space where we move from one type of LSP to another as this indicates a degeneracy in the mass of the LSP and NLSP. We also expect to see the standard large coannihilation contributions for a predominantly MSSM-like higgsino LSP or predominantly wino LSP.

## 4 Elastic scattering of neutralinos from nuclei

The direct cold dark matter search experiments, such as DAMA/LIBRA, CDMS, ZEPLIN, EDELWEISS, CRESST, XENON, WARP [39], aim at detecting dark matter particles through their elastic scattering with nuclei. This is complementary to indirect detection efforts, such as GLAST, EGRET, H.E.S.S. [40], which attempt to observe the annihilation products of dark matter particles trapped in celestial bodies.

Since we assume the LSP to be the lightest neutralino , we consider the elastic scattering of the lightest neutralino from nuclei. The elastic scattering is mediated by the -channel and Higgs exchange, as well as the -channel squark exchange, as depicted in Fig.5 for scattering. There are also important contributions from interactions of neutralinos with gluons at one loop [42, 43].

The extended particle content and new couplings present in the USSM model have also a direct effect on the elastic cross section calculations, as discussed in the previous chapter.

The elastic cross section for neutralino scattering from a nucleus can be broken into a spin-independent (SI) and a spin-dependent (SD) part,

 σ=σSI+σSD, (33)

each of which can be expressed in terms of the elastic scattering of neutralino from individual nucleons in the nuclei. In the limit of zero-momentum transfer they can be written as [41]

 σSI=4m2rπ[Zfp+(A−Z)fn]2, (34) σSD=32m2rπG2FJ(J+1)Λ2, (35)

where and are atomic number and mass of the nucleus, is the total nucleus angular momentum and is the reduced neutralino-nucleus mass. Note that the spin-independent part benefits from coherent effect of the scalar couplings, which leads to cross section and rates proportional to the square of the atomic mass of the target nuclei.

The spin-dependent quantity is given by

 Λn=1J[⟨Sp⟩∑q=u,d,sAq√2GFΔpq+⟨Sn⟩∑q=u,d,sAq√2GFΔnq] (36)

where and are the expectation values of the spin content of the proton and neutron group in the nucleus, while and are the quark spin content of the proton and neutron, respectively.

For the spin-independent part, the effective couplings of the LSP neutralino to proton and neutron and are more complicated. In the limit of and , which we will later confine to, they simplify and can be approximated as:

 fp,nmp,n=∑q=u,d,sfp,nTqBqmq+227fp,nTG∑q=c,b,tBqmq (37)

The first term in Eq.(37) corresponds to interactions with the quarks in the target nuclei, while the second term corresponds to interactions with the gluons in the target through a quark/squark loop diagram, and

 fp,nTG=1−∑q=u,d,sfp,nTq. (38)

Finally, the effective Lagrangian for elastic scattering of neutralinos from quarks in the non-relativistic limit can be written as a sum of axial-vector (spin-dependent) and scalar (spin-independent) terms:

 Leff=Aq(¯χ1γμγ5χ1)(¯qγμγ5q)+Bq(¯χ1χ1)(¯qq) (39)

The effective couplings and are given by:

 Aq=g2216∑i=1,2∣∣BiLq∣∣2+∣∣BiRq∣∣2m2~qi−(m~χ0−mq)2−GF√2[|N13|2−|N14|2]I3q −g′214m2Z′[Q1|N13|2+Q2|N14|2+Qs|N15|2](QQ+Q¯q) (40) Bq=−g228∑i=1,2Re(BiLqBiRq∗)m2~qi−(m~χ0−m
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9616909027099609, "perplexity": 1062.1488927791543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662539101.40/warc/CC-MAIN-20220521112022-20220521142022-00092.warc.gz"}