1 Introduction

LAPTH-1327/09

Dark Matter with Dirac and Majorana Gaugino Masses

G. Belanger 111belanger@lapp.in2p3.fr, K. Benakli 222kbenakli@lpthe.jussieu.fr, M. Goodsell 333goodsell@lpthe.jussieu.fr, C. Moura 444moura@lpthe.jussieu.fr and A. Pukhov 555pukhov@lapp.in2p3.fr

LAPTH, Univ. de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux, France

LPTHE, Université Pierre et Marie Curie - Paris VI, France

SINP, Moscow State University, Moscow 119992, Russia

We consider the minimal supersymmetric extension of the Standard Model allowing both Dirac and Majorana gauginos. The Dirac masses are obtained by pairing up extra chiral multiplets: a singlet for , a triplet for and an octet for with the respective gauginos. The electroweak symmetry breaking sector is modified by the couplings of the new fields and to the Higgs doublets. We discuss two limits: i) both the adjoint scalars are decoupled with the main effect being the modification of the Higgs quartic coupling; ii) the singlet remaining light, and due to its direct coupling to sfermions, providing a new contribution to the soft masses and inducing new decay/production channels. We discuss the LSP in this scenario; after mentioning the possibility that it may be a Dirac gravitino, we focus on the case where it is identified with the lightest neutralino, and exhibit particular values of the parameter space where the relic density is in agreement with WMAP data. This is illustrated for different scenarios where the LSP is either a bino (in which case it can be a Dirac fermion) or bino-higgsino/wino mixtures. We also point out in each case the peculiarity of the model with respect to dark matter detection experiments.

## 1 Introduction

A remarkable fact of nature is that the light fundamental fermions appear in the smallest representations, the singlet or fundamental, of the Standard Model symmetry group. Could larger representations be present at higher energies and be discovered at the LHC? Such particles are in fact predicted by low energy supersymmetry as gauginos, superpartners of the gauge vector bosons. These gauginos can appear as fermions of Majorana (with only two degrees of freedom) or Dirac (with four degrees of freedom) type. In the Minimal Supersymmetric extension of the Standard Model (MSSM) the gauginos are Majorana. Obtaining Dirac masses would require pairing them with additional Dirac Gaugino adjoint (henceforth DG-adjoint) states: a singlet for , a triplet for and an octet for .

Construction of models with spontaneous breaking of supersymmetry may lead to non-minimal extensions of the Standard Model that include Dirac gauginos. The presence of the required DG-adjoints in the light spectrum is often motivated by the presence of an underlying supersymmetry, that pairs them with the vector multiplets. Such a scenario was first introduced by Fayet [1] as a way to give masses for gluinos while preserving R-symmetry111 In [1] -symmetry was later broken by Majorana masses for the DG-adjoints in order to avoid tachyonic masses for their scalar components. An issue that was recently solved in [22] and independently for an explicit model in [17].. The soft nature of these masses requires a modification of the interaction as was shown using -term breaking by [2, 3]. More recently, Dirac gauginos have arisen in models with an extra dimension where supersymmetry is broken by a Scherk-Schwarz mechanism (see for example [4, 5]). More precisely, they are a combination of two Majorana fermions with mass given by (half of the compactification scale ), one given by the (mass shifted) massless mode and the other from the (mass shifted) first Kaluza-Klein state, thus, the DG-adjoints originate as “half of the first Kaluza-Klein excitation”. It was noted there that the soft masses are UV finite and do not exhibit the usual logarithmic sensitivity to the UV cut-off [6]. This property was shown to persist in four dimensional models as being peculiar to the Dirac nature of the gaugino masses, and denoted as supersoft in [7, 8, 9, 10, 11] where the important phenomenological implications of -term supersymmetry breaking were first outlined. They were further studied in constructions of non-supersymmetric intersecting brane models [12, 13, 14]. More recent examples arise from the possibility of using calculable -symmetric models [3, 15, 16, 17, 18, 19]. For instance, it was pointed out in [16] that they lessen the flavor problem in supersymmetric theories. This renewal of interest in such models has been motivated by the work of ISS [20]. Furthermore, deforming the ISS model with explicit breaking of -symmetry could leave states in adjoint representations [21] and allow the simultaneous presence of both Majorana and Dirac masses for the gauginos. The generation of Dirac gaugino masses can also be included [22] in the framework of “general gauge mediation” [23, 24, 25, 26, 27, 28].

In this work, we will study a minimal extension of the MSSM that incorporates both Majorana and Dirac gaugino masses. The field content is that of the MSSM, supplemented with the DG-adjoints. The MSSM renormalisable Lagrangian is then supplemented by i) the DG-adjoint kinetic and mass terms, ii) the Dirac gaugino masses, iii) coupling of the singlet and the triplet to the Higgs doublets with strength and respectively, iv) the DG-adjoint scalar soft masses and trilinear terms.

For phenomenological issues the strength of the coupling of the Higgs to the DG-adjoint is of particular importance. In general such couplings are arbitrary and subject to diverse phenomenological bounds as discussed in [8]. Inspired by extra dimensional models, and in order to make the role of manifest, one can assume that the two MSSM Higgs doublets originate from a single hypermultiplet of an underlying supersymmetry. These models with combined and sectors were introduced in [14, 29]. In this case, the couplings and are related to the gauge couplings by supersymmetry. In this work, we will arbitrarily take the values of both couplings to go from zero to their tree level value to illustrate the model.

Some particular signatures of these models at collider experiments have been studied in [30, 31, 32, 33, 34, 35]. They stressed that the Dirac/Majorana nature of the gluinos affects the distribution of produced squark states. It was pointed out in [32] that the pair creation of scalar octets at the LHC will have a peculiar signature through cascade decays giving rise to a burst of eight or more jets together with four LSPs as well as through a resonance due to the decays into gluons or a pair at the one-loop level.

In this work we will focus mainly on the fate of dark matter in this class of models. It is by now an important quality of the -parity preserving versions of the MSSM that they provide a natural candidate for Dark Matter, the lightest supersymmetric particle (LSP). A particular case is when the LSP is identified with the lightest neutralino. While in the MSSM the latter is a linear combination of the four neutral fermions, given by the bino, the wino and the two Higgsinos, it is now a linear combination of six states, the singlet and triplet fermions adding to the previous four. In this work we shall not try to give an exhaustive discussion of such a situation but try to answer such questions as: Are there parameter regions where the neutralino is a good dark matter candidate? How does the situation compare to the MSSM? An early study [36] of dark matter in a related model to that considered here focussed on the bino LSP, for a very particular case with vanishing and , and assumed dominance of the exchange of sfermions and thus neglecting, for example, the exchange of Higgs or gauge bosons. It concluded that the bino annihilation cross section can be enhanced which might help to obtain a smaller relic abundance than in the MSSM.

The present paper is organized as follows: Section 2 presents the model and defines our notations and conventions. Section 3 gives a comprehensive discussion, which we believe to be missing in the present literature, of the electroweak breaking sector. The discussion follows the same lines as in [14] which specializes to a model with combined and sectors in the limit of very large soft masses for the scalar components of and . It differs from the usual extension of the MSSM by the couplings to the Higgs of a singlet, whose vacuum expectation value (vev) is not necessarly related to the supersymmetric Higgs mass term , and a triplet (see [37, 38, 39, 40, 41, 42]). While the triplet scalar is required to be heavy by electroweak precision tests, the singlet can be either very heavy and integrated out or remain light with sensible mixing with the ordinary MSSM Higgs states. We discuss both limits. In section 5, we briefly discuss the gravitino LSP and then focus on the case of a neutralino LSP. The corresponding mass matrix is exhibited and the nature of the lightest eiganstates is studied for some particular limits, in particular the necessary condition for having a Dirac fermion LSP is given. Section 6 presents numerical results for the relic adundance and the corresponding signature at direct/indirect detection experiments are stressed.

## 2 The model

The particle content of the model is presented in table 1. The MSSM matter fields acquire masses through the Yukawa superpotential:

 WYukawa=yijuuciQj⋅Hu−yijddciQj⋅Hd−yijeeciLj⋅Hd (2.1)

and the usual soft breaking terms:

 L0soft = ~Q†im2Qij~Qj+~L†im2Lij~Lj+~u†im2uij~uj+~d†im2dij~dj+~e†im2eij~ej (2.2) +Aiju~uci~Qj⋅Hu−Aijd~dci~Qj⋅Hd−Aije~eci~Lj⋅Hd+c.c.

where the bold characters denote superfields. Here, are family indices and run from to . The matrices are the Yukawa couplings. The “” denotes invariant couplings, for example: .

In order to have Dirac masses for the gauginos, additional fields in the adjoint representations, the “DG-adjoints”, are introduced. We define the superfields:

 S = S+√2θχS+⋯ (2.3) T = T+√2θχT+⋯ (2.4) Og = Og+√2θχg+⋯ (2.5)

where is a singlet and an triplet parametrized as:

 T(1)=T1σ12,T(2)=T2σ22, T(3)=T0σ32, T=12(T0√2T+√2T−−T0), T0=1√2(TR+iTI),T+=1√2(T+R+iT+I), T−=1√2(T−R+iT−I), (2.6)

and are the Pauli matrices. Their quantum numbers are presented in Table 1.

Due to the presence of these extra fields, the gauge kinetic terms are modified to become:

 Lgauge= ∫d4xd2θ[14M1Wα1W1α+12M2% tr(Wα2W2α)+12M3tr(Wα3W3α) (2.7) +√2mα1DW1αS+2√2mα2Dtr(W2αT)+2√2mα3Dtr(W3αOg)] + ∫d4xd2θd2¯θ(∑ijΦ†iegjVjΦi+h.c.)

where are the vector and the corresponding field strength superfields associated to , and for respectively. Here, we have introduced spurion superfields to take into account the generation of gaugino masses:

 Mi = 1+2θθMi (2.8) mαiD = θαmiD (2.9)

The Dirac gaugino spurion superfield can be written as . This mass originates as a -term if is identified as a vector field , or as non vanishing -term by writing with , where is the appropriate supersymmetry breaking mediation scale.

The DG-adjoints may also modify the Higgs superpotential, since new relevant and marginal operators are now allowed:

 ∫d4xd2θ[μHu⋅Hd+MS2S2+λSSHd⋅Hu+MTtr(TT)+2λTHd⋅THu] (2.10)

with the definition .

Finally, the soft supersymmetry breaking terms for the scalars are:

 −ΔLsoft = m2Hu|Hu|2+m2Hd|Hd|2+Bμ(Hu⋅Hd+h.c.) (2.11) +m2S|S|2+12BS(S2+h.c.)+2m2T% tr(T†T)+BT(tr(TT)+h.c.) +ASλS(SHd⋅Hu+h.c.)+2ATλT(Hd⋅THu+h.c.)

Note that we did not include in the superpotential a cubic term as this identically vanishes. Neither did we include linear and cubic terms in the singlet. The latter is due to the fact that we assume that the DG-adjoint appears due to some underlying supersymmetry that forbids these terms. Of course a microscopic model explaining the origin of the soft terms should also address the fact that the supersymmetric and are assumed to take values of order of the electroweak scale, introducing an issue of scale hierarchy as does the Higgs -term.

Below, we will give special attention to the scenario where the DG-states arise as a result of an extension of the gauge sector. In this case, if the Higgs multiplets and are assumed to form an hypermultiplet then and are related to the gauge couplings, at the scale, by:

 λS=√2g′12,λT=√2g12, (2.12)

where and are the and gauge couplings respectively. The factor in arises from the charge of the Higgs doublets.

## 3 Electroweak scalar potential

We turn now to the electroweak scalar potential. This receives contributions from three sources:

 VEW=Vgauge+VW+Vsoft (3.1)

The first is a contribution from the gauge kinetic term (2.7). Integration on the spinor coordinates, and going on-shell, leads to the and -terms:

 D1 = −2m1DSR+D(0)YwithD(0)Y=−g′∑jYjφ∗jφj (3.2) Da2 = −√2m2D(Ta+Ta†)+Da(0)2% withDa(0)2=−g∑jφ∗jσa2φj (3.3)

where are the scalar components of matter chiral superfields, whereas and are the -terms in absence of Dirac masses. The resulting Lagrangian contains terms of the form:

 Lgauge→−m1Dλα1χSα −m2Dtr(λα2χTα)−12D21−12tr (Da2Da2) (3.4)

where we can identify the Dirac components of the gauginos and as given in Table 1.

The contribution from the DG-triplet to is:

 D2∝12((|T−|2−|T+|2)√2(T+T∗0−T0T∗−)√2(T0T∗+−T−T∗0)−(|T−|2−|T+|2)) (3.5)

which vanishes in electrically neutral vacuum, where:

 = ===0. (3.6)

The contribution to the scalar potential of the neutral fields is then given by:

 Vgauge=2m21DS2R−2m1DSRD(0)Y+12D(0)2Y+2m22DT2R−2m2DTRD(0)2+12D(0)22 (3.7)

where we have dropped the generator label, , for the only non-vanishing component.

The second contribution comes from the superpotential (2.10):

 W = (−μ+λSS)(H0uH0d−H+uH−d)+MS2S2+MT2(T0T0+2T+T−) (3.8) −λT(H0dT0H0u+H−dT0H+u)−√2λT(H−dT+H0u−H0dT−H+u).

Keeping only the neutral components, it reads:

 VW=|MSS+λSH0dH0u|2+|MTT0−λTH0dH0u|2+|μ−λSS+λTT0|2(|H0d|2+|H0u|2) (3.9)

The third source is due to soft supersymmetry breaking terms for the scalars (2.11). We define:

 H0u=H0uR+iH0uI√2,H0d=H0dR+iH0dI√2 (3.10)

then, all together, the scalar Lagrangian for the neutral fields is now given by:

 VEW = (m2Hu+μ2)|H0u|2+(m2Hd+μ2)|H0d|2−Bμ(H0uH0d+h.c.)+g2+g′28(|H0u|2−|H0d|2)2 +(λ2S+λ2T)|H0uH0d|2 +12(M2S+m2S+4m21D+BS)S2R+12(M2S+m2S−BS)S2I +12(M2T+m2T+4m22D+BT)T2R+12(M2T+m2T−BT)T2I +[ λ2S2(S2R+S2I)+ λ2T2(T2I+T2R)−√2μ(λSSR−λTTR)−λSλT(SITI+SRTR)] ×[|H0u|2+|H0d|2] +g′m1DSR(|H0u|2−|H0d|2)+gm2DTR(|H0d|2−|H0u|2) +λS√2(MS+AS)SR(H0dRH0uR−H0dIH0uI)+λS√2(MS−AS)SI(H0dRH0uI+H0dIH0uR) −λT√2(MT+AT)TR(H0dRH0uR−H0dIH0uI)−λT√2(MT−AT)TI(H0dRH0uI+H0dIH0uR)

Note that the MSSM potential is given by the first line. All the parameters are chosen to be real. We are left with four neutral fields satisfying equations of type:

 M2xaxa+XSTya=Vxa,a=R,Ifor  (x=S,y=T)  or  (x=T,y=R) (3.12)

with solutions of the form:

 xa=VxaM2ya−VyaXSTM2xaM2ya−X2ST,a=R,Ifor  (x=S,y=T)  or  (x=T,y=R) (3.13)

where:

 M2SR = M2S+m2S+4m21D+BS+λ2S(|H0u|2+|H0d|2) (3.14) M2TR = M2T+m2T+4m22D+BT+λ2T(|H0u|2+|H0d|2) (3.15) M2SI = M2S+m2S−BS+λ2S(|H0u|2+|H0d|2) (3.16) M2TI = M2T+m2T−BT+λ2T(|H0u|2+|H0d|2) (3.17)

while

 XST=−λSλT(|H0u|2+|H0d|2) (3.18)

and:

 VSR = √2μλS(|H0u|2+|H0d|2)−g′m1D(|H0u|2−|H0d|2) (3.19) −λS√2(MS+AS)(H0dRH0uR−H0dIH0uI) VTR = −√2μλT(|H0u|2+|H0d|2)+gm2D(|H0u|2−|H0d|2) (3.20) +λT√2(MT+AT)(H0dRH0uR−H0dIH0uI) VSI = −λS√2(MS−AS)(H0dRH0uI+H0dIH0uR) (3.21) VTI = +λT√2(MT−AT)(H0dRH0uI+H0dIH0uR) (3.22)

We are not going to pursue exact computations. Instead, we will consider the case with in which case the formulae simplify as we can neglect all terms proportional to , in particular . This gives , i.e.

 SR ≃ −g′m1D(|H0u|2−|H0d|2)−√2μλS(|H0u|2+|H0d|2)+λS√2(MS+AS)(H0dRH0uR−H0dIH0uI)M2S+m2S+4m21D+BS TR ≃ −gm2D(|H0d|2−|H0u|2)+√2μλT(|H0u|2+|H0d|2)−λT√2(MT+AT)(H0dRH0uR−H0dIH0uI)M2T+m2T+4m22D+BT SI ≃ −λS(MS−AS)(H0dRH0uI+H0dIH0uR)√2(M2S+m2S+4m21D+BS) TI ≃ λT(MT−AT)(H0dRH0uI+H0dIH0uR)√2(M2T+m2T+4m22D+BT) (3.23)

We are interested by the case of CP neutral vacuum, i.e. which implies . From now on, we will drop the indices and define222With this convention GeV, .:

 = vu=vsβ,=vd=vcβ,0⩽β⩽π2 = vs
=vt (3.24)

where we denote:

 cβ ≡ cosβ,sβ≡sinβ,tβ≡tanβ c2β ≡ cos2β,s2β≡sin2β (3.25)
 vs ≃ v22(M2S+m2S+4m21D+BS)  [g′m1Dc2β+√2μλS−λS√2(MS+AS)s2β] vt ≃ v22(M2T+m2T+4m22D+BT)  [−gm2Dc2β−√2μλT+λT√2(MT+AT)s2β]. (3.26)

Electroweak precision data give strong bounds on the expectation value of the DG-triplet as it contributes to  [43]. Thus we require:

 Δρ≃4v2tv2≲8⋅10−4 (3.27)

which is satisfied for GeV. Here we will allow the Dirac and Majorana masses to vary arbitrarily and will satisfy this bound by taking large enough. For instance, for GeV and all couplings of order one, this is satisfied for TeV.

Integrating out the DG-adjoints and , leads then to the effective tree-level scalar potential at first order in

 VEW = (m2Hu+μ2)2h2u+(m2Hd+μ2)2h2d−Bμhuhd+g2+g′232(h2u−h2d)2 +λ2S+λ2T4h2uh2d −18[g′m1D(h2u−h2d)−√2μλS(h2u+h2d)+√2λS(MS+AS)hdhu]2M2S+m2S+4m21D+BS −18[−gm2D(h2u−h2d)+√2μλT(h2u+h2d)−√2λT(MT+AT)hdhu]2M2T+m2T+4m22D+BT

This decomposes into three parts:

 VEW=V0+V1+V2 (3.29)

The first part:

 V0=(m2Hu+μ2)2h2u+(m2Hd+μ2)2