Aspects of the Exceptional Supersymmetric Standard Model

Transcription

1 Nuclear Physics B (Proc. Suppl.) (2010) Aspects of the Exceptional Supersymmetric Standard Model Peter Athron a, Jonathan P. Hall b, Richard Howl b, Stephen F. King b, D. J. Miller c, S. Moretti bd, R. Nevzorov c a Institut für Kern- und Teilchenphysik, TU Dresden, Dresden, D-01062, Germany b School of Physics and Astronomy, University of Southampton, SO17 1BJ, United Kingdom c University of Glasgow, Glasgow, G12 8QQ, UK d Dipartimento di Fisica Teorica, Università di Torino, Via Pietro Giuria 1, Torino, Italy In this talk we review aspects of the Exceptional Supersymmetric Standard Model 1. Introduction The existence of weak scale supersymmetry (SUSY) is theoretically well motivated, because of its ability to stabilise the electro-weak symmetry breaking (EWSB) scale. One of the benefits of weak scale SUSY with conserved R-parity is that the lightest supersymmetric particle (LSP) is absolutely stable and provides a weakly interacting massive particle (WIMP) candidate capable of accounting for the observed cold dark matter (CDM) relic density Ω CDM h In particular, the lightest neutralino in SUSY models is an excellent such candidate, providing its mass, composition and interactions are suitably tuned to result in the correct value of Ω CDM h 2. The Minimal Supersymmetric Standard Model (MSSM) [1] provides the simplest supersymmet- p.athron@physik.tu-dresden.de jonathan.hall@soton.ac.uk rhowl@soton.ac.uk king@soton.ac.uk. The speaker is grateful to the Royal Society, London, UK. for a travel grant which made his attendance at this conference possible. d.miller@physics.gla.ac.uk stefano@phys.soton.ac.uk. SM is financially supported in part by the scheme Visiting Professor - Azione D - Atto Integrativo tra la Regione Piemonte e gli Atenei Piemontesi nevzorov@physics.gla.ac.uk ric extension of the Standard Model (SM) in which the superpotential contains the bilinear term μh d H u, where H d,u are the two Higgs doublets whose neutral components Hd,u 0 develop vacuum expectation values (VEVs) at the weak scale and the μ parameter has the dimensions of mass. However, since this term respects supersymmetry, there is no reason for μ to be of order the weak scale, leading to the so-called μ problem [2]. Also, the MSSM suffers a fine-tuning of parameters at the per cent level [3]. To address the above shortcomings of the MSSM one may replace the μ term of the MSSM by the low energy VEV of a singlet field S via the interaction λsh d H u. For example, such a singlet coupling can be enforced by a low energy U(1) gauge symmetry arising from a high energy E 6 GUT group [4]. Within the class of E 6 models there is a unique choice of Abelian gauge group, referred to as U(1) N, which allows zero charges for right-handed neutrinos. This choice of U(1) N, which allows large right-handed neutrino Majorana masses, and hence a high scale see-saw mechanism, defines the so-called Exceptional Supersymmetric Standard Model (E 6 SSM) [5,6]. In the E 6 SSM, in order to cancel gauge anomalies involving U(1) N, the low energy (TeV scale) /$ see front matter 2010 Elsevier B.V. All rights reserved. doi: /j.nuclphysbps

2 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) theory must contain the matter content of three complete 27 representations of E 6 (minus the neutral right-handed neutrinos which acquire intermediate scale masses). It is clear that the E 6 SSM predicts a rich spectrum of new states at the TeV scale corresponding to the matter content of three 27 component families. Since each 27 includes a pair of Higgs doublets plus a SM singlet, the E 6 SSM predicts in total three families of Higgs doublets and three families of Higgs singlets. Each 27 component family also includes a pair of vector-like charged ±1/3 coloured states D, D which are readily produced at the LHC and provide a clear signature of the model. The two Higgs doublets familiar from the MSSM are denoted as H d and H u, while the two further replicas of these Higgs doublets predicted by the E 6 SSM are denoted as H1 d, H1 u and H2 d, H2 u. Each 27 representation also contains a separate SM singlet. There is the singlet S whose VEV yields an effective μ term, plus two further copies of this singlet, S 1 and S 2. In the E 6 SSM the extra Higgs doublets, H1 d, H1 u, H2 d, H2 u, and singlets, S 1, S 2, are not supposed to develop VEVs and the scalar components of these superfields are consequently called inert. From the perspective of dark matter, of particular interest are the fermionic partners of these inert Higgs doublet and singlet superfields, which we refer to as inert Higgsinos/singlinos. Such inert Higgsinos/singlinos will in general mix with the other neutralinos and therefore change the nature of lightest neutralino. If the LSP is the lightest neutralino, identified as a WIMP CDM candidate, then the calculation of the thermal relic density will necessarily be affected by the presence of such inert Higgsinos/singlinos. Neutralino dark matter has been studied in the USSM [7] which, in addition to the states of the MSSM, also includes a singlet, S, plus a Z, together with their fermionic superpartners, namely the singlino S and an extra gaugino B. In the USSM the neutralino LSP may have components of the extra gaugino B and singlino S in addition to the usual MSSM neutralino states, which can have interesting consequences for the calculation of the relic density Ω CDM h 2. Recently neutralino dark matter has been studied in the E 6 SSM [8], including all the states of the USSM, plus the extra inert Higgsino doublets predicted by the E 6 SSM but not included in the USSM, namely H d 1, Hu 1, Hd 2, H u 2, and the singlinos, S 1 and S The E 6 SSM One of the most important issues in models with additional Abelian gauge symmetries is the cancellation of anomalies. In E 6 theories, if the surviving Abelian gauge group factor is a subgroup of E 6, and the low energy spectrum constitutes a complete 27 representation of E 6, then the anomalies are cancelled automatically. The 27 i of E 6, each containing a quark and lepton family, decompose under the SU(5) U(1) N subgroup of E 6 as follows: 27 i (10, 1) i +(5, 2) i +(5, 3) i +(5, 2) i +(1, 5) i +(1, 0) i. (1) The first and second quantities in the brackets are the SU(5) representation and extra U(1) N charge while i is a family index that runs from 1 to 3. From Eq. (1) we see that, in order to cancel anomalies, the low energy (TeV scale) spectrum must contain three extra copies of 5 +5 of SU(5) in addition to the three quark and lepton families in To be precise, the ordinary SM families which contain the doublets of left-handed quarks Q i and leptons L i, right-handed up- and down-quarks (u c i and dc i ) as well as right-handed charged leptons, are assigned to (10, 1) i +(5, 2) i. Right-handed neutrinos Ni c should be associated with the last term in Eq. (1), (1, 0) i. The nextto-last term in Eq. (1), (1, 5) i, represents SMtype singlet fields S i which carry non-zero U(1) N charges and therefore survive down to the EW scale. The three pairs of SU(2)-doublets (Hi d and Hi u) that are contained in (5, 3) i and (5, 2) i have the quantum numbers of Higgs doublets, and we shall identify one of these pairs with the usual MSSM Higgs doublets, with the other two pairs being inert Higgs doublets which do not get VEVs. The other components of these SU(5) multiplets form colour triplets of exotic quarks D i and D i with electric charges 1/3 and +1/3 respectively.

3 122 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) We also require a further pair of superfields H and H with a mass term μ H H from incomplete extra 27 and 27 representations to survive to low energies to ensure gauge coupling unification. Because H and H originate from 27 and 27 these supermultiplets do not spoil anomaly cancellation in the considered model. Our analysis reveals that the unification of the gauge couplings in the E 6 SSM can be achieved for any phenomenologically acceptable value of α 3 (M Z ), consistent with the measured low energy central value, unlike in the MSSM which requires significantly higher values of α 3 (M Z ), well above the central measured value [9] 8. Since right handed neutrinos have zero charges they can acquire very heavy Majorana masses. The heavy Majorana right-handed neutrinos may decay into final states with lepton number L = ±1, thereby creating a lepton asymmetry in the early Universe. Because the Yukawa couplings of exotic particles are not constrained by the neutrino oscillation data, substantial values of CP violating lepton asymmetries can be induced even for a relatively small mass of the lightest right handed neutrino (M GeV) so that successful thermal leptogenesis may be achieved without encountering any gravitino problem [14]. The superpotential of the E 6 SSM involves a lot of new Yukawa couplings in comparison to the SM. In general these new interactions violate baryon number conservation and induce nondiagonal flavour transitions. To suppress baryon number violating and flavour changing processes one can postulate a Z2 H symmetry under which all superfields except one pair of Hi d and Hi u (say H d H3 d and H u H3 u ) and one SMtype singlet field (S S 3 ) are odd. The Z2 H symmetry reduces the structure of the Yukawa interactions, and an assumed hierarchical structure of the Yukawa interactions allows to simplify the form of the E 6 SSM superpotential substantially. Keeping only Yukawa interactions whose couplings are allowed to be of order unity leaves 8 The two superfields H and H may be removed from the spectrum, thereby avoiding the μ problem, leading to unification at the string scale [16]. However we shall not pursue this possibility in this paper. us with W E6 SSM λs(h d H u )+λ α S(HαH d α) u + κ i S(D i D i )+h t (H u Q)t c + h b (H d Q)b c + h τ (H d L)τ c + μ (H H ), (2) where α, β =1, 2 and i =1, 2, 3, and where the superfields L = L 3, Q = Q 3, t c = u c 3, b c = d c 3 and τ c = e c 3 belong to the third generation and λ i, κ i are dimensionless Yukawa couplings with λ λ 3. Here we assume that all right handed neutrinos are relatively heavy so that they can be integrated out 9. The SU(2) L doublets H u and H d, which are even under the Z2 H symmetry, play the role of Higgs fields generating the masses of quarks and leptons after EWSB. The singlet field S must also acquire a large VEV to induce sufficiently large masses for the Z boson. The couplings λ i and κ i should be large enough to ensure the exotic fermions are sufficiently heavy to avoiding conflict with direct particle searches at present and former accelerators. They should also be large enough so that the evolution of the soft scalar mass m 2 S of the singlet field S results in negative values of m 2 S at low energies, triggering the breakdown of the U(1) N symmetry. However the Z2 H can only be approximate (otherwise the exotics would not be able to decay). To prevent rapid proton decay in the E 6 SSM a generalised definition of R parity should be used. We give two examples of possible symmetries that can achieve that. If Hi d, Hu i, S i, D i, D i and the quark superfields (Q i, u c i, dc i ) are even under a discrete Z2 L symmetry while the lepton superfields (L i, e c i, Ni c ) are odd (Model I) then the allowed superpotential is invariant with respect to a U(1) B global symmetry. The exotic D i and D i are then identified as diquark and anti-diquark, i.e. B D = 2/3 and B D =2/3. An alternative possibility is to assume that the exotic quarks D i and D i as well as lepton superfields are all odd under Z2 B whereas the others remain even. In this case (Model II) the D i and D i are leptoquarks [5]. After the breakdown of the gauge symmetry, H u, H d and S form three CP even, one CP-odd 9 We shall ignore the presence of right-handed neutrinos in the subsequent RG analysis.

4 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) and two charged states in the Higgs spectrum. The mass of one CP even Higgs particle is always very close to the Z boson mass M Z. The masses of another CP even, the CP odd and the charged Higgs states are almost degenerate. Furthermore, like in the MSSM and NMSSM, one of the CP even Higgs bosons is always light irrespective of the SUSY breaking scale. However, in contrast with the MSSM, the lightest Higgs boson in the E 6 SSM can be heavier than GeV even at tree level. In the two loop approximation the lightest Higgs boson mass does not exceed GeV [5]. Thus the SM like Higgs boson in the E 6 SSM can be considerably heavier than in the MSSM and NMSSM, since it contains a similar F-term contribution as the NMSSM but with a larger maximum value for λ(m t )asitis not bounded as strongly by the validity of perturbation theory up to the GUT scale [5]. 3. The Constrained E 6 SSM The simplified superpotential of the E 6 SSM involves seven extra couplings (μ, κ i and λ i )as compared with the MSSM with μ = 0. The soft breakdown of SUSY gives rise to many new parameters. The number of fundamental parameters can be reduced drastically though within the constrained version of the E 6 SSM (ce 6 SSM) [13]. Constrained SUSY models imply that all soft scalar masses are set to be equal to m 0 at some high energy scale M X, taken here to be equal to the GUT scale, all gaugino masses M i (M X ) are equal to M 1/2 and trilinear scalar couplings are such that A i (M X )=A 0. Thus the ce 6 SSM is characterised by the following set of Yukawa couplings, which are allowed to be of the order of unity, and universal soft SUSY breaking terms, λ i (M X ), κ i (M X ), h t (M X ), h b (M X ), h τ (M X ), m 0, M 1/2, A 0, (3) where h t (M X ), h b (M X ) and h τ (M X ) are the usual t quark, b quark and τ lepton Yukawa couplings, and λ i (M X ), κ i (M X ) are the extra Yukawa couplings defined in Eq. (2). The universal soft scalar and trilinear masses correspond to an assumed high energy soft SUSY breaking potential of the universal form, V soft = m i 27 i + A 0 Y ijk 27 i 27 j 27 k + h.c., (4) where Y ijk are generic Yukawa couplings from the trilinear terms in Eq. (2) and the 27 i represent generic fields from Eq. (1), and in particular those which appear in Eq. (2). To simplify our analysis we assume that all parameters in Eq. (3) are real and M 1/2 is positive. In order to guarantee correct EWSB m 2 0 has to be positive. The set of ce 6 SSM parameters in Eq. (3) should in principle be supplemented by μ and the associated bilinear scalar coupling B. However, since μ is not constrained by the EWSB and the term μ H H in the superpotential is not suppressed by E 6, the parameter μ will be assumed to be 10 TeV so that H and H decouple from the rest of the particle spectrum. As a consequence the parameters B and μ are irrelevant for our analysis. To calculate the particle spectrum within the ce 6 SSM one must find sets of parameters which are consistent with both the high scale universality constraints and the low scale EWSB constraints. To evolve between these two scales we use two loop renormalisation group equations (RGEs) for the gauge and Yukawa couplings together with two loop RGEs for M a (Q) and A i (Q) as well as one loop RGEs for mi 2 (Q). Q is the renormalisation scale. The details of the procedure are discussed in [13]. An early LHC discovery benchmark point is shown in Fig. 1. Note that this ce 6 SSM analysis 4. LHC Predictions of the E 6 SSM 4.1. SUSY spectrum and signatures In the ce 6 SSM m 0 >M 1/2 for each value of s and also that lower M 1/2 is weakly correlated with lower s and thus lower Z masses. As is discussed in detail in Ref. [6] this bound is caused, depending on the value of tan β, either by the inert Higgs masses being driven below their experimental limit from negative D-term contributions canceling the positive contribution from m 0 or the light Higgs mass going below the LEP2 limit. Another remarkable feature of the ce 6 SSM is that the low energy gluino mass parameter M 3 is

5 124 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) Figure 1. A spectrum for an early discovery LHC benchmark point with the following parameters: tan β = 10, λ 3 (M X )= 0.378, λ 1 (M X )= λ 2 (M X ) = 0.1, κ 3 (M X ) = 0.42, κ 1 (M X ) = κ 2 (M X )=0.06, s =2.7TeV, M 1/2 = 388GeV, m 0 = 681GeV, A 0 = 645GeV. driven to be smaller than M 1/2 by RG running. The reason for this is that the E 6 SSM has a much larger (super)field content than the MSSM (three 27 s instead of three 16 s), so much so that at one loop order the QCD beta function (accidentally) vanishes in the E 6 SSM, and at two loops it loses asymptotic freedom (though the gauge couplings remain perturbative at high energy). This implies that the low energy gaugino masses are all less than M 1/2 in the ce 6 SSM, being given by roughly M 3 0.7M 1/2, M M 1/2, M M 1/2. These should be compared to the corresponding low energy values in the MSSM, M 3 2.7M 1/2, M 2 0.8M 1/2, M 1 0.4M 1/2. Thus, in the ce 6 SSM, since the low energy gaugino masses M i are driven by RG running to be small, the lightest SUSY states will generally consist of a light gluino of mass M 3, a light wino-like neutralino and chargino pair of mass M 2, and a light binolike neutralino of mass M 1, which are typically all much lighter than the Higgsino masses of order μ = λs/ 2, where λ cannot be too small for correct EWSB. Since m 0 > M 1/2 and the squarks and sleptons also have larger M 1/2 coefficients than the gauginos these are also much heavier than the light gauginos. Thus, throughout all ce 6 SSM regions of parameter space there is the striking prediction that the lightest sparticles always include the gluino g, the two lightest neutralinos χ 0 1,χ 0 2, and a light chargino χ ± 1. Therefore pair production of χ0 2χ 0 2, χ 0 2χ ± 1, χ± 1 χ 1 and g g should always be possible at the LHC irrespective of the Z mass. Due to the hierarchical spectrum, the gluinos can be relatively narrow states with width Γ g M 5 g /m4 q, comparable to that of W ± and Z bosons. They will decay through g q q q q + ET miss, so gluino pair production will result in an appreciable enhancement of the cross section for pp q qq q + ET miss + X, where X refers to any number of light quark/gluon jets. The second lightest neutralino decays through χ 0 2 χ l l and so would produce an excess in pp l ll l + ET miss + X, which could be observed at the LHC. Since all squarks and sleptons, as well as new exotic particles, turn out to be rather heavy compared to the low energy wino mass, the calculation of the branching ratio Br(χ 0 2 χ l l) is very similar to that in the MSSM. This branching ratio in the MSSM is known to be very sensitive to the choice of fundamental parameters of the model. For the type of the neutralino spectra presented later, in which the second lightest neutralino is approximately wino, the lightest neutralino is approximately bino, and where the other sparticles are much heavier, Br(χ 0 2 χ l l) is known to vary from 1.5% to 6% Exotic spectrum and signatures Other possible manifestations of the E 6 SSM at the LHC are related to the presence of a Z and exotic multiplets of matter. The production of a TeV scale Z will provide an unmistakable and spectacular LHC signal even with first data [5]. At the LHC, the Z boson that appears in the E 6 inspired models can be discovered if it has a mass below TeV. The determination of its

6 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) couplings should be possible if M Z < TeV. When the Yukawa couplings κ i of the exotic fermions D i and D i have a hierarchical structure, some of them can be relatively light so that their production cross section at the LHC can be comparable with the cross section of t t production [5]. In the E 6 SSM, the D i and D i fermions are SUSY particles with negative R parity so they must be pair produced and decay into quark squark (if diquarks) or quark slepton, squark lepton (if leptoquarks), leading to final states containing missing energy from the LSP. The lifetime and decay modes of the exotic coloured fermions are determined by the Z2 H violating couplings. If Z2 H is broken significantly the presence of the light exotic quarks gives rise to a remarkable signature. Assuming that D i and D i fermions couple most strongly to the third family (s)quarks and (s)leptons, the lightest exotic D i and D i fermions decay into tb, t b, t b, t b (if they are diquarks) or tτ, t τ, bν τ, bν τ (if they are leptoquarks). This can lead to a substantial enhancement of the cross section of either pp t tb b + ET miss + X (if diquarks) or pp t tτ τ +ET miss +X or pp b b+e T miss +X (if leptoquarks). Notice that SM production of t tτ + τ is (α W /π) 2 suppressed in comparison to the leptoquark decays. Therefore light leptoquarks should produce a strong signal with low SM background at the LHC. In principle the detailed LHC analyses are required to establish the feasibility of extracting the excess of t tb b or t tτ + τ production induced by the light exotic quarks predicted by our model. We have already remarked that the lifetime and decay modes of the exotic coloured fermions are determined by the Z2 H violating couplings. If Z2 H is only very slightly broken exotic quarks may be very long lived, with lifetimes up to about 1 sec. This is the case, for example, in some minimal versions of the model [16]. In this case the exotic D i and D i fermions could hadronize before decaying, leading to spectacular signatures consisting of two low multiplicity jets, each containing a single quasi-stable heavy D-hadron, which could be stopped for example in the muon chambers, before decaying much later. The observation of the D fermions might be possible if they have masses below about TeV [5]. Similar considerations apply to the case of exotic D i and D i scalars except that they are non SUSY particles so they may be produced singly and decay into quark quark (if diquarks) or quark lepton (if leptoquarks) without missing energy from the LSP. It is possible to have relatively light exotic coloured scalars due to mixing effects. The RGEs for the soft breaking masses, m 2 Di and m 2 Di, are very similar, with d dt m (m2 Di 2 Di )=g 1 2 M 1 2, resulting in comparatively small splitting between these soft masses. Consequently, mixing can be large even for moderate values of the A 0, leading to a large mass splitting between the two scalar partners of the exotic coloured fermions. 10 Recent, as yet unpublished, results from Tevatron searches for dijet resonances rule out scalar diquarks with mass less than 630 GeV. However, scalar leptoquarks may be as light as 300 GeV since at hadron colliders they are pair produced through gluon fusion. Scalar leptoquarks decay into quark lepton final states through small Z2 H violating terms, for example D tτ, and pair production leads to an enhancement of pp t tτ τ (without missing energy) at the LHC. In addition, the inert Higgs bosons and Higgsinos (i.e. the first and second families of Higgs doublets predicted by the E 6 SSM which couple weakly to quarks and leptons and do not get VEVs) can be light or heavy depending on their free parameters. The light inert Higgs bosons decay via Z2 H violating terms which are analogous to the Yukawa interactions of the Higgs superfields, H u and H d. One can expect that the couplings of the inert Higgs fields would have a similar hierarchical structure as the couplings of the normal Higgs multiplets, therefore we assume the Z2 H breaking interactions predominantly couple the inert Higgs bosons to the third genera- 10 Note that the diagonal entries of the exotic squark mass matrices have substantial negative contributions from the U(1) N D term quartic interactions in the scalar potential. These contributions reduce the masses of exotic squarks and also contribute to their mass splitting since the U(1) N charges of D i and D i are different.

7 126 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) tion. So the neutral inert Higgs bosons decay predominantly into 3rd generation fermion antifermion pairs like H1,i 0 b b. The charged inert Higgs bosons also decay into fermion antifermion pairs, but in this case it is the antiparticle of the fermions EW partner, e.g. H 1,i τ ν τ. The inert Higgs bosons may also be quite heavy, so that the only light exotic particles are the inert Higgsinos. Similar couplings govern the decays of the inert Higgsinos; the electromagnetically neutral Higgsinos predominantly decay into fermion-anti-sfermion pairs (e. g. H0 i t t, H i 0 τ τ ). The charged Higgsinos decays similarly but in this case the sfermion is the SUSY partner of the EW partner of the fermion (e.g. H + i t b, H i τ ν τ ). 5. Inert Neutralino Dark Matter Neutralino dark matter in the E 6 SSM naturally arises from the inert neutralino sector as discussed in [8]. The most important couplings for neutralino dark matter are the trilinear couplings between the three generations of up- and down-type Higgs doublets and Higgs SM singlets contained in the superpotential of the E 6 SSM, λ ijk S i H dj H uk = λ ijk (S i H dj H+ uk S ih 0 dj H 0 uk).(5) The trilinear coupling tensor λ ijk in Eq. (5) consists of 27 numbers, which play various roles. The purely third family coupling λ 333 λ is very important, because it is the combination μ = λs/ 2 that plays the role of an effective μ term in this theory (where s/ 2 is the VEV of the third family singlet scalar S 3 S). Some other neutralino mass terms, such as those involving S, are also proportional to λ. The couplings of the inert (first and second generation) Higgs doublet superfields to the third generation Higgs singlet superfield λ 3αβ λ αβ (where α, β, γ index only the first and second generations) directly contribute to neutralino and chargino mass terms for the inert Higgsino doublets. λ α3β f dαβ and λ αβ3 f uαβ directly contribute to neutralino mass terms involving an inert doublet Higgsino and singlino. The full neutralino mass matrix is of the form M ME n USSM n B 2 B 1 6 SSM = B2 T A 22 A 21, (6) B1 T A T 21 A 11 where MUSSM n is the neutralino mass matrix in the USSM [7] in the basis: χ 0 USSM =( B W 3 H0 d H0 u S B ) T. (7) The matrix blocks denoted as A ij refer to the inert neutralinos. For example, for the first inert generation, the neutralino mass sub-matrix is, 0 λ s f u v sin β A 11 = c λ s 0 f d v cos β (8) f u v sin β f d v cos β 0 in the basis χ 0 int =( H0 d1 H0 u1 S1 ) T (9) where c = 1 2, λ = λ 11 λ 311, f d = f d11 λ 131 and f u = f u11 λ 113. It is natural to assume that λ s fv and this will lead to a light, mostly first-generation-singlino lightest neutralino. Note that the ce 6 SSM analysis discussed in the previous section did not consider the full neutralino mass matrix in Eq. (6), but only considered the MUSSM n part, plus the parameters λ 1 λ 11 and λ 2 λ 22 from a completely decoupled inert neutralino sector, i.e. B 1 = B 2 =0 and f dαβ = f uαβ =0. (λ 21 and λ 12 were set to zero in Eq. (2) by a rotation of the inert Higgs basis.) Dark matter relic abundance from the inert sector was therefore not considered and inert fermions were not present in fig. 1. Numerical results in Fig. 2 show that regions with successful relic abundance are possible in this model, assuming A 11 = A 22 and A 21 = ɛa 22 and λ = λ 22 = λ 11, f = f u = f d. The results for the relic abundance in the E 6 SSM are radically different from those for both the MSSM and the USSM. This is because the two families of inert Higgsinos and singlinos predicted by the E 6 SSM provide an almost decoupled neutralino sector with a naturally light LSP which can account for the cold dark matter relic abundance independently of the rest of the model. Although the E 6 SSM has two inert families, the presence of

8 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) tan(β) GeV mh1 /2 70 GeV mw ( 80 GeV) mz ( 91 GeV) 50 GeV Ωχ > ΩCDM mz/2 Ωχ < ΩCDM 40 GeV λ 30 GeV Ωχ > ΩCDM Figure 2. Contour plot of the LSP mass and relic density Ω χ h 2 regions in the (λ, tan β)-plane with s = 3000 GeV, ɛ =0.1 and f = 1. The red region is where the prediction for Ω χ h 2 is consistent with the measured 1-sigma range of Ω CDM h 2. Where the LSP is lower than half of the Z-boson mass, the region to the right of the hatched line is ruled out by Z decay data. the second inert family is not crucial for achieving successful dark matter relic abundance. In the successful regions where the observed dark matter relic density is reproduced the neutralino mass spectrum is well described by the LSP being mostly inert singlino and with a mass approximately proportional to v 2 /s, and, as λ s is decreased, the LSP becomes heavier and also less inert singlino dominated, picking up significant inert doublet Higgsino contributions. To avoid conflict with high precision LEP data on the Z-pole, the LSP, which necessarily must couple significantly to the Z-boson in order to achieve a successful relic abundance, should have a mass which exceeds half the Z-boson mass. Since the LSP mass is proportional to f d f u sin(2β), we find that regions of parameter space in which the dark matter relic density prediction is consistent with observation require low values of tan β, less than about 2. Depending on the value of the singlet VEV s, the f u,d trilinear Higgs coupling parameters should also be reasonably large compared to the λ αβ ones. Note that these f u,d couplings did not appear in Eqs. (2) or (3). In general it is difficult for the true neutralino LSP to be heavier than about 100 GeV. In the successful regions we find the lightest chargino mass could be as low as the experimental lower limit of 94 GeV, although it could also be as high as about 300 GeV. Thus, neutralino dark matter could arise from an almost decoupled sector of inert Higgsinos and singlinos, and if it does then the parameter space of the rest of the model is completely opened up. For example if such a model is regarded as an extension of the MSSM, then the lightest MSSM-like SUSY particle is not even required to be a neutralino, and could even be a sfermion which would be able to decay into the true LSP coming from the almost decoupled inert Higgsino/singlino sector. This is because the mostly inert-higgsino/singlino LSP would have admixtures of MSSM neutralino states. The size of these components are set by Z2 H breaking λ ijk couplings and need not be extremely small. In collider experiments it may be possible that the E 6 SSM could be distinguished from other supersymmetric models due to long neutralino/chargino decay chains, since the E 6 SSM has more of these states. However this has not been studied. The unique make-up of this model s LSP would also have implications for direct detection, but this has also not yet been studied. 6. Beyond the E 6 SSM 6.1. The Minimal E 6 SSM As already mentioned, the E 6 SSM requires a further pair of superfields H and H with a mass term μ H H from incomplete extra 27 and 27 representations to survive to low energies to ensure gauge coupling unification. Moreover, in the E 6 SSM, the superfields H and H are not identified with Higgs fields, and do not develop VEVs. If these fields couple to right-handed neutrinos then they may alternatively be regarded a fourth family of lepton doublets L 4 H and may play a role in leptogenesis [14]. Clearly, although the E 6 SSM solves the μ problem, it introduces a new μ problem, since μ 10 TeV is a theoretically undetermined mass scale in the superpotential, unrelated (in principle) to the SUSY breaking scale. Accordingly, as for the μ

9 128 P. Athron et al. / Nuclear Physics B (Proc. Suppl.) (2010) term in the MSSM, there is no reason why the scale μ should not be of order the Planck scale. The difference here is that, since the scale μ is unrelated to the EWSB scale, there is no reason why we cannot consider μ M P as a possibility. In other words, it is possible to contemplate a version of the model in which the low energy theory (below the Planck scale) does not contain the pair of superfields H and H. Such a version of the model is called the minimal E 6 SSM [15]. In such ame 6 SSM the Standard Model gauge couplings can be embedded into a Pati-Salam gauge group at the GUT scale GeV, leading to a full unification with gravity at the string or Planck scale GeV [15]. The phenomenology of the me 6 SSM is very similar to that of the E 6 SSM, apart from the couplings of the exotic coloured D fermions being highly suppressed in the me 6 SSM, leading to long lived (lifetimes of order 1 second) D-hadrons at the LHC which provides a striking signature of the me 6 SSM [15] An E 6 SSM theory of flavour The E 6 SSM (or me 6 SSM) is designed to permit a conventional see-saw mechanism, since by construction the right-handed neutrinos are neutral under the extra U(1) N gauge group charges. As such, it is possible to account for neutrino tribimaximal mixing in the E 6 SSM (or me 6 SSM) by using the idea of discrete family symmetry, as in flavour extensions of the MSSM. For example, by introducing a discrete Δ 27 family symmetry it is possible to account for neutrino masses and tri-bimaximal lepton mixing, as well as all the charged fermion masses and mixing [16]. Such theories of flavour also explain the origin of the Z2 H symmetry which distinguishes the third family of Higgs doublets leading to suppressed flavour changing neutral currents from Higgs exchange [16]. REFERENCES 1. For a recent review see e.g. D. J. H. Chung, L. L. Everett, G. L. Kane, S. F. King, J. Lykken, L. T. Wang, Phys. Rept. 407 (2005) J. E. Kim and H. P. Nilles, Phys. Lett. B 138 (1984) 150. For a recent discussion of the μ problem see T. Cohen and A. Pierce, arxiv: [hep-ph]. 3. G. L. Kane and S. F. King, Phys. Lett. B 451 (1999) 113 [arxiv:hep-ph/ ]; 4. P. Binetruy, S. Dawson, I. Hinchliffe, M. Sher, Nucl. Phys. B 273 (1986) 501; J. R. Ellis, K. Enqvist, D. V. Nanopoulos, F. Zwirner, Mod. Phys. Lett. A 1 (1986) 57; L. E. Ibanez, J. Mas, Nucl. Phys. B 286 (1987) 107; J. F. Gunion, H. E. Haber, L. Roszkowski, Phys. Lett. B 189 (1987) 409; H. E. Haber, M. Sher, Phys. Rev. D 35 (1987) 2206; J. R. Ellis, D. V. Nanopoulos, S. T. Petcov, F. Zwirner, Nucl. Phys. B 283 (1987) 93; M. Drees, Phys. Rev. D 35 (1987) 2910; H. Baer, D. Dicus, M. Drees, X. Tata, Phys. Rev. D 36 (1987) 1363; J. F. Gunion, H. E. Haber, L. Roszkowski, Phys. Rev. D 38 (1988) 105; D. Suematsu, Y. Yamagishi, Int. J. Mod. Phys. A 10 (1995) 4521; E. Keith, E. Ma, Phys. Rev. D 56 (1997) 7155; Y. Daikoku, D. Suematsu, Phys. Rev. D 62 (2000) ; E. Ma, Phys. Lett. B 380 (1996) S. F. King, S. Moretti, R. Nevzorov, Phys. Rev. D 73 (2006) , Phys. Lett. B 634 (2006) 278 and arxiv:hep-ph/ ; S. Kraml et al. (eds.), Workshop on CP studies and non-standard Higgs physics, CERN [arxiv:hep-ph/ ]; S. F. King, S. Moretti, R. Nevzorov, AIP Conf. Proc. 881 (2007) P. Athron, S. F. King, D. J. Miller, S. Moretti and R. Nevzorov, arxiv: [hep-ph]; P. Athron, S. F. King, D. J. 2. Miller, S. Moretti and R. Nevzorov, arxiv: [hep-ph]. 7. J. Kalinowski, S. F. King, J. P. Roberts, arxiv: [hep-ph]. 8. J. P. Hall and S. F. King, arxiv: [hep-ph]. 9. S. F. King, S. Moretti, R. Nevzorov, Phys. Lett. B 650 (2007) R. Howl, S. F. King, JHEP 0801 (2008) S. F. King, R. Luo, D. J. Miller, R. Nevzorov, JHEP 0812 (2008) R. Howl and S. F. King, JHEP 0805 (2008)

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