Non-Standard Higgs Decays in U(1) Extensions of the MSSM

In U(1) extensions of the Minimal Supersymmetric extension of the Standard Model there is a simple mechanism that leads to a heavy Z' boson with a mass which is substantially larger than the supersymmetry breaking scale. This mechanism may also result in a pseudoscalar state that is light enough for decays of the 125 GeV Standard Model-like Higgs boson into a pair of such pseudoscalars to be kinematically allowed. We study these decays within E6 inspired supersymmetric models with an exact custodial symmetry that forbids tree-level flavor-changing transitions and the most dangerous baryon and lepton number violating operators. We argue that the branching ratio of the lightest Higgs boson decays into a pair of the light pseudoscalar states may not be negligibly small.


Introduction
Such scenarios may be realised when the extra U (1) gauge symmetry, U (1) , is broken by two VEVs coming from the superfields S and S, which are both singlets under the SM gauge group but have opposite U (1) charges. In this case the U (1) D-term contribution to the scalar potential may force the minimum of this potential to be along the D-flat direction (see, for example, [3]). As a consequence the VEVs S and S can be much larger than the SUSY breaking scale.
The simplest renormalisable superpotential of the SUSY model of the type discussed above can be written as where φ is a scalar superfield that does not participate in the gauge interactions. When the coupling σ goes to zero the corresponding tree-level scalar potential takes the form where m 2 S , m 2 S and m 2 φ are soft SUSY breaking mass parameters squared, while g 1 is the U (1) gauge coupling and Q S is the U (1) charge of the SM singlet superfields S and S.
In Eq. (2) the last term is associated with the extra U (1) D-term contribution. In the limit S = S this quartic term vanishes. If (m 2 S + m 2 S ) < 0 then there is a run-away direction in this model, so that S = S → ∞. When the F -terms from the interaction in the superpotential Eq. (1) are included this stabilizes the run-away direction and for small values of the coupling σ the SM singlet superfields tend to acquire large VEVs, i.e.
resulting in an extremely heavy Z boson.
Although the SUSY model mentioned above looks rather simple and elegant it also possesses an additional accidental global U (1) symmetry which can be associated with the Peccei-Quinn (PQ) symmetry [4]. This symmetry is spontaneously broken by the VEVs of the SM singlet superfields resulting in a massless axion [5]. To avoid the appearance of this axion one needs to include in the superpotential of Eq. (1) polynomial terms with respect to the superfield φ which explicitly break the global U (1) symmetry. If the couplings that violate the PQ symmetry are very small, then the particle spectrum of this SUSY model should contain a pseudo-Goldstone boson which can be considerably lighter than all sparticles and Higgs bosons. In fact, the corresponding pseudoscalar Higgs state may be so light that the decay of the SM-like Higgs boson into a pair of these states can be kinematically allowed.
In this article we consider such non-standard Higgs decays within well motivated E 6 inspired extensions of the MSSM, with the particular model described in section 2. We focus on scenarios with an approximate global U (1) symmetry that leads to a pseudo-Goldstone boson in the particle spectrum. The pseudo-Goldstone state in these scenarios is mainly a linear superposition of the imaginary parts of the scalar components of the SM singlet superfields φ, S and S. The SM-like Higgs boson on the other hand is predominantly a linear superposition of the neutral components of the Higgs doublets H u and H d , so that the coupling of the pseudo-Goldstone state to the SM-like Higgs boson can be expected to be somewhat suppressed. However it can still lead to a non-negligible branching ratio of the lightest Higgs decays into a pair of pseudo-Goldstone bosons.
In this context it is worth noting that the decay rate of the SM-like Higgs state into a pair of pseudoscalars was intensively studied within the simplest extension of the MSSM, i.e. the Next-to-Minimal Supersymmetric extension of the SM (NMSSM). For reviews of non-standard Higgs boson decays see [6] and for a more recent work see e.g. [7]. The NMSSM superpotential is given by [8]: where W MSSM (µ = 0) is the MSSM superpotential with the bilinear mass µ set to zero, and λ and κ are new NMSSM-specific couplings 2 . In the limit where the cubic coupling 2 One of the motivations of the NMSSM is the dynamic generation of the supersymmetric Higgs mass parameter µ through the coupling term SH d H u when the singlet field S acquires a vacuum expectation value S , i.e. µ = λ S . κ = 0 the Lagrangian of the NMSSM is invariant under the transformations of the PQ symmetry which leads to the massless axion when it is spontaneously broken by the VEV S . If κ is rather small then the NMSSM particle spectrum involves one light scalar state and one light pseudoscalar state. In addition, if κ → 0 and the SUSY breaking scale is of the order of a TeV, then the lightest SUSY particle (LSP) is the lightest neutralinoχ 0 1 and is predominantly singlino. In this case the LSP couplings to the SM particles are quite small resulting in a relatively small annihilation cross section forχ 0 1χ 0 1 → SM particles, which gives rise to a relic density that is typically much larger than its measured value.
As a consequence it seems to be rather problematic to find phenomenologically viable scenarios with a light pseudoscalar in the case of the NMSSM with approximate PQ symmetry. Nevertheless a sufficiently light pseudoscalar can always be obtained by tuning the parameters of the NMSSM.
In contrast to the NMSSM the mass of the lightest neutralino in the SUSY model considered here does not become small when the PQ symmetry violating couplings vanish.
Moreover, even when all PQ symmetry violating couplings are negligibly small, the LSP can be higgsino-like. This allows a reasonable value for the dark matter density to be obtained if the LSP has a mass below 1 TeV (see, for example [9] The paper is organised as follows. In section 2 we briefly review E 6 inspired SUSY models with exact custodialZ H 2 symmetry. In section 3 we study the breakdown of the gauge symmetry and the implications for Higgs phenomenology. In section 4 we discuss a set of benchmark scenarios that lead to the decays of the lightest Higgs boson into a pair of pseudoscalar states. Our results are summarized in section 5. In Appendix A the spectrum of the neutralino states is examined.
2. E 6 inspired SUSY models with exactZ H 2 symmetry In this section, we briefly review the E 6 inspired SUSY models with exact custodial Z H 2 symmetry [10], which we then use to demonstrate how light pseudoscalar states can appear in SUSY models with an extra U (1) gauge symmetry. We also consider what kind of Higgs decay rates they can lead to.
The breakdown of the E 6 symmetry at high energies may lead to models based on rank-5 gauge groups with an additional U (1) factor in comparison to the SM. In this case the extra U (1) gauge symmetry is a linear combination of U (1) χ and U (1) ψ , which are defined by the breakdown of the exceptional Lie group E 6 into SO(10), E 6 → SO(10) × U (1) ψ , and the subsequent breakdown of SO(10) into SU (5), SO(10) → SU (5) × U (1) χ (for a review see Refs. [11,12] [12,13]. A few years ago the Tevatron and early LHC Z mass limits in these models were discussed in Ref. [14]. Collider signatures associated with the exotic quarks and squarks have been considered in [15]. Previously, the implications of E 6 inspired SUSY models with an additional U (1) gauge symmetry have been studied for electroweak symmetry breaking (EWSB) [16][17][18][19][20], neutrino physics [21,22], leptogenesis [23,24], electroweak (EW) baryogenesis [25], the muon anomalous magnetic moment [26], the electric dipole moment of the electron [27] and of the tau lepton [28], for lepton flavor violating processes like µ → eγ [29] and for CP-violation in the Higgs sector [30]. The neutralino sector in E 6 inspired SUSY models was analysed in [19,[27][28][29][31][32][33][34][35]. Such models have also been proposed as the solution to the tachyon problems of anomalymediated SUSY breaking, via U (1) D-term contributions [36], and have been used in combination with a generation symmetry to construct a model explaining the fermion mass hierarchy and mixing [37]. The Higgs sector and the theoretical upper bound on the lightest Higgs boson mass in the E 6 inspired SUSY models were examined in [20,35,[38][39][40][41].
Here we focus on the E 6 inspired SUSY extension of the SM based on the low-energy SM gauge group together with an extra U (1) N gauge symmetry in which right-handed neutrinos do not participate in the gauge interactions. This corresponds to θ = arctan √ 15 in Eq. (5). In this Exceptional Supersymmetric Standard Model (E 6 SSM) [38,39] righthanded neutrinos may be superheavy, shedding light on the origin of the mass hierarchy in the lepton sector and providing a mechanism for the generation of the baryon asymmetry in the Universe via leptogenesis [23,24]. E 6 inspired SUSY models with an additional U (1) N gauge symmetry have been studied in a variety of contexts. Thus they have been investigated in [22] in the context of non-standard neutrino models with extra singlets, in [31] from the point of view of Z − Z mixing, in Refs. [19,31,32] the neutralino sector was explored, in [19,42] the renormalisation group (RG) flow of the couplings was examined, and in [18][19][20] EWSB was studied. The presence of a Z boson and of exotic quarks as predicted by the E 6 SSM provides spectacular new physics signals at the LHC, which were analysed in [38][39][40]43]. The existence of light exotic particles also leads to the non-standard decays of the SM-like Higgs boson that were discussed in detail in [44,45].
Within the constrained version of the E 6 SSM the particle spectrum and associated collider signatures were studied in [46] and the degree of fine tuning has recently been examined in [47]. The threshold corrections to the running gauge and Yukawa couplings in the E 6 SSM and their numerical impact in the cE 6 SSM were studied in detail in Ref. [48].
Alternative boundary conditions that take account of D-terms from the other U (1) gauge symmetry broken at the GUT scale were considered in [49], using the first or second generation sfermion masses to constrain the GUT scale parameters. The renormalisation of the VEVs in the E 6 SSM was considered in [50].
The presence of exotic matter in the E 6 SSM generically leads to non-diagonal flavor transitions and rapid proton decay. A set of discrete symmetries can be imposed in order to suppress these processes [38,39]. In this article we study the non-standard Higgs decays mentioned above within the E 6 inspired SUSY models with the extra U (1) N factor in which a single discreteZ H 2 symmetry forbids tree-level flavor-changing transitions and the most dangerous baryon and lepton number violating operators [10]. These models imply that E 6 or its subgroup is broken down to SU (3) C × SU (2) L × U (1) Y × U (1) ψ × U (1) χ near the Grand Unification Theory (GUT) scale, which we denote as M X . Below this scale M X the particle content of these SUSY models involves three copies of 27-plets and a set of M l and M l supermultiplets from the incomplete 27 l and 27 l representations of E 6 , where l runs over the different multiplets that are summarized below in Table   1. All components of the complete 27 i -plets are odd under the discrete symmetryZ H 2 , while the supermultiplets M l can be either odd or even. The supermultiplets M l are even under theZ H 2 symmetry, and as a consequence they can be used for the breakdown of the gauge symmetry, while preserving the discrete symmetry. To ensure that the in these E 6 inspired models, where Z M 2 = (−1) 3(B−L) is a matter parity. This is possible because matter parity is a discrete subgroup of U (1) ψ and U (1) χ . With such a breakdown into U (1) N × Z M 2 the right-handed neutrino mass can be generated without breaking the remaining gauge symmetry and all exotic states which originate from the 27 i representa-to be considerably larger than the age of the Universe. Long-lived exotic quarks would have been produced during the very early epochs of the Big Bang and those that survive annihilation would subsequently have been confined in heavy hadrons forming nuclear isotopes that would be present in terrestrial matter. Various theoretical estimates [51] show that if such stable relics in the mass range from 1 GeV to 10 TeV would exist in nature, today their concentration would be O(10 −10 ) per nucleon. At the same time different experiments set strong upper limits on the relative concentrations of such nuclear isotopes from 10 −15 to 10 −30 per nucleon [52]. Therefore E 6 inspired models with very long-lived exotic quarks are ruled out.
To ensure that the lightest exotic quarks decay within a reasonable time in the simplest scenario, we supplement the set ofZ H 2 even supermultiplets M l with L 4 , where L 4 and L 4 are lepton SU (2) L doublet supermultiplets that originate from a pair of additional 27 L and 27 L . The supermultiplets L 4 and L 4 should form TeV scale vectorlike states to break the U (1) D symmetry and render the lightest exotic quark unstable 3 . Therefore L 4 and L 4 both have to be even under theZ H 2 symmetry. In this case the baryon and lepton number conservation implies that the exotic quarks are leptoquarks.
Here we assume that, in addition to H u , H d , S, L 4 , L 4 N c H and N c H , the particle spectrum below the scale M X involvesZ H 2 -even superfields S and φ. The superfield φ does not participate in the SU (3) C ×SU (2) L ×U (1) Y ×U (1) ψ ×U (1) χ gauge interactions but its scalar component acquires a non-zero VEV. Taking into account that the components of the superfields S and φ are expected to gain TeV scale masses whereas the right-handed neutrino superfields are superheavy, the low-energy matter content in the E 6 inspired SUSY models discussed above involves where α = 1, 2 runs over the first two generations and i = 1, 2, 3 runs over all three.
We have denoted here the left-handed quark and lepton doublets by Q i and L i , respectively, and the right-handed up-and down-type quarks and charged leptons by u c i , d c i and e c i . Neglecting all suppressed non-renormalisable interactions, the low-energy effective superpotential of these models can be written as 3 The appropriate mass term µ L L 4 L 4 in the superpotential can be induced within SUGRA models just after the breakdown of local SUSY if the Kähler potential contains an extra term (Z L (L 4 L 4 ) + h.c) [53].
The gauge group and field content of the E 6 inspired SUSY models under consideration can originate from the 5D and 6D orbifold GUT models in which the splitting of GUT multiplets can be naturally achieved [10]. In these orbifold GUT models all GUT relations between the Yukawa couplings can get spoiled while the approximate unification of the SM gauge couplings still takes place. From Eq. (7) it follows that extra matter beyond the MSSM fills in complete SU (5) representations in these models. As a consequence the gauge coupling unification remains almost exact in the one-loop approximation. It was also shown that in the two-loop approximation the unification of the gauge couplings in the SUSY models under consideration can be achieved for any phenomenologically acceptable value of the strong coupling α 3 (M Z ) at the scale M Z , consistent with the measured central low-energy value [10,42]. A '+' denotes that the field is even under the discrete symmetry, a '−' means that the field is odd, while '±' denotes that the field may be even or odd depending on which construction is considered.
For the analysis of the phenomenological implications of the SUSY models discussed above it is convenient to introduce the Z E 2 symmetry, which is defined such that The transformation properties of different components of the 27 i , 27 l and 27 l supermultiplets under theZ H 2 , Z M 2 and Z E 2 symmetries are summarized in Table 1. Since the low-energy effective Lagrangian of the E 6 inspired SUSY models studied here is invariant under the transformations of the Z M 2 andZ H 2 symmetries, the Z E 2 symmetry associated with the exotic states is also conserved. The invariance of the Lagrangian under the Z E 2 symmetry implies that in collider experiments the exotic particles, which are odd under this symmetry, can only be created in pairs and the lightest exotic state should be absolutely stable. Using the method proposed in [54] it was argued that the masses of the lightest inert neutralino states 4 , which are predominantly linear superpositions of the fermion components of the superfields S i from complete 27 i representations of E 6 , do not exceed 60 − 65 GeV [45]. Because of this the corresponding states tend to be the lightest exotic particles in the spectrum.
The presence of lightest exotic particles with masses below 60 GeV gives rise to new decay channels of the SM-like Higgs boson. Moreover, if these states are heavier than 5 − 10 GeV (i.e. approximately above the bottom quark pair threshold) then the SMlike Higgs state decays predominantly into the lightest inert neutralinos while the total branching ratio into SM particles gets strongly suppressed. Nowadays such scenarios are basically ruled out. On the other hand if the lightest exotic particles have masses below 5 GeV their couplings to the SM-like Higgs state are small so that problems with non-standard Higgs decays can be avoided. However, as their couplings to the gauge bosons, quarks and leptons are also very small this results in a cold dark matter density that is much larger than its measured value because the corresponding annihilation cross section tends to be small. The simplest phenomenologically viable scenarios imply that the lightest inert neutralinos are substantially lighter than 1 eV 5 . This can be achieved if f αβ ∼f αβ 10 −6 . In this case the lightest exotic particles form hot dark matter (dark radiation) in the Universe but give only a very minor contribution to the dark matter density.
The Z M 2 symmetry conservation ensures that R-parity is also conserved in the SUSY models discussed above. As also the Z E 2 symmetry is conserved there are two states possible that can be stable. This is either the lightest R-parity even exotic state or the lightest R-parity odd state with Z E 2 = +1. In the E 6 inspired models studied here the stable state tends to be the lightest ordinary neutralino state (i.e. the lightest neutralino state with Z E 2 = +1). Like in the MSSM, this state may account for all or for some of the observed cold dark matter density.
As mentioned before, in the simplest case the sector responsible for the breakdown of For this case the Higgs sector of the E 6 inspired SUSY models with the extra U (1) N factor was explored in [38]. If CP-invariance is preserved then the Higgs spectrum in these models contains three CP-even, one CP-odd and two charged states. The singlet dominated CP-even state is always almost degenerate with the Z gauge boson. In contrast to the MSSM, the lightest Higgs boson in these models can be heavier than 110 − 120 GeV even at tree-level. In the two-loop approximation the lightest Higgs boson mass does not exceed develop VEVs. The fermionic components of these supermultiplets form inert neutralino and chargino states. 5 The presence of very light neutral fermions in the particle spectrum might have interesting implications for neutrino physics (see, for example [55]).
150 − 155 GeV [38]. Recently, the RG flow of the Yukawa couplings and the theoretical upper bound on the lightest Higgs boson mass in these models were analysed in the vicinity of the quasi-fixed point [56] that appears as a result of the intersection of the invariant and quasi-fixed lines [57]. It was argued that near the quasi-fixed point the upper bound on the mass of the SM-like Higgs boson is rather close to 125 GeV [56].
The qualitative pattern of the Higgs spectrum in the E 6 inspired SUSY models with the extra U (1) N gauge symmetry and minimal Higgs sector is determined by the Yukawa coupling λ. When λ < g 1 , where g 1 is the gauge coupling associated with the U (1) N gauge symmetry, the singlet dominated CP-even state is very heavy and decouples from the rest of the spectrum, which makes the Higgs spectrum indistinguishable from the one in the MSSM. If λ g 1 the Higgs spectrum has an extremely hierarchical structure, which is rather similar to the one that arises in the NMSSM with the approximate PQ symmetry [58,59]. As a consequence the mass matrix of the CP-even Higgs sector can be diagonalised using a perturbative expansion [59,60]. In this case the mass of the second lightest CP-even Higgs state is set by the Z boson mass, while the heaviest CP-even, CP-odd and charged states are almost degenerate and lie beyond the multi-TeV range.

The Higgs Potential and Gauge Symmetry Breaking
As was mentioned in the previous section, the sector responsible for breaking the gauge symmetry in the SUSY model under consideration includes two Higgs doublets, H u and H d , as well as the SM singlet fields S, S and φ.
The interactions between these fields are determined by the structure of the gauge interactions and by the superpotential in Eq. (8). The resulting Higgs potential reads where denote the three Pauli matrices. At tree-level the Higgs potential in Eq. (9) is described by the sum of the first three terms. V F and V D contain the F -and D-term contributions that do not violate SUSY. The terms in the expression for V D are proportional to the SU (2) L , U (1) Y and U (1) N gauge couplings, i.e. g 2 , g and g 1 , respectively. The values of the gauge couplings g 2 and g at the EW scale are well known, whereas the low-energy value of the extra U (1) N coupling g 1 and the effective U (1) N charges of H d , H u and S can be calculated assuming gauge coupling unification [38]. The soft SUSY breaking terms are collected in V sof t and include the soft masses m 2 H d , m 2 Hu , m 2 S , m 2 S and m 2 φ , the trilinear couplings A λ , A σ and A κ , the bilinear coupling B and a linear coupling ξ. The term ∆V in Eq. (9) contains the loop corrections to the Higgs effective potential. In SUSY models the most significant contribution to ∆V comes from the loops involving the top-quark and its superpartners, the stops.
At the physical minimum of the scalar potential, Eq. (9), the Higgs fields develop Using the short-hand notation ∂V /∂Φ (Φ= Φ ) ≡ ∂V /∂ Φ , the minimum conditions for the Higgs potential of Eq. (9) read, Instead of specifying v 1 , v 2 , s 1 and s 2 , it is more convenient to use The VEV v is given by the electroweak scale, v = v 2 1 + v 2 2 ≈ 246 GeV, and s = s 2 1 + s 2 2 sets the Z mass, as discussed below.
Initially the Higgs sector involves fourteen degrees of freedom. However four of them are massless Goldstone modes. They are swallowed by the W ± , Z and Z gauge bosons.
The charged W ± bosons gain masses via the interaction with the neutral components of the Higgs doublets H u and H d just in the same way as in the MSSM, resulting in On the other hand the mechanism of the neutral gauge boson mass generation differs substantially. Let the Z and Z states be the gauge bosons associated with the group U (1) N and with the SM-like Z boson, respectively. Then the Z − Z mass-squared matrix is given by where The fields S and S must acquire large VEVs, i.e. s 1 s 2 1 TeV, to ensure that the extra U (1) N gauge boson is sufficiently heavy. Then the mass of the lightest neutral gauge boson Z 1 is very close to M Z =ḡv/2, whereas the mass of Z is set by M Z ≈ g 1Q S s.

The Higgs Boson Spectrum
For the analysis of the Higgs boson spectrum we use Eq. (11) for the extrema to express the soft masses m 2 H d , m 2 Hu , m 2 S , m 2 S and m 2 φ in terms of s, v, ϕ, β, θ and other parameters. Because of the conversation of the electric charge, the charged components of the Higgs doublets are not mixed with the neutral Higgs fields. They form a separate sector, the spectrum of which is described by a 2 × 2 mass matrix. The determinant of this matrix is zero and results in the appearance of two Goldstone states, i.e.
and its charge conjugate (that are absorbed into the longitudinal degrees of freedom of the W ± gauge bosons) and of two charged Higgs states, with mass where ∆ ± denotes the loop corrections to m 2 H ± . The imaginary parts of the neutral components of the Higgs doublets and the imaginary parts of the two SM singlet fields S and S compose two neutral Goldstone states which are swallowed by the Z and Z bosons, as well as three physical states. In Eq. (17) we have introduced In the field basis (P 1 , P 2 , P 3 ), where P 1 = √ 2(Im H 0 u cos β + Im H 0 d sin β) cos γ + √ 2(Im S cos θ − Im S sin θ) sin γ , P 2 = √ 2 Im S sin θ + Im S cos θ , the mass matrix of the CP-odd Higgs sector takes the form In Eqs. (21) the∆ ij (i, j = 1, 2, 3) denote loop corrections. Since in the models under consideration s must be much larger than v, it follows that γ goes to zero. Moreover, since in phenomenologically acceptable SUSY models the supersymmetry breaking scale also tends to be considerably larger than v, the mixing between P 1 and the other two pseudoscalar states P 2 and P 3 is somewhat suppressed. So one CP-odd mass eigenstate is predominantly P 1 . The other two CP-odd mass eigenstates are mainly made up of linear superpositions of the imaginary parts of the SM singlet fields S, S and φ, i.e. of P 2 and P 3 .
In other words, as the off-diagonal entriesM 2 12 ,M 2 13 M 2 11 , the mass matrix, Eqs. (20) and (21), can be diagonalised analytically. In particular, the mass of the CP-odd Higgs eigenstate, that is predominantly P 1 , is set byM 2 11 . As a consequence this CP-odd state and the charged physical Higgs states are expected to be approximately degenerate.
The mass matrix, Eqs. (20) and (21), is diagonalised by means of a unitary transformation U that relates the components of the CP-odd Higgs basis Eq. (19) to the corresponding Higgs mass eigenstates A i (i = 1, 2, 3), The pseudoscalar mass eigenstates are labeled according to increasing absolute value of mass, where A 1 is the lightest CP-odd Higgs state and A 3 the heaviest. At tree-level and neglecting all terms proportional to λv one obtains where we have defined tan δ s 2ϕ sin 2θ .
In this case the lightest CP-odd mass eigenstate is a linear combination of P 2 and P 3 , In the limit where the previously discussed global U (1) symmetry violating couplings κ, µ and Λ vanish, the mass of the lightest CP-odd Higgs boson goes to zero.
The CP-even Higgs sector involves the Re H 0 d , Re H 0 u , Re S, Re S and Re φ. In the field space basis (S 1 , S 2 , S 3 , S 4 , S 5 ), where the mass matrix of the CP-even Higgs sector takes the form where In Eq. (28)   M 2 Z ∼ M 2 S /σ 2 , whereas cos 2θ almost vanishes in this case. Indeed, combining the first and the second equations for the extrema Eq. (11) one obtains the following tree-level expression for cos 2θ which becomes vanishingly small when M Z M S and/or when σ ∼ λ → 0. In this case the hierarchical structure of the mass matrix, Eqs. (27)- (28), implies that the mass of the Z boson and the mass of the heaviest CP-even Higgs particle associated with S 1 are almost degenerate. Thus the heaviest CP-even Higgs state can be integrated out.
The mass of another CP-even state that is predominantly S 4 , the mass of the CP-odd state that corresponds to P 1 , and the masses of the charged Higgs states are also almost degenerate in this limit. Assuming that the Higgs state that is mainly S 4 is the second heaviest CP-even Higgs state, neglecting all terms that are proportional to the global U (1)-violating couplings (κ, µ and Λ) and setting cos 2θ = 0, one obtains the following approximate analytic expressions for the tree-level masses of the three lightest CP-even Higgs bosons, Note that in the scenario under consideration the tree-level expression for the SM-like Higgs mass m 2 h 1 is essentially the same as in the MSSM.

Non-Standard Higgs Decays
We now focus on that region of the parameter space that corresponds to the approximate global U (1) symmetry mentioned above, where we have a light pseudoscalar. Since our primary concern in our numerical investigation is to study non-standard Higgs decays and we do not assume any breaking pattern among the soft masses, most of the sfermion masses do not play a significant role. We therefore choose the SUSY breaking parameters that control the masses of the sfermions to be well above the TeV scale thus comfortably evading limits set by the LHC and decoupling them from the spectrum. We do, however, adjust the stop mass parameters to get a Higgs mass of 125 − 126 GeV. The gaugino masses are also chosen to be heavy enough to give a Higgsino dark matter candidate and to evade the LHC limit on the gluino mass.
Additionally we assume that the SM singlet superfields S, S and φ acquire very large VEVs inducing multi-TeV masses of the Z boson. Our analysis of the Higgs sector in the previous section indicates that the Higgs spectrum in general has a very hierarchical structure when all SUSY breaking parameters are sufficiently large, i.e. above about 1 TeV. In with the trilinear Higgs couplings G h 1 A 1 A 1 which is given by the rather lengthy expression v Q H d cos 2 β +Q Hu sin 2 β × × Q H d sin 2 β cos 2 γ +Q Hu cos 2 β cos 2 γ +Q S sin 2 γ cos 2θ +U 11 U 21 λ 2 2 v sin 2θ sin γ + g 2 1Q S v Q H d cos 2 β +Q Hu sin 2 β sin γ sin 2θ + λA λ √ 2 sin θ + λσ 2 ϕ cos θ cos γ + λσ 2 sin θ U 11 U 31 (s cos γ + v sin 2β sin γ) ϕ sin 2β cos 2 γ cos θ + λ 2 4 s cos 2 γ sin 2θ + λA λ 2 √ 2 sin 2β cos 2 γ sin θ + σ 2 4 s sin 2 γ sin 2θ ϕ sin 2β cos 2 γ sin θ + λ 2 2 s cos 2 γ cos 2 θ + λA λ 2 √ 2 sin 2β cos 2 γ cos θ S s Q H d sin 2 β cos 2 γ +Q Hu cos 2 β cos 2 γ +Q S sin 2 γ cos 2θ If m A 1 60 GeV then the CP-even Higgs boson with mass around 125 GeV can decay into a pair of two lightest pseudoscalar Higgs bosons A 1 , through the interaction given in Eq. (32). The corresponding partial decay width is given by To compare the partial width of the non-standard Higgs decay of the SM-like Higgs state, Eq. (34), with the Higgs decay rates into the SM particles, we specify a set of benchmark points (see Table 3). For each benchmark scenario a code that is automatically generated by FlexibleSUSY  [70][71][72][73][74] are included by using files provided by Pietro Slavich. The additional corrections that may arise due to our model are expected to be small, as either the new particles do not couple directly to the involved particles or their contributions are small due to suppressed couplings and/or large masses.
The couplings and branching ratios of the lightest CP-even Higgs state were also obtained by calling routines generated by FlexibleSUSY in a small extension of the automatically generated code from FlexibleSUSY. FlexibleSUSY uses SARAH-4.2.1 to derive 6 We used an adapted version of FlexibleSUSY-1.0.2 which contains updates that will appear in the new version FlexibleSUSY-1.0.3. The generated (and modified) code can be supplied on request.
analytical expressions, which is independent of the derivation used to obtain the expressions presented here. Therefore we were able to do an independent check of the mass matrices presented in the previous section and of the coupling G h 1 A 1 A 1 given in Eq. (33) by comparing our code from FlexibleSUSY numerically against an alternative Mathematica code based on those expressions. Additionally, we cross-checked the thus obtained total width and branching ratios for the lightest CP-even Higgs boson against the ones obtained from the code HDECAY [75], respectively its extension eHDECAY [76]. The Fortran program HDECAY, originally written for the calculation of the decay widths and branching ratios of the SM and the MSSM Higgs bosons, has been extended to allow for the possiblity to change the couplings of the SM Higgs boson by global modification factors. Using the modification factors, i.e. the ratios of the couplings of the lightest CP-even Higgs boson in our model to the SM particles with respect to the corresponding couplings of the SM Higgs boson of same mass, as inputs in HDECAY, we are able to compute the decay rates of h 1 into SM particle final states. Implementing in addition the partial decay width h 1 → A 1 A 1 generated by FlexibleSUSY, HDECAY can be used to also calculate all branching ratios. 7 This procedure allows us to profit from the state-of-the-art QCD corrections implemented in HDECAY, which can be taken over to our model. 8 It should be noted, that in the loop-induced couplings to gluons and photons, respectively, the SUSY-related loops are not taken into account, however, as the option of applying coupling modifications in HDECAY only applies to the SM Higgs boson. As in our scenarios the sfermion and charged Higgs boson masses are heavy, the change should be marginal only. The branching ratios for the decay h 1 → A 1 A 1 given for our benchmark scenarios in Table 2, are the ones obtained by this procedure.
To simplify our analysis we set B = µ = ξ = 0 and Λ = 0. This does not change the physics we are investigating here and still leaves us with one crucial Peccei-Quinn violating coupling κ. We also fix σ = 0.1 and tan β 10. A large value of tan β allows us to maxime the tree-level mass value of the lightest CP-even Higgs boson, so that the experimental value of ∼ 125 GeV can be obtained more easily. The small value for σ leads to VEVs that can be much heavier than the SUSY breaking scale, cf. Eq. (3). Then, in order to find an appropriate set of benchmark points, we vary λ, κ, A κ , A λ , A σ , A t , m 2 Q 3 , m 2 u c 3 , ϕ, s and θ. In all our benchmark scenarios m 2 S and m 2 Hu are negative which ensures that the Higgs fields S and H u acquire VEVs that result in non-zero VEVs for the other 7 Note, that also the decay h 1 → ZA 1 is in principle possible. In all benchmarks scenarios, however, this decay is kinematically closed or strongly suppressed. 8 The electroweak corrections cannot be taken over. They are consistently included only for the SM part of the decay widths, cf. [76]. To construct benchmark scenarios which are consistent with cosmological observations, it is important to guarantee that they lead to relic densities Ω CDM h 2 that are not larger than the result given by PLANCK [77]: A theory predicting a greater relic density than the PLANCK result is basically ruled out, assuming standard pre-BBN cosmology. A theory that predicts less dark matter cannot be ruled out in the same way, but would require to have other contributions to the dark matter relic density. Since the dark matter density is inversely proportional to the annihilation cross section at the freeze-out temperature, this cross section has to be sufficiently large. In the E 6 inspired SUSY models considered here, the cold dark The qualitative pattern of the neutralino spectrum discussed above reveals that for sufficiently small values of λ the lightest neutralino tends to be a higgsino dominated state. If this higgsino-like state is lighter than 1 TeV, then it leads to a cold dark matter density that is less than the observed value [9]. Therefore, in all benchmark scenarios specified in Table 3, the value of the coupling λ is chosen such that the lightest neutralino is predominantly higgsino with mass below 1 TeV.
In summary, we choose the following values for our benchmark scenarios: The soft SUSY breaking left-and right-handed mass parameters: The soft SUSY breaking gaugino mass parameters: The coupling values: The mixing angle: tan β DR (M Z ) = 10 , where the masses in Eq. (37) are associated with the soft scalar masses of the scalar components of superfields listed in Eq. (7) and δ ij in Eq. (40) denotes the Kronecker δ.
All other values are specified in Table 3 Table 4 for the various benchmarks scenarios. The . We follow the procedure of Ref. [80] and combine the signal rates and errors of the two experiments according to Eq. (5) in [81]. We require the h 1 µ-values to be within 2 times the 1σ interval around the respective best fit value. The combined signal rates and errors are given in Table 2  is the production channel, they are based on the inclusive production cross section. Details can be found in Refs.
the calculation of the µ-values we have assumed the dominant production cross section to be given by gluon fusion. Subsequently, we have approximated the ratios of the h 1 and the SM gluon fusion production cross sections by the ratio of their decay widths Γ gg into a gluon pair. The gluon decay widths have been calculated with HDECAY in both cases, as outlined above. The program includes the higher order QCD corrections to this decay. 9 The approximation of the production cross section ratios by the decay width ratios is valid within about 10-20% depending on the scenario [7]. The µ-value for h 1 into the final state XX is hence given by Let us now turn to the discussion of the five benchmark scenarios BMA-BME, summarized in Tables 3 and 4. Table 3 Table 4 we give for each benchmark scenario the coupling G h 1 A 1 A 1 , relevant for the non-standard decay h 1 → A 1 A 1 , and the ratios R XXh 1 of the couplings of the SM-like h 1 with respect to the ones of the SM Higgs boson for the couplings to a pair of massive gauge bosons V , of up-type quarks f u and of down-type quarks f d . The Table contains the µ-values in the LHC Higgs search final states and finally the partial decay width and the branching ratio for the non-standard decay of h 1 into a pair of lightest pseudoscalars, along with the h 1 total decay width. For all scenarios, the µ-values are within 2 times the 1σ error interval around the experimentally measured values. 10 As can be inferred from Table 4, the branching ratios of the nonstandard Higgs decays h 1 into A 1 A 1 vary from 10 −3 % to 21% for the various benchmarks.
The table shows, that their size depends rather strongly on the absolute value of the coupling κ. Reasonably large branching ratios ( > ∼ 1%) of these Higgs decays can be obtained 9 For details and a recent discussion, see e.g. [76,82]. 10 The difference in the µ-values for the W W and ZZ final states arises from different branching ratios.
Although the coupling modifications are the same for these final states, the branching ratios differ, as in eHDECAY electroweak corrections are included in the decay width in the term linear in the SM amplitude. For details, see [76].   mass, we are getting closer to the SM-limit, as can be seen from the coupling ratios to the SM particles, which are almost one. Therefore, the non-standard coupling G h 1 A 1 A 1 is smaller compared to BMA and BMC and hence also the corresponding branching ratio.
Both in scenario BMB and BME the κ value is chosen to be small, κ = 0.001. This results in BR(h 1 → A 1 A 1 ) = 4.4 · 10 −5 in scenario BMB. In BME the VEV s has been increased and hence the mass of the Z which is M Z ≈ 6 TeV here, resulting in an even smaller branching ratio. With h 1 non-standard branching ratios of O(10 −5 ) and µ rates that are compatible with the SM, this scenario will not be quickly ruled out by the LHC. 11 For the non-standard decays to take place, the pseudoscalar mass must be small enough. As follows from Eqs. (23) such a small m A 1 can always be obtained by tuning the parameter A κ . The degree of tuning is illustrated in Fig. 1, where the region of the parameter space that leads to a sufficiently small m A 1 for non-standard Higgs decays, is shown in the κ-A κ plane. One can see that with decreasing κ, the range of A κ that results in a sufficiently small m A 1 becomes considerably wider. This corresponds to a An adapted version of FlexibleSUSY was used to calculate the mass spectra and the branching ratios. All other parameters are fixed to the values of BMA. For each value of κ there is a lower limit on A κ where the the BR is zero because the pseudoscalar is too heavy and an upper limit above which there is a pseudoscalar tachyon.
smaller degree of fine-tuning required to obtain a sufficiently light pseudoscalar. 12 Therefore, in order to get a pseudoscalar with mass around 40 − 60 GeV for the κ values of O(0.03) of BMA, BMC and BMD, a fine-tuning of order 1% is needed. It turns out that BME is strongly fine-tuned (∼ 1%) as well. This is because the SUSY breaking scale is doubled compared to the other scenarios and the value of A κ is twice the value of A κ in benchmark BMB. For scenario BMB, on the other hand, the fine-tuning is < ∼ 10%. Both Fig. 1 and Tables 3 and 4 demonstrate that the branching ratios of the decays of the SM-like Higgs boson into a pair of pseudoscalar Higgs states become smaller when κ decreases. This is not a surprising result. Indeed, in the PQ symmetric limit, when the global PQ symmetry is only broken spontaneously, the coupling G h 1 A 1 A 1 is set by (see for example [83]) 12 The branching ratios for this figure have been estimated using generated routines from FlexibleSUSY, which allows for a fast scan over the parameter range. We have checked for a few points that the branching ratios from this simple estimate agree with the ones from HDECAY well enough for our purposes. The deviations are due to approximation made for the h 1 decays into SM particles, which in our simple estimate were set equal to the ones of a SM Higgs boson with same mass. Further differences arise due to the inclusion of higher order corrections in HDECAY.
where M P Q is the PQ symmetry breaking scale and ε represents a suppression associated with the mixing between the SM-like Higgs state and heavy CP-even Higgs states that induce the breakdown of the PQ symmetry. Equation (47) also determines the size of G h 1 A 1 A 1 when the PQ violating couplings are rather small and m A 1 is naturally very light.
A simple estimate using the values of the VEVs of the SM singlet fields given in Table 3 indicates that the absolute value of G h 1 A 1 A 1 is expected to be considerably smaller than 1 GeV when κ → 0. Thus one can expect that small values of κ lead to small branching ratios of the non-standard Higgs decays. When κ ∼ 0.001 a pseudoscalar state with mass around 40 − 60 GeV can be obtained with very little fine-tuning. However in this case, the branching ratios of the non-standard Higgs decays are of the order of 10 −4 and below. At the same time when the PQ violating parameters are sufficiently large so that one should fine-tune m A 1 for h 1 → A 1 A 1 to be kinematically allowed then there can be additional contributions to G h 1 A 1 A 1 from the explicit PQ violating terms which can make its absolute value even larger than m 2 h 1 /(2M P Q ). This is why we can obtain a large partial width for this decay when κ 0.01.
We can hence summarize that in the model, that we introduced in order to reduce the fine-tuning, from which simple U (1)-extended SUSY models suffer, the non-standard decay of the SM-like Higgs state h 1 into a pair of pseudoscalars is possible. Small κ values lead naturally to light pseudoscalar masses without fine-tuning. In this case, however, the non-standard branching ratios are tiny, and it will be difficult to test these exotic decays at the LHC. For larger absolute values of κ, the corresponding branching ratio can reach the level of a few per cent up to O(20%). This can be achieved, however, only at the price of fine-tuning, in order to get a low enough pseudoscalar mass in this case for the decay to be kinematically allowed.

Conclusions
In this paper we considered the non-standard decays of the SM-like Higgs state within U (1) extensions of the MSSM in which the extra U (1) gauge symmetry is broken by two VEVs of the SM singlet fields S and S with opposite U (1) charges. Because in these models S and S can acquire very large VEVs along the D-flat direction, the Z -boson can naturally be substantially heavier than the SUSY particles. This allows us to satisfy experimental constraints and to alleviate the fine-tuning associated with the Z -boson.
Such U (1) extensions of the MSSM can possess an approximate global U (1) symmetry that gets spontaneously broken by the VEVs of S and S leading to a pseudo-Goldstone boson in the particle spectrum. If this pseudo-Goldstone state is considerably lighter than the lightest CP-even Higgs boson then it may give rise to the decays of the SM-like Higgs particle into a pair of pseudoscalars.
Here we studied such decays within well motivated SUSY extensions of the SM based on the SU (3) C × SU (2) L × U (1) Y × U (1) N × Z M 2 group which is a subgroup of E 6 . The low-energy matter content of these E 6 inspired models includes three 27 representations of E 6 , a pair of SU (2) L doublets L 4 and L 4 , a pair of the SM singlets S and S with opposite U (1) N charges, as well as an SM singlet φ that does not participate in the gauge interactions. To suppress flavor changing processes at tree-level and to forbid the most dangerous baryon and lepton number violating operators an extraZ H 2 discrete symmetry is imposed. We analysed the spectrum of the CP-even and CP-odd Higgs bosons in these E 6 inspired models assuming that all SUSY breaking parameters are of the order of the TeV scale. Expanding the Higgs potential we obtained an analytical expression for the coupling of the lightest CP-even Higgs state h 1 to a pair of the lightest Higgs pseudoscalars A 1 . The dependence of the branching ratio of the non-standard Higgs decay, h 1 → A 1 A 1 , on the parameters in these SUSY models was examined. For simplicity, we assumed that there is only one dimensionless coupling κ in the superpotential that explicitly violates the global U (1) symmetry. When κ vanishes the global U (1) symmetry is restored.
In order to illustrate the results of our analysis we specified a set of benchmark scenarios with the SM-like Higgs mass around 125 − 126 GeV. To ensure that the obtained benchmark points are consistent with the measured value of the cold dark matter density we chose the parameters such that the lightest neutralino is mainly higgsino with a mass below 1 TeV. In this case the dark matter density tends to be smaller than its observed value. The results of our analysis indicate that the couplings of the lightest pseudoscalar Higgs to the SM particles are always quite small. As a consequence, although this pseudoscalar state can be rather light, it could escape detection at former and present collider experiments.
We argued that the branching ratio of the decays of the lightest CP-even Higgs boson into a pair of the lightest pseudoscalars depends rather strongly on the absolute value of the coupling κ. For absolute values of κ which are substantially smaller than 0.01 the branching ratio of the non-standard Higgs decay in a light pseudoscalar pair decreases considerably. Indeed, in the limit when κ goes to zero the global U (1) symmetry is spontaneously broken and the coupling of the lightest Higgs pseudoscalar to the SM-like Higgs becomes extremely suppressed. As a result the branching ratio of the non-standard Higgs decay tends to be negligibly small. Therefore although for κ ∼ 0.001 the lightest Higgs pseudoscalar with a mass of 40 − 60 GeV can be obtained without fine-tuning the branching ratio of the SM-like Higgs decays into a pair of the lightest CP-odd states is smaller than 10 −4 . Decays with such small branching ratios will be difficult to be tested at the LHC.
When κ 0.01 the branching ratio of the non-standard Higgs decays can be larger than 1%. Nonetheless a fine tuning of at least 1% is required in this case to obtain a lightest pseudoscalar state with mass of 40 − 60 GeV. After being produced from the decay of the lightest CP-even Higgs boson, the lightest CP-odd Higgs states sequentially decay into a pair of either b-quarks or τ -leptons. Thus these decays of the lightest CP-even Higgs boson result in four fermion final states.
We have found that with a modest fine tuning of A κ one can obtain scenarios with h 1 → A 1 A 1 branching ratios of 1% and acceptable values of dark matter relic density.

A The Neutralino Mass Matrix
After the breaking of the gauge symmetry in the E 6 inspired SUSY models under consideration all superpartners of the gauge and Higgs bosons get non-zero masses. Because the extra vector superfield associated with the Z boson and the extra SM singlet Higgs superfields S, S and φ are electromagnetically neutral they do not contribute any extra particles to the chargino spectrum. As a consequence the chargino mass matrix and its eigenvalues remain almost the same as in the MSSM, i.e.