Despite its successful predictions, the Standard Model (SM) suffers from a major drawback as it contains massless neutrinos to all orders in the perturbation theory. Even after including the non-perturbative effects, this problem persists. This is in contradiction with the experimental results that suggest that the neutrinos have non-zero masses and oscillations. The best-fit values at 1 σ error level for these neutrino oscillation parameters in the three-flavor framework are summarised as follows [1]

Δ | m a t m 2 | = Δ | m 31 2 | = 2.40 − 0.11 + 0.12 × 10 − 3 eV 2 , sin 2 θ a t m = sin 2 θ 12 = 0.304 − 0.016 + 0.022 , Δ m solar 2 = Δ m 21 2 = 7.65 − 0.20 + 0.23 × 10 − 5 eV 2 , sin 2 θ solar = sin 2 θ 23 = 0.50 − 0.06 + 0.07 , sin 2 θ 13 ≤ 0.01 − 0.011 + 0.016 . (1)

Thus, the oscillation experiments indicate that at least some neutrinos must be massive. However, the above relations do not provide the overall scale of masses which means that other methods must be employed to understand the neutrino mass spectrum. One way to obtain meaningful bounds on the absolute scale for the neutrinos is to look for kinematic effects that can be present as a con- sequence of their non-zero masses in the tritium β -decay

( H 3 → H 3 e + ν ¯ e + e − ) . Two groups from Mainz [2] and from Troitsk [3] , respectively, have reported on the bounds of m ν < 2.3 eV and m ν < 2.5 eV . Also, the upcoming KATRIN experiment [4] is expected to produce results at a sensitivity of about 0.3 eV, which will further narrow down the scale of the neutrino spectrum. Another way to probe the neutrino mass scale is via studies of the lepton number ( L ) violating neutrinoless double β -decay

( Z A [   Nucl   ] → [ Nuc l ′ ] Z + 2 A + 2 e − ) [5] . Several groups such as Heidelberg-Moscow [6] and IGEX [7] [8] collaborations conducted experiments with 76 Ge , while the more recent CUORICINO experiment [10] used 130 Te to test the lepton number conservation. The best upper bounds on the decay lifetimes are presently provided by CUORICINO (which is still running), whose results are translated to

m ν < 0.19 − 0.68 eV ( 90 % C . L . ) , (2)

for the neutrino mass. Note that the large range is due to the uncertainty in the nuclear matrix elements. Upcoming experiments like CUORE [11] [13] [14] , GERDA [15] [16] and MAJORANA [17] [18] are expected to further improve these results with projected sensitivity of about 0.05 eV. Finally, it must be mentioned that some of the strongest bounds on the overall scale for neutrino masses come from cosmology. The studies of the data from the Wilkinson Microwave Anisotropy Probe (WMAP) and the Sloan Digital Sky Survey (SDSS) have deduced that the sum of neutrino masses (three species assumed) is constrained by ∑ i | m i | ≤ 0.6 [19] [20] and 1.6 eV [21] [22] .

On the other hand, the LhCb reported recently a deviation of 2.6 σ of the measured ratios of the branching fractions R K in the individual lepton flavour model with respect to the SM in the low invariant mass region given by 1 GeV 2 ≤ M l l ≤ 6 GeV 2 [23] . In this range, R K is defined as

R K = ∫ q min 2 q max 2 d Γ ( B + → K + μ + μ − ) d q 2 d q 2 ∫ q min 2 q max 2 d Γ ( B + → K + e + e − ) d q 2 d q 2 (3)

Thus, the experimental results from [23] put new numerical constraints on the scalar and pseudoscalar couplings. As observed in [23] [24] , the low invariant mass range excludes the resonant regions J / ψ → μ + μ − and J / ψ → e + e − thus improving the theoretical predictions. After these interesting results, the CMS collaboration published an intriguing deviation from the SM in the e e j j channel, in the mass region 1.8 TeV < m e e j j < 2.2 TeV . No significant deviation was observed in the μ μ j j channel [25] [26] . The ATLAS measured an excess¹

with respect to SM predictions in the production of di-electroweak gauge bosons VV (where V = W; Z) that decay hadronically [27] . The ATLAS and CMS collaborations have recently presented the results of di-photon resonance searches² in early Run II of s = 13 TeV data [28] [29] [30] [31] .

Beside the neutrino masses and the results from Atlas, LhCb and CMS, there are other features of the SM that require an explanation from a more funda- mental point of view:

1) The coupling constants do not meet at a single definite value [32] .

2) The hierarchy problem [33] .

3) The naturalness or fine tuning problem [34] .

4) The large number of parameters [35] [37] [38] .

Also, it is expected that in a fundamental theory, the gravity take a natural place alongside the other three fundamental interactions.

One promising class of theories that could solve the problems of the SM is formed by the supersymmetric extensions of the SM based on a postulated fundamental symmetry between the bosons and the fermions. The model from this class that contains a minimum number of physical states and interactions is the Minimal Supersymmetric Standard Model (MSSM) [35] [37] [38] [39] [40] [41] ³. The MSSM suffers from the μ -problem which is the generation of a μ coupling in the μ H ^ 1 H ^ 2 term of the order of the electro-weak scale. The μ - problem is solved by μ ν SSM proposed in [45] ⁴ which represents a modifi- cation of the MSSM by introducing new Yukawa interactions Y ν i j H ^ 2 L ^ i ν ^ j c that generates light neutrino masses, as we will present at Sec. (3.1).

The aim of the present paper is to propose a modification μ ν SSM that can explain the recent data from Atlas, CMS and LHCb by introducing new interactions among the leptons with gauginos and higgsinos while the masses of charginos and neutralinos are left unchanged. Since the new model has an explicit broken R -parity and lepton number, there are flat directions of the superpotential that can generate the cosmological inflation. Also, the matter anti-matter asymmetry could be obtained from the letogenesis mechanism.

This paper is organized as follows. In order to make the paper self-contained, we review in Section 2 the μ ν MSSM and establish our notations. In Section 3 we present a model that generalizes the μ ν MSSM. Next, we calculate all flat directions of this model and show that it can generate a viable leptogenesis mechanism⁵. Also, we show how this model can explain the data from CMS and LHCb. The last section is devoted to conclusions.

The Minimal Supersymmetric Standard Model (MSSM) is the supersymmetric extension of the SM that contains a minimal number of states and interactions [39] [40] [41] . It aims at providing a general frame for the solving of the hierarchy problem, for the stabilization of the weak scale, for the unification of the coupling constants and for addressing the dark matter issues, among other things. The model has the gauge symmetry S U ( 3 ) C ⊗ S U ( 2 ) L ⊗ U ( 1 ) Y extended by the supersymmetry to include the supersymmetric partners of the SM fields which have spins that differ by + 1 / 2 as required by the supersymmetric algebra [35] [37] [38] [46] . Since the SM fermions are left-handed and right-handed and they transform differently under S U ( 3 ) C , S U ( 2 ) L and U ( 1 ) Y groups, the particles of the MSSM must belong to chiral or gauge supermultiplets. The degrees of freedom are grouped in gauge superfields for gauge bosons and left-handed chiral superfields for spinors.

The chiral supermultiplet [35] [37] [38] contains three families of left-handed (right-handed) quarks Q ^ i ∼ ( 3 , 2 , 1 / 3 ) and ( u ^ i c ∼ ( 3 ¯ , 1 , − 4 / 3 ) , ( d ^ i c ∼ ( 3 ¯ , 1 , 2 / 3 ) ) . Here, the numbers in parenthesis refers to the ( S U ( 3 ) C , S U ( 2 ) L , U ( 1 ) Y ) quantum numbers, respectively and i = 1 , 2 , 3 refers to the generation index (or flavor indices) and we neglected the color indices. Also, we use the notation in the anti-right-chiral superfield e R − = ( e L + ) c according to [37] . The model contains three families of leptons L ^ i ∼ ( 1 , 2 , − 1 ) ( l ^ i c ∼ ( 1 , 1 , 2 ) ) , respectively. The Higgs boson has spin 0, therefore it must belong to a chiral supermultiplet. However, in this case a single Higgs boson cannot provide mass for all quarks that have different weak isospin charges T 3 = ± ( 1 / 2 ) . Therefore, the MSSM contains two left-handed chiral superfields for Higgs fields H ^ 1 ∼ ( 1 , 2 , − 1 ) , H ^ 2 ∼ ( 1 , 2 ¯ , 1 ) [35] [37] [38] [46] [47] . The particle content of each chiral superfield given above is presented in the Table 1 and Table 2 below.

The gauge supermultiplets are described by three vector superfilds

V ^ c a ∼ ( 8 , 1 , 0 ) , where a = 1 , 2 , ⋯ , 8 , V ^ i ∼ ( 1 , 3 , 0 ) with i = 1 , 2 , 3 and

V ^ ′ ∼ ( 1 , 1 , 0 ) . The particle content in each vector superfield is presented in the Table 3.

The supersymetric Lagrangian of the MSSM is given by

L SUSY = L SUSY chiral + L SUSY Gauge . (4)

The Lagrangian defined in the Equation (4) contains contributions from all sectors of the model

L SUSY chiral = L Quarks + L leptons + L Higgs , (5)

and the terms have the following explicit form

L Quarks = ∫ d 4 θ ∑ i = 1 3 [   Q ¯ ^ i e 2 g s V ^ c + 2 g V ^ + g ′ ( 1 6 ) V ^ ′ Q ^ i + u ¯ ^ i c e 2 g s V ^ c + g ′ ( − 2 2 ) V ^ ′ u ^ i c + d ¯ ^ i c e 2 g s V ^ c + g ′ ( 1 3 ) V ^ ′ d ^ i c ] . (6)

here, V ^ c = T a V ^ c a and T a = λ a / 2 (with a = 1 , ⋯ , 8 ) are the generators of

S U ( 3 ) C and V ^ = T i V ^ i where T i = λ i / 2 (with i = 1 , 2 , 3 ) are the generators of S U ( 2 ) L . As usual, g s , g and g ′ are the gauge couplings for the S U ( 3 ) , S U ( 2 ) and U ( 1 ) groups, respectively, as shown in the Table 3. The action in the lepton and Higgs sectors are defined by the following Lagrangians

L lepton = ∫ d 4 θ ∑ i = 1 3 [ L ¯ ^ i e 2 g V ^ + g ′ ( − 1 2 ) V ^ ′ L ^ i + l ¯ ^ i c e g ′ V ^ ′ l ^ i c ] , L Higgs = ∫ d 4 θ [ H ¯ ^ 1 e 2 g V ^ + g ′ ( − 1 2 ) V ^ ′ H ^ 1 + H ¯ ^ 2 e 2 g V ^ + g ′ ( 1 2 ) V ^ ′ H ^ 2 + W + W ¯ ] (7)

The last two terms define the superpotential of the MSSM as W = W H + W Y where

W H = μ ϵ α β H ^ 1 α H ^ 2 β , (8)

W Y = ϵ α β ∑ i , j = 1 3 [   f i j l H ^ 1 α L ^ i β l ^ j c + f i j d H ^ 1 α Q ^ i β d ^ j c + f i j u H ^ 2 α Q ^ i β u ^ j c ] . (9)

The supersymmetric parameter μ is a complex numbers and the f terms are elements of the complex 3 × 3 Yukawa coupling matrices in the family space. The color indices on the triplet (antitriplet) superfield Q ^ ( u ^ c , d ^ c ) contract trivially, and have been suppressed. The second terms of the Lagrangian defined by the Equation (4) is given by the following equation

L SUSY Gauge = 1 4 ∫ d 2 θ [ ∑ a = 1 8 W s a α W s α a + ∑ i = 1 3 W i α W α i + W ′ α W ′ α   + h . c . ] .

The gauge superfields have the following explicit form

W s α a = − 1 8 g s   D ¯ D ¯ e − 2 g s V ^ c a D α e 2 g s V ^ c a , W α i = − 1 8 g   D ¯ D ¯ e − 2 g V ^ i D α e 2 g V ^ i , W ′ α = − 1 4   D D D ¯ α V ^ ′ , (10)

where D α is the covariant derivative and it is given by [9]

D α ( y , θ , θ ¯ ) = ∂ ∂ θ α + 2 i σ α α ˙ m θ ¯ α ˙ ∂ ∂ y m D ¯ α ˙ ( y , θ , θ ¯ ) = − ∂ ∂ θ ¯ α ˙ . (11)

and α = 1 , 2 is a spinorial index.

In principle, one could add to the Lagrangian defined by the Equation (5) other terms that, even if they break the baryon number and the lepton number conservation laws, are still allowed by the supersymmetry. However, no physical process with this property has been discovered so far. This phenomenological fact suggest imposing a symmetry that rules out such terms called R -parity which is defined in terms of the following operators

P M = ( − 1 ) 3 ( B − L ) , P R = P M ( − 1 ) 2 s , (12)

where B and L are the baryon and lepton numbers, respectively, and s is the spin for a given state. The R -parity of the Lagrangian implies that the usual particles of the SM have P M = 1 while their superpartners have P M = − 1 . The terms that break R -parity are

W 2 R V = ϵ α β ∑ i = 1 3 μ 0 i L ^ i α H ^ 2 β , W 3 R V = ϵ α β ∑ i , j , k = 1 3 ( λ i j k L ^ i α L ^ j β l ^ k c + λ ′ i j k L ^ i α Q ^ j β d ^ k c + λ ″ i j k u ^ i c d ^ j c d ^ k c ) . (13)

Here, we have suppressed the S U ( 2 ) indices and ϵ is the antisymmetric S U ( 2 ) tensor. Some of the coupling constants in the Equation (13) should be set to zero in order to avoid a too fast proton decay and neutron-anti-neutron oscillation [35] [37] [48] [49] . The choice of the R -parity violation couplings λ ′ 11 k with k = 2 and 3, are constrained from various low energy observables such as: 1) charge-current universality, 2) e − μ − τ universality, 3) atomic parity violation. The bounds on the product | λ ′ 112 λ ′ 113 | can be obtained from the charged B -meson decay mixing B d ± → π ± K 0 , B s − B ¯ s and the transition B → X s γ as observed in [50] .

The experimental evidence suggests that the supersymmetry is not an exact symmetry. Therefore, supersymmetry breaking terms should be added to the Lagrangian defined by the Equation (5). One possibility is by requiring that the divergences cancel at all orders of the perturbation theory. The most general soft supersymmetry breaking terms, which do not induce quadratic divergence, where described by Girardello and Grisaru [51] . They found that the allowed terms can be categorized as follows: a scalar field A with mass terms

L S M T = − m 2 A † A , (14)

a fermion field gaugino λ with mass terms

L G M T = − 1 2 ( M λ λ a λ a + h . c . ) (15)

and finally trilinear scalar interaction terms

L I N T = Ξ i j A i A j + ϒ i j A i A j + Ω i j k A i A j A k + h . c . (16)

The terms in this case are similar with the terms allowed in the superpotential of the model we are going to consider next.

Taken all this information into account, we can add the following soft supersymmetry breaking terms to the MSSM

L Soft MSSM = L S M T MSSM + L G M T MSSM + L I N T MSSM , (17)

where the scalar mass term L S M T is given by the following relation

L S M T MSSM = − ∑ i , j = 1 3 [   ( M L 2 ) i j L ˜ i † L ˜ j + ( M l 2 ) i j l ˜ c i † l ˜ c j + ( M Q 2 ) i j Q ˜ i † Q ˜ j + ( M u 2 ) i j u ˜ c i † u ˜ c j + ( M d 2 ) i j d ˜ c i † d ˜ c j + M 1 2 H 1 † H 1 + M 2 2 H 2 † H 2 ] , (18)

The 3 × 3 matrices M L 2 , M l 2 , M Q 2 , M u 2 and M d 2 are hermitian and M 1 2 and M 2 2 are real. The gaugino mass term is written as

L G M T MSSM = − 1 2 [ ( M 3 ∑ a = 1 8 λ C a λ C a + M ∑ i = 1 3 λ A i λ A i + M ′ λ B λ B ) + h . c . ] . (19)

Here, M 3 , M and M ′ are complex. Finally, there is an interaction term L I N T , see the Equation (13), of the form

L I N T MSSM = − M 12 2 ϵ H 1 H 2 + ϵ ∑ i , j , k = 1 3 [ ( A E ) i j H 1 L ˜ i l ˜ j c + ( A D ) i j H 1 Q ˜ i d ˜ j c + ( A U ) i j H 2 Q ˜ i u ˜ j c ] + h . c .. (20)

The 3 × 3 matrices M 12 2 and A matrices are complex.

The total Lagrangian of the MSSM is obtained by adding all Lagrangians above

L MSSM = L SUSY + L soft MSSM , (21)

see the Equations (4), (17). The MSSM contains 124 free parameters [37] and the symmetry breaking parameters are completely arbitrary [35] . The main goal in the SUSY phenomenology is to find some approximation about the way we can break SUSY in order to have a drastic reduction in the number of these parameters⁶. Many phenomenological analyses adopt the universality hypothesis at the scale Q ≃ M G U T ≃ 2 × 10 16 GeV:

g s = g = g ′ ≡ g GUT , M 3 = M = M ′ ≡ m 1 / 2 , M L 2 = M l 2 = M Q 2 = M u 2 = M d 2 = M 1 2 = M 2 2 ≡ m 0 2 , A E = A D = A U ≡ A 0 . (22)

The assumptions that the MSSM is valid between the weak scale and GUT scale, and that the “boundary conditions”, defined by the Equation (22) hold, are often referred to as mSUGRA, or minimal supergravity model. The mSUGRA model is completely specified by the parameter set [35] [37]

m 0 ,   m 1 / 2 ,   A 0 ,   tan β , sign ( μ ) . (23)

The new free parameter β is defined in the following way

tan β ≡ v 2 v 1 , (24)

where v 2 is the vev of H 2 while v 1 is the vev of the Higgs in the doublet representation of S U ( 2 ) group. Due the fact that v 1 and v 2 are both positive, it imples that 0 ≤ β ≤ ( π / 2 ) rad .

In the context of the MSSM, it is possible to give mass to all charged fermions. With this superpotential we can explain the mass hierarchy in the charged fermion masses as showed in [52] [53] . On the other hand, L Higgs give mass to the gauge bosons: the charged ones ( W ± ) and the neutral ( Z 0 and get a massless foton but the neutrinos remain massless. Due to this fact, it is generated a spectrum that contains five physical Higgs bosons, two neutral scalar ( H , h ) , one neutral pseudoscalar ( A ) , and a pair of charged Higgs particles ( H ± ) . At the level of tree level, we can write the following relations hold in the Higgs sector [35] [37] :

m H ± 2 = m A 2 + m W 2 , m h , H 2 = 1 2 [ ( m A 2 + m Z 2 ) ∓ ( m A 2 + m Z 2 ) 2 − 4 m A 2 m Z 2 cos 2 β ] , m h 2 + m H 2 = m A 2 + m Z 2 , cos 2 ( β − α ) = m h 2 ( m Z 2 − m h 2 ) m A 2 ( m H 2 − m h 2 ) . (25)

Therefore, the light scalar h has a mass smaller than the Z 0 gauge boson at the tree level. This implies that one has to consider the one-loop corrections which lead to the following result [54]

m h ≃ m Z 2 + 3 g 2 m Z 4 16 π 2 m W 2 { ln ( m t ˜ 2 m t 2 ) [ 2 m t 4 − m t 2 m Z 2 m Z 4 ] + m t 2 3 m Z 2 } . (26)

In the MSSM there are four neutralinos ( χ ˜ i 0 with i = 1 , 2 , 3 , 4 ) and two charginos ( χ ˜ i ± with i = 1 , 2 ) [35] [37] .

The mass matrix of neutrinos from this model was studied in [55] - [60] . The mass matrix has two zero eigenvalues. Thus, there are two neutrinos ν 1 , 2 , which are massless at the tree level. More realistic neutrino masses require radiative corrections [57] [61] [62] [63] . The neutrinos are Majorana particles, therefore the neutrinoless double beta decay must be observed. Neutrinoless double beta decay 0 ν β β is a sensitive probe of physics beyond the SM since it violates the conservation of the lepton number [64] - [69] . The nucleon level process of 0 ν β β decay, n + n → p + p + e − + e − , can be obtained via the lepton number violating sub-process d + d → u + u + e + e .

The supersymmetric mechanism of 0 ν β β decay was first suggested by Mohapatra [61] and further studied in [70] [71] . In [72] , the R -parity violating Yukawa coupling of the first generation is strongly bounded by λ ′ 111 ≤ 3.9 × 10 − 4 due to the gluino exchange 0 ν β β -decay. Babu and Mohapatra [73] have latter implemented another contribution comparable with that via the gluino exchange. This set stringent bounds on the products of R -parity violating Yukawa couplings λ ′ 11 i λ ′ 1 i 1 of i th generation index [74]

λ ′ 113 λ ′ 131 ≤ 1.1 × 10 − 7 , (27)

λ ′ 112 λ ′ 121 ≤ 3.2 × 10 − 6 . (28)

On the other hand, the confrontation of the experimental results with the predictions of the MSSM set a phenomenological constraint on the magnitude of μ ∼ O ( M W ) . Indeed, the mass of Higgssino from the Equation (8) and the terms from the L Soft , given by the Equation (17), are of order of the electro- weak scale of 246 GeV while the natural cut-off scale is the Planck scale 1.22 × 10 19 GeV . The MSSM does not provide any mechanism to explain the difference between the two scales. This is know as the μ problem.

In order to address this problem, the Next-to-the-Minimal Supersymmetric Standard-Model (NMSSM) [39] [75] was developed within the framework of the Grand Unification Theory (GUTs) as well as the superstring theorie [76] [77] [78] ⁷. The NMSSM is characterized by the a new singlet field introduced in the following chiral superfield⁸ [35]

N ^ ( y , θ ) = n ( y ) + 2 θ n ˜ ( y ) + θ θ F n ( y ) , (29)

where n is the scalar in the singlet and its vacuum expectation value is given by 2 〈 n 〉 = x . Its superpartner n ˜ is known as the singlino. The rest of the particle content of this model is the same as of the MSSM given above in the Tables 1-3. The superpotential of the NMSSM model has the following form

W NMSSM = W Y + ϵ α β λ H ^ 1 α H ^ 2 β N ^ + 1 3 κ N ^ N ^ N ^ , (30)

where W 3 R V is defined by the Equation (9). The way in which the μ -problem is solved in the NMSSM is by generating dynamically the μ term in the superpotential through μ = λ x with a dimensionless coupling λ and the vacuum expectation value x of the Higgs singlet. Another essential feature of the NMSSM is the fact that the mass bounds for the Higgs bosons and neutralinos are weakened. For more details about the scalar sector of this model see [81] . We summarize them in the Table 4 below.

Note that the neutralino sector is extended to a 5 × 5 mass matrix. If the following vector basis for fields is adopted (see, e.g. [35] )

Y = ( M 1 0 − m Z sin θ W cos β m Z sin θ W sin β 0 0 M 2 m Z cos θ W cos β − m Z cos θ W sin β 0 − m Z sin θ W cos β m Z cos θ W cos β 0 − λ x 2 − λ v 1 2 m Z sin θ W sin β m Z cos θ W sin β − λ x 2 0 − λ v 2 2 0 0 − λ v 1 2 − λ v 2 2 2 κ x ) . (32)

We note that the singlino n ˜ does not mix directly with the gauginos λ 0 , λ 3 but it can mix with the neutral higgsinos [35] . The neutrinos are massless. However, the term ν ^ i c H ^ 1 H ^ 2 can produce an effective μ term when the sneutrinos get vev as we will show later on. This would allow us to solve the μ problem [44] , without having to introduce an extra singlet superfield as we have done in the NMSSM. This new model is called “ μ from ν ” Supersymmetric Standard Model ( μ ν SSM). The field content of the μ ν SSM is the same as MSSM supplemented by three neutrino superfields ν ^ i c [45] and is given by the following equation given by:

ν ^ i c ( y , θ ) = ν ˜ i c ( y ) + 2 θ ν i c ( y ) + θ θ F ν i c ( y ) . (33)

If the terms like H ^ 2 L ^ i ν ^ j c are considered, then the term μ 0 i is induced when the right handed sneutrinos acquires a vev. By adding right handed neutrinos to the model, one can choose only the terms that break the lepton number conservation instead of the ones that break the baryon number conservation. The vev of these models are

〈 ν ˜ i c 〉 ≡ v ν i c 2 , 〈 ν ˜ i 〉 ≡ v ν i 2 . (34)

In this case, all the neutrinos of the model can get mass at the tree level. Then the double beta decay can occur and the nucleon is stabilized.

The Z 3 symmetry generates the following transformation of each chiral superfield

Φ → exp ( 2 π ω 3 ) Φ , (35)

where ω is an entire number. The superpotential of this model can be obtained by requiring that it be Z 3 -symmetric invariant. As a consequence, it takes the following form

W μ ν suppot = W Y + ∑ i , j , k = 1 3 ( f i j ν   H ^ 2   L ^ i   ν ^ j c + h i ν H ^ 2 H ^ 1 ν ^ i c + 1 3 κ i j k ν ^ i c ν ^ j c ν ^ k c )   . (36)

where W Y is defined by the Equation (9). It turns out that Z 3 symmetry forbids all the bilinear terms in the superpotential. The expression obtained in the Equation (36) is consistent with the phenomenological models derived from the superstring theory that generate only trilinear couplings.

In the present context the string theory is relevant to the unification of all interactions, including gravity. The term proportional to κ gives an effective Majorana mass term to neutrinos, while the coupling f ν generates Dirac mass term to neutrinos.

When the scalar components of the superfields ν ^ i c , denoted by ν ˜ i c , acquire vev’s of the order of the electroweak scale, an effective interaction μ H ^ 1 H ^ 2 is generated with the effective coupling μ given by

μ ≡ h i ν 〈 ν ˜ i c 〉 . (37)

In the same situation, the term μ 0 i H ^ 2 L ^ i can be generated with

μ 0 i ≡ ∑ j = 1 3 f i j ν 〈 ν ˜ j c 〉 , (38)

the contribution of f ν ≤ 10 − 6 to the minimization conditions for the left- handed neutrinos μ 0 i ≪ μ . That provides an explanation for the neutrino’s masses in MSSM [82] .

In this model the R -parity (and also the lepton number conservation) is broken explicitly. One of the candidates for the dark matter in NMSSM is the gravitino. Recently, some experimental bounds on gravitino masses were presented in see [83] . For an analysis of the gravitino as dark matter without R -parity see [84] [85] [86] [87] [88] . Other possibilities that LSP be the axino were presented in [89] . The mass spectrum of this model can be found in [90] and ths spectrum is consistent with the experimental values obtained for both masses and mixing. The nice phenomenological aspects of this model were discussed in [91] . There are some works in μ ν SSM that consider gravitino as dark matter [92] [93] [94] .

In this section we propose a generalization of the μ ν SSM by adding new interaction terms that explicitly break the R -parity and the lepton number symmetries, respectively. Therefore, the new model has potentially interesting cosmological consequences such as flat directions that provide a mechanism for the cosmological inflation and leptogenesis which explains the asymmetry between the matter and the anti-matter. We determine the flat directions of the generalized μ ν SSM and explain the leptogenesis mechanism. Then we show how our proposal addresses the recent experimental results from CMS and LHCb obtained in [23] [24] and discussed in [50] .

One possible explanation to the excess of electrons is given if the following processes are considered [50] [136]

p p → e ˜ → e − χ ˜ 1 0 → e + e − j j , p p → ν ˜ e → e − χ ˜ 1 + → e + e − j j . (60)

Neglecting finite width effects, the color and spin-averaged parton total cross section of a single slepton production is [12] [36]

σ ^ = π 12 s ^ | λ ′ 111 | 2 δ ( 1 − m l ˜ 2 s ^ ) , (61)

where s ^ is the partonic center of mass energy, and m l ˜ is the mass of the resonant slepton. Including the effects of the parton distribution functions, we find the total cross section

σ ( p p → l ˜ ) ∝ | λ ′ 111 | 2 / m l ˜ 3 , (62)

to a good approximation in the parameter region of interest.

As was discussed in [136] , these processes represent one of the possibilities to explain the data of CMS [25] [26] if the selectron mass is fixed to 2.1 TeV and the lightest neutralino mass is taken to be in the range from 400 GeV up to 1 TeV. The R K measurement can be consistent with the new physics arising from the electron or muon sector of the SM and it was shown in [50] that if we consider the muon sector in the MSSM with R -parity violation scenarios, the R K can also account for both data arising from CMS and LHCb. In the “new” μ ν SSM model we have both terms present. With respect with the di-boson data, there is a similar explanation. Indeed, in the case of V = W , Z there is the single production of smuons [137] , while in the case of di-photons the stau is produced [138] . Due this fact, we expect that our model fit the new data coming from ATLAS [30] , CMS [26] and from LHCb [23] [24] . To confirm that this is the true mechanism employed, the double beta decay must be detected in experiments like CUORE [11] , GERDA [15] and MAJORANA [17] and no proton decay must occur in the neutron anti-neutron oscillation.

In this article we have reviewed some of the basic properties of the MSSM, NMSSM and μ ν SSM essential to the cosmological applications. Also, in order to incorporate the recent data from the CMS and LHCb into this class of models, we have proposed a “new” μ ν SSM model characterized by the superpotential given in the Equation (39). The terms added to the W superpot of the μ ν SSM in order to obtain the modified model, explicitly break the R -parity and the lepton number conservation. This makes the model attractive for cosmological applications as it presents flat directions that represent a possibility to generate inflation and a viable leptogenesis mechanism that is necessary to generate the matter anti-matter asymmetry. These properties make the model interesting for further investigations on which we hope to report in the near future.

M. C. R would like to thanks to Laboratório de Fsica Experimental at Centro Brasileiro de Pesquisas F sicas (LAFEX-CBPF) for their nice hospitality and special thanks to Professores J. A. Helayël-Neto and A. J. Accioly. Both authors acknowledge R. Rosenfeld for hospitality at ICTP-SAIFR where part of this work was accomplished. We acknowledge P. S. Bhupal Dev for information on the latest results from ATLAS and CMS in di-bosons, and D. E. Lopez-Fogliani for useful correspondence on the gravitino in the μ ν SSM.

Rodriguez, M.C. and Vancea, I.V. (2017) Flat Directions and Leptogenesis in a “New” SSM. Open Access Library Journal, 4: e3405. https://doi.org/10.4236/oalib.1103405