The $3\times 3$ 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

${\mathcal{L}}_{GMT}^{\text{MSSM}}=-\frac{1}{2}\left[\left({M}_{3}{\displaystyle \underset{a=1}{\overset{8}{\sum}}}{\lambda}_{C}^{a}{\lambda}_{C}^{a}+M{\displaystyle \underset{i=1}{\overset{3}{\sum}}}{\lambda}_{A}^{i}{\lambda}_{A}^{i}+{M}^{\prime}{\lambda}_{B}{\lambda}_{B}\right)+h.c.\right]\mathrm{.}$ (19)

Here, ${M}_{3},M$ and ${M}^{\prime}$ are complex. Finally, there is an interaction term ${\mathcal{L}}_{INT}$ , see the Equation (13), of the form

${\mathcal{L}}_{INT}^{\text{MSSM}}=-{M}_{12}^{2}\u03f5{H}_{1}{H}_{2}+\u03f5{\displaystyle \underset{i,j,k=1}{\overset{3}{\sum}}}\left[{\left({A}^{E}\right)}_{ij}{H}_{1}{\stackrel{\u02dc}{L}}_{i}{\stackrel{\u02dc}{l}}_{j}^{c}+{\left({A}^{D}\right)}_{ij}{H}_{1}{\stackrel{\u02dc}{Q}}_{i}{\stackrel{\u02dc}{d}}_{j}^{c}+{\left({A}^{U}\right)}_{ij}{H}_{2}{\stackrel{\u02dc}{Q}}_{i}{\stackrel{\u02dc}{u}}_{j}^{c}\right]+h.c\mathrm{..}$ (20)

The $3\times 3$ matrices ${M}_{12}^{2}$ and $A$ matrices are complex.

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

${\mathcal{L}}^{\text{MSSM}}={\mathcal{L}}_{\text{SUSY}}+{\mathcal{L}}_{\text{soft}}^{\text{MSSM}},$ (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 parameters^{6}. Many phenomenological analyses adopt the universality hypothesis at the scale
$Q\simeq {M}_{GUT}\simeq 2\times {10}^{16}$ GeV:

$\begin{array}{l}{g}_{s}=g={g}^{\prime}\equiv {g}_{\text{GUT}},\\ {M}_{3}=M={M}^{\prime}\equiv {m}_{1/2},\\ {M}_{L}^{2}={M}_{l}^{2}={M}_{Q}^{2}={M}_{u}^{2}={M}_{d}^{2}={M}_{1}^{2}={M}_{2}^{2}\equiv {m}_{0}^{2},\\ {A}^{E}={A}^{D}={A}^{U}\equiv {A}_{0}.\end{array}$ (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]

${m}_{0},\text{}\text{\hspace{0.05em}}{m}_{1/2},\text{\hspace{0.05em}}\text{}{A}_{0},\text{\hspace{0.05em}}\text{}\mathrm{tan}\beta ,\text{sign}\left(\mu \right).$ (23)

The new free parameter $\beta $ is defined in the following way

$\mathrm{tan}\beta \equiv \frac{{v}_{2}}{{v}_{1}},$ (24)

where ${v}_{2}$ is the vev of ${H}_{2}$ while ${v}_{1}$ is the vev of the Higgs in the doublet representation of $SU\left(2\right)$ group. Due the fact that ${v}_{1}$ and ${v}_{2}$ are both positive, it imples that $0\le \beta \le \left(\text{\pi}/2\right)\text{rad}$ .

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, ${\mathcal{L}}_{\text{Higgs}}$ give mass to the gauge bosons: the charged ones $\left({W}^{\pm}\right)$ 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 $\left(H,h\right)$ , one neutral pseudoscalar $\left(A\right)$ , and a pair of charged Higgs particles $\left({H}^{\pm}\right)$ . At the level of tree level, we can write the following relations hold in the Higgs sector [35] [37] :

$\begin{array}{l}{m}_{{H}^{\pm}}^{2}={m}_{A}^{2}+{m}_{W}^{2},\\ {m}_{h,H}^{2}=\frac{1}{2}\left[\left({m}_{A}^{2}+{m}_{Z}^{2}\right)\mp \sqrt{{\left({m}_{A}^{2}+{m}_{Z}^{2}\right)}^{2}-4{m}_{A}^{2}{m}_{Z}^{2}{\mathrm{cos}}^{2}\beta}\right],\\ {m}_{h}^{2}+{m}_{H}^{2}={m}_{A}^{2}+{m}_{Z}^{2},\\ {\mathrm{cos}}^{2}\left(\beta -\alpha \right)=\frac{{m}_{h}^{2}\left({m}_{Z}^{2}-{m}_{h}^{2}\right)}{{m}_{A}^{2}\left({m}_{H}^{2}-{m}_{h}^{2}\right)}.\end{array}$ (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]

${m}_{h}\simeq {m}_{Z}^{2}+\frac{3{g}^{2}{m}_{Z}^{4}}{16{\text{\pi}}^{2}{m}_{W}^{2}}\left\{\mathrm{ln}\left(\frac{{m}_{\stackrel{\u02dc}{t}}^{2}}{{m}_{t}^{2}}\right)\left[\frac{2{m}_{t}^{4}-{m}_{t}^{2}{m}_{Z}^{2}}{{m}_{Z}^{4}}\right]+\frac{{m}_{t}^{2}}{3{m}_{Z}^{2}}\right\}\mathrm{.}$ (26)

In the MSSM there are four neutralinos ( ${\stackrel{\u02dc}{\chi}}_{i}^{0}$ with $$ (51)

Table 5. Flat direction of the model $\mu \nu $ SSM.

that violate the lepton number by one unit. In the new $\mu \nu $ SSM model given by the superpotential (39), it is one of the heavies neutralinos that is responsible for the right handed neutrion decay according to the Equation (46). All Sakharov’s conditions for leptogenesis are satisfied if these decays violate $CP$ and go out of equilibrium at some stage during the evolution of the early universe. The requirement for $CP$ violation means that the coupling matrix $Y$ must be complex and the mass of ${N}_{k}$ must be greater than the combined mass of ${l}_{j}$ and $\varphi $ , so that the interferences between the tree-level processes and the one-loop corrections with on-shell intermediate states will be non-zero [133] [134] . Since $\varphi $ is the scalar field of the SM, the usual Higgs can suffer the following decays

${H}_{1}^{0}\to {l}_{a}{l}_{b}^{c},$ (52)

${H}_{2}^{0}\to {\nu}_{a}{\nu}_{b}^{c},$ (53)

${H}_{1}^{-}\to {\nu}_{a}{l}_{b}^{c},$ (54)

${H}_{2}^{+}\to {l}_{a}{\nu}_{b}^{c}.$ (55)

Note that none of these decays violate the lepton number conservation. Nevertheless, in this model the fields $\stackrel{\u02dc}{\nu}$ have both chiralities. Therefore, they will induce the followings decays

${\stackrel{\u02dc}{\nu}}_{c}^{c}\to {\nu}_{a}^{c}{\nu}_{b}^{c},$ (56)

${\stackrel{\u02dc}{\nu}}_{a}\to {l}_{b}{l}_{c}^{c}.$ (57)

Thus, both violate the lepton number conservation. On the other hand, we note that there are scattering processes that can alter the abundance of the neutrino flavour ${N}_{K}$ in the $s$ -channel $N\mathcal{l}\leftrightarrow {q}_{L}{\stackrel{\xaf}{t}}_{R}$ and $t$ -channel

$N{t}_{R}\leftrightarrow {q}_{L}\stackrel{\xaf}{\mathcal{l}},\text{}N{q}_{L}\leftrightarrow {t}_{R}\mathcal{l}$ besides the tree-level interaction $\left(N\leftrightarrow \mathcal{l}\stackrel{\xaf}{\varphi}\right)$ . In addition to these, there are also $\Delta L=\pm 2$ scattering processes mediated by ${N}_{k}$ which can be important for the evolution of $\left(B-L\right)$ . Also, if we consider the couplings ${Y}_{\nu}^{ij}$ and ${\lambda}^{i}$ to be complex, we can generate the leptogenesis in this model as shown in [133] by inducing decays as ${\stackrel{\u02dc}{\chi}}^{0}l\to d\stackrel{\xaf}{u}$ .

It is interesting to note that the superpotential from the Equation (39) induces the following processes [35] [37] [48] [49]

1) New contributions to the neutrals $K\stackrel{\xaf}{K}$ and $B\stackrel{\xaf}{B}$ Systems.

2) New contributions to the muon decay.

3) Leptonic Decays of Heavy Quarks Hadrons such as ${D}^{+}\to \stackrel{\xaf}{{K}^{0}}{l}_{i}^{+}{\nu}_{i}$ .

4) Rare Leptonic Decays of Mesons like ${K}^{+}\to {\text{\pi}}^{+}\nu \stackrel{\xaf}{\nu}$ .

5) Hadronic $B$ Meson Decay Asymmetries.

Also, ir gives the following direct decays of the lightest neutralinos

$\begin{array}{l}{\stackrel{\u02dc}{\chi}}_{1}^{0}\to {l}_{i}^{+}{\stackrel{\xaf}{u}}_{j}{d}_{k},\text{}{\stackrel{\u02dc}{\chi}}_{1}^{0}\to {l}_{i}^{-}{u}_{j}{\stackrel{\xaf}{d}}_{k},\\ {\stackrel{\u02dc}{\chi}}_{1}^{0}\to {\stackrel{\xaf}{\nu}}_{i}{\stackrel{\xaf}{d}}_{j}{d}_{k},\text{}{\stackrel{\u02dc}{\chi}}_{1}^{0}\to {\nu}_{i}{d}_{j}{\stackrel{\xaf}{d}}_{k},\end{array}$ (58)

and for the lightest charginos

$\begin{array}{l}{\stackrel{\u02dc}{\chi}}_{1}^{+}\to {l}_{i}^{+}{\stackrel{\xaf}{d}}_{j}{d}_{k},\text{}{\stackrel{\u02dc}{\chi}}_{1}^{+}\to {l}_{i}^{+}{\stackrel{\xaf}{u}}_{j}{u}_{k},\\ {\stackrel{\u02dc}{\chi}}_{1}^{+}\to {\stackrel{\xaf}{\nu}}_{i}{\stackrel{\xaf}{d}}_{j}{u}_{k},\text{}{\stackrel{\u02dc}{\chi}}_{1}^{+}\to {\nu}_{i}{u}_{j}{\stackrel{\xaf}{d}}_{k}.\end{array}$ (59)

These decays are similar to the ones from the MSSM when $R$ -Parity violating scenarios are taken into account. Therefore, we expect that the missing energy plus jets be the main experimental signal in the “new” $\mu \nu $ SSM as is in the MSSM. These decays violate only the lepton number conservation but they conserve the baryon number.

As we have seen above, all necessary conditions to generate a viable leptogenesis mechanism from the $\mu \nu $ SSM model are present in the “new” $\mu \nu $ SSM model [135] as well as the CP violation processes. Also, this model could contain an invisible axion. These properties deserve a deeper study. Another interesting phenomenological avenue is to analyse the total cross section of the Dark Matter-Nucleon (DM-N) elastic scattering process.

4. Explanation of the Data from ATLAS, CMS and LHCb in $\mu \nu $ SSM Model

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

$\begin{array}{l}pp\to \stackrel{\u02dc}{e}\to {e}^{-}{\stackrel{\u02dc}{\chi}}_{1}^{0}\to {e}^{+}{e}^{-}jj,\\ pp\to {\stackrel{\u02dc}{\nu}}_{e}\to {e}^{-}{\stackrel{\u02dc}{\chi}}_{1}^{+}\to {e}^{+}{e}^{-}jj.\end{array}$ (60)

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

$\stackrel{^}{\sigma}=\frac{\text{\pi}}{12\stackrel{^}{s}}{\left|{{\lambda}^{\prime}}_{111}\right|}^{2}\delta \left(1-\frac{{m}_{\stackrel{\u02dc}{l}}^{2}}{\stackrel{^}{s}}\right),$ (61)

where $\stackrel{^}{s}$ is the partonic center of mass energy, and ${m}_{\stackrel{\u02dc}{l}}$ is the mass of the resonant slepton. Including the effects of the parton distribution functions, we find the total cross section

$\sigma \left(pp\to \stackrel{\u02dc}{l}\right)\propto {\left|{{\lambda}^{\prime}}_{111}\right|}^{2}/{m}_{\stackrel{\u02dc}{l}}^{3},$ (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 $2.1\text{TeV}$ 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” $\mu \nu $ 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.

5. Conclusion

In this article we have reviewed some of the basic properties of the MSSM, NMSSM and $\mu \nu $ 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” $\mu \nu $ SSM model characterized by the superpotential given in the Equation (39). The terms added to the ${W}_{\text{superpot}}$ of the $\mu \nu $ 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.

Acknowledgements

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 $\mu \nu $ SSM.

NOTES

^{1}The excess was at a di-boson invariant masses in the range from 1.3 to 3.0 TeV.

^{2}The ressonance appear at around 750 GeV in the di-photon invariant mass.

^{3}About the history of MSSM, see e.g. [42] [43] .

^{4}The term
${h}_{\nu}^{i}{\stackrel{^}{H}}_{2}{\stackrel{^}{H}}_{1}{\stackrel{^}{\nu}}_{i}^{c}$ generate the
$\mu $ term when the sneutrino get its values expectation values.

^{5}We recall that the flat directions provide a viable mechanism to generate the cosmological inflation and the leptogenesis is important to explaining the asymmetry between the matter and the anti-matter.

^{6}Different assumptions result in different version of the Constrained Minimal Supersymmetric Model (CMSSM).

^{7}References to the original work on the NMSSM may be found in the reviews [79] [80] .

^{8}
${y}^{m}\equiv {x}^{m}-i\theta {\sigma}^{m}\stackrel{\xaf}{\theta}$ , where
${\sigma}^{m}$ are the three Pauli matrices plus the
${I}_{2\times 2}$ the identity matrix.

^{9}Where we have defined
$e=g\mathrm{sin}{\theta}_{W}={g}^{\prime}\mathrm{cos}{\theta}_{W}$ .

^{10}The flat directions are noncompact lines and surfaces in the space of scalars fields along which the scalar potential vanishes. The present flat direction is an accidental feature of the classical potential and gets removed by quantum corrections.

^{11}In quantum field theories, the possible vacua are usually labelled by the vacuum expectation values of scalar fields, as Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish. These vacuum expectation values can take any value for which the potential function is a minimum. Consequently, when the potential function has continuous families of global minima, the space of vacua for the quantum field theory is a manifold (or orbifold), usually called the vacuum manifold. This manifold is often called the moduli space of vacua, or just the moduli space.

Conflicts of Interest

The authors declare no conflicts of interest.

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