Neutrino/sneutrino mass spectrum
The \(\mu \nu \mathrm{SSM}\) [26] is a natural extension of the MSSM where the \(\mu \) problem is solved and, simultaneously, neutrino data can be reproduced [26, 52,53,54,55,56]. This is obtained through the presence of trilinear terms in the superpotential involving right-handed neutrino superfields \(\hat{\nu }^c_i\), which relate the origin of the \(\mu \)-term to the origin of neutrino masses and mixing angles. The simplest superpotential of the \(\mu \nu \mathrm{SSM}\) [26, 52, 57] with three right-handed neutrinos is the following:
$$\begin{aligned} W= & {} \epsilon _{ab} \left( Y_{e_{ij}} \, \hat{H}_d^a\, \hat{L}^b_i \, \hat{e}_j^c + Y_{d_{ij}} \, \hat{H}_d^a\, \hat{Q}^{b}_{i} \, \hat{d}_{j}^{c} + Y_{u_{ij}} \, \hat{H}_u^b\, \hat{Q}^{a} \, \hat{u}_{j}^{c} \right) \nonumber \\&+ \epsilon _{ab} \left( Y_{{\nu }_{ij}} \, \hat{H}_u^b\, \hat{L}^a_i \, \hat{\nu }^c_j - \lambda _i \, \hat{\nu }^c_i\, \hat{H}_u^b \hat{H}_d^a \right) + \frac{1}{3} \kappa _{ijk} \hat{\nu }^c_i\hat{\nu }^c_j\hat{\nu }^c_k,\nonumber \\ \end{aligned}$$
(2)
where the summation convention is implied on repeated indices, with \(a,b=1,2\) \(SU(2)_L\) indices and \(i,j,k=1,2,3\) the usual family indices of the SM.
The simultaneous presence of the last three terms in Eq. (2) makes it impossible to assign R-parity charges consistently to the right-handed neutrinos (\(\nu _{iR}\)), thus producing explicit RPV (harmless for proton decay). Note nevertheless, that in the limit \(Y_{{\nu }_{ij}} \rightarrow 0\), \(\hat{\nu }^c\) can be identified in the superpotential as a pure singlet superfield without lepton number, similar to the next-to-MSSM (NMSSM) [58], and therefore R parity is restored. Thus, the neutrino Yukawa couplings \(Y_{\nu _{ij}}\) are the parameters which control the amount of RPV in the \(\mu \nu \mathrm{SSM}\), and as a consequence this violation is small. After the electroweak symmetry breaking induced by the soft SUSY-breaking terms of the order of the TeV, and with the choice of CP conservation, the neutral Higgses (\(H_{u,d}\)) and right (\(\widetilde{\nu }_{iR}\)) and left (\(\widetilde{\nu }_i\)) sneutrinos develop the following vacuum expectation values (VEVs):
$$\begin{aligned} \langle H_{d}\rangle = \frac{v_{d}}{\sqrt{2}},\quad \langle H_{u}\rangle = \frac{v_{u}}{\sqrt{2}},\quad \langle \widetilde{\nu }_{iR}\rangle = \frac{v_{iR}}{\sqrt{2}},\quad \langle \widetilde{\nu }_{i}\rangle = \frac{v_{i}}{\sqrt{2}},\nonumber \\ \end{aligned}$$
(3)
where \(v_{iR}\sim \) TeV, whereas \(v_i\sim 10^{-4}\) GeV because of the small contributions \(Y_{\nu } \lesssim 10^{-6}\) whose size is determined by the electroweak-scale seesaw of the \(\mu \nu \mathrm{SSM}\) [26, 52]. Note in this sense that the last term in Eq. (2) generates dynamically Majorana masses, \({\mathcal M}_{ij}={2}\kappa _{ijk} \frac{v_{kR}}{\sqrt{2}}\sim \) TeV. On the other hand, the fifth term in the superpotential generates the \(\mu \)-term, \(\mu =\lambda _i \frac{v_{iR}}{\sqrt{2}}\sim \) TeV.
The new couplings and sneutrino VEVs in the \(\mu \nu \mathrm{SSM}\) induce new mixing of states. The associated mass matrices were studied in detail in Refs. [52, 54, 57]. Summarizing, there are eight neutral scalars and seven neutral pseudoscalars (Higgses-sneutrinos), eight charged scalars (charged Higgses-sleptons), five charged fermions (charged leptons-charginos), and ten neutral fermions (neutrinos-neutralinos). In the following, we will concentrate in briefly reviewing the neutrino and neutral Higgs sectors, which are the relevant ones for our analysis.
The neutral fermions have the flavor composition \((\nu _{i},\widetilde{B},\widetilde{W},\widetilde{H}_{d},\widetilde{H}_{u},\nu _{iR})\). Thus, with the low-energy bino and wino soft masses, \(M_1\) and \(M_2\), of the order of the TeV, and similar values for \(\mu \) and \(\mathcal {M}\) as discussed above, this generalized seesaw produces three light neutral fermions dominated by the left-handed neutrino (\(\nu _i\)) flavor composition. In fact, data on neutrino physics [29,30,31,32] can easily be reproduced at tree level [26, 52,53,54,55,56], even with diagonal Yukawa couplings [53, 55], i.e. \(Y_{{\nu }_{ii}}=Y_{{\nu }_{i}}\) and vanishing otherwise. A simplified formula for the effective mass matrix of the light neutrinos is [55]:
$$\begin{aligned}&(m_{\nu })_{ij} \simeq \frac{Y_{{\nu }_{i}} Y_{{\nu }_{j}} v_u^2}{6\sqrt{2} \kappa v_{R}} (1-3 \delta _{ij})-\frac{v_{i} v_{j}}{4M^{\text {eff}}}\nonumber \\&\quad -\frac{1}{4M^{\text {eff}}}\left[ \frac{v_d\left( Y_{{\nu }_{i}}v_{j} +Y_{{\nu }_{j}} v_{i}\right) }{3\lambda } +\frac{Y_{{\nu }_{i}}Y_{{\nu }_{j}} v_d^2}{9\lambda ^2 }\right] , \end{aligned}$$
(4)
with
$$\begin{aligned} M^{\text {eff}}\equiv & {} M -\frac{v^2}{2\sqrt{2} \left( \kappa v_R^2+\lambda v_u v_d\right) \ 3 \lambda v_R}\nonumber \\&\times \left( 2 \kappa v_R^{2} \frac{v_u v_d}{v^2} +\frac{\lambda v^2}{2}\right) , \end{aligned}$$
(5)
and
$$\begin{aligned} \frac{1}{M} = \frac{g'^2}{M_1} + \frac{g^2}{M_2}, \end{aligned}$$
(6)
where \(v^2 = v_d^2 + v_u^2 + \sum _i v^2_{i}={4 m_Z^2}/{(g^2 + g'^2)}\approx \) (246 GeV)\(^2\). For simplicity, we are also assuming in these formulas, and in what follows, \(\lambda _i = \lambda \), \(v_{iR}= v_{R}\), and \(\kappa _{iii}\equiv \kappa _{i}=\kappa \) and vanishing otherwise. We are then left with the following set of variables as independent parameters in the neutrino sector:
$$\begin{aligned} \lambda , \, \kappa ,\, Y_{\nu _i}, \tan \beta , \, v_{i}, \, v_R, \, M_1, \, M_2, \end{aligned}$$
(7)
and the \(\mu \)-term is given by
$$\begin{aligned} \mu =3 \lambda \frac{v_{R}}{\sqrt{2}}. \end{aligned}$$
(8)
In Eq. (7), we have defined \(\tan \beta \equiv v_u/v_d\) and since \(v_{i} \ll v_d, v_u\), we have \(v_d\approx v/\sqrt{\tan ^2\beta +1}\). For the discussion, hereafter we will use indistinctly the subindices (1, 2, 3) \(\equiv \) (\(e,\mu ,\tau \)). In the numerical analyses of the next sections, it will be enough for our purposes to consider the sign convention where all these parameters are positive. Of the five terms in Eq. (4), the first two are generated through the mixing of \(\nu _i\) with \(\nu _{iR}\)-Higgsinos, and the rest of them also include the mixing with the gauginos. These are the so-called \(\nu _{R}\)-Higgsino seesaw and gaugino seesaw, respectively [55].
As we can understand from these equations, neutrino physics in the \(\mu \nu \mathrm{SSM}\) is closely related to the parameters and VEVs of the model, since the values chosen for them must reproduce current data on neutrino masses and mixing angles.
Concerning the neutral scalars and pseudoscalars in the \(\mu \nu \mathrm{SSM}\), although they have the flavor composition (\(H_d, H_u, \widetilde{\nu }_{iR},\widetilde{\nu }_{i}\)), the off-diagonal terms of the mass matrix mixing the left sneutrinos with Higgses and right sneutrinos are suppressed by \(Y_{\nu }\) and \(v_{iL}\), implying that scalar and pseudoscalar left sneutrino states will be almost pure. In addition scalars have degenerate masses with pseudoscalars \(m_{\widetilde{\nu }^{\mathcal {R}}_{i}} \approx m_{\widetilde{\nu }^{\mathcal {I}}_{i}} \equiv m_{\widetilde{\nu }_{i}}\). From the minimization equations for \(v_i\), we can write their approximate tree-level values as
$$\begin{aligned} m_{\widetilde{\nu }_{i}}^2 \approx \frac{Y_{{\nu }_i}v_u}{v_i} \frac{v_R}{\sqrt{2}} \left[ \frac{-T_{{\nu }_i}}{Y_{{\nu }_i}} + \frac{v_R}{\sqrt{2}} \left( -\kappa + \frac{3\lambda }{\tan \beta }\right) \right] , \end{aligned}$$
(9)
where \(T_{{\nu }_i}\) are the trilinear parameters in the soft Lagrangian, \(-\epsilon _{ab} T_{{\nu }_{ij}} H_u^b \widetilde{L}^a_{iL} \widetilde{\nu }_{jR}^*\), taking for simplicity \(T_{{\nu }_{ii}}=T_{{\nu }_i}\) and vanishing otherwise. Therefore, left sneutrino masses introduce in addition to the parameters of Eq. (7), the
$$\begin{aligned} T_{{\nu }_i}, \end{aligned}$$
(10)
as other relevant parameters for our analysis. In the numerical analyses of Sects. 4 and 5, we will use negative values for them in order to avoid tachyonic left sneutrinos.
Let us point out that if we follow the usual assumption based on the breaking of supergravity, that all the trilinear parameters are proportional to their corresponding Yukawa couplings, defining \(T_{\nu }= A_{\nu } Y_{\nu }\) we can write Eq. (9) as:
$$\begin{aligned} m_{\widetilde{\nu }_{i}}^2 \approx \frac{Y_{{\nu }_i}v_u}{v_i} \frac{v_R}{\sqrt{2}} \left[ -A_{{\nu }_i} + \frac{v_R}{\sqrt{2}} \left( -\kappa + \frac{3\lambda }{\tan \beta }\right) \right] , \end{aligned}$$
(11)
and the parameters \(A_{{\nu }_i}\) substitute the \(T_{{\nu }_i}\) as the most representative. We will use both type of parameters throughout this work.
Using diagonal sfermion mass matrices, from the minimization conditions for Higgses and sneutrinos one can eliminate the corresponding soft masses \(m_{H_{d}}^{2}\), \(m_{H_{u}}^{2}\), \(m^2_{\widetilde{\nu }_{iR}}\) and \(m^2_{\widetilde{L}_{iL}}\) in favor of the VEVs. Thus, the parameters in Eqs. (7) and (10), together with the rest of soft trilinear parameters, soft scalar masses, and soft gluino masses
$$\begin{aligned} T_{\lambda }, \, T_{\kappa }, \, T_{u_{i}}, \, T_{d_{i}}, \, T_{e_{i}}. \, m_{\widetilde{Q}_{iL}},\, m_{\widetilde{u}_{iR}}, \, m_{\widetilde{d}_{iR}}, \, m_{\widetilde{e}_{iR}}, \, M_3, \end{aligned}$$
(12)
constitute our whole set of free parameters. Given that we will focus on a light \(\widetilde{\nu }_{\mu }\), we will use negative values for \(T_{u_3}\) in order to avoid cases with too light left sneutrinos due to loop corrections.
The neutral Higgses and the three right sneutrinos, which can be substantially mixed in the \(\mu \nu \mathrm{SSM}\), were discussed recently in detail in Ref. [59]. The tree-level mass of the SM-like Higgs can be written in an elucidate form for our discussion below as
$$\begin{aligned} m^{2}_{0h}= & {} {m^2_Z} \left\{ \left( \frac{1 - \mathrm{tan}^2\beta }{1 + \mathrm{tan}^2\beta }\right) ^2 + \left( \frac{v/\sqrt{2}}{m_Z}\right) ^2 \right. \nonumber \\&\times \left. ({\sqrt{3}\lambda })^2 \left( \frac{\mathrm{2\ tan\beta }}{1 + \mathrm{tan}^2\beta }\right) ^2 \right\} , \end{aligned}$$
(13)
where the factor \(({v/\sqrt{2} m_Z})^2\approx 3.63\), and we have neglected for simplicity the mixing of the SM-like Higgs with the other states in the mass squared matrix. We see straightforwardly that the second term grows with small tan\(\beta \) and large \({\lambda }\). If \(\lambda \) is not large enough, a contribution from loops is essential to reach the target of a SM-like Higgs in the mass region around 125 GeV as in the case of the MSSM. In Refs. [33, 60, 61], a full one-loop calculation of the corrections to the neutral scalar masses was performed. Supplemented by MSSM-type corrections at the two-loop level and beyond (taken over from the code FeynHiggs [62,63,64]) it was shown that the \(\mu \nu \)SSM can easily accommodate a SM-like Higgs boson at \(\sim 125\) GeV, while simultaneously being in agreement with collider bounds and neutrino data. This contribution is basically determined by the soft parameters \(T_{u_3}, m_{\widetilde{u}_{3R}}\) and \(m_{\widetilde{Q}_{3L}}\). Clearly, these parameters together with \(\lambda \) and \(\tan \beta \) are crucial for Higgs physics. In addition, the parameters \(\kappa \), \(v_R\) and \(T_{\kappa }\) are the key ingredients to determine the mass scale of the right sneutrino states [52, 53]. For example, for \(\lambda \lesssim 0.01\) they are basically free from any doublet contamination, and the masses can be approximated by [57, 65]:
$$\begin{aligned} m^2_{\widetilde{\nu }^{\mathcal {R}}_{iR}} \approx \frac{v_R}{\sqrt{2}} \left( T_{\kappa } + \frac{v_R}{\sqrt{2}}\ 4\kappa ^2 \right) , \quad m^2_{\widetilde{\nu }^{\mathcal {I}}_{iR}}\approx - \frac{v_R}{\sqrt{2}}\ 3T_{\kappa }. \end{aligned}$$
(14)
Given this result, we will use negative values for \(T_{\kappa }\) in order to avoid tachyonic pseudoscalar right sneutrinos. Finally, the parameters \(\lambda _i\) and \(T_{\lambda _i}\) (\(A_{\lambda _i}\) assuming the supergravity relation \(T_{\lambda _i}= \lambda _i A_{\lambda _i}\)) also control the mixing between the singlet and the doublet states and hence, contribute in determining the mass scale. We conclude that the relevant independent low-energy parameters in the Higgs-right sneutrino sector are the following subset of the parameters in Eqs. (7), (10), and (12):
$$\begin{aligned} \lambda ,\,\, \kappa ,\,\, \tan \beta ,\,\, v_R,\,\, T_\kappa ,\,\, T_\lambda ,\,\, T_{u_{3}},\,\, m_{\widetilde{Q}_{3L}},\,\, m_{\widetilde{u}_{3R}}. \end{aligned}$$
(15)
Neutrino/sneutrino physics
Since reproducing neutrino data is an important asset of the \(\mu \nu \mathrm{SSM}\), as explained above, we will try to establish here qualitatively what regions of the parameter space are the best in order to be able to obtain correct neutrino masses and mixing angles. Although the parameters in Eq. (7), \(\lambda \), \(\kappa \), \(v_{R}\), \(\tan \beta \), \(Y_{\nu _i}\), \(v_{i}\), \(M_1\) and \(M_2\), are important for neutrino physics, the most crucial of them are \(Y_{\nu _i}\), \(v_{i}\) and M, where the latter is a kind of average of bino and wino soft masses (see Eq. (6)). Thus, we will first determine natural hierarchies among neutrino Yukawas, and among left sneutrino VEVs.
Considering the normal ordering for the neutrino mass spectrum, and taking advantage of the dominance of the gaugino seesaw for some of the three neutrino families, representative solutions for neutrino physics using diagonal neutrino Yukawas were obtained in Ref. [41]. In particular, the so-called type 3 solutions, which have the following structure:
$$\begin{aligned} M>0, \quad \text {with}\ Y_{\nu _2}< Y_{\nu _1}< Y_{\nu _3}, \ \text {and} \ v_1<v_2\sim v_3, \end{aligned}$$
are especially interesting for us, since, as will be argued below, they are able to produce the left muon-sneutrino as the lightest of all sneutrinos. In this case of type 3, it is easy to find solutions with the gaugino seesaw as the dominant one for the second family. Then, \(v_2\) determines the corresponding neutrino mass and \(Y_{\nu _2}\) can be small. On the other hand, the normal ordering for neutrinos determines that the first family dominates the lightest mass eigenstate implying that \(Y_{\nu _{1}}< Y_{\nu _{3}}\) and \(v_1 < v_2,v_3\), with both \(\nu _{R}\)-Higgsino and gaugino seesaws contributing significantly to the masses of the first and third family. Taking also into account that the composition of the second and third families in the third mass eigenstate is similar, we expect \(v_3 \sim v_2\).
In addition, left sneutrinos are special in the \(\mu \nu \mathrm{SSM}\) with respect to other SUSY models. This is because, as discussed in Eq. (9), their masses are determined by the minimization equations with respect to \(v_i\). Thus, they depend not only on left sneutrino VEVs but also on neutrino Yukawas, and as a consequence neutrino physics is very relevant. For example, if we work with Eq. (11) assuming the simplest situation that all the \(A_{{\nu }_i}\) are naturally of the order of the TeV, neutrino physics determines sneutrino masses through the prefactor \({Y_{{\nu }_i}v_u}/{v_i}\). Thus, values of \({Y_{{\nu }_i}v_u}/{v_i}\) in the range of about 0.01–1, i.e. \(Y_{{\nu }_i}\sim 10^{-8}{-}10^{-6}\), will give rise to left sneutrino masses in the range of about 100-1000 GeV. This implies that with the hierarchy of neutrino Yukawas \(Y_{{\nu }_{2}}\sim 10^{-8}{-}10^{-7}<Y_{{\nu }_{1,3}}\sim 10^{-6}\), we can obtain a \(\widetilde{\nu }_{\mu }\) with a mass around 100 GeV whereas the masses of \(\widetilde{\nu }_{e,\tau }\) are of the order of the TeV, i.e. we have \(m_{\widetilde{\nu }_{2}}\) as the smallest of all the sneutrino masses. Clearly, we are in the case of solutions for neutrino physics of type 3 discussed above.
Let us finally point out that the crucial parameters for neutrino physics, \(Y_{\nu _i}\), \(v_{iL}\) and M, are essentially decoupled from the parameters in Eq. (15) controlling Higgs physics. Thus, for a suitable choice of \(Y_{\nu _i}\), \(v_{iL}\) and M reproducing neutrino physics, there is still enough freedom to reproduce in addition Higgs data by playing with \(\lambda \), \(\kappa \), \(v_R\), \(\tan \beta \), etc., as shown in Ref. [59]. As a consequence, in Sect. 5 we will not need to scan over most of the latter parameters, relaxing our demanding computing task. We will discuss this issue in more detail in Sect. 4.3.