One Right-handed Neutrino to Generate Complete Neutrino Mass Spectrum in the Framework of NMSSM

The see-saw mechanism is usually applied to explain the lightness of neutrinos. The traditional see-saw mechanism introduces at least two right-handed neutrinos for the realistic neutrino spectrum. In the case of supersymmetry, loop corrections can also contribute to neutrino masses, which lead to the possibility to generate the neutrino spectrum by introducing just one right-handed neutrino. To be realistic, MSSM suffers from the mu problem and other phenomenological difficulties, so we extend NMSSM (the MSSM with a singlet S) by introducing one single right-handed neutrino superfield (N) and relevant phenomenology is discussed


I. INTRODUCTION
Up to now, experiments have established the fact that neutrinos oscillate among each other. Two mass-squared differences (δm 2 12 , δm 2 23 ), together with all three mixing angles(θ 12 , θ 23 , θ 13 ) have been measured [1], prompting us that at least two generations of neutrinos among the three are massive. Type I see-saw Mechanism is a way to generate small Majorana neutrino masses [2] by introducing lepton number violating ∆L = 2 Majorana mass term for right-handed neutrinos. On the other hand, in the framework of supersymmetry, after the lepton number is violated, one-loop radiative corrections naturally generate non-zero neutrino mass terms [3].
It is possible to generate realistic neutrino spectrum by introducing just one right-handed neutrino in supersymmetry. Minimal supersymmetry standard model (MSSM) extended with one right-handed neutrino is discussed in [4]. Considering both tree-level see-saw mechanism and the one-loop radiative corrections [3], one right-handed neutrino is enough to generate masses for three generations of neutrinos. However, MSSM suffers from the µ problem. Next to minimal supersymmetry standard model (NMSSM) [5] extended with one right-handed neutrino was originally discussed in [7], [6] [8], where right-handed neutrinos acquire TeV-scale Majorana mass terms through their couplings with the singlet Higgs superfield S introduced in NMSSM. This coupling also establishes the connection between the right-handed neutrinos with the Higgs sectors, which may influence the phenomenology of the Higgs bosons. The possibility that right-handed neutrino in the framework of NMSSM may contribute to the Higgs boson mass is also discussed in [9].
In this paper, we will show that NMSSM extended with a single right-handed neutrino superfield can generate the complete neutrino mass spectrum considering both contributions from the tree-level Type-I see-saw mechanism and one-loop radiative corrections. We will show that this model contains all the possibilities of the size of mixing angles. It is because once the correct mass spectrum is generated, it is almost free for us to rotate the mass matrix, with little experimental limitations. We also considered other experimental constraints and calculated the corrections of the Higgs Boson mass from the contributions of the right-handed neutrino [9], and gave some numerical results.

II. MODEL AND CALCULATION
Here we impose a global Z 3 symmetry and keep the R-parity conservation, just as usual NMSSM with Z 3 symmetry [5]. The Z 3 quantum number assigned to the right-handed neutrino N is the same as Singlet Higgs S, while the R-parity of N is set as positive, just as other MSSM matter superfields. Thus the involved superpotential is strongly limited into the form where S is the singlet Higgs superfield originally existed in NMSSM, and N is the righthanded neutrino. Here we only listed the terms involving lepton and Higgs fields.
The relevant soft terms breaking the supersymmetry are listed below, where m l2 ij = m l2 ji . One might consider a more compact model that N just plays the role of S [10]. However, this model breaks R-parity and do not contain dark matter.
If scalarS acquires a vacuum expectation value(vev) v s , terms likeÑÑ appears, which supplies the ∆L = 2 quadratic terms in the sneutrino sector, contributing to the one-loop neutrino mass corrections.
The vevs of the doublet Higgs fields are defined as Neutrinos then acquire tree-level Majorana mass terms after integrating out the fermionic N through Figure 1 [9],  Now that we are considering a supersymmetric theory, each particle is paired up with a super-partner, so is the right-handed neutrino. Thus, right-handed scalar-neutrino contribute into the mass terms of neutrino through radiative corrections in Figure 2. As mentioned in [4], in order for the loop-level neutrino mass terms not to be aligned with the tree level ones, we also need cross terms in the Sneutrino soft mass-squared matrix m νij which result in flavor-changing neutral current (FCNC) processes in the lepton sector.
From Figure 2, we can estimate the loop-level neutrino mass, where M is the typical mass scale of the propagators in the loop, m l2 ij is the off-diagonal elements of the soft mass-square matrix of the left-handed leptons. Fortunately, for TeV see-saw mechanics, y N tend to be rather small and are ∼ 10 −6 , which is comparable with the electron Higgs Yukawa coupling term, thus allows relatively large A yN , which significantly increase the m oneloop , and its phenomenological effects are also highly suppressed by the factor y N which always appear together with A yN . Figure 2 is based upon the "Mass Insertion Method", which is clear in concept, however is complicated to be calculated when the number of "crosses" inserted into a propagator are many. Unlike [4], in this paper we only use "Mass Insertion Method" to analyse qualitatively however calculate directly in mass-eigenstate basis quantitatively through Figure 3.
Calculating in mass-eigenstate basis according to Figure 3 involves a summation over a group of graphs with different real scalar propagators. Each graph in Figure 3 Figure 2. If we use the dimensional regularization scheme, all 1 ǫ terms appear in each diagram should be accurately cancelled, the non-zero remains of the finite part is just due to difference of the masses and the mixing between the real part and the imaginary part of each scalar field. The mixing among scalar fields cannot be omitted even though they're small, exceeding the capability of ordinary computational numeric float-point data types, however, we use gmp/mpfr to deal with it.
Calculating one single diagram in Figure 3 generates the result particularly, when m s → m f , where m f , m s separately indicates the mass of the Majorana particle and the real scalar particle running in the loop. µ can be any mass-scale and must be accurately cancelled into disappearance after summing over all corresponding diagrams. In (7) and (8), we also dropped the divergent 1 ǫ for simplicity, which we know that should also be cancelled finally. We defineÑ Notice that according to (1) (2), whenS acquires vacuum expectation value v s ,ÑÑ terms are generated and thus split the mass spectrum ofÑ R andÑ I , and influence the spectrum ofν R andν I through mixture between right-handed and left-handed sneutrinos. The result of the 8 × 8 mass-square matrix of sneutrinos is showed below, where M ν3×3 is just the ordinary mass matrix of sneutrinos in NMSSM, where [m l ij ] is the supersymmetry breaking soft mass matrix of left-handed leptons. In addition, From observing (10) In numerical calculation, we rotate (3) into mass eigenstate by multiplying the matrices with the neutrilino transforming matrix supplied by NMSSMtools, and the sneutrino transforming matrix is calculated according to (10).
We have to mention that this model do not have the ability to predict any mixing angles, that is to say, any mixing angle is permitted if only the correct mass-squared difference is acquired. If we get an example of neutrino mass matrix M ν with the correct mass spectrum, then we can always find a unitary matrix V to rotate M ν into the "correct" matrix M correct ν with the "correct" mixing angles, that is to say, Use the same V to operate the all of the sneutrino soft mass-square matrix, the H u L i N Yukawa coupling constants y N i , and the A-terms A yN i y N i altogether into a new group of parameters to input into the theory, Then we can always acquire "correct" M ′ ν . What we only need to consider is that whether these operations involving m l2 ij disturb the off-diagonal terms which generate large FCNC. In fact, as |V ij | ≤ 1 always hold, so the magnitude of m ij (i = j) do not change. Therefore, in the processes of numeric calculation, we only concern about the neutrino mass-square hierarchy.

III. NUMERICAL RESULTS AND ANALYSE
We modified NMSSMTools-4.2.1 [13] [14] by adding the effects of our extended sectors.
The H u LN Yukawa coupling constants and the corresponding A terms are so small that their phenomenology effects are highly suppressed, thus we needn't consider them. We checked and followed [9] together with [11] to calculate the loop contribution to Higgs mass, and see appendix for the process and formulae in detail. We also assume that only R-parity oddÑ can be the candidate of dark matter, thus we modified the model files contained in the MicroOMEGAS [15] inside the NMSSMTools.
We opened the constraint on muon anomalous g µ − 2 when scanning, which is so sensitive to the supersymmetry breaking soft masses of sleptons. Though we need non-zero off-diagonal elements of the soft mass-square matrix of the left-handed leptons, they are constrained by experimental limits such as the branching ratio of µ → e + γ [12]. In order to avoid such constraints, we need either relatively large slepton masses or small off-diagonal terms. However, the constraint of muon's g µ − 2 prefers smaller supersymmetry breaking soft terms of sleptons, so we set the range (500GeV ) 2 < m l 11 = m l 22 = m l 33 < (1500GeV ) 2 , and m l ij < 1 10 m l ii for each i = j. This scale cannot avoid FCNC µ → e + γ completely, however, which will be discussed in the next section.
When calculating neutrino mass matrix, cases are that accuracy of beyond 10 −20 is needed, so we used the numerical tools gmp/mpfr. However, this technique extremely slows down the speed, so we scanned avoiding to consider the neutrino masses at first, then calculate the neutrino mass matrices by testing in y N -A yN -m l ij (i = j) parameter space for each parameter point passed the previous constraints. We should note that if the lepton-number U(1) L symmetry does not break, diagrams in Fig. 3 cancel with each other strictly. It is due to the existence of the lepton-number violating terms λ N SNN and λ N A N SÑÑ that different diagrams in Fig. 3 cannot be cancelled out strictly, leaving a small finite value, looking as if we are tuning something. As mentioned before, we needn't care about the mixing angles as they can always be acquired after exerting the mentioned process on each parameter point without disturbing the phenomenology.
The scanning processes are divided into three steps. First of all, scan from parameter During this step, NMSSMTools is hardly modified except the lower-bound of the lightest Higgs mass. The Higgs mass bound is temporarily set as 112.7GeV < M Higgs < 128.7GeV .
Just because we opened the anomalous g µ − 2 constraints, we can see from Figure 6 that m l ii concentrate below 600GeV. Then the second step to calculate the modification of the Higgs mass by the right-handed neutrino sectors is applied. We modified NMSSMTools-4.2.1 by changing the part of the Higgs sectors considering the effects from the right-handed neutrino sectors. Here, the Higgs mass bound is set back to 122.7GeV < M Higgs < 128.7GeV in order to filter the consequences output from the previous step.
The final and the most important step is to decide the remaining parameters inside the for each of the parameter point obtained from the last step. We scanned randomly in this area at first, and tested each point to see whether it can lead to the correct neutrino mass-squared difference, then start from the nearest point to "jog" near the correct position 7.12 × 10 −5 < ∆m 2 21 < 8.20 × 10 −5 2.31 × 10 −3 < |∆m 2 31 | < 2.74 × 10 −3 .
This process takes most of the time.
If we rotate the basis of L i by a unitary matrix V , the parameters y N i , y N i A yN i (i is not summed up) correspondingly behave like a "vector-like representation" of V , so define the scalar-like norm of these two parameters Though m l2 ij (i = j) do not transform like a vector, we still define a "scalar-like" as mentioned in the Ref. [6]. However, in this situation, we should also consider the gauge quartic terms such as 1 8 (g 2 + g ′2 )(|H 0 u | 2 − |H 0 d | 2 ) 2 , and the λ 2 N N 2 N * 2 from other yukawa terms. After a some calculation, we can see these quartic terms actually stabilize the correct vacuum. In fact, when we calculate the contribution to higgs mass from the right-handed neutrino sectors, we omitted these A-terms. By the way, Figure 7 shows that Higgs mass does have the possibility to recieve a relatively large correction, which is compatible with that in [9].

IV. PHENOMENOLOGY
Although we would like the A yN terms to be large enough in order for the loop corrections to be comparable with the tree-level terms, loop-corrections still tend to be smaller. They only give smaller masses to the two neutrinos which are originally massless up to treelevel see-saw mechanism. Thus, we predict a normal-hierarchy mass spectrum of neutrinos without any degeneracy.
We have mentioned that we need off-diagonal terms in the soft mass-squared terms of sleptons, which may lead to visible µ → e + γ decay. The branching ratio of this chain is estimated in [12] Br From the data showed in Figure 6, we can estimate that Br(µ → e + γ) varies from 10 −12 − 10 −10 . Compared with the PDG data Br(µ → e + γ) < 2.4 × 10 −12 [16], it means at least some of the points have passed the constraint and some are on the edge of the experimental bound. As mentioned before, if we assume that the source of muon anomalous g µ − 2 comes from the NMSSM sectors, the soft masses of sleptons are strictly constrained to be much smaller than 1TeV, and we imposed this constraint during our scanning procedure. If we release such a constraint, the slepton mass can reach above 1TeV so that µ → e + γ decay is totally invisible.
In Ref. [16], bounds on other muon FCNC decay channels are listed, such as µ → 3e, µ → e + 2γ. These bounds on branching ratios are roughly of the similar order of magnitude of µ → e + γ, but their Feynmann-diagrams usually involve a higher-order perturbative expansion, thus the effects are depressed, so we did not discuss them. On the other hand, the τ -FCNC bounds are much looser, so we did not talk about them either.
As mentioned in Appendix I, TeV scale right-handed neutrinos have no hope to become dark matter, as they usually decay quickly. If R-parity is conserved, the scalar partner of N might become the LSP, thus the candidate of the dark matter, and the corresponding phenomenology is discussed in [20]. As we have noted, we added the effects of the righthanded scalar neutrinos in our theory when calculating the dark matter relic density.
Generally, the possibility to discover a right-handed neutrino directly on a collider is significantly suppressed due to the rather small Yukawa coupling y N in the case of TeV see-saw mechanism, unless other physical sectors beyond the Standard Model which appear to interact with right-handed neutrinos exist [19]. In the circumstances of NMSSM, the interaction between the right-handed neutrino and the singlet Higgs (characterized by the magnitude of y N and A yN ) can be relatively large. If we are able to observe the singlet Higgs directly in the future, we can take a glimpse of right-handed neutrinos by watching the properties of the singlet, e.g. an invisible decay chain in the case that the right-handed (s)neutrino is lighter than the singlet higgs, or its correction to the propagator when it is heavier.

V. CONCLUSION
In this article, we have shown that NMSSM with a Z 3 symmetry extended with only one right-handed neutrino superfield can generate a complete spectrum of three massive light Majorana left-handed neutrinos. The tree-level see-saw mechanism can only generate one massive neutrino, with the remaining two acquiring masses from radiative one-loop corrections. To accumulate the loop effects in order it can be comparable with the tree-level in quantity, we need relatively large A yN ∼ 10 2 T eV , however other phenomenological effects from them are suppressed by the Yukawa y N ∼ 10 −7 . Though off-diagonal terms are needed in the soft mass-square terms of the sleptons, we are still able to acquire the correct neutrino mass differences without conflicting with the phenomenology constraints. We also showed that once the correct mass-difference is acquired, any figure of mixing angles is allowed, and of course so is the one measured by experiments. We also confirmed that the right-handed neutrino can contribute to Higgs mass by its coupling with the Higgs sectors. 13 Appendix: I In order to consider the effects of N,Ñ during the calculation of dark matter decay modes, we should give all of the corresponding vertices.
where the definition of the diagonalized field h i , a i , together with their diagonalizing matrix S ij , P ij is similar to the tree-level ones in Appendix II. However, unlike appendix II, we should note that when applying these vertices to calculate the dark matter relic density, we should use the renormalized version of h i , a i , S ij , P ij . Part of the vertices listed here is copied and modified from [9].
All of them are input into the MicroOMEGAS [15] model files inside the NMSSMTools, and most of the vertices will also be used when calculating the Higgs mass. BecauseÑ is assigned with the odd R-parity,Ñ rather than fermionic N is set as the candidate of the dark matter. One may ask the question that whether right-handed neutrinos can play the role of dark matter if they decay slowly enough. According to [17][18], right-handed neutrinos heavier than 1 GeV usually decay less than one second, so it is impossible for them to become the dark matter.  should be solved. We also need to calculate Σ A i A j (k 2 ), in order to extract some divergent terms. These constants determine the counterterms listed in (A.5), and we list them in the following text. where Σ P,11 = P i1 Σ A i A j P j1 . where and β B denotes the tree-level β.
After the determination of the counter-terms, the Higgs mass sectors are differentiated and the related terms are replaced with the counter terms acquired through the previous where c X , s X , t X denote respectively cos X, sin X, tan X. The M 2 S i j are mass-square matrix elements in the basis (Re(H 0 u ), Re(H 0 d ), S R ), and M 2 P i j are the elements in the basis (A, S I , G).
Theoretically, all divergent 1 ǫ should be precisely cancelled after the renormalization scheme in the final results. We checked this carefully. Though the existence of matrices P ij , S ij blurred the final expressions, divergent terms proportional to 1 ǫ must be independent of field basis, so we can directly calculate the divergent part by setting P ij = S ij = δ ij , which is much easier to operate. We checked and modified the formulae listed in [9] by verifying whether the infinite parts of the diagrams can be automatically cancelled. Only when confirming this, can we calculate on.