Natural emergence of neutrino masses and dark matter from R-symmetry

# Natural emergence of neutrino masses and dark matter from $R$-symmetry

## Abstract

We propose a supersymmetric extension of the Standard Model (SM) with a continuous global symmetry. The -charges of the SM fields are identified with that of their lepton numbers. As a result, both bilinear and trilinear ’-parity violating’ (RPV) terms could be present at the superpotential. However, -symmetry is not an exact symmetry as it is broken by supergravity effects. Hence, sneutrinos acquire a small vacuum expectation value in this framework. However, a suitable choice of basis ensures that the bilinear RPV terms can be completely rotated away from the superpotential and the scalar potential. On the other hand, the trilinear terms play a very crucial role in generating neutrino masses and mixing at the tree level. This is noticeably different from the typical -parity violating Minimal Supersymmetric Standard Model. Also, gravitino mass turns out to be the order parameter of -breaking and is directly related to the neutrino mass. We show that such a gravitino, within the mass range can be an excellent dark matter candidate. Finally, we looked into the collider implications of our framework.

\preprint

TIFR/TH/17-04 a]Sabyasachi Chakraborty, b]Joydeep Chakrabortty \affiliation[a]Department of Theoretical Physics, Tata Institute of Fundamental Research,
1, Homi Bhabha Road, Mumbai 400005, India \affiliation[b]Department of Physics, Indian Institute of Technology, Kanpur-208016, India \emailAddsabya@theory.tifr.res.in \emailAddjoydeep@iitk.ac.in \keywordsSupersymmetry Phenomenology \arxivnumber1701.04566 \toccontinuoustrue

## 1 Introduction

Neutrino oscillation experiments have firmly established the existence of tiny non-zero masses of active neutrinos and non-trivial mixing [1] in the lepton sector. Since neutrinos are massless within the paradigm of Standard Model (SM), therefore, neutrino physics is a natural testing ground for physics beyond the SM (BSM). The most popular way to generate neutrino masses is through the see-saw mechanism. This also predicts Majorana nature of the neutrinos which signifies lepton number violation. The basic idea behind the see-saw mechanism is to integrate out the heavy modes leading to higher-dimensional neutrino mass operators. Depending on the choice of the heavy particles one can classify variants of this mechanism, namely Type-I, -II, -III, Inverse, Double see-saw etc., both in supersymmetric (SUSY) and non-SUSY scenarios. Apart from neutrino masses and mixing, the deviation from the galactic rotation curve and bullet clusters provide concrete evidences in favor of dark matter (DM). Cosmological observations have also measured the relic density [2, 3] of the DM with very high precision. But unfortunately the DM characteristics e.g. mass, spin and its nature i.e., cold, warm, single or multi component are yet to be determined.

All these shortcomings of the SM can be explained quite efficiently in SUSY. For example, the lightest supersymmetric particle (LSP) is an excellent cold DM candidate in the paradigm of the minimal supersymmetric standard model (MSSM) with -parity conservation. On the other hand, MSSM with -parity violation (RPV) is an intrinsic SUSY way to generate neutrino masses and mixing both at the tree level as well as at the one-loop [4] level. However, if -parity is broken, the LSP becomes unstable and hence fails to explain the observed relic density of the universe. In such cases, the DM candidate could be a gravitino, axino, axion and keV sterile neutrino [5].

Concerning the experimental verification of proposed models, unfortunately, the early 13 TeV run of the LHC has not found any signals [6, 7] in favor of SUSY. Although the non-observation of superpartners does not invalidate the idea of SUSY, it certainly questions the ability of MSSM to resolve the naturalness problem. So far, LHC has already ruled out gluinos lighter than 2 TeV when the gluino and LSP masses are well separated. This in turn makes the whole colored sector heavy due to the logarithmic sensitivity to the ultra-violet (UV) scale through renormalisation group evolutions. Interestingly, this correlation is not generic and can be avoided within the models of -symmetry and Dirac gauginos. One needs to extend the gauge sector of MSSM to representation to construct a Dirac gaugino mass. This require chiral superfields such as a singlet , a triplet under and an octet under , in the adjoint representation of the SM gauge group. These fields couple with bino, wino and gluinos respectively to generate Dirac masses for the gauginos. The presence of additional adjoint scalars cancel the UV logarithmic divergence for squark masses which results in a finite correction [8] only. Hence, the Dirac gluino masses can easily be made heavy. An immediate consequence of having heavy Dirac gluinos is the suppression of the gluino pair or squark-gluino associated pair production cross-sections due to kinematic suppression. However, it is important to note that gluino pair production proceeds through QCD, and the production cross-section for Dirac gluinos would be twice as large compared to the Majorana gluinos with equal masses [9]. This is based on the fact that Dirac gluinos have twice as many degrees of freedom than the Majorana gluinos. In addition, the pair production cross-section of squarks are also suppressed as it requires chirality flipping Majorana gluino masses in the propagator which is absent in these scenarios. This invariably weakens [10] the stringent bound on the first two generation squark masses. Also, trilinear scalar couplings (-terms) are absent in an -symmetric framework and as a result flavor and CP violating interactions are suppressed [11].

Motivated by the issues pertaining to neutrino masses, DM and the non-observation of superpartners at the LHC, we propose a SUSY framework with an -symmetry. Our prime aim in this paper is to generate neutrino masses within the -symmetric Dirac gaugino framework. The -charges are now identified with the lepton number of the SM fields. In general, we have the freedom to cast these charges and one can look into other assignments in [12, 13, 14, 15, 16, 17, 18, 19, 20]. An outcome of our -charge assignment is the presence of bilinear and trilinear “-parity violating” terms in the superpotential which are -symmetric. When -symmetry gets broken, the bilinear RPV terms in the superpotential generate bilinear RPV terms in the scalar potential. As a result, sneutrinos can acquire tiny vacuum expectation values (VEVs) proportional to the order parameter of -breaking, i.e., gravitino mass. However, the superpotential and scalar potential have related sources of bilinear RPV. Therefore, such terms can be simultaneously rotated away. Nevertheless, the trilinear RPV terms will be present in the superpotential resulting in a mixing between light neutrinos and Dirac gauginos. Hence, one generates light neutrino masses at the tree-level. The RPV couplings will also allow the gravitino to decay to a neutrino and a photon. Such a gravitino is an excellent decaying dark matter candidate provided its lifetime is greater than the age of the universe and is consistent with diffuse gamma ray fluxes.

We categorize our paper in the following manner. We propose the model and discuss the basic features of our scenario in section 2. We emphasize on the specific choice of -charges leading to the presence of bilinear and trilinear “RPV” terms in the superpotential. In section 2.2, we discuss soft SUSY breaking, -symmetry breaking and also the generation of sneutrino VEVs. Before proceeding to the fermionic sector, we choose a particular basis to remove the redundancy of the RPV operators in this model. Physics should be independent of the choice of basis. Therefore, we define two basis independent parameters and through which bilinear RPV is manifested. In section 2.4, we show that within the paradigm of our construction, the bilinear RPV terms can be rotated away completely from the superpotential and the scalar potential. However, the trilinear terms will remain. In section 3, we explore the neutral fermion mass matrix. There we illustrate how successfully neutrino masses and mixing can be generated in a simplistic scenario through the trilinear superpotential couplings. While considering normal as well as inverted hierarchies, we obtain the constraints on the relevant superpotential parameters. In section 4 we explore the possibility of gravitino dark matter in our model satisfying necessary constraints. We conclude by providing a direction to explore this scenario in collider experiments in section 5.

## 2 The Model

Our scenario is based on a SUSY framework with an added global -symmetry where the gauginos are Dirac type unlike the MSSM. The choice of -charges are shown in table 1.

It is rather straightforward to show that the scalars share the same -charges with their corresponding superfields whereas for fermions they are one less. Similarly, the gauginos have -charge one and the corresponding gauge bosons have zero -charge. From table 1, we note that the lepton number of the SM particles can be identified with their -charges. The lepton number of the superpartners can be non-standard. The definition of our -charge assignment can also be understood as our choice of equals  [14]. Here is the traditional -charge assignment in MRSSM [21, 22, 23, 24] 1 and stands for the lepton number. In addition, an invariant action demands the superpotential to have -charge of two units. Hence, the allowed terms in superpotential are:

 WMSSM = yijuˆUciˆQjˆHu−yijdˆDciˆQjˆHd−yijeˆEciˆLjˆHd−yirˆEciˆRdˆHd, Wadj = λuSˆSˆRdˆHu−λuTˆHuˆTˆRd+λdSˆSˆHdˆRu−λdTˆRuˆTˆHd, Wbi-RPV" = μiˆHuˆLi, Wtri-RPV" = ξiˆSˆLiˆHu−ηiˆHuˆTˆLi, Wμ = μuˆRdˆHu+μdˆHdˆRu. (1)

The total superpotential is then . The triplet under is parametrised as where , ’s being the Pauli matrices. We denote the components of the triplet field as , and . In eq. (1), are the usual MSSM fields. Further, are trilinear/Yukawa couplings and , and are couplings with mass dimension one. The traditional higgsino mass term () is forbidden in -symmetric models. To generate higgsino-like chargino and neutralino masses, it is mandatory to include two additional -doublet chiral superfields and carrying non-zero -charges. The presence of an -charge for these two fields imply that -symmetry cannot be spontaneously broken in the visible sector. Otherwise one has to encounter massless -axions. These doublets are also known as inert doublets in the literature.

We like to stress that both the bilinear and trilinear “RPV” but -symmetric terms are present in the superpotential due to the assignment of -charges. Also -symmetry prohibits baryonic “RPV” terms in the superpotential and in the process the stringent constraints from proton decays can be circumvented. Before discussing neutrino mass generation mechanism, we would like to first address soft SUSY breaking, -symmetry breaking and the generation of sneutrino VEVs.

### 2.1 Soft (super-soft) SUSY breaking interactions

We choose to work in a scenario where SUSY (global) breaking is not associated with -symmetry breaking. This can be achieved through both – and –type spurions. For example, Dirac gaugino masses can be generated with the help of a spurion superfield as [25, 26]:

 LgauginoDirac = ∫d2θW′αΛ[κ1W1αˆS+κ2Tr(W2αˆT)+κ3Tr(W3αˆO)]+h.c., (2)

where and have -charge 1, i.e., and . The integration over the Grassmann co-ordinates generates the Dirac gaugino masses . Here, represent masses for , and gauginos and refers to the messenger scale. After the discovery of Higgs boson with mass around 125 GeV, it is important to address the status of Higgs mass within the given scenario. In a purely supersoft scenario, the scalar masses are one-loop suppressed compared to the gaugino masses. Consequently, because of -flatness of the scalar potential [8], the tree level quartic term for the Higgs field is vanishingly small. This is challenging from the perspective of fitting the Higgs mass around 125 GeV. Thus instead of working in the generalized supersoft supersymmetry framework [27], we consider -type breaking  [14, 16] also. In our model, the scalar masses can be generated through such -type spurion defined as  [14] which allows the following preserving operators

 ∫d4θˆX†ˆXΛ2[∑iˆΦ†iˆΦi+{ˆHuˆHd+ϵΛˆS+ˆS2+ˆT2+(1Λ×cubic)+h.c.}], ∫d2θXΛ(ˆSˆT2+ˆSˆO2+ˆS3)+h.c., (3)

and can automatically generate the following preserving renormalizable soft SUSY breaking terms

 Lsoft = ∑im2iϕ†iϕi+[tSS+12bSS2+BμHuHd+....]. (4)

The soft mass squared terms are proportional to where we consider same magnitude for the - and -type VEVs. Such a mechanism also generates a scalar singlet tadpole . However, as long as , such a tadpole is not expected to destabilize the hierarchy [28]. Nevertheless, due to the absence of -breaking terms in the scalar sector, sneutrinos cannot acquire VEV. This is an important ingredient for neutrino mass generation in traditional bilinear RPV scenarios.

### 2.2 R-symmetry breaking

It is well established that our universe is associated with a vanishingly small vacuum energy or cosmological constant. To explain this from the perspective of spontaneously broken supergravity theory in the hidden sector, the superpotential needs to acquire a non-zero VEV. Since the superpotential carries non-zero -charge, therefore, implies breaking of -symmetry. As a result, the gravitino would also acquire a mass which turns out to be the order parameter of -symmetry breaking. The -breaking information is then communicated to the visible sector through anomaly mediation and in the process the following -symmetry breaking terms are generated

where the Majorana gaugino masses are generated through small -breaking effects as

 Mi=g2i16π2bim3/2   (i=1,2,3), (6)

with beta functions

 b1=33/5,b2=1,b3=−3 . (7)

The small  symmetry-breaking trilinear scalar interactions are as follows

 Aτ = m3/216π2(−95g21−3g22+3Y2b+4Y2τ), At = m3/216π2(−1315g21−3g22−163g23+6Y2t+Y2b), Ab = m3/216π2(−715g21−3g22−163g23+Y2t+6Y2b+Y2τ). (8)

It is also important to note that the presence of a conformal compensator field invariably generates a term [29] in the superpotential through the following operator:

 L=∫d2θ Σ3μiˆHuˆLi. (9)

After scaling out this compensator field with where is a chiral superfield, we generate a bilinear term () in the scalar potential

 Bμi=μim3/2. (10)

Hence, the term is always aligned with the term. Such terms are -breaking effects and proportional to the gravitino mass. The presence of this small effect would generate tiny sneutrino VEVs which might become important for neutrino mass–mixing as we discuss in the next section.

### 2.3 Sneutrino VEV

To compute the sneutrino VEVs, one has to include the contributions from -, - and soft SUSY breaking terms. The additional pieces associated with and in the -terms are

 Da2 = g(H†uτaHu+˜ℓ†iτa˜ℓi+T†λaT)+√2(MD2Ta+MD2Ta†), (11)

where and ’s represent the generators in the fundamental and adjoint representations respectively. Similarly, the weak hyper-charge contribution is given by

 DY = g′2(H†uHu−˜ℓ†i˜ℓi)+√2MD1(S+S†). (12)

The tree level scalar potential terms which participate in the sneutrino field minimization equations are

 VF = |μi|2|˜ν0i|2, Vsoft = ˜m2|˜ν0i|2+BμiH0u˜νi, VD = [(g2+g′2)8|˜ν0i|2−√2g′MD1vS+√2gMD2vT]|˜ν0i|2. (13)

In the limit , the sneutrino VEVs can be well approximated as

 ⟨˜νi⟩ = −Bμivu˜m2i+μ2i. (14)

Such a choice of the singlet and triplet VEVs also ensure that these fields are very heavy through their respective minimization equations. Assuming i.e., at the electroweak scale, we find . Off course, in the same manner the inert scalars () would also acquire a VEV and as a result would mix with the Higgs fields. However, that mixing is also suppressed by the -breaking parameter and does not play any important role in the phenomenological description.

### 2.4 Choice of basis

In the usual framework of bilinear RPV-MSSM [30, 31, 32, 33], the lepton and the Higgs superfields are at the same footing [34] as they carry the same gauge charges. The lepton number violating couplings depend on the choice of basis. Thus it is important to explicitly mention the choice of basis in which the analysis is being performed. However, physics should not depend on such choices. Therefore, two basis independent parameters and are often introduced in the literature [35, 36, 37, 38] which encapsulate the total lepton number violation in the superpotential as well as in the scalar potential respectively.

In our scenario, both the superfields and carry the same charges as can be seen from table 1. Hence, in terms of the four-vector , where , the renormalizable superpotential can be written as:

 W = yijuˆUciˆQjˆHu−yijdˆDciˆQjˆHd−yiαeˆEciˆLαˆHd+ξαˆSˆLαˆHu−ηαˆHu.ˆT.ˆLα+λdsˆSˆHdˆRu (15) − λdTˆRuˆTˆHd+μdˆHdˆRu+μαˆLαˆHu.

Similarly, the scalar potential consisting of soft and super-soft terms reduces to the following form:

 Vsoft = ∑im2iϕ†iϕi+MD1˜b˜S+MD2˜W˜T+MD3˜g˜O+M1˜b˜b+M2˜W˜W+M3˜g˜g+BαHu˜Lα (16) + [tSS+12bSS2+BμHuHd+...]+Au˜uR˜u∗LH0u+Ad˜dR˜d∗LH0d+Al˜lR˜l∗LH0d.

In the zero-sneutrino VEV basis we define in terms of the superfields as

 ˆRd = 1vR∑αvαˆLα, (17)

where and gets generated due to -symmetry breaking. Likewise, the four vector superfield can now be defined in terms of the usual slepton superfields with vanishing VEVs and in the following way

 ˆLα = vαvRˆRd+∑ieαiˆLi. (18)

Even then, there is a freedom to rotate the lepton () superfields arbitrarily. We choose that only a single lepton superfield couples to in the superpotential. This allows us to rewrite the superpotential in terms of basis independent quantities by plugging eqs. (17) and (18) in eq. (15) as

 W = yijuˆUciˆQjˆHu−yijdˆDciˆQjˆHd−˜yieˆEciˆRdˆHd−˜yijeˆEciˆLjˆHd+˜λuS˜S˜Rd˜Hu+˜ξiˆSˆLiˆHu (19) − ˜λuTˆHuˆTˆRd−˜ηiˆHuˆTˆLi+λdSˆSˆHdˆRu−λdTˆRuˆTˆHd+μdˆHdˆRu + μucosκ ˆRdˆHu+μusinκ ˆL3ˆHu,

where

 ˜ξi = ∑αξαeαi,     ˜λuS=∑αvαvRξα,     ˜yie=∑αyαievαvR, ˜ηi = ∑αηαeαi,     ˜λuT=∑αvαvRηα,     ˜yiej=∑αyαieeαj, (20)

and

 cosκ≡1μuvR∑αμαvα, (21)

with  [4]. is the angle between the four-vectors and . It is evident from eq. (14) that , i.e., and therefore, . Though and are basis dependent quantities their relative angle does not depend on the choice of basis for superfields. As a result, the effect of bilinear RPV terms can be rotated away completely from the superpotential.

Similarly, the bilinear -parity violation in the scalar sector can be parametrised in terms of the angle formed by four-vectors and as:

 cosζ ≡ 1BvR∑αBαvα, (22)

where . But it is clear from the earlier section that are also aligned together. This implies and thus allows us to rotate away the bilinear terms from the scalar sector also. Therefore, in reality, the bilinear RPV terms do not play any role in generating neutrino masses and mixing. However, trilinear terms in the superpotential can not be rotated away simultaneously. These terms play crucial role for neutrino mass generation as discussed in the following section.

## 3 The fermion sector

In this section we will consider both neutral and charged fermion sectors. The mixing between neutral fermions and neutrinos lead to neutrino masses at the tree level. Similarly, chargino mixes with the charged leptons which may potentially give rise to lepton number violating processes.

### 3.1 The neutral fermion sector

The Lagrangian corresponding to the neutral fermion sector after -symmetry breaking contain the following terms

 Lf0 = MD1˜b˜S+MD2˜w0˜T+μu˜H0u˜R0d+μd˜H0dR0u+˜λuSvu˜S˜R0d+˜λdSvd˜S˜R0u+˜λuTvu˜T0˜R0d (23) + ˜λdTvd˜R0u˜T0+M1˜b˜b+M2˜w0˜w0+MS˜S˜S+MT˜T0˜T0+g′vu√2˜b˜H0u−g′vd√2˜b˜H0d − gvu√2˜w0˜H0u+gvd√2˜w0˜H0d+˜ξivu˜Sνi+˜ηivu√2˜T0νi+(˜ξivS+˜ηivT)˜H0uνi.

Here, stands for , and .

The Lagrangian mass terms expressed in the basis ,,,,,,, can be written schematically as

 Lmassf0 = 12(f0)TMNf0, (24)

where

 MN = (m˜f0mDmTD0), (25)

with

 m˜f0 = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝M1MD100g′vu√20−g′vd√20MD1MS000˜λuSvu0˜λdSvd00M2MD2−gvu√20gvd√2000MD2MT0˜λuTvu√20˜λdTvd√2g′vu√20−gvu√200μ000˜λuSvu0˜λuTvu√2μ000−g′vd√20gvd√20000μ0˜λdSvd0˜λdTvd√200μ0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (26)

and

 mTD = ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝0˜ξ1vu0˜η1vu√200000˜ξ2vu0˜η2vu√200000˜ξ3vu0˜η3vu√20000⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (27)

Here, , with and . We also consider , and . For simplicity, we have chosen , . We note, an order one value of the superpotential couplings provide substantial one-loop corrections to the up-type Higgs [39, 40] boson mass . These corrections are ‘stop-like’ and an 125 GeV Higgs boson can be obtained without requiring too heavy top squarks. Hence, we kept but assumed to be small. In the next section, we carry out a simplified analysis to explain neutrino masses and mixings. For brevity, we shall also make the following transformations: , .

### 3.2 Neutrino mass and mixing

In order to obtain a quantitative estimation of the relevant parameters which satisfy neutrino masses and mixing, we choose all the -symmetry preserving masses are of the same order, i.e., . The structure of the effective neutrino mass matrix follows from the typical Type-I seesaw expression and represented as

 (mν)ij≃m3/216π2[a ˜ηi˜ηj+b2(˜ηi˜ξj+˜ηj˜ξi)+c ˜ξi˜ξj], (28)

where

 a = −[2−2√2λg′+(g2+g′2)λ2][2+λ(g−g′){2√2+λ(g−g′)}], b = −2[√2λ(g−g′)+(g2+g′2)λ2][2+λ(g−g′){2√2+λ(g−g′)}], c = −[2+2√2λg′+(g2+g′2)λ2][2+λ(g−g′){2√2+λ(g−g′)}]. (29)

In principle, the parameters and can be varied within the validated range to fit the observed neutrino masses and mixing as showed in [41] for bilinear RPV scenario in MSSM. However, in that framework such a form of the neutrino mass matrix arises only after taking the loop corrections into account. Here we adopt a rather simplified approach [42] to estimate the values of these superpotential parameters such that neutrino masses and mixing can be successfully generated. The co-efficient of vanishes for distinct values of which can be obtained by setting in eq. (29). Under such approximation, the neutrino mass expression in eq. (28) turns out to be:

 (mν)ij|λ=0 ≃ m3/216π2[˜ηi˜ηj+˜ξi˜ξj], (mν)ij|λ=−√2(g−g′)(g2+g′2) ≃ m3/216π2[(1+tan2θW)2˜ηi˜ηj+(1+cot2θW)2˜ξi˜ξj]. (30)

Further assuming (), we can decompose the full neutrino mass matrix as

 (mν)ij≃m3/216π2⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣c0⎛⎜ ⎜⎝˜η21˜η