LHC signatures of sterile neutrinos in a Minimal Radiative Extended Seesaw framework

The type-I seesaw mechanism for light neutrino masses, popularly known as canonical seesaw was originally proposed by Minkowski:1977sc ; Mohapatra:1979ia ; Yanagida:1979as ; Schechter:1980gr . It is the simplest mechanism that explains the lightness of SM neutrino masses with the introduction of heavy right-handed neutrinos. The right-handed neutrinos being singlet under SM, are used to write down a Dirac mass term between SM neutrinos $\nu_{L}$ and right-handed neutrinos $\nu_{R}$ along with a bare Majorana mass term for right-handed neutrinos $\nu_{R}$.
The mass matrix for type-I seesaw in the $(\nu_{L},\nu^{c}_{R})$ basis is given by

With the type-I seesaw approximated mass hierarchy $M_{D}\ll M_{R}$, the contribution to the light neutrino masses is given by

Here, the expression for heavy right-handed neutrino mass is $m_{N}=M_{R}$. The $\mathcal{O}(0.1)$eV scale light neutrinos requires $M_{R}$ to be of the order of $10^{14}$ GeV with natural choice of $M_{D}$ around $100$ GeV. The other outcome of the type-I seesaw is the mixing between left-handed and a right-handed neutrino as $V_{\ell N}\propto\sqrt{m_{\nu}/m_{N}}$. To detect right-handed neutrinos, we need their masses in the experimentally accessible range and also need their mixing to be large with active neutrinos. Considering $100$ GeV right-handed neutrinos, the left-right neutrino mixing is constrained from oscillation data as, $V_{\ell N}\leq 10^{-6}\left(m_{\nu}/\mbox{0.1\,eV}\right)\left(\mbox{100\,GeV}/m_{N}\right)$ which makes the type-I seesaw framework experimentally inaccessible in the current detector limit. This motivates researchers to think of alternative seesaw mechanisms where firstly the seesaw scale could be brought down to the experimentally verifiable range and also the mixing between left- and right-handed neutrinos could be made large. Thus, in the next subsection, we introduce the model framework by adding two types of neutrinos along with the known SM particles which allow us to overcome the above-discussed constraints.

The idea of extended seesaw is to allow the seesaw scale operative at experimentally accessible range and provide large light-heavy neutrino mixing. This is implemented with the introduction of two types of sterile neutrinos (denoted by $\nu_{R}$ and $\nu_{S}$). The extensive work on extended seesaw structure can be found in the literature Ma:1989tz ; Johnson:1986gt ; Zhang:2011vh . One of the highly promising framework among them is the minimal inverse seesaw mohapatra:1986aw ; Nandi:1985uh ; mohapatra:1986bd that comes with a mass matrix structure given as,

With the mass hierarchy $\mu_{S}$, $M_{D}\ll M_{RS}$, the light neutrino mass expression in minimal inverse seesaw is given as

and the heavy neutrino forms a pseudo-Dirac pair with the mass of the order of $M_{RS}$. The expression for light neutrino masses in inverse seesaw has a direct proportionality to the lepton number violating scale $\mu_{S}$ leading to suppressed LNV signals. To have large LNV signals, one can add a large Majorana mass term to this seesaw structure without affecting the light neutrino masses at tree level. However, one has to be careful with radiative correction terms for light neutrino masses mediated with the exchange of heavy neutrinos at loop as discussed in ref Pilaftsis:1991ug ; Dev:2012sg .

Similarly, minimal linear seesaw has been explored within extended seesaw scheme with the following structure,

with $\mu_{L}$, $M_{D}\ll M_{RS}$.

The light neutrino masses in this case is given by

In the present work, we wish to discuss the extended seesaw model while emphasizing the radiative contribution to light neutrino masses only and its implications in the following section. Our work is different from the inverse and linear see-saw models because the mass matrix in our framework doesn’t allow a Majorana mass term for sterile neutrino $\nu_{s}$ like in inverse see-saw or a mixing term between the SM left-handed neutrino and the sterile neutrino $\nu_{s}$ like in linear see-saw. The allowed mass term is a bare Majorana mass term for right-handed neutrino $\nu_{R}$ and a mixing term between $\nu_{s}$ and $\nu_{R}$. So , although the model is based on the extended scheme but is different in matrix structure when compared with linear and inverse models. A recent work on the phenomenology of KeV-scale sterile neutrino in a Minimally Extended see-saw(MES) framework seems interesting Das:2019kmn . MERS model in our work has a scope to allow for a eV-scale sterile neutrino, which has its effects in $\beta\text{-decay}$, neutrino-less double $\beta\text{-decay}$, and cosmology Giunti:2019aiy .

The relevant leptonic Yukawa interaction terms for "Minimal Extended Radiative Seesaw" is given by

Here $\mathcal{L}_{\text{SM}}$ is the well known SM lagrangian, next two terms correspond to the kinetic energy terms for the right-handed and sterile neutrinos respectively. The fourth term represents generic Dirac Yukawa interaction between left-handed lepton doublet with right-handed neutrinos while the fifth term is for bare Majorana mass for $\nu_{R}$ and finally, the last term corresponds to mixing between right-handed neutrinos and sterile neutrinos. After electroweak symmetry breaking, the resulting neutral fermion mass matrix in the basis the basis $\left(\nu_{L},\nu^{c}_{R},\nu_{S}\right)$ is given by

with $M_{D}=Y_{D}v/\sqrt{2}$ is the Dirac neutrino mass in which $v\equiv v_{\rm EW}=174~{}$GeV is the VEV of electroweak symmetry breaking scale.

It has been shown in the appendix A.2 that the light neutrinos are massless at tree level while both right-handed and sterile neutrinos have Majorana masses within the "Minimal Extended Radiative Seesaw" with mass hierarchy $M_{R}\gg M_{RS}>M_{D}$. After integrating out the heaviest right-handed neutrinos, the effective tree-level contribution to light active and sterile neutrinos ($\mathcal{M}^{\prime}_{\nu_{L}\nu_{S}}\equiv{\cal M}_{\rm eff}^{\rm tree}$) can be written as,

where ${\bf 0}_{3}$ is a null matrix of order 3 and the $6\times 6$ unitary matrix $V$ diagonalizes the light active and sterile neutrinos after right-handed neutrinos are already integrated out and is expressed in terms of a approximated mixing $\zeta$, Grimus:2000vj (upto the order ot $\mathcal{O}(1/M_{RS})$) as,

From (11), it is straightforward to obtain

It has been shown in the appendix A.2 that there are no tree level contributions to light neutrino masses in MERS scheme and type-I seesaw contribution is exactly cancelled out at leading order. The interesting feature of the model is that the heaviest right-handed neutrinos generate light active neutrino masses at one loop level via exchange of SM $W$ and $Z$ gauge bosons as shown in the Feynman diagram figure 1. The one loop contribution to light neutrino masses in the limit $\|M_{R}\|\gg\|M_{D}\|,\|M_{RS}\|$ has been shown in the Appendix section A.3 and is given by

After simplication by using the same seesaw approximations, it can be shown that the light active neutrino masses are governed by radiative seesaw mechanism and the same unitary mixing matrix diagonalizes it as,

For illustrating that how one can get eV scale sterile neutrinos and large mixing with sub-eV scale active neutrinos, let us consider one flavor case where these matrices reduce to complex numbers and thus a subscript of "1" is used in the notation for neutrino masses to denote the same. Using relations given in (75) and (76), the eigenvalues for one mass eigenstates of light active and sterile neutrino are given as

and the mass eigenvectors are given by

• •

  $|M_{RS}|/M_{R}=10^{-4}$, one can have a sterile neutrinos of keV mass which can be dark matter candidate.

• •

  $|M_{RS}|/M_{R}=10^{-5}$, we get eV scale sterile neutrino explanining LSND anomalies or have large impact on neutrino oscillation probabilities.

A Recent Review on the sterile neutrino search experiments can be found in the reference Seo:2020ehv .

The detailed discussion on block diagonalization proceedue is presented in appendix A.3 for all neutrinos and the results are presented below as,  59  15  14

and for the sterile and right-handed neutrinos, respectively, as

The physical masses for these neutral leptons are obtained by respective unitary mixing matrices as follows,

The full $9\times 9$ mixing matrix within the allowed mass hierarchy $M_{R}\gg M_{RS}>M_{D}$ following the standard block diagonalization procedures is given by

Here the parameters used for mixing between different neutral fermion states are, $X=M_{RS}M^{-1}_{R}$, $Y=M_{RS}M^{-1}_{R}$ and $Z=M_{D}M^{-1}_{R}$.

The mass matrices of the left-handed, right-handed and sterile neutrinos are proportional to each other,

For three flavour scenario, one can consider Dirac neutrino mass matrix propertional to identity i.e, $M_{D}\propto k_{d}\,({\bf 1}_{3\times 3})$ (Here, $k_{d}$ is the proportionality constant with ’d’ pointing to the Dirac nature of the term) and then the structure of $M_{R}$ will be decided by light neutrino masses and mixings. The complex symmetric right-handed neutrino mass matrix, $M_{R}$ is diagonalized as $\widehat{M_{R}}=U^{\dagger}_{R}M_{R}U^{*}_{R}$ and the light neutrino mass matrix is diagonalized as, $\widehat{m_{\nu}}=U^{\dagger}m_{\nu}U^{*}$. The interesting point here is that the unitary transformation matrices diagonalizing the active light left-handed and heavy right-handed neutrinos are themselves related in the following way

Similarly, ignoring the suppressed mixing factor $\zeta\propto M_{D}/M_{RS}$ and assuming $M_{RS}\propto k_{rs}\,({\bf 1}_{3\times 3})$(Here, $k_{rs}$ is the proportionality constant with ’rs’ pointing to the mixing between heavy right-handed neutrino and sterile neutrino), one can verify that unitary matrices diagonalizing light sterile neutrinos $\nu_{S}$ and light active neutrinos are related as

Thus, all the masses and mixing are fully deterministic for a fixed values of $k_{d}$ and $k_{rs}$.The mass eigenvalues for light as well as heavy neutrinos are related as

where $M_{i}$ and $m_{i}$ are physical masses for light active and heavy Majorana neutrinos. Fixing $M_{\rm max}$( $M_{3}$ for NH and $M_{2}$ for IH), the mass relation can be expressed as

The MERS framework in its neutral lepton sector consists of the usual three generations of left-handed SM neutrinos along with the introduction of three families of two right-handed sterile neutrinos $\nu_{s}$ and $\nu_{R}$. The added sterile leptons are called so, because other than the mixing composition with each other and the usual left-handed neutrinos, these do not directly interact with any other SM particles. At the LHC, sterile neutrinos with masses around $5\sim 20GeV$ will be mainly produced from the decay of on-shell $W$ bosons. Thus, when the mass of the sterile neutrino is less than the mass of W bosons, then W boson decays into the sterile neutrino and a lepton, which thereafter decays into a lepton plus other particles like pions. This scenario can be referred from the figure 3. In reference from Dib:2019ztn , it is inferred that most promising modes should be $W^{+}\rightarrow\mu^{+}N$ followed by a displaced decay either into $N\rightarrow\pi\mu^{+}$,$2\pi\mu^{+}$ or $3\pi\mu^{+}$ for a Majorana sterile neutrino as can be seen from the figure 4, or $W^{+}\rightarrow\mu^{+}N$ followed by $N\rightarrow\pi\mu^{-}$,$2\pi\mu^{-}$ or $3\pi\mu^{-}$ for a Dirac neutrino as can be seen from the figure 5. The decay rate $W^{+}\to\ell^{+}N$ has been derived in the reference Dib:2019ztn by calculating the branching fraction of $W^{+}\to\ell^{+}N$ concerning the total decay width of the W boson:

Here, $\Gamma_{W}\simeq 2.085$ GeV is the total decay width of the $W$ boson Patrignani:2016xqp .

From here, the sterile neutrino $N$ can decay in several modes, depending on its mass. The case to look for is when the sterile neutrino decays into pions, namely $N\to\pi^{\mp}\ell^{\pm}$, $N\to\pi^{0}\pi^{\mp}\ell^{\pm}$ and $N\to\pi^{\mp}\pi^{\mp}\pi^{\pm}\ell^{\pm}$. Both charged modes will occur for a Majorana $N$, while for a Dirac $N$, only the mode where $N$ decays into a negatively charged lepton will be produced as shown in the table[6]

With the expressions described above we can estimate the exclusive semileptonic decay rates of $N$ into $\pi\ell$, $2\pi\ell$ and $3\pi\ell$, for a neutrino $N$ with mass in the range 5 to 20 GeV, produced at the LHC in the process $W\to\ell N$.

Reference Dib:2019ztn introduce a term called “Canonical” decay rates for $N\to\pi^{-}\ell^{+}$, $N\to\pi^{0}\pi^{-}\ell^{+}$, $N\to\pi^{-}\pi^{-}\pi^{+}\ell^{+}$ and “canonical” branching ratios for the full processes $W^{+}\to\ell^{+}\ell^{+}n\pi$ which are basically the decay rates and branching ratios with all mixing factors $|V_{\ell N}|$ removed. To obtain the actual values, the “canonical” values must be multiplied by the factor $|V_{\ell N}|^{2}$, and $|V_{\ell N}|^{4}/\sum_{l_{i}}|V_{\ell_{i}N}|^{2}$, respectively.

The decay modes with final leptons as muons instead of electrons or tau particles are the most promising decay modes Dib:2018iyr .

The expected number of $W\to N\ell\to n\pi+\ell\ell$ events at the LHC, or equivalently the minimal value of the lepton mixing element that would generate 5 events or more, for a benchmark value of $m_{N}=10$ GeV is given by Dib:2019ztn :

At this mass range, a canonical branching ratio is given as Dib:2019ztn :

According to Ref. Aad:2016naf , at the end of the LHC Run II one may expect a sample of ${\cal N}_{W}\sim 10^{9}$ $W$ decays. Therefore, to obtain more than 5 events, we must have:

which implies $|V_{\mu N}|^{2}\gtrsim 6.2\times 10^{-6}$, provided other mixing elements are smaller. If instead, all mixing elements are comparable, then this lower bound increases by a factor 3, i.e. $|V_{eN}|^{2},\,|V_{\mu N}|^{2},\,|V_{\tau N}|^{2}\gtrsim 1.9\times 10^{-5}$. Now, following the framework of MERS, the table  4 and 5 shows the two sterile neutrino candidates $\nu_{s}$ and $\nu_{R}$ where the mass of $\nu_{R}$ and coupling between $\nu_{l}$ and $\nu_{R}$ in the first row and the mass of $\nu_{s}$ and coupling between $\nu_{l}$ and $\nu_{s}$ in the second row, are within the bounds required to have 5 or more significant events at the LHC, respectively. These results are in ideal conditions, with no cuts or backgrounds.

One last important point in the observability of these processes is the long lifetime of $N$, which would cause an observable displacement in the detector between the production and decay vertices of $N$. This displacement will drastically help reduce the possible backgrounds. For a sterile neutrino $N$ with mass in the range 5 GeV to 20 GeV, the total width can be estimated as Dib:2015oka ; Drewes:2019fou :

We see from the above-written expression that, lighter the mass of $N$ and smaller the mixing $|V_{\ell N}|^{2}$ are, the longer $N$ will live. Using the current upper bound $|V_{\ell N}|^{2}\sim 19\times 10^{-5}$, the characteristic displacement $\tau c\equiv\hbar c/\Gamma_{N}$ is in the range $\sim 20\ \mu m$ to 20 $mm$ (for 20 GeV and 5 GeV respectively). For smaller $|V_{\ell N}|^{2}$, the displacements will be proportionally larger. Moreover, the relativistic $\gamma$ factor will increase the displacement as well.

The case in this sub-section is something that we in our present work are not interested in. However, LHC searches for sterile neutrino masses greater than 100GeV ATLAS:2015gtp are based on inclusive processes $pp\rightarrow W^{*}X$, $W^{*}\rightarrow l^{\pm}l^{\pm}jj$ as shown in the references Keung:1983uu ; delAguila:2007qnc . Taking these processes as a possibility, we can hope to extend our model in future works to include displaced vertex signals from processes involving decays of particles into neutrinos having masses beyond 100GeV. Then, further studies and implications of those heavier neutrinos in the field of astrophysics, cosmology and collider physics can be subsequently explored.