Conventions

On the Constraints Summary and Plots and Data pages we use a model-independent phenomenological approach, parametrised by a single heavy neutrino mass scale \(m_N\) and a single flavor light-heavy neutrino mixing \(|U_{\ell N}|^2\), assuming the mixing effects in other flavors to be subdominant. Although this assumption may not be strictly valid for a realistic seesaw model satisfying the observed neutrino oscillation data, it enables us to consider generic bounds on the mixing strength which can be translated or scaled appropriately in the context of a specific neutrino mass model (see e.g. [138]).

We nevertheless outline two models explaining the smallness of the left-handed neutrino (LH) masses and predicting an admixture of the active and sterile neutrinos.

Canonical Seesaw

The simplest renormalisable extension of the Standard Model (SM) for understanding the smallness of the left-handed (LH) neutrino masses is defined by the interaction Lagrangian

(1)\[\begin{aligned} -\mathcal{L}_{Y}=h_{\ell \alpha} \bar{L}_{\ell} \widetilde{\Phi} N_{R \alpha}+\mathrm{h.c.}\,, \end{aligned}\]

where

(2)\[\begin{split}\begin{aligned} L_{\ell}=\left(\begin{array}{c} \nu_{\ell} \\ l_{\ell} \end{array}\right), \quad \Phi=\left(\begin{array}{c} \phi^{-} \\ \frac{1}{\sqrt{2}}\left(v+h+i \phi^{3}\right) \end{array}\right)\,, \end{aligned}\end{split}\]

are \(SU(2)_{L}\) lepton and Higgs doublets respectively, \(\widetilde\Phi = i \sigma_2 \Phi^*\) and \(N_{R\alpha}\) are SM singlet neutral fermions, also known as the sterile neutrinos, since they cannot directly participate in the SM charged-current (CC) and neutral-current (NC) interactions in the absence of any mixing with the active neutrino sector. In Eq. (1), \(\alpha=1,2, \cdots, \mathcal{N}\) is the sterile neutrino flavor index and \(h_{\ell \alpha}\) are the dimensionless complex Yukawa couplings. From the structure of Eq. (1), we see that the fermions \(N_{\alpha}\) must necessarily be right-chiral; hence, they are also known as right-handed (RH) neutrinos. Assuming that Eq. (1) is the only source of neutrino masses and oscillations, we need at least two or three RH neutrinos, depending on whether the lightest active neutrino is massless or not. After Electroweak Symmetry Breaking (EWSB) the term Eq. (1) generates a Dirac mass \(M_{D}=hv\).

Since the RH neutrinos carry no SM gauge charges, one can also write a Majorana mass term

(3)\[\begin{aligned} -\mathcal{L}_{M}=\frac{1}{2}\left(M_{N}\right)_{\alpha \beta} \bar{N}_{R \alpha}^{C} N_{R \beta}+\mathrm{h.c.}\,, \end{aligned}\]

while preserving gauge invariance. The term Eq. (3) implies that the hypercharge of \(N_{R\alpha}\) is zero, and therefore, from Eq. (1), we deduce that the hypercharges of the lepton and Higgs doublets are the same. Thus, the requirement of cancellation of gauge chiral anomalies implies charge quantisation, provided that the neutrino mass eigenstates are Majorana fields [139][140][141][142].

The terms Eq. (1) and Eq. (3) together lead to the following neutrino mass matrix in the flavor basis \(\left\{\nu_{L \ell}^{C}, N_{R \alpha}\right\}\):

(4)\[\begin{split}\begin{aligned} \mathcal{M}_{\nu}=\left(\begin{array}{cc} \mathbf{0} & M_{D} \\ M_{D}^{\mathrm{T}} & M_{N} \end{array}\right)\,, \end{aligned}\end{split}\]

For \(\left\|M_{D} M_{N}^{-1}\right\| \ll 1\) (with \(\|M\| \equiv \sqrt{\operatorname{Tr}\left(M^{\dagger} M\right)}\) being the norm of matrix \(M\)), the light neutrino masses and mixing are given by the diagonalisation of the effective mass matrix

(5)\[\begin{aligned} M_{\nu} \simeq-M_{D} M_{N}^{-1} M_{D}^{\top}\,, \end{aligned}\]

and the left-right neutrino mixing parameter is given by \(U_{\ell N_{\alpha}} \sim M_{D} M_{N}^{-1}\). This is the Type-I seesaw mechanism [143][144][145][146][147].

Due to the smallness of the light neutrino mass \(M_{\nu}\sim 0.1 \,\mathrm{eV}\) [136], for a low seesaw scale in the sub-TeV to TeV range, the experimental effects of the light-heavy neutrino mixing are expected to be too small, unless the RH neutrinos have additional interactions, e.g. when they are charged under \(U(1)_{B-L}\). However, there exists a class of minimal SM plus low-scale Type-I seesaw scenarios [148][149][150][151][152][153][154][155][156][157][158][159], where \(U_{\ell N_{\alpha}}\) can be sizable while still satisfying the light neutrino data. This is made possible by assigning specific textures to the Dirac and Majorana mass matrices in the seesaw formula Eq. (5). The stability of these textures can in principle be guaranteed by enforcing some symmetries in the lepton sector [150][151][157][160][161]. We will generically assume this to be the case for constraints from collider signatures on low-scale minimal seesaw mixing, without referring to any particular texture or model-building aspects.

Inverse Seesaw

Another natural realisation of a low-scale seesaw scenario with large active-sterile neutrino mixing is the inverse seesaw model [162], where one introduces two sets of SM singlet fermions \(\{N_{R\alpha}, S_{L\rho}\}\) with opposite lepton numbers, i.e. \(L(N_R) = +1 = −L(S_L)\). In this case, the neutrino Yukawa sector of the Lagrangian is in general given by

(6)\[\begin{aligned} -\mathcal{L}_{Y}=h_{l \alpha} \bar{L}_{\ell} \widetilde{\Phi} N_{R \alpha}+\left(M_{S}\right)_{\rho \alpha} \bar{S}_{L \rho} N_{R \alpha}+\frac{1}{2}\left[\left(\mu_{R}\right)_{\alpha \beta} \bar{N}_{R \alpha}^{C} N_{R \beta}+\left(\mu_{S}\right)_{\rho \lambda} \bar{S}_{L \rho} S_{L \lambda}^{C}\right]+\mathrm{h.c.} \end{aligned}\]

where \(M_{S}\) is a Dirac mass term and \(\mu_{R,S}\) are Majorana mass terms. After EWSB, the Lagrangian Eq. (6) gives rise to the following neutrino mass matrix in the flavor basis \(\{(\nu_L\ell)^C, N_{R\alpha},(S_{L\rho})^C\}\):

(7)\[\begin{split}\begin{aligned} \mathcal{M}_{\nu}=\left(\begin{array}{ccc} \mathbf{0} & M_{D} & \mathbf{0} \\ M_{D}^{\mathrm{T}} & \mu_{R} & M_{S}^{\mathrm{T}} \\ \mathbf{0} & M_{S} & \mu_{S} \end{array}\right) \equiv\left(\begin{array}{cc} \mathbf{0} & \mathcal{M}_{D} \\ \mathcal{M}_{D}^{\mathrm{T}} & \mathcal{M}_{N} \end{array}\right) \end{aligned}\end{split}\]

which has a form similar to the Type-I seesaw matrix (4), with \(M_D = (M_D, 0)\) and

(8)\[\begin{split}\begin{aligned} \mathcal{M}_{N}=\left(\begin{array}{cc} \mu_{R} & M_{S}^{\top} \\ M_{S} & \mu_{S} \end{array}\right)\, . \end{aligned}\end{split}\]

Here we have not considered the dimension-4 lepton-number breaking term \(\bar{L} \tilde{\Phi} S_L^C\) which appears, for instance, in linear seesaw models [163][164][165][166], since the mass matrix in presence of this term can always be rotated to the form given by Eq. (8) [167]. Also observe that the inverse seesaw model discussed originally in [162] set the RH neutrino Majorana mass \(\mu_R = 0\) in Eq. (8). At the tree-level, the light neutrino mass is directly proportional to the Majorana mass term \(\mu_S\) for \(||\mu_S||\ll ||M_{S}||\):

(9)\[\begin{aligned} M_{\nu}=M_{D} M_{S}^{-1} \mu_{S} M_{S}^{-1^{\top}} M_{D}^{\top}+\mathcal{O}\left(\mu_{S}^{3}\right) \end{aligned}\]

whereas at the one-loop level, there is an additional contribution proportional to \(\mu_{R}\) [168][84], arising from standard electroweak radiative corrections [169]. The smallness of \(\mu_{R,S}\) is ‘technically natural’ in the ’t Hooft sense [170], i.e. in the limit of \(\mu_{R,S} \rightarrow 0\), lepton number symmetry is restored and the light neutrinos \(\nu_L\) are massless to all orders in perturbation theory, as in the SM.

The freedom provided by the small parameter \(\mu_S\) in Eq. (9) is the key feature of the inverse seesaw mechanism, allowing us to fit the light neutrino data for any value of active-sterile neutrino mixing, without introducing any fine-tuning or cancellations in the light neutrino mass matrix Eq. (9) [171][172]. In essence, the magnitude of the neutrino mass becomes decoupled from the heavy neutrino mass, thus allowing for a large mixing

(10)\[\begin{aligned} U_{\ell N} \simeq \sqrt{\frac{M_{\nu}}{\mu_{S}}} \approx 10^{-2} \sqrt{\frac{1 \mathrm{keV}}{\mu_{S}}} \end{aligned}\]

The heavy neutrinos \(N_R\) and \(S_L\) have opposite CP parities and form a pseudo-Dirac state with relative mass splitting \(\Delta m_{N}\) of the order \(\kappa = \mu_S/M_S\). All lepton number violating processes are usually suppressed by this small mass splitting. For instance, in the one-generation case, the light neutrino mass in Eq. (9) can be conveniently expressed as \(M_{\nu} \simeq \kappa U_{\ell N} M_{D}\), in contrast with \(U_{\ell N} M_{D}\) in the Type-I seesaw case.

As pointed out in [52], the spectrum of massive states caused by the small lepton number violating perturbation \(\mu_{S}\) depends on the number of \(N_{R}\) and \(S_{L}\) fields. If there are an equal number \(a\) of \(N_{R}\) and \(S_{L}\) fields, in the \(\mu_S\ll 1\) limit we obtain \(a\) pseudo-Dirac pairs and 3 light Majorana states. If there are \(a\) \(N_{R}\) fields and \(b\) \(S_{L}\) fields, we obtain \(a\) pseudo-Dirac pairs and \(|3+b-a|\) Majorana states. In the pseudo-Dirac case it is likely that experiments will only be able to resolve what appears to be a Dirac neutrino with mixing \(\left|U_{\ell N}\right|^{2}=\left|U_{\ell N_{1}}\right|^{2}+\left|U_{\ell N_{2}}\right|^{2}\).