.. _constraints-label: Constraints Summary ======================================================== We summarise the results of experimental searches for sterile neutrinos and constraints on the active-sterile mixing :math:`|U_{e N}|^2` over the sterile neutrino mass range :math:`1 ~\mathrm{eV}`, while :numref:`Fig. %s ` illustrates the expected sensitivities of future experiments and observations. We also list and plot the current and future constraints on :math:`|U_{\mu N}|^2` and :math:`|U_{\tau N}|^2` in :ref:`data-label`. For earlier summary plots showing a partial list of these constraints, see e.g. Refs. :cite:`Atre:2009rg,Deppisch:2015qwa,deGouvea:2015euy,Chrzaszcz:2019inj`. Most constraints given in :ref:`data-label` were derived assuming a single heavy Majorana neutrino :math:`N`. For a pseudo-Dirac pair :math:`N_1` and :math:`N_2` with small splitting, the limits are also roughly applicable for :math:`|U_{\ell N}|^2=|U_{\ell N_{1}}|^2+|U_{\ell N_{2}}|^2`. Collider Searches ------------------ Heavy states are produced in charged-current (CC) and neutral-current (NC) processes through their admixture with the active states, and thus their decay products can be searched for at high-energy colliders which copiously produce :math:`W` and :math:`Z` bosons. For sufficiently small mixing angles, the macroscopic decay length of the heavy neutrinos can result in displaced vertices with distinct detector signatures. We consider the following searches: Direct Production at the LHC ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The LHC collaborations **ATLAS** and **CMS** have searched for :math:`N` production and decay through various channels. Both have recently searched for decays of :math:`W`-produced :math:`N` to three charged leptons, :math:`W^{\pm}\rightarrow\ell^{\pm}N, N\rightarrow\ell^{\pm}\ell^{\mp}\nu_{\ell}` (:math:`\ell = e, \mu`), either in the LNC or LNV mode. ATLAS used the prompt final state of three isolated leptons and no opposite-charge same-flavour lepton pairs (LNV channel) to reject Drell-Yan, :math:`W` + jets and :math:`t\bar{t}` backgrounds. CMS broadened the search to the LNC channel with a sensitivity to displaced decays. The analyses impose the limits :math:`|U_{e N}|^2,\,|U_{\mu N}|^2 < 10^{-5}-10^{-4}` over the mass range :math:`5 ~\mathrm{GeV} < m_{N} < 50~\mathrm{GeV}` :cite:`Aad:2019kiz,Sirunyan:2018mtv`. ATLAS and CMS have also conducted searches for the LNV same-sign dilepton + jets channel, :math:`W^{\pm}\rightarrow\ell^{\pm}N, ~N\rightarrow\ell^{\pm}jj` :cite:`Aad:2015xaa,Sirunyan:2018xiv`. Above the :math:`Z` boson mass limits can be improved in future by ATLAS and CMS during the high luminosity (:math:`\mathcal{L} = 3~\mathrm{ab}^{-1}`) LHC phase (**HL-LHC**) and by a future :math:`\sqrt{s} = 27` or :math:`100\,\mathrm{TeV}` Future Circular Collider (**FCC-hh**) :cite:`Benedikt:2018csr,Pascoli:2018heg`. Around the Higgs mass, limits can also be set from the SM Higgs decay to sterile neutrinos :cite:`Das:2017zjc`. Displaced Vertices ^^^^^^^^^^^^^^^^^^ In the future, **ATLAS**, **CMS** and **LHCb** are expected to probe smaller :math:`|U_{\ell N}|^2` through displaced vertex searches. For a given mixing, :math:`m_{N}` must lie in a specific range in order to avoid :math:`N` decaying promptly or outside the detector. The best projected limit is :math:`|U_{eN}|^2,\,|U_{\mu N}|^2 \lesssim 10^{-9}` for :math:`m_{N} \approx 30\,\mathrm{GeV}` :cite:`Aad:2019kiz`. Proposed detectors placed near existing LHC interaction points have been designed specifically to search for displaced vertex signatures. These include **AL3X** :cite:`Dercks:2018wum`, **CODEX-b** :cite:`Gligorov:2017nwh`, **FASER2** :cite:`Feng:2017uoz`, **MATHUSLA** :cite:`Chou:2016lxi` and the **MoEDAL** experiment's MAPP detector :cite:`Frank:2019pgk`. In :numref:`Fig. %s `, we show the expected sensitivity of AL3X, FASER--2 :cite:`Kling:2018wct` and MATHUSLA :cite:`Curtin:2018mvb` for illustration. The best projected limits of MATHUSLA are :math:`|U_{eN}|^2,\,|U_{\mu N}|^2 \lesssim 10^{-9}` for :math:`1~\mathrm{GeV}m_{Z}` :cite:`Mondal:2016kof,Das:2018usr,Antusch:2019eiz`. An overview of proposed collider sensitivities is given in Ref. :cite:`Antusch:2016ejd`. LNV Signature at Colliders ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ As for the LNV signature at colliders, in a natural seesaw scenario with approximate lepton number conservation, the LNV amplitude for the on-shell production of heavy neutrinos at average four-momentum squared :math:`\bar{s}=(m_{N_1}^2+m_{N_2}^2)/2` can be written as :cite:`Bray:2007ru,Dev:2013wba` .. math:: :label: ampLNV \begin{aligned} \mathcal{A}_\text{LNV}(\bar s) \ = \ U_{\ell N}^2\frac{2\Delta m_N}{\Delta m_N^2 + \Gamma_N^2} + \mathcal{O}\left(\frac{\Delta m_N}{m_N}\right), \end{aligned} for :math:`\Delta m_N\lesssim \Gamma_N`, i.e. for a small mass splitting :math:`|\Delta m_N| = |m_{N_2} - m_{N_1}|` between the heavy neutrinos compared to their average decay width :math:`\Gamma_N\equiv (\Gamma_{N_1} + \Gamma_{N_2})/2`. Thus, the LNV amplitude in Eq. :eq:`ampLNV` will be suppressed by the small mass splitting, except for the case :math:`\Delta m_N\simeq \Gamma_N` when it can be resonantly enhanced :cite:`Pilaftsis:1997dr,Bray:2007ru`. For the :math:`5 - 50 ~ \mathrm{GeV}` range of sterile neutrino masses probed by the ATLAS and CMS same-sign trilepton and dilepton + jets analyses, the total sterile neutrino decay width, if decays only takes to place to SM leptonic and hadronic degrees of freedom, is given by .. math:: \begin{aligned} \Gamma_{N} & \ = \ \sum_{\ell }a_{\ell}(m_{N})\,|U_{\ell N}|^2\,, \end{aligned} where the complete expressions for the factors :math:`a_{\ell}(m_{N})` are given in Refs. :cite:`Atre:2009rg,Helo:2010cw`. The factors :math:`a_{\ell}(m_N)` include the contributions from two-body semi-leptonic and three-body leptonic decays, and are approximately given by .. math:: \begin{aligned} a_{\ell}(m_N) \ \approx \ N^{\mathrm{2-body}}\cdot\Gamma^{\mathrm{2-body}}+N^{\mathrm{3-body}}\cdot\Gamma^{\mathrm{3-body}}\,, \end{aligned} where :math:`N^{\mathrm{2-body}}` and :math:`N^{\mathrm{3-body}}` are the number of decay channels open for each decay topology. :math:`\Gamma^{\mathrm{2-body}}` and :math:`\Gamma^{\mathrm{3-body}}` are given roughly by .. math:: \begin{aligned} \Gamma^{\mathrm{2-body}} \ \sim \ \frac{G_{F}^2f_{M}^2m_{N}^3}{5\pi}\,, \qquad \Gamma^{\mathrm{3-body}}\ \sim \ \frac{G_{F}^2m_{N}^5}{200\pi^3}\,, \end{aligned} where :math:`f_{M}` is the order of magnitude of the meson decay constants :cite:`Atre:2009rg`. For :math:`m_{N} \approx 50\,\mathrm{GeV}` all three-body leptonic decays and two-body semi-leptonic decays to pseudoscalar mesons (:math:`\pi^0`, :math:`\eta`, :math:`\eta'`, :math:`\eta_c`, :math:`\eta_b`, :math:`\pi^\pm`, :math:`K^{\pm}`, :math:`D^{\pm}`, :math:`D_{s}^{\pm}`, :math:`B^{\pm}`, :math:`B_{c}^{\pm}`) and vector mesons (:math:`\rho^0`, :math:`\omega`, :math:`\phi`, :math:`J/\psi`, :math:`\Upsilon(4S)`, :math:`K^{*0}`, :math:`D^{*0}`, :math:`B^{*0}`, :math:`B^{*0}_s`, :math:`\rho^{\pm}`, :math:`K^{*\pm}`, :math:`D^{*\pm}`, :math:`D^{*\pm}_s`, :math:`B^{*\pm}`, :math:`B_{c}^{*\pm}`) are open, and so the total decay width (for :math:`|U_{\mu N}|^2 = |U_{\tau N}|^2 = 0`) is approximately .. math:: :label: approxwidth \begin{aligned} \Gamma_{N} & \ \sim \ \left(30 \cdot \Gamma^{\mathrm{2-body}}+10\cdot\Gamma^{\mathrm{3-body}}\right)|U_{eN}|^2 %\nonumber\\& \ \sim \ (30\cdot10^{-8}+10\cdot 10^{-5})~|U_{e N}|^2 ~\mathrm{GeV} \ \sim \ 10^{-4}~|U_{e N}|^2 ~\mathrm{GeV}\,. \end{aligned} For small splittings, e.g. :math:`\Delta m_{N}/m_{N} = 10^{-4}` and hence :math:`\Delta m_{N} \approx 5 ~\mathrm{MeV}` for :math:`m_{N} \approx 50 ~\mathrm{GeV}`, and the :math:`|U_{eN}|^2 \sim 10^{-5}` mixing probed by the LNV analyses, Eq. :eq:`approxwidth` implies that :math:`\Gamma_N/\Delta m_{N} \sim 10^{-6}`. Collider searches specifically looking for an LNV signal in :numref:`Fig. %s ` are therefore still valid for this splitting and splittings down to :math:`\Delta m_{N}/m_{N} \sim 10^{-10}`. As will be discussed later, this is important for the comparison with :math:`0\nu\beta\beta` decay in this mass range. We finally note that the analysis of Ref. :cite:`Drewes:2019byd` gives an estimate for the regions of the :math:`(m_{N},|U_{\ell N}|^2)` parameter space where the ratio .. math:: :label: Rll \begin{aligned} R_{\ell \ell}=\frac{\Delta m_{N}^{2}}{2 \Gamma_{N}^{2}+\Delta m_{N}^{2}} \end{aligned} is less than or greater than a third. Comparing with Fig. 1 of that work, we again confirm that LNV signals searched for by colliders below the electroweak scale remain unsuppressed, particularly for :math:`\Delta m_{N}` of order the light neutrino mass splittings (motivated by naturalness). Meson Decays and Beam Dump Experiments --------------------------------------- At the intensity frontier :math:`N` can be produced abundantly in beam-dump experiments and through various meson decays. We consider the following limits: - The TRIUMF **PIENU** experiment :cite:`PhysRevD.84.052002` conducted a search for :math:`N` produced in pion decays at rest. Utilising the helicity suppression of the :math:`\pi \rightarrow e \nu` decay channel in comparison to :math:`\pi \rightarrow \mu \nu` channel, the presence of :math:`N` induces extra peaks in the lower positron energy region. Improving on previous results limited by the background :math:`\mu^{+}\rightarrow e^{+}\nu_{e}\bar{\nu}_{\mu}`, the collaboration set limits at the level of :math:`|U_{\ell N}|^2 \lesssim 10^{-8}` in the range :math:`60~\mathrm{MeV} < m_{N} < 129 ~\mathrm{MeV}` :cite:`Aguilar-Arevalo:2017vlf,Bryman:2019ssi,Bryman:2019bjg`. - The **NA62** experiment :cite:`CortinaGil:2017mqf` used a secondary :math:`75 ~\mathrm{GeV}` hadron beam containing a fraction of kaons, and has been able to probe the decays :math:`K^{+}\rightarrow \ell^{+}N` (:math:`\ell = e, \mu)`. For small :math:`|U_{\ell N}|^2` the :math:`N` decay length is much longer than the :math:`156 \mathrm{m}` detector volume and the process is characterised by a single detected track -- a positive signal is a peak in the missing mass distribution. Limits :math:`|U_{eN}|^2,~|U_{\mu N}|^2< 10^{-8}-10^{-9}` in the range :math:`170~\mathrm{MeV} < m_{N}< 450~\mathrm{MeV}` (up to the kaon mass) have been made. In future NA62 will be converted to a beam-dump configuration and will be able to probe hadronic decays to :math:`N`, followed by :math:`N` decays, up to the :math:`D` meson mass. The projected sensitivity is :math:`|U_{eN}|^2,~|U_{\mu N}|^2< 10^{-8}` for :math:`1~\mathrm{GeV} < m_{N} < 2 ~\mathrm{GeV}` :cite:`Drewes:2018gkc`. A recent recalculation of the impact of sterile neutrinos on kaon decays was conducted in Ref. :cite:`Abada:2016plb`. - The **Belle** experiment :cite:`Abashian:2000cg` was a :math:`B` factory that extended the peak search method to higher energies -- using :math:`B\overline{B}` pairs collected at the :math:`\Upsilon(4S)` resonance, the decay mode :math:`B \rightarrow (X) \ell N`, with :math:`X` a charmed meson :math:`D^{(*)}` or light meson, could be followed by :math:`N\rightarrow \ell \pi` (:math:`\ell = e, \mu`). Constraints were made between the :math:`K` and :math:`B` meson masses and at best were :math:`|U_{\ell N}|^2 \lesssim 3\times 10^{-5}` for :math:`m_{N}\approx 2\,\mathrm{GeV}` :cite:`Liventsev:2013zz`. - The **NA3** experiment :cite:`Badier:1986xz` collided a secondary :math:`300 ~\mathrm{GeV}` :math:`\pi^{-}` beam with an iron absorber, producing hadronic states which subsequently decayed to leptonic, semi-leptonic or fully hadronic final states. :math:`N` decays producing leptonic or semi-leptonic final states could be produced from the decays of :math:`\pi`, :math:`K`, :math:`D` and :math:`B` mesons. NA3 was most sensitive up to the :math:`D` meson mass, setting limits of :math:`|U_{eN}|^2,~|U_{\mu N}|^2< 10^{-4}` for :math:`1~\mathrm{GeV} < m_{N} < 2 ~\mathrm{GeV}`. - Accelerated neutrino beam experiments have conducted a variety of parallel searches. The **CHARM** :cite:`Bergsma:1985is,Vilain:1994vg` and **PS191** :cite:`Bernardi:1987ek` experiments and the **IHEP-JINR** neutrino detector :cite:`Baranov:1992vq,Aguilar-Arevalo:2017vlf` searched for a small fraction of :math:`N` in a predominantly :math:`\nu_{\mu}` beam. The beams were produced by colliding a primary beam of protons with an iron or copper fixed target, with the hadronic products decaying as :math:`\pi/K/D \rightarrow \ell \nu (N)` (:math:`\ell = e,\mu`). If sufficiently massive, :math:`N` may decay before reaching the detector via the channel :math:`N\rightarrow \ell^{+}\ell^{-}\nu_{\ell}`. CHARM also used a wide-band neutrino beam to constrain the NC process :math:`\nu_{\mu}n(p)\rightarrow N X` followed by :math:`N\rightarrow \mu X` within the detector. IHEP-JINR and PS191 provide constraints (down to :math:`|U_{eN}|^2 \lesssim 10^{-7}` and :math:`|U_{eN}|^2 \lesssim 10^{-9}`, respectively) up to the kaon mass. CHARM provides constraints up to the :math:`D` meson mass, at best :math:`|U_{eN}|^2, |U_{\mu N}|^2 \lesssim 10^{-7}` for :math:`1~\mathrm{GeV}< m_{N}<2~\mathrm{GeV}`. - The long-baseline neutrino oscillation experiment **T2K** :cite:`Abe:2019kgx` searched for an admixture of :math:`N` in its initial neutrino beam flux, produced by colliding :math:`30\, \mathrm{GeV}` protons with a graphite target at J-PARC. Daughter :math:`K^{\pm}` of a given charge are focused and decay via :math:`K \rightarrow \ell \nu (N)`. The off-axis near-detector at a baseline of :math:`280\, \mathrm{m}` searched for :math:`N` decays via the channel :math:`N\rightarrow \ell\pi`, improving on the constraints made by PS191. In future, the near detector of the oscillation experiment **DUNE** will be highly sensitive for :math:`m_{N}` up to the :math:`D_{s}` meson mass :cite:`Krasnov:2019kdc,Ballett:2019bgd`. - The future beam-dump experiment **SHiP** :cite:`Alekhin:2015byh` is purposely designed to look for exotic long-lived particles. Utilising a :math:`400\, \mathrm{GeV}` proton beam from the CERN Super Proton Synchrotron, it is expected to be sensitive to sterile neutrinos with :math:`m_{N}` up to the :math:`B_{c}` meson mass (:math:`\sim 6\,\mathrm{GeV}`). In a benchmark scenario where the electron-sterile coupling dominates, SHiP is expected to be sensitive down to :math:`|U_{eN}|^2 \lesssim 10^{-10}` for :math:`m_{N}\approx 1.6\,\mathrm{GeV}` :cite:`SHiP:2018xqw`. - In parallel with collider searches it is possible to look for **LNV Decays** of tau leptons and pseudoscalar mesons as discussed in Refs. :cite:`Kovalenko:2009td,Atre:2009rg,Helo:2010cw,Abada:2017jjx`. One issue is that if the LNV process is mediated by the light neutrinos the amplitude is proportional to and suppressed by the small :math:`m_{\nu}^2`, while if mediated by heavy neutrinos it is suppressed by :math:`1/m_{N}` and :math:`|U_{\ell N}|^2`. LNV decay widths however can be strongly enhanced if a sterile state is produced on-shell. The sensitivity of NA62 to three-body LNV light mesons decays (:math:`K^{+}\rightarrow\ell^{+}\ell^{'+}\pi^{-}`), BESIII to charmed meson decays (:math:`D^{+}/D_{s}^{+}\rightarrow\ell^{+}\ell^{'+}\pi^{-}/K^{-}`) and BaBar, Belle and LHCb for :math:`B` meson decays (:math:`B^{+}\rightarrow\ell^{+}\ell^{'+}\pi^{-}/K^{-}/D^{-}/\rho^{-}/K^{*-}`) for :math:`\ell,\,\ell' = e,\,\mu` were estimated most recently by Ref. :cite:`Abada:2017jjx`. The **BESIII** has also conducted its own analysis on the (:math:`D^{+}\rightarrow\ell^{+}\ell^{'+}\pi^{-}/K^{-}`) decay channel :cite:`Ablikim:2019gvd`. Finally, the **Future LNV** decay sensitivities of NA62, LHCb, Belle-II, MATHUSLA, SHiP and FCC-ee have been explored in Ref. :cite:`Chun:2019nwi`. Beta Decays and Sterile Neutrino Decays --------------------------------------- Active neutrinos are produced in the :math:`\beta`-decays of unstable isotopes and in nuclear fission processes. Heavy sterile neutrinos can also be produced via the active-sterile mixing if the sterile mass is smaller than the energy release (:math:`Q`-value) of the relevant nuclear process. The production of a sterile state results in a distortion or 'kink' in the :math:`\beta`-decay spectrum and associated Kurie plot. It is also possible for the sterile state to decay before detection. We include the following searches: - Heavy neutrinos produced in :math:`\beta`-decays significantly alter the energy spectrum of the emitted :math:`\beta` electron. In order to be kinematically accessible the sterile neutrino mass must be smaller than the :math:`Q`-value of the process, :math:`m_{N}< Q_{\beta}`. If this is satisfied and the sterile states are sufficiently more massive than the active states, the :math:`\beta`-decay spectrum becomes the incoherent sum .. math:: \begin{align} \frac{d\Gamma}{dE} \ = \ \left(1-\sum_{i}|U_{ eN_i}|^2\right)\,\frac{d\Gamma}{dE}(m^2_{\beta})+\sum_{i}|U_{e N}|^2\,\frac{d\Gamma}{dE}(m^2_{N_i})\,\Theta(Q_{\beta}-m_{N_i})\,, \end{align} where :math:`m^2_{\beta} = \sum_{k}|U_{e k}|^2m^2_k` is the usual scale probed by :math:`\beta`-decay :cite:`Shrock:1980vy`. This expression can give rise to multiple kinks in the spectrum of relative size :math:`|U_{e N_i}|^2` and at energies :math:`E_{\mathrm{kink}}=Q_{\beta} - m_{N_i}`. Such an effect for a single sterile neutrino has been probed for a variety of isotopes with a range of different :math:`Q`-values, and therefore sensitive to different :math:`m_{N}`. Isotopes include :math:`^3\mathrm{H}` :cite:`Hiddemann:1995ce,Kraus:2012he,Belesev:2013cba,Abdurashitov:2017kka`, :math:`^{20}\mathrm{F}` :cite:`PhysRevC.27.1175`, :math:`^{35}\mathrm{S}` :cite:`Holzschuh:2000nj`, :math:`^{45}\mathrm{Ca}` :cite:`Derbin:1997ut`, :math:`^{63}\mathrm{Ni}` :cite:`Holzschuh:1999vy`, :math:`^{64}\mathrm{Cu}` :cite:`Schreckenbach:1983cg`, :math:`^{144}\mathrm{Ce}-^{144}\mathrm{Pr}` :cite:`Derbin2018` and :math:`^{187}\mathrm{Re}` :cite:`PhysRevLett.86.1978`. In the future, strongly improved limits by the operating tritium :math:`\beta`-decay experiment **KATRIN** and the proposed **TRISTAN** upgrade are expected :cite:`Mertens:2018vuu`. The capability of the PROJECT 8 experiment, which uses the alternative method of cyclotron radiation emission spectroscopy, has also been briefly explored :cite:`Adhikari:2016bei`. - Reactor neutrino experiments are sensitive to sterile neutrinos with masses in the range :math:`1~\mathrm{MeV}< m_N < 10~\mathrm{MeV}`. At these masses it is possible for :math:`N` to decay within the detector via the channel :math:`N\rightarrow e^{+}e^{-}\nu`. Limits have been set by searches at the **Rovno** :cite:`Derbin:1993wy` and **Bugey** :cite:`PhysRevD.52.1343` reactors. This effect was also searched for by the **Borexino** experiment :cite:`PhysRevD.88.072010`, which detected neutrinos produced by the fission processes in the Sun -- heavy neutrinos with masses up to :math:`14\,\mathrm{MeV}` can be produced in the decay of :math:`^8`B. Borexino has set the best limits; :math:`|U_{eN}|^2\lesssim 10^{-6} - 10^{-5}` for :math:`m_{N}\sim 10 ~\mathrm{MeV}`. Neutrinoless Double Beta Decay ------------------------------- Persistent anomalies in neutrino oscillation experiments are still providing intriguing hints for the existence of an additional mass squared splitting :math:`\Delta m^2 \sim 1 ~\mathrm{eV}^2` to the well-established solar and atmospheric mass squared splittings :math:`\Delta m_{\mathrm{sol}}^2 = 7.55\times 10^{-5} ~\mathrm{eV}^2` and :math:`|\Delta m_{\mathrm{atm}}^2| = 2.5 \times 10^{-3}~\mathrm{eV}^2`, respectively :cite:`deSalas:2017kay,Diaz:2019fwt`. This apparent splitting has been established in the measurement of multiple oscillation processes, including :math:`\nu_{\mu}\rightarrow\nu_e` accelerator neutrino appearance (LSND anomaly), :math:`\bar{\nu}_{e}\rightarrow\bar{\nu}_e` reactor neutrino disappearance (reactor anomaly) and the :math:`\nu_{e}\rightarrow\nu_e` disappearance of :math:`^{37}\,\mathrm{Ar}` and :math:`^{51}\,\mathrm{Cr}` electron capture decay neutrinos (gallium anomaly). Attempts have been made to fit the data to models with additional :math:`\mathrm{eV}`-scale neutrinos, e.g. (3+1) and (3+2) phenomenological models. While recent reactor experiments such as **DANSS** :cite:`Alekseev:2018efk` and **NEOS** :cite:`Ko:2016owz` have improved the statistical significance of an additional :math:`\mathrm{eV}`-scale sterile state, when combined with the :math:`\nu_e` appearance data of MiniBooNE they are in strong tension with the observed :math:`\nu_{\mu}\rightarrow\nu_{\mu}` accelerator neutrino disappearance of the MINOS, NOvA and IceCube experiments. In the context of the single-generation simplification of this work we interpret the mass squared splitting to be :math:`\Delta m_{41}^2 = m^2_{N}-m_{\nu}^2`. As we are focused on the electron-sterile coupling it is thus only the :math:`\nu_e\rightarrow\nu_e` and :math:`\bar{\nu}_e\rightarrow\bar{\nu}_e` experiments sensitive to :math:`\sin^2 2\theta_{ee} \approx 4|U_{eN}|^2` that are relevant. For sub-:math:`\mathrm{eV}` sterile neutrino masses the **Daya Bay** :cite:`An:2016luf`, **KamLAND** :cite:`Cirelli:2004cz` and upcoming **JUNO** :cite:`Berryman:2019nvr` experiments can probe the mixing down to :math:`|U_{eN}|^2 \lesssim 10^{-3}`. However it should be noted that if one wants to fit the solar and atmospheric mass splittings in a minimal (3+1) or (3+2) extension, solar data excludes the region :math:`10^{-9} ~\mathrm{eV}< m_{N}< 0.6~\mathrm{eV}` :cite:`Donini:2011jh,Donini:2012tt`. Below this region is the pseudo-Dirac scenario and above the mini-seesaw extending to the conventional high-scale seesaw. Light sterile neutrinos can be implemented in the context of an inverse seesaw as considered in Ref. :cite:`Barry:2011wb,Dev:2012bd,Abada:2017ieq`. In :numref:`Fig. %s ` and :numref:`Fig. %s ` we therefore start :math:`m_{N}` at the :math:`\mathrm{eV}`-scale. Above this the DANSS and NEOS experiments provide limits down to :math:`|U_{eN}|^2 \lesssim 10^{-2}` (as both exclusions are similar, :numref:`Fig. %s ` shows NEOS only) while the operating **PROSPECT** :cite:`Ashenfelter:2018iov` experiment provides constraints up to :math:`m_{N} = \sqrt{\Delta m_{41}^2+m^2_{\nu}}\sim 5\,\mathrm{eV}`. Over the same mass range Super-Kamiokande, IceCube and DeepCore (**SK+IC+DC**) provide complementary limits :cite:`Dentler:2018sju`. We note that the above limits are from oscillations conserving total lepton number. While it is in principle possible to observe LNV in oscillations, this requires new physics beyond sterile neutrinos such as right-handed currents :cite:`Bolton:2019wta`. Active-Sterile Neutrino Oscillations --------------------------------------- Any mixing between active and sterile neutrinos necessarily induces non-unitarity effects among the active neutrinos visible in CC and NC processes :cite:`Abada:2007ux,Fernandez-Martinez:2016lgt,Blennow:2016jkn`. This is most easily parametrised by a non-unitary light neutrino mixing matrix .. math:: \begin{align} U_\nu = (1 - \eta)\cdot U_\text{PMNS}, \end{align} where the matrix :math:`\eta` measures deviations from unitarity. The elements of :math:`\eta` are given in a generic seesaw model by :math:`\sqrt{2|\eta_{\ell\ell'}|} = \sum_{i}\sqrt{U_{\ell N_{i}}U^*_{\ell' N_{i}}}` and alter electroweak precision data (**EWPD**) observables. These include leptonic and hadronic measurements of the weak mixing angle :math:`s^2_W`, the :math:`W` boson mass :math:`m_{W}`, ratios of fermionic :math:`Z` boson decay rates :math:`R_{l}`, :math:`R_{c}`, :math:`R_{b}` and :math:`\sigma_{\mathrm{had}}^{0}`, the :math:`Z` invisible decay width :math:`\Gamma^{\mathrm{inv}}_{Z}` and ratios of leptonic weak decays testing EW universality :math:`R^{\pi}_{\ell\ell'}`, :math:`R^{W}_{\ell\ell'}`, :math:`R^{K}_{\ell\ell'}` and :math:`R^{l}_{\ell\ell'}`. Furthermore, by modifying :math:`G_{F}`, the non-unitarity of :math:`U_{\nu}` impacts the values of CKM mixing matrix elements extracted from experiments. Numerous weak decays have been used to pin down the CKM elements :math:`V_{ud}`, :math:`V_{us}`, :math:`V_{ub}` and the unitarity condition :math:`|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1`. Assuming a single sterile state coupling to just the first generation, all of these measurements enforce a constant bound of :math:`\sqrt{2|\eta_{ee}|} = |U_{eN}| < 0.050` for :math:`m_N \gtrsim 1\,\mathrm{GeV}` :cite:`delAguila:2008pw,Akhmedov:2013hec,deBlas:2013gla,Antusch:2014woa,Blennow:2016jkn,Flieger:2019eor`. Another indirect measurement of :math:`\eta_{\ell\ell'}` and hence different combinations of the active-sterile mixings comes from the non-observation of lepton flavour violating (LFV) processes :math:`\ell\rightarrow \ell'\gamma` and :math:`\mu-e` conversion in nuclei :cite:`Deppisch:2012vj`. Due to the different flavours of charged leptons involved in these processes, active-sterile mixings to at least two active generations are required. For the purpose of our single active generation picture we may convert the constraint on :math:`|U_{eN} U^*_{\mu N}|` obtained from the limits :math:`\mathrm{Br}(\mu\rightarrow e\gamma) < 4.2 \times 10^{-13}` :cite:`Tanabashi:2018oca` and :math:`R^{\mathrm{Ti}}_{\mu\rightarrow e}< 4.3\times 10^{-12}` :cite:`Dohmen:1993mp` to a constraint in the :math:`(m_{N},|U_{eN}|^2)` parameter space by assuming :math:`|U_{\mu N}| = |U_{eN}|`. We find :math:`|U_{eN}|^2 \lesssim 10^{-3}` for :math:`m_{N}\approx 10\,\mathrm{GeV}`, improving to :math:`|U_{eN}|^2 \lesssim 10^{-5}` for :math:`100~\mathrm{GeV} \lesssim m_{N}\lesssim 10~\mathrm{TeV}`. In making the assumption :math:`|U_{\mu N}| = |U_{eN}|` however, the constraints in the :math:`(m_{N},|U_{\mu N}|^2)` parameter space equally apply for :math:`|U_{e N}|^2`. For clarity and consistency we therefore do not show the LFV constraints in :numref:`Fig. %s `. Cosmological and Astrophysical Constraints ------------------------------------------- The presence of sterile states with masses :math:`m_{N}` and mixings :math:`|U_{\ell N}|^2` (and therefore predicted production rates, decay lengths and active-sterile oscillations) can have drastic consequences on early-universe observables, and have been explored extensively in the literature :cite:`Abazajian:2012ys`. These include the abundances of light nuclei formed during Big Bang Nucleosynthesis (BBN), temperature anisotropies in the Cosmic Microwave Background (CMB) radiation and the large-scale clustering of galaxies. Deviations from the standard smooth, isotropic background evolution and perturbations around this background impose severe constraints, especially for sterile states with masses :math:`m_{N}\lesssim 100\,\mathrm{MeV}`. The limits are however highly sensitive to the production and decay mechanism of the sterile state and can be relaxed in certain models. For the purpose of comparison we consider the following scenarios: Nucleosynthesis ^^^^^^^^^^^^^^^^ Sterile neutrinos with masses :math:`m_{N}\lesssim 1 ~\mathrm{GeV}` can be sufficiently long-lived to disrupt the standard formation of light nuclei :math:`^{4}\mathrm{He}`, :math:`\mathrm{D}`, :math:`^{3}\mathrm{He}` and :math:`^{7}\mathrm{Li}` during **BBN** :cite:`Boyarsky:2009ix,Ruchayskiy:2012si`. For larger masses the decay products from the accessible two-body and three-body decays have enough time to thermalise with the plasma. For decay times :math:`\tau\gtrsim 1` s occuring below :math:`T \lesssim 1\,\mathrm{MeV}`, i.e. roughly after :math:`\nu/N` -- :math:`e^{\pm}` decoupling and the onset of BBN, both the modified background expansion due to the presence of non-relativistic :math:`N` and the altered weak processes :math:`n+\nu\leftrightarrow p+e^{-}` and :math:`p+\bar{\nu}\leftrightarrow n+e^{+}` involving non-thermal decay product neutrinos lead to modified nuclei abundances. The condition :math:`\tau = \Gamma_{N}^{-1} \gtrsim 1` s naively translates to a lower limit of :math:`|U_{eN}|^2\gtrsim 10^{-11}\,(\mathrm{GeV}/m_{N})^5` for :math:`N\rightarrow 3\nu`, :math:`N\rightarrow\nu e^{+}e^{-}` and the sub-dominant radiative decay :math:`N\rightarrow \nu\gamma`. Above the pion mass threshold the already considerably less stringent constraints are made even weaker by including the decays :math:`N\rightarrow \nu\pi^{0}` and :math:`N\rightarrow e^{\pm}\pi^{\mp}`. Sterile Neutrino Decays ^^^^^^^^^^^^^^^^^^^^^^^^^ Sterile neutrinos decaying at later times (with :math:`\tau \lesssim t_{\mathrm{rec}} \approx 1.2\times 10^{13}` s) to non-thermally distributed active neutrinos can modify the amount of dark radiation measured (beyond the usual value including active neutrino oscillations, :math:`N_{\mathrm{eff}} \simeq 3.046`) at recombination, :math:`\Delta N_{\mathrm{eff}}`. Decays after recombination but before the current epoch (:math:`t_{\mathrm{rec}} \lesssim \tau \lesssim t_0 \approx 4.3\times 10^{17}` s) can also be important. Useful probes of these effects on the smooth, isotropic expansion history include the CMB shift parameter :math:`R_{\mathrm{CMB}}` (related to the position of the first acoustic peak in the CMB temperature power spectrum), the first peak of Baryon Acoustic Oscillation (BAO) sound waves imprinted on the large-scale distribution of galaxies and finally the value of the Hubble parameter :math:`H(z)` inferred from Type Ia supernova, BAO and Lyman-`\alpha` survey data. These exclude values of :math:`m_{N}` and :math:`|U_{eN}|^2` corresponding to lifetimes up to :math:`t_0`, where the condition that :math:`N` does not make up more than the observed matter density :math:`\Omega_{\mathrm{sterile}}<\Omega_{\mathrm{DM}} \approx 0.12 \,h^{-2}` and thus overcloses the Universe also applies. This constraint can naturally be evaded in exotic models :cite:`Bezrukov:2009th,Nemevsek:2012cd,El-Zant:2013nta,Biswas:2018iny`, for example those that inject additional entropy and dilute the dark matter (DM) energy density. We indicate the combined constraints from Ref. :cite:`Vincent:2014rja` in :numref:`Fig. %s ` as **CMB+BAO+** :math:`\mathbf{H_0}`. Sterile Neutrino as Dark Matter ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Sterile neutrinos with masses :math:`1 ~\mathrm{keV} \lesssim m_{N} \lesssim 100 ~\mathrm{keV}` can avoid the global constraints above if the active-sterile mixing is sufficiently small, i.e :math:`|U_{eN}|^2\lesssim 10^{-10} - 10^{-8}`. With lifetimes longer than the current age of the Universe these sterile states are viable DM candidates if efficiently produced :cite:`Abada:2014zra,Adhikari:2016bei,Abazajian:2017tcc`. Depending on the size of the lepton-antilepton asymmetry :math:`\eta_{L} \equiv n_{L}/n_{\gamma}`, population can occur either through resonant (:math:`\eta_{L} > 10^{6}\,\eta_{b}`) or non-resonant (:math:`\eta_{L} \approx 0`) active-sterile oscillations. The former (Shi-Fuller mechanism :cite:`Shi:1998km`) is independent of :math:`|U_{\ell N}|^2` while the latter (Dodelson-Widrow mechanism :cite:`Dodelson:1993je`) requires values of :math:`|U_{\ell N}|^2` now excluded by the global constraints. If DM is composed entirely of :math:`\mathrm{keV}` sterile neutrinos their fermionic nature limits the phase space density of DM-rich dwarf galaxies and imposes the Tremaine-Gunn bound, :math:`m_{N}\gtrsim 0.4\,\mathrm{keV}`. It is also possible to search for anomalous **X-ray** lines from the radiative decays :math:`N\rightarrow\nu\gamma` in the diffuse X-ray background and from DM-rich astrophysical objects. An intriguing signal at :math:`E \simeq 3.55\,\mathrm{keV}` implying a sterile neutrino with a mass of :math:`7.1\,\mathrm{keV}` has continued to persist in observations of stacked galaxy clusters :cite:`Bulbul:2014sua`, the Perseus galaxy cluster and Andromeda M31 galaxy :cite:`Boyarsky:2014jta` and the centre bulge of the Milky Way :cite:`Boyarsky:2014ska`. In :numref:`Fig. %s ` we include the most recent observations of M31 and the Milky Way by NuSTAR :cite:`Ng:2019gch,Roach:2019ctw`. In :numref:`Fig. %s ` we show the slightly improved future sensitivity of **ATHENA** :cite:`Neronov:2015kca`. These constraints assume :math:`\Omega_{\mathrm{DM}}=\Omega_{\mathrm{sterile}}`, but can be multiplied by :math:`\Omega_{\mathrm{sterile}}/\Omega_{\mathrm{DM}}` to account for other DM species :cite:`Vincent:2014rja`. Supernovae ^^^^^^^^^^^ Active-sterile mixings can be excluded for sterile neutrinos in the mass range :math:`10~\mathrm{eV} \lesssim m_{N}\lesssim 10~\mathrm{keV}` by examining their impact on Type II **Supernovae**. Active-sterile neutrino oscillations hinder the standard neutrino reheating of the reflected shock wave which becomes stalled in the first fraction of a second after the core bounce. For the explosion to proceed and additionally produce the observed SN1987A :math:`\bar{\nu}_e` signal of terrestrial detectors such as Kamioka :cite:`hirata:1987hu` and IMB :cite:`bionta:1987qt`, a certain region of the :math:`(m_{N},|U_{\ell N}|^2)` parameter space must be excluded. Refs. :cite:`Kainulainen:1990bn,Shi:1993ee,Nunokawa:1997ct,Hidaka:2006sg,Hidaka:2007se,Tamborra:2011is,Warren:2014qza` have studied in detail the resonant conversion :math:`\nu_{e}\rightarrow N` in the dense medium of collapsing stars and the necessary conditions to prevent impeding the supernova explosion. Refs. :cite:`Fuller:2009zz,Raffelt:2011nc,Arguelles:2016uwb` have similarly investigated :math:`\nu_{\mu,\tau}\rightarrow N` conversions for which the Mikheyev–Smirnov–Wolfenstein resonance conditions are different. An open question is whether the conditions for :math:`r`-process nucleosynthesis to produce heavy elements in the supernova outflows are met in these cases :cite:`Nunokawa:1997ct,Tamborra:2011is`. Lastly, sterile neutrinos that escape supernovae can subsequently decay radiatively via :math:`N\to \nu_e\gamma` and :math:`N\to \nu_e e^+ e^- \gamma`, producing an excess of gamma rays arriving soon after the detection of the :math:`\nu_e`. The non-observation of such an excess for SN1987A provides a stringent limit in the mass range :math:`1~\mathrm{MeV} \lesssim m_{N} \lesssim 30~\mathrm{MeV}` :cite:`Oberauer:1993yr`. Given the various assumptions and calculational differences of the constraints discussed we show for illustration in :numref:`Fig. %s ` the excluded region from Ref. :cite:`Shi:1993ee`. Active-Sterile Neutrino Oscillations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Sufficiently stable and light sterile neutrinos with masses :math:`m_{N}\lesssim 50\,\mathrm{eV}` can be produced with quasi-thermal temperatures before the decoupling of active neutrinos via active-sterile oscillations :cite:`Adhikari:2016bei,Gariazzo:2015rra,Asaka:2006nq`. While relativistic they contribute themselves towards the extra effective number of light fermionic degrees of freedom :math:`\Delta N_{\mathrm{eff}}`. Once becoming non-relativistic they contribute towards the matter density as :math:`\Omega_{\mathrm{sterile}}\,h^2 = (m_{\mathrm{eff}}^{\mathrm{sterile}}/94.1\,\mathrm{eV})` while also damping density perturbations below a mass-dependent free-streaming scale. The most simple case of a single sterile neutrino thermalising through oscillations at the active neutrino temperature has :math:`\Delta N_{\mathrm{eff}} = 1` and :math:`m_{\mathrm{eff}}^{\mathrm{sterile}} \simeq m_{N}` :cite:`Dolgov:2003sg,Cirelli:2004cz,Hannestad:2015tea` which is now likely excluded :cite:`Mirizzi:2013gnd`. The Planck collaboration has made fits of CMB (TT+lowP+lensing+BAO) data to the parameters (:math:`\sum m_{\nu}`, :math:`N_{\mathrm{eff}}`) and (:math:`m_{\mathrm{eff}}^{\mathrm{sterile}}`, :math:`\Delta N_{\mathrm{eff}}`) :cite:`Aghanim:2018eyx`. In Refs. :cite:`Vincent:2014rja` and :cite:`Bridle:2016isd` these constraints are mapped to the (:math:`\Delta m_{41}^2,\sin^2{2\theta_{ee}}`) parameter space which we use to plot the grey dot-dashed **CMB** constraints in :numref:`Fig. %s `.