Light Sterile Neutrinos: A White Paper
In memoriam
Ramaswami “Raju” S. Raghavan
1937 – 2011
Contents

I Theory and Motivation
 I.1 Introduction: What is a Sterile Neutrino?
 I.2 Theoretical Motivations and Symmetries Behind the Existence of Light Sterile Neutrinos
 I.3 The LowEnergy Seesaw and Minimal Models
 I.4 Sterile Neutrino Dark Matter
 I.5 Light Sterile Neutrinos as Messengers of New Physics
 I.6 NonStandard Neutrino Interactions (NSI)
 I.7 Extra Forces
 I.8 Lorentz Violation
 I.9 CPT Violation in Neutrino Oscillations and the Early Universe as an Alternative to Sterile Neutrinos
 II Astrophysical Evidence

III Evidence from Oscillation Experiments
 III.1 The LSND Signal
 III.2 The KARMEN Constraint
 III.3 Joint Analysis of LSND and KARMEN Data
 III.4 Sterile Neutrino Analysis of SuperK
 III.5 The MiniBooNE and Appearance Searches
 III.6 Disappearance Results from Accelerator Experiments
 III.7 The Gallium Anomaly
 III.8 The Reactor Antineutrino Anomaly
 III.9 Limit on Disappearance Derived from KARMEN and LSND Carbon Cross Sections
 III.10 Constraints from the MINOS LongBaseline Experiment
 III.11 Conclusion

IV Global Picture
 IV.1 3+1 Global Fit of ShortBaseline Neutrino Oscillation Data
 IV.2 3+1 and 3+2 Fits of ShortBaseline Experiments
 IV.3 Discussion of the LSND and MiniBooNE Results
 IV.4 Impact of Sterile Neutrinos for Absolute Neutrino Mass Measurments
 IV.5 Sterile Neutrinos and IceCube
 IV.6 Sterile Neutrinos and Dark Matter Searchs
 IV.7 Brief Summary
 V Requirements for Future Measurements

A Future Experiments
 A.1 LENSSterile^{23}^{23}23Proposed by the LENS Collaboration http://www.phys.vt.edu/k̃imballton/lens/public/collab/
 A.2 RICOCHET: Coherent Scattering and Oscillometry Measurements with Lowtemperature Bolometers^{26}^{26}26Proposed by K. Scholberg (Duke University), G. Karagiorgi, M. Shaevitz (Columbia University), M. Pyle (UC Berkeley), E. FigueroaFeliciano, J. Formaggio, J. Conrad, K. Palladino, J. Spitz, L. Winslow, A. Anderson, N. Guerrero (MIT)
 A.3 Very Short Baseline Oscillation Search with a Dual Metallic Ga Target at Baksan and a Cr Neutrino Source ^{28}^{28}28Proposed by: B. T. Cleveland, R. G. H. Robertson (University of Washington), H. Ejiri (Osaka University), S. R. Elliott (Los Alamos National Laboratory), V. N. Gavrin, V. V. Gorbachev, D. S. Gorbunov, E. P. Veretenkin, A. V. Kalikhov, T. V. Ibragimova (Institute for Nuclear Research of the Russian Academy of Sciences), W. Haxton (University of California, Berkeley), R. Henning, J. F. Wilkerson (University of North Carolina, Chapel Hill), V. A. Matveev (Joint Institute for Nuclear Research, Dubna) J. S. Nico (National Institute of Standards and Technology) A. Suzuki (Tohoku University)
 A.4 Proposed search of sterile neutrinos with the Borexino detector using neutrino and antineutrino sources^{30}^{30}30Proposed by the Borexino Collaboration (http://borex.lngs.infn.it).
 A.5 CeLAND: A proposed search for a fourth neutrino with a PBq antineutrino source^{32}^{32}32 Proposed by M. Cribier, M. Fechner, Th. Lasserre, A. Letourneau, D. Lhuillier, G. Mention, D. Franco, V. Kornoukhov, S. Schoenert, M. Vivier
 A.6 Search for Sterile Neutrinos with a Radioactive Source at Daya Bay^{35}^{35}35Proposed by: D.A. Dwyer, P. Vogel (Caltech), K. M. Heeger, B.R. Littlejohn (University of Wisconsin)
 A.7 SNO+Cr^{37}^{37}37Proposed by P. Huber and J.M. Link(Virginia Tech)
 A.8 Reactors with a small core^{40}^{40}40Contributed by Osamu Yasuda (Tokyo Metropolitan University)
 A.9 SCRAAM: A reactor experiment to rapidly probe the Reactor Antineutrino Anomaly
 A.10 Nucifer: a Small Detector for ShortDistance Reactor Electron Antineutrino Studies^{43}^{43}43 Proposed by J. Gaffiot, D. Lhuillier, Th. Lasserre, A. Cucoanes, A. Letourneau, G. Mention, M. Fallot, A. Porta, R. Granelli, L. Giot, F. Yermia, M. Cribier, J. L. Sida, M. Fechner, M. Vivier
 A.11 Stereo Experiment^{45}^{45}45Proposed by D. Lhuillier, A. Collin, M. Cribier, Th. Lasserre, A. Letourneau, G. Mention, J. Gaffiot (CEASaclay, DSM/Irfu), A. Cucoanes, F. Yermia (CNRSIN2P3, Subatech), D. Duchesneau, H. Pessard (LAPP, Universit de Savoie, CNRS/IN2P3).
 A.12 A Very ShortBaseline Study of Reactor Antineutrinos at the National Institute of Standards and Technology Center for Neutron Research^{47}^{47}47Proposed by H. P. Mumm (National Institute of Standards and Technology) and K. M. Heeger(University of Wisconsin)
 A.13 OscSNS: A Precision Neutrino Oscillation Experiment at the SNS^{49}^{49}49 Proposed by: S. Habib, I. Stancu (University of Alabama), M. Yeh (Brookhaven National Laboratory), R. Svoboda (University of California, Davis), B. Osmanov, H. Ray (University of Florida, Gainesville), R. Tayloe (Indiana University, Bloomington), G. T. Garvey, W. Huelsnitz, W. C. Louis, G. B. Mills, Z. Pavlovic, R. Van de Water, D. H. White (Los Alamos National Laboratory), R. Imlay (Louisiana State University), B. P. Roe (University of Michigan), M. Chen, (Oak Ridge National Laboratory), Y. Efremenko (University of Tennessee), F. T. Avignone (University of South Carolina), J. M. Link (Virginia Tech).
 A.14 LSND Reloaded^{51}^{51}51Proposed by Sanjib K. Agarwalla and Patrick Huber (Virginia Tech).
 A.15 Kaon DecayatRest for a Sterile Neutrino Search ^{53}^{53}53Proposed by J. Spitz (MIT)
 A.16 The MINOS+ Project^{55}^{55}55Proposed by the MINOS Collaboration (http://wwwnumi.fnal.gov/)
 A.17 The BooNE Proposal^{57}^{57}57Proposed by the BooNE Collaboration
 A.18 Search for anomalies with muon spectrometers and large LAr TPC imaging detectors at CERN^{59}^{59}59Proposed by the ICARUS + NESSiE Collaborations Antonello et al. (2012).
 A.19 Liquid Argon Time Projection Chambers^{61}^{61}61Proposed by the MicroBooNE Collaboration (http://wwwmicroboone.fnal.gov/public/collaboration.html).
 A.20 VeryLow Energy Neutrino Factory (VLENF)^{63}^{63}63Proposed by M. Ellis, P. Kyberd (Brunel University), C. M. Ankenbrandt, S. J. Brice, A. D. Bross, L. Coney, S. Geer, J. Kopp, N. V. Mokhov, J. G. Morfin, D. Neuffer, M. Popovic, T. Roberts, S. Striganov, G. P. Zeller (Fermi National Accelerator Laboratory), A. Blondel, A. Bravar (University of Geneva), R. Bayes, F. J. P. Soler (University of Glasgow), A. Dobbs, K. R. Long, J. Pasternak, E. Santos, M. O. Wascko (Imperial College, London), S. A. Bogacz (Jefferson Laboratory), J. B. Lagrange, Y. Mori (Kyoto University) A. P. T. Palounek (Los Alamos National Laboratory), A. de Gouv a (Northwestern University), Y. Kuno, A. Sato (Osaka University), C. D. Tunnell (University of Oxford), K. T. McDonald (Princton University), S. K. Agarwalla (Universitat de València), P. Huber, J. M. Link (Virginia Tech), and W. Winter (Universität Würzburg)
 A.21 Probing activesterile oscillations with the atmospheric neutrino signal in large iron/liquid argon detectors^{65}^{65}65Contributed by Pomita Ghoshal and Raj Gandhi.
Executive summary
This white paper addresses the hypothesis of sterile neutrinos based on recent anomalies observed in neutrino experiments. It is by no means certain that sterile neutrinos are responsible for the set of anomalies which have triggered the current effort, but the extraordinary consequence of such a possibility justifies a detailed assessment of status of the field. Decades of experimentation have produced a vast number of results in neutrino physics and astrophysics, some of which are in perfect agreement with only three active^{1}^{1}1Active neutrinos are those which couple to and bosons. neutrinos, while a small subset calls for physics beyond the standard model^{2}^{2}2Here, the standard model is to be understood to include massive neutrinos.. The first, and individually still most significant, piece pointing towards new physics is the LSND result, where electron antineutrinos were observed in a pure muon antineutrino beam. The most straightforward interpretation of the LSND result is antineutrino oscillation with a mass squared difference, , of about . Given that solar neutrino oscillations correspond to and atmospheric neutrino oscillations correspond to , the LSND requires a fourth neutrino. However, the results from the Large Electron Positron collider (LEP) at CERN on the invisible decay width of the boson show that there are only three neutrinos with a mass below one half of the mass of the boson, which couple to the boson, and therefore the fourth neutrino, if it indeed exists, can not couple to the boson and hence is a sterile neutrino, i.e. a Standard Model gauge singlet.
A new anomaly supporting the sterile neutrino hypothesis emerges from the recent reevaluations of reactor antineutrino fluxes, which find a 3% increased flux of antineutrinos relative to the previous calculations. At the same time, the experimental value for the neutron lifetime became significantly smaller, which in turn implies a larger inverse decay cross section. In combination with the previouslyneglected effects from longlived isotopes which do not reach equilibrium in a nuclear reactor, the overall expectation value for antineutrino events from nuclear reactors increased by roughly 6%. As a result, more than 30 years of data from reactor neutrino experiments, which formerly agreed well with the flux prediction, have become the observation of an apparent 6% deficit of electron antineutrinos. This is known as the reactor antineutrino anomaly and is compatible with sterile neutrinos having a .
Another hint consistent with sterile neutrinos comes from the source calibrations performed for radiochemical solar neutrino experiments based on gallium. In these calibrations very intense sources of Cr and Ar, which both decay via electron capture and emit monoenergetic electron neutrinos, were placed in proximity to the detector and the resulting event rate were measured. Both the source strength and reaction cross section are known with some precision and a 520% deficit of the measured to expected count rate was observed. Again, this result would find a natural explanation by a sterile neutrino with , which would allow some of the electron neutrinos from the source to “disappear” before they can interact. This anomaly persists even if one used the minimum cross section compatible with the precisely know Ga lifetime.
The aforementioned results suggesting a sterile neutrino with a mass around have to be contrasted with a number of results which clearly disfavor this interpretation. The strongest constraints derive from the nonobservation of muon neutrino disappearance by accelerator experiments like CDHSW or MINOS. Bounds on the disappearance of electron neutrinos are obtained from KARMEN and LSND, as well. The MiniBooNE neutrino result, a nonobservation of electron neutrino appearance in a muon neutrino beam, is incompatible with the LSND appearance result, if CP is conserved. On the other hand, the antineutrino result from the same experiment is fully compatible with the LSND result. A further difficulty in interpreting experimental evidence in support of a light sterile neutrino is that the effects are purely in count rates. The energy and distancedependent characteristic of the oscillation phenomena associated with sterile neutrinos remains to be observed.
The existing data from oscillation experiments, both for and against the sterile neutrino hypothesis, are summarized in Section III of this report, while Section IV discusses the compatibility of and tensions between the various oscillation data sets.
Cosmological data, mainly from observations of the cosmic microwave background and large scale structure favor the existence of a fourth light degreeoffreedom which could be a sterile neutrino. At the same time the standard cosmological evolution model prefers this neutrino to be lighter than . The relevant cosmological and astrophysical evidence is discussed in Section II.
From a theoretical point of view (see Section I) the existence of sterile neutrinos is a rather natural consequence of neutrinos having a nonzero mass. Sterile neutrinos are gauge singlets and as a result no a priori scale for their mass is set. Once neutrino mass generation via the seesaw mechanism is put into the wider context of grand unification and leptogenesis, light sterile neutrinos are slightly less natural. However, in this context, expectations for the mass of sterile neutrinos are ultimately based on an attempt to predict the eigenvalues of a 66 matrix in which we do not know any entries and therefore, having one or several sterile neutrinos in the 1 eV mass range is certainly far from being a surprise. The same is true for keV sterile neutrinos, which are warm dark matter candidates.
In summary, there are a number of experimental results that appear anomalous in the context of the standard 3 neutrino framework, and can be explained by a sterile neutrino with mass around 1 eV. At the same time, there are a number of results which are in conflict with this interpretation. The data collected to date present an incomplete, perhaps even contradictory picture, where 23 agreement in favor of and in contradiction to the existence of sterile neutrinos is present. The need thus arises to provide a more ardent and complete test of the sterile neutrino hypothesis, which will unambiguously confirm ir refute the interpretation of past experimental results. This white paper documents the currently available evidence for and against sterile neutrinos, it highlights the theoretical implications of their existence and provides an overview of possible future experiments (see the appendix) to test the LSND result and the sterile neutrino hypothesis. The overriding goal is to provide the motivation for a new round of measurements so that the questions laid out here can be definitively answered.
I Theory and Motivation
In this Chapter we provide an overview of the theoretical background of sterile neutrinos. We will set the stage by defining the term “sterile neutrino”, and will present model building aspects of these entities. Sterile neutrinos are naturally present in many theories beyond the standard model, in particular in several manifestations of the seesaw mechanism. We will be often interested in directly observable sterile neutrinos, and will categorize different possibilities for naturally accommodating them. More concretely, the socalled low energy seesaw will be analyzed in some detail. Particle physics aspects of keV scale neutrinos as warm dark matter candidates will be focussed on as well. Finally, for the sake of completeness, other nonstandard neutrino physics fields, often connected to sterile neutrinos or used as an alternative explanation to some of the current experimental hints, are summarized.
i.1 Introduction: What is a Sterile Neutrino?
A sterile neutrino is a neutral lepton with no ordinary weak interactions except those induced by mixing. They are present in most extensions of the standard model, and in principle can have any mass. Very heavy sterile neutrinos are utilized in the minimal type I seesaw model Minkowski (1977); GellMann et al. (1979); Mohapatra and Senjanovic (1980); Yanagida (1979); Schechter and Valle (1980) and play a pivotal role in leptogenesis Fukugita and Yanagida (1986); Davidson et al. (2008). However, here we are especially concerned with relatively light sterile neutrinos that mix significantly with ordinary neutrinos and those that are relevant to neutrino oscillation experiments, astrophysics, dark matter, etc. We first introduce the necessary terminology and formalism to discuss sterile neutrinos and their possible mixing with ordinary neutrinos, following the notation of Langacker (2010).
A massless fermion can be described by a Weyl spinor, which necessarily has two components of opposite chirality related (up to Dirac indices) by Hermitian conjugation. Chirality coincides with helicity for a massless particle and is associated with the chiral projections . Thus, a leftchiral Weyl spinor , which annihilates a leftchiral particle, is related to its rightchiral CP conjugate , which annihilates a rightchiral antiparticle, by
(1) 
In (1) is the charge conjugation matrix^{3}^{3}3 in the PauliDirac representation. and . exists even if CP is not conserved. One can just as well define a rightchiral Weyl spinor and its associated leftchiral antispinor . It is a matter of convention which is called the particle and which the antiparticle.
A Weyl neutrino is said to be active (or ordinary or doublet) if it participates in the conventional charged and neutral current weak interactions. This means that it transforms as a member of a doublet together with a charged lepton partner under the weak gauge group of the standard model. There are three light active neutrinos in nature, , , and , which are associated respectively with , , and in charged current transitions. Their CP conjugates, , are the weak partners of the . Including small neutrino masses, the (and ) are linear combinations of the mass eigenstates. The observed boson decay width implies that any additional active neutrinos are quite heavy, .
A sterile (or singlet or righthanded) neutrino is an singlet which does not take part in weak interactions except those induced by mixing with active neutrinos. It may participate in Yukawa interactions involving the Higgs boson or in interactions involving new physics. Many extensions of the standard model introduce a rightchiral sterile , which has a leftchiral CP conjugate .
Chirality can be a conserved quantum number for massless fermions, depending on the nature of their interactions. However, fermion mass terms violate chirality and also break its equivalence to helicity (which is not a Lorentz invariant quantity). For example, a weak transition involving, e.g., a massive but relativistic , may produce the “wrong helicity” state, with amplitude . The chirality violation is due to the fact that a fermion mass term describes a transition between right and left chiral Weyl spinors:
(2) 
where we have taken to be real and nonnegative by choosing an appropriate relative phase. There are two important types of fermion mass terms, Dirac and Majorana, depending on whether or not is distinct from , the CP conjugate of .
A Dirac neutrino mass term relates two distinct Weyl spinors:
(3) 
where, in the second form, is the Dirac field. (and its conjugate ) have four components, , and . One can define a global lepton number ( for and for ), which is conserved by . In most theories and in a Dirac mass term are respectively active and sterile. In this case, violates the third component of weak isospin by . can be generated by the vacuum expectation value (VEV) of the neutral component of a Higgs doublet, as illustrated in Figure 1, yielding
(4) 
where is the Yukawa coupling and GeV is the weak scale. This is analogous to the generation of quark and charged lepton masses in the standard model. In order to accommodate the observed neutrino masses, must be extremely small, e.g. for eV, as compared to for the electron. These small Dirac neutrino mass are often considered unnatural, or as evidence that some more subtle underlying mechanism is at play. For example, one may interpret the tiny neutrino Yukawa couplings as evidence that the elementary Yukawa couplings are required to vanish by some new discrete, global, or gauge symmetry. A small effective Yukawa coupling can then be generated by higherdimension operators Langacker (1998); ArkaniHamed et al. (2001) or by nonperturbative effects such as string instantons Cvetic and Langacker (2008); Blumenhagen et al. (2009). Another possibility is that a small Dirac coupling is due to wave function overlaps between and in theories involving large and/or warped extra dimensions Dienes et al. (1999); ArkaniHamed et al. (2002).
A Majorana mass term requires only one Weyl spinor. Majorana masses are forbidden by color and electromagnetic gauge invariance for quarks and charged leptons, but are possible for both active (after electroweak symmetry breaking) and sterile neutrinos if there is no conserved lepton number. For an active neutrino ,
(5) 
where is a (selfconjugate) twocomponent Majorana field. violates lepton number by two units, , and weak isospin by one unit, (hence one sometimes writes as , with for triplet). can be generated by a Higgs triplet, with a small Yukawa coupling and/or a small VEV , as illustrated in Figure 1. It could instead be associated with a higherdimensional operator (the Weinberg operator Weinberg (1980)) involving two Higgs doublets, with a coefficient . The scale represents some heavy new physics which has been integrated out, such as the exchange of a very heavy Majorana sterile neutrino (the type I seesaw) Minkowski (1977); GellMann et al. (1979); Mohapatra and Senjanovic (1980); Yanagida (1979); Schechter and Valle (1980), a heavy scalar triplet (the type II seesaw) Hambye et al. (2001), a fermion triplet (the type III seesaw) Foot et al. (1989), or new degrees of freedom in a string theory Langacker (2011). The second form in (5) emphasizes that also describes the creation or annihilation of two neutrinos. A sterile neutrinos can also have a Majorana mass term,
(6) 
where is a Majorana field. also violates lepton number by , but does not violate weak isospin ( denotes singlet, one often writes as ) and can in principle occur as a bare mass term. However, in some models a bare mass for the righthanded neutrinos is forbidden by new physics, and is instead generated by the VEV of, say, a SM Higgs singlet field , or by a higherdimension operator.
When Dirac and Majorana mass terms are both present, one must diagonalize the resulting mass matrix in order to identify the masseigenstates, which will, in general, be linear combinations of and . For one active neutrino and one sterile neutrino (the superscripts imply weak eigenstates)
(7) 
The mass matrix can be diagonalized by a unitary matrix ,
(8) 
because the matrix is symmetric. The mass eigenvalues can be taken to be real and positive by appropriate phase choices. The corresponding eigenvectors represent two Majorana mass eigenstates, , where
(9) 
There are a number of important special cases and limits of (7):

The pure Majorana case, . There is no mixing between the active and sterile states, and the sterile neutrino decouples (unless there are new interactions);

The pure Dirac case, . This leads to two degenerate Majorana neutrinos which can be combined to form a Dirac neutrino with a conserved lepton number;

The seesaw limit, . There is one, mainly sterile, state with , which decouples at low energy, and one light, mainly active, state, with mass . For , this yields an elegant explanation for why . The sterile state may be integrated out, leading to an effective higherdimensional operator for the active Majorana mass with ;

The pseudoDirac limit, , which leads to a small shift in the mass eigenvalues (we are taking the masses here to be real and positive for simplicity). Another possibility for the pseudoDirac limit occurs when ;

The activesterile mixed case, in which is comparable to and/or . The mass eigenstates contain significant admixtures of active and sterile.
The mass matrix in (7) can be generalized to 3 active and sterile neutrinos, where need not be 3. In case (a), and in cases (b) and (c) with , the observable spectrum consists of three active or essentially active neutrinos. The leftchiral components are related to the weak eigenstate fields by , where is the unitary PMNS matrix Pontecorvo (1968); Maki et al. (1962), and similarly for their CP conjugates .
In the case of singlet neutrinos, the full neutral fermion mass matrix takes the form
(10) 
where can, without loss of generality, be made diagonal , while is a generic complex matrix. This matrix is diagonalized by a unitary matrix . From now on we will neglect , unless otherwise noted. This scenario is the result of the most general renormalizable Lagrangian consistent with the standard model augmented by gaugesinglet (righthanded neutrino) fermion fields , written as
(11) 
For sterile neutrinos to be relevant to neutrino oscillations, as suggested by LSND and MiniBooNE, or for most astrophysical and cosmological implications, there has to be nonnegligible mixing between active and sterile states of the same chirality. That does not occur in the pure Majorana, pure Dirac, or very high energy seesaw limits, but only for the pseudoDirac and activesterile cases (of course, hybrid possibilities can occur for three families and ). In fact, the pseudoDirac case is essentially excluded unless eV. Otherwise, there would be significant oscillations of solar neutrinos into sterile states, contrary to observations de Gouvêa et al. (2009) (masssquared differences as small as eV could be probed with neutrino telescopes Beacom et al. (2004)). An elegant method to parameterize for pseudoDirac neutrinos can be found in Kobayashi and Lim (2001). Therefore, significant activesterile mixing requires that at least some Dirac masses () and some Majorana masses ( and/or ) are simultaneously very small but nonzero^{4}^{4}4More accurately, two distinct types of neutrino masses must be simultaneously small but nonzero. For example, there could be simultaneous Dirac mass terms between active and sterile neutrinos, as well as between sterile left and rightchiral neutrinos, as occurs in some extradimensional theories. See, e.g. Davoudiasl et al. (2002)., presenting an interesting challenge to theory. One possibility is that some new symmetry forbids all of the mass terms at the perturbative level, but allows, e.g. eV and eV due to higherdimensional operators (as in the miniseesaw). A list of models and some of the general aspects associated with building them will be discussed below.
In the case , the socalled seesaw limit, which is an excellent approximation for , it is instructive to express in terms of 4 submatrices,
(12) 
where is a matrix, is an matrix, are matrices, and we will refer to as the “active–sterile” mixing matrix. In the case of interest, the elements of are very similar to those of the standard neutrino mixing matrix, which are measured with different degrees of precision Nakamura et al. (2010). The elements of are parametrically smaller than those of . In the seesaw limit, is approximately unitary: . In the absence of interactions beyond those in Eq. (11), is not physical, while the elements of are proportional to those of . For more details on how to parameterize in the case at hand see, for example, de Gouvêa and Jenkins (2008); de Gouvêa et al. (2011); Asaka et al. (2011); Blennow and FernandezMartinez (2011).
Eq. (10) is expressed in the socalled flavor basis: . This is related to the neutrino mass basis – – via the neutrino unitary mixing matrix :
(13) 
, such that
(14) 
where are the neutrino masses.
In the seesaw limit, there are 3 light, mostly active neutrinos with mass matrix and heavy, mostly sterile, neutrinos with mass matrix . The distribution of masses is parametrically bimodal: . It is easy to show (see, for example, de Gouvêa (2005)) that
(15) 
Our ability to observe the mostly sterile states is proportional to . Given eV, the mostly sterile states are very weakly coupled unless eV and . This generic behavior of the active–sterile mixing angles is also depicted in Fig. 2. In general, in a theory with sterile neutrinos, the leptonic neutral current is nondiagonal in mass eigenstates Schechter and Valle (1980); Lee and Shrock (1977). This is a special case of a theorem Lee and Shrock (1977) that the necessary and sufficient condition that the leptonic neutral current is diagonal in mass eigenstates is that all leptons of the same chirality and charge must have the same weak total and third component isospin. This condition is violated in a theory with sterile neutrinos.
In the context of the sterile neutrino parameters we are dealing with in this paper, there are two important statements one can make from Eq. (15): if there is a sterile neutrino of mass around eV and mixing around 0.1, then its contribution to the active neutrino masses is of order eV, and can lead to interesting effects, see e.g. Smirnov and Zukanovich Funchal (2006) for a detailed analysis. In case a sterile neutrino with mass around keV and mixing around exists, then its contribution to the active neutrino masses is of order eV, and completely negligible. This also implies that there is a massless active neutrinos in this case.
An example of a theory with sterile neutrinos and the seesaw mechanism is the grand unified theory (GUT), in which a sterile neutrino is the singlet member of the 16dimensional spinor representation for each SM generation, which has decomposition . In the original GUT, the number of sterile neutrinos, , is thus equal to the number of SM fermion generations, although in other models, may differ from (there have been, and continue to be searches for quarks and leptons in a possible fourth SM generation, but since these are negative so far, we take ). In an analogous manner, sterile neutrinos appear naturally as the spinor component of the corresponding chiral superfields in supersymmetric GUTs. At the simplest level, in the theory, SM fermion mass terms involve the bilinear , with the ClebschGordon decomposition and arise from Yukawa couplings to corresponding Higgs fields transforming as 10, 120, and 126 dimensional representations. Dirac masses arise from terms transforming according to each of these representations, while Majorana mass terms for the sterile neutrinos occur in the terms transforming according to the 126dimensional representation. Obviously, another aspect of the seesaw mechanism which makes it even more appealing is the remarkable fact that the decays of the righthanded neutrinos in the early Universe can provide an explanation of the baryon asymmetry of the Universe, in the leptogenesis mechanism Fukugita and Yanagida (1986).
i.2 Theoretical Motivations and Symmetries Behind the Existence of Light Sterile Neutrinos
In this Section the theoretical aspects entering models of sterile neutrinos will be discussed. It is important to stress that none of these imply that light sterile neutrinos must exist, and none specify the sterile neutrino mass scale. Most models allow for the possibility of light, mostly sterile states, and can accommodate a subset of the shortbaseline anomalies and/or provide a (warm) dark matter candidate. This does not imply, of course, that sterile neutrinos do not exist. Recall that large mixing in the lepton sector, now a well established fact, was not predicted by theorists.
Sterile neutrinos in “standard approaches”
We have seen that sterile neutrinos are a natural ingredient of the most popular and appealing mechanism to generate neutrino masses, the conventional or type I seesaw mechanism. No equally appealing alternative has emerged in the thirty years since it was discovered. In what follows, we will attempt to give a flavor of the required model building ingredients that can be introduced in order to generate or accommodate (light) sterile neutrinos in the mass range of eV or keV. Note that these scales are not far away from each other, and the same techniques apply for both cases.

The split seesaw mechanism
The split seesaw model Kusenko et al. (2010) consists of the Standard Model with three righthanded neutrinos and a spontaneously broken gauge symmetry. It is also assumed that spacetime is a fivedimensional space compactified to four dimensions at some scale GeV. If one of the righthanded neutrinos is localized on a brane separated from the standardmodel brane by a distance , then the wave function overlap between this righthanded neutrino and the other fields is very small. This leads to a suppression of both the Yukawa coupling and the sterile neutrino mass in the lowenergy effective fourdimensional theory, but the successful seesaw formula is preserved. This opens a possibility for one of the righthanded, or sterile, neutrinos to serve as the darkmatter particle, and the observed abundance can be achieved for some values of parameters.
More specifically, in this model, one can express the effective mass and the Yukawa coupling of the sterile neutrino in terms of the fundamental (5dimensional) Planck mass , the distance between the branes, and the righthanded neutrino bulk mass :
(16) where and are couplings in the underlying fivedimensional theory. Clearly, the exponential suppression in is the square of the term which appears in , so the ratio is unchanged, while both the Yukawa coupling and the mass are much smaller than or . The idea has been extended with an flavor symmetry in Adulpravitchai and Takahashi (2011).
The split seesaw makes vastly different scales equally likely, because a small difference in the bulk masses or a small change in the distance between the two branes can lead to exponentially different results for the sterile neutrino mass.^{5}^{5}5It is also possible to have multiple light sterile neutrinos because small changes in the bulk masses can result in big changes in and . In particular, it is possible to reproduce the mass spectrum of the model MSM Asaka et al. (2005); Boyarsky et al. (2009a), in which the lightest darkmatter sterile neutrino is accompanied by two GeV sterile neutrinos which are nearly degenerate in mass. This provides a strong motivation for considering the sterile neutrino as a dark matter candidate. Indeed, the split seesaw model provides at least two ways of generating the correct darkmatter abundance Kusenko et al. (2010). Furthermore, each of these production scenarios generates dark matter that is substantially colder than the warm dark matter produced in neutrino oscillations Dodelson and Widrow (1994); Shi and Fuller (1999); Laine and Shaposhnikov (2008). The reason this dark matter is cold enough to agree with the structure formation constraints is simple: sterile neutrinos produced out of equilibrium at temperatures above the electroweak scale suffer redshifting when the standard model degrees of freedom go out of equilibrium and entropy is produced Kusenko (2006).
The emergence of a sterile neutrino as a darkmatter particle gives a new meaning to the fact that the standard model has three generations of fermions. Three generations of fermions allow for CP violation in the CabibboKobayashiMaskawa matrix, but, quantitatively, this CP violation is too small to play any role in generating the matterantimatter asymmetry of the Universe. In the context of leptogenesis, CP violation in the neutrino mass matrix would have been possible even with two generations, thanks to the Majorana phase. Hence, the existence of three families of leptons, apparently, has no purpose in the standard model, except for anomaly cancellation. However, if the lightest sterile neutrino is the darkmatter particle (responsible for the formation of galaxies from primordial fluctuations), while the two heavier sterile neutrinos are responsible for generating the matterantimatter asymmetry via leptogenesis, the existence of three lepton families acquires a new, significant meaning.

Symmetries leading to a vanishing mass
An appealing way to have a strong hierarchy in the neutrino masses is to introduce a symmetry that implies one vanishing sterile neutrino mass. A slight breaking of this symmetry will lead to a naturally small mass compared to those allowed by the symmetry. A very popular Ansatz is the flavor symmetry Petcov (1982).
The first suggestion to apply the symmetry in the context of keV sterile neutrinos was made in Shaposhnikov (2007a), and a concrete model was presented later on in Ref. Lindner et al. (2011). The basic idea is the following: the symmetry leads to a very characteristic mass pattern for active neutrinos, namely to one neutrino being exactly massless, while the other two are exactly degenerate, denoted by . Applying the symmetry to three righthanded neutrinos, results in an analogous pattern for the heavy neutrino masses.
The flavor symmetry must be broken, which can be conveniently parameterized by soft breaking terms of the order of a new mass scale . Any explicit or spontaneous symmetry breaking will result in terms of that form, and their general effect is to lift the degeneracies and to make the massless state massive, , cf. left panel of Fig. 3. A similar effect is observed for the light neutrinos. Since the symmetry breaking scale must be smaller than the symmetry preserving scale (we would not speak of a symmetry otherwise), this mechanism gives a motivation for or eV, while or heavier. Note, however, that symmetry tends to predict bimaximal mixing, which is incompatible with data Schwetz et al. (2011a). This problem can be circumvented by taking into account the mixing from the charged lepton sector Lindner et al. (2011); Petcov and Rodejohann (2005). In this case, an additional bonus is the prediction of a nonzero value of the small mixing angle , in accordance with recent experimental indications Ahn et al. (2012); Abe et al. (2011a); An et al. (2012).
Figure 3: The mass shifting schemes for the and the FroggattNielsen models (figures taken from Ref. Merle and Niro (2011)). The potential of to accommodate eVscale neutrinos within the type I seesaw was noted earlier in Mohapatra (2001). Suppose we have a symmetry in the theory that leads to det with only one of the eigenvalues zero. If the same symmetry also guarantees that det, then one can “take out” the zero mass eigenstates from both and and then use the seesaw formula for the subsystem.^{6}^{6}6The case of vanishing determinant of is called singular seesaw Glashow (1991), and leads to two light, two intermediate and two heavy masses Chun et al. (1998); Chikira et al. (2000); Liu and Song (1999); McKellar et al. (2001); McDonald et al. (2004); McDonald and McKellar (2007). The spectrum will consist of two light Majorana neutrinos (predominantly lefthanded), two heavy righthanded Majorana neutrinos; one massless righthanded neutrino, which will play the role of the sterile neutrino of the model, and a massless lefthanded neutrino. Breaking the symmetry very weakly either by loop effects or by higher dimensional operators can lead to a nonzero mass for the light sterile neutrino as well as its mixing with the , so that one can have oscillations between sterile and active states. The simplest realization makes use of an times a symmetry for the righthanded sector, and yields Mohapatra (2001)
(17) It is easy to see that in this case the two massless neutrino states are: and , where we are using the symbol for the SM singlet RH neutrino states. The second of the above states is a sterile state and can be identified as . Excluding these two modes, the mass matrix reduces to a seesaw form and one gets an inverted mass hierarchy for the active neutrino with being the lightest active state. Note that a similar method for generating a suitable mass pattern is to use discrete instead of continuous symmetries, which do not automatically suffer from problems with Goldstone bosons. One example from the literature is Araki and Li (2011), where the authors have used a symmetry to forbid a leading order mass for one of the sterile neutrinos.

The FroggattNielsen mechanism
Another idea to associate small righthanded neutrino masses to ultrahigh new physics scales is to use the FroggattNielsen (FN) mechanism Froggatt and Nielsen (1979). The FN mechanism introduces an implicit highenergy sector of scalars and fermions, which are all suitably charged under a new global symmetry. Although this has the drawback of lacking an explicit UVcompletion Mavromatos (2011), it has the big advantage of leading to a very strong (exponential) generationdependent suppression of mass eigenvalues, and thus to strong hierarchies, cf. right panel of Fig. 3. This is the reason why this mechanism is so popular when it comes to explaining the pattern of quark masses and mixing, and it is equally well applicable to the problem of generating a keV or eV scale for righthanded neutrinos.
Recent applications of the FN mechanism to sterile states were presented in Merle and Niro (2011); Barry et al. (2011a, b), the latter work combining it with an explicit flavor symmetry in order to generate (closeto) tribimaximal mixing for the neutrinos. Effects of the leading and nexttoleading order breaking terms have been carefully analyzed and it was shown that, e.g. the strong constraints on the activesterile mixing as well as a nonzero mixing angle are consistent with the model. Corrections to the leading order seesaw formula Grimus and Lavoura (2000); Hettmansperger et al. (2011) have also been addressed. In addition, many constraints arising from different requirements, such as anomaly cancellation, embedding into Grand Unified Theories, or bounds on lepton flavor violation have been taken into account Merle and Niro (2011). The combination of all of these requirements may render the FN models predictive: For example, they would be incompatible with the leftright symmetry used for keV neutrino DM production in Bezrukov et al. (2010). Finally, in Barry et al. (2011b) the FN charges of all 3 righthanded neutrinos were varied such that several different results could be achieved: two super heavy neutrinos for leptogenesis and one for keV warm dark matter; one eVscale neutrino for the reactor anomaly/LSND and one for warm dark matter, etc.
Similar to the split seesaw case, the FN models presented accommodate only slightly modified versions of the high scale seesaw mechanism, even though keV or eV scale righthanded neutrinos are involved. The reason for this is that the charges of the righthanded neutrinos under any global – be lepton number or the – drop out of the seesaw formula: if a righthanded neutrino has charge under the , then its mass is suppressed by , where is the ratio of the FN field VEV and the cutoff scale of the theory. The respective column of is suppressed by , hence the seesaw formula is constant.

Extended seesaw mechanisms
Other approaches to the problem extend the seesaw mechanism. One possibility is similar to the “take out zero mass eigenstates” idea discussed above, and was put forward in Mohapatra et al. (2005). Considering the gauge group and various new weak singlets leads to an extended (inverse) seesaw mass matrix Mohapatra (1986); Mohapatra and Valle (1986)
(18) where
(19) The matrix has an arbitrary (symmetric) form. Decoupling first the fields associated with gives
(20) The determinant of the mass matrix vanishes and the zero mass eigenstate is given by . We “take out” this state and the remaining righthanded neutrino mass matrix is involving only the states . Similar to the discussion above, the seesaw matrix is now a two generation matrix yielding two light Majorana neutrino states. The other active neutrino state is massless, together with .
Another model is given in Ref. Fong et al. (2011), where an attempt was made to understand the tiny Majorana mass required in the inverse seesaw mechanism from warped extradimensional models. Parity anomaly cancellation requires the total number of bulk fermions that couple to gauge and gravity fields to be even, thus a fourth singlet fermion was needed. Three of the singlet fields pair up with the three righthanded Majorana neutrino fields to make the three pseudoDirac fermions of inverse seesaw, and the remaining singlet in the end becomes the sterile neutrino with mass in the eV range.
A related approach, first discussed in Chun et al. (1996) and studied in detail in Zhang (2011), introduces in addition to the three generations of lefthanded active neutrinos and three generations of righthanded sterile neutrinos , another singlet righthanded field . This field carries a nontrivial charge under a new discrete auxiliary symmetry, different from the charges of the . The effect of these chirality and charge assignments is to forbid any direct Dirac or Majorana masses for the field , which only obtains a mass by coupling to one generation of ordinary righthanded neutrinos, resulting in a mixed mass term of size . The full mass matrix is
(21) With and , one blockdiagonalizes this matrix and finds active neutrinos roughly of order and a sterile neutrino of order . In such a framework, both active and sterile neutrino masses are suppressed via the seesaw mechanism, and thus an eV scale sterile neutrino together with sizable activesterile mixing is accommodated without the need of inserting small mass scales or Yukawa couplings.
Very often the mass matrix for the active plus sterile neutrinos obeys an approximate – symmetry, which naturally generates small and closeto maximal . For instance Mohapatra et al. (2005),
(22) 
Due to small mixing with the active states the are typically smaller than the , and one can diagonalize the mass matrix with
(23) 
where and .
There are other related approaches capable of “explaining” or at least accommodating light sterile sterile neutrinos. For example, a general effective approach is possible: in many flavor symmetry models to generate the unusual lepton mixing scheme (see Refs. Altarelli and Feruglio (2010); Ishimori et al. (2010) for recent reviews) the mass terms are effective. Simply adding a sterile state to the particle content, and making it a singlet under the flavor symmetry group as well as the SM group, does the job Barry et al. (2011a). Other models which accommodate light sterile neutrinos include Berezhiani and Mohapatra (1995); Ma (1996); Langacker (1998); Bando and Yoshioka (1998); ArkaniHamed and Grossman (1999); Shafi and Tavartkiladze (1999); Babu and Seidl (2004); Sayre et al. (2005); Chen et al. (2007a); Barger et al. (2011a); Ghosh et al. (2011); McDonald (2011); Araki and Li (2011); Chen and Takahashi (2011); Geng and Takahashi (2012).
Sterile neutrinos in “nonstandard approaches”
A variety of other ways to explain light sterile neutrinos exists, for instance mirror models. Starting from the famous Lee and Yang parity violation paper Lee and Yang (1956) attempts were made to generalize the concepts of mirror symmetry and parity by assuming the existence of mirror images of our particles. According to Lee and Yang (1956), these mirror particles were supposed to have strong and electromagnetic interactions with our particles. For a few years it seemed plausible that the role of these mirror particles is played by antiparticles of ordinary matter and that the true mirror symmetry is the CP symmetry. But in 1964 violation of CP symmetry was discovered. Then Kobzarev, Okun and Pomeranchuk Kobzarev et al. (1966) returned to the idea of mirror particles, which are different from the ordinary matter, and came to the conclusion that mirror particles cannot have strong and electromagnetic interaction with ordinary matter and hence could appear as dark matter. This “dark matter mirror model” was discussed by many authors, for instance Okun and Rubbia (1967); Pavsic (1974); Blinnikov and Khlopov (1982); Foot et al. (1991, 1992); Silagadze (1997). For a review of some two hundred papers see Okun (2007).
Mirror model were invoked to understand neutrino puzzles in Foot and Volkas (1995a); Berezhiani and Mohapatra (1995); Berezinsky et al. (2003) after the LSND results were announced. In this picture, the mirror sector of the model has three new neutrinos which do not couple to the boson and would therefore not have been seen at LEP, even if these are light. We will refer to the as sterile neutrinos of which we now have three. The lightness of is dictated by the mirror symmetry in a manner parallel to what happens in the standard model. The masses of the mirror (sterile) neutrinos could arise for example from a mirror analog of the seesaw mechanism.
The two “Universes” communicate only via gravity or other forces that are very weakly coupled or associated to very heavy intermediate states. This leads to a mixing between the neutrinos of the two Universes and can cause neutrino oscillation between, say, the of our Universe to the of the parallel one in order to explain the LSND results without disturbing the three neutrino oscillation picture that explains the solar and the atmospheric data.
Such a picture appears quite natural in superstring theories which lead to gauge theories below the Planck scale, where both ’s living on two separate branes are connected by gravity.
There are two classes of mirror models: (i) after symmetry breaking, the breaking scales in the visible sector are different from those in the mirror sector (this is called asymmetric mirror model); or (ii) all the scales are the same (symmetric mirror model). If all scales in both sectors are identical and the neutrinos mix, this scenario is a priori in contradiction with solar and atmospheric data since it leads to maximal mixing between active and sterile neutrinos. In the asymmetric mirror model, one can either have different weak scales or different seesaw scales or both. Either way, these would yield different neutrino spectra in the two sectors, along with different mixing between the two sectors.
As far as neutrino masses and mixings go, they can arise from a seesaw mechanism, with the mixings between sterile and active neutrinos given by higher dimensional operators of the form after both electroweak symmetries are broken. The mixing between the two sectors could arise from a mixing between RH neutrinos in the two sectors i.e., operators of the form In this case the activesterile neutrino mixing angles are of order implying that if this is the dominant mechanism for neutrino masses, the mirror weak scale should be within a factor 1030 of the known weak scale. In this case, all charged fermion masses would be scaled up by a common factor of .
Regarding warm dark matter, in the mirror model there is no need to use oscillation to generate dark matter keV steriles. Here the inflaton reheating produces primordial sterile neutrinos. However, due to asymmetric inflation, the density of the keV steriles is down by a factor , where () is the reheating temperature of our (the mirror) sector. One has
(24) 
To get , with , one needs keV. Note that this is regardless of whether the active and sterile neutrinos mix at all. Since the Xray constraints depend on the activesterile mixing angles, the mirror model for warm dark matter need not be seriously constrained by Xray data. In this picture, one of the two other s could be in the eV range to explain LSND if needed.
The bottom line here is that in this case, there are three sterile
neutrinos with arbitrary masses and mixings. For example, one could
accommodate a 3+2 solution to the LSND puzzle as well as a keV warm dark
matter.
A completely different sterile “neutrino” candidate is the axino. It has been argued that this particle, arising from the supersymmetric version of the axion solution to the strong CP problem, is a natural candidate for a sterile neutrino in the framework of gauge mediated supersymmetry breaking.
A longstanding puzzle in the standard model is the smallness of the QCD vacuum angle which appears in the Lagrangian
(25) 
where is the QCD gauge field strength. The current upper limits of the neutron electric dipole moment severely constrain , which is a priori a free parameter. The apparently arbitrary “smallness” of is referred to as the strong CP problem Kim and Carosi (2010). An elegant solution is to promote the parameter to a dynamical degree of freedom, the axion , which is a pseudoGoldstone boson of the PecceiQuinn (PQ) symmetry. The QCD potential of the axion sets its vacuum expectation value to zero; , and thus the strong CP problem is solved dynamically. Here is the scale of the PQ symmetry breaking which is constrained to be Kim and Carosi (2010).
In supersymmetric theories, the axion is accompanied by its fermionic partner, the axino , which can remain very light due to its quasi Goldstone fermion character Chun et al. (1996). In the following we argue that the axino can naturally be a sterile neutrino in gauge mediated supersymmetry breaking models Chun (1999); Choi et al. (2001) implementing the PQ symmetry through the KimNilles mechanism Kim and Nilles (1984); Chun et al. (1992a).
We first discuss how the axino can be as light as . As the superpartner of a Goldstone boson, the axino is massless as long as supersymmetry is unbroken. When the supersymmetry breaking scale is higher than the PQ symmetry breaking scale as in gravitymediated supersymmetry breaking, the axino is expected to have generically a supersymmetry breaking mass, that is the gravitino mass Chun et al. (1992b); Chun and Lukas (1995):
(26) 
However, if the PQ symmetry breaking occurs before supersymmetry breaking as in gauge mediated supersymmetry breaking, the axino mass is suppressed by the high PQ scale and thus can remain very light even after supersymmetry breaking. In the effective Lagrangian below the PQ scale, the axion (axino) appears in combination of () due to its Goldstone nature. This implies that the axino would get a mass of order
(27) 
where is a supersymmetry breaking scale Chun (1999). In the framework of gauge mediated supersymmetry breaking Giudice and Rattazzi (1999), a hidden sector field breaking supersymmetry can be charged under the PQ symmetry. In this case, is the hidden sector supersymmetry breaking scale. Taking roughly GeV, one gets eV for GeV. Otherwise, the axino mass would get a generic contribution from the supersymmetry breaking scale GeV of the Supersymmetric Standard Model (SSM) sector: eV for GeV. There exists also a supergravity contribution
(28) 
where with GeV and GeV. This will be the main contribution to the axino mass in the scenario Choi et al. (2001). If the axino is light enough, appropriate mixing with active neutrinos and flavor structure can be constructed Choi et al. (1999) and hence the supersymmetric axion solution to the strong CP problem can provide a candidate of a sterile neutrino.
i.3 The LowEnergy Seesaw and Minimal Models
General Aspects
Let us focus in more detail on the low energy seesaw. Recall the most general renormalizable Lagrangian
(29) 
where is the standard model Lagrangian in the absence of gauge singlet fermions, are the neutrino Yukawa couplings, and are the righthanded neutrino Majorana mass parameters. Eq. (29) is expressed in the weak basis where the Majorana mass matrix for the righthanded neutrinos is diagonal.
The seesaw formula allows the mass of singlet neutrinos to be a free parameter: Multiplying by any number and by does not change the righthand side of the formula. Therefore, the choice of is a matter of theoretical prejudice that cannot be fixed by activeneutrino experiments alone. A possible approach is to choose these parameters so that they explain certain phenomena and aspects beyondthestandard model, for example, provide a dark matter candidate or a mechanism of baryogenesis. The most often considered standard approach takes Yukawa couplings and the Majorana masses in the range GeV. Models with this choice of parameters give rise to baryogenesis through leptogenesis Fukugita and Yanagida (1986). For a review of the GUTscale seesaw and the thermal leptogenesis scenario associated with it see e.g. Davidson et al. (2008). Here we would like to focus on variants at lower energy scales.
Figure 4 summarizes various choices of combination of mass/Yukawa couplings of sterile neutrinos in seesaw models. The right panel summarizes properties of resulting seesaw models, their ability to solve various beyondtheSM problems and anomalies, and their testability.
The main generic prediction of Eq. (29) is the existence of Majorana neutrinos, most of them massive. All of these “contain” the three active neutrino flavors and hence can, in principle, be observed experimentally. One exception is the case . In this case, the massive Majorana neutrinos ‘‘pair up’’ into at most three massive Dirac fermions.^{7}^{7}7In the case , there are two massive Dirac neutrinos and one massless neutrino. In the case , there are three massive Dirac neutrinos and massless gauge singlet, bona fide sterile, neutrinos that do not mix with any of the active states and are completely unobservable. The neutrino data can determine all physically observable values of – the neutrino masses and the elements of the neutrino mixing matrix, three angles and one CPodd Dirac phase. Qualitatively, the neutrino data require eV to eV. The case , as far as observations are concerned, is similar to the Dirac neutrino case. Here the neutrinos are Majorana fermions but they still “pair up” into pseudoDirac fermions. These can be distinguished from Dirac fermions if one is sensitive to the tiny mass splitting between the components of the pseudoDirac pair, which are proportional to . Solar neutrino data require eV de Gouvêa et al. (2009), see also Donini et al. (2011).
One can also deduce an upper bound for . Theoretical considerations allow one to rule out GeV Maltoni et al. (2001a), while a simple interpretation of the gauge hierarchy problem leads one to favor GeV, assuming there are no other new states at or above the electroweak symmetry breaking scale Casas et al. (2004). Naturalness is not a good guide when it comes to picking a value for – all values for are technically natural in the sense that in the limit where all vanish the nonanomalous global symmetries of Eq. (11) are augmented by , global baryon number minus lepton number (see, for example, de Gouvêa (2005)). Finally, direct experimental probes of Eq. (11) are possible for values below the TeV scale – the reach of collider experiments. We argue, however, that Eq. (29) can only be unambiguously tested for values less than, very roughly, 10 eV, mostly via searches for the effects of light sterile neutrinos in neutrino oscillations.
A few important points are worthy of note. Eq. (15), describing the mixing between active and sterile states, can be severely violated and much larger values of are possible (see, for example, de Gouvêa (2007)). These are, however, not generic and require special choices for the entries in the neutrino mass matrix. This implies that, under extraordinary circumstances, one may be able to observe notsolight seesaw sterile neutrinos for, say, MeV or GeV, but it is not possible to falsify Eq. (29) if all are much larger than 10 eV. On the flip side, the structure of the active–sterile mixing matrix is not generic. This means that Eq. (29) can be ruled out (or ruled “in”) if enough information concerning hypothetical sterile neutrinos is experimentally collected. For all the details see de Gouvêa and Huang (2011).
eVscale Seesaw
Here we will focus on the case that the seesaw scale corresponds to the scale of light sterile neutrinos responsible for various short baseline anomalies.
These light sterile neutrinos should be observable in shortbaseline neutrino oscillation experiments sensitive to disappearance at the few percent level or appearance at the level de Gouvêa and Huang (2011). It is interesting that these can also accommodate and solutions (see, for example, Kopp et al. (2011); Giunti and Laveder (2011a)) to the socalled short baseline neutrino anomalies from LSND Aguilar et al. (2001), MiniBooNE AguilarArevalo et al. (2007, 2009a, 2010a) and reactor data Mention et al. (2011). If that is indeed the case, Eq. (11) further predicts that appearance at short baselines is just beyond the current experimental upper bounds de Gouvêa and Huang (2011).
Nonoscillation experiments sensitive to the low energy seesaw include searches for doublebeta decay de Gouvêa (2005); de Gouvea et al. (2007), precision measurements of the ray spectrum in nuclear decay de Gouvêa (2005); de Gouvea et al. (2007); Barrett and Formaggio (2011), and cosmological bounds on the number of relativistic particle species in the early Universe and bounds on the fraction of hot dark matter de Gouvêa (2005); de Gouvea et al. (2007); Hamann et al. (2010a, 2011). An interesting consequence of the lowenergy seesaw (all masses well below 100 MeV) is that while the neutrinos are Majorana fermions, the rate for neutrinoless doublebeta decay and all other potentially observable searches for leptonnumber violation, vanishes de Gouvêa (2005); de Gouvea et al. (2007). It is also possible that there is partial cancellation: this means that the contribution to neutrinoless doublebeta decay of a light sterile neutrino, which generates an active neutrino via seesaw, cancels the contribution to doublebeta decay from this active state Barry et al. (2011b). The observation of a finite lifetime for neutrinoless doublebeta decay would rule out Eq. (29) unless at least one of the righthanded neutrino mass parameters is above a few tens of MeV. A recent reanalysis of sterile neutrino effects in double beta decay can be found in Mitra et al. (2012). This is assuming that no other mechanisms Rodejohann (2011) that can lead to double beta decay is realized in nature.
If a global symmetry (e.g. lepton number) is imposed on the Lagrangian (29), the general matrices , as well as the in general nondiagonal have a more constrained structure which depends on how the different fields transform under the global symmetry. It is clear that the complexity of these models increases very fast with the number of extra Weyl fermions, . This number is often taken to be the same as the number of families, but there is no fundamental reason why this should be the case. In Table 1 we summarize the neutrino mass spectrum and the number of leptonic mixing parameters as a function of , with and without an exact global lepton number symmetry. Note that various lepton charge assignments are possible.
zero modes  masses  angles  CP phases  
1    2  2  2  0 
+1  2  1  2  0  
2    1  4  4  3 
(+1,+1)  1  2  3  1  
(+1,)  3  1  3  1  
3    0  6  6  6 
(+1,+1,+1)  0  3  3  1  
(+1,,+1)  2  2  6  4  
(+1,,)  4  1  4  1 
A first step towards a systematic exploration of the phenomenology of such models, in increasing order of complexity, in order to quantify the constraints imposed by data, was presented in Donini et al. (2011), see also Blennow and FernandezMartinez (2011); Fan and Langacker (2012). For previous work on the subject, see also de Gouvêa (2005); de Gouvea et al. (2007); de Gouvêa et al. (2009).
The requirement that two distinct mass splittings exist implies that the second, fifth and last entries on the table are excluded by neutrino oscillation data. The fourth and seventh entries correspond to two and three massive Dirac neutrinos, i.e., to the standard three neutrino scenario. Here we consider the remaining cases for . The addition of just one additional singlet Weyl fermion, , has in principle enough free parameters (two mixing angles and two mass differences) to fit the solar and atmospheric oscillation data. The nexttominimal choice requires two Weyl fermions, . Such a possibility is, of course, well known to give a good fit to the data, in the limits and , where the standard three neutrino scenario is recovered. In both cases, the physics spectrum contains one massless neutrino and two massive ones. The main goal will now be the exploration of the parameter space in between these two limits, in search for other viable solutions that could accommodate at least the solar and atmospheric oscillation, and maybe explain some of the outliers (e.g. LSND).
In the literature, several authors have devoted a lot of effort to study the implications of neutrino oscillation data on models with extra sterile neutrinos (some recent analyses are Sorel et al. (2004); Akhmedov and Schwetz (2010); Kopp et al. (2011); Giunti and Laveder (2010a)), usually referred to as . Most of these studies have been done with the motivation of trying to accommodate LSND Aguilar et al. (2001), and MiniBooNE data AguilarArevalo et al. (2009a, 2010a). It is important to stress that these phenomenological models usually correspond to a generic model with mass eigenstates, but do not correspond in general to a model with extra Weyl fermions. The number of free parameters for is typically much larger than what is shown in Table 1, either because the number of Weyl fermions involved is larger (e.g. Dirac fermions correspond in our context to and not to ) or because couplings that are forbidden by gauge invariance in our model, such as Majorana mass entries for the active neutrinos, are included, effectively, in the phenomenological models. Models for any can be parameterized as the phenomenological models with , but in that case there are generically correlations between parameters (e.g. between mixings and masses). The analyses performed in the context of the phenomenological models do not take such correlations into account, and usually restrict the number of parameters by assuming instead some hierarchies between neutrino masses that could accommodate LSND. In order to distinguish the phenomenological models from those in Eq. (29), we refer from now on to the latter as minimal models.
There are many possible parametrizations of the mass matrix. A good choice will usually be one that satisfies two properties: 1) it contains all independent parameters and no more, 2) it is convenient for imposing existing constraints. Without loss of generality, we can choose a basis where the mass matrix takes a diagonal form, while is a generic complex matrix. In the case when , a convenient parametrization is the one first introduced by CasasIbarra Casas and Ibarra (2001), which exploits the approximate decoupling of the light and heavy sectors, using as parameters the light masses and mixings that have already been measured. When this condition is not satisfied, we will use the following parametrization for the cases:
(30) 
where is a unitary matrix, with the same structure as the PMNS matrix, while is a unitary matrix, depending on two phases and one angle. In the degenerate limit for , i.e., when , the matrix becomes unphysical and coincides with the PMNS matrix in the two limiting cases .
It turns out that the is excluded Donini et al. (2011), even though in principle it has sufficient parameters to fit two mass splittings and two mixing angles. Thus, the most minimal working model is the 3+2 model, which contains one massless neutrino, four massive states, four angles and 2 CP phases. As a first simplification, the degenerate case in the CP conserving limit was explored, where the two eigenvalues of were assumed to be the degenerate, and no phases. The number of extra parameters with respect to the standard threeneutrino scenario is just one extra mass, the common Majorana mass, . As mentioned above, this model in fact reduces to the standard three neutrino scenario in the two limits: , when the four eigenstates degenerate into two Dirac pairs, and the opposite , when the two heavier states decouple. It is clear therefore that the fit to neutrino data will be good for below the quasiDirac limit, ( being the scale where the two massive neutrinos become PseudoDirac particles) and also for above the seesaw limit ( being the scale where the mass of the active neutrinos is given by the seesaw formula), see Fig. 5. As it turns out, only the expected region and survive the fits to oscillation data. The value of is essentially fixed by solar data to be very small. The seesaw limit is mostly determined by longbaseline data.
Above and near the seesaw limit , the spectrum has two almost degenerate states with mass (eV). The heavylight mixings in the minimal model with degenerate Majorana masses are completely fixed in terms of the parameters measured in oscillations and ( are fixed in terms of and the atmospheric/solar mass splittings):