Sterile neutrino oscillations: the global picture
Abstract
Neutrino oscillations involving eVscale neutrino mass states are investigated in the context of global neutrino oscillation data including short and longbaseline accelerator, reactor, and radioactive source experiments, as well as atmospheric and solar neutrinos. We consider sterile neutrino mass schemes involving one or two masssquared differences at the scale denoted by 3+1, 3+2, and 1+3+1. We discuss the hints for eVscale neutrinos from disappearance (reactor and Gallium anomalies) and and neutralcurrent disappearance data. An explanation of all hints in terms of oscillations suffers from severe tension between appearance and disappearance data. The best compatibility is obtained in the 1+3+1 scheme with a pvalue of 0.2% and exceedingly worse compatibilities in the 3+1 and 3+2 schemes. appearance (LSND and MiniBooNE) searches, and we present constraints on sterile neutrino mixing from
Keywords:
neutrino oscillations, sterile neutrinosIFTUAM/CSIC13026
1 Introduction
Huge progress has been made in the study of neutrino oscillations Fukuda:1998mi ; Ahmad:2002jz ; Araki:2004mb ; Adamson:2008zt , and with the recent determination of the last unknown mixing angle Abe:2011sj ; Adamson:2011qu ; Abe:2011fz ; An:2012eh ; Ahn:2012nd ; Abe:2012tg a clear firstorder picture of the threeflavor lepton mixing matrix has emerged, see e.g. GonzalezGarcia:2012sz . Besides those achievements there are some anomalies which cannot be explained within the threeflavor framework and which might point towards the existence of additional neutrino flavors (socalled sterile neutrinos) with masses at the eV scale:

The LSND experiment Aguilar:2001ty reports evidence for transitions with , where and are the neutrino energy and the distance between source and detector, respectively.

This effect is also searched for by the MiniBooNE experiment AguilarArevalo:2007it ; AguilarArevalo:2010wv ; MBnu2012 ; AguilarArevalo:2012va ; AguilarArevalo:2013ara , which reports a yet unexplained event excess in the lowenergy region of the electron neutrino and antineutrino event spectra. No significant excess is found at higher neutrino energies. Interpreting the data in terms of oscillations, parameter values consistent with the ones from LSND are obtained.

Radioactive source experiments at the Gallium solar neutrino experiments SAGE and GALLEX have obtained an event rate which is somewhat lower than expected. This effect can be explained by the hypothesis of disappearance due to oscillations with Acero:2007su ; Giunti:2010zu (“Gallium anomaly”).

A recent reevaluation of the neutrino flux emitted by nuclear reactors Mueller:2011nm ; Huber:2011wv has led to somewhat increased fluxes compared to previous calculations Schreckenbach:1985ep ; Hahn:1989zr ; VonFeilitzsch:1982jw ; Vogel:1980bk . Based on the new flux calculation, the results of previous shortbaseline ( m) reactor experiments are in tension with the prediction, a result which can be explained by assuming disappearance due to oscillations with Mention:2011rk (“reactor anomaly”).
Sterile neutrino oscillation schemes have been considered for a long time, see e.g. GomezCadenas:1995sj ; Goswami:1995yq ; Bilenky:1996rw ; Okada:1996kw for early references on fourneutrino scenarios. Effects of two sterile neutrinos at the eV scale have been considered first in Peres:2000ic ; Sorel:2003hf , oscillations with three sterile neutrinos have been investigated in Maltoni:2007zf ; Conrad:2012qt .
Thus, while the phenomenology of sterile neutrino models is well known, it has also been known for a long time that the LSND and MiniBooNE appearance signals are in tension with bounds from disappearance experiments Maltoni:2002xd ; Strumia:2002fw ; Cirelli:2004cz , challenging an interpretation in terms of sterile neutrino oscillations. This problem remains severe, and in the following we will give a detailed discussion of the status of the disappearance from the reactor and Gallium anomalies, which are not in direct conflict with any other data. This somewhat ambiguous situation asks for an experimental answer, and indeed several projects are under preparation or under investigation, ranging from experiments with radioactive sources, shortbaseline reactor experiments, to new accelerator facilities. A recent review on light sterile neutrinos including an overview on possible experimental tests can be found in Abazajian:2012ys . appearance hints from LSND and MiniBooNE in the light of recent global data. The situation is better for the hints for
In this paper we provide an extensive analysis of the present situation of sterile neutrino scenarios. We discuss the possibility to explain the tentative positive signals from LSND and MiniBooNE, as well as the reactor and Gallium anomalies in terms of sterile neutrino oscillations in view of the global data. New ingredients with respect to our previous analysis Kopp:2011qd are the following.

We use latest data from the MiniBooNE MBnu2012 ; AguilarArevalo:2012va ; AguilarArevalo:2013ara . Our MiniBooNE appearance analysis is now based on Monte Carlo events provided by the collaboration taking into account realistic event reconstruction, correlation matrices, as well as oscillations of various background components in a consistent way. appearance searches

We include the constraints on the appearance probability from E776 Borodovsky:1992pn and ICARUS Antonello:2012pq .

We include the Gallium anomaly in our fit.

We take into account constraints from solar neutrinos, the KamLAND reactor experiment, and LSND and KARMEN measurements of the reaction .

The treatment of the reactor anomaly is improved and updated by taking into account small changes in the predicted antineutrino fluxes as well as an improved consideration of systematic errors and their correlations.

We take into account chargedcurrent (CC) and neutralcurrent (NC) data from the MINOS longbaseline experiment Adamson:2010wi ; Adamson:2011ku .

We include data on disappearance from MiniBooNE AguilarArevalo:2009yj as well as disappearance from a joint MiniBooNE/SciBooNE analysis Cheng:2012yy .

In our analysis of atmospheric neutrino data, we improve our formalism to fully take into account the mixing of with other active or sterile neutrino states.
All the data used in this work are summarized in Tab. 1. For other recent sterile neutrino global fits see Conrad:2012qt ; Giunti:2011hn ; Archidiacono:2013xxa . We are restricting our analysis to neutrino oscillation data; implications for kinematic neutrino mass measurements and neutrinoless double betadecay data have been discussed recently in Li:2011ss ; Barry:2011wb ; Giunti:2011cp .
Experiment  dof  channel  comments 
Shortbaseline reactors  76  SBL  
Longbaseline reactors  39  LBL  
KamLAND  17  
Gallium  4  SBL  
Solar neutrinos  261  + NC data  
LSND/KARMEN C  32  SBL  
CDHS  15  SBL  
MiniBooNE  15  SBL  
MiniBooNE  42  SBL  
MINOS CC  20  LBL  
MINOS NC  20  LBL  
Atmospheric neutrinos  80  + NC matter effect  
LSND  11  SBL  
KARMEN  9  SBL  
NOMAD  1  SBL  
MiniBooNE  11  SBL  
MiniBooNE  11  SBL  
E776  24  SBL  
ICARUS  1  LBL  
total  689 
Sterile neutrinos at the eV scale also have implications for cosmology. If thermalized in the early Universe they contribute to the number of relativistic degrees of freedom (effective number of neutrino species ). A review with many references can be found in Abazajian:2012ys . Indeed there might be some hints from cosmology for additional relativistic degrees of freedoms ( bigger than 3), coming mainly from CMB data, e.g. Hamann:2010bk ; Giusarma:2011ex ; GonzalezGarcia:2010un ; Archidiacono:2012ri ; NeffLunardini:2013 ; Archidiacono:2013xxa . Recently precise CMB data from the PLANCK satellite have been released Ade:2013zuv . Depending on which additional cosmological data are used, values ranging from to (uncertainties at 95% CL) are obtained Ade:2013zuv . Constraints from Big Bang Nucleosynthesis on have been considered recently in Mangano:2011ar . Apart from their contribution to , thermalized eVscale neutrinos would also give a large contribution to the sum of neutrino masses, which is constrained to be below around 0.5 eV. The exact constraint depends on which cosmological data sets are used, but the most important observables are those related to galaxy clustering Hamann:2010bk ; Giusarma:2011ex ; GonzalezGarcia:2010un ; Archidiacono:2012ri . In the standard CDM cosmology framework the bound on the sum of neutrino masses is in tension with the masses required to explain the aforementioned terrestrial hints Archidiacono:2012ri . The question to what extent such sterile neutrino scenarios are disfavored by cosmology and how far one would need to deviate from the CDM model in order to accommodate them remains under discussion Hamann:2011ge ; Joudaki:2012uk ; Archidiacono:2013xxa . We will not include any information from cosmology explicitly in our numerical analysis. However, we will keep in mind that neutrino masses in excess of few eV may become more and more difficult to reconcile with cosmological observations.
The outline of the paper is as follows. In Sec. 2 we introduce the formalism of sterile neutrino oscillations and fix the parametrization of the mixing matrix. We then consider disappearance data in Sec. 3, discussing the reactor and Gallium anomalies. Constraints from disappearance as well as neutralcurrent data are discussed in Sec. 4, and global 5. The global fit of all these data combined is presented in Sec. 6 for scenarios with one or two sterile neutrinos. We summarize our results and conclude in Sec. 7. Supplementary material is provided in the appendices including a discussion of complex phases in sterile neutrino oscillations, oscillation probabilities for solar and atmospheric neutrinos, as well as technical details of our experiment simulations. appearance data including the LSND and MiniBooNE signals in Sec.
2 Oscillation parameters in the presence of sterile neutrinos
In this work we consider the presence of or 2 additional neutrino states with masses in the few eV range. When moving from 1 to 2 sterile neutrinos the qualitative new feature is the possibility of CP violation already at shortbaseline Karagiorgi:2006jf ; Maltoni:2007zf .^{1}^{1}1Adding more than two sterile neutrinos does not lead to any qualitatively new physical effects and as shown in Maltoni:2007zf the fit does not improve significantly. Therefore, we restrict the present analysis to sterile neutrinos. The neutrino mass eigenstates are labeled such that , , contribute mostly to the active flavor eigenstates and provide the mass squared differences required for “standard” threeflavor oscillations, and , where . The mass states , are mostly sterile and provide masssquared differences in the range . In the case of only one sterile neutrino, denoted by “3+1” in the following, we always assume , but the oscillation phenomenology for would be the same. For two sterile neutrinos, we distinguish between a mass spectrum where and are both positive (“3+2”) and where one of them is negative (“1+3+1”). The phenomenology is slightly different in the two cases Goswami:2007kv . We assume that the linear combinations of mass states which are orthogonal to the three flavor states participating in weak interactions are true singlets and have no interaction with Standard Model particles. Oscillation physics is then described by a rectangular mixing matrix with and , and .^{2}^{2}2In this work we consider socalled phenomenological sterile neutrino models, where the neutrino mass eigenvalues and the mixing parameters are considered to be completely independent. In particular we do not assume a seesaw scenario, where the Dirac and Majorana mass matrices of the sterile neutrinos are the only source of neutrino mass and mixing. For such “minimal” sterile neutrino models see e.g. Blennow:2011vn ; Fan:2012ca ; Donini:2012tt .
We give here expressions for the oscillation probabilities in vacuum, focusing on the 3+2 case. It is trivial to recover the 3+1 formulas from them by simply dropping all terms involving the index “5”. Formulas for the 1+3+1 scenario are obtained by taking either or negative. Oscillation probabilities relevant for solar and atmospheric neutrinos are given in appendices C and D, respectively.
First we consider the socalled “shortbaseline” (SBL) limit, where the relevant range of neutrino energies and baselines is such that effects of and can be neglected. Then, oscillation probabilities depend only on and with . We obtain for the appearance probability
(1) 
with the definitions
(2) 
Eq. (1) holds for neutrinos; for antineutrinos one has to replace . Since Eq. (1) is invariant under the transformation and , we can restrict the parameter range to , or equivalently , without loss of generality. Note also that the probability Eq. (1) depends only on the combinations and . The only SBL appearance experiments we are considering are in the channel. Therefore, the total number of independent parameters is 5 if only SBL appearance experiments are considered.
The 3+2 survival probability, on the other hand, is given in the SBL approximation by
(3) 
In this work we include also experiments for which the SBL approximation cannot be adopted, in particular MINOS and ICARUS. For these experiments is of order one. In the following we give the relevant oscillation probabilities in the limit of and . We call this the longbaseline (LBL) approximation. In this case we obtain for the neutrino appearance probability ()
(4) 
The corresponding expression for antineutrinos is obtained by the replacement . The survival probability in the LBL limit can be written as
(5) 
Note that in the numerical analysis of MINOS data neither the SBL nor the LBL approximations can be used because , and can all become of order one either at the far detector or at the near detector Hernandez:2011rs . Moreover, matter effects cannot be neglected in MINOS. All of these effects are properly included in our numerical analysis of the MINOS experiment.
Sometimes it is convenient to complete the rectangular mixing matrix by rows to an unitary matrix, with . For we use the following parametrization for :
(6) 
where represents a real rotation matrix by an angle in the plane, and represents a complex rotation by an angle and a phase . The particular ordering of the rotation matrices is an arbitrary convention which, however, turns out to be convenient for practical reasons.^{3}^{3}3Note that the ordering chosen in Eq. (6) is equivalent to , where the standard threeflavor convention appears to the right (apart from an additional complex phase), and mixing involving the mass states and appear successively to the left of it. We have dropped the unobservable rotation matrix which just mixes sterile states. There is also some freedom regarding which phases are removed by field redefinitions and which ones are kept as physical phases. In appendix A we give a specific recipe for how to remove unphysical phases in a consistent way. Throughout this work we consider only phases which are phenomenologically relevant in neutrino oscillations. Under certain approximations, more phases may become unphysical. For instance, if an angle which corresponds to a rotation which can be chosen to be complex is zero the corresponding phase disappears. In practical situations often one or more of the masssquared differences can be considered to be zero, which again implies that some of the angles and phases will become unphysical. In Tab. 2 we show the angle and phase counting for the SBL and LBL approximations for the 3+2 and 3+1 cases.
A/P  LBL approx.  (A/P)  SBL approx.  (A/P)  

3+2  9/5  (8/4)  (6/2)  
3+1  6/3  (5/2)  (3/0) 
In the notation of Eqs. (1), (3), (4), (5), it is explicit that only appearance experiments depend on complex phases in a parametrization independent way. However, in a particular parametrization such as Eq. (6), also the moduli may depend on cosines of the phase parameters , leading to some sensitivity of disappearance experiments to the in a CPeven fashion. Our parametrization Eq. (6) guarantees that disappearance experiments are independent of .
3 and disappearance searches
Disappearance experiments in the sector probe the moduli of the entries in the first row of the neutrino mixing matrix, . In the shortbaseline limit of the 3+1 scenario, the only relevant parameter is . For two sterile neutrinos, also is relevant. In this section we focus on 3+1 models, and comment only briefly on 3+2. For 3+1 oscillations in the SBL limit, the survival probability takes an effective two flavor form
(7) 
where we have defined an effective disappearance mixing angle by
(8) 
This definition is parametrization independent. Using the specific parametrization of Eq. (6) it turns out that .
3.1 SBL reactor experiments
experiment  [m]  obs/pred  unc. error [%]  tot. error [%] 

Bugey4 Declais:1994ma  15  0.926  1.09  1.37 
Rovno91 Kuvshinnikov:1990ry  18  0.924  2.10  2.76 
Bugey3 Declais:1994su  15  0.930  2.05  4.40 
Bugey3 Declais:1994su  40  0.936  2.06  4.41 
Bugey3 Declais:1994su  95  0.861  14.6  1.51 
Gosgen Zacek:1986cu  38  0.949  2.38  5.35 
Gosgen Zacek:1986cu  45  0.975  2.31  5.32 
Gosgen Zacek:1986cu  65  0.909  4.81  6.79 
ILL Kwon:1981ua  9  0.788  8.52  1.16 
Krasnoyarsk Vidyakin:1987ue  33  0.920  3.55  6.00 
Krasnoyarsk Vidyakin:1987ue  92  0.937  19.8  2.03 
Krasnoyarsk Vidyakin:1994ut  57  0.931  2.67  4.32 
SRP Greenwood:1996pb  18  0.936  1.95  2.79 
SRP Greenwood:1996pb  24  1.001  2.11  2.90 
Rovno88 Afonin:1988gx  18  0.901  4.24  6.38 
Rovno88 Afonin:1988gx  18  0.932  4.24  6.38 
Rovno88 Afonin:1988gx  18  0.955  4.95  7.33 
Rovno88 Afonin:1988gx  25  0.943  4.95  7.33 
Rovno88 Afonin:1988gx  18  0.922  4.53  6.77 
Palo Verde Boehm:2001ik  820  1 rate  
Chooz Apollonio:2002gd  1050  14 bins  
DoubleChooz Abe:2012tg  1050  18 bins  
DayaBay DBneutrino  6 rates – 1 norm  
RENO Ahn:2012nd  2 rates – 1 norm  
KamLAND Gando:2010aa  17 bins 
The data from reactor experiments used in our analysis are summarized in Tab. 3. Our simulations make use of a dedicated reactor code based on previous publications, see e.g. Grimus:2001mn ; Schwetz:2011qt . We have updated the code to include the latest data and improved the treatment of uncertainties, see appendix B for details. The code used here is very similar to the one from Ref. GonzalezGarcia:2012sz , extended to sterile neutrino oscillations. The reactor experiments listed in Tab. 3 can be divided into shortbaseline (SBL) experiments with baselines m, longbaseline (LBL) experiments with , and the very longbaseline experiment KamLAND with an average baseline of 180 km. SBL experiments are not sensitive to standard threeflavor oscillations, but can observe oscillatory behavior for , . On the other hand, for longbaseline experiments, oscillations due to , are most relevant, and oscillations due to scale masssquared differences are averaged out and lead only to a constant flux suppression. KamLAND is sensitive to oscillations driven by and , whereas all with lead only to a constant flux reduction.
For the SBL reactor experiments we show in Tab. 3 also the ratio of the observed and predicted rate, where the latter is based on the flux calculations of Huber:2011wv for neutrinos from U, Pu, Pu fission and Mueller:2011nm for U fission. The ratios are taken from Abazajian:2012ys (which provides and update of Mention:2011rk ) and are based on the Particle Data Group’s 2011 value for the neutron lifetime, s Nakamura:2010zzi .^{4}^{4}4This number differs from their 2012 value 880.1 s by less than 0.2% pdg . The neutron lifetime enters the calculation of the detection cross section and therefore has a direct impact on the expected rate. The quoted uncertainties of are small compared to the uncertainties on the predicted flux, see RevModPhys.83.1173 for a discussion of the neutron lifetime determination. We observe that most of the ratios are smaller than one. In order to asses the significance of this deviation, a careful error analysis is necessary. In the last column of Tab. 3, we give the uncorrelated errors on the rates. They include statistical as well as uncorrelated experimental errors. In addition to these, there are also correlated experimental errors between various data points which are described in detail in appendix B. Furthermore, we take into account the uncertainty on the neutrino flux prediction following the prescription given in Huber:2011wv , see also appendix B for details.
Fitting the SBL data to the predicted rates we obtain which corresponds to a value of 2.4%. When expressed in terms of an energyindependent normalization factor , the best fit is obtained at
(9) 
Here denotes the improvement in compared to a fit with . Clearly the value increases drastically when is allowed to float, leading to a preference for at the confidence level. This is our result for the significance of the reactor anomaly. Let us mention that (obviously) this result depends on the assumed systematic errors. While we have no particular reason to doubt any of the quoted errors, we have checked that when an adhoc additional normalization uncertainty of 2% (3%) is added, the significance is reduced to (). This shows that the reactor anomaly relies on the control of systematic errors at the percent level.
/dof (GOF)  /dof (CL)  

SBL rates only  0.13  0.44  11.5/17 (83%)  11.4/2 (99.7%) 
SBL incl. Bugey3 spectr.  0.10  1.75  58.3/74 (91%)  9.0/2 (98.9%) 
SBL + Gallium  0.11  1.80  64.0/78 (87%)  14.0/2 (99.9%) 
SBL + LBL  0.09  1.78  93.0/113 (92%)  9.2/2 (99.0%) 
global disapp.  0.09  1.78  403.3/427 (79%)  12.6/2 (99.8%) 
The flux reduction suggested by the reactor anomaly can be explained by sterile neutrino oscillations. In Tab. 4 we give the best fit points and values obtained by fitting SBL reactor data in a 3+1 framework. The allowed regions in and are shown in Fig. 1 (left) for a rateonly analysis as well as a fit including also Bugey3 spectral data. Both analyses give consistent results, with the main difference being that the spectral data disfavors certain values of around and 1.3 . The right panel of Fig. 1 shows the predicted rate suppression as a function of the baseline compared to the data. We show the prediction for the two best fit points from the left panel as well as one point located in the island around , which will be important in the combined fit with SBL appearance data. We observe that for the rateonly best fit point with the prediction follows the tendency suggested by the ILL, Bugey4, and SRP (24 km) data points. This feature is no longer present for , somewhat preferred by Bugey3 spectral data, where oscillations happen at even shorter baselines. However, from the GOF values given in Tab. 4 we conclude that also those solutions provide a good fit to the data.
3.2 The Gallium anomaly
The response of Gallium solar neutrino experiments has been tested by deploying radioactive Cr or Ar sources in the GALLEX Hampel:1997fc ; Kaether:2010ag and SAGE Abdurashitov:1998ne ; Abdurashitov:2005tb detectors. Results are reported as ratios of observed to expected rates, where the latter are traditionally computed using the best fit cross section from Bahcall Bahcall:1997eg , see e.g. Giunti:2010zu . The values for the cross sections weighted over the 4 (2) neutrino energy lines from Cr (Ar) from Bahcall:1997eg are , . While the cross section for into the ground state of is well known from the inverse reaction there are large uncertainties when the process proceeds via excited states of Ge at 175 and 500 keV. Following Bahcall:1997eg , the total cross section can be written as
(10) 
with , Ar. The coefficients , , , are phase space factors. The ground state cross sections are precisely known Bahcall:1997eg : , . BGT denote the GamovTeller strength of the transitions, which have been determined recently by dedicated measurements Frekers:2011zz as
(11) 
In our analysis we use these values together with Eq. (10) for the cross section.
This means that the ratios of observed to expected rates based on the Bahcall prediction have to be rescaled by a factor 0.982 (0.977) for the Cr (Ar) experiments, so that we obtain for them the following updated numbers for our fits:
(12) 
Here, we have symmetrized the errors, and we have included only experimental errors, but not the uncertainty on the cross section (see below).
We build a out of the four data points from GALLEX and SAGE and introduce two pulls corresponding to the systematic uncertainty of the two transitions to excited state according to Eq. (11). The determination of BGT is relatively poor, with zero being allowed at . In order to avoid unphysical negative contributions from the 175 keV state, we restrict the domain of the corresponding pull parameter accordingly. Fitting the four data points with a constant neutrino flux normalization factor we find
(13) 
Because of the different cross sections used, these results differ from the ones obtained in Giunti:2010zu , where the best fit point is at , while the significance is comparable, around . An updated analysis including also a discussion of the implications of the measurement in Frekers:2011zz can be found in Giunti:2012tn .
The event deficit in radioactive source experiments can be explained by assuming mixing with an eVscale state, such that disappearance happens within the detector volume Acero:2007su . We fit the Gallium data in the 3+1 framework by averaging the oscillation probability over the detector volume using the geometries given in Acero:2007su . The resulting allowed region at 95% confidence level is shown in orange in Fig. 2. Consistent with the above discussion we find mixing angles somewhat smaller than those obtained by the authors of Giunti:2010zu . The best fit point from combined Gallium+SBL reactor data is given in Tab. 4, and the nooscillation hypothesis is disfavored at 99.9% CL (2 dof) or compared to the 3+1 best fit point.
(GOF)  (CL)  (CL)  

SBLR  0.46  0.87  0.12  0.13  53.0/(764) (95%)  5.3 (93%)  14.3 (99.3%) 
SBLR+gal  0.46  0.87  0.12  0.14  60.2/(804) (90%)  3.8 (85%)  17.8 (99.9%) 
Let us consider now the Gallium and SBL reactor data in the framework of two sterile neutrinos, in particular in the 3+2 scheme. SBL and disappearance data depend on 4 parameters in this case, , , and the two mixing angles and (or, equivalently, the moduli of the two matrix elements and ). We report the best fit points from SBL reactor data and from SBL reactor data combined with the Gallium source data in Tab. 5. For these two cases we find an improvement of 5.3 and 3.8 units in , respectively, when going from the 3+1 scenario to the 3+2 case. Considering that the 3+2 model has two additional parameters compared to 3+1, we conclude that there is no improvement of the fit beyond the one expected by increasing the number of parameters, and that SBL data sets show no significant preference for 3+2 over 3+1. This is also visible from the fact that the confidence level at which the no oscillation hypothesis is excluded does not increase for 3+2 compared to 3+1, see the last columns of Tabs. 4 and 5. There the is translated into a confidence level by taking into account the number of parameters relevant in each model, i.e., 2 for 3+1 and 4 for 3+2.
3.3 Global data on and disappearance
Let us now consider the global picture regarding disappearance. In addition to the shortbaseline reactor and Gallium data discussed above, we now add data from the following experiments:

The remaining reactor experiments at a long baseline (“LBL reactors”) and the very longbaseline reactor experiment KamLAND, see table 3.

Global data on solar neutrinos, see appendix C for details.

LSND and KARMEN measurements of the reaction Auerbach:2001hz ; Armbruster:1998uk . The experiments have found agreement with the expected cross section Fukugita:1988hg , hence their measurements constrain the disappearance of with eVscale masssquared differences Reichenbacher:2005nc ; Conrad:2011ce . Details on our analysis of the scattering data are given in appendix E.1.
So far the LBL experiments DayaBay and RENO have released only data on the relative comparison of near ( m) and far ( km) detectors, but no information on the absolute flux determination is available. Therefore, their published data are essentially insensitive to oscillations with eVscale neutrinos and they contribute only indirectly via constraining . In our analysis we include a free, independent flux normalization factor for each of those two experiments. Chooz and DoubleChooz both lack a near detector. Therefore, in the official analyses performed by the respective collaborations the Bugey4 measurement is used to normalize the flux. This makes the official Chooz and DoubleChooz results on also largely independent of the presence of sterile neutrinos. However, the absolute rate of Bugey4 in terms of the flux predictions is published (see Tab. 3) and we can use this number to obtain an absolute flux prediction for Chooz and DoubleChooz. Therefore, in our analysis Chooz and DoubleChooz (as well as Palo Verde) by themselves also show some sensitivity to sterile neutrino oscillations. In a combined analysis of Chooz and DoubleChooz with SBLR data the official analyses are recovered approximately. Previous considerations of LBL reactor experiments in the context of sterile neutrinos can be found in Refs. Bandyopadhyay:2007rj ; Bora:2012pi ; Giunti:2011vc ; Bhattacharya:2011ee .
We show in Tab. 4 a combined analysis of the SBL and LBL reactor experiments (row denoted by “SBL+LBL”), where we minimize with respect to . We find that the significance of the reactor anomaly is not affected by the inclusion of LBL experiments and finite . The even slightly increases from 9.0 to 9.2 when adding LBL data to the SBL data (“noosc” refers here to ). Hence, we do not agree with the conclusions of Ref. Zhang:2013ela , which finds that the significance of the reactor anomaly is reduced to when LBL data and a finite value of is taken into account.
Solar neutrinos are also sensitive to sterile neutrino mixing (see e.g. Giunti:2009xz ; Palazzo:2011rj ; Palazzo:2012yf ). The main effect of the presence of mixing with eV states is an overall flux reduction. While this effect is largely degenerate with , a nontrivial bound is obtained in the combination with DayaBay, RENO and KamLAND. KamLAND is sensitive to oscillations driven by and , whereas sterile neutrinos affect the overall normalization, degenerate with . The matter effect in the sun as well as SNO NC data provide additional signatures of sterile neutrinos, beyond an overall normalization. As we will show in Sec. 4 solar data depend also on the mixing angles and , controlling the fraction of transitions, see e.g. Giunti:2009xz . As discussed in appendix C, in the limit for , solar data depends on 6 real mixing parameters, 1 complex phase and . Hence, in a 3+1 scheme all six mixing angles are necessary to describe solar data in full generality. However, once other constraints on mixing angles are taken into account the effect of , , and the complex phase are tiny and numerically have a negligible impact on our results. Therefore we set for the solar neutrino analysis in this section. In this limit solar data becomes also independent of the complex phase.
The results of our fit to global disappearance data are shown in Fig. 2 and the best fit point is given in Tab. 4. For this analysis the masssquared differences and have been fixed, whereas we marginalize over the mixing angles and . We see from Fig. 2 that the parameter region favored by shortbaseline reactor and Gallium data is well consistent with constraints from longbaseline reactors, KARMEN’s and LSND’s rate, and with solar and KamLAND data.
Recently, data from the Mainz Kraus:2012he and Troitsk Belesev:2012hx tritium betadecay experiments have been reanalyzed to set limits on the mixing of with new eV neutrino mass states. Taking the results of Belesev:2012hx at face value, the Troitsk limit would cutoff the highmass region in Fig. 2 at around 100 Giunti:2012bc (above the plotrange shown in the figure). The bounds obtained in Kraus:2012he are somewhat weaker. The differences between the limits obtained in Kraus:2012he and Belesev:2012hx depend on assumptions concerning systematic uncertainties and therefore we prefer not to explicitly include them in our fit. The sensitivity of future tritium decay data from the KATRIN experiment has been estimated in Riis:2010zm . Implications of sterile neutrinos for neutrinoless double betadecay have been discussed recently in Li:2011ss ; Barry:2011wb ; Giunti:2011cp .
Let us now address the question whether the presence of a sterile neutrino affects the determination of the mixing angle (see also Bhattacharya:2011ee ; Zhang:2013ela ). In Fig. 3 we show the combined determination of and for two fixed values of . The left panel corresponds to a relatively large value of 10 , whereas for the right panel we have chosen the value favored by the global disappearance best fit point, 1.78 . The masssquared differences and have been fixed, whereas we marginalize over the mixing angle . We observe a clear complementarity of the different data sets: SBL reactor and Gallium data determine , since oscillations are possible only via , all other masssquared differences are effectively zero for them. For LBL reactors can be set to infinity, is finite, and is effectively zero; therefore they provide an unambiguous determination of by comparing near and far detector data. The upper bound on from LBL reactors is provided by Chooz, Palo Verde, DoubleChooz, since for those experiments also information on the absolute flux normalization can be used, as mentioned above. In contrast, for solar neutrinos and KamLAND, both and are effectively infinite, and and affect essentially the overall normalization and are largely degenerate, as visible the figure.
In conclusion, the determination is rather stable with respect to the presence of sterile neutrinos. We note, however, that its interpretation becomes slightly more complicated. For instance, in the 3+1 scheme using the parametrization from Tab. 2, the relation between mixing matrix elements and mixing angles is and . Hence, the onetoone correspondence between and as in the threeflavor case is spoiled.
4 , , and neutralcurrent disappearance searches
In this section we discuss the constraints on the mixing of and with new eVscale mass states. In the 3+1 scheme this is parametrized by and , respectively. In terms of the mixing angles as defined in Eq. (6) we have and . In the present paper we include data sets from the following experiments to constrain and mixing with eV states:

SBL disappearance data from CDHS Dydak:1983zq . Details of our simulation are given in Grimus:2001mn .

SuperKamiokande. It has been pointed out in Bilenky:1999ny that atmospheric neutrino data from SuperKamiokande provide a bound on the mixing of with eVscale mass states, i.e., on the mixing matrix elements , . In addition, neutralcurrent matter effects provide a constraint on , . A discussion of the effect is given in the appendix of Maltoni:2007zf . Details on our analysis and references are given in appendix D.

MiniBooNE AguilarArevalo:2009yj ; Cheng:2012yy . Apart from the appearance search, MiniBooNE can also look for SBL disappearance. Details on our analysis are given in appendix E.4.

MINOS Adamson:2010wi ; Adamson:2011ku . The MINOS longbaseline experiment has published data on charged current (CC) disappearance as well as on the neutral current (NC) count rate. Both are based on a comparison of near and far detector measurements. In addition to providing the most precise determination of (from CC data), those data can also be used to constrain sterile neutrino mixing, where CC (NC) data are mainly relevant for , (, ). See appendix E.5 for details.
Additional constrains on mixing with eVscale states (not used in this analysis) can be obtained from data from the Ice Cube neutrino telescope Nunokawa:2003ep ; Choubey:2007ji ; Razzaque:2011ab ; Barger:2011rc ; Razzaque:2012tp ; Esmaili:2012nz .
Limits on the row of the mixing matrix come from disappearance experiments. In a 3+1 scheme the SBL disappearance probability is given by
(14) 
where we have defined an effective disappearance mixing angle by
(15) 
i.e., in our parametrization (6) the effective mixing angle depends on both and . In contrast to the disappearance searches discussed in the previous section, experiments probing disappearance have not reported any hints for a positive signal. We show the limits from the data listed above in the left panel of Fig. 4. Note that the MINOS limit is based on the comparison of the data in near and far detectors. For oscillation effects become relevant at the near detector, explaining the corresponding features in the MINOS bound around that value of , whereas the features around emerge from oscillation effects in the far detector. The roughly constant limit in the intermediate range corresponds to the limit in the near (far) detector adopted in Adamson:2010wi ; Adamson:2011ku . In that range the MINOS limit on is comparable to the one from SuperK atmospheric data. For the limit is dominated by CDHS and MiniBooNE disappearance data.
In Fig. 4 (left) we show also the region preferred by the hints for eVscale oscillations from LSND and MiniBooNE appearance data (see next section) combined with reactor and Gallium data on disappearance. For fixed we minimize the corresponding with respect to to show the projection in the plane of and . The tension between the hints in the 6. data is clearly visible in this plot. We will discuss this conflict in detail in section channels compared to the limits from and
Limits on the mixing of with eVscale states are obtained from data involving information from NC interactions, which allow to distinguish between ^{5}^{5}5The searches for appearance at NOMAD Astier:2001yj and CHORUS Eskut:2007rn at short baselines are sensitive only to specific parameter combinations like or and therefore do not provide a constraint on by itself. The relevant data samples are atmospheric and solar neutrinos (via the NC matter effect) and MINOS NC data. Furthermore, the parameter transitions. and