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Searches for sterile component with solar neutrinos and KamLAND

Possible mixing of the active and sterile neutrinos has been considered both in the single Delta m^2 approximation and in the case of more than one Delta m^2. We perform global fit of the available solar neutrino data with free boron neutrino flux in the single Delta m^2 context. The best fit value corresponds to zero fraction of sterile component eta=0. We get the upper bounds: eta<0.26 (0.64) at 1\sigma (3 sigma). Due to degeneracy of parameters no one individual experiment restricts eta. The bound appears as an interplay of the SNO and Gallium as well as SuperKamiokande data. Future measurements of the NC/CC ratio at SNO can strengthen the bound down to eta<0.5 (3 sigma). If KamLAND confirms the LMA solution with its best fit point a combined analysis of the KamLAND and solar neutrino results will lead to eta<0.19 (0.56) at 1 sigma (3 sigma). We find that existence of sterile neutrino can explain the intermediate value of suppression of the KamLAND event rate: R_{KL}~0.75-0.90 in the case when more than one Delta m^2 is involved.


Introduction
Sterile neutrino, ν s , is a panacea of various problems which appear from time to time in neutrino physics. Oscillation interpretation of the LSND result [1] in the context of schemes which describe the solar and atmospheric neutrino problems [2] is the long term motivation for existence of ν s .
There are, however, other motivations to search for the sterile component. It might happen that ν s is the right-handed component of neutrino which turns out to be light (or massless) in the context of the see-saw mechanism, as a consequence of certain symmetry [3]. Such a symmetry (L e −L µ −L τ as in [3] ) can also be responsible for large or maximal lepton mixing. Sterile neutrino can be a component of multiplet of the extended gauge group, e.g., E 6 [4]. Sterile neutrino -light singlet fermion -can originate from completely different sector of theory. Many extensions of the standard model predict existence of singlet fermions: ν s can be a mirror neutrino [5], goldstino in SUSY [6], modulino of the superstring theories [7], bulk fermions related to existence of extra dimensions [8], etc..
Neutrinos are unique particles, since only they can mix with singlets of the Standard Model. This mixing can lead to coherent effects which provide high sensitivity to existence of ν s . The mixing terms as small as (10 −10 − 10 −6 ) eV can produce the observable effects and in the case of supernova neutrinos the masses can be even smaller. In a sense, neutrino mixing is a window to physics beyond the SM.
Another motivation to search for and restrict sterile component is that interpretation of certain results can be changed strongly if one admits even very small admixture of sterile neutrinos. This concerns with the neutrinoless double beta decay, CP-violation, generation of maximal/large flavor mixing [9], etc.. So, we deal with long term program of searches for sterile neutrinos and improvements of the bounds on mixing of sterile neutrinos with all possible masses.
In this paper we will consider in details searches for sterile component in the solar neutrino flux. First detailed discussion of signatures of the sterile neutrinos have been done in [10,11,12] where it was marked, in particular, that studies of the spectrum distortion can reveal existence of ν s .
There were a number of fits of the solar neutrino data in terms of pure ν e → ν s sterile conversion, but recent SNO results [13,14] exclude pure ν e → ν s conversion at high confidence level. Still partial transformation of ν e in ν s is possible. The effects, and allowed fraction of sterility depend on specific scheme of mixing. Most of recent studies have been done in a single ∆m 2 context: according to which the electron neutrino mixes with the state being the combination of the active (ν µ and ν τ ) and sterile neutrino. Fraction of sterility is described by the parameter η. In [15] it was marked that η is weakly restricted provided that the original boron neutrino flux is substantially larger than the SSM flux. Global fit of the solar neutrino data [16] gives η < 0.25 (0.6) at 1σ (3σ) level. If the boron neutrino flux is fixed according to the SSM with corresponding uncertainties the bound becomes stronger at the 3σ level: η < 0.5 [17].
In this paper we will consider effects of sterile neutrinos and bounds on sterility from the existing and future experiments. In particular, we study how combined analysis of the KamLAND and solar neutrino data can improve the bound. We generalize the analysis for the case when more that one ∆m 2 is relevant. We discuss how presence of sterile component can change predictions for KamLAND rate.
The paper is organized as follows. In section 2 we consider the mixing the sterile neutrino in a single ∆m 2 context. In section 3 we perform global fit of all solar neutrino data, and put bounds on the sterile fraction η. In section 3.2 we identify the data sensitive to η. In section 4 we study how future experiments can improve the bound. We consider precision NC measurements at SNO (section 4.1) and combined analysis of KamLAND and solar neutrino data (section 4.2). In section 5 we consider the ν s mixing in the case when more than one ∆m 2 is relevant. We discuss how the presence of sterile component can change predictions for the KamLAND experiment. We present our conclusions in section 6.

Mixing of sterile neutrinos
In what follows we will consider the case of one sterile neutrino. In general, fourth neutrino may have an arbitrary mass and sterile component can mix with all three active neutrinos. So, the scheme will have 4 new real parameters: the mass m 4 and 3 mixing angles. Situation can, however, be simplified if one takes into account existing bounds on mixing of active neutrinos. We will consider the 4-neutrino schemes which explain the solar and atmospheric neutrino data. We assume that two mass eigenstates, ν 1 and ν 2 , are splited by the solar ∆m 2 < several × 10 −4 eV 2 . We will call them the "solar pair". Then the state ν 3 , is splited from the solar pair or from ν 4 by the atmospheric mass split: ∆m 2 atm ∼ (2 − 3) · 10 −3 eV 2 . We define the mixing matrix, ||U αi || as the unitary matrix which connects the flavor and the mass eigenstates: ν α = i U αi ν i , α = e, µ, τ, s, i = 1, 2, 3, 4.

Single ∆m 2 case
As far as solar neutrinos are concerned, mixing of the electron neutrinos plays crucial role.
Let us first assume that U e3 = U e4 = 0, so that the electron flavor is distributed only in the "solar pair of states": In fact, there is a strong upper bound on U e3 from CHOOZ experiment [19], and strong bound on U e4 from BUGEY experiment [20] if corresponding ∆m 2 is large.
Since only U e1 and U e2 are non-zero the orthogonality conditions for ν e and other neu-trinos, ν µ , ν τ , ν s can be written as From these equations we get immediately: That is, all non-electron flavor components enter ν 1 and ν 2 in the same combination. Indeed, let ν x be the combination of the non-electron neutrinos which mixes with ν e in ν 1 : where sin θ 12 = 1 − U 2 e1 . Similarly, let ν y be the combination of the non-electron neutrinos which mixes with ν e in ν 2 : Then according to (2), we have: Thus, the scheme is reduced exactly to the two two neutrino case: ν e mixes with ν x in the states ν 1 and ν 2 and the mixing angle equals θ 12 .
The state ν x can be written as where ν a is the combination of active (non-electron) neutrinos, ν µ and ν τ : So, √ η describes admixture of sterile neutrino in the state ν x to which ν e can be transformed.
According to (3), (4) and (6), admixtures of the sterile component in ν 1 and ν 2 equal and consequently, That is, η gives total fraction of the sterile component in the solar pair.
We have proven that if the electron neutrino is distributed in two mass eigenstates only, then independently of other mixings and masses, the effect of sterile component in solar neutrinos is described by one parameter only, which is the total amount of sterile component in these two mass eigenstates.

Conversion probabilities and observable fluxes
Let us consider the oscillation effects. We introduce -the ν e survival probability in the system of mixed neutrinos ν e −ν x .
is the effective potential. According to (3) and (6) the transition probabilities of the electron neutrino to the sterile, P es , and active, P ea , components in terms of P ee equal Using these probabilities we can write the fluxes of neutrinos which determine the rates of events of different types. Let us introduce f B , the original boron neutrino flux in units of the SSM flux, : Here the SSM boron neutrino flux is taken to be F SSM B = 5.05 · 10 6 cm −2 s −1 . Then the elastic scattering events (ES) (detected by SuperKamiokande and SNO), the neutral current (NC) and charged current (CC) event rates at SNO are determined by the following fluxes: where r = 0.16 is the ratio of the ν µ,τ e − and ν e e − elastic scattering cross sections. The equations (14) depend on three parameters, f B , P ee and η via two combinations In terms of variables x and y, the fluxes can be rewritten as Excluding x and y we get relation between the fluxes [15]: which has a simple interpretation (see the first equality): the flux measured in the neutrinoelectron scattering equals the flux of electron neutrinos at the detector plus the flux of nonelectron active neutrinos, Φ N C − Φ CC , suppressed by r. The equality (18) is well satisfied in the experiment.
The equalities (17) imply the degeneracy of parameters [15]: different values of the original parameters f B , η, P ee lead to the same observables provided that they keep to be constant the combinations x and y. In particular, changes of η can be compensated by corresponding variations of P ee and f B , so that the combinations and therefore the observables are not changed. Clearly this is possible if P ee does not depend on the energy or time. Therefore the energy and the time variations of the probability break the degeneracy. In other words, time variations and distortion of the spectrum are, in general, sensitive to the fraction of sterile neutrinos. Since no significant distortion or variations have been found experimentally, the present data have no strong sensitivity to the fraction η as we will see from exact calculations in the sect. 3.
3 Global fit and bounds on fraction of sterile component

Global analysis
We have performed a global analysis of all available solar neutrino data taking into account possible presence of the sterile component. We follow the procedure described in our previous publication [21] where details can be found, and here we summarize the main ingredients of the analysis.
We use the same data sample as in [21], which consists of -three total rates: 1) the Ar-production rate, Q Ar , from Homestake [22], 2) the Ge−production rate, Q Ge from SAGE [24] and 3) the combined Ge−production rate from GALLEX and GNO [25]; -44 data points from the zenith-spectra measured by Super-Kamiokande [26] during 1496 days of operation; -38 day-night spectral points from SNO [13,14].
All solar neutrino fluxes (but the boron neutrino flux) are taken according to SSM BP2000 [28]. We use the boron neutrino flux as free parameter. For the hep−neutrino flux we take fixed value F hep = 9.3 × 10 3 cm −2 s −1 [28,29] .
In contrast to [21], the neutrino scheme includes now mixing with sterile component which is described by the parameter η (6). So, there are three oscillation parameters: the mass squared difference, ∆m 2 , and two mixings: tan 2 θ and η. Consequently, in the free boron neutrino flux fit we have four parameters: ∆m 2 , tan 2 θ, η and f B , and therefore there are 81(data points) -4 = 77 d.o.f..
We perform the χ 2 test of the oscillation solution by calculating where χ 2 rate , χ 2 SK and χ 2 SN O are the contributions from the total rates, the Super-Kamiokande zenith spectra and the SNO day and night spectra correspondingly. Each of the entries in (19) is the function of four parameters (∆m 2 , tan 2 θ, f B , η), in particular, We perform the global analysis for various fixed values of η. We first find the best fit points for a given η minimizing χ 2 global with respect to ∆m 2 , tan 2 θ and f B . This gives χ 2 min (η). The results are shown in the Table 1. The best fit corresponds to zero value of sterile fraction and f B = 1.05. So, the absolute minimum is characterized by χ 2 min (0). According to the Table 1, χ 2 increases with η, and moreover, the increase is fast for η > 0.3. ∆χ 2 = χ 2 min (0.4) − χ 2 min (0) = 2.4; for η = 0.6 this difference is ∆χ 2 = 7.5. Then we construct the 3σ regions in the ∆m 2 − tan 2 θ plane ( Fig. 1) minimizing χ 2 (∆m 2 , tan 2 θ, f B ) with respect to f B : we denote the corresponding minimal value of χ 2 by χ 2 f (∆m 2 , tan 2 θ). Then 3σ contours are determined by χ 2 f (∆m 2 , tan 2 θ) − χ 2 min (η) = 11.73, where χ 2 min (η) is the minimum χ 2 for a given value of η. We have found also lines of constant values of f B which minimize χ 2 for a given η. Fig. 1 and the Table 1, with increase of η the allowed region shifts to smaller values of tan 2 θ and ∆m 2 . At the same time f B and χ 2 min increase. These features can be well understood using Eqs. (15 -16). Increase of η [decrease of (1 − η)] can be compensated by increase of f B and decrease of P ee . Since P ee ∼ sin 2 θ, the decrease of P ee implies decrease of mixing angle. With decrease of mixing the distortion of the spectrum (mainly the turn up of the probability at low energies) becomes stronger. This leads to shift of the allowed region to smaller values of ∆m 2 (shift of the spectrum from the adiabatic edge).

According to
Let us consider breaking of parameter degeneracy. Taking P ee ≈ sin 2 θ in expressions (16) and using results of the Table 1 we find values of parameters x(η) and y(η) for different η in the best fit points (see Table 3.1). As follows from the Table 3.1 in the best fit points the parameters x(η) and y(η) are not invariant under changes of η: x slightly increases (by 5%) whereas y decreases significantly: almost by 50%. Another way to see these variations  Table 2: Values of parameters x and y in the best fit points of global analysis for different values of η.
is to keep x = Φ CC = constant and to exclude P ee from the combination y: Using Φ CC = 0.32 we find y(η) shown in the third line of the Table 3.1. Thus, the invariance of x and y (and therefore a degeneracy) is broken which leads the dependence of χ 2 min on η and therefore to sensitivity of the analysis to the presence of sterile neutrinos.
For fixed values of η and f B we minimize χ 2 with respect to ∆m 2 , tan 2 θ which gives Using this function we construct contours of constant confidence level in η, f B plane ( fig. 2). From this figure we get an upper limit for the sterile fraction The boron neutrino flux which correspond to maximal allowed value of η equals f B = 1.25 (1σ) and f B = 1.68 (3σ).
Imposing the SSM bound on the neutrino flux further strengthen the bound on "sterility". It was found in [17]: η < 0.25 at the 1σ and η < 0.5 at the 3σ level. Notice that in fact 1σ bound is unchanged since at this level the required boron neutrino flux is within 1σ SSM prediction. In contrast, the limit becomes stronger at the 3σ level, where substantially larger than in SSM flux is needed.

Who does not like sterile neutrino?
Let us identify observables which are sensitive to η. As we have discussed in sect. 2.1 these observables should contain information about the time variations or/and the energy dependence of the conversion effect. These dependences remove the degeneracy and therefore restrict η.
No statistically significant time variations have been found although some indications of the Day-Night asymmetry, A N D , exist [26,14]. Let us consider if A N D can restrict the admixture of sterile neutrino. The high energy neutrino data can be described by the ν e -survival probability during the day and during the night: P D ee , P N ee , or equivalently, by ∆P ee ≡ P N ee − P D ee andP ee ≡ (P N ee + P D ee )/2. Consequently, in Eq. (13 -15) we should substitute P ee →P ee and add to analysis an additional observable: In the LMA region the average suppression is determined basically by mixing angle:P ee ∼ sin 2 θ, whereas the DN asymmetry depends mainly on ∆m 2 . This can be seen from lines of constant ratio CC/NC ∼ P ee and lines of constant A N D which are nearly orthogonal each other (see figs. 5 and 6 from [21]). Therefore ∆P ee andP ee do not correlate and can be considered as independent fit parameters. For a givenP ee , the difference ∆P ee can change in the wide range, and the data on the asymmetry can be easily fitted. So, adding experimental information about the Day and Night asymmetry does not help to resolve degeneracy and therefore to improve the bound on η. Precise measurements of the zenith angle dependence of signal will change this conclusion.
No statistically significant distortion of the energy spectrum is found for E > 5 MeV, which supports the degeneracy problem. The distortion (in experiments sensitive to E > 5 MeV) is expected for higher values of ∆m 2 and small mixing. However, substantial distortion is expected over wider detectable energy range which includes also the pp− neutrinos. Therefore, the interplay of the high energy data and results from the low energy experiments should break the degeneracy.
To illustrate this effect, let us describe the distortion by two (for simplicity) different values of probabilities: at high energies, P H ee , and at low energies, P L ee (E < 0.5 MeV). The signal in the Gallium experiment is determined essentially by P L ee : so that the Geproduction rate equals Q Ge ∝ P L ee , and obviously, no substantial dependence on f B appears. In contrast to the Day-Night asymmetry case, the probabilities P L ee and P H ee are strongly (anti) correlated. Indeed, in the range of pp-neutrinos: and since P H ee ≈ sin 2 θ we get With decrease of P H ee the low energy probability P L ee increases. Consequently, the predicted value of the Ge-production rate increases. Thus, combination of the Gallium and SNO data should break the degeneracy of parameters.  Notice that with increase of η the quality of the fit does not change: χ 2 even slightly decreases. So, as far as SNO data alone are concerned, no bound on η appears. Variations of η can be completely compensated by changes of P ee and f B . The day-night asymmetry has low statistical significance.
The SNO experiment has higher sensitivity to the NC-events. For this reason the allowed region provided by SNO is restricted in tan 2 θ from both sides. Increase of η has the effect of moving the allowed region to smaller values of tan 2 θ, as can be seen in fig. 3.
Indeed, to keep the combinations (16) unchanged with increases of η one needs (i) to diminish P ee , which implies decrease of θ since in the LMA region P ee ∼ sin 2 θ, (ii) to decrease ∆m 2 to avoid distortion (turn up) of the spectrum at small energies and (iii) to increase f B to compensate decrease of P ee .
2). In fig. 3 we show also lines of constant Ge-production rate. The rate increases with decrease of mixing. According to the figure, in the SNO allowed region for η = 0.6 the rate Q Ge > 75 SNU and in the best fit point Q Ge = 83 SNU. For η = 0.4 we get Q Ge = 75 SNU in the best fit point. The combined results from SAGE, Gallex and GNO experiments is Q Ge = (70 ± 4) SNU. So, the Gallium results prevent from further shift of the allowed region to smaller θ 12 and therefore forbid larger η. To illustrate the role of Gallium experiments, we find the bounds on η that can be derived by taking only SNO and Gallium data: which are weaker than those obtained by taking all the data (23).
In principle, further more precise measurements of the Ge-production rate could improve the bound on the sterile component.
3). Let us comment on the impact of SuperKamiokande. In fig. 4 we show result of analysis of the SuperKamiokande data only for different values of η. The excluded (at 1σ) regions in the oscillation parameter space are due to the effects of spectrum distortion and the Earth ν e regeneration. They are only slightly modified by the presence of sterile component. With increase of η the excluded region shifts to larger mixing angles. This is related to the fact that with increase of sterile fraction the distortion of the spectrum increases due to decrease of the damping effect related to active neutrinos. The exclusion due to earth regeneration becomes weaker. Since for active-sterile system the effective matter potential is smaller the regeneration region shifts to smaller ∆m 2 . Correspondingly, lower bound of the allowed region shifts to smaller ∆m 2 .
In the allowed region, where distortion and time variations are small the data can be reproduced for different values of η by adjusting f B and P ee (see (15)).
In fig. 4 we show also f B that gives the best fit to the Super-Kamiokande data. With increase of η the lines f B = const shift to larger tan 2 θ and ∆m 2 (where the survival probability is larger) to compensate decrease of muon and tau neutrino contribution. Large values of η are allowed, when the boron neutrino flux is much larger than the SSM flux.
Similarly to the Gallium experiments, SuperKamiokande prevents from the shift of the allowed region to smaller mixings. Consequently, the combined fit of the SNO and SK will also give the bound on the sterile fraction. Comparison of the bounds (23) with and (27) without the SK result allows to evaluate the role of SuperKamiokande. 4). The Homestake experiment alone is not sensitive to η. However, it has some impact since its result restrict the oscillation parameters. The Ar-production rate decreases with mixing angle and the agreement with the Homestake result becomes even better with increase of η. Comparing the SNO and Homestake results we find that in the best fit SNO point for η = 0.6 the rate equals Q Ar = 2.75 SNU which is within 1σ of the Homestake result. So, the SNO and Homestake data favor presence of sterile component.
Summarizing, the bound on sterility appears as combined effect of several experiments. In the global analysis the constrain on η comes from the SNO measurements of the spectrum which contains information about the NC and CC event rate from the one hand side and Gallium as well as the SK experiment from the other side. With increase of η, SNO pushes the allowed region to smaller tan 2 θ, whereas Gallium and SK experiments put lower bound on tan 2 θ. For instance, the best SNO fit point for η = 0.6 lies in the excluded Gallium region and at the border of the SK excluded region.

N C measurements at SNO
SNO is taking data with much higher sensitivity to the NC current events, and therefore precision of measurements of Φ N C will be significantly improved. Clearly, distortion of the spectrum of the NC events will be signature of the sterile neutrino conversion. Although if LMA is the dominant solution, one does not expect observable effect. As far as total rate is concerned, the parameters can be adjusted in such a way that even more precise data can be reproduced for large η. Basically we will get the same shift of the allowed region as in fig. 1 but the size of the region around the the best fit point will be smaller. For this reason no drastic improvements of the bound on sterility is expected.
To evaluate the impact of future NC-measurements at SNO we show in fig. 5 the 1σ region of predictions of Φ N C as a function of η from the global fit. The results have been obtained in the following way. We use the global analysis of the solar neutrino data for different values of η, as is shown in fig. 1. Then we calculate Φ N C in the best fit points (solid line) as well as the spread of predictions for Φ N C within 1σ allowed region. Using the fig. 5, we find that the present 1σ experimental result on Φ N C , gives the upper bound η < 0.7 which agrees with the 3σ bound from the global analysis. Assuming three times smaller error bars for the NC measurements, Φ N C = 1 ± 0.03, we get the bound η < 0.5 which should be compared with present bound η < 0.64.

KamLAND and sterile fraction
The signal at KamLAND is determined by the vacuum oscillation survival probability, so that the rate of events is where F i is the flux from i reactor, σ is the cross-section ofνp → e + n reaction, A is the energy resolution function. The sum is taken over all reactors contributing to the flux at Kamioka and the integral denotes schematically the integrations over the neutrino energy, and the true energy of produced positron. This rate does not depend on η or f B . Being combined with the solar neutrino results it breaks degeneracy. KamLAND will fix the oscillation parameters ∆m 2 12 , tan 2 θ 12 , and consequently, will allow to predict P ee which describes the solar neutrino conversion.
Let us evaluate how combined analysis of the solar data and KamLAND can improve the bound on sterile component.
Let us introduce the suppression factor of total event rate above certain threshold: where N is given in (28) and N 0 is the event rate in absence of oscillations.
Since predictions for KamLAND do not depend on the state to which the electron neutrino converts, we can use results of calculations of R KL for the pure active case [21]. In fig. 8 of paper [21] we have presented lines of constant R KL in (∆m 2 , tan 2 θ) plane. In the best fit points we find R KL (0) = 0.66, R KL (0.2) = 0.63, R KL (0.4) = 0.55, R KL (0.6) = 0.52. Due to decrease of ∆m 2 in the best fit points, the expected value of the suppression in the KamLAND experiment becomes stronger.
To quantify impact of the event rate measurements at KamLAND we have constructed the χ 2 (∆m 2 , tan 2 θ, f B , η) as the function R KL . Practically we impose the following condition R KL (∆m 2 , tan 2 θ) = const which implies relation between ∆m 2 and tan 2 θ. So, for a given value R KL the χ 2 is the function of ∆m 2 or tan 2 θ only: For each value of R KL we minimize χ 2 with respects to tan 2 θ, f B , η thus finding For η = 0 the value R KL ∼ 0.725 is allowed. So, the introduction of sterile neutrino in the single ∆m 2 approach does not lead to significant increase of maximal allowed value of R KL . The goodness of the fit decreases sharply: e.g. for R KL = 0.8 we get ∆χ 2 = 25 even in presence of sterile component.
Let us analyze how the KamLAND spectral measurements can help in constraining η. We have simulated the KamLAND spectral data for 4 different points a), b), c), d) in the ∆m 2 − tan 2 θ plane (marked by stars in fig. 7a)). In simulations we use statistics which can be collected during 3 years in 600 tons above the energy threshold for visible energy E vis = 2.6 MeV. In absence of oscillations this would correspond to 1170 events. We assumed a 4% uncertainty in the overall normalization and a 2% uncertainty in the energy scale. We have simulated the energy spectrum (for each point) using 12 bins of 0.5 MeV size in the visible energy.
We perform χ 2 analysis of simulated spectra assuming a full correlation of systematics uncertainties. We minimize and construct contours of constant 1σ, 2σ, 3σ confidence level around selected points in the ∆m 2 , tan 2 θ plane (see fig. 7a). Notice that our allowed regions are somehow larger than those obtained in [31,32,35], since we include systematic errors and also use higher threshold (smaller number of events). In agreement with results of calculations of other groups [31,32,35] we find that KamLAND will be able to measure ∆m 2 rather precisely (if ∆m 2 < 2 · 10 −4 eV 2 ). At the same time determination of the mixing angle will not be very accurate. For the present best fit point (b) we get from the figure tan 2 θ = 0.28 − 0.60 at the 1σ level, and tan 2 θ = 0.25 − 0.85 at the 3σ level.
Then we have performed combined analysis of the existing solar and simulated Kam-LAND data (notice that by the time KamLAND will have 3 years statistics new solar neutrino data will appear). For each selected point of the oscillation parameters in fig. 7a we calculate a global χ 2 tot , which includes contributions from the solar neutrino analysis, χ 2 Sun , and the KamLAND, χ 2 KL ,: For each value of η we minimize χ 2 tot with respect to ∆m 2 , tan 2 θ and f B . Results of this minimization as a function of η are presented in fig. 7b. Let us comment on results for four selected points.
The point b) is the best fit point of the global fit which corresponds to η = 0. As follows from the figure, η < 0.19 (0.56) 1σ (3σ), (34) thus improving the bound (23) from the solar data analysis only.
The points c) and d) are at larger ∆m 2 and tan 2 θ 12 as compared with b) thus further removed from the region preferred in the case of non-zero sterile component (small ∆m 2 and tan 2 θ 12 ). If KamLAND selects these points the bound on sterile component will be stronger than in a): η < 0.09 (0.45) at 1σ (3σ).
The point a) is in the region preferable by sterile component. Selecting this point KamLAND will favor non-zero η. From the lower panel we get: η = 0.36 +0. 16 −0.12 at 1σ and η > 0 at 2σ level. The 1σ lower bound is η > 0.24.
For the points a) -d) we get the upper bounds on η: η < 0.08 (c and d), 0.19 (b) and 0.52 (a). We have performed similar analysis for all the points of the oscillation parameter space. We show in fig. 8 the results as lines of constant upper bounds in the oscillation parameter plane. As a tendency, the bound becomes weaker with decrease of mixing and ∆m 2 . For sufficiently small tan 2 θ the best fit corresponds to a non-zero value of η and the lower bound on η appears. In the lower panel of fig. 8 we show such a lower limit for all points in (∆m 2 , tan 2 θ) plane.

Beyond single ∆m 2 context
The results described in the previous sections can change if the electron neutrino mixes in the other mass eigenstates beyond the solar pair. Now more than one ∆m 2 is involved, and effect depends on specific values of additional ∆m 2 .
In what follows we will assume that mixing of ν e in the states ν 3 and ν 4 is small. Then if additional ∆m 2 are outside the range of the MSW enhancement (e.g. ∆m 2 14 is large) the effect of new ν e mixing is small being suppressed by the mixing itself. If, however, ∆m 2 14 is in the range of the MSW enhancement, the effect of additional mixing can be large. We will consider both possibilities.

Large additional ∆m 2
Suppose ν e has non-zero admixture in ν 3 characterized by by U e3 = s 13 . (Recall that ν 3 is separated from the solar neutrino pair by the mass gap which corresponds to the atmospheric neutrino split.) We assume that ν e -admixture in ν 4 is negligible so that the mass of ν 4 can be arbitrary within the allowed range by non-solar neutrino experiment. We assume also that in the limit s 13 → 0, the state ν 3 coincides with ν ′ a -the combination of ν µ and ν τ , which is orthogonal to ν a (7). In turn, ν 4 is the combination of ν s and ν a . As a result, we get the following mixing pattern: ν 1 = cos θ 12 (c 13 ν e − s 13 ν ′ a ) − sin θ 12 ν x , ν 2 = sin θ 12 (c 13 ν e − s 13 ν ′ a ) + cos θ 12 ν x , ν 3 = c 13 ν ′ a + s 13 ν e , where ν x is defined in (6) and ν ′ x is the orthogonal to ν x combination of ν s and ν a . The mixing (35) differs from the generic case since ν e and ν ′ a do not mix in ν 4 , and ν a and ν s do not mix in ν 3 .
The dynamics of propagation is similar to that of the case of three neutrinos. Indeed, let us represent ν e in the following form: where ν ′ e = cos θ 12 ν 1 + sin θ 12 ν 2 coincides with ν x , as can be easily checked. For the energies of solar neutrinos the matter effect on s 13 is negligible. Furthermore, the oscillations of solar neutrinos related to ν 3 are averaged out. At the same time, ν ′ e and ν x form standard two neutrino system with vacuum mixing and mater effects.
Using these features of dynamics, as well as projection of original flavor states onto ν ′ e , ν x , ν 3 one gets immediately the conversion probabilities: whereP ee is the ν ′ e -survival probability in the ν ′ e − ν x system:P ee =P ee (θ 12 , ∆m 2 12 ,Ṽ ) and η). As in the previous case, the parameter η determines fraction of ν s in the state ν x to which ν ′ e converts, and also it gives total fraction of the ν s in the solar pair (9).
Using (37), we get fluxes (in the units of the SSM flux) measurable by different reactions: Comparing with Eq. (13,14,15) we find that corrections are indeed of the order s 2 13 . For the allowed values of s 2 13 these corrections have small impact on the data fit, as one can see from our analysis in [21].
The fluxes Φ CC , Φ N C , Φ ES depend on two combinations of parameters, as in Eq. (17), with substitution x → x ′ , y → y ′ , where and x and y are defined in (16). So, the problem of degeneracy is not resolved with s 13 corrections. Notice also that the fluxes in (38 -40) satisfy the relation (18).
The KamLAND signal is determined by the survival probability where P 2 is the two neutrino vacuum oscillation probability determined by ∆m 2 12 and θ 12 . The factor c 4 13 leads to additional suppression of the KamLAND signal and shift of the allowed regions to smaller θ 12 (see detailed analysis in the 3ν context in [34]).

Small additional ∆m 2
Let us consider the case when even small ν e mixing beyond the solar pair can produce a strong effect. Suppose that, (i) ν e mixes with ν x in the solar pair, ν 1 and ν 2 , which has ∆m 2 12 and tan 2 θ 12 in the LMA region, as in (35); (ii) the state ν 4 ≈ ν ′ x (see (35)) is heavier than ν 1 and the mass splitting is ∆m 2 41 ∼ 10 −6 eV 2 . (The mass m 1 can be very small or even zero). Furthermore, ν e has small admixture in ν 4 : tan 2 θ 14 ∼ 10 −3 , so that ν e − ν ′ x oscillation parameters are in the SMA region. (iii) the admixture of ν e in the ν 3 is zero (for simplicity).
Then, similarly to (35) we can write the mixing as Here ν ′ x is the combination of ν a and ν s orthogonal to ν x (6), and ν ′ a is the combination of ν µ and ν τ orthogonal to ν a (7).
The system has two resonances associated with ∆m 2 12 and ∆m 2 41 . Let us estimate qualitatively the conversion effect in different energy ranges.
In the first resonance related to ∆m 2 12 the conversion is completely adiabatic. So, at the high energies, E > 7 − 8 MeV, where crossing of the resonance occurs the survival probability equals P H ee ≈ sin 2 θ 12 . At low energies, E < 0.5 MeV, (no 12-resonance crossing) effect driven by ∆m 2 12 is reduced to averaged vacuum oscillations. The result of conversion is similar to the one in usual three neutrino system: where P ′ 2 is the survival probability in the resonance related to ∆m 2 14 . It has a typical form of the survival probability of the SMA solution and characterized by ∆m 2 14 and mixing parameter tan 2 θ = tan 2 θ 14 / cos 2 θ 12 . Appearance of P ′ 2 in this expression leads to an additional suppression of the pp-neutrino flux.
In the intermediate energy range, E = (0.5 − 7) MeV, the conversion driven by ∆m 2 14 will lead to flattening of the adiabatic edge of the suppression pit due to ∆m 2 12 .

KamLAND result and LMA
In the best fit point of the LMA region we predict R KL = 0.66 for zero η [21]. At 3σ level R KL < 0.73. In presence of the sterile component this bound can be slightly relaxed: The following comments are in order: If solution of the solar neutrino problem is in the LMA region, a strong suppression is expected in KamLAND. In this case, introduction of additional sterile states will not help in the single ∆m 2 context, as we have found in the Sect. 3.
Let us discuss what happens if two or more ∆m 2 contributes.
Oscillation parameters can be taken beyond the LMA region in such a way that suppression in the KamLAND experiment is weaker. In this case, however, the conversion of solar neutrinos will not describe data in the two neutrino context. Then the parameters (mass, flavor mixing) of the ν 4 can be selected in such a way that conversion driven by ν 4 will not change KamLAND result but correct description of the solar data.
Such a possibility can be realized in the scheme described in the previous section 5.2 with ∆m 2 12 ∼ 6 · 10 −5 eV 2 . Indeed the KamLAND result is determined by the oscillations driven by ∆m 2 12 and θ 12 . For tan 2 θ 12 ∼ 0.10 − 0.15 we get R KL = 0.80 − 0.85. As far as the solar neutrino signal is concerned, the high energy signal will be suppressed by sin 2 θ, so that large boron neutrino flux: f B ∼ 1.6 is required to explain the SNO and SK results. According to our previous consideration, in the single ∆m 2 context this point would be excluded by the Gallium experiment (large Q Ge ) and by SK (distortion of spectrum). The effect of ∆m 2 14 changes a situation: appearance of P ′ 2 in expression for probability at low energies (44) leads to an additional suppression of the pp-neutrino flux, and consequently, Q Ge rate. In the intermediate energy range, E = (0.5 − 7) MeV, the conversion driven by ∆m 2 14 will relax the SuperKamiokande lower bound on mixing which follows from absence of spectrum distortion. The detailed analysis of this possibility will be given elsewhere [36].
Another possibility to reconcile an intermediate suppression of signal in KamLAND and explanation of the solar neutrino data is to split two problems. Suppose the solar pair has ∆m 2 12 in the LOW or VO regions, so that it does not produce any effect in KamLAND. The state ν 4 which consists predominantly of the sterile component has the mass split: ∆m 2 14 > 10 −4 eV 2 . Then the suppression of the KamLAND signal is determined by admixture of ν e in ν 4 : R KL = 0.5 sin 2 2θ 14 . For instance, R KL = 0.85 can be achieved at sin 2 θ 14 = 0.09. Effect in solar neutrinos will be determined by P sun ee = c 4 14 P ee + s 4 14 , where P ee is the survival probability for the conversion driven by ∆m 2 12 . Additional factor c 4 14 ∼ 0.8 can even improve the fit of the data for LOW and SMA [21].

Conclusions
1. We have studied properties of mixing of the sterile and active neutrinos. In the single ∆m 2 context (when electron neutrino is present in two mass eigenstates only) the problem is reduced precisely to two neutrino problem with mixing of ν e with ν x in the solar pair (ν x is the combination of the active and sterile neutrinos).

2.
We have performed the global fit of the solar neutrino data in the single ∆m 2 context, treating the boron neutrino flux as a free parameter. The best fit corresponds to zero admixture of sterile component, η = 0. We get the upper bound η < 0.26 (0.64) at 1σ (3σ). With increase of η the best fit point shifts to smaller ∆m 2 and tan 2 θ 12 .
3. Due to degeneracy of parameters, no one single observable gives the bound on η. The bound on sterility appears as an interplay of the SNO results whose allowed region of the oscillation parameters shifts to smaller mixings and ∆m 2 with increase of η and Gallium as well as SK results which restrict the mixing and ∆m 2 from below. 4. Further more precise measurements of the NC event rate at SNO will strengthen the bound: η < 0.5 (3σ).

Implications of KamLAND depend on specific results of KamLAND measurements. If
KamLAND result (rate, spectrum) corresponds to the best fit point of the solar neutrino analysis, then after 3 years of the KamLAND operation the limit will be strengthened down to η < 0.19 (0.55) at 1σ (3σ). The limit will be stronger if the KamLAND best fit point will be at larger tan 2 θ 12 and ∆m 2 than the solar neutrino data give at present. In contrast, if KamLAND selects smaller tan 2 θ 12 and ∆m 2 , the limit will be weaker, and moreover, the data may prefer non-zero sterile admixture. 6. The presence of sterile neutrinos (in the single ∆m 2 context) relaxes the upper bound on the predicted rate R KL only weakly from 0.73 to 0.76 (3σ). Larger values of R KL can be reached in the two ∆m 2 context. 7. If electron neutrino mixes also in other mass eigenstates, the effects depend substantially on values of other ∆m 2 .
If additional ∆m 2 14 ≫ ∆m 2 12 the small corrections appear to formulas with a single ∆m 2 which are proportional to the mixing of ν e in additional mass eigenstates, e.g., s 2 13 . If ∆m 2 , is in the MSW range an additional mixing can be resonantly enhanced in matter producing much larger effect. In this case intermediate values of R KL can be obtained.