Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses
MiniBooNE Collaboration, A. A. Aguilar-Arevalo, C. E. Anderson, A. O. Bazarko, S. J. Brice, B. C. Brown, L. Bugel, J. Cao, L. Coney, J. M. Conrad, D. C. Cox, A. Curioni, R. Dharmapalan, Z. Djurcic, D. A. Finley, B. T. Fleming, R. Ford, F. G. Garcia, G. T. Garvey, J. Grange, C. Green, J. A. Green, T. L. Hart, E. Hawker, W. Huelsnitz, R. Imlay, R. A. Johnson, G. Karagiorgi, P. Kasper, T. Katori, T. Kobilarcik, I. Kourbanis, S. Koutsoliotas, E. M. Laird, S. K. Linden, J. M. Link, Y. Liu, Y. Liu, W. C. Louis, K. B. M. Mahn, W. Marsh, C. Mauger, V. T. McGary, G. McGregor, W. Metcalf, P. D. Meyers, F. Mills, G. B. Mills, J. Monroe, C. D. Moore, J. Mousseau, R. H. Nelson, P. Nienaber, J. A. Nowak, B. Osmanov, S. Ouedraogo, R. B. Patterson, Z. Pavlovic, D. Perevalov, C. C. Polly, E. Prebys, J. L. Raaf, H. Ray, B. P. Roe, A. D. Russell, V. Sandberg, R. Schirato, D. Schmitz, M. H. Shaevitz, F. C. Shoemaker, D. Smith, M. Soderberg, M. Sorel, P. Spentzouris, J. Spitz, I. Stancu, R. J. Stefanski, M. Sung, H. A. Tanaka, R. Tayloe, M. Tzanov, R. G. Van de Water, M. O. Wascko, D. H. White, M. J. Wilking, H. J. Yang, G. P. Zeller, E. D. Zimmerman
aa r X i v : . [ h e p - e x ] J un Test of Lorentz and CPT violation with Short BaselineNeutrino Oscillation Excesses
A. A. Aguilar-Arevalo , C. E. Anderson , A. O. Bazarko , S. J. Brice ,B. C. Brown , L. Bugel , J. Cao , L. Coney , J. M. Conrad ,D. C. Cox , A. Curioni , R. Dharmapalan , Z. Djurcic , D. A. Finley ,B. T. Fleming , R. Ford , F. G. Garcia , G. T. Garvey , J. Grange ,C. Green , , J. A. Green , , T. L. Hart , E. Hawker , , W. Huelsnitz ,R. Imlay , R. A. Johnson , G. Karagiorgi , P. Kasper , T. Katori , ,T. Kobilarcik , I. Kourbanis , S. Koutsoliotas , E. M. Laird ,S. K. Linden ,J. M. Link , Y. Liu , Y. Liu , W. C. Louis ,K. B. M. Mahn , W. Marsh , C. Mauger , V. T. McGary ,G. McGregor , W. Metcalf , P. D. Meyers , F. Mills , G. B. Mills ,J. Monroe , C. D. Moore , J. Mousseau , R. H. Nelson , P. Nienaber ,J. A. Nowak , B. Osmanov , S. Ouedraogo , R. B. Patterson ,Z. Pavlovic , D. Perevalov , , C. C. Polly , E. Prebys , J. L. Raaf ,H. Ray , B. P. Roe , A. D. Russell , V. Sandberg , R. Schirato ,D. Schmitz , M. H. Shaevitz , F. C. Shoemaker , D. Smith ,M. Soderberg , M. Sorel , P. Spentzouris , J. Spitz , I. Stancu ,R. J. Stefanski , M. Sung , H. A. Tanaka , R. Tayloe , M. Tzanov ,R. G. Van de Water , M. O. Wascko , D. H. White , M. J. Wilking ,H. J. Yang , G. P. Zeller , E. D. Zimmerman (The MiniBooNE Collaboration) University of Alabama; Tuscaloosa, AL 35487 Argonne National Laboratory; Argonne, IL 60439 Bucknell University; Lewisburg, PA 17837 University of Cincinnati; Cincinnati, OH 45221 University of Colorado; Boulder, CO 80309 Columbia University; New York, NY 10027 Embry Riddle Aeronautical University; Prescott, AZ 86301 Fermi National Accelerator Laboratory; Batavia, IL 60510 University of Florida; Gainesville, FL 32611 Present address: University of California; Riverside, CA 92521 Corresponding author. Teppei Katori: [email protected] deceased Present address: IFIC, Universidad de Valencia and CSIC, Valencia 46071, Spain Present address: Imperial College; London SW7 2AZ, United Kingdom
Preprint submitted to Physics Letters B October 31, 2018 Indiana University; Bloomington, IN 47405 Los Alamos National Laboratory; Los Alamos, NM 87545 Louisiana State University; Baton Rouge, LA 70803 Massachusetts Institute of Technology; Cambridge, MA 02139 Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, D.F.04510, M´exico University of Michigan; Ann Arbor, MI 48109 Princeton University; Princeton, NJ 08544 Saint Mary’s University of Minnesota; Winona, MN 55987 Virginia Polytechnic Institute & State University; Blacksburg, VA 24061 Yale University; New Haven, CT 06520
Abstract
The sidereal time dependence of MiniBooNE ν e and ¯ ν e appearance data areanalyzed to search for evidence of Lorentz and CPT violation. An unbinnedKolmogorov-Smirnov test shows both the ν e and ¯ ν e appearance data are com-patible with the null sidereal variation hypothesis to more than 5%. Usingan unbinned likelihood fit with a Lorentz-violating oscillation model derivedfrom the Standard Model Extension (SME) to describe any excess eventsover background, we find that the ν e appearance data prefer a sidereal time-independent solution, and the ¯ ν e appearance data slightly prefer a siderealtime-dependent solution. Limits of order 10 − GeV are placed on combina-tions of SME coefficients. These limits give the best limits on certain SMEcoefficients for ν µ → ν e and ¯ ν µ → ¯ ν e oscillations. The fit values and limits ofcombinations of SME coefficients are provided. Keywords:
MiniBooNE Neutrino oscillation Lorentz violation PACS:11.30.Cp 14.60.Pq 14.60.St
1. Introduction of Loremtz violation
Violation of Lorentz invariance and CPT symmetry is a predicted phe-nomenon of Planck scale physics, especially with a spontaneous violation [1],and it does not require any modifications in quantum field theory or generalrelativity. Since neutrino oscillation experiments are natural interferome-ters, they can serve as sensitive probes of space-time structure. Neutrinooscillations have the potential to provide the first experimental evidence for2orentz and CPT violation through evidence of oscillations that deviate fromthe standard
L/E dependence [2], or that show sidereal time dependent os-cillations as a consequence of a preferred direction in the universe [3].In this paper, we test the MiniBooNE ν µ → ν e and ¯ ν µ → ¯ ν e oscillationdata [4, 5] for the presence of a Lorentz violation signal. Similar analyseshave been performed in other oscillation experiments, including LSND [6],MINOS [7], and IceCube [8]. Naively, experiments with longer baselinesand higher energy neutrinos would be expected to have better sensitivity toLorentz violation, because small Lorentz violating terms are more prominentat high energy, where neutrino mass terms are negligible. However, someLorentz violating neutrino oscillation models mimic the standard massiveneutrino oscillation energy dependence [9]. Then, in this case, the signalmay only be seen in sidereal variations of oscillation experiments.
2. MiniBooNE experiment
MiniBooNE is a ν e (¯ ν e ) appearance short baseline neutrino oscillationexperiment at Fermilab. Neutrinos are created by the Booster NeutrinoBeamline (BNB), which produces a 93% (83%) pure ν µ (¯ ν µ ) beam in neutrino(anti-neutrino) mode, determined by the polarity of the magnetic focusinghorn. The MiniBooNE Cherenkov detector, a 12.2 m diameter sphere filledwith mineral oil, is used to detect charged particles from neutrino interactionsand is located 541 m from the neutrino production target. It is equipped with1,280 8” PMTs in an optically separated inner volume, and 240 8” veto PMTsin an outer veto region. Details of the detector and the BNB can be foundelsewhere [10, 11]. Charged leptons created by neutrino interactions in thedetector produce Cherenkov photons, which are used to reconstruct chargedparticle tracks [12]. The measured angle and kinetic energy of charged leptonsfrom neutrino interactions are used to reconstruct the neutrino energy, E QEν ,for each event, under the assumption that the target nucleon is at rest insidethe nucleus and the interaction type is a charged current quasielastic (CCQE)interaction [13].For this analysis, we use the background and error estimates from [14](neutrino mode) and [15] (antineutrino mode). For neutrino mode, data from6 . × protons on target (POT) are used. An excess in the “low energy”region (200 < E QEν (MeV) < . ± . ± . L/E energy dependence of a simple3wo massive neutrino oscillation model. Additionally, it is not consistent withthe energy region expected for the “LSND” signal [16]. For the anti-neutrinomode analysis (5 . × POT), MiniBooNE observed a small excess in thelow energy region, and an excess in the region 475 < E
QEν (MeV) < < E QEν (MeV) < ν e candidate events as compared to the prediction, 200 . ± . ± .
3. Analysis
We use the Standard Model Extension (SME) formalism for the generalsearch for Lorentz violation [21]. The SME is an effective quantum fieldtheory, and the minimum extension of the Standard Model including particleLorentz and CPT violation [21]. A variety of data have been analyzed underthis formalism [22], including neutrino oscillations [6, 7, 8]. In the SMEformalism for neutrinos, the evolution of a neutrino can be described by aneffective Hamiltonian [3],( h ν eff ) ab ∼ E [( a L ) µ p µ − ( c L ) µν p µ p ν ] ab . (1)Here, E and p µ are the energy and the 4-momentum of a neutrino, and ( a L ) µab and ( c L ) µνab are CPT-odd and CPT-even SME coefficients in the flavor basis.Under the assumption that the baseline is short compared to the oscillationlength [23], the ν µ → ν e oscillation probability takes the form, P ≃ L ( ~ c ) | ( C ) eµ + ( A s ) eµ sin ω ⊕ T ⊕ + ( A c ) eµ cos ω ⊕ T ⊕ ( B s ) eµ sin 2 ω ⊕ T ⊕ + ( B c ) eµ cos 2 ω ⊕ T ⊕ | . (2) This probability is a function of sidereal time T ⊕ . Four parameters, ( A s ) eµ ,( A c ) eµ , ( B s ) eµ , and ( B c ) eµ are sidereal time dependent, and ( C ) eµ is a siderealtime independent parameter. We use a baseline distance of L =522.6 m,where the average pion decay length is subtracted from the distance betweenthe neutrino production target and detector. And ω ⊕ is the sidereal timeangular frequency described shortly. These parameters are expressed in termsof SME coefficients and directional factors [23]. The same formula describesthe ¯ ν µ → ¯ ν e oscillation probability by switching the signs of the CPT-oddSME coefficients. We neglect the standard neutrino mass term, m eµ /E ≪ − GeV, which is well below our sensitivity which is discussed later.For this analysis, we convert the standard GPS time stamp for each eventto local solar time (period 86400 . . ω ⊙ = π . (rad/s)and the sidereal time angular frequency ω ⊕ = π . (rad/s). The time origincan be arbitrary, but we follow the standard convention with a Sun-centeredcoordinate system [6]. We choose a time-zero of 58 min after an autumnalequinox of 2002 (September 23, 04:55GMT), so that this serves not onlyas the sidereal time-zero, but also as the solar time-zero since Fermilab ison the midnight point at this time. The local coordinates of the BNB arespecified by three angles [23], colatitude χ = 48 . ◦ , polar angle θ = 89 . ◦ ,and azimuthal angle φ = 180 . ◦ .Any time dependent background variation, such as the time variationof detector and BNB systematics, are important. To evaluate these, weuse our high statistics CCQE samples (Figure 1). These data are from our ν µ CCQE double differential cross section measurement sample [24] composedof 146 ,
070 events (5 . × POT), and our ¯ ν µ CCQE candidate sample [25]composed of 47 ,
466 events (5 . × POT). The ν µ (¯ ν µ )CCQE local solartime distribution exhibits ± ν µ CCQE sidereal time distribution, however, it persists in ∼ ν µ CCQE sidereal time distribution. We evaluate the impact ofthis variation on our analysis by correcting POT variation event by eventin ν e (¯ ν e ) candidate data. It turned out the correction only has a negligibleeffect. Thus we decided to use unweighted events. This also simplifies the5 ocal solar time (sec) CC Q E eve n t s mn CCQE variationPOT variation (arbitrary)flat distribution local solar time (sec) CC Q E eve n t s mn a n t i - Figure 1: The top (bottom) plot shows the ν µ CCQE (¯ ν µ CCQE) local solar time distri-bution. The solid curves are fit functions extracted from the CCQE event distributions,and the dotted curves are from the POT distributions (arbitrary units) during the sameperiod of data taking. The dashed line shows a flat distribution. unbinned likelihood function used in later analysis. Figures 2 and 3 showthe ν e and ¯ ν e oscillation candidates sidereal time distributions both withand without the POT correction. These plots verify that time-dependentsystematics are negligible in this analysis.To check for a general deviation from a flat distribution (null sidereal vari-ation hypothesis), we perform an unbinned Kolmogorov-Smirnov test (K-Stest) [26] as a statistical null hypothesis test for both the ν e and ¯ ν e sam-ples. The K-S test is suitable in our case because it is sensitive to runs indistributions, which may be a characteristic feature of the sidereal time de-pendent hypothesis. Table 1 gives the result. The K-S test is applied tothe low energy, high energy, and combined regions, for both neutrino, andanti-neutrino mode data. To investigate the time dependent systematics, wealso apply the K-S test to the local solar time distribution. The test shows6 ow energy high energy combinedsolar sidereal solar sidereal solar siderealNeutrino mode < E ν > .
36 GeV 0 .
82 GeV 0 .
71 GeV P (KS) 0.42 0.13 0.81 0.64 0.64 0.14Anti-neutrino mode < E ν > .
34 GeV 0 .
78 GeV 0 .
60 GeV P (KS) 0.62 0.15 0.79 0.39 0.69 0.08 Table 1: A summary of K-S test results on the sidereal and local solar time distributions.The top table is for ν e candidate data, and the bottom table is for ¯ ν e candidate data. Thethree rows show the average neutrino energy of each sample, number of events, and theK-S test compatibility with the null hypothesis. The test is performed in three energyregions, and for both solar local time and sidereal time distributions. none of the twelve samples has less than 5% compatibility ( ∼ σ ), whichwe chose as a benchmark prior to the analysis. Hence, all samples are com-patible with the null sidereal variation hypothesis. Interestingly, the siderealtime distributions tend to show lower compatibility with a flat hypothesis,but not by a statistically significant amount. These results indicate that anysidereal variation extracted from our data, discussed below, is not expectedto be statistically significant.To fit the data with the sidereal time-dependent model, we use a gen-eralized unbinned maximum likelihood method [27]. This method finds thebest fit model parameters by fitting data with a log likelihood function ℓ . Itis suitable for our analysis because this method has the highest statisticalpower for a low statistics sample. In this method, the log likelihood function ℓ is constructed by adding ℓ i from each event. After dropping all constants, ℓ i has the following expression, ℓ i = − N ( µ s + µ b ) + ln [ µ s F is + µ b F ib ] − N (cid:18) µ b − µ b σ b (cid:19) . (3) Here, N is the number of observed candidate events, µ s is the predicted num-ber of signal events which is given by the time integral of Eq. 2 together withthe estimated efficiency, µ b is the predicted number of background events, F s is the probability density function (PDF) for the signal and is a function7f sidereal time and the fitting parameters (Eq. 2 with proper normaliza-tion), F b is the PDF for the background, σ b is the 1 σ error on the predictedbackground, and µ b is the central value of the predicted total backgroundevents. Two sources contribute equally to the background: intrinsic beambackground and mis-identification (mainly π ◦ s). Their total variation is as-signed as the systematic error, assumed to be time independent. Details canbe found in [4, 5]. The parameter space is scanned (grid search method)to find the largest ℓ , or the maximum log likelihood (MLL) point, and thisMLL point provides the combination of the best fit (BF) parameters. The loglikelihood function includes six parameters, five that are functions of SMEcoefficients, and one for the background. However the background term isconstrained within a ± σ range. Neither the neutrino nor the anti-neutrinomode data allow us to extract errors if we fit all five parameters at once, dueto the high correlation of parameters. Therefore, we set ( B s ) eµ and ( B c ) eµ tozero and concentrate on three parameters (( C ) eµ , ( A s ) eµ , and ( A c ) eµ ) for theuncertainty estimates. Since the five parameter fit is quantitatively similarto the three parameter fit, we will focus the discussion on the results on thethree parameter fits. This three parameter fit also corresponds to the casewith only CPT-odd SME coefficients [23].
4. Results
Figure 2 shows the neutrino mode low energy region fit results. The topthree plots show the three projections of three dimensional parameter space.Because of the square of fitting parameters in the PDF, the BF point has asign ambiguity and is always duplicated. The 1 σ and 2 σ contours are formedfrom a constant slice of the log likelihood function in the three dimensionalparameter space. To avoid under coverage, these slices are expanded untilthey enclose 68% (1 σ ) or 95% (2 σ ) of BF points for the three parameter fitof simulated, or “fake”, data sets with the signal. Note that because fittingparameters are not linear in the PDF, twice the 1 σ error does not yield the2 σ error.A null sidereal variation hypothesis, or a flat solution, is equivalent to athree or five parameter fit solution where only the ( C ) eµ parameter is nonzero.The fit to neutrino data favors a nonzero solution only for the ( C ) eµ term.The bottom plot in Figure 2 shows data plotted against curves correspondingto the flat solution and the best fits for three and five parameter functions.Since all three curves are close to each other, the solution of neutrino mode is8 − mode BF 2 σ limit ¯ ν − mode BF 2 σ limit | ( C ) eµ | . ± . ± . < . . ± . ± . < . | ( A s ) eµ | . ± . ± . < . . ± . ± . < . | ( A c ) eµ | . ± . ± . < . . ± . ± . < . − GeV) | ( C ) eµ | ± [( a L ) Teµ + 0 . a L ) Zeµ ] − < E > [1 . c L ) T Teµ + 1 . c L ) T Zeµ + 0 . c L ) ZZeµ ] | ( A s ) eµ | ± [0 . a L ) Yeµ ] − < E > [1 . c L ) T Yeµ + 0 . c L ) Y Zeµ ] | ( A c ) eµ | ± [0 . a L ) Xeµ ] − < E > [1 . c L ) T Xeµ + 0 . c L ) XZeµ ]Table 2: The fit parameters for the neutrino mode low energy region and the anti-neutrinomode combined region. The BF points are the MLL points of the log likelihood function,here top rows from left to right, BF values, 1 σ statistical, and systematic errors. The2 σ limits are also shown. Bottom rows show detailed expressions of each sidereal fitparameter in terms of SME parameters, and directional factors [23]. The upper (lower)sign of ( a L ) λeµ terms is applied for neutrino (anti-neutrino) results, due to the CPT-oddnature. The average neutrino energy “ < E > ” is 0 .
36 GeV for the neutrino mode lowenergy region and 0 .
60 GeV for the anti-neutrino mode combined region (Tab. 1). dominated by the sidereal time-independent component. To find the signifi-cance of time dependence over the flat distribution, fake data sets without asignal are formed where the ν e candidate events are simulated without anytime structure. The MLL difference between the three parameter fit andthe flat solution is used to form a ∆ χ , and the expected ∆ χ distributionis determined by testing 500 random distributions from the fake data sets.This test shows that there is a 26 .
9% chance that a random distribution of ν e candidate events would yield a ∆ χ value equal to, or greater than, thevalue observed for the data. This result is consistent with the sensitivity ofthis experiment. We estimate our sensitivity to the limitted case. First, a2 σ threshold is set from this ∆ χ distribution. Then, time dependent ampli-tudes were incrementally increased in the model until the 2 σ threshold wasexceeded. When we assume ( C ) eµ = 0 and ( A s ) eµ = ( A c ) eµ = 0 the 2 σ dis-covery threshold of sidereal time dependent amplitudes from ν e (¯ ν e ) candidatedata statistics are 0 . . × − GeV.Figure 3 shows the analogous fit results for the anti-neutrino mode com-bined energy region. Due to lower statistics, the combined region is usedrather than dividing the data into two subsets. Unlike the neutrino modelow-energy region, the ( C ) eµ parameter no longer significantly deviates fromzero. The fit to anti-neutrino data favors a non-zero solution for the ( A s ) eµ and ( A c ) eµ parameters at the nearly 2 σ level. Performing the same ∆ χ test9 GeV) -20 (10 m e (C) -5 -4 -3 -2 -1 0 1 2 3 4 5 G e V ) - ( m e ( A s ) -5-4-3-2-1012345 BF points s s GeV) -20 (10 m e (C) -5 -4 -3 -2 -1 0 1 2 3 4 5 G e V ) - ( m e ( A c ) -5-4-3-2-1012345 GeV) -20 (10 m e (As) -5 -4 -3 -2 -1 0 1 2 3 4 5 G e V ) - ( m e ( A c ) -5-4-3-2-1012345 sidereal time (sec) - o sc ca nd i d a t e eve n t s n dataPOT corrected databackground flat solution3 parameter fit5 parameter fit Figure 2: (color online) Three parameter fit results for the neutrino mode low energyregion. The top three plots show the projection of three dimensional parameter space.The dark (light) shaded area shows the 1 σ (2 σ ) contour in each projection. The starsshow the BF points. The bottom plot shows the curves corresponding to the flat solution(dotted), three parameter fit (solid), and five parameter fit (dash-dotted), together withbinned data (solid marker). The POT corrected data are also shown in open circle marker.Here, the fitted background is shown as a dashed line, and the BF value is 1.00 ( i.e. ,equivalent to the central value of the predicted background). as is outlined above results in only 3 .
0% of the random distributions fromthe ¯ ν e candidate events having a ∆ χ value exceeding the value observed forthe data. Note that this is consistent with the 8% compatibility with a flathypothesis found with the K-S test (Tab. 1).Table 2 shows fit parameters for the neutrino mode low energy regionand anti-neutrino mode combined region. All errors are estimated from 1 σ contours of parameter space projections. Errors are asymmetric, but wechoose the larger excursions as the symmetric errors for simplicity. The 2 σ contours provide the limits. In principle, these fit parameters are complex10 GeV) -20 (10 m e (C) -5 -4 -3 -2 -1 0 1 2 3 4 5 G e V ) - ( m e ( A s ) -5-4-3-2-1012345 BF points s s GeV) -20 (10 m e (C) -5 -4 -3 -2 -1 0 1 2 3 4 5 G e V ) - ( m e ( A c ) -5-4-3-2-1012345 GeV) -20 (10 m e (As) -5 -4 -3 -2 -1 0 1 2 3 4 5 G e V ) - ( m e ( A c ) -5-4-3-2-1012345 sidereal time (sec) - o sc ca nd i d a t e eve n t s n dataPOT corrected databackground flat solution3 parameter fit5 parameter fit Figure 3: Three parameter fit results for the anti-neutrino mode combined region. Nota-tions are the same as Fig. 2. Here, the BF value for the fitted background is 0.97 (3%lower than the central value of the predicted background). numbers. Here, all parameters are assumed to be real. A naive estimationfrom Tab. 2 indicates possible SME coefficients to satisfy the MiniBooNEdata are of order 10 − GeV (CPT-odd), and 10 − to 10 − (CPT-even).However, these SME coefficients are too small to produce a visible effect forLSND [6]. On the other hand, any SME coefficients extracted from LSND [6]predict too large of a signal for MiniBooNE. Therefore, a simple pictureusing Lorentz violation to explain both data sets leaves some tension, and amechanism to cancel the Lorentz violating effect at high energy [3, 19, 20] isneeded.
5. summary
In summary, we performed a sidereal time variation analysis for Mini-BooNE ν e and ¯ ν e appearance candidate data. For the neutrino mode low11nergy region, K-S test statistics indicate the null hypothesis is compatibleat the 13% level, and the relative improvement in the likelihood betweenthe null hypothesis and the three parameter fit occurs 26 .
9% of the time inrandom distributions from a null hypothesis. Analysis of the combined en-ergy region in anti-neutrino mode results in a K-S test that indicates a 8%compatibility with the null hypothesis, however the relative improvement inthe likelihood between the null hypothesis and the three parameter fit onlyoccurs 3 .
0% of the time in random distributions from a null hypothesis. Thelimits of fit parameters, 10 − GeV, are consistent with Planck scale sup-pressed physics. This is the first sidereal variation test for an anti-neutrinobeam of ∼ ∼
500 m base line. These limits are currentlythe best limits on the sidereal-time independent ( a L ) eµ and ( c L ) eµ SME co-efficients. These limits can be significantly improved by long baseline ν e (¯ ν e )appearance experiments, such as T2K [28] and NOvA [29]. acknowledgment This work was conducted with support from Fermilab, the U.S. Depart-ment of Energy, the National Science Foundation and the Indiana UniversityCenter for Spacetime Symmetries.
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