A Short Travel for Neutrinos in Large Extra Dimensions
PPrepared for submission to JHEP
A Short Travel for Neutrinos in Large ExtraDimensions
G. V. Stenico , , D. V. Forero and O. L. G. Peres Instituto de F´ısica Gleb Wataghin - UNICAMP, 13083-859, Campinas, SP, Brazil Northwestern University, Department of Physics & Astronomy, 2145 Sheridan Road, Evanston,IL 60208, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Neutrino oscillations successfully explain the flavor transitions observed in neu-trinos produced in natural sources like the center of the sun and the earth atmosphere, andalso from man-made sources like reactors and accelerators. These oscillations are drivenby two mass-squared differences, solar and atmospheric, at the sub-eV scale. However,longstanding anomalies at short-baselines might imply the existence of new oscillation fre-quencies at the eV-scale and the possibility of this sterile state(s) to mix with the threeactive neutrinos. One of the many future neutrino programs that are expected to providea final word on this issue is the Short-Baseline Neutrino Program (SBN) at FERMILAB.In this letter, we consider a specific model of Large Extra Dimensions (LED) which pro-vides interesting signatures of oscillation of extra sterile states. We started re-creatingsensitivity analyses for sterile neutrinos in the 3+1 scenario, previously done by the SBNcollaboration, by simulating neutrino events in the three SBN detectors from both muonneutrino disappearance and electron neutrino appearance. Then, we implemented neutrinooscillations as predicted in the LED model and also we have performed sensitivity analysisto the LED parameters. Finally, we studied the SBN power of discriminating between thetwo models, the 3+1 and the LED. We have found that SBN is sensitive to the oscillationspredicted in the LED model and have the potential to constrain the LED parameter spacebetter than any other oscillation experiment, for m D < . Keywords: neutrino oscillation, large extra dimension, short-baseline a r X i v : . [ h e p - ph ] A ug ontents Our knowledge of the three neutrino oscillation paradigm has substantially improved in thelast decade mainly thanks to the reactor and accelerator-based experiments [1, 2]. Nowa-days, the neutrino oscillation parameters have been measured with certain precision [3, 4],except for the Dirac phase encoding the possibility that leptons violate the charge-parity(CP) symmetry. In this so-called three active neutrino framework , the neutrino mass or-dering, whether the third mass eigenstate is the upper (normal ordering) or the lower(inverted ordering) of the three states, is also unknown. Future neutrino oscillation ex-periments are expected to resolve both important missing pieces and also to improve overthe current precision of the neutrino oscillation parameters. In particular, there is a questfor establishing if the atmospheric mixing angle is maximal, and if not, what would be itscorrect octant. Besides providing information on the unknowns, in the precision era, newphysics signals might emerge as subleading effects of the three neutrino paradigm or as anew oscillation phase(s). This last scenario is mainly motivated by results of short-baselineexperiments [5–8] which call for a new neutrino flavor state that has to be sterile, i.e. itcan not interact with the Standard Model gauge bosons. So far, there is no indicationof a new oscillation phase and running experiments have constrained a large part of theparameter space, at least in the economical 3 + 1 oscillation framework [9–16]. Severalefforts are devoted to discover a sterile oscillation at the eV mass scale or to completelyrule out this hypothesis. For instance, at FERMILAB, there is a Short-Baseline NeutrinoOscillation Program (SBN) [17], which is expected to provide a definitive answer to thismatter. However, there are several beyond the standard three-neutrino oscillation scenar-ios, which might be considered as a the subleading effect, that can be probed in futurelong and short-baseline neutrino experiments. Here we focus on Large Extra Dimensions– 1 –LED) and the possibility that its signals be differentiated from the sterile hypothesis atthe SBN program. Other proposals can be tested in SBN facility for instance the searchfor multiple sterile states [18–20] and MeV-scale sterile decay [21–23].Initially, the main motivation for introducing extra space-time dimensions was to lowerhigh energy scales, as for instance the GUT [24, 25] or the Planck scale, even to the TeVenergy scale [26–28]. This appeared as an alternative to the usual seesaw mechanism that inits natural form calls for a high energy scale to suppress the active neutrino masses. Sinceright-handed neutrinos are singlets under the Standard Model (SM) gauge group, theyare one of the candidates that can experience extra space-time dimensions and thereforecollect an infinite number of Kaluza-Klein excitations [29, 30]. The other SM fermionsare restricted to a brane and therefore experiencing four dimensions only. In this way, theYukawa couplings between the right-handed neutrinos and the active ones are suppressed bythe volume factor after compactification of the extra dimensions. In this context, neutrinosacquire a Dirac mass that is naturally small, however, other alternatives violating leptonnumber are possible [29]. It is phenomenological appealing to considered an asymmetriccase where one of the extra dimensions is ‘large’ respect to the others, effectively reducingthe problem to be five dimensional [29–31]. In this letter, we consider the model for LargeExtra Dimensions (LED) from Ref. [31] (which is based on previous works in Refs. [29, 30,32]), and recently considered in the context of DUNE in Ref. [33], with three bulk neutrinos(experiencing extra space-time dimensions) coupled to the three active brane neutrinos.In this letter, we consider neutrino oscillations within the LED model with three bulkneutrinos coupled to the three active brane neutrinos, which effectively act as a largenumber of sterile neutrinos in contrast to the usual oscillation of light sterile neutrinosat the eV energy scale. Our goal is to establish the sensitivity of the SBN program toneutrino oscillations in the LED model. This letter is organized in the following way. Wefirst introduce the LED formalism in Section 2. The SBN program and the experimentaldetails used in our numerical simulations are condensed in Section 3. Our results arepresented in Section 4. Finally, we conclude and summarize in Section 5.
In general, it is assumed the right-handed neutrino (bulk fermions [31]) can propagate inmore than four dimensions while the left-handed neutrino ν L , and the SM Higgs H , areconfined to the four-dimensional brane. It is also assumed that one of the extra space-time dimensions is larger than the others so that effectively it is enough to consider fivedimensions in total. A Dirac fermion Ψ α in five dimensions can be decomposed into twocomponent spinors (Weyl fermions), ψ L and ψ R and after the extra dimension is compact-ified a natural coupling with ν L emerges [29] and, as a result, Dirac neutrino masses areobtained [29–32]. Along this letter we follow the model with three bulk neutrinos coupledvia Yukawa couplings to the three active brane neutrinos, the so-called (3 ,
3) model inRef. [31]. Other formulations for large extra dimension models are possible as described inRef. [34].The action in the (3 ,
3) model is given by:– 2 – = (cid:90) d xdy ¯Ψ α Γ A i∂ A Ψ α + (cid:90) d x (cid:104) ¯ ν αL γ µ i∂ µ ν αL + λ αβ H ¯ ν αL ψ βR ( x,
0) + H.c. (cid:105) (2.1)where y is the coordinate of the extra compactified dimension, Γ A are the five-dimensionalDirac matrices for A = 0 , ...,
4, and λ αβ the Yukawa couplings. To compactify the actionin Eq. (2.1) one need to expand the the five-dimensional Weyl fields ψ L,R in Kaluza-Klein(KK) modes ψ ( n ) L,R (with n = 0 , ± , ..., ±∞ ) and also to impose suitable periodic boundaryconditions [29]. It is convenient to define the following linear combinations: ν α ( n ) R = 1 √ (cid:16) ψ α ( n ) R + ψ α ( − n ) R (cid:17) ν α ( n ) L = 1 √ (cid:16) ψ α ( n ) L + ψ α ( − n ) L (cid:17) , (2.2)for n >
0, and also ν α (0) R ≡ ψ α (0) R . Therefore, after electroweak symmetry breaking, theLagrangian mass terms that results from Eq. (2.1) are given by: L mass = m Dαβ (cid:32) ¯ ν α (0) R ν βL + √ ∞ (cid:88) n =1 ¯ ν α ( n ) R ν βL (cid:33) + ∞ (cid:88) n =1 nR LED ¯ ν α ( n ) R ν β ( n ) L + H.c. , (2.3)Where m D is the Dirac mass matrix that is proportional to the Yukawa couplingsand can be written in terms of the fundamental mass scales of the theory [29, 31], and R LED is the compactification radius. It is useful to consider a basis in which the Diracmass is diagonal [31] r † m D l = diag { m Di } , by defining pseudo mass eigenstates N iL,R = (cid:0) ν i (0) , ν i (1) , ν i (2) , ... (cid:1) TL,R [35], such that the mass Lagrangian in Eq. (2.3) can be written L mass = (cid:80) i =1 ¯ N iR M i N iL + H.c. where M i is the infinite-dimensional matrix given by [30,31]: M i = m Di . . . √ m Di /R ED . . . √ m Di /R ED . . . ... ... ... ... . . . . (2.4)To find the neutrino masses and the relevant unitary matrices L i ( R i ) that relate the masseigenstates N (cid:48) iL ( R ) with the pseudo eigenstates N iL ( R ) , N (cid:48) iL ( R ) = L ( R ) † i N iL ( R ) , one needsto perform the bi-diagonalization R † i M i L . However, since we are mostly interested in therelation of the active brane neutrino states ν αL with the mass eigenstates, it is enough toconsider only the left matrices l and L i . L i is obtained from the diagonalization of theHermitian matrix M † i M i while l is the unitary 3 × m D diagonal-ization. – 3 –ffectively the active neutrino flavor states, can be finally written in terms of the masseigenstates (as composed of the KK n -modes of the fermion field), as follows: ν α L = (cid:88) i =1 l αi ν (0) i L = (cid:88) i =1 l αi ∞ (cid:88) n =0 L ni ν (cid:48) ( n ) i L ≡ (cid:88) i =1 ∞ (cid:88) n =0 W ( n ) αi ν (cid:48) ( n ) i L . (2.5)where W ( n ) αi is the amplitude in the LED case. We recover the usual three-neutrino casewhen W ( n ) αi → l αi .Formally, the mass eigenvalues and the eigenvectors of M i in Eq. (2.4) are obtainedfrom the diagonalization of the matrix R M † i M i by assuming a maximum integer valuefor the KK-modes k m ax and then taking the limit k m ax → ∞ [29, 30]. The L ni matrix inEq. (2.5) is explicitly given by: (cid:0) L ni (cid:1) = 21 + π (cid:0) R ED m Di (cid:1) + (cid:104) λ ( n ) i / (cid:0) R ED m Di (cid:1)(cid:105) , (2.6)where the neutrino mass eigenstates are equal to λ ( n ) i /R ED and therefore each one of themis composed of n -KK modes. λ ( n ) i in Eq. (2.6) corresponds to the eigenvalues of the full n × n neutrino mass matrix and can be calculated from the following transcendental equation: λ ( n ) i − π (cid:0) R ED m Di (cid:1) cot (cid:16) πλ ( n ) i (cid:17) = 0 , (2.7)and the roots λ ( n ) i are constrained such that they belong to the range [ n, n + 1 /
2] [29].In order to make a physical sense of the formalism, one should assume that the mostactive state is obtained for n = 0. Additionally, if we go to the limit R ED m Di (cid:28) λ (0) → R ED m Di , and following Eq. (2.6) L i →
1, therefore recovering the standardresult where l αi → U αi is the lepton mixing matrix that is usually parametrized by threerotations , through the three mixing angles θ ij , and the Dirac CP phase δ .Assuming the mostly active mass state is related with the lightest mass state in theKK-tower implies a relation between the eigenvalues of this LED framework, obtainedby Eq. (2.7), with the square mass differences obtained in the three-neutrino case. Thisrelation can be written as: (cid:16) λ (0) k (cid:17) − (cid:16) λ (0)1 (cid:17) R = ∆ m k (2.8)with ∆ m k is the solar ( k = 2) and the atmospheric ( k = 3) squared mass differences.Therefore, the existing values on the squared mass differences of the active neutrino masseigenstates ∆ m k , Ref. [4, 36], constrain the parameter space ( m Di , R − ) of the LED model.Thus, a good strategy is to use this information before scanning the parameter space.Basically, λ (0) i , i=1,2,3 are fixed by the m Di in Eq. (2.7), and using Eq. (2.8) for k=2,3 wegot a constrain between m D , m D and m D [37]. With these constraints, we have now only The three rotations are in general complex, accounting for the three physical CP phases. However,neutrino oscillations are insensitive to the two Majorana phases, and therefore, only sensitive to the DiracCP phase. In this case the more used parametrization is written as two real rotations plus a complex one. – 4 –wo independent parameters m D and R ED that we will rename from now on as m D → m for normal mass ordering. Similarly, one can follow the same procedure for the invertedmass ordering, and this case the two independent parameters are m D → m and R ED . Inthe cases where the condition in Eq. (2.8) is not fulfilled by the ( m D , R − ) combination,we quoted the excluded region as excluded by squared mass differences constraints . We willcomeback to this point in Section 3.In the LED framework the neutrino mixing matrix W , as defined in Eq. (2.5), is ingeneral different to the standard three neutrino mixing matrix U . To avoid spoiling theneutrino oscillations observations, condensed in part as constraints on the mixing angles θ ij ( i, j = 1 , ,
3) in scenario of three-neutrino scheme (with values in Ref. [3, 4, 36]), themixing angles in the LED framework have to be redefine. Following the procedure fromRef. [33] we have defined new mixing angles φ ij ( i, j = 1 , ,
3) in the LED scenario suchthat the lowest mass state in KK tower, n = 0, have the W (0) αi amplitude equal to thenumerical value of U αi : U αi = W (0) αi = l αi L i . From this relation we can get the mixingangles in the LED framework, φ ij , related with the solar and atmospheric mixing angles, θ ij . Explicitly we have using the elements of mixing matrix | U e | , | U e | and | U µ | sin φ = sin θ (cid:0) L (cid:1) cos φ sin φ = cos θ sin θ (cid:0) L (cid:1) cos φ sin φ = cos θ sin θ (cid:0) L (cid:1) . (2.9)From now on, the mixing angles φ ij in the LED formalism are given by the values inEq. (2.9). For some values of m Di and R ED the L i value can be smaller than the numeratorin Eq. (2.9) such that sin φ ij > m Di and R ED that results in this unphysical φ ij will be disregarded and we have quoted them as excludedby mixing angle constraints . We will comeback to this point in Section 3. In this section, we describe the experimental set-up and our working assumptions that wefollowed in the sensitivity analyses presented in Section 4. The SBN experimental proposalwill align three liquid argon detectors in the central axis of the Booster Neutrino Beam(BNB), located at FERMILAB [17]. Table 1 gives the SBN detector names, active masses,locations, and protons on target POT. We computed the expected number of events ofSBN facility by implementing the detectors in the GLoBES [38, 39] c-library, followingthe proposal description. The flux information for both neutrino and anti-neutrino modeswas taken from Ref. [40], and the neutrino-argon cross section was taken from inputs toGLoBES prepared for Deep Underground Neutrino Experiment (DUNE) simulation [41],with the cross section inputs, originally generated using GENIE 2.8.4 [42].The SBN facility will search for oscillations in two channels: 1) electron neutrino ap-pearance from muon neutrino conversion ( ν µ → ν e ) and 2) muon neutrino disappearance( ν µ → ν µ ) from muon neutrino survival. We considered a Gaussian detector energy resolu-tion function with a width of σ ( E ) = 6% / (cid:112) E [GeV] for muons and σ ( E ) = 15% / (cid:112) E [GeV]– 5 – etector Active Mass Distance from BNB target POTLar1-ND 112 t 110 m 6 . × MicroBooNE 89 t 470 m 1 . × ICARUS-T600 476 t 600 m 6 . × Electron Neutrino Appearance Channel Muon Neutrino Disappearance Channel
Energy Bin Size (GeV) Energy Range (GeV) Energy Bin Size (GeV) Energy Range (GeV)0.15 0.2-1.10 0.10 0.2-0.40.20 1.10-1.50 0.05 0.4-1.00.25 1.50-2.00 0.25 1.0-1.51.00 2.00-3.00 0.50 1.5-3.0
Table 1 . Upper: SBN detector active masses and distances from local of neutrino production.Lower: Energy range and energy bin size of the electron and muon sample used in this analysis. for electrons, according to Ref. [20]. The energy range for the neutrino event reconstructionextends from 0.2 GeV to 3 GeV where each channel has different bin widths, as describedin the Table 1. We simulated three years of operation for the neutrino beam in Lar1-NDand ICARUS-T600 detectors and six years in MicroBooNE detector. It is important to em-phasize that the detectors do not make a distinction between neutrinos and anti-neutrinos,so neutrinos and anti-neutrinos events are added in our simulations. After event recon-struction, we included an efficiency factor for each channel in order to mimic event ratesfrom collaboration proposal [17].In the presence of LED, the relations in Eq. (2.8) and Eq. (2.9) gives the squared massdifferences and the mixing angles in terms of the standard oscillation parameters. Whensimulating neutrino event rates, to perform the different studies along this letter, we usedthe best-fit values for the oscillation parameters in the standard three-neutrino frameworkpresented in Nu-Fit 3.2 (2018) [4, 36]. The LED parameters are the lightest neutrino mass m (for normal ordering m = m D while for inverted ordering m = m D ) and the radiusof extra dimension R ED .In Figure 1 the behavior of the oscillation probability for different m and R ED values isshown, considering an L/E ν of 1 . L/E ν value was calculated usingthe ICARUS baseline L = 0 . E ν = 0 . R − − m plane, the appearance probability isnot larger than 10 − and almost all survival probability is larger than 0.9. The gray shadedregion is excluded by neutrino oscillation data, with the relations Eq. (2.8) and Eq. (2.9),as described in Section 2.In the following, we assume forward horn current (FHC) beam mode and we definedsignal and background for each one of the SBN oscillation channels as follows: • Muon neutrino disappearance channel :1.
Signal:
Survival of muon neutrinos ( ν µ → ν µ ) from the beam which interact– 6 – igure 1 . Iso-probability regions for different values of LED parameters, m and R ED . In the left(right) panels we have ν µ → ν µ ( ν µ → ν e ). In the top (bottom) we show the normal (inverted)ordering. We chose here a typical short-baseline L/E ν of 1.2 km/GeV, see text for details, andwe compute probabilities using the first 40 KK modes. The gray shaded region is excluded due toneutrino oscillation data (see Sec.2). with liquid argon through weak charged-current (CC) producing muons in thedetectors.2. Background : The only background contribution considered by the collabora-tion comes from neutral-current (NC) charged pion production, where the pionproduced in the BNB target interacts with argon and can be mistaken for amuon [17]. This contribution is small due to the track cutting imposed in theevent selections and we did not consider it in our simulations. • Electron neutrino appearance channel :– 7 –.
Signal: electron neutrinos coming from muon neutrino conversion ( ν µ → ν e )which interacts through CC producing electrons in the detectors.2. Background : The main background contribution comes from the survival of in-trinsic electron neutrinos ( ν e → ν e ) in the beam, beam contamination. We alsoconsidered muons (muon neutrinos from the CC interaction), which can be mis-taken for electrons. NC photon emission, cosmic particles and dirty events werenot considered in our simulation, which corresponds to a background reductionof 8 .
4% for Lar1-ND, 14% for MicroBooNE and 13% for ICARUS-T600, respectto the total number of background events expected by the collaboration in theelectron neutrino channel [17].The information on the neutrino fluxes, neutrino cross section, energy resolution ofleptons and backgrounds used in the analysis were compiled using the AEDL format (tobe used with the GLOBES c-library), in order to perform the different sensitivity analysesof SBN program at FERMILAB. These files are available under request following Ref. [43].Since one of the main goals of the SBN program is to detect or rule out sterile neutrinooscillations, we introduce the generalities of the 3 + 1 case right now. Later, we will notonly take it as a reference but also we will quantify the discrimination power of the SBNprogram between the two models, the 3 + 1 and the LED. Several neutrino experimentshave performed a sensitivity analysis in the specific scenario of the so-called 3+1 model,where one sterile neutrino is added to the three active neutrino framework. In this 3+1framework, active and sterile neutrinos mix and three new oscillation frequencies appear,thanks to the four mass eigenstates, which can be written in terms of only ∆ m , the solar,and the atmospheric splittings. The additional mass eigenstate is the source of short-baseline oscillations mainly driven by the square mass difference ∆ m , and the effectiveamplitudes sin θ µe ≡ | U e | | U µ | and sin θ µµ ≡ (cid:0) − | U µ | (cid:1) | U µ | defined by theelements of the 4 × χ function for the different models: ‘data’ simulated assuming an energy spectrum defined by the three-neutrino case antesting the LED hypothesis, i.e., the usual sensitivity analysis, ‘data’ simulated assumingan energy spectrum distributed with the LED model and testing the standard oscillationscenario. Here we investigated the SBN potential of measuring the LED parameters R ED and m . Finally, ‘data’ simulated assuming an energy spectrum distributed with the 3+1model, where we evaluated the discrimination power of SBN to distinguish LED hypothesisfrom other models accommodating light sterile neutrino oscillations. We also performedsensitivity calculations for the 3+1 model in appearance and disappearance channels in– 8 – igure 2 . Left (Right) panel: Sensitivity limits for the LED parameters, R ED and m , consid-ering normal (inverted) ordering of neutrino masses. The regions for LED sensitivity, consideringboth channels, muon disappearance and electron appearance channels, are to the top-left of thecurves. Here, we show our 90% C. L. line from SBN limit (green), the 95% C. L. lines from DUNE(black) [33], ICECUBE-40 (magenta) and ICECUBE79 (blue) [44], and 95% C. L. from combinedanalysis of T2K and Daya Bay (gold) [45]. The 90% C. L. line from KATRIN sensitivity analysis isalso shown (brown) [37] and the pink regions are preferred at 95% C. L. by the reactor and Galliumanomaly [46]. The light and dark gray regions are excluded due to neutrino oscillation data. order to explore relations between LED and 3+1 signatures. The results are shown in thenext section. For the sensitivity analysis, total normalization errors in signal and background were set to10%, and all parameters that were not shown in the plots were fixed to their best-fit values.We tested that our sensitivity results are independent of the δ CP value. For simplicity, weset δ CP = 234 o for normal ordering and δ CP = 278 o for inverted ordering, according toRef. [4].Figure 2 shows SBN sensitivity limit with 90% of confidence level (C.L.) in the greencurve for normal (left panel) and inverted ordering (right panel), compared with otherlimits: Sensitivity limits at 95 % of C. L. for DUNE experiment (black-dashed curve)presented in Ref. [33], as well as ICECUBE-40 data and ICECUBE-79 data (dot-dashedmagenta and blue curves, respectively) from Ref. [44], and the combined analysis of T2Kand Daya Bay data (dot-dashed gold curve) presented in Ref. [45] are shown. The preferredregion (in pink) at 95% C. L. by Gallium and Reactor anti-neutrino experiments from theanalysis in Ref. [46] is also included. Finally, sensitivity limits for KATRIN at 90% C. L.(dashed brown curve) due to kinematic limits in beta decay estimated in Ref. [37] are shown.The gray shaded regions are the parameters excluded by measurements of square mass– 9 –ifferences ∆ m and ∆ m (light gray) and of mixing angles θ , θ and θ (dark gray).It is important to mention that excluded region due to mixing angle measurements alsocovers excluded region due to square mass differences. An additional constrain to the LEDparameters comes from MINOS analysis in Ref. [47] where a similar restriction curve tothe one from ICECUBE was obtained. When m →
0, MINOS constrains R ED < . µ m(or R − > .
44 eV) for normal ordering.We can see that the SBN program is sensitive to the LED parameters and this sen-sitivity is very competitive, respect to other facilities shown in the plot. This happensspecifically for the lower m region and particularly for normal ordering. Comparing withthe constraints from other experiments, the SBN sensitivity for LED mechanism is the bet-ter than any other constraints in the region when m < × − for normal ordering, andin this region, the maximum sensitivity of our analysis for R ED is better than any other os-cillation experiment which we trace to the fact that we are testing LED in a short-baselineexperiment for the first time, all other sensitivity results corresponds to long-baseline ex-periments. With respect to the reactor anomaly allowed region, the SBN program has thepotential to ruled out completely this anomaly for any value of m < × − . For highervalues of m , the DUNE experiment [33] have the potential to exclude the reactor anomalyallowed region, complementing SBN. In order to investigate the potential of SBN to measure the LED parameters, neutrinoevents were calculated in the same fashion than for the previous sensitivity analysis, butassuming now the LED model with m = 0 .
05 eV and 1 /R ED = 0 .
398 eV as the ‘true’values, and testing the LED scenario. All the standard oscillation parameters (which areincluded in the LED parameters) were fixed to their best-fit values from Refs. [4, 36] asdescribed in Section 2. Figure 3 shows the allowed regions consistent with the computedevents with the true value (black dot) at 68 .
3% of C.L. (blue curve), 95% of C.L. (orangecurve) and 99% of C.L. (purple curve) for both normal ordering (left panel) and invertedordering (right panel).We also included in Figure 3 the sensitivity result obtained in Figure 2 (dashed greenline), which we called
Blind Region , i.e., the region that agrees with the standard three-neutrino scenario, being in this way, ‘blind’ to LED effects. Any point inside the
BlindRegion will have a null result either for the muon disappearance channel or for the electronneutrino appearance channel. The ν e Ch. Blind Region presented in Figure 3 (dashedbrown line) is the result of the sensitivity analysis performed only with the computedevents from electron neutrino appearance channel. Any point inside the ν e Ch. BlindRegion will have a null result for the electron neutrino appearance channel. The ‘true’LED parameters were chosen around the ν e Ch. Blind Region, but outside the BlindRegion for both mass ordering.It is worth noticing that since the electron neutrino appearance probability is smallerthan 10 − for LED, as shown in Figure 1, one might not expect a sensitivity exclusion limitfrom the appearance channel, i.e., all the obtained sensitivity is shown in Figure 2 wouldcome from the muon disappearance channel. However, when we computed the sensitivity– 10 – igure 3 . Left (right) panel: Allowed regions for the ‘true’ LED parameters m = 0 .
05 eV and1 /R ED = 0 .
398 eV and assuming as test model the LED scenario for normal (inverted) ordering.All the other oscillation parameters were fixed to their best fit values. The dashed green (dashedbrown) curve show respectively the SBN sensitivity to the both muon disappearance and electronappearance channel (only electron appearance channel). The region denoted by
Blind region ( ν e Ch.Blind region ) is the region were are not sensitive respectively to the muon neutrino disappearance(electron neutrino appearance). curve only considering electron appearance channel, we obtained the exclusion limit showedin Figure 3 (dashed brown line). In fact, we have a sensitivity curve from electron appear-ance channel when we consider changes in background profile due to LED effects. Theelectron neutrino survival probability induced by the LED parameters decreases the intrin-sic electron neutrinos from the beam, which is the majority contribution to our background.In other words, we have sensitivity due to the decrease in the number of backgrounds andnot by the increase in the signal. A similar effect was found in Ref. [20].Although not shown in Figure 3, we repeated the same analysis with other LED truevalues located inside the exclusion region for both electron and muon neutrino channels(outside the
Blind Region and the ν e Ch. Blind Region ). In this case, we have a non-null result in both muon disappearance and electron neutrino appearance channels, andtherefore the LED parameters that explain this results are unique. As a consequence ofthis, and due to the logarithmic scale in the plot, we obtained small and concentratedregions around the chosen ‘true’ values, which results in a precision of SBN experiment tothe LED parameters below 1%.
In the standard three-neutrino scenario, we expect no oscillations in SBN due to its short-baseline and the energies considered. Now, if SBN ‘sees’ an oscillation, it will correspondsto a beyond the standard three-neutrino scenario signal that might be interpreted as an– 11 – igure 4 . Left (Right) panel: the sensitivity limit with 90% C.L. for the 3+1 model for respectivelythe muon neutrino disappearance channel (electron neutrino appearance channel), in the parameterspace which depends on sin θ µµ (sin θ µe ) and ∆ m . Exclusion (sensitivity) regions are to top-right of the black dashed curves in both panels. The solid black curve (solid blue curve) respectivelyshown our sensitivity (the SBN sensitivity was taken from Ref. [17]). sterile neutrino oscillation. In the 3+1 scenario, the neutrino probabilities for short-baselinedistances are given by [48]: P ν µ → ν e = sin θ µe sin (cid:0) ∆ m L/ (4 E ν ) (cid:1) (4.1) P ν µ → ν µ = 1 − sin θ µµ sin (cid:0) ∆ m L/ (4 E ν ) (cid:1) , (4.2) P ν e → ν e = 1 − sin θ ee sin (cid:0) ∆ m L/ (4 E ν ) (cid:1) , (4.3)where sin θ αα ≡ (cid:0) − | U α | (cid:1) | U α | , with α = e, µ and sin θ µe ≡ | U e | | U µ | arethe oscillation amplitudes, defined by the elements of the 4 × U e and U µ , and ∆ m is the squared mass difference between the fourth massstate m (which is made majority by the sterile component of neutrino flavor basis) and thefirst mass state m . The probabilities in Eqs. (4.1), (4.2), (4.3) at short-baselines dependon the three parameters U e , U µ , and ∆m [49].We now test the two following cases in the 3+1 scenario:1. Assuming the ‘true’ event energy distribution as compatible with the three-neutrinoscenario and testing the 3+1 model. This gives the sensitivity of SBN to the 3+1scenario that can be seen in Figure 4. Exclusion regions are to the right of the blackcurves for both appearance (right panel) and disappearance (left panel) channels. Wehave a very good agreement with the SBN sensitivity, comparing the blue and solidcurves in Figure 4. – 12 –. Assuming as the ‘true’ event energy distribution as compatible with the 3+1 scenarioand testing the 3+1 model. This will give the accuracy of SBN facility to the pa-rameters of the 3+1 scenario that can be seen in Figure 5. For illustration purposes,we show the sensitivity as dashed black curves for the 3+1 model at the SBN fromFigure 4. The allowed regions assuming the ‘true’ 3+1 parameters sin θ µµ = 0 . θ µe = 0 .
01 and ∆ m = 1 eV and also fitting 3+1 hypothesis. Notice that SBNis very sensitive to the square mass difference around 1 eV and the precision that wecan get for this value are very good and below 1%. Even though not shown in thefigure, large values of sin θ µµ and sin θ µe gets more precise determined than thelower values shown in the plot. The fast oscillations ∆ m >
10 eV were handledassuming a low-pass filter in our analysis using GLoBES 3.2.17 [38, 39], otherwise wewill have spurious results in our sensitivity for 3+1 model. Figure 5 . Left (Right) panel: Allowed Regions considering the ‘true’ neutrino event spectrumgiven by the 3 + 1 model with the values sin θ µµ = 0 .
02 and ∆ m = 1 eV ( sin θ µe = 0 . m = 1 eV ) in the muon neutrino disappearance channel (the electron neutrino appearancechannel). The dashed curve in both plots is the sensitivity curve for the respective channels. One question that remains is, in the case SBN finds a departure from the three neutrinoframework, is it possible to identify which of the two scenarios analyzed in this letter wouldbe responsible for the new signal (assuming is not something else)? In the following, weanalyze the discrimination power of the SBN experiment comparing both the LED and the3+1 scenarios. Regarding the 3+1 fit to the LED scenario, we calculated events with the‘true’ LED parameters m = 0 .
05 eV and 1 /R ED = 0 .
398 eV assuming normal ordering.With this ‘true’ events, both appearance and disappearance channels were fitted separately,fixing the parameters not shown in the plots. Figure 6 shows the result of the fit in the disappearance channel (left panel) with allowed curves of 68 .
3% of C.L. (blue), 95% of C.L.– 13 – igure 6 . Left panel, sensitivity results fitting the 3+1 model parameters assuming the ‘true’ LEDparameters m = 0 .
05 eV and 1 /R ED = 0 .
398 eV, for normal ordering. Right panel, sensitivityresults fitting the LED parameters for the ‘true’ 3+1 parameters sin θ µµ = 0 . m =0 . , also for normal ordering. The allowed sensitivity regions correspond to the 68 .
3% of C.L.(blue), 95% of C.L. (orange) and 99% of C.L. (purple), the best-fit points appear as black dots. (orange) and 99% of C.L. (purple). The number of degrees of freedom (d.o.f.) was equal to17 (19 energy bins minus 2 free parameters). The best-fit of the test values is representedin the black dot and has values of sin θ µµ = 0 . m = 0 . . We have not founda good fit, where ∆ χ = χ − χ = 8 for the best-fit point, giving more than 2 σ ofdeviation between the two models.We have also checked that when using the new set of parameters m = 0 .
316 eV and1 /R ED = 1 eV for the muon disappearance case, we have obtained a ∆ χ ≈
104 for thebest-fit (of the test values) point, implying a bad fit. This result can be explained due tothe fact that for some values of the LED parameters, as in this case, more sterile statesstart to contribute in the oscillation probability and the 3+1 model cannot emulate theLED model.Following a similar procedure, this time fitting the LED model for some ‘true’ valuesfor the 3+1 parameters, we could not obtain good fits. The analysis is shown in theright panel of Figure 6. In fact, if we consider the amplitude sin θ µµ = 0 .
01 and thesame ∆ m = 0 . , the allowed regions would be almost entire inside the Blind Region (bottom-right part from the dashed green curve in the right panel of Figure 6). From thisanalysis, we obtained the value ∆ χ ≈ θ µµ = 0 . m = 3 eV and we obtained the value ∆ χ ≈ electron neutrino appearance channel , we repeated the same proce-dure done for the muon channel: we calculated events for a given ‘true’ values for the LEDparameters and we fitted the electron neutrino appearance parameters in the 3+1 model.– 14 – µ Disappearance ν e Appearance (cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)
Test model True hypothesis LED ( m , /R ED ) LED ( m , /R ED )3+1 (sin θ µµ or sin θ µ e , ∆m ) True: (0.05 eV, 0.398 eV) True: (0.05 eV, 0.398 eV)best fit test Values: (0.1, 0.5 eV ) -∆ χ ≈ χ ≈ θ µµ or sin θ µ e , ∆m ) True: (0.316 eV, 1 eV) True: (0.316 eV, 1 eV)- -∆ χ ≈
104 ∆ χ ≈ (cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104) Test model True hypothesis 3+1 (sin θ µµ , ∆ m ) 3+1 (sin θ µe , ∆ m )LED ( m , /R ED ) True: (0.1, 0.5 eV )best fir test values: (0.017 eV, 0.22 eV) *∆ χ ≈ m , /R ED ) True: (0.01, 0.5 eV )- *∆ χ ≈ m , /R ED ) True: (0.1, 3 eV )- *∆ χ ≈ ( - ) Best-Fit Test Value is outside Exclusion Region, ( * ) LED does not expect positive signal of ν e appearance in SBN. Table 2 . Discrimination power of SBN facility for 3+1 model and LED model.
The summary of the results are the following: • For m = 0 .
05 eV and 1 /R ED = 0 .
398 eV, the best-fit and the allowed regions werelocated outside the Sensitivity Region with the value ∆ χ ≈ . • For m = 0 .
316 eV and 1 /R ED = 1 eV, the best-fit and allowed regions were locatedoutside the sensitivity region, with ∆ χ ≈
538 for the best-fit point, implying a verypoor fit.The previous results (for the electron appearance case) were somehow expected since wecould only obtain LED sensitivity from electron neutrino channel in Figure 3 with effectsof the LED parameters in the background. Then, we should not expect that the signal ofthe electron neutrino conversion can be fitted with the 3+1 parameters. In other words,evidence of electron appearance in short-baseline experiments would be inconsistent withLED hypothesis. Similar conclusion was made in Ref. [34].The right panel of Figure 6 also shows the LED fit for a given set of ‘true’ parametersof the 3 + 1 model considering only muon disappearance. We fixed the 3+1 parameterssin θ µµ = 0 . m = 0 . and fitted the LED parameters for normal ordering.The allowed curves corresponds to the 68 .
3% of C.L. (blue), 95% of C.L. (orange) and 99%– 15 –f C.L. (purple). The best-fit point obtained is m = 0 .
017 eV and 1 /R ED = 0 .
22 eV.Following the same procedure, we found ∆ χ ≈ . ν e of the beam) are considered. In this way, LED is notcontributing to the signal ( ν e conversion) in the electron neutrino channel. Therefore,when regarding the LED fit under 3+1 scenario on these conditions, we would not expectto accommodate LED parameters for any set of ‘true’ parameters of the 3 + 1 modelconsidering only the signal of electron neutrino appearance channel.Finally, all the results obtained for the discrimination power of LED and the 3+1model are summarized in Table 2. In the dawn of the new era of high precision neutrinos experiments, the search for BeyondStandard Model (BSM) physics will bring an understanding of the mechanism beyondneutrino masses and neutrino mixing. The possibility to have in Nature the presence oflarge extra dimension is intriguing and it has several consequences for the phenomenologyof neutrino physics, such as the existence of infinite tower of Kaluza-Klein states of sterileneutrinos. The Short-Baseline Neutrino Program SBN at FERMILAB will fully test thepresence of large extra dimension (LED) in neutrino oscillations.We have developed GLoBES simulation files [43] that include the three detectors atSBN facility where information of the two main channels of SBN program, the ν µ muonneutrino disappearance channel and the ν e electron neutrino appearance channel, are in-cluded. In the paradigm of three neutrino oscillation, we expect to see no oscillation inany of SBN detectors. With the assumption that we measure no oscillations in any of SBNdetectors, we can put bounds on the LED scenario. In the LED scenario, the non-standardoscillations are accounted for with two parameters, the lightest Dirac neutrino mass m and the radius of large extra dimension R ED . We have shown in Figure 1 the regions withsizable muon neutrino disappearance probability and electron neutrino appearance proba-bility in the presence of LED, for either normal or inverted hierarchy of active states. Thetypical values that we can test are P ( ν µ → ν µ ) ∼ .
90 and P ( ν µ → ν e ) ∼ − − − fora L/E ν = 1 . the strongest bound for almost all parameter region, with exception of the valuesof m > × − eV and 1 /R ED > σ − σ . The worst scenario was shownin Figure 6, where we get a 2 σ − σ discrimination using the muon disappearance channelonly. For other choices of parameters, as detailed in Table 2, we can easily discriminatethe source of new physics in the SBN experiment, the large extra dimension or the 3+1scenario. Acknowledgments
G.V.S is thankful for the support of FAPESP funding Grant No. 2016/00272-9 and No.2017/12904-2. G.V.S. thanks the useful discussions with Pedro Pasquini and Andr´e deGouvˆea. D. V. F. is thankful for the support of FAPESP funding Grant No. 2017/01749-6. O.L.G.P. is thankful for the support of FAPESP funding Grant No. 2014/19164-6,No. 2016/08308-2, FAEPEX funding grant No. 519.292, CNPQ research fellowship No.307269/2013-2 and No. 304715/2016-6.
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