Large Extra Dimensions and neutrino experiments

The existence of Large Extra Dimensions can be probed in various neutrino experiments. We analyze several neutrino data sets in a model with a dominant large extra dimension. We show that the Gallium anomaly can be explained with neutrino oscillations induced by the large extra dimension, but the region of parameter space which is preferred by the Gallium anomaly is in tension with the bounds from reactor rate data, as well as the data of Daya Bay and MINOS. We also present bounds obtained from the analysis of the KATRIN data. We show, that current experiments can put strong bounds on the size $R_{\text{ED}}$ of the extra dimension: $R_{\text{ED}}<0.20~\mu\text{m}$ and $R_{\text{ED}}<0.10~\mu\text{m}$ at 90\% C.L. for normal and inverted ordering of the standard neutrino masses, respectively.

these parameters obtained from the analysis of several neutrino oscillation experiments and from the recent results of the KATRIN experiment on the search for the effects of sub-eV neutrino masses on the spectrum of electrons emitted in tritium decay.
We discuss the implications for the LED model from the results of reactor short-baseline neutrino oscillation experiments and the results of the Gallium source experiments in the context of the reactor antineutrino anomaly [29] and the Gallium neutrino anomaly [30][31][32] (see, e.g., the reviews in Refs. [33][34][35][36][37]). In particular, we take into account the recent results of the BEST Gallium source experiment [38,39].
The plan of the paper is as follows: in Section II we review the relevant aspects of neutrino mixing in the LED model under consideration; in Section III we describe the method of analysis of the data and the results of the separate analyses of the data of shortbaseline experiments (Subsection III A), long-baseline experiments (Subsection III B), and the KATRIN experiment (Subsection III C); in Section IV we present the results for the combined bounds on the LED parameters; finally, in Section V we present a summary of our results and the conclusions.

II. NEUTRINO OSCILLATIONS IN PRESENCE OF LARGE EXTRA DIMEN-SIONS
The possible existence of Large Extra Dimensions (LED) was originally proposed as a solution of the hierarchy problem [5,6]. In the LED model, the ordinary four-dimensional space-time is a brane embedded in a space-time with 4 + N ED dimensions, having N ED large space-like extra dimensions. The fields which are charged under the gauge symmetries of the Standard Model (SM) are restricted to the four-dimensional brane, whereas the fields which are singlets under the SM gauge symmetries propagate in the (4 + N ED )-dimensional bulk. In particular, right-handed sterile neutrino fields are SM gauge singlets which propagate in the bulk. The Yukawa couplings with the SM left-handed neutrinos are suppressed by the LED volume, leading to naturally small Dirac neutrino masses [7][8][9][10][11][12]. As most phenomenological studies of neutrino physics in a LED model [12,13,[17][18][19][20][21][22][23][24][25][26][27], we consider a LED model with one of the extra dimensions which is compactified on a circle with radius R ED which is much larger than the size of the other extra dimensions. Therefore, we consider an effective five-dimensional space-time and we assume that there are three five-dimensional right-handed singlet fermion fields associated with the three active left-handed flavor neutrino fields ν αL , with α = e, µ, τ . Each of the five-dimensional right-handed singlet fermion fields can be decomposed as an infinite tower of four-dimensional Kaluza-Klein (KK) fields. After diagonalization of the mass matrix, the mixing of the three active neutrinos is given by where U is the ordinary 3 × 3 unitary neutrino mixing matrix and each ν (n) iL is a neutrino field with mass m are the solutions of the transcendental equation where m D i are the three eigenvalues of the Dirac neutrino mass matrix, which are naturally much smaller than the electroweak scale because of the LED volume suppression. The components of the mixing matrix V are given by [8][9][10] The neutrino oscillation probability is given by where E is the neutrino energy and L is the source-detector distance. The transcendental equation (2) has an infinite number of solutions λ (n) i for n = 0, 1, . . . , ∞ in the intervals [n, n + 1/2]. To get a feeling of the behavior of these solutions, one can solve the transcendental equation (2) analytically for m D i R ED 1 [12] and find the leading expressions Therefore, for k > 0 the masses m (k) i increase with k and the mixing V ik decreases with k. Since the standard three-neutrino mixing describes well the oscillations observed in solar, atmospheric and long-baseline neutrino experiments, the LED model must be considered as a perturbation of three-neutrino mixing, which corresponds to V in = δ n0 for i = 1, 2, 3. Hence, we require that the zero-mode masses m (0) i generate the standard mass-squared differences [19]: We consider [14]: where NO and IO indicate, respectively, Normal Ordering and Inverted Ordering. In this way, the two independent squared-mass mass-squared differences ∆m 2 21 and ∆m 2 31 allow us to fix two of the independent Dirac mass parameters m D i of the LED model. It is convenient to choose as the remaining free mass parameter, denoted by m 0 , the lightest Dirac mass, which depends on the ordering: m 0 = m D 1 (m 0 = m D 3 ) in the normal (inverted) neutrino mass ordering. With this method, the LED model depends on two parameters: R ED and m 0 . For fixed values of R ED and m 0 , the determination of the masses and mixing is done as follows.    The constraints (9) restrict the physical region of the LED parameters m 0 and R ED , that must allow the λ (0) i 's to be smaller than 1/2 in order to have a solution of the transcendental equation (2). Denoting with r and s the indices of the largest and smallest zero-mode mass (r = 3 and s = 1 in NO; r = 2 and s = 3 in IO) the physical region is determined by the inequality Note that there is a bound even for m 0 R ED 1, which can be found using the approximation in Eq. (5): Since the largest mass-squared difference is the atmospheric one, which is about 2.5 × 10 −3 eV 2 in both NO and IO, we have the physical upper bound for the radius of the extra dimension R ED 2 µm.
The behavior of the masses is illustrated in Fig. 1 for the zero-mode and the first seven   Fig. 1. One can see that for small values of m 0 the mixing elements V i0 are dominant and the mixing elements V ik decrease with k, in agreement with Eq. (8). Therefore, for m 0 R −1 ED the phenomenology of the LED model is determined by a few low KK modes. On the other hand, for m 0 R −1 ED it is necessary to take into account a sufficient number of high KK modes. We have verified that for the oscillation experiments considered in this paper adding more than 5 modes does not effect our results. For the analyses of oscillation data we therefore use 5 modes. In the case of KATRIN one needs to calculate the number of relevant modes dependent on the values of m 0 and R ED , as will be explained below in Sec. III C.

III. DATA ANALYSIS
In this section we present the results of our analyses. Subsection III A discusses the analysis of reactor rate and Gallium experiments, Subsection III B is dedicated to MI-NOS/MINOS+ and Daya Bay, and in Subsection III C we outline the analysis procedure of KATRIN data. In Section IV we will discuss the combined bound from all experiments.
When performing the analyses we take into account possible correlations between LED and standard oscillation parameters by marginalizing over them. In particular, marginalizing over θ 13 and θ 23 affects the bounds obtained in the analyses of Daya Bay and MI-NOS/MINOS+ data, respectively. Since current data only slightly prefer normal over inverted neutrino mass ordering [40] of the three mostly active states, we perform all of our analyses for both orderings.
For electron neutrino experiments (ν e disappearance and β-decay experiments), the effects of the LED KK modes are quite different in the two orderings for small values of m 0 . As one can see from Fig. 1(a), in the NO case m . Therefore in the NO case the dominant LED effects are given by the terms with i = 3 in the neutrino oscillation probability (4) for ν e disappearance and in β-decay spectra as that of the KATRIN experiment that we consider in Section III C. Since these terms are suppressed by the smallness of |U e3 | 2 , the LED effects are small. On the other hand, in the IO case m In this case, the dominant LED contributions appear in the terms with i = 1 and i = 2, which are not suppressed. Therefore, ν e disappearance and β-decay experiments give stronger constraints on the LED parameters for inverted ordering than for normal ordering (see, e.g., Refs [19,27]). Since this occurs for m 0 0.1 eV, the result is a significant decrease of the upper bounds for R ED for small values of m 0 in the case of inverted ordering with respect to normal ordering. Instead, in the case of ν µ disappearance all the relevant |U µi | 2 are large and there is no big difference between normal and inverted ordering.

A. LED at short baseline experiments
In this section we present the results of our analyses of the data of the reactor rate experiments and the Gallium source experiments. It has been shown in Ref. [18] that the Gallium and reactor anomalies can be resolved if LED induced neutrino oscillations are present in nature. In this section we perform an updated analysis using the most up-to-date reactor rate and Gallium data.
The importance of reactor fluxes and its impact on the statistical significance of the anomaly has been recently discussed in Refs. [41,42]. Here we repeat the reactor rate analysis as in Ref. [41], but with the 3+1 oscillation probability replaced by its LED counterpart. The flux models that are considered in this work are the Huber-Mueller model [43,44] (HM), the model by Estienne, Fallot et al [45] (EF), the model by Hayen et al [46] (HKSS) and the recent model from the Kurchatov institute [47] (KI). For a detailed description of the models and for the details of the statistical analysis the interested reader is referred to Ref. [41].
The results for the four flux models are shown in Fig. 3 for normal (left panel) and inverted (right panel) neutrino mass ordering. The results are similar to those in Ref. [41]. For the KI and EF models no anomaly is found and we can only set bounds on the LED parameter space. In the case of the HM model we find an elongated region at 90% C.L., but no closed regions at 99% C.L. In the case of the HKSS model the anomaly is the strongest and then also the 99% C.L. contour is closed.
The GALLEX [48][49][50] and SAGE [30,[51][52][53] experiments were constructed to detect solar neutrinos. They have been tested by placing intense artificial 51 Cr and 37 Ar radioactive sources inside of the detectors. The resulting ratios of observed to expected events are significantly smaller than unity. This deficit became known as the Gallium anomaly [30][31][32]. The results of GALLEX and SAGE have been recently confirmed by the BEST collaboration [38,39], pushing the combined significance of the Gallium anomaly to the 5σ level. We analyze the data of the Gallium experiments following Ref. [54], but including also the BEST data. We also consider a 3% uncertainty on the Gallium cross section. We use only the cross section model of Bahcall [55], since other models produce a similar Gallium anomaly [56]. In Fig. 3 we show the preferred region obtained from the combined analysis of the data of the Gallium experiments (orange lines). This region updates the region obtained in Ref. [18] by including also the recent results from the BEST experiment [38]. One sees, that the results of the reactor rate analysis, for all the considered flux models, and the result from the Gallium analysis are in tension. The same feature has already been observed in the context of 3+1 oscillations, see Ref. [41].

B. LED at long baseline experiments
The Main Injector Neutrino Oscillation Search (MINOS) was an accelerator-based neutrino oscillation experiment. Unlike the former experiments considered in this paper, MINOS uses a beam of muon neutrinos, instead of electron neutrinos. The neutrinos were produced at the NuMI beam facility at Fermilab and detected at the near and far detectors of the experiment, located at 1.04 km and 735 km, respectively. During the MINOS data taking period, the neutrino beam peaked at an energy of 3 GeV and was later tuned to cover larger energies peaking at 7 GeV for the upgraded version of the experiment, MINOS+. Traces of Large Extra Dimensions in the data have been sought for by the MINOS collaboration [57]. We update these results, considering the data corresponding to an exposure of 10.56 × 10 20 POT in MINOS (mostly in neutrino mode, only 3.36 × 10 20 POT were gathered in antineutrino mode) and 5.80 × 10 20 POT in MINOS+ (in neutrino mode). The data are the same as those used for the search of light sterile neutrinos [58]. We adopt the analysis procedure followed by the experimental collaboration for the search of active-sterile neutrino oscillations in Ref. [58] by adapting the public MINOS/MINOS+ code to account for LED neutrino oscillations instead of active-sterile oscillations.
In addition to the data collected by MINOS/MINOS+, we also analyze data from the Daya Bay reactor neutrino experiment [59]. Daya Bay uses several nuclear reactors summing up a total thermal power of ∼ 17 GW th and measures the antineutrinos at 8 identical detectors, each with 20 ton fiducial mass, situated at three different sites (experimental halls). For this analysis, we consider the data set corresponding to 1958 days [59]. First, we reproduced the three-neutrino analysis performed by the collaboration using information from Refs. [59][60][61] through an implementation of the experiment in the GLoBES C-library [62,63]. Instead of a far-over-near ratio analysis, the spectral information at the three halls was used. Systematic uncertainties were also included in the analysis, in the same way as in the analysis in Ref. [14]. After that, we modified GLoBES in order to include the LED oscillation probability in Eq. (4).
The results of our analysis are shown in Fig. 4. The red and black lines correspond to the bounds at 90% (dashed) and 99% (solid) C.L. for two degrees of freedom for MI-NOS/MINOS+ and Daya Bay, respectively. The results of our analysis of MINOS/MINOS+ data are in reasonable agreement with the preliminary results obtained by the MINOS collaboration in Ref. [64]. Note, however, that we perform a simple χ 2 analysis, while the results from Ref. [64] are obtained from a Monte Carlo analysis. We find that Daya Bay sets the strongest bounds on the LED parameters for inverted ordering, while for normal ordering the bound by MINOS/MINOS+ is the strongest for small values of m 0 . It should be noted, that these bounds exclude the LED explanation of the Gallium anomaly and are in even stronger tension with the Gallium region than the regions preferred from the analysis of reactor rate data. It should also be noted that for small values of R ED the bounds become very weak, since in this case LED effects become very small and then m 0 corresponds to the overall neutrino mass scale to which oscillation experiments are not sensitive.

C. LED at KATRIN
In this subsection we present the results of our analysis of the data of the KATRIN experiment. KATRIN is an experiment for direct neutrino mass measurement. It measures the single beta-decay of molecular Tritium, T 2 → 3 HeT + + e − + ν e , near the endpoint of the spectrum. The KATRIN collaboration presented their first two mass measurements in Refs. [65] and [66]. From the first (second) campaign, they obtained an upper bound of 1.1 eV (0.9 eV) at 90% C.L. on the effective electron neutrino mass m β in the standard three-neutrino mixing framework, which is given by The combined upper bound from both data sets is 0.8 eV at 90% C.L. [66]. A best fit of m 2 β = 0.26 eV 2 was obtained in the second campaign. The KATRIN data has also been used to search for light sterile neutrinos, see Refs. [67][68][69]. Using the current data KATRIN is sensitive to neutrino masses up to 40 eV, because the data span the last 40 eV of the integral spectrum.
In our analysis we use the KATRIN data given in Ref. [69] (corresponding to the second campaign) to search for the effects of large extra dimensions. In the presence of LED, the Kurie function is [19] K(E, m 0 , where j = E 0 −V j −E is the neutrino energy, which depends on the endpoint of the spectrum E 0 and the energies of the different final states V j (which occur with probability p j ). In our analysis we include several sources of systematic uncertainties, following the discussion in Ref. [69]. In particular, we fit the data with a free total normalization of the spectrum, a free flat background component, a hypothetical retarding-potential-dependent background, and a small time-dependent background contribution from electrons stored in the Penning trap. Moreover, we constrain the Q-value to be 18575.72 ± 0.07 eV, as determined from the precise measurement of the atomic mass difference of Tritium and 3 He in Ref. [70]. This Q-value implies the endpoint of the spectrum E 0 = 18574.21 ± 0.6 eV [66]. We first verified our analysis method by repeating the standard three-neutrino analysis using the data from the second campaign. We obtained an upper bound of m β < 0.83 eV at 90% C.L. and a best fit m 2 β = 0.1 eV 2 . These results are consistent with the results of the KATRIN collaboration presented in Ref. [66]. Next, we exchanged the standard Kurie function with Eq. (15) in our analysis code in order to bound the LED parameters. In the calculation of the Kurie function all KK modes with masses less than 40 eV have to be included. The results of our analyses are shown in Fig. 5, where we show the 90% (dashed orange lines) and 99% (solid orange lines) C.L. bounds for the LED parameters for normal (left) and inverted (right) neutrino mass ordering. For R ED 5 × 10 −3 µm the upper bound on m 0 is the same as the upper bound that we obtained for m β . The difference of the upper bound 0.9 eV at 90% C.L. in Fig. 5, with the 0.83 eV bound declared above is due to the fact that the contours in Fig. 5 are calculated with two degrees of freedom, instead of one. The coincidence of the two bounds is due to the negligible contribution of the KK modes for R ED 5 × 10 −3 µm. In this case, only the zero mode is relevant and m 0 is equivalent to m β .
For R ED 5 × 10 −3 µm the KK modes start to be relevant and for R ED ∼ 1 µm more than 200 KK modes must be considered.
An interesting feature that can be noticed in Fig. 5 is the bump of the bound that occurs around R ED ∼ 3 × 10 −2 µm. We think that this feature is due to the spacing of the KATRIN data points. We checked this explanation with a fit of simulated data having a different spacing. We generated a mock data sample using 100 bins of the retarding potential which are evenly spaced in the 40 eV interval below the endpoint (compared to the 23 slightly unevenly spaced points used by KATRIN), and we found a bound without the bump. This implies that the bump is caused by the particular retarding potentials at which the KATRIN measurements have been done. A similar bump, in the same mass range, was found in the sterile neutrino analysis of the KATRIN collaboration, see Ref. [69].
Note that the bound on the LED parameters obtained from the analysis of the KATRIN data is not very strong in comparison to the bounds obtained in the previous subsections. It should be noted, however, that the KATRIN bound does not vanish for small values of R ED , where it simply corresponds to the bound on the neutrino mass scale of the standard three-neutrino analysis. Therefore, the analysis of KATRIN data allows us to reduce the volume of the allowed parameter space of the LED parameters, which from the analysis of oscillation data alone is unbound for small values of R ED . In Fig. 5 we also show the expected final sensitivity * of KATRIN (magenta lines). We see that the KATRIN bound will improve significantly in the near future. * Note that our sensitivity estimation is a bit weaker than that of Ref. [19]. The difference is due to the fact that our estimation uses more up-to-date information of the experimental details than the estimation of Ref. [19] done in 2012.

IV. COMBINED BOUNDS ON LED PARAMETERS
In this section we discuss the combined analysis of the experiments considered individually in Section III. We do not consider the Gallium data in this section, since it is in tension with the other data. Regarding the reactor rate data, we consider only the analysis using the KI fluxes. Note that since the combined analysis is dominated by MINOS and Daya Bay, using another model of the reactor antineutrino fluxes would not affect the combined bound on the LED parameters.
The result of our analysis is shown in Figs. 6 and 7. In Fig. 6 we show the 2-dimensional allowed regions for the individual experiments (below the corresponding lines) and the allowed regions obtained from the combined analysis (colored regions). As it can be seen in the right panel of the figure, where the black Daya Bay lines nearly coincide with the boundaries of the blue and yellow regions obtained from the combined analysis, for inverted ordering the combination is dominated by Daya Bay for R ED 7 × 10 −3 µm. Instead, for normal ordering both MINOS and Daya Bay give essential contributions to the definition of the combined allowed region for R ED 7 × 10 −3 µm. As it can be seen, MINOS is dominating for large values of R ED , while Daya Bay is dominating for intermediate values of R ED . For both orderings the contribution of KATRIN is to close the allowed region for small values of R ED . For small values of R ED the LED oscillation probability becomes basically the standard three-neutrino oscillation probability and then the oscillation experiments can not bound m 0 , which in this case is simply the neutrino mass scale. In Fig. 5 we showed the projected final sensitivity of KATRIN. Comparing it with the bounds in Fig. 6, we see that the contribution of KATRIN to bound the LED parameter space will become much more relevant in the near future.
In Fig. 7 we show the 1-dimensional ∆χ 2 profiles for the compactification radius of the large extra dimension. Also from this figure one sees that for normal (inverted) ordering the bound on R ED is dominated by the analysis of data from MINOS/MINOS+ (Daya Bay). From our combined analyses we find that R ED < 0.20 µm at 90% C.L. for NO , (16) R ED < 0.10 µm at 90% C.L. for IO .
These bounds are quite strong for both orderings. In spite of the potentialities of the next generation of neutrino oscillation experiments, unfortunately it is unlikely that they will be able to improve these bounds according to Ref. [27] for JUNO and TAO, Refs. [22,25,26] for DUNE, and Ref. [24] for SBN. Let us finally emphasize that the results of our analysis improve all the bounds on the LED parameters obtained previously from neutrino oscillation data [12,13,17,20,57,71].

V. CONCLUSIONS
We have performed an analysis of the data of several neutrino oscillation experiments and the KATRIN experiment in the framework of a Large Extra Dimensions model with a dominant large extra dimension. We obtained the strongest bounds on the LED parameter space that have been attained so far from the data of neutrino experiments. We have shown that the LED explanation of the Gallium anomaly is excluded by the analyses of the data of the other experiments. This behavior has also been observed in the context of  neutrino mixing with a light sterile neutrino [41]. Combining all data we obtained a bound of R ED < 0.20 (0.10) µm at 90% C.L. for NO (IO). Using KATRIN data we are able to close the region at small values of R ED , as it can be seen in Fig. 6. The KATRIN bound on m 0 is expected to improve by approximately one order of magnitude, according to the final KATRIN sensitivity. Hence, the role of KATRIN will become more crucial in the future.