Limits and correlations of T and Z components of CPT-odd coefficients of the Standard Model Extension at DUNE
L. A. Delgadillo O. G. Miranda G. Moreno-Granados C. A. Moura





 ldelgadillof2100@alumno.ipn.mx
Departamento de Física, Escuela Superior de
  Física y Matemáticas del Instituto Politécnico Nacional, Unidad Adolfo López Mateos, Edificio 9, 07738 Ciudad de México, Mexico
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  
 omar.miranda@cinvestav.mx
Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN
  Apdo. Postal
  14-740 07000 Ciudad de México, Mexico
 guadalupe.moreno@cinvestav.mx
Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN
  Apdo. Postal
  14-740 07000 Ciudad de México, Mexico
Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061, USA

celio.moura@ufabc.edu.br
Centro de Ciências Naturais e Humanas, Universidade Federal do ABC - UFABC, Av. dos Estados, 5001, 09210-580, Santo André-SP, Brazil







§ ABSTRACT 
We consider the possible effect of the Standard Model Extension (SME) coefficient (a_L)^Z in the neutrino propagation and discuss how this can affect DUNE limits on the coefficient (a_L)^T found elsewhere. Based on an analysis that considers both coefficients, we find new constraints for them coming from a DUNE-like experiment. Furthermore, we investigate the correlations of the standard oscillation parameters, the leptonic CP-violating phase, δ_CP, and the atmospheric mixing angle, sin^2 θ_23, with respect to the SME coefficients (a_L)^T and (a_L)^Z.

Limits and correlations of T and Z components of CPT-odd coefficients of the Standard Model Extension at DUNE
    C. A. Moura
    September 9, 2024
=============================================================================================================





§ INTRODUCTION


On the neutrino oscillation frontier, where nearly all of the available data is compatible with the standard three-flavor oscillation paradigm, precision measurements can be performed to investigate Beyond Standard Model (BSM) physics that may disturb the oscillation pattern. Looking for perturbations on the neutrino oscillation standard model brings new opportunities to access, for example, the Planck scale.

In the three neutrino oscillation picture, NOvA <cit.>, T2K <cit.>, and recently IceCube <cit.> collaborations are leading the precision measurements on both the atmospheric oscillation parameters, Δ m^2_32 and sin^2 θ_23, and the CP-violating phase, δ_CP <cit.>.
The current measurement of the angle θ_23 demonstrates consistency with maximal mixing (i.e., θ_23=π/4). 

This has implications for our understanding of neutrino masses and the nature of neutrino flavor mixing.
Furthermore, the precise determination of sin^2 θ_23 is crucial for refining our understanding of the neutrino mass hierarchy, the ordering of neutrino mass eigenstates, and for probing possible sources of CP violation in the neutrino sector.

On the other hand, accurately determining δ_CP  is a primary goal of several ongoing and planned long-baseline neutrino oscillation experiments. Present-day measurements of δ_CP have been reported, mainly by NOvA and T2K collaborations. Their results seem to differ, and might be a challenge to interpret them if this difference persists in the future. While the NOvA collaboration best-fit agrees with a value of δ_CP≃ 0.8 π <cit.>, a similar analysis from the T2K collaboration reports δ_CP≃ 1.4 π <cit.>, under the neutrino mass Normal Ordering (NO). 
A recent joint oscillation analysis of the Super-Kamiokande and T2K collaborations is consistent with a best-fit value of δ_CP≃ 1.4 π <cit.>.
Nevertheless, some proposals to alleviate such discrepancy on the measurement of δ_CP include: neutrino Non-Standard Interactions (NSI) <cit.> and sterile neutrinos <cit.>, among other BSM scenarios <cit.>.


The Standard Model of particle physics (SM) is thought to represent a low-energy effective gauge theory of a more fundamental theory in which the Planck scale (M_Pl∼ 10^28 eV) would be the natural mass scale. Likewise, the Standard Model Extension (SME) framework <cit.> parameterizes the possible breakdowns of the Lorentz and CPT symmetries that might arise from a fundamental theory that incorporates both, the gravitational force from one side, and the strong, weak, and electromagnetic interactions of the SM from the other side. 
Such potential violations of the CPT and Lorentz symmetries may arise at very high energies, well above the electroweak scale, for instance, within the context of string theory <cit.>.
Furthermore, it has been argued that potential Lorentz Invariance Violation (LIV) and CPT symmetry breakdown in the neutrino sector might result from neutrino NSI with scalar fields <cit.>.

For reviews regarding CPT violation and LIV in the neutrino sector, see, e.g., Refs. <cit.>.
As far as neutrino oscillation experiments are concerned, CPT violation in the neutrino sector was suggested in order to explain the Liquid Scintillator Neutrino Detector (LSND) and  MiniBooNE anomalies <cit.>.
For studies regarding possible LIV and CPT-violating effects in various neutrino oscillation experiments, we refer to <cit.>.
Moreover, non-oscillatory probes of Lorentz and CPT breakdowns in the neutrino sector include: beta decay <cit.>, double beta decay <cit.>, and cosmic neutrino background <cit.>.

In this paper, we examine the particular BSM case of potential violations of Lorentz invariance and CPT symmetries within the context of accelerator-based long-baseline neutrino oscillation experiments, such as the Deep Underground Neutrino Experiment (DUNE) <cit.>, a next-generation long-baseline neutrino oscillation experiment.
Long-baseline neutrino experiments have sensitivity to CPT symmetry and Lorentz invariance violations through SME coefficients proportional to their baseline. One can see this through the oscillation probability expression, which can be parameterized as <cit.>,


    P_ν_β→ν_α^(1) = 2L(P_𝒞^(1))_αβ ,

where L is the baseline and

    (P_𝒞^(1))_αβ= Im((S^(0)_αβ)^*(𝒞^(1))_αβ) .

In Eq. (<ref>),

    |(S^(0)_αβ)|^2 = P^(0)_ν_β→ν_α ,

is the standard three neutrino flavor oscillation probability. Considering only the CPT-odd coefficients (a_L)^A_αβ, for A=T,Z and α,β=e,μ,τ:

    (𝒞^(1))_αβ = (a_L)^T_αβ - N̂^Z (a_L)^Z_αβ ,

where

    N̂^Z = -sinχsinθcosϕ+cosχcosθ,

is the Z component of the vector representing the neutrino propagation direction in the Sun-centered frame in terms of local spherical coordinates at the detector (see Fig. 2 of Ref. <cit.> for an illustration). In Eq. (<ref>), χ is the colatitude of the detector, θ is the angle at the detector between the beam direction and vertical, and ϕ the angle between the beam and east of south. Notice that we are not taking into account sidereal variations so the components A=X,Y of (a_L)^A_αβ are not relevant. The aim of this work is to investigate the consequences of the possible coexistence of the time (T) and spatial (Z) components of the CPT-odd coefficients of the SME.





§ EXPERIMENTAL CONTEXT AND THEORETICAL FRAMEWORK


Long-baseline neutrino oscillation experiments are crucial for solving puzzles in the conventional three-neutrino paradigm as well as investigating different new physics scenarios, such as possible breakdowns of the Lorentz and CPT symmetries. 

In recent neutrino oscillation analyzes <cit.>, the possible violation of CPT symmetry, that would imply violation of the Lorentz invariance principle, was studied in the framework of the SME, in such a way that CPT-odd coefficients can have their time-component constrained through the neutrino propagation in long baseline experiments. However, from equations (<ref>)-(<ref>), we see that if the Z-component of the same coefficient is considered in the analysis, a correlation arises between them which affects the predicted limits on the time component alone. 


We introduced the SME phenomenology of CPT symmetry violation and LIV through the coefficients (ã_L)^A_αβ found in Eq. (<ref>) as part of the oscillation probability. Henceforth, we will focus on the Hamiltonian picture of neutrino propagation, which is directly related to our computing methodology explained in the next section, Section <ref>.

We describe the neutrino propagation from the source to the detector considering the Hamiltonian containing three components,

    Ĥ=Ĥ_vacuum + Ĥ_matter + Ĥ_LIV ,

where Ĥ_vacuum describes the neutrino propagation in vacuum, Ĥ_matter contains the Mikheyev-Smirnov-Wolfenstein (MSW) matter potential, and Ĥ_LIV includes the perturbative effects from LIV and possibly CPT symmetry violation. Explicitly, considering only the relevant CPT-odd coefficients of the SME neutrino sector:

    Ĥ_LIV =   (
    [    a_ee   a_e μ   a_e τ; a_e μ^*    a_μμ    a_μτ; a_e τ^*  a_μτ^*    a_ττ;         ]) - N̂^Z (
    [   a_ee^Z  a_e μ^Z  a_e τ^Z; a_e μ^Z*   a_μμ^Z   a_μτ^Z; a_e τ^Z*  a_μτ^Z*   a_ττ^Z;          ]) ,

where a_αβ represent the time component and a_αβ^Z the Z-spatial component of the coefficients for each neutrino oscillation channel (α,β=e,μ,τ). 

From now on, we neglect the script T from the time component and simplify the notation so that (a_L)^T_αβ = a_αβ and (a_L)^Z_αβ = a^Z_αβ.






§ METHODOLOGY



In this study, we use GLoBES <cit.> and its additional NSI plugin snu.c <cit.> which was modified to implement the CPT-odd SME coefficients at the Hamiltonian. Moreover, to simulate a DUNE-like experimental configuration, we use the available DUNE configuration ancillary files <cit.> for GLoBES and specifications from DUNE technical design report (TDR) <cit.>, where can be found more details and information on the experiment. DUNE will consist of up to four modules, each containing approximately 10 kton liquid argon detectors, with a mean neutrino energy around 2.5 GeV, located at 1285 km from the beam source (on-axis) with a 1.2 MW power in the first phase of operations. We consider a time exposure of 10 years, evenly distributed between neutrino and antineutrino modes. 

We employ a minimum square analysis to quantify the statistical significance to CPT-odd SME coefficients, using both neutrino and antineutrino data sets. The total χ^2-function is given as <cit.>

    χ^2 = ∑_ℓχ̃^2_ℓ + χ^2_prior,

where the corresponding χ̃^2_ℓ-function stands for each channel ℓ, with ℓ= ( ν_μ(ν_μ)→ν_e (ν_e), ν_μ(ν_μ)→ν_μ (ν_μ) ), and is provided as in Ref. <cit.>

    χ̃_ℓ^2= min_ξ_j[  ∑ _i^n_bin 2 { N_i,test^3 ν+LIV( Ω, Θ, {ξ_j})-N_i,true^3ν + N_i,true^3νlogN_i,true^3ν/N_i,test^3 ν+LIV( Ω, Θ, {ξ_j})}
                       + ∑_j^n_syst(ξ_j/σ_j)^2 ],

where N_i, true^3ν are the simulated events at the i-th energy bin, considering the standard three neutrino oscillations framework, N_i, test^3ν +LIV( Ω, Θ, {ξ_j}) are the computed events at the i-th energy bin including CPT-odd SME coefficients (one parameter at a time). In addition, Ω = {θ_12, θ_13, θ_23, δ_CP, Δ m_21^2, Δ m^2_31} is the set of neutrino oscillation parameters, while Θ = {|a_αβ|, ϕ_αβ, a_αα, ⋯} is the set of either isotropic, a_αβ, or Z-spatial, a_αβ^Z, SME coefficients, and {ξ_j} are the nuisance parameters to account for the systematic uncertainties. Moreover, σ_j are the systematic uncertainties as reported in the DUNE TDR <cit.>. To obtain our simulated events, we consider the neutrino oscillation parameters from Salas et al. <cit.> as true values, shown in Table <ref>.

 

Furthermore, the implementation of external input for the standard oscillation parameters on the χ^2 function is performed via Gaussian priors <cit.>

    χ^2_prior= ∑_k^n_priors(Ω_k,true-Ω_k,test)^2/σ^2_k .

The central values of the oscillation parameter priors, Ω_k,true, are fixed to their best-fit value from Ref. <cit.>, considering NO and σ_k corresponding to the 68.27% confidence level (C.L.) for each parameter k.








§ RESULTS AND DISCUSSIONS


In this section, we present our results of the projected limits and sensitivities to the CPT-odd SME coefficients a_αβ and a_αβ^Z at a DUNE-like experimental configuration. 

First, we examine the correlations among the SME coefficients a_αβ and a_αβ^Z. 
The correlation between the time and the Z-component of the CPT-odd coefficients for DUNE is depicted in Fig. <ref>. Because the Z-component is proportional to N̂^Z, which is relatively small for the DUNE location (N̂^Z ≃ 0.16), constraints on the time component are stronger in this case. We  obtain N̂^Z from the values of the angles relative to the DUNE location found in Table <ref>, Appendix <ref>.


Second, in  Figs. <ref> and <ref>, we examine the correlations of the coefficients a_αβ and a_αβ^Z with δ_CP. 
Moreover, we assess the impact of the flavor-changing a_αβ and a_αβ^Z on the future determination of δ_CP. A robust measurement of δ_CP can be accomplished even in the presence of Lorentz violating effects, as shown in Fig. <ref>.
The correlations of a_αβ and a_αβ^Z with sin^2 θ_23 are presented in Figs. <ref> and <ref>.
Just as importantly, the LIV phases, ϕ_αβ and ϕ_αβ^Z, might compromise the determination of δ_CP, and similarly for the atmospheric mixing angle, θ_23 <cit.>. 
In Fig. <ref>, we show the projected sensitivities to the isotropic coefficients a_αβ, as well as the anisotropic Z-spatial coefficients a_αβ^Z.
Finally, in Fig. <ref>, we display the correlations of the LIV phases, ϕ_αβ  and ϕ_αβ^Z, with respect to δ_CP.




 §.§ Correlations of time and Z-spatial components the CPT-odd SME coefficients


Previous assessments on the isotropic SME coefficients a_αβ considered  
a_αβ^Z=0 <cit.>. However, in this paper, we consider the possibility of correlations between a_αβ and a_αβ^Z.


In Fig. <ref>, we display in dashed lines the expected 95% C.L. sensitivity regions in the |a_αβ| vs. |a_αβ^Z| and (a_αα-a_ττ ) vs. (a_αα^Z-a_ττ^Z) projection planes, respectively on the left and right panels. The projected 95% C.L. sensitivities to the isotropic non-diagonal SME coefficients are |a_e τ|≲ 0.92× 10^-23 GeV (|a_e τ^Z|=0), |a_e μ|≲ 0.75× 10^-23 GeV (|a_e μ^Z|=0), and |a_μτ|≲ 0.68× 10^-23 GeV (|a_μτ^Z|=0), accordingly. However, existing bounds on the isotropic SME coefficients from the Super-Kamiokande experiment set |a_e τ| < 2.8 × 10^-23 GeV, |a_e μ| < 1.8 × 10^-23 GeV, and |a_μτ| < 5.1 × 10^-24 GeV <cit.>; all bounds at 95% C.L. Hence, a DUNE-like experimental configuration could improve the limits set on the isotropic SME coefficients |a_e μ| and |a_e τ| with respect to those from Super-Kamiokande.



 §.§ Correlations of the SME coefficients with δ_CP



The correlations among δ_CP and the off-diagonal SME coefficients, |a_αβ| and |a_αβ^Z|, are displayed in Fig. <ref>. In both situations, we have marginalized over the atmospheric mixing angle, θ_23, around its 1σ uncertainty <cit.>, as well as the CPT-odd phases, ϕ_αβ and ϕ_αβ^Z, in the interval [0-2π]. Furthermore, we find that the presence of the flavor-changing (e-μ) and (e-τ) channels have a greater impact on the determination of δ_CP .


Regarding the diagonal SME coefficients, taking advantage of the freedom to redefine the diagonal elements up to a global constant, we conduct an analysis of the elements a_αα-a_ττ and a_αα^Z-a_ττ^Z. The correlations with respect to δ_CP and sin^2θ_23 are shown in Figs. <ref> and <ref>, accordingly. 


In Fig. <ref>, we display our results of the projected 95% C.L. sensitivity regions (a_αα-a_ττ)-δ_CP (left panel) and (a_αα^Z-a_ττ^Z)-δ_CP (right panel). We notice that the determination of δ_CP is more affected by the coefficients a_ee-a_ττ and a_ee^Z-a_ττ^Z than it is by the μμ-ττ coefficients. Here, have marginalized over the atmospheric mixing angle θ_23 around its 1σ uncertainty <cit.>. Besides, all the remaining oscillation parameters were fixed to their best-fit values from Table <ref>.

In Fig. <ref>, we show the precision sensitivity to the leptonic CP-phase in the presence of LIV. We observe that the precision sensitivity of δ_CP is modified, at high significance (√(Δχ^2)≳ 7σ C.L.), by the inclusion of non-zero SME coefficients (|a_αβ| or |a_αβ^Z|) from the off-diagonal (e-μ) and (e-τ) channels, while remaining practically unchanged by the presence of the (μ-τ) channel. Hence, a robust measurement of δ_CP can be accomplished even in the presence of Lorentz violating effects from the non-diagonal channels.





 §.§ Correlations of the SME coefficients with sin^2 θ_23






In Fig. <ref>, we show the correlations between the off-diagonal SME coefficients, |a_αβ| and |a_αβ^Z|, with the atmospheric mixing angle, sin^2θ_23. In this case, we marginalized over δ_CP around its 1σ uncertainty <cit.>, as well as corresponding CPT-odd phases, ϕ_αβ and ϕ_αβ^Z, in the interval [0-2π]. In both cases, we notice that the determination of sin^2θ_23 is mostly impacted by the SME coefficients a_e τ and a_e τ^Z, while the (e-μ) and (μ-τ) channels have a mild influence on the determination of the atmospheric mixing angle.




In Fig. <ref>, we observe that the diagonal SME coefficients have a similar impact on the determination of the mixing angle, showing a positive (negative) correlation for the isotropic SME coefficients, a_ee-a_ττ (a_μμ-a_ττ), and negative (positive) correlation for the Z-spatial ones, a_ee^Z-a_ττ^Z (a_μμ^Z-a_ττ^Z).



 §.§ Sensitivities and correlations of the LIV phases (ϕ_αβ, ϕ_αβ^Z) with δ_CP





In Fig. <ref>, we present our results of the expected sensitivities to the SME coefficients a_αβ and a_αβ^Z at DUNE, the projected 95% C.L. sensitivities to the isotropic SME coefficients are |a_e τ|≲ 1.6 × 10^-23 GeV, |a_e μ|≲ 0.9 × 10^-23 GeV, and |a_μτ|≲ 0.8 × 10^-23 GeV, respectively, while for the Z-spatial coefficients, the expected 95% C.L. sensitivities are |a_e τ^Z|≲ 7.8 × 10^-23 GeV, |a_e μ|≲ 5.7 × 10^-23 GeV, and |a_μτ|≲ 5.2 × 10^-23 GeV, accordingly. 




In Fig. <ref>, we display the correlations of δ_CP with the corresponding LIV phases ϕ_αβ and ϕ_αβ^Z. The magnitude of the non-diagonal SME coefficients were fixed to |a_αβ| = 1.0 × 10^-23 GeV and |a_αβ^Z| = 8.0 × 10^-23 GeV. We notice that the LIV phases, ϕ_αβ and ϕ_αβ^Z, have a similar impact on the reconstructed δ_CP phase. Furthermore, the expected allowed value of the leptonic CP phase is δ_CP / π= 1.08 ± 0.6 at 95% C.L. The (e-μ) and (e-τ) channels show higher correlations.







§ CONCLUSIONS



High energy neutrino oscillations provide an excellent opportunity to investigate potential breakdowns of the Lorentz and CPT symmetries.
In this study, we have analyzed the impact of either the isotropic or anisotropic Z-spatial coefficients for CPT-odd Lorentz violation on the determination of the mixing angle θ_23 and leptonic CP-phase δ_CP, as well as the projected sensitivities and correlations, considering a DUNE-like setup.


The determination of the aforementioned atmospheric mixing angle θ_23 and phase δ_CP will be mostly impacted by effects from either the isotropic or Z-spatial CPT-odd SME coefficients from the (e-τ) channel; effects from the (e-μ) channel are moderate, and the impact from the (μ-τ) channel is muted: Figs. <ref> and <ref>. In addition, we observed that the presence of the corresponding LIV phases, ϕ_αβ and ϕ_αβ^Z, will have a similar effect on the reconstructed CP-phase, showing a significant correlation among the flavor-changing channels, (e-μ) and (e-τ), accordingly: Fig. <ref>. Regarding the diagonal elements, (ee-ττ) and (μμ-ττ), the determination of the atmospheric mixing angle θ_23 will be slightly influenced by the presence of either isotropic or anisotropic coefficients: Fig. <ref>. Effects from the diagonal (ee-ττ) channel will have a considerable impact on the leptonic CP-phase determination: Fig. <ref>.

On the other hand, the expected 95% C.L. sensitivities to the isotropic SME coefficients are |a_e τ|≲ 1.6 × 10^-23 GeV, |a_e μ|≲ 0.9 × 10^-23 GeV, and |a_μτ|≲ 0.8 × 10^-23 GeV (left panel of Fig. <ref>), while for the Z-spatial coefficients, the projected 95% C.L. sensitivities are |a_e τ^Z|≲ 7.8 × 10^-23 GeV, |a_e μ|≲ 5.7 × 10^-23 GeV, and |a_μτ|≲ 5.2 × 10^-23 GeV, accordingly (right panel of Fig. <ref>). Besides, we have explored the correlations among the isotropic and Z-spatial sectors (Fig. <ref>), and limits on the isotropic a_αβ can be relaxed depending on the limits set on the a_αβ^Z coefficients. While current limits set on the (e-τ) and (e-μ) channels from the Super-Kamiokande experiment can be improved considering a DUNE-like configuration, that is not the case for the (μ-τ) channel, confirming a complementarity between the two experiments.


This paper represents the views of the authors and should not be considered a DUNE collaboration paper.




§ ACKNOWLEDGEMENTS

We would like to acknowledge J. S. Diaz for useful discussions. This work was partially supported by SNII-México and CONAHCyT research Grant No. A1-S-23238.








§ DUNE LOCATION DEFINITIONS



In our analysis we consider a fixed value of N̂^Z. Here we show how the value of N̂^Z change in accordance with the determination of the experiment location.





As displayed in Fig. <ref>, and from Eqs. (<ref>) and (<ref>), experiment location resulting in bigger values of N̂^Z could enhance the sensitivity of the Z-spatial SME coefficients (ã_L)^Z_αβ.


84


NOvA:2021nfi
M. A. Acero et al. [NOvA],

Phys. Rev. D 106 (2022) no.3, 032004
doi:10.1103/PhysRevD.106.032004
[arXiv:2108.08219 [hep-ex]].



NOvA:2023iam
M. A. Acero et al. [NOvA],

Phys. Rev. D 110 (2024) no.1, 1
doi:10.1103/PhysRevD.110.012005
[arXiv:2311.07835 [hep-ex]].



T2K:2023smv
K. Abe et al. [T2K],

Eur. Phys. J. C 83 (2023) no.9, 782
doi:10.1140/epjc/s10052-023-11819-x
[arXiv:2303.03222 [hep-ex]].



T2K:2023mcm
K. Abe et al. [T2K],

Phys. Rev. D 108 (2023) no.7, 072011
doi:10.1103/PhysRevD.108.072011
[arXiv:2305.09916 [hep-ex]].



IceCube:2024xjj
R. Abbasi et al. [IceCube],

[arXiv:2405.02163 [hep-ex]].



T2K:2024wfn
K. Abe et al. [T2K and Super-Kamiokande],

[arXiv:2405.12488 [hep-ex]].



Denton:2020uda
P. B. Denton, J. Gehrlein and R. Pestes,

Phys. Rev. Lett. 126 (2021) no.5, 051801
doi:10.1103/PhysRevLett.126.051801
[arXiv:2008.01110 [hep-ph]].




Chatterjee:2020kkm
S. S. Chatterjee and A. Palazzo,

Phys. Rev. Lett. 126 (2021) no.5, 051802
doi:10.1103/PhysRevLett.126.051802
[arXiv:2008.04161 [hep-ph]].




Delgadillo:2023lyp
L. A. Delgadillo and O. G. Miranda,

Phys. Rev. D 108 (2023) no.9, 095024
doi:10.1103/PhysRevD.108.095024
[arXiv:2304.05545 [hep-ph]].



Chatterjee:2020yak
S. S. Chatterjee and A. Palazzo,

[arXiv:2005.10338 [hep-ph]].




deGouvea:2022kma
A. de Gouvêa, G. Jusino Sánchez and K. J. Kelly,

Phys. Rev. D 106 (2022) no.5, 055025
doi:10.1103/PhysRevD.106.055025
[arXiv:2204.09130 [hep-ph]].




Lin:2023xyk
H. X. Lin, J. Tang and S. Vihonen,

[arXiv:2312.11704 [hep-ph]].



Colladay:1998fq
D. Colladay and V. A. Kostelecky,

Phys. Rev. D 58 (1998), 116002
doi:10.1103/PhysRevD.58.116002
[arXiv:hep-ph/9809521 [hep-ph]].



Kostelecky:1988zi
V. A. Kostelecky and S. Samuel,

Phys. Rev. D 39 (1989), 683
doi:10.1103/PhysRevD.39.683




Kostelecky:1991ak
V. A. Kostelecky and R. Potting,

Nucl. Phys. B 359 (1991), 545-570
doi:10.1016/0550-3213(91)90071-5




Kostelecky:1995qk
V. A. Kostelecky and R. Potting,

Phys. Lett. B 381 (1996), 89-96
doi:10.1016/0370-2693(96)00589-8
[arXiv:hep-th/9605088 [hep-th]].




Colladay:1996iz
D. Colladay and V. A. Kostelecky,

Phys. Rev. D 55 (1997), 6760-6774
doi:10.1103/PhysRevD.55.6760
[arXiv:hep-ph/9703464 [hep-ph]].




Gu:2005eq
P. H. Gu, X. J. Bi and X. m. Zhang,

Eur. Phys. J. C 50 (2007), 655-659
doi:10.1140/epjc/s10052-007-0217-7
[arXiv:hep-ph/0511027 [hep-ph]].




Ando:2009ts
S. Ando, M. Kamionkowski and I. Mocioiu,

Phys. Rev. D 80 (2009), 123522
doi:10.1103/PhysRevD.80.123522
[arXiv:0910.4391 [hep-ph]].




Simpson:2016gph
F. Simpson, R. Jimenez, C. Pena-Garay and L. Verde,

Phys. Dark Univ. 20 (2018), 72-77
doi:10.1016/j.dark.2018.04.002
[arXiv:1607.02515 [astro-ph.CO]].





Klop:2017dim
N. Klop and S. Ando,

Phys. Rev. D 97 (2018) no.6, 063006
doi:10.1103/PhysRevD.97.063006
[arXiv:1712.05413 [hep-ph]].





Capozzi:2018bps
F. Capozzi, I. M. Shoemaker and L. Vecchi,

JCAP 07 (2018), 004
doi:10.1088/1475-7516/2018/07/004
[arXiv:1804.05117 [hep-ph]].




Farzan:2018pnk
Y. Farzan and S. Palomares-Ruiz,

Phys. Rev. D 99 (2019) no.5, 051702
doi:10.1103/PhysRevD.99.051702
[arXiv:1810.00892 [hep-ph]].




Gherghetta:2023myo
T. Gherghetta and A. Shkerin,

Phys. Rev. D 108 (2023) no.9, 9
doi:10.1103/PhysRevD.108.095009
[arXiv:2305.06441 [hep-ph]].




Arguelles:2023wvf
C. A. Argüelles, K. Farrag and T. Katori,

PoS ICRC2023 (2023), 1415
doi:10.22323/1.444.1415
[arXiv:2402.18126 [hep-ph]].



Arguelles:2023jkh
C. A. Argüelles, K. Farrag and T. Katori,

doi:10.1142/9789811275388_0047
[arXiv:2401.15716 [hep-ph]].



Lambiase:2023hpq
G. Lambiase and T. K. Poddar,

JCAP 01 (2024), 069
doi:10.1088/1475-7516/2024/01/069
[arXiv:2307.05229 [hep-ph]].



Cordero:2023hua
R. Cordero and L. A. Delgadillo,

Phys. Lett. B 853 (2024), 138687
doi:10.1016/j.physletb.2024.138687
[arXiv:2312.16320 [hep-ph]].



Arguelles:2024cjj
C. A. Argüelles, K. Farrag and T. Katori,

[arXiv:2404.10926 [hep-ph]].



Kostelecky:2003cr
V. A. Kostelecky and M. Mewes,

Phys. Rev. D 69 (2004), 016005
doi:10.1103/PhysRevD.69.016005
[arXiv:hep-ph/0309025 [hep-ph]].





Kostelecky:2011gq
A. Kostelecky and M. Mewes,

Phys. Rev. D 85 (2012), 096005
doi:10.1103/PhysRevD.85.096005
[arXiv:1112.6395 [hep-ph]].




Diaz:2016xpw
J. S. Diaz,

Symmetry 8 (2016) no.10, 105
doi:10.3390/sym8100105
[arXiv:1609.09474 [hep-ph]].




Torri:2020dec
M. D. C. Torri,

Universe 6 (2020) no.3, 37
doi:10.3390/universe6030037
[arXiv:2110.09186 [hep-ph]].



Moura:2022dev
C. A. Moura and F. Rossi-Torres,

Universe 8 (2022) no.1, 42
doi:10.3390/universe8010042




Barenboim:2022rqu
G. Barenboim,

Front. in Phys. 10 (2022), 813753
doi:10.3389/fphy.2022.813753




Murayama:2000hm
H. Murayama and T. Yanagida,

Phys. Lett. B 520 (2001), 263-268
doi:10.1016/S0370-2693(01)01136-4
[arXiv:hep-ph/0010178 [hep-ph]].




Barenboim:2001ac
G. Barenboim, L. Borissov, J. D. Lykken and A. Y. Smirnov,

JHEP 10 (2002), 001
doi:10.1088/1126-6708/2002/10/001
[arXiv:hep-ph/0108199 [hep-ph]].




Barenboim:2004wu
G. Barenboim and N. E. Mavromatos,

JHEP 01 (2005), 034
doi:10.1088/1126-6708/2005/01/034
[arXiv:hep-ph/0404014 [hep-ph]].




Kostelecky:2004hg
V. A. Kostelecky and M. Mewes,

Phys. Rev. D 70 (2004), 076002
doi:10.1103/PhysRevD.70.076002
[arXiv:hep-ph/0406255 [hep-ph]].



MiniBooNE:2011pix
A. A. Aguilar-Arevalo et al. [MiniBooNE],

Phys. Lett. B 718 (2013), 1303-1308
doi:10.1016/j.physletb.2012.12.020
[arXiv:1109.3480 [hep-ex]].




Katori:2012pe
T. Katori [MiniBooNE],

Mod. Phys. Lett. A 27 (2012), 1230024
doi:10.1142/S0217732312300248
[arXiv:1206.6915 [hep-ex]].



Bahcall:2002ia
J. N. Bahcall, V. Barger and D. Marfatia,

Phys. Lett. B 534 (2002), 120-123
doi:10.1016/S0370-2693(02)01714-8
[arXiv:hep-ph/0201211 [hep-ph]].





DoubleChooz:2012eiq
Y. Abe et al. [Double Chooz],

Phys. Rev. D 86 (2012), 112009
doi:10.1103/PhysRevD.86.112009
[arXiv:1209.5810 [hep-ex]].





MINOS:2008fnv
P. Adamson et al. [MINOS],

Phys. Rev. Lett. 101 (2008), 151601
doi:10.1103/PhysRevLett.101.151601
[arXiv:0806.4945 [hep-ex]].




MINOS:2010kat
P. Adamson et al. [MINOS],

Phys. Rev. Lett. 105 (2010), 151601
doi:10.1103/PhysRevLett.105.151601
[arXiv:1007.2791 [hep-ex]].




IceCube:2010fyu
R. Abbasi et al. [IceCube],

Phys. Rev. D 82 (2010), 112003
doi:10.1103/PhysRevD.82.112003
[arXiv:1010.4096 [astro-ph.HE]].



Super-Kamiokande:2014exs
K. Abe et al. [Super-Kamiokande],

Phys. Rev. D 91 (2015) no.5, 052003
doi:10.1103/PhysRevD.91.052003
[arXiv:1410.4267 [hep-ex]].




SNO:2018mge
B. Aharmim et al. [SNO],

Phys. Rev. D 98 (2018) no.11, 112013
doi:10.1103/PhysRevD.98.112013
[arXiv:1811.00166 [hep-ex]].




Barenboim:2017ewj
G. Barenboim, C. A. Ternes and M. Tórtola,

Phys. Lett. B 780 (2018), 631-637
doi:10.1016/j.physletb.2018.03.060
[arXiv:1712.01714 [hep-ph]].




Barenboim:2018ctx
G. Barenboim, M. Masud, C. A. Ternes and M. Tórtola,

Phys. Lett. B 788 (2019), 308-315
doi:10.1016/j.physletb.2018.11.040
[arXiv:1805.11094 [hep-ph]].




KumarAgarwalla:2019gdj
S. Kumar Agarwalla and M. Masud,

Eur. Phys. J. C 80 (2020) no.8, 716
doi:10.1140/epjc/s10052-020-8303-1
[arXiv:1912.13306 [hep-ph]].





Rahaman:2021leu
U. Rahaman,

Eur. Phys. J. C 81 (2021) no.9, 792
doi:10.1140/epjc/s10052-021-09598-4
[arXiv:2103.04576 [hep-ph]].



Ngoc:2022uhg
T. V. Ngoc, S. Cao, N. T. H. Van and P. T. Quyen,

Phys. Rev. D 107 (2023) no.1, 016013
doi:10.1103/PhysRevD.107.016013
[arXiv:2210.13044 [hep-ph]].





Sarker:2023mlz
A. Sarker, A. Medhi and M. M. Devi,

Eur. Phys. J. C 83 (2023) no.7, 592
doi:10.1140/epjc/s10052-023-11785-4
[arXiv:2302.10456 [hep-ph]].





Raikwal:2023lzk
D. Raikwal, S. Choubey and M. Ghosh,

Phys. Rev. D 107 (2023) no.11, 115032
doi:10.1103/PhysRevD.107.115032
[arXiv:2303.10892 [hep-ph]].




Sahoo:2021dit
S. Sahoo, A. Kumar and S. K. Agarwalla,

JHEP 03 (2022), 050
doi:10.1007/JHEP03(2022)050
[arXiv:2110.13207 [hep-ph]].



Agarwalla:2023wft
S. K. Agarwalla, S. Das, S. Sahoo and P. Swain,

JHEP 07 (2023), 216
doi:10.1007/JHEP07(2023)216
[arXiv:2302.12005 [hep-ph]].




Mishra:2023nqf
S. Mishra, S. Shukla, L. Singh and V. Singh,

[arXiv:2309.01756 [hep-ph]].





Barenboim:2018lpo
G. Barenboim, C. A. Ternes and M. Tórtola,

Eur. Phys. J. C 79 (2019) no.5, 390
doi:10.1140/epjc/s10052-019-6900-7
[arXiv:1804.05842 [hep-ph]].




Sahoo:2022nbu
S. Sahoo, A. Kumar, S. K. Agarwalla and A. Dighe,

Phys. Lett. B 841 (2023), 137949
doi:10.1016/j.physletb.2023.137949
[arXiv:2205.05134 [hep-ph]].




Majhi:2022fed
R. Majhi, D. K. Singha, M. Ghosh and R. Mohanta,

Phys. Rev. D 107 (2023) no.7, 075036
doi:10.1103/PhysRevD.107.075036
[arXiv:2212.07244 [hep-ph]].




Barenboim:2023krl
G. Barenboim, P. Martínez-Miravé, C. A. Ternes and M. Tórtola,

Phys. Rev. D 108 (2023) no.3, 035039
doi:10.1103/PhysRevD.108.035039
[arXiv:2305.06384 [hep-ph]].





Pan:2023qln
S. Pan, K. Chakraborty and S. Goswami,

Eur. Phys. J. C 84 (2024) no.4, 354
doi:10.1140/epjc/s10052-024-12541-y
[arXiv:2308.07566 [hep-ph]].



Arguelles:2015dca
C. A. Argüelles, T. Katori and J. Salvado,

Phys. Rev. Lett. 115 (2015), 161303
doi:10.1103/PhysRevLett.115.161303
[arXiv:1506.02043 [hep-ph]].




Fiza:2022xfw
N. Fiza, N. R. Khan Chowdhury and M. Masud,

JHEP 01 (2023), 076
doi:10.1007/JHEP01(2023)076
[arXiv:2206.14018 [hep-ph]].




Cordero:2024nho
R. Cordero and L. A. Delgadillo,

[arXiv:2407.02729 [hep-ph]].



Shukla:2024fnw
S. Shukla, S. Mishra, L. Singh and V. Singh,

[arXiv:2408.01520 [hep-ph]].



IceCube:2021tdn
R. Abbasi et al. [IceCube],

Nature Phys. 18 (2022) no.11, 1287-1292
doi:10.1038/s41567-022-01762-1
[arXiv:2111.04654 [hep-ex]].



Testagrossa:2023ukh
F. Testagrossa, D. F. G. Fiorillo and M. Bustamante,

[arXiv:2310.12215 [astro-ph.HE]].




Telalovic:2023tcb
B. Telalovic and M. Bustamante,

[arXiv:2310.15224 [astro-ph.HE]].




Diaz:2013saa
J. S. Díaz, A. Kostelecký and R. Lehnert,

Phys. Rev. D 88 (2013) no.7, 071902
doi:10.1103/PhysRevD.88.071902
[arXiv:1305.4636 [hep-ph]].



Lehnert:2021tbv
R. Lehnert,

Phys. Lett. B 828 (2022), 137017
doi:10.1016/j.physletb.2022.137017
[arXiv:2112.13803 [hep-ph]].



KATRIN:2022qou
M. Aker et al. [KATRIN],

Phys. Rev. D 107 (2023) no.8, 082005
doi:10.1103/PhysRevD.107.082005
[arXiv:2207.06326 [nucl-ex]].





EXO-200:2016hbz
J. B. Albert et al. [EXO-200],

Phys. Rev. D 93 (2016) no.7, 072001
doi:10.1103/PhysRevD.93.072001
[arXiv:1601.07266 [nucl-ex]].




CUPID:2019kto
O. Azzolini et al. [CUPID],

Phys. Rev. D 100 (2019) no.9, 092002
doi:10.1103/PhysRevD.100.092002
[arXiv:1911.02446 [nucl-ex]].




Diaz:2015aua
J. S. Diaz and F. R. Klinkhamer,

Phys. Rev. D 93 (2016) no.5, 053004
doi:10.1103/PhysRevD.93.053004
[arXiv:1512.00817 [hep-ph]].




DUNE:2020jqi
B. Abi et al. [DUNE],

Eur. Phys. J. C 80 (2020) no.10, 978
doi:10.1140/epjc/s10052-020-08456-z
[arXiv:2006.16043 [hep-ex]].




Diaz:2009qk
J. S. Diaz, V. A. Kostelecky and M. Mewes,

Phys. Rev. D 80, 076007 (2009)
doi:10.1103/PhysRevD.80.076007
[arXiv:0908.1401 [hep-ph]].




Bailey:2006fd
Q. G. Bailey and V. A. Kostelecky,

Phys. Rev. D 74 (2006), 045001
doi:10.1103/PhysRevD.74.045001
[arXiv:gr-qc/0603030 [gr-qc]].




Huber:2004ka
P. Huber, M. Lindner and W. Winter,

Comput. Phys. Commun. 167 (2005), 195
doi:10.1016/j.cpc.2005.01.003
[arXiv:hep-ph/0407333 [hep-ph]].




Huber:2007ji
P. Huber, J. Kopp, M. Lindner, M. Rolinec and W. Winter,

Comput. Phys. Commun. 177 (2007), 432-438
doi:10.1016/j.cpc.2007.05.004
[arXiv:hep-ph/0701187 [hep-ph]].




Kopp:2006wp
J. Kopp,

Int. J. Mod. Phys. C 19 (2008), 523-548
doi:10.1142/S0129183108012303
[arXiv:physics/0610206 [physics]].



Kopp:2007rz
J. Kopp, M. Lindner, T. Ota and J. Sato,

[arXiv:0710.1867 [hep-ph]].





DUNE:2021cuw
B. Abi et al. [DUNE],

[arXiv:2103.04797 [hep-ex]].





DUNE:2020lwj
B. Abi et al. [DUNE],

JINST 15 (2020) no.08, T08008
doi:10.1088/1748-0221/15/08/T08008
[arXiv:2002.02967 [physics.ins-det]].




DUNE:2020ypp
B. Abi et al. [DUNE],

[arXiv:2002.03005 [hep-ex]].






Huber:2002mx
P. Huber, M. Lindner and W. Winter,

Nucl. Phys. B 645 (2002), 3-48
doi:10.1016/S0550-3213(02)00825-8
[arXiv:hep-ph/0204352 [hep-ph]].




deSalas:2020pgw
P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola and J. W. F. Valle,

JHEP 02 (2021), 071
doi:10.1007/JHEP02(2021)071
[arXiv:2006.11237 [hep-ph]].










