Neutrino Lorentz invariance violation from the $CPT-$even SME coefficients through a tensor interaction with cosmological scalar fields

The discovery of the Universe’s accelerating expansion Riess et al. (1998); Perlmutter et al. (1999) is one of the most important, captivating, and puzzling open questions in cosmology Sahni (2004); Alam et al. (2004). Several proposals to architecture Dark Matter (DM) and Dark Energy (DE) have been investigated in the literature. The Lambda Cold Dark Matter ($\Lambda$CDM) model is among the most popular explanations for cosmological observations, although recent results from the Dark Energy Spectroscopic Instrument (DESI) Adame et al. (2024) show some tension with this model. Different candidates for DM particles have been considered, ranging in a very wide mass spectrum, making the Weakly Interacting Massive Particle (WIMP) paradigm one of the best motivated from a theoretical point of view Arbey and Mahmoudi (2021). However, despite thorough searches for WIMPs, they all have failed to find any signature.

Other DM and DE candidates have also been studied in an effort to find a plausible explanation to the cosmological observations. For instance, ultralight scalars are well-motivated proposals for cosmological DM Magana and Matos (2012); Ferreira (2021); Suárez et al. (2014); Hui et al. (2017); Matos et al. (2024). Moreover, several ultralight scalars become apparent within the context of string theories Arvanitaki et al. (2010); Cicoli et al. (2022); Acharya et al. (2010); Marsh (2016). ¹¹1Some strategies to hunt for ultralight scalars as cosmological DM involve atomic clocks Arvanitaki et al. (2015), resonant-mass detectors Arvanitaki et al. (2016) and atomic gravitational wave detectors Arvanitaki et al. (2018). Scalar fields minimally coupled to gravity are sufficiently justified models of DE Li et al. (2011), namely quintessence Wetterich (1988); Zlatev et al. (1999), symmetrons Kading:2023hdb ; Burrage:2018zuj , and k-essence Chiba et al. (2000); Armendariz-Picon et al. (2000, 2001); Melchiorri et al. (2003); Chiba (2002); Chimento and Feinstein (2004); Chimento (2004).

In k-essence models, the scalar field plays a significant role in describing the DE puzzle. This field could have adequate behavior at early epochs and can reproduce the dynamical effects of the cosmological constant at late times. Classes of k-essence Lagrangians were introduced in several settings, for example, as a possible model for inflation Armendariz-Picon et al. (1999); Garriga and Mukhanov (1999). Subsequently, k-essence models were used as another possibility to describe the characteristics of DE and as an alternative mechanism for unifying DE and DM Scherrer (2004). Purely kinetic k-essence models de Putter and Linder (2007); Gao and Yang (2010) are, in a way, as simple as quintessence models, because they rely only on one function ($F$) through the expression of the Lagrangian density ${\mathcal{L}}=F(X)$, where $X$ is the kinetic term.

Among other proposals, there are second-order derivative scalar field models, or generalized galileons Nicolis et al. (2009); de Rham and Tolley (2010); Goon et al. (2011a, b), with the property that their equations of motion are second-order.

Interesting examples of this class of models are the so-called kinetic gravity braiding models, that are formulated from a Lagrangian that includes a D’Alambertian operator and an arbitrary function of a non-canonical kinetic term giving rise to appealing cosmological effects Deffayet et al. (2010); Pujolas et al. (2011); Kimura and Yamamoto (2011); Maity (2013).

At the cosmological level, all the aforementioned models of DM and DE can be described as a perfect fluid through their energy-momentum tensor $T_{\varphi}^{\mu\nu}$ Weinberg (2008). At present, discrimination among them could be done at the level of the energy-momentum tensor perturbations. On the other hand, the search for signatures confirming the existence of scalars is an important challenge. Proposals considering the interaction of scalar fields with neutrinos have demonstrated that a possible signature might be detected in long-baseline neutrino experiments Berlin (2016); de Salas et al. (2016); Krnjaic et al. (2018); Brdar et al. (2018); Smirnov and Xu (2019); Dev et al. (2021); Losada et al. (2022, 2023); Huang et al. (2022); Cordero et al. (2023); Sen and Smirnov (2024) or via modifications to the ultrahigh energy neutrino fluxes Barranco et al. (2011); Reynoso and Sampayo (2016). Besides, it has also been argued that interactions of neutrinos with scalars, if they exist, could induce an apparent violation of Lorentz and $CPT$ symmetries in the neutrino sector Gu et al. (2007); Ando et al. (2009); Klop and Ando (2018); Capozzi et al. (2018); Farzan and Palomares-Ruiz (2019); Ge and Murayama (2019); Gherghetta and Shkerin (2023); Argüelles et al. (2023a, b); Lambiase and Poddar (2024); Cordero and Delgadillo (2024); Argüelles et al. (2024). These scalars can be identified as either DM or DE candidates. We refer to Kostelecky and Mewes (2004a); Diaz (2016); Torri (2020); Moura and Rossi-Torres (2022); Barenboim (2022) for comprehensive reviews concerning Lorentz and $CPT$ symmetry violations in the neutrino sector, within the Standard Model Extension (SME) framework Colladay and Kostelecky (1998). An initial proposal for $CPT$ violation in the neutrino sector was discussed in Ref. Barenboim and Lykken (2003).

Recently, there has been an increased interest in searches for Lorentz invariance and $CPT-$breakdowns in neutrino oscillation experiments Barenboim et al. (2019); Abbasi et al. (2022); Sahoo et al. (2022); Agarwalla et al. (2023); Raikwal et al. (2023); Crivellin:2020oov ; Testagrossa et al. (2023); Shukla:2024fnw . For instance, comprehensive studies of the isotropic $CPT-$even SME coefficient, $(c_{L})^{TT}$, at different long-baseline experiments can be found in Refs. Agarwalla et al. (2023); Raikwal et al. (2023). ²²2As shown in Díaz et al. (2020), it is possible to relate the isotropic SME coefficients $(c_{L})^{TT}$ with the effects of the violation of the equivalence principle (VEP) in the neutrino sector. Besides, there is a correspondence between quantum-decoherence effects in neutrinos and the effective coefficients of the SME Barenboim and Mavromatos (2005); De Romeri et al. (2023); Barenboim and Gago (2024). In Ref. Delgadillo:2024bae , we examine the phenomenology of isotropic and anisotropic $CPT-$odd SME coefficients, $(a_{L})^{T}$ and $(a_{L})^{Z}$, considering the Deep Underground Neutrino Experiment (DUNE) Abi et al. (2020a) configuration.

In this paper, we consider the possibility of tensorial neutrino non-standard interactions with scalar fields; such as quintessence and other related DE models, as well as the case of ultralight dark matter (ULDM). We focus on the scenario where the $CPT-$even SME coefficients could correspond to a tensorial interaction of neutrinos with cosmological scalar fields. In this scenario, an apparent Lorentz$-$violating signature may manifest and give a sizable signal at upcoming and present neutrino oscillation experiments. Besides, as a case of study, we asses the projected sensitivities of a DUNE-like setup to these types of LIV scenarios. This is the first time sensitivities for the $CPT-$even SME coefficients $(c_{L})^{ZZ}$ at DUNE are presented.

The manuscript is organized as follows: In Section II, we introduce the theoretical foundations, where the basics of Lorentz invariance violations are presented together with the necessary Lagrangian formalism to describe the several models of scalar fields useful to model DM and DE. In Section III, we describe the phenomenology of a tensorial neutrino interaction with scalar fields as either DM or DE candidates and its connection with the $CPT-$even SME coefficients $(c_{L})_{\alpha\beta}^{\mu\nu}$. In Section IV, we assess the sensitivities to the $CPT-$even SME coefficients, $(c_{L})_{\alpha\beta}^{\mu\nu}$, considering the DUNE configuration. Finally, we give our conclusions.

Within the SME framework, violations of Lorentz invariance (LIV) in the fermion sector, are parameterized by the effective Lagrangian Barenboim (2022)

$$\mathcal{L}_{\text{eff}}^{\psi}=\mathcal{L}_{\text{SM}}^{\psi}+\mathcal{L}_{\text{LIV}}+\text{h.c.}\,,$$

(1)

where $\mathcal{L}_{\text{SM}}^{\psi}$ is the Standard Model (SM) fermion Lagrangian, $\mathcal{L}_{\text{LIV}}$ being the Lorentz and $CPT-$violating Lagrangian term,

$$-\mathcal{L}_{\text{LIV}}=\frac{1}{2}\Big{\{}p^{\mu}_{\alpha\beta}\bar{\psi}_{\alpha}\gamma_{\mu}\psi_{\beta}+q^{\mu}_{\alpha\beta}\bar{\psi}_{\alpha}\gamma_{5}\gamma_{\mu}\psi_{\beta}-ir^{\mu\nu}_{\alpha\beta}\bar{\psi}_{\alpha}\gamma_{\mu}\partial_{\nu}\psi_{\beta}-is^{\mu\nu}_{\alpha\beta}\bar{\psi}_{\alpha}\gamma_{5}\gamma_{\mu}\partial_{\nu}\psi_{\beta}\Big{\}}\,.$$

(2)

In the case of neutrinos, it is convenient to define

$$(a_{L})^{\mu}_{\alpha\beta}=(p+q)^{\mu}_{\alpha\beta}\,\,\,\,\,{\rm and}\,\,\,\,\,(c_{L})^{\mu\nu}_{\alpha\beta}=(r+s)^{\mu\nu}_{\alpha\beta}\,,$$

(3)

where the $(a_{L})^{\mu}_{\alpha\beta}$ coefficients are $CPT-$odd, while the $(c_{L})^{\mu\nu}_{\alpha\beta}$ coefficients are $CPT-$even. These type of $CPT-$conserving terms, $(c_{L})^{\mu\nu}_{\alpha\beta}$, were studied in the context of a Finslerian Geometrical model Antonelli et al. (2018), violations of diffeomorphism invariance Reyes:2024hqi ; Santos:2024iyc , and neutrino interactions with a k-essence field Gauthier et al. (2010). For other scenarios of neutrino$-$dark energy interactions, we refer the reader to Refs. Khalifeh and Jimenez (2021a, b, 2022).

Lately, it has been suggested that violations of Lorentz invariance and $CPT$ symmetry in the neutrino sector may arise from a neutrino$-$current coupled with a dynamical cosmic field, $\varphi(t)$ (see, e.g., Refs. Gu et al. (2007); Ando et al. (2009); Klop and Ando (2018); Capozzi et al. (2018); Argüelles et al. (2023b); Lambiase and Poddar (2024); Cordero and Delgadillo (2024))

$$(a_{L})^{\mu}_{\alpha\beta}\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta}\rightarrow y_{\alpha\beta}\frac{\partial^{\mu}\varphi}{\Lambda}\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta},$$

(4)

being $y_{\alpha\beta}$ some coupling constants, $\Lambda$ the energy scale of the interaction, and $\varphi=\varphi(t)$ a time$-$varying scalar field, which could be identified as either a dark matter or dark energy candidate. ³³3For ultra-relativistic neutrinos, the phenomenology with axion-like dark matter is the same as the ultralight dark matter scenario Lambiase and Poddar (2024); Gherghetta and Shkerin (2023); Huang and Nath (2018). Here, violations of the Lorentz and $CPT$ symmetries emerge via the $CPT-$odd SME coefficients $(a_{L})_{\alpha\beta}^{\mu}\rightarrow y_{\alpha\beta}\partial^{\mu}\varphi\Lambda^{-1}$.

Motivated by the aforementioned proposals, let us consider the corresponding free scalar field Lagrangian and its energy-momentum tensor, which are relevant for describing both, scalar field dark matter and dark energy quintessence Weinberg (2008); Ferreira (2021); Wetterich (1988),

$$\begin{split}&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathcal{L}_{\varphi}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+V(\varphi),\\ &\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ T^{\mu\nu}_{\varphi}=\frac{2}{\sqrt{-g}}\frac{\delta}{\delta g_{\mu\nu}}\Big{[}\sqrt{-g}\Big{(}\frac{1}{2}g^{\lambda\sigma}\partial_{\lambda}\varphi\partial_{\sigma}\varphi+V(\varphi)\Big{)}\Big{]}\,.\end{split}$$

(5)

Therefore, the scalar field energy-momentum tensor $T^{\mu\nu}_{\varphi}$ can be expressed as

$$T^{\mu\nu}_{\varphi}=g^{\mu\nu}\Big{(}\frac{1}{2}g^{\lambda\sigma}\partial_{\lambda}\varphi\partial_{\sigma}\varphi+V(\varphi)\Big{)}-g^{\lambda\mu}g^{\sigma\nu}\partial_{\lambda}\varphi\partial_{\sigma}\varphi,$$

(6)

and we identify the density and pressure of the scalar field $\varphi$ as

$$\begin{split}&\rho_{\varphi}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+V(\varphi)\,\,\,\,\,{\rm and}\\ &p_{\varphi}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V(\varphi)\,.\end{split}$$

(7)

The scalar field four-velocity, $U^{\mu}$, is given as

$$U^{\mu}=\frac{-\partial_{\mu}\varphi}{\sqrt{-g^{\lambda\sigma}\partial_{\lambda}\varphi\partial_{\sigma}\varphi}}\,.$$

(8)

The isotropy and homogeneity of space-time require the stress-energy-momentum tensor for a free scalar field, $\varphi$, to be that of a perfect fluid,

$$\begin{split}&T^{ij}_{\varphi}=p_{\varphi}\delta^{i}_{j};\leavevmode\nobreak\ \leavevmode\nobreak\ T^{i0}_{\varphi}=T_{\varphi 0i}=0;\leavevmode\nobreak\ \leavevmode\nobreak\ T^{00}_{\varphi}=\rho_{\varphi},\\ &T^{\mu\nu}_{\varphi}=p_{\varphi}g^{\mu\nu}+(p_{\varphi}+\rho_{\varphi})U^{\mu}U^{\nu},\end{split}$$

(9)

subject to the constraint $g_{\mu\nu}U^{\mu}U^{\nu}=-1$. The conservation of $T^{\mu\nu}_{\varphi}$ in a Friedmann-Lemaitre-Robertson-Walker (FLRW) background leads to the continuity equation,

$$\partial_{0}T^{00}_{\varphi}=\dot{\rho}_{\varphi}+3\frac{\dot{a}}{a}(p_{\varphi}+\rho_{\varphi})=0\,,$$

(10)

which gives the equation of motion of the scalar field,

$$\ddot{\varphi}+3H\dot{\varphi}+\frac{dV(\varphi)}{d\varphi}=0\,.$$

(11)

Nevertheless, dark energy can be modeled by a k-essence field $\varphi$ which has very appealing properties and is described by the Lagrangian density $\mathcal{L}=K(\varphi,X)$ Armendariz-Picon et al. (2000) which is associated with the following energy-momentum tensor,

$\displaystyle T_{\mu\nu}^{\varphi}=K,_{X}\nabla_{\mu}\varphi\nabla_{\nu}\varphi-g_{\mu\nu}K\,,$

(12)

where $X=\nabla_{\alpha}\varphi\nabla^{\alpha}\varphi/2$ is the kinetic term and $K,_{X}=\partial K/\partial X$. The energy density and pressure are given by

$\displaystyle\rho_{\varphi}=2XK_{,X}-K\,\,\,\,\,{\rm and}\,\,\,\,\,$

$\displaystyle p_{\varphi}=K\,\,,$

(13)

respectively, and the equation of motion can be obtained from the continuity equation,

$$(2XK_{,XX}+K_{,X})\ddot{\varphi}+K_{,X\varphi}\dot{\varphi}^{2}+3HK_{,X}\dot{\varphi}=K_{,\varphi}\,,$$

(14)

where $K_{,X\varphi}=\partial^{2}K/\partial X\partial\varphi$ and $K_{,\varphi}=\partial K/\partial\varphi$.

Another possible interesting scalar field used to describe the dark energy evolution is the braided scalar field described by the Lagrangian:

$\displaystyle\mathcal{L}=\square\;\varphi G\left(X,\varphi\right),$

(15)

where $\square\;\varphi=\nabla^{\mu}\nabla_{\mu}\varphi$ and the function $G$ is arbitrary Pujolas et al. (2011). The energy-momentum tensor, $T_{\mu\nu}^{\varphi}$, in covariant form, is given by

$\displaystyle T_{\mu\nu}^{\varphi}=\frac{2}{\sqrt{-g}}\frac{\delta S_{\varphi}}{\delta g^{\mu\nu}}=\mathcal{L}_{X}\nabla_{\mu}\varphi\nabla_{\nu}\varphi-g_{\mu\nu}P_{\varphi}-\nabla_{\mu}G\nabla_{\nu}\varphi-\nabla_{\nu}G\nabla_{\mu}\varphi\,\,\,,$

(16)

where $P_{\varphi}=\nabla^{\lambda}\varphi\nabla_{\lambda}G_{\varphi}$, and $G_{\varphi}=\partial G/\partial\varphi$. This energy-momentum tensor can be described in terms of an imperfect fluid Pujolas et al. (2011) . We start defining some quantities to describe relativistic fluids. A local rest frame is set defining the normalized four-velocity $U_{\mu}$

	$\displaystyle U_{\mu}\equiv\frac{\nabla_{\mu}\varphi}{\sqrt{2X}},\quad U_{\mu}U^{\mu}=1,$		(17)
	$\displaystyle a_{\mu}\equiv U^{\nu}\nabla_{\nu}U_{\mu},$		(18)

with $a_{\mu}$ is the four-acceleration which is orthogonal to velocity, $U_{\mu}a^{\mu}=0$. The expansion, $\vartheta$, and the diffusivity, $\Omega$, are written as

$\displaystyle\vartheta=\nabla_{\mu}U^{\mu},\quad\Omega=2XG_{X},$

(19)

where $G_{X}=\partial G/\partial X$. The energy-momentum tensor also can be expressed in this way:

$\displaystyle T_{\mu\nu}^{\varphi}=\rho_{\varphi}U_{\mu}U_{\nu}-\perp_{\mu\nu}p_{\varphi}+U_{\mu}q_{\nu}+U_{\nu}q_{\mu},$

(20)

where $m=\sqrt{2X}=\dot{\varphi}$ is the chemical potential, $q_{\mu}=-m\Omega a_{\mu}$ is the heat flux (purely spatial, $U_{\mu}q^{\mu}=0$) and $\perp_{\mu\nu}=g_{\mu\nu}-U_{\mu}U_{\nu}$ is the transverse projector. The energy density and isotropic pressure are:

	$\displaystyle\rho_{\varphi}$	$\displaystyle\equiv$	$\displaystyle T_{\mu\nu}^{\varphi}U^{\mu}U^{\nu}=-2XG_{\varphi}+\vartheta m\Omega,$		(21)
	$\displaystyle p_{\varphi}$	$\displaystyle\equiv$	$\displaystyle-\frac{1}{3}T^{\mu\nu}_{\varphi}\perp_{\mu\nu}=-2XG_{\varphi}-\Omega\dot{m}.$		(22)

The conservation of $T_{\mu\nu}^{\varphi}$ using these definitions can be expressed as:

$\displaystyle U_{\nu}\nabla_{\mu}T^{\mu\nu}_{\varphi}=\dot{\rho_{\varphi}}+\vartheta\left(\rho_{\varphi}+p_{\varphi}\right)-\nabla_{\lambda}\left(m\Omega a^{\lambda}\right)+m\Omega a_{\lambda}a^{\lambda}=0.$

(23)

The equation of motion for $\varphi$ can be obtained from the last equation

$\displaystyle\nabla_{\mu}\left[2G_{\varphi}\nabla^{\mu}\varphi-\square\varphi G_{X}\nabla^{\mu}\varphi+G_{X}\nabla^{\mu}X\right]=\nabla^{\lambda}\varphi\nabla_{\lambda}G_{\varphi}.$

(24)

In the cosmological background, the evolution of the scalar field reduces to the dynamics of a perfect fluid which is described only through its energy density and pressure.

Consider the effective Lagrangian (see Appendix A for further details)

$$-\mathcal{L}_{\text{eff}}=-i\frac{\lambda_{\alpha\beta}}{M_{*}^{4}}T^{\mu\nu}_{\varphi}\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\partial_{\nu}\nu_{\beta},$$

(25)

being $\lambda_{\alpha\beta}$ a coupling constant matrix, $M_{*}$ the energy scale of the interaction, and $T^{\mu\nu}_{\varphi}$ is the tensor associated to the scalar field $\varphi$, for instance those of Eqs. (6), (12), and (16). Hence, we can identify the $CPT-$even LIV coefficients of the SME $(c_{L})^{\mu\nu}_{\alpha\beta}=c^{\mu\nu}_{\alpha\beta}$ as

$$c^{\mu\nu}_{\alpha\beta}\rightarrow\frac{\lambda_{\alpha\beta}}{M_{*}^{4}}T^{\mu\nu}_{\varphi}.$$

(26)

Here, the scalar field $\varphi$ could be one of the DM or DE candidates described in Section II. Besides, the effective interaction (Eq. 25) may induce potential scattering among the neutrinos and the scalar particles. However, such interactions would be expected to be negligible (see Appendix B of Ref. Klop and Ando (2018)). Henceforth, we discuss the case of a tensorial neutrino-scalar field interaction. ⁴⁴4A study of the the back-reaction effects from the tensorial neutrino-scalar field interaction is beyond the scope of this work.

Long-baseline neutrino oscillation experiments play a significant role in both deciphering mysteries within the traditional three-neutrino oscillation picture and exploring other novel physics scenarios, including the potential breaking of the Lorentz and $CPT$ symmetries. Hence, as a case of study, we will focus on a long-baseline experimental configuration, the Deep Underground Neutrino Experiment (DUNE), which is a next-generation accelerator-based neutrino oscillation experiment that will consist of up to 40 kt of liquid argon (far) detector located at the Sanford underground research facility (SURF) in South Dakota Abi et al. (2020b). Moreover, this configuration is expected to deliver a neutrino flux with a mean neutrino energy $E_{\nu}\sim$ 3 GeV, which will be located at a distance of $L\sim$ 1300 km from the beam source (on-axis) at Fermilab (we refer the reader to Refs. Agarwalla et al. (2023); Abi et al. (2020a), for a detailed discussion regarding the experimental configuration).

In order to obtain sensitivities to the LIV coefficients at DUNE, we use the GLoBES software Huber et al. (2005, 2007) and its additional NSI tool snu.c Kopp (2008); Kopp et al. (2007) which was modified to implement the $CPT-$even coefficients of the SME at the Hamiltonian level. Moreover, to simulate the DUNE configuration, we employ the available GLoBES ancillary files Abi et al. (2021) and specifications from the Technical Design Report (TDR) configuration Abi et al. (2020a). Furthermore, in this work, we have contemplated a $10-$year running time, evenly distributed among neutrino and anti-neutrino modes. To simulate the DUNE event spectra, we consider the reconstructed neutrino and anti-neutrino energy range from 0 to 18 GeV for both appearance and disappearance channels. While elaborating our sensitivity plots, we performed a full spectral analysis with a total of 70 bins in the aforementioned energy range (having non-uniform bin widths). We have 64 bins each having a width of 0.125 GeV in the energy range of 0 to 8 GeV, and 6 bins with variable widths beyond 8 GeV Abi et al. (2021).

In this analysis, we have considered the following Hamiltonian

$$H=H_{0}+H_{\text{MSW}}+H_{\text{LIV}},$$

(38)

here, $H_{0}$ and $H_{\text{MSW}}$ are the standard neutrino Hamiltonian in vacuum and matter, respectively. Besides, $H_{\text{LIV}}$ is the contribution from the Lorentz invariance violation (LIV) sector, which can be parameterized as Kostelecky and Mewes (2004b); Diaz et al. (2009); Mishra et al. (2023)

$$H_{\text{LIV}}=-\frac{E_{\nu}}{2}\big{[}(3-\hat{N}_{Z}^{2})(c_{L})_{\alpha\beta}^{TT}+(3\hat{N}_{Z}^{2}-1)(c_{L})_{\alpha\beta}^{ZZ}-2\hat{N}_{Z}(c_{L})_{\alpha\beta}^{TZ}\big{]},$$

(39)

where $E_{\nu}$ is the neutrino energy, $(c_{L})_{\alpha\beta}^{\mu\nu}$ are the $CPT-$even LIV coefficients of the SME (being $\alpha,\beta=e,\mu,\tau$, and $\mu,\nu=T,X,Y,Z$), for non-diagonal $\alpha\neq\beta$, $c_{\alpha\beta}^{\mu\nu}=|c_{\alpha\beta}^{\mu\nu}|e^{i\phi_{\alpha\beta}^{\mu\nu}}$ (from now on, we will denote $(c_{L})^{\mu\nu}_{\alpha\beta}=c_{\alpha\beta}^{\mu\nu}$, and $(c_{L})^{TT}_{\alpha\beta}=c_{\alpha\beta}$ ), and

$$\hat{N}^{Z}=-\sin\chi\sin\theta\cos\phi+\cos\chi\cos\theta,$$

(40)

is the spatial $Z-$component factor, expressed in terms of local spherical coordinates at the detector, that represents the direction of neutrino propagation in the Sun-centered frame. Being $\chi$ the colatitude of the detector, $\theta$ the angle at the detector between the beam direction and vertical, and $\phi$ the angle between the beam and east of south Kostelecky and Mewes (2004b). In the particular case of the DUNE location, $\hat{N}^{Z}\simeq 0.16$ Delgadillo:2024bae .