Neutrino Lorentz Invariance Violation from Cosmic Fields

From a cosmological perspective, scalar fields are well-motivated dark matter and dark energy candidates. Several possibilities of neutrino couplings with a time-varying cosmic field have been investigated in the literature. In this work, we present a framework in which violations of Lorentz invariance (LIV) and $CPT$ symmetry in the neutrino sector could arise from an interaction among neutrinos with a time-varying scalar field. Furthermore, some cosmological and phenomenological aspects and constraints concerning this type of interaction are discussed. Potential violations of Lorentz and $CPT$ symmetries at present and future neutrino oscillation experiments such as IceCube and KM3NeT can probe this scenario.


I. INTRODUCTION
Possible Lorentz invariance violations (LIV) and violations of the CP T symmetry in the neutrino sector could arise from the interactions among scalar fields with neutrinos via exotic currents [1][2][3][4].For comprehensive treatments of Lorentz and CP T violations in neutrinos, we refer to Refs.[5][6][7].In a model-independent framework, LIV effects are parameterized by the effective Lagrangian [8] in the case of neutrinos, it is convenient to define where the contribution from the LIV sector is being E the neutrino energy, the LIV coefficients (a L ) µ αβ also violate CP T in the neutrino sector.From a theoretical point of view, violations of the CP T symmetry manifest themselves within the framework of string theory [9][10][11].Regarding neutrino oscillation experiments, CP T violations in the neutrino sector were proposed as a solution to the Liquid Scintillator Neutrino Detector (LSND) anomaly [12][13][14].Furthermore, it was shown in Ref. [15] that besides neutrino oscillation experiments, CP T violation can have consequences in neutrinoless double-beta decay experiments.
Recently, it has been considered the case where violations of Lorentz invariance and CP T at neutrino oscillation experiments, 1 arise from a neutrino current coupled with a time-dependent cosmic field ϕ(t) (see e.g., Refs.[2][3][4]) being λ αβ some coupling constant, Λ the energy scale of the interaction, and ϕ a scalar field, which could be identified as a dark matter (DM) or dark energy (DE) candidate.Here, violations of Lorentz invariance emerge via CP T -odd LIV coefficients a µ αβ → λ αβ ∂ µ ϕΛ −1 .In the scenario explored by the authors of Refs.[1][2][3], the scalar field ϕ is considered to be a DE candidate (namely quintessence) being H 0 ∼ 10 −33 eV the Hubble parameter at present, M Pl the Planck mass, and the equation of state parameter w ≃ −0.95 [17].Hence, φ0 Furthermore, the isotropic CP T -odd LIV coefficients are related to the scalar field ϕ as Moreover, in this scenario, experiments such as IceCube and the KM3NeT proposal could search for directional-dependent LIV effects at neutrino energies E ν ≳ 10 5 GeV [3,20], the sensitivity reach to the mediator mass scale Λ would be Λ ≲ 10 5 GeV or Λ ≲ 10 8 GeV, depending on the neutrino event exposure (see, e.g., Fig. 5a and Fig. 5b of Ref. [3] for a detailed explanation).

II. THEORETICAL FRAMEWORK
Let us consider the following phenomenological Lagrangian 3 which can emerge, for instance, from the interaction among a real scalar field ϕ and a new vector boson Z ′ µ in a broken approximate global symmetry 4 and a neutrino current where g αβ is the 3 × 3 coupling matrix.Hence, by integrating out the neutrinophilic Z ′ µ boson from Eqs. (11) and (12) we can obtain Eq. (10).There exist some models in which the neutrinos are the only standard model fermions that couple to the Z ′ µ boson at tree level (see, e.g.[34][35][36] and references therein).
Besides, a related scenario of neutrino-ULDM interaction was proposed by [37], the model preserves the flavor composition of high-energy astrophysical neutrinos at production; it includes a complex ultralight scalar field as a DM candidate and a light Z ′ µ boson.Recently, the authors of Ref. [38] investigated the interaction of an O(eV) sterile neutrino coupled with an ultralight scalar field m ϕ ∼ 10 −15 eV as DM.

III. COSMOLOGY
The cosmological evolution of the interacting scalar field with neutrinos can be described through energy exchange between the two fluids [39,40] ρν where the explicit form of the energy exchange term Q depends on the model.In some models [28], the explicit form of this term is where g is a coupling constant.However, Q ∝ φ [39,40] is a possible scenario as well.
The evolution of the scalar field can be determined from its equation of motion where H = ȧ/a is the Hubble parameter and a is the scale factor of the Universe.The evolution of the Hubble parameter can be obtained from the Friedmann equation Here, ρ r , ρ b , ρ DM , and ρ DE are the energy densities associated with radiation, nonrelativistic baryonic matter, dark matter, and dark energy, respectively.In some cases, we can identify ρ ϕ as ρ DM and in another case as ρ DE (see below).

Scalar field as dark energy
Besides, the equation of motion for the scalar field is modified by the presence of the neutrino−scalar field energy exchange.If we consider the scenario where Q ∝ φ [28], we The evolution of the field may tend to be constant near the region when the derivative of the scalar field is null. 5This could imply avoiding the oscillations of the field and preventing it from being considered DM.It would be interesting to study a possible mechanism to start an oscillatory behavior by perturbing the field or considering quantum fluctuations.
The scenario when the scalar field could be frozen (when at some point and behave as DE was studied in [28].Furthermore, the phenomenology of the scalar field ϕ as DE, LIV and CP T violations in neutrino oscillations was studied in Refs.[1][2][3].
As an exemplification, within the M -theory axiverse [44,45] an ultralight (m ∼ 10 −15 eV) axion-like particle arises, with decay constant F ∼ 10 16 GeV. 7 5 See Appendix B for a discussion. 6If the scalar field mass is lighter than about m ϕ ≲ 10 −20 eV, the scalar field density ρ ϕ is not allowed to be the total component of cosmological DM ρ DM , see e.g.Ref. [41]. 7Searches of ultralight scalars as dark matter candidates include atomic clocks [46], resonant-mass detectors [47] and atomic gravitational wave detectors [48].
During the early epochs, an effective back-reaction potential V ( φ) develops due to the neutrino−scalar field interaction in Eq. (10), where V ( φ) ≲ V (ϕ) ∼ m 2 ϕ ϕ 2 in such way that the field ϕ ≃ ϕ(t) behaves as DM.For instance, prior to matter-radiation equality (T ∼ eV), the Universe was highly isotropic and neutrinos were relativistic.Thus, the corresponding average of the neutrino current was ⟨ν α γ µ (1 − γ 5 )ν β ⟩ ∼ ⟨ν α γ 0 (1 − γ 5 )ν β ⟩ ∼ n ν , with neutrino number density proportional to the temperature n ν ∼ 0.1 T 3 , therefore the effective backreaction potential was For the case of a non-interacting scalar field ϕ with mass m ϕ ∼ 10 −15 eV, it will transition to its oscillation phase around a temperature T osc,ϕ ∼ M Pl m ϕ ∼ MeV8 (z osc ∼ 6 × 10 9 ), the corresponding scalar field amplitude around that time was roughly |ϕ| ∼ 10 24 eV, with potential therefore, the contribution from the effective back-reaction potential was hence at that epoch, the back-reaction potential was a sub-leading contribution to the scalar In contrast, if we consider an energy exchange term, (with neutrino mass m ν ∼ 0.1 eV), we obtain the following equation of motion Thus, the energy exchange term (Q ∝ φ) has to satisfy in order for ϕ to start its rapid oscillations and act as DM.For instance, around matterradiation equality (T ∼ eV), an order of magnitude estimate gives In addition, once neutrinos enter the nonrelativistic regime (T ≲ m ν ), the spatial terms from the neutrino−scalar field interaction in Eq. ( 10) become relevant, modifying the form of Eqs. ( 18) and (23).
On the other hand, in terms of the background quantities, one could have either Q = χκρ ν φ or Q = χHρ ν (with χ a dimensionless constant, see e.g.Ref. [39]) which leads to in the case of energy exchange Q ∝ ρ ν M −1 Pl , the neutrino-scalar field interaction is suppressed by the Planck mass.However, the last term with energy exchange Q = χHρ ν might be problematic if we consider ϕ as DM candidate.
Hence, if the energy exchange term, Q ∝ φ, the scalar field could freeze at some point and behave as DE [28], while if the energy exchange term, Q ∝ φ, the scalar field could avoid freezing and act as DM.Besides, in the case where the energy exchange among the neutrinos and the scalar field is Q ∝ φ, the beginning of the rapid scalar field oscillations can be delayed (see Eqs. 24 and 25).Therefore, as long as the fast oscillations occurred after BBN but before the CMB formation, ϕ can account for all DM [41,49].
Henceforth, we will discuss some consequences for neutrino oscillation experiments in the case where the scalar field ϕ can be considered a DM candidate.

IV. IMPLICATIONS IN NEUTRINO OSCILLATION EXPERIMENTS
The phenomenological Lagrangian in Eq. (10)  where the isotropic LIV coefficients a αβ (t) are with a αβ ≲ 10 −25 GeV at our benchmark values.In a similar fashion as the scenario explored in Ref. [3], directional-dependent effects (a X αβ ) will be sub-leading due to the small anisotropic components of the scalar field amplitude, which are proportional to the virialized ULDM these coefficients have been studied in the context of active-sterile neutrino oscillations and supernovae neutrino emission [50].
Imprints of the aforementioned Lorentz/CP T effects can be traced in neutrino oscillation experiments.For instance, the effective neutrino Hamiltonian H(t) in the presence of the where H 0 is the neutrino Hamiltonian in vacuum, V MSW is the MSW potential, and the last terms correspond to the CP T -odd LIV Hamiltonian a µ αβ (t), for high energy atmospheric neutrinos (E ν ∼ TeV), due to the presence of the V MSW and a αβ (t), a X αβ (t) potentials, propagation of neutrinos can be implemented by dividing the trajectory into N -layers of thickness ∆L = L/N .The oscillation probability in the flavor base, with t n = t 0 + n∆L, given by [51] Hence, the observed oscillation probability will be the time average of P αβ (t 0 , L) [49] ⟨P where τ ϕ = 2π/m ϕ is the period of oscillation of ϕ(t).
The corresponding CP T -odd LIV isotropic coefficients (a αβ ≲ 10 −25 GeV) 9 can be probed in the near future, for instance at the IceCube and KM3NeT experiments with neutrino 9 Directional-dependent effects a X αβ ≲ 10 −28 GeV (which are four orders of magnitude stronger than current limits [52]), can be searched at IceCube and KM3NeT with 0.1−100 EeV ultra-high-energy (UHE) neutrinos [3].

A. Other phenomenological implications
In addition to the previous considerations, the interaction term in Eq. ( 12) induces a non-standard four-neutrino interaction where the effective coupling [57].Moreover, neutrino neutral current (NC) non-standard interactions (NSI) with quarks and leptons can be generated via Z − Z ′ mixing, however, suppressed by the mixing angle θ ZZ ′ ≲ 0.1g αβ [35] here f is the corresponding quark or lepton, 2P C = 1 ± γ 5 , C = R, L is the chiral projector, and For some experimental limits on the nonstandard parameters ϵ αβ , see Tab.I, bounds from charged current (CC) NSI processes do not apply in this scenario.
On the other hand, the phenomenological Lagrangian in Eq. ( 10) induces CP T violation in electrons at one loop [1] via the weak interaction (W µ − neutrino loop) here α is the fine-structure constant and θ W is the weak mixing angle.Experimental bounds set ã0 e < 5×10 −25 GeV [1] and ãX e < 10 −25 GeV [52], for the isotropic and spatial coefficients, respectively.Therefore, at our reference values, we obtain which are within the experimental constraints, as long as M Z ′ ≲ TeV.Furthermore, at earlier times, the corresponding interaction in Eq. ( 35) could induce an electron-scalar field back-reaction potential V (n e , φ) ∝ n e φ, which is expected to be negligible after electron decoupling.
Regarding the scalar field ϕ, radiative corrections could destabilize its mass.For instance, contributions to the scalar field mass m ϕ ∼ 10 −15 eV can be generated from the effective LIV neutrino interaction in Eq. ( 10) Besides, the corresponding Lagrangian in Eq. ( 10) may produce scattering between the neutrinos and the scalar particles.However, these interactions are expected to be negligible [3].
Moreover, the neutrino current in Eq. ( 12) induces radiative corrections to the neutrino mass V. CONCLUSIONS In the cosmological context, scalar fields have been used to describe the behavior of dark matter and dark energy.The interaction of these fields with matter could be an important way to determine its existence by searching for its possible observational and experimental consequences.
In this paper, we have studied the interactions between the active neutrinos with a timevarying scalar field, which might lead to possible signals of Lorentz invariance and CP T violation in future and present neutrino oscillation experiments such as IceCube and the KM3NeT proposal.Furthermore, we discussed how those interactions induce a back-reaction potential as well as energy exchange among the neutrinos and the scalar field, modifying its cosmological evolution.The scalar field can act as either cosmological dark energy or dark matter, depending on the form of the neutrino-scalar field energy exchange term.
In addition, we outlined some phenomenological consequences and constraints in the scenario where the scalar field ϕ can be considered a dark matter candidate.
No. 20230732 and No. 20241624.We acknowledge the anonymous referee for the illuminating comments and suggestions.
Appendix A: Boson current in a non-linear realization The interaction term from Eq. ( 11), could arise from a non-linear CCWZ realization [30][31][32]60], in analogy with that of pions and kaons.For instance, considering a non-linear Nambu-Goldstone field being ϕ a real scalar field and f ϕ some energy scale, with an effective Lagrangian where the corresponding derivative f ϕ D µ Φ is given by [60] the effective interacting Lagrangian, including the scalar field ϕ and vector boson Z ′ µ is with periodicity for the scalar field ϕ ≃ ϕ + nπf ϕ .The minimum of the scalar field potential For ϕ ≪ f ϕ , we can expand the scalar field potential as and the dominant term is the mass term m ϕ = µ 2 ϕ /f ϕ .Hence, for a scalar field with mass m ϕ ∼ 10 −15 eV and f ϕ ∼ 10 25 eV, the quartic coupling is λ ϕ ∼ m 2 ϕ f −2 ϕ ∼ 10 −80 .Therefore, a free scalar field was roughly ϕ ∼ f ϕ at early epochs until H ∼ m ϕ .

Appendix B: Scalar field evolution
The equation of motion for the scalar field, including the neutrino−scalar field energy Besides, in the radiation epoch, the Hubble parameter H is related to the temperature T as which can be written in terms of the temperature with the help of the previous relations (in Eq.B6, instead of the neutrino number density n ν ∼ 0.1 T 3 , a term such as ρ ν m −1 ν , could be of phenomenological interest as well) in the following form (a L ) µ αβ coefficients are CP T -odd, while the (c L ) µν αβ coefficients are CP T -even.Besides, if we are interested to study the isotropic LIV contributions at neutrino oscillation experiments, namely the CP T -even (c 00 αβ = c αβ ) and CP T -odd (a 0 αβ = a αβ ) coefficients, i.e. (µ = ν = 0), we can consider the following Hamiltonian

d 3 ϕ dT 3
= 0 when dϕ dT = 0.In general d n ϕ dT n = 0 once the scalar field has dϕ dT = 0.The last relations may imply a frozen scalar field, ϕ = constant, which is a possible solution to the differential equation as well.It would be interesting to study a possible mechanism to escape from this state employing quantum fluctuations or perturbations of the scalar field.
The former expression is suitable to extract information about the dynamics of the scalar field.When dϕ dT = 0, then d 2 ϕ dT 2 = 0.Moreover, if we derive the former equation, we get * T6