Light scalar dark matter at neutrino oscillation experiments

Couplings between light scalar dark matter (DM) and neutrinos induce a perturbation to the neutrino mass matrix. If the DM oscillation period is smaller than ten minutes (or equivalently, the DM particle is heavier than 0.69×10−17 eV), the fast-averaging over an oscillation cycle leads to a modification of the measured oscillation parameters. We present a specific μ − τ symmetric model in which the measured value of θ13 is entirely generated by the DM interaction, and which reproduces the other measured oscillation parameters. For a scalar DM particle lighter than 10−15 eV, adiabatic solar neutrino propagation is maintained. A suppression of the sensitivity to CP violation at long baseline neutrino experiments is predicted in this model. We find that DUNE cannot exclude the DM scenario at more than 3σ C.L. for bimaximal, tribimaximal and hexagonal mixing, while JUNO can rule it out at more than 6σ C.L. by precisely measuring both θ12 and θ13.


Introduction
The existence of DM has been well established through various cosmological and astrophysical observations. However, after decades of experimental searches for DM, the particle nature of DM is still unknown, and viable DM particle candidates span an enormous mass range from fuzzy DM [1] to primordial black holes [2]. Among the DM candidates, fuzzy DM with a mass range 1-10 × 10 −22 eV has attracted much attention recently since it can resolve the small scale crisis for standard cold DM due to its large de Broglie wavelength; see ref. [3] and references therein. Constraints on fuzzy DM can be obtained from Lyman-α forest data and a lower limit of 20 ×10 −22 eV at 2σ C.L. has been set from a combination of XQ-100 and HIRES/MIKE data [4], although a proper handling of the effect of quantum pressure and systematic uncertainties may relax the limit [5]. Nevertheless, light scalar DM candidate of mass below a few keV are generally expected in many extensions of the Standard Model (SM). Examples include a QCD axion [6][7][8], moduli [9][10][11][12], dilatons [13,14], and Higgs portal DM [15]. Constraints on hot DM require that light scalar DM cannot be produced thermally in the early universe. A popular production mechanism for generating light scalar DM is the misalignment mechanism, in which the fields take on some initial nonzero value in the early universe, and as the Hubble expansion rate becomes comparable to the light scalar mass, the DM field starts to oscillate as a coherent state with a single macroscopic wavefunction [16].
The properties of light scalar DM can be probed if they are coupled to SM fermions, which induce a time variation to the masses of the SM fermions due to the oscillation of JHEP04(2018)136 the DM field. Here we consider the couplings between the light scalar DM and the SM neutrinos, which were first studied in ref. [17] by using the nonobservation of periodicities in solar neutrino data. Constraints on light scalar DM couplings were also considered in refs. [18,19] by using the data from various atmospheric, reactor and accelerator neutrino experiments. In general, the interactions between DM and neutrinos provide a small perturbation to the neutrino mass matrix; a generic treatment of small perturbations on the neutrino mass matrix is provided in refs. [20,21]. If the DM oscillation period is much smaller than the periodicity to which an experiment is sensitive, the oscillation probabilities get averaged, and a modification of the oscillation parameters can be induced if the data are interpreted in the standard three-neutrino framework.
Evidence of time varying signals has been searched for in many neutrino oscillation experiments. Super-Kamiokande finds no evidence for a seasonal variation in the atmospheric neutrino flux [22], and the annual modulation of the atmospheric neutrino flux observed at IceCube is correlated with the upper atmospheric temperature [23]. Monthlybinned data from KamLAND indicate time variations in reactor powers [24]. Tests of Lorentz symmetry via searches for sidereal variation in LSND [25], MINOS [26][27][28], Ice-Cube [29], MiniBooNE [30], Double Chooz [31], and T2K [32] data, are negative. Also, Super-Kamiokande [34] and SNO [35,36] find no significant temporal variation in the solar neutrino flux with periods ranging from ten minutes to ten years. We therefore take the DM oscillation period to be smaller than ten minutes.
In this work, we study the modification of neutrino oscillation parameters due to light scalar DM-neutrino interactions. Since the predictions are flavor structure-dependent, we present a specific µ − τ symmetric model in which the symmetry is broken by light scalar DM interactions thus generating a nonzero mixing angle θ 13 . We first examine the effects of this model on data from various neutrino experiments. We then study the potential to distinguish this model from the standard three-neutrino oscillation scenario at the future long-baseline accelerator experiment, DUNE, and the medium-baseline reactor experiment, JUNO.
The paper is organized as follows. In section 2, we present a model in which µ − τ symmetry is broken by the DM interactions. In section 3, we examine the implications of this model for the measured neutrino oscillation parameters. In section 4, we simulate future neutrino oscillation experiments to study the potential to distinguish this model from the standard scenario. We summarize our results in section 5. In appendix A we calculate how the scalar DM interactions with neutrinos affect neutrino mass and mixing parameters, and in appendix B we determine how the DM oscillations cause a shift in the effective neutrino oscillation parameters measured in experiments.

The model
The Lagrangian describing the interactions between light scalar DM and neutrinos can be written in the flavor basis as [18,19]

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where α, β = e, µ, τ , m 0 is the initial neutrino mass matrix, and λ is the coupling constant matrix. Since the light scalar DM can be treated as a classical field, the nonrelativistic solution to the classical equation of motion can be approximated as [17] φ(x) where m φ is the mass of the scalar DM particle, ρ φ ∼ 0.3 GeV/cm 3 is the local DM density, and v ∼ 10 −3 is the virialized DM velocity. Since v 1, we neglect the spatial variation in φ for neutrino oscillation experiments. In the presence of scalar DM interactions, the effective Hamiltonian for neutrino oscillations can be written as where N e is the number density of electrons. The effective mass matrix can be treated as the sum of an initial mass matrix and a small perturbation [21], i.e., where U 0 is the initial mixing matrix, m 0 i 's are the initial neutrino eigenmasses, and the elements of the perturbation matrix are (2.5) Note that the bounds in figure 1 of ref. [17] only apply to a specific combination of λ αβ and m φ . Planck measurements yield m ν < 0.23 eV at the 95% C.L. [33], which is much larger than the size of perturbation considered here. We consider a model in which the initial mixing angle θ 0 13 = 0 and the measured θ 13 value is generated by the DM interactions. In order to simplify our calculations, we specialize to models in which the DM interactions only affect the masses at higher orders in the perturbation, leaving them effectively unchanged. From the generalized perturbation results of ref. [21], we find that the most general perturbation satisfying the latter requirement is of the form,  As a further simplification, we assume the model is µ − τ symmetric, i.e., θ 0 23 = π/4. Then the perturbation becomes (2.7)

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With this perturbation, the shifts in all three angles are first order in the small quantities , , and δm 21 , where δm ij ≡ m i − m j ; since the eigenmasses are not shifted at leading order, we drop the superscript '0' hereafter. In appendix A, we show that the leading order corrections have amplitudes Note that δθ 12 is second-order in the 's and is therefore proportional to cos 2 (m φ t), while δ CP depends only on the phase of and is constant, i.e., it is not affected by the DM oscillation. Both δθ 13 and δθ 23 are dependent linearly on cos(m φ t).

Effects on neutrino oscillation parameters
In this section, we study how the neutrino oscillation parameters are modified in our model, assuming the period of the DM oscillation (τ φ = 2π/m φ ) is short compared to the experimental resolution of periodicity. Here we use the superscript '0' to denote the initial oscillation parameters, and the superscript 'eff' to denote the effective parameters measured at neutrino oscillation experiments if the data are interpreted in the standard three-neutrino framework. For the parameters obtained after incorporating the DM perturbation, no superscript is used.

Long-baseline appearance experiments
For long-baseline experiments, the formulas are more complicated. From ref. [37],

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where x = sin θ 23 sin 2θ 13 , y = α cos θ 23 sin 2θ 12 Assuming θ 0 13 = 0, and before averaging, where C = cos(m φ t). After averaging, the leading term for x 2 f 2 is which is similar to the reactor case, i.e., the effective θ 13 is δθ 13 / √ 2. We can write yg as where y 0 = α cos θ 0 23 sin 2θ 0 12 . After explicitly putting in the perturbation, yg becomes Combining eqs. (3.5) and (3.7) and after averaging, the xyf g term is where the term in parentheses replaces x in the standard expression. Note that this term is suppressed compared to the usual case since it is proportional to two factors of the 's (assuming ∼ ), instead of just one -the term proportional to one factor of was linear in C and averaged to zero. For µ−τ symmetry, θ 0 23 = π/4 and the term vanishes completely. The upshot is that the effect of the Dirac CP phase on P (ν µ → ν e ) and P (ν µ →ν e ) is suppressed in long-baseline neutrino oscillation appearance experiments. Also, as shown in appendix B, this model predicts a suppression of the sensitivity to CP violation in all types of neutrino oscillation experiments.

Solar neutrinos
In the SM scenario, solar neutrinos created in the center of the Sun undergo adiabatic evolution to the surface of the Sun, and travel to the Earth as an incoherent sum of the mass eigenstates. To not spoil the adiabatic evolution inside the Sun, we require that the period of the DM oscillation τ φ be much larger than the time in which neutrinos travel through the Sun, which is about 2.3 seconds. This requirement restricts the mass of the scalar field: m φ 1.8 × 10 −15 eV. The three-neutrino survival probability for adiabatic propagation is  2 21 , and N 0 e is the electron number density at the point in the Sun where the neutrino was created. Here θ ij = θ 0 ij + δθ ij cos(m φ t) andθ ij = θ 0 ij + δθ ij cos(m φ t + m φ t 0 ) are the mixing angles at the production point in the Sun and at the Earth, respectively. They differ by a phase factor m φ t 0 , where t 0 is the time traveled by neutrinos from the production point to the Earth.

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and the corresponding barred quantities can be obtained by replacing the phase m φ t with m φ t + m φ t 0 . Also, to the leading order, we have cos 2θ m ≈ cos 2θ 0 m + F δθ 12 cos 2 (m φ t) , (3.20) where cos 2θ 0 m has the same form as eq. (3.18), and If P 0 is the probability without the perturbation, i.e., then keeping the leading correction for each δθ, we have Because there is no interference term between cos 2 (m φ t) and cos 2 (m φ t + m φ t 0 ), we can average over them separately. Hence, By a similar calculation, the effective shifts in θ eff 12 and θ eff 13 lead to P ≈ P 0 + 1 2 δθ eff 12 (F cos 2θ 0 12 − 2 sin 2θ 0 12 cos 2θ 0 m ) − 2(δθ eff 13 ) 2 P 0 , and we see that δθ eff 12 = δθ 12 /2 and δθ eff 13 = δθ 13 / √ 2, the same as for medium-baseline reactor experiments.

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for the normal hierarchy, and for the inverted hierarchy. Since the correction to θ 23 is doubly suppressed in the oscillation probabilities in this model, θ eff 23 remains maximal. We first study the sensitivity of long-baseline accelerator experiments to this model. Since the currently running experiments, T2K and NOνA, have large experimental uncertainties, we consider the next-generation DUNE program. In our simulation, we use the GLoBES software [38,39] with the same experimental configurations as in ref. [40]. For the oscillation probabilities in the DM scenario, we modify the probability engine in the GLoBES software by averaging the probabilities over a DM oscillation cycle numerically. We also use the Preliminary Reference Earth Model density profile [41] with a 5% uncertainty for the matter density.
To obtain the sensitivities to the DM parameters at future long-baseline neutrino experiments, we simulate the data with the SM scenario in the normal hierarchy. Since the sensitivity to the Dirac CP phase is suppressed at such experiments, we conservatively choose δ CP = 0. Also, due to the double suppression of the correction to θ 23 , we choose θ 23 = π 4 , which is within the 1σ range of the global fit [42]. We also adopt the other mixing angles and mass-squared differences from the best-fit values in the global fit, which are  underlying discrete symmetries. Namely, θ 0 12 = 45 • for bimaximal (BM) mixing [43][44][45], θ 0 12 = 35.3 • for tri-bimaximal (TBM) mixing [46][47][48], and θ 0 12 = 30 • for hexagonal (HG) mixing [49,50]. Since the masses are not affected at the leading order, we adopt the central values and uncertainties for the mass-squared differences from the global fit, i.e., Also, since the long baseline experiments are not sensitive to θ 12 , we impose a prior on θ eff 12 to account for constraints from the current global fit, i.e., sin 2 θ eff 12 = 0.307 ± 0.013. We use eq. (4.2) to calculate the predicted value of θ eff 12 . Then for a given θ 0 12 and the lightest mass m 1 (m 3 ) for the normal (inverted) hierarchy, we scan over the magnitudes and phases of and . We find that the phases of and only have a small effect on the χ 2 value, which agrees with the analytical expectation that the measurement of the CP violation is suppressed. We also marginalize over both the normal and inverted hierarchy for the tested DM scenario. We find that the χ 2 value for the inverted mass hierarchy is always larger than that for the normal hierarchy for the same lightest mass. This is because the masses are not affected at the leading order and the mass hierarchy can be resolved with high confidence at DUNE [51].
The minimum value of χ 2 as a function of m 1 is shown in figure 1 for the three benchmark values of θ 0 12 . As an illustrative example, we show the oscillation probabilities for θ 0 12 = 35.3 • and m 1 = 0.1 eV in the neutrino and antineutrino appearance channels in figure 2. We see that the DM oscillation curves overlap the SM curves sufficiently in both modes that a clear discrimination is not possible. From figure 1 we see that DUNE alone cannot distinguish the DM scenario from the SM scenario at more than the 3σ C.L. if m 1 JHEP04(2018)136 is greater than about 0.05 eV. We also see that as m 1 decreases, χ 2 min increases. This can be understood from eqs. (4.1) and (4.3). For a smaller m 1 , the magnitude of required to explain the measured θ 13 becomes larger, and higher order corrections then break the degeneracies between the SM and DM scenarios.
Since future medium-baseline reactor experiments can make a high precision measurement of both θ 12 and θ 13 , we study the sensitivity reach at JUNO. We use the GLoBES software to simulate the JUNO experiment. The experimental configuration is the same as that in ref. [52], which reproduces the results of ref. [53]. We use the same procedure for the long-baseline accelerator experiments except with no prior on θ eff 12 , since JUNO can measure θ 12 more precisely than the current experiments. For the lightest mass between 0 and 0.2 eV, we find that the minimum value of χ 2 at JUNO is 47.6, 46.9 and 57.0, with the initial mixing being BM, TBM and HG, respectively. Hence, JUNO can rule out this model with the three initial mixings at more than 6σ C.L.

Summary
We studied the effects of light scalar DM-neutrino interactions at various neutrino oscillation experiments. For a light scalar DM field oscillating as a coherent state, the coupling between DM and neutrinos induces a small perturbation to the neutrino mass matrix. We consider the case in which the DM oscillation period is smaller than the experimental resolution of periodicity, i.e., ten minutes. After averaging the oscillation probabilities over a DM oscillation cycle, the perturbation to the neutrino mass matrix leads to a modification of the effective neutrino oscillation parameters if the experimental data are interpreted in the standard three-neutrino oscillation framework.
Since the results depend on the flavor structure of the initial mass matrix and the perturbation matrix, we presented a specific µ − τ symmetric model with DM interactions that do not affect the eigenmasses at the leading order. We examined the effects of this model on the effective oscillation parameters measured at various neutrino experiments. If the mass of the scalar field is lighter than 1.8 × 10 −15 eV, then solar neutrinos propagate adiabatically. We find that all existing neutrino oscillation results can be explained in this JHEP04(2018)136 model with a shift of the effective mixing angles -the measured value of θ 13 arises wholly from DM-neutrino interactions. The model also predicts a suppression of the CP violation at neutrino oscillation experiments.
We then studied the potential of DUNE and JUNO to discriminate between this model and the standard three-neutrino oscillation scenario. We find that DUNE cannot make a distinction at more than 3σ C.L. for bimaximal, tribimaximal and hexagonal mixing, while JUNO can rule out the DM scenario at more than 6σ C.L. by making high-precision measurements of both θ 12 and θ 13 .

Acknowledgments
We diagonalize the above mass matrix by the unitary matrix, where R 0 ij is the rotation matrix in the i − j plane with a rotation angle θ 0 ij , U δ is and R 12 is From eq. (A.1), the leading order corrections in the 1-3 and 2-3 sector are We see that after the rotations of R 0 23 , U δ and R 0 12 , the mass matrix in the 1-2 sector is . (A.6)

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Combining these 2-D rotations together in the full 3-D rotation matrix and making some phase changes in rows and columns so that the 1-1, 1-2, 2-3, and 3-3 elements are real we get where η = φ 23 −φ 23 −φ 12 −φ 12 . This is not quite in the standard form, but we can multiply the second and third rows by e −iη and the third column by e iη to get the standard form for U with δ CP = −(φ 13 + η). Since the phases in η are all small, δ CP is primarily given by −φ 13 , i.e., arg( ).
In the more general case with (∂P/∂θ 13 ) 0 = 0 (such as when there is a single factor of s 13 or sin 2θ 13 ), there is no single power of δθ 13 in eq. (B.2) that matches the single power of δθ eff 13 in eq. (B.3), and the simple correspondence between δθ 13 and δθ eff 13 breaks down. The only measurement that appears to have this problem is the appearance measurement at long-baseline experiments. Also, since the Dirac CP phase is always associated with s 13 in an oscillation probability, the absence of a single power of δθ 13 in eq. (B.2) indicates a reduced sensitivity to the Dirac CP phase in neutrino oscillation experiments.
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