# Neutrinophilic axion-like dark matter

## Abstract

The axion-like particles (ALPs) are very good candidates of the cosmological dark matter, which can exist in many extensions of the standard model (SM). The mass of the ALPs can be as small as \({\mathcal {O}}(10^{-22})~\mathrm{eV}\). On the other hand, the neutrinos are found to be massive and the SM must be extended to explain the sub-eV neutrino masses. It becomes very interesting to consider an exclusive coupling between these two low scale frontiers that are both beyond the SM. The propagation of neutrinos inside the Milky Way would undergo the coherent forward scattering effect with the ALP background, and the neutrino oscillation behavior can be modified by the ALP-induced potential. Assuming a derivative coupling between the ALP and the three generations of active neutrinos, possible impacts on the neutrino oscillation experiments have been explored in this paper. In particular, we have numerically studied the sensitivity of the deep underground neutrino experiment (DUNE). The astrophysical consequences of such coupling have also been investigated systematically.

## 1 Introduction

As one of the most promising dark matter candidates, the ALP with a tiny mass that can be as small as \({\mathcal {O}}(10^{-22})~\mathrm{eV}\)^{1} is drawing more and more attention (see [5, 6, 7, 8] for recent reviews). The typical QCD axion [9, 10] was first identified as the pseudo-Nambu–Goldstone (pNG) boson in the Peccei–Quinn (PQ) mechanism [11, 12] to solve the strong CP problem of QCD. It was later realized [13, 14, 15, 16, 17] to be a very good candidate of the cold dark matter which can acquire an effective anomalous mass by interacting with the gluons. The mass of the QCD axion is settled by the QCD phase transition scale \(\Lambda ^{}_{\mathrm{QCD}}\) and the PQ scale \(f_a\), e.g. \(m_a \approx \Lambda ^2_{\mathrm{QCD}}/f_a \approx 10^{-5}~\mathrm{eV}~(10^{12}~\mathrm{GeV}/f_a)\) with \(\Lambda ^{}_{\mathrm{QCD}} \approx {\mathcal {O}}(10^2)~\mathrm{MeV}\) and \(f_a \approx 10^{9}\)–\(10^{12}~\mathrm{GeV}\). Moreover, the axion can interact with the standard model (SM) fermions through a derivative type of coupling whose strength is proportional to its mass \(m^{}_{a}\). On the other hand, the ALPs with very similar properties to the QCD axion are well motivated in many other extensions of the SM. For example, generating the tiny neutrino mass by spontaneously breaking the lepton number [18, 19, 20, 21, 22, 23, 24] predicts the existence of a Nambu–Goldstone (NG) boson, the Majoron, which inevitably couples with the neutrinos. To spontaneously break the family symmetries will lead to the familons [25, 26]. In the string theory framework, many different ALPs can appear naturally [27, 28, 29, 30, 31, 32], and one of them could just be the QCD axion. In some unified models [33, 34, 35, 36, 37, 38, 39, 40, 41], the pseudoscalar particle can even play multiple roles among the QCD axion, the majoron and the familon at the same time.

The neutrino propagating in the Milky Way can coherently interact with the neutrinophilic ALP field, and the mixings among the neutrino flavors will be modified effectively. In this note, we will study systematically the impacts of the ALP-neutrino coupling on the neutrino oscillation experiments as well as the astrophysical phenomenology. In particular, we will take DUNE as a typical example, and numerically study its sensitivity. The influences of the possible ultralight dark matter field on the neutrino phenomenology have been discussed from various perspectives in the literature [43, 44, 45, 46, 47, 48, 49]. However, it is still very worthwhile to conduct a work like this one. In the existing works, the ultralight scalar [43, 44, 45, 46, 47, 48, 49], vector [44, 47] and tensor [47] DMs have been considered for the neutrino oscillation experiments. The pseudoscalar ALP with a derivative coupling is now being investigated in this work. We will see the derivative coupling in Eq. (2) can have very different laboratory and astrophysical consequences. The impacts on the neutrino oscillation experiments greatly depend on the duration of one ALP oscillation cycle. The time-dependent perturbation analysis should be adopted when the ultralight DM oscillates a number of cycles during the neutrino propagation. If the ALP field loses its coherence within a single neutrino flight, the forward scattering effect which is coherently enhanced should vanish stochastically. Various astrophysical constraints on the coupling are considered systematically in this work. The free-streaming constraint of the cosmic microwave background (CMB) for the neutrino decay process should be the most stringent one. The influences on the propagation of supernova neutrinos and ultrahigh-energy (UHE) neutrinos are also discussed in detail.

This paper is organized as follows. In Sect. 2, the influence of the ALP dark matter on the neutrino oscillation behavior has been explored in detail, and the impact on DUNE has been elaborated with a numerical sensitivity study. In Sect. 3, we will generally discuss the existing constraints from astrophysical observations. We make our conclusion in Sect. 2.

## 2 Impacts on neutrino oscillations

*U*stands for the flavor mixing matrix of the three generations of neutrinos in vacuum, \(\Delta m^{2}_{21}\) and \(\Delta m^{2}_{31}\) are the neutrino mass-squared differences in vacuum. Here

*V*is a potential characterizing the MSW effect with the possible ordinary matter around the neutrino, and \(\xi ^{}_{\alpha \beta }\) for \(\alpha ,\beta =(e,\mu ,\tau )\) is the potential contributed by the ALP background. \(\xi ^{}_{\alpha \beta }\) can be derived from Eq. (3), which at the leading-order reads

- (i)
Case I: \(t^{}_{\mathrm{opr}} \ll t^{}_{a}\). In this case, neutrinos will experience an approximately constant ALP-induced potential throughout the operation time of the experiment. The influence to neutrino oscillations is very similar to the non-standard interactions (NSIs) of neutrinos

^{2}with a constant matter density profile. But the effect of the ALP field is irreducible for all oscillation experiments in our galaxy, even with no ordinary matter surrounding the neutrino. For an ALP mass \(m^{}_{a} \approx 10^{-22}~\mathrm{eV}\) which might be the minimal feasible mass to form the dark matter, the corresponding period is \(t^{}_{a} \approx 1.3~\mathrm{year}\). However, the operation time of most oscillation experiments is longer than \(1.3~\mathrm{year}\). For them, the ALP-induced potential is no longer constant during the operation. - (ii)
Case II: \(t^{}_{\mathrm{rsv}} \lesssim t^{}_{a} \lesssim t^{}_{\mathrm{opr}}\). The operation time is longer than the period of the ALP field, so it can cover a number of ALP cycles during the operation. To detect the modulation effect of the oscillating potential, the duration of each ALP cycle must be longer than the experimental resolution limit of the periodicity. There are several experiments that have looked for the periodic modulation effect of the solar neutrinos, e.g. Super-Kamiokande [53, 54, 55, 56] and SNO [56, 57, 58, 59]. The period of possible periodic signals have been scanned from \(\sim 10~\mathrm{min}\) to \(\sim 10~\mathrm{year}\). No anomalous modulation effect is found in the data sets of Super-Kamiokande and SNO using various statistical methods. Based on this, Ref. [43] has set a strong constraint on the ultralight scalar coupling. The minimal recognizable period \(t^{}_{\mathrm{rsv}}\) of an experiment depends on many factors: the time binning of the data, the events number, the statistical approach etc. In principle, \(t^{}_{\mathrm{rsv}}\) can be as small as the precision of the time measurement, e.g. \(\sim 100~\mathrm{ns}\) for SNO [58]. The statistical analysis in the solar case can be similarly applied to other types of oscillation experiments, to locate the ALP-induced signal. Moreover, in this case one can also choose to integrate over the data-taking time to get a smaller averaged effect [44, 45, 46].

- (iii)
Case III: \(t^{}_{\mathrm{flt}} \ll t^{}_{a} \lesssim t^{}_{\mathrm{rsv}}\). The ALP field oscillates so fast that the experiment can no longer resolve the periodic modulation effect. But during each single neutrino flight from source to detector, the ALP field is still approximately constant. There is no other way but to integrate over the data-taking time and obtain the averaged distortion effect.

- (iv)
Case IV: \(t^{}_{\mathrm{a}} \lesssim t^{}_{\mathrm{flt}} \lesssim t^{\prime }_{\mathrm{a}}\). Throughout a single neutrino flight, the ALP field has been oscillating for a number of cycles. In this case, the neutrino flavor evolution is driven by a varying potential. The time-dependent perturbation theory is required for the perturbative expansion analysis. However, if the neutrino flight time is even longer than the decoherence time of the ALP \(t^{\prime }_{\mathrm{a}} \approx 1000 \times t^{}_{\mathrm{a}}\), i.e. \(t^{}_{\mathrm{flt}} > t^{\prime }_{a}\), the coherent potential induced by the ALP field should vanish stochastically.

### 2.1 Perturbative expansion

^{3}

*U*and \({\tilde{U}}(t)\) respectively, one can formally expand the effective quantities to the second-order as

*i*th-order result. Using the time-dependent perturbation theory, one can work out the following first-order amplitude correction,

*i*,

*j*run over (1, 2). Here \(t^{}_{1}\) and \(t^{}_{2}\) denote the initial time and the final time of the neutrino flight, respectively, \({\hat{H}}^{}_{0,11} = 0\) and \({\hat{H}}^{}_{0,22} = \Delta m^2_{}/(2E^{}_{\nu })\) are the eigenvalues for the Hamiltonian in vacuum \(H^{}_{0}\), and we define \(\Lambda ^{}_{i j} \equiv \sum _{\eta \gamma } U^{}_{e i} U^{\dagger }_{i\eta } U^{}_{\gamma j} U^{\dagger }_{j\mu } g^{}_{\eta \gamma }\sqrt{2\rho } \approx {\mathcal {O}}(\xi )\) with \(\eta ,\gamma \) running over \((e,\mu )\). \(\Lambda ^{}_{i j}\) sums over flavor indices and in general is not suppressed by \(\theta ^{}_{13}\). One can notice a singularity at the point \(m^{}_{a} = \Delta m^2_{}/(2E^{}_{\nu })\) in the denominator. However, the singularity can be cancelled by the numerator and reduce to a factor \(\sim t^{}_{\mathrm{flt}}\). The baseline of an oscillation experiment is usually selected around the oscillation maximum, i.e. \(t^{}_\mathrm{flt} \equiv (t^{}_{2} - t^{}_{1}) \approx 2\pi E^{}_{\nu }/\Delta m^2_{}\). For Case IV with \(t^{}_{a} \equiv 2\pi /m^{}_{a} < t^{}_\mathrm{flt}\), we should have \( \Delta m^2_{}/E^{}_{\nu } \lesssim m^{}_{a}\). One might as well further set \(\Delta m^2_{}/(2E^{}_{\nu }m^{}_{a}) \lesssim {\mathcal {O}}(0.1)\), such that Eq. (16) can be approximated as

In conclusion, the ALP field will induce a potential \(\xi \approx g \sqrt{2\rho } \sin {m^{}_{a} t}\) in the neutrino effective Hamiltonian, with *g* denoting the general order of magnitude of the coupling strength \(g^{}_{\alpha \beta }\). If the ALP field is approximately constant for a single neutrino flight and its periodicity is also resolvable for the concerned experiment, the potential will shift \(\Delta m^2\) and \(\theta \) by \(E^{}_{\nu } \xi \) and \(\xi E^{}_{\nu } / \Delta m^2\) respectively. No matter whether the ALP periodicity is resolvable for the experiment, one can always average over the observation time. In this case, the shifts of \(\Delta m^2\) and \(\theta \) are \((E^{}_{\nu } \xi )^2\) and \((\xi E^{}_{\nu } / \Delta m^2)^2\) respectively. If the ALP field oscillates very rapidly along the neutrino course but still remains in coherence, the oscillation probability will also be modified. The probability can either fluctuate with the amplitude \(\xi /m^{}_{a}\) or averagely get distorted by the magnitude \((\xi /m^{}_{a})^2\).

### 2.2 General sensitivities

*P*is in the order of \({\mathcal {O}}(\xi /m^{}_{a})\) or \({\mathcal {O}}(\xi /m^{}_{a})^2\). One can use the approximation \(\delta (P) \approx \delta (\theta ) \approx \delta (\Delta m^2)/\Delta m^2\) to get the corrections to \(\theta \) and \(\Delta m^2\). However, as has been mentioned before, these corrections can be either enhanced or suppressed by considering the effect of the small mixing angle \(\theta ^{}_{13}\), depending on the flavor structure of the ALP coupling and the concerned experimental channels. Therefore, one should keep in mind that our general estimation has a factorial error of \(\theta ^{}_{13} \approx 0.1\) for some special case. When the ALP potential is either constant or periodically resolvable, the sensitive parameter space for all four cases reads

Experimental parameters

Daya Bay | T2K | JUNO | DUNE | atm. | |
---|---|---|---|---|---|

Baseline (\( t^{}_{\mathrm{flt}}\)) | \(1.6~\mathrm{km}\) | \(295~\mathrm{km}\) | \(53~\mathrm{km}\) | \(1300~\mathrm{km}\) | 10–\(10^4~\mathrm{km}\) |

Energy (\(E^{}_{\nu }\)) | \(4~\mathrm{MeV}\) | \(0.6~\mathrm{GeV}\) | \(4~\mathrm{MeV}\) | \(2.5~\mathrm{GeV}\) | 1–\(100~\mathrm{GeV}\) |

Error (\(\sigma ^{}_{\mathrm{min}}\)) | \(\sim 0.3\%\) | \(\sim 3\%\) | \(\sim 0.2\%\) | \(\sim 0.6\%\) | \(\sim 4\%\) |

\(\theta \) | \(\theta ^{}_{13}\) | \(\theta ^{}_{23}\) | \(\theta ^{}_{12}\) | \(\theta ^{}_{13}\) | \(\theta ^{}_{23}\) |

\(\Delta m^2\) | \(\Delta m^{2}_{ee}\) | \(\Delta m^{2}_{32}\) | \(\Delta m^{2}_{21}\) | \(\Delta m^{2}_{31}\) | \(\Delta m^{2}_{31}\) |

\(t^{}_{\mathrm{flt}} / (2\pi E^{}_{\nu }/\Delta m^2)\) | 0.81 | 0.99 | 0.80 | 0.87 | – |

### 2.3 Impact on DUNE

The expected sensitivities of DUNE on the ALP-neutrino coupling terms \( g_{\alpha \beta } \) at the 2\( \sigma \) and the 3\( \sigma \) C.L. Here the second (third) column represents the sensitivities for the non-averaged (averaged) case

Parameters (\( 10^{-11} \) eV\( ^{-1} \)) | Non-averaged | Averaged | ||
---|---|---|---|---|

2\( \sigma \) | 3\( \sigma \) | 2\( \sigma \) | 3\( \sigma \) | |

\(| g_{e \mu } |\) | < 0.25 | < 0.34 | < 0.95 | < 1.14 |

\( |g_{e \tau } |\) | < 0.58 | < 1.16 | < 1.46 | < 1.71 |

\( |g_{\mu \tau } | \) | < 0.95 | < 1.23 | < 1.65 | < 2.65 |

\( g_{ee} \) | (\(-13.62, -9.30\)) | (\(-16.30, -8.71\)) | (\(-8.90, 9.30\)) | (\(-11.10, 11.30\)) |

\( \oplus \)(\(-1.11, 2.33\)) | \( \oplus \)(\(-1.63, 6.15\)) | |||

\( g_{\mu \mu } \) | (\(-1.42, 2.02\)) | (\(-1.95, 2.56\)) | (\(-2.34, 2.34\)) | (\(-2.90, 2.88\)) |

\( g_{\tau \tau } \) | (\(-2.01, 0.94\)) | (\(-2.60, 1.37\)) | (\(-2.44, 2.50\)) | (\(-3.18, 3.07\)) |

To study the impact of the ALP-neutrino coupling on DUNE, we use the GLoBES extension file *snu.c* as has been presented in Refs. [73, 74]. In our analysis, we have modified the NSI matter potential in *snu.c* with the potential induced by the ALP field. We simulate the fake data of DUNE using the standard oscillation parameters \( \sin ^2 \theta _{12} = 0.307,~\sin ^2 \theta _{13} = 0.022,~\sin ^2 \theta _{23} = 0.535,~\delta = - \pi /2,~\Delta m^{2}_{21} = 7.40 \times 10^{-5} ~\mathrm{eV^2}\) and \(\Delta m^{2}_{31} = 2.50 \times 10^{-3} ~\mathrm{eV^2}\) which are compatible with the latest global-fit results [75, 76, 77]. We marginalize the standard parameters \(\theta _{23}\), \(\delta \) and the mass hierarchy over their current 3\(\sigma \) ranges in the test. As the remaining standard oscillation parameters have been measured with very high precisions, we keep them fixed in the analysis. On the other hand, for the ALP-neutrino coupling parameters, we fix their true values as zero during the statistical analysis. For the fit we switch on only one of them at one time for numerical simplicity. While performing the sensitivity study of the diagonal ALP-neutrino coupling parameters, we marginalize over the standard parameters in the fit. Moreover, for the off-diagonal parameters, we also marginalize their phase in the range (\( 0 \rightarrow 2\pi \)). All along this work, we perform our analysis considering two different scenarios for the ALP-induced potential, namely non-averaged and averaged cases. For the non-averaged case, we take the ALP-induced potential \(\xi ^{}_{\alpha \beta }\) as a constant over time for simplicity. A more careful treatment would require the statistical analysis of the modulation on the time-binned data, which might be interesting for a future work. For the averaged case, we have modified the *snu.c* file by averaging the oscillation probability over the time points within one ALP cycle.

In Fig. 2, we show the oscillation probabilities of the appearance channels (\( {\nu _{\mu } \longrightarrow \nu _{e} } \) and \( {{\overline{\nu }}_{\mu } \longrightarrow {\overline{\nu }}_{e} } \)) for DUNE. For illustration, we perform this study with a single non-zero representative value (\( g_{ee} = 5.6 \times 10^{-11} \) eV\( ^{-1} \)) of the ALP-neutrino coupling term. The green (dashed and dotted) curves show the case where the ALP-induced potential is not averaged over the observation time (see the figure legend for more details). Whereas, the solid blue curves describe the scenario where the ALP-induced potential is averaged over time. Besides this, the standard case without the ALP-induced potential is shown as the gray solid curves for comparison. For the non-averaged case, one can observe significant deviations of the probabilities around the peak energy \(\sim 2.5~\mathrm{GeV}\) compared to the standard case. When the ALP-induced potential oscillates as \(g^{}_{ee} \sqrt{2\rho } \sin {m^{}_{a}t}\), the associated probability will vary roughly in between the two green curves. After averaging over the time, one would obtain the distortion effect as the blue curve in each panel, which is smaller than the non-averaged case. As our intention at the probability level is for demonstration, hence we illustrate this analysis with a single non-zero ALP-neutrino coupling term for the appearance channel. Nevertheless, we will perform a detailed sensitivity study considering both the appearance and disappearance channels for all the coupling terms at the \( \chi ^{2} \) level.

In Fig. 3, we show the sensitivity of DUNE to the six ALP-neutrino coupling parameters \( g_{\alpha \beta } \) in the \( \chi ^{2} \)–\( g_{\alpha \beta } \) plane. The top and bottom panels represent our results for the non-diagonal and the diagonal parameters, respectively. The green solid curves represent the non-averaged case where a constant potential has been taken during the numerical simulation. The blue dotted curves signify the averaged case where the ALP-induced potential is averaged over time in the final probability. Note that the black dotted and black dashed horizontal lines stand for the \( \chi ^{2} \) values corresponding to the 2\( \sigma \) and the 3\( \sigma \) C.L., respectively. In Fig. 3, one can clearly observe that the averaged case has looser sensitivity than the non-averaged case. This can be easily understood from the analytical results before and from the probability demonstration in Fig. 2. It can be ascribed as the loss of information after averaging. From the first plot of the bottom row, we notice that for the green curve there is a local minimum around \( g_{ee} = -10.1 \times 10^{-11}\) eV\( ^{-1} \) other than a global minimum at zero. This shows a degenerate solution for \( g_{ee} \) and it arises from the unknown hierarchy (as we have marginalized over the mass hierarchy in the fit). In our careful analysis by fixing the hierarchy to the normal one both in data and theory we find no degenerate solution for \( g_{ee} \). This tells that the lack of knowledge of the hierarchy may lead to a degenerate solution. Thus, for all these different cases we perform our numerical analysis over the marginalized hierarchy. Also, comparing the top and the bottom row, we notice that overall sensitivity of the off-diagonal parameters is better than that of the diagonal ones. Table 2 summarizes the expected sensitivities for all the ALP-neutrino coupling parameters \( g_{\alpha \beta } \) for DUNE, where the 2\( \sigma \) and 3\( \sigma \) intervals are presented respectively. Note that the second and third panels of the table show the sensitivity limit for the non-averaged and averaged cases, respectively.

## 3 Astrophysical bounds

### 3.1 Early universe

The cosmological evolution will be more or less modified, if neutrinos have strong interactions with an ultralight degree of freedom. During the expansion of the Universe, the thermal ALPs can be produced via the processes like \(\nu +\nu \leftrightarrow a+a\) and \(\nu _i \leftrightarrow \nu _j +a\), where \(\nu _{i}\) and \(\nu _{j}\) are the neutrino mass eigenstates with \(m^{}_i > m^{}_j\). We assume the mass of the ALP is much smaller than that of the neutrinos such that the process like \({\nu }+\nu \leftrightarrow a\) is kinematically forbidden. If the ALP mass is larger than the neutrino masses,^{4} the ALP-neutrino coupling must be severely suppressed to maintain the dark matter abundance. Suppose that the ALPs are fully thermalized before the neutrino decoupling around \(1~\mathrm{MeV}\), they will contribute to the extra effective neutrino number by an amount of \(\Delta N_\mathrm{eff} \approx 0.57\) [79]. The extra \(N_{\mathrm{eff}}\) is strongly constrained by the observations of BBN and CMB. The current constraint of BBN is \(\Delta N_{\mathrm{eff}} \lesssim 1\) at \(95\%\) C.L. [80], while CMB can give a stronger constraint with the latest Planck 2018 result [81]: \(\Delta N_\mathrm{eff} \lesssim 0.52\) (\(95\%\) C.L., *Planck* TT+lowE). On the other hand, the strong interaction during the CMB epoch will also suppress the free-streaming length of neutrinos [82, 83], such that the evolution of CMB perturbations is severely disturbed. This can lead us to a very strong constraint on the coupling strength of the secret neutrino interactions between neutrinos and ALPs.

*T*being the temperature of the neutrino plasma.

^{5}During the epoch of radiation domination, the Hubble expansion rate is given by \(H \approx 1.66 \sqrt{g^{}_*} T^2/M^{}_{\mathrm{Pl}}\), with \(g^{}_*\) denoting the effective number of relativistic degrees of freedom at

*T*(refer to [85, 86] for its values at various temperatures) and \(M^{}_{\mathrm{Pl}} \simeq 1.221\times 10^{19}~\mathrm{GeV}\) being the Planck mass. Because of the ratio \(\Gamma / H \propto T\), the thermal ALPs are first copiously produced at high temperatures in the very beginning of the Universe. They should eventually freeze out at some low temperature \(T^{}_{\mathrm{fo}}\) to evade the CMB constraint. The extra \(N^{}_{\mathrm{eff}}\) contributed by the thermal ALPs is found to be

### 3.2 SN1987A explosion

### 3.3 Galactic propagation of neutrinos from SN1987A and blazar TXS 0506+056

The propagation of astrophysical neutrino fluxes in the dark matter halo of our galaxy will be affected when the ALP-neutrino interaction is very strong [44, 49, 90]. These neutrinos will suffer the energy loss and the direction change, due to the frequent scattering with the ALPs which forms the dark matter. The observed neutrino flux of SN1987A agrees well with the standard supernova neutrino theory, which should in turn give us a constraint on the ALP-neutrino coupling. On the other hand, the recent UHE neutrino event IceCube-170922A observed by IceCube coincides with a flaring blazar TXS 0506+056 with \(\sim 3\sigma \) level [91]. The estimated neutrino luminosities are similar to that of the associated \(\gamma \)-rays, consistent with the prediction of blazar models [92, 93]. However, if there is a large attenuation effect for the UHE neutrino flux by scattering with ALPs, the original flux from TXS 0506+056 must be much stronger than the estimated one, which is inconsistent with the blazar observation. For the diffuse UHE neutrinos, the observed flux is around the Waxman–Bahcall (WB) bound [94]. The large attenuation effect of ALP would require a diffuse flux to be much larger than the WB bound, which can in turn set a limit on the ALP-neutrino coupling. However, the original diffuse UHE neutrino flux might be well above the WB bound, since the complete sources of the UHE neutrino still remain unknown for us.

## 4 Conclusion

Assuming the derivative coupling between the ALP and the neutrinos, we have investigated the impacts of the ALP dark matter on neutrino oscillation experiments. Depending on the mass of the ALP, there are two different scenarios regarding the data analysis: the modulation effect induced by the oscillating ALP field can be resolved for the experiment (*Non-Averaged*); the modulation effect is simply averaged to a distortion effect (*Averaged*). Based on the simple argument, we find that the existing experiment like T2K can already exclude \(g \gtrsim 3\times 10^{-10}~\mathrm{eV^{-1}}\) for \(10^{-22}~\mathrm{eV} \lesssim m^{}_{a} \lesssim 10^{-11}~\mathrm{eV}\). The projected experiments like DUNE are sensitive to the coupling strength \(g \gtrsim 10^{-12}~\mathrm{eV^{-1}}\) in the ALP mass range \(10^{-22}~\mathrm{eV} \lesssim m^{}_{a} \lesssim 10^{-12}~\mathrm{eV}\) for the non-averaged case. The \(1\sigma \) sensitivity is reduced by a factor of \({\mathcal {O}}(10)\) for the averaged case due to the cancellation of the oscillating ALP-induced potential. Using the GLoBES package, we have numerically simulated the data of DUNE. The sensitivity results on the six coupling parameters \(g^{}_{\alpha \beta }\) for \(\alpha ,\beta =(e,\mu ,\tau )\) have been yielded and summarized in Table 2. The numerical results agree well in the order of magnitude with the simple estimation for DUNE. The impact of such coupling on the evolution of the early Universe has been discussed. A very stringent bound from the free-streaming of CMB can be made as \(g^{}_{ij} \lesssim 10^{-10}~\mathrm{eV^{-1}}\). The propagation of neutrinos from SN1987A can put a constraint \(g \lesssim 4\times 10^{-8}~\mathrm{eV^{-1}}\), while the IceCube observation can put a much stronger one \(g \lesssim 7\times 10^{-10}~\mathrm{eV^{-1}}\). The next generation of neutrino experiments can probe the parameter range two orders of magnitude beyond the astrophysical limits.

## Footnotes

- 1.
The ultralight dark matter with a mass \(\sim 10^{-22}~\mathrm{eV}\) is often called “fuzzy dark matter” [1]. Its macroscopic wavelength can suppress the power of the galactic clustering and resolve the small scale tension of the cold dark matter paradigm [2]. A lower bound on the ALP mass can be set by the observation of the Lyman-\(\alpha \) forest at \({\mathcal {O}}(10^{-22})~\mathrm{eV}\) [3, 4].

- 2.
Wolfenstein in Ref. [50] first proposed that dimension-six four-fermion operators in the form of NSI can potentially impact the neutrino propagation.

- 3.
The ordinary matter effect is temporarily ignored for the perturbation analysis. It is very straightforward to include it by replacing the vacuum quantities with the matter-corrected ones.

- 4.
Like in Ref. [78], the mass of the ALP dark matter can reach almost \({\mathcal {O}}(\mathrm MeV)\).

- 5.
The derivative coupling in Eq. (2) is equivalent to the pseudoscalar coupling \(h^{}_{\alpha \beta } a {\nu }^{}_{\alpha }\gamma ^{}_{5}\nu ^{}_{\beta }\) with a dimensionless coupling constant \(h^{}_{\alpha \beta }=\sum ^{}_{ij}U^{}_{\alpha i}U^{*}_{\beta j}(m^{}_{i}+m^{}_{j})g^{}_{\alpha \beta }\), only if each neutrino line in the Feynman diagram is attached to only one NG boson line [82, 84]. However, this is not the case for the process \({\nu }+\nu \leftrightarrow a+a\). The reaction rate for the pseudoscalar coupling case is proportional to

*T*instead of \(T^3\).

## Notes

### Acknowledgements

The authors are indebted to Prof. Shun Zhou for carefully reading this manuscript and for many valuable comments and suggestions. The authors thank Prof. Zhi-zhong Xing, Jing-yu Zhu and Xin Wang for insightful discussions. GYH would like to thank Qin-rui Liu for helpful discussions. NN also thanks Dr. Sushant K. Raut and Dr. Mehedi Masud for useful discussion. GYH is supported by the National Natural Science Foundation of China under Grant no. 11775232. The research work of NN was supported in part by the National Natural Science Foundation of China under Grant no. 11775231.

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