Eur. Phys. J. C (2018) 78:922 https://doi.org/10.1140/epjc/s10052-018-6391-y
Regular Article - Theoretical Physics
Neutrinophilic axion-like dark matter
Guo-yuan Huang1,2,a, Newton Nath1,2,b 1 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 2 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Received: 18 September 2018 / Accepted: 27 October 2018 © The Author(s) 2018
Abstract The axion-like particles (ALPs) are very good a very good candidate of the cold dark matter which can candidates of the cosmological dark matter, which can exist acquire an effective anomalous mass by interacting with in many extensions of the standard model (SM). The mass of the gluons. The mass of the QCD axion is settled by the O( −22) the ALPs can be as small as 10 eV. On the other QCD phase transition scale QCD and the PQ scale fa, hand, the neutrinos are found to be massive and the SM ≈ 2 / ≈ −5 ( 12 / ) e.g. ma QCD fa 10 eV 10 GeV fa with must be extended to explain the sub-eV neutrino masses. It ≈ O( 2) ≈ 9 12 QCD 10 MeV and fa 10 Ð10 GeV. Moreover, becomes very interesting to consider an exclusive coupling the axion can interact with the standard model (SM) fermions between these two low scale frontiers that are both beyond through a derivative type of coupling whose strength is pro- the SM. The propagation of neutrinos inside the Milky Way portional to its mass ma. On the other hand, the ALPs with would undergo the coherent forward scattering effect with very similar properties to the QCD axion are well motivated the ALP background, and the neutrino oscillation behavior in many other extensions of the SM. For example, gener- can be modified by the ALP-induced potential. Assuming a ating the tiny neutrino mass by spontaneously breaking the derivative coupling between the ALP and the three gener- lepton number [18Ð24] predicts the existence of a NambuÐ ations of active neutrinos, possible impacts on the neutrino Goldstone (NG) boson, the Majoron, which inevitably cou- oscillation experiments have been explored in this paper. In ples with the neutrinos. To spontaneously break the family particular, we have numerically studied the sensitivity of the symmetries will lead to the familons [25,26]. In the string deep underground neutrino experiment (DUNE). The astro- theory framework, many different ALPs can appear naturally physical consequences of such coupling have also been inves- [27Ð32], and one of them could just be the QCD axion. In tigated systematically. some unified models [33Ð41], the pseudoscalar particle can even play multiple roles among the QCD axion, the majoron and the familon at the same time. 1 Introduction The mass spectrum and the coupling strength with the SM particles of the generic ALPs are not limited as in the As one of the most promising dark matter candidates, the QCD axion model, greatly enriching their phenomenology O( −22) 1 ALP with a tiny mass that can be as small as 10 eV and the experimental efforts to hunt them. The ALPs can be is drawing more and more attention (see [5Ð8] for recent non-thermally produced in the early Universe by the mis- reviews). The typical QCD axion [9,10] was first identi- alignment mechanism or the topological defect decay. After fied as the pseudo-NambuÐGoldstone (pNG) boson in the the ALPs form the dark matter halo of our Milky Way, their de PecceiÐQuinn (PQ) mechanism [11,12] to solve the strong Broglie wavelength λdB can be as long as several kpc for an CP problem of QCD. It was later realized [13Ð17]tobe −22 ALP mass ma ≈ 10 eV, almost comparable to the scale ∼ O( ) 1 ∼ −22 of the Milky Way 10 kpc. The local number density of The ultralight dark matter with a mass 10 eV is often called = ρ / ≈ 30 −3 “fuzzy dark matter” [1]. Its macroscopic wavelength can suppress the the ultralight particles is na a ma 10 cm if they −3 power of the galactic clustering and resolve the small scale tension of saturate the dark matter energy density ρ ≈ 0.3GeV· cm . the cold dark matter paradigm [2]. A lower bound on the ALP mass The huge particle number within the de Broglie wavelength α O( −22) can be set by the observation of the Lyman- forest at 10 eV makes them oscillate coherently as a single classical field [3,4]. with a e-mail: [email protected] b e-mail: [email protected] a(t, x) = a0 cos (mat −p ·x), (1)
0123456789().: V,-vol 123 922 Page 2 of 13 Eur. Phys. J. C (2018) 78:922 √ where a0 = 2ρ/ma is the amplitude of the ALP field, p been elaborated with a numerical sensitivity study. In Sect. is the momentum of the ALP, and ma denotes the mass of 3, we will generally discuss the existing constraints from the ALP. The ALP field maintains its coherence during the astrophysical observations. We make our conclusion in Sect. expansion of the universe. However, after the formation of 2. the Milky Way, it develops a slight stochastic decoherence due to the gravitational clustering effect. The typical velocity − dispersion of the ALP dark matter is σv ∼ 10 3c [42], cor- 2 Impacts on neutrino oscillations λ ≈ λ × 3 responding to a coherence length of coh a 10 with λ = π/ a 2 ma being its Compton wavelength. In our work, When neutrinos propagate inside our galaxy, which is always we consider an ALP-neutrino interacting Lagrangian which true for the current ground-based oscillation experiment, they preserves the shift symmetry as will inevitably scatter with the ambient ALP background μ under our consideration. The coherent forward scattering − L = gαβ ∂μa ναγ γ νβ , (2) int 5 process which can be analogous to the MikhyevÐSmirnovÐ where gαβ with α, β = (e,μ,τ)is the dimensional coupling Wofenstein (MSW) [50Ð52] matter effect is dominant. Under ∗ strength in the neutrino flavor basis with gαβ = gβα, and the interaction form in Eq. (2), the dispersion relation of the να(β) represents the neutrino flavor eigenstate. In the QCD three generations of neutrinos will be modified as axion model, the coupling strength g is suppressed by the αβ ( · ˆ + ∂ · )2 − ( · ˆ + ∂ · )2 = 2, Eν I 0a gm pν I a gm mν (3) PQ scale fa, which is not necessarily the case in the generic ALPs context. ˆ × × where Iisthe3 3 identity matrix, gm stands for the 3 3 The neutrino propagating in the Milky Way can coherently Hermitian matrix of the coupling constant gαβ , mν stands for interact with the neutrinophilic ALP field, and the mixings the mass matrix of neutrinos in the flavor basis, Eν and pν are among the neutrino flavors will be modified effectively. In the energy and momentum of the neutrino, respectively. The this note, we will study systematically the impacts of the modified neutrino oscillation behavior can be conveniently ALP-neutrino coupling on the neutrino oscillation experi- described by the following effective Hamiltonian: ments as well as the astrophysical phenomenology. In par- ⎛ ⎞ 00 0 ticular, we will take DUNE as a typical example, and numer- 1 H(t) = U ⎝0 m2 0 ⎠ U † ically study its sensitivity. The influences of the possible 2E 21 ν 00 m2 ultralight dark matter field on the neutrino phenomenology ⎛ 31 ⎞ + ξ ( )ξ ( )ξ ( ) have been discussed from various perspectives in the litera- V ee t eμ t eτ t + ⎝ ξ ∗ ( )ξ( )ξ ( )⎠ , ture [43Ð49]. However, it is still very worthwhile to conduct eμ t μμ t μτ t (4) ξ ∗ ( )ξ∗ ( )ξ ( ) a work like this one. In the existing works, the ultralight eτ t μτ t ττ t scalar [43Ð49], vector [44,47] and tensor [47]DMshave where U stands for the flavor mixing matrix of the three gen- been considered for the neutrino oscillation experiments. The erations of neutrinos in vacuum, m2 and m2 are the neu- pseudoscalar ALP with a derivative coupling is now being 21 31 trino mass-squared differences in vacuum. Here V is a poten- investigated in this work. We will see the derivative cou- tial characterizing the MSW effect with the possible ordinary plinginEq.(2) can have very different laboratory and astro- matter around the neutrino, and ξ for α, β = (e,μ,τ) is physical consequences. The impacts on the neutrino oscilla- αβ the potential contributed by the ALP background. ξ can be tion experiments greatly depend on the duration of one ALP αβ derived from Eq. (3), which at the leading-order reads oscillation cycle. The time-dependent perturbation analysis should be adopted when the ultralight DM oscillates a num- ξαβ ≈−gαβ ∂0a + gαβ ∂a ·pν/|pν|, (5) ber of cycles during the neutrino propagation. If the ALP field /| | loses its coherence within a single neutrino flight, the forward where pν pν stands for the direction of the neutrino flux. scattering effect which is coherently enhanced should vanish Providing the velocity of the ALP field in the Milky Way is O( −3) stochastically. Various astrophysical constraints on the cou- of the order 10 c, the temporal term should dominate pling are considered systematically in this work. The free- the potential in Eq. (5). The further approximation can be made to give streaming constraint of the cosmic microwave background (CMB) for the neutrino decay process should be the most ξ ( ) ≈− ∂ ≈ ρ , αβ t gαβ 0a gαβ 2 sin mat (6) stringent one. The influences on the propagation of supernova √ neutrinos and ultrahigh-energy (UHE) neutrinos are also dis- with 2ρ ≈ 2.15×10−3 eV2 ρ/(0.3GeV· cm−3). As has cussed in detail. been mentioned before, the ALP potential oscillates with a ≈ π/ ≈ . ( −20 / ) This paper is organized as follows. In Sect. 2, the influence period ta 2 ma 4 8 day 10 eV ma . The impact of the ALP dark matter on the neutrino oscillation behavior on the neutrino oscillation strongly depends on the magni- has been explored in detail, and the impact on DUNE has tude relation between the ALP period and several time scales 123 Eur. Phys. J. C (2018) 78:922 Page 3 of 13 922 of the experiment. The relevant experimental time scales modulation effect. But during each single neutrino flight include: topr, the operation time of the experiment, typically from source to detector, the ALP field is still approxi- over several years for an oscillation experiment; trsv, the min- mately constant. There is no other way but to integrate imal period that the experiment can resolve for a periodic over the data-taking time and obtain the averaged dis- modulation effect; tflt, the flight time of each single neutrino tortion effect. ∼ . from the source to the detector, 0 33 ms for a baseline (iv) Case IV: ta tflt ta. Throughout a single neutrino length of 100 km. There are four different cases that should flight, the ALP field has been oscillating for a number be addressed: of cycles. In this case, the neutrino flavor evolution is driven by a varying potential. The time-dependent per- turbation theory is required for the perturbative expan- (i) Case I: topr ta. In this case, neutrinos will experi- ence an approximately constant ALP-induced potential sion analysis. However, if the neutrino flight time is throughout the operation time of the experiment. The even longer than the decoherence time of the ALP ≈ × > influence to neutrino oscillations is very similar to the ta 1000 ta, i.e. tflt ta, the coherent potential non-standard interactions (NSIs) of neutrinos2 with a induced by the ALP field should vanish stochastically. constant matter density profile. But the effect of the ALP field is irreducible for all oscillation experiments in our 2.1 Perturbative expansion galaxy, even with no ordinary matter surrounding the ≈ −22 We will work under the two-neutrino-flavor scheme to see neutrino. For an ALP mass ma 10 eV which might be the minimal feasible mass to form the dark matter, the influence of the ALP potential analytically. Let us assume ≈ . the ALP-induced potential is very small, such that one can the corresponding period is ta 1 3 year. However, the operation time of most oscillation experiments is longer perturbatively expand the effective mixing parameters and than 1.3 year. For them, the ALP-induced potential is the oscillation probability around the standard ones. Taking ν ν no longer constant during the operation. e and μ as the two neutrino flavors under our consideration, 3 (ii) Case II: trsv ta topr. The operation time is longer the effective Hamiltonian in Eq. (4) is then reduced to than the period of the ALP field, so it can cover a num- 1 ˜ 00 ˜ † ˆ ber of ALP cycles during the operation. To detect the U(t) U (t) + V · I 2E 0 m˜ 2(t) 0 modulation effect of the oscillating potential, the dura- ν ξ ( )ξ ( ) tion of each ALP cycle must be longer than the exper- = 1 00 † + ee t eμ t , U 2 U 2Eν ξ ∗ ( )ξ ( ) (7) imental resolution limit of the periodicity. There are 2Eν 0 m eμ t μμ t several experiments that have looked for the periodic where V · Iˆ is a diagonal potential term to be determined modulation effect of the solar neutrinos, e.g. Super- 0 which is irrelevant for neutrino oscillations, m˜ 2(t) is the Kamiokande [53Ð56] and SNO [56Ð59]. The period effective mass-squared difference which has been shifted by of possible periodic signals have been scanned from the ALP background, and U˜ (t) is the effective mixing matrix ∼ 10 min to ∼ 10 year. No anomalous modulation that diagonalizes the total Hamiltonian. Setting θ and θ˜ as the effect is found in the data sets of Super-Kamiokande mixing angles of the matrices U and U˜ (t) respectively, one and SNO using various statistical methods. Based on can formally expand the effective quantities to the second- this, Ref. [43] has set a strong constraint on the ultra- order as light scalar coupling. The minimal recognizable period ˜ 2 = 2 + δ ( 2) + δ ( 2), trsv of an experiment depends on many factors: the time m m 1 m 2 m binning of the data, the events number, the statistical θ˜ = θ + δ (θ) + δ (θ). 1 2 (8) approach etc. In principle, t can be as small as the rsv δ δ precision of the time measurement, e.g. ∼ 100 ns for Here 1 and 2 represent the first-order and the second-order ALP perturbations respectively, which read as follows SNO [58]. The statistical analysis in the solar case can be similarly applied to other types of oscillation exper- 2 R δ ( m ) = 2 ξ cos 2θ + 2ξ μ sin 2θ Eν, (9) iments, to locate the ALP-induced signal. Moreover, in 1 e R this case one can also choose to integrate over the data- 2ξ μ cos 2θ − ξ sin 2θ δ (θ) = e , 1 Eν (10) taking time to get a smaller averaged effect [44Ð46]. m2 (iii) Case III: t t t . The ALP field oscillates so fast flt a rsv δ ( 2) = (ξ I )2 + (ξ R )2 2 θ that the experiment can no longer resolve the periodic 2 m 4 eμ 4 eμ cos 2
2 Wolfenstein in Ref. [50] first proposed that dimension-six four- 3 The ordinary matter effect is temporarily ignored for the perturbation fermion operators in the form of NSI can potentially impact the neutrino analysis. It is very straightforward to include it by replacing the vacuum propagation. quantities with the matter-corrected ones. 123 922 Page 4 of 13 Eur. Phys. J. C (2018) 78:922
2 2 2Eν The above discussion is under the assumption that the +ξ sin2 2θ − 2ξ R ξ sin 4θ , (11) eμ 2 m ALP field is approximately constant during a single neutrino flight. For Case IV, the above results do not hold any more. δ (θ) = − ξ R ξ θ + (ξ I )2 θ 2 4 eμ cos 4 4 eμ cot 2 In the case of the time-dependent potential, the transition amplitude M ≡ ν |ν can be formally expanded as M = E2 e μ t +ξ 2 θ − (ξ R )2 θ ν , M + M + M +··· with M being the ith-order result. sin 4 4 eμ sin 4 2 (12) 0 1 2 i m2 Using the time-dependent perturbation theory, one can work ξ ≡ ξ − ξ ξ R(I) out the following first-order amplitude correction, with μμ ee, and eμ being the real (imaginary) part ξ 2 θ M = − ˆ + ˆ of eμ. One can observe that m and have been shifted 1 i ij exp i H0,iit2 i H0, jjt1 2 by amounts of ξ Eν and ξ Eν/ m in general respectively, ij with ξ denoting the general order of magnitude of ξαβ .Ifwe ( ˆ − ˆ + ) exp i H0,ii H0, jj ma t have originally a very small mixing angle, e.g. θ ≈ 9◦,the × 13 ( ˆ − ˆ + ) ξ ξ 2 H , H , ma shift contribution of or eμ could be suppressed, but there 0 ii 0 jj = are always unsuppressed terms survived. Assuming the ALP ( ˆ − ˆ − ) t t2 exp i H0,ii H0, jj ma t t − , (16) potential is constant within flt (for Case I, Case II or Case ( ˆ − ˆ − ) ν → ν 2 H0,ii H0, jj ma t=t III), the survival probability of μ e reads 1 where the subscripts i, j run over (1, 2).Heret and t P˜ = P + δ (P ) + δ (P ), 1 2 μe μe 1 μe 2 μe denote the initial time and the final time of the neutrino = 2 φ 2 θ, ˆ = ˆ = 2/( ) Pμe sin sin 2 (13) flight, respectively, H0,11 0 and H0,22 m 2Eν are the eigenvalues for the Hamiltonian in vacuum H , and φ ≡ 2 /( ) √ 0 where m L 4E is a dimensionless phase factor and ≡ † † ρ ≈ O(ξ) we define ij ηγ UeiUiηUγ j U jμgηγ 2 with the corrections are η, γ ( ,μ) running over e . ij sums over flavor indices and δ ( m2) in general is not suppressed by θ . One can notice a sin- δ (P ) = φ sin 2φ sin2 2θ × 1 13 1 μe 2 = 2/( ) m gularity at the point ma m 2Eν in the denominator. + 2 φ θ × δ (θ), 2sin sin 4 1 (14) However, the singularity can be cancelled by the numerator and reduce to a factor ∼ t . The baseline of an oscillation δ ( 2) 2 flt 2 2 1 m δ (Pμ ) = (φ) cos 2φ sin 2θ × experiment is usually selected around the oscillation maxi- 2 e 2 2 m mum, i.e. t ≡ (t − t ) ≈ 2π Eν/ m . For Case IV with flt 2 1 2 t ≡ π/m < t m2/E m +4sin2 φ cos 4θ × δ (θ) a 2 a flt, we should have ν a.One 1 2/( ) O( . ) might as well further set m 2Eνma 0 1 , such that δ ( m2)δ (θ) +2φ sin 2φ sin 4θ × 1 1 Eq. (16) can be approximated as 2 m δ ( m2) M ≈ ij − ˆ + ˆ +φ sin 2φ sin2 2θ × 2 1 i exp i H0,iit2 i H0, jjt1 m2 ma ij +2sin2 φ sin 4θ × δ (θ). (15) × ( ) − ( ) . 2 cos mat2 cos mat1 (17) When the ALP field oscillates fast enough like in Case II ν → ν ≈ The final survival probability of μ e is given by Pμe δ ( ) ∝ ∗ and Case III, the first-order term 1 Pμe sin mat can |M |2 + (M M ) +|M |2 M ≈ O(ξ/ ) 0 2Re o 1 1 with 1 ma . be averaged to 0. The second-order term after being aver- As has been mentioned before, the flight time for a base- δ ( ) ∝ 2 = / aged, 2 Pμe sin mat 1 2, will lead to the dis- = ≈ . line length of L 100 km is tflt 0 3ms.ForCaseIV tortion effect on the final probability. It should be empha- under the discussion, the oscillating period of ALP should sized that the second-order corrections in Eq. (8)areas be in the range of (0.3ms, 0.3ms). If the experiment can δ ( ) important as the first-order ones when we derive 2 Pμe resolve the periodic fluctuation in this time range (that would δ ( )/ ≈ ξ / 2 in Eq. (15). We notice that 1 Pμe Pμe Eν m and require a very refined scanning in the experimental data set), δ ( )/ ≈ (ξ / 2)2 φ,θ ≈ O( ) O(ξ/ ) 2 Pμe Pμe Eν m assuming 1 .How- the modulation pattern with an amplitude ma can be θ ever, if the concerned mixing angle is 13 for the oscillation identified. Regardless of the practical periodicity resolving channel, most of the higher-order terms in Eqs. (14) and (15) power, one can always choose to average over the observa- θ would be suppressed by 13. It is in line with our expecta- tion time, obtaining the probability correction in the order of tion, since the original survival probability is already highly O(ξ/ )2 (M M∗) ma . Note that the cross term Re o 1 vanishes suppressed. But in Eq. (15) one can notice an unsuppressed for the averaged probability. When the oscillation channel is δ (θ) 2 θ term for 1 which will become important when the driven by the mixing angle 13, the magnitude of the impact first-order correction is averaged out. is more or less modified. The first-order correction will be 123 Eur. Phys. J. C (2018) 78:922 Page 5 of 13 922
M ∝ θ suppressed by 0 sin 2 13, while there is no suppression ent enhancement effect of the ALP field is lost during the ≈ for the second-order term. flight of neutrinos. When tflt ta, we expect the sensitiv- ξ ≈ √In conclusion, the ALP field will induce a potential ity should be in between Eqs. (18) and (19), whose exact ρ g 2 sin mat in the neutrino effective Hamiltonian, with value however can only be derived by solving Eq. (16)ina g denoting the general order of magnitude of the coupling given oscillation context. To join the two cases in Eqs. (18) strength gαβ . If the ALP field is approximately constant for and (19), we can simplify the result by using the approx- ≈ π / 2 ≈ a single neutrino flight and its periodicity is also resolvable imation tflt 2 Eν m , such that when ta tflt (i.e., 2 ≈ 2/ for the concerned experiment, the potential will shift m ma m Eν) they yield the same constraint. We will find 2 and θ by Eνξ and ξ Eν/ m respectively. No matter whether in Table I that this is a good approximation for most of the the ALP periodicity is resolvable for the experiment, one can oscillation experiments that are going to be investigated. The always average over the observation time. In this case, the errors due to the approximation should be negligible for our 2 2 2 2 shifts of m and θ are (Eνξ) and (ξ Eν/ m ) respec- purpose. Based on the above consideration, the constraints tively. If the ALP field oscillates very rapidly along the neu- can be summarized into the following conclusive results for trino course but still remains in coherence, the oscillation two different scenarios: probability will also be modified. The probability can either m2 σ ξ/ g √min max 2π, m t (Non-Averaged) (20) fluctuate with the amplitude ma or averagely get distorted π ρ a flt (ξ/ )2 2 Eν 2 by the magnitude ma . when the ALP potential is either approximately constant or 2.2 General sensitivities periodically resolvable; and 2 σ m min π, ( v ) In this section, we will roughly estimate the sensitivities (or g max 2 matflt A eraged (21) 2π Eν 2ρ constraints) of the oscillation experiments based on the ana- lytical results of the last section. Suppose that an oscillation when the oscillating potential is averaged to the distortion experiment can measure m2 and θ with the 1σ experimen- effect. With higher energy the oscillation experiment can tal errors of σ( m2) and σ(θ). The experiment should be have stronger sensitivities. One can observe that the sensitiv- sensitive to the ALP coupling strength whose corrections to ity (constraint) can be made by an experiment depends on: (1) m2 and θ are bigger than the experimental errors. To obtain its baseline; (2) its beam energy; (3) the resolution power of the rough sensitivity, one can demand that the ALP-induced the periodicity; (4) the statistics. We list in Table 1 for the nec- corrections should be smaller than errors of the relevant m2 essary information that will be used in Eqs. (20) and (21)of and θ. This argument should be good enough as an order-of- several representative experiments: Daya Bay [60,61], T2K magnitude estimate which is the main purpose of this section. [62], JUNO [63,64], DUNE [65] and the future atmospheric experiments [66Ð69]. For Daya Bay, the latest analysis [61] For Cases I, II and III with tflt ta, the ALP corrections to 2 2 θ = . ± . 2 = the mixing angle θ and the mass-squared difference m can has yielded sin 2 13 0 0841 0 0033 and mee (2.5 ± 0.085) × 10−3 eV2, corresponding to σ(θ ) ≈ be directly read from Eqs. (9Ð12). For Case IV with t ta, 13 flt . σ( 2 )/ 2 ≈ . the correction to the oscillation probability P is in the order 0 3% and mee mee 3 4%. The T2K experi- + . O(ξ/ ) O(ξ/ )2 ment [62] has given their latest result sin2 θ = 0.526 0 032 of ma or ma . One can use the approximation 23 −0.036 δ( ) ≈ δ(θ) ≈ δ( 2)/ 2 θ 2 = ( . ± . ) × −3 2 P m m to get the corrections to and and m32 2 463 0 065 10 eV correspond- 2 σ(θ ) ≈ . σ( 2 )/ 2 ≈ . m . However, as has been mentioned before, these correc- ing to 23 3 4% and m32 m32 2 6%. For tions can be either enhanced or suppressed by considering the JUNO experiment, after 6 years of running the 1σ relative θ . . . effect of the small mixing angle 13, depending on the flavor precisions of 0 54%, 0 24% and 0 27% can be achieved 2 θ 2 2 structure of the ALP coupling and the concerned experimen- for sin 12, m21 and mee [64], which correspond to σ(θ ) ≈ . σ( 2 )/ 2 ≈ . tal channels. Therefore, one should keep in mind that our 12 0 18% and m21 m21 0 24%. With an θ ≈ . · · general estimation has a factorial error of 13 0 1forsome exposure of 150 kt MW year for DUNE, the resolutions 2 θ 2 θ 2 . . special case. When the ALP potential is either constant or of sin 2 13,sin 23 and m31 can reach 0 007, 0 012 and . × −5 2 σ(θ ) ≈ periodically resolvable, the sensitive parameter space for all 1 6 10 eV [65], respectively, corresponding to 13 . σ(θ ) ≈ . σ( 2 )/ 2 ≈ . four cases reads 0 63%, 23 1 2% and m31 m31 0 64%. For the atmospheric neutrino experiment, we will take its base- ξ Eν σ when t t , (18) line and energy as 7×103 km and 10 GeV as an example. It is m2 min flt a able to push the 1σ errors of sin2 θ and m2 down to 0.05 ξ 23 31 σ when t t t , (19) . × −3 2 m min a flt a and 0 1 10 eV respectively [66,68], corresponding to a σ(θ ) ≈ σ( 2 )/ 2 ≈ 23 5% and m31 m31 4%. These informa- σ ≡ σ( 2)/ 2,σ(θ) with min min m m . When tflt ta,the tion that gives the strongest constraint has been summarized behavior of the ALP potential is stochastic, and the coher- in Table 1. 123 922 Page 6 of 13 Eur. Phys. J. C (2018) 78:922
Table 1 Experimental Daya Bay T2K JUNO DUNE atm. parameters . 4 Baseline (tflt)16 km 295 km 53 km 1300 km 10Ð10 km Energy (Eν )4MeV0.6 GeV 4 MeV 2.5 GeV 1Ð100 GeV σ ∼ . ∼ ∼ . ∼ . ∼ Error ( min) 0 3% 3% 0 2% 0 6% 4% θθθ θ θ θ 13 23 12 13 23 2 2 2 2 2 2 m mee m32 m21 m31 m31 /( π / 2) tflt 2 Eν m 0.81 0.99 0.80 0.87 Ð
In Fig. 1, sensitivities or constraints of different oscillation considering the 40 kton liquid argon detector. The flux corre- experiments are demonstrated as the solid curves. From left sponding to an 1.07 megawatt beam power gives 1.47×1021 = −22 = −2 (ma 10 eV) to right (ma 10 eV), the first inflec- protons on target (POT) per year due to an 80 GeV pro- tion points in these curves are due to that ta becomes smaller ton beam energy has been considered. The remaining exper- / than tflt.Afterta further decreases to tflt 1000, the coherence imental details like the signal and background normaliza- of the ALP field cannot be maintained during the neutrino tion uncertainties for both the appearance and disappearance flight. We cut the constraint curves with the vertical lines channels have been taken from DUNE CDR [70]. In addi- to represent this effect. For comparison, the existing astro- tion, we consider a 3.5 years running time for each of the physical bounds which will be discussed in Sect. 3 have also neutrino and antineutrino modes unless otherwise stated. been given. In the left panel, the sensitivities and constraints To study the impact of the ALP-neutrino coupling on are given based on Eq. (21) for the averaged case, while DUNE, we use the GLoBES extension file snu.c as has the right panel is based on Eq. (20) for the non-averaged been presented in Refs. [73,74]. In our analysis, we have case. The red, blue and green curves demonstrate the future modified the NSI matter potential in snu.c with the poten- sensitivity of DUNE, JUNO, and the atmospheric neutrino tial induced by the ALP field. We simulate the fake data of 2 experiment respectively. They can exceed the most stringent DUNE using the standard oscillation parameters sin θ12 = 2 2 astrophysical bound given by the CMB free-streaming argu- 0.307, sin θ13 = 0.022, sin θ23 = 0.535,δ = −π/ , 2 = . × −5 2 2 = . × ment. One should keep in mind that the sensitivity study here 2 m21 7 40 10 eV and m31 2 50 is based on the rough arguments, while the exact one can 10−3 eV2 which are compatible with the latest global-fit only be done with the dedicated experimental simulation. results [75Ð77]. We marginalize the standard parameters θ23, We will take DUNE as an example with detailed numer- δ and the mass hierarchy over their current 3σ ranges in ical simulations and analysis in the next section. We will the test. As the remaining standard oscillation parameters find the rough estimation in this section is consistent in the have been measured with very high precisions, we keep them order of magnitude with the result given by the detailed fixed in the analysis. On the other hand, for the ALP-neutrino simulation. 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 2.3 Impact on DUNE the sensitivity study of the diagonal ALP-neutrino coupling parameters, we marginalize over the standard parameters in In this section, we will perform a detailed study of the ALP- the fit. Moreover, for the off-diagonal parameters, we also neutrino coupling parameters considering DUNE both at the marginalize their phase in the range (0 → 2π). All along this χ 2 probability as well as the level. DUNE is a proposed work, we perform our analysis considering two different sce- long baseline superbeam neutrino oscillation experiment at narios for the ALP-induced potential, namely non-averaged Fermilab [65,70]. It will have two neutrino detectors. The and averaged cases. For the non-averaged case, we take the near detector will be placed near the source at Fermilab, while ALP-induced potential ξαβ as a constant over time for sim- the far one will be installed 1300 km away from the neutrino plicity. A more careful treatment would require the statistical source at Sanford Underground Research Facility (SURF) in analysis of the modulation on the time-binned data, which Lead, South Dakota. The far detector will utilize four 10 kton might be interesting for a future work. For the averaged case, liquid argon time projection chambers (LArTPCs). Also, it we have modified the snu.c file by averaging the oscillation will use the existing Neutrinos at the Main Injector (NuMI) probability over the time points within one ALP cycle. beamline design at Fermilab as the neutrino source. In Fig. 2, we show the oscillation probabilities of the For the numerical analysis of DUNE, we use the GLoBES appearance channels (νμ −→ νe and νμ −→ νe) for DUNE. package [71,72] along with the required auxiliary files given For illustration, we perform this study with a single non- in Ref. [70]. Throughout this work we perform our simulation 123 Eur. Phys. J. C (2018) 78:922 Page 7 of 13 922
Fig. 1 The future sensitivities or constraints of the oscillation experi- DUNE, JUNO, and the future atmospheric experiment respectively. In ments as well as the astrophysical bounds. The left panel stands for the the left panel (right panel), the light red and light blue curves stand for case where the ALP-induced field is averaged over the observation time, the constraints (sensitivities) that can be made by T2K and Daya Bay, while the right panel is for the non-averaged case. On the top axis, the respectively. The dash-dotted curves are the astrophysical constraints corresponding periods of the ALPs with masses in the bottom axis have including CMB (gray curves), IceCube-170922A (magenta curve), and been marked. The red, blue and green curves represent the sensitivity of SN1987A (purple curves)
Fig. 2 The oscillation probabilities of the appearance channels νμ → observation time, whereas the green (dashed and dotted) curves sig- ν ν → ν e (left panel) and μ e (right panel) for DUNE. The blue curves nify the non-averaged case. Also, the gray curves show the standard stand for the case where the ALP-induced field is averaged over the oscillation scenario without the ALP impact
−11 −1 zero representative value (gee = 5.6 × 10 eV )ofthe parison. For the non-averaged case, one can observe signif- ALP-neutrino coupling term. The green (dashed and dotted) icant deviations of the probabilities around the peak energy ∼ . curves show the case where the ALP-induced potential is 2 5 GeV compared to the standard√ case. When the ALP- ρ not averaged over the observation time (see the figure legend induced potential oscillates as gee 2 sin mat, the associ- for more details). Whereas, the solid blue curves describe ated probability will vary roughly in between the two green the scenario where the ALP-induced potential is averaged curves. After averaging over the time, one would obtain the over time. Besides this, the standard case without the ALP- distortion effect as the blue curve in each panel, which is induced potential is shown as the gray solid curves for com- smaller than the non-averaged case. As our intention at the
123 922 Page 8 of 13 Eur. Phys. J. C (2018) 78:922
Fig. 3 Sensitivities of ALP-neutrino coupling parameters gαβ for DUNE. The dotted blue curves stand for the case where the ALP-induced field is averaged over the observation time, whereas the solid green curves signify the non-averaged case
Table 2 The expected sensitivities of DUNE on the ALP-neutrino coupling terms gαβ at the 2σ and the 3σ C.L. Here the second (third) column represents the sensitivities for the non-averaged (averaged) case Parameters (10−11 eV−1) Non-averaged Averaged 2σ 3σ 2σ 3σ
|geμ| < 0.25 < 0.34 < 0.95 < 1.14
|geτ | < 0.58 < 1.16 < 1.46 < 1.71 |gμτ | < 0.95 < 1.23 < 1.65 < 2.65 gee (−13.62, −9.30) (−16.30, −8.71) (−8.90, 9.30) (−11.10, 11.30) ⊕(−1.11, 2.33) ⊕(−1.63, 6.15) gμμ (−1.42, 2.02) (−1.95, 2.56) (−2.34, 2.34) (−2.90, 2.88) gττ (−2.01, 0.94) (−2.60, 1.37) (−2.44, 2.50) (−3.18, 3.07) probability level is for demonstration, hence we illustrate this from the analytical results before and from the probability analysis with a single non-zero ALP-neutrino coupling term demonstration in Fig. 2. It can be ascribed as the loss of for the appearance channel. Nevertheless, we will perform information after averaging. From the first plot of the bot- a detailed sensitivity study considering both the appearance tom row, we notice that for the green curve there is a local −11 −1 and disappearance channels for all the coupling terms at the minimum around gee =−10.1 × 10 eV other than a χ 2 level. global minimum at zero. This shows a degenerate solution In Fig. 3, we show the sensitivity of DUNE to the six for gee and it arises from the unknown hierarchy (as we have ALP-neutrino coupling parameters gαβ in the χ 2Ðgαβ plane. marginalized over the mass hierarchy in the fit). In our care- The top and bottom panels represent our results for the ful analysis by fixing the hierarchy to the normal one both in non-diagonal and the diagonal parameters, respectively. The data and theory we find no degenerate solution for gee.This green solid curves represent the non-averaged case where a tells that the lack of knowledge of the hierarchy may lead constant potential has been taken during the numerical sim- to a degenerate solution. Thus, for all these different cases ulation. The blue dotted curves signify the averaged case we perform our numerical analysis over the marginalized where the ALP-induced potential is averaged over time in hierarchy. Also, comparing the top and the bottom row, we the final probability. Note that the black dotted and black notice that overall sensitivity of the off-diagonal parameters dashed horizontal lines stand for the χ 2 values correspond- is better than that of the diagonal ones. Table 2 summarizes ingtothe2σ and the 3σ C.L., respectively. In Fig. 3, one can the expected sensitivities for all the ALP-neutrino coupling clearly observe that the averaged case has looser sensitivity parameters gαβ for DUNE, where the 2σ and 3σ intervals are than the non-averaged case. This can be easily understood presented respectively. Note that the second and third panels
123 Eur. Phys. J. C (2018) 78:922 Page 9 of 13 922 of the table show the sensitivity limit for the non-averaged evade the CMB constraint. The extra Neff contributed by the and averaged cases, respectively. thermal ALPs is found to be
4/3 1 8 g∗ (Tν) N = × s (22) eff 2 7 g (T ) 3 Astrophysical bounds ∗s fo > for Tfo Tν, where Tν is the neutrino decoupling tempera- ≈ 3.1 Early universe ture around 1 MeV, and g∗s g∗ holds before Tν. To satisfy . Neff 0 52, one can find the freeze-out temperature of The cosmological evolution will be more or less modified, if thermal ALPs should be Tfo 20 MeV [87]. Demanding neutrinos have strong interactions with an ultralight degree of H at 20 MeV, we obtain the following constraint freedom. During the expansion of the Universe, the thermal ν + ν ↔ + − − 0.05 eV ALPs can be produced via the processes like a a g 10 8 eV 1 . (23) and νi ↔ ν j + a, where νi and ν j are the neutrino mass mν > eigenstates with mi m j .WeassumethemassoftheALPis ν → ν + The Neff generated by the freeze-in process i j a much smaller than that of the neutrinos such that the process − − yields a negligible bound g 10 2 eV 1, therefore not like ν + ν ↔ a is kinematically forbidden. If the ALP mass elaborated here. is larger than the neutrino masses,4 the ALP-neutrino cou- During the CMB era, the process ν + ν ↔ a + a is sup- pling must be severely suppressed to maintain the dark mat- pressed by the low plasma temperature. There are mainly two ter abundance. Suppose that the ALPs are fully thermalized processes that can reduce the neutrino free-streaming length: before the neutrino decoupling around 1 MeV, they will con- ν + ν ↔ ν + ν and ν ↔ ν + a. For these two processes, tribute to the extra effective neutrino number by an amount of i j the derivative coupling is equivalent to the pseudoscalar cou- N ≈ 0.57 [79]. The extra N is strongly constrained by eff eff pling. By requiring (ν + ν ↔ ν + ν) H at the photon the observations of BBN and CMB. The current constraint decoupling temperature Tγ ≈ 0.256 eV [34,82], one can of BBN is N 1 at 95% C.L. [80], while CMB can eff obtain the constraint give a stronger constraint with the latest Planck 2018 result . − − 0.05 eV [81]: Neff 0 52 (95% C.L., Planck TT+lowE). On the g 10 6 eV 1 . (24) other hand, the strong interaction during the CMB epoch will mν also suppress the free-streaming length of neutrinos [82,83], The decay process νi ↔ ν j + a is only relevant for the off- such that the evolution of CMB perturbations is severely dis- = diagonal coupling gij with i j in the mass basis. The decay turbed. This can lead us to a very strong constraint on the rate differs for different choices of neutrino mass patterns. If coupling strength of the secret neutrino interactions between the three masses of neutrinos are very hierarchical, the decay neutrinos and ALPs. 2 4 ∗ rate reads ≈ g m /(48πT ) with g ≡ Uα Uβ gαβ ij ij i ij ij i j First, let us focus on the possible influence on Neff .We and m being the mass of the heavier neutrino ν .Tomakesure ν + ν ↔ + i i will find the binary process a a is the dominant the free propagation of neutrinos is sufficiently disrupted, one one for generating ALPs. The reaction rate of the binary ( / )2 should include an angular factor mi 3T [82]. Requiring process can be approximated as ≈ g4m2 · T 3 with T 2 ν t ≡ (m /3T ) H at Tγ , the following bound can be 5 ij i being the temperature of the neutrino plasma. During the derived [82]: epoch of radiation domination, the Hubble expansion rate √ 3 ≈ . 2/ − − 0.05 eV is given by H 1 66 g∗T MPl, with g∗ denoting the 10 1 . gij 10 eV (25) effective number of relativistic degrees of freedom at T (refer mi to [85,86] for its values at various temperatures) and MPl However, if the neutrino masses are degenerate, one should . × 19 1 221 10 GeV being the Planck mass. Because of the ratio instead have /H ∝ T , the thermal ALPs are first copiously produced at 3/2 −3 2 high temperatures in the very beginning of the Universe. They − − 2.5 × 10 eV g 10 10 eV 1 , (26) should eventually freeze out at some low temperature T to ij 2 fo mij
4 Like in Ref. [78], the mass of the ALP dark matter can reach almost which is in the same order of magnitude as the former one O(MeV). in Eq. (25). Overall, the most stringent constraint can be 5 The derivative coupling in Eq. (2) is equivalent to the pseudoscalar placed on the off-diagonal couplings in the mass basis gij − − coupling h aναγ ν with a dimensionless coupling constant h = 10 1 αβ 5 β αβ 10 eV based on the free-streaming of CMB. The Neff U U ∗ (m + m )g , only if each neutrino line in the Feynman ij αi β j i j αβ limit of CMB can place bounds on all elements of gαβ as diagram is attached to only one NG boson line [82,84]. However, this −8 −1 is not the case for the process ν + ν ↔ a + a. The reaction rate for the g 10 eV . These early Universe bounds have been pseudoscalar coupling case is proportional to T instead of T 3. included in Fig. 1 as the dash-dotted gray curves. 123 922 Page 10 of 13 Eur. Phys. J. C (2018) 78:922
3.2 SN1987A explosion ALP-neutrino coupling. However, the original diffuse UHE neutrino flux might be well above the WB bound, since the The ALPs can also play an important role in the evolution of complete sources of the UHE neutrino still remain unknown the supernovae. The constraints on the ultralight scalar cou- for us. pling with neutrinos from the supernova neutrino observation A very simple argument would be that the mean free path SN1987A [88,89] have been extensively studied in the liter- of neutrinos λν should be smaller than the radius of the Milky ∼ ature, see Refs. [83,84] and the references therein for more Way RMW 26 kpc. However, this is not true in general details. The presence of the ALP-neutrino coupling can mod- because the energy loss for each ALP-neutrino collision is ify the standard evolution of supernova explosion by the fol- unclear, which might be negligible compared with the initial ν + lowing processes: (1) The ALPs can be thermally generated neutrino energy in some case. For the scattering process i → ν + inside the core, carrying energy away from the core-collapse a j a, the average relative energy loss and angular supernovae; (2) The ALPs can transport energy inside the change under one collision are estimated as supernova core and reduce the cooling time or the duration of the neutrino burst signal; (3) The spectrum of the observed neutrino flux might be distorted; (4) If the neutrino is the m2 Eν ≈ ma Eν + ij , Majorana particle, the ALP-neutrino coupling can cause the Eν s 2s excessive deleptonization which will disable the supernova 2 m mij explosion. However, if the coupling is so strong that the θ ≈ √a + √ , (28) ALPs is severely trapped inside the core, no constraint can 2 s 4 sEν be made on the contrary. The most stringent upper bound can be placed on the coupling g from the energy loss argument ee s ≈ m E + m2 + m2 as [84] where 2 a ν i a is the square of the center- of-mass energy. Eν/Eν and θ are defined as halves 0.05 eV −5 −1 , of their possible maximum values for the collision. One can gee 10 eV (27) mν easily find θ Eν/Eν for our case, so we focus on 2 which is much weaker than the limits of CMB. We have used the energy loss of neutrinos. Assuming 2m a Eν mν such ≈ / ≈ O( ) Eq. (27) to represent the constraining power of the SN1987A that s 2ma Eν, we can have Eν Eν 1 .Inthis explosion in Fig. 1. case, the energy loss of one single collision is significant, so λ one can adopt the argument ν RMW. However, in the case of 2m E m2 with i = j, one would have a negligible 3.3 Galactic propagation of neutrinos from SN1987A and a ν ν energy loss for each collision. It is difficult to transfer energy blazar TXS 0506+056 to a static ALP when the ALP mass is too small. Integrating over the energy loss along the neutrino course in our galaxy The propagation of astrophysical neutrino fluxes in the dark and in order not to violate the observations of SN1987A and matter halo of our galaxy will be affected when the ALP- blazar TXS 0506+056, a general requirement would be neutrino interaction is very strong [44,49,90]. These neutri- nos will suffer the energy loss and the direction change, due to the frequent scattering with the ALPs which forms the dark −1 Eν matter. The observed neutrino flux of SN1987A agrees well λν R , (29) MW E with the standard supernova neutrino theory, which should ν in turn give us a constraint on the ALP-neutrino coupling. On the other hand, the recent UHE neutrino event IceCube- − where λ = σ n 1.Hereσ ∼ g4m2 is the approximated 170922A observed by IceCube coincides with a flaring blazar ν a ν cross section for ν + a ↔ ν + a, which should be good TXS 0506+056 with ∼ 3σ level [91]. The estimated neutrino enough for an order-of-magnitude estimation. The general luminosities are similar to that of the associated γ -rays, con- bound can be formally written as sistent with the prediction of blazar models [92,93]. How- ever, if there is a large attenuation effect for the UHE neu- trino flux by scattering with ALPs, the original flux from 1/4 m Eν/ Eν TXS 0506+056 must be much stronger than the estimated g a . (30) 2ρ one, which is inconsistent with the blazar observation. For mν DM RMW the diffuse UHE neutrinos, the observed flux is around the WaxmanÐBahcall (WB) bound [94]. The large attenuation 2/( ) effect of ALP would require a diffuse flux to be much larger For an ALP mass ma mν 2Eν , one can obtain the fol- than the WB bound, which can in turn set a limit on the lowing constraint 123 Eur. Phys. J. C (2018) 78:922 Page 11 of 13 922 1/4 1 Xin Wang for insightful discussions. GYH would like to thank Qin-rui g Liu for helpful discussions. NN also thanks Dr. Sushant K. Raut and Dr. ρ R Eν DM MW Mehedi Masud for useful discussion. GYH is supported by the National / / 1 4 . · −3 1 4 Natural Science Foundation of China under Grant no. 11775232. The ≈ −9 −1 290 TeV 0 3GeV cm . 10 eV ρ research work of NN was supported in part by the National Natural Eν DM Science Foundation of China under Grant no. 11775231. (31) Open Access This article is distributed under the terms of the Creative Note that the constraint given by IceCube is almost two orders Commons Attribution 4.0 International License (http://creativecomm of magnitude stronger than that of the supernova under the ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit same argument. This is simply due to the higher energy of to the original author(s) and the source, provide a link to the Creative the IceCube event (∼ 290 TeV) compared with the supernova Commons license, and indicate if changes were made. 3 one (∼ 30 MeV). The constraint with mν ≈ 0.05 eV over the Funded by SCOAP . whole ALP mass range for IceCube can be found in Fig. 1 as the magenta curve. The corresponding supernova constraint along with the explosion one in the last section are shown as the purple curves in Fig. 1. References
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