MITP/17-037

Fuzzy Dark Matter and Non-Standard Neutrino Interactions

Vedran Brdar,a Joachim Kopp,b Jia Liu,c Pascal Prass,d and Xiao-Ping Wange

PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics,Johannes Gutenberg-Universitat Mainz, 55099 Mainz, Germany

(Dated: May 24, 2017)

We discuss novel ways in which neutrino oscillation experiments can probe dark matter. Inparticular, we focus on interactions between neutrinos and ultra-light (“fuzzy”) dark matter particleswith masses of order 10−22 eV. It has been shown previously that such dark matter candidates arephenomenologically successful and might help ameliorate the tension between predicted and observedsmall scale structures in the Universe. We argue that coherent forward scattering of neutrinos onfuzzy dark matter particles can significantly alter neutrino oscillation probabilities. These effectscould be observable in current and future experiments. We set new limits on fuzzy dark matterinteracting with neutrinos using T2K and solar neutrino data, and we estimate the sensitivity ofreactor neutrino experiments and of future long-baseline accelerator experiments. These results arebased on detailed simulations in GLoBES. We allow the dark matter particle to be either a scalar ora vector boson. In the latter case, we find potentially interesting connections to models addressingvarious B physics anomalies.

Our ignorance about the particle physics nature ofdark matter (DM) is so vast that viable candidate parti-cles span more than 90 orders of magnitude in mass. Atthe heavy end of the spectrum are primordial black holes[1–5]. On the low end of the DM mass spectrum are mod-els of “Fuzzy Dark Matter” with a mass of order mφ ∼10−22 eV. The term “fuzzy” refers to the huge Comptonwave length λ = 2π/mφ ' 0.4 pc×(10−22 eV/mφ) of suchDM particles. Fuzzy DM has been studied mostly in thecontext of axions or other extremely light scalar fields [6–13]. Such DM candidates can be searched for in lab-oratory experiments using cavity-based haloscopes [14–16], helioscopes [17–20], LC circuits [21], atomic clocks[22, 23], atomic spectroscopy [24] and interferometry [25],as well as accelerometers [26] and magnetometry [27–29].Constraints on their parameter space can also be set us-ing current gravitational wave detectors [30–32]. How-ever, ultra-light vector bosons are also conceivable fuzzyDM candidates [7, 33–44]. The tightest constraints onthe mass of Fuzzy DM come from observations of largescale structure in the Universe, and very recent studiessuggest that mφ > 10−21 eV may be required [10, 12, 45].

Because of its macroscopic delocalization, Fuzzy DMhas the potential to resolve several puzzles related tostructure formation in the Universe: (i) DM delocal-ization can explain the observed flattening of (dwarf)galaxy rotation curves towards their center [6], which isin tension with predictions from N -body simulations [46–48] (“cusp vs. core problem”); (ii) the lower than ex-pected abundance of dwarf galaxies [49] (“missing satel-lites problem”) can be understood in Fuzzy DM scenar-ios because of the higher probability for tidal disruptionof DM subhalos and because of the suppression of thematter power spectrum at small scales [9, 50]; (iii) theapparent failure of many of the most massive Milky Waysubhalos to host visible dwarf galaxies [51, 52] (“too big

to fail problem”) is ameliorated since Fuzzy DM predictsfewer such subhalos [9, 50]; While it is conceivable thatthese galactic anomalies will disappear with a more re-fined treatment of baryonic physics in simulations [53–55], the possibility that DM physics plays a crucial roleis far from excluded.

Our goal in the present paper is to highlight thetremendous opportunities for probing interactions offuzzy DM in current and future neutrino oscillation ex-periments. These opportunities exist in particular inscenarios in which DM–neutrino interactions are flavornon-universal or flavor violating. In this case, even veryfeeble couplings between neutrinos and dark matter aresufficient for coherent forward scattering to induce a non-negligible potential for neutrinos, which affects neutrinooscillation probabilities and will thus alter the expectedevent rates and spectra in current and future neutrinooscillation experiments [56, 57]. We will in particular de-rive constraints from T2K and solar neutrino neutrinodata, and we will determine the sensitivities of DUNEand RENO. Similar effects have been considered previ-ously in ref. [56], where the focus has been on anomaloustemporal modulation of neutrino oscillation probabilities.

Dark Matter–Neutrino Interactions. Fuzzy DM canconsist either of scalar particles φ or of vector bosons φµ.In the scalar case, the relevant terms in the Lagrangianare given by [58]

Lscalar = ναLi/∂ναL − 1

2mαβν (νcL)ανβL −

12yαβφ (νcL)ανβL ,

(1)

where α, β are flavor indices and yαβ are the couplingconstants. For vector DM, the Lagrangian is

Lvector = ναLi/∂ναL − 1

2mαβν (νcL)ανβL + gQαβφµναLγµν

βL ,(2)

with the coupling constant g and the charge matrix Qαβ .

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In both Lagrangians, mν is the effective Majorana neu-trino mass matrix. The interaction term in eq. (1) can begenerated in a gauge invariant way by coupling the scalarDM particle φ to heavy right-handed neutrinos in a see-saw scenario [58]. The interaction in eq. (2) could arisefor instance if the DM is the feebly coupled gauge bosoncorresponding to a local Lµ − Lτ lepton family numbersymmetry, defined via Qee = 0, Qµµ = 1, Qττ = −1.Alternatively, the DM particle could couple to the SMvia mixing with a much heavier gauge boson Z ′ with fla-vor non-universal couplings. If the Z ′ boson has a massof order mZ′ ∼ TeV, we expect the mixing-induced cou-pling g in eq. (2) to be of order g ∼ mφ/mZ′ . Intrigu-ingly, we will see below that such tiny couplings maybe within reach of neutrino oscillation experiments. In-teresting candidates for a TeV-scale Z ′ boson mediat-ing interactions of ultra-light vector DM and neutrinosinclude an Lµ − Lτ gauge boson, or a new gauge bo-son coupled predominantly to the second family of lep-tons. The latter possibility is of particular interest assuch a particle could explain several recent anomalies inB physics [59]. We defer a detailed discussion of possi-ble UV completions of eqs. (1) and (2) to a forthcomingpublication [60]. The mass of φµ can be generated eitherthrough the Stuckelberg mechanism [61] or from sponta-neous symmetry breaking in a dark Higgs sector.

Production of Ultra-light DM Particles. DM particleswith masses mφ � keV must have been produced non-thermally in the early Universe to avoid constraints onhot (i.e. relativistically moving) DM. The most popu-lar way to achieve this is the misalignment mechanism,which was first introduced in the context of QCD axionmodels [62–64], but can also be applied to other ultra-light fields. For vector bosons, the misalignment mecha-nism has been discussed in refs. [7, 34, 39]. In this case,the mechanism may require also a non-minimal couplingof φµ to the Ricci scalar to avoid the need for super-Planckian field excursions [7, 39]. It might be possibleto avoid these extra couplings in certain UV completionsof the model [39]. The misalignment mechanism for vec-tor bosons is more constrained if the bosons obtain theirmass through a Higgs mechanism than in models withStuckelberg masses. In particular, it is required that theboson is massive at the temperature Tosc at which φµ

begins to oscillate about its minimum [34]. This tem-perature is given by Tosc ∼

√mφMPl, where MPl is the

Planck mass. We see that the dark Higgs boson thusneeds to acquire a vacuum expectation value (vev) v at acritical temperature Tc much larger than mφ ' gv. Since

typically Tc ' v [65], this implies g .√mφ/MPl. We

will, however, see that neutrino oscillation experimentsare sufficiently sensitive to probe the relevant parameterregion. As an alternative to the misalignment mecha-nism, the authors of ref. [39] propose production of vec-tor DM from quantum fluctuations during inflation, butargue that this mechanism can only account for all the

DM in the Universe if mφ > 10−6 eV.Coherent Forward Scattering of Neutrinos on Fuzzy

DM. By inspecting eqs. (1) and (2), we observe thatscalar DM φ, treated as a classical field, alters the neu-trino mass matrix, mν → mν + yφ, while vector DMφµ alters their effective 4-momenta, pµ → pµ + gQφµ.This can be seen as dynamical Lorentz violation [66].For implementing these effects in simulation codes, weparameterize them in terms of a Mikheyev–Smirnov–Wolfenstein-like potential Veff [67–70]. To do so, we usethe equations of motion derived from eqs. (1) and (2)(treating φ and φµ as classical fields) to derive a modi-fied neutrino dispersion relation in the form

(Eν − Veff)2 = p2ν +m2

ν . (3)

Here, Eν , Veff, and mν should be understood as 3 × 3matrices. Neglecting the V 2

eff term in eq. (3), we read offthat

Veff =1

2Eν

(φ (ymν +mν y) + φ2y2

), (scalar DM)

(4)

Veff = − 1

2Eν

(2(pν · φ)gQ+ g2Q2φ2

). (vector DM)

(5)

These expressions for Veff should now be added to theHamiltonian on which the derivation of neutrino oscilla-tion probabilities is based. The classical DM field canbe expressed as φ = φ0 cos(mφt) for scalar DM and asφµ = φ0ξ

µ cos(mφt) for vector DM, where ξµ is a po-larization vector. The oscillation amplitude φ0 is re-lated to the local DM energy density ρφ ∼ 0.3 GeV/cm3

via [7, 34, 71]

φ0 =

√2ρφ

mφ. (6)

For the tiny DM masses we are interested in here, theperiod τ of field oscillations is macroscopic, τ ' 1.3 yrs×(10−22 eV/mφ) [72–75]. Note that in eqs. (4) and (5), theterms linear in the couplings constants are valid when theDM mass is so low that the DM field can be treated asclassical; the quadratic terms are approximately valid forany DM mass.

In deriving numerical results, we will for definitenessassume the neutrino–DM couplings to have a flavor struc-ture given by y = y0(mν/0.1 eV ) for scalar DM, wherey0 is a constant for scalar DM. This choice is moti-vated by the assumption of universal couplings of φto right-handed neutrinos. For vector DM, we assumeQ = diag(0, 1,−1), as motivated by Lµ − Lτ symmetry.We will moreover assume that contributions to the neu-trino oscillation probabilities proportional to powers ofcos(mφt) are averaged. In other words, we assume therunning time of the experiment to be much larger than

3

Log1�[PN�

PS�]

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

FIG. 1. Impact of neutrino–DM interactions on neutrino os-cillation probabilities in the νµ → νe, shown as a functionof baseline L and energy Eν . The color-code shows the ra-tio PNP/PSM, where PNP is the oscillation probability in thepresence of unpolarized fuzzy vector DM coupled to neutrinoswith Qee = 0, Qµµ = 1, Qττ = −1, and PSM is the standardoscillation probability.

τ . Equation (5) shows that for vector DM, Veff dependson the polarization of the field. As it is unclear whetherthe initial polarization survives structure formation or iscompletely randomized even on scales ∼ 1000 km rele-vant to long-baseline experiments, we will consider boththe case of fully polarized and fully unpolarized DM. Inthe former case, we assume the polarization axis to beparallel to the ecliptic plane for definiteness. For fullypolarized DM, the leading contribution to Veff is linear inthe small coupling g, while for unpolarized DM ξµ variesrandomly along the neutrino trajectory, so the leadingcontribution to Veff is O(g2). The same would be truefor DM polarized in a direction transverse to the neu-trino trajectory.

Modified Neutrino Oscillation Probabilities. We haveimplemented the potential from eqs. (1) and (2) inGLoBES [76–79]. To facilitate integration of the pre-dicted event rates over time, we evaluate the oscillationprobabilities at several fixed times and interpolate themusing a second order polynomial in cos(mφt). The lat-ter can then be integrated analytically. We do not in-clude long-term temporal modulation effects in our fitsbecause the available long-baseline data is presented intime-integrated form. We have checked that includingmodulation with time in the fit does not significantly im-prove our results [60].

In fig. 1 we show the impact of neutrino–DM inter-actions on the oscillation probabilities as a function ofneutrino energy Eν and baseline L. We see that even fortiny couplings, substantial modifications are possible.

Signals in Long Baseline Experiments. In fig. 2, wecollect various limits and future sensitivities on neutrino–DM interactions. For the T2K experiment, we have de-veloped a new GLoBES implementation [80, 81], whichwe use to fit data based on an integrated luminosity of6.6 × 1020 protons on target (pot) [82, 83]. We haveverified that we reproduce T2K’s standard oscillationresults to high accuracy before setting limits on DM.For the projected sensitivity of DUNE [84], we use thesimulation code released with ref. [85], corresponding to14.7× 1020 pot for neutrinos and anti-neutrinos each. Todetermine the sensitivity of RENO, we rely on a simula-tion based on refs. [86–88] and corresponding to 3 yrs ofdata taking.

We observe that experimental sensitivities are superb,thanks to the scaling of Veff with 1/mφ, see eqs. (4) to (6).For vector DM, the sensitivity is more than ten ordersof magnitude better in the polarized case (left panel offig. 2) than in the unpolarized case. In the former case,the sensitivity comes from the term linear in g, which isenhanced by Eν/(gφ) compared to the quadratic one. Ingeneral, experiments exclude values of the coupling con-stant for which Veff is much larger than the oscillation fre-quency ∼ m2

ν/(2E). For scalar or polarized vector DM,long baseline experiments have a significant edge overreactor experiments, while for unpolarized DM, RENOis able to compete even with DUNE. The reason is thescaling of Veff with 1/E according to eqs. (4) and (5).

Signals in Solar Neutrino Experiments. Since solarneutrinos evolve adiabatically as they propagate out ofthe Sun, their survival probability in the electron flavoris given by

Pee(Eν) =∑i

|U�ei |2 |U⊕ei |

2, (7)

where U� and U⊕ are the effective leptonic mixing ma-trices at the center of the Sun and at Earth, respectively.U� is strongly affected by both SM matter effects andDM–neutrino interactions, while U⊕ differs from the vac-uum mixing matrix mainly through the DM term in ourscenario. We neglect Earth matter effects as their impacton our results would be negligible [89].

We fit solar neutrino data from Borexino [90], Super-Kamiokande [91], and SNO [92, 93], as collected inref. [89]. We illustrate in fig. 3 how the presence ofDM–neutrino interactions could improve the fit to so-lar neutrino data. For standard oscillations, we findχ2/dof ' 22/20, while the best fit point for polarizedvector DM yields χ2/dof ' 12/19. Even though we findin both cases an acceptable goodness of fit, standard os-cillations are disfavored compared to the new physics hy-pothesis. This is a reflection of the fact that the upturnof the survival probability at low energy has not beenobserved yet [89]. As the preference for new physics inour fit is somewhat stronger than in a fit including full

4

10-1 100 101 1020

5

10

15

20

25

30

y0/mϕ[ eV-1]

Δχ2

1σ

2σ

3σ

Scalar

CMB

T2KExclusion

DUNESensitivity

SOLARExclusion

RENOSensitivity

10-11 10-10 10-9 10-8 10-7 10-60

5

10

15

20

25

30

g/mϕ[ eV-1]

Δχ2

1σ

2σ

3σ

Polarized Vector

RENOSensitivity

T2KExclusion

DUNESensitivity

SOLARExclusion

SOLAR

(inflatederrors)

10-1 100 101 1020

5

10

15

20

25

30

g/mϕ[ eV-1]

Δχ2

1σ

2σ

3σ

Unpolarized Vector

T2KExclusion

DUNESensitivity

SOLARExclusion

RENOSensitivity

(a) (b) (c)

FIG. 2. ∆χ2 curves from existing (thick solid curves) and future (thin dotted curves) analyses of neutrino oscillation data.Shaded parameter regions are excluded by the current data. Panel (a) applies to scalar DM, panel (b) is for vector DM withfixed polarization parallel to the ecliptic plane, and panel (c) is for unpolarized vector DM. For scalar DM, we have assumedy = y0(mν/0.1 eV ), while for vector DM, we use a coupling structure inspired by Lµ−Lτ symmetry, namely Qee = 0, Qµµ = 1,Qττ = −1. In panel (a), we show also a limit based on the cosmological constraint on

∑mν .

0.002 0.004 0.006 0.008 0.010 0.012 0.0140.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Eν[GeV]

surv

iva

lp

rob

ab

ilit

yP

ee

BorexinoSuper-KSNO

Without DMBFP polarized

sin2θ12=0.31, Δm122 =3.5⨯10-5eV2

sin2θ12=0.30, Δm122 =3.5⨯10-5eV2

g/mϕ=5⨯10-9eV-1

FIG. 3. Predicted νe survival probability for solar neutrinosat the SM best fit point (blue) and at the best fit point in-cluding neutrino interactions with polarized vector DM (red).The best fit curves for unpolarized vector DM and for scalarDM are similar to the SM curve. We compared to data fromBorexino [90], Super-Kamiokande [91], and SNO [92, 93].

spectral data [94], we show in fig. 2 also conservative con-straints obtained by artificially inflating the error bars ofall solar data points by a factor of two.

Comparing limits from solar neutrino observations tothose from long-baseline experiments, we see from fig. 2that for unpolarized vector DM, solar neutrinos offer themost powerful constraints. This is once again due to the1/Eν dependence of Veff in this case. Even though thesame scaling applies to scalar DM, solar limits are much

weaker because in our benchmark scenario, neutrino–DMinteractions alter only neutrino masses (to which solarneutrinos have poor sensitivity), but not the mixing an-gles. This is also the reason why the limits from ref. [56],which rely on variations in the mixing angle θ12, are notapplicable here.

Cosmological Constraints on∑mν . As pointed out in

ref. [58], interactions between neutrinos and ultra-lightscalar DM are constrained by the requirement that theDM-induced contribution to the neutrino mass term doesnot violate the cosmological limit on the sum of neutrinomasses,

∑mν . We estimate this constraint in fig. 2 (a)

by requiring that, at recombination (redshift z = 1 100),the correction to the heaviest neutrino mass (taken at0.05 eV) should not be larger than 0.1 eV.

Astrophysical Neutrinos. One may wonder whetherneutrino–DM interactions could inhibit the propagationof astrophysical neutrinos [95] from distant sources [96].The optical depth for such neutrinos is given by [97]τν(Eν) = σνφ(Eν)Xφm

−1φ , with the DM column den-

sity Xφ ≡∫

l.o.sdl ρφ, where the integral runs along the

line of sight. For both galactic and extragalactic neutrinosources, we have typically Xφ ∼ 1022–1023 GeV/cm2 [97].The scattering cross section for vector DM is approxi-mately

σTνφ 'g4

8π

m2ν

E2νm

2φ

, (vector DM) (8)

where the superscript T indicates that, for simplicity, wehave only considered the transverse polarization states ofDM. For scalar DM, the corresponding expression is

σνφ 'g4

36πm2ν

, (scalar DM) (9)

5

Requiring τν < 1, we obtain the constraints

g

mφ< 3 · 108 eV−1

(Eν

PeV

) 12(

0.1 eV

mν

) 12(

10−22 eV

mφ

) 14

,

(vector DM) (10)

y

mφ< 1.3 · 1011 eV−1

(mν

0.1 eV

) 12(

10−22 eV

mφ

) 34

.

(scalar DM) (11)

We see that these limits are much weaker than the con-straints imposed by oscillation experiments (see fig. 2)except for DM masses much larger than the ones con-sidered here and for very low neutrino energies At lowenergy, however, astrophysical neutrinos cannot be ob-served because of prohibitively large atmospheric back-grounds.

Summary. To conclude, we have demonstrated thatunique opportunities exist at current and future neu-trino oscillation experiments to probe interactions be-tween neutrinos and ultra-light DM particles. The lat-ter are an interesting alternative to WIMP (Weakly In-teracting Massive Particle) DM, avoiding many of thephenomenological challenges faced by WIMPs. A par-ticularly interesting possibility, which we plan to explorefurther in an upcoming publication [60], is a possible con-nection to flavor non-universal new physics at the TeVscale, as motivated by recent anomalies in quark flavorphysics.

Note added. While we were finalizing this paper,ref. [58] appeared on the arXiv, addressing similar ques-tions. While the main focus of ref. [58] (and also of theearlier ref. [56]) is on scalar DM, we consider also DMin the form of ultra-light gauge bosons. The authors ofref. [58] have considered a larger range of experimentsfor setting limits than us, while our results are based onmore detailed numerical simulations of the few most rel-evant experiments. Where our results are comparable tothose of ref. [58], they are in good agreement.

Acknowledgments. We would like to thank PedroMachado, Georg Raffelt, and Felix Yu for very help-ful discussions. This work has been funded by theGerman Research Foundation (DFG) under Grant Nos.EXC-1098, KO 4820/1–1, FOR 2239, GRK 1581, and bythe European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme (grant agreement No. 637506, “νDirections”).

a vbrdar@uni-mainz.deb jkopp@uni-mainz.dec liuj@uni-mainz.ded pprass@students.uni-mainz.de

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