Fuzzy Dark Matter from Infrared Confining Dynamics

0.37 MB·

by  Hooman Davoudiasl

Fuzzy Dark Matter from Infrared Confining Dynamics
Hooman Davoudiasl and Christopher W. Murphy
Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA
AverylightbosonofmassO(10−22)eVmaypotentiallybeaviabledarkmatter(DM)candidate
which can avoid phenomenological problems associated with cold DM. Such “fuzzy DM (FDM)”
may naturally be an axion with a decay constant f ∼ 1016÷1018 GeV, and a mass m ∼ µ2/f
a a a
with µ ∼ 102 eV. Here we propose a concrete model where µ arises as a dynamical scale from
infraredconfiningdynamics,analogoustoQCD.Ourmodelisanalternativetotheusualapproach
of generating µ through string theoretic instanton effects. We outline the features of this scenario
that result from various cosmological constraints. We find that those constraints are suggestive
of a period of mild of inflation, perhaps from a strong first order phase transition, that reheats
the Standard Model (SM) sector only. A typical prediction of our scenario, broadly speaking, is a
larger effective number of neutrinos compared to the SM value N ≈3, as inferred from precision
eff
7 measurements of the cosmic microwave background. Some of the new degrees of freedom may
1 be identified as “sterile neutrinos,” which may be required to explain certain neutrino oscillation
0 anomalies. Hence, aspects of our scenario could be testable in terrestrial experiments, which is a
2 novelty of our FDM model.

r
p
A The dark matter (DM) riddle – the question of what associatedwithgravityandsupersymmetrybreaking[8].
composes nearly 85% of all matter in the Universe – has Indeed, this fact can be used to entertain the intriguing
4
yet to find a resolution. What is known about this mys- possibilitythatDMismadeofultralightbosonsofmass

] terious substance, which has no viable candidate in the ∼10−22 eV,withaDeBrogliewavelengthofO(kpc)[9].
h
Standard Model (SM), is only gleaned from its gravita- Such DM could address the shortcomings of the CDM
p
tional effects on cosmology and astrophysics. Generally scenario, typically related to its implications for small
-
p speaking, DM seems to interact only feebly if at all with scale galactic structure and dynamics. This type of DM
e any form of matter, dark or otherwise. has been dubbed “fuzzy DM (FDM)” [10] and can be
h
[ Variousmodelsofnewphysicshavebeenconsideredfor naturally identified with an almost massless axion, see
DM, often also addressing other open questions of parti- for instance [11–20]. If the scale µ∼100 eV is generated
2 cle physics. Many such models lead to cold DM (CDM), from ultraviolet scales, its manifestation does not entail
v
6 a particle often characterized by mass scales m >∼ GeV. any dynamics in the low energy theory.
3 An interesting alternative is axion DM, which initially Alternatively, one may also assume that the above
1 arose in the extensions of the Peccei-Quinn (PQ) reso- scale µ is related to non-trivial dynamics at very low
1
lution of the strong charge conjugation parity symmetry energies, in analogy with QCD (as alluded to in [9]). In
0
(CP) problem [1–3] (related to why the CP violating pa- thiswork,weadoptthispossibilityandpioneerthestudy
.
1 rameter in QCD seems to be <∼10−9 [4]). of its typical phenomenology. To the best of our knowl-
0
If the decay constant for the PQ axion, f , is of order edge, this is the first time a concrete model of this type
7 a
1 1012 GeV,itcanalsobeagoodCDMcandidate,address- hasbeenproposedforFDM.Giventhehierarchyofscales
: ing two problems at once [5–7]. Yet, one is faced with ofO(106)betweenthisnewdynamicsandQCD,werefer
v
i thequestionofwhysuchananomaloussymmetryshould to it as micro-QCD (µ-QCD for short) whose confine-
X exist in Nature. Fortunately, string theory, a promising ment scale is denoted by µ. For simplicity and in order
r framework for quantum gravity, seems to offer a multi- to take advantage of the well-known QCD features, we
a
tude of axion candidates, see for example Ref. [8]. How- will choose SU(3)µ as the gauge group and further as-
ever, the decay constant of axions in these models are sume that there are “µ-quarks” that get masses from a
typicallyf ∼1016÷1018 GeV,betweenthetypicalscale low energy, O(keV), Higgs mechanism. As a toy ana-
a
of grand unified theories and the reduced Planck mass logue of the SM case, we will also assume that there are
M¯ ≈2.4×1018 GeV. newlight“colorless”fermionsthattheµ-hadronscande-
P
cay into; these “µ-leptons” may be identified as “sterile”
The above string theory predictions for f , without
a
neutrinos. We emphasize that – in comparison to the
severe tuning of initial conditions, would yield excessive
stringtheoreticgenerationofthescaleµ–ourmodelen-
amountsofDM,inseriousconflictwithcosmologicalob-
tails not only extra cosmological signatures, but also the
servations. Thissituationcanbechangedifthepotential
novel possibility of terrestrial probes at neutrino oscilla-
fortheaxionisnotassumedtobegeneratedbyQCD,and
tion experiments.
instead non-perturbative mechanisms from string theory
itself are employed. Here, various instanton effects may Thecurrentconstraintsimposedbysuccessfulbigbang
be used to generate a small scale, µ ∼ 102 eV, for the nucleosynthesis (BBN) suggest that no more than one
axionpotentialfromunderlyinghighscalesofthetheory new degree of freedom may be present aside from the
SM content at temperatures of O(MeV). Given that have masses >∼ 100 eV in our scenario, and may come
we would like to introduce new gauge dynamics with a to dominate the energy density before the usual matter-
confining scale of O(102eV), we need to suppress the ef- radiationequalityatT ∼1eV[24]. Toavoidunwanted
SM
fect of those new states on the evolution of the Universe effects we will assume that the lightest µ-hadrons can
during BBN. Similar constraints are also present from annihilate quickly into light, colorless fermions n which
precision measurements of the cosmic microwave back- get masses of <∼ 1 eV from their coupling to φ given by
ground(CMB).Theseconstraintscanbelargelyavoided y φn¯ n +H.C., whereZ (n )=+1andZ (n )=−1.
n L R 2 L 2 R
if the new degrees of freedom do not contribute more IncloseanalogywiththeQCDaxion,non-perturbative
than ∆N ∼ 0.1. This can be achieved if the tempera- effectssetupapotentialfortheFDMaxiona. Following
eff
ture of the µ-sector T(cid:48) is is sufficiently smaller than that thewell-knowncaseofQCDaxion, theenergydensityof
of the SM sector T . By “SM-sector” we mean all the the FDM axion zero mode is given by [5–7]
SM
states that are in the SM or its extensions that decouple
from the new infrared particles, which we will refer to as 1 (cid:18)g (cid:19)(cid:18)T (cid:19)3
ρ = m m A2 s,f f , (1)
the µ-sector. As we will see, O(30) new degrees of free- a 2 a,i a,f i g T
s,i i
dom need to be added, and at low energies would make
a sufficiently small contribution if T(cid:48) <∼TSM/4. where ma is the axion mass, the subscripts i and f refer
The cooler temperature of the µ-QCD sector may be to initial and final values. In the above, Ai is the initial
due to its decoupling from the SM sector at some high amplitude of oscillations, gs denotes the relativistic de-
scale, withsubsequenttransferofentropytotheSMsec- greesoffreedominequilibriumintheSMsector,andT is
toronly. Thisissimilartothewaybackgroundneutrinos its corresponding temperature: T ≡TSM. Here, Ti is the
areassumedtobecolderthanphotonsforTSM <∼1MeV, temperature at which the axion FDM starts to oscillate,
in standard cosmology. After the two sectors decouple, given by
decoupling of the SM sector states would only increase
TSM and not T(cid:48). To have T(cid:48) <∼ TSM/4, the number of ma,i ≈3H(Ti), (2)
theSMsectorstatesthatleavethethermalbathpriorto
BBN must be >∼ 10×43, where we have approximated where H(Ti) = 1.66gi1/2Ti2/MP is the Hubble scale at
the relativistic degrees of freedom during BBN as ∼10. T =Ti andMP ≈1.2×1019 GeV.Sinceweareinterested
There is another possibility that could provide a suffi- in the regime T (cid:28) 1 MeV, we will assume gs,i = gs,f =
ciently small T(cid:48), which is a period of mild inflation [21] gs. We also have Tf ≈ 2.3×10−4 eV, corresponding to
(see also Ref. [22]). Such a possibility may be realized the CMB temperature today.
through thermal effects (“thermal inflation” [23]) or else The mass ma as a function of temperature T(cid:48) in the
as a consequence of a strong first order phase transi- µ-sector can be written as [26, 27]
tion that briefly lingers in a false vacuum before tun-
nneelnitngo.faInbafarycot,gesuncehsisampercohcaenssismma,yprboeviadinnagttuhrealrecqoumispiote- ma(T(cid:48))≈ξN2fκNµ(cid:18)c2Nη(cid:19)12 (cid:18)mµψ(cid:19)N2f (cid:16)πµT(cid:48)(cid:17)η (3)
Sakharov condition of departure from equilibrium. The (cid:18)µ2(cid:19)
required number of e-foldings is >∼ 1, which will univer- × f lnNµ(πT(cid:48)/µ),
sally cool all sectors. As long as the consequent reheat- a
ing involves only the SM sector and its possible exten- where ξ ≈ 1.34, N is the number of µ-colors, which we
µ
sions, the µ-sector would remain at the cooler temper- will set at N =3 for the rest of our discussion, and N
µ f
ature. However, further decoupling, as in after µ-QCD is the number of µ-quark flavors. We also have c ≈
N
confinement, will reheat the µ-sector somewhat, which 0.26/[ξNµ−2(Nµ −1)!(Nµ −2)!], κ ≡ (11Nµ −2Nf)/6,
will be considered in our discussion later. and η ≡ κ + N /2 − 2. The finite-temperature mass,
f
We will assume an SU(3) interaction and at least m (T(cid:48)),shouldbeamonotonicallydecreasingfunctionof
µ a
oneflavorofµ-quarks,denotedbyψ,inthefundamental T(cid:48). Eq. (3) is the leading high temperature approxima-
representation. WewillfurtherassumeaZ parityunder tion to m (T(cid:48)), and as such it only gives sensible results
2 a
which ψ has a chiral representation, with Z (ψ ) = +1 for T(cid:48) (cid:29)µ/π, a point which we will return to later.
2 L
and Z (ψ ) = −1. In order to have a non-trivial axion Togofurtherwewillchooseaminimalmattercontent
2 R
potential for the FDM, we cannot have massless quarks. with N = 1. (See Ref. [28] for a review of N = 1
f f
We hence introduce a real scalar φ with Z (φ) = −1, QCD.) In this case, due to a U(1) anomaly, no chiral
2
whosevacuumexpectationvalue(vev)breakstheZ par- symmetries of the µ-QCD will survive and there are no
2
ity and endows µ-quarks with mass through the interac- light Goldstone boson counterparts to pions in the SM.
tion y φψ¯ ψ +H.C., with (cid:104)φ(cid:105)=(cid:54) 0. Byanalogywiththemassofη(cid:48) intheSM,wemayexpect
ψ L R
Assuming that the lightest mesons are lighter than that the mass of the lightest meson, denoted here by Π,
the baryons, the baryons and anti-baryons will annihi- is given by m ≈ 4µ. We will assume that there is only
Π
late into those mesons. The mesons are expected to one flavor of n, for simplicity.
2
In our scenario, µ-hadrons are assumed to annihilate
into a Π meson population, which will then in turn an- 7.5
nihilate via ΠΠ → nn¯ or else decay, Π → nn¯, promptly.
It is interesting that the parameters of the FDM axion 7.0
allowforΠ→n¯ntobeefficient,aswewillexplainbelow.
Curiously, for a FDM axion of mass ma ∼ 10−22 eV, 6.5
the period of oscillation is ∼ 1 year (see also Refs. [29– V
e
31]. Hence, if the lifetime of Π is short compared to this G 6.0
6
time scale, the Π population can decay into nn¯ while 10
1
the magnitude of the effective CP violating angle θeff ∼ /
µ fa 5.5
a/f ∼1. We will later show that this is the case in our
a
typical parameter space.
5.0
Thetotalnumberofdegreesoffreedomintheµ-sector,
giventheaboveminimalassumptions,isN(cid:48) =(4×3+4)×
4.5
7/8+16+1=31. Hence, as a reasonable value, we may
choose T(cid:48) = T/4, which would yield ∆N ≈ 0.12, at
eff 4.0
temperatures>∼µ. ThisissufficientlysmallduringBBN, -22.0 -21.8 -21.6 -21.4 -21.2 -21.0
however we should also make sure that at T ∼1 eV, we
log10[ma/eV]
do not end up with an unacceptable ∆N . Assuming
eff
thatthe31degreesoffreedomannihilateintonandn¯,we FIG. 1: FDM density in the f vs. m plane with
a a
would end up with N(cid:48) =3.5 and T(cid:48) =(31/3.5)1/3T/4≈ N =1, T(cid:48) =T/4, and m =0.5µ. The blue (bottom)
f ψ
0.5T which yields ∆Neff ≈ 0.25, which is a reasonable and red (top) bands reproduce the co√rrect value of ΩDM
value given current uncertainties [32]. at the 95% CL for A =f and f / 2, respectively,
i a a
To obtain the contribution of FDM to the cosmic en- with the dashed lines indicating the central value.
ergy budget, we need to know the T = 0 value of the
axionmass,heredenotedbym . Onecanshowthat[33]
a
(cid:104)0|ψ¯ψ|0(cid:105)
m2 ≈− , (4) trino physics. In fact, there have been a number of mea-
a f2Tr(M−1)
a surements over the years that have challenged the three
where the µ-quark condensate (cid:104)0|ψ¯ψ|0(cid:105) ≈ −µ3 and M neutrino paradigm. These include the short baseline ex-
periments LSND [34, 35] and MiniBooNE [36], which
is the diagonal µ-quark mass matrix. We will typically
reported evidence for ν¯ → ν¯ oscillations outside the
set m = 10−22 eV in the rest of our analysis, as moti- µ e
a
realmofexplanationinathreeflavorneutrinomodel. In
vated by the FDM scenario. Note that the above equa-
addition, the so-called reactor [37] and gallium [38, 39]
tion would then yield a value for µ given a choice for
anomalieshaveanaturalinterpretationintermsof“ster-
M.
ile” neutrinos. The preferred parameter space is m ∼
Cosmological observations yield Ω ≈0.258±0.008, n
DM 1 eV with a mixing angle θ ∼ 0.1. These parameters
where Ω ≡ ρ /ρ and ρ ≈ (2.6×10−3 eV)4 is the
DM a c c are used as benchmark points in searches for sterile neu-
critical energy density [32]. In the regime of interest,
trinos in other experiments including ICARUS [40] and
T (cid:28)1MeV,wehaveg ≈3.36(onlyconsideringtheSM
i IceCube [41].
sector). With N = 1, T(cid:48) = T/4, m = 0.5µ, and using
f ψ
Eqs. (1), (2), (3), and (4), we find the observed value of
A major model building challenge associated with
Ω can be reproduced. This is illustrated in Fig. 1,
DM “sterile”neutrinoswithasizablemixingangleisexplain-
which shows the FDM density in the f vs. m plane.
a a ing the precise measurement of N =3.15±0.23 by the
eff
The blue (bottom) and red (top) bands indicate the cor-
√ Planckcollaboration[32]. Themostcommonproposalto
rectvalueofΩ atthe95%CLforA =f and f / 2,
DM i a a solvethisproblemisthatthe“sterile”neutrinosactually
respectively, with the dashed lines indicating the central
have interactions of some sort [42–49]. The possibility of
value.
a late phase transition has been explored as well [50].
Note that as T(cid:48) →0, the expression in Eq. (3), which
corresponds to the leading high temperature approxima- Ourmechanismforevadingthisboundproceedsasfol-
tion, does not yield the expected zero temperature value lows. We introduce a second scalar field with an approx-
ofm . Hence,weonlyconsidersolutionstoEq.(2)where imate shift symmetry S → S +S that makes all of in-
a 0
T(cid:48) > 2.27µ/π, which corresponds to the temperature at teractions with other particles very weak, and some new
which the right-hand side of (3) is maximal. right-handed neutrinos, N .[51] Furthermore, we assign
R
The presence of new, neutral degrees of freedom with S and N odd parity under a new Z symmetry under
R 2
massesofordereVissuggestiveofaconnectionwithneu- which all the other particles are even. The relevant in-
3
teractions are satisfiedforthischoiceofparameters. Additionally,after
S gets a vev there is an O(1) correction to m as well as
L⊃−λφ(cid:0)φ2−vφ2(cid:1)2+λφ2S2−λSS4 (5) a small mixing between φ and S. However noφne of these
1 effects change our discussion in any significant way.
−gSn¯ N − SH˜L¯ N +H.C.,
L R M L R Any Πs that do not annihilate through the above pro-
cess would decay into nn¯ with a lifetime
with (cid:104)φ(cid:105) ≡ v , and where g, λ , λ, λ , and M are pa-
φ φ S
rameters to be determined. The classical stability of the (cid:32) (cid:33)2
16π v2m2 1
scalar potential requires τ(Π→nn¯)∼ φ φ , (8)
mΠ µ2mψmn sin2(θµeff)
λφλS >∼λ2. (6)
where,theeffectiveCPviolatingangleθeff ∼a/f isex-
µ a
In addition to the previously introduced φ, n , and N ,
L R pected to be ∼1 initially. Using the same parameters as
we see that S interacts with the active SM neutrinos above, we find τ(Π→nn¯)∼1018csc2(θeff) eV−1. Note
contained in L = (ν ,(cid:96) )T and the SM Higgs boson, µ
L L L that the Hubble time at T ∼ 100 eV, near the onset of
H˜ = (cid:15)ijHj†. For simplicity we take M = fa as the FDM oscillations, is ∼10 yr. Hence, for m ∼10−22 eV
a
FDMaxiondecayconstantissufficientlyhightosuppress ∼2π/ yr, the angle θeff ∼1 does not get redshifted ap-
µ
the active-sterile neutrino interactions. We assume the
preciably over many cycles. We then expect that the
renormalizableinteractionsofS withH aswellasahard
Π population will decay efficiently, with a lifetime of
mass term for the S are small enough that they can be
∼10 min, into nn¯ and turn into µ-sector radiation, long
ignored.
beforethestandarderaofmatterdominationcorrespond-
The idea is that the active neutrinos do not have ing to ∼105 yr.
masses and do not mix with the sterile neutrinos until
In this Letter we demonstrated that a “fuzzy Dark
S getsavev, andS doesnotgetavevuntiltemperature
Matter” axion can obtain its mass of O(10−22 eV) from
of the universe cools such that µ-Higgs can settle down
infraredconfiningdynamicsatascaleofO(102 eV)while
into its vev. Demanding no mixing until after the active
being consistent with various cosmological constraints.
neutrinos decouple from the electron-photon bath places
The scenario we proposed typically leads to a deviation
the bound vφ <∼ 100 keV. Once the temperature of the from the standard cosmology prediction for the effective
universe cools sufficiently, i.e. below 100 keV, φ gets a
(cid:112) number of neutrinos, and is potentially testable in neu-
vev, which then leads to a vev (cid:104)S(cid:105)≡v ∼ λ/λ v .
S S φ trino oscillation experiments that search for sterile neu-
ThevevofS,onceitturnsonafterevadingthebound
trinos.
fromN ,givesrisetothemassesoftheactiveneutrinos,
eff WethankMichaelCreutzandRobertPisarskiforuse-
and the mixing between the active and sterile neutrinos.
ful discussions. This work is supported by the United
Since (cid:104)H(cid:105)/f ∼ 10−14, to achieve the correct value for
a States Department of Energy under Grant Contract de-
the active neutrino mass, m ∼ 0.1 eV, we need v ∼
ν S sc0012704.
10TeV.Thisleadstothefollowingrelationλ ∼10−16λ,
S
inwhichcase(6)impliesλφ >∼1016λ. Thecorrectmixing
angle, θ ∼0.1, is then obtained for g ∼10−13, consistent
with the assumption of an approximate shift symmetry.
Ifmφ islightenoughthepopulationofΠcanbedepleted [1] R.D.PecceiandH.R.Quinn,Phys.Rev.Lett.38,1440
through the φ mediated process ΠΠ → nn¯. Assuming (1977).
T <∼mΠ (cid:28)mφ we find roughly [2] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
[3] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
(cid:32) (cid:33)2 [4] C. Patrignani et al. (Particle Data Group), Chin. Phys.
1 m µm m
Γ(ΠΠ→nn¯)∼T(cid:48)3σ ∼T(cid:48)3 ψ n Π , C40, 100001 (2016).
64πm2 v2m2 [5] J. Preskill, M. B. Wise, and F. Wilczek, Phys. Lett.
Π φ φ
B120, 127 (1983).
(7)
[6] L. F. Abbott and P. Sikivie, Phys. Lett. B120, 133
where (64πm2)−1 is our estimate for the phase space
Π (1983).
of the cross section. Here, we have also assumed m ∼
ψ [7] M.DineandW.Fischler,Phys.Lett.B120,137(1983).
100eV,mn ∼1eV,andvφ ∼100keV.Thisprocessstays [8] P. Svrcek and E. Witten, JHEP 06, 051 (2006),
efficient well below T ∼ 100 eV, for λφ <∼ 10−3, which arXiv:hep-th/0605206 [hep-th].
implies λ <∼ 10−19 and λS <∼ 10−35. These values are [9] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten,
also consistent with a shift symmetry for S. With these Phys.Rev.D95,043541(2017),arXiv:1610.08297[astro-
choices of parameters, mφ <∼few k√eV. These parameters [10] Wph..CHOu],.R. Barkana, and A. Gruzinov, Phys. Rev. Lett.
lead to a small mass for S, m ∼ λv ∼10−5 eV.
S φ 85, 1158 (2000), arXiv:astro-ph/0003365 [astro-ph].
Quantumconsistencyrequiresλφ,λS >∼λ2/16π2,λφ >∼ [11] W.H.Press,B.S.Ryden, andD.N.Spergel,Phys.Rev.
yψ4,n/16π2, and λS >∼g4/16π2; conditions that are easily Lett. 64, 1084 (1990).
4
[12] S.-J. Sin, Phys. Rev. D50, 3650 (1994), arXiv:hep- A13 (2016), arXiv:1502.01589 [astro-ph.CO].
ph/9205208 [hep-ph]. [33] H. Georgi, D. B. Kaplan, and L. Randall, Phys. Lett.
[13] P. J. E. Peebles, Astrophys. J. 534, L127 (2000), B169, 73 (1986).
arXiv:astro-ph/0002495 [astro-ph]. [34] C. Athanassopoulos et al. (LSND), Phys. Rev. Lett. 75,
[14] J. Goodman, New Astron. 5, 103 (2000), arXiv:astro- 2650 (1995), arXiv:nucl-ex/9504002 [nucl-ex].
ph/0003018 [astro-ph]. [35] A. Aguilar-Arevalo et al. (LSND), Phys. Rev. D64,
[15] L. Amendola and R. Barbieri, Phys. Lett. B642, 192 112007 (2001), arXiv:hep-ex/0104049 [hep-ex].
(2006), arXiv:hep-ph/0509257 [hep-ph]. [36] A. A. Aguilar-Arevalo et al. (MiniBooNE), Phys. Rev.
[16] M.Viel,G.D.Becker,J.S.Bolton, andM.G.Haehnelt, Lett. 110, 161801 (2013), arXiv:1303.2588 [hep-ex].
Phys. Rev. D88, 043502 (2013), arXiv:1306.2314 [astro- [37] G. Mention, M. Fechner, T. Lasserre, T. A. Mueller,
ph.CO]. D.Lhuillier,M.Cribier, andA.Letourneau,Phys.Rev.
[17] H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature D83, 073006 (2011), arXiv:1101.2755 [hep-ex].
Phys. 10, 496 (2014), arXiv:1406.6586 [astro-ph.GA]. [38] J. N. Abdurashitov et al., Phys. Rev. C73, 045805
[18] R. Hlozek, D. Grin, D. J. E. Marsh, and P. G. Ferreira, (2006), arXiv:nucl-ex/0512041 [nucl-ex].
Phys. Rev. D91, 103512 (2015), arXiv:1410.2896 [astro- [39] C. Giunti, M. Laveder, Y. F. Li, Q. Y. Liu, and H. W.
ph.CO]. Long, Phys. Rev. D86, 113014 (2012), arXiv:1210.5715
[19] J. E. Kim and D. J. E. Marsh, Phys. Rev. D93, 025027 [hep-ph].
(2016), arXiv:1510.01701 [hep-ph]. [40] M. Antonello et al. (ICARUS), Eur. Phys. J. C73, 2599
[20] D. J. E. Marsh, Phys. Rept. 643, 1 (2016), (2013), arXiv:1307.4699 [hep-ex].
arXiv:1510.07633 [astro-ph.CO]. [41] M. G. Aartsen et al. (IceCube), Phys. Rev. Lett. 117,
[21] H.Davoudiasl,D.Hooper, andS.D.McDermott,Phys. 071801 (2016), arXiv:1605.01990 [hep-ex].
Rev. Lett. 116, 031303 (2016), arXiv:1507.08660 [hep- [42] S. Hannestad, R. S. Hansen, and T. Tram, Phys. Rev.
ph]. Lett.112,031802(2014),arXiv:1310.5926[astro-ph.CO].
[22] E. D. Carlson, M. E. Machacek, and L. J. Hall, Astro- [43] B.DasguptaandJ.Kopp,Phys.Rev.Lett.112,031803
phys. J. 398, 43 (1992). (2014), arXiv:1310.6337 [hep-ph].
[23] D. H. Lyth and E. D. Stewart, Phys. Rev. D53, 1784 [44] T. Bringmann, J. Hasenkamp, and J. Kersten, JCAP
(1996), arXiv:hep-ph/9510204 [hep-ph]. 1407, 042 (2014), arXiv:1312.4947 [hep-ph].
[24] For example, stable hadrons associated with a new con- [45] M. Archidiacono, S. Hannestad, R. S. Hansen,
fining dynamics could themselves be dark matter can- and T. Tram, Phys. Rev. D91, 065021 (2015),
didates, which would call into question the need for an arXiv:1404.5915 [astro-ph.CO].
axion in the first place. See Ref. [25] for an example of [46] N. Saviano, O. Pisanti, G. Mangano, and A. Mirizzi,
such a scenario with Nµ =2 and Nf =1. Phys. Rev. D90, 113009 (2014), arXiv:1409.1680 [astro-
[25] A. Francis, R. J. Hudspith, R. Lewis, and S. Tulin, ph.CO].
inProceedings, 34th International Symposium on Lattice [47] A. Mirizzi, G. Mangano, O. Pisanti, and N. Saviano,
FieldTheory(Lattice2016): Southampton,UK,July24- Phys. Rev. D91, 025019 (2015), arXiv:1410.1385 [hep-
30, 2016 (2016) arXiv:1610.10068 [hep-lat]. ph].
[26] D. J. Gross, R. D. Pisarski, and L. G. Yaffe, Rev. Mod. [48] X. Chu, B. Dasgupta, and J. Kopp, JCAP 1510, 011
Phys. 53, 43 (1981). (2015), arXiv:1505.02795 [hep-ph].
[27] H.Davoudiasl,Phys.Lett.B646,172(2007),arXiv:hep- [49] J.F.Cherry,A.Friedland, andI.M.Shoemaker, (2016),
ph/0607111 [hep-ph]. arXiv:1605.06506 [hep-ph].
[28] M. Creutz, Annals Phys. 322, 1518 (2007), arXiv:hep- [50] L. Vecchi, Phys. Rev. D94, 113015 (2016),
th/0609187 [hep-th]. arXiv:1607.04161 [hep-ph].
[29] Y. V. Stadnik and V. V. Flambaum, Phys. Rev. Lett. [51] Ref. [52] considers the case where an ultra-light scalar
115, 201301 (2015), arXiv:1503.08540 [astro-ph.CO]. field, such as an FDM axion, mediates the neutrino in-
[30] Y. V. Stadnik and V. V. Flambaum, Phys. Rev. A94, teractions.
022111 (2016), arXiv:1605.04028 [physics.atom-ph]. [52] A. Berlin, Phys. Rev. Lett. 117, 231801 (2016),
[31] D. Blas, D. L. Nacir, and S. Sibiryakov, (2016), arXiv:1608.01307 [hep-ph].
arXiv:1612.06789 [hep-ph].
[32] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594,
5