#### Light scalar dark matter at neutrino oscillation experiments

HJE
Light scalar dark matter at neutrino oscillation
Jiajun Liao 0 1 3
Danny Marfatia 0 1 3
Kerry Whisnant 0 1 2
0 2323 Osborn Drive , Ames, IA, 50011 U.S.A
1 2505 Correa Rd. , Honolulu, HI, 96822 U.S.A
2 Department of Physics and Astronomy, Iowa State University , USA
3 Department of Physics and Astronomy, University of Hawaii-Manoa , USA
Couplings between light scalar dark matter (DM) and neutrinos induce a perturbation to the neutrino mass matrix. If the DM oscillation period is smaller than ten minutes (or equivalently, the DM particle is heavier than 0:69 10 17 eV), the fast-averaging over an oscillation cycle leads to a modi cation of the measured oscillation parameters. We present a speci c symmetric model in which the measured value of 13 is entirely generated by the DM interaction, and which reproduces the other measured oscillation parameters. For a scalar DM particle lighter than 10 15 eV, adiabatic solar neutrino propagation is maintained. A suppression of the sensitivity to CP violation at long baseline neutrino experiments is predicted in this model. We nd that DUNE cannot exclude the
Neutrino Physics; Beyond Standard Model; Solar and Atmospheric Neutrinos
DM scenario at more than 3
C.L. for bimaximal, tribimaximal and hexagonal mixing,
while JUNO can rule it out at more than 6 C.L. by precisely measuring both 12 and 13.
1 Introduction The model 2 3
E ects on neutrino oscillation parameters
Tests of the model in future neutrino experiments
Summary
A Second-order corrections
B
A general treatment of oscillation probabilities
10 22 eV has attracted much attention recently since it can
resolve the small scale crisis for standard cold DM due to its large de Broglie wavelength;
see ref. [3] and references therein. Constraints on fuzzy DM can be obtained from
Lymanforest data and a lower limit of 20
10 22 eV at 2
C.L. has been set from a combination of
XQ-100 and HIRES/MIKE data [4], although a proper handling of the e ect of quantum
pressure and systematic uncertainties may relax the limit [5]. Nevertheless, light scalar DM
candidate of mass below a few keV are generally expected in many extensions of the
Standard Model (SM). Examples include a QCD axion [6{8], moduli [9{12], dilatons [13, 14],
and Higgs portal DM [15]. Constraints on hot DM require that light scalar DM cannot be
produced thermally in the early universe. A popular production mechanism for generating
light scalar DM is the misalignment mechanism, in which the elds take on some initial
nonzero value in the early universe, and as the Hubble expansion rate becomes comparable
to the light scalar mass, the DM
eld starts to oscillate as a coherent state with a single
macroscopic wavefunction [16].
The properties of light scalar DM can be probed if they are coupled to SM fermions,
which induce a time variation to the masses of the SM fermions due to the oscillation of
{ 1 {
the DM
eld. Here we consider the couplings between the light scalar DM and the SM
neutrinos, which were rst studied in ref. [17] by using the nonobservation of periodicities
in solar neutrino data. Constraints on light scalar DM couplings were also considered in
refs. [18, 19] by using the data from various atmospheric, reactor and accelerator neutrino
experiments. In general, the interactions between DM and neutrinos provide a small
perturbation to the neutrino mass matrix; a generic treatment of small perturbations on the
neutrino mass matrix is provided in refs. [20, 21]. If the DM oscillation period is much
smaller than the periodicity to which an experiment is sensitive, the oscillation
probabilities get averaged, and a modi cation of the oscillation parameters can be induced if the
data are interpreted in the standard three-neutrino framework.
Evidence of time varying signals has been searched for in many neutrino oscillation
experiments. Super-Kamiokande
nds no evidence for a seasonal variation in the
atmospheric neutrino
ux [22], and the annual modulation of the atmospheric neutrino
ux
observed at IceCube is correlated with the upper atmospheric temperature [23].
Monthlybinned data from KamLAND indicate time variations in reactor powers [24]. Tests of
Lorentz symmetry via searches for sidereal variation in LSND [25], MINOS [26{28],
IceCube [29], MiniBooNE [30], Double Chooz [31], and T2K [32] data, are negative. Also,
Super-Kamiokande [
34
] and SNO [35, 36] nd no signi cant temporal variation in the solar
neutrino
ux with periods ranging from ten minutes to ten years. We therefore take the
DM oscillation period to be smaller than ten minutes.
In this work, we study the modi cation of neutrino oscillation parameters due to
light scalar DM-neutrino interactions. Since the predictions are avor structure-dependent,
we present a speci c
symmetric model in which the symmetry is broken by light
scalar DM interactions thus generating a nonzero mixing angle
13.
We rst examine
the e ects of this model on data from various neutrino experiments. We then study the
potential to distinguish this model from the standard three-neutrino oscillation scenario at
the future long-baseline accelerator experiment, DUNE, and the medium-baseline reactor
experiment, JUNO.
The paper is organized as follows. In section 2, we present a model in which
symmetry is broken by the DM interactions. In section 3, we examine the implications
of this model for the measured neutrino oscillation parameters. In section 4, we simulate
future neutrino oscillation experiments to study the potential to distinguish this model
from the standard scenario. We summarize our results in section 5. In appendix A we
calculate how the scalar DM interactions with neutrinos a ect neutrino mass and mixing
parameters, and in appendix B we determine how the DM oscillations cause a shift in the
e ective neutrino oscillation parameters measured in experiments.
2
The model
The Lagrangian describing the interactions between light scalar DM and neutrinos can be
written in the avor basis as [18, 19]
1
2
m0
c
L
L
1
2
c
L
L + h.c. ;
(2.1)
{ 2 {
= e; ; , m0 is the initial neutrino mass matrix, and
is the coupling constant
matrix. Since the light scalar DM can be treated as a classical eld, the nonrelativistic
solution to the classical equation of motion can be approximated as [17]
(x) '
p2 (x)
m
cos(m t
~v ~x) ;
(2.2)
(2.3)
(2.4)
(2.6)
(2.7)
where U0 is the initial mixing matrix, mi0's are the initial neutrino eigenmasses, and the
elements of the perturbation matrix are
E
=
p2
m
= 0:0021 eV
We consider a model in which the initial mixing angle 103 = 0 and the measured
13 value is generated by the DM interactions. In order to simplify our calculations, we
specialize to models in which the DM interactions only a ect the masses at higher orders
in the perturbation, leaving them e ectively unchanged. From the generalized
perturbation results of ref. [21], we nd that the most general perturbation satisfying the latter
requirement is of the form,
0
0
E = BBp
2 s023 0 sin 2 203
2 c023 0 cos 2 203
1
p
0 c0osisn22 202033CAC :
and v
in
where m is the mass of the scalar DM particle,
10 3 is the virialized DM velocity. Since v
0:3 GeV=cm3 is the local DM density,
1, we neglect the spatial variation for neutrino oscillation experiments. In the presence of scalar DM interactions, the e ective Hamiltonian for neutrino oscillations can be written as
where Ne is the number density of electrons. The e ective mass matrix can be treated as
the sum of an initial mass matrix and a small perturbation [21], i.e.,
H =
1
2E
M yM + p
2GF Ne B@0 0 0C ;
A
0
M = U0 B@ 0 m02 0 CA U0y + E cos(m t) ;
As a further simpli cation, we assume the model is the perturbation becomes
0
0
E = B
0
{ 3 {
symmetric, i.e., 203 = =4. Then
13
23
12
CP
p
mj ; since the eigenmasses are not shifted at leading
order, we drop the superscript `0' hereafter. In appendix A, we show that the leading order
corrections have amplitudes
(2.8)
(2.9)
(2.10)
(2.11)
(3.1)
(3.2)
(3.3)
(3.4)
Note that
12 is second-order in the 's and is therefore proportional to cos2(m t), while
CP depends only on the phase of
and is constant, i.e., it is not a ected by the DM
oscillation. Both
23 are dependent linearly on cos(m t).
3
E ects on neutrino oscillation parameters
In this section, we study how the neutrino oscillation parameters are modi ed in our
model, assuming the period of the DM oscillation (
= 2 =m ) is short compared to the
experimental resolution of periodicity. Here we use the superscript `0' to denote the initial
oscillation parameters, and the superscript `e ' to denote the e ective parameters measured
at neutrino oscillation experiments if the data are interpreted in the standard three-neutrino
framework. For the parameters obtained after incorporating the DM perturbation, no
superscript is used.
3.1
Short-baseline reactor experiments
The leading oscillation probability for reactor antineutrinos (at a Daya Bay-like distance)
is P = 1
13 and averaging over a DM oscillation cycle, we get
as found previously in ref. [18]. If 103 = 0, then sin2 2 1e3 = 2( 13)2, so the angle being
measured in these experiments is
e
13 '
p
13= 2 :
3.2
Long-baseline appearance experiments
For long-baseline experiments, the formulas are more complicated. From ref. [37],
P (
P (
! e) = x2f 2 + 2xyf g cos( 31 + CP) + y2g2 ;
! e) = x2f 2 + 2xyf g cos( 31
CP) + y2g2 ;
{ 4 {
where x = sin 23 sin 2 13, y =
A^), g = sin(A^ 31)=A^, A^ = jA= m231j, and A
before averaging,
= j m221= m231j, f; f = sin[(1
^
where C = cos(m t). After averaging, the leading term for x2f 2 is
which is similar to the reactor case, i.e., the e ective 13 is
13=p2.
We can write yg as
x f
cos 203 sin 2 102. After explicitly putting in the perturbation, yg becomes
Combining eqs. (3.5) and (3.7) and after averaging, the xyf g term is
yg
y0g 1 + 2C2
12 cot 2 102
is suppressed compared to the usual case since it is proportional to two factors of the 's
(assuming
0), instead of just one | the term proportional to one factor of
was linear
in C and averaged to zero. For
symmetry, 203 =
=4 and the term vanishes completely.
The upshot is that the e ect of the Dirac CP phase on P (
!
e) and P (
!
e) is
suppressed in long-baseline neutrino oscillation appearance experiments. Also, as shown
in appendix B, this model predicts a suppression of the sensitivity to CP violation in all
types of neutrino oscillation experiments.
3.3
Medium-baseline reactor experiments
For KamLAND and JUNO, the oscillation probability is
P ( e ! e) = 1
For 103 = 0, the angular factors after averaging over the DM oscillations are
Equations (3.11) and (3.12) are identical to the standard case with sin2 2 13 replaced by
2( 13)2, the same as for short-baseline reactor and long-baseline accelerator experiments.
In eq. (3.13), the sin2 2 102-dependent term on the right-hand side has a coe cient,
1
shift in the measured value of 12. To determine how the shift depends on
sin2 2 102 + 2 sin 2 102 cos 2 102 12 = sin2 2 1e2
12=2, and what one measures in this type of experiment is
be much larger than the time in which neutrinos travel
through the Sun, which is about 2:3 seconds. This requirement restricts the mass of the
scalar eld: m
1:8
10 15 eV.
The three-neutrino survival probability for adiabatic propagation is 2
P ( e ! e) = cos2 13 cos2 13 cos2
m cos2 12 + sin2 m0 sin2 12 + sin2 13 sin2 13
=
cos2 13 cos2 13 1 + cos 2 m cos 2 12 + sin2 13 sin2 13 ;
where
cos 2 m =
q
cos 2 12
^
A0
(cos 2 12
A^0)2 + sin2 2 12
;
p
with A^0 = 2 2GF Ne0E= m221, and Ne0 is the electron number density at the point in
the Sun where the neutrino was created. Here ij =
i0j +
ij cos(m t) and ij =
i0j +
ij cos(m t + m t0) are the mixing angles at the production point in the Sun and at the
Earth, respectively. They di er by a phase factor m t0, where t0 is the time traveled by
neutrinos from the production point to the Earth.
Since 103 = 0, expanding to the leading term, we have
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
1
2
1
2
1
2
F
2
1
2
123P0[cos2(m t) + cos2(m t + m t0)] :
P
P0 +
12[cos 2 102F cos2(m t)
2 sin 2 102 cos 2 m0 cos2(m t + m t0)] (3.23)
and we see that
reactor experiments.
3.5
Atmospheric neutrinos
The survival probability of atmospheric neutrinos is
:
(3.21)
(3.20)
(3.22)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
Because there is no interference term between cos2(m t) and cos2(m t + m t0), we can
average over them separately. Hence,
hP i
P0 +
12
By a similar calculation, the e ective shifts in 1e2 and 1e3 lead to
P
P0 +
1e2 (F cos 2 102
2 sin 2 102 cos 2 m0)
2( 1e3 )2P0 ;
13=p2, the same as for medium-baseline
and the corresponding barred quantities can be obtained by replacing the phase m t with
m t + m t0. Also, to the leading order, we have
where cos 2 m0 has the same form as eq. (3.18), and
F =
q
If P0 is the probability without the perturbation, i.e.,
then keeping the leading correction for each
, we have
For 103 = 0, after averaging,
P (
!
) = 1
(cos4 13 sin2 2 23 + sin2 23 sin2 2 13) sin2
Since the ( 23)2 term is doubly suppressed, we have 1e3
p
13= 2 and
e
23
This also applies to the long-baseline
survival probability. Also, as shown in appendix B,
matter e ects do not change these results.
e
23
From the analytic analysis of the last section, we see that the constraints on this model
mainly come from the measurement of 1e3 and 1e2 . From eqs. (2.8) and (3.2), we have
e
and from eqs. (2.10) and (3.16), we have
for the normal hierarchy, and
for the inverted hierarchy. Since the correction to 23 is doubly suppressed in the oscillation
probabilities in this model, 2e3 remains maximal.
We rst study the sensitivity of long-baseline accelerator experiments to this model.
Since the currently running experiments, T2K and NO A, have large experimental
uncertainties, we consider the next-generation DUNE program. In our simulation, we use
the GLoBES software [38, 39] with the same experimental con gurations as in ref. [40].
For the oscillation probabilities in the DM scenario, we modify the probability engine in
the GLoBES software by averaging the probabilities over a DM oscillation cycle
numerically. We also use the Preliminary Reference Earth Model density pro le [41] with a 5%
uncertainty for the matter density.
23 = 4
To obtain the sensitivities to the DM parameters at future long-baseline neutrino
experiments, we simulate the data with the SM scenario in the normal hierarchy. Since
the sensitivity to the Dirac CP phase is suppressed at such experiments, we conservatively
choose CP = 0. Also, due to the double suppression of the correction to 23, we choose
, which is within the 1 range of the global t [42]. We also adopt the other mixing
angles and mass-squared di erences from the best- t values in the global t, which are
203 = 4 and 103 = 0. For 102, we consider three benchmark values that are inspired by
{ 8 {
(4.1)
(4.2)
(4.3)
(4.4)
TBM
HG
χ
6
4
2
0
correspond to 102 = 45 , 35:3 , and 30 , respectively.
underlying discrete symmetries. Namely, 102 = 45 for bimaximal (BM) mixing [43{45],
0 0
12 = 35:3 for tri-bimaximal (TBM) mixing [46{48], and 12 = 30 for hexagonal (HG)
mixing [
49, 50
]. Since the masses are not a ected at the leading order, we adopt the central
values and uncertainties for the mass-squared di erences from the global t, i.e.,
m221 = (7:40
0:21)
to account for constraints from the current global t, i.e., sin2 e
Also, since the long baseline experiments are not sensitive to 12, we impose a prior on 1e2
12 = 0:307
0:013. We use
eq. (4.2) to calculate the predicted value of 1e2 . Then for a given 102 and the lightest mass
m1 (m3) for the normal (inverted) hierarchy, we scan over the magnitudes and phases of
and 0. We nd that the phases of
and 0 only have a small e ect on the
2 value,
which agrees with the analytical expectation that the measurement of the CP violation
is suppressed. We also marginalize over both the normal and inverted hierarchy for the
tested DM scenario. We nd that the
2 value for the inverted mass hierarchy is always
larger than that for the normal hierarchy for the same lightest mass. This is because the
masses are not a ected at the leading order and the mass hierarchy can be resolved with
high con dence at DUNE [51].
The minimum value of
2 as a function of m1 is shown in
gure 1 for the three
benchmark values of 102. As an illustrative example, we show the oscillation probabilities
for 102 = 35:3 and m1 = 0:1 eV in the neutrino and antineutrino appearance channels in
gure 2. We see that the DM oscillation curves overlap the SM curves su ciently in both
modes that a clear discrimination is not possible. From
gure 1 we see that DUNE alone
cannot distinguish the DM scenario from the SM scenario at more than the 3
C.L. if m1
{ 9 {
HJEP04(218)36
are:
0
12 = 35:3 , m1 = 0:1 eV,
for the other parameter values.
= 0:0024
ei0:7502 eV and 0 = 0:00041
ei1:278 eV. See the text
is greater than about 0.05 eV. We also see that as m1 decreases, 2
min increases. This can
be understood from eqs. (4.1) and (4.3). For a smaller m1, the magnitude of
required
to explain the measured 13 becomes larger, and higher order corrections then break the
degeneracies between the SM and DM scenarios.
Since future medium-baseline reactor experiments can make a high precision
measurement of both 12 and 13, we study the sensitivity reach at JUNO. We use the GLoBES
software to simulate the JUNO experiment. The experimental con guration is the same
as that in ref. [52], which reproduces the results of ref. [53]. We use the same procedure
for the long-baseline accelerator experiments except with no prior on 1e2 , since JUNO can
measure 12 more precisely than the current experiments. For the lightest mass between
0 and 0.2 eV, we nd that the minimum value of 2 at JUNO is 47.6, 46.9 and 57.0, with
the initial mixing being BM, TBM and HG, respectively. Hence, JUNO can rule out this
model with the three initial mixings at more than 6
C.L.
5
Summary
We studied the e ects of light scalar DM-neutrino interactions at various neutrino
oscillation experiments. For a light scalar DM
eld oscillating as a coherent state, the coupling
between DM and neutrinos induces a small perturbation to the neutrino mass matrix. We
consider the case in which the DM oscillation period is smaller than the experimental
resolution of periodicity, i.e., ten minutes. After averaging the oscillation probabilities over a
DM oscillation cycle, the perturbation to the neutrino mass matrix leads to a modi cation
of the e ective neutrino oscillation parameters if the experimental data are interpreted in
the standard three-neutrino oscillation framework.
Since the results depend on the
avor structure of the initial mass matrix and the
perturbation matrix, we presented a speci c
symmetric model with DM interactions
that do not a ect the eigenmasses at the leading order. We examined the e ects of this
model on the e ective oscillation parameters measured at various neutrino experiments. If
the mass of the scalar eld is lighter than 1:8
10 15 eV, then solar neutrinos propagate
adiabatically. We nd that all existing neutrino oscillation results can be explained in this
model with a shift of the e ective mixing angles | the measured value of 13 arises wholly
from DM-neutrino interactions. The model also predicts a suppression of the CP violation
at neutrino oscillation experiments.
We then studied the potential of DUNE and JUNO to discriminate between this model
and the standard three-neutrino oscillation scenario. We
nd that DUNE cannot make a
distinction at more than 3
C.L. for bimaximal, tribimaximal and hexagonal mixing, while
JUNO can rule out the DM scenario at more than 6
C.L. by making high-precision
measurements of both 12 and 13.
Acknowledgments
This research was supported in part by the U.S. DOE under Grant No. DE-SC0010504.
A
Second-order corrections
For 203 = 45 and 103 = 0, the mass matrix can be rewritten as
HJEP04(218)36
M = m1I + R203 BB m21c102s12
0
We diagonalize the above mass matrix by the unitary matrix,
m21c102s102
m21(c102)2
0
p
2
m31
1
A
0 CC (R203)T :
where Ri0j is the rotation matrix in the i
j plane with a rotation angle i0j , U is
and R102 is
From eq. (A.1), the leading order corrections in the 1{3 and 2{3 sector are
We see that after the rotations of R203, U and R102, the mass matrix in the 1{2 sector is
0
0
0 m21
!
1
m31
2 2(c012)2 + 02(s012)2
p
2 0 sin(2 102)
p
2 0 cos(2 102) + 2 22 02 sin(2 102) !
p
2 0 cos(2 102) + 2 22 02 sin(2 102) 2 2(s012)2 + 02(c012)2
p
2 0 sin(2 102)
U = R203U R102R102 ;
0
U = B 0
1
13
23CA ;
0
1
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
:
(A.6)
Hence, the next-to-leading order correction in the 1{2 sector is
p
2
Since the DM perturbation potentially introduces additional complex phases in
and
12, we must recast the parameters to put them in the standard form. We combine
the initial rotation with the in nitesimal one to get (e.g., in the 2{3 sector),
23
!
=
To rst order, the magnitude of the 1-1 element is
Likewise for the o -diagonal element,
Therefore the rotation in the 2{3 sector is now
arg((R23)12)
023 ' cot 203 Im 23 :
c23ei 23
where 23 includes a shift in Re( 23). Since j 23j
similar manipulation can be done for R12, with j 12j; j 012j
23
1, 23 and 023 are small. A
1. In the 1{3 sector, we get
R13 =
1
where 13 = arg( 13). Note that the cosine terms in R13 do not get a phase at rst order
because their shifts are at second order (due to the fact that 103 = 0). Hence, the leading
corrections to the three mixing angles are
12
(A.7)
23
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
Combining these 2-D rotations together in the full 3-D rotation matrix and making
some phase changes in rows and columns so that the 1{1, 1{2, 2{3, and 3{3 elements are
real we get
0
c13c12
c13s12
s12s23e
i
012. This is not quite in the standard form, but we can multiply
the second and third rows by e i and the third column by ei to get the standard form
for U with CP =
( 13 + ). Since the phases in
are all small, CP is primarily given by
A general treatment of oscillation probabilities
A general way to look at the oscillation probabilities is to do a Taylor series expansion
about the standard expression:
P
P0 +
+
+
1
2
13C +
123C2 +
12 23C3 +
23C
223C2
Using hCi = 0, hC2i = 1=2, hC3i = 0, and hC4i = 3=8, where h i indicates averaging over
the DM oscillation, and after averaging,
On the other hand, the expansion in terms of e ective parameter shifts is
P
223 +
2
12
1
2
1e2 +
( 1e2 )2 +
12 1e3 +
e
each
)
hP i
12 +
1
4
123 +
1
4
223 ;
(B.1)
(B.2)
(B.3)
(B.4)
and eq. (B.3) reduces to (again keeping only the leading correction for each
)
P
1e2 +
(neglecting the small, second-order correction to 23, which is acceptable since the
leading order terms involving 23 are generally not zero). Note that since the period of DM
oscillation considered here is much larger than the neutrino travel time at a terrestrial
experiment, the expansions in eqs. (B.2) and (B.3) are not a ected by matter e ects and
the shifts in the e ective angles remain the same.
In the more general case with (@P=@ 13)0 6= 0 (such as when there is a single factor of
s13 or sin 2 13), there is no single power of
13 in eq. (B.2) that matches the single power
of
1e3 in eq. (B.3), and the simple correspondence between
The only measurement that appears to have this problem is the appearance measurement
at long-baseline experiments. Also, since the Dirac CP phase is always associated with s13
in an oscillation probability, the absence of a single power of
13 in eq. (B.2) indicates a
reduced sensitivity to the Dirac CP phase in neutrino oscillation experiments.
Open Access.
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