Neutrino interactions in liquid scintillators including active-sterile neutrino mixing
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Neutrino interactions in liquid scintillators including
active-sterile neutrino mixing
arXiv:2109.06244v1 [hep-ph] 13 Sep 2021
M. M. Saez1 , M. E. Mosquera 1 2
and O. Civitarese2 3
1
Facultad de Ciencias Astronómicas y Geofı́ sicas, University of La Plata. Paseo del
Bosque S/N
1900, La Plata, Argentina.
2
Dept. of Physics, University of La Plata, c.c. 67
1900, La Plata, Argentina
3
Dept. of Physics, University of La Plata, c.c. 67
1900, La Plata, Argentina
E-mail: msaez@fcaglp.unlp.edu.ar
Abstract. Neutrinos play an important role in core-collapse supernova events since
they are a key piece to understand the explosion mechanisms. The analysis of the
neutrino fluxes can bring answers to neutrino’s related problems e.g.: mass hierarchy,
spectral splitting, sterile neutrinos, etc. In this work we study the impact of neutrino
oscillations and the possible existence of eV sterile neutrinos upon the supernova
neutrino flux (Fν (E)). We have calculated the energy distribution of the neutrino
flux from a supernova and the total number of events which would be detected in a
liquid scintillator. We also present an analysis for the conversion probabilities as a
function of the active-sterile neutrino mixing parameters. Finally, we have carried out
a statistical analysis to extract values for the mixing parameters of the model.
Keywords: Neutrino oscillations, sterile neutrinos, supernovae
Submitted to: J. Phys. G: Nucl. Part. Phys.Neutrino interactions in liquid scintillators 2
1. Introduction
Core-collapse supernovae (SN) represents the final evolutionary stage of stars with
masses heavier than 8 M⊙ . To explain these events one needs to combine nuclear
and particle physics with astrophysics. Neutrinos are important to the study of the
energy balance involved in SN collapses, since only about 1% of the gravitational
binding energy is released as kinetic energy in the compact object formation, while
the remaining 99% is carried out by neutrinos with energies of several MeV [1]. In
the SN core, a medium of high temperature and density, neutrinos of all flavors
are produced. The neutrino production mechanisms are, mainly, pair annihilation
e+ + e− → νe + ν¯e , flavor-conversion νe + ν¯e → ντ,µ + ν̄τ,µ , and nucleon bremsstrahlung
N + N → N + N + ντ,µ + ν̄τ,µ [2]. Neutrinos, that travel through stellar material and
space, can reach a neutrino detector on Earth, giving information from deep inside the
stellar core. During the accretion phase, which takes place few tens to hundreds of
milliseconds after the bounce, the expected neutrino energy spectrum would exhibit a
flavor hierarchy hEνe i < hEν̄e i < hEνx i [3]. However, the presence of neutrino oscillations
can alter the composition of the flux that reaches the Earth [4, 5, 6, 7].
Neutrino oscillations have been studied during the last decades, and different
detectors on Earth have measured neutrino fluxes generated by reactions which take
place in the sun, the Earth atmosphere, reactors, and supernovae like the SN 1987A
[8, 9, 10, 11, 12, 13]. The analysis and reconstruction of SN neutrino fluxes is a powerful
tool to clarify the role of neutrinos in stellar explosions and nucleosynthesis [14], as well
as studying physics beyond the standard model [15, 16, 17, 18].
The possibility of the existence of a light sterile neutrino, motivated by some
experimental anomalies detected in short-baseline neutrinos oscillation experiments
[19, 20], in reactor experiments [21] and gallium detectors [22, 23], is currently under
investigation. This extra neutrino does not participate in weak interactions, and
interacts only gravitationally, thus it can participate in the mixing processes with
active neutrinos [24]. The consequences of the existence of sterile neutrinos in different
astrophysical scenarios have being examined in previous works [25, 26], among others. In
particular, in the context of SN, several authors have already analysed the effects of the
inclusion of a sterile neutrino upon the fraction of free neutrons, the baryonic density,
the electron fraction of the material, and nucleosynthesis processes [27, 14, 28, 5, 29, 30].
In particular, in reference [31], the SN events produced via proton and electron elastic
scattering in scintillators are studied.
In this work, we focus on the study of SN neutrino signals produced by inverse beta
decays in a liquid scintillator and how they are affected by the inclusion of the oscillations
and by the inclusion of a light sterile neutrino in the formalism. We have analyzed the
effects of different mixing parameters in the 3 + 1 scheme, upon some observables as
well as the possibility of having sterile neutrinos in the initial composition. In addition,
we have studied the adiabaticity of the transition between active and sterile ν-species
as a function of the mixing parameters and performed a statistical analysis in energyNeutrino interactions in liquid scintillators 3
bins to find values for the unknown parameters of the model.
This paper is organized as follows. In Sec. 2 we introduce the formalism needed
to calculate SN neutrino fluxes and the neutrino interactions in liquid scintillators.
In Sec. 3, we present and discuss the results of the calculation of the neutrinos
fluxes, number of events and crossing probabilities when active-active and active-sterile
neutrino oscillations are included in the formalism. The conclusions are drawn in Sec.
4.
2. Formalism
2.1. Neutrino fluxes and crossing probabilities
We follow [32] and consider the standard energy released by the supernova (SN) neutrino
outflow similar to the SN1987A [33, 34], that is Eνtot = 3 × 1053 erg distributed between
different neutrino’s flavors. The luminosity flux, for each flavor, is time dependent and
can be written as
E tot
Lνβ (t) = ν e−t/3 . (1)
18
From the previous expression one can write the spectral flux for neutrinos of each
flavor which are produced in the SN explosion and which are detected at a distance D
from the SN, in units of MeV−1 cm−2 , as
Lνβ (t) fνβ (E, η)
Fν0β (E, t) = , (2)
4πD 2 hEνβ i
where fνβ is the neutrino distribution function, E is the neutrino energy and hEνβ i is
the mean energy of the β-flavor neutrino eigenstate.
There are two different distribution functions used in the computation of SN
neutrino spectrum, i) the Fermi-Dirac distribution
1
fνβ (E, ηνβ ) = , (3)
1 + exp[E/Tνβ − ηνβ ]
where the neutrino temperature is related with its mean energy as hEνβ i ∝ Tνβ if ηνβ = 0
and hEνβ i ∝ Tν2β if ηνβ 6= 0, being ηνβ the pinching parameter; and ii) the power law
distribution [35]
!α
(α + 1)(α+1) E
fνβ (E) = e−(α+1)E/hEi . (4)
Γ(α + 1) hEi hEi
In the previous equations both distribution functions are normalized, therefore F2 is the
Fermi integral of order 2 and hE 2 i / hEi2 = (α + 2)/(α + 1).
If the neutrino flux is thermalized one can used a Fermi Dirac distribution function
with η = 0, however, in a SN neutrino flux this condition in not always fulfilled.
Therefore one uses the power law distribution function instead [5, 35, 36, 37, 32].Neutrino interactions in liquid scintillators 4
2.1.1. Time evolution and mixing scheme The time evolution of neutrinos interacting
with electrons, in the flavor basis, is given by the equation
d
i ψνβ = (Hvac + V )ψνβ , (5)
dt
where ψνβ is the neutrino wave function, V represents the interactions between neutrinos
√
and electrons, V = 2GF diag(Ne , 0, 0), GF is the Fermi constant and Ne is the electron
number density [31]. For antineutrinos, the potential has opposite sign. The vacuum
Hamiltonian Hvac in the flavor basis can be obtained from the mass Hamiltonian as
UM 2 U †
Hvac = . (6)
2E
In the last expression M 2 is the matrix of square mass differences, M 2 =
diag (0, ∆m221 , ∆m231 ) for the 3-scheme, and M 2 = diag (0, ∆m221 , ∆m231 , ∆m241 ) for
the 3 + 1-scheme, with the standard notation ∆m2ij = m2i − m2j , being mj the mass of
the neutrino in the j-eigenstate. The unitary matrix U is the PMNS mixing matrix,
[38, 39], which for the standard 3-mass scheme is written
c c s12 c13 s13
12 13
U = α
β ,
s23 c13 (7)
δ ω c23 c13
where cij (sij ) stands for cos θij (sin θij ) (i,j=1,2,3), and α = −s12 c23 − c12 s23 s13 ,
β = c12 c23 − s12 s23 s13 , δ = s12 s23 − c12 c23 s13 and ω = −c12 s23 − s12 c23 s13 .
The U-matrix for the 3+1 scheme has the form [40, 41]
c12 c13 c14 s12 c13 c14 s13 c14 s14
α β s23 c13 0
U = , (8)
δ ω c23 c13 0
−c12 c13 s14 −s12 c13 s14 −s13 s14 c14
where the parameter θ14 is added in order to account for the mixing between the sterile
and the lightest neutrino mass eigenstate.
In both cases, one can perform a rotation in this subspace to diagonalize the
submatrix µ τ of Eq.(6) [42, 5], and since the νµ and ντ fluxes in the SN are similar
we denote them by Fνx . Neutrino flavor conversions inside a supernova are possible
and their effectiveness depends on the matter density. Since the matter density (and
therefore the potencial) decreases with the star radius, active neutrinos exhibit two
Mikheyev-Smirnov-Wolfenstein (MSW) resonances called H (high density) and L (low
density) where the flavor conversion mechanism is amplified [42]. One can solve Eq.
(5) and plot the neutrino level crossing scheme of Fig. 1 (the half-plane with positive
values of density corresponds to neutrinos and the half-plane with negative values of the
density to antineutrinos) [42, 8, 43]. If we consider sterile neutrinos, there is an inner
extra resonance, the S resonance, between the electron type neutrino and the sterile
one. It is worth mentioning that in other works, the resonance is considered to occur at
low densities (outer resonance) [31]
The neutrino flux that arrives at Earth can be computed as a superposition of the
neutrino fluxes produced in the neutrinosphere inside the SN.Neutrino interactions in liquid scintillators 5
Figure 1. Crossing diagram. Left hand-side: normal hierarchy; right hand-side:
inverted hierarchy. Top panel: 3 active neutrinos; bottom panel: 3 + 1 neutrino
scheme.
2.1.2. Neutrino fluxes in the 3-active scheme Following the top row diagrams of Fig.
1, the flux for νe and νx neutrinos (anti-neutrinos) is written:
Fνe = Pe Fν0e + (1 − Pe )Fν0x ,
Fν̄e = P̄e Fν̄0e + (1 − P̄e )Fν̄0x ,
Fνx = (1 − Pe )Fν0e + (1 + Pe )Fν0x ,
Fν̄x = (1 − P̄e )Fν̄0e + (1 + P̄e )Fν̄0x . (9)
In the previous equations Pe and P̄e are the survival probabilities for νe and ν̄e
respectively, which depend on the hierarchy. That is:
Pe = |Ue1 |2 PH PL + |Ue2 |2 PH (1 − PL )
+ |Ue3 |2 (1 − PH ) ,
P̄e = |Ue1 |2 , (10)
for normal hierarchy, and
Pe = |Ue1 |2 PL + |Ue2 |2 (1 − PL ) ,
P̄e = |Ue1 |2 P̄H + |Ue3 |2 (1 − P̄H ) , (11)
for inverse hierarchy, respectively.
In last equations Uek (k = 1, 2, 3) are the components of the first row of the
U-matrix (see Eq.7). PH (P̄H ) and PL (P̄L ) are the neutrino (antineutrino) crossing
probabilities for the H and L resonances respectively (see Section 2.1.4).Neutrino interactions in liquid scintillators 6
2.1.3. Neutrino fluxes in the 3+1 scheme Following the bottom row of Fig. 1, the
fluxes for the 3 + 1 scheme are
Fνe = Θee Fν0e + Θxe Fν0x + Θse Fν0s ,
Fν̄e = Ξee Fν̄0e + Ξxe Fν̄0x + Ξse Fν̄0s ,
Fνx = Θeµ + Θeτ Fν0e + Θxµ + Θxτ Fν0x
+ Θsµ + Θsτ Fν0s ,
Fν̄x = Ξeµ + Ξeτ Fν̄0e + Ξsµ + Ξsτ Fν̄0x
+ Ξsµ + Ξsτ Fν̄0s ,
Fνs = Θes Fν0e + Θxs Fν0x + Θss Fν0s ,
Fν̄s = Ξes Fν̄0e + Ξxs Fν̄0x + Ξss Fν̄0s , (12)
where for normal hierarchy we have defined
Θeα = |Uα1 |2 PH PL (1 − PS ) + |Uα3 |2 PS
+ |Uα2 |2 PH (1 − PL )(1 − PS )
+ |Uα4 |2 (1 − PS )(1 − PH ) ,
Θxα = |Uα1 |2 (1 − PH PL ) + |Uα2 |2 (1 − PH + PH PL )
+ |Uα4 |2 PH ,
Θsα = |Uα1 |2 PS PH PL + |Uα2 |2 PS PH (1 − PL ) ,
+ |Uα3 |2 (1 − PS ) + |Uα4 |2 PS (1 − PH ) ,
Ξeα = |Uα1 |2 ,
Ξxα = |Uα2 |2 + |Uα3 |2 ,
Ξsα = |Uα4 |2 . (13)
For the inverse hierarchy we have called
Θeα = |Uα1 |2 PL (1 − PS ) + |Uα2 |2 PS
+ |Uα4 |2 (1 − PS )(1 − PL ) ,
Θxα = |Uα1 |2 (1 − PL ) + |Uα3 |2 + |Uα4 |2 PL ,
Θsα = |Uα1 |2 PS PL + |Uα2 |2 (1 − PS )
+ |Uα4 |2 PS (1 − PL ) ,
Ξeα = |Uα2 |2 P̄H + |Uα3 |2 (1 − P̄H ) ,
Ξxα = |Uα1 |2 + |Uα2 |2 (1 − P̄H ) + |Uα3 |2 P̄H ,
Ξsα = |Uα4 |2 . (14)
In the last expressions Uαk (k = 1, 2, 3, 4) are the elements of the mixing matrix
(see Eq.(8)) and PS stands for the probability of crossing the S-resonance.
2.1.4. Crossing probabilities As we have seen in the previous sections, the fluxes can
be expressed in terms of the crossing probabilities PH , PL and PS . These are related toNeutrino interactions in liquid scintillators 7
the adiabatic parameter γ as [44, 45]
P = e−πγ/2 . (15)
The adiabatic parameter depends on the change of the density with the radius as
∆m2 sin2 (2θ)
γ= d ln(Ne )
t, (16)
2E cos(2θ) dr res
where θ is the neutrino mixing angle and ∆m2 is the square mass difference. In order to
describe the SN environment, we have considered Ne = ρ/mN = A/(mN r 3 ) where A is
a constant and mN is the neutron mass [15, 46, 47, 42]. The MSW resonance condition
can be written as [48, 31]
2Vee E
cos(2θ) = , (17)
∆m2
therefore
!2/3 √ !1/3
1 ∆m2 sin2 (2θ) 2GF AYe
γ= . (18)
3 2E cos4/3 (2θ) mN
If this parameter is larger than 1, the crossing probability of Eq.(15) is almost zero and
the regime becomes adiabatic.
In order to compute the different probabilities associated to the three resonances,
we have considered the mixing parameters of atmospheric neutrinos (∆m231 and θ13 )
for the H resonance (PH ), of the solar neutrinos (∆m221 and θ12 ) for the L resonance
(PL ) and for the probability of the S resonance (PS ) we used the sterile active neutrino
oscillation parameters (∆m241 and θ14 ). The position of these resonances are shown
schematically in Fig. 1.
2.2. Neutrino detection
Once the SN neutrino burst arrives to the Earth it can be detected by several
experimental arrays, such as GALLEX/GNO [10, 49], SAGE [50], Kamiokande [51],
Super-Kamiokande [52], MiniBooNE [19], KamLAND [53], Borexino [54] , LSND [55],
SNO+ [56].
For the case of neutrino detection in liquid scintillators, such as KamLand, Borexino
and SNO+, the interaction between neutrinos and nuclei are inverse beta decay
(ν¯e + p → n + e+ ) or elastic scattering (ν + p → ν + p). The first kind of interactions is
dominant since its cross section is larger than the one for the scattering, therefore the
number of events can be computed as [57]
Z ∞
N = Np dEFν̄e (E)σν̄e (E) , (19)
Emin
where Np is the number of free protons in the detector, E is the energy and Emin =
1.806 MeV is the threshold energy. For example, a 3 kT detector of C6 H5 C12 H25 (SNO
[56]) has Np = 2.2×1032 free protons, the KamLAND detector, made of 80% C12 H26 and
20% C9 H12 , has Np = 1.7 × 1033 free protons [53] and the detector located in BorexinoNeutrino interactions in liquid scintillators 8
(C9 H12 ) has Np = 1.81 × 1031 [54, 58]. As our reference detector, we adopt a 3 kT one
based on alkyl benzene since it represents a suitable option for the class of detectors
to be installed in the ANDES laboratory. The results presented hereinafter correspond
to this choice [18, 59]. The cross section for the inverse beta decay σν̄ep , in units of
10−43 cm2 , can be approximated as [57]
3
σν̄e (E) = pe Ee E − 0.07056 + 0.02018 y − 0.001953 y , (20)
for energies lower than 300 MeV. In the previous equation pe stands for the positron
momentum related to the neutrino energy E (in MeV) as p2e = (E −∆)2 −m2e , where the
neutron to proton mass difference is ∆ = mn − mp = 1.293 MeV and me is the positron
mass. The neutrino and positron energy are related by Ee = E − ∆, and y = ln E in
the exponent of Eq.(20).
3. Results and Discussion
3.1. Active-active neutrino mixing
In order to calculate the SN neutrino flux and the number of events in a detector, we
have adopted the parameters given in Ref. [42], and fixed the density at the value
AYe = 2 × 1016 kg/km3 . We have performed the integration of the luminosity flux of
Eq. (1), in a time-interval of 20 seconds [33, 34] and the SN-detector distance was fixed
at D = 10 kpc [60, 18]. For the active-active neutrino oscillation parameters needed to
compute Pe and P̄e we have used the values provided by the Particle Data Group [61]
listed in Table 1.
Parameter Normal hierarchy Inverse hierarchy
sin2 (θ12 ) 0.307 0.307
∆m221 7.53 × 10−5 eV2 7.53 × 10−5 eV2
sin2 (θ23 ) 0.545 0.547
∆m232 2.46 × 10−3 eV2 −2.53 × 10−3 eV2
sin2 (θ13 ) 0.0218 0.0218
Table 1. Active-active neutrino mixing parameters [61].
To compute the SN neutrino flux which arrives to Earth (see Eq. (9)), we use the
two previously introduced distribution functions as initial conditions, the Fermi-Dirac
(FD) distribution of Eq. (2), and the power law (PL) distribution of Eq. (4). For the
FD distribution functions we have considered three different sets of values for ηνβ :
i) FD0: ηνβ = 0 for all the flavors;
ii) FD1: ηνe = 1.7, ην̄e = 3 and ηνx = ην̄x = 0.8 [35, 36];
iii) FD2: ηνe = 3, ην̄e = 2 and ηνx = ην̄x = 0.8 [32].Neutrino interactions in liquid scintillators 9
For the PL distribution we have used the value α = 3.
The mean energies used for the calculation of neutrino fluxes are hEνe i = 12 MeV,
hEν̄e i = 15 MeV and hEνx i = hEν̄x i = 18 MeV [18]. If one uses the values for the mixing
parameters given in Table 1, for the adopted density profile, the crossing probabilities
PH and PL are quite small, indicating an adiabatic crossing.
In Fig. 2 we show the SN neutrino’s fluxes at the detector, as a function of the
neutrino energy, for different initial conditions (PL, FD0, FD1, and FD2) and for both
hierarchies. The inclusion of non-zero values for the chemical potentials (that is FD1
and FD2) reduces all of the neutrino fluxes. The flux of the electron antineutrino
(the one that can be detected in scintillator detectors) depends strongly upon the mass
hierarchy. For normal hierarchy (NH), the electron-antineutrino flux dominates over the
electron-neutrinos but for the inverse hierarchy (IH), a swap between electronic fluxes
occurs. Furthermore, for IH the peaks of the electron antineutrinos fluxes locate at lower
energies with respect to the case corresponding to the normal hierarchy. If the neutrino
PL νe FD0 FD1 FD2
16
Fν(E) [109 MeV−1 cm−2]
νe
−
14
νx
12
νx
−
10
8
6
4
2
16
Fν(E) [109 MeV−1 cm−2]
14
12
10
8
6
4
2
5 15 25 35 45 55 5 15 25 35 45 55 5 15 25 35 45 55 5 15 25 35 45 55
E [MeV] E [MeV] E [MeV] E [MeV]
Figure 2. Neutrino’s fluxes at the detector as a function of the neutrino energy, for
the active-active neutrino mixing scenario. Top row: normal hierarchy; bottom row:
inverse hierarchy. Form left to right: first column: power law distribution function
(PL); second column: Fermi Dirac with ηνβ = 0 (FD0); third column: Fermi Dirac
distribution function with ηνe = 1.7, ην̄e = 3 and ηνx = ην̄x = 0.8 (FD1); fourth
column: Fermi Dirac distribution function with ηνe = 3, ην̄e = 2 and ηνx = ην̄x = 0.8
(FD2).
mean energies are larger, for instance hEi = 22 MeV, the neutrino fluxes decrease, as
expected from other studies [35, 18].
In Fig. 3 we present our result for the neutrino event number as a function of
the neutrino energy for all the scenarios considered in Fig. 2. As one can see for theNeutrino interactions in liquid scintillators 10
normal hierarchy, different distribution functions give different results, however, for the
inverse hierarchy, the events are quite similar in all the cases. We have also computed
40
PL
FD0
30 FD1
dN/dE [MeV−1]
FD2
20
10
0
30
dN/dE [MeV−1]
20
10
0
10 20 30 40 50 60
E [MeV]
Figure 3. Energy distribution of neutrino events, for the active-active neutrino mixing
scenario. Top row: normal hierarchy; bottom row: inverse hierarchy. Solid line: PL;
dashed line: FD0; dotted line: FD1; dash-dotted line: FD2.
the total event number (see Table 2) for each scenario. As one can see, the normal
hierarchy provides lower counts than the inverse hierarchy. We have computed the total
event-number without considering neutrino oscillations, for comparison, and found out
that it is smaller than the one obtained in presence of active neutrino oscillations. Also,
the power law distribution function gives lower counts than the ones obtained using any
of the Fermi Dirac distribution function.
Distribution function no/osc. NH IH
PL 760 803 901
FD0 786 830 930
FD1 920 930 952
FD2 856 886 952
Table 2. Calculated number event in a detector for active-active neutrino oscillations,
for the different distribution functions discussed in the text. The second column shows
the values obtained without taking oscillations between neutrino flavors into account,
the third and fourth columns give the values obtained with the normal (NH) and
inverse (IH) mass hierarchies, respectively.Neutrino interactions in liquid scintillators 11
3.2. Active-sterile neutrino mixing
We have fixed the SN environment parameters and the active-active neutrino oscillation
parameters in the previous section and studied the crossing probability for the active-
sterile mixing. From Figs. 4 and 5 one can see that the crossing probability is larger
for larger values of the neutrino energy and that it depends on the active-sterile mixing
parameter. These results are in complete agreement with the ones obtained by other
authors [42, 30].
0.8
0.6
PS
0.4
0.2
0
θ14=π/500
θ14=π/200
θ14=π/100
0.8 θ14=π/50
θ14=π/10
0.6
PS
0.4
0.2
0
10 20 30 40 50 60 70 80 90
E [MeV]
Figure 4. Crossing probability for the S resonance as a function of the neutrino energy
for different active-sterile neutrino mixing parameters. Top figure: ∆m241 = 10−3 eV2 ;
bottom figure: ∆m241 = 1 eV2 .
From our results we notice that the adiabatic approximation is quite good for
∆m241 ∼ 1 eV2 except for small mixing angles, that is θ14 < π/200. For smaller
square mass difference ∆m241 < 10−2 eV2 , the adiabatic approximation is not suitable
for θ14 < π/50. If one fixes the neutrino energy, for instance at E = 20 MeV as in Fig.
6, one can see that the crossing probability is one for zero active-sterile neutrino mixing
angle and for very small values of ∆m241 .
To compute the SN neutrino fluxes with active-sterile neutrino mixing, we use the
values for the mean energies given in the previous section. The active-sterile square
mass difference was fixed to ∆m214 = 1.3 eV2 [62, 63, 20, 64, 65, 66]. In Fig. 7 we show
the SN neutrino fluxes that arrive at the detector
as a function of the neutrino energy,
0
in the absence of sterile neutrinos in the SN Fνs = 0 and for different mixing angles.
In all of the following cases we have considered, for both active and sterile neutrinoNeutrino interactions in liquid scintillators 12
0.8
0.6
PS
0.4
0.2
0
2 −4 2
∆m 14=1x10 eV
2 −3 2
∆m 14=1x10 eV
2 −2 2
∆m 14=1x10 eV
0.8 2 −1
∆m 14=1x10 eV
2
2 2
∆m 14=1 eV
0.6
PS
0.4
0.2
0
10 20 30 40 50 60 70 80 90
E [MeV]
Figure 5. Crossing probability for the S resonance as a function of the neutrino
energy for different active-sterile neutrino mixing parameters. Top figure: θ14 = π/500;
bottom figure: θ14 = π/50.
fluxes, a power-law distribution function. As one can see, the larger is θ14 , the lower is
the electron antineutrino flux.
When a possible non-vanishing initial flux of sterile neutrinos Fν0s 6= 0 is
considered, the fluxes are modified according to Fig. 8. The flux for the electron
type antineutrino changes if one changes the active-sterile mixing or the mean energy
of the sterile neutrino. The number of events in the detector also would change.
The neutrino number events in the detector as a function of the neutrino energy
are presented in Fig. 9. In particular, we show the energy distribution of the number
event without initial sterile neutrinos and different mixing angles (left column), and for
non-vanishing initial flux of sterile neutrinos, for different mean energies and mixing
angles (right column). As one can see, the inclusion of a sterile neutrino with a large
mixing angle and a small mean energy affects the distribution of events in the detector,
specially the energy at which the maximum number of events is predicted. This is
translated into the total number of events (see Table 3).
In Fig. 10 we show the number of events in the detector as a function of the active-
sterile mixing angle. We have found that the number of events does not depend on the
square mass difference, but strongly depends on the mixing angle. In all the studied
cases, the neutrino inverse hierarchy produces more events than the normal hierarchy.
The number of events for Fν0s = 0 decreases for larger mixing angles. If we considerNeutrino interactions in liquid scintillators 13
1
2 −5 2
∆m 14=1x10 eV
0.8 ∆m
2
14=1x10
−3
eV
2
2 2
∆m 14=1.3 eV
0.6
PS
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
θ14
1
θ14=π/500
0.8 θ14=π/100
θ14=0.1
0.6
PS
0.4
0.2
0
0 0.5 1 1.5 2 2.5
2 2
∆m 14 [eV ]
Figure 6. Crossing probability for the S resonance at E = 20 MeV. Top figure: PS as
a function of the mixing angle and for different values of the square mass difference;
bottom figure: PS as a function of ∆m214 for different values of θ14 in radians.
Fν0s θ14 hEνs i NH IH
0 0.1 – 796 889
0 0.6 – 527 587
10 MeV 801 894
6= 0 0.1 12 MeV 804 897
17 MeV 805 898
10 MeV 699 760
6= 0 0.6 12 MeV 789 850
17 MeV 822 883
Table 3. Calculated number event in a scintillator detector for active-sterile neutrino
oscillation.
sterile neutrinos inside the supernova, the mean energy of these neutrinos affects the
value of N. For Fν0s 6= 0 the value of N also decreases with the mixing angle except for
NH with sterile neutrino mean energy larger than the electron neutrino mean energy.
3.3. Departure from Poisson’s statistics
Since the data from SN1987A do not allow to study the neutrino energy distribution
in detail [67, 68, 69] and there is a lack of a quantitative estimation of theoreticalNeutrino interactions in liquid scintillators 14
νe
18 νe
−
cm ]
−2
16 νx
ν
−
x
14
νs
−1
MeV
12 νs
−
10
9
8
Fν(E) [10
6
4
2
cm ] 18
−2
16
14
−1
MeV
12
10
9
8
Fν(E) [10
6
4
2
18
cm ]
−2
16
14
−1
MeV
12
10
9
8
Fν(E) [10
6
4
2
5 15 25 35 45 55 5 15 25 35 45 55
E [MeV] E [MeV]
Figure 7. Neutrino’s fluxes at the detector as a function of the neutrino energy, for
the active-sterile neutrino mixing scenario. Top row: θ14 = 0.1; middle row: θ14 = 0.6;
bottom row: θ14 = 0.006. Left column: normal hierarchy; right column: inverse
hierarchy. All the cases ∆m214 = 1.3 eV2 and no initial sterile neutrino Fν0s = 0 .
The calculation was performed by using a power-law distribution function for active
neutrinos.
uncertainties, it is worth exploring deviations of our theoretical data with respect to the
Poisson statistics
µk
P (k; µ) = exp (−µ) (21)
k!
where µ is the average number of occurrences in a given interval and k is the discrete
random variable. For µ we have adopted the value corresponding to the 3 active neutrino
mixing scenario following Ref. [31]. The statistical indicator for Poisson distributions
is [70, 71]
N
ni
2
(θ14 , ∆m241 )
X
∆χ =2 mi − ni + ni ln (22)
i=1 mi
The sum runs over energy bins, mi are the theoretical values for the number of events for
the i-th bin, which depend on θ14 and ∆m241 and ni are the number of events derived from
√
the above mentioned Poisson distribution for each energy bin with an error σi = ni .
Following Refs. [72, 73] we have studied the number of events along 32 bins of 2 MeV
width. We have performed the analysis for an active-sterile mixing angle in the rangeNeutrino interactions in liquid scintillators 15
νe νe
νe νe
− −
22 22
cm ]
cm ]
−2
−2
νx νx
18 ν
−
x 18 ν
−
x
νs νs
−1
−1
νs νs
MeV
MeV
− −
14 14
9
9
10 10
Fν(E) [10
Fν(E) [10
6 6
2 2
22 22
cm ]
cm ]
−2
−2
18 18
−1
−1
MeV
MeV
14 14
9
9
10 10
Fν(E) [10
Fν(E) [10
6 6
2 2
22 22
cm ]
cm ]
−2
18 −2 18
−1
−1
MeV
MeV
14 14
9
9
10 10
Fν(E) [10
Fν(E) [10
6 6
2 2
5 15 25 35 45 55 5 15 25 35 45 55 5 15 25 35 45 55 5 15 25 35 45 55
E [MeV] E [MeV] E [MeV] E [MeV]
Figure 8. Neutrino’s fluxes at the detector as a function of the neutrino energy, for
the active-sterile neutrino mixing scenario and ∆m214 = 1.3 eV2 and with initial sterile
neutrino. Top row: hEνs i = 10 MeV; middle row: hEνs i = hEνe i = 12 MeV; bottom
row: hEνs i = 17 MeV. For each inset: left column: normal hierarchy; right column:
inverse hierarchy. The computation was performed by using a power law distribution
function for all neutrinos. Left panel: θ14 = 0.1; right panel: θ14 = 0.6.
0 ≤ θ14 ≤ π/4, and fixed square mass difference ∆m241 = 1.3 eV2 . The best value of the
mixing angle is determined by minimizing Eq. (22). For all the studied cases, we have
obtained a best fit value for θ14 shown in Table 4.
∆χ2
Hierarchy Fν0s hEνs i θ14 ± σ N −1
NH 0 – 0.044 ± 0.021 1.32
10 MeV 0.032 ± 0.016 1.32
6= 0 12 MeV 0.033 ± 0.015 1.32
17 MeV 0.044 ± 0.025 1.30
IH 0 – 0.016 ± 0.009 1.32
10 MeV 0.016 ± 0.009 1.37
6 0 12 MeV
= 0.016 ± 0.009 1.32
17 MeV 0.016 ± 0.010 1.37
Table 4. Best-fit values of the mixing angle θ14 in radians at 1σ. The reduced χ-square
value of each of the fits are given in the last column.Neutrino interactions in liquid scintillators 16
θ14=0.1 =12 MeV
s
35 θ14=0.6 =10 MeV
s
θ14=0.006 =12 MeV
s
30 =17 MeV
dN/dE [MeV−1]
s
25
20
15
10
5
35
30
dN/dE [MeV−1]
25
20
15
10
5
10 20 30 40 50 10 20 30 40 50
E [MeV] E [MeV]
Figure 9. Energy distribution of neutrino events, for the active-sterile neutrino
mixing scenario. Top row: normal hierarchy; bottom row: inverse hierarchy. Left
column: with out sterile neutrino in the SN Fν0s = 0 ; right column: with initial
sterile-neutrinos flux for several mean energies. Red lines: θ14 = 0.1; blue lines:
θ14 = 0.6; green lines: θ14 = 0.006
Although the analysis carried out indicates a preference for the 3 + 1 model with
a small but non-zero angle θ14 , it should be noted that all cases are consistent with
θ14 = 0 at 3σ. This indicates that the data can be represented by considering only
active-active neutrino oscillations in the formalism or by the inclusion of active-sterile
neutrino mixing with a small mixing angle.
So far, the data and the analysis presented in this section are those provided by
measurements coming from liquid sctinllators. To complete the picture about the mixing
with νs it would be necessary to perform a similar analysis for data gathered by different
types of neutrino detectors and complement this kind of study with the determination
of the neutrino spectral shape.
4. Conclusions
In this work, we have included massive neutrinos, neutrino oscillation, and a light sterile
neutrino in the formalism of SN neutrino fluxes and their detection in liquid scintillators
through inverse beta decay reactions.
We have computed the neutrino fluxes and the number of detected events with
active neutrino mixing. We have found that the neutrino fluxes are sensitive toNeutrino interactions in liquid scintillators 17
900
800
700
N
600
Fνs0= 0
=10 MeV
500
=12 MeV
=17 MeV
900
800
700
N
600
500
0.1 0.2 0.3 0.4 0.5 0.6 0.7
θ14
Figure 10. Neutrino number events as a function of the active-sterile neutrino mixing
angle. Top row: normal hierarchy; bottom row: inverse hierarchy. Solid line: no sterile
neutrino in the SN; small dashed line: hEνs i = 10 MeV; dotted line: hEνs i = 12 MeV;
long dashed line: hEνs i = 17 MeV. In all cases ∆m214 = 1.3 eV2 .
differences between the normal and inverse mass hierarchies. The power-law distribution
generates a lower number of total events than the Fermi-Dirac distribution function.
Furthermore, the number of expected events increases with respect to the case for which
oscillations are not included.
Then, we have computed the crossing probabilities, neutrino fluxes and the number
of detected events as a function of the mixing parameters for the active-sterile sector.
As the energy increases, the crossing of the resonances becomes less adiabatic, and when
both mixing parameters are small, the non-adiabaticity is more pronounced. We have
found that the adiabatic approximation for PS is no longer valid for ∆m241 ∼ 1eV2 and
0 < θ14 < π/200 and for ∆m241 < 1 × 10−2 eV2 with θ14 < π/50.
When there is not initial sterile neutrino flux, that is Fν0s = 0, the number of events
decreases for large mixing angles θ14 . However, for a non-zero initial sterile neutrino
flux, the number of events depends on the mean energy used to calculate the spectral
function Fν0s . This event detection does not depend on the mass-square difference (since
the flux ν̄e does not depend on PS ). In all the studied cases, the IH produces a larger
number of events than the NH.
Finally, we have performed a statistical test to set limits on the mixing angle
between active and sterile neutrinos. We have found that in all the cases the best
value of the active-sterile mixing angle is small but not null. Studying neutrinos from
SN in different types of detectors can be useful to provide a definitive conclusion aboutNeutrino interactions in liquid scintillators 18
the existence of eV-scale sterile neutrinos and the preferred mixing scheme.
We hope that a better understanding of the expected spectrum of events and fluxes
will help to the analysis of the next supernova event in our galaxy, as well as to set
prospects for the construction of future detectors, as the one planned to be hosted by
the ANDES lab [18, 59, 74].
Acknowledgments
This work was supported by a grant (PIP-616) of the National Research Council of
Argentina (CONICET), and by a research-grant (PICT 140492) of the National Agency
for the Promotion of Science and Technology (ANPCYT) of Argentina. O. C. and M.
E. M. are members of the Scientific Research Career of the CONICET, M. M. S. is a
Post Doctoral fellow of the CONICET.
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