IFIC/07-03
Probing non-standard neutrino interactions with supernova neutrinos
A. Esteban-Pretel, R. Tomàs and J. W. F. Valle1
1AHEP Group, Institut de F́ısica Corpuscular - C.S.I.C/Universitat de València
Edifici Instituts d’Investigació, Apt. 22085, E-46071 València, Spain
(Dated: November 4, 2018)
We analyze the possibility of probing non-standard neutrino interactions (NSI, for short) through
the detection of neutrinos produced in a future galactic supernova (SN). We consider the effect of
NSI on the neutrino propagation through the SN envelope within a three-neutrino framework, paying
special attention to the inclusion of NSI-induced resonant conversions, which may take place in the
most deleptonised inner layers. We study the possibility of detecting NSI effects in a Megaton water
Cherenkov detector, either through modulation effects in the ν̄e spectrum due to (i) the passage
of shock waves through the SN envelope, (ii) the time dependence of the electron fraction and (iii)
the Earth matter effects; or, finally, through the possible detectability of the neutronization νe
burst. We find that the ν̄e spectrum can exhibit dramatic features due to the internal NSI-induced
resonant conversion. This occurs for non-universal NSI strengths of a few %, and for very small
flavor-changing NSI above a few×10−5.
PACS numbers: 13.15.+g, 14.60.Lm, 14.60.Pq, 14.60.St, 97.60.Bw
I. INTRODUCTION
The very first data of the KamLAND collaboration [1]
have been enough to isolate neutrino oscillations as the
correct mechanism explaining the solar neutrino prob-
lem [2, 3], indicating also that large mixing angle (LMA)
was the right solution. The 766.3 ton-yr KamLAND data
sample further strengthens the validity of the LMA os-
cillation interpretation of the data [4].
Current data imply that neutrino have mass. For an
updated review of the current status of neutrino oscil-
lations see [5]. Theories of neutrino mass [6, 7] typ-
ically require that neutrinos have non-standard prop-
erties such as neutrino electromagnetic transition mo-
ments [8, 9, 10] or non-standard four-Fermi interactions
(NSI, for short) [11, 12, 13]. The expected magnitude of
the NSI effects is rather model-dependent.
Seesaw-type models lead to a non-trivial structure of
the lepton mixing matrix characterizing the charged and
neutral current weak interactions [6]. The NSI which
are induced by the charged and neutral current gauge
interactions may be sizeable [14, 15, 16, 17, 18]. Alter-
natively, non-standard neutrino interactions may arise in
models where neutrinos masses are radiatively “calcula-
ble” [19, 20]. Finally, in some supersymmetric unified
models, the strength of non-standard neutrino interac-
tions may arise from renormalization and/or threshold
effects [21].
We stress that non-standard interactions strengths are
highly model-dependent. In some models NSI strengths
are too small to be relevant for neutrino propagation,
because they are either suppressed by some large mass
scale or restricted by limits on neutrino masses, or both.
However, this need not be the case, and there are many
theoretically attractive scenarios where moderately large
NSI strengths are possible and consistent with the small-
ness of neutrino masses. In fact one can show that
NSI may exist even in the limit of massless neutri-
nos [14, 15, 16, 17, 18]. Such may also occur in the
context of fully unified models like SO(10) [22].
We argue that, in addition to the precision determi-
nation of the oscillation parameters, it is necessary to
test for sub-leading non-oscillation effects that could arise
from non-standard neutrino interactions. These are nat-
ural outcome of many neutrino mass models and can be of
two types: flavor-changing (FC) and non-universal (NU).
These are constrained by existing experiments (see be-
low) and, with neutrino experiments now entering a pre-
cision phase [23], an improved determination of neutrino
parameters and their theoretical impact constitute an im-
portant goal in astroparticle and high energy physics [5].
Here we concentrate on the impact of non-standard
http://arxiv.org/abs/0704.0032v1
neutrino interactions on supernova physics. We show
how complementary information on the NSI parame-
ters could be inferred from the detection of core-collapse
supernova neutrinos. The motivation for the study is
twofold. First, if a future SN event takes place in our
Galaxy the number of neutrino events expected in the
current or planned neutrino detectors would be enor-
mous, O(104 − 105) [24]. Moreover, the extreme con-
ditions under which neutrinos have to travel since they
are created in the SN core, in strongly deleptonised re-
gions at nuclear densities, until they reach the Earth,
lead to strong matter effects. In particular the effect of
small values of the NSI parameters can be dramatically
enhanced, possibly leading to observable consequences.
This paper is planned as follows. In Sec. II we summa-
rize the current observational bounds on the parameters
describing the NSI, including previous works on NSI in
SNe. In Sec. III we describe the neutrino propagation
formalism as well as the SN profiles which will be used.
In Sec. IV we analyze the effect of NSI on the ν propaga-
tion in the inner regions near the neutrinosphere and in
the outer regions of the SN envelope. In Sec. V we discuss
the possibility of using various observables to probe the
presence of NSI in the neutrino signal of a future galactic
SN. Finally in Sec. VI we present our conclusions.
II. PRELIMINARIES
A large class of non-standard interactions may be
parametrized with the effective low-energy four-fermion
operator:
LNSI = −εfPαβ 2
2GF (ν̄αγµLνβ)(f̄γ
µPf) , (1)
where P = L, R and f is a first generation fermion:
e, u, d. The coefficients ε
αβ denote the strength of the
NSI between the neutrinos of flavors α and β and the
P−handed component of the fermion f .
Current constraints on ε
αβ come from a variety of dif-
ferent sources, which we now briefly list.
A. Laboratory
Neutrino scattering experiments [25, 26, 27, 28,
29] provide the following bounds, |εfPµµ | . 10−3 −
10−2, |εfPee | . 10−1 − 1, |εfPµτ | . 0.05, |εfPeτ | . 0.5 at
90 % C.L [30, 31, 32]. On the other hand the analysis
of the e+e− → νν̄γ cross section measured at LEP II
leads to a bound on |εePττ | . 0.5 [33]. Future prospects
to improve the current limits imply the measurement of
sin2 ϑW leptonically in the scattering off electrons in the
target, as well as in neutrino deep inelastic scattering in
a future neutrino factory. The main improvement would
be in the case of |εfPee | and |εfPeτ |, where values as small
as 10−3 and 0.02, respectively, could be reached [31].
The search for flavor violating processes involving
charged leptons is expected to restrict corresponding neu-
trino interactions, to the extent that the SU(2) gauge
symmetry is assumed. However, this can at most give
indicative order-of-magnitude restrictions, since we know
SU(2) is not a good symmetry of nature. Using radiative
corrections it has been argued that, for example, µ − e
conversion on nuclei like in the case of µ−T i also con-
strains |εqPµe | . 7.7× 10−4 [31].
Non-standard interactions can also affect neutrino
propagation through matter, probed in current neutrino
oscillation experiments. The bounds so obtained apply to
the vector coupling constant of the NSI, ε
αβ = ε
since only this appears in neutrino propagation in mat-
ter [91].
B. Solar and reactor
The role of neutrino NSIs as subleading effects on the
solar neutrino oscillations and KamLAND has been re-
cently considered in Ref. [34, 35, 36] with the following
bounds at 90 % CL for ε ≡ − sinϑ23εdVeτ with the al-
lowed range −0.93 . ε . 0.30, while for the diagonal
term ε′ ≡ sin2 ϑ23εdVττ − εdVee , the only forbidden region is
[0.20, 0.78] [36]. Only in the ideal case of infinitely pre-
cise solar neutrino oscillation parameters determination,
the allowed range would “close from the left” for negative
NSI parameter values, at −0.6 for ε and −0.7 for ε′.
C. Atmospheric and accelerator neutrinos
Non-standard interactions involving muon neutrinos
can be constrained by atmospheric neutrino experiments
as well as accelerator neutrino oscillation searches at
K2K and MINOS. In Ref. [37] Super-Kamiokande and
MACRO observations of atmospheric neutrinos were con-
sidered in the framework of two neutrinos. The limits ob-
tained were −0.05 . εdVµτ < 0.04 and |εdVττ − εdVµµ | . 0.17
at 99 % CL. The same data set together with K2K were
recently considered in Refs. [38, 39] to study the nonstan-
dard neutrino interactions in a three generation scheme
under the assumption εeµ = εµµ = εµτ = 0. The al-
lowed region of εττ obtained for values of εeτ smaller
than O(10−1) becomes Σf=u,d,eεfVαβNf/Ne . 0.2 [39] ,
where Nf stands for the fermion number density.
D. Cosmology
If non-standard interactions with electrons were large
they might also lead to important cosmological and as-
trophysical implications. For instance, neutrinos could
be kept in thermal contact with electrons and positrons
longer than in the standard case, hence they would share
a larger fraction of the entropy release from e± annihi-
lations. This would affect the predicted features of the
cosmic background of neutrinos. As recently pointed out
in Ref [40] required couplings are, though, larger than
the current laboratory bounds.
E. NSI in Supernovae
According to the currently accepted supernova (SN)
paradigm, neutrinos are expected to play a crucial role
in SN dynamics. As a result, SN physics provides
a laboratory to probe neutrino properties. Moreover,
many future large neutrino detectors are currently be-
ing discussed [41]. The enormous number of events,
O(104 − 105) that would be “seen” in these detectors in-
dicates that a future SN in our Galaxy would provide a
very sensitive probe of non-standard neutrino interaction
effects.
The presence of NSI can lead to important conse-
quences for the SN neutrino physics both in the highly
dense core as well as in the envelope where neutrinos
basically freely stream.
The role of non-forward neutrino scattering processes
on heavy nuclei and free nucleons giving rise to flavor
change within the SN core has been recently analyzed in
Ref. [42, 43]. The main effect found was a reduction in
the core electron fraction Ye during core collapse. A lower
Ye would lead to a lower homologous core mass, a lower
shock energy, and a greater nuclear photon-disintegration
burden for the shock wave. By allowing a maximum
∆Ye = −0.02 it has been claimed that εeα . 10−3, where
α = µ, τ [43].
On the other hand it has been noted since long ago
that the existence of NSI plays an important role in the
propagation of SN neutrinos through the envelope lead-
ing to the possibility of a new resonant conversion. In
contrast to the well known MSW effect [44, 45] it would
take place even for massless neutrinos [13]. Two basic
ingredients are necessary: universal and flavor changing
NSI. In the original scheme neutrinos were mixed in the
leptonic charged current and universality was violated
thanks to the effect of mixing with heavy gauge singlet
leptons [6, 14]. Such resonance would induce strong neu-
trino flavor conversion both for neutrinos and antineutri-
nos simultaneously, possibly affecting the neutrino sig-
nal of the SN1987A as well as the possibility of having
r−process nucleosynthesis. This was first quantitatively
considered within a two-flavor νe−ντ scheme, and bounds
on the relevant NSI parameters were obtained using both
arguments [46].
One of the main features of the such “internal” or
“massless” resonant conversion mechanism is that it re-
quires the violation of universality, its position being
determined only by the matter chemical composition,
namely the value of the electron fraction Ye, and not by
the density. In view of the experimental upper bounds
on the NSI parameters such new resonance can only take
place in the inner layers of the supernova, near the neu-
trinosphere, where Ye takes its minimum values. In this
region the values of Ye are small enough to allow for
resonance conversions to take place in agreement with
existing bounds on the strengths of non-universal NSI
parameters.
The SN physics implications of another type of NSI
present in supersymmetric R-parity violating models
have also been studied in Ref. [47], again for a system
of two neutrinos. For definiteness NSI on d−quarks were
considered, in two cases: (i) massless neutrinos without
mixing in the presence of flavor-changing (FC) and non-
universal (NU) NSIs, and (ii) neutrinos with eV masses
and FC NSI. Different arguments have been used in
order to constrain the parameters describing the NSI,
namely, the SN1987A signal, the possibility to get suc-
cessful r−process nucleosynthesis, and the possible en-
hancement of the energy deposition behind the shock
wave to reactivate it.
On the other hand several subsequent articles [48, 49,
50] considered the effects of NSI on the neutrino propa-
gation in a three–neutrino mixing scenario for the case
Ye > 0.4, typical for the outer SN envelope. Together
with the assumption that εdVαβ . 10
−2 this prevents the
appearance of internal resonances in contrast to previous
references.
Motivated by supersymmetric theories without R par-
ity, in Ref. [48] the authors considered the effects of
small-strength NSI with d−quarks. Following the for-
malism developed in Refs. [51, 52] they studied the cor-
rections that such NSI would have on the expressions
for the survival probabilities in the standard resonances
MSW-H and MSW-L. A similar analysis was performed
in Ref. [49] assuming Z-induced NSI interactions orig-
inated by additional heavy neutrinos. A phenomeno-
logical generalization of these results was carried out in
Ref. [50]. The authors found an analytical compact ex-
pression for the survival probabilities in which the main
effects of the NSI can be embedded through shifts of the
mixing angles ϑ12 and ϑ13. In contrast to similar expres-
sions found previously these directly apply to all mixing
angles, and in the case with Earth matter effects. The
main phenomenological consequence was the identifica-
tion of a degeneracy between ϑ13 and εeα, similar to the
analogous “confusion” between ϑ13 and the correspond-
ing NSI parameter noted to exist in the context of long-
baseline neutrino oscillations [53, 54].
We have now re-considered the general three–neutrino
mixing scenario with NSI. In contrast to previous
work [48, 49, 50], we have not restricted ourselves to
large values of Ye, discussing also small values present
in the inner layers. This way our generalized descrip-
tion includes both the possibility of neutrinos having the
“massless” NSI-induced resonant conversions in the in-
ner layers of the SN envelope [13, 46, 47], as well as the
“outer” oscillation-induced conversions [48, 49, 50] [92].
III. NEUTRINO EVOLUTION
In this section we describe the main ingredients of our
analysis. Our emphasis will be on the use of astrophys-
ically realistic SN matter and Ye profiles, characterizing
its density and the matter composition. Their details,
in particular their time dependence, are crucial in deter-
mining the way the non-standard neutrino interactions
affect the propagation of neutrinos in the SN medium.
A. Evolution Equation
As discussed in Sec. II in an unpolarized medium the
neutrino propagation in matter will be affected by the
vector coupling constant of the NSI, ε
αβ = ε
αβ [93].
The way the neutral current NSI modifies the neu-
trino evolution will be parametrized phenomenologically
through the effective low-energy four-fermion operator
described in Eq. (1). We also assume ε
αβ ∈ ℜ, neglect-
ing possible CP violation in the new interactions.
Under these assumptions the Hamiltonian describing
the SN neutrino evolution in the presence of NSI can be
cast in the following form [94]
να = (Hkin +Hint)αβ νβ , (2)
where Hkin stands for the kinetic term
Hkin = U
U † , (3)
with M2 = diag(m21,m
3), and U the three-neutrino
lepton mixing matrix [6] in the PDG convention [55] and
with no CP phases.
The second term of the Hamiltonian accounts for the
interaction of neutrinos with matter and can be split into
two pieces,
Hint = H
int +H
int . (4)
The first term, Hstd
describes the standard interaction
with matter and can be written asHstd
= diag (VCC , 0, 0)
up to one loop corrections due to different masses of the
muon and tau leptons [56]. The standard matter poten-
tial for neutrinos is given by
VCC =
2GFNe = V0ρYe , (5)
where V0 ≈ 7.6×10−14 eV, the density is given in g/cm3,
and Ye stands for the relative number of electrons with
respect to baryons. For antineutrinos the potential is
identical but with the sign changed.
The term in the Hamiltonian describing the non-
standard neutrino interactions with a fermion f can be
expressed as,
(Hnsiint )αβ =
f=e,u,d
)αβ , (6)
with (V
)αβ ≡
2GFNfε
αβ. For definiteness and mo-
tivated by actual models, for example, those with broken
R parity supersymmetry we take for f the down-type
quark. However, an analogous treatment would apply
to the case of NSI on up-type quarks, the existence of
NSI with electrons brings no drastic qualitative differ-
ences with respect to the pure oscillation case (see be-
low). Therefore the NSI potential can be expressed as
follows,
(V dnsi)αβ = ε
αβV0ρ(2− Ye) . (7)
From now on we will not explicitely write the superindex
d. In order to further simplify the problem we will rede-
fine the diagonal NSI parameters so that εµµ = 0, as one
can easily see that subtracting a matrix proportional to
the identity leaves the physics involved in the neutrino
oscillation unaffected.
B. Supernova matter profiles
Neutrino propagation depends on the supernova mat-
ter and chemical profile through the effective potential.
This profile exhibits an important time dependence dur-
ing the explosion. Fig. 1 shows the density ρ(t, r) and the
electron fraction Ye(t, r) profiles for the SN progenitor as
well as at different times post-bounce.
Progenitor density profiles can be roughly
parametrized by a power-law function
ρ(r) = ρ0
, (8)
where ρ0 ∼ 104 g/cm3, R0 ∼ 109 cm, and n ∼ 3. The
electron fraction profile varies depending on the matter
composition of the different layers. For instance, typical
values of Ye between 0.42 and 0.45 in the inner regions
are found in stellar evolution simulations [57]. In the in-
termediate regions, where the MSW H and L-resonances
take place Ye ≈ 0.5. This value can further increase in
the most outer layers of the SN envelope due to the pres-
ence of hydrogen.
After the SN core bounce the matter profile is affected
in several ways. First note that a front shock wave starts
to propagate outwards and eventually ejects the SN enve-
lope. The evolution of the shock wave will strongly mod-
ify the density profile and therefore the neutrino propa-
gation [58, 59]. Following Ref. [60] we shall assume that
the structure of the shock wave is more complicated and
an additional “reverse wave” appears due to the collision
of the neutrino-driven wind and the slowly moving mate-
rial behind the forward shock, as seen in the upper panel
of Fig. 1 [95].
On the other hand, the electron fraction is also affected
by the time evolution as the SN explosion proceeds. Once
the collapse starts the core density grows so that the neu-
trinos become eventually effectively trapped within the
so called “neutrinosphere”. At this point the trapped
electron fraction has decreased until values of the order of
0.33 [61]. When the inner core reaches the nuclear density
it can not contract any further and bounces. As a con-
sequence a shock wave forms in the inner core and starts
propagating outwards. When the newly formed super-
nova shock reaches densities low enough for the initially
trapped neutrinos to begin streaming faster than the
shock propagates [62], a breakout pulse of νe is launched.
In the shock-heated matter, which is still rich of elec-
trons and completely disintegrated into free neutrons and
protons, a large number of νe are rapidly produced by
electron captures on protons. They follow the shock on
its way out until they are released in a very luminous
flash, the breakout burst, at about the moment when
the shock penetrates the neutrinosphere and the neutri-
nos can escape essentially unhindered. As a consequence,
the lepton number in the layer around the neutrinosphere
decreases strongly and the matter neutronizes [63]. The
value of Ye steadily decreases in these layers until val-
ues of the order of O(10−2). Outside the neutrinosphere
there is a steep rise until Ye ≈ 0.5. This is a robust
feature of the neutrino-driven baryonic wind. Neutrino
heating drives the wind mass loss and causes Ye to rise
within a few 10 km from low to high values, between 0.45
and 0.55 [64], see bottom panel of Fig. 1. Inspired in the
numerical results of Ref. [60] we have parametrized the
behavior of the electron fraction near the neutrinosphere
phenomenologically as,
Ye = a+ b arctan[(r − r0)/rs] , (9)
where a ≈ 0.23− 0.26 and b ≈ 0.16− 0.20. The param-
eters r0 and rs describe where the rise takes place and
how steep it is, respectively. As can be seen in Fig. 1
both decrease with time.
FIG. 1: Density (upper panel) and electron fraction (bottom
panel) profiles for the SN progenitor and at different instants
after the core bounce, from Ref. [60]. The regions where the
H (yellow) and the L (cyan) resonance take place are also
indicated, as well as the NSI-induced I (gray) resonance for
the parameters εee = 0, εττ . 0.07 and |εµτ | . 0.05
IV. THE TWO REGIMES
In order to study the neutrino propagation through
the SN envelope we will split the problem into two differ-
ent regions: the inner envelope, defined by the condition
VCC ≫ ∆m2atm/(2E) with ∆m2atm ≡ m23 − m22, and the
outer one, where ∆m2atm/(2E) & VCC . From the upper
panel of Fig. 1 one can see how the boundary roughly
varies between r ≈ 108 cm and 109 cm, depending on the
time considered. This way one can fully characterize all
resonances that can take place in the propagation of su-
pernova neutrinos, both the outer resonant conversions
related to neutrino masses and indicated as the upper
bands in Fig. 1, and the inner resonances that follow
from the presence of non-standard neutrino interactions,
indicated by the band at the bottom of the same figure.
Here we pay special attention to the use of realistic mat-
ter and chemical supernova profiles and three-neutrino
flavors thus generalising previous studies.
A. Neutrino Evolution in the Inner Regions
Let us first write the Hamiltonian in the inner layers,
where Hint ≫ Hkin. In this case the Hamiltonian can be
written as
H ≈ Hint = V0ρ(2− Ye)
+ εee εeµ εeτ
εeµ 0 εµτ
εeτ εµτ εττ
When the value of the εαβ is of the same order as the
electron fraction Ye internal resonances can arise [13].
Taking into account the current constraints on the ε’s
discussed in Sec. II one sees that small values of Ye are
required [46, 47]. As a result, these can only take place
in the most deleptonised inner layers, close to the neu-
trinosphere, where the kinetic terms of the Hamiltonian
are negligible.
Given the large number of free parameters εαβ in-
volved we consider one particular case where |εeµ| and
|εeτ | are small enough to neglect a possible initial mixing
between νe and νµ or ντ . Barring fine tuning, this basi-
cally amounts to |εeµ|, |εeτ | ≪ 10−2. According to the
discussion of Sec. II εeµ automatically satisfies the condi-
tion, whereas one expects that the window |εeτ | & 10−2
will eventually be probed in future experiments.
Since the initial fluxes of νµ and ντ are expected to be
basically identical, it is convenient to redefine the weak
basis by performing a rotation in the µ− τ sector:
= U(ϑ′23)
1 0 0
0 c23′ s23′
0 −s23′ c23′
where c23′ and s23′ stand for cos(ϑ
23) and sin(ϑ
23), re-
spectively. The angle ϑ′23 can be written as
tan(2ϑ′23) ≈
. (12)
The Hamiltonian becomes in the new basis
H ′αβ = U
†(ϑ′23)HαβU(ϑ
23) (13)
= V0ρ(2− Ye)
+ εee ε
ε′eµ ε
ε′eτ 0 ε
,(14)
where
ε′eµ = εeµc23′ − εeτs23′ (15)
ε′eτ = εeµs23′ + εeτ c23′ (16)
ε′µµ = (εττ −
ε2ττ + 4ε
µτ )/2 (17)
ε′ττ = (εττ +
ε2ττ + 4ε
µτ )/2 . (18)
With our initial assumptions on εeα one notices that
the new basis ν′α basically diagonalizes the Hamiltonian,
and therefore coincides roughly with the matter eigen-
state basis. A novel resonance can arise if the condition
H ′ee = H
ττ is satisfied, we call this I-resonance, I stand-
ing for “internal” [96]. The corresponding resonance con-
dition can be written as
Y Ie =
1 + εI
, (19)
where εI is defined as ε′ττ − εee. In Fig. 2 we represent
the range of εee and ε
ττ leading to the I-resonance for
an electron fraction profile between different Y mine ’s and
Y maxe = 0.5. It is important to notice that the value
of Y mine depends on time. Right before the collapse the
minimum value of the electron fraction is around 0.4.
Hence the window of NSI parameters that would lead
to a resonance would be relatively narrow, as indicated
by the shaded (yellow) band in Fig. 2. As time goes on
Y mine decreases to values of the order of a few %, and
as a result the region of parameters giving rise to the I-
resonance significantly widens. For example, in the range
|εee| ≤ 10−3 possibly accessible to future experiments one
sees that the I-resonance can take place for values of ε′ττ
of the order of O(10−2). This indicates that the potential
sensitivity on NSI parameters that can be achieved in su-
pernova studies is better than that of the current limits.
FIG. 2: Contours of Y Ie as function of εee and ε
ττ accord-
ing to Eq. (19) for different values of Ye. The region in yel-
low represents the region of parameters that gives rise to I-
resonance before the collapse. The arrows indicate how this
region widens with time.
As seen in Fig. 1 in order to fulfill the I-resonance con-
dition for such small values of the NSI parameters the
values of Ye must indeed lie, as already stated, in the
inner layers.
Several comments are in order: First, in contrast to
the standard H and L-resonances, related to the kinetic
term, the density itself does not explicitly enter into the
resonance condition, provided that the density is high
enough to neglect the kinetic terms. Analogously the en-
ergy plays no role in the resonance condition, which is
determined only by the electron fraction Ye. Moreover,
in contrast to the standard resonances, the I-resonance
occurs for both neutrinos and antineutrinos simultane-
ously [13]. Finally, as indicated in Fig. 3 the νe’s (ν̄e) are
not created as the heaviest (lightest) state but as the in-
termediate state, therefore the flavor composition of the
neutrinos arriving at the H-resonance is exactly the op-
posite of the case without NSI. As we show in Sec. V, this
fact can lead to important observational consequences.
In order to calculate the hopping probability between
matter eigenstates at the I-resonance we use the Landau-
ν m2 ν
FIG. 3: Level-crossing schemes, first panel is for the case of
normal hierarchy (oscillations only), the second includes the
NSI effect. The two lower panels correspond to the inverse hi-
erarchy, oscillations only and oscillations + NSI, respectively.
Zener approximation for two flavors
P ILZ ≈ e−
γI , (20)
where γI stands for the adiabaticity parameter, which
can be generally written as
Em2 − Em1
, (21)
where ϑ̇m ≡ dϑm/dr. If one applies this for-
mula to the e − τ ′ box of Eq. (14) assuming that
tan 2ϑmI = 2H
eτ/(H
ττ − Hee) and Em2 − Em1 =
(H ′ττ −Hee)2 + 4H ′eτ
one gets
4H ′2eτ
(Ḣ ′ττ − Ḣee)
16V0ρε
(1 + εI)3Ẏe
≈ 4× 109rs,5ρ11ε′2eτf(εI) , (22)
where the parametrization of the Ye profile has been de-
fined as in Eq. (9) with b = 0.16. The density ρ11 rep-
resents the density in units of 1011 g/cm3, rs,5 stands
for rs in units of 10
5 cm, and f(εI) is a function whose
value is of the order O(1) in the range of parameters we
are interested in. Taking all these factors into account
it follows that the internal resonance will be adiabatic
provided that ε′eτ & 10
−5, well below the current limits,
in full numerical agreement with, e. g., Ref. [47].
In Fig. 4 we show the resonance condition as well as
the adiabaticity in terms of εττ and εeτ assuming the
other εαβ = 0. In order to illustrate the dependence on
time we consider profiles inspired in the numerical profiles
of Fig. 1 at t = 2 s (upper panel) and 15.7 s (bottom
panel). For definiteness we take Y mine as the electron
fraction at which the density has value of 5× 1011g/cm3.
For comparison with Fig. 2 we have assumed Y mine =
10−2 in the case of 15.7 s. We observe how the border
of adiabaticity depends on εττ through the value of the
density at rI which in turn depends on time.
Before moving to the discussion of the outer resonances
a comment is in order, namely, how does the formalism
change for other non-standard interaction models. First
note that the whole treatment presented above also ap-
plies to the case of NSI on up-type quarks, except that
the position of the internal resonance shifts with respect
to the down-quark case. Indeed, in this case the NSI
potential
(V unsi)αβ = ε
αβV0ρ(1 + Ye) , (23)
 0.001  0.01  0.1  1
 0.001  0.01  0.1  1
FIG. 4: Contours of constant jump probability at the I-
resonance in terms of εττ and εeτ for two profiles correspond-
ing to Fig. 1 at 2 s with a = 0.235 and b = 0.175 (upper panel)
and 15.7 s with a = 0.26 and b = 0.195 (bottom panel). For
simplicity the other ε’s have been set to zero.
would induce a similar internal resonance for the condi-
tion Ye = ε
I/(1− εI).
In contrast, for the case of NSI with electrons, the
NSI potential is proportional to the electron fraction, and
therefore no internal resonance would appear.
B. Neutrino Evolution in the Outer Regions
In the outer layers of the SN envelope neutrinos can un-
dergo important flavor transitions at those points where
the matter induced potential equals the kinetic terms.
In absence of NSI this condition can be expressed as
VCC ≈ ∆m2/(2E). Neutrino oscillation experiments in-
dicate two mass scales, ∆m2atm and ∆m
⊙ ≡ m22−m21 [5],
hence two different resonance layers arise, the so-called
H-resonance and the L-resonance, respectively.
The presence of NSI with values of |εαβ| . 10−2 modi-
fies the properties of the H and L transitions [48, 49, 50].
In particular one finds that the effects of the NSI can be
described as in the standard case by embedding the ε’s
into effective mixing angles [50]. An analogous “confu-
sion” between sinϑ13 and the corresponding NSI param-
eter εeτ has been pointed out in the context of long-
baseline neutrino oscillations in Refs. [53, 54].
In this section we perform a more general and com-
plementary study for slightly higher values of the NSI
parameters: |εαβ | & few 10−2, still allowed by current
limits, and for which the I-resonance could occur.
The phenomenological assumption of hierarchical
squared mass differences, |∆m2atm| ≫ ∆m2⊙, allows, for
not too large ε’s, a factorization of the 3ν dynamics
into two 2ν subsystems roughly decoupled for the H
and L transitions [65]. To isolate the dynamics of the
H transition, one usually rotates the neutrino flavor ba-
sis by U †(ϑ23), and extracts the submatrix with indices
(1,3) [48, 50]. Whereas this method works perfectly for
small values of εαβ it can be dangerous for values above
10−2. In order to analyze how much our case deviates
from the simplest approximation we have performed a
rotation with the angle ϑ′′23 ≡ ϑ23 − α instead of just
ϑ23. By requiring that the new rotation diagonalizes the
submatrix (2,3) at the H-resonance layer one obtains the
following expression for the correction angle α
tan(2α) =
∆⊙s212s13 + V
ττ s223 − 2V NSIµτ c223
(∆atm +
∆⊙c212(−3 + c213)
+V NSIττ c223 + 2V
µτ s223
, (24)
where ∆atm ≡ ∆m2atm/(2E) and ∆⊙ ≡ ∆m2⊙/(2E). In
our notation sij and s2ij represent sinϑij and sin(2ϑij),
respectively. The parameters cij and c2ij are analogously
defined. In the absence of NSI α is just a small correction
to ϑ23 [97],
tan(2α) ≈ ∆⊙s212s13/∆atmc213 . O(10−3) . (25)
In order to calculate α we need to know the H-
resonance point. To calculate it one can proceed as in
the case without NSI, namely, make the ϑ′′23 rotation and
analyze the submatrix (1, 3). The new Hamiltonian H ′′αβ
has now the form
H ′′ee = V0ρ[Ye + εee(2− Ye)] + ∆atms213
+∆⊙(c
12 + s
13) ,
H ′′ττ = V0ρ(2− Ye)ε′′ττ +∆atmc213c2α
c213c
α + (sαc12 + cαs12s13)
H ′′eτ = V0ρ(2− Ye)ε′′eτ +
∆atms213cα
∆⊙(−c13sαs212 + c212cαs213) . (26)
We have defined ε′′ττ = εττc
23−α + εµτs223−α, and ε
εeτ c23−α+εeµs23−α, where s23−α ≡ sin(ϑ23−α), c23−α ≡
cos(ϑ23 − α), and s223−α ≡ sin(2ϑ23 − 2α), c223−α ≡
cos(2ϑ23 − 2α). The resonance condition for the H tran-
sition, H ′′ee = H
ττ can be then written as
H [Y He + (εee − ε′′ττ )(2− Y He )] = ∆atm(c213c2α − s213)
+∆⊙[c
13 − c2αs213)− s2αs212 + 12s2αs212s13] .(27)
It can be easily checked how in the limit of εαβ → 0 one
recovers the standard resonance condition,
HY He ≈ ∆atmc213 . (28)
In the region where the H-resonance occurs Y He ≈ 0.5.
Taking into account Eqs. (24) and (27) one can already
estimate how the value of α changes with the NSI param-
eters. In Fig. 5 we show the dependence of α on the εττ
after fixing the value of the other NSI parameters. One
can see how for εττ & 10
−2 the approximation of neglect-
ing α significantly worsens. Assuming ϑ23 = π/4 and a
fixed value of εµτ one can easily see that εττ basically
affects the numerator in Eq. (24). Therefore one expects
a rise of α as the value of εττ increases, as seen in Fig. 5.
The dependence of α on εµτ is correlated to the rela-
tive sign of the mass hierarchy and εµτ . For instance,
for normal mass hierarchy and positive values of εµτ the
dependence is inverse, namely, higher values of εµτ lead
to a suppression of α. Apart from this general behav-
ior, α also depends on the diagonal term εee as seen in
Fig. 5. This effect occurs by shifting the resonance point
through the resonance condition in Eq. (27).
One can now calculate the jump probability be-
tween matter eigenstates in analogy to the I-resonance
by means of the Landau-Zener approximation, see
Eqs. (20), (21), and 22,
PHLZ ≈ e−
γH , (29)
where γH represents the adiabaticity parameter at the
FIG. 5: Angle α as function of εττ for different values of εee
and εµτ , in the case of neutrinos of energy 10 MeV, with
normal mass hierarchy, and s213 = 10
−5. The other NSI pa-
rameters take the following values: εeµ = 0 and εeτ = 10
H-resonance, which can be written as
4H ′′2eτ
(Ḣ ′′ττ − Ḣ ′′ee)
, (30)
where the expressions for H ′′αβ are given in Eqs (26).
Let us first consider the case |εαβ| . 10−2. In this case
α ≈ 0 and one can rewrite the adiabaticity parameter as
∆atm sin
cos(2ϑ
)|d ln V/dr|rH
, (31)
where
= ϑ13 + ε
eτ (2− Ye)/Ye (32)
in agreement with Ref. [50]. For slightly larger ε’s there
can be significant differences. In Fig. 6 we show PHLZ in
the εeτ -εττ plane for antineutrinos with energy 10 MeV
in the case of inverse mass hierarchy, using Eq. (29) with
(upper panel) and without (bottom panel) the α cor-
rection. The values of ϑ13 and εeτ have been chosen so
that the jump probability lies in the transition regime be-
tween adiabatic and strongly non adiabatic. In the limit
of small εττ , α becomes negligible and therefore both re-
sults coincide. From Eq. (31) one sees how as the value of
εeτ increases γH gets larger and therefore the transition
becomes more and more adiabatic. For negative values
of εeτ there can be a cancellation between εeτ and ϑ13,
and as a result the transition becomes non-adiabatic.
An additional consequence of Eq. (32) is that a degen-
eracy between εeτ and ϑ13 arises. This is seen in Fig. 7,
which gives the contours of PH
in terms of εeτ and ϑ13
for εττ = 10
−4. One sees clearly that the same Landau-
Zener hopping probability is obtained for different com-
binations of εeτ and ϑ13. This leads to an intrinsic “con-
fusion” between the mixing angle and the corresponding
NSI parameter, which can not be disentangled only in
the context of SN neutrinos, as noted in Ref. [50].
We now turn to the case of |εττ | ≥ 10−2. As |εττ |
increases the role of α becomes relevant. Whereas in the
bottom panel PHLZ remains basically independent of εττ ,
one can see how in the upper panel PHLZ becomes strongly
sensitive to εττ for |εττ | ≥ 10−2.
One sees that for positive values of εττ it tends to adi-
abaticity whereas for negative values to non-adiabaticity.
This follows from the dependence of H ′′eτ on α, essen-
tially through the term −∆⊙c13sαs212, see Eq. (26). For
|εττ | ≥ 10−2 one sees that sinα starts being important,
and as a result this term eventually becomes of the same
order as the others in H ′′eτ . At this point the sign of
εττ , and so the sign of sinα, is crucial since it may con-
tribute to the enhancement or reduction of H ′′eτ . This
directly translates into a trend towards adiabaticity or
non-adiabaticity, seen in Fig. 6. Thus, for the range of
εττ relevant for the NSI-induced internal resonance the
adiabaticity of the outer H resonance can be affected in
a non-trivial way.
Turning to the case of the L transition a similar expres-
sion can be obtained by rotating the original Hamiltonian
by U(ϑ13)
†U(ϑ23)
† [48, 50]. However, in contrast to the
case of the H-resonance, where the mixing angle ϑ13 is
still unknown, in the case of the L transition the angle
ϑ12 has been shown by solar and reactor neutrino exper-
iments to be large [5]. As a result, for the mass scale ∆⊙
this transition will always be adiabatic irrespective of the
values of εαβ, and will affect only neutrinos.
FIG. 6: Landau-Zener jump probability isocontours at the H-
resonance in terms of εeτ and εττ for 10 MeV antineutrinos
in the case of inverted mass hierarchy. Upper panel: α given
by Eq. (24). Bottom panel: α set to zero. The remaining
parameters take the following values: sin2 ϑ13 = 10
, εeτ =
10−3, εee = εeµ = 0. See text.
V. OBSERVABLES AND SENSITIVITY
As mentioned in the introduction one of the major mo-
tivations to study NSI using the neutrinos emitted in a
SN is the enhancement of the NSI effects on the neutrino
propagation through the SN envelope due to the specific
extreme matter conditions that characterize it. In this
section we analyze how these effects translate into ob-
servable effects in the case of a future galactic SN.
Schematically, the neutrino emission by a SN can be di-
vided into four stages: Infall phase, neutronization burst,
accretion, and Kelvin-Helmholtz cooling phase. During
the infall phase and neutronization burst only νe’s are
emitted, while the bulk of neutrino emission is released
in all flavors in the last two phases. Whereas the neutrino
emission characteristics of the two initial stages are basi-
FIG. 7: Landau-Zener jump probability isocontours at the
H-resonance in terms of εeτ and ϑ13 for εττ = 10
−4. An-
tineutrinos with energy 10 MeV and inverted mass hierarchy
has been assumed.
cally independent of the features of the progenitor, such
as the core mass or equation of state (EoS), the details
of the neutrino spectra and luminosity during the ac-
cretion and cooling phases may significantly change for
different progenitor models. As a result, a straightfor-
ward extraction of oscillation parameters from the bulk
of the SN neutrino signal seems hopeless. Only features
in the detected neutrino spectra which are independent
of unknown SN parameters should be used in such an
analysis [66].
The question then arises as to how can one obtain in-
formation about the NSI parameters. Taking into ac-
count that the main effect of NSI is to generate new in-
ternal neutrino flavor transitions, one possibility is to in-
voke theoretical arguments that involve different aspects
of the SN internal dynamics.
In Ref. [47] it was argued that such an internal flavor
conversion during the first second after the core bounce
might play a positive role in the so-called SN shock re-
heating problem. It is observed in numerical simula-
tions [67, 68, 69, 70] that as the shock wave propa-
gates it loses energy until it gets stalled at a few hun-
dred km. It is currently believed that after neutrinos
escape the SN core they can to some extent deposit en-
ergy right behind and help the shock wave continue out-
wards. On the other hand it is also believed that due to
the composition in matter of the protoneutronstar (PNS)
the mean energies of the different neutrino spectra obey
〈Eνe〉 < 〈Eν̄e〉 < 〈Eνµ,ντ 〉. This means that a reso-
nant conversion between νe(ν̄e) and νµ,τ (ν̄µ,τ ) between
the neutrinosphere and the position of the stalled shock
wave would make the νe(ν̄e) spectra harder, and there-
fore the energy deposition would be larger, giving rise to
a shock wave regeneration effect.
Another argument used in the literature was the pos-
sibility that the r−process nucleosynthesis, responsible
for synthesizing about half of the heavy elements with
mass number A > 70 in nature, could occur in the region
above the neutrinosphere in SNe [71, 72]. A necessary
condition is Ye < 0.5 in the nucleosynthesis region. The
value of the electron fraction depends on the neutrino
absorption rates, which are determined in turn by the
νe(ν̄e) luminosities and energy distribution. These can
be altered by flavor conversion in the inner layers due to
the presence of NSI. Therefore by requiring the electron
fraction be below 0.5 one can get information about the
values of the NSI parameters.
While it is commonly accepted that neutrinos will play
a crucial role in both the shock wave re-heating as well
as the r−process nucleosynthesis, there are still other as-
trophysical factors that can affect both. While the issue
remains under debate we prefer to stick to arguments
directly related to physical observables in a large water
Cherenkov detector. There are several possibilities.
(A) the modulations in the ν̄e spectra due to the pas-
sage of shock waves through the supernova [58, 59,
(B) the modulation in the ν̄e spectra due to the time
dependence of the electron fraction, induced by the
I-resonance
(C) the modulations in the ν̄e spectra due to the Earth
matter [73, 74, 75, 76]
(D) detectability of the neutronization νe burst [77, 78]
Three of these observables, 1, 3 and 4 have already been
considered in the literature in the context of neutrino
oscillations. Here we discuss the potential of the above
promising observables in providing information about the
Scheme Hierarchy sin2 ϑ13 NSI Psurv P̄surv
A normal & 10−4 No 0 cos2 ϑ12
B inverted & 10−4 No sin2 ϑ12 0
C any . 10−6 No sin2 ϑ12 cos
AI normal & 10−4 Yes sin2 ϑ12 sin
BI inverted & 10−4 Yes cos2 ϑ12 cos
CIa normal . 10−6 Yes 0 sin2 ϑ12
CIb inverted . 10−6 Yes cos2 ϑ12 0
TABLE I: Definition of the neutrino schemes considered in
terms of the hierarchy, the value of ϑ13, and the presence
of NSI, as described in the text. The values of the survival
probabilities for νe (Psurv) and ν̄e (P̄surv) for each case are
also indicated.
NSI parameters. It is important to pay attention to the
possible ocurrence of the internal I-resonance and to its
effect in the external H and L-resonances. The first can
induce a genuinely new observable effect, item 2 above.
Here we concentrate on neutral current-type non-
standard interactions, hence there will be not effect in the
main reaction in water Cherenkov and scintillator detec-
tors, namely the inverse beta decay, ν̄e+p → e++n [98].
For definiteness we take NSI with d (down) quarks, in
which case the NSI effects will be confined to the neu-
trino evolution inside the SN and the Earth, through the
vector component of the interaction.
From all possible combinations of NSI parameters we
will concentrate on those for which the internal I tran-
sition does take place, namely |εI | & 10−2, see Fig. 2.
Concerning the FC NSI parameters we will consider |ε′eτ |
between few × 10−5 and 10−2, range in which the I-
resonance is adiabatic, see Fig. 4. In the following discus-
sion we will focus on the extreme cases defined in Table I.
One of the motivations for considering these cases is the
fact that the resonances involved become either adiabatic
or strongly non adiabatic, and hence the survival prob-
abilities in the absence of Earth effects or shock wave
passage, become energy independent. This assumption
simplifies the task of relating the observables with the
neutrino schemes.
A. Shock wave propagation
During approximately the first two seconds after the
core bounce, the neutrino survival probabilities are con-
stant in time and in energy for all cases mentioned in Ta-
ble I. Only the Earth effects could introduce an energy
dependence. However, at t ≈ 2 s the H-resonance layer is
reached by the outgoing shock wave, see Fig. 1. The way
the shock wave passage affects the neutrino propagation
strongly depends on the neutrino mixing scenario. In the
absence of NSI cases A and C will not show any evidence
of shock wave propagation in the observed ν̄e spectrum,
either because there is no resonance in the antineutrino
channel as in scenario A, or because the H-resonance is
always strongly non-adiabatic as in scenario C. How-
ever, in scenario B, the sudden change in density breaks
the adiabaticity of the resonance, leading to a time and
energy dependence of the electron antineutrino survival
probability P̄surv(E, t). In the upper panel of Fig. 8 we
show P̄surv(E, t) in the particular case that two shock
waves are present, one forward and a reverse one [60].
The presence of the shocks results in the appearance of
bumps in survival probability at those energies for which
the resonance region is passed by the shock waves. All
these structures move in time towards higher energies, as
the shock waves reach regions with lower density, leading
to observable consequences in the ν̄e spectrum.
We now turn to the case where NSI are present, which
opens the possibility of internal resonances. When such
I-resonance is adiabatic the situation will be similar to
the case without NSI. For normal mass hierarchy, AI and
CIa, ν̄e will not feel the H-resonance and therefore the
adiabaticity-breaking effect will not basically alter their
propagation. In contrast, for inverted mass hierarchy
and large ϑ13, case BI, the H-resonance occurs in the
antineutrino channel and therefore ν̄e will feel the shock
wave passage. However, in contrast to case B now ν̄e will
reach the H-resonance in a different matter eigenstate:
ν̄m1 instead of ν̄
3 , see Fig. 3. That means that before
the shock wave reaches the H-resonance the ν̄e survival
probability will be P̄surv ≈ cos2 ϑ12 ≈ 0.7. Once the
adiabaticity of the H-resonance is broken by the shock
wave then ν̄e will partly leave as ν̄
3 and therefore the
survival probability will decrease. As a consequence one
expects a pattern in time and energy for the survival
FIG. 8: Survival probability P̄surv(E, t) for ν̄e as function
of energy at different times averaged in energies with the en-
ergy resolution of Super-Kamiokande; for the profile shown in
Fig. 1. Upper panel: case B is assumed for sin2 ϑ13 = 10
Bottom panel: case BI , with εττ = 0.07, εeτ = 10
−4 and the
rest of NSI parameters put to zero.
probability in the case BI to be roughly opposite than in
the case B, see bottom panel of Fig. 8. The position of
the peaks and dips en each panel do not exactly coincide
as the value of εττ roughly shifts the position of the H-
resonance.
In the left panels of Fig. 9 we represent in light-shaded
(yellow) the range of εeτ and εττ for which this opposite
shock wave imprint would be observable. In the upper
panels we have assumed a minimum value of the electron
fraction of 0.06, based on the numerical profiles at t =
2 s of Fig. 1. In the bottom panels Y mine is set to 0.01,
inspired in the profiles at t = 15.7 s. It can be seen how as
time goes on the range of εττ ’s for which the I-resonance
takes place widens towards to smaller and smaller values.
This is a direct consequence of the steady deleptonization
of the inner layers.
For smaller ϑ13, case CIb, the situation is different.
Except for relatively large εeτ values theH-resonance will
be strongly non-adiabatic, as in case C. Therefore the
passage of the shock waves will not significantly change
the ν̄e survival probability and will not lead to any ob-
servable effect. In the right panels of Fig. 9 we show
the same as in the left panels but for sin2 ϑ13 = 10
Whereas for large values of ϑ13, left panels, the H-
resonance is always adiabatic and one has only to ensure
the adiabaticity of the I-resonance, for smaller values of
ϑ13 the adiabaticity of the H-resonance strongly depends
on the values of εeτ and εττ , as discussed in Sec. IVB.
This can be seen as a significant reduction of the yel-
low area. Only large values of either εeτ or εττ would
still allow for a clear identification of the opposite shock
wave effects. In dark-shaded (cyan) we show the region
of parameters for which PH lies in the transition region
between adiabatic and strongly non-adiabatic, and there-
fore could still lead to some effect.
A useful observable to detect effects of the shock prop-
agation is the average of the measured positron energies,
〈Ee〉, produced in inverse beta decays. In Fig. 10, we
show 〈Ee〉 together with the one sigma errors expected for
a Megaton water Cherenkov detector and a SN at 10 kpc
distance, with a time binning of 0.5 s, for different neu-
trino schemes: caseB and caseBI with different values of
εττ . For the neutrino fluxes we assumed the parametriza-
tion given by Refs. [79, 80] with 〈E0(ν̄e)〉 = 15 MeV and
〈E0(ν̄µ,τ )〉 = 18 MeV and the following ratio of the total
neutrino fluxes Φ0(ν̄e)/Φ0(ν̄µ,τ ) = 0.8 [99].
One can see how the features of the average positron
energy are a direct consequence of the shape of the sur-
vival probability, where dips have to be translated into
bumps and vice-versa.
Thus, it is important to stress that whereas in case
B one expects the presence of one or two dips (depend-
ing on the structure of the shock wave, see Ref [60]), or
nothing in the other cases, one or two bumps are ex-
pected in case BI, as seen in the upper left panel of
Fig. 10. As discussed in Ref. [60] the details of the
dips/bump will depend on the exact shape of the neu-
trino fluxes, but as long as general reasonable assump-
tions like 〈Eν̄e〉 . 〈Eν̄µ,τ 〉 are considered the dips/bumps
should be observed.
B. Time variation of Ye
We have just seen how the distorsion of the density
profile due to the shock wave passage through the outer
FIG. 9: Range of εττ and εeτ for which the effect of the shock
wave will be observed. In the upper panels a minimum value
of Y mine = 0.06 based on the numerical profiles at t = 2 s
has been assumed, see Fig. 1. In the lower panels we have
considered a case with Y mine = 0.01 inspired in the profile
at t = 15.7 s. The value of sin2 ϑ13 has been assumed to
be 10−2 and 10−7 in the left and right panels, respectively.
We have also superimposed isocontours of constant hopping
probability 0.1 (blue) and 0.9 (red) in the I (solid lines) and
H (dashed lines) resonances for inverted mass hierarchy and
E = 10 MeV and antineutrinos. The area in yellow represents
the parameter space where both resonances will be adiabatic.
In the cyan area the I-resonance is assumed to be adiabatic
whereas H lies in the transition region.
SN envelope can induce a time-dependent modulation in
the ν̄e spectrum in cases B and BI. However the time
dependence of the electron fraction Ye can also reveal the
presence of NSI leaving a clear imprint in the observed
ν̄e spectrum, as we now explain.
As discussed in Sec. IVA the region of NSI parame-
ters leading to I-resonance is basically determined by the
minimum and maximum values of the electron fraction,
Y mine and Y
e . The crucial point is that as the delep-
tonization of the proto-neutron star goes on, the value of
Y mine steadily decreases with time. As a result, the range
of NSI strengths for which the I-resonance takes place
FIG. 10: The average energy of ν̄p → ne+ events binned in
time for case B (dashed blue) and BI (solid red). In each
panel different values of εττ have been assumed. The error
bars represent 1 σ errors in any bin. εeτ = 10
increases with time, as can be seen in Fig. 2.
Let us first discuss the observational consequences of
the time dependence of the electron fraction in case BI.
If εττ (ε
I in general) is large enough the I-resonance will
take place right after the core bounce. In this case, as
seen in the upper left panel of Fig. 10 the two bumps we
have just discussed in Sec. VA would be clearly observed.
However for smaller NSI parameter values it could hap-
pen that the I-resonance occurs only after several sec-
onds. In particular for the specific Ye profile considered
we show how this delay could be of roughly 2, 4 or 9 sec
for values of εττ of 0.025, 0.02 or 0.015, respectively, see
last three panels Fig. 10. As can be inferred from the
figure this delay effect can lead to misidentification of
the pure NSI effect. So, for instance, in the upper right
panel, one sees how the two bumps might also be inter-
preted as two dips, given the astrophysical uncertainties.
This subtle degeneracy can only be solved by extra in-
formation on, for example, the time dependence of the
spectra or the velocity of the shock wave. Given the su-
pernova model, however, the time structure of the signal
could eventually not only point out the presence of NSI
but even potentially indicate a range of NSI parameters.
Let us now turn to the normal mass hierarchy sce-
nario (cases AI and CIa). In analogy to the BI case,
if εI is relatively large the onset of the I-resonance will
take place early on. As can be inferred from Fig. 3 that
implies that ν̄e will escape the SN as ν̄2. For smaller
values, though, it may happen that the I-resonance be-
comes effective only after a few seconds. This means
that during the first seconds of the neutrino signal ν̄e
would leave the star as ν̄1 (cases A and C). Then,
after some point, the electron fraction would be low
enough to switch on the I-resonance, and consequently
ν̄e would enter the Earth as ν̄2. This would result in
a transition in the electron antineutrino survival prob-
ability from P̄surv ≈ cos2 ϑ12 = 0.7 to sin2 ϑ12 = 0.3.
Given the expected hierarchy in the average neutrino en-
ergies 〈Eν̄e〉 . 〈Eν̄µ,τ 〉, it follows that the change in Ye
would lead to a hardening of the observed positron spec-
trum. The effect is quantified in Fig. 11 for different
values of εττ . The figure shows the average energy of
the ν̄p → ne+ events for the case of a Megaton water
Cherenkov detector exactly as in Fig. 10, but for scenar-
ios AI and CIa. One can see how for εττ = 0.07 the I-
resonance condition is always fulfilled and therefore there
is no time dependence. However for smaller values one
can see a rise at a certain moment which depends on the
magnitude of εττ . A similar effect would occur in case
C. Earth matter effects
Before the shock wave reaches the H-resonance layer
the dependence of the neutrino survival probability in the
cases we are considering, on the neutrino energy E is very
weak. However, if neutrinos cross the Earth before reach-
ing the detector, the conversion probabilities may become
energy-dependent, inducing modulations in the neutrino
energy spectrum. These modulations may be observed
in the form of local peaks and valleys in the spectrum of
the event rate σFDν̄e plotted as a function of 1/E. These
modulations arise in the antineutrino channel only when
ν̄e leave the SN as ν̄1 or ν̄2. In the absence of NSI this
happens in cases A and C, where ν̄e leave the star as ν̄1.
FIG. 11: The average energy of ν̄p → ne+ events binned in
time for case AI and CIa and different values of εττ . The
error bars represent 1 σ errors in any bin. εeτ = 10
In the presence of NSI ν̄e will arrive at the Earth as ν̄1
in cases BI, and as ν̄2 in case AI and CIa. Therefore
its observation would exclude cases B and CIb. This
distortion in the spectra could be measured by compar-
ing the neutrino signal at two or more different detectors
such that the neutrinos travel different distances through
the Earth before reaching them [73, 74]. However these
Earth matter effects can be also identified in a single de-
tector [75, 76].
By analyzing the power spectrum of the detected neu-
trino events one can identify the presence of peaks located
at the frequencies characterizing the modulation. These
do not dependend on the primary neutrino spectra, and
can be determined to a good accuracy from the knowl-
edge of the solar oscillation parameters, the Earth matter
density, and the position of the SN in the sky [76]. The
latter can be determined with sufficient precision even if
the SN is optically obscured using the pointing capability
of water Cherenkov neutrino detectors [81].
This method turns out to be powerful in detecting the
modulations in the spectra due to Earth matter effects,
and thus in ruling out cases B and CIb. However, the po-
sition of the peaks does not depend on how ν̄e enters the
Earth, as ν̄1 or ν̄2. Hence it is not useful to discriminate
case AI and CIa from the cases A, C, and BI.
The time dependence of Ye, however, can transform
case B into BI, and C with inverse hierarchy into CIb,
leading respectively to an appearance and disappearance
of these Earth matter effects. In case BI the presence of
the shock wave modulation can spoil a clear identification
of the Earth matter effects. Nevertheless, the disappear-
ance of the Earth matter effects in the transition from
case C to CIb allows us to pin down case CIb.
D. Neutronization burst
The prompt neutronization burst takes place during
the first ∼ 25 ms after the core bounce with a typical
full width half maximum of 5–7ms and a peak luminos-
ity of 3.3–3.5×1053 erg s−1. The striking similarity of the
neutrino emission characteristics despite the variability
in the properties of the pre-collapse cores is caused by
a regulation mechanism between electron number frac-
tion and target abundances for electron capture. This
effectively establishes similar electron fractions in the in-
ner core during collapse, leading to a convergence of the
structure of the central part of the collapsing cores, with
only small differences in the evolution of different pro-
genitors until shock breakout [77, 78].
Taking into account that the SN will be likely to be
obscured by dust and a good estimation of the distance
will not be possible, the time structure of the detected
neutrino signal should be used as signature for the neu-
tronization burst. In Ref. [78] it was shown that such a
time structure can be in principle cleanly seen in the
case of a Megaton water Cherenkov detector. It was
also shown how the time evolution of the signal depends
strongly on the neutrino mixing scheme. In the absence
of NSI the νe peak could be observed provided that the
νe survival probability Pνeνe is not zero. As can be seen
in Table I this happens for cases B and C. However
for case A (normal mass hierarchy and “large” ϑ13), νe
leaves the SN as ν3. This leads to a survival probability
Pνeνe ≈ sin2 ϑ13 . 10−1, and therefore the peak remains
hidden.
Let us now consider the situation where NSI are
prensent. For normal mass hierarchy νe, which is born
as νm2 passes through three different resonances, I, H
and L. Whereas I and L will be adiabatic, the fate of
H will depend on the value of ϑ13. For “large” values,
case AI, the H-resonance will also be adiabatic. This
implies that νe’s will leave as ν2, the survival probability
will be Pνeνe ≈ sin2 ϑ12 ≈ 0.3, and therefore the peak
will be seen, as in cases B and C. If ϑ13 happens to
be very small, case CIa, then H will be strongly non-
adiabatic and therefore νe will leave the star as ν3. As a
consequence the neutronization peak will not be seen.
For inverse mass hierarchy, νe is born as ν
1 and tra-
verses adiabatically I and L. This implies that they will
leave the star as ν1 and therefore the peak will also be
observed. However now the survival probability will be
larger, Pνeνe ≈ cos2 ϑ12 ≈ 0.7. Thus for a given known
normalization, i.e. the distance to the SN, one expects a
larger number of events during the neutronization peak
in this case. In Fig. 12 we show the expected number
of events per time bin in a water Cherenkov detector in
the case of a SN exploding at 10 kpc, for two different
neutrino schemes, C and BI, and for different SN pro-
genitor masses. One can see how the difference due to
the larger survival probability is bigger than the typi-
cal error bars, associated to the lack of knowledge of the
progenitor mass.
Two comments are in order. The neutronization νe
burst takes place during the first milliseconds, before
strong deleptonization takes place. As a result, in con-
trast to other observables we have considered in this pa-
per, here the I-resonance will only occur for εI & 10−1.
On the other hand in the presence of additional NSI with
electrons this would significantly affect the ν − e cross
sections, and consequently the results presented here.
VI. SUMMARY
We have analyzed the possibility of observing clear sig-
natures of non-standard neutrino interactions from the
detection of neutrinos produced in a future galactic su-
pernova.
In Secs. III and IV we have re-considered effect of ν−d
non-standard interactions on the neutrino propagation
through the SN envelope within a three-neutrino frame-
work. In contrast to previous works we have analyzed
the neutrino evolution in both the more deleptonized in-
FIG. 12: Number of events from the elastic scattering on elec-
trons, per time bin in a Megaton water Cherenkov detector for
a SN at 10 kpc for cases C (dashed lines) and BI (solid lines).
Different progenitor masses have been assumed: 13 M⊙ (n13)
in red, 15 M⊙ (s15s7b2) in black, and 25 M⊙ (s25a28) in blue.
1-sigma errors are also shown for the 15 M⊙ case.
ner layers and the outer regions of the SN envelope. We
have also taken into account the time dependence of the
SN density and electron fraction profiles.
First we have found that the small values of the elec-
tron fraction typical of the former allows for internal NSI-
induced resonant conversions, in addition to the standard
MSW-H and MSW-L resonances of the outer envelope.
These new flavor conversions take place for a relatively
large range of NSI parameters, namely |εαα| between
10−2 − 10−1, and |εeτ | & few × 10−5, currently allowed
by experiment. For this range of strengths, in particu-
lar εττ , non-standard interactions can significantly affect
the adiabaticity of the H-resonance. On the other hand
the NSI-induced resonant conversions may also lead to
the modulation of the ν̄e spectra as a result of the time
dependence of the electron fraction.
In Sec. V we have studied the possibility of detecting
NSI effects in a Megaton water Cherenkov detector us-
ing the modulation effects in the ν̄e spectrum due to (i)
the passage of shock waves through the SN envelope, (ii)
the time dependence of the electron fraction and (iii) the
Earth matter effects; and, finally, through the possible
detectability of the neutronization νe burst. Note that
observable (ii) turns out to be complementary to the ob-
servation of the shock wave passage, (i), and offers the
possibility to probe NSI effects also for normal hierarchy
neutrino spectra.
In Table II we summarize the results obtained for dif-
ferent neutrino schemes. We have found that observable
(i) can clearly indicate the existence of NSI in the case
of inverse mass hierarchy and large ϑ13 (case BI). On
the other hand, observable (ii) allows for an identification
of NSI effects in the other cases, normal mass hierarchy
(cases AI and CIa) and inverse mass hierarchy and small
ϑ13 (case CIb). Therefore a positive signal of either ob-
servable (i) or (ii) would establish the existence of NSI. In
the latter case this would, however, leave a degeneracy
among cases AI, CIa, and CIb. Such degeneracy can
be broken with the help of observables (iii) and the ob-
servation of the neutronization νe burst. The detection
of Earth matter effects during the whole supernova neu-
trino signal would rule out case CIb since, as discussed in
Sec. VC, a disappearance of Earth matter effects would
take place due to a transition from C to CIb. Finally,
the (non) observation of the neutronization burst can be
used to distinguish between cases AI and CIa.
Similarly, other degeneracies in Table II may be lifted
by suitably combining different observables. For exam-
ple, a negative of observable (ii) could mean either neg-
ligible NSI strengths or (NU) NSI parameter values so
large that the internal resonance is always present. In
this case one could use the observation of the neutron-
ization burst in order to establish the presence of NSI for
the case of inverse mass hierarchy. In addition the ob-
servation of the shock wave imprint in the ν̄e spectrum
would provide additional information on ϑ13.
In conclusion, by suitably combining all observables
one may establish not only the presence of NSI, but also
the mass hierarchy and probe the magnitude of ϑ13.
Acknowledgments
The authors wish to thank H-Th. Janka, O. Miranda,
S. Pastor, Th. Schwetz, and M. Tórtola for fruitful discus-
sions. Work supported by the Spanish grant FPA2005-
Scheme Hierarchy sin2 ϑ13 NSI shock Ye Earth νe burst
A normal & 10−4 No No No Yes No
B inverted & 10−4 No Yes No No Yes
C any . 10−6 No No No Yes Yes
AI normal & 10−4 Yes No Yes Yes Yes
BI inverted & 10−4 Yes Yes⋆ No Yes Yes⋆
CIa normal . 10−6 Yes No Yes Yes No
CIb inverted . 10−6 Yes No Yes No Yes⋆
TABLE II: Expectations for the observables discussed in the
text: modulation of the ν̄e spectrum due to the shock wave
passage, the time variation of Ye, the Earth effect, and the
observation of the νe burst within various neutrino schemes.
Asterisks indicate that the effect differs from that expected
in the absence of NSI. See text.
01269 and European Network of Theoretical Astroparti-
cle Physics ILIAS/N6 under contract number RII3-CT-
2004-506222. A. E. has been supported by a FPU grant
from the Spanish Government. R. T. has been supported
by the Juan de la Cierva program from the Spanish Gov-
ernment and by an ERG from the European Commission.
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agation.
[95] Here we neglect the possible effects of density fluctua-
tions [87, 88] taking place during the shock wave prop-
agation. For a detailed study of the phenomenological
consequences see Refs. [89, 90].
[96] The alternative condition H ′ee = H
µµ would give rise to
another internal resonance which can be studied using
the same method. For brevity, we will not pursue this in
this paper.
[97] Note that, in the limit of high densities one recovers
the rotation angle obtained for the internal I-resonance
23 → ϑ
23 after neglecting the kinetic terms.
[98] For the case of NSI with electrons both the vector and
axial components of εeαβ will contribute to the ν−e cross
section.
[99] We assume that for the values of the NSI parameters con-
sidered the initial neutrino spectra do not significantly
change.
http://arxiv.org/abs/astro-ph/0608695
http://arxiv.org/abs/hep-ph/0701182
http://arxiv.org/abs/astro-ph/9604061
http://arxiv.org/abs/hep-ph/9602307
http://arxiv.org/abs/hep-ph/0603033
http://arxiv.org/abs/astro-ph/0607244

  We analyze the possibility of probing non-standard neutrino interactions
(NSI, for short) through the detection of neutrinos produced in a future
galactic supernova (SN).We consider the effect of NSI on the neutrino
propagation through the SN envelope within a three-neutrino framework, paying
special attention to the inclusion of NSI-induced resonant conversions, which
may take place in the most deleptonised inner layers. We study the possibility
of detecting NSI effects in a Megaton water Cherenkov detector, either through
modulation effects in the $\bar\nu_e$ spectrum due to (i) the passage of shock
waves through the SN envelope, (ii) the time dependence of the electron
fraction and (iii) the Earth matter effects; or, finally, through the possible
detectability of the neutronization $\nu_e$ burst. We find that the $\bar\nu_e$
spectrum can exhibit dramatic features due to the internal NSI-induced resonant
conversion. This occurs for non-universal NSI strengths of a few %, and for
very small flavor-changing NSI above a few$\times 10^{-5}$.

IFIC/07-03
Probing non-standard neutrino interactions with supernova neutrinos
A. Esteban-Pretel, R. Tomàs and J. W. F. Valle1
1AHEP Group, Institut de F́ısica Corpuscular - C.S.I.C/Universitat de València
Edifici Instituts d’Investigació, Apt. 22085, E-46071 València, Spain
(Dated: November 4, 2018)
We analyze the possibility of probing non-standard neutrino interactions (NSI, for short) through
the detection of neutrinos produced in a future galactic supernova (SN). We consider the effect of
NSI on the neutrino propagation through the SN envelope within a three-neutrino framework, paying
special attention to the inclusion of NSI-induced resonant conversions, which may take place in the
most deleptonised inner layers. We study the possibility of detecting NSI effects in a Megaton water
Cherenkov detector, either through modulation effects in the ν̄e spectrum due to (i) the passage
of shock waves through the SN envelope, (ii) the time dependence of the electron fraction and (iii)
the Earth matter effects; or, finally, through the possible detectability of the neutronization νe
burst. We find that the ν̄e spectrum can exhibit dramatic features due to the internal NSI-induced
resonant conversion. This occurs for non-universal NSI strengths of a few %, and for very small
flavor-changing NSI above a few×10−5.
PACS numbers: 13.15.+g, 14.60.Lm, 14.60.Pq, 14.60.St, 97.60.Bw
I. INTRODUCTION
The very first data of the KamLAND collaboration [1]
have been enough to isolate neutrino oscillations as the
correct mechanism explaining the solar neutrino prob-
lem [2, 3], indicating also that large mixing angle (LMA)
was the right solution. The 766.3 ton-yr KamLAND data
sample further strengthens the validity of the LMA os-
cillation interpretation of the data [4].
Current data imply that neutrino have mass. For an
updated review of the current status of neutrino oscil-
lations see [5]. Theories of neutrino mass [6, 7] typ-
ically require that neutrinos have non-standard prop-
erties such as neutrino electromagnetic transition mo-
ments [8, 9, 10] or non-standard four-Fermi interactions
(NSI, for short) [11, 12, 13]. The expected magnitude of
the NSI effects is rather model-dependent.
Seesaw-type models lead to a non-trivial structure of
the lepton mixing matrix characterizing the charged and
neutral current weak interactions [6]. The NSI which
are induced by the charged and neutral current gauge
interactions may be sizeable [14, 15, 16, 17, 18]. Alter-
natively, non-standard neutrino interactions may arise in
models where neutrinos masses are radiatively “calcula-
ble” [19, 20]. Finally, in some supersymmetric unified
models, the strength of non-standard neutrino interac-
tions may arise from renormalization and/or threshold
effects [21].
We stress that non-standard interactions strengths are
highly model-dependent. In some models NSI strengths
are too small to be relevant for neutrino propagation,
because they are either suppressed by some large mass
scale or restricted by limits on neutrino masses, or both.
However, this need not be the case, and there are many
theoretically attractive scenarios where moderately large
NSI strengths are possible and consistent with the small-
ness of neutrino masses. In fact one can show that
NSI may exist even in the limit of massless neutri-
nos [14, 15, 16, 17, 18]. Such may also occur in the
context of fully unified models like SO(10) [22].
We argue that, in addition to the precision determi-
nation of the oscillation parameters, it is necessary to
test for sub-leading non-oscillation effects that could arise
from non-standard neutrino interactions. These are nat-
ural outcome of many neutrino mass models and can be of
two types: flavor-changing (FC) and non-universal (NU).
These are constrained by existing experiments (see be-
low) and, with neutrino experiments now entering a pre-
cision phase [23], an improved determination of neutrino
parameters and their theoretical impact constitute an im-
portant goal in astroparticle and high energy physics [5].
Here we concentrate on the impact of non-standard
http://arxiv.org/abs/0704.0032v1
neutrino interactions on supernova physics. We show
how complementary information on the NSI parame-
ters could be inferred from the detection of core-collapse
supernova neutrinos. The motivation for the study is
twofold. First, if a future SN event takes place in our
Galaxy the number of neutrino events expected in the
current or planned neutrino detectors would be enor-
mous, O(104 − 105) [24]. Moreover, the extreme con-
ditions under which neutrinos have to travel since they
are created in the SN core, in strongly deleptonised re-
gions at nuclear densities, until they reach the Earth,
lead to strong matter effects. In particular the effect of
small values of the NSI parameters can be dramatically
enhanced, possibly leading to observable consequences.
This paper is planned as follows. In Sec. II we summa-
rize the current observational bounds on the parameters
describing the NSI, including previous works on NSI in
SNe. In Sec. III we describe the neutrino propagation
formalism as well as the SN profiles which will be used.
In Sec. IV we analyze the effect of NSI on the ν propaga-
tion in the inner regions near the neutrinosphere and in
the outer regions of the SN envelope. In Sec. V we discuss
the possibility of using various observables to probe the
presence of NSI in the neutrino signal of a future galactic
SN. Finally in Sec. VI we present our conclusions.
II. PRELIMINARIES
A large class of non-standard interactions may be
parametrized with the effective low-energy four-fermion
operator:
LNSI = −εfPαβ 2
2GF (ν̄αγµLνβ)(f̄γ
µPf) , (1)
where P = L, R and f is a first generation fermion:
e, u, d. The coefficients ε
αβ denote the strength of the
NSI between the neutrinos of flavors α and β and the
P−handed component of the fermion f .
Current constraints on ε
αβ come from a variety of dif-
ferent sources, which we now briefly list.
A. Laboratory
Neutrino scattering experiments [25, 26, 27, 28,
29] provide the following bounds, |εfPµµ | . 10−3 −
10−2, |εfPee | . 10−1 − 1, |εfPµτ | . 0.05, |εfPeτ | . 0.5 at
90 % C.L [30, 31, 32]. On the other hand the analysis
of the e+e− → νν̄γ cross section measured at LEP II
leads to a bound on |εePττ | . 0.5 [33]. Future prospects
to improve the current limits imply the measurement of
sin2 ϑW leptonically in the scattering off electrons in the
target, as well as in neutrino deep inelastic scattering in
a future neutrino factory. The main improvement would
be in the case of |εfPee | and |εfPeτ |, where values as small
as 10−3 and 0.02, respectively, could be reached [31].
The search for flavor violating processes involving
charged leptons is expected to restrict corresponding neu-
trino interactions, to the extent that the SU(2) gauge
symmetry is assumed. However, this can at most give
indicative order-of-magnitude restrictions, since we know
SU(2) is not a good symmetry of nature. Using radiative
corrections it has been argued that, for example, µ − e
conversion on nuclei like in the case of µ−T i also con-
strains |εqPµe | . 7.7× 10−4 [31].
Non-standard interactions can also affect neutrino
propagation through matter, probed in current neutrino
oscillation experiments. The bounds so obtained apply to
the vector coupling constant of the NSI, ε
αβ = ε
since only this appears in neutrino propagation in mat-
ter [91].
B. Solar and reactor
The role of neutrino NSIs as subleading effects on the
solar neutrino oscillations and KamLAND has been re-
cently considered in Ref. [34, 35, 36] with the following
bounds at 90 % CL for ε ≡ − sinϑ23εdVeτ with the al-
lowed range −0.93 . ε . 0.30, while for the diagonal
term ε′ ≡ sin2 ϑ23εdVττ − εdVee , the only forbidden region is
[0.20, 0.78] [36]. Only in the ideal case of infinitely pre-
cise solar neutrino oscillation parameters determination,
the allowed range would “close from the left” for negative
NSI parameter values, at −0.6 for ε and −0.7 for ε′.
C. Atmospheric and accelerator neutrinos
Non-standard interactions involving muon neutrinos
can be constrained by atmospheric neutrino experiments
as well as accelerator neutrino oscillation searches at
K2K and MINOS. In Ref. [37] Super-Kamiokande and
MACRO observations of atmospheric neutrinos were con-
sidered in the framework of two neutrinos. The limits ob-
tained were −0.05 . εdVµτ < 0.04 and |εdVττ − εdVµµ | . 0.17
at 99 % CL. The same data set together with K2K were
recently considered in Refs. [38, 39] to study the nonstan-
dard neutrino interactions in a three generation scheme
under the assumption εeµ = εµµ = εµτ = 0. The al-
lowed region of εττ obtained for values of εeτ smaller
than O(10−1) becomes Σf=u,d,eεfVαβNf/Ne . 0.2 [39] ,
where Nf stands for the fermion number density.
D. Cosmology
If non-standard interactions with electrons were large
they might also lead to important cosmological and as-
trophysical implications. For instance, neutrinos could
be kept in thermal contact with electrons and positrons
longer than in the standard case, hence they would share
a larger fraction of the entropy release from e± annihi-
lations. This would affect the predicted features of the
cosmic background of neutrinos. As recently pointed out
in Ref [40] required couplings are, though, larger than
the current laboratory bounds.
E. NSI in Supernovae
According to the currently accepted supernova (SN)
paradigm, neutrinos are expected to play a crucial role
in SN dynamics. As a result, SN physics provides
a laboratory to probe neutrino properties. Moreover,
many future large neutrino detectors are currently be-
ing discussed [41]. The enormous number of events,
O(104 − 105) that would be “seen” in these detectors in-
dicates that a future SN in our Galaxy would provide a
very sensitive probe of non-standard neutrino interaction
effects.
The presence of NSI can lead to important conse-
quences for the SN neutrino physics both in the highly
dense core as well as in the envelope where neutrinos
basically freely stream.
The role of non-forward neutrino scattering processes
on heavy nuclei and free nucleons giving rise to flavor
change within the SN core has been recently analyzed in
Ref. [42, 43]. The main effect found was a reduction in
the core electron fraction Ye during core collapse. A lower
Ye would lead to a lower homologous core mass, a lower
shock energy, and a greater nuclear photon-disintegration
burden for the shock wave. By allowing a maximum
∆Ye = −0.02 it has been claimed that εeα . 10−3, where
α = µ, τ [43].
On the other hand it has been noted since long ago
that the existence of NSI plays an important role in the
propagation of SN neutrinos through the envelope lead-
ing to the possibility of a new resonant conversion. In
contrast to the well known MSW effect [44, 45] it would
take place even for massless neutrinos [13]. Two basic
ingredients are necessary: universal and flavor changing
NSI. In the original scheme neutrinos were mixed in the
leptonic charged current and universality was violated
thanks to the effect of mixing with heavy gauge singlet
leptons [6, 14]. Such resonance would induce strong neu-
trino flavor conversion both for neutrinos and antineutri-
nos simultaneously, possibly affecting the neutrino sig-
nal of the SN1987A as well as the possibility of having
r−process nucleosynthesis. This was first quantitatively
considered within a two-flavor νe−ντ scheme, and bounds
on the relevant NSI parameters were obtained using both
arguments [46].
One of the main features of the such “internal” or
“massless” resonant conversion mechanism is that it re-
quires the violation of universality, its position being
determined only by the matter chemical composition,
namely the value of the electron fraction Ye, and not by
the density. In view of the experimental upper bounds
on the NSI parameters such new resonance can only take
place in the inner layers of the supernova, near the neu-
trinosphere, where Ye takes its minimum values. In this
region the values of Ye are small enough to allow for
resonance conversions to take place in agreement with
existing bounds on the strengths of non-universal NSI
parameters.
The SN physics implications of another type of NSI
present in supersymmetric R-parity violating models
have also been studied in Ref. [47], again for a system
of two neutrinos. For definiteness NSI on d−quarks were
considered, in two cases: (i) massless neutrinos without
mixing in the presence of flavor-changing (FC) and non-
universal (NU) NSIs, and (ii) neutrinos with eV masses
and FC NSI. Different arguments have been used in
order to constrain the parameters describing the NSI,
namely, the SN1987A signal, the possibility to get suc-
cessful r−process nucleosynthesis, and the possible en-
hancement of the energy deposition behind the shock
wave to reactivate it.
On the other hand several subsequent articles [48, 49,
50] considered the effects of NSI on the neutrino propa-
gation in a three–neutrino mixing scenario for the case
Ye > 0.4, typical for the outer SN envelope. Together
with the assumption that εdVαβ . 10
−2 this prevents the
appearance of internal resonances in contrast to previous
references.
Motivated by supersymmetric theories without R par-
ity, in Ref. [48] the authors considered the effects of
small-strength NSI with d−quarks. Following the for-
malism developed in Refs. [51, 52] they studied the cor-
rections that such NSI would have on the expressions
for the survival probabilities in the standard resonances
MSW-H and MSW-L. A similar analysis was performed
in Ref. [49] assuming Z-induced NSI interactions orig-
inated by additional heavy neutrinos. A phenomeno-
logical generalization of these results was carried out in
Ref. [50]. The authors found an analytical compact ex-
pression for the survival probabilities in which the main
effects of the NSI can be embedded through shifts of the
mixing angles ϑ12 and ϑ13. In contrast to similar expres-
sions found previously these directly apply to all mixing
angles, and in the case with Earth matter effects. The
main phenomenological consequence was the identifica-
tion of a degeneracy between ϑ13 and εeα, similar to the
analogous “confusion” between ϑ13 and the correspond-
ing NSI parameter noted to exist in the context of long-
baseline neutrino oscillations [53, 54].
We have now re-considered the general three–neutrino
mixing scenario with NSI. In contrast to previous
work [48, 49, 50], we have not restricted ourselves to
large values of Ye, discussing also small values present
in the inner layers. This way our generalized descrip-
tion includes both the possibility of neutrinos having the
“massless” NSI-induced resonant conversions in the in-
ner layers of the SN envelope [13, 46, 47], as well as the
“outer” oscillation-induced conversions [48, 49, 50] [92].
III. NEUTRINO EVOLUTION
In this section we describe the main ingredients of our
analysis. Our emphasis will be on the use of astrophys-
ically realistic SN matter and Ye profiles, characterizing
its density and the matter composition. Their details,
in particular their time dependence, are crucial in deter-
mining the way the non-standard neutrino interactions
affect the propagation of neutrinos in the SN medium.
A. Evolution Equation
As discussed in Sec. II in an unpolarized medium the
neutrino propagation in matter will be affected by the
vector coupling constant of the NSI, ε
αβ = ε
αβ [93].
The way the neutral current NSI modifies the neu-
trino evolution will be parametrized phenomenologically
through the effective low-energy four-fermion operator
described in Eq. (1). We also assume ε
αβ ∈ ℜ, neglect-
ing possible CP violation in the new interactions.
Under these assumptions the Hamiltonian describing
the SN neutrino evolution in the presence of NSI can be
cast in the following form [94]
να = (Hkin +Hint)αβ νβ , (2)
where Hkin stands for the kinetic term
Hkin = U
U † , (3)
with M2 = diag(m21,m
3), and U the three-neutrino
lepton mixing matrix [6] in the PDG convention [55] and
with no CP phases.
The second term of the Hamiltonian accounts for the
interaction of neutrinos with matter and can be split into
two pieces,
Hint = H
int +H
int . (4)
The first term, Hstd
describes the standard interaction
with matter and can be written asHstd
= diag (VCC , 0, 0)
up to one loop corrections due to different masses of the
muon and tau leptons [56]. The standard matter poten-
tial for neutrinos is given by
VCC =
2GFNe = V0ρYe , (5)
where V0 ≈ 7.6×10−14 eV, the density is given in g/cm3,
and Ye stands for the relative number of electrons with
respect to baryons. For antineutrinos the potential is
identical but with the sign changed.
The term in the Hamiltonian describing the non-
standard neutrino interactions with a fermion f can be
expressed as,
(Hnsiint )αβ =
f=e,u,d
)αβ , (6)
with (V
)αβ ≡
2GFNfε
αβ. For definiteness and mo-
tivated by actual models, for example, those with broken
R parity supersymmetry we take for f the down-type
quark. However, an analogous treatment would apply
to the case of NSI on up-type quarks, the existence of
NSI with electrons brings no drastic qualitative differ-
ences with respect to the pure oscillation case (see be-
low). Therefore the NSI potential can be expressed as
follows,
(V dnsi)αβ = ε
αβV0ρ(2− Ye) . (7)
From now on we will not explicitely write the superindex
d. In order to further simplify the problem we will rede-
fine the diagonal NSI parameters so that εµµ = 0, as one
can easily see that subtracting a matrix proportional to
the identity leaves the physics involved in the neutrino
oscillation unaffected.
B. Supernova matter profiles
Neutrino propagation depends on the supernova mat-
ter and chemical profile through the effective potential.
This profile exhibits an important time dependence dur-
ing the explosion. Fig. 1 shows the density ρ(t, r) and the
electron fraction Ye(t, r) profiles for the SN progenitor as
well as at different times post-bounce.
Progenitor density profiles can be roughly
parametrized by a power-law function
ρ(r) = ρ0
, (8)
where ρ0 ∼ 104 g/cm3, R0 ∼ 109 cm, and n ∼ 3. The
electron fraction profile varies depending on the matter
composition of the different layers. For instance, typical
values of Ye between 0.42 and 0.45 in the inner regions
are found in stellar evolution simulations [57]. In the in-
termediate regions, where the MSW H and L-resonances
take place Ye ≈ 0.5. This value can further increase in
the most outer layers of the SN envelope due to the pres-
ence of hydrogen.
After the SN core bounce the matter profile is affected
in several ways. First note that a front shock wave starts
to propagate outwards and eventually ejects the SN enve-
lope. The evolution of the shock wave will strongly mod-
ify the density profile and therefore the neutrino propa-
gation [58, 59]. Following Ref. [60] we shall assume that
the structure of the shock wave is more complicated and
an additional “reverse wave” appears due to the collision
of the neutrino-driven wind and the slowly moving mate-
rial behind the forward shock, as seen in the upper panel
of Fig. 1 [95].
On the other hand, the electron fraction is also affected
by the time evolution as the SN explosion proceeds. Once
the collapse starts the core density grows so that the neu-
trinos become eventually effectively trapped within the
so called “neutrinosphere”. At this point the trapped
electron fraction has decreased until values of the order of
0.33 [61]. When the inner core reaches the nuclear density
it can not contract any further and bounces. As a con-
sequence a shock wave forms in the inner core and starts
propagating outwards. When the newly formed super-
nova shock reaches densities low enough for the initially
trapped neutrinos to begin streaming faster than the
shock propagates [62], a breakout pulse of νe is launched.
In the shock-heated matter, which is still rich of elec-
trons and completely disintegrated into free neutrons and
protons, a large number of νe are rapidly produced by
electron captures on protons. They follow the shock on
its way out until they are released in a very luminous
flash, the breakout burst, at about the moment when
the shock penetrates the neutrinosphere and the neutri-
nos can escape essentially unhindered. As a consequence,
the lepton number in the layer around the neutrinosphere
decreases strongly and the matter neutronizes [63]. The
value of Ye steadily decreases in these layers until val-
ues of the order of O(10−2). Outside the neutrinosphere
there is a steep rise until Ye ≈ 0.5. This is a robust
feature of the neutrino-driven baryonic wind. Neutrino
heating drives the wind mass loss and causes Ye to rise
within a few 10 km from low to high values, between 0.45
and 0.55 [64], see bottom panel of Fig. 1. Inspired in the
numerical results of Ref. [60] we have parametrized the
behavior of the electron fraction near the neutrinosphere
phenomenologically as,
Ye = a+ b arctan[(r − r0)/rs] , (9)
where a ≈ 0.23− 0.26 and b ≈ 0.16− 0.20. The param-
eters r0 and rs describe where the rise takes place and
how steep it is, respectively. As can be seen in Fig. 1
both decrease with time.
FIG. 1: Density (upper panel) and electron fraction (bottom
panel) profiles for the SN progenitor and at different instants
after the core bounce, from Ref. [60]. The regions where the
H (yellow) and the L (cyan) resonance take place are also
indicated, as well as the NSI-induced I (gray) resonance for
the parameters εee = 0, εττ . 0.07 and |εµτ | . 0.05
IV. THE TWO REGIMES
In order to study the neutrino propagation through
the SN envelope we will split the problem into two differ-
ent regions: the inner envelope, defined by the condition
VCC ≫ ∆m2atm/(2E) with ∆m2atm ≡ m23 − m22, and the
outer one, where ∆m2atm/(2E) & VCC . From the upper
panel of Fig. 1 one can see how the boundary roughly
varies between r ≈ 108 cm and 109 cm, depending on the
time considered. This way one can fully characterize all
resonances that can take place in the propagation of su-
pernova neutrinos, both the outer resonant conversions
related to neutrino masses and indicated as the upper
bands in Fig. 1, and the inner resonances that follow
from the presence of non-standard neutrino interactions,
indicated by the band at the bottom of the same figure.
Here we pay special attention to the use of realistic mat-
ter and chemical supernova profiles and three-neutrino
flavors thus generalising previous studies.
A. Neutrino Evolution in the Inner Regions
Let us first write the Hamiltonian in the inner layers,
where Hint ≫ Hkin. In this case the Hamiltonian can be
written as
H ≈ Hint = V0ρ(2− Ye)
+ εee εeµ εeτ
εeµ 0 εµτ
εeτ εµτ εττ
When the value of the εαβ is of the same order as the
electron fraction Ye internal resonances can arise [13].
Taking into account the current constraints on the ε’s
discussed in Sec. II one sees that small values of Ye are
required [46, 47]. As a result, these can only take place
in the most deleptonised inner layers, close to the neu-
trinosphere, where the kinetic terms of the Hamiltonian
are negligible.
Given the large number of free parameters εαβ in-
volved we consider one particular case where |εeµ| and
|εeτ | are small enough to neglect a possible initial mixing
between νe and νµ or ντ . Barring fine tuning, this basi-
cally amounts to |εeµ|, |εeτ | ≪ 10−2. According to the
discussion of Sec. II εeµ automatically satisfies the condi-
tion, whereas one expects that the window |εeτ | & 10−2
will eventually be probed in future experiments.
Since the initial fluxes of νµ and ντ are expected to be
basically identical, it is convenient to redefine the weak
basis by performing a rotation in the µ− τ sector:
= U(ϑ′23)
1 0 0
0 c23′ s23′
0 −s23′ c23′
where c23′ and s23′ stand for cos(ϑ
23) and sin(ϑ
23), re-
spectively. The angle ϑ′23 can be written as
tan(2ϑ′23) ≈
. (12)
The Hamiltonian becomes in the new basis
H ′αβ = U
†(ϑ′23)HαβU(ϑ
23) (13)
= V0ρ(2− Ye)
+ εee ε
ε′eµ ε
ε′eτ 0 ε
,(14)
where
ε′eµ = εeµc23′ − εeτs23′ (15)
ε′eτ = εeµs23′ + εeτ c23′ (16)
ε′µµ = (εττ −
ε2ττ + 4ε
µτ )/2 (17)
ε′ττ = (εττ +
ε2ττ + 4ε
µτ )/2 . (18)
With our initial assumptions on εeα one notices that
the new basis ν′α basically diagonalizes the Hamiltonian,
and therefore coincides roughly with the matter eigen-
state basis. A novel resonance can arise if the condition
H ′ee = H
ττ is satisfied, we call this I-resonance, I stand-
ing for “internal” [96]. The corresponding resonance con-
dition can be written as
Y Ie =
1 + εI
, (19)
where εI is defined as ε′ττ − εee. In Fig. 2 we represent
the range of εee and ε
ττ leading to the I-resonance for
an electron fraction profile between different Y mine ’s and
Y maxe = 0.5. It is important to notice that the value
of Y mine depends on time. Right before the collapse the
minimum value of the electron fraction is around 0.4.
Hence the window of NSI parameters that would lead
to a resonance would be relatively narrow, as indicated
by the shaded (yellow) band in Fig. 2. As time goes on
Y mine decreases to values of the order of a few %, and
as a result the region of parameters giving rise to the I-
resonance significantly widens. For example, in the range
|εee| ≤ 10−3 possibly accessible to future experiments one
sees that the I-resonance can take place for values of ε′ττ
of the order of O(10−2). This indicates that the potential
sensitivity on NSI parameters that can be achieved in su-
pernova studies is better than that of the current limits.
FIG. 2: Contours of Y Ie as function of εee and ε
ττ accord-
ing to Eq. (19) for different values of Ye. The region in yel-
low represents the region of parameters that gives rise to I-
resonance before the collapse. The arrows indicate how this
region widens with time.
As seen in Fig. 1 in order to fulfill the I-resonance con-
dition for such small values of the NSI parameters the
values of Ye must indeed lie, as already stated, in the
inner layers.
Several comments are in order: First, in contrast to
the standard H and L-resonances, related to the kinetic
term, the density itself does not explicitly enter into the
resonance condition, provided that the density is high
enough to neglect the kinetic terms. Analogously the en-
ergy plays no role in the resonance condition, which is
determined only by the electron fraction Ye. Moreover,
in contrast to the standard resonances, the I-resonance
occurs for both neutrinos and antineutrinos simultane-
ously [13]. Finally, as indicated in Fig. 3 the νe’s (ν̄e) are
not created as the heaviest (lightest) state but as the in-
termediate state, therefore the flavor composition of the
neutrinos arriving at the H-resonance is exactly the op-
posite of the case without NSI. As we show in Sec. V, this
fact can lead to important observational consequences.
In order to calculate the hopping probability between
matter eigenstates at the I-resonance we use the Landau-
ν m2 ν
FIG. 3: Level-crossing schemes, first panel is for the case of
normal hierarchy (oscillations only), the second includes the
NSI effect. The two lower panels correspond to the inverse hi-
erarchy, oscillations only and oscillations + NSI, respectively.
Zener approximation for two flavors
P ILZ ≈ e−
γI , (20)
where γI stands for the adiabaticity parameter, which
can be generally written as
Em2 − Em1
, (21)
where ϑ̇m ≡ dϑm/dr. If one applies this for-
mula to the e − τ ′ box of Eq. (14) assuming that
tan 2ϑmI = 2H
eτ/(H
ττ − Hee) and Em2 − Em1 =
(H ′ττ −Hee)2 + 4H ′eτ
one gets
4H ′2eτ
(Ḣ ′ττ − Ḣee)
16V0ρε
(1 + εI)3Ẏe
≈ 4× 109rs,5ρ11ε′2eτf(εI) , (22)
where the parametrization of the Ye profile has been de-
fined as in Eq. (9) with b = 0.16. The density ρ11 rep-
resents the density in units of 1011 g/cm3, rs,5 stands
for rs in units of 10
5 cm, and f(εI) is a function whose
value is of the order O(1) in the range of parameters we
are interested in. Taking all these factors into account
it follows that the internal resonance will be adiabatic
provided that ε′eτ & 10
−5, well below the current limits,
in full numerical agreement with, e. g., Ref. [47].
In Fig. 4 we show the resonance condition as well as
the adiabaticity in terms of εττ and εeτ assuming the
other εαβ = 0. In order to illustrate the dependence on
time we consider profiles inspired in the numerical profiles
of Fig. 1 at t = 2 s (upper panel) and 15.7 s (bottom
panel). For definiteness we take Y mine as the electron
fraction at which the density has value of 5× 1011g/cm3.
For comparison with Fig. 2 we have assumed Y mine =
10−2 in the case of 15.7 s. We observe how the border
of adiabaticity depends on εττ through the value of the
density at rI which in turn depends on time.
Before moving to the discussion of the outer resonances
a comment is in order, namely, how does the formalism
change for other non-standard interaction models. First
note that the whole treatment presented above also ap-
plies to the case of NSI on up-type quarks, except that
the position of the internal resonance shifts with respect
to the down-quark case. Indeed, in this case the NSI
potential
(V unsi)αβ = ε
αβV0ρ(1 + Ye) , (23)
 0.001  0.01  0.1  1
 0.001  0.01  0.1  1
FIG. 4: Contours of constant jump probability at the I-
resonance in terms of εττ and εeτ for two profiles correspond-
ing to Fig. 1 at 2 s with a = 0.235 and b = 0.175 (upper panel)
and 15.7 s with a = 0.26 and b = 0.195 (bottom panel). For
simplicity the other ε’s have been set to zero.
would induce a similar internal resonance for the condi-
tion Ye = ε
I/(1− εI).
In contrast, for the case of NSI with electrons, the
NSI potential is proportional to the electron fraction, and
therefore no internal resonance would appear.
B. Neutrino Evolution in the Outer Regions
In the outer layers of the SN envelope neutrinos can un-
dergo important flavor transitions at those points where
the matter induced potential equals the kinetic terms.
In absence of NSI this condition can be expressed as
VCC ≈ ∆m2/(2E). Neutrino oscillation experiments in-
dicate two mass scales, ∆m2atm and ∆m
⊙ ≡ m22−m21 [5],
hence two different resonance layers arise, the so-called
H-resonance and the L-resonance, respectively.
The presence of NSI with values of |εαβ| . 10−2 modi-
fies the properties of the H and L transitions [48, 49, 50].
In particular one finds that the effects of the NSI can be
described as in the standard case by embedding the ε’s
into effective mixing angles [50]. An analogous “confu-
sion” between sinϑ13 and the corresponding NSI param-
eter εeτ has been pointed out in the context of long-
baseline neutrino oscillations in Refs. [53, 54].
In this section we perform a more general and com-
plementary study for slightly higher values of the NSI
parameters: |εαβ | & few 10−2, still allowed by current
limits, and for which the I-resonance could occur.
The phenomenological assumption of hierarchical
squared mass differences, |∆m2atm| ≫ ∆m2⊙, allows, for
not too large ε’s, a factorization of the 3ν dynamics
into two 2ν subsystems roughly decoupled for the H
and L transitions [65]. To isolate the dynamics of the
H transition, one usually rotates the neutrino flavor ba-
sis by U †(ϑ23), and extracts the submatrix with indices
(1,3) [48, 50]. Whereas this method works perfectly for
small values of εαβ it can be dangerous for values above
10−2. In order to analyze how much our case deviates
from the simplest approximation we have performed a
rotation with the angle ϑ′′23 ≡ ϑ23 − α instead of just
ϑ23. By requiring that the new rotation diagonalizes the
submatrix (2,3) at the H-resonance layer one obtains the
following expression for the correction angle α
tan(2α) =
∆⊙s212s13 + V
ττ s223 − 2V NSIµτ c223
(∆atm +
∆⊙c212(−3 + c213)
+V NSIττ c223 + 2V
µτ s223
, (24)
where ∆atm ≡ ∆m2atm/(2E) and ∆⊙ ≡ ∆m2⊙/(2E). In
our notation sij and s2ij represent sinϑij and sin(2ϑij),
respectively. The parameters cij and c2ij are analogously
defined. In the absence of NSI α is just a small correction
to ϑ23 [97],
tan(2α) ≈ ∆⊙s212s13/∆atmc213 . O(10−3) . (25)
In order to calculate α we need to know the H-
resonance point. To calculate it one can proceed as in
the case without NSI, namely, make the ϑ′′23 rotation and
analyze the submatrix (1, 3). The new Hamiltonian H ′′αβ
has now the form
H ′′ee = V0ρ[Ye + εee(2− Ye)] + ∆atms213
+∆⊙(c
12 + s
13) ,
H ′′ττ = V0ρ(2− Ye)ε′′ττ +∆atmc213c2α
c213c
α + (sαc12 + cαs12s13)
H ′′eτ = V0ρ(2− Ye)ε′′eτ +
∆atms213cα
∆⊙(−c13sαs212 + c212cαs213) . (26)
We have defined ε′′ττ = εττc
23−α + εµτs223−α, and ε
εeτ c23−α+εeµs23−α, where s23−α ≡ sin(ϑ23−α), c23−α ≡
cos(ϑ23 − α), and s223−α ≡ sin(2ϑ23 − 2α), c223−α ≡
cos(2ϑ23 − 2α). The resonance condition for the H tran-
sition, H ′′ee = H
ττ can be then written as
H [Y He + (εee − ε′′ττ )(2− Y He )] = ∆atm(c213c2α − s213)
+∆⊙[c
13 − c2αs213)− s2αs212 + 12s2αs212s13] .(27)
It can be easily checked how in the limit of εαβ → 0 one
recovers the standard resonance condition,
HY He ≈ ∆atmc213 . (28)
In the region where the H-resonance occurs Y He ≈ 0.5.
Taking into account Eqs. (24) and (27) one can already
estimate how the value of α changes with the NSI param-
eters. In Fig. 5 we show the dependence of α on the εττ
after fixing the value of the other NSI parameters. One
can see how for εττ & 10
−2 the approximation of neglect-
ing α significantly worsens. Assuming ϑ23 = π/4 and a
fixed value of εµτ one can easily see that εττ basically
affects the numerator in Eq. (24). Therefore one expects
a rise of α as the value of εττ increases, as seen in Fig. 5.
The dependence of α on εµτ is correlated to the rela-
tive sign of the mass hierarchy and εµτ . For instance,
for normal mass hierarchy and positive values of εµτ the
dependence is inverse, namely, higher values of εµτ lead
to a suppression of α. Apart from this general behav-
ior, α also depends on the diagonal term εee as seen in
Fig. 5. This effect occurs by shifting the resonance point
through the resonance condition in Eq. (27).
One can now calculate the jump probability be-
tween matter eigenstates in analogy to the I-resonance
by means of the Landau-Zener approximation, see
Eqs. (20), (21), and 22,
PHLZ ≈ e−
γH , (29)
where γH represents the adiabaticity parameter at the
FIG. 5: Angle α as function of εττ for different values of εee
and εµτ , in the case of neutrinos of energy 10 MeV, with
normal mass hierarchy, and s213 = 10
−5. The other NSI pa-
rameters take the following values: εeµ = 0 and εeτ = 10
H-resonance, which can be written as
4H ′′2eτ
(Ḣ ′′ττ − Ḣ ′′ee)
, (30)
where the expressions for H ′′αβ are given in Eqs (26).
Let us first consider the case |εαβ| . 10−2. In this case
α ≈ 0 and one can rewrite the adiabaticity parameter as
∆atm sin
cos(2ϑ
)|d ln V/dr|rH
, (31)
where
= ϑ13 + ε
eτ (2− Ye)/Ye (32)
in agreement with Ref. [50]. For slightly larger ε’s there
can be significant differences. In Fig. 6 we show PHLZ in
the εeτ -εττ plane for antineutrinos with energy 10 MeV
in the case of inverse mass hierarchy, using Eq. (29) with
(upper panel) and without (bottom panel) the α cor-
rection. The values of ϑ13 and εeτ have been chosen so
that the jump probability lies in the transition regime be-
tween adiabatic and strongly non adiabatic. In the limit
of small εττ , α becomes negligible and therefore both re-
sults coincide. From Eq. (31) one sees how as the value of
εeτ increases γH gets larger and therefore the transition
becomes more and more adiabatic. For negative values
of εeτ there can be a cancellation between εeτ and ϑ13,
and as a result the transition becomes non-adiabatic.
An additional consequence of Eq. (32) is that a degen-
eracy between εeτ and ϑ13 arises. This is seen in Fig. 7,
which gives the contours of PH
in terms of εeτ and ϑ13
for εττ = 10
−4. One sees clearly that the same Landau-
Zener hopping probability is obtained for different com-
binations of εeτ and ϑ13. This leads to an intrinsic “con-
fusion” between the mixing angle and the corresponding
NSI parameter, which can not be disentangled only in
the context of SN neutrinos, as noted in Ref. [50].
We now turn to the case of |εττ | ≥ 10−2. As |εττ |
increases the role of α becomes relevant. Whereas in the
bottom panel PHLZ remains basically independent of εττ ,
one can see how in the upper panel PHLZ becomes strongly
sensitive to εττ for |εττ | ≥ 10−2.
One sees that for positive values of εττ it tends to adi-
abaticity whereas for negative values to non-adiabaticity.
This follows from the dependence of H ′′eτ on α, essen-
tially through the term −∆⊙c13sαs212, see Eq. (26). For
|εττ | ≥ 10−2 one sees that sinα starts being important,
and as a result this term eventually becomes of the same
order as the others in H ′′eτ . At this point the sign of
εττ , and so the sign of sinα, is crucial since it may con-
tribute to the enhancement or reduction of H ′′eτ . This
directly translates into a trend towards adiabaticity or
non-adiabaticity, seen in Fig. 6. Thus, for the range of
εττ relevant for the NSI-induced internal resonance the
adiabaticity of the outer H resonance can be affected in
a non-trivial way.
Turning to the case of the L transition a similar expres-
sion can be obtained by rotating the original Hamiltonian
by U(ϑ13)
†U(ϑ23)
† [48, 50]. However, in contrast to the
case of the H-resonance, where the mixing angle ϑ13 is
still unknown, in the case of the L transition the angle
ϑ12 has been shown by solar and reactor neutrino exper-
iments to be large [5]. As a result, for the mass scale ∆⊙
this transition will always be adiabatic irrespective of the
values of εαβ, and will affect only neutrinos.
FIG. 6: Landau-Zener jump probability isocontours at the H-
resonance in terms of εeτ and εττ for 10 MeV antineutrinos
in the case of inverted mass hierarchy. Upper panel: α given
by Eq. (24). Bottom panel: α set to zero. The remaining
parameters take the following values: sin2 ϑ13 = 10
, εeτ =
10−3, εee = εeµ = 0. See text.
V. OBSERVABLES AND SENSITIVITY
As mentioned in the introduction one of the major mo-
tivations to study NSI using the neutrinos emitted in a
SN is the enhancement of the NSI effects on the neutrino
propagation through the SN envelope due to the specific
extreme matter conditions that characterize it. In this
section we analyze how these effects translate into ob-
servable effects in the case of a future galactic SN.
Schematically, the neutrino emission by a SN can be di-
vided into four stages: Infall phase, neutronization burst,
accretion, and Kelvin-Helmholtz cooling phase. During
the infall phase and neutronization burst only νe’s are
emitted, while the bulk of neutrino emission is released
in all flavors in the last two phases. Whereas the neutrino
emission characteristics of the two initial stages are basi-
FIG. 7: Landau-Zener jump probability isocontours at the
H-resonance in terms of εeτ and ϑ13 for εττ = 10
−4. An-
tineutrinos with energy 10 MeV and inverted mass hierarchy
has been assumed.
cally independent of the features of the progenitor, such
as the core mass or equation of state (EoS), the details
of the neutrino spectra and luminosity during the ac-
cretion and cooling phases may significantly change for
different progenitor models. As a result, a straightfor-
ward extraction of oscillation parameters from the bulk
of the SN neutrino signal seems hopeless. Only features
in the detected neutrino spectra which are independent
of unknown SN parameters should be used in such an
analysis [66].
The question then arises as to how can one obtain in-
formation about the NSI parameters. Taking into ac-
count that the main effect of NSI is to generate new in-
ternal neutrino flavor transitions, one possibility is to in-
voke theoretical arguments that involve different aspects
of the SN internal dynamics.
In Ref. [47] it was argued that such an internal flavor
conversion during the first second after the core bounce
might play a positive role in the so-called SN shock re-
heating problem. It is observed in numerical simula-
tions [67, 68, 69, 70] that as the shock wave propa-
gates it loses energy until it gets stalled at a few hun-
dred km. It is currently believed that after neutrinos
escape the SN core they can to some extent deposit en-
ergy right behind and help the shock wave continue out-
wards. On the other hand it is also believed that due to
the composition in matter of the protoneutronstar (PNS)
the mean energies of the different neutrino spectra obey
〈Eνe〉 < 〈Eν̄e〉 < 〈Eνµ,ντ 〉. This means that a reso-
nant conversion between νe(ν̄e) and νµ,τ (ν̄µ,τ ) between
the neutrinosphere and the position of the stalled shock
wave would make the νe(ν̄e) spectra harder, and there-
fore the energy deposition would be larger, giving rise to
a shock wave regeneration effect.
Another argument used in the literature was the pos-
sibility that the r−process nucleosynthesis, responsible
for synthesizing about half of the heavy elements with
mass number A > 70 in nature, could occur in the region
above the neutrinosphere in SNe [71, 72]. A necessary
condition is Ye < 0.5 in the nucleosynthesis region. The
value of the electron fraction depends on the neutrino
absorption rates, which are determined in turn by the
νe(ν̄e) luminosities and energy distribution. These can
be altered by flavor conversion in the inner layers due to
the presence of NSI. Therefore by requiring the electron
fraction be below 0.5 one can get information about the
values of the NSI parameters.
While it is commonly accepted that neutrinos will play
a crucial role in both the shock wave re-heating as well
as the r−process nucleosynthesis, there are still other as-
trophysical factors that can affect both. While the issue
remains under debate we prefer to stick to arguments
directly related to physical observables in a large water
Cherenkov detector. There are several possibilities.
(A) the modulations in the ν̄e spectra due to the pas-
sage of shock waves through the supernova [58, 59,
(B) the modulation in the ν̄e spectra due to the time
dependence of the electron fraction, induced by the
I-resonance
(C) the modulations in the ν̄e spectra due to the Earth
matter [73, 74, 75, 76]
(D) detectability of the neutronization νe burst [77, 78]
Three of these observables, 1, 3 and 4 have already been
considered in the literature in the context of neutrino
oscillations. Here we discuss the potential of the above
promising observables in providing information about the
Scheme Hierarchy sin2 ϑ13 NSI Psurv P̄surv
A normal & 10−4 No 0 cos2 ϑ12
B inverted & 10−4 No sin2 ϑ12 0
C any . 10−6 No sin2 ϑ12 cos
AI normal & 10−4 Yes sin2 ϑ12 sin
BI inverted & 10−4 Yes cos2 ϑ12 cos
CIa normal . 10−6 Yes 0 sin2 ϑ12
CIb inverted . 10−6 Yes cos2 ϑ12 0
TABLE I: Definition of the neutrino schemes considered in
terms of the hierarchy, the value of ϑ13, and the presence
of NSI, as described in the text. The values of the survival
probabilities for νe (Psurv) and ν̄e (P̄surv) for each case are
also indicated.
NSI parameters. It is important to pay attention to the
possible ocurrence of the internal I-resonance and to its
effect in the external H and L-resonances. The first can
induce a genuinely new observable effect, item 2 above.
Here we concentrate on neutral current-type non-
standard interactions, hence there will be not effect in the
main reaction in water Cherenkov and scintillator detec-
tors, namely the inverse beta decay, ν̄e+p → e++n [98].
For definiteness we take NSI with d (down) quarks, in
which case the NSI effects will be confined to the neu-
trino evolution inside the SN and the Earth, through the
vector component of the interaction.
From all possible combinations of NSI parameters we
will concentrate on those for which the internal I tran-
sition does take place, namely |εI | & 10−2, see Fig. 2.
Concerning the FC NSI parameters we will consider |ε′eτ |
between few × 10−5 and 10−2, range in which the I-
resonance is adiabatic, see Fig. 4. In the following discus-
sion we will focus on the extreme cases defined in Table I.
One of the motivations for considering these cases is the
fact that the resonances involved become either adiabatic
or strongly non adiabatic, and hence the survival prob-
abilities in the absence of Earth effects or shock wave
passage, become energy independent. This assumption
simplifies the task of relating the observables with the
neutrino schemes.
A. Shock wave propagation
During approximately the first two seconds after the
core bounce, the neutrino survival probabilities are con-
stant in time and in energy for all cases mentioned in Ta-
ble I. Only the Earth effects could introduce an energy
dependence. However, at t ≈ 2 s the H-resonance layer is
reached by the outgoing shock wave, see Fig. 1. The way
the shock wave passage affects the neutrino propagation
strongly depends on the neutrino mixing scenario. In the
absence of NSI cases A and C will not show any evidence
of shock wave propagation in the observed ν̄e spectrum,
either because there is no resonance in the antineutrino
channel as in scenario A, or because the H-resonance is
always strongly non-adiabatic as in scenario C. How-
ever, in scenario B, the sudden change in density breaks
the adiabaticity of the resonance, leading to a time and
energy dependence of the electron antineutrino survival
probability P̄surv(E, t). In the upper panel of Fig. 8 we
show P̄surv(E, t) in the particular case that two shock
waves are present, one forward and a reverse one [60].
The presence of the shocks results in the appearance of
bumps in survival probability at those energies for which
the resonance region is passed by the shock waves. All
these structures move in time towards higher energies, as
the shock waves reach regions with lower density, leading
to observable consequences in the ν̄e spectrum.
We now turn to the case where NSI are present, which
opens the possibility of internal resonances. When such
I-resonance is adiabatic the situation will be similar to
the case without NSI. For normal mass hierarchy, AI and
CIa, ν̄e will not feel the H-resonance and therefore the
adiabaticity-breaking effect will not basically alter their
propagation. In contrast, for inverted mass hierarchy
and large ϑ13, case BI, the H-resonance occurs in the
antineutrino channel and therefore ν̄e will feel the shock
wave passage. However, in contrast to case B now ν̄e will
reach the H-resonance in a different matter eigenstate:
ν̄m1 instead of ν̄
3 , see Fig. 3. That means that before
the shock wave reaches the H-resonance the ν̄e survival
probability will be P̄surv ≈ cos2 ϑ12 ≈ 0.7. Once the
adiabaticity of the H-resonance is broken by the shock
wave then ν̄e will partly leave as ν̄
3 and therefore the
survival probability will decrease. As a consequence one
expects a pattern in time and energy for the survival
FIG. 8: Survival probability P̄surv(E, t) for ν̄e as function
of energy at different times averaged in energies with the en-
ergy resolution of Super-Kamiokande; for the profile shown in
Fig. 1. Upper panel: case B is assumed for sin2 ϑ13 = 10
Bottom panel: case BI , with εττ = 0.07, εeτ = 10
−4 and the
rest of NSI parameters put to zero.
probability in the case BI to be roughly opposite than in
the case B, see bottom panel of Fig. 8. The position of
the peaks and dips en each panel do not exactly coincide
as the value of εττ roughly shifts the position of the H-
resonance.
In the left panels of Fig. 9 we represent in light-shaded
(yellow) the range of εeτ and εττ for which this opposite
shock wave imprint would be observable. In the upper
panels we have assumed a minimum value of the electron
fraction of 0.06, based on the numerical profiles at t =
2 s of Fig. 1. In the bottom panels Y mine is set to 0.01,
inspired in the profiles at t = 15.7 s. It can be seen how as
time goes on the range of εττ ’s for which the I-resonance
takes place widens towards to smaller and smaller values.
This is a direct consequence of the steady deleptonization
of the inner layers.
For smaller ϑ13, case CIb, the situation is different.
Except for relatively large εeτ values theH-resonance will
be strongly non-adiabatic, as in case C. Therefore the
passage of the shock waves will not significantly change
the ν̄e survival probability and will not lead to any ob-
servable effect. In the right panels of Fig. 9 we show
the same as in the left panels but for sin2 ϑ13 = 10
Whereas for large values of ϑ13, left panels, the H-
resonance is always adiabatic and one has only to ensure
the adiabaticity of the I-resonance, for smaller values of
ϑ13 the adiabaticity of the H-resonance strongly depends
on the values of εeτ and εττ , as discussed in Sec. IVB.
This can be seen as a significant reduction of the yel-
low area. Only large values of either εeτ or εττ would
still allow for a clear identification of the opposite shock
wave effects. In dark-shaded (cyan) we show the region
of parameters for which PH lies in the transition region
between adiabatic and strongly non-adiabatic, and there-
fore could still lead to some effect.
A useful observable to detect effects of the shock prop-
agation is the average of the measured positron energies,
〈Ee〉, produced in inverse beta decays. In Fig. 10, we
show 〈Ee〉 together with the one sigma errors expected for
a Megaton water Cherenkov detector and a SN at 10 kpc
distance, with a time binning of 0.5 s, for different neu-
trino schemes: caseB and caseBI with different values of
εττ . For the neutrino fluxes we assumed the parametriza-
tion given by Refs. [79, 80] with 〈E0(ν̄e)〉 = 15 MeV and
〈E0(ν̄µ,τ )〉 = 18 MeV and the following ratio of the total
neutrino fluxes Φ0(ν̄e)/Φ0(ν̄µ,τ ) = 0.8 [99].
One can see how the features of the average positron
energy are a direct consequence of the shape of the sur-
vival probability, where dips have to be translated into
bumps and vice-versa.
Thus, it is important to stress that whereas in case
B one expects the presence of one or two dips (depend-
ing on the structure of the shock wave, see Ref [60]), or
nothing in the other cases, one or two bumps are ex-
pected in case BI, as seen in the upper left panel of
Fig. 10. As discussed in Ref. [60] the details of the
dips/bump will depend on the exact shape of the neu-
trino fluxes, but as long as general reasonable assump-
tions like 〈Eν̄e〉 . 〈Eν̄µ,τ 〉 are considered the dips/bumps
should be observed.
B. Time variation of Ye
We have just seen how the distorsion of the density
profile due to the shock wave passage through the outer
FIG. 9: Range of εττ and εeτ for which the effect of the shock
wave will be observed. In the upper panels a minimum value
of Y mine = 0.06 based on the numerical profiles at t = 2 s
has been assumed, see Fig. 1. In the lower panels we have
considered a case with Y mine = 0.01 inspired in the profile
at t = 15.7 s. The value of sin2 ϑ13 has been assumed to
be 10−2 and 10−7 in the left and right panels, respectively.
We have also superimposed isocontours of constant hopping
probability 0.1 (blue) and 0.9 (red) in the I (solid lines) and
H (dashed lines) resonances for inverted mass hierarchy and
E = 10 MeV and antineutrinos. The area in yellow represents
the parameter space where both resonances will be adiabatic.
In the cyan area the I-resonance is assumed to be adiabatic
whereas H lies in the transition region.
SN envelope can induce a time-dependent modulation in
the ν̄e spectrum in cases B and BI. However the time
dependence of the electron fraction Ye can also reveal the
presence of NSI leaving a clear imprint in the observed
ν̄e spectrum, as we now explain.
As discussed in Sec. IVA the region of NSI parame-
ters leading to I-resonance is basically determined by the
minimum and maximum values of the electron fraction,
Y mine and Y
e . The crucial point is that as the delep-
tonization of the proto-neutron star goes on, the value of
Y mine steadily decreases with time. As a result, the range
of NSI strengths for which the I-resonance takes place
FIG. 10: The average energy of ν̄p → ne+ events binned in
time for case B (dashed blue) and BI (solid red). In each
panel different values of εττ have been assumed. The error
bars represent 1 σ errors in any bin. εeτ = 10
increases with time, as can be seen in Fig. 2.
Let us first discuss the observational consequences of
the time dependence of the electron fraction in case BI.
If εττ (ε
I in general) is large enough the I-resonance will
take place right after the core bounce. In this case, as
seen in the upper left panel of Fig. 10 the two bumps we
have just discussed in Sec. VA would be clearly observed.
However for smaller NSI parameter values it could hap-
pen that the I-resonance occurs only after several sec-
onds. In particular for the specific Ye profile considered
we show how this delay could be of roughly 2, 4 or 9 sec
for values of εττ of 0.025, 0.02 or 0.015, respectively, see
last three panels Fig. 10. As can be inferred from the
figure this delay effect can lead to misidentification of
the pure NSI effect. So, for instance, in the upper right
panel, one sees how the two bumps might also be inter-
preted as two dips, given the astrophysical uncertainties.
This subtle degeneracy can only be solved by extra in-
formation on, for example, the time dependence of the
spectra or the velocity of the shock wave. Given the su-
pernova model, however, the time structure of the signal
could eventually not only point out the presence of NSI
but even potentially indicate a range of NSI parameters.
Let us now turn to the normal mass hierarchy sce-
nario (cases AI and CIa). In analogy to the BI case,
if εI is relatively large the onset of the I-resonance will
take place early on. As can be inferred from Fig. 3 that
implies that ν̄e will escape the SN as ν̄2. For smaller
values, though, it may happen that the I-resonance be-
comes effective only after a few seconds. This means
that during the first seconds of the neutrino signal ν̄e
would leave the star as ν̄1 (cases A and C). Then,
after some point, the electron fraction would be low
enough to switch on the I-resonance, and consequently
ν̄e would enter the Earth as ν̄2. This would result in
a transition in the electron antineutrino survival prob-
ability from P̄surv ≈ cos2 ϑ12 = 0.7 to sin2 ϑ12 = 0.3.
Given the expected hierarchy in the average neutrino en-
ergies 〈Eν̄e〉 . 〈Eν̄µ,τ 〉, it follows that the change in Ye
would lead to a hardening of the observed positron spec-
trum. The effect is quantified in Fig. 11 for different
values of εττ . The figure shows the average energy of
the ν̄p → ne+ events for the case of a Megaton water
Cherenkov detector exactly as in Fig. 10, but for scenar-
ios AI and CIa. One can see how for εττ = 0.07 the I-
resonance condition is always fulfilled and therefore there
is no time dependence. However for smaller values one
can see a rise at a certain moment which depends on the
magnitude of εττ . A similar effect would occur in case
C. Earth matter effects
Before the shock wave reaches the H-resonance layer
the dependence of the neutrino survival probability in the
cases we are considering, on the neutrino energy E is very
weak. However, if neutrinos cross the Earth before reach-
ing the detector, the conversion probabilities may become
energy-dependent, inducing modulations in the neutrino
energy spectrum. These modulations may be observed
in the form of local peaks and valleys in the spectrum of
the event rate σFDν̄e plotted as a function of 1/E. These
modulations arise in the antineutrino channel only when
ν̄e leave the SN as ν̄1 or ν̄2. In the absence of NSI this
happens in cases A and C, where ν̄e leave the star as ν̄1.
FIG. 11: The average energy of ν̄p → ne+ events binned in
time for case AI and CIa and different values of εττ . The
error bars represent 1 σ errors in any bin. εeτ = 10
In the presence of NSI ν̄e will arrive at the Earth as ν̄1
in cases BI, and as ν̄2 in case AI and CIa. Therefore
its observation would exclude cases B and CIb. This
distortion in the spectra could be measured by compar-
ing the neutrino signal at two or more different detectors
such that the neutrinos travel different distances through
the Earth before reaching them [73, 74]. However these
Earth matter effects can be also identified in a single de-
tector [75, 76].
By analyzing the power spectrum of the detected neu-
trino events one can identify the presence of peaks located
at the frequencies characterizing the modulation. These
do not dependend on the primary neutrino spectra, and
can be determined to a good accuracy from the knowl-
edge of the solar oscillation parameters, the Earth matter
density, and the position of the SN in the sky [76]. The
latter can be determined with sufficient precision even if
the SN is optically obscured using the pointing capability
of water Cherenkov neutrino detectors [81].
This method turns out to be powerful in detecting the
modulations in the spectra due to Earth matter effects,
and thus in ruling out cases B and CIb. However, the po-
sition of the peaks does not depend on how ν̄e enters the
Earth, as ν̄1 or ν̄2. Hence it is not useful to discriminate
case AI and CIa from the cases A, C, and BI.
The time dependence of Ye, however, can transform
case B into BI, and C with inverse hierarchy into CIb,
leading respectively to an appearance and disappearance
of these Earth matter effects. In case BI the presence of
the shock wave modulation can spoil a clear identification
of the Earth matter effects. Nevertheless, the disappear-
ance of the Earth matter effects in the transition from
case C to CIb allows us to pin down case CIb.
D. Neutronization burst
The prompt neutronization burst takes place during
the first ∼ 25 ms after the core bounce with a typical
full width half maximum of 5–7ms and a peak luminos-
ity of 3.3–3.5×1053 erg s−1. The striking similarity of the
neutrino emission characteristics despite the variability
in the properties of the pre-collapse cores is caused by
a regulation mechanism between electron number frac-
tion and target abundances for electron capture. This
effectively establishes similar electron fractions in the in-
ner core during collapse, leading to a convergence of the
structure of the central part of the collapsing cores, with
only small differences in the evolution of different pro-
genitors until shock breakout [77, 78].
Taking into account that the SN will be likely to be
obscured by dust and a good estimation of the distance
will not be possible, the time structure of the detected
neutrino signal should be used as signature for the neu-
tronization burst. In Ref. [78] it was shown that such a
time structure can be in principle cleanly seen in the
case of a Megaton water Cherenkov detector. It was
also shown how the time evolution of the signal depends
strongly on the neutrino mixing scheme. In the absence
of NSI the νe peak could be observed provided that the
νe survival probability Pνeνe is not zero. As can be seen
in Table I this happens for cases B and C. However
for case A (normal mass hierarchy and “large” ϑ13), νe
leaves the SN as ν3. This leads to a survival probability
Pνeνe ≈ sin2 ϑ13 . 10−1, and therefore the peak remains
hidden.
Let us now consider the situation where NSI are
prensent. For normal mass hierarchy νe, which is born
as νm2 passes through three different resonances, I, H
and L. Whereas I and L will be adiabatic, the fate of
H will depend on the value of ϑ13. For “large” values,
case AI, the H-resonance will also be adiabatic. This
implies that νe’s will leave as ν2, the survival probability
will be Pνeνe ≈ sin2 ϑ12 ≈ 0.3, and therefore the peak
will be seen, as in cases B and C. If ϑ13 happens to
be very small, case CIa, then H will be strongly non-
adiabatic and therefore νe will leave the star as ν3. As a
consequence the neutronization peak will not be seen.
For inverse mass hierarchy, νe is born as ν
1 and tra-
verses adiabatically I and L. This implies that they will
leave the star as ν1 and therefore the peak will also be
observed. However now the survival probability will be
larger, Pνeνe ≈ cos2 ϑ12 ≈ 0.7. Thus for a given known
normalization, i.e. the distance to the SN, one expects a
larger number of events during the neutronization peak
in this case. In Fig. 12 we show the expected number
of events per time bin in a water Cherenkov detector in
the case of a SN exploding at 10 kpc, for two different
neutrino schemes, C and BI, and for different SN pro-
genitor masses. One can see how the difference due to
the larger survival probability is bigger than the typi-
cal error bars, associated to the lack of knowledge of the
progenitor mass.
Two comments are in order. The neutronization νe
burst takes place during the first milliseconds, before
strong deleptonization takes place. As a result, in con-
trast to other observables we have considered in this pa-
per, here the I-resonance will only occur for εI & 10−1.
On the other hand in the presence of additional NSI with
electrons this would significantly affect the ν − e cross
sections, and consequently the results presented here.
VI. SUMMARY
We have analyzed the possibility of observing clear sig-
natures of non-standard neutrino interactions from the
detection of neutrinos produced in a future galactic su-
pernova.
In Secs. III and IV we have re-considered effect of ν−d
non-standard interactions on the neutrino propagation
through the SN envelope within a three-neutrino frame-
work. In contrast to previous works we have analyzed
the neutrino evolution in both the more deleptonized in-
FIG. 12: Number of events from the elastic scattering on elec-
trons, per time bin in a Megaton water Cherenkov detector for
a SN at 10 kpc for cases C (dashed lines) and BI (solid lines).
Different progenitor masses have been assumed: 13 M⊙ (n13)
in red, 15 M⊙ (s15s7b2) in black, and 25 M⊙ (s25a28) in blue.
1-sigma errors are also shown for the 15 M⊙ case.
ner layers and the outer regions of the SN envelope. We
have also taken into account the time dependence of the
SN density and electron fraction profiles.
First we have found that the small values of the elec-
tron fraction typical of the former allows for internal NSI-
induced resonant conversions, in addition to the standard
MSW-H and MSW-L resonances of the outer envelope.
These new flavor conversions take place for a relatively
large range of NSI parameters, namely |εαα| between
10−2 − 10−1, and |εeτ | & few × 10−5, currently allowed
by experiment. For this range of strengths, in particu-
lar εττ , non-standard interactions can significantly affect
the adiabaticity of the H-resonance. On the other hand
the NSI-induced resonant conversions may also lead to
the modulation of the ν̄e spectra as a result of the time
dependence of the electron fraction.
In Sec. V we have studied the possibility of detecting
NSI effects in a Megaton water Cherenkov detector us-
ing the modulation effects in the ν̄e spectrum due to (i)
the passage of shock waves through the SN envelope, (ii)
the time dependence of the electron fraction and (iii) the
Earth matter effects; and, finally, through the possible
detectability of the neutronization νe burst. Note that
observable (ii) turns out to be complementary to the ob-
servation of the shock wave passage, (i), and offers the
possibility to probe NSI effects also for normal hierarchy
neutrino spectra.
In Table II we summarize the results obtained for dif-
ferent neutrino schemes. We have found that observable
(i) can clearly indicate the existence of NSI in the case
of inverse mass hierarchy and large ϑ13 (case BI). On
the other hand, observable (ii) allows for an identification
of NSI effects in the other cases, normal mass hierarchy
(cases AI and CIa) and inverse mass hierarchy and small
ϑ13 (case CIb). Therefore a positive signal of either ob-
servable (i) or (ii) would establish the existence of NSI. In
the latter case this would, however, leave a degeneracy
among cases AI, CIa, and CIb. Such degeneracy can
be broken with the help of observables (iii) and the ob-
servation of the neutronization νe burst. The detection
of Earth matter effects during the whole supernova neu-
trino signal would rule out case CIb since, as discussed in
Sec. VC, a disappearance of Earth matter effects would
take place due to a transition from C to CIb. Finally,
the (non) observation of the neutronization burst can be
used to distinguish between cases AI and CIa.
Similarly, other degeneracies in Table II may be lifted
by suitably combining different observables. For exam-
ple, a negative of observable (ii) could mean either neg-
ligible NSI strengths or (NU) NSI parameter values so
large that the internal resonance is always present. In
this case one could use the observation of the neutron-
ization burst in order to establish the presence of NSI for
the case of inverse mass hierarchy. In addition the ob-
servation of the shock wave imprint in the ν̄e spectrum
would provide additional information on ϑ13.
In conclusion, by suitably combining all observables
one may establish not only the presence of NSI, but also
the mass hierarchy and probe the magnitude of ϑ13.
Acknowledgments
The authors wish to thank H-Th. Janka, O. Miranda,
S. Pastor, Th. Schwetz, and M. Tórtola for fruitful discus-
sions. Work supported by the Spanish grant FPA2005-
Scheme Hierarchy sin2 ϑ13 NSI shock Ye Earth νe burst
A normal & 10−4 No No No Yes No
B inverted & 10−4 No Yes No No Yes
C any . 10−6 No No No Yes Yes
AI normal & 10−4 Yes No Yes Yes Yes
BI inverted & 10−4 Yes Yes⋆ No Yes Yes⋆
CIa normal . 10−6 Yes No Yes Yes No
CIb inverted . 10−6 Yes No Yes No Yes⋆
TABLE II: Expectations for the observables discussed in the
text: modulation of the ν̄e spectrum due to the shock wave
passage, the time variation of Ye, the Earth effect, and the
observation of the νe burst within various neutrino schemes.
Asterisks indicate that the effect differs from that expected
in the absence of NSI. See text.
01269 and European Network of Theoretical Astroparti-
cle Physics ILIAS/N6 under contract number RII3-CT-
2004-506222. A. E. has been supported by a FPU grant
from the Spanish Government. R. T. has been supported
by the Juan de la Cierva program from the Spanish Gov-
ernment and by an ERG from the European Commission.
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[91] Axial couplings would affect neutrino propagation in po-
larized media, see Ref. [82].
[92] However we have confined ourselves to values of εeα small
enough not to lead to drastic consequences during the
core collapse.
[93] For the sake of simplicity we will omit the superindex V .
[94] The importance of collective flavor neutrino conversions
driven by neutrino-neutrino interactions has been re-
cently noted in Refs. [83, 84, 85, 86]. Here we consider
only the case where the effective potential felt by neutri-
nos comes from their interactions with electrons, protons
and neutrons. In a future work we plan to include this
effect and have a complete picture of the neutrino prop-
agation.
[95] Here we neglect the possible effects of density fluctua-
tions [87, 88] taking place during the shock wave prop-
agation. For a detailed study of the phenomenological
consequences see Refs. [89, 90].
[96] The alternative condition H ′ee = H
µµ would give rise to
another internal resonance which can be studied using
the same method. For brevity, we will not pursue this in
this paper.
[97] Note that, in the limit of high densities one recovers
the rotation angle obtained for the internal I-resonance
23 → ϑ
23 after neglecting the kinetic terms.
[98] For the case of NSI with electrons both the vector and
axial components of εeαβ will contribute to the ν−e cross
section.
[99] We assume that for the values of the NSI parameters con-
sidered the initial neutrino spectra do not significantly
change.
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