arXiv:hep-ph/0501184v2 6 May 2005
Supernova neutrinos can tell us the neutrino mass
hierarchy independently of flux models
V. Barger1, Patrick Huber1and Danny Marfatia2
1Department of Physics, University of Wisconsin, Madison, WI 53706
2Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045
We demonstrate that the detection of shock modulations of the neutrino spectra
from a galactic core-collapse supernova is sufficient to obtain a high significance de-
termination of the neutrino mass hierarchy if the supernova event is observed in both
a Mton-class water Cherenkov detector and a 100 kton-class liquid argon detector.
Neither detailed supernova neutrino flux modelling nor observation of Earth matter
effects is needed for this determination. As a corollary, a nonzero value of θxwill be
The current status of neutrino oscillation parameter estimations can be very briefly sum-
marized  as follows: Atmospheric (solar) neutrinos oscillate with |δm2
a| ∼ 0.002 eV2and
θa∼ π/4  (δm2
s∼ 8 × 10−5eV2, θs∼ π/6 )1. All we presently know about θxis that
sin2θx<∼0.05 at the 2σ C. L. . A long-standing hope is that neutrinos from a core-collapse
supernova (SN) may shed light on two of the unknown oscillation parameters, sgn(δm2
Only a handful of neutrinos from a Type II SN have ever been detected. The detection
of 11 neutrinos from SN 1987A in Kamiokande II  and 8 neutrinos in the Irvine Michigan
Brookhaven experiment  have been of great importance for understanding core-collapse .
It is evident that the physics potential offered by a future galactic SN event is immense.
With cognizance of this potential, experiments dedicated to SN neutrino detection have
been proposed  even though only a few galactic SN are expected per century.
Attempts have been made to extract neutrino oscillation parameters from the 19 SN
1987A events. However, conclusions drawn from such analyses are highly dependent on
the neutrino flux model adopted and are far from robust. For example (and within the
context of this paper), it was claimed that the data favor the normal hierarchy (δm2
over the inverted hierarchy (δm2
a< 0) provided sin2θx>∼10−4, but this conclusion was
contradicted in Ref. .
Neutrinos from a galactic SN could in principle provide a wealth of information on neu-
trino oscillations. A determination of θxand the neutrino mass hierarchy from SN neutrinos
is unique in that ambiguities [11, 12] arising from the unknown CP phase δ and the devia-
tion of atmospheric neutrino mixing from maximality do not corrupt it. The absence of the
eight-fold parameter degeneracies that are inherent in long baseline experiments  results
because (i) nonelectron neutrino fluxes2do not depend on δ independently of neutrino con-
version , and so SN neutrinos directly probe θx, and (ii) whether atmospheric mixing is
1In our notation, δm2
a) is the solar (atmospheric) mass-squared difference and θs, θxand θaare the
mixing angles conventionally denoted by θ12, θ13and θ23, respectively .
2We focus on detection via charged current νeand ¯ νeinteractions, which cannot distinguish between the
different nonelectron neutrino species (that we denote by νxwith x = µ,τ, ¯ µ, ¯ τ).
maximal or not is immaterial since θadoes not affect the oscillation dynamics.
Investigations of the effect of neutrino oscillations on SN neutrinos in the context of a
static density profile (i.e., neglecting shock effects) have been made in Refs. [14, 15, 16].
Whether or not the mass hierarchy can be determined and θxbe constrained depends sensi-
tively on the strength of the hierarchy between ?E¯ νe? and ?Eνx?. The higher ?Eνx?/?E¯ νe?
is above unity, the better the possible determinations .Unfortunately, modern SN
models that include all relevant neutrino interaction effects like nuclear recoil and nucleon
bremsstrahlung indicate that the hierarchy of average energies is likely smaller than ex-
pected from traditional predictions; ?Eνx?/?E¯ νe? is expected to be about 1.1, and no larger
than 1.2  as opposed to ratios above 1.5 from older SN codes . Another relevant
uncertainty is that different SN models predict different degrees to which equipartitioning
of energy between νe, ¯ νe and νx is violated. For example, in Ref.  an almost perfect
equipartitioning is obtained while according to Refs. [20, 21], equipartitioning holds only to
within a factor of 2.
Given these uncertainties, it is not a simple task to determine θxand the mass hierarchy
simultaneously from SN data . A significant improvement would be a determination of
the mass hierarchy independently of predictions for ?Eνx?/?E¯ νe? and equipartitioning from
SN models. In this paper we propose a new method using SN neutrinos to determine the
mass hierarchy that exploits recent advances in the understanding of shock propagation in
At densities ∼ 103g/cm3, neutrino oscillations are governed by δm2
aand sin2θx .
Neutrinos (antineutrinos) pass through a resonance if δm2
a> 0 (δm2
a< 0). As the shock tra-
verses the resonance, adiabaticity is severely affected causing oscillations to be temporarily
suppressed, as first pointed out in Ref. . After the shock moves beyond the resonance,
oscillations are restored. Then one expects a dip in the time evolution of the average neutrino
energy and the number of events3. This modulation is visible in the neutrino (antineutrino)
3Recently, this idea was taken one step further in Ref. . A reverse shock caused by the collision
between a neutrino-driven baryonic wind and the more slowly moving primary ejecta may also form. The
direct and reverse shocks yield a “double dip” signature . In the present work we restrict our attention
to the effects of the forward shock which is a generic feature of SN models and whose existence is better
established than that of the reverse shock.
channel for a normal (inverted) mass hierarchy and only if tan2θx≫ 10−5i.e., only for oscil-
lations that would occur adiabatically for a static density profile. Previous work exploiting
the dip to obtain information about oscillation parameters can be found in Ref. .
Within the first 4 seconds or so, the violation of adiabaticity caused by the shock is felt
only by neutrinos with energy less than about 20 MeV. At these energies some models predict
the νxflux to be larger than the νeand ¯ νefluxes  and others predict the converse .
At later times, the shock affects higher energy neutrinos for which all models predict the νx
flux to be dominant. The dip is observable even for ?Eνx? = ?E¯ νe? because the fluxes are
flavor-dependent . Thus, a signature in high energy neutrinos a few seconds after bounce
is quite model-independent. Through our analysis we show that the signal is so robust that
a restriction to high-energy events is unnecessary.
We investigate the significance with which the mass hierarchy can be determined by
measurement of the νespectrum at a 100 kton liquid argon detector and the ¯ νespectrum at
a 1 Mton water Cherenkov detector from a galactic SN at a distance of 10 kpc with binding
energy 3 ×1053ergs. The number of unoscillated events in the liquid Ar (water Cherenkov)
detector is expected to be O(105) (O(106)). (Although a liquid argon detector can measure
both the νeand ¯ νespectra, it is an order of magnitude more sensitive to the νeflux than to
the ¯ νeflux ). We are interested in the detectability of a dip in the time evolution of either
the νeor the ¯ νespectrum, but not both. By correlating the two spectra, it should be possible
to establish the mass hierarchy and that tan2θx≫ 10−5. The presence of a dip will also
provide further evidence that SN simulations correctly depict shock propagation. If a dip is
not found in either channel, it would suggest that tan2θx<∼10−5, since the salient features
of the theory of core-collapse SN have already received strong support from SN 1987A .
2 Shock density profile and neutrino oscillation prob-
Realistic time-dependent density profiles of SN are obtained from detailed numerical siu-
mulations. As one moves in towards the neutron star, the profile at a given instant has a
sharp density rise followed by a rarefaction region where the density can drop significantly
below that at the outer edge of the shock. The authors of Ref.  have provided a generic
time-dependent density profile that mimics those of supernova simulations. The shock front
is steepened artificially to reintroduce the physical requirement of a density discontinuity
which is lost in hydrodynamic simulations due to the limited (few 100 km) spatial resolu-
tion. We adopt the empirical parameterization of this profile (which is continuous in the
supernova radius and time) developed in Ref. .
Because of the dip in density in the rarefaction region, neutrinos may hop between mass
eigenstates up to 3 times before leaving the SN envelope.Under the assumptions that
the transitions factorize and the neutrino phases can be averaged away, a simple analytic
expression for the overall hopping probability PH(|δm2
a|,θx) was obtained in Ref. , which
agrees remarkably well with phase-averaged results of Runge-Kutta evolution of the neutrino
flavor propagation equations. We employ the analytic expression for PHto calculate the νe
and ¯ νesurvival probabilities. These survival probabilities do not depend on δm2
are neglecting Earth matter effects :
PN(νe→ νe) = sin2θscos2θxPH+ sin2θx(1 − PH), (1)
PN(¯ νe→ ¯ νe) = cos2θscos2θx, (2)
PI(νe→ νe) = sin2θscos2θx, (3)
PI(¯ νe→ ¯ νe) = cos2θscos2θxPH+ sin2θx(1 − PH). (4)
Here, the N and I subscripts denote normal and inverted hierarchy, respectively. We note
that the factorization of the 3-neutrino dynamics into two 2-neutrino subsystems continues
to hold with the subsystem governed by δm2
sand θsremaining adiabatic (for the now well-
established Large Mixing Angle solution ) as in the case of a static density profile .
We use the parameterization of Ref.  for the primary unoscillated neutrino spectra given