Sterile neutrinos at the CNGS
IFT-UAM/CSIC-07-16
Sterile neutrinos at the CNGS
Andrea Donini, a Michele Maltoni, a Davide Meloni, b
Pasquale Migliozzi, c Francesco Terranova d
aInstituto F́ısica Teórica UAM/CSIC, Cantoblanco, E-28049 Madrid, Spain
bI.N.F.N., Sezione di Roma I and Dip. Fisica, Univ. Roma “La Sapienza”,
Pl. A. Moro 2, I-00185, Rome, Italy
cI.N.F.N., Sezione di Napoli, I-80126, Naples, Italy
dI.N.F.N., Laboratori Nazionali Frascati, Via E. Fermi 40, I-00044, Frascati, Italy
PACS: 14.60.Pq, 14.60.Lm
Abstract
We study the potential of the CNGS beam in constraining the parameter space of a
model with one sterile neutrino separated from three active ones by an O(eV2) mass-
squared difference, ∆m2
. We perform our analysis using the OPERA detector as
a reference (our analysis can be upgraded including a detailed simulation of the
ICARUS detector). We point out that the channel with the largest potential to
constrain the sterile neutrino parameter space at the CNGS beam is νµ → ντ . The
reason for that is twofold: first, the active-sterile mixing angle that governs this
oscillation is the less constrained by present experiments; second, this is the signal
for which both OPERA and ICARUS have been designed, and thus benefits from
an extremely low background. In our analysis we also took into account νµ → νe
oscillations. We find that the CNGS potential to look for sterile neutrinos is limited
with nominal intensity of the beam, but it is significantly enhanced with a factor 2
to 10 increase in the neutrino flux. Data from both channels allow us, in this case,
to constrain further the four-neutrino model parameter space. Our results hold for
any value of ∆m2
& 0.1 eV2, i.e. when oscillations driven by this mass-squared
difference are averaged. We have also checked that the bound on θ13 that can be
put at the CNGS is not affected by the possible existence of sterile neutrinos.
http://arxiv.org/abs/0704.0388v2
1 Introduction
The results of solar [1,2,3,4,5,6], atmospheric [7,8], reactor [9,10,11,12] and accelera-
tor [13,14,15] neutrino experiments show that flavour mixing occurs not only in the
hadronic sector, as it has been known for long, but in the leptonic sector as well. The
full understanding of the leptonic mixing matrix constitutes, together with the dis-
crimination of the Dirac/Majorana character of neutrinos and with the measurement
of their absolute mass scale, the main goal of neutrino physics for the next decade.
The experimental results point to two very distinct mass-squared differences, ∆m2
7.9 × 10−5 eV2 and |∆m2
| ≈ 2.4 × 10−3 eV2. On the other hand, only two out of
the four parameters of the three-family leptonic mixing matrix UPMNS [16,17,18,19] are
known: θ12 ≈ 34
◦ and θ23 ≈ 43
◦ [20]. The other two parameters, θ13 and δ, are still
unknown: for the mixing angle θ13 direct searches at reactors [9,10,11] and three-family
global analysis of the experimental data give the upper bound θ13 ≤ 11.5
◦, whereas for
the leptonic CP-violating phase δ we have no information whatsoever (see, however,
Ref. [20]).
The LSND data [21,22,23], on the other hand, would indicate a ν̄µ → ν̄e oscillation with
a third neutrino mass-squared difference: ∆m2
∼ 0.3 − 6 eV2, about two orders of
magnitude larger than ∆m2
. Given the strong hierarchy among the solar, atmospheric
and LSND mass-squared splittings, ∆m2
≪ ∆m2
≪ ∆m2
, it is not possible to
explain all these data with just three massive neutrinos, as it has been shown with
detailed calculations in Ref. [24]. A necessary condition to explain the whole ensemble
of data in terms of neutrino oscillations is therefore the introduction of at least a
fourth light neutrino state. This new light neutrino must be an electroweak singlet [18]
in order to comply with the strong bounds on the Z0 invisible decay width [25,26]. For
this reason, the LSND signal has often been considered as an evidence of the existence
of a sterile neutrino.
In recent years, global analyses of solar, atmospheric, short-baseline [27,28,29,30] exper-
iments and LSND data have been performed to establish whether four-neutrino models
can really reconcile the data and solve the puzzle [31,32,33,34,35,36,37,38]. The point
is that providing a suitable mass-squared difference to each class of experiments is not
enough: it is also necessary to show that the intrinsic structure of the neutrino mixing
matrix is compatible with all the data. This turned out to be very hard to accomplish.
In Ref. [39] it was shown that four-neutrino models were only marginally allowed, with
best fit around ∆m2
≃ 1 eV2 and sin2 2θLSND ≃ 10
−3. Generically speaking, the
global analysis indicated that a single sterile neutrino state was not enough to reconcile
LSND with the other experiments. For this reason, to improve the statistical compat-
ibility between the LSND results and the rest of the oscillation data, models with
two sterile neutrino states have been tested (see, for example, Ref. [40] and references
therein). Although a slightly better global fit was achieved, a strong tension between
the LSND data and the results from atmospheric and short-baseline experiments was
still present.
So far, the LSND signal has not been confirmed by any other experiment [41]. It is
therefore possible that the LSND anomaly arises from some some yet unknown problem
in the data set itself. To close the issue, the MiniBooNE collaboration [42] at FermiLab
has recently performed a search for νµ → νe appearance with a baseline of 540 m and a
mean neutrino energy of about 700 MeV. The primary purpose of this experiment was
to test the evidence for ν̄µ → ν̄e oscillation observed at LSND with a very similar L/E
range. No evidence of the expected signal has been found, hence ruling out once and for
all the four-neutrino interpretation of the LSND anomaly. However, MiniBooNE data
are themselves not conclusive: although no evidence for νµ → νe oscillation has been
reported in the spectrum region compatible with LSND results, an unexplained excess
has been observed for lower energy neutrinos. Furthermore, within a five-neutrino model
this excess can be easily explained, and even reconciled with LSND and all the other
appearance experiments [43]. On the other hand, a post-MiniBooNE global analysis
including also disappearance data show that five-neutrino models suffer from the same
problems as four-neutrino schemes, and in particular they are now only marginally
allowed – a situation very similar to that of four-neutrino models before MiniBooNE
data. Adding a third sterile neutrino 1 does not help [43], and in general global analyses
seem to indicate that sterile neutrinos alone are not enough to reconcile all the data.
Models with sterile neutrinos and exotic physics have been therefore proposed (see, for
example, Ref. [46]).
In summary, the present experimental situation is still confused. It is therefore worth-
wile to understand if, aside of MiniBooNE, new neutrino experiments currently running
or under construction can investigate the existence of sterile neutrinos separated from
the active ones by O(eV2) mass-squared differences. In this paper we explore in detail
the capability of the CNGS beam to perform this search. For definiteness we focus on
the simplest case with only one extra sterile neutrino. Note that this model is perfectly
viable once the LSND result is dropped, as it contains as a limiting case the usual
three-neutrino scenario. Furthermore, it is easily generalizable by adding new sterile
neutrino states, and it can be used as a basis for models with extra “sterile” states
strongly decoupled from active neutrinos (such as in extra-dimensions models with a
right-handed neutrino in the bulk [47]).
The CNGS beam [48] has been built to test the (supposedly) dominant oscillation in at-
mospheric neutrino data, νµ → ντ . In order to make possible τ production through CC
interactions, the mean neutrino energy, 〈Eν〉 = 17 GeV, is much above the atmospheric
oscillation peak for the CERN to Gran Sasso baseline, L = 732 Km. Two detectors are
illuminated by the CNGS beam: OPERA (see Ref. [49] and refs. therein) will start data
taking with the lead-emulsion target in 2007; ICARUS-T600 (see Ref. [50] and refs.
therein) will start operating in 2008. Both detectors have been especially designed to
look for τ ’s produced through νµ → ντ oscillation and to minimize the corresponding
1 A quite interesting scenario is, in our opinion, that in which three right-handed Majorana
neutrinos are added to the three weakly-interacting ones. If the Majorana mass term M is
O(eV), (3+3) light Majorana neutrinos are present at low-energy [44,45].
backgrounds. The expected number of τ events after signal selection in an experiment
such as OPERA (after five years of data taking with nominal CNGS luminosity) is
O(10) events with O(1) background event.
At the CNGS distance and energy, neutrino oscillations mediated by an O(eV2) mass
difference will appear as a constant term in the oscillation probability. In four-neutrino
models, fluctuations induced by this term over the atmospheric νµ → ντ oscillation can
be as large as 100% for specific points of the allowed parameter space. This is due to the
fact that the leading angle for this oscillation is the less constrained one. The νµ → ντ
channel, therefore, is extremely promising as a “sterile neutrino” smoking gun, as it has
been commented elsewhere (see, for example, Refs. [51,52] and refs. therein). To test
the model we will also make use of the νµ → νe channel. Notice that the background to
this signal coming from τ → e decay is modified in four-neutrino models with respect
to standard three-family oscillations. In fact, since νµ → ντ oscillations are depleted by
active-sterile mixing with respect to standard ones, the τ → e background to νµ → νe
oscillations gets depleted, too. A combined analysis of the two channels in four-neutrino
models at the OPERA detector has been performed, taking into account properly all
of the backgrounds. We stress, however, that the same analysis could be performed
at ICARUS, as well. The previous considerations hold for any facility operating well
beyond the kinematical threshold for τ production.
In the specific case of the CNGS beam, the limited flux implies a modest improvement
in the parameter space exclusion, see Sec. 6. An increase in the exposure of such
facilities, however, would permit to improve the present bounds on the parameters of
four-neutrino models and, in particular, to constrain the leading angle in νµ → ντ
oscillations at the level of the other mixing parameters.
The paper is organized as follows. In Sec. 2 we briefly review the main features of
four-neutrino models and we introduce our parametrization of the mixing matrix. In
Sec. 3 we compute the vacuum oscillation probabilities in the atmospheric regime and
we review the present bounds on the active-sterile mixing angles. In Sec. 4 we recall the
most relevant parameters of CNGS. In Sec. 5 we study theoretically the expectations
of the νµ → ντ and νµ → νe channels at the CNGS. In Sec. 6 we present our results
using these channels at the OPERA detector and the CNGS beam. Finally, in Sec. 7
we draw our conclusions.
2 Four neutrino mass schemes
In four-neutrino models, one extra sterile state is added to the three weakly interacting
ones. The relation between the flavor and the mass eigenstates is then described by a
4×4 unitary matrix U , which generalizes the usual 3×3 matrix UPMNS [16,17,18,19]. As
stated in the introduction, in this work we only consider the case when the fourth mass
eigenstate is separated by the other three by an O(eV2) mass-squared gap. There are
six possible four-neutrino schemes, shown in Fig. 1, that can accommodate the results
                 
(3+1) (2+2)
Fig. 1. The two classes of four–neutrino mass spectra, (3+1) and (2+2).
from solar and atmospheric neutrino experiments and contain a third much larger ∆m2.
They can be divided in two classes: (3+1) and (2+2). In the (3+1) schemes, there is
a group of three close-by neutrino masses that is separated from the fourth one by
the larger gap. In (2+2) schemes, there are two pairs of close masses separated by the
large gap. While different schemes within the same class are presently indistinguishable,
schemes belonging to different classes lead to very different phenomenological scenarios.
A characteristic feature of (2+2) schemes is that the extra sterile state cannot be
simultaneously decoupled from both solar and atmospheric oscillations. To understand
why, let us define
i∈ sol
|Usi|
2 and cs =
j ∈ atm
|Usj|
2 (1)
where the sums in i and j run over mass eigenstates involved in solar and atmospheric
neutrino oscillations, respectively. Clearly, the quantities ηs and cs describe the fraction
of sterile neutrino relevant for each class of experiment. Results from atmospheric and
solar neutrino data imply that in both kind of experiments oscillation takes place
mainly between active neutrinos. Specifically, from Fig. 46 of Ref. [20] we get ηs ≤ 0.30
and cs ≤ 0.36 at the 3σ level. However, in (2+2) schemes unitarity implies ηs + cs = 1,
as can be easily understood by looking at Fig. 1. These models are therefore ruled out
at a very high confidence level [53], and in the rest of this work we will not consider
them anymore.
On the other hand, (3+1) schemes are not affected by this problem. Although the
experimental bounds on ηs and cs quoted above still hold, the condition ηs + cs = 1 no
longer applies. For what concerns neutrino oscillations, (3+1) models are essentially
unfalsifiable, since they reduce to the conventional three-neutrino scenario when the
mixing between active and sterile states are small enough.
The mixing matrix U can be conveniently parametrized in terms of six independent
rotation angles θij and three (if neutrinos are Dirac fermions) or six (if neutrinos are
Majorana fermions) phases δi. In oscillation experiments, only the so-called “Dirac
phases” can be measured, the effect of the “Majorana phases” being suppressed by
factors of mν/Eν . The Majorana or Dirac nature of neutrinos can thus be tested only
in ∆L = 2 transitions such as neutrino-less double β-decay [54] or lepton number
violating decays [25]. In the following analysis, with no loss in generality, we will restrict
ourselves to the case of 4 Dirac-type neutrinos only.
A generic rotation in a four-dimensional space can be obtained by performing six dif-
ferent rotations along the Euler axes. Since the ordering of the rotation matrices Rij
(where ij refers to the plane in which the rotation takes place) is arbitrary, plenty of
different parametrizations of the mixing matrix U are allowed. The large parameter
space (6 angles and 3 phases, to be compared with the standard three-family mixing
case of 3 angles and 1 phase) is however reduced to a subspace whenever some of
the mass differences become negligible. If the eigenstates i and j are degenerate, ro-
tations in the ij-plane become unphysical and the corresponding mixing angle should
drop from oscillation probabilities. If the matrix Rij is the rightmost one the angle
θij automatically disappears, since the matrix commutes with the vacuum hamilto-
nian. The parameter space gets therefore reduced to the physical angles and phases.
If a different ordering of the rotation matrices is taken, no angle explicitly disappears
from the oscillation formulas, but the physical parameter space is still smaller than the
original one. In this case, a parameter redefinition is needed to reduce the parameter
space to the observable sector. In Refs. [55,56] it was shown how the one-mass domi-
nance (∆sol → 0 and ∆atm → 0, where ∆ = ∆m
2L/4E [57]) and two-mass dominance
(∆sol → 0) approximations can be implemented in a transparent way (in the sense that
only the physical parameters appear in oscillation probabilities) using a parametriza-
tion in which rotations are performed in the planes corresponding to smallest mass
difference first:
USBL = R14(θ14) R24(θ24) R34(θ34) R23(θ23, δ3) R13(θ13, δ2) R12(θ12, δ1) . (2)
This parametrization was shown to be particularly useful when maximizing oscilla-
tions driven by a O(eV2) mass difference. The analytical expressions for the oscillation
probabilities in the (3+1) model in the one-mass dominance approximation in this
parametrization have been presented in Ref. [51].
In this paper, however, we are interested in a totally different regime: the “atmospheric
regime”, with oscillations driven by the atmospheric mass difference, ∆m2
L/E ∼
π/2. We will then make use of the following parametrization, adopted in Ref. [43]:
Uatm = R34(θ34) R24(θ24) R23(θ23, δ3) R14(θ14) R13(θ13, δ2) R12(θ12, δ1) . (3)
It is convenient to put phases in R12 (so that it automatically drops in the two-mass
dominance regime) and R13 (so that it reduces to the “standard” three-family Dirac
phase when sterile neutrinos are decoupled). The third phase can be put anywhere;
we will place it in R23. Note that in the one-mass dominance regime all the phases
disappear from the oscillation probabilities.
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
Fig. 2. Allowed regions at 90%, 95%, 99% and 3σ CL in the (θ13, θ14) plane (left) and in the
(θ24, θ34) plane (right) from the results of present atmospheric, reactor and LBL neutrino
experiments. The undisplayed parameters θ23 and δ3 are marginalized.
3 Oscillation probabilities and allowed parameter space
Let us first consider νe disappearance at L/E such that ∆sol can be safely neglected
with respect to ∆atm and ∆sbl. We get for this probability in vacuum:
Pee = 1− sin
2 2θ14 sin
∆sblL
sin2 2θ13 sin
∆atmL
, (4)
where cij = cos θij and sij = sin θij . It is clear from Eq. (4) that reactor experiments
such as Bugey and Chooz can put stringent bounds to θ13 and θ14, in this parametriza-
tion. This is depicted in Fig. 2(left), where 90%, 95%, 99% and 3σ CL contours in
the (θ13 − θ14)-plane are shown for ∆sol → 0 and ∆m
= 2.4 × 10−3 eV2. The third
mass difference, ∆m2
, is free to vary above 0.1 eV2. Notice that the νe disappearance
probability does not depend on θ23, θ24 and θ34. It can be clearly seen that the three-
family Chooz bound on θ13 is slightly modulated by θ14. Both angles, however, cannot
be much larger than 10◦. We will therefore expand in these two parameters to deduce
the other relevant oscillation probabilities.
At the CNGS beam atmospheric oscillations are large, solar oscillations can be ne-
glected and O(eV2) oscillations are extremely fast and can be averaged. It is useful to
write down the oscillation probability (in vacuum) at typical atmospheric L/E, in the
approximation ∆sol → 0, ∆sbl → ∞. In this regime:
P (να → νβ) = δαβ − 4ℜ
β3 (δαβ − U
α3Uβ3 − U
α4Uβ4)
δαβUα4U
β4 − |Uα4|
2|Uβ4|
α4Uβ4
sin∆23L ,
where + stands for neutrinos and − for antineutrinos, respectively. Up to second order
in θ13 and θ14 we get for the νµ disappearance oscillation probability:
Pµµ = 1− 2c
(1− c2
)− s2
− 2c3
s23(1− 2s
)s13s14s24 cos(δ2 − δ3)
∆atmL
A “negative” result in an atmospheric L/E νµ disappearance experiment (such as, for
example, K2K), in which νµ oscillations can be very well fitted in terms of three-family
oscillations, will put a stringent bound on the mixing angle θ24. The bound from such
experiments on θ24 can be seen in Fig. 2(right), where 90%, 95%, 99% and 3σ CL
contours in the (θ24 − θ34)-plane are shown for ∆sol → 0 and ∆m
= 2.4× 10−3 eV2.
The third mass difference, ∆m2
, is free to vary above 0.1 eV2. The mixing angles not
shown have been fixed to: θ23 = 45
◦; θ13 = θ14 = 0 (in this hypothesis, Pµµ does not
depend on phases). Notice that the νµ disappearance probability does not depend on
From the figure, we can see that θ24 cannot be much larger than 10
◦, either. We will
consider, therefore, the three mixing angles θ13, θ14 and θ24 being of the same order and
expand in powers of the three. At second order in θ13, θ14 and θ24, we get:
Pµµ = 1− 2s
− 4s2
(1− 2s2
)− s2
∆atmL
. (7)
Since both νe and νµ disappearance do not depend on θ34, we should ask which mea-
surements give the upper bound to this angle that can be observed in Fig. 2(right).
This is indeed a result of indirect searches for νµ → νs conversion in atmospheric exper-
iments, using the different interaction with matter of active and sterile neutrinos. This
can be understood from the (vacuum) νµ → νs oscillation probability at atmospheric
L/E for which, at second order in θ13, θ14 and θ24, we get:
Pµs = 2c
sin2 2θ23(c
+ 2c34 sin 2θ23s34
s24(1− 2s
) cos δ3 + 2s23s13s14 cos δ2
∆atmL
± c34 sin 2θ23s24s34 sin δ3 sin∆atmL .
As it can be seen, the bound on θ34 arises from a measurement of spectral distortion
(i.e., from the “atmospheric” term proportional to sin2∆atmL/2). On the other hand,
bounds on θ13, θ14 and θ24 are mainly drawn by a flux normalization measurement.
As a consequence, the bound on θ34 that we can draw by non-observation of νµ → νs
oscillation in atmospheric experiments is less stringent than those we have shown before.
For this reason, θ34 can be somewhat larger than θ13, θ14 and θ24, but still bounded to
be below 40◦. In the following, we will expand in powers of the four mixing angles
θ13, θ14, θ24 and θ34, that will be considered to be comparably small.
Up to fourth-order in θ13, θ14, θ24 and θ34, the νµ → νe appearance probability in the
atmospheric regime is:
Pµe = 4
[1− s2
] + s23s13s14s24 cos(δ2 − δ3)
∆atmL
± 2s23s13s14s24 sin(δ2 − δ3) sin∆atmL+ 2s
Eventually, the νµ → ντ appearance probability up to fourth-order in θ13, θ14, θ24 and
θ34 in the atmospheric regime is:
Pµτ = 2s
sin2 2θ23[c
− 4 sin 2θ23s13s14[s23s34 cos δ2 + c23s24 cos(δ2 − δ3)]
+ 2 sin 2θ23s24s34c
c34[c
− 2c2
] cos δ3
∆atmL
∓ sin 2θ23s24s34c
c34 sin δ3 sin∆atmL .
As it was shown in Refs. [51,52], the νµ → ντ appearance channel is a good place to
look for sterile neutrinos. This can be understood as follows: consider the νµ → ντ
three-family oscillation probability in the atmospheric regime, up to fourth-order in
P 3νµτ = Pµτ (θi4 = 0) ≃ c
sin2 2θ23 sin
∆atmL
. (11)
The genuine active-sterile neutrino mixing effects are:
∆Pµτ ≡ Pµτ − P
) sin2 2θ23 + 2 sin 2θ23(1− 2s
)s24s34 cos δ3
∆atmL
∓ sin 2θ23s24s34 sin δ3 sin∆atmL+ . . .
that is second-order in small angles θ13, θ14, θ24 and θ34. We would get a similar result
for νµ disappearance, also. On the other hand, computing the corresponding quantity
in the νµ → νe channel, we get:
∆Pµe ≡ Pµe − Pµe(θi4 = 0)
= s23s13s14s24 cos(δ2 − δ3) sin
∆atmL
± 2s23s13s14s24 sin(δ2 − δ3) sin∆atmL+ . . .
that is third-order in the same parameters.
Notice, eventually, that all oscillation probabilities start with an energy-independent
term and are, therefore, non-vanishing for L = 0, a result of our assumption that
∆sbl → ∞.
0 10 20 30 40 50
Eν(GeV)
anti−νµ
anti−νe
Fig. 3. CNGS neutrino fluxes (in arbitrary units) as a function of the neutrino energy. Both
muon and electron neutrino fluxes are illustrated.
4 The CNGS facility
The CNGS is a conventional neutrino beam in which neutrinos are produced by the
decay of secondary pions and kaons, obtained from collisions of 400 GeV protons from
the CERN-SPS onto a graphite target. The resulting neutrinos are aimed to the under-
ground Gran Sasso Laboratory (LNGS), located at 730 Km from CERN. This facility
provided the first neutrinos in August 2006 [49]. Differently from other long baseline
experiments, the neutrinos from CNGS can be exploited to search directly for νµ → ντ
oscillations, since they have a mean energy well beyond the kinematic threshold for τ
production. Moreover, the prompt ντ contamination (mainly fromDs decays) is negligi-
ble. The expected νe contamination is also relatively small compared to the dominant νµ
component, thus allowing for the search of sub-dominant νµ → νe oscillations through
an excess of νe CC events.
The energy spectra of the CNGS neutrino beam are shown (in arbitrary units) in
Fig. 3 [58]. In the present paper we assume the nominal intensity for the CNGS,
corresponding to 4.5× 1019 pot/year.
OPERA has been designed to search for τ appearance through identification of the
ντ CC interaction on an event-by-event basis. In particular, τ ’s are tagged identifying
explicitly their decay kink through high resolution nuclear emulsions interleaved with
lead sheets. For this detector, we can take advantage of the detailed studies of the
νµ → ντ signal (see Ref. [59]) and of the νµ → νe signal (see Ref. [60]).
The total non-oscillated CC event rates for a 1 Kton lead target with a neutrino flux
νµ ν̄µ νe ν̄e
669.0 13.7 5.9 0.3
Table 1
Nominal performance of the CNGS reference beam [58]. The total non-oscillated CC event
rates are calculated assuming 1019 pot and 1 Kton lead target mass.
normalized to 1019 pot are shown in Tab. 1 and are evaluated according to
dφνα(E)
σνα(E) dE , (14)
in which φνα is the flux of the neutrino flavour να and σνα the corresponding cross
section on lead.
5 Appearance channels at the CNGS
5.1 νµ → ντ oscillations
Since the CNGS experiments have been designed to search for νµ → ντ oscillation
in the parameter region indicated by the atmospheric neutrino data, we can take full
advantage of them in order to constrain (and, possibly, study) the four-family parameter
space.
The number of taus from νµ → ντ oscillations is given by the convolution of the νµ
flux dφνµ(E)/dE with the ντ charged-current cross-section on lead, σ
(E), weighted
by the νµ → ντ oscillation probability, Pµτ (E), times the efficiency for the OPERA
detector, εµτ :
Nµτ = A
dφνµ(E)
Pµτ (E)σ
(E) εµτ dE . (15)
A is a normalization factor which takes into account the target mass and the nor-
malization of the νµ flux in physical units. Specializing our analysis for the OPERA
detector, we have considered an overall efficiency εµτ ∼ 13%, [59]. This efficiency takes
into account that OPERA is able to exploit several decay modes of the final state τ ,
using both so-called short and long decays.
The dominant sources of background for the νµ → ντ signal are charm decays and
hadronic reinteractions. Both of them only depend on the total neutrino flux and not
on the oscillation probabilities. The OPERA experiment at the CNGS beam has been
designed precisely to measure this channel, and thus the corresponding backgrounds
are extremely low.
In Tab. 2 we report the expected number of τ events in the OPERA detector, according
to Eq. (15), for different values of θ13, θ14, θ24 and θ34. Input points have been chosen
according to the allowed regions in the parameter space shown in Sec. 3. The other
(θ13; θ14; θ24; θ34) Nτ background (θ13; θ14; θ24; θ34) Nτ background
(5◦; 5◦; 5◦; 20◦) 8.9 1.0 (10◦; 5◦; 5◦; 20◦) 8.5 1.0
(5◦; 5◦; 5◦; 30◦) 6.9 1.0 (10◦; 5◦; 5◦; 30◦) 6.5 1.0
(5◦; 5◦; 10◦; 20◦) 8.3 1.0 (10◦; 5◦; 10◦; 20◦) 7.9 1.0
(5◦; 5◦; 10◦; 30◦) 10.5 1.0 (10◦; 5◦; 10◦; 30◦) 10.3 1.0
3 families 15.1 1.0 3 families 14.4 1.0
Table 2
Event rates and expected background for the νµ → ντ channel in the OPERA detector, for
different values of θ14, θ24 and θ34 in the (3+1) scheme. The other unknown angle, θ13 has
been fixed to: θ13 = 5
◦, 10◦. The CP-violating phases are: δ1 = δ2 = 0; δ3 = 90
◦. As a
reference, the expected value in the case of standard three-family oscillation (i.e., for θi4 = 0)
is shown for maximal CP-violating phase δ. The rates are computed according to Eq. (15).
parameters are: θ12 = 34
◦; θ23 = 45
◦; ∆m2
= 7.9× 10−5 eV2; ∆m2
= 2.4× 10−3 eV2
and ∆m2
= 1 eV2 (all mass differences are taken to be positive). Eventually, phases
have been fixed to: δ1 = δ2 = 0; δ3 = 90
◦. The expected background is also shown. Rates
refer to a flux normalized to 4.5× 1019 pot/year (the nominal intensity of the CNGS),
an active lead target mass of 1.8 Kton and 5 years of data taking. For comparison, we
also report the expected number of events in the usual 3-family scenario.
As it can be seen, in most part of the parameter space we expect a significant depletion
of the signal with respect to standard three-neutrino oscillations. However, the differ-
ence between (3+1) model νµ → ντ oscillations and standard ones is much bigger than
the expected background. A good signal/noise separation can therefore be used to test
the model.
5.2 νµ → νe oscillations
The number of electrons from the νµ → νe oscillation is given by the convolution of
the νµ flux dφνµ(E)/dE with the νe charged-current cross-section on lead, σ
weighted by the νµ → νe oscillation probability, Pµe(E), times the efficiency for the
OPERA detector, εµe(E) [60]:
Nµe = A
dφνµ(E)
Pµe(E)σ
(E) εµe(E) dE , (16)
where A is defined as above. The overall signal efficiency εµe is the convolution of the
kinematic efficiency εkinµe (that ranges from 60% to 80% for neutrino energies between
5 to 20 GeV) and several (nearly factorizable) contributions. Among them, the most
relevant are trigger efficiencies, effects due to fiducial volume cuts, vertex and brick
finding efficiencies and the electron identification capability. They result in a global
constant factor εfactµe ∼ 48%.
(θ13; θ14; θ24; θ34) Ne ν
µ τ → e ν
(5◦; 5◦; 5◦; 20◦) 3.5 19.4 5.3 2.8 0.9
(5◦; 5◦; 5◦; 30◦) 3.5 19.4 5.3 2.1 0.9
(5◦; 5◦; 10◦; 20◦) 2.4 19.4 5.3 2.3 0.9
(5◦; 5◦; 10◦; 30◦) 2.4 19.4 5.3 2.4 0.9
3 families 3.7 19.7 5.3 4.6 0.9
(10◦; 5◦; 5◦; 20◦) 10.6 19.4 5.3 2.7 0.9
(10◦; 5◦; 5◦; 30◦) 10.4 19.4 5.3 2.0 0.9
(10◦; 5◦; 10◦; 20◦) 8.8 19.4 5.3 2.2 0.9
(10◦; 5◦; 10◦; 30◦) 8.6 19.4 5.3 2.4 0.9
3 families 15.1 19.7 5.3 4.8 0.9
Table 3
Event rates and expected background for the νµ → νe channel in the OPERA detector, for
different values of θ14, θ24 and θ34 in the (3+1) scheme. The other unknown angle, θ13, has
been fixed to: θ13 = 5
◦, 10◦. The CP-violating phases are: δ1 = δ2 = 0; δ3 = 90
◦. As a
reference, the expected value in the case of standard three-family oscillation(i.e., for θi4 = 0)
is shown for maximal CP-violating phase δ. The rates are computed according to Eq. (16).
Backgrounds have been computed following Ref. [60].
The dominant sources of background to the νµ → νe signal are, in order of importance:
(1) νe beam contamination;
(2) fake electrons due to π0 decays from νµ NC interactions;
(3) electrons produced through τ decay, where the τ comes from νµ → ντ oscillations;
(4) CC νµ events where the muon is lost and a track mimics an electron.
Backgrounds (1), (2) and (4) depend very little on the oscillation parameters. On the
other hand, the τ → e background depends strongly on the active-sterile mixing angles.
As we have seen in Sec. 5.1, in the allowed region of the parameter space νµ → ντ
oscillations are significantly depleted with respect to the standard three-neutrino ones.
As a consequence, this background gets depleted, too.
In Tab. 3 we report the expected number of electrons in the OPERA detector, according
to Eq. (16), for different values of θ13, θ14, θ24 and θ34. Input points have been chosen
according to the allowed regions in the parameter space shown in Sec. 3. The other
parameters are: θ12 = 34
◦; θ23 = 45
◦; ∆m2
= 7.9× 10−5 eV2; ∆m2
= 2.4× 10−3 eV2
and ∆m2
= 1 eV2 (all mass differences are taken to be positive). Eventually, phases
have been fixed to: δ1 = δ2 = 0; δ3 = 90
◦. Backgrounds have been computed accordingly
to Ref. [60]. Rates refer to a flux normalized to 4.5×1019 pot/year (the nominal intensity
of the CNGS), an active lead target mass of 1.8 Kton and 5 years of data taking. For
comparison, we also report the expected number of events in the usual 3-family scenario.
As it can be seen from Tab. 3, the difference between the (3+1) model and the stan-
dard three-neutrino oscillations are smaller in this channel than in the νµ → ντ one.
Moreover, they linearly depends on θ13, as it is clear from Eq. (13). For θ13 = 5
◦, this
channel will be of no help to test the allowed parameter space of the (3+1) model. On
the other hand, for θ13 saturating the Chooz-Bugey bound, both νµ → ντ and νµ → νe
might cooperate. However, notice that backgrounds to this signal are much larger than
the difference between (3+1) model and standard three-neutrino oscillations for any
value of θ13.
6 Sensitivity to (3 + 1) sterile neutrinos at OPERA
In this section we study the sensitivity to θ13 and to the active-sterile mixing angles
θ14, θ24 and θ34 at the CNGS beam, using both the νµ → ντ and νµ → νe appearance
channels at the OPERA detector. In the rest of this section, the known three-family
subspace angles have been fixed to: θ12 = 34
◦; θ23 = 45
◦. The mass differences have
been fixed to: ∆m2
= 7.9× 10−5 eV2 and ∆m2
= 2.4× 10−3 eV2. The CP-violating
phases δ1 and δ2 have been kept fixed to δ1 = δ2 = 0. On the contrary, the CP-violating
phase δ3 is fixed to two values: δ3 = 0 or 90
◦. Notice that this phase is still present in
the oscillation probabilities even when θ12 and θ13 vanish, see Eq. (10). At atmospheric
L/E, oscillations driven by an O(eV2) mass difference are averaged. We have checked
that our results apply for any value of ∆m2
≥ 0.1 eV2.
In Fig. 4 we show the sensitivity limit at 99% CL in the (θ13, θ14) plane (left) and
in the (θ24, θ34) plane (right) from a null result of the OPERA experiment, assuming
1, 2, 3, 5 and 10 times the nominal intensity of 4.5 × 1019 pot/year. The coloured
regions show the present bounds at 90% and 99% CL. We assume θ23 = 45
◦ and
δ3 = 0
◦ (top) or δ3 = 90
◦ (bottom). The sensitivity is defined as the region for which a
(poissonian) 2 d.o.f.’s χ2 is compatible with a “null result” at the 99% CL. We refer to
“null result” when θ13 and the three active-sterile mixing angles, θ14, θ24 and θ34 vanish
simultaneously. Both νµ → ντ and νµ → νe oscillations have been considered, with the
corresponding backgrounds treated properly as in Sec. 5. An overall systematic error
of 10% has been taken into account.
In the left panels of Fig. 4 we can see that OPERA can improve only a little the bound
on θ13 after 5 years of data taking working at nominal CNGS beam intensity, both
for δ3 = 0 (top panel) or δ3 = 90
◦ (bottom panel). Increasing the nominal intensity,
however, a significant improvement on the bound is achieved for any value of θ14. Notice
that the limit on θ14 is almost unaffected by the OPERA data. This is because for the
νµ → ντ and νµ → νe oscillation probabilities at atmospheric L/E, the θ14-dependence
always arises at third-order in the small parameters θ13, θ14, θ24 and θ34 (see Eqs. (9)
and (10) for the explicit expression in the adopted parametrization, Eq. (3)). On the
contrary, the θ13-, θ24- and θ34-dependences in the same oscillation probabilities are
quadratic in the small parameters. In case of vanishing active-sterile mixing angles,
0 2 4 6 8 10 12 14
 = 45°, δ
 = 0°
0 2 4 6 8 10 12 14
 = 45°, δ
 = 0°
0 2 4 6 8 10 12 14
 = 45°, δ
 = 90°
0 2 4 6 8 10 12 14
 = 45°, δ
 = 90°
× 5× 10
Fig. 4. Sensitivity limit at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane
(right) from a null result of the OPERA experiment, assuming 1, 2, 3, 5 and 10 times the
nominal intensity of 4.5 × 1019 pot/year. The coloured regions show the present bounds at
90% and 99% CL. We assume θ23 = 45
◦ and δ3 = 0
◦ (top) or δ3 = 90
◦ (bottom).
θi4 = 0, see Ref. [60].
In the right panels of Fig. 4 the sensitivity of OPERA to θ24 and θ34 is shown. First
of all, notice that the sensitivity is strongly affected by the intensity of the beam.
No improvement on the existing bounds on these two parameters is achieved after 5
years of data taking at nominal CNGS beam intensity, for any of the considered value
of δ3. Already with a doubled flux intensity, some sensitivity to θ24, θ34 is achievable.
The sensitivity enhancement strongly depends on the value of the CP-violating phase
δ3, however. For δ3 = 0, OPERA can exclude a small part of the 99% CL allowed
region, only. On the other hand, for δ3 = 90
◦ twice the nominal CNGS flux suffices
0 2 4 6 8 10 12 14
 = 45°, δ
 = 0°
0 2 4 6 8 10 12 14
 = 45°, δ
 = 0°
0 2 4 6 8 10 12 14
 = 45°, δ
 = 90°
0 2 4 6 8 10 12 14
 = 45°, δ
 = 90°
Fig. 5. Sensitivity limit at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane
(right) from the combined analysis of present data and a null result of the OPERA experiment,
assuming 1, 2, 3, 5 and 10 times the nominal intensity of 4.5 × 1019 pot/year. The coloured
regions show the present bounds at 90% and 99% CL. We assume θ23 = 45
◦ and δ3 = 0
(top) or δ3 = 90
◦ (bottom).
to put a bound on θ34 ≤ 25
◦ for θ24 ≥ 4
◦ at 99% CL. For maximal CP-violating
δ3, increasing further the CNGS flux can significantly constrain the (θ24, θ34) allowed
parameter space. Notice, eventually, the strong correlation between θ24 and θ34 in the
right panels of Fig. 4. This is an indication that the dominant channel that constrains
these angles is νµ → ντ . As it can be seen in Eq. (10), the two angles always appear in
combination, with an approximate exchange symmetry θ24 ↔ θ34.
The allowed regions at 99% CL in the (θ13, θ14) plane (left) and in the (θ24, θ34) plane
(right) from the combined analysis of present data and a null result of the OPERA
experiment after 5 years of data taking (assuming 1, 2, 3, 5 and 10 times the nominal
CNGS intensity of 4.5 × 1019 pot/year) are eventually shown in Fig. 5. The coloured
regions refer to the present bounds at 90% and 99% CL, for θ23 = 45
◦ and δ3 = 0
◦ (top)
or δ3 = 90
◦ (bottom). As it can be seen, the sensitivity of OPERA strongly benefits
from the complementary information on the neutrino parameters provided by other
experiments. In this case, even with the nominal beam intensity the extension of the
allowed regions is reduced by a moderate but non-negligible amount.
7 Conclusions
The results of atmospheric, solar, accelerator and reactor neutrino experiments show
that flavour mixing occurs not only in the quark sector, as it has been known for
long, but also in the leptonic sector. Experimental data well fit into a three-family
scenario. The existence of new “sterile” neutrino states with masses in the eV range is
not excluded, however, provided that their couplings with active neutrinos are small
enough.
In this paper, we have tried to test the potential of the OPERA experiment at the
CNGS beam to improve the present bounds on the parameters of the so-called four-
neutrino models. The model, in which only one sterile neutrino is added to the three
active ones responsible for solar and atmospheric oscillations, is the minimal extension
of the standard three-family oscillation scenario.
We have determined the presently allowed regions for all active-sterile mixing angles
and studied the OPERA capability to constrain them further using both the νµ → νe
and νµ → ντ channels. We have performed our analysis using the OPERA detector as
a reference. It can be extended including a detailed simulation of the ICARUS detector
at the CNGS beam.
Our conclusions are the following: if the OPERA detector is exposed to the nominal
CNGS beam intensity, a null result can improve a bit the present bound on θ13, but
not those on the active-sterile mixing angles, θ14, θ24 and θ34. If the beam intensity is
increased by a factor 2 or beyond, not only the sensitivity to θ13 increases accordingly,
but a significant sensitivity to θ24 and θ34 is achievable. The (θ24, θ34) sensitivity strongly
depends on the value of the CP-violating phase δ3, however, with stronger sensitivity for
values of δ3 approaching π/2. Only a marginal improvement is achievable on the bound
on θ14, that should be constrained by high-intensitiy νe disappearance experiments.
Notice that our results hold for any value of ∆m2
≥ 0.1 eV2, i.e. in the region of L/E
for which oscillations driven by this mass difference are effectively averaged.
Acknowledgements
We acknowledge E. Fernández-Mart́ınez, P. Hernández, J. López-Pavón, M. Sorel and
P. Strolin for useful discussions and comments. We thank T. Schwetz for pointing out
to us an error in the first version of the paper and for useful comments on it. The
work has been partially supported by the E.U. through the BENE-CARE networking
activity MRTN-CT-2004-506395. A.D. received partial support from CiCYT through
the project FPA2006-05423. M.M. received partial support from CiCYT through the
project FPA2006-01105 and the MCYT through the Ramón y Cajal program. A.D. and
M.M. acknowledge also financial support from the Comunidad Autónoma de Madrid
through the project P-ESP-00346. D.M. would like to thank CERN, where part of this
work has been accomplished.
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	Introduction
	Four neutrino mass schemes
	Oscillation probabilities and allowed parameter space
	The CNGS facility
	Appearance channels at the CNGS
	 oscillations
	e oscillations
	Sensitivity to (3+1) sterile neutrinos at OPERA
	Conclusions
	References
