Table Of ContentNeutral-Current Atmospheric Neutrino Flux Measurement Using Neutrino-Proton
Elastic Scattering in Super-Kamiokande
John F. Beacom1 and Sergio Palomares-Ruiz1,2
1 NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500, USA
2 Departamento de F´ısica Te´orica, Universidad de Valencia, 46100 Burjassot, Valencia, Spain
[email protected], [email protected]
(Dated: January 9, 2003)
Recent results show that atmospheric νµ oscillate with δm2 ≃3×10−3 eV2 and sin22θatm ≃1,
andthatconversionintoνeisstronglydisfavored. TheSuper-Kamiokande(SK)collaboration,using
a combination of three techniques, reports that their data favor νµ → ντ over νµ → νsterile. This
distinction is extremely important for both four-neutrino models and cosmology. We propose that
neutrino-protonelastic scattering (ν+p→ν+p)in water Cˇerenkov detectors can also distinguish
betweenactiveandsterileoscillations. Thiswasnotpreviouslyrecognized asausefulchannelsince
3 onlyabout2%ofstruckprotonsareabovetheCˇerenkovthreshold. Nevertheless,inthepresentSK
0 data there should be about 40 identifiable events. We show that these events have unique particle
0 identification characteristics, point in the direction of the incoming neutrinos, and correspond to a
2 narrow range of neutrino energies (1−3 GeV, oscillating near the horizon). This channel will be
n particularly important in Hyper-Kamiokande, with ∼40 times higher rate. Our results have other
a important applications. First, for a similarly small fraction of atmospheric neutrino quasielastic
J events,theproton is relativistic. This uniquelyselects νµ (not ν¯µ) events, useful for understanding
9 matter effects, and allows determination of the neutrino energy and direction, useful for the L/E
dependenceofoscillations. Second,usingaccelerator neutrinos,bothelastic andquasielastic events
1 with relativistic protons can be seen in the K2K 1-kton near detector and MiniBooNE.
v
0 PACSnumbers: 13.15.+g,25.30.Pt,29.40.Ka FERMILAB-Pub-03/003-A,FTUV-03-0109,IFIC/03-01
6
0
1 I. INTRODUCTION ton is above the Cˇerenkov threshold (see also a related
0 detectionapplicationatlowenergiesinRef.[12]). Thisis
3 High-statistics atmosphericneutrino data from Super- aneutral-current(NC)channel,andsodirectlymeasures
0
Kamiokande(SK)showν vacuumoscillationdisappear- the total active neutrino flux. The number of charged-
/ µ
h ance with δm2 ≃ 3×10−3 eV2 and sin22θ ≃ 1 [1]. current(CC)atmosphericneutrinoeventsinSKis∼104,
atm
p and the ratio of cross sections σ /σ ≃ 0.2 for both
Both atmospheric [1] and reactor[2] data strongly disfa- NC CC
- neutrinos and antineutrinos in this energy range [13], so
p vor νµ →νe oscillations, so one of the crucial remaining we expect ∼103 NC events. It has alwaysbeen assumed
e questions is whether the oscillations are of ν → ν or
h ν →ν ,oramixture. Adefiniteanswerwµouldhτave thatprotonsabovethe Cˇerenkovthresholdaretoo small
µ sterile
: in number and too difficult to identify. We show that
v important implications for four-neutrino mixing models
the number of events is reasonable and that the pro-
i designed to accommodate [3] the LSND signal [4] and
X tons are identifiable. The protons are directional, and
also for the role of neutrinos in cosmology [5].
r an up-down ratio discriminates between active and ster-
A variety of techniques have been proposed to distin-
a
ileoscillationssinceupgoingν arefullydetectablewhile
guish between atmospheric ν → ν and ν → ν τ
µ τ µ sterile
upgoing ν are invisible.
oscillations [6, 7, 8, 9, 10]. The SK collaborationreports sterile
that their data favor pure active oscillations ν → ν This detection channel for atmospheric neutrinos in
µ τ
over pure sterile oscillations ν →ν at better than SK was mentioned by Vissani and Smirnov, but was
µ sterile dismissed as having too few events above the Cˇerenkov
the 99% CL [11], though an appreciable sterile admix-
threshold [6]. Some estimates for the event rate were
ture remains possible. This claim is based on a com-
mentioned in SK Ph.D. theses [14]. In those studies, all
bination of three techniques: (i) matter effects on the
of the protons were classified by the Monte Carlo as ei-
partially-containedandthroughgoing-muonsamples,(ii)
ther e-like or µ-like, and were assumed to be buried by
the contribution of inelastic neutral-current events with
themuchmorenumerousrelativisticelectronsandmuons
pionstothefully-containedmulti-ringeventssample,and
from CC channels. It is unknown what fraction of rela-
(iii) statistically-identified tau lepton decays. Individu-
tivistic protons failed general cuts designed for electrons
ally, these techniques do not offer strong evidence, and
and muons, and hence were not classified at all.
the latter two are made difficult by large backgrounds.
We propose a new technique for active-sterile discrim- In what follows, we develop the case that atmo-
ination: neutrino-proton elastic scattering, spheric neutrino-proton elastic scattering events above
theCˇerenkovthresholdshouldbeinthepresentSKdata
ν+p→ν+p, (1.1) set, that they can be identified, and that they will be
very useful in studying neutrino oscillations. Finally, we
frombothfreeandboundprotons,wherethestruckpro- identify some other applications of our results.
2
0.2 0.08
0.15 0.06
-1V]
2m Ge -1GeV]
-380 c 0.1 2cm 0.04
p [1 -380
d 1
σ/el [
d 0.05 0.02
0 0
0.1 1 5 1 1.2 1.4 1.6 1.8 2
Struck proton momentum, p [GeV] Struck proton momentum, p [GeV]
FIG.1: Differentialν+p→ν+pcrosssectionasafunction FIG. 2: The components of the differential ν +p → ν +p
ofprotonmomentum(notelogscaleusedfordisplay). Upper cross section: the sum (solid line), A term (dot-dashed line),
linescorrespondtoneutrinosandlowerlinestoantineutrinos, B term (dotted line) and C term (dashed line) for Eν = 2
atEν =1GeV(solid),3GeV(dotted),and50GeV(dashed), GeV, as a function of the recoil proton momentum p. For
terminatedatthemaximumallowedprotonmomentum(only antineutrinos,theB termcontributeswiththeoppositesign,
visible in the 1 GeV case). The Cˇerenkov threshold in water and so the differential cross section is much smaller. Note
at p=1.07 GeV is shown with a thin dotted line. that only p>1 GeV is shown, in contrast to Fig. 1.
II. CROSS SECTION FOR ν+p→ν+p Themostimportantprotonrecoilmomentaliebetween
about1and2GeV.Thelowerlimitisdeterminedbythe
Cˇerenkovthresholdin waterat 1.07GeV, andthe upper
A. Free Proton Targets
limitbythefallingdifferentialcrosssection(andneutrino
spectrum). In Fig. 2, we show the separate terms of the
The neutrino-proton elastic scattering cross section is
differential cross section for E =2 GeV (other relevant
an important prediction of the Standard Model that has ν
energiesgivesimilar results). The mostimportantterms
been confirmedatGeV energies,e.g., in the E734exper-
in Eq. (2.1) are the B and C terms, which are compa-
iment [13]. The differential cross section on free proton
rable. For neutrinos, they add constructively, and for
targets in terms of the struck proton momentum p (and
antineutrinos, they add destructively. The suppression
corresponding mass M , kinetic energy T , and total en-
p p of the differential cross section at large proton momenta
ergy E ) and neutrino energy E is given by [15]
p ν iscausedbythe decreaseofthe formfactorsatlargeQ2.
The vector form factors F and F that couple the
dσel = G2FMp3p A∓B(s−u) +C(s−u)2 . (2.1) exchanged Z◦ to the proton h1ave been2well measured in
dp 4πE2E M2 M4
ν p (cid:20) p p (cid:21) electron-nucleon scattering, and are
The minus (plus) is for neutrinos (antineutrinos), and F = (1−2sin2θ )(G3 −F3)
1 W V V
2
A = 4τ G2A(1+τ)−F12(1−τ) − 3sin2θW(G0V −FV0) (2.5)
+ F22(cid:2)(1−τ)τ +4F1F2τ (2.2) F = (1−2sin2θ )F3 − 2sin2θ F0 , (2.6)
B = 4τG (F +F ) (2.3) 2 W V 3 W V
A 1 2 (cid:3)
C = 1 G2 +F2+F2τ , (2.4) wheresin2θW =0.231. The remainingformfactorshave
4 A 1 2 a dipole form (vector mass M =0.84 GeV), and are
V
(cid:0) (cid:1)
wheres−u=4M (E −M τ),τ =Q2/4M2,andQ2 = 1 1+κ −κ
p ν p p G3 = p n (2.7)
−q2 =2M T . The differential cross section is shown in V 2(1+Q2/M2)2
p p V
Fig. 1. Almost all of the struck protons are below the 3 1+κ +κ
Cˇerenkov threshold; the subject of this paper are those G0V = 2(1+Qp2/M2n)2 (2.8)
few above it. Since this is a neutral-current channel, all V
1 κ −κ
activeflavorsofneutrinoscontributeequally(thoughthe F3 = p n (2.9)
antineutrino cross section is smaller at the relevant Eν). V 2(1+τ)(1+Q2/MV2)2
We consider backgrounds to the detection of protons in F0 = 3 κp+κn , (2.10)
the elastic channel in detail below. V 2(1+τ)(1+Q2/M2)2
V
3
where κ = 1.793 and κ = −1.913 are the proton and 0.3
p n
neutron anomalous magnetic moments.
Theaxialformfactorisassumedtohaveadipoleform 0.25
(axial mass M =1.03 GeV), given by
A
1 1.267 2cm] 0.2
GA =−2(1+Q2/M2)2 . (2.11) -380
A 1
θ [p0.15
The main uncertainty in the differential cross section, os
c
Etrqin.o(2s.c1a),ttiesricnaguseexdpberyimuennctesrtsauingtgieesstinMGAA=. W1.h0i3leGneeVu-, σd/del 0.1
charged-pion electroproduction data [16] suggest M =
A
0.05
1.08 GeV; all of the data are reviewed in Ref. [17]. The
larger value of M would increase the differential cross
A
sectionbyabout10%intheintervalweareinterestedin. 0
0 0.2 0.4 0.6 0.8 1
There can also be strange sea quark contributions to all cos θ
of the form factors, especially the spin contribution ∆s p
that modifies the axialformfactor [18], possibly increas-
ingthedifferentialcrosssectionbyabout10%. However, FIG. 3: Thedifferential cross section dσel/dcosθp (θp is the
angle of the struck proton with respect to the incoming neu-
the effects of changing ∆s and M are correlated. We
A trinodirection)fordifferentneutrinoenergies,Eν =1(dashed
do not assume enhancements to the cross section from
line),2(dottedline),3GeV(dot-dashedline). Thesolid-line
either a larger MA or a ∆s contribution. segment of each curve indicates protons that are above the
Cˇerenkov threshold. Since we are neglecting proton reinter-
actionsinthenucleus,onlytheforwardhemisphereisshown.
B. Nuclear Effects
We have so far only considered free protons, which
are 2 out of 10 targets in water. Bound protons have C. Angular Distribution
nonzero initial momenta (Fermi motion), and the struck
protons cannot make transitions to already-filled states
In order to discriminate between active and sterile os-
at low energies (Pauli blocking). In typical Fermi-gas
cillations, the struck protons must be directional. The
models (p ≃ 220 MeV) for neutrino interactions at a
F anglesofthefinalparticlesrelativetotheinitialneutrino
few GeV, these effects reduce the total cross section by
direction are given by
about 20% [19]. However, these effects can be neglected
whenthe struckprotonis abovetheCˇerenkovthreshold.
Inthislimit,thestruckprotonisejectedfromthenucleus
E +M T
and the momentum transfer greatly exceeds the initial cosθ = ν p p (2.12)
p
momentum. This maybe seenfromthe differentialcross Eν sTp+2Mp
section results (for the CC channel) in Refs. [19, 20]. M T
cosθ = 1− p p . (2.13)
The struck proton may reinteract as it leaves the nu- ν E (E −T )
ν ν p
cleus[20];attherelevantmomenta,theinteractionprob-
ability is about 1/2, corresponding mostly to forward
elastic collisions [21]. There are neutrino interaction The maximum proton momentum is obtained when the
codes [22] that take nuclear reinteractions into account, neutrino reverses its direction and the proton goes for-
but we do not. The average momentum loss for protons ward. The most important neutrino energies for this
boundinoxygeninquasielasticscatteringusingtheK2K channel are 1 − 3 GeV, so for δm2 ≃ 3 × 10−3 eV2,
neutrino beam is only ≃90 MeV [21]; it is reasonable to the oscillation length corresponds to the direction of the
assume a similar average value for elastic atmospheric horizon. Thusdowngoingν havenotoscillatedyet,and
µ
neutrino events. Since the spectrum dN/dp is so steep upgoing ν have oscillated into either ν (same rate as
µ τ
(see below), taking this into account could reduce the downgoing) or ν (a reduced rate). As shown in
sterile
numberofeventsabovetheCˇerenkovthresholdbyabout Fig.3,mostprotonsemergeatratherlargeanglesrelative
20%. Weneglectthiseffectbecausewearealsoneglecting to the neutrino direction. However, these are the same
the fact that there would be some compensation to this majority of protons that are below the Cˇerenkov thresh-
loss from ν+n→ν+n (about 1.5 times largerthan for old. The relevant protons above the Cˇerenkov threshold
protons)followedbyan+p→p+nnuclearreinteraction are in fact quite forward, but not perfectly so (we show
that transfers most of the momentum to the proton. A belowthatthelowestneutrinoenergiesarethemostrele-
full detector Monte Carlo will be needed to model these vant). Comparedtotheintrinsicangularvariation,angu-
effects, along with the details of the Cˇerenkov particle lardeflectionsfromnuclearreinteractions[21]canalmost
identification and backgrounds. always be ignored.
4
100
1000
100 10
-1dN/dp [yr GeV] 101 -1dN/dp [yr GeV] 0.11
0.1
0.01
0.01
0 1 2 3 4 5 0.0011 2 3 4 5
Struck proton momentum, p [GeV] Struck proton momentum, p [GeV]
FIG. 4: Proton spectrum, dN/dp (yr GeV)−1, as a function FIG. 5: Proton spectrum, dN/dp (yrGeV)−1, as a function
ofprotonmomentum(solidline)inthe22.5ktonmassofSK. of proton momentum for different neutrino energy intervals
Notethatthespectrumfalls veryquickly(verticallog scale). contributing to it: all neutrinos (solid line), up to 1 GeV
The Cˇerenkov threshold for protons in water is also shown (dashedline),[1,2]GeV(dot-dashedline),[2,3]GeV(dotted
(thin dotted vertical line). line), [3, 4] GeV (doubly-dot-dashedline), [4, 5] GeV (short-
dashed line). Note that only p>1 GeV is shown.
III. PROTON RECOIL SPECTRUM
about 0.7 (approximately the official SK number). Thus
A. No-Oscillation Prediction
in the present data there should be about 40 elastically-
scattered protons above the Cˇerenkov threshold. This
The struck proton momentum spectrum in SK for no
number is small, but it should be noted that much more
oscillations (or oscillations among active flavors only) is
data is expected from SK in the future. And indeed,
∞ possibly also from a future 1 Mton Hyper-Kamiokande
dN dNν dσel detector with ∼40 times higher rate [24] (or UNO, with
(p)=Z dE (E ) (E ,p) (3.1)
ν ν ν
dp Z dEν dp ∼ 20 times higher rate [25]); a high-statistics sample of
(Eν)min neutrino-proton elastic scattering events could then be
quickly collected.
where Z = 7.5 × 1033 is the number of protons (free
and bound) in the 22.5 kton fiducial mass of SK and The small number of protons above the Cˇerenkov
dσ /dp is the differential elastic cross section, Eq. (2.1). threshold is a consequence of the large proton mass and
el
The atmospheric neutrino flux [23] dN /dE has been the shape of the differential cross section, which falls
ν ν
integratedover4π(thisisdoneonlytocalculatethetotal steeply above a peak at p ≃ 400 MeV, as shown in
yields, as in practice the directionality of the protons Fig. 1. Additionally, the atmospheric neutrino spectrum
can be used). We sum over all three flavors of neutrinos is steeply falling with neutrino energy, and this is not
and antineutrinos, taking into account the reduced cross compensated by growth in the cross section, since both
sectionfor antineutrinos. The minimum neutrino energy the differential and total elastic cross sections become
for a given proton momentum is independent of neutrino energy above a few GeV.
About 95% of the protons above threshold are in the
1
(E ) = (E +p−M ). (3.2) interval between p =1.07 GeV and 2 GeV. This plays a
ν min p p
2
crucialrole in distinguishing protons from other charged
In Fig. 4, we show the complete momentum spectrum particles, as well as the details of how they are stopped.
forprotonselasticallyscatteredbyatmosphericneutrinos InFig.5,weshowhowdifferentrangesofneutrinoenergy
and antineutrinos in SK, per year of detector livetime. contributeto the protonspectruminthis momentumin-
The spectrum falls very steeply, and the fraction of terval. About 90% of the protons above the Cˇerenkov
protons above the Cˇerenkov threshold at p = 1.07 GeV threshold are produced by neutrinos with E < 5 GeV,
ν
is very small, about 2%. For the present exposure time and in fact, the majority are produced by much lower
of SK, 1489 days, we predict about 60 protons above neutrino energies. The kinks in Fig. 5 arise because we
the Cˇerenkov threshold (and about 2000 below). In or- considerbothminimumandamaximumneutrinoenergy
der to normalize our results, we calculated the number to draw each curve; e.g., the dotted line for neutrino en-
of quasielastic events in SK; we agree with the SK no- ergies between 2 and 3 GeV has a kink at p ≃ 2.4 GeV
oscillation numbers if we assume a detector efficiency of becausewedonotincludeneutrinoenergiesbelow2GeV.
5
B. Effect of Neutrino Oscillations IV. PROTON PARTICLE IDENTIFICATION
The vacuum oscillation length is A. Electron and Muon PID
4πE E
L = ν ∼1000 ν km (3.3)
osc δm2 1 GeV In this section, we show how relativistic protons from
(cid:18) (cid:19)
neutrino-protonelastic scattering can be separated from
where we have used δm2 ≃3×10−3 eV2 [1, 11]. This is other single-ring events in a Cˇerenkov detector like SK.
close to the distance to the horizon, so that downgoing
Quasielasticinteractionsofatmosphericneutrinoscreate
neutrinoshavenotoscillatedandupgoingneutrinoshave relativistic electrons and muons that produce Cˇerenkov
oscillated several times. Since the mixing is maximal,
radiation, which is seen by phototubes as rings on the
half of the upgoing ν remain ν and half oscillate to
µ µ walls of the detector. The rates are large, of order 103
either ν or ν .
τ sterile eventsperyear,tobecomparedtooforder10relativistic
As shown in Fig. 3, the initial neutrino direction is
protons per year. However, the unique particle identifi-
largely maintained by the proton direction, the latter to
cation(PID) properties of protons will allow rejectionof
be measured from its Cˇerenkov cone. We show below
these backgrounds.
that the neutrino energy can also be estimated from the
proton Cˇerenkov information, so that L /E can be es- Electrons and muons are stopped by continuous elec-
ν ν
tromagneticenergylosses(mostlyionization,butalsora-
timated on an event-by-event basis, which improves the
diative lossesfor electrons); Cˇerenkovradiationdoes not
ability to study neutrino oscillations. Even in the ab-
cause significantenergyloss. The continuousenergy loss
sence of a neutrino energy estimate, Fig. 5 shows that
−dE/dx is given by the Bethe-Bloch equation, reviewed
only a narrow range of neutrino energies contributes to
thesignalabovetheCˇerenkovthresholdbutbelowwhere inRef.[26]. Therangeofachargedparticle,thedistance
required to bring it to rest, is obtained immediately by
the proton spectrum is greatly diminished.
integration. In Fig. 6, we show the range of muons in
There are uncertainties in the neutrino-proton elastic
water as a function of momentum. Electrons, because
scattering cross section, e.g., from the axial form fac-
of their small mass, have higher −dE/dx for the same
tor as well as from nuclear correctionsto the free-proton
momentum and are stopped in less distance; in addi-
cross section. There are also uncertainties introduced
tion,multiplescatteringchangestheirdirection. Wealso
by our simple modeling of SK. For example, the number
of protons above the Cˇerenkov threshold is quite sensi- show a range curve for protons, which would be correct
if protons only lost energy electromagnetically (at high
tive to the index of refraction; we assumed 1.33, but in
momenta, where β = 1, the proton and muon range are
a more careful treatment one would have to model the
wavelengthdependence of the Cˇerenkovemission, atten- nearly the same; at low momenta, the proton velocity
is less and hence the electromagnetic losses are higher).
uation, index of refraction, and phototube quantum ef-
However, for protons in the relevant momentum range,
ficiency. Finally, there is also a 20% uncertainty on the
discrete nuclear collisions are more important than con-
atmosphericneutrinofluxnormalization. Inlightofthese
tinuous electromagnetic energy losses.
facts, we must focus ona normalization-independentob-
servablesuchthezenithanglespectrumshape,oratleast Since most atmospheric neutrinos are at energies of
an up-down ratio. at least a GeV, the electrons and muons created in
Consideraninitialatmosphericneutrinoflavorratioof quasielastic reactions almost all have an initial velocity
ν : ν : ν = 1 : 2 : 0, which is a good approximation. of β ≃ 1. This corresponds to the maximum Cˇerenkov
e µ τ
Downgoingneutrinoshavenotoscillated,andhavethese intensity,anda Cˇerenkovemissionangle of41◦ inwater.
flavor ratios. However,the upgoing neutrinos have oscil- Formuons,theouteredgeoftheCˇerenkovringsissharp,
lated several times. For maximal mixing of ν to either but for electrons, which suffer changes in direction due
µ
ν or ν , the flavorratios for the upgoing events are to multiple scattering, the outer edge has a more fuzzy
τ sterile
either 1 : 1 : 1 or 1 : 1 : 0 (we ignore mixing with ν appearance. As the velocity decreases,the Cˇerenkov an-
e
as well as matter effects). Since this is a neutral-current gleandintensityboth decrease. Inanidealdetector,the
cross section, equally sensitive to all flavors,the upgoing rings on the wall would fill in completely as the particle
flux divided by the downgoing flux would be 1 for pure sloweddown(and approachedthe wall). In practice, the
active oscillations and 2/3 for pure sterile oscillations. rapid decrease in the Cˇerenkov angle and intensity once
Assuming 40 events in the present SK data, this corre- the particle falls below β ≃ 1 mean that the inner edge
spondsto20downgoingeventsandeither20(activecase) of the Cˇerenkov rings is typically undersampled. This is
or 13 (sterile case) upgoing events, the latter reflecting especially true for electrons, which lose velocity in less
a 1.5 sigma deviation. Thus with the present data this distance than muons. Thus muon rings have a sharp
technique could not be decisive, but none of the three outer edge and are partially filled in, whereas electron
techniques used by SK to distinguish ν from ν is rings are fuzzy and not filled in. Electrons and muons
τ sterile
individually decisive [11]. The advantage of neutrino- can be very reliably distinguished in SK (to about 1%),
proton elastic scattering is that it could be rather clean, as has been confirmed by a variety of means, including
both in concept and in practice. direct beam tests at KEK [27].
6
B. Proton PID: Nuclear Collisions Only
104
Unlikeelectronsandmuonsatthesemomenta,protons
have a large cross section for nuclear collisions. If elec-
tromagnetic energy losses can be ignored, as we assume 2m]103
in this subsection, then the fraction of protons surviving c
g/
a distance x without undergoing a nuclear collision is e [
c
N(x)/N(0)=exp(−x/λN). (4.1) Distan102
For the nuclear attenuation length λ in water, we use
N 101
λ =[(ρ/M ) σ ]−1 , (4.2)
N H2O p+H2O
100
whereρ/M isthenumberdensityofwatermolecules. 0.1 1 10 100
H2O
For the cross section, we use Initial momentum [GeV]
σp+H2O =σp+16O+2σp+pfree. (4.3) FIG. 6: Distance (g/cm2) traveled in water by protons and
muons as a function of momentum. We show the range (the
Note that we cannot simply use the number density of distancetocomecompletelytorestbyelectromagneticlosses
nucleonsinwater,sincenucleonsboundinnucleishadow alone)formuons(dashedline)andprotons(dot-dashedline).
eachother(sincenucleardensitiesareapproximatelycon- For protons, we also show thenuclear attenuation length λN
stant,thecrosssectionalareaofanucleusscalesasA2/3). in water (solid line), the distance over which a fraction 1/e
Forscatteringfrom16O,weuseonlythereaction(inelas- ofprotonstravelwithoutscattering. TheCˇerenkovthreshold
for protons is also shown (thin dotted vertical line).
tic)crosssection,takenfromRef.[28](whichcorrectsthe
earlierRef.[29]). Wehavenotincludedtheelasticpartof
the cross section on 16O, as it is very strongly peaked in
from no collision at all). Thus we assume that after a
theforwarddirection[30],correspondingtominimalmo-
nuclearcollision,therearenorelativisticprotons,neither
mentum loss (for a proton with p =2 GeV, a scattering
the original proton nor any accelerated target protons.
angle less than 20 degrees corresponds to less than 1% NowweconsidertheCˇerenkovsignaturesofthestruck
changein momentum). For scatteringfromfree protons,
protons, assuming that a single nuclear collision brings
wedousethetotalcrosssection,takenfromRef.[26,31]; the proton below the Cˇerenkov threshold. Protons have
since the target and projectile have the same mass, it is
several unique PID characteristics. Since the proton ve-
easier to have substantial momentum transfer. Above locity is constantuntil thatcollision,the Cˇerenkovangle
1 GeV momentum, the proton nuclear cross section on
and intensity are constant until they abruptly vanish.
water is nearly constant at about 390 mb; the oxygen
Whiletheprotonsarerelativistic,theydohaveβ <1,so
reaction cross section is about 300 mb, and the proton their Cˇerenkov angle is less than the 41◦ for relativistic
cross section is about 45 mb.
electrons and muons. Just as for muons, the outer edge
In Fig. 6, we show the nuclear attenuation length for of the Cˇerenkov rings is sharp for protons. However,
protonsinwater,ignoringelectromagneticlosses. Above since their Cˇerenkov angle is both smaller and constant,
theCˇerenkovthreshold,thislengthisalwaysshorterthan
proton rings are filled in very densely and at a constant
theelectromagneticrangeofprotonscalculatedifnuclear
rate. The proton path length is rather short, of order
collisionsareignored. Thusinthissubsectionweconsider
λ ≃ 80 cm, compared to the several meters typical of
N
that only nuclear collisions are important, and in the
muons, and thus the proton events are always fully con-
next subsection we calculate the corrections due to elec-
tained. When the proton is abruptly stopped, the inner
tromagneticenergy losses. In either case,protons havea edgeoftheCˇerenkovringsisalsosharp,unlikeformuons
short path length and will be fully-contained events.
or electrons.
Since 95% of the proton events are in the momentum The number of Cˇerenkov photons produced per unit
range of 1−2 GeV, with a steeply falling spectrum (see
pathlengthandphotonwavelengthintervalbyaparticle
Fig.5),asinglenuclearcollisionwithevenmoderatemo-
of unit charge and velocity β is
mentum transfer brings the proton below the Cˇerenkov
threshold. Inaninelasticcollisionwithanoxygennucleus d2N 2πα 1
ph = 1− (4.4)
that breaks it into several fragments, it is very unlikely dxdλ λ2 β2n2(λ)
than any of them are above the Cˇerenkov threshold. In (cid:18) (cid:19)
a collision with a free proton, the initial momentum of In Fig. 7 we compare how the Cˇerenkov intensity and
less than 2 GeV is shared between both protons, leav- angle vary with the distance traveled for muons (elec-
ing them both below the Cˇerenkov threshold (a forward tromagnetic losses only) and protons (nuclear collisions
elastic collision mightexchange the projectile andtarget only). We choose the same initial velocities (and hence
protons, causing little change in the Cˇerenkov pattern the same initial Cˇerenkov angle and intensity), using
7
350 1
300
1] 250 0.8
-m
c 200
[
x
/dh 150 0.6
p
dN 100 FN
50
0.4
0
40
0.2
] 30
s
e
e
gr 0
de 20 1 1.2 1.4 1.6 1.8 2
θ [c Struck proton momentum, p [GeV]
10
FIG. 8: The fraction of protons that go below theCˇerenkov
threshold by a nuclear collision as a function of proton mo-
00 20 40 60 80 100 mentum. TheCˇerenkovthresholdforprotonsinwaterisalso
Distance [cm] shown (thin dotted vertical line)
FIG. 7: Cˇerenkov intensity (upper plot) and angle (lower
plot) as a function of the traveled path length in water for samples. Most were classified as µ-like, which is why we
muons(solidlines)andprotons,consideringonlynuclearcol- have emphasized distinguishing protons and muons. In
lisions (dashed lines) or only electromagnetic losses (dotted fact, protons should be quite distinguishable from elec-
lines). From bottom to top, theinitial velocities are β =0.8, trons as well.
0.9,and0.95. TheCˇerenkovintensityiscalculatedforvisible Since the nuclear cross section is nearly constant, the
light, without attenuation or detection efficiency. With elec- proton track length is a poor estimator of the proton
tromagnetic losses neglected, individual protons always stop
momentum,andevent-by-eventtracklengthfluctuations
abruptly, but with a distribution of path lengths, Eq. (4.1);
from Eq. (4.1) are more important. However, the pro-
we used the average path length λN in thefigure. tonmomentumcanbereliablyestimatedbytheconstant
Cˇerenkov angle and intensity, since those vary apprecia-
bly in the momentum range considered, where β <1.
β = 0.8, 0.9, and 0.95, to highlight how the muon and
proton stopping mechanisms differ. These correspondto
muon momenta of 140 MeV, 220 MeV, and 320 MeV; C. Proton PID: Inclusion of −dE/dx Effects
andtoprotonmomentaof1.25GeV,1.95GeV,and2.85
GeV. Muons of a given β travel a well-defined distance
before they fall below the Cˇerenkov threshold. However, From Fig. 6, it is clear that for protons with momen-
tum somewhat above the Cˇerenkov threshold, electro-
protons of a given β travel a variety of distances, sam-
magnetic losses can be neglected, as we have assumed.
pledfromthe distributioninEq.(4.1). InFig.7wehave
However, right at the Cˇerenkov threshold, this is no
adopted λ as the path length for protons, since this is
N longer true, since the −dE/dx losses are greater and the
the average value.
range is as small as the nuclear interaction length. Thus
MostmuonsintheSKatmosphericneutrinodatahave
theprotonbehaviorisdifferent,asitiscontinuouslyslow-
much longer track lengths than shown here, and hence
ingdown. ItmayfinallygobelowtheCˇerenkovthreshold
are distinguishable. Those muons shown in Fig. 7 have
by either further electromagnetic losses or a nuclear col-
path lengths short enough to be confused with protons,
lision. In Fig. 7, we show results for nuclear collisions
buthaveverydifferentCˇerenkovcharacteristics. Inaddi-
only (dashed lines), as well as for electromagnetic losses
tion, excepting the 20% of negative muons that capture
only (dotted lines).
onoxygen,fully containedmuons canbe taggedby their
Since the nuclear interaction length is nearly indepen-
subsequent decay to an electron or positron of up to 53
dent of momentum, the fraction F of protons brought
MeV. Thus it should be possible to distinguish protons N
belowtheCˇerenkovthresholdbyasinglenuclearcollision
from muons with very high efficiency on an event-by-
is:
event basis. In some SK Ph.D. theses [14] the neutrino-
protonelasticscatteringcrosssectionwasincludedinthe F (p)≃1−e−r(p)/λN (4.5)
N
atmospheric neutrino Monte Carlo. Those events were
automatically classified as e-like or µ-like and then con- where r(p) is the distance a proton travels before going
sideredtobeburiedbythemuchlargerquasielasticevent below the Cˇerenkov threshold, considering only electro-
8
magnetic losses, and λ = 80 cm. Note r(p) is not the elastic scatteringsignal. It shouldbe less here since that
N
full range, and a 1 GeV proton still travels about a me- tracking calorimeter had a lower (sub-Cˇerenkov) thresh-
ter,thoughinvisibly. ThefractionF isshowninFig.8. old for protons. Note also that such events would par-
N
Lower-momentum protons are affected most by electro- tiallycompensatethelossofneutrino-protonelasticscat-
magnetic energy losses. Protons with p >∼ 1.25 GeV are tering events from nuclear reinteractions.
more likely to be brought below Cˇerenkov threshold by We ignore the production of pions, in the initial inter-
a nuclear collision than by electromagnetic losses. In action, by proton reinteraction in the initial nucleus, or
fact, if we convolve this curve with the falling spectrum in the final nuclear collision. Pions produced in the ini-
of struck protons, Fig. 4, about half of all protons are tial interaction are not part of the neutrino-proton elas-
broughtbelowCˇerenkovthresholdby anuclearcollision. tic scattering channel, and cause multi-ring events. We
For the protons brought below threshold by a nu- are only considering single-ring events. Neutral-current
clear collision, the discussion above about proton PID single-pion events with the pion absorbed in the nucleus
neglecting electromagnetic losses is the most appropri- could be a background to the elastic channel; however,
ate, though in some cases there will be some slight de- the fraction of these events with a proton above the
crease in the Cˇerenkov angle and intensity before they Cˇerenkovthresholdshouldbeevenlowerthan2%,dueto
abruptly vanish. The remaining protons fall below the kinematics. MonteCarloandrealdataononquasielastic
Cˇerenkov threshold more gradually, e.g., the lowest mo- scattering using accelerator neutrinos in the K2K SciFi
mentum case (β = 0.8) in Fig. 7, for which electromag- detector suggest that secondary pion contributions are
netic losses dominate. Nevertheless, the Cˇerenkov be- minimal [21]. In their Monte Carlo results, which have
havior is still quite different from muons. For muons at a much more complete treatment of the physics than we
the same initial β, protons go much farther and produce have presented here, the track multiplicity was always 1
much more Cˇerenkov light. For muons that travel the (muon) or 2 (muon and proton), and never 3 (including
same distance above the Cˇerenkov threshold, the proton secondarypions). Additionally,pionscreatedinthefinal
Cˇerenkovangle is much smaller and falls off more slowly nuclearcollisionwouldbedelayedfromtheinitialproton
with distance (the proton velocity is also less, leading to byseveralnanoseconds. ThefiguresinRef.[21]alsosup-
a higher Cˇerenkov ring density on the wall). Thus even portourassumptionthatwhenahigh-momentumproton
low-momentum protons should be quite distinct from has a nuclear collision, it suffers a large momentum loss
muons (and electrons). Finally, when electromagnetic without accelerating new protons.
lossesdominate,theprotonmomentumcanbeestimated Another possible source of background is atmospheric
by conventional Cˇerenkov techniques. muons interacting with the surrounding rock and pro-
ducing fast neutrons that can enter the detector with-
out triggering the veto. These neutrons could in princi-
D. Possible Complications ple scatter protons above the Cˇerenkov threshold. Most
neutrons are far too low in energy to be effective [33]
Since the number of neutrino-protonelastic scattering andneutronsarestronglyattenuatedbythe 4.5mwater
events above the Cˇerenkov threshold is small, of order shielding. Incidentally,neutrino-protonelasticscattering
10 per year in SK, careful consideration of backgrounds events might be visible in the Soudan-2 experiment [34],
will be necessary. Above, we have motivated the case whichhasmuchlessmassandshieldingthanSKbutcan
thatprotonsaredistinguishablefromthemorenumerous detect lower-energy (sub-Cˇerenkov) protons in a track-
electrons and muons from the quasielastic channel; we ing calorimeter. The number of events could be a few
now consider other possible backgrounds. tens, but the neutron backgrounds could be compara-
Onepossiblebackgroundisfromthequasielasticchan- ble [6]. We are not aware of any official analysis of these
nel, ν + n → µ− + p, but with the muon below the eventsbytheSoudan-2collaboration. Fastneutronsfrom
µ
Cˇerenkov threshold and the proton above it. How- the walls can also produce neutral pions by nuclear col-
ever, the total number of quasielastic events with a sub- lisions in the detector; if the two photon rings from the
Cˇerenkovmuonisverylow,about35peryearinSK[32]. decay are overlapping,this can resemble an atmospheric
Most of these are produced by low-energy neutrinos; for νe event [35]. However, the SK collaboration has shown
high-energyneutrinos,thefractionwitharelativisticpro- that such events contribute less than 0.1% of the atmo-
tonisonlyabout2%. Thusthisbackgroundisnegligible. spheric νe signal [36, 37].
There are also neutrino-neutron elastic scattering In summary, a full Monte Carlo study will be needed
events(infact,thecrosssectionisabout1.5timeslarger to correctly implement the initial neutrino interactions,
than for protons) in which the struck neutron carries a possible nuclear reinteractions, pions, nuclear stopping
large momentum. Such events are of course invisible in and electromagnetic losses, backgrounds, and most im-
SK. However, the struck neutrons can sometimes scat- portantly the PID in a realistic detector. Nevertheless,
ter a proton with enough momentum transfer that the we believe thatit lookspromising thatthe relativelyfew
protonis above the Cˇerenkov threshold. In the E734ac- (oforder10peryearinSK)neutrino-protonelasticscat-
celeratorneutrinoexperiment,itwasestimatedthatsuch tering events above the Cˇerenkov threshold can be de-
events were about 15% of the measured neutrino-proton tected with little background.
9
V. RELATED APPLICATIONS ofatmosphericneutrinooscillations,asitwouldallowthe
determinationoftheneutrinoenergyanddirectiononan
A. Accelerator Neutrinos: NC and CC Channels event-by-event basis, allowing a better measurement of
the L/E dependence ofoscillations. Note alsothatthese
quasielastic events with a proton are produced only by
We are not aware of any experiments with p = 1−2
GeVprotonbeamsinCˇerenkovdetectorsthatwouldtest neutrinos,andnotantineutrinos,usefultostudyingmat-
ter effects and the neutrino/antineutrino ratio [41].
thePIDtechniquesintroducedabove. However,itshould
be possible to use accelerator neutrino beams to initiate
neutrino-protonelasticscatteringeventsintherightmo-
mentum range. The spectrum of accelerator neutrinos VI. CONCLUSIONS
doesnotextendashighinenergyasforatmosphericneu-
trinos (though note Fig. 5 shows that most of the signal We propose that neutrino-proton elastic scattering,
comesfromlowenergyneutrinos),butthetotalnumbers ν+p→ν+p,couldbeausefuldetectionreactionforat-
of events expected are much larger. mosphericneutrinosinSK.Thefractionofprotonsabove
The K2K 1-kton near detector would be a good place the Cˇerenkov threshold is not large, only about 2%, but
to start, as this detector is designed to mimic SK [38]. thereshouldbeabout40identifiableeventsinthepresent
ThisdatacouldbeveryusefulfordevelopingprotonPID 1489-day data. We have shown that it should be possi-
techniques. It would also be useful to study quasielastic ble to separate protons from electrons and muons, since
eventsinwhichthe protonisabovetheCˇerenkovthresh- the relevant protons are not fully relativistic and will
old; these are about 8 times more numerous than the typicallybestoppedbyasinglenuclearcollision. Proton
comparable elastic events (though σ /σ ≃ 0.2, the Cˇerenkovringshavesharpouterandinneredges,arevery
NC CC
ratioofthedifferentialcrosssectionsforp≃1−2GeVis denselyfilled, andcorrespondto ashortpathlengthand
smaller). Measuringboththeoutgoingleptonandproton small Cˇerenkov angle. These are fully-contained, single-
wouldallowreconstructionofthe neutrinoenergy,useful ring events. In order to test our proposal, a detailed
formeasuringtheneutrinospectrum. With∼105 events detector Monte Carlo simulation will be needed.
expected, we estimate ∼103 quasielasticand∼102 elas- Neutrino-proton elastic scattering is a neutral-current
tic events with a relativistic proton. reaction and so measures the total active neutrino flux.
TheMiniBooNEdetector[39]couldalsobeused. Since For the relevant neutrino energies, oscillations occur at
itisdesignedtotesttheLSNDsignal(smallmixingangle the distance to the horizon. In addition, protons above
and large δm2) [4], it can be considered a near detector the Cˇerenkov threshold preserve the neutrino direction.
for oscillations with the atmospheric δm2. It contains Thesefactsmeanthatthisdatacanbeusedtotestatmo-
0.68ktonsofmineraloil,andisprimarilyaCˇerenkovde- spheric ν → ν versus ν → ν oscillations. Since
µ τ µ sterile
tector (about 3:1 Cˇerenkov to scintillation light). Mini- there are normalizationuncertainties in the atmospheric
BooNE has unique characteristics that will help proton neutrinoflux,thecrosssection,andaspectsofthedetec-
PID. The index of refractionin oil (≃1.5) is larger than tion,anup-downasymmetrytestshouldbe used. Letus
inwater,allowingalowerCˇerenkovthreshold(andlarger assume 40 identifiable events in the present SK data (no
angle and intensity). The density of oil (≃ 0.8 g/cm3) oscillations). Withoscillations,thereshouldbe20down-
is less than for water, which means longer tracks. And goingeventsandeither20(ν →ν )or13(ν →ν )
µ τ µ sterile
once a proton falls below the Cˇerenkov threshold, it still upgoing events. While not decisive, other techniques for
produces scintillation light. With ∼ 5×105 events ex- active-sterilediscrimination are not individually decisive
pected, we estimate ∼5×103 quasielasticand∼5×102 either;theyobtaintheirpowerincombination. Neutrino-
elastic events with a relativistic proton. These studies protonelasticscatteringhastheadvantageofbeingclean
would be an appealing complement to plans to measure inconcept. TherateintheproposedHyper-Kamiokande
the elastic scattering cross section at Q2 = 0, a test of detector would be about 40 times larger [24].
thestrangequarkcontribution∆stotheprotonspin[40]. Our results on neutrino-proton elastic scattering have
Thecombinedelasticandquasielasticdatacouldmeasure other immediate and important applications. First, us-
the Q2-dependence of the uncertain axial form factor. ingacceleratorneutrinos,this channelcanbe seeninthe
K2K1-ktonnear detector andin MiniBooNE.This data
will reduce the cross section uncertainties and develop
B. Atmospheric Neutrinos: CC Channel the proton PID techniques. Tagging relativistic protons
willbesimilarlyusefulinthequasielasticchannelinthese
As noted, we expect about 8 times more quasielastic detectors. Second,forasmallfractionoftheatmospheric
than elastic events with a proton above the Cˇerenkov neutrinoquasielasticevents,theprotonisrelativisticand
threshold. Thus we estimate about ∼ 300 such events can be tagged using the techniques presented here. This
in the 1489-day SK data. Taking ν oscillations into ac- uniquelyselectsν (notν¯ )events,usefulforunderstand-
µ µ µ
count would reduce this number since ν oscillate to ei- ing matter effects, and allows determination of the neu-
µ
therν (mostlybelowtheCCthreshold)orν . This trino energy and direction, useful for studying the L/E
τ sterile
hasaveryimportantconsequencefromthepointofview dependence of oscillations.
10
ACKNOWLEDGMENTS tract No. DE-AC02-76CH03000), and additionally by
NASA under NAG5-10842. SPR is supported by the
We thank Steve Brice, Takaaki Kajita, Paolo Li- Spanish MCED with a graduate FPU fellowship, by the
pari,Mark Messier,Georg Raffelt, Michael Smy, Manuel Spanish MCYT and FEDER European Funds through
Vicente-Vacas,andSamZellerforusefuldiscussions. We the project FPA2002-00612, and by the OCYT of the
are especially grateful to Mark Vagins for extensive dis- Generalitat Valenciana under the Grant GV01-94. SPR
cussions. JFB is supported as the David N. Schramm thanks the Fermilab Theoretical Astrophysics Group for
Fellow by Fermilab (operated by URA under DOE con- hospitality.
[1] Y.Fukudaetal.,Phys.Lett.B433,9(1998);Y.Fukuda (1986).
et al., Phys. Lett. B 436, 33 (1998); Y. Fukuda et al., [20] C. Bleve, G. Co, I. De Mitri, P. Bernardini, G. Mancar-
Phys. Lett. B 467, 185 (1999); Y. Fukuda et al., Phys. ella, D.MartelloandA.Surdo,Astropart.Phys.16,145
Rev.Lett. 82, 2644 (1999). (2001); G. Co’, C. Bleve, I. De Mitri and D. Martello,
[2] M. Apollonio et al., Phys. Lett. B 466, 415 (1999); Nucl. Phys. Proc. Suppl.112, 210 (2002).
F. Boehm et al., Phys.Rev.D 64, 112001 (2001). [21] C.W.Walter,Nucl.Phys.Proc.Suppl.112,140(2002),
[3] M. Maltoni, T. Schwetz, M. A. Tortola and J. W. Valle, http://neutrino.kek.jp/nuint01/.
Nucl. Phys. B 643, 321 (2002); H. Pas, L. G. Song and [22] D. Casper, Nucl. Phys. Proc. Suppl. 112, 161 (2002);
T. J. Weiler, hep-ph/0209373;and references therein. Y. Hayato, Nucl. Phys. Proc. Suppl. 112, 171 (2002);
[4] A.Aguilar et al.,Phys. Rev.D 64, 112007 (2001). H. Gallagher, Nucl. Phys. Proc. Suppl. 112, 188 (2002);
[5] P.DiBari,Phys.Rev.D65,043509(2002);K.N.Abaza- http://neutrino.kek.jp/nuint01/.
jian, astro-ph/0205238;and references therein. [23] M. Honda, T. Kajita, K. Kasahara and S. Midorikawa,
[6] F. Vissani and A. Y. Smirnov, Phys. Lett. B 432, 376 Phys. Rev.D 52, 4985 (1995).
(1998). [24] Y. Itow et al.,hep-ex/0106019.
[7] Q. Y. Liu and A. Y. Smirnov, Nucl. Phys. B 524, 505 [25] C. K.Jung, hep-ex/0005046.
(1998); R. Foot, R. R. Volkas and O. Yasuda, Phys. [26] K. Hagiwara et al.,Phys. Rev.D 66, 010001 (2002).
Rev. D 58, 013006 (1998); P. Lipari and M. Lusignoli, [27] S. Kasuga et al.,Phys. Lett.B 374, 238 (1996).
Phys.Rev.D58,073005(1998).M.C.Gonzalez-Garcia, [28] R. E. Prael and M. B. Chadwick, Los Alamos National
H. Nunokawa, O. L. Peres and J. W. Valle, Nucl. Phys. Laboratory Report LA-UR-97-1745 (May 6, 1997),
B 543, 3 (1999); G. L. Fogli, E. Lisi and A. Marrone, http://www-xdiv.lanl.gov/XCI/PEOPLE/rep/plist.html.
Phys.Rev.D 63, 053008 (2001). [29] H. P. Wellisch and D. Axen, Phys. Rev. C 54, 1329
[8] L. J. Hall and H. Murayama, Phys. Lett. B 436, 323 (1996).
(1998). [30] D. F. Measday and C. Richard-Serre, CERN-69-17;
[9] L. J. Hall and H. Murayama, Phys. Lett. B 463, 241 H. Feshbach, Theoretical Nuclear Physics: Nuclear Re-
(1999); T. Stanev, Phys.Rev.Lett. 83, 5427 (1999). actions (John Wiley & Sons, Inc., New York,1992).
[10] S. Nussinov and R. Shrock, Phys. Rev. Lett. 86, 2223 [31] R.K.Tripathi,J.W.WilsonandF.A.Cucinotta,Nucl.
(2001); E.Kolbe, K.Langanke andP. Vogel, Phys.Rev. Instrum. Meth. B 152, 425 (1999).
D 66, 013007 (2002). [32] M. Malek et al., hep-ex/0209028.
[11] S. Fukuda et al., Phys. Rev. Lett. 85, 3999 (2000); [33] F.Boehmetal.,Phys.Rev.D62,092005(2000);J.Wolf,
A.Habig, hep-ex/0106025. hep-ex/0211032.
[12] J. F. Beacom, W. M. Farr and P. Vogel, Phys. Rev. D [34] W. W. Allison et al.,Phys. Lett.B 449, 137 (1999).
66, 033001 (2002). [35] O. G. Ryazhskaya, JETP Lett. 60, 617 (1994);
[13] L. A. Ahrens et al., Phys. Rev. Lett. 56, 1107 (1986) O. G. Ryazhskaya,JETP Lett. 61, 237 (1995).
[Erratum-ibid. 56, 1883 (1986)]; L. A. Ahrens et al., [36] Y. Fukudaet al., Phys.Lett. B 388, 397 (1996).
Phys. Rev. D 35, 785 (1987); L. A. Ahrens et al., Phys. [37] T.KajitaandY.Totsuka,Rev.Mod.Phys.73,85(2001).
Lett.B 202, 284 (1988). [38] M. H. Ahn et al., hep-ex/0212007; Y. Oyama,
[14] M. D. Messier, Ph. D. Thesis, UMI-99-23965; hep-ex/0210030.
Jun Kameda, Ph. D. Thesis (2002); [39] E. Church et al., “A proposal for an experiment to
http://www-sk.icrr.u-tokyo.ac.jp/sk/pub/. measure νµ → νe oscillations and νµ disappearance at
[15] C. H. Llewellyn Smith,Phys. Rept. 3, 261 (1972). the Fermilab Booster: BooNE,” FERMILAB-P-0898; I.
[16] A.Liesenfeld et al.,Phys. Lett. B 468, 20 (1999). Stancu et al., “The MiniBooNE Detector Technical De-
[17] V.Bernard, L.Elouadrhiriand U.G. Meissner, J.Phys. sign Report”; see http://www-boone.fnal.gov/.
G 28, R1(2002). [40] R. Tayloe, Nucl.Phys. Proc. Suppl. 105, 62 (2002).
[18] G.T.Garvey,W.C.LouisandD.H.White,Phys.Rev. [41] M. C. Ban˜uls, G. Barenboim and J. Bernab´eu, Phys.
C 48, 761 (1993); W. M. Alberico, S. M. Bilenky and Lett.B513,391(2001);J.Bernab´eu,S.Palomares-Ruiz,
C. Maieron, Phys. Rept. 358, 227 (2002); R. D. McKe- A.P´erezandS.T.Petcov,Phys.Lett.B531,90(2002);
own and M. J. Ramsey-Musolf, hep-ph/0203011. T. K.Gaisser and T. Stanev,astro-ph/0210512.
[19] T.K.Gaisser andJ.S.O’Connell,Phys.Rev.D 34,822
