Neutrino oscillations
Oleg Lychkovskiy
ITEP
2008
Plan

Lecture I





Introduction
Two-flavor oscillations
Three- flavor oscillations
Matter effect
Lecture II

Overview of experiments and observations.
Introduction: acquaintance with neutrinos
Typical energies: MeV-PeV >> m:
always ultrarelativistic!
SM interactions:
Low energy (E<<100 GeV) interactions:
β – decay:
(Z, A)  (Z+1,A) + e- + ve
π – decay:
Deep inelastic
scattering:
… and so on
ve – capture:
ve + p  n + e +
Two-flavor oscillations
Key feature: flavor eigenstates, in which neutrinos are created
and detected, do not coincide with mass eigenstates!
m1 and m2 - masses of v1 and v2
Two-flavor oscillations, wave packet formalism
(at given t only x=Vt ± a/2 are relevant)
Two-flavor oscillations, wave packet formalism
Two-flavor oscillations, plane wave formalism
Final oscillation probability does not depend on the specific
form of the wave packet F(x)!
Thus we may put F(x)=1, x=L and drop the integration over x!
We get the same final result with less calculations:
Three-flavor mixing
νe , νμ , ντ - flavor eigenstates
ν1 , ν2 , ν3 - mass eigenstates with masses
m1, m2, m3
• 3 angles: θ12 , θ13 , θ23
• 1 CP-violating Dirac phase: δ
• 2 CP-violating Majorana phases: α1 , α2
(physical only if ν’s are Majorana fermions)
Three-flavor mixing
Unknown: absolute values of masses, θ13 , δ, α1 , α1 ,
sign of Δm232 , octet of θ23
Three-flavor mixing
sin213
3
| m232 |
or
(Mass)2
2
1
}m
2
21
}m
2
21
| m232 |
3
sin213
inverted hierarchy
normal hierarchy
e
2
1


Three-flavor oscillations
3
3
i
j
P( l   l ' )  U liU U U l ' j e

l 'i

lj
iLm 2ji / 2 E
  ll '  2 Re U liU U U l ' j (1  e
i j

l 'i

lj
iLm 2ji / 2 E
In particular, one can see that Majorana phases do not enter
the oscillation probability
)
Three-flavor oscillations: νμ  νl’
L Δm221 /4E<< π, sin213 neglected
Assume
Then, neglecting
Relevant for the majority
of accelerator experiments
and for atmospheric neutrinos
and
one obtains
Example: K2K (E=1GeV, L=250km)
Three-flavor oscillations: νe  νe ,
sin213 neglected
Assume the detector registers only electron neutrinos

P( e   e )  1  4 Re  U ei U ej sin L m / 4 E
2
2
i j
2
2
ji
Neglecting |Ue3|2 = |s13|2 < 0.05 , one obtains
The same result one can get in a more illuminating way

Three-flavor oscillations: νe  νe ,
sin213 neglected
Two-flavor mixing effectively!
=12 , m2m221
Relevant for KamLAND
Three-flavor oscillations: νe  νe ,
small baselines, 13 in play
If one does not neglect s132 ,
oscillations with small
amplitude ~ s132 and small period
Losc = 4E/Δm231 are
superimposed on the Δm21–
related oscillations.
If in addition
one comes to
http://dayawane.ihep.ac.cn/docs/experiment.html
Relevant for
Double Chooz, Daya Bay
Example: Double Chooz (E=4 MeV, L=1 km)
Matter (MSW) effect in neutrino
oscillations
νe-e interaction (through W-boson exchange):
averaging of this Lagrangian over the matter electrons
gives an effective matter potential:
νl-e interaction through Z-boson exchange does not depend on
flavor and thus does not influence oscillations
Matter (MSW) effect
Vacuum Hamiltonia n is diagonal in the mass eigenstate basis  1 , 2 , 3  :
3
 
ˆ
H 0   (  i  0   m i  0 )  i  i
i 1


Matter interactio n term is diagonal in the flavor eigenstate basis  e ,  ,  :
Vˆ  2G F ne  e  e

ˆ
ˆ
Diagonaliz ation of the total Hamiltonia n H  H 0  V

matter eigenstate basis  1m , 1m , 1m
for the details see lecture notes by Y.Nir, arXiv:0708.1872

Neutrinos in matter, two-flavor case, ne=const
Resonance:
Oscillations with the maximal
amplitude!
Overwhelming
matter effect:
No oscillations!
Relevance of matter effect
Key parameter:
Earth: ρ =(1-10) g/cm3
V = (0.4-4) 10-13 eV
Reactors: E ~ few MeV
Δm212 /2E ~ (1-10)10-11 eV
Δm312 /2E ~ (3-30)10-10 eV
Matter effect is irrelevant
Supernova core:
ρ ~ 1012 g/cm3
Sun core:
E ~10 MeV
~ 100 g/cm3
V ~ 0.1 eV
V ~0.5 · 10-11eV
2 /2E ~0.5 · 10-11 eV
Δm
21
E ~ (0.5-20) MeV
Δm312 /2E ~ 10-10 eV
Overwhelming
Δm212 /2E ~(0.2-8)10-11 eV
relevant
matter effect!
Accelerators, atmospheric
neutrinos: E ~ few GeV
Δm212 /2E ~ (0.1-1)10-13 eV
Δm312 /2E ~ (0.6-24) 10-10 eV
2
-12
Δm31 /2E ~ (0.3-3)10 eV
irrelevant
Matter effect may be relevant
Remarks upon the previous lecture



Misprint: tree-flavor
three-flavor
MSW effect = Mikheyev-Smirnov-Wolfenstein
effect
“octant”=… = 1/4 of the coordinate plane
Lecture II.
Neutrino oscillations.
Overview of experiments and
observations.
Based on the review by
O.Lychkovskiy, A.Mamonov, L.Okun, M.Rotaev,
to be published in UFN (УФН).
Three-flavor mixing
νe , νμ , ντ - flavor eigenstates
ν1 , ν2 , ν3 - mass eigenstates with masses
m1, m2, m3
• 3 angles: θ12 , θ13 , θ23
• 1 CP-violating Dirac phase: δ
• 2 CP-violating Majorana phases: α1 , α2
(physical only if ν’s are Majorana fermions)
SOURSE
ν/ν,
flavor
relevant
energy
MSW
what was (can
be) extracted
Sun
νe
0.5-19 MeV
of major
importance
θ12 , m221
irrelevant
m221, θ12
θ13
relevant
θ23 , m232
octant of θ23
Reactors
νe
Cosmic rays
(atmospheric
ν’s)
νμ, νμ,
minor fraction
of other
flavors
1-6 MeV
0.1 GeV 10 TeV
m232, θ23
νμ, νμ,
Accelerators
Supernova
minor fraction
of other
flavors
0.5-50 GeV
all species 1-40 MeV
relevant
θ13 , δ
hierarchy, octant
of major
importance
hierarchy, θ13
Solar neutrinos
Neutrino oscillations in the matter of the Sun
We are interested in νe  νe oscillations and we neglect θ13
Effectively two-flavor case with 1-2 mixing:
θ =θ12 , m2=m221
ne=ne(r),
r is the distance from
the center of the Sun
adiabaticity condition holds:
, m=m(r), θ= θ(r)
Neutrino oscillations in the matter of the Sun
At the Earth (r=R)
where averaging over the production point r0 is performed
Neutrino oscillations in the matter of the Sun
Probability weakly depends on m221 , but, nevertheless,
is sensitive to its sign!
Radiochemical experiments
Homestake:
SAGE, GALLEX/GNO:
νe + 37Cl 37Ar + e-
νe + 71Ga 71Ge + e-
37Ar 37Cl
71Ge
+ e+ + νe
 71Ga + e+ + νe
Eth=0.86 MeV
Eth=0.23 MeV
t1/2=35 days
t1/2=11.4 days
Result: ~ 4 times less
neutrinos, than predicted
by the SSM
Result: ~ 2 times less
neutrinos, than predicted by
the SSM
Cherenkov detector experiments
Kamiokande ((1-3) kt of H2O) and Super-Kamiokande (50 kt of H2O):
νl + e νl + e
SNO: (1 kt of D2O):
νe + d p + p + e
νl + d p + n + νl
νl + e νl + e
Eth>5 MeV
The total flux was measured, and it coincided with the SSM prediction!
SSM verified
the νe deficite is due to oscillations!
Borexino
Main goal: mono-energetic (E= 862 кэВ) 7Be neutrinos
Scintillation detector:
low threshold (Eth= 0.5 MeV),
but no direction measured
!!!First real-time low-energy
solar neutrinos:
47 ± 7stat ± 12syst
7Be ν / (day · 100 t)
(arXiv:0708.2251)
Reactor experiments

oscillations
νe:
• produced in β-decays in nuclear reactors:
(A,Z)  (A,Z+1) + e- + νe
• detected through
νe + p  n + e+
• scintillation detectors used
• antineutrino energy: few MeV
Long-baseline, L=O(100) km:
KamLAND
Short-baseline, L=O(1) km:
Chooz, Double Chooz,
Daya Bay
KamLAND
• Sources of : 55 Japanese reactors
• Baselines: L=(140 - 210) km
•
energies: 1.7 MeV < E < 9.3 MeV
• Probability of survival:
Status: running
Sensitive to
Δm221
and
θ12
KamLAND
!!!The latest result:
arXiv: 0801.4589v2
Also 70± 27 geo-neutrinos registered!
Chooz
• Source: Chooz nuclear station
• Baseline: L=1.05 km
• energies: 3 MeV < E < 9 MeV
• Probability of survival:
Status: finished
The final result: sin22θ13 < 0.2
90%CL
Future experiments: Double Chooz
and Daya Bay
Goal: measuring θ13
Double Chooz
sin22θ13 < 0.03
by 2012
Daya Bay
sin22θ13 < 0.01
by 2013
near detectors will be built
Double Chooz sensitivity evolution
arXiv:hep-ex/0701020v3
the initial spectrum will be measured,
not calculated
Double Chooz and Daya Bay sensitivities
Atmospheric neutrinos
• Source: cosmic rays, interacting with the atmosphere.
Major fraction:
Minor fraction:
Negligible fraction:
• Detection reactions: deep inelastic scattering
νμ + N  μ + hadrons
• Experiments:
Kamiokande, IMB, Super-Kamiokande, Amanda, Baikal, MACRO,
Soudan, IceCube, …
• “Baselines”: L=(0 - 13000) km
• Energies: 0.1 GeV < E < 10 TeV
Atmospheric neutrinos
Approximate expressions:
Original flux and energy spectrum
are poorly known
MSW-effect and 3-flavor oscillations
in play, extended source
large theoretical
flux uncertainties
no simple precise
expressions!
Atmospheric neutrino fluxes
SK atmospheric neutrino results
sin22θ23 > 0.92
1.5 · 10-3 < m232 < 3.4 · 10-3 eV2
90% CL
Evidence for
appearance!
Phys.Rev.Lett.97:171801,2006,
hep-ex/0607059
Prospects for resolving
hierarchy ambiguity
arXiv:0707.1218
Phys.Rev. D71 (2005) 112005, arXiv:hep-ex/0501064v2
Accelerator neutrino experiments

oscillations
• νμ and νμμ are produced in meson decays
• energies: few GeV
• baselines: hundreds of kilometers
Main goals:
appearance observations: search for   e or   τ
measuring 13
precise measurement of m223 , 23
mass hierarchy
CP
Accelerator neutrino experiments
К2К
MINOS
OPERA
MiniBooNE
Т2К
NOVA
  
  
LSND
  e
m232, sin2223
sterile 
13
For К2К, MINOS (?)
and OPERA (?)
L Δm221 /4E<< π, 13=0
approximation is valid
T2К, NOvA and, probably, OPERA and MINOS,
will go beyond this approximation!
CP(?)
Accelerator neutrino experiments
Next several slides are from the talk by Yury
Kudenko at NPD RAS Session
ITEP, 30 November 2007
First LBL experiment К2К
 disappearance
1999-2005

e
L/E  200
L=250 km
<E>  1.3 GeV
Predictions of  flux and interactions
at Far Detector by Far/Near ratio
98.2%
1.3%
Signal of  oscillation at K2K
Reduction of  events
Distortion of  energy spectrum
~1 event/2 days at SK
K2K final result
- # Events
+
PRD74:072003,2006
- Shape distortion
Expected: 158.1 + 9.2 – 8.6
Observed: 112
Expected shape
(no oscillation)
Best fit
Null oscillation probability
(shape + # events) = 0.0015% (4.3)
Best fit values
sin22 = 1.00
m2 [eV2] = (2.80  0.36)10-3
Kolmogorov-Smirnov test
Best fit probability = 37%
MINOS
Precise study of “atmospheric”
neutrino oscillations, using the
NUMI beam and two detectors
Far Det:
5400 tons
735 km
Near Det:
980 tons
Beam: NuMI beam, 120 GeV
Protons  - beam
Detectors: ND, FD
Far Det: 5.4 kton magnetized
Fe/Sci Tracker/Calorimeter at
Soudan, MN (L=735 km)
Near Det: 980 ton version of
FD, at FNAL (L  1 km)
New MINOS result
2.50 POT analyzed ≈ 2x statistics of 2006 result
Improved analysis
J.Thomas, talk at Lepton-Photon2007
# expected (no osc.) 73830
# observed
563
Comparison of
new and old
MINOS results
m223 =(2.38 +0.20 -0.16) x 10-3
sin2223=1.00 -0.08
m223 and 23: SK/K2K/MINOS
|m223|| m213|= (2.4  0.2)x10-3 eV2 23 ~ 45o
MINOS: projected sensitivity
M.Ishitsuka, talk at NNN07
After 5 years running: expected accuracy of m232 and sin2223 10%
chance for first indication of non-zero 13
OPERA
   direct search
P(  ) = cos413sin223sin2[1.27m223L(km)/E(GeV) ]
High energy, long baseline  beam
( E  17 GeV
kink
Target mass

1 mm



L ~ 730 km )
~1300t
E/L ~ 2.310-2  10m223 (atm)

pure beam: 2% anti  <1% e


Pb
Emulsion layers
after 5 years data taking:
~22000  interactions
~120  interactions
~12  reconstructed
<1 background event
OPERA:    sensitivity
M.Spinetti, talk at NNN07
full mixing,
5 years run
4.5 x1019pot/y
New MINOS
Second generation
LBL experiments
Off Axis Neutrino Beams
• Increases flux on oscillation maximum
• Reduces high-energy tail and NC backgrounds
• Reduces e contamination from K and  decay
T2K
NOVA
T2K (Tokai to Kamioka)
JPARC facility
~1GeV  beam (100 of K2K)
 beam
off-axis
E(GeV)
Int(1012 ppp)
Rate (Hz)
Power (MW)
JPARC
50
330
0.29
0.77
on-axis
MINOS
120
40
0.53
0.41
Opera
400
24
0.17
0.5
K2K
12
6
0.45
0.0052
Statistics at SK
OAB 2.5 deg, 1 yr = 1021 POT, 22.5 kt
~ 2200  tot
~ 1600  charged current
e < 0.5% at  peak
T2K off-axis beam
OA2°
SuperK
0o
Target Horns
Decay Pipe
OA2.5°
OA3°
0 deg

Principle Goals of T2K
- Search for e appearance
13 sensitivity  1o (90% c.l.)
Background uncertainty
CP = 0
CP = /2
CP = - /2
CP = 
-Measurement m223
with accuracy of 1%
(sin2223)  0.01
(m223) < 110-4 eV2
10%
m2=2.5x10-3
T2K sensitivity to 13
CHOOZ limit
ambiguities: CP - 13
sign m223
23
NOA
P(  e) depends on
sin2213 sign m223
CP
matter effects
increase (decrease) oscillations
for normal (inverted) hierarchy
for 
Mass hierarchy can be resolved
if 13 near to present limit
using both  anti- beams and
sin2213 from T2K + reactor experiments
13 sensitivities vs time
A.Blondel et al.,
hep-ph/0606111
Daya Bay goal
Short baseline reactor experiments
Double-Chooz and Daya Bay
13 ( insensitive to CP)
Summary for accelerator experiments
K2K
confirmation of atmospheric neutrino
oscillations discovered by SK
MINOS
confirmed the SK и K2K results
high precision measurements of oscillation parameters
MiniBooNe rules out (98% cl) the LSND result as   e
oscilations with m2 ~ 1 eV2
new anomaly appears
run with anti- beam
OPERA
data taking begun in 2007
T2K-I
neutrino beam in 2009
Main goal for next 5 years: 13
Neutrino production in SN
Matter effect in Supernova


Adiabaticity almost everywhere, resonant layers are possible
exeptions
Three flavors in play, two different resonanses
H-резонанс:
L-резонанс:
m312
2GF ne (rH ) 
cos 213
2 E
2
m21
2GF ne (rL ) 
cos 212
2 E
rH  (3  5)  104 km
rL  (8  15)  104 km
Adiabaticity conditions
In resonance layer the adiabaticity parameter reads
m sin 2 tan 2

E
2
 d ln ne 


 dr 
1
 L  2.5 104  (10 МэВ/E ) 2 / 3  1
L- resonance is always adiabatical!
H
2
sin
213
3
2/3
 10 

(
10
МэВ/
E
)

4/3
(cos 213 )
Adiabaticity of H-resonance depends on θ13 !
Level crossing scheme for SN
Mass hierarchy and θ13
NH, L IH, L NH and IH, S
PH
0
1
1
PH
1
0
1
NH=Normal Hierarchy, IH=Inverted Hierarchy
L=Large θ13 : θ13 >0.03
S=Small θ13 : θ13 < 0.003
Future SN neutrino signal in SK
R=10 kpc
Takahashi, Sato,
hep-ph/0205070
θ13 measurment with SN
If
0.003  13  0.015
(0.06o  13  1o )
and the hierarchy is
inverted, than
θ13 is measurable!
Takahashi, Sato, hep-ph/0205070
Conclusions
Present knowledge:
central value 2 interval
m212 (10-5 eV2)
7.6
7.1 - 8.3
m231 (10-3eV2)
2.4
2.0 - 2.8
sin212
0.32
0.26 - 0.40
sin223
0.50
0.34 - 0.67
sin213
0.0
<0.05
5-year goals:
• to increase the sensitivity for
m212 , m231 , sin212 , sin223 up to (1-10)%
• sin213 sensitivity at the level 0.003
• mass hierarchy,  (?)