1. Oscillation formalism
Neutrino mixing
Oscillations of 2 flavors
Experimental sensitivities
Oscillations in 3 flavors
Discovery of oscillations:
atmospheric neutrinos in Super-Kamiokande

2. Quark mixing in Standard Model
States partcipating in strong
interactions with well defined
masses (mass matrix eigenstates):
u c t
d s b
u c t
d` s` b`
States participating in weak
interactions:
Quark
mixing:
„From neutrinos to cosmic sources”, DK & ER

3. Neutrino mixing NOT in Standard Model
IF neutrinos are massive:
States with well defined
masses (mass matrix eigenstates):
States participating in weak
interactions:
Lepton
mixing:

4. Neutrino oscillation – 2 flavors
mixing angle:
mass states: ϑ
are defined as different proportions of ν1 ,ν2 states
ν1 ,ν2
states have different masses different velocities
The ratio
changes during propagation, hence

5. Oscillation probability – 2 flavors
A state of mass mk, energy and momentum Ek,pk propagates:
with phase:
Let’s assume an initial state:

6. Oscillation probability – 2 flavors
During propagation the contribution of ν1,ν2 components changes:
A probability that after t,x the state α is still in its initial α state:
Finally:

7. Oscillation probability – 2 flavors
Probability of transition from a state α to a state β:
oscillation
parameters
m - mass (in eV)
ϑ - mixing angle
experimental
conditions:
Eν – neutrino energy (in GeV)
L - distance from a neutrino source to detector (km)
Oscillation length:

8. Appearance and disappearance experiments
In an appearance experiment one searches for
neutrinos νβ in an initial beam of να :
In a disappearance experiment one counts how many of the initial
neutrinos να are left after passing a distance L:
Note: Neutrino oscillate only if masses are non-zero and not the same

9. Sensitivity to oscillations
Eν (MeV)
L (m)
Supernovae
<100
>1019
Solar
<14
1011
10-10
Atmospheric
>100
10-4
Reactor
<10
<106
10-5
Accelerator with
short baseline
103
10-1
long baseline
10-3

10. Graphic illustration of neutrino oscillations
For max mixing ϑ=π/4
and
at a distance
L=Losc/2
all the initial flavor νa
are transformed to
another flavor νb

11. Transitions between 3 mass states
With 3 generations there are 3 Δm2’s but only two are independent.

12. Mixing of 3 flavors For 3 flavors we need 3x3 matrix.
In quark case the corresponding matrix
is called CKM (Cabibo-Kobayashi-Maskava).
For neutrinos MNS (Maki-Nakagava-Sakata)

13. Mixing of 3 flavors (part 2)
The 3x3 matrix has 4 independent real parameters:
where:
4 independent parameters:
Current experiments are not sensitive to φ. It’s assumed φ=0

14. Mixing of 3 flavors (part 3)
The mixing matrix can be written:
φ=0
rotation by:
rotation by:
rotation by:
„From neutrinos to cosmic sources”, D.Kiełczewska, E.Rondio

15. Oscillation Probability – 3 flavors (part 1)
Per analogy with 2 flavor case the amplitude for the neutrino
oscillation:

16. How do Neutrinos Oscillate?
Amplitude
Amplitude

17. Oscillation Probability – 3 flavors
In a general case, with at least one non-zero complex phase:
Note here: if α=β then the imaginary components disappear
CP phase cannot be measured in disappearance experiments

18. Oscillation Probability – 3 flavors (φ=0)

19. Oscillation Probability – 3 flavors (φ=0)
Let’s assume:
Then we have 2 types of experiments:
Case A – „atmospheric” - small L/E:
Tu Ewa doszla 16/04/2008
Case B – „solar” - large L/E

20. Oscillation probability – 3 flavors (φ=0)
∆m >>δm
Case A – „atmospheric” - small L/E:
Note:
for ϑ13=0
all formulas
are the same
as for 2 flavors
Case B – „solar” - large L/E

21. More exact formula: one gets: By expanding in: + neutrinos
- antineutrinos
solar term
CP violation
L – baseline;
We will introduce later:
Mena v1.pdf
matter effects
 sensitivity to
mass hierarchy
If LA<<1:
The above formula is necessary for future, more exact studies

22. Let’s try to understand atmospheric neutrino puzzle

23. Neutrino events in Super-K
All have to be separated
from „cosmic” muons
(3Hz)
Contained events:
Fully contained FC
Partially contained PC
Upward through-going μ μ μ
Upward stopping μ μ
e/μ identification
all assumed
to be μ
interactions
in rocks below
the detector
different energy scale
different analysis technique
different systematics

24. Neutrino energy spectra
Fully contained FC
Partially contained PC μ
e/μ identification
all assumed
to be μ
Upμ thru
Upμ stop
Interactions in rocks

25. Super-Kamiokande results (contained)
Sub-GeV (Fully Contained)
Multi-GeV
Evis < 1.33 GeV,
Pe > 100 MeV, Pμ > 200 MeV
Fully Contained (Evis > 1.33 GeV)
Data MC
1-ring e-like
μ-like
Data MC
1ring e-like
μ-like
Partially Contained (assigned as m-like)
We take ratios to cancel out errors on absolute neutrino fluxes:
Too few muon neutrinos observed!

26. Super-Kamiokande I results - upward going muons
Up through-going μ, (1678days)
Data: (x10-13 cm-2s-1sr-1)
MC:
Up stopping μ, (1657days)
Data: (x10-13cm-2s-1sr-1)
MC:
Again one observes a muon deficit

27. Super-Kamiokande evidence for neutrino oscillations

28. Interpretation of the zenith angle distributions
Let’s try to find interpretation of the deficit
of νμ after passing the Earth
Looks like νμ disappearance...
What happens to muon neutrinos?
Let’s suppose an oscillation:
but what is
We see that ne angular distribution is as expected

29. Oscillations of muon neutrinos
Looks like νμ oscillates:..
Remember that we identify neutrinos by the corresponding
charged lepton which they produce:
But look at the masses:
μ MeV
τ MeV
Does neutrino have enough
energy to produce τ ?

30. ντ cross sections Total CC cross sections for: compared with νμ
masses: μ MeV
τ MeV

31. We don’t see the neutrinos after oscillations!
The cross section for CC interaction (with τ ) too small
NC interactions possible but
then we cannot tell
the neutrino flavor!

32. Rough estimate of Oscillation length: Eν – neutrino energy (inGeV)
L - distance (km)
Oscillation length:
Max probability of oscillation for L=Losc/2
Find corresponding
Down, L=15 km Up, L=12000 km
For Eν=1 GeV
For Eν=10 GeV
For Eν=100 GeV
The trouble is – we don’t know precisely Eν

33. Sensitivity of angular distribution to neutrino oscillations
Expected angular distributions for oscillations with various mass parameters for the neutrino energy distribution of
multi-GeV sample.
No oscillations
10-2
Δm2 = 10-4
10-3
10-1
Sensitivity from ~10-4 to 10-1 eV2
Up-going
Down-going

34. Zenith angle distributions
e-like
1 ring
μ-like
1 ring
μ-like
multi- ring
upward going μ
Sub-GeV
Multi-GeV
Red: MC expectations
Black points: Data
Green: oscillations
Missing are the muon
neutrinos passing
through the Earth!
down up

35. Super-K up-down asymmetry
expected- no oscil
Data
with -oscillations
Fig The up-to-down asymmetry for muon (2486) and electron (2531) single
ring fully contained and partially contained (665) events in SuperK, from
days of live time (analyzed by 6/00), as a function of observed charged particle mo-
mentum. The muon data include a point for the partially contained events (PC)
with more than about 1 GeV . The hatched region indicates no-oscillation expec-
tations, and the dashed line µ - oscillations with m2 = 3.2 × 10-3eV 2 and
maximal mixing[21].
Learned

36. Rough estimate of the mixing angle
Probability of νμ disappearance...
For large distances L oscillations happen for a variety of Eν,
so that one should take:
Then:
(at some angles we see half of neutrinos disappearing)
i.e. maximal mixing

37. Definition of 2 for oscillation analysis
A fit is performed i.e. a minimum of χ2 is found.
The corresponding
are the best fit oscillation parameters

38. Results of combined fit
c2 vs Dm2
flat between
and

39. Contours for different subsamples
Sub-GeV 1-ring e-like 3353
Sub-GeV 1-ring μ-like 3227
Multi-GeV 1-ring e-like
Multi-GeV 1-ring μ-like
PC μ-like
Multi-ring Upward muons
------All

40. Oscillation parameters from different experiments (atmospheric)
J. Goodman, LP2001

41. L/E analysis of the atmospheric neutrino data from Super-Kamiokande

42. Hypotheses other than oscillation
Neutrino oscillation :
Neutrino decay:
Neutrino decoherence :
Przez oddz z jakims polem w prozni moze zostac zaburzony warunek koherencji
Idea: use events with the best resolution
in L/E
From neutrinos to cosmic sources, DK&ER

43. Reconstruction of Eν and L
Neutrino energy
Neutrino direction
Eobserved  Eν
Zenith angle  Flight length
Neutrino energy is reconstructed from observed energy using relations based on MC simulation
Neutrino flight length is estimated from zenith angle of particle direction
From neutrinos to cosmic sources, DK&ER

44. Neutrino path-length L vs angle
Very bad
Close to the horizon
From neutrinos to cosmic sources, DK&ER

45. Survival probability divide DATA/MC
Null oscillation MC
Best-fit expectation
days FC+PC
A dip just where oscil. min expected
From neutrinos to cosmic sources, DK&ER

46. Test for neutrino decay & neutrino decoherence
Oscillation
Decay
Decoherence
Alternative hypotheses excluded.
From neutrinos to cosmic sources, DK&ER

47. Oscillation analysis – fitting L/E distribution
From neutrinos to cosmic sources, DK&ER

48. Atmospheric neutrino experiments
The largest statistics of atmospheric neutrino events
were collected in Super-Kamiokande.
The results showed: a deficit of muon neutrinos
passing long distances through the Earth.
first evidence of neutrino oscillatons
Atmospheric neutrinos were also measured in MACRO
and SOUDAN detectors. The results were consistent
with neutrino oscillations.

49. Summary: evidence of oscillations in atmospheric neutrinos
Missing
Effect observed in different Super-K event samples
and also by other experiments
Stat. significance above 10 sigmas
Angular distributions
probability of disappearance depends on its path-length
and energy in a way consistent with oscillation
survival dependnce on L/E
only oscillations can produce a dip
Oscillation parameters from comparison
between data and MC simulations:
From neutrinos to cosmic sources, DK&ER