2/19/01 HEP lunch talk: George Gollin 1
Neutrino mixing
Can νe νµ ντ ?↔ ↔
If this happens:
•neutrinos have mass
•physics beyond the (perturbative) Standard Model participates
Outline:
•description/review of mixing phenomenology
•possible experimental signatures
•short review of existing experimental results
Analogy with mixing
…but the mass eigenstates are what propagate “sensibly” withoutmixing:
0 0K K↔
ud
dn
Λss
Κ0
strong interaction produces a K0
d
W+uπ+
µ+
νµ
weak interaction produces a νµ
We produce flavor eigenstates...
( ) ( ) ( ) ( )2 2
0 0S S L Lim c im cS S L LK K e e K K e eτ τ τ ττ τ−Γ − −Γ −= =h h
( ) ( ) ( ) ( )2 21 2
1 1 2 20 0im c im ce eτ τν τ ν ν τ ν− −= =h h
( ) ( ) 23
3 3 0 im ce τν τ ν −= h
2/19/01 HEP lunch talk: George Gollin 3
Analogy with mixing0 0K K↔
We know KL ≠ K0, etc. Perhaps ν1 ≠ νe or ν2 ≠ νµ or ν3 ≠ ντ?
Rewrite the production eigenstates as linear combinations ofmass eigenstates:
0
2 20
S
L
K KU
KK×
= ( )
0
2 2 0
S T
L
KKU
K K
∗
×
=
1
2
3
e
Uµ
τ
ν νν νν ν
=
( )1
2
3
eTU µ
τ
ν νν νν ν
∗ =
U is the Maki-Nakagawa-Sakata matrix.
2/19/01 HEP lunch talk: George Gollin 4
12 12sin ...s θ≡12 12cos ,c θ≡ 0 : violationCPδ ≠
Maki-Nakagawa-Sakata mixing matrix
Parameterize mixing with three angles and one phase:
This form is convenient if only two neutrino species mix.(CP violation requires that all three mix.)
13 13 12 12
23 23 12 12
23 23 13 13
1 0 0 0 0
0 0 1 0 00 0 0 0 1
i
i
c s e c s
U c s s cs c s e c
δ
δ
−
+
= −
− −
e τν ν↔µ τν ν↔ e µν ν↔
Analogy with mixing0 0K K↔
d
s
Κ0 Κ0W
s
dW
u,c,t
u,c,t???
Κ0 Κ0 νµ νe
( ) 00 +S LK K K Kτ = = ∼
( ) 2 2
+S S L Lim c im cS LK K e e K e eτ τ τ ττ −Γ − −Γ −= h h
2
~ + ( )S L i mcS L L SK e K e e m m mτ τ τ−Γ −Γ − ∆ ∆ = −h
KL phase rotates (relative to KS phase) ~30° per 10-10 sec.
(maximal mixing!)
Recall: and .0 +S LK K K∼ 0 -S LK K K∼
2/19/01 HEP lunch talk: George Gollin 6
Analogy with mixing0 0K K↔
ν1, ν3 phases rotate relative to ν2 phase if masses are unequal.
( ) 22 231 2
21 1 22 2 23 3im cim c im cU e U e U e ττ τν τ ν ν ν −− −∗ ∗ ∗= + + hh h
( ) 21 1 22 2 23 30 U U Uµν τ ν ν ν ν∗ ∗ ∗= = = + +
d
s
Κ0 Κ0W
s
dW
u,c,t
u,c,t???
Κ0 Κ0 νµ νe
n1 n2 n3m1 = 0.1 eV m2 = 0.3 eV m3 = 0.4 eV
P = 8.73 ´10-34 P = 1. P = 1.1´10-32ne nm nt
q12=10. deg q13=20. deg q23=30. deg
t = 0. sec
Highly contrived example:m1 = 0.1 eV, m2 = 0.3 eV, m3 = 0.4 eVθ12 = 10°, θ13 = 20°, θ23 = 30°, δ = 0°
Radius of circle = |amplitude|; line indicates Arg(amplitude)
( ) ( ) ( )1 2 30 0.32 0 0.82 0 0.47ν ν ν ν ν ν= − = =
t = 0
( )0 1µν ν =
n1 n2 n3m1 = 0.1 eV m2 = 0.3 eV m3 = 0.4 eV
P = 0.0341 P = 0.961 P = 0.00445ne nm nt
q12=10. deg q13=20. deg q23=30. deg
t = 1.654´10-15 sec
ν1, ν2, ν3 phases have changed: heavier species phase-rotate more rapidly.
t = 1.65×10-15 sec ( ) 1tµν ν ≠
probabilities
2-15 -15 -1531 1 2 dd d
= 8.7 10 sec; 26.1 10 sec; 34.8 10 secd d d
m c ϕϕ ϕτ τ τ
= ° = ° = °h
2/19/01 HEP lunch talk: George Gollin 9
Mathematica animation form1 = 0.1 eV, m2 = 0.3 eV, m3 = 0.4 eVθ12 = 10°, θ13 = 20°, θ23 = 30°, δ = 0°(http://web.hep.uiuc.edu/home/g-gollin/neutrinos/amplitudes1.nb)
2/19/01 HEP lunch talk: George Gollin 10
ν oscillations in a beam with energy Eν ...x = distance from production t = time since production
( ) ( ) 210 im c
i i e τν τ ν −= h
( ) ( )( ) ( )
2
, rest frame: 0,
, rest frame: 0,
all frames (Lorentz scalar) lab frame
i
x x ct c
p p E c mc
p x m cpx E t
µ
µ νµ
µ
ν
τ
τ
≡≡
= −= −
rr
( ) ( ) ( )2 2 2 3 2 ; i ip E c m c E c m c E x ctν ν ν= − ≈ − ≈
( ) ( )2 32 2ii i px E t im c x Eim ce e eν ντ − −−∴ = ≈h hh
( ) ( ) ( )2 3 20 iim c x Ei ix e νν ν −= h
2/19/01 HEP lunch talk: George Gollin 11
( ) 21 1 22 2 23 30x U U Uµν ν ν ν ν∗ ∗ ∗= = = + +
Relative phases of ν1, ν2, ν3 coefficients change: νµ→ other stuff
2 2221 2 1m m m∆ = − 2 22
31 3 1m m m∆ = −
( ) ( ) ( )2 3 2 321 312 2
21 1 22 2 23 3i m c x E i m c x Ex U U e U eν ν ν ν− ∆ − ∆∗ ∗ ∗+ +h h∼
νµ oscillations in a beam...
(factored out .)( )2 31 2im c x Ee− h
( ) ( )2 3 2
2221 21 2121
d 145 per km for in eV , in GeV
d 2m c m
m Ex E E
ϕ ∆ ∆= ≈ ⋅ ° ∆h
2/19/01 HEP lunch talk: George Gollin 12
Mathematica animation for∆m21
2 = 0.3 eV2 , ∆m312 = 0.6 eV2
θ12 = 5°, θ13 = 10°, θ23 = 15°, δ = 0° (http://web.hep.uiuc.edu/home/g-gollin/neutrinos/amplitudes2.nb)
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( ) 12 1 12 20 sin cosx µν ν θ ν θ ν= = = − +
( ) ( )2 321 2
12 1 12 2sin cos i m c x Ex eν θ ν θ ν − ∆= − + h
Two-flavor mixingResults are often analyzed with the simplifying assumption thatonly two of the three ν species mix.
For example: νe νµ ...↔
( ) ( )( )
( ) ( )
2 321
2 321
2
22
12 1 12 2 12 1 12 2
22 212 12
;
cos sin sin cos
cos sin 1
e e
i m c x E
i m c x E
P x x
e
e
µν ν ν ν
θ ν θ ν θ ν θ ν
θ θ
− ∆
− ∆
→ =
= + − +
= −
h
h
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Two-flavor mixing
units: ∆m2 in eV2, x in km, E in GeV.
( ) ( ) ( )
( )
( )
( )
2 321
22 212 12
2 32 21
12
2 32 2 21
12
2 2 212 21
; cos sin 1
1sin 2 1 cos
2 2
sin 2 sin4
sin 2 sin 1.27
i m c x EeP x e
m c xE
m c xE
xm
E
µν ν θ θ
θ
θ
θ
− ∆→ = −
∆= −
∆
= ≈ ∆
h
h
h
( ) ( ); 1 ;e e eP x P xµν ν ν ν→ = − →
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Mathematica animation for∆m21
2 = 0.3 eV2
θ12 = 15°, θ13 = 0°, θ23 = 0°, δ = 0° (http://web.hep.uiuc.edu/home/g-gollin/neutrinos/amplitudes3.nb)
Those confusing plots for two-flavor mixing...
( ) ( )2 2 212 21; sin 2 sin 1.27e
LP L m
Eµν ν θ → ≈ ∆
GreenGreen curve is contour of fixedprobability to observe oscillation
A search experiment sets a limit (or measures!!) the mixingprobability P, but little else, at the present time.
L for an event is uncertain due tolength of π→µ decay/drift region.
Large Large ∆∆mm22: uncertainty in L/Ecorresponds to several oscillations.
P determined by experiment is anaverage over several oscillations andis insensitive to ∆m2 in this case...
More on those confusing plots...
( ) ( )2 2 212 21; sin 2 sin 1.27e
LP L m
Eµν ν θ → ≈ ∆
Medium Medium ∆∆mm22:
•uncertainty in L/E corresponds to afraction of an oscillation.
•1.27∆m2L/E ~ π/2 is possible forsome of the detected events
P (limit) determined by experimentcorresponds to smallest sin2(2θ)when 1.27∆m2L/E = π/2.
Even more on those confusing plots...
( ) ( )2 2 212 21; sin 2 sin 1.27e
LP L m
Eµν ν θ → ≈ ∆
Small Small ∆∆mm22:
•uncertainty in L/E corresponds to afraction of an oscillation.
•sin2(1.27∆m2L/E) < 1 since L/E isalways too small
P (limit) determined by experimentcorresponds to larger and largersin2(2θ) as ∆m2L/E shrinks.
(GreenGreen curve is contour of fixedprobability to observe oscillation)
2/19/01 HEP lunch talk: George Gollin 19
What could be happening?
•Nothing
•Two- or three-flavor oscillations
•Oscillations into “sterile” neutrinos (ν’s just “disappear”)
•Neutrinos decay (into what??)
•Matter-enhanced (Mikheyev-Smirnov-Wolfenstein mechanism)oscillations
•Extra dimensions
•Something else
νe νµ ντ↔ ↔
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Is anything happening?
1. Atmospheric neutrinos
•cosmic ray interactions in the atmosphere produce ν’s throughdecays of π, K, µ.
•expect νµ/ νe ratio for upwards- and downwards-goingneutrinos to be equal if no oscillations.
•expect νµ/ νe ratio for upwards-going neutrinos to shrinkrelative to downwards-going if νµ oscillates into ντ or νsterile butνe doesn’t.
•Super-Kamikande finds νµ(up)/ νµ(down) = 0.52 ± 0.05 butνe(up)/ νe (down) ~ 1.
•suggestive of νµ→ ντ oscillations
Is anything happening?
2. Solar neutrinos...
Is anything happening?
... Solar neutrinos...
Is anything happening?
... Solar neutrinos...
Not enough neutrinos.
MSW mechanism?
(more on this next week)
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Is anything happening?
3. LSND result
•1 mA beam of 800 MeV (kinetic) energy protons produces π+,µ+ which decay in flight (most π-, µ- captured in shielding)
• in beam
•Look for appearing in liquid scintillator volume
•find 82.8 ± 23.7 excess events
•also see weaker evidence for νµ→ νe
44 10e µν ν −≈ ×
eν
eν
Is anything happening?
...LSND result
LSND 90%
LSND 95%
BugeyKarmen
BNL E776
LSND has theonly appearanceresult.
2/19/01 HEP lunch talk: George Gollin 26
Next week
More about the existing results and planned experiments.
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