neutrinos theory
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March 7, 2005 Benasque
Neutrinos Theory Neutrinos Theory
Carlos Pena Garay
IAS, Princeton
~
PlanPlan
I. Neutrinos as Dirac Particles
II. Neutrinos from earth and heavens
III. Neutrinos as Majorana Particles
PlanPlan
I. Neutrinos as Dirac Particles
II. Making Neutrinos
III. Neutrinos as Majorana Particles
Nuclear Nuclear -decay-decay
Neutrinos are left-handedNeutrinos are left-handed
All neutrinos left handed => massless
Neutrinos must be masslessNeutrinos must be massless
If neutrinos are massive
neutrino right-handed => Contradiction!
Helicity = Chirality (massless)
Standard Model Standard Model
Exact Lepton number symmetry : terms never arise in perturbation theoryRenormalizability : All gauge currents satisfy anomaly cancellationWhat about global symmetries : L is broken non perturbatively through the anomaly, but B-L is conserved to all orders : terms never arise in the Standard Model
annihilate , create ,
create , annihilate ,c
c
Tc C
cm
cm
CPT : L => R
Anti-neutrinos are right-handedAnti-neutrinos are right-handed
-
3 neutrino families (flavors) 3 neutrino families (flavors)
Standard Model Standard Model
...
h.c. 2
-
cos2i 0
ii
iiW
ii
Wlg
Zg
L
Global symmetriesGlobal symmetries
0,...,;1,;1, eeeee LeLeL
LLLL e
0,...,;1,;1, eeLLL
The results of charged-current experiments (until recently) were consistent with the existence of three different neutrino species with the separate conservation of Le, L and L.
Massive Neutrinos : Dirac Massive Neutrinos : Dirac
(p,h) (p,-h)CPT
(p,h)(p,-h)CPT
Boost, if m ≠ 0
Boost, if m ≠ 0
Standard Model + right handed Standard Model + right handed
... h.c. m
h.c. cos2
-
cos2i
i
0
ii
iijiW
iiW
ii
WUlg
Zg
L
)1(2
1
)(2
vm
5,
LR
RLLR
P
vvvvy
Standard Model + right handed Standard Model + right handed
... h.c. m
h.c. cos2
-
cos2i
i
0
ii
iijiW
iiW
ii
WUlg
Zg
L
)1(2
1
)(2
vm
5,
i
LR
RLLRii
P
vvvvy
Standard Model + right handedStandard Model + right handed
Masses : Several distinctive features I am going to concentrate today is neutrino oscillations
Parameters : Dirac Parameters : Dirac
100
0
0
0
010
0
0
0
001
1212
1212
1313
1313
2323
2323
321
321
321
cs
sc
ciδes
iδesc
cs
sc
UUU
UUU
UUU
U
τττ
μμμ
eee
i
im
20
ij 002 mΔ 02 MΔ0m
Exercise
Neutrino Oscillations Neutrino Oscillations
tiEjkkji
j
ijeUUllA *
li
W X
lk
j
Uij Ujk
Neutrino Oscillations in Vacuum Neutrino Oscillations in Vacuum
tiEjkkji
j
ijeUUllA *
Due to different masses, different phase velocities
2
2
2
2
i 211v
i
i
i
i
i
i
E
m
p
m
p
E
2
2
i 2v
E
mi
2osc
4l
m
E
Osc. Length : distance to come back to the initial state
Production, propagation and detection as a unique process, where neutrinos are virtual particles propagating between production and detection : Neutrinos are described by propagators S (xP-xD)
Consider finite production and detection regions and finite energy and momentum resolution
pxx PD
1
||If neutrinos can be considered real (on shell) and production, propagation, and detection can be treated separately
Match initial and final conditionsWave packet formalism
)(v
|| )(
2
2v
||)(
2
22
2
PDg
PDPD
g
PDPD
ttxx
EttE
m
p
mEm
dm
dpE
dE
dpp
xxpttE
Correct calculation of the phaseCorrect calculation of the phase
Phases calculated in the same space-time point
Second summand is small by either one and/or the other term => standard result Oscillation effects disappear if neutrino masses are all equal
t
E
mi
jjii
tiEii
ji
i
eUUUU
eUUvP
2**
2*
2
||
100
0
0
0
010
0
0
0
001
1212
1212
1313
1313
2323
2323
321
321
321
cs
sc
ciδes
iδesc
cs
sc
UUU
UUU
UUU
U
τττ
μμμ
eee
i
Exercise : Write Pee, Pe
Exercise : Minimal number of Pto know all the others
One example : One example :
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 3 4 5 6 7 8
"pee_kland.txt"
E
LmP f
4sin2sin
2222
Some real cases :Some real cases :
Neutrino Oscillations in matter Neutrino Oscillations in matter
After a plane wave pass through a slab, the phase is shifted : p (x+(n-1)R)
Net effect :
p
NRfe
edrNRfee
ipx
ipx
Rx
ipxRnxip
)0( 211
)0( 2 ))1((
2
)0( 21
p
fNn
Only the difference of potential is relevant
Net effect :
eFF NGeevv
GH 2)1( )1(
255int
i
0
vei
e
Nee
Nee
eF NG2
2lmatt
cosVesin 2
sin 2
m2
m2
m2
i =
ddt
e
e
Two Neutrino Oscillations in matterTwo Neutrino Oscillations in matter
sin22m =sin22
( cos2GFNeE/m2)2 + sin 22
Difference of the eigenvalues
H2 - H1 = m2
2E( cos2GFneE/m2)2 + sin 22
l / l0
sin2 2m
sin2 2 = 0.08 sin2 2 = 0.825
~ n E
Resonance width: nR = 2nR tan2
Resonance layer: n = nR + nR
ResonanceResonance
sin2 2m = 1
Flavor mixing is maximalLevel split is minimal
In resonance:
l = l0 cos 2
Vacuumoscillation length
Refractionlength~~
l/ l0
l/ l0
2m
2m
1m
e
1m
e
sin2 2 = 0.825 sin2 2 = 0.08
Large mixing
Small mixing
Level crossingLevel crossing
(t) = cosa 1m + sina2m e-i(t)
= (Re e
+, Im e+, e
+e - 1/2) = (Re e+, Im e
+, e+e - 1/2)
B = (sin 2m, 0, cos2m) 2 lm
elements of density matrix
lm = 2H oscillation length
= ( B x ) d dt
Evolution equation
Coincides with equation for the electron spin precession in the magnetic field
= 2t/ lm - phase of oscillations
P = e+e = Z + 1/2 = cos2Z/2P = e+e = Z + 1/2 = cos2Z/2
Graphical interpretationGraphical interpretation
A real case: solar neutrinosA real case: solar neutrinos
The Neutrino Matrix :SM + The Neutrino Matrix :SM + massmassALL oscillation neutrino data BUT LSND
)2.8( 3.9eV10
3.725
2m
Δ
)2.2( 6.3eV10
M6.1
23
2
Δ
3 ranges
81.056.074.044.053.020.0
82.058.073.042.052.019.0
20.000.061.047.088.079.0
||||||
||||||
||||||
||
321
321
321
τττ
μμμ
eee
i
UUU
UUU
UUU
U
(0.39) 60.0tan0.28 122
(1.0) 1.2tan0.51 232
(0.005) 041.0sin 132
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