Physics of Massive
Neutrinos
Mu-Chun Chen
Fermilab
NuFact 2006 Summer School, UCLA/UC Irvine, August 15-17, 2006
The Plan
•
•
•
Lecture I:
•
•
•
History
•
•
Quantum mechanics of oscillation (con’t)
•
•
Theoretical Implications
Neutrinos in the Standard Model
Quantum mechanics of oscillation
Lecture II:
Evidence of flavor mixing
Lecture III:
Precision Measurements
References
•
•
Yuval Grossman, TASI 2002 lectures
S. Parke, Cinvesta 2006 lectures
Beta Decay
Heavy Atom decay into light atom, emitting electron
(N,Z)→(N-1,Z+1) + eenergy-momentum conservation
electrons emitted have
He3 (1,2)
Tritium (2,1)
e-
why is the spectrum continuous?
why the
electron energy
not
1914:
Chadwick
β spectrum
equal to difference between
energies of the two atoms?
Number of events
constant energy
observed spectrum implies
missing energy
Expected
Observed
Beta particle energy
Energy conservation in question
Wolfgan Pauli,1930 Propsed a “desperate” remedy to save the
law of energy conservation in nuclear beta decay by
introducing a new neutral particle with spin-1/2 dubbed the
“neutron”
He3 (1,2)
(N, Z) → (N − 1, Z + 1) + e− + ν
e-
ν
Tritium (2,1)
Number of events
pectrum
Brief History
Expected
Observed
Beta particle energy
nservation in question
electron and anti-neutrino
share the available energy
Brief History
Neutretto: “The little neutral o
1932 James Chadwick discovered what we now call the
There are
three,
neutron, but it was clear that this• particle
was
too heavy to be
the “neutron”
and no more
1933 Enrico Fermi formulated the first theory of nuclear beta
Have
no electric or
decay and invented a new name:• the
“neutrino”
strong charge
1956 Fred Reines (UCI) and Clyde Cowan
! interact very weakly:
first detected antineutrinos fromonly
a nuclear
at
throughreactor
weak force
Savannah River in South Carolinaand gravity
1957 Goldhaber, Grodzins, Sunyar ! hard to detect
measured the “handedness” of neutrinos in a ingenious
• They are all left-handed
experiment at BNL
They are all left handed!
They had to be massless!
!
Brief History
1968 Ray Davis and colleague first radiochemical solar neutrino
results using cleaning fluid in Homestake Mine in South Dakota,
leading to the observed deficit known as the “solar neutrino
problem”
1971 SM with massless neutrinos
1987 the atmospheric neutrino anomaly was observed at IMB and
Kamiokande
1995 Nobel Prize to Cowan & Reines
1998 SuperKamiokande reports finding oscillations of muon
neutrinos: new data on the deficit in muon neutrinos produced in the
Earth's atmosphere.The data suggest that the deficit varies
depending on the distance neutrinos travel—an indication that
neutrinos oscillate and have mass.
Brief History
2001 SNO announced observation of neutral currents from solar
neutrinos providing convincing evidence that neutrino oscillations are
the cause of the solar neutrino problem
2002 Nobel Prize to Davis & Koshiba
First Hint that SM is not enough
2003 KamLand confirmed LMA solution to solar neutrino problem
post-2003:
many great discoveries yet to come;
discovery phase into precision phase for some oscillation parameters
Standard Model of Particle Physics
The theory is based on the group:
SU(3)c x SU(2)L x U(1)Y
SU(3)
Quantum Chromodynamics
strong force (quarks, gluons)
SU(2)xU(1)
electroweak interactions
broken to U(1)EW by the Higgs
(Tevatron, LHC, ...)
The Standard Model
The
Standard
Model
of
Particle
Physics
SU
×
U
(1)
⇒Ugroup
U
• The⇒
gauge
is (1)(1)
L(2)
Y (1)
EM
SU
(2)
×
U
L
Y
EM
SU
×
U
(1)
⇒
U
(1)
SU(3)
! SU(2)
!(1)
U(1)
L(2)
Y (1)
EM
SU
(2)
×
U
⇒
U
SUSU
(2)
×
U
(1)
⇒
U
(1)
(2)
×
U
(1)
⇒
U
(1)
L
Y
EM
LSU
Y
EM
L (2) × U
Y (1) ⇒ EM
SU
(2)
×
UY•Yγ The
(1)
⇒ UU
(1)(1)
±
0
L
EM
there W
areL,three
generations
of
fermions
EM
“VEV” of
the
Higgs
boson
Force •Carriers:
Z and
masses:
80,
91
and
0 GeV
C
±
L
Y
0
Force
Carriers:
γ masses:
80,91
91and
and
0 GeV
GeV
±W0 0, Z andbreaks
the electroweak
symmetry
±
Force
Carriers:
W
,
Z
and
γ
masses:
80,
0
±
0
Force Carriers:
W ,WZ±
and
γ and
masses:
80,
91
and
GeV
0 and
Carriers:
W, Z
,Z
80,
91for
and
0 GeV
generate
masses
charged
ForceForce
Carriers:
and
γγ"masses:
masses:
80,
91
and
0 0GeV
!
!
"
!
"
" (Higgs
! "BEC)
! !
"
! !u!fermions
"
!
"
"
!
"
!
""
c
t
u
c
t
quark, quark,
SU(2)SU(2)
doublets:
cc ,, bb t t
doublets: u u , ,
d
s
d
s
SU(2)
!
doublets:
quark,quark,
SU(2)SU(2)
doublets:
, EM
LL
L L, , L U(1)
LLY " ,U(1)
d
s
bb L
d LL s L
L
L
up-quark,
SU(2)
singlets:
u, c
, cR
R,,ttR
R
• There
areuthree
generations
of
up-quark,
SU(2)
singlets:
RR
fermions uR , cR , tR
up-quark,
SU(2)
singlets:
down-quark,
SU(2)
singlets:udR
, scRR
, b,RtR
up-quark,
SU(2)
singlets:
R,
down-quark, SU(2) singlets:
d
,
s
,
R
!
" !R R" b!
"
down-quark, SU(2)
d
,
s
,
b
ν"
ν
ν
! singlets:
!
"
!
R R
e
µR
τ "
lepton,
SU(2)
doublets:
,
,
down-quark, SU(2)!singlets:
νe e" !νdµµR , s"
R , bν
R
τ
"
τ!
lepton, SU(2) doublets:
,
,
! νe"
!
e
µ νµ " , τ! ντ "
lepton, SU(2) doublets:
,
νe e L νµ L
τντ L
u
c
t
g
d
s
b
W
e
µ
"
Z
!ee
!µµ
!""
#
II
III
I
lepton, SU(2) doublets:
e
,
µ
,
L
neutrino, SU(2) singlets:
− − −L
τ
Force
carriers
4
L
neutrino,
SU(2)
singlets:
−
charge lepton,
SU(2)
singlets:−e−
R , µR , τ R
charge lepton, SU(2) singlets: eR, µR, τR
– Typeset
by FoilT
– Typeset
by FoilT
XE–X –
4
4
The Standard Model of Particle Physics
$%&' (%))*+,-.#$/+0
the vev of the Higgs H(1,2)
•
breaks the EW
symmetry: SU(2)L x U(1)Y → U(1)EW
+1/2
ase 2: !2 < 0
2
" #
V ($ ) ( ' ! $ & % $
2
2 2
nimum energy state at:
!2
v
+ $ *( '
)
%
2L
2
SU
×
U
(1)
⇒
U
(1)
Y
EM
SU•(2)
(2)
×
U
(1)
⇒
U
(1)
Lforce carriers:
Y
EM
SUSU
××UUYY(1)
⇒UUEM
(1) ⇒
(1)(1)
L(2)
EM
L(2)
Vacuum breaks U(1) symmetry
Force Carriers: W ±, Z 0 and γ masses: 80, 91 and 0 GeV
± 00
±
Force
Carriers:
W
, Z and
and2γ
γ masses:
80,80,
91 91
andand
0 GeV
ce Carriers:
W
,
Z
masses:
0 GeV
Aside: What fixes sign (!
! )? " ! " ! "
! u " !c " t! "
quark, SU(2) doublets:
,
u
c , b t
d
s
quark, SU(2) doublets:
, L
L ,
L
d
L
s
L
up-quark, SU(2) singlets: uR, cR, tR
b
L
Electron mass
mass
Electron
Electron mass
comes
from
a
term
of
the
form
comes from a term of the form
comes from aL̄φe
term of the form
L̄φeRR
L̄φeRof ν
Absence
R
Absence of νR
Absence
νR (dim 4)
forbids such
a massofterm
forbids such a mass term (dim 4)
forbids such
a mass
term (dim 4)
for the
Neutrino
for the Neutrino
for the Neutrino
Therefore in the SM neutrinos are massless
and hence travel at speed of light.
Fermion Mass Generation
•
Two types of mass terms:
•
Dirac masses
•
•
•
•
couple left and right handed fields
mD ψ L ψ R
it always involve two different fields
the additive quantum numbers of the two
fields are opposite
there are four d.o.f. with the same mass
Fermion Mass Generation
•
Majorana masses
•
couple identical left and right handed fields
c
T
c
mM ψ L ψ R , ψ R
= Cψ R
•
•
•
the two fields may be same fields or not
•
can only be written for neutral fermions
there can be only two d.o.f. with the same
mass
additive quantum numbers of the two fields
are the same break all the U(1) symmetries
Neutrino Mass in the SM
•
SM implies exactly massless neutrinos
•
•
•
no νR
•
U(1) is an accidental non-anomalous global
symmetry of the SM no radiative generated
Majorana mass term HHLL
neutrinos are massless
no Higgs triplet
no Majorana mass ∆LL
SM renormalizable no Majorana mass term
from dim-5 operator HHLL
Unlike mγ = 0 prediction, the mν = 0 prediction is
somewhat accidental
Interactions:
Majorana
Neutrinos or Di
Interactions
Charge Current (CC)
Interactions:
Interactions:
Interactions:
W → lor
+ ν̄Dirac
α = e, µ, Neutrinos?
or τ
Majorana
Neutrinos
The S(tandard)
− M(odel)
−
W
→
l
+ ν̄α α = e, µ, o
α
Interactions:
Neutral Current
(NC)
Interactions:
Charge Current (CC)
–
0
l
νταor+τ ν̄α α = e, µ, o
α
=
e,orµ,
→ ν →+lν̄ + ν̄α = Z
e,Interactions:
µ,→
The
S(tandard)
M(odel)
W →
l + Interactions:
ν̄
α = e, µ,
or τ Z W
−
−
−
α
0
α
−
α
−
α
−
αα
α
−
−
W
→
l
ν̄αµ, or
α τ= e, µ, or τ
0
α +
+ =−e,
Z →Z ν0Interactions:
+lν̄α−−
α→
α+ lα
W →α lα + ν̄α α = e, µ, or τ
!
−
−
W 0→ lα + ν̄α α = e, µ, or τ
= e, ν
µ, or
Z0 α →
+τνν̄α+ ν̄α =α e,
Z 0α→
= µ,
e, µ,ororττ
α
Majorana
Neutrinos or Dirac
W
Z
Interactions: and
!α
Z →
α = Interactions:
e, µ, or τ The S(tandard) M(odel)
− ν + ν̄
α
ν̄α W
l
+ ν̄ Z α = e, µ, or τ
W α
−Z → ν−
Interactions:
and
W
→
l
e, µ, +or τ
α + ν̄α0 α =−
Z →l +l
–
−
–
Neutral Current (NC)
Interactions:
ν̄α W −llα−
0
α
α
0
−
α
α
+
α
α
+
Interactions:
Interactions:
Z−0 → Z
lα− + l→
!ν̄α W W
Interactions:
−
llαµ, +
α
l
→
l
+
ν̄
α
= e,
or lτα !
α
α
α
0
α l=
τ e,−µ, or
− e, µ, or
Z
→
+
ν̄conserve
or τLepton
−
−
α
W− →
+
ν̄
α
=
τ
=
µ,
orα
τ =
couplings
α
lα−0e,
+ν
ν̄α
α τe,
=
e, µ,
or
τ e, µ,the
W → l + ν̄W
α=
µ,
α
ααor
α →
W
Z → να + ν̄α α = e, µ, or τand
αe, µ,=−ore,τ µ,
or τ
0
0
0
+
Z
→
ν
+
ν̄
α
=
Z → να + ν̄α the
α =αLepton
e, µ,α Z
or τby—
→ Number
lα + !lα
conserve
L
defined
α
Z
couplings
Z → l + lZ → l + l
– =–
–)τ = – L(!)
defined by—
α
=
e,
µ,
or
L(!)
=
L(l
α = e, µ, or τ
couplings
conserve
the
Lepton
Nu
α = e, µ,
or τ –
+) = 1.
L(!) = L(l–) = – L(!)
=
–
L
(l
defined
by— charged-lepto
So do
the Dirac
–– = – L (
–) = – L(!)
L(!) = L(l
So do the Dirac charged-lepton
mass
terms
mllLlR+
–
+
l
l
m
l
Solldo
the
Dirac
charged-lepton
m
L R
X
0
−
α
+
α
0
−
α
+
α
– Typeset by FoilTEX –
(— )
(— )
–
Qualitative successes of SU (2)L ⊗ U (1)Y theory:
Number of Neutrinos
! neutral-current interactions
! necessity of charm
±
0
! existencewidth
and properties
of Wother
and Z
invisible
of Z and
data
from LEP
Decade of precision tests EW (one-per-mille)
MZ
ΓZ
0
σhadronic
Γhadronic
Γleptonic
Γinvisible
e+ e− → Z o → νν
91 187.6 ± 2.1 MeV/c2
2495.2 ± 2.3 MeV
e+
41.541 ± 0.037 nb
1744.4 ± 2.0 MeV
e
83.984 ± 0.086 MeV
499.0 ± 1.5 MeV
!
#
Z
#
+
"
"
e
!
e
"
e
W
"
e
FIG. 1: Leading order Feynman diagrams contributing
useful to estimate the ultimate sensitivity for LEP (including all four exp
around the Z-boson mass). We do this by naively rescaling the L3 resu
Assuming that both the statistical error and the systematic error will de
δΓinvleptonic
= ±5 ± 5 MeV.
The relatively large (compared to the indirect result) error of the direc
statistical sample of e+ e− → γν ν̄ events available at LEP. Therefore, a sign
a high-luminosity
linear collider running around the Z-boson mass. At such
i i
within 100 days of running, a sample of 109 Z-boson decays can be collecte
Assuming 50 fb−1 of e+ e− data collected around the Z-boson mass,
measurement of the invisible Z-width. We are mostly interested in the res
γ+ missing energy, but will first briefly present the improvement that can be
width indirectly, as discussed in the previous subsection. We assume [16, 17
µa factor of roughly
τ
two times more precisely, δΓtot = ±1 MeV, while R
measured with uncertainties δ(R! ) = ±0.018 and δ(σh0 ) = ±0.03 nb (most
and δ(σh0 ) = ±0.015 nb (most optimistic scenario). We refer readers to [1
that the correlation matrix between the observables is identical to the one o
where Γinvisible ≡ ΓZ − Γhadronic − 3Γ
light neutrinos Nν = Γinvisible /ΓSM (Z → ν ν̄ )
Current value: Nν = 2.994 ± 0.012
. . . excellent agreement with νe , ν , and ν
Z Boson
Fig. 1.Number
of Neutrinos
Measurements
of the hadron production cross-section around the Z
implies
N
ν = 2.9840 ± 0.0082
resonance. The curves indicate the predicted cross-section for two,
and four neutrino species with Standard Model couplings and
Threethree
active
neutrinos! (sterile neutrinos don’t couple to Z)
negligible mass.
Back to Article
Copyright © IOP
Publishing Ltd 1998-
Majorana Neutrinos or Dirac Neutrino
The S(tandard)
M(odel)
Interactions:
Note That
Interactions: ν̄α W −ll–α−
!
Interactions:
ν̄ Come
W
Neutrinos
W l in at Least Three Flavors
Implies
α
−
α
and
Observed
Interactions: !ν̄α W − lα
small L/E
−
W − → lα
+ ν̄α
Not Observed
Z
!
α = e, µ, or τ
The known
neutrino
flavors:
!
,
!
,
!
Note
That
e
µ
couplings conserve the Lepton" Number L
defined
by—
Implies
Each of these
is associated
–
–
with the corresponding
L(!) = L(l ) = – L(!) = – L (l+) = 1.
Observed e , µ , "
charged-lepton flavor:
So do the Dirac
charged-lepton mass terms
small L/E
–
+
l
m l l Ll R
X
(— )
Not Observed
ml
Majorana
Neutrinos
or Di
Majorana Neutrinos
or Dirac
Neutrinos?
Interactions:
The
S(tandard)
− M(odel)
−
The S(tandard)
M(odel)
Standard
Model
of Particle
Physics
Interactions:
W
→
l
α
Interactions:
Majorana Neutrinos or Dirac
Neutrinos? + ν̄α α = e, µ,
–
Gauge
Theory
based
on
the
group:
−l–
−
Interactions:
0
l
ν̄α W M(odel)
lα
ναor+τ ν̄α !α = e, µ, o
W → l + ν̄ Z
αInteractions:
=→
e, µ,
The S(tandard)
SUInteractions:
(3)l × SU (2) Z× U→(1)
−
ν + ν̄− α!=−e, µ, or τ
ν̄α W
l
W α
W
W Z → lα + ν̄α α =
e, µ, or τ Z
and
and
W SU (3) ⇒ Quantum
−Z
andChromodynamics
−
–
0
−
α
α
α
α
!
!
lα Z 0 → !ν + ν̄
! ν̄α W
α
α
Strong Force (Quarks and Gluons)
α=!
e, µ, or τ
the Interactions
Lepton Number
SUL(2)couplings
× U (1) ⇒ conserve
ElectroWeak
brokenLto UEM (1)
couplings
conserve
the Leptonconserve
Number the
L Lepton
defined
by—
couplings
by HIGGS
(Tevatron,
– = –LHC)
–) = – L(!)
L(!)
=
L(l
L (l+) = 1.
defined by—
defined
by—
So do the Dirac charged-lepton
mass
terms
–
L(!) = L(l–) m=l––l L(!) = –l L (ll +–) = 1. –
L(!)
Actually L , lLL R, and
L = L(l ) = – L(!) = –
(— )
+
(— )
e
µ
τ
+
X
m
So
do
the
Dirac
charged-lepton
mass
terms
Actually Le, Lµ, and Lτ separately
So do
– the Dirac charged-lepto
+
+
l
l
m l l Ll R
X–
l
2
(— )
(— )
mlmlLlR
l
– Typeset by FoilTEX –
– Typeset by FoilTEX –
4
2
Mysteries of Families and Masses in the Standard Mo
Mass and Mixing in Quark
Sector
of Families
and Masses in the
Mysteries
Mixing among quarks
•
d
u
b
c
s
t
Mass states: d’, s’, b
Mysteries of Families
and Masses
the Standard Model
• Mixing
amonginquarks
W+
Mixing among quarks
d
u
d
u
•
Vcd
Vcs
Vcb
!
s
t
# d "& #cVud W
Vcd+ Vtd &# d &
+ (% (
%W+( % +
W
W
Vus Vcsquarks
Vts (% s (
Mixing
% s" ( = %among
% ( %
(% (
$ b"' $Vub Vcb Vtb '$ b '
c
Vcb
W
t
W+
Charged current weak states: d, s, b
Vtb '$ b '
mass CKM matrix weak
• eigenstates
Neutrinos are exactly massless
eigenstates
!
mass
s
Mass states:
+ d’, s’, b’
Vtd &# d &
# d "& #Vud Vcd Vtd &# d &
(% (
% ( matrix
%
Vts (mass
weak (% (
% s(
CKM
"
s
=
V
V
Vts (% s ( 0
(
%
(
us
cs
%
(
%
eigenstates
eigenstates (% (K
V
tb '$ b '
% ( %
$ b"' $Vub
Charged current we
W+
W+
b
b
W+
# d "& #Vud
% ( %
% s" ( = % Vus
% ( %
$ b"' $Vub
W+
M
C
W+
K0
!
weak
CKM
matrix
-- mass eigenstates
= weak
eigenstates
eigenstates
eigenstates
Neutrinos are exactly massless
-- accidental symmetries in SM: lepton flavor numbers Le , Lµ , L!
-- mass eigenstates!= weak eigenstates
=> no processes cross family lines in lepton sector
-- accidental symmetries•in SM:
lepton flavor
numbers
Le , massless
Lµ , L!
Neutrinos
are
exactly
-- no neutrino oscillations
=> no processes cross family lines in lepton sector
-- --mass
eigenstates
= weakdecays
eigenstates
no lepton
flavor violating
-- no neutrino oscillations
!
"& #Vud
( %
" ( = % Vus
( %
"' $Vub
ss
states
Vtd &# d &
(% (
Vts (% s (
Mass
(% (
Vtb '$ b '
Vcd
Vcs
Vcb
•
Mass states: d’, s’, b’
and Mixing
in
Quark
Sector
Charged current weak states: d, s, b
weak
among quarks
eigenstates
CKM matrix Mixing
3 mixin angles
1 phase
K0
ss eigenstates =
xture of weak
eigenstates
!
6
Mass
and
Mixing
in
Lepton
Sector
Mysteries of masses and families in SM
areare
degenerate
(all zeros)
Neutrinomasses
masses
degenerate
(all zeros)
•• Neutrino
eigenstates
= weak
eigenstates
mass
eigenstates
= weak
eigenstates
• !mass
symmetries in SM
• • Accidental
accidental symmetries in SM
• --nonoprocesses
processes cross
cross family
familylines
linesininthe
thelepton
leptonsector
sector
leptonflavor
flavornumbers
numbers L , Lµ , L!
• --lepton
e
As a result,
! no neutrino oscillation
+
+
no neutrino oscillation
! Decay forbidden: µ " e + #
decays forbidden:µ + " 2e +µ++ e$→
•
•
e+ + γ
µ+ → $ 2e+ + e−
$
µ + N(n, p) " e + N(n, p)
µ− + N (n, p) → e− + N (n, p)
µ−$ +
p) "→e + e++N(n
µ
+ N(n,
N (n, p)
+ N+
(n2,+p2,$p2)
− 2)
+
+ µ+ → e+ + ν e + νµ
µ "e +%e +%
µ
!
7
Missing
Solar
Neutrinos
Missing
Solar
Neutrinos?
Missing
Solar
Neutrinos?
Missing
Solar
Neutrinos?
Nuclear
theSolar
sun Neutrinos?
Nuclear
fusionfusion
inMissing
thein
sun
+
+
Nuclear fusion in the sun
!
+
+MeV
25 MeV
!
+
+
2e
"e25
2e
2"e++ +225
+
produces
a lot
of energy
produces
a lot of
energy
and !eand !e!
+
+
MeV
2e
2
"
e
produces
a lot in
ofthe
energy
Nuclear fusion
sun and !e
! proton proton
+ 2e4+ + 2"4He
e + 25 MeV
produces a lot of energy and !e
4 He
proton
He
4
proton
He
Solar Solar
Solar
Solar
core
core
core
core
Standard
Solar Model
Predictions
Standard
Solar Model
Predictions
Predictions
Standard
Solar
Model
Standard
Solar Model
Predictions
!e produced
!e!produced
e produced
!e produced
108 km
108 km108 km
10 km
8
earth
Under ground
!e detector
earth earth
earth
Under ground
Under ground
Under ground
!e detector
!e detector
!e detector
8
8
8
8
What if Neutrinos Have Mass?
What if neutrinos have mass?
•
Weak interaction eigenstates:
!e
e
W-
" e, " µ, "#
!µ
µ
!
W-
Mass eigenstates:
•
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) Matrix:
•
W-
"1, " 2 , " 3
•
!
!"
!
$" e ' $U e1 U e 2 U e 3 '$ "1 '
& ) &
)& )
"
=
U
U
U
µ2
µ 3 )&" 2 )
& µ ) & µ1
& ) &
)& )
"
U
U
U
% # ( % #1
#2
# 3 (%" 3 (
13
!
Leptonic Mixing Matrix
Leptonic Mixing Matrix
Two mass differences:
•
"ma2 , "ms2
threemixing:
neutrinos
Three families
U MNS
!
case:
! 0&# c
"i)
#1
0
0
s
e
x
x
%
(%
= %0 c a sa ( % 0
1
0
%
%$0 "sa c a (' "sx e i) 0
cx
%$
!
atm
reactor
&
(
(
(
('
# cs
%
%"ss
%$ 0
ss
cs
0
0&
(
0(
1('
solar
#1
0
0
%
i 1*
%0 e 2 12
0
1
%
i * +)
%$0
0
e 2 13
Majorana phases
(
)
(
3 mixing angles: " a , " s, " x
", #12 , #13 !
3 complex
phases:
three mixing angles: ca ,
cs , cx
•
CP
violation
in neutrino oscillations
sensitive to "
three complex
phases (majorana):
•Neutrinoless
! double-beta decay sensitive to
"12 , "13
!
δ, φ12 , φ13
14
)
&
(
(
(
('
αj
lTEX –
Neutrino Flavor Change (Oscillation)
( B.K.Approach
"of )
in!Vacuum
& Stodolsky
cos θ l-sin
(e.g. $)θ
− sin θ cos θ
l"+(e.g. µ)
Uαj =
Amp
!
Amplitude
W
Amplitude
Uαj =
=
=
"
!
(!")
−im2j L/2E(!#)
e
Target
#
cos θ sin θ l+
"
− sin θ cos θ
!
e
W
Source
l#-
−im2j L/2E
$Amp
!i
W
i
"
#
Source
U"i*
U#i W
Prop(!i)
#
Target
cos θ sin θ
− sin θ ! cos
θ
"
cos θ sin θ
Uαj =
−im2j L/2E
− sin θ cos θ
e
17
1
Neutrino Oscillation
•
neutrino flavor is identified by charged current
interactions
Expressed in terms of the mass eigenstates
•
•
!
"
!να (t)
! " # ∗! "
! να =
!
i Uαi νi
A
that was produced at t = 0 evolves in time
according to
!
! "
"
!να (t) = # e−iEi t U ∗ !νi
αi
i
For relativistic neutrinos (pi = p)
Ei ≈ p +
m2i
2p
Oscillation mechanism
Neutrino
Oscillation
•
Simplified two-flavor analysis:
#" e & # cos) sin ) &# "1 &
% (=%
(% ( !µ
$" µ ' $*sin ) cos) '$" 2 '
• transition probability from
!µ to !e
!2
sin "
cos "
!e
P(" µ # " e ) = " e " µ (t)
"
!
" e = "1 cos # + " 2 sin #
= sin 2 2$ sin 2 (%L / & )
"
• Survival probability for !µ
" µ = " 2 cos # $ "1 sin #
!
!
•
In vacuum: " µ
2
!
P(" µ # " µ )
= 1$ sin 2 2% sin 2 (&L / ' )
evolves in time
2
" µ (t) = " 2 e#im 2 t / 4 p cos$ # "1 e#im1 t / 4 p sin $
!
!
!1
• Oscillation length
!
"=
"m2 must be non-zero
to have neutrino
oscillation!!
!
!
2.5E #
$m 2
"m 2 = m12 # m22
9
Vacuum oscillation: E!=1 GeV, !m2=10-3 eV2, # = $ /6
!µ
!µ
!e
!µ
!µ
!e
!µ
!µ
!e
!µ
!µ
!e
!µ
P( !µ" !µ )
Probability
P( !µ" !e )
%osc=2.5 E! / !m2
L (km)
10
Neutrino Oscillation
•
The probability to observe flavor oscillation α→β
is
Pαβ
! " ! #! 2
%
& 2
$
∗
∗
!
!
!
= νβ να
= δαβ − i<j Re Uαi Uβi Uαj Uβj sin xij
where
xij =
•
∆m2ij t
2p
For two flavors (α≠β)
U=
!
cos θ
− sin θ
sin θ
cos θ
"
⇒ Pαβ = sin2 2θ sin2 x (α #= β)
Neutrino Oscillation
Pαβ = sin2 2θ sin2 x (α != β)
•
For relativistic neutrinos: L = t, E = p and
x=
•
2πL
Losc
4πE
∆m2
It is also convenient to use
xij = 1.27
•
Losc =
!
∆m2ij
eV 2
"#
L
km
$ # GeV $
E
more sophisticated derivations give the same result
Neutrino Oscillation
Pαβ = sin 2θ sin 2x,
2
•
2
x=
∆m2 L
2E
x=
2πL
Losc
longer baseline L and smaller energy E
smaller ∆m2
‣ L ! Losc : sin2 x ∼ x2
‣
very small signal
: due to energy spread of the beam and
decoherence effects, the oscillation is averaged
L ! Losc
!
"
sin x = 0.5
2
Matter Effects
•
Like photons, when neutrinos travel in medium, they
acquire an effective mass
•
The CC interaction for νe gives,
VC =
√
2GF Ne ≈ 7.6Ye
!
ρ
14
10 g/cm3
"
eV
Ne : electron density ; Ye : relativistic electron density
Ye = (Ne)/(Nn+Np)
✤ at the earth core: Vc ~ 10-13 eV
✤ at the solar core:
Vc ~ 10-12 eV
✤ at a SN core:
Vc ~ 1 eV
Matter Effects
•
•
non-universal matter effects affect the oscillation
for one flavor: a vector interaction (Vc, 0,0,0)
(E − VC )2 = p2 + m2
•
⇒
E ≈ p + VC +
m2
2p
the effective mass squared is enhanced by E
m2m = m2 + A ,
A = 2EVC
Matter Effects
For more than one flavor:
✴ only νe has CC interactions
the mixing matrix is modified by the matter
•
∆m2m
tan 2θm
•
•
•
=
=
!
(∆m2 cos 2θ − A)2 + (∆m2 sin 2θ)2
∆m2 sin 2θ
∆m2 cos 2θ − A
Pαβ = sin2 2θ sin2 2x,
x=
∆m2 L
2E
the vacuum result is reproduced for A = 0
vacuum mixing is needed in order to get mixing in
matter
For xm << 1 matter effects cancel
Matter Effects
∆m2m
tan 2θm
✤ For
!
=
(∆m2 cos 2θ − A)2 + (∆m2 sin 2θ)2
=
∆m2 sin 2θ
∆m2 cos 2θ − A
! !
∆m2 cos 2θ ! !A!
:
‣ the matter effects are small perturbation
! !
✤ For ∆m2 cos 2θ ! !A! :
‣ the mass is a small perturbation
‣ oscillation suppressed
! !
✤ For ∆m2 cos 2θ = !A! :
‣ mixing is maximal (resonance)
Varying Density
•
The effective masses and mixing angles vary with
distance
Q = 8π tan 2θ AA! ,
A! =
dA
dr
‣ adiabatic: Q >> 1
❖
the variation is very slow. no transition
between effective mass eigenstates
‣ non-adiabatic: Q << 1
❖
transition between effective mass eigenstates
possible
The MSW Effect
tan 2θm =
•
Consider small ϑ and
ν(0) = νe ,
•
•
∆m2 sin 2θ
∆m2 cos 2θ−A
A(0) ! ∆m2 ,
‣ initially:
θm → π/2 ⇒ ν2 ≈ νe
‣ finally:
θm → θ ⇒ ν2 ≈ νµ
A(r) → 0
neutrinos pass through resonance
for adiabatic transition: almost full conversion
MSW Effect
FIG. 3. The mixing angle in matter for a system of two massive neutrinos as a function of the
potential A for two different values of the mixing angle in vacuum [see Eq. (61)].