Elementary Particle Physics III
(素粒子物理学 III)
Time Monday 14:55-16:40 (105min/lecture) R207
Junichi Tanaka(田中) and Satoru Yamashita(山下)
E-mail:
satoru@icepp.s.u-tokyo.ac.jp
Junichi.Tanaka@cern.ch
for Yamashita
for Tanaka
Website for materials : (to be uploaded after the each lecture)
http://www.icepp.s.u-tokyo.ac.jp/~satoru/lecture/pp3/
http://www.icepp.s.u-tokyo.ac.jp/~jtanaka/lecture/pp3/
Schedule •
•
•
•
•
•
•
•
•
4/6 Introduction, definition of Unit (1) JT
4/13 Quark and Hadrons, Weak and EW unification (2) SY 4/20 QCD, parton model (3) JT
4/27 CKM Matrix and CP Violation (4) JT
5/11, 18 Higgs Mechanism, Higgs Search and Measurements (5,6) SY
6/1 Higgs Measurements and Supersymmetry(7) JT
6/8 Supersymmetry(8) JT
6/15 Neutrino Physics (9)SY
6/22, 6/29 New Physics Search at the Energy Frontier Experiments (10,11) JT
• 7/6 Grand Unified Theories (12) SY
• 7/13 Search for LFV and Summary of this Course (13) SY –Final Report Gauge sector
U(1)
SU(2)L
Gauge bosons
B0
W0, W+, W-­‐
charge
Hyper Charge Y
fermions Quark (L & R)
Lepton (L & R)
Coupling α1
constant
SU(3)
(2x2-­‐1=3)
Gluons (8)
(3x3-­‐1 = 8)
Weak Isospin T
Color charge
Quark (L only)
Lepton (L only)
α2
Quark (L & R)
No lepton
(3 bosons)
Higgs Mechanism à MIXING
B0, W0 à photon, Z0
αs
Summary of the Standard Model
• Particles and SU(3) × SU(2) × U(1) quantum numbers:
• Lagrangian:
gauge interactions
matter fermions
Yukawa interactions
Higgs potential
Weak Interactions
•
Interactions of lepton doublets:
•
Charged-current interactions:
•
Neutral-current interactions:
•
Mixing between quark flavours:
Parameters of the Standard Model
• Gauge sector:
– 3 gauge couplings: g3, g2, g ́
Unification?
– 1 strong CP-violating phase
• Yukawa interactions: (related to Higgs?)
– 3 charge-lepton masses
Flavour?
– 6 quark masses
– 4 CKM angles and phase
• Higgs sector:
– 2 parameters: µ, λ
Mass?
• Total: 19 parameters
Parameters of the Standard Model ++
• Gauge sector:
– 3 gauge couplings: g3, g2, g ́
Unification?
– 1 strong CP-violating phase
• Yukawa interactions: (related to Higgs?)
–
–
–
–
–
3 charge-lepton masses
6 quark masses
4 CKM angles and phase for quarks
3 neutrino masses
4 MNS angles and phase for leptons
• Higgs sector:
– 2 parameters: µ, λ
• Total: 26 parameters
Flavour?
Mass?
小柴先生のスライド（2009）より
u, u, u,
d, d, d,
L
L
L
L
L
L
eL
νeL
c, c, c,
s, s, s,
L
L
L
L
L
L
t, t, t,
b, b, b,
L
L
µL
νµL
u, u, u,
d, d, d,
R
R
R
R
R
R
eR
νeR
1st%Family%or%
e.Family%
L
L
L
L
τL
ντL
c, c, c,
s, s, s,
R
R
R
R
R
R
µΡ
νµR
2nd%%Family%or%
µ.Family%
t, t, t,
b, b, b,
R
R
R
R
τR
ντR
3rd%Family%or%
τ.Family%
R
R
4"Interac*ons"and"
4"media*ng"Bosons"
%
%
Strong% Interac7on% on%
colored%Quarks%and%%
Gluons,%g."
%
E.M% interac7on% on%
charged% par7cles% and%
Photon,%γ.%
%
Weak% Interac7on% and%
Z0,W+8."
%
G r a v i t a 7 o n a l%
Interac7on%on%all%and%
Gravitons.%%
%
%
Generations
u, u, u,
d, d, d,
L
L
L
L
L
L
eL
νeL
c, c, c,
s, s, s,
L
L
L
L
L
L
µL
νµL
u, u, u,
d, d, d,
R
R
R
R
R
R
eR
νeR
1st%Family%or%
e.Family%
Higgs Fields
c, c, c,
s, s, s,
R
R
R
R
R
R
µΡ
νµR
2nd%%Family%or%
µ.Family%
t t t
b b b
4"Interac*ons"and"
,
,
,
4"media*ng"Bosons"
%
L,
L,
L,
%
τL
Strong% Interac7on% on%
L
ντ
Left-­Right（Yukawa）
colored%Quarks%and%%
Gluons,%g."
%
E.M% interac7on% on%
R,
R,
R,
charged% par7cles% and%
R,
R,
R,
Photon,%γ.%
%
τR
New Force
Weak% Interac7on% and%
ντR
Z0,W+8."
%
3rd%Family%or%
G r a v i t a 7 o n a l%
τ.Family%
Interac7on%on%all%and%
Gravitons.%%
%
Self-­coupling
%
L
L
L
t t t
b b b
e+e− Energy Frontier :
LEP (√s=90−209 GeV, 1989-2000) & SLC (√s=90−100 GeV, 1989-1998)
LEP measurements
Martin, hep-­ph/9709356
60
without SUSY
U(1)
50
40
SU(2)
α
-1
30
SUSY
around 1 TeV
20
Unification
10
0
2
with SUSY
SU(3)
4
6
8
10
12
Log
(Q/GeV)
Energy
10 Scale 10 (GeV)
14
16
18
N
Unification with supersymmetry (SUSY)
Precise measurement of the gauge couplings with RGE calculations indicate: if supersymmetry exists around 1 TeV, the three forces unify at around 1016 GeV.
10
DIRECTIONS
1019GeV
Planck scale
1016GeV?
1014GeV?
Composite?
(New Interaction)
GUT? SUSY breaking?
Inflation?
Right-­handed?
Heavy neutrino?
SUSY?
100GeV
SM
Dark matter?
H, W, Z, leptons, quarks
Neutrino Physics
12
“Neutrino” hypothesis
Electron energy from particles
does not have a sharp peak
but a board distribution.
(Z, N) -­> (Z+1, N-­1) + e-­
2-­body decay -­> a sharp peak
(Z, N) -­> (Z+1, N-­1) + e-­ + “n”
3-­body decay -­> a board dist.
(Phosphorus = 燐)
W. Pauli introduced “n” in 1930.
13
Neutrino Discovery
at Nuclear Reactor in 1956 by Reines and Cowan
Inverse b-­decay
ν e + p → n + e+
O(ns)
Neutron capture by cadmium
-­> Excited state -­> emit γs
14
(Revisited) Neutrino in the SM
–
–
–
–
–
Lepton, Spin 1/2
3 generations … Electron neutrino, muon neutrino and tau neutrino ν e ν µ ν τ
Massless
Neutral particle -­> Only the neutral “matter” particles in the SM
Interact via only the weak interaction: W-­>lν, Z-­>νν
• Left-­handed neutrino and right-­handed anti-­neutrino
– Left-­handed neutrino: helicity -­
– Right-­handed anti-­neutrino: helicity + = “Left-­handed neutrino with E<0”
• No “right-­handed neutrino and left-­handed anti-­neutrino” (=massless)
– Such particles (nR) have no charge (T3=Y=Q=0) in the SM
EM, weak and strong
-­> Even if they exist, we cannot observe them with the SM interactions.
Since in this case neutrinos can have mass, we can observe them via
Yukawa interaction.
15
Neutrino Mass Measurements
•
Information from the PDG web site (http://www-­pdg.lbl.gov/)
Each measurements
3
3
+
m(ne) < 2 eV (95%CL) … Tritium (三重水素) beta-­decay 1T → 2 He
+ e − +ν e
π + → µ + +ν µ
m(nm) < 190 keV (90%CL) … m(nt) < 18.2 GeV (95%CL) …
τ − → π − + π − + π + +ν τ
Total mass … constraints from cosmology (many results/calculations exist. See PDG.)
Sm < 0.24eV (68%CL)
• Constraints the total mass of neutrinos from observational Hubble parameter data with seven-­year WMAP data and the most recent estimate of H0.
-­> The absolute values have not been measured yet because of its smallness.
16
Observation of Neutrino Oscillation
•
We observed neutrino oscillation in many experiments.
– Several experiments will be shown later.
it means that neutrinos have finite mass!!!
Anyway, 2
−5
−3
2
Δm = 10 − 10 (eV )
-­ How do we describe such massive neutrino in theories?
-­ Why does the neutrino oscillation indicate “massive” neutrino? 17
PMNS Matrix
•
Neutrino oscillation -­> mixture of neutrino flavors like quarks
•
Pontecorvo-­Maki-­Nakagawa-­Sakata matrix (PMNS or MNS Matrix) was introduced.
Weak eigenstates
⎛ν eʹ′ ⎞
⎛ν e ⎞
⎜ ⎟
⎜ ⎟
⎜ν µʹ′ ⎟ = U PMNS ⎜ν µ ⎟
⎜ν ʹ′ ⎟
⎜ν ⎟
⎝ τ ⎠
⎝ τ ⎠
Mass eigenstates
=VCKM (exactly the same parameterization)
-­> 3 real parameters/angles and 1 complex phase (Dirac neutrino)
-­> CP-­violation is possible!
If neutrinos are majorana, we need two additional phases, which might cause CPV.
-­> 3 real parameters/angles and 3 complex phases (Majorana neutrino)
18
Experiments and/or Observatories
• Very Old days experiments using Neutrinos from nuclear decay • Solar Neutrino using neutrinos from Sun
• Atmospheric Neutrino using cosmic ray neutrinos • Reactor Neutrino experiments using Neutrinos from Nuclear Reactors
• Accelerator-­based experiments using Neutrinos produced by Accelerators
19
PMNS vs CKM
(d=0 is used.)
(I calculated elements by using the following values,
which are obtained from PDG web site)
+0.018
2
+0.024
12
− 0.016
23
− 0.021
sin 2 θ = 0.307
, sin θ = 0.386
,
sin 2θ13 = 0.0241 ± 0.0025 (m1 < m 2 < m 3 )
θ12 = 33.6o , θ 23 = 38.4o , θ13 = 8.9o
The mixing size of the neutrinos are relatively larger.
(but we don’t know this reason.)
20
Oscillation
For simplicity, we consider a two-­generation oscillation.
weak
mass
See the evolution of a neutrino created as na at t=0
Two mass states can travel with different frequencies.
Then get the transition na(t=0) à nb
The probability of na(t=0) à nb
21
Oscillation/transition probability of na to nb
Survival probability of na
We should optimize the parameters of experiments
to get the maximum oscillation/transition/survival probability.
22
L = 180km
Δm 2 = 7 ×10 −5 eV 2
sin 2 2θ = 0.84
(PDG)
1.27Δm 2 L 3π
=
→ E = 3.3 [MeV]
E
2
1.27Δm 2 L π
= → E = 10 [MeV]
E
2
23
Neutrino Experiments •
Dm232 and q23 : nm -­> nt
à Atmospheric neutrino and accelerator neutrino experiments
•
Dm122 and q12 : ne -­> nm
à Solar neutrino and reactor neutrino experiments
Δm 2 [eV 2 ]L[km]
~1
E[GeV]
• q13 : ne -­> nt
à Reactor neutrino and accelerator neutrino experiments
PDG
24
Neutrino Experiments (a part of them…)
•
Dm232 and q23 : nm -­> nt
à Atmospheric neutrino and accelerator neutrino experiments
Atmospheric exp: (Super-­)Kamiokande, Soudan-­2, MACRO, MINOS, … -­> q23
Accelerator exp: K2K, MINOS, T2K, … -­> Dm232
•
Dm122 and q12 : ne -­> nm
à Solar neutrino and reactor neutrino experiments
Solar exp: Homestack, (Super-­)Kamiokande, Gallex, SAGE, SNO,
Borexino, … -­> q12
Reactor neutrino exp: KamLAND, … -­> Dm122 and q12
•
q13 : ne -­> nt
à Reactor neutrino experiments and accelerator neutrino experiments
Reactor exp: Daya Bay, Chooz, Double Chooz, RENO, …
Accelerator exp: T2K, …
25
Atmospheric n
Atmospheric Neutrino
So-­called “atmospheric neutrinos” are produced from cosmic rays.
±
±
(−)
π → µ +ν µ
(−)
(−)
µ → e +ν e +ν µ
±
±
ν µ :ν e = 2 : 1
is naively expected.
A nominal n energy range is 1-­100GeV.
(-­> The flux is maximum around 1GeV.) In 1988, Kamiokande reported
a kind of anomaly in the atmospheric
neutrinos. (See the next slide)
26
Atmospheric n
Phys.Lett.B 205 (1988) 416
Kamionkande
Muon-­like events 85/144 = 59%
Historically they (including other experiments in 1980’s) would observe “Proton decay”.
-­> They observed many atmospheric neutrinos, which are background for the proton
decay search.
+ 0
+
p → e π → e γγ
p → K +ν
27
Atmospheric n
Kamiokande and Super-­K
Kamiokande
1983~1996
1996~
15.5m
3000 ton (~1032 proton)
1000 PMT
28
29
cos θ c =
1
nβ
nwater = 1.33 (T = 20o C)
c Emit Cherenkov light (300 photons/cm of λ=300-­600nm)
① cos θ c ≤ 1 v ≥
n (àthis is very small comparing to MIP.)
mc 2
mc 2
nmc 2
1
No Cherenkov light E =
< Eth =
=
②β <
2
2
n
1− β
1−1 n
n2 −1
Eth = 0.77MeV for electron, 158MeV for muon
30
Atmospheric n
n
“Fully contained” neutrino events
with single Cherenkov ring
(“partially contained” -­> not discussed)
(“multi rings” -­> not discussed)
Electron-­like
Single Cherenkov ring
A ring can be observed but
due to EM shower, it is not so clear.
lepton
Muon-­like
Single Cherenkov ring
A clear ring can be observed.
31
Atmospheric n
Atmospheric Neutrino
q zenith angle
Diameter=~104km
32
Atmospheric n
nm
Result of 1489-­day SK exposure
nt at SK
PRD71 (2005) 112005
Non-­oscillation
Δm23 = 2.1×10-3 eV 2 , sin 2 2θ23 = 1.00
33
Solar n
Solar Neutrino
Standard Solar Model (SSM)
Main chain
0.26MeV
1.44MeV
0.86MeV (90%)
0.38MeV (10%)
7.2MeV
4 p → 4He + 2e + + 2ν e + 27[MeV]
Solar neutrino observation à “nuclear fusion in the Sun”
34
Solar n
Solar Neutrino Energy
35
Solar n
1968
Homestake
37Cl
(Chlorine)
PRL 20 (1968) 1205
Half-­life of Argon = 35days
-­ Keep the detector (C2Cl4) at least 35 days
-­> The number of Ar becomes stable.
-­ Count Ar by using X-­ray from Ar.
-­> Can calculate neutrino flux
Measured = 2.56±0.23 SNU
SSM predictions
9.3±1.3 SNU
6.36 SNU
7.64 SNU
C2Cl4 615ton
APJ 496 (1998) 505
36
Solar n
•
Solar Neutrino Problem/Puzzle
The Homestake result shows a deficit of solar neutrino.
~1/3 of SSM predictions
•
At that time one did not understand this result. à “Solar Neutrino Problem/Puzzle”
Did some mistake in the Homestake experiment ?
SSM is wrong ?
or anything else à Oscillation ?
(Gallium)
Similar type of experiments with Ga
37
Solar n
Solar neutrino at Super Kamiokande
1496 days
(1996-­2001)
ν e + e− →ν e + e−
22400±230 events
46.5% of the SSM prediction
-­> Confirm the deficit of the Solar neutrinos
but no one understood its reason.
38
Solar n
Canada
SNO experiment
We observed the deficit of ne. We need to show such neutrinos
are converted into other neutrinos (nm/nt). SNO exp can test it.
39
Solar n
SNO exp.
6
-2 −1
)
φCC (ν e ) = (1.68 ± 0.06 +−00..08
×
10
cm
s
09
φES (ν X ) = (2.35 ± 0.22 ± 0.15)×106 cm -2 s −1
38
)×106 cm-2 s −1
φ NC (ν X ) = (4.94 ± 0.21+−00..34
Phys. Rev. C 72, 055502 (2005) Why so important?
① φCC
(ν e ) < φES (ν X )
ne (solar neutrino) à
Something(nm, nt?)
This shows still only the “deficit”.
② Three measurements are consistent in term of e and other n flux.
φCC (ν e ), φES (ν X ) ↔ φ NC (ν X )
Solar neutrinos really change into other neutrinos.
“Solar neutrino puzzle” was
solved.
40
Matter effect
•
Neutrino oscillation in matter is different from that in the vacuum:
Light in a matter has different wave length due to the refraction (屈折).
Neutrino also has a similar effect in matter.
This is called “MSW effect”.
(Mikheyev–Smirnov–Wolfenstein)
m2 n
m2nm
m2ne
Surface
Center
Electron Density in the Sun Electron neutrinos produced
in the center of the Sun is changed
into muon neutrinos due to the matter
effect when they are emitted from the Sun.
We need to take into account this effect
properly in our calculations.
This is the case for the earth etc.
(T2K etc à See it later)
41
Survival probability of electron neutrinos in the Sun
We can see contours in case of small angles.
42
Solar n
Large Mixing Angle
Electron recoil energy at SK
Small Mixing Angle
Low Δm2
Quasi-­Vaccum
Still LMA and LOW solutions
4 possibilities from solar neutrino exp.
à Wait for KamLAND!
43
Reactor n
Reactor Neutrino exp. KamLAND
1000tons Liquid Scintillator (à low energy neutrino)
Inverse b-­decay reaction
ν e + p → e + + n (Eνth = 1.806 MeV)
à Take coincidence
44
Reactor n
Reactors around KamLAND
Lν = 180km (with ~70GW)
Eν = a few MeV
2
−5
2
Δm ~ 10 [eV ]
45
Reactor n
Evidence of reactor anti-­neutrino disappearance (2002)!
46
Reactor n
Determination of Solution
2
Δm 12 , θ12
Without KamLAND results, we had still a few regions for parameters of .
à By combining both solar and KamLAND we selected one of them.
PRL 94 (2005) 081801
LMA
2
Δm = 7.9
+0.6
− 0.5
−5
2
2
×10 [eV ] , tan θ = 0.40
+0.10
− 0.07
47
Reactor n
Oscillation observed at KamLAND
Reactors are turned off for the maintenance etc.
à A “effective distance” between KamLAND
and neutrino
L0 Eν e
sources are changed.
48
Neutrino Mass Hierarchy
What we have measured is relative mass size.
We cannot measure the absolute values.
Note: cosmology
Sm < 0.24eV (68%CL)
+
reactor
+
reactor
49
Θ13
The last phase to be observed (expect CP phase) is q13.
There are (at least) two ways to measure this phase:
① ne disappearance
Reactor neutrino experiments
② ne appearance
Accelerator neutrino experiments
(Atmospheric neutrino experiments)
50
Reactor n
NuFact2013 Talk
Daya Bay Experiment
GPS and modern theodolites
-­> relative detector-­core positions ~ 3cm
51
Reactor n
Daya Bay
52
Why can Daya Bay measure q13?
Short baseline
Reactor neutrino En~MeV
Dm2~10-­3 with L~km
Long baseline
Dm2~10-­5 with L~100km
53
Reactor n
NuFact2013 Talk by Soeren Jetter
Daya Bay
(mee~m13 ß A term for the short baseline is more complicated.)
54
Reactor n
NuFact2013 Talk by Soeren Jetter
March 2012
DC=Double Chooz
55
Accelerator n
T2K Experiment
Long baseline neutrino experiment
Muon neutrino beam produced by accelerators
56
Accelerator n
2
sin 2 2θ13 = 0.140 +−00..038
(
Δ
m
032
32 > 0)
045
2
sin 2 2θ13 = 0.170 +−00..037
(Δm 32
< 0)
2
(| Δm32
|= 2.4 ×10 −3 [eV 2 ],
2
sin θ 23 = 0.5, δ CP = 0)
PRL 112 (2014) 061802
57
Future Neutrino Experiments
•
What we need to understand in (near) future is
–
–
–
–
–
Mass Hierarchy
CP phase(s)
Dirac vs Majorana
Absolute mass size
Existence of other neutrinos
• More than 3 types
• “Sterile” neutrino (not interact like the left-­handed neutrino (= the weak interaction).)
58
Reactor n
MC simulation for RENO-­50 exp.
Reactor Experiment for Neutrino Oscillations in Korea
We can observe small differences in, for example, P(ne-­>ne).
(see PLB 533 (2002), 94)
Solid … Normal Hierarchy
Dashed … Inverted Hierarchy
@1MeV
Require better energy resolution à 3% or better
(also need statistics)
59
Accelerator n
CP Violation at Neutrino Sector
More complete ne appearance probability including the matter effects
Leading term with the matter effects
Matter
effects
2
⎛
⎞ ⎛ E ⎞
ρ
⎟ ⎜
⎟⎟
a = ±7.56 × 10 −5 eV 2 ⎜⎜
3 ⎟ ⎜
⎝ [g cm ] ⎠ ⎝ [GeV] ⎠
[ ]
By measuring this probability precisely with nm beam, we can extract CP phase.
Ideally it is better to measure this probability with both nm and nm beam.
In addition, we may separate the mass hierarchy with this measurement.
(see the next)
60
Accelerator n
ne appearance probability including the matter effects at the oscillation maximum
(sin22q13=0.1 and sin22q23=1)
(Ichikawa-­san’s slide)
This value can be obtained
from the 1st term
of the expression
shown in the previous page
with the present
measurements.
61
Hyper Kamiokande
•
•
Size : 48m(W) x 54m(H) x 250m(L) x 2 à Huge detector!
Water volume
– Total : 0.5M ton x 2 = 1Mton (~20 x Super Kamiokande)
– Fiducial volume : 0.56M ton (~25 x SK)
•
Photo-­detectors
– Inner detector : ~99000 20inch PMTs (~8 x SK)
– Outer detector : ~25000 8inch PMTs (~12 x SK)
Operate from ~2023(plan)
Don’t forget the “proton decay”…
62
Double b-­decay
2ν mode : (Z ) → (Z + 2 ) + 2e − + 2ν e
0ν mode : (Z ) → (Z + 2 ) + 2e −
n
e
-­
e-­
n p
n p
bb2n
n
e
-­
Must be n=n
e-­
n
n
n p
n p
bb0n
2n mode is possible in the SM (with both Dirac and Majorana neutrino).
0n mode is possible in case of Majorana neutrino.
Also this mode violates the lepton number: 0 à 2
The search for 0n mode is very interesting.
à If we’ll observe it, it means that neutrino is not “Dirac” but “Majorana”.
63
Normalized “sum of two electron energy” In case of the “neutrino double beta decay”, ideally we can observe
a peak instead of a broad energy distribution.
64
The type of isotopes is limited.
~ 30 kinds of isotopes could be candidates. Why?
Because the next type of “Energy level structure” is required.
76
76
2.04 MeV
Ge
As
Double b-­decay
76 Se
The bb2n decay itself is rare. Typical half-­life of bb2n is 1018-­1024 or longer.
-­> We have observed bb2n decay, for example,
76
Ge→76 Se + 2e − + 2ν e
2ν
T1/2
= (1.5 ± 0.1) ×10 21 year
For the bb0n decay, we have not observe such decay yet (except one?).
The Heidelberg-­Moscow group set a limit on it with 1.9x1025 year
with a limit on <mn> of 0.3-­0.6 eV by using 76Ge. But a part of
the same collaboration also claimed the observation of bb0n decay,
that is, 1.5x1025 year and <mn> of 0.39 eV in 2001.
-­> In the final paper (2006), this observation is still supported.
76
0ν
25
Ge→76 Se + 2e − T1/2
= (2.23+−00..44
31 ) × 10 year
MPLA Vol21 No20 (2006) 1547
One need to check it in future. (PDG quotes but not adopt it so far.)
65
Many experiments are on-­going and also planed.
CUORICINO … 130Te
EXO-­200 … 136Xe
KamLAND-­Zen … 136Xe
NEMO3 … 48Ca, 82Se, 100Mo, 116Cd, 130Te
etc
(Not MeV but meV)
m(majorana)<140-­380meV
EXO-­200
EXO-­200 (SLAC, US)
Signal region arXiv:1205.5608
2nbb … grey region
232 Th … dotted magenta
>1.6x1025 year
66
Introduction
A Possibility of New physics
Majorana and See-­Saw
67
Dirac vs Majorana mass term
Dirac mass
Add nR to the SM like a quark part. We can get the next mass term from Lyukawa.
(
)
(
mD ν Rν L +ν Lν R = mD ψ Rψ L +ψ Lψ R
)
To avoid confusion of “-­” on neutrinos (some of you may
think this is anti-­neutrino but this is wrong. This is a Dirac conjugate.), we use y instead of n for neutrino’s spinor.
Majorana mass
Use anti-­particle instead of different chiral particle.
-­> This type of form can be Lorentz invariant. (also Hermite)
(
m ((ψ
c
)
))
c
mL (ψ L ) ψ L +ψ L (ψ L )
R
c
R
) ψ R +ψ R (ψ R
c
68
Dirac mass
(
mD ψ Rψ L +ψ Lψ R
Y
Y
)
L
m
R
Y
m
L
No lepton number violation
Majorana mass
(
c
c
mL (ψ L ) ψ L +ψ L (ψ L )
(
c
R
= mL ψ ψ L +ψ Lψ
c
R
)
)
Yc
Y
L
m
R
Y
m
L
Lepton number violation
(particle and anti-­particle have different lepton number
if such conservation might exist.)
69
Charge conjugate of Y is written by the Charge conjugate operator C as follows.
)T
ψ c = C (ψ
c
(ψ L )
( ) = C ((ψ ) γ )
1
1
= C (((1 − γ )ψ ) γ ) = C (ψ
2
2
=CψL
T
T
†
5
L
0
†
T
0
(1 − γ 5 ) γ 0 )
T
1
1
T
†
= C ψ γ 0 (1 + γ 5 ) = C (1 + γ 5 ) ψ †γ 0
2
2
T
1
1
= (1 + γ 5 )C ψ = (1 + γ 5 )ψ c = ψ Rc
2
2
(
)
T
†
†
(
T
)
()
ψ = ψ †γ 0
†
* T
( )
A = A
70
(
mD ψ Rψ L +ψ Lψ R
Dirac mass
)
This type of mass is not preferred by many physicists. Why?
This term can be generated with the Higgs mechanism
by adding the right-­handed neutrino to the SM.
(Personally) in term of uniqueness of theory approach,
this looks to be OK except for the fact that there is “no interaction with other particles” (=all the charges are ZERO).
However, when we looked at the measured/known masses of fermions,
the values of Yukawa couplings are much different among them even in
a single generation.
àSuch differences must be explained.
Too small!!!
BTW, concerning the right-­handed neutrinos, since they cannot
interact with the SM interactions, we call such particles “sterile”.
71
Majorana mass
(
c
R
mL ψ ψ L +ψ Lψ
c
R
)
Write a “general” mass term,
which is a Lorentz invariant (+Hermite)
c
Lmass = mDψ ψ + mDψ ψ
(
c
R
+ mL ψ ψ L +ψ Lψ
➀
➁
†
c
c
R
)+ m (ψ ψ
R
c
L
R
+ψ Rψ
c
L
)
∗
ψψ = ψ γ 0ψ ∝ ψ ψ
c
c†
ψ ψ = ψ γ 0ψ = C(ψ
iα
ψ →e ψ
a … (any) charge
T†
)
γ 0ψ ∝ ψψ
We can have conserved charges in ➀ but not in ➁.
à consider electronic charges
à ➀ … for all fermions, ➁ … only for neutral fermions
72
Majorana mass
(
c
R
mL ψ ψ L +ψ Lψ
c
R
)
Write a general mass term,
which is a Lorentz invariant (+Hermite)
c
Lmass = mDψ ψ + mDψ ψ
(
c
R
+ mL ψ ψ L +ψ Lψ
c
c
R
)+ m (ψ ψ
R
c
L
R
+ψ Rψ
c
L
)
c
N1 = ψ L + (ψ L ) = ψ L +ψ Rc
c
N 2 = ψ R + (ψ R ) = ψ R +ψ Lc
N1 N1 = ψ Lψ Rc +ψ Rcψ L
N1 N 2 = ψ Lψ R +ψ Rcψ Lc
N 2 N 2 = ψ Rψ Lc +ψ Lcψ R
N 2 N1 = ψ Rψ L +ψ Lcψ Rc
73
Lmass = mD (N1 N 2 + N 2 N1 ) + mL N1 N1 + mR N 2 N 2
c
N1 = ψ L + (ψ L ) = ψ L +ψ Rc
c
N 2 = ψ R + (ψ R ) = ψ R +ψ Lc
à N1 and N2 are Majorana neutrinos
74
Lmass = mD (N1 N 2 + N 2 N1 ) + mL N1 N1 + mR N 2 N 2
= (N1
⎛ mL
N 2 ) ⎜⎜
⎝ mD
mR + mL
m± =
±
2
mD ⎞ ⎛ N1 ⎞
⎟⎟ ⎜⎜ ⎟⎟
mR ⎠ ⎝ N 2 ⎠
mR − mL
2
2
D
4m
1+
2
(mR − mL )
mR + mL mR − mL
mD2
~
±
±
2
2
mR − mL
mD2
mD2
m− = mL −
~−
mR − mL
mR
mD2
m+ = mR +
~ mR
mR − mL
(mR >> mL , mD )
mD2
(mR >>
>> mL ~ 0)
mR
75
mD
N1ʹ′ = N1 −
N2
mR
mD
N 2ʹ′ = N 2 +
N1
mR
Lmass
2
D
m
~−
N1ʹ′N1ʹ′ + mR N 2ʹ′ N 2ʹ′
mR
See-­saw mechanism
à Two majorana neutrinos mass
mD2
mν =
mR
… observed small mass
mD2
and mR
mR
mR … heavy mass
mν = 10 −3 [eV], mD = 0.1 [GeV] → m R = 1010 [GeV]
-­> Need a new physics in this scale
76