Reflections on Symmetries and Neutrinos Adam Para.
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Transcript of Reflections on Symmetries and Neutrinos Adam Para.
![Page 1: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/1.jpg)
Reflections onSymmetries and Neutrinos
Adam Para
![Page 2: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/2.jpg)
• Special Relativity• General Relativity• Quantum Mechanics• Quantum Field Theory• Standard Model of Elementary Particles and their Interactions
• Role of symmetry in physics
Revolutionary Developments in XX Century Physics
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Mathematics – Physics Connection• Mathematics is the language of physics. Mathematical concepts adopted
or invented by physicists to express precisely and concisely the physical ideas:– Calculus– Complex numbers/functions– Differential geometry– Group theory– Hilbert spaces, hermitian operators– Differential forms
• Symmetries: mathematical concept with profound physics consequences– Conservation laws– Interactions
• Spontaneously broken (hidden) symmetries [Is parity really violated ???]
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Symmetry• What is a symmetry?
– Symmetry operation leaves the system unchanged/undistinguishable
– Property of a system having symmetry operations– Examples:
• Space is uniform: translations and/or rotations do not change the laws of physics
• Red/blue/green quarks have the same strong interactions: we can change(rotate, mix) the quarks definition without any physical consequences
• CPT: if we replace particles by antiparticles, reflect the space and time coordinated then all the process will proceed with the same rates/probabilities
• Symmetry of what?– ‘World we live in’
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What is (the Mathematical Model of?) Our World
We live/move here, on our Home Manifold: 3+1+N dimensional space-time, N≥0
Internal Space:•Phase of the wave function U(1)•SU(2) rotation phase•SU(3) (color) rotation phase•Quark mixing matrix•Lepton Mixing Matrix• . . .
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Where are the Extra Dimensions???
• Everywhere, but they are compactified to a tiny Calabi-Yau manifold at every point of ‘our’ space-time
• Excitations corresponding to the quantized momentum in these dimenions → Kaluza-Klein ‘tower’
• We (all particles of a Standard Model) are confined to a 3+1 dimensional hyperspace (brane) embedded in 3+1+N dimensional Universe
• Only graviton is allowed to travel in ‘bulk’. And this is the reason why gravity appears to be so weak on our brane
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Symmetries of Our World• Continuous (translations,
rotations)• Discrete (reflections, C,T)
• 3+1 space-time:– Lorentz transformations– Reflections
• Internal space:– U(1) phase– SU(2) phase (weak
interactions)– SU(3) strong interactions– Families (KM, MNS mixing
matrix)– Lepton-quark symmetry?
• Extra dimensions (Calabi-Yau manifolds)
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Global Symmetries ↔Conservation Laws (Noether 1915)
• If the Euler-Lagrange equation of motion is invariant under a coordinate transformation
then there exist an integral of motion, i.e. a conserved quantity
• Every continuous global symmetry operation has an associated conserved quantity
• Examples/consequences of Neother’s theorem:– Invariance under space translation ↔ momentum conservation– Invariance under time translation ↔ energy conservation– Invariance under space rotations ↔ angular momentum conservation
', ( ), ( , )t q t t q q t
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Connection Between Symmetries and Conservation Laws in Quantum Mechanics
• What took long time to discover in classical physics is self-evident in Quantum Mechanics:
• Operators related to conserved physical quantities are generators of the corresponding symmetry operations
;
;
( )
x x
z z
p p ix
E H it
L L i x yy x
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Local Symmetries ↔ Interactions(Weyl 1919, Yang-Mills 1952)
• ‘Physics’ ↔ equations of motions ↔ lagrangian
Is evidently invariant under the global phase transformation ↔ charge conservation
• But it requires that the phase is redefined instantaneously in the whole universe ↔ problems with special relativity. Can we propagate the phase redefinition ?
( ) L i m
iqe
( )iq xe
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• No, the lagrangian is not invariant under the local phase transformation
• How to define a lagrangian invariant under the local transformation? Need additional vector field
↔ new interaction
( )L L iq x
( ) ; ( )L i iq A m A A x
int
L iq A
Profound consequences:Maxwell equationsMassless photonsInteractions of matter with radiation
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Geometry and Interactions
• Geometry of spacetime ↔ gravitation (general relativity)• Local Symmetry of the Internal Space (Gauge Symmetry)
↔ electromagnetic, weak and strong interactions – Universal couplings – Massless vector bosons (long range interactions)
• Spontaneous symmetry breaking ↔ mass of intermediate vector bosons
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Neutrinos• SU(2) partners of charged leptons• The only elementary fermions with Q=0• Left-handed only (no longer)• Only weakly interactions• Incredibly light (formerly massless)• Self-conjugate (own antiparticles)? Or not?
• What do they have to do with symmetries??:• Saviors/daughters of symmetries• Symmetries killers• Symmetries messengers• Symmetries probes• … examples follow…
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Two body decay
m1 m2
2 2 22 12
222
2
M m mE p
Mm
Energy-momentum conservation =>
Energy of the decay products always the same
M
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1913-1930: Puzzle of decay
• (Bohr)
• Continuous spectrum of particles
• Energy is not conserved?? (Bohr)
• No translational symmetry of space-time?
•… or ?
• Conflict between ‘theory’ and ‘Experiment’
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Dec 1930: An Act of Faith in Theory:Incomplete Experiment?
P h y s i k a l i s c h e s I n s t i t u t D e r E i d g . T e c h n i s c h e n H o c h s h u l e Z u r i c h
Z u r i c h 4 d e c . 1 9 3 0 G l o a r i a s t r .
D e a r R a d i o a c t i v e L a d i e s a n d G e n t l e m e n A s t h e b e a r e r p f t h e s e l i n e s w i l l e x p l a i n t o y o u i n m o r e d e t a i l – a n d I b e g y o u t o l i s t e n t o h i m w i t h b e n e v o l e n c e – I h a v e c o n s i d e r e d , i n c o n n e c t i o n w i t h t h e ‘ w r o n g ’ s t a t i s t i c s o f 1 4 N a n d 6 L i a s w e l l a s w i t h t h e c o n t i n u o u s s p e c t r u m , a w a y o u t f o r s a v i n g t h e ‘ l a w o f c h a n g e ’ o f s t a t i s t i c s a n d t h e c o n s e r v a t i o n o f e n e r g y : i . e . t h e p o s s i b i l i t y t h a t i n s i d e t h e n u c l e i t h e r e a r e p a r t i c l e s e l e c t r i c a l l y n e u t r a l , t h a t I w i l l c a l l n e u t r o n s , w h i c h h a v e s p i n ½ a n d f o l l o w t h e e x c l u s i o n p r i n c i p l e a n d t h a t i n a d d i t i o n d i f f e r f r o m p h o t o n s b e c a u s e t h e y d o n o t m o v e w i t h t h e v e l o c i t y o f l i g h t . T h e m a s s o f n e u t r o n s s h o u l d b e o f t h e s a m e o r d e r o f m a g n i t u d e o f t h a t o f t h e e l e c t r o n s a n d a n y h o w n o t g r e a t e r t h a n 0 . 0 1 p r o t o n i c m a s s e s . T h e c o n t i n u o u s s p e c t r u m w o u l d t h e n b e u n d e r s t a n d a b l e , a s s u m i n g t h a t i n t h e d e c a y t o g e t h e r w i t h t h e e l e c t r o n , i n a l l c a s e s , a l s o a n e u t r o n i s e m i t t e d , i n s u c h a w a y t h a t t h e s u m o f t h e e n e r g y o f t h e n e u t r o n a n d o f t h e e l e c t r o n r e m a i n s c o n s t a n t . T h e q u e s t i o n i s n o w t o s e e w h i c h f o r c e s a c t o n t h e n e u t r o n s . T h e m o s t p r o b a b l e m o d e l a p p e a r s t o m e t o b e , f o r w a v e m e c h a n i c a l r e a s o n s ( t h e d e t a i l c a n b e g i v e n t o y o u b y t h e b e a r e r o f t h e s e l i n e s ) , f o r t h e n e u t r o n a t r e s t t o b e a m a g n e t i c d i p o l e p f a c e r t a i n m o m e n t . T h e e x p e r i m e n t a l d a t a c e r t a i n l y r e q u i r e f o r t h e i o n i z i n g p o w e r o f s u c h a n e u t r o n t o b e n o t g r e a t e r t h a n t h a t o f a g a m m a r a y a n d t h e r e f o r e s h o u l d n o t b e g r e a t e r t h a n
1310 e c m . I d o n o t c o n s i d e r a d v i s a b l e , f o r t h e m o m e n t , t o p u b l i s h s o m e t h i n g a b o u t t h e s e i d e a s a n d f i r s t I a p p l y t o w i t h c o n f i d e n c e , d e a r R a d i o a c t i v e s , w i t h t h e q u e s t i o n : w h a t d o y o u t h i n k a b o u t t h e p o s s i b i l i t y o f p r o v i d i n g t h e e x p e r i m e n t a l p r o o f o f s u c h a n e u t r o n , i f i t w o u l d p o s s e s s a p e n e t r a t i n g p o w e r e q u a l o r t e n t i m e s g r e a t e r o f t h a t o f g a m m a r a y s ? I a d m i t t h a t m y s o l u t i o n m a y a p p e a r t o y o u n o t v e r y p r o b a b l e , b e c a u s e i t t h e n e u t r o n w o u l d e x i s t , t h e y w o u l d h a v e b e e n o b s e r v e d l o n g s i n c e . B u t o n l y w h o d a r e s w i n s , a n d t h e g r a v i t y o f t h e s i t u a t i o n i n r e g a r d t o t h e c o n t i n u o u s ? s p e c t r u m i s e n l i g h t e n e d b y t h e o p i n i o n o f m y p r e d e c e s s o r i n t h e c h a i r M r . D e b y e , w h o l o n g s i n c e t o l d m e i n B r u s s e l s : ‘ O h , t h e b e s t t h i n g t o d o i s n o t t o t a l k a b o u t , l i k e f o r n e w t a x e s ’ . F o r t h i s r e a s o n o n e s h o u l d c o n s i d e r s e r i o u s l y a n y w a y t o w a r d s s a f e t y . T h u s , d e a r R a d i o a c t i v e s , c o n s i d e r a n d j u d g e . U n f o r t u n a t e l y I c a n n o t c o m e p e r s o n a l l y t o T u b i n g e n , b e c a u s e I a m n e c e s s a r y h e r e f o r a b a l l t h a t w i l l t a k e p l a c e i n Z u r i c h t h e n i g h t f r o m 6 t o 7 D e c e m b e r . W i t h m a n y g r e e t i n g s t o y o u a s w e l l a s t o M r . B a c k . Y o u r d e v o t e d s e r v a n t , W . P a u l i
“I have done something very bad today by proposing a particle that cannot be detected; it is something no theorist should ever do.”
W. Pauli
Neutrino – a daughter of symmetry
A
A’
e
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1956: A Fateful Year. For Neutrinos Too.
• 1956, Savannah River, Reines and Cowen:”We are happy to inform you (Pauli) that we have definitely detected neutrinos…”
• A downfall of parity: T.D. Lee, C.N. Yang. • Two –component neutrino theory: a neutrino has no mirror image.
(A vampire-neutrino?) Massless neutrino.• V-A theory of weak interactions (Feynmann, Gell-Mann)
![Page 18: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/18.jpg)
• A search for new symmetries: Sakata/Nagoya/Nagoya-Kyoto model
• Symmetry of strong interactions (SU(3)?) – Baryons are bound states of 3 ‘fundamental’ baryons/ur-baryons: p,n,– Mesons are baryon-antibaryon bound states
• Leptons-baryons connection/symmetry: baryons are bound states of new field B+ and leptons: p = < B+ >, n = < eB+ >, p = < B+ >
At the Same Time in Japan…
p
e
n
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1962: Lederman, Schwartz, Steinberger
• Two different kinds of neutrinos! If Sakata symmetry holds – there must be a new heavy baryon X
• Prediction of a new heavy baryon (==charm!). Niu 1971 – discovery of charm particle (m(C)=1.78 GeV)
• Maki-Nakagawa-Sakata: two neutrinos should, in general, mix. MNS neutrino mixing matrix!
• Neutrinos a guide in postulating/establishing symmetries of Nature, predicting new particles.
X
e
n p
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Conception of the Standard Model
• Glashow, Weinberg, Salam + many others: local SU(2)xU(1) gauge symmetry (local redefinition of up-down members of the weak doublets) as a possible model for unified electromagnetic and weak interactions
• Spontaneous symmetry breaking (Higgs) as a way to avoid problems with massless bosons/long range
• Predictions: – weak neutral current interactions– Intermediate vector bosons, W/Z– Mass of IVB ~ 80 GeV
• Free parameter: weak mixing angle
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1973: Golden Event (Gargamelle)• Heavy liquid bubble chamber exposed to a medium energy beam of muon antineutrinos:
• a single energetic electron appearing in the middle of the fiducial volume• the only plausible interpretation
+ e- → + e- • Existence of neutral currents established. • Explanation of previously reported ‘unexplained background events’.
•Later, in 1984, SPS collider: existence of W± and Z0 established. Neutrinos (a.k.a. missing energy) an important signature
Neutrino (beam) a tool to discover new interactions.Neutrino a signature for weak decays of new particles.
![Page 22: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/22.jpg)
1970-ies, Quantum Chromodynamics
• The success of the gauge symmetry approach for weak/electromagnetic interactions
• Spectacular success of QED• Asymptotic freedom (Gross, Politzer, Wilczek)
Quantum Chromodynamics: Strong interactions related to a local symmetry of SU(3) color phase
Strong interactions: neutrinos not involved (?)
![Page 23: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/23.jpg)
Probing Nucleon with Neutrinos
q p p
2 2 2
22
q
q
m x P
m xP q
2 2
02 2 N
q qx
Pq M q
Probe momentum distribution of partons inside the nucleon
Neutrino scatters off a parton inside the nucleon
![Page 24: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/24.jpg)
Strong Interactions of Partons
2
122
2
2
( )
log
( , )
)
,
2
( ,
( )
s
x
qq qg
Q dy
x xP
q x Q
q y g y Q
Q
y yQ P
y
•Pqq(x/y) = probability of finding a quark with momentum x within a quark with momentum y
•Pgq(x/y) = probability of finding a q with momentum x within a gluon with momentum y
2
22
4 1( ) 2 (1 )
3 (1 )
1( ) 1
2
gq
zP z z
z
P z z z
![Page 25: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/25.jpg)
Establishing the QCD
Observed quark distributions vary with Q2
In a quantitative agreement with the QCD predictions
Neutrinos: a tool to establish a theory of strong interactions/ local gauge SU(3) symmetry
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Keep Unifying?
• Given the success of electroweak unification: do all forces become one at some high energies? And is this unified force a consequence of a local gauge symmetry? Try :
• A single coupling constant , g5, for all interactions/vector bosons
(5) (3) (2) (1)SU SU SU U
5( ) D m i g A m
24 8 20 23
241 1 9 21
1 1
2 2A A G X A B
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Electroweak sector in SU(5)
2
gD i gTA YB
5 5
3
5g g g g
22
2 2
3sin
8W
g
g g
with
hence At the GUT energy scale!
22
2 2 2
1 1ln
( ) ( ) 4i
i ii i
gqb
q M M
But the coupling constants ‘run’ with energy
2
2 2
2
e
2
xp
22
2
sin (
3 55sin ( ) ( ) ln
8
) 0.2
sin 0.2325 0.0008
06
48
W
W
W
W
M
qq q
M Neutrino experiments a severe test for putative Grand Unifications. exclude SU(5) as a GUT
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Families
• Who ordered them?? • Perhaps nobody, but here they are, inviting a number of
interesting questions,or trying to tell us something– What is the origin of quark-lepton relation/symmetry?
(Anomalies cancellation: qi=0)– Which quark families relate to which lepton families?
(u,d,e,e? or perhaps u,d,? Or perhaps ? Re: proton decay/stability)
– How many familes?
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How Many Families?
2 2
12( ) e fpeak
e e fZ Z
fs
M
3Z had l N Neutrinos: families counter
![Page 30: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/30.jpg)
Are Neutrinos Different?
>45 years ago…A. n -> p + e- + B. - ->
Question: are the neutrinos produced in A and B the same? Or different? How can they be different???
Answer: Lederman, Schwartz, Steinberger (Nobel Prize 1987)
+
- only, never e-
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Three Kinds of Neutrinos
X
Y
e+,
e,
neutrino born in conjunction with electron, muon, tau is called an electron, muon, tau neutrino.
When it interacts it will produce an electron, muon, tau.
Family lepton number conservation Le,L,L↔ some new underlying symmetry??
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Massive Neutrinos Revolution
X
Y
e+
e1 1 2 2 3 3e e e e
U U U
• Electron (muon,tau) neutrino is not a mass eigenstate
• Electron (muon, tau) neutrino is a coherent mixture of mass eigenstates
1 2 3
1 2
1
2
3
2
1 2 3
e
e
e eU U U
U U U
U U U
![Page 33: Reflections on Symmetries and Neutrinos Adam Para.](https://reader033.fdocuments.us/reader033/viewer/2022051516/56649ea05503460f94ba38da/html5/thumbnails/33.jpg)
Neutrinos Oscillations
i
Amplitude
Amplitude
2
* 2imi
i i
LE
i
U Ue
A
Components of the initial state have different time evolution => (t) ≠ (0)
3-slit interference Experiment: mass difference difference in optical path length
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Neutrino Oscillations: Lessons and New Questions
• No family lepton numbers conservation/symmetry• Neutrinos have mass: right handed neutrinos exist, after all• Downfall of two-component neutrino theory?New questions • Do neutrinos violate CP? Origin of leptogenesis and baryon
number asymmetry of the Universe and our own existemce? Note: CP violation possibly responsible for leptogenesis has nothing to do with the CP measured in oscillation experiments.
• Is there a lepton number at all? Or are neutrinos Majorana particles?
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35
Parameterization of Mixing Matrix
1
2
13 13
13
12
23 23
23
13
12
1
2
2
3
12
1
0 0
0 0
0
0 1 0
0
0
0 0 1
0 0
0
0
0
0 0 1
i
i
i
i
c
c s
s c
s e
s e c
e
U
e
c s
s c
•Three mixing angles (like Euler rotation angles
•One complex phase (CP violation)
•Two Majorana phases
1311 12 1
21 22 23 2
31 32 31 3
1
2
3
e
s
UU U
U U U
U U U
B B
B B B
B B B
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Surprising Pattern of Mixing Angles: New Symmetries of Nature?
• Where do mixing angles come from?• Why are they so different (even pattern-wise) from quark
mixing angles• sin2223 very close to 1. Is it = 1.00? Maximal mixing
some new symmetry??• Sin2213 small. How small? If tiny, or zero, as opposed to the
other mixing angles – why?? Protected by some new symmetry??
Neutrinos as indicators of new symmetries?
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Are There any Discrete Symmetries Left?
• P, C is violated (maximally) in weak interactions• CP is violated, but• Lorentz invariance + local quantum field theory → CPT invariance• Wait… local quantum field theory?? Aren’t we made of strings? Non-local!• Suppose there is a weak violation (at our energy/distance scale) of CPT
communicated to all Standard Model particles. It must be extremely small ~ 10-14 (from Ko
L-Kos mass difference)
• But… perhaps the source of the CPT violation is located in the ‘bulk’. The only particle which would be subject to CPT violation is the right-handed neutrino!
• CPT violation in the neutrino sector? – Do neutrinos/antineutrinos oscillate in the same way?? – The same masses? – The same mixing angles? Limits rather poor so far, but be on lookout
(MINOS!)
Neutrinos messengers of CPT violations?
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(Kind of) Summary
• Symmetries (especially continuous ones) play a very important role in our understanding of our world
• We may not know the complete list of symmetries, yet
• Neutrinos are a surprisingly powerful tool for learning about symmetries of the Nature