Reflections onSymmetries and Neutrinos

Adam Para

• 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

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 ???]

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’

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• . . .

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

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)

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

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

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

• 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

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

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…

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

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’

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

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)

• 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

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

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

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.

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 (?)

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

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

qq

gq

zP z z

z

P z z z

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

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

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

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?

How Many Families?

2 2

12( ) e fpeak

e e fZ Z

fs

M

3Z had l N Neutrinos: families counter

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-

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??

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

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

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?

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

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?

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?

(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