# Presentation: 002159

## Reflections onSymmetries and Neutrinos

- Adam Para

## Revolutionary Developments in XX Century Physics

- Special Relativity
- General Relativity
- Quantum Mechanics
- Quantum Field Theory
- Standard Model of Elementary Particles and their Interactions
- Role of symmetry in 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

## 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

## 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 ?

- 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

- Profound consequences:
- Maxwell equations
- Massless photons
- Interactions 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

- 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?

- “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)

## At the Same Time in Japan...

- 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+ >

## 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.

## 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

- 

- 

- Probe momentum distribution of partons inside the nucleon

- Neutrino scatters off a parton inside the nucleon

## Strong Interactions of Partons

- 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

## 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

## Electroweak sector in SU(5)

- with

- hen**c**e

- At the GUT energy scale!

- But the coupling constants ‘run’ with energy

- 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? 

- Neutrinos: families counter

## Are Neutrinos Different? 

- >45 years ago...
- n -> p + e- + 
- - -> 
- 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+

- e

-  Electron (muon,tau) neutrino is not a mass eigenstate
-  Electron (muon, tau) neutrino is a coherent mixture of mass eigenstates

## Neutrinos Oscillations

- Amplitude

- Amplitude

- 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?

## Parameterization of Mixing Matrix

- Three mixing angles (like Euler rotation angles
- One complex phase (CP violation)
- Two Majorana phases

## 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
- sin2223 very close to 1. Is it = 1.00? Maximal mixing  some new symmetry??
- Sin2213 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 KoL-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