1. PHYS 3446 – Lecture #20 Elementary Particle Properties
Monday, Nov. 14, 2016
Dr. Jaehoon Yu
Elementary Particle Properties
Forces and their relative magnitudes
Elementary particles
Quantum Numbers
Strangeness
Isospin
Gell-Mann-Nishijima Relations
Production and Decay of Resonances
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

2. Announcements Reading assignments: 9.6 and 9.7 Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

3. Reminder: Homework #9 End of chapter problems 9.1, 9.2 and 9.3
Due for these assignments is Monday, Nov. 14
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

4. Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

5. Elementary Particles
Before the quark concepts, all known elementary particles were grouped in four depending on the nature of their interactions
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

6. Elementary Particles How do these particles interact??
All particles, including photons and neutrinos, participate in gravitational interactions
Photons can interact electromagnetically with any particles with electric charge
All charged leptons participate in both EM and weak interactions
Neutral leptons do not have EM couplings
All hadrons (Mesons and baryons) responds to the strong force and appears to participate in all the interactions
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

7. Elementary Particles: Bosons and Fermions
All particles can be classified as bosons or fermions
Bosons follow Bose-Einstein statistics
Quantum mechanical wave function is symmetric under exchange of any pair of bosons
xi: space-time coordinates and internal quantum numbers of particle i
Fermions obey Fermi-Dirac statistics
Quantum mechanical wave function is anti-symmetric under exchange of any pair of Fermions
Pauli exclusion principle is built into the wave function
For xi=xj,
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

8. Bosons, Fermions, Particles and Antiparticles
All have integer spin angular momentum
All mesons are bosons
Fermions
All have half integer spin angular momentum
All leptons and baryons are fermions
All particles have anti-particles
What are anti-particles?
Particles that have the same masses as particles but with opposite charges and quantum numbers
What is the anti-particle of
A p0?
A neutron?
A K0?
A Neutrinos?
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

9. Quantum Numbers When can an interaction occur?
If it is kinematically allowed
If it does not violate any recognized conservation laws
Eg. A reaction that violates charge conservation will not occur
In order to deduce conservation laws, a full theoretical understanding of forces are necessary
Since we do not have full theory for all the forces
Many of general conservation rules for particles are based on experiments
One of the clearest conservation is the lepton number conservation
While photon and meson numbers are not conserved
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

10. Baryon Numbers Can the decay occur?
Kinematically??
Yes, proton mass is a lot larger than the sum of the two masses
Electrical charge?
Yes, it is conserved
But this decay does not occur (<10-40/sec)
Why?
Must be a conservation law that prohibits this decay
What could it be?
An additive and conserved quantum number, Baryon number (B)
All baryons have B=1
Anti-baryons? (B=-1)
Photons, leptons and mesons have B=0
Since proton is the lightest baryon, it does not decay.
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

11. Lepton Numbers Quantum number of leptons
All leptons carry L=1 (particles) or L=-1 (antiparticles)
Photons or hadrons carry L=0
Lepton number is a conserved quantity
Total lepton number must be conserved
Lepton numbers by species must be conserved
This is an empirical law necessitated by experimental observation (or lack thereof)
Consider the decay
Does this decay process conserve energy and charge?
Yes
But it hasn’t been observed, why?
Due to the lepton number conservation
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

12. Lepton Number Assignments
Leptons (anti-leptons) Le Lm Lt
L=Le+Lm+Lt
e- (e+)
1 (-1)
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

13. Lepton Number Conservation
Can the following decays occur?
Case 1: L is conserved but Le and Lm not conserved
Case 2: L is conserved but Le and Lm not conserved
Case 3: L is conserved, and Le and Lm are also conserved
Decays Le Lm Lt
L=Le+Lm+Lt
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

14. Quantum Numbers Baryon Number Lepton Number
An additive and conserved quantum number, Baryon number (B)
All baryons have B=1
Anti-baryons? (B=-1)
Photons, leptons and mesons have B=0
Lepton Number
Quantum number assigned to leptons
All leptons carry L=1 (particles) or L=-1 (antiparticles)
Photons or hadrons carry L=0
Total lepton number must be conserved
Lepton numbers by species must be conserved
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

15. Strangeness From cosmic ray shower observations Consider the reaction
K-mesons and S and L0 baryons are produced strongly w/ large x-sec
But their lifetime typical of weak interactions (~10-10 sec)
Are produced in pairs – a K w/ a S or a K w/ a L0
Gave an indication of a new quantum number
Consider the reaction
K0 and L0 subsequently decay
and
Observations on L0
Always produced w/ K0 never w/ just a p0
Produced w/ K+ but not w/ K-
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

16. Strangeness Consider reactions and Observations from S+
With the subsequent decays and
Observations from S+
S+ is always produced w/ a K+ never w/ just a p+
S+ is also produced w/ a K0 but w/ an additional p+ for charge conservation
Observations from S-
S- is always produced w/ a K+ never w/ K-
Thus,
Observed:
Not observed:
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

17. Strangeness Further observation of cross section measurements
The cross section for reactions and w/ 1GeV/c pion momenta are ~ 1mb
Whereas the total pion-proton scattering cross section is ~ 30mb
The interactions are strong interactions
L0 at v~0.1c decays in about 0.3cm
Lifetime of L0 baryon is
The short lifetime of these strange particles indicate weak decay
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

18. Strangeness Strangeness quantum number
Murray Gell-Mann and Abraham Pais proposed a new additive quantum number that are carried by these particles
Conserved in strong interactions
Violated in weak decays
S=0 for all ordinary mesons and baryons as well as photons and leptons
For any strong associated-production reaction w/ the initial state S=0, the total strangeness of particles in the final state should add up to 0.
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

19. Strangeness
Based on experimental observations of reactions and w/ an arbitrary choice of S(K0)=1, we obtain
S(K+)=S(K0)=1 and S(K-)=S(`K0)=-1
S(L0)=S(S+)=S(S0)=S(S-)=-1
Does this work for the following reactions?
For strong production reactions and
cascade particles if
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

20. More on Strangeness Let’s look at the reactions again
This is a strong interaction
Strangeness must be conserved
S:  +1 -1
How about the decays of the final state particles?
and
These decays are weak interactions so S is not conserved
S: -1  and  0 + 0
A not-really-elegant solution
S only conserved in Strong and EM interactions  Unique strangeness quantum numbers cannot be assigned to leptons
Leads into the necessity of strange quarks
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

21. Isospin Quantum Number
Strong force does not depend on the charge of the particle
Nuclear properties of protons and neutrons are very similar
From the studies of mirror nuclei, the strengths of p-p, p-n and n-n strong interactions are essentially the same
If corrected by EM interactions, the x-sec between n-n and p-p are the same
Since strong force is much stronger than any other forces, we could imagine a new quantum number that applies to all particles
Protons and neutrons are two orthogonal mass eigenstates of the same particle like spin up and down states
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

22. Isospin Quantum Number
Protons and neutrons are degenerate in mass because of some symmetry of the strong force
Isospin symmetry  Under the strong force these two particles appear identical
Presence of Electromagnetic or Weak forces breaks this symmetry, distinguishing p from n
Isospin works just like spins
Protons and neutrons have isospin ½  Isospin doublet
Three pions, p+, p- and p0, have almost the same masses
X-sec by these particles are almost the same after correcting for EM effects
Strong force does not distinguish these particles  Isospin triplet
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

23. Isospin Quantum Number
This QN is found to be conserved in strong interactions
But not conserved in EM or Weak interactions
Third component of the isospin QN is assigned to be positive for the particles with larger electric charge
Isospin is not a space-time symmetry
Cannot be assigned uniquely to leptons and photons since they are not involved in strong interactions
There is something called weak-isospin for weak interactions
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

24. Gell-Mann-Nishijima Relation
Strangeness assignment is based on Gell-Mann-Nishijima relation
Electric charge of a hadron can be related to its other quantum numbers
Where
Q: hadron electric charge
I3: third component of isospin
Y=B+S, strong hypercharge
Quantum numbers of several long lived particles follow this rule
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

25. Gell-Mann-Nishijima Relation
With the discovery of new flavor quantum numbers, charm and bottom, this relationship was modified to include these new additions (Y=B+S+C+B)
Since the charge and the isospin are conserved in strong interactions, the strong hypercharge, Y, is also conserved in strong interactions
This relationship holds in all strong interactions
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

26. Quantum numbers for a few hadrons
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

27. Violation of Quantum Numbers
The QN we learned are conserved in strong interactions are but many of them are violated in EM or weak interactions
Three types of weak interactions
Hadronic decays: Only hadrons in the final state
Semi-leptonic decays: both hadrons and leptons are present
Leptonic decays: only leptons are present
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

28. Hadronic Weak Decays |D| 1/2 1 1/2 1 1/2 1 1/2 1
These decays follow selection rules: |DI3|=1/2 and |DS|=1
|D| QN
L0  p- p I3 -1 S
1/2 1 QN
S+  p0 p I3 1 S -1
1/2 1 QN
K0  p+ p- I3
- ½ 1 -1 S
1/2 1 QN
X-  L0 p- I3
- ½ -1 S -2
1/2 1
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

29. Semi-leptonic Weak Decays
These decays follow selection rules: |DI3|=1 and |DS|=0 or |DI3|= ½ and |DS|=1
|D| QN n p
e-+`ne I3
-1/2
1/2 S 1 QN
p-  m-
`nm I3 -1 S 1 QN
K+  p0
m++nm I3 S 1
1/2 1 QN
S-  n
e-+`ne I3 -1
-1/2 S
1/2 1
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

30. Summary of Weak Decays Hadronic weak-decays Semi-leptonic weak-decays
Selection rules are |DI3|=1/2 and |DS|=1
|DI3|=3/2 and |DS|=2 exists but heavily suppressed
Semi-leptonic weak-decays
Type 1: Strangeness conserving
Selection rules are: |DS|=0, |DI3|=1 and |DI|=1
Type 2: Strangeness non-conserving
Selection rules are: |DS|=1, |DI3|= ½ and |DI|= ½ or 3/2
DI=3/2 and |DS|=1 exist but heavily suppressed
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

31. EM Processes |D| Strangeness is conserved but the total isospin is notQN
p0  g I3 S QN
h0  g I3 S QN
S0  L0 g I3 S -1
Strangeness is conserved but the total isospin is not
Selection rules are: |DS|=0, |DI3|=0 and DI= 1or 0
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

32. Quantum Numbers Baryon Number Lepton Number Strangeness Numbers
An additive and conserved quantum number, Baryon number (B)
This number is conserved in general but not absolute
Lepton Number
Quantum number assigned to leptons
Lepton numbers by species and the total lepton numbers must be conserved
Strangeness Numbers
Conserved in strong interactions
But violated in weak interactions
Isospin Quantum Numbers
But violated in weak and EM interactions
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

33. Quantum Numbers
We’ve learned about various newly introduced quantum numbers as a patch work to explain experimental observations
Lepton numbers
Baryon numbers
Isospin
Strangeness
Some of these numbers are conserved in certain situation but not in others
Very frustrating indeed….
These are due to lack of quantitative description by an elegant theory
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

34. Why symmetry?
Some quantum numbers are conserved in strong interactions but not in electromagnetic and weak interactions
Inherent reflection of underlying forces
Understanding conservation or violation of quantum numbers in certain situations is important for formulating quantitative theoretical framework
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

35. Why symmetry? When is a quantum number conserved?
When there is an underlying symmetry in the system
When the quantum number is not affected (or is conserved) by (under) the changes in the physical system
Noether’s theorem: If there is a conserved quantity associated with a physical system, there exists an underlying invariance or symmetry principle responsible for this conservation.
Symmetries provide critical restrictions in formulating theories
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

36. Symmetries in Lagrangian Formalism
Symmetry of a system is defined by any set of transformations that keep the equation of motion unchanged or invariant
Equations of motion can be obtained through
Lagrangian formalism: L=T-V where the Equation of motion is what minimizes the Lagrangian L under changes of coordinates
Hamiltonian formalism: H=T+V with the equation of motion that minimizes the Hamiltonian under changes of coordinates
Both these formalisms can be used to discuss symmetries in non-relativistic (or classical cases) or relativistic cases and quantum mechanical systems
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

37. Symmetries in Lagrangian Formalism?
Consider an isolated non-relativistic physical system of two particles interacting through a potential that only depends on the relative distance between them
EM and gravitational force
The total kinetic and potential energies of the system are: and
The equations of motion are then
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

38. Symmetries in Lagrangian Formalism
If we perform a linear translation of the origin of coordinate system by a constant vector
The position vectors of the two particles become
But the equations of motion do not change since is a constant vector
This is due to the invariance of the potential V under the translation
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

39. Symmetries in Lagrangian Formalism?
This means that the translation of the coordinate system for an isolated two particle system defines a symmetry of the system (remember Noether’s theorem?)
This particular physical system is invariant under spatial translation
What is the consequence of this invariance?
From the form of the potential, the total force is
Since
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

40. Symmetries in Lagrangian Formalism?
What does this mean?
Total momentum of the system is invariant under spatial translation
In other words, the translational symmetry results in linear momentum conservation
This holds for multi-particle system as well
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

41. Symmetries in Lagrangian Formalism
For multi-particle system, using Lagrangian L=T-V, the equations of motion can be generalized
By construction,
As previously discussed, for the system with a potential that depends on the relative distance between particles, The Lagrangian is independent of particulars of the individual coordinate and thus
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

42. Symmetries in Lagrangian Formalism
Momentum pi can expanded to other kind of momenta for the given spatial translation
Rotational translation: Angular momentum
Time translation: Energy
Rotation in isospin space: Isospin
The equation says that if the Lagrangian of a physical system does not depend on specifics of a given coordinate, the conjugate momentum is conserved
One can turn this around and state that if a Lagrangian does not depend on some particular coordinate, it must be invariant under translations of this coordinate.
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

43. Translational Symmetries & Conserved Quantities
The translational symmetries of a physical system give invariance in the corresponding physical quantities
Symmetry under linear translation
Linear momentum conservation
Symmetry under spatial rotation
Angular momentum conservation
Symmetry under time translation
Energy conservation
Symmetry under isospin space rotation
Isospin conservation
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

44. Symmetries in Quantum Mechanics
In quantum mechanics, an observable physical quantity corresponds to the expectation value of the Hermitian operator in a given quantum state
The expectation value is given as a product of wave function vectors about the physical quantity (operator)
Wave function ( )is the probability distribution function of a quantum state at any given space-time coordinates
The observable is invariant or conserved if the operator Q commutes with Hamiltonian
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

45. Types of Symmetry
All symmetry transformations of the theory can be categorized in
Continuous symmetry: Symmetry under continuous transformation
Spatial translation
Time translation
Rotation
Discrete symmetry: Symmetry under discrete transformation
Transformation in discrete quantum mechanical system
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

46. Isospin
If there is isospin symmetry, proton (isospin up, I3= ½) and neutron (isospin down, I3= -½) are indistinguishable
Let’s define new neutron and proton states as some linear combination of the proton, , and neutron, , wave functions
Then the finite rotation of the vectors in isospin space by an arbitrary angle q/2 about an isospin axis leads to a new set of transformed vectors
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

47. Isospin
What does the isospin invariance mean to nucleon-nucleon interaction?
Two nucleon quantum states can be written in the following four combinations of quantum states
Proton on proton (I3=+1)
Neutron on neutron (I3=-1)
Proton on neutron or neutron on proton for both symmetric or anti-symmetiric (I3=0)
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

48. Impact of Isospin Transformation
For I3=+1 wave function w/ isospin transformation:
Can you do the same for the other two wave functions of I=1?
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

49. Isospin Tranformation
For I3=0 anti-symmetric wave function
This state is totally insensitive to isospin rotation singlet combination of isospins (total isospin 0 state)
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016

50. Isospin Tranformation
The other three states corresponds to three possible projection state of the total isospin =1 state (triplet state)
If there is an isospin symmetry in strong interaction all these three substates are equivalent and indistinguishable
Based on this, we learn that any two nucleon system can be in an independent singlet or triplet state
Singlet state is anti-symmetric under n-p exchange
Triplet state is symmetric under n-p exchange
Monday, Nov. 14, 2016
PHYS 3446, Fall 2016