1 Invariant Principles and Conservation laws Kihyeon Cho April 26, 2011 HEP

2 Contents Symmetry Parity (P) Gauge invariance Charge (C) CP Violation

3 Line Symmetry Shape has line symmetry when one half of it is the mirror image of the other half. Symmetry exists all around us and many people see it as being a thing of beauty.

4 Is a butterfly symmetrical?

5 Line Symmetry exists in nature but you may not have noticed.

6 At the beach there are a variety of shells with line symmetry.

7 Under the sea there are also many symmetrical objects such as these crabs and this starfish.

8 Animals that have Line Symmetry Here are a few more great examples of mirror image in the animal kingdom.

9 THESE MASKS HAVE SYMMETRY These masks have a line of symmetry from the forehead to the chin. The human face also has a line of symmetry in the same place.

10 Human Symmetry The 'Proportions of Man' is a famous work of art by Leonardo da Vinci that shows the symmetry of the human form.

11 REFLECTION IN WATER

12 The Taj Mahal Symmetry exists in architecture all around the world. The best known example of this is the Taj Mahal.

13 This photograph shows 2 lines of symmetry. One vertical, the other along the waterline. (Notice how the prayer towers, called minarets, are reflected in the water and side to side).

14 2D Shapes and Symmetry After investigating the following shapes by cutting and folding, we found:

15 an equilateral triangle has 3 internal angles and 3 lines of symmetry.

16 a square has 4 internal angles and 4 lines of symmetry.

17 a regular pentagon has 5 internal angles and 5 lines of symmetry.

18 a regular hexagon has 6 internal angles and 6 lines of symmetry.

19 a regular octagon has 8 internal angles and 8 lines of symmetry.

20

21 Conservation Rules Conserved Quantity WeakElectromagneticStrong I(Isospin)No (  I=1 or ½) NoYes (No in 1996) S(Strangeness)No (  S=1,0) Yes C(charm)No (  C=1,0) Yes P(parity)NoYes C(charge)NoYes CPNoYes

22 Conserved Quantities and Symmetries Every conservation law corresponds to an invariance of the Hamiltonian (or Lagrangian) of the system under some transformation. We call these invariances symmetries. There are 2 types of transformations: continuous and discontinuous Continuous  give additive conservation laws x  x+dx or   +d  examples of conserved quantities: electric charge momentum baryon # Discontinuous  give multiplicative conservation laws parity transformation: x, y, z  (-x), (-y), (-z) charge conjugation (particle  antiparticle): e -  e + examples of conserved quantities: parity (in strong and EM) charge conjugation (in strong and EM) parity and charge conjugation (strong, EM, almost always in weak)

23 Conserved Quantities and Symmetries Example of classical mechanics and momentum conservation. In general a system can be described by the following Hamiltonian: H=H(p i,q i,t) with p i =momentum coordinate, q i =space coordinate, t=time Consider the variation of H due to a translation q i only. For our example dp i =dt=0 so we have: Using Hamilton’s canonical equations: We can rewrite dH as: If H is invariant under a translation (dq) then by definition we must have: This can only be true if: Thus each p component is constant in time and momentum is conserved. Read Perkins: Chapters 3.1

24 Conserved Quantities and Quantum Mechanics In quantum mechanics quantities whose operators commute with the Hamiltonian are conserved. Recall: the expectation value of an operator Q is: How does change with time? Recall Schrodinger’s equation: Substituting the Schrodinger equation into the time derivative of Q gives: H + = H *T = hermitian conjugate of H Since H is hermitian ( H + = H ) we can rewrite the above as: So then is conserved. Read Perkins: Chapters 3.1

25 Three Important Discrete Symmetries Parity, P –Parity reflects a system through the origin. Converts right-handed coordinate systems to left-handed ones. –Vectors change sign but axial vectors remain unchanged x   xL  L Charge Conjugation, C –Charge conjugation turns a particle into its anti-particle e   e  K   K   Time Reversal, T –Changes the direction of motion of particles in time t  t CPT theorem –One of the most important and generally valid theorems in quantum field theory. –All interactions are invariant under combined C, P and T transformations. –Implies particle and anti-particle have equal masses and lifetimes  

Discrete Symmetries An example of a discrete transformation is the operation of inverting all angles:   -  In contrast a rotation by an amount  is a continuous transformation. Reminder: Discrete symmetries give multiplicative quantum numbers. Continuous symmetries give additive quantum numbers. The three most important discrete symmetries are: Parity (P)(x,y,z)  (-x,-y,-z) Charge Conjugation (C)particles  anti-particles Time Reversal (T)time  -time Other not so common discrete symmetries include G parity: G=CR=C exp (i π I 2 ) = (-1) l+S+I G parity is important for pions under the strong interaction. Note: discrete transformations do not have to be unitary transformations ! P and C are unitary transformations T is not a unitary transformation, T is an antiunitary operator! Read Perkins: Chapters 3.3, 3.4 Read Perkins: Chapters 4.5.1

27 Read Perkins: Chapters 3.3, 3.4

28 Parity Quantum Number

Parity Let us examine the parity operator (P) and its eigenvalues. The parity operator acting on a wavefunction is defined by: P  (x, y, z) =  (-x, -y, -z) P 2  (x, y, z) = P  (-x, -y, -z) =  (x, y, z) Therefore P 2 = I and the parity operator is unitary. If the interaction Hamiltonian (H) conserves parity then [H,P]=0, and: P  (x, y, z) =  (-x, -y, -z) = n  (x, y, z) with n = eigenvalue of P P 2  (x, y, z) = PP  (x, y, z) = nP  (x, y, z) = n 2  (x, y, z)  (x, y, z) = n 2  (x, y, z)  n 2 =  1 so, n=1 or n=-1. The quantum number n is called the intrinsic parity of a particle. If n= 1 the particle has even parity. If n= -1 the particle has odd parity. In addition, if the overall wavefunction of a particle (or system of particles) contains spherical harmonics (Y L m ) then we must take this into account to get the total parity of the particle (or system of particles). The parity of Y L m is: PY L m = (-1) L Y L m. For a wavefunction  (r, ,  )=R(r)Y L m ( ,  ) the eigenvalues of the parity operator are: P  (r, ,  )=PR(r)Y L m ( ,  ) = (-1) L R(r)Y L m ( ,  ) The parity of the particle would then be: n(-1) L Note: Parity is a multiplicative quantum number M&S pages Read Perkins: Chapters 3.3, 3.4

Parity The parity of a state consisting of particles a and b is: (-1) L n a n b where L is their relative orbital momentum and n a and n b are the intrinsic parity of each of the two particle. Note: strictly speaking parity is only defined in the system where the total momentum (p) =0 since the parity operator (P) and momentum operator anticommute, (Pp=-p). How do we know the parity of a particle ? By convention we assign positive intrinsic parity (+) to spin 1/2 fermions: +parity: proton, neutron, electron, muon (  - ) Anti-fermions have opposite intrinsic parity -parity: anti-proton, anti-neutron, positron, anti-muon (  + ) Bosons and their anti-particles have the same intrinsic parity. What about the photon? Strictly speaking, we can not assign a parity to the photon since it is never at rest. By convention the parity of the photon is given by the radiation field involved: electric dipole transitions have + parity magnetic dipole transitions have - parity We determine the parity of other particles ( , K..) using the above conventions and assuming parity is conserved in the strong and electromagnetic interaction. Usually we need to resort to experiment to determine the parity of a particle. Read Perkins: Chapters 3.3, 3.4

Parity Example: determination of the parity of the  using  - d  nn. For this reaction we know many things: a) s  =0, s n =1/2, s d =1, orbital angular momentum L d =0, => J d =L d +s d =0+1=1 b) We know (from experiment) that the  is captured by the d in an s-wave state. Thus the total angular momentum of the initial state is just that of the d (J=1). c) The isospin of the nn system is 1 since d is an isosinglet and the  - has I=|1,-1> note: a |1,-1> is symmetric under the interchange of particles. (see below) d) The final state contains two identical fermions and therefore by the Pauli Principle the wavefunction must be anti-symmetric under the exchange of the two neutrons. Let’s use these facts to pin down the intrinsic parity of the . i) Assume the total spin of the nn system =0. Then the spin part of the wavefunction is anti-symmetric: |0,0> = (2) -1/2 [|1/2,1/2>|1/2-1/2>-|1/2,-1/2>|1/2,1/2>] To get a totally anti-symmetric wavefunction L must be even (0,2,4…) Cannot conserve momentum (J=1) with these conditions! ( since J=L+s => 1  0+0,2,) ii) Assume the total spin of the nn system =1. Then the spin part of the wavefunction is symmetric: |1,1> = |1/2,1/2>|1/2,1/2> |1,0> = (2) -1/2 [|1/2,1/2>|1/2-1/2>+|1/2,-1/2>|1/2,1/2>] |1,-1> = |1/2,-1/2>|1/2,-1/2> ( since J=L+s => 1=1+1) To get a totally anti-symmetric wavefunction L must be odd (1, 3, 5…) L=1 consistent with angular momentum conservation: nn has s=1, L=1, J=1  3 P 1 The parity of the final state is: n n n n (-1) L = (+)(+)(-1) 1 = - The parity of the initial state is: n  n d (-1) L = n  (+)(-1) 0 = n  Parity conservation gives: n n n n (-1) L = n  n d (-1) L  n   = - Read Perkins: Chapters 3.3.1

Parity There is other experimental evidence that the parity of the  is -: the reaction  - d  nn  0 is not observed the polarization of  ’s from  0  Some use “spin-parity” buzz words: buzzwordspinparityparticle pseudoscalar 0 - , k scalar 0 +higgs (none observed) vector 1 -  pseudovector 1 +A1 (axial vector) How well is parity conserved? Very well in strong and electromagnetic interactions ( ) not at all in the weak interaction! The  puzzle and the downfall of parity in the weak interaction In the mid-1950’s it was noticed that there were 2 charged particles that had (experimentally) consistent masses, lifetimes and spin = 0, but very different weak decay modes:  +  +  0  +  +  -  + The parity of  +  = + while the parity of  +  = - Some physicists said the  +  and  + were different particles, and parity was conserved. Lee and Yang said they were the same particle but parity was not conserved in weak interaction! Lee and Yang win Nobel Prize when parity violation was discovered. Note:  +  + is now known as the K +. M&S pages Read Perkins: Chapters 3.3, 3.4

Discrete Symmetries, Parity Parity and nature: The strong and electromagnetic interactions conserve parity. The weak interaction does not. Thus if we consider a Hamiltonian to be made up of several pieces: H = H s + H EM + H W Then the parity operator (P) commutes with H s and H EM but not with H W. The fact that [P, H W ]  0 constrains the functional form of the Hamiltonian. What does parity do to some common operations ? vector or polar vector x  - x or p  - p. axial or pseudo vectors J = x  p  J. time (t) t  t. nameformparity scalarrr+ pseudoscalarx(y  z)- vectorr- axial vectorr x p+ TensorF uv indefinite According to special relativity, the Hamiltonian or Lagrangian of any interaction must transform like a Lorentz scalar. Read Perkins: Chapters 3.5

Parity Violation in  -decay Co J=5 60 Ni* J=4 60 Ni* J=4 B-field J z =1 pvpv pvpv p e- YES NO Classic experiment of Wu et. al. (Phys. Rev. V105, Jan. 15, 1957) looked at  spectrum from: followed by: Note: 3 other papers reporting parity violation published within a month of Wu et. al.!!!!!   detector   detector   detector  counting rate depends on  p e which is – under a parity transformation Read Perkins: Chapters 3.5

35 References Class P by Richard Kass (2003) B.G Cheon’s Summer School (2002) S.H Yang’s Colloquium (2001) Class by Jungil Lee (2004) PDG home page (