1. 1 Symmetries and conservation laws Introduction Parity Charge conjugation Time reversal Isospin Hadron quantum numbers G parity Quark diagrams

2. 2 Symmetry and Group Theory Symmetries – linked closely to dynamics of system. Symmetries – the most fundamental explanation for the way things behave (laws of physics). Symmetry – described by group theory. A group = collection of elements with specific interrelationships defined by the group transformations. Demand: repeated transformations between elements of the group equivalent to another group transformation from the initial to the final elements. Example: equilateral triangle. A CB a R + - clockwise rotation through 120° R - - counterclockwise rotation through 120° R a, R b, R c – flipping about axis Aa, Bb, Cc, respectively I – doing nothing Rotation clockwise through 240°  rotation counterclockwise by 120° : R + 2 =R -.

3. 3 Properties of a group closure – if R i and R j in set  R i R j =R k also in set identity – there is element I so that IR i =R i I=R i inverse – for every R i there is an inverse R i -1 : R i R i -1 =R i -1 R i =I associativity – R i (R j R k )=(R i R j )R k The group elements need not commute. If all elements do commute, the group is called Abelian. Translation in space and time – Abelian. Rotation – non Abelian. Groups can be finite (like in triangle example) or infinite. There are continuous groups (rotation) and discrete groups (finite groups).

4. 4 Nöther’s theorem If the Lagrangian governing the phenomenon does not change under the group transformation  a conserved quantity exists. Nöther’s theorem: to every symmetry of a Lagrangian, there corresponds a quantity which is conserved by its dynamics, and vice versa. SymmetryConservation law Translation in timeEnergy Translation in spaceMomentum RotationAngular momentum Gauge transformationElectric charge

5. 5 Special unitary transformations Most of the groups of interest in physics are groups of matrices. In particle physics the most common groups are of the type U(n): the collection of all unitary n x n matrices. A unitary matrix:  (inverse = transpose conjugate) If the group is restricted to all unitary matrices with determinant 1, the group is called : (“special unitary”)

6. 6 Geometrical Symmetries Translation Rotation Reflection (parity) R R R R R R R R R R R R R R R R R R R R R R continuous discrete

7. 7 Parity operator Parity operator P reflects the system through origin of coor system  left-handed coor system  right-handed one. Parity operation  mirror reflection followed by a rotation through 180°: A system is said to be invariant under parity operation if the Hamiltonian remains unchanged by the transformation: x i – position vector of particle i. Consider single particle wavefunction where a identifies particle, P a – constant phase factor. Two successive parity operations leave system unchanged 

8. 8 Intrinsic Parity Eigenfunction of momentum given as Thus particle at rest (p = 0) is eigenstate of parity operator with eigenvalue P a – which is called the intrinsic parity of particle a. Particle with definite orbital angular momentum l, also eigenstate of P: r, ,  spherical polar coordinates eigenvalue:

9. 9 Parity conservation If Hamiltonian H is invariant under parity transformation  In strong and electromagnetic interactions, parity is conserved. Thus parity of the final state, P f, equals parity of initial sate P i. Example: atomic bound states, parity is a good quantum number. Atomic states s,d,g,… have even parity, while p,f,h,… have odd parity. Electric dipole transitions between states have the selection rule  l =  1  parity of atomic state changes. However, since a  is emitted in the transition (P  = -1), parity of the system is conserved.

10. 10 Parity of Particles Intrinsic parity – Fermions –Consider electrons and positrons represented by a wave function . –Dirac equation is satisfied by a wave function representing both electrons and positrons  intrinsic parity related: –Strong and EM reactions always produce e + e - pairs. –Arbitrarily have to set one =1 and the other = -1.

11. 11 Parity of Fermions Assign positive parity state to particles, negative to antiparticles: Make same assumption about quarks to be consistent:

12. 12 Example: parapositronium The ground state of an e+e- bound system is called parapositronim. The initial state has 0 orbital momentum (s state). The system can decay into a state of two photons: Parity of initial state: Parity of final state: Conclusion: relative angular momentum between the two  ’s has to be odd, to conserve parity. This was confirmed experimentally.

13. 13 Parity of Mesons Mesons are quark antiquark pairs : Since the  meson in the lowest energy bound state of a q-qbar system, it has L=0, and thus expected to have negative parity. This was confirmed from completely different considerations (before the quark model was introduced).

14. 14 Parity of Baryons Baryons contain three quarks: q1q1 q2q2 q3q3 L 12 L3L3 The lowest energy state baryons, like proton, neutron, ,  C, have L 12 +L 3 =0, and therefore positive parities.

15. 15 Naming convention The  has spin 0 and negative parity  called pseudoscalar meson. The  has spin 1 and negative parity  called vector meson.

16. 16 Charge conjugation Charge conjugation operator, C, replaces all particles by their antiparticles in the same state, so that momenta, positions, spins, etc., are unchanged. C changes charge, and other additive quantum numbers like Baryon number or Lepton number.  only those particles with additive quantum numbers = 0, like  or  0, are eigenstates of C. Denote these states by  : C  are called C-parities, and ifC-parity is conserved. The electromagnetic field is produced by a moving charge, which changes sign under charge conjugation  C  = -1. C is multiplicative  C(  0   ) = +1. This is true for strong and electromagnetic interactions.

17. 17 C of particle-antiparticle system Particle-antiparticle system has additive quantum numbers = 0  is eigenstate of C. However, since the operation changes particle into antiparticle, this means also a spatial interchange and thus will depend on the relative orbital momentum L and total spin S of the system:  0 is the lowest energy state of u-ubar or d-dbar, which are fermion- antifermion pairs with L=0 and S=0, therefore C(  0 )=+1, as before. stateJPC n=1 1S01S0 01 3S13S1 1 n=2 1S01S0 01 3S13S1 1 1P11P1 11 3P03P0 011 3P13P1 111 3P23P2 211 A hydrogen-like electron-positron bound state is called positronium. The total spin of the electron-positron state can be 0 or 1. L is restricted to be ≤ n-1. The following J, P, C values are for n≤2:

18. 18 C of particle-antiparticle system x 1,  1 x 2,  2 Need to interchange space and spin coordinates to make an eigenvalue equation. Interchange of spin: (-1) s+1. Interchange of space: (-1) l from orbital momentum and another (-1) from the opposite intrinsic parities of the proton and antiproton.

19. 19 Time reversal Time reversal makes the transformation t  t’ = -t. Invariance of time reversal means that the probability of finding a particle at position x and time t is the same as finding it at position x and time –t: The operation of time reversal changes the sign of momentum p and of the direction of the total angular momentum J: Time reversal on a+b  c+d  c+d  a+b. However, p and J opposite to that of the original reaction. Operation with parity operator on the reversed reaction, all momenta will reverse sign. Average over spin  back to original configuration.

20. 20 Principle of detailed balance If time reversal and parity operation are both invariants of a given reaction, the rate of the original one and of the time reversed one is the same, provided the spin states of the initial states are averaged and that of the final state are summed over. This principle, called principle of detailed balance, was used to determine that the spin of the  meson is 0, using the reactions: And the time reversed one A study of the ratio of the differential cross sections of both reactions, which is a function of the spin of the pion, led to the experimental measurement of S  ≈ 0.

21. 21 Isospin Isospin invariance follows from the fact that strong interactions are independent of quark type, and so do not distinguish up quarks from down quarks. Furthermore, the masses of the up and down quarks are small compared to their energy in the proton or neutron, and thus protons and neutrons have close to equal masses. As far as strong interactions are concerned, protons and neutrons behave identically. Isospin is the invariance that relates strong interaction processes or states that differ only by replacing some number of protons by equal number of neutrons. Isospin is a compound word which suggests two ideas – isotopes and spin. Isospin is only very loosely related to the concept of an isotope and has nothing whatever to do with spin! However, the mathematics of isospin is similar to the mathematics of spin for spin ½ particles, and this is where the spin part of the name arose.

22. 22 Isospin (2) Heisenberg (1932) suggested that the similarity of the neutron and proton (M~938 MeV, spin ½) indicated that they were two states of the same particle – the nucleon The formalism was the same as for spin-1/2, thus the name “isospin” (self spin) Isospin was found to be conserved in strong interactions, not in weak & EM –Only the magnitude I matters, not particular I z Evidence –Equivalence of nn, np, pp interactions –Equivalence of “mirror” nuclei

23. 23 Mirror nuclei

24. 24 Isospin in the Quark Model Isospin can be understood in the quark model of hadrons The only difference between n and p is interchange d  u If d & u had the same mass, this would be a true symmetry –No feature except charge would distinguish p/n But it’s not perfect –We know that M n -M p = 1.3 MeV –Coulomb effects should be stronger for p than n So we conclude that  M q =M d -M u ~2-3 MeV –And  M q /M N ~.2% so a small effect!

25. 25 Isospin in other hadrons We now know the fundamental doublet for isospin is The comparable doublet for anti-particles is We can combine two isospin-1/2 states just like spin Bigger charge has larger I 3

26. 26 Note on  wavefunction Following the Condon-Shortly convention : (arrow denotes C-operation) Isospin shift operators : Example:

27. 27 Note on  wavefunction  wavefunction:

28. 28 Note on  wavefunction  wavefunction: Symmetry determined under interchanges:  Triplet antisymmetric singlet - symmetric

29. 29 Isospin in the N-N system Consider a system of 2 nucleons –Nucleon is p or n We again combine the isospins into triplet and singlet combinations Symmetric Anti-symmetric Symmetry under label interchange 1 2

30. 30 Some additive quantum numbers In the simple quark model, only two types of quark bound states (and their anti-states) are allowed: baryons, made up of 3-quarks mesons, made up of quark-antiquark For each state we can associate several quantum numbers, which refer to the quark content: strangeness S, charm C, beauty B, truth T : S =-1 for s-quark, +1 for anti-s-quark, 0 – rest. SC B T Number of u and d quarks have no special names: Baryon number B: baryon: B=1, antibaryon: B=-1 rest: B=0

31. 31 Baryons and Mesons particleMass (MeV) quarksQ SCB B P938uud10001 n940udd00001  1116uds0001 cc 2285udc10101 ++ 140ud10000 K-K- 494su 000 D-D- 1869dc0 00 D+sD+s 1969cs11100 B-B- 5278bu00 0  9460bb00000

32. 32 Hypercharge Hypercharge Y is defined as follows: Y  B + S + C + B + T The hypercharge has the same value for each family member. Gel-Mann and Nishijima showed the relation between the third component of the isospin, I 3, the electric charge, Q, and the hypercharge, Y: Examples:

33. 33 Some quantum numbers of quarks quark BYQI3I d1/3 -1/3-1/21/2 u1/3 2/31/2 s1/3-2/3-1/300 c1/34/32/300 b1/3-2/3-1/300 t1/34/32/300

34. 34 Hadron quantum numbers Assuming only light quarks (u,d,s), the following states are allowed: SQISQI 02,1,0,-13/2,1/211,01/2 1,0,-11,001,0,-11,0 -20,-11/20,-11/2 -30 BaryonsMesons

35. 35 J, P, C of hadrons Each hadron characterized by mass, spin, parity, charge-conjugation, isospin, etc. Can obtain its quantum numbers by knowing its quark content and the relative angular momentum between them. Notation of particle quantum numbers: J PC. Example: mesonic states LSJP state 000- 1S01S0 011- 3S13S1 101+ 1P11P1 110+ 3P03P0 111+ 3P13P1 112+ 3P23P2 Q = +1  I 3 = +1  I = 1 Since Q ≠ 0  not C eigenstate

36. 36 Hadrons from quarks C I3I3 Y

37. 37 Penta-quarks Are there states like: T. Nakano et al., PRL 91 (2003) 012002

38. 38 Hadron quantum numbers from decay Example:

39. 39 G parity Only few particles are eigenstates of C. In strong interactions, one can use rotation in isospin state to create an operator for which more particles will become eigenstates. A rotation of 180° in isospin space around e.g. y-axis, I y, will change I 3 to –I 3, converting for instance a  + into a  -. Applying now C will change the  - back to  +. All non-strange (or charm, beauty, top) mesons are eigenstates of G. For a multiplet of isospin I:

40. 40 Examples with G parity has to decay into even number of pions. has to decay into odd number of pions. has to decay into even number of pions. But J=0 state of two pions has to have even parity, whilenot enough energy: violating G parity  decay is electromagnetic.

41. 41 Summary of conservation rules conserved quantitySIEMIWI Energy/momentumyes Chargeyes Baryon numberyes Lepton numberyes I-isospinyesno G-parityyesno S-strangenessyes no C,B,T (charm, bottom, top)yes no P-parityyes no C-charge conjugationyes no CP (or T)yes yes* CPTyes * 10 -3 violation in K 0 decay

42. 42 Quark diagrams uuuuuu duuduu C-parity not conserved (see also Clebsh coeff.)

43. 43 Quark diagrams (2)

44. 44 Quark diagrams (3) “ Exotic ”

45. 45

46. 46

47. 47 Okubo-Zweig-Iizuka (OZI) rule Problem: Why so narrow? Q-value is 30 MeV for KK decay, while 605 MeV for 3  decay. Why is BR(KK)»BR(3  )? Answer: OZI rule – quark diagram of 3  decay not continuous.