Quantum Numbers. F. Di Lodovico 1 EPP, SPA6306. Queen Mary University of London. Quantum Numbers. F. Di Lodovico. Quantum Numbers.

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1 1 1 School of Physics and Astrophysics Queen Mary University of London EPP, SPA6306

2 Outline :

3 Number Conservation Rules Based on the experimental observation of particle interactions a number of particle properties (quantum numbers) were seen to be conserved. Not all interactions conserve all quantum numbers. The quantum numbers and their rules of conservation are useful to predict which interactions can occur. The rules of conservation were observed experimentally.

4 Number: Q Electric charge, Q, is conserved in all interactions, that is the total charge of the particles before and after the interaction takes place is always conserved. Examples: π + p K 0 + Λ 0 Strong Interaction Q Q is Conserved K 0 π + + π Weak Interaction Q Q is Conserved

5 Number: B Baryon number is B = (n q n q )/3 where n q is the total number of quarks and n q the total number of antiquarks. Quarks are assigned B = 1/3, antquarks B = 1/3. Baryons have 3 quarks and so B = +1, antibaryons have 3 antiquarks and so B = 1, mesons have a quark and antiquark and so B = 0, leptons contain no quarks and have B = 0. Baryon number is conserved in Strong, Electromagnetic and weak interactions as the total number of quarks and anti-quarks is constant. Leptons Quarks Baryons Mesons Number (Anti- (Anti- (Anti- (Anti- Leptons) Quarks) Baryons) Mesons) B 0 (0) 1/3 ( 1/3) +1 ( 1) 0 (0)

6 Number: B Examples π + p K 0 + Λ 0 Strong Interation B B is Conserved Λ 0 p + π Weak Interaction B B is Conserved p e + + π 0 B B is Violated The last interaction is allowed energetically, but violates B. The proton is the lightest baryon so it cannot decay into any other baryon. There is a lot of interest in the search for the proton decay as this can herald new physics.

7 Number: L Lepton number is given by L = n l n l where n l is the number of leptons and n l the number of anti-leptons. Since each lepton is not made of other leptons we see L = +1 for leptons and L = 1 for antileptons and L = 0 for other particles. L is conserved in Strong, Electromagnetic and Weak interactions. Leptons can be grouped into families each with a conserved lepton quantum number. Each generation lepton number needs to be conserved. Le = +1 for e, ν e and L e = 1 for e +, ν e. L µ = +1 for µ +, ν µ and L µ = 1 for µ, ν µ. L τ = +1 for τ +, ν τ and L τ = 1 for τ, ν τ. L e, L µ and L τ are separately conserved.

8 Number: L Examples: Pair production L e conserved γ + N e + + e + N L e Pion decay L µ conserved π + µ + + ν µ L µ Muon decay L e and L µ conserved µ + e + + ν e + ν µ L e L µ

9 : L Radiative decay µ + e + + γ L e L e is violated L µ L µ is violated L L is conserved Although L is conserved the lepton number for each family is violated making the decay. To date this decay has not been observed and a limit on its branching ratio is less than

10 : Number S is given by S = n qs n qs where n qs is the number of strange quarks and n qs the number of strange antiquarks. The strange quark has S = 1 and its antiquark has S = +1. is conserved in strong and electromagnetic interactions, but not in weak interactions. Weak interactions change a quark of one type (flavor) into another. Further experiments produced similar behaviours for heavier particles that required the introduction of more quantum numbers: Charm, Bottom and Top and their corresponding quarks.

11 The spin is an intrinsic property of the particle itself. Similar to the angular momentum, for the spin of a particle: S: it can assume the values s = 0, 1 2, 1, 3 2, 2, 5 2, 3, 7 2,... Sz : it can assume the values m s, m s = s, s + 1,..., 1, 0, +1,..., s 1, s for a total of 2s + 1 possibilities. For each particle the value of the spin is fixed.

12 Examples Adding A quark and an antiquark are bound together in a state of zero orbital angular momentum to form a meson. What are the possible values of the meson s spin? (Anti)quarks spin = 1 2 Addition of spins: = 1 (vector) or = 0 (scalar) Repeating the exercise with three quarks (baryons) Adding to the vector = 3 2 or = 1 2 Adding to the scalar = 1 2 All mesons (2 quarks) carry integer spins (bosons). All baryons (3 quarks) carry half-integer spin (fermions).

13 I Used mostly in Nuclear Physics from charge independence of nuclear force p p = n n = p n sometimes called Isobaric /Isotopic T (or t). is represented by a spin vector I with a component I 3 along some axis. Baryons have half-integer isospin and mesons have integer isospin.

14 I The quarks (antiquarks) u (u) and d (d) have I = 1/2 and: I 3 = +1/2 for u quarks and d and antiquarks. I 3 = 1/2 for d quarks and u antiquarks. All other types of quarks have I = 0 and I 3 = 0. I 3 is additive. A proton (uud) has I 3 = (+1/2) + (+1/2) + ( 1/2) = +1/2. A π (ud) has I 3 = ( 1/2) + ( 1/2) = 1.

15 I There is a relationship between the charge Q, Baryon number B and I 3 : Q = I 3 + B/2. For a proton B = 1, I 3 = +1/2, so Q = (+1/2) + 1/2 = +1. is conserved in strong interactions since it does not distinguish between p and n (i.e. u and d quarks. is not conserved in weak interactions. In Electromagnetic interactions I is not conserved, but I 3 is conserved.

16 Each particle has an antiparticle. In some cases (some neutral mesons) the antiparticle is the same as the particle. An antiparticle has the same mass and spin as its corresponding particle.

17 Antileptons For the leptons each particle has a distinct antilepton. Generation First Second Third +e e + (positron) µ + (antimuon) τ + (antitau) 0 ν e (electron ν µ (muon ν tau (tau antineutrino) antineutrino) antineutrino)

18 Antiquarks The quarks also have a corresponding antiparticle with the same mass, spin and opposite charge. Quarks Antiquarks Type Type u +2/3 u -2/3 d -1/3 d +1/3 s -1/3 s +1/3 c +2/3 c -2/3 b -1/3 b +1/3 t +2/3 t -2/3 The proton consists of uud and has a charge of +2/3 + 2/3 1/3 = +1. The antiproton consists of uud and has a charge of 2/3 2/3 + 1/3 = 1

19 Antibaryons In the case of baryons the antibaryons have the same quantum number, but with opposite sign. Quantity Particle Antiparticle Q q -q B b -b L l -l I I I 3rd Cpt I 3 I 3 -I 3 S s -s E.g. the proton has baryon number 1, the antiproton has baryon number -1.

20 Antimesons Mesons are made from a quark and antiquark. In some cases a meson is its own antiparticle. E.g the π 0 because it contains the same type of quark and antiquark: uu. But, this is not true for all neutral mesons. E.g. the K 0 (us) has antiparticle K 0 (us). The strangeness and I 3 for an antiparticle have the opposite sign of the values for the particle (e.g. S = +1 for the K + but S = 1 for the K ).

21 Examples A hadronic strong interaction obeys the conservation of all quantum numbers: π + p K 0 + Λ 0 Particle π p K 0 Λ 0 Q B S I /2-1/2 0 A hadronic weak interaction violates some quantum numbers: K 0 π + + π Particle K 0 π + π Q B S strangeness not conserved I 3-1/ I 3 not conserved

22 P is a quantum mechanical concept. It reverses the spatial coordinates r r. takes the values ±1. If P = +1 a state is said to have even parity. If P = 1 a state is said to have odd parity. Figure: A reflection in the xy-plane followed by a rotation about the z-axis is the same as a parity operation.

23 P is a multiplicative quantum number. The parity of a composite system Ψ = Φ A Φ B... is equal to the product of the parity of each component: P Ψ = P A P B... A state with angular momentum l has parity P = ( 1) l. For a system of particles: P(overall) = P(relative motion) + P(intrinsic). For fermions P(antiparticle) = ( 1) P(particle). For Bosons P(antiparticle) = P(particle). For particles n, p have P = +1 (this is arbitrarily assigned). For the antiparticles p, n have P = 1. Other particles have parity determined by experiment.

24 P is conserved in Strong and Electromagnetic interactions, but not weak interactions. Mesons and Baryons are classified according to their spin-parity (J P ). The diagram below shows the operation in a plane for a Vector (J P = 1 ). r r, P = 1.

25 P The diagram below shows the operation in a plane for an Axial Vector (J P = 1 + ). r r, P = +1.

26 C conjugation, C, is a quantum mechanical operation that changes a particle into its antiparticle. For example: C(particle) = antiparticle. The mass, energy, momentum and spin do not change under C. As in the case of, C is a multiplicative quantum number.

27 C Most particles are not eigenstates of C as their particle and antiparticle masses differ. Only particles that are their own antiparticles (e.g. π 0, γ, η, φ etc) are eigenstates of C. For N photons: C = ( 1) N. In the electromagnetic decay π 0 γ + γ we must have C = ( 1)( 1) = +1 for the π 0 (using the fact that C = 1 for the photon). C is conserved in Strong and Electromagnetic interactions.

28 T, T, is q quantum mechanical operation that reverses the time component of processes. It has been tested in the process p + n d + γ which transforms to the process d + γ n + p under time reversal. The reactions have been studied and no violation of T-invariance has been observed. Time reversal is conserved in Strong and Electromagnetic interactions.

29 The operations of - (C), (P) and Time-Reversal (T) act on different parts of a particles state. P and T act on extrinsic or external properties such as the spatial or temporal coordinates whereas C acts on intrinsic or internal properties (the quantum numbers). Operation Extrinsic Intrinsic P Invariant r r Invariant T Invariant t t Invariant C q q Invariant q numbers -(q numbers)

30 Noether s theorem: Symmetries Conservation laws. Symmetry is an operation performed on a system that leaves it invariant, i.e. it is indistinguishable from the original one. Symmetry Conservation law Translation in time Energy Translation in space Momentum Rotation Angular momentum Gauge transformations

31 : Rules of Conservation of the quantum numbers and their conservation for the different forces. Quantity Strong EM Weak Q B L I X X 3rd Cpt I 3 X S X P X C X T X