Gian Gopal Particle Attributes Quantum Numbers 1

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1 Particle Attributes Quantum Numbers Intro Lecture Quantum numbers (Quantised Attributes subject to conservation laws and hence related to Symmetries) listed NOT explained. Now we cover Electric Charge Baryon Number, Lepton Number Spin Parity Isospin Strange, Charm, Bottom & Top Charge Conjugation Colour etc. Gian Gopal Particle Attributes Quantum Numbers 1

2 Electric Charge (Q) & Baryon Number (B) conserved in all interactions Strong, Weak & Electromagnetic! Baryons: B = +1 Anti-Baryons: B = -1 Everything else: B = 0 Baryons made up of 3 quarks: B = 1/3 For an anti-quark B = -1/3 N N q q Constant Strong Interaction: π - p K 0 Λ 0 B: Weak Interaction: n p e - υ e B: Proton being the lightest Baryon cannot decay i.e p e + π 0 B: Gian Gopal Particle Attributes Quantum Numbers 2

3 Lepton Number: Leptons have L = +1 Anti-Leptons L = -1 All Else L = 0 Leptons of different type do NOT mix! Unique Lepton Number for each Type + e, ν e have Le = + 1 e, ν e have L e = 1 + µ, ν µ havelµ = + 1 µ, ν µ havelµ = τ, have L = + 1 τ, ν τ have Lτ = 1 ν τ τ L e, L µ & L τ separately conserved X µ ± e ± γ (Both e & µ lepton numbers violated!!) X e + e - µ ± τ (Both µ & τ lepton numbers violated!!) Each Lepton Number is conserved in strong, weak & electromagnetic interactions Gian Gopal Particle Attributes Quantum Numbers 3

4 SPIN ( Angular Momentum) Particles can have 2 types of Angular Momenta Orbital Classical Analogue subject to Quantum Conditions. A particle can be in any Orbital Angular Momentum State Spin Intrinsic Angular momentum specific attribute of a particle discrete states (values) i.e. a Quantum Mechanical effect for Spin S - (2S+1) states of different S Z S z = +1 S z = +½ S Bosons - 0 h, 1h, 2 h... Fermions - h, h, h... For S = ½ - 2 states with S z = +1/2 & -1/2 For S = 1-3 States with S z = +1, 0, S z = -1 Gian Gopal Particle Attributes Quantum Numbers 4 z S z = -½ S z = 0

5 Angular Momentum Classically r r r L= p But Quantum Mechanically r r and p are quantised by the Uncertainty Principle L r Can only take certain values and assume defined orientation wrt to a given direction in Space Particle Wave Function has both Orbital & Spin angular momentum components besides the position & momentum components ih r Ψ = p Ψ 2 2 L z J z p p L Ψ lm Ψ lm Gian Gopal Particle Attributes Quantum Numbers 5 = l ( l + 1) h Ψ = mh Ψ lm lm For a given l m = -l, -l+1, -1,0,1 With l = 0,1,2,3 l-1,l = j Ψ( j) (2j+1) states for spin-j z

6 Parity Parity is an entirely Quantum Mechanical Concept. The operator (P) acts on the space part of the wave function reversing the space coordinate r to -r P Ψ( r) Ψ( r) = λ Ψ( r) But 2 P Ψ( r) = 2 λ Ψ( r) = Ψ( r), 2 λ = 1 i.e. λ = ± 1 Eigenvalues of P are +1 (even) or -1 (Odd) Gian Gopal Particle Attributes Quantum Numbers 6

7 Parity (cont.) Parity Operator P reverses r to -r Equivalent to a Reflection in the x-y plane Followed by a Rotation about the z-axis 2004 S. Lloyd Gian Gopal Particle Attributes Quantum Numbers 7

8 Gian Gopal Particle Attributes Quantum Numbers 8 Parity (cont.) For a given state of orbital angular momentum l, P = (-1) l Consider the H-atom bound by a central potential wave function a product of the spatial and angular functions φ θ π φ θ φ θ χ ϕ ϑ im e m l P m l m l l m l Y m l Y r r ) (cos )! ( 4 )! 1)( (2 ), ( ), ( ) ( ),, ( + + = = Ψ Changing r to r implies θ π - θ & φ π+ φ ), ( 1) ( ), ( φ θ φ θ m l Y l m l Y = φ θ φ θ,, ( 1) (,, ( r l r P Ψ = Ψ 1) ( φ imφ e m im e ) (cos 1) ( ) (cos and θ θ m l P m l m l P +

9 Parity (cont.) For Fermions, P (anti-particle) = -1 x P (particle) For Bosons, P (anti-particle) = P (particle) Arbitrarily define P = 1 for nucleons and then determine P for other particles from experiment (angular distributions in interactions) Parity for π +, π 0, π - P = -1 N.B. P is conserved in Strong & electromagnetic interactions but not in Weak Particles (& Physical attributes) are labelled by their total spin (J) and Parity (P) J P Gian Gopal Particle Attributes Quantum Numbers 9

10 J P Name of Object Example 0 - PseudoScalar Pressure 0 + Scalar M, T, λ 1 - Vector p, x 1 + Axial Vector S, L 2 + Tensor Spin-spin coupling Vector P = -1 Axial Vector P = S.Lloyd Gian Gopal Particle Attributes Quantum Numbers 10

11 Isospin (I) 1932 Heisenberg suggested n & p sub-states of nucleon ascribed new quantum number I = 1/2 isospin analogous to spin with Cartesian coordinates I 1, I 2, I 3, in an imaginary Isospin space Related to charge & Baryon Number Q/e = B/2 + I 3 this gives the Proton I 3 = +1/2 and the neutron I 3 = -1/2 I, I 3 are conserved in Strong interactions BUT not in electromagnetic interactions due to coupling to Q. Quark Model p = (uud) & n = (udd) => u-quark I 3 = +1/2 while d- quark has I 3 = -1/2. So knowing the quark-content of a particle we can determine the isospin Gian Gopal Particle Attributes Quantum Numbers 11

12 Isospin (I) Cont. + π = ud 0 1 -, I3 = + 1 π = ( dd uu ), I3 = 0 π = du, I3 = 1 2 Isospin represented by a vector with component I 3 designating upness and downness I 3 :: Up (u) Isospin predicts: σ ( p + p d + π + ) ~ 2σ (p + n d + π 0 ) As observed! I 3 :: Down (d) Gian Gopal Particle Attributes Quantum Numbers 12

13 Strangeness Associated Production of long-lived neutral particles π - p K 0 Λ 0 K 0 & Λ 0, decaying weakly, were labelled Strange Gian Gopal Particle Attributes Quantum Numbers 13

14 Strangeness (cont.) Associated production explained by introduction of a 3 rd quark (strange s) s-quark is assigned Strangeness S = -1, anti-quark (s) has S = +1 Gian Gopal Particle Attributes Quantum Numbers 14

15 Strangeness (cont.) Quark Content of some Strange Particles Λ 0 ( uds), Λ 0 ( uds), K 0 ( ds), K 0 ( ds), K + ( us) & K ( us) Q, B, I & S What s the Connection? I 3 B + Gellman Nishijima Relation: 2 Q = + S Strong & EM forces Conserve S Weak does not. Charm (C), Bottom (B) & Top (T) analogous to Strange. u,d,s,c,b & t are the 6 Quarks Weak force changes quark flavour Gian Gopal Particle Attributes Quantum Numbers 15

16 Charge Conjugation (C) Charge Conjugation Operator (C) changes particle to anti-particle e.g. e - e + Only if no Conservation Law is violated. C has definite Eigenvalues for a neutral boson which is its own anti-particle C π = η π And C π 0 = η π 0 = π 0 η = ± 1 EM fields are produced by moving charges. But Q -Q under C so C γ = -1 And C is a multiplicative Quantum Number, so C γγ = 1. π 0 = π 0 With π 0 γγ, C i.e. η = 1 If EM forces are C-invariant π 0 γγγ is forbidden 0 π 0 π 3 γ 8 < 2γ C is conserved in Strong & EM Interactions but not in Weak Gian Gopal Particle Attributes Quantum Numbers 16

17 Quark Colour The Interaction π + p p π + π 0 the final state π + p form a resonance of mass 1232 MeV ++ has I 3 = 3/2, J= 3/2, P = +1, Wave function is totally symmetric BUT It is a Fermion Gian Gopal Particle Attributes Quantum Numbers 17

18 Quark Colour (Cont.) qqq = space l, m s flavour All components are symmetric l=0, s=3/2 and flavour (uuu) Need extra anti-symmetric component Each Quark is assigned a new quantum number Colour can have 3 values Red, Green & Blue and the particle wave function has an extra factor Ψ color > Which is anti-symmetric All particles baryons & mesons are colourless so each of the 3 quarks in a baryon has a different colour (r, g, b)! In a meson the quark and the anti-quark carry opposite colour to be colourless! Strong force between quarks carried by Gluons carry colour! Gian Gopal Particle Attributes Quantum Numbers 18

19 Quark-Parton Model Hadronic ( Baryons & mesons) states classified into Groups by spin & parity e.g. 10 excited baryons with J P = 3/2 + and a ground state 8 with J P = 1/2 + d u Decuplet d s missing Gian Gopal Particle Attributes Quantum Numbers 19

20 QPM Ground State Baryons J P = 1/2 + each with 3 quarks with spins 1/2, -1/2,1/2 ( ) 939 MeV Λ 1318 MeV Gian Gopal Particle Attributes Quantum Numbers 20

21 Baryons With 3 different flavour quarks get 27 possible combinations SU(3) symmetry arguments: = = 1(Singlet) + 8(Octet) + 8(Octet) + 10(Decuplet) uud Combinations Flavour symmetric (uud + udu + duu) or anti-symmetric (uud) Spin symmetric ( + + ) spin anti- ( ) J = 3/2 J =1/2 A (uud) baryon in both the decuplet & the octet! Gian Gopal Particle Attributes Quantum Numbers 21

22 Baryons Note: No (uuu), (ddd) or (sss) states in the ½ + Baryons!! (uuu/ddd/sss) Combinations (uuu) Flavour symmetric And aligned spins ( ) with J = 3/2 Spin Symmetric giving a flavour> spin> product symmetric. Colour component has to be anti-symmetric. ( ) with J= ½ makes the flavour-spin anti-symmetric BUT colour component must be anti-symmetric. Overall product symmetric!! So J=1/2 not possible (uuu), (ddd) and (sss) states only in the Decuplet NOT in an Octet Gian Gopal Particle Attributes Quantum Numbers 22

23 QPM - Mesons With 3 different flavour quarks get 9 possible combinations SU(3) symmetry arguments: 3 3 = 1+ 8 Pseudo-Scalar (J P = 0 + ) Mesons (qq) pairs with opposite spins ( ) 0 S + K ( ds ) K ( us) π ( ud) π 0 ( dd uu) 2 η uu + dd 2ss 2 π + ( ud ) I 3 K ( us) 0 K ( ds) Gian Gopal Particle Attributes Quantum Numbers 23

24 QPM Mesons (Cont.) Vector Mesons quark anti-quark pairs with aligned spins ( ) S *0 * + K ( ds ) K ( us) ρ ( ud) 0 ρ ( uu dd ) φ ω ( ss) ( u u + dd ) ρ + ( ud ) I K * ( us) *0 K ( ds) Gian Gopal Particle Attributes Quantum Numbers 24

25 QPM Higher Mass states Hadron states seen so far are all with 0 orbital angular momentum between the component quarks. When the partons are excited into higher orbital angular momenta heavier excited states are produced. J = 2,3 for mesons & J = 5/2, 7/2 for Baryons Gian Gopal Particle Attributes Quantum Numbers 25

26 QPM Evidence 1 st evidence from di-muon production in pion- Carbon scattering 12 C Is an Isoscalar target i.e. N u = N d = 18 6 If QPM then q q annihilate & the photon materialises into a d-muon pair. σ α (charge of q)(charge of q) α (charge of q) 2 With π - (ud) & π + (ud) π π + C C + µ + µ - µ - µ In good agreement with data Confirming fractional quark charges Gian Gopal Particle Attributes Quantum Numbers 26