6.6 Charge Conjugation Charge Conjugation in Electromagnetic Processes Violation of C in the Weak Interaction

Transcription

1 Chapter 6

2 6.1 Introduction 6.2 Invariance Principle Reminder Invariance in Classical Mechanics Invariance in Quantum Mechanics Continuous Transformations: Translations and Rotations 6.3 Spin-Statistics Connection 6.4 Parity 6.5 Spin-Parity of the π Meson Spin of the π Meson Parity of the π Meson Particle Antiparticle Parity 6.6 Charge Conjugation Charge Conjugation in Electromagnetic Processes Violation of C in the Weak Interaction 6.7 Time Reversal 6.8 CP and CPT 6.9 Electric Charge and Gauge Invariance

3 Symmetries and invariance properties of underlying interactions play important role in physics often lead to universal conservation laws (Space) translational invariance à momentum conservation (Time) translational invariance à energy conservation Rotational invariance à angular momentum conservation Gauge invariance restricts form of fundamental interactions Discrete symmetries Parity, Charge conjugation, Time reversal à very useful for classification and when we want to know whether a process is allowed or not Symmetries are so important that even broken ones are useful Electroweak Symmetry Breaking, CP-violation, 24/02/17 F. Ould-Saada 3

4 Equations ruling dynamic evolution of a system (Schrödinger or Lagrange equations) First order differential equations in time and second order in space coordinates Every first integral of motion gives rise to a conservation law Each fundamental interaction obeys various conservation laws Interaction formalism obeys invariance requirements, which limit mathematical form of interaction Transformations Continuous can be achieved by applying successive infinitesimal transformations (rotation, ) Correspond to additive conservation laws Discrete mirror reflection in space (Parity), Charge conjugation, Correspond to multiplicative conservation laws 24/02/17 F. Ould-Saada 4

5 Lagrange equations Lagrangian L=T-V = Kinetic energy potential energy n generalised coordinates q i, n conjugated momenta p i Motion of a system is described by the Euler-Lagrange equation p i = L q i L q i d dt # L % $ q i & (=0 ' dp i dt L q i =0 L q i =0 dp i dt =0 p i = constant 24/02/17 F. Ould-Saada 5

6 L q i d dt # L % $ q i & (=0 ' dp i dt L q i =0 Translation along x Linear momentum conserved Roatation Angular momentum conserved Time translation Energy conserved Poincare invariance (Lorentz + space-time translation) Lagrangian relativistically transforms as scalar Noether s theorem L = T V = 1 2 mx2 p x = L x = mx = constant L= 1 2 m ϕ 2 r 2 = 1 2 mv2 p ϕ = L ϕ = mvr = constant in a Lagrangian field theory, a conserved quantity is associated to a continuous symmetry (and vice versa) 24/02/17 F. Ould-Saada 6

7 Reminder Read in book Hamilton equations Hamiltonian H=T+V q i = H p i ; p i = H q i Invariance means that H does not change under a given transformation Space translations Infinitesimal translation along x If momentum is conserved, then H is invariant x x + dx dh = dx H x = dx p x p x = 0 dh =0 27/02/17 F. Ould-Saada 7

8 Classically, L(q, q,t) = T( q) U(q) = 1 2 m q2 U(q) T:kinetic energy; U:potential energy (U = mgq for gravity) Euler Lagrange equations lead to equationsof motion: L q d dt # % $ L q & (=0 F = du ' dq = ma In QM or QFT, dealing with wavefunctions or fields, define Lagrange density as a functional of the field Φ(x µ ). The integral over the 3-dimensional space leads to: L = (Φ, µ Φ) The Euler-Lagrange equations become: Φ µ ( µ Φ) =0

9 Lagrangian density of a free electron in terms of its wavefunction or field it is trivial to go from the Lagrangian to the equations of motions à For scalar fields (neutral pion, Higgs), = 1 2 ( µφ)( µ Φ) 1 2 M 2 Φ 2 leads to the Klein-Gordon equation ( µ µ + M 2 )Φ = 0; where µ µ = c 2 t 2 For a spin 1 Dirac field (electron) 2 =Ψ(iγ µ µ m)ψ leads to the Dirac equation : (iγ µ µ m)ψ(x)=0 The situation is more complex in case of elementary particles in interaction equations in general not known when known, they are difficult to solve à To describe elementary particle interactions, it is necessary to propose an expression for the Lagrangian of the interacting quantum fields

10 Same formalism for spin (and Isospin we will see later) # L = r p % = % % $ yp z zp y zp x xp z xp y yp x & # ( % ( ˆL % = % ( ' % $ ˆL x = ŷˆp z ẑˆp y ˆL y = ẑˆp x ˆxˆp z ˆL z = ˆxˆp y ŷˆp x & ( ( ( ( ' with. 0 0 / * + * + * + ˆL x, ˆL, y - = i" ˆL z ˆL y, ˆL, z - = i" ˆL x ˆL z, ˆL, x - = i" ˆL y ˆL 2 = ˆL x 2 + ˆL y 2 + ˆL z 2 ˆL + = ˆL x + i ˆL y ˆL = ˆL x i ˆL y 24/02/17 F. Ould-Saada 10 with & ( ( ' ( ( ) & ( ( ' ( ( ) " ˆL 2, ˆL $ # x % = 0 " ˆL 2, ˆL $ # y % = 0 " ˆL 2, ˆL $ # z % = 0 " ˆL 2, ˆL $ # ± % = 0 " ˆL z, ˆL $ # ± % = ± ˆL ± ˆL 2 = ˆL ˆL+ + ˆL z + ˆL z 2

11 Pictorial representation of the 2l+1 states of l=2 ˆL z, ˆL 2 common eigenstates l, m ˆL z l, m = m l, m ˆL 2 l, m = l(l +1) l, m l m = l, l +1,...,+l 1,+l ˆL + l, m = l(l +1) m(+1) l, m +1 ˆL l, m = l(l +1) m( 1) l, m 1 From Thomson Coupling of 2 am / spins Clebsh-Gordon coefficients l 1, m 1 l 2, m 2 l, m % l = l 1 + l ' 2 & l 1 l 2 l l 1 + l 2 ' ( m = m 1 + m 2 l, m = C(m 1, m 2 ;l, m) l 1, m 1 l 2, m 2 m 1,m 2 24/02/17 F. Ould-Saada 11

12 Coupling of ½X½, 1X½ Spin multiplicity: 2l+1 Symmetric, anti-symmetric and mixed configurations 1 1 : 2 2 = : 3 2 = : = 4 S 2 MS 2 MA 24/02/17 F. Ould-Saada 12

13 Particles ½ integer spin (1/2, 3/2, ), Fermi-Dirac-statistics à fermions Integer spin (1,2, ), Bose-Einstein statistics à bosons Statistics fix symmetry properties of WF for a pair of identical particles wrt their exchange I (1, 2) (2,1) Operator I reverses position of 2 particles Eigen values: I 2 =1 ; I=±1 I ψ(1, 2) = ψ(2,1) I 2 ψ(1, 2) = I ψ(2,1) = ψ(1, 2) I ψ(1, 2) = ±ψ(1, 2) ψ(2,1) = ±ψ(1, 2) 24/02/17 F. Ould-Saada 13

14 Under exchange, WF for 2 identical Bosons must be symmetric Fermions must be anti-symmetric 1 2 : ψ(1, 2) = ψ(2,1) 1 2 : ψ(1, 2) = ψ(2,1) Total WF product of 2 functions Spatial function α describes orbital motion of particle wrt to the other à spherical harmonics Spin function β Symmetric if the 2 spins are parallel Anti-symmetric if anti-parallel Y l m (θ,φ) ( 1) l Identical bosons must have both α and β sym or anti-symmetric fermions must have α sym and β anti-symmetric or vice-versa 24/02/17 F. Ould-Saada 14

15 Behavior of a state under a spatial reflection P reverses spatial coordinates r and p Left-Right symmetry Application on wave function r t ˆP r ˆP t' = t ; p ˆP ψ( r,t) P ψ( r,t) ˆP p; J ˆP J Parity applied twice Eigenvalue equation ˆP 2 ψ( r,t) =P ˆPψ( r,t) = P 2 ψ( r,t) P = ±1 ˆPψ = Pψ = ±1ψ Examples of WF with Positive parity: cos x Negative parity: sin x Undefined parity: sin x + cos x Particle at rest (p=o) is eigenstate of parity with eigenvalue P=±1 (intrinsic parity) Partity conserved à[h,p]=0 27/02/17 F. Ould-Saada 15

16 ψ nlm (r,θ,ϕ) = χ nl (r)y m l (θ,φ) n, l, m : principal, orbital, magnetic QNs Y m l : spherical harmonics, P m l : Legendre polynomials Y l m (θ,φ) = (2l +1)(l m) P m l (cosθ)e imφ 4π (l + m) Y 0 0 = Y 1 0 = Y 1 ±1 = 1 4π 3 4π cosθ 3 sinθ e iφ 8π Space inversion x = rsinθ cosφ # y = rsinθ sinφ " z = r cosθ # $ r r θ π θ φ π +φ ˆP e imφ = ( 1) m e imφ " $ # ˆPY m ˆP P m l (cosθ) = ( 1) l+m P m l (θ,φ) = ( 1) l Y m l (θ,φ) l (cosθ)%$ ˆP ψ lmn ( r) = Pψ lmn ( r) = P( 1) l ψ lmn ( r) 24/02/17 F. Ould-Saada 16

17 π Parity Particle-Antiparticle parity Dirac P(fermion-antifermion)=-1 P(boson-antiboson)=+1 24/02/17 F. Ould-Saada 17

18 Pion capture in deuterium at very low energy l πd =0 π d nn π d : s d =1, s π = 0, J i = s π + s d + L πd =1 nn : J f = s nn + L nn =1 WF describing the nn system ψ tot Must be antisymmetric under exchange of 2 identical fermions ψ tot = α space β spin 1 2 r r,θ π θ,φ π +φ Y m 1 2 l (θ,φ) ( 1) l Y m l (θ,φ) (-1) l gives symmetry of α space under exchange /02/17 F. Ould-Saada 18

19 β spin combination of β 1 and β 2 β spin : β 1 (s 1 = 1 2, s 1z = ± 1 2 ) ; β 2 (s 2 = 1 2, s 2z = ± 1 2 ) " β(1,1) = β 1 (1/ 2,+1/ 2)β 2 (1/ 2,+1/ 2) $ s = s 1 + s 2 =1; s z = 0,±1# $ % β(1, 1) = β 1 (1/ 2, 1/ 2)β 2 (1/ 2, 1/ 2) { β(1, 0) =1/ 2 [ β 1 (1/ 2,+1/ 2)β 2 (1/ 2, 1/ 2)+ β 2 (1/ 2,+1/ 2)β 1 (1/ 2, 1/ 2) ] s = 0; s z = 0 β(0, 0) =1/ 2 [ β 1 (1/ 2,+1/ 2)β 2 (1/ 2, 1/ 2) β 2 (1/ 2,+1/ 2)β 1 (1/ 2, 1/ 2) ] Symmetry of β spin : (-1) s+1 combination Symmetry of ψ tot : (-1) l+s+1 = Symmetry arguments 2. Angular momentum conservation nn in p-wave state 3 P 1 l = 0 # J =1 " s =1 # $ l + s = odd 24/02/17 F. Ould-Saada 19 ψ tot 1 2 """ ( 1) l+s+1 ψ tot = ψ tot 1. l + s +1 odd, l + s even 2. J =1 l =1 l =1 # # " s = 0 " s =1 # $ l + s = odd # $ l + s = even l = 2 # " s =1 # $ l + s = odd

20 Parity of final state nn Parity conserved in strong interactions P(nn) = P(n) P(n) ( 1) l = (+1)(+1)( 1) 1 = 1 P(π d) = P(nn) = 1 P(d) = P(p) P(n) ( 1) l=0 = +1 P(π d) = P(π ) P(d) ( 1) l=0 = P(π ) = 1 Pion π - has negative parity N-pion system has parity (-1) N P(π ) = P(π + ) = P(π 0 ) = 1 P(Nπ ) = ( 1) N Various meson families Pions are pseudo-scalars: J P =0 - Scalar mesons : J P =0 + Vector mesons : J P =1 - Axial mesons: : J P =1 + 24/02/17 F. Ould-Saada 20

21 Parity multiplicative QN à P(ψ 1 ψ 2 )=P 1.P 2 SI, EM invariant under Parity à [H,P]=0 Invariance under parity of Dirac equation à + P( e e ) = 1 P( f f ) = 1 Convention: P=+1 for leptons and quarks and P=-1 for anti-fermions Parity of photon: -1 (see last Slide) Intrinsic parities of hadrons follow structure in terms of quarks + orbital l between constituent quarks: P( 1) l Meson = quark-antiquark: P=(-1)(-1) l =(-1) l+1 Pion (l=0): P=-1 Proton (uud,l=0): P=+1 neutron (udd,l=0): P=+1 WI violates Parity (maximally) Not observed 24/02/17 F. Ould-Saada 21

22 C q = -q C ψ(q) = ψ (-q) c 2 =1 à c=±1 Effect of C- parity on proton and electron 24/02/17 F. Ould-Saada 22

23 Operation changing particle à antiparticle Multiplicative QN conserved in SI, EM not in WI Distinguish cases where (a) Particle = antiparticle: γ, π 0 à Ĉ a,ψ a C a = ±1 : = C a a,ψ a C parity (b) Particle diff. antiparticle: π + àπ -, nà anti-n only linear combinations are relevant Ĉ b,ψ b Ĉ π + = C b b,ψ b = π π 0 Ĉ π + = π Ĉ π 0 = ± π 0 EM fields produced by moving electric charges, which change sign under C, so C γ =-1 Ĉ γ = γ # % Ĉ nγ = ( 1) n nγ % $ C π 0 π 0 = +1 γγ (99%)% % C γ = 1 & / γγγ C invariance π 0 γγγ π 0 γγ < π 0 J PC = /02/17 F. Ould-Saada 23

24 (b) mesons (spin-less): π + à π - Interchanging position of particles à reverses relative position in WF à (-1) L (b) fermions anti-fermions Interchanging positions à (-1) L Interchange fermion-antifermion à (-1) Interchanging spins à (-1) S+1 (b) mesons with spin: 0,1,2 Ĉ m + m ;L, S = ( 1) L+S m + m ;L, S Ĉ π + π ;L = ( 1) L π + π ;L Ĉ f f ; L, S = ( 1) L+S ( ) 1 2 = + 1 = 0 = 1 S = 1 ( ) S = 0 S = 0 S S z S z z z f f ; L, S ( ) = ( pp π π + π π +...) ( ) = ( e e + π π + π π +...) Ĉ pp π + π π + π... Ĉ e + e π + π π + π... Average number and energy spectra of π + and π- must be equal à verified experimentally 24/02/17 F. Ould-Saada 24

25 SI and EM invariant under C operation C eigenvalues are conserved quantum numbers WI violate C (and P) but conserve CP to a good precision neutrinos 24/02/17 F. Ould-Saada 25

26 T-Symmetry of SI & EM à Invariance under transformation but violated in WI No conserved quantum number associated to time reversal (neglect WI), unlike P and C t T t ' = t ; r T r ; p T p; J T J If system invariant under T, probability ψ( r,t) 2 T ψ ' ( r,t) 2 = ψ( r, t) 2 Schrödinger equation not invariant under T i ψ(" r,t) t Ψ( r,t) = e i( p r Et) = H( " r, " p)ψ( " r,t) " Tψ( r,t) = ψ ' ( r,t) = e i( p r Et) " 24/02/17 F. Ould-Saada 26

27 Introduce T-operator by analogy with P: ψ( r,t) T ψ ' ( r,t) = ψ * ( r, t) ˆTψ( r,t) Then Schrödinger equation invariant i ψ * ( " r,t) t = H( " r, " p)ψ * ( " r,t) t t i ψ * ( " r, t) t = H( " r, " p)ψ * ( " r, t) Same form as for Ψ ψ( r,t) = e i( p r Et) " ˆTψ( r,t) = ψ * ( r, t) = e i( p r+et) Time-reversed wave function describes a particle with momentum -p " = e i( p r Et) " 24/02/17 F. Ould-Saada 27

28 Quantum Mechanics operators corresponding to physical observables must be Linear to ensure superposition principle holds O ˆ ( α 1 ψ 1 + α 2 ψ 2 ) = α 1 ( O ˆ ψ 1 ) + α 2 O ˆ ψ 2 and Hermitian eigenvalues (observed values ) are real dx ( ) = α 1 * ˆ ˆ T α 1 ψ 1 + α 2 ψ 2 dx Ψ( r,t) ( T ˆ ψ 1 ) * * ψ 2 dxψ 1 T # Ψ * ( r, t) T ˆ Ψ( r,t) * ( T ψ 1 ) + α ˆ 2 T ψ 2 ( T ˆ ψ 2 ) ( ) ( O ˆ ψ 1 ) * * ψ 2 = dxψ ˆ 1 ( O ψ 2 ) ( ) α ˆ 1 ( T ψ 1 ) + α ˆ 2 ( T ψ 2 ) Time reversal operator does not correspond to a physical observable No observable conserved as a consequence of T invariance 24/02/17 F. Ould-Saada 28

29 24/02/17 F. Ould-Saada 29 T-invariance leads to a relation between process and its time-reversed Reactions and time-reversed counter parts are related ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b b a a d d c c d d c c b b a a m p b m p a m p d m p c m p d m p c m p b m p a,,,,,,,, Reactions proceed with equal rates if WI neglected m i : magnetic quantum number

30 Time reversal & Parity Combination of T & P Same rate of reactions if P and T invariance hold (neglect WI) a ( p a, m a ) + b ( p b, m b ) c ( p c, m c ) + d ( p d, m d ) T c( p c, m c ) + d ( p d, m d ) a( p a, m a ) + b( p b, m b ) ˆP c ( p c, m c ) + d ( p d, m d ) a ( p a, m a ) + ( p b, m b ) If averaged over all possible spin projections Principle of detailed balance Confirmed experimentally in a variety of Strong and EM processes m i = s i, s i +1,... s i (i = a, b, c, d) i a p a ( ) + b ( p b ) c ( p c ) + d ( p d ) f 24/02/17 F. Ould-Saada 30

31 pp π + d vs π + d pp pp ßà π + d Principle of detailed balance Assume strong interaction invariant under time reversal and parity Deuteron d=pn= 3 S 1 l pn =0; S wave; s d =1 ; 2s d +1=3 Factor ½ for identical pp in final state Cross section measurements at same energy Velocity factors equal Measure momenta Deduce s π+ =0 Pions (π +, π -,π 0 ) produced with same abundance in e + e -, ppbar, pp colliders (E cm >10GeV) è s π =0 σ (pp π + d) = M if 2 matrix element v i = v p v p' ; v f = v d v π σ (π + d pp) = 1 2 M fi same E cm v ' f v' i = v i v f (2s π +1)(2s d +1) π v i v f phase space & flux factor 2 (2s p +1) 2 πv ' f v' i detailed balance principle M if 2 = M fi 2 σ (pp π + d) σ (π + d pp) = 2 (2s +1)(2s +1) 2 π d p π (2s p +1) 2 2 p p = 3 2 (2s +1) p 2 π π 2 p p p p 2 2 p π 27/02/17 F. Ould-Saada 31

32 If P or T is conserved, Hamiltonian of interaction must not contain terms that change sign after the operation Magnetic dipole moments allowed Not electric DM µ Εn 0 would imply èp and T violated Longitudinal polarization only through WI 24/02/17 F. Ould-Saada 32

33 CP violation played an important role in the earliest moments of the Universe. It is believed that at the beginning, all quantum numbers of the Universe were equal to zero, with an equal number of particles and antiparticles. Probably after t~10-35 s, a phase transition took place, after which particles began to decay with a small CP violation Leading to a slight predominance of particles with respect to antiparticles (<1/10 9 ). When later particle antiparticle annihilated, that little excess of particles produced the matter-dominated Universe which we observe today. The small amount of CP violation observed in the weak interaction (B and K meson decays) seems not enough to fully explain this scenario. The CP violation involves a violation of Time reversal as well because all interactions are invariant under CPT in any order the three transformations are applied. 24/02/17 F. Ould-Saada 33

34 24/02/17 F. Ould-Saada 34

35 Although C & P violated in weak interactions (100%) CP violated in some weak processes (~0.1 %) T violated in some weak processes (~0.1%) there is a general result CPT (Lüders) theorem: Any Quantum Theory that (i) obeys the postulates of Special Relativity, (ii) admits a state with minimum energy and (iii) respects causality is invariant under CPT Causality of physical events requires that the fields obey commutation or anticommutation relations, implying the correct statistics according to the spin of particles: Fermi-Dirac statistics for fermions and Bose-Einstein statistics for bosons CPT invariance predicts that particles and antiparticles must have exactly same masses and lifetimes, opposite magnetic moments, 24/02/17 F. Ould-Saada 35

36 Consequences of CPT invariance Particle & antiparticle have same mass, lifetime, opposite magnetic moments, Particle in state a> = m,τ, > [ CPT, H ] = 0 # " $# a H a = a H CPT a H a = a H a m a = m a ( CPT ) 2 =1 ( ) 2 a = a CPT H CPT a q p m p q p < ± , "m # e + m $ e % < m e m p τ µ + τ µ < ± , " µ e + µ $ # e % < ( 0.5± 2.1) µ e 24/02/17 F. Ould-Saada 36

37 Additional material (slides not shown) 27/02/17 F. Ould-Saada 37

38 Reminder Read in book Average value (or expectation value) of an operator Q associated with an observed quantity q, acting on a wave function ψ Q =Q (hermiticity) q = τ ψ Q ψ dτ ψ Q ψ Time evolution of <q> described either by SR or HR ψ(t) à Schrödinger representation i t ψ s(t) = H s ψ s (t) Q(t) à Heisenberg representation If Q does not explicitely depend on time, q and associated quantum numbers are conserved if Q commutes with H i dq dt = i Q t + Q, H [ ] Relation between the 2 representations à see book 27/02/17 F. Ould-Saada 38

39 Translations ψ(x') = ψ(x + dx) = ψ(x)+ dx ψ(x) x = 1+ dx ψ(x) = dd x ψ(x) x dd x =1+ dx x =1+ i p dx x Δx = n dx, n : Infinitesimal translation along x: x =x+dx dd x : operator generating infinitesimal translation Finite translation Δx= series of infinitesimal translations dx p x is generator of unitary operator D x, associated with spatial translation along x D x = lim 1+ i n p dx x n = exp i p Δx x Equivalent statements: 1. H invariant under space translations [D x,h]=0 2. [p x,h]=0 3. p x conserved D x = D x 27/02/17 F. Ould-Saada 39

40 Rotations in space Infinitesimal rotation around z axis dr z : operator associated with infinitesimal rotations φ φ + dφ dr z =1+ dφ φ =1+ i L z dφ L z = i φ Finite rotation: Δφ = n dφ, n L z is generator of unitary operator R z associated with spatial rotation around z R z = lim 1+ i n L zdφ n = exp i L zδφ Equivalent statements: 1. H invariant under space rotations [R z,h]=0 2. [L z,h]=0 3. Angular momentum L z conserved 40

41 Gauge invariance discovered in EM interaction Deeply correlated to electric charge (Q) conservation Extension to local gauge invariance Classical electrodynamics Potential φ defined up to an arbitrary constant Measurable quantities (E) depend on potential difference, not on absolute value E, B field expressed in terms of scalar and vector potentials Invariant under transformation of scalar and vector potential of type A µ A' µ (x' µ ) = A µ (x µ )+ Λ 1 A' = A + Λ ; φ ' = φ x µ c As consequence of this symmetry Photon is massless Conservation of Q à invariance for local group of gauge transformations Λ t % ' E = ϕ 1 & c ' ( B = A à gauge field A (quantum=photon) required which couples to Q More in chapter 11 A t 24/02/17 F. Ould-Saada 41

42 (Classical) Poisson s equation: Electric field E vs charge ρ density EM field A Parity: P γ =-1; C- parity: C γ =-1 π 0 =q-qbar P π0 =-1 J PC =0 -+ E ( r,t) = 1 ε 0 ρ( r,t) : P invariant r P r ρ( r,t) P ρ( r,t) ; P, E = ( φ) A A ( r,t) = N ε ( k )e i( t k r Et ) r P r A ( r,t) P P γ A ( r,t) ; E ( r,t) P P γ E ( q C q A ( r,t) C C γ A ( r,t) q C q E ( r,t) C E ( r,t); φ( r,t) C φ( r,t) ) + * + E ( r,t) E ( r,t) ) + * + P γ = 1 r,t), ) + *, + C γ = 1 24/02/17 F. Ould-Saada 42