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Lecture 7 ● Parity ● Charge conjugation ● G-parity ● CP FK7003 1 Already done a lot to understand the basic particles of nature Lepton universality Strong, weak,em ? Isospin singlets 175000 Isospin multiplets small small small Neutrino oscillations/ mass Quark composition Decay modes Symmetry and QM demands much of what is observed! FK7003 2 Transforming under parity Convert the "handedness" of a particle. Eg right-handed particle left-handed particle. Wu's experiment which showed parity violation in weak processes since left-handed antineutrinos don't exist! FK7003 3 Ways of thinking about parity Transformation: r x, y, z r x, y, z Further complementary ways to think about parity invariance. (a) Parity transformation is equivalent to making a reflection and then a rotation. Since nature is invariant to a rotation (angular momentum conservation), is the mirror image of a process/particle equally possible. + v real (b) Not possible. ms=-1/2 ms=1/2 Possible. ”virtual” ˆ H ; Hˆ x, y, z Hˆ x, y, z ; x, y, z , t x, y, z , t t Is x, y, z , t x, y, z , t 2 2 ? Is the Hamiltonian invariant ? Look for a conserved quantity (this lecture). FK7003 4 Intrinsic Parity Consider closed system of particles. Parity transformation is: ri ri ' ri ; ri position vector of a particle at the origin. Consider single particle (easily generalised to many particles) Parity operator: Pˆ r , t Pa r , t (7.01) r , t "mirror image" of particle Pa phase factor, a particle, eg e , u etc. i p r Et i p r Et Eigen function of momentum: r , t e Pˆ r , t P e (7.02) P P For a particle at rest p 0, P r , t e iEt a ˆ iEt P e iEt Pˆ P r , t Pe a (7.03) Pˆ 2 P r , t Pa2 P r , t Pa 1 (7.04) eigen state of Pˆ with eigen value Pa . A particle has an intrinsic parity Pa . This is an intrinsic property of a particle, like baryon number or lepton number. Showed this for particles which can be at rest. General - photons can also have an intrinsic parity (to come) FK7003 5 Intrinsic Parity Let's give our particle a some definite orbital angular momentum: nlm is a parity eigenstate. nlm r Rnl r Yl m , (7.05) 1 3 3 ; Y10 , cos ; Y11 , sin ei (7.06) 4 8 4 r r r r ; ; (7.07) 1 3 3 Y00 , ; Y10 , cos cos 4 4 4 Y00 , 3 3 3 i i sin e ei sin ei sin e 8 8 8 Y11 , - Yl m , 1 Yl m , (7.08) l l Pˆ nlm r Pa nlm r Pa 1 r nlm r Parity=Pa 1 l (7.09). Generalise to two particles c, d rotating in centre-of-mass frame: Pc Pd 1 , l =relative orbital angular momentum. (7.10) l For a Hamiltonian invariant under parity: Pˆ , Hˆ 0 (7.11) We have a conserved quantity with which we can study parity violation (weak)/invariance (strong and em). FK7003 6 Calculating parity Parity is a multiplicative quantum number. It is, like spin, an intrinsic property of a particle. i.e. System of particles A, B with orbital angular momentum l : PABC PA PB 1 (7.12) l 1 fermions: Pfermion Pantifermion 1 (7.13) 2 1, Pantifermion 1 (7.14) From Dirac equation: spinConvention: Pfermion Ground state mesons (no orbital angular momentum): Pmes -1 (7.15) Check and measure with data. Particle Pseudoscalar mesons (s=0) ,K,h,D,B Vector mesons (s=1) K*,w, Parity (P) -1 -1 FK7003 7 Parity of the photon Decay of atomic energy level with single photon emission. Selection rules (dipole): l 1, S 0 (7.16) Eg electron in excited H -atom. Recall (7.09): l=+-1 Pˆ nlm r Pa 1 nlm r l Parity before photon emission=Pa 1 Parity after photon emission=Pa 1 Pa 1 Pa 1 l l 1 l 1 l Atomic energy levels P P 1 1 (7.17) 1 FK7003 8 Parity of the electron 1 fermions: Pfermion Pantifermion 1 (7.13) 2 e e form positronium (short-lived state - forthcoming lecture) Parapositronium L 0, S 0 From Dirac equation: For spin e e Parity conservation: LHS: Pe Pe : RHS: P 1 2 Pe Pe 1 l l positronium (7.18) Measurements of l confirm (7.13). Pair production of e , e makes it impossible to ever determine e or e absolutely. Have to define Pe 1, Pe 1 FK7003 9 Intrinsic parity Particle Parity (P) Electron, muon, tau +1 -1 +1 -1 -1 -1 -1 +1 Positron,antimuon,antitau Quark Antiquark Photon Pseudoscalar mesons (s=0,l=0) ,K,h,D,B.. Vector mesons (s=1,l=0) K*,w, Ground state baryons (l=0) p,n,S.. FK7003 10 Question ● Before Wu’s experiment the t and particles were observed with the same spin, mass, charge. They were thought to be different particles because they decayed into states with different parities P 1 P ? t ● 0 0 0 Calculate the parity of the pion system from the second decay. Assume no orbital angular momentum in the final pion system. ● State which particle t and is. ● Which of the decays violates parity ? FK7003 11 Charge conjugation: C Charge conjugation Cˆ converts each particle to its antiparticle, to . particle which is its own antiparticle, eg 0 , , 0 .. Cˆ | C | ˆ ˆ | | CC C 1 (7.19) a particle which is not its own antiparticle, eg , K , ˆ ˆ | a C C | a C C 1 (7.20) Cˆ | a Ca | a ; Cˆ | a Ca | a CC a a a a C =C -parity is a useful quantum number for particles which are their own antiparticles and are eigenstates of Cˆ , eg 0 , , . Can also be extended into G-parity (later). Ca ,Ca are arbitary phase factors with no physical significance. Cˆ changes the sign of all "internal" quantum numbers: charge, lepton number, baryon number, strangeness, charm,bottomness, I 3 . Mass, energy, momentum, spin unchanged. Cˆ , Hˆ 0 (7.21) when charge conjugation symmetry is respected: em and strong forces. FK7003 12 Question Show, with the example of a neutrino, that charge conjugation is not a symmetry of the weak force. FK7003 13 C-parity (C) • Same game as for parity! We’ve found a symmetry of the em and strong forces, but not of the weak. • Find a quantity conserved in strong and em processes. • Most particles are not eigenstates of Ĉ • Particles which are eigenstates are their own particles, eg 0,,0 • Can also construct eigenstates using particle 0 , antiparticle pairs, eg • Particles or multiparticle states have eigen value known as an intrinsic C-parity quantum number, eg has C1 • C is a multiplicative quantum number like parity. FK7003 14 Calculating C Eigenvalue of Cˆ for pion-pion pair ( ) given by Cˆ | ; l (1)l | ; l (7.22) 1 ˆ Eigenvalue of C for spin- fermion-antifermion pair (ff ) 2 given by Cˆ | ff ; l , s (1)l s | ff ; l , s (7.23) l orbital angular momentum, s total spin angular momentum for combined ff state. C for the lowest mass hadron states (l=0 ) 0 uu dd ff (spin 0) C 1 0 uu dd ff (spin 1) C 1 Unless otherwise stated, we will be dealing in this lecture with systems of particles for which l=0. FK7003 15 Question Using the decay calculate C 0 . 0 FK7003 16 C and P Particle P C Pseudoscalar mesons (s=0) ,K,h,D,B -1 +1 Vector mesons -1 -1 -1 +1 -1 N/A Charged Leptons 1 N/A Charged antileptons -1 N/A C value only applicable for particles which are their own anti-particles (s=1) K*,w, Photon Ground state baryons FK7003 17 Formalism A particle is characterised by the form JPC, eg 1-J=total angular momentum, P=parity(+ = +1,- = -1), C=charge conjugation number (+ = +1, - = -1) In certain situations C is not a useful quantum number – most particles are not eigenstates of C: JP is used. Eg : 0 , 0 : 0 , K : 0 , :1. P=1 C=1 ”even” parity, P=-1 ”even” C-parity, C=-1 FK7003 ”odd” parity ”odd” C-parity 18 G-parity The strong force is invariant under isospin and respects charge conjugation. Can we combine these symmetries to get another useful symmetry ? Combine Cˆ charge conjugation with Rˆ rotation around " y " (or 2) axis in isospin space. For pions Rˆ 0 0 ; Rˆ (7.24) Same algebra of angular momentum as for isospin. 3 3 3 0 0 0 Spherical harmonic Y1 , cos , Y1 , cos cos Y1 , (7.25) 4 4 4 3 3 3 i i 1 i 1 Y11 , sin e , Y1 , sin e sin e =Y1 , (7.26) 8 8 8 3 3 3 i 1 i 1 i 1 Y1 , sin e , Y1 , sin e sin e =Y1 , (7.27) 8 8 8 For neutral pion Cˆ 0 0 (7.28) ˆ ˆ (7.29) Gˆ 0 G 0 0 Define Gˆ CR G 1. (7.30) Eigenvalue of Gˆ G-parity or G - conserved in strong interactions. Choose Cˆ (7.31). Obs! Results don't depend on our choice of arbitary phase Choose that all isomultiplet members have same G. Gˆ (7.32) Generally for an isomultiplet, eg , , 0 : G (-1) I C (7.33) C C -parity for neutral member of isomultiplet, I isospin. FK7003 19 Particles for which G-parity is relevant G G parity, a useful quantum number. Which particles are eigenstates of Gˆ ? Particles carrying no flavour quantum numbers (S , C , B ) or baryon number. Gˆ G , Gˆ G Gˆ K a K 0 Gˆ 0 b 0 Some eigenvalues (not exclusive list): w , ,, '... C 1 ; I 0 From (7.30) G (-1)0 (1) 1 , , ... C 1 ; I 1 G (-1)1 (1) 1 (7.34) Since G is a multiplicative conserved quantity. states of N -pions have G 1 (7.35) N explains why some particles decay strongly to 2 and others to 3 . FK7003 20 Some decays explained with G-parity Eg 2 LHS: G 1, 3 LHS: G 1, RHS: G 1 1 ok!! 2 RHS: G 1 1 forbidden! 3 w 3 LHS : G 1, RHS: G 1 1 ok!! 3 w 2 LHS : G 1, RHS: G 1 1 forbidden!! 2 , "Explains" decays of many other particles: ',,h,h '..... FK7003 21 Conserved quantities/symmetries Quantity Strong Weak Electromagnetic Energy Linear momentum Angular momentum Baryon number Lepton number Isospin - - Flavour (S,C,B) - Charges (em, strong and weak forces) Parity (P) - C-parity (C) - G-parity (G) - - CP - T CPT Coming up - FK7003 22 CP • C and P are not separately respected in weak decays • What about CP ? ( left - handed) ( left - handed! ! ) X ( right - handed) Apply C Now apply P Combined CP transforme d decay ok! • Original and CP-transformed decays occur with same rate. CP symmetry is respected in many weak processes. FK7003 23 FK7003 24 Neutral kaons We define a neutral K0 by its quark content (sd), mass (498 MeV), spin (0), isospin (I=1/2,I3=-1/2) - a normal particle ! Consider production of K 0 : eg . p n K 0 K p p Strong decays forbidden (no flavour violation!) It decays weakly with time: Non-exponential decay time! K 0 seems to comprise two other particles (K1 and K 2 ) with definite lifetimes t 1 ,t 2 t 1 1010 s, t 2 5 108 s FK7003 25 Strategy Test the hypothesis that CP is a good symmetry of the weak force. Try to form CP eigenstates from K0 and K0 and check they decay in CP-conserving ways (recall and C conservation.) Pˆ | K 0 | K 0 Cˆ | K 0 | K 0 Pˆ | K 0 | K 0 (7.36) Cˆ | K 0 | K 0 (7.37) ˆ ˆ | K 0 | K 0 CP ˆ ˆ | K 0 | K 0 (7.38) CP ˆ ˆ: Normalised eigenstates of CP 1 1 0 0 0 0 0 | K | K | K | K | K | K (7.39) 2 2 2 ˆ ˆ | K 0 | K 0 ˆ ˆ | K 0 | K 0 (7.40) CP CP 1 1 2 2 0 1 FK7003 26 1 | K 0 | K 0 2 1 | K 0 (| K10 | K 20 ) 2 | K10 1 | K 0 | K 0 (7.39) 2 1 | K 0 (| K10 | K 20 ) (7.41) 2 | K 20 Hypothesise that the CP eigenstates are the two neutral states observed in the K 0 decay. If CP is conserved then K10 decays into CP 1 states , 0 0 and K 20 decays into CP 1 states 0 , 0 0 0 . K10 and K 20 . Two neutral kaons states had been observed decaying weakly: t 1 1010 s, t 2 5 108 s The 2 decay will happen faster because of greater phase space - associate K10 ,t 1:CP 1 and K 20 ,t 2 : CP 1 FK7003 (7.42) 27 Neutral Kaons and Strangeness Oscillations 1 (| K 0 | K 0 ) 2 1 0 | K (| K10 | K 20 ) 2 1 (| K 0 | K 0 ) (7.39) 2 1 0 | K (| K10 | K 20 ) (7.41) 2 | K10 | K 20 Consider in the kaon rest frame and allow a decay t t i i i ( m ) t i ( m )t 1 2 1 1 im1t 2t1 im2 2t1 2t1 2t 2 o 0 0 0 K (t ) K1 e e K2 K1 e K 20 e e e 2 2 (7.43) Consider a particle produced at t=0 as a K0 . Amplitude that it is still a K0 at a later time t: i i i ( m ) t i ( m )t 1 2 1 2t1 2t 2 o o 0 0 0 K K (t ) K1 K 2 ) e K1 e K 20 2 1 1 i ( m ) t i ( m )t 1 2 1 2t1 2t 2 o o K K (t ) e e (7.45) 2 FK7003 (7.44) 28 Probability that it is still a K0 at a later time t: K o K o (t ) 2 i ( m2 ) t i ( m1 )t i ( m2 )t 1 i ( m1 2t1 )t 2t 2 2t1 2t 2 e e e e 4 K o K o (t ) i o o 2 K K (t ) Similarly: K o K o (t ) 2 2 i i t t i i i i i ( m m ) t i ( m m )t 2 1 1 2 1 t1 t2 2t1 2t 2 2t1 2t 2 e e e e 4 2 K o K o (t ) i t t t 1 1 ( ) 1 t1 t2 2 t1 t 2 i ( m2 m1 ) t i ( m1 m2 ) t e e e e e 4 t t t 1 1 ( ) 1 t1 t2 2 t1 t 2 e e 2e cos (m2 m1)t 4 (7.46) t t t 1 1 ( ) 1 t1 t2 2 t1 t 2 e e 2e cos (m2 m1)t (7.47) 4 Strangeness oscillation!! Beam of particles, initially pure K 0 . After time t (in particles rest frame) study intensity of the beam and composition of K 0 , K 0 . Eg compare rates of K 0 p uds (not possible as strong reaction for K 0 , why??) and K 0 p K 0 , K 0 p K 0 . FK7003 29 Kaon oscillations Intensity (Niebergall et al., 1974) K Oscillations observed. 0 From many experiment: m2 m1 3.5 106 eV (7.44) (Obs! m2 m1 mK 1014 -tiny, m2 m1 from other experiments. ) 0.25 K 0 2 50% of the beam | K10 (short lived) has died away leaving the 0 1 | K 0 | K 0 with apparently 2 equal contributions of | K 0 ,| K 0 . | K 20 4 6 8 Time in K0 rest frame (x10-10 s) The | K 20 decays very slowly (t 5 108 ). FK7003 30 Question To measure the oscillations of a beam of neutral kaons of energy 10 GeV how large should an experiment be ? (Niebergall et al., 1974) Intensity ● Oscillation pattern visible over time K 0 scale tosc 109 s in kaon rest frame. E 10 20 . m 0.5 Distance travelled in the lab Speed v c ; ctosc 20 3 108 109 6m. Pretty small compared with neutrino oscillations! K 0 2 0 4 6 8 Time in K0 rest frame (x10-10 s) FK7003 31 Some interpretation and comparisons K 0 oscillation (1) Particles are produced in a flavour eigenstate by (s ). (2) As they pass through space the proportion of Neutrino oscillation (2-component approximation) (1) Particles are produced in a beam in a flavour eigenstate, eg . of flavour s , s oscillates (K 0 K 0 ). (3) This is a simple quantum mechanical effect. (3) This is a simple quantum mechanical effect. (2) As they pass through space the proportion of flavour , t oscillates ( t ). The is not a mass eigenstate, rather a mixture 0 The K is not a mass eigenstate, rather a mixture 0 1 of two mass eigenstates: 1 , 2 . 0 2 of two mass and CP eigen states K , K . (4) The beam intensity drops since the K 0 decays. (5) Oscillations seen over distance scales m. (6) The K 0 K 0 transformation can be understood by the Feynman diagram of a weak process in which strangeness is violated (next lecture). K0 (4) The beam intensity stays constant since the neutrinos do not decay (nothing to decay down to perhaps!). (5) Oscillations seen over distance scales 1000 km. (6) The t transformation can't be understood by any Feynman diagram since lepton number violation "can't" happen at a vertex. K0 |S|=2 Strangeness violated t L L t 1 Lepton number violated! FK7003 32 Summary ● Discrete symmetries Parity (P) Charge conjugation (C-parity) G-parity Fundamental symmetries of nature constrain the behaviour of particles CP ● ● ● Neutral kaons Strangeness oscillations Next lecture – CP violation FK7003 33