* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Elementary Particle Physics
Ferromagnetism wikipedia , lookup
Double-slit experiment wikipedia , lookup
Renormalization group wikipedia , lookup
Tight binding wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum entanglement wikipedia , lookup
Nitrogen-vacancy center wikipedia , lookup
Hydrogen atom wikipedia , lookup
Scalar field theory wikipedia , lookup
Quantum state wikipedia , lookup
Matter wave wikipedia , lookup
Ising model wikipedia , lookup
EPR paradox wikipedia , lookup
Technicolor (physics) wikipedia , lookup
Wave–particle duality wikipedia , lookup
Light-front quantization applications wikipedia , lookup
Atomic theory wikipedia , lookup
Bell's theorem wikipedia , lookup
Identical particles wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Wave function wikipedia , lookup
Spin (physics) wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
The quark model Goal - to demonstrate that the basic properties of the observed mesons and baryons can be understood with a quark model: (a) Composition of multiplets (b) Spins (c) Magnetic moments Show the theory and ideas which won Gell-Mann the Nobel prize in 1969. FK7003 1 The quark model Start with ground state hadrons containing only u or d quarks which have no orbital angular momentum. Postulates: (1) Mesons consist of qq and baryons of qqq. (2) Quarks carry a quantum number: isospin. The u, d u , d quarks represent different projections of the same quark (antiquark) in an internal isospin space, as p, n. When adding different isospins to form a hadron nature uses SU(2) symmetry to determine the allowed flavour combinations. (3) Quarks (antiquarks) carry a colour quantum number: r , g , b (r , g , b ) When adding colours nature has chosen an SU(3) symmetry to determine the possible colour combinations. Nature always chooses hadrons to be colour singlets. With these postulates can we support the quark picture of hadrons ? FK7003 2 Forming hadrons When combining two particles to form a composite particle we need to know what combinations could occur in nature given the various conservation laws. Out of everything which can exist our theory should predict only those do exist!! How do the various properties add up ? (1) Angular momentum (2) Isospin (3) Colour Do we satisfy Pauli's exclusion principle ? - is the wave function for a hadron consisting of identical particles antisymmetric to the interchange of two fermions ? FK7003 3 Combining particles (1) – angular momentum Combine a quark and an antiquark Angular momentum: Add in usual quantum mechanics way (Clebsch-Gordon co-efficients) Symmetric spin-1 triplet: 11 11 11 = 22 22 11 1 1 1 1 2 2 2 2 1 11 1 1 1 1 11 1 22 2 2 2 2 22 2 2 Antisymmetric spin-0 singlet 10 00 1 11 1 1 1 1 11 1 22 2 2 2 2 22 2 2 An alternative way to do this is to use group theory: Invariance to a rotation SU(2) symmetry in nature SU(2) group theory language: 2 2 1 3 gives the same result. FK7003 4 Quarks and isospin Same mathematics as angular momentum. The up and down quarks form a doublet: 1 1 u 2 2 iso 1 1 ; d 2 2 iso Anti-up and down quarks form a doublet: 1 1 u 2 2 iso 1 1 ; d 2 2 iso (- sign is a technical and (for us) unimportant detail) The other quarks carry no isospin. FK7003 5 Isospin of antiquarks (not for lecture or exam) 1 1 2 2 Light quarks form an isospin doublet: u iso 1 0 iso d 1 1 2 2 iso 0 1 iso From spin questions in lecture 5 and eqn. 523. 1 0 1 1 U ( ) Rˆ 2 2 0 iso 1 iso 0 1 1 1 1 1 ; U ( ) Rˆ 2 2 iso 2 2 iso 1 iso 0 iso iso iso Rˆ rotation of around "y"-axis (or 2-axis) in isospin space Rˆ u d ; Rˆ d u (6.08) 1 1 2 2 Define charge conjugation phase factors: Cˆ u u ; Cˆ d d Cˆ u u ; Cˆ d d ˆˆ u u CC Apply charge conjugation transformation to rotation operations: Rˆ u d ; Rˆ d u Rˆ u d ; Rˆ d u (6.10) ˆ ˆ d d (6.09) ; CC Use this info to define antiquark isospin doublet. Desire that antiquark doublet transforms in the same way as the quark doublet (necessary when we combine quarks and antiquarks together in mesons and want to transform the whole thing by a rotation). i.e. Rˆ upper lower ; Rˆ lower upper 1 0 Then Rˆ 1 iso 0 iso Rˆ d Without the negative sign: d u set: d 1 1 2 2 iso 1 0 iso 0 1 Rˆ 0 iso 1 iso 0 1 1 u - 2 2 1 , ok! 1 1 1 2 2 0 u Rˆ u ; 1 1 2 2 d iso 0 (6.11) 1 iso , ok! (6.12) 1 0 0 1 Then Rˆ Rˆ d u ,not ok! ; Rˆ Rˆ u d ,not ok! (6.13) 0 1 1 0 iso iso iso iso The minus signs are a way ensuring symmetry under charge conjugation and are defined after a convention. FK7003 6 Combining particles (2) – isospin Combine a quark and an antiquark but only u, d quarks Isospin: u 11 22 ,u iso 1 1 2 2 ,d iso 1 1 2 2 iso 11 22 1 1 2 2 iso 1 1 2 2 iso iso 1 1 2 2 iso iso 1 1 2 2 iso ,d iso Symmetric spin-1 triplet: ud 11 iso 11 22 1 du ud 10 2 iso iso 11 22 du 1 1 iso iso 1 11 2 22 iso 1 1 2 2 iso iso 1 1 2 2 11 22 iso 11 22 Antisymmetric spin-0 singlet 1 du ud 00 2 iso 1 11 2 22 FK7003 7 Combining particles (3) – colour Combine a quark and antiquark with colour and anticolour 3 3 8 1. Singlet: singlet 1 RR BB GG 3 Octet: octet RB ; RG ; BR ; GR ; BG ; 1 RR GG ; 2 1 RR GG 2 BB . 6 We've never observed a particle with naked colour so nature clearly takes the singlet. Obs if you're unconvinced - we also test SU(3) when combining three quarks for baryons. FK7003 8 Can we describe the observed mesons Need to make sure our model leads to isomultiplets of light hadrons. Spin 0: ,0, (isospin 1) and 0 (isospin 0) -1 0 +1 r r0 r Spin 1: r ,0, (isospin 1) and w 0 (isospin 0) Can we form wave functions ? w? -1 FK7003 0 +1 9 Meson wave functions The total wave function consists of different parts: spatial, spin, flavour and colour. space iso flav col Consider ground state hadrons - no orbital angular momentum Lq 2 q Central potential: space R(r12 )Yl m r12 L 0 Yl m r12 1, space R(r12 ) space symmetric for exchange of quarks i.e. r12 r12 r12 1 (symmetry only important for baryons since q, q are not identical particles). wave function: 1 1 space ud RR BB GG 3 2 1 uR d R u B d B uG d G = space 6 uR d R u B d B uG d G q Lq 1 of the time the up quark would be red and spin-up. 6 Its straightforward to form the meson wave functions and thus demonstrate that these states are quantum mechanically possible. if you could pull apart a , FK7003 10 Baryons Same procedure as before. Which spin wave functions can be formed ? 3 S Spin spin symmetric with change of a pair of quarks 2 33 3 3 22 2 2 31 1 3 1 1 22 2 2 3 3 1 PA Spin spin partially antisymmetric with change of a pair of quarks. 2 12 12 23 23 11 1 1 1 1 22 2 2 2 2 antisymmetric upon exchange of particles 1 and 2 11 1 1 1 1 22 2 2 2 2 antisymmetric upon exchange of particles 2 and 3. FK7003 12 23 11 Can also have 13 13 11 1 1 1 1 22 2 2 2 2 antisymmetric upon exchange of particles 1 and 3. 13 Not independent of the other two 11 22 13 11 22 12 11 22 23 ; 1 1 2 2 13 1 1 2 2 12 1 1 2 2 23 Could obtain the combined spin states either from Clebsch-Gordon co-efficients or group theory/SU(2) symmetry. 222 422 One quadruplet of symmetric spin combinations and 2 independent doublets of partially antisymmetric spin combinations. FK7003 12 Baryons Now do the same for isospin. Same as before - isospin and angular momentum have same algebra. 3 S Isospin iso symmetric with change of a pair of quarks 2 33 3 3 31 1 3 1 uuu ; ddd ; uud udu duu ; 2 2 iso 2 2 iso 2 2 iso 2 2 3 12 12 1 1 1 1 ud du u ud du d 2 2 2 2 iso iso antisymmetric upon exchange of particles 1 and 2 11 22 iso 1 udd dud ddu 3 1 PA iso partially antisymmetric with change of a pair of quarks. 2 Isospin 11 22 23 23 1 1 1 1 u ud du d ud du 2 2 2 2 iso iso antisymmetric upon exchange of particles 2 and 3. 13 12 iso 23 iso 13 1 1 1 1 Also: uud duu ; udd ddu iso13 2 2 iso 2 2 iso Antisymmetric upon exchange of p articles 1 and 3. Not independent of the other two 11 22 11 22 13 iso 11 22 12 11 22 iso 23 ; iso 1 1 2 2 13 iso 1 1 2 2 12 1 1 2 2 iso FK7003 23 iso 13 Combining colour Follow the postulates - combine colours according to SU (3) symmetry. We have 3 colours R, G, B 3 projections of one quark 3 3 3 10 8 8 1 According to the postulate, the colour singlet is chosen: 1 col rgb rbg gbr grb brg bgr 6 FK7003 14 Baryon wave function space spin iso col Spatial wave function. Depends on orbital angular momentum. General: two contributions: L12 angular momentum of 1 and 2 in "subsystem" L3 angular momentum of 3 about c.m of 1 and 2 in the overall c.m. frame: Ltot L12 L3 We only consider ground state baryons: angular momentum comes only from spin: L12 L3 0 space depends only the distance between quarks no change if r 12 r12 spin iso must be completely symmetric to satisfy Pauli's exclusion principle. FK7003 15 Can we describe the observed baryons Need to make sure our model leads to isomultiplets of light hadrons. -1/2 1 1 Spin : p, n (isospin ) 2 2 3 3 ,0, , Spin : D (isospin ) 2 2 D Can we form wave functions ? FK7003 D0 0 +1/2 D D I3 -3/2 -1/2 +1/2 3/2 16 Test the quark model – Spin 3/2 baryon 3 baryons should be present in nature 2 with the same properties except those governed by quark content,eg charge. If our theory is correct then all four spin- S S D spin iso (ignore colour and space part) symmetric ! Try and form a spin spin 3 3 1 wave function for D uud 2 2 2 3 1 1 2 2 3 D iso 1 uud udu duu 3 u u d u u d u u d 1 u d u u d u u d u 3 d u u d u u d u u 1 ? 2 No ! It is imposible to form a completely symmetric spin-flavour wave function and thus satisfy Pauli's exclusion principle. 1 no multiplet of four baryons for spin- observed in data ! 2 Can we form a wave function for a D uud if it was spin FK7003 17 Test the quark model – spin-1/2 baryons Ignore the spatial and colour parts. Need a symmetric flavour - spin wave function. 12 12 13 13 23 23 A spin iso spin iso spin iso partially antisymmetric partially antisymmetric symmetric! A normalisation factor 23 is not independent of 13 , 12 1 ). 2 Consider spin-up projection and form the wave function. Consider the proton (spin 1 1 1 A udu duu uud udu uud duu 2 2 2 Left as an exercise…. 1 2u d u u d u u d u d u u 2d u u 18 d u u u u d u u d 2u u d Can form wave functions for two spin 1 states ( p, n). 2 3 ; iso is symmetric and spin partially antisymmetric 2 3 can't form doublet from spin baryons. Agreement with data! 2 If baryons are spin FK7003 18 So far so good Can understand the existence and properties of multiplets the very lightest hadrons with SU(2)-isospin symmetry. SU(2) symmetry is ”good” - all hadrons in an isospin multiplet have similar masses since u,d have similar masses. FK7003 19 Extending the model These multiplets of the lowest mass hadrons are observed. Our model should able to predict these (and nothing more). -1 -½ ½ +1 -1 I3 Meson nonet (spin 0) Baryon decuplet (spin 1/2) -½ ½ +1 I3 Meson nonet (spin 1) FK7003 Baryon decuplet (spin 3/2) 20 Extending the quark model Start with ground state hadrons containing only u , d , s quarks which have no orbital angular momentum. Postulates: (1) Mesons consist of qq and baryons of qqq. (2) Quarks carry a flavour quantum number. The u, d , s u , d , s quarks represent different projections of the same quark (antiquark) in an internal flavour space. When adding different isospins to form a hadron nature uses SU(3) symmetry to determine the allowed flavour combinations. (3) Quarks (antiquarks) carry a colour quantum number: r , g , b (r , g , b ) When adding colours nature has chosen an SU(3) symmetry to determine the possible colour combinations. Nature always chooses hadrons to be colour singlets. With these postulates can we support the quark picture of hadrons ? FK7003 21 Combining quarks (3) - flavour SU (3): 3 3 8 1 Octet: 1 1 dd uu 2 ss uu - dd , us , ds , us, ds, ud , du , 6 2 1 dd uu ss Singlet: 3 Isospin multiplets form part of pattern. Meson nonet (octet+singlet) (spin 0) -1 -½ +1 ½ FK7003 I3 22 Identifying the states Put together the flavour states into observed particles: K 0 sd K - su K 0 ds K us (Mass 500 MeV) 1 0 du (uu dd ) du (Mass 140 MeV) 2 Two neutral combinations with same quantum numbers: K 0 ds 1 1 8 (uu dd 2ss ) and 0 (uu dd ss ) 6 3 Nature uses a linear combination of them as observable particles: (if it can happen, it will happen!) 100 1 (uu dd 2 ss ) 6 1 ' 8 sin 0 cos (uu dd ss ) 3 0 8 cos 0 sin FK7003 (Mass 550 MeV) (Mass 950 MeV) 23 SU(3) Flavour – mesons of spin 1 (vector) K *0 ds K *0 sd K *- su K *0 ds K * us (Mass 890 MeV) 1 (uu dd ) r du (Mass 780 MeV) 2 8 , 0 in analogy with 0 ,8 r - du r 0 35o 1 (uu dd ) (Mass 550 MeV) 2 8 cos 0 sin ss (Mass 950 MeV). w 8 sin 0 cos FK7003 24 Meson wave functions The total wave function consists of different parts: spatial, spin, flavour and colour. space spin flav col As earlier K wave function: space 1 1 us RR BB GG 2 3 1 uR sR uB sB uG sG = space 6 uR sR uB sB uG s G 1 if you could pull apart a K , of the time the antistrange quark would be antired 6 and spin-down. FK7003 25 Baryons SU (3) : 3 3 3 10 8 8 1 uuu ddd 1 ddu dud udd 3 1 dds dsd sdd 3 1 duu udu uud 3 1 uus usu suu 3 1 uds usd dus dsu sud sdu 6 1 uss sus ssu 3 1 dss sds ssd 3 sss -3/2 -1 -1/2 0 1/2 1 Decuplet: S - completely symmetric states. FK7003 3/2 I3 26 1 ud du d 2 1 ud du u 2 1 2 ud du s us su d ds sd u 12 Octet: 12 : antisymmetric in 1 and 2 1 us su u 2 1 ds sd d 2 1 us su d ds sd u 2 1 ds sd s 2 1 us su s 2 1 d ud du 2 1 u ud du 2 1 2s ud du d us su u ds sd 12 Octet: 23 : antisymmetric in 2 and 3 1 d ds sd 2 1 u us su 2 1 d us su u ds sd 2 1 s ds sd 2 1 s us su 2 1 uds usd dsu dus sud sdu 6 Singlet: A : completely antisymmetric FK7003 27 1 udd ddu 2 1 uud duu 2 1 2 usd dsu uds sdu dus sud 12 Octet: 13 : antisymmetric in 1 and 3 1 uus suu 1 dds sdd 2 2 1 uds sdu dus sud 2 1 dss ssd 2 1 uss ssu 2 13 12 23 - not independent of the other two. FK7003 28 Test the quark model – decuplet If our theory is correct then all ten decuplet baryons should be present in nature with the same properties except those governed by quark content,eg charge. S D spin Sflavour (ignore colour and spa ce part) symmetric ! Try and form a spin spin 3 3 1 wave function for dds 2 2 2 3 1 1 1 ; flavour dds dsd sdd 2 2 3 3 d d s d d s d d s 1 d s d d s d d s d 3 d s d d s d d s d 1 ? 2 No ! It is imposible to form a completely symmetric spin-flavour wave function and thus satisfy Pauli's exclusion principle. Can we form a wave function for a dds if it was spin no decuplet of baryons for spin- 1 observed in data ! 2 FK7003 29 The quark model works Possible mutliplets of ground state baryons with similar properties Spin 1/2 octet Observed Predicted to in nature ? exist by the quark model Comments PA spin PA flav (part antisym part antisym) symmetric. Spin 3/2 octet X X S spin PA flav (sym part. antisym) can't be symmetric. Spin 1/2 singlet X X PA A spin flav (part antisym full antisym ) can't be symmetric. Spin 3/2 singlet X X S A spin flav (sym full antisym) can't be symmetric. Spin 1/2 decuplet X X PA spin Sflav part antisym sym can't be symmetric. Spin 3/2 decuplet S spin Sflav (sym sym) symmetric. FK7003 30 Success of SU(3) With SU(3) symmetry and u,d,s quark model can understand the existence and basic properties of all light ground state hadrons SU(2) symmetry is ”good” - all hadrons in an isospin multiplet have similar masses since u,d have similar masses. SU(3) symmetry is ”approximate” – larger mass differences owing to difference of s and u,d, masses. To include charm, use SU(4) symmetry – useful to enumerate states but a poor symmetry owing to the heavy charm mass. FK7003 31 A historical note Like all good theories, the theory of quarks came with a prediction. 3 The sss (spin ) state had not 2 been measured when the theory was developed. When measured this helped to confirm the quark model. If the sss was not present in nature the whole theory would have been dead. FK7003 32 Baryon magnetic moments Need to check if quarks are not just a simple mathematical model. Wave functions tell us combinations of quarks and spins. predict magnetic moments and test against octet baryons. Magnetic dipole moment of a point-like particle: qSˆz 1 1 ˆ z ; Spin- ; S z . m 2 2 Spin-up baryon B: B aq1 q 2 q 3 +bq1 q 2 q 3 ... ˆ z B a 1z z2 z3 q1 q 2 q 3 b 1z z2 z3 q1 q 2 q 3 ..... i.e. not an eigenstate need to find expectation value: zB B | ˆ z | B Proton: 1 2u d u u d u u d u d u u 2d u u 18 d u u u u d u u d 2u u d 2 2 u 2 d 2 u u d u u d u d u u 2 d 2 u 2 u 1 2 z 2 z 2 z z z z z z z z z z 2 z 2 z 2 z d u u u u d u u d 2 u 2 u 2 d 18 z z z z z z z z z 2 z 2 z 2 z 1 1 zp = 24 zu 6 zd 4 zu zd 18 3 p z FK7003 33 Baryon magnetic moments qS z q m 2S z u 3mu Sz ; d 3md Sz ; s 3ms Quark constituent masses in baryons: mu md 363 MeV ; ms 538 MeV Predict all octet baryon magnetic moments. Baryon p n L 0 0 Moment 4 u 1 d z z 3 3 4 d 1 u z z 3 3 zs 4 u 1 s z z 3 3 2 u 1 z zd zs 3 3 4 d 1 s z z 3 3 4 s 1 u z z 3 3 4 s 1 d z z 3 3 Prediction Experiment 2.79 2.793 -1.86 1.913 -0.58 -0.61 2.68 2.33 +/- 0.13 0.82 No measurement -1.05 -1.41 +/- 0.25 -1.40 -1.253 +/- 0.014 -0.47 -0.69 +/- 0.04 Good agreement with data! FK7003 34 Summary ● ● ● ● Hadrons with similar properties occur in multiplets in nature The quark model is based on a few simple postulates The quark model successfully describes the observed multiplet structure Still to show evidence that ”quarks” are physical objects (next lecture) FK7003 35