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Fundamental principles of particle physics G.Ross, CERN, July03 Fundamental principles of particle physics G.Ross, CERN, July03 Outline Introduction - Fundamental particles and interactions Symmetries I - Relativity Quantum field theory Cross sections and decay rates - theory confronts experiment Symmetries – Gauge symmetries and QED Mainly fermions and the weak interactions Fundamental principles of particle physics G.Ross, CERN, July03 Fundamental Interactions Strength Strong s 4 c 1† 2 1 Electromagnetic em 4e c 137 Weak GF m 2p 105† Gravitational GN m 2p 1036 g s2 † Short range Fundamental principles of particle physics G.Ross, CERN, July03 Fundamental Interactions Strength Strong s 4 c 1† 2 1 Electromagnetic em 4e c 137 Weak GF m 2p 105† Gravitational GN m 2p 1036 g s2 Fundamental Particles Fundamental principles of particle physics G.Ross, CERN, July03 Fundamental Interactions Strength Strong s 4 c 1† 2 1 Electromagnetic em 4e c 137 Weak GF m 2p 105† Gravitational GN m 2p 1036 g s2 Fundamental Particles Fundamental principles of particle physics G.Ross, CERN, July03 Fundamental Interactions Fundamental Particles Strength Strong s 4 c 1† 2 1 Electromagnetic em 4e c 137 Weak GF m 2p 105† Gravitational GN m 2p 1036 g s2 Strength Size E PE + KE = - Energy e2 4 r p2 2 me p.r Heisenberg's Uncertainty principle r 1 2 3 5 E- -5 -10 E 4 -15 e2 4 r 2 mr 2 2 -20 -25 E r 0 em e2 4 c me rc Units c 3.10 m/sec 8 1034 kg m2 /sec Natural Units Length : L Time : T Energy : E or Mass : m } Choose units such that : c 1 L /T 1 E.T 1 unit left : choose E 1 GeV ( M .L2 / T ) ( 109 electron volts = 1.6 10-10 J) Natural Units 1 c 3.108 m/sec = 3.1023 fm/sec 1 10 kg m /sec = 10 J/sec 34 2 1.6 10-10 J energy of 1 GeV length of 1 GeV 1 time of 1 GeV 1 me † GeV fm ...typical of elementary particles 0.2 fm 0.7 1024 sec em 1010 m 105 fm 3 1 5 1.8 10-27 kg mass of 1 GeV ratom 34 0.510 GeV † e2 4 c me rc 1 me r em e2 4 2 10 Fundamental Interactions and sizes Strength Strong s 4 c 1 2 1 Electromagnetic em 4e c 137 Weak GF m 2p 105 Gravitational GN m 2p 1036 g s2 m1r (dimensionless - same in any units) ratom me rnucleus mp 1010 m 105 fm 0.5103 GeV 1015 m 1 fm 1 GeV em e2 4 strong 102 g2 4 1 Elementary particles Leptons : e , , e , , Quarks : 2/ 3 2/ 3 u ,c , t 2/ 3 d 1/ 3 , s 1/ 3 , b 1/ 3 Elementary particles Leptons : e , , e , , Quarks : u, c, t u, c, t u, c, t Hadrons : 3 strong “colour” charges d , s, b d , s, b d , s, b Elementary forces Exchange forces e Electromagnetism Vem (r ) e1 e2 1 4 r Nucleus Experiments conducted in momentum space : Vem ( q ) ik .r 3 V ( r ) e d r em k 2 The strength of the force Exchange forces e Electromagnetism Vem (r ) (Q ) 2 e1 e2 1 4 r Q 1 R Nucleus Experiments conducted in momentum space : Vem ( q ) ik .r 3 V ( r ) e d r em k 2 Q 2 k 2 " virtual photon " Exchange forces q Strong interactions Vstrong (r ) g s2 1 4 r g(Q2) q In momentum space : Vs ( q ) V ( r )e s ik .r 3 d r s k 2 " virtual gluon " Exchange forces e Weak force Vweak (r ) g1 g2 1 4 r e M Z r Z (Q2) q In momentum space : Vweak ( k ) ik .r 3 V ( r ) e d r weak weak k 2 M Z2 " virtual Z boson " GF 1 M Z2 Exchange forces m1 Gravitational force G (Q2) Vgravity (r ) G m1 m2 N r m2 Vgravity ( k ) V gravity ( r )e ik .r 3 d r GN m1m2 k 2 " virtual graviton " Exchange forces m1 Gravitational force G (Q2) Vgravity (r ) G m1 m2 N r m2 GN 6.6 1011 m3 / kg.sec 2 ....could provide fundamental scale : mass ( c / GN )1/ 2 1.2 1019 GeV length 1033 cm or maybe M Planck not fundamental ! Units c 3.10 m/sec 8 1034 kg m2 /sec c 1 L /T 1 E.T ( M .L2 / T ) Symmetries Central to our description of the fundamental forces : Relativity - translations and Lorentz transformations Lie symmetries - SU (3) SU (2) U (1) Copernican principle : “Your system of co-ordinates and units is nothing special” Physics independent of system choice Special relativity a (ct , x, y, z ) Space time point Space-time vector not invariant under translations (a a) a a (ct , x, y, z ) Invariant under translations …but not invariant under rotations or boosts Einstein postulate : the real invariant distance is a 0 2 a 1 2 a 2 2 a 3 2 3 , 0 g a a a a a 2 g diag (1, 1, 1, 1) Physics invariant under all transformations that leave all such distances invariant : Translations and Lorentz transformations Lorentz transformations : 3 x x x x 0 g x x g x x g g (Summation assumed) Solutions : 0 1 0 cos 0 0 0 sin 3 rotations R 3 boosts B cosh sinh 0 0 0 0 0 sin 1 0 0 cos Space reflection – parity P 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 sinh cosh 0 0 0 0 1 0 0 0 0 1 Time reflection, time reversal T 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 The Lorentz transformations are a group g1 g2 L g1 , g 2 L if Rotations R( ) e iJ . / [ Ji , J j ] i ijk , Angular momentum operator (c. f . J r p) 3 k 1 ijk Jk totally antisymmetric Levi-Civita symbol, 123 1 Representations e.g. J 1/ 2 J i i / 2 J J z i ( x y y x ).. z 1/ 2 0 1 J 0, 12 ,1,... 0 1 0 i 1 0 2 3 1 0 i 0 0 1 1 Demonstration that RZ ( ) e iJ z x cos RZ ( ) y sin sin x x ' cos y y ' For small = , x' x y y ' y x RZ ( ) (t , x, y, z ) (t , x y, y x, z ) (t , x, y, z ) ( y x ) x y (1 i ( xp y ypx )) (t , x, y, z ) i.e. RZ ( ) (1 i ( xp y ypx )) 1 i J z Hence iJ z RZ ( n ) (1 i J Z )n e n Derivation of the commutation relations of SU(2) Rx ( ) Ry ( ) Rx ( ) Ry ( ) 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 Rz ( ) (1 i J z ) 0 0 1 Rx ( ) Ry ( ) Rx ( ) Ry ( ) (1 i J x )(1 i J y )(1 i J x )(1 i J y ) ( J x J y J y J x ) Equating the two equations implies [ J x , J y ] iJ z QED Rotations Ji Boosts Ki The Lorentz group [J i , J j ] i ijk J k } [J i , K j ] i ijk K k [K i , K j ] i ijk J k (M i( x x x x ) Generate the group SO(3,1) J i 12 ijk M jk Ki M 0i ) To construct representations a more convenient (non-Heritian) basis is Ni 12 ( J i iKi ) [N i , N j ] i ijk N k [N i† , N j† ] i ijk N k† [N i , N j† ] 0 } SU (2) SU (2) representation (n, m) Rotations Ji Boosts Ki The Lorentz group [J i , J j ] i ijk J k } [J i , K j ] i ijk K k [K i , K j ] i ijk J k (M i( x x x x ) Generate the group SO(3,1) J i 12 ijk M jk Ki M 0i ) To construct representations a more convenient (non-Hermitian) basis is Ni 12 ( J i iKi ) [N i , N j ] i ijk N k Representations (n, m) J i N i N i† J nm [N i† , N j† ] i ijk N k† (0, 0) scalar J=0 [N i , N ] 0 ( 12 , 0), (0, 12 ) LH and RH spinors J= 12 † j ( 12 , 12 ) vector J=1, etc ( 12 , 0) (0, 12 ) Weyl spinors L R 2-component spinors of SU(2) Rotations and Boosts L( R) SL( R) L( R) SL( R) e i σ2 .ω SL( R) e σ2 . ν : Rotations : Boosts Dirac spinor Can combine L, R Lorentz transformations where Note : 0 0 I to form a 4-component “Dirac” spinor ei , I 0 , i 0 σi L ( R ) 12 (1 5 ) L = R 2i , σi I 0 3 0 1 2 , 5 i o 0 I Weyl basis Relativistic quantum field theory Fundamental division of physicist’s world : speed slow A c large fast Classical Newton Classical relativity (QM amplitude e t i o n S i( ) small Classical Quantum mechanics Quantum Field theory (S ) c ) The Klein Gordon equation (1926) Scalar field (J=0) : E p m 2 2 2 Energy eigenvalues (x) 2 t 2 2 m2 E (p2 m2 )1/ 2 ??? 1927 Dirac tried to eliminate negative solutions by writing a relativistic equation linear in E 1934 Pauli and Weisskopf revived KG equation with E<0 solutions as E>0 Solutions for particles of opposite charge (antiparticles). Unlike Dirac’s hole theory this interpretation is applicable to bosons (integer spin) as well as to fermions (half integer spin). Physical interpretation of Quantum Mechanics i t 21m 2 0 Schrödinger equation (S.E.) t i ( S .E.) i (S .E.) * * = .j 0 continuity eq. j 2im ( * * ) 2 “probability density” “probability current” i( * t t ) * Klein Gordon equation Neip. x , 2 E N V } 3 dV d x 2E Lorentz invariant 2 f p e j i( * * ) j ( , j) ip. x 1 2 p0V Normalised free particle solutions Field theory of Scalar particle – satisfies KG equation ( 1c t , ), ( 1c t , ) ( m ) 0 2 Classical electrodynamics, motion of charge –e in EM potential Is obtained by the substitution : Quantum mechanics : A (A0 , A) p p eA i i eA The Klein Gordon equation becomes: ( m2 ) V where V ie( A A ) e2 A2 em 4e 2 The smallness of the EM coupling, 1 137 , Make a “perturbation” expansion of V in powers of means that it is sensible to em ( m2 ) V Want to solve : ( x) ( x) d 4 x ' F ( x ' x)V ( x ') ( x ') Solution : ( m2 ) F ( x ' x) 4 ( x ' x) where Dirac Delta function Feynman propagator Simplest to solve for propagator in momentum space by taking Fourier transform 1 (2 2 2) 2 ip .( x ' x ) e ( m2 ) F ( x ' x)d 4 ( x ' x) (221 )2 e ip.( x ' x ) 4 ( x ' x)d 4 ( x ' x) 2 ( p 2 m 2 ) F ( p) F ( p) (2 )2 1 1 p2 m2 i , 11 2(22) 2 F ( x) (2 )4 d p e 1 4 ip. x 1 p2 m2 i The Born series ( x) ( x) d x ' F ( x ' x)V ( x ') ( x ') 4 Since V(x) is small can solve this equation iteratively : ( x) ( x) d 4 x ' F ( x ' x)V ( x ') ( x ') d 4 x ' d 4 x '' F ( x ' x)V ( x ') F ( x '' x ')V ( x '') ( x '') .... The Born series ( x) ( x) d x ' F ( x ' x)V ( x ') ( x ') 4 Since V(x) is small can solve this equation iteratively : ( x) ( x) d 4 x ' F ( x ' x)V ( x ') ( x ') d 4 x '' d 4 x ''' F ( x '' x)V ( x '') F ( x ''' x '')V ( x ''') ( x ''') .... x Interpretation : V ( x ') x' x ''' x '' F ( x ' x) But energy eigenvalues E (p2 m2 )1/ 2 ??? Feynman – Stuckelberg interpretation ( E 0) e ( E 0) ei ( E )( t ) iEt Two different time orderings giving same observable event : t1 t2 time space t1 t2 But energy eigenvalues E (p2 m2 )1/ 2 ??? Feynman – Stuckelberg interpretation ( E 0) e ( E 0) ei ( E )( t ) iEt Two different time orderings giving same observable event : t1 t2 time space t1 t2 t1 t2 t1 t2 time space F ( x ' x) (2 )4 d p e 4 1 ip.( x ' x ) 1 2 p m2 i i (2 )3 2 p0 e d3 p ip0 t ' t ip.( x ' x ) F ( x) i d p f ( x ') f ( x) (t ' t ) i d p f ( x ') f p* ( x ) (t t ') 3 p where f p e ip. x 1 2 p0V * p 3 p are positive and negative solutions to free KG equation Scattering in Quantum Mechanics Prepare state at t | in (t ) | i Time evolution (possibly scattering) | in (t ) S | in (t ) Observe resulting system in state | out (t ) | f QM : probability amplitude : out (t ) | in (t ) out (t ) | S | in (t ) f | S | i S fi S fi fi iT fi S matrix for Klein Gordon scattering Relativistic probability density 3 * d x f S p ' , p out (t ) | in (t ) lim p ' i 0 ( x) t ( x) ( x) d 4 x ' F ( x ' x)V ( x ') ( x ') ... F ( x) i d p f ( x ') f ( x) (t ' t ) i d p f ( x ') f p* ( x) (t t ') 3 * p p 3 p S p ' , p ( p p ) i d y f 3 ' 4 * p ( y ) V ( y ) f p ( y ) ... Feynman rules iT fi i d 4 y f p* ( y ) V ( y ) f p ( y ) i d 4 y f p* ( y ) ie( A A ) f p ( y ) i d y j A 4 fi f p e jfi ie f p* ( y) f p ( y) f p* ( y) f p ( y ) e( p f pi ) e ip. x i ( p f pi ). x k, ie( p p ' ) p p' Feynman rule associated with Feynman diagram 1 2 p0V Fundamental experimental objects (a1 b1b2 ...bn ) Decay width = 1/lifetime (a1a2 b1b2 ...bn ) Cross section Units “barn” 1 barn 102 fm2 1 mb 101 fm2 " milli " 1 b 104 fm2 " micro " 1 nb 107 fm 2 " nano " 1 pb 1010 fm2 " pico " 1 fb 1013 fm 2 " fempto " (Natural Units 1GeV 2 0.39mb) (Dimension 1/T=M) (Dimension L2=M-2) Fundamental experimental objects (a1 b1b2 ...bn ) Decay width = 1/lifetime (a1a2 b1b2 ...bn ) Cross section (Dimension 1/T=M) (Dimension L2=M-2) Transition rate x Number of final states Cross section = Initial flux Fundamental experimental objects (a1 b1b2 ...bn ) Decay width = 1/lifetime (a1a2 b1b2 ...bn ) Cross section (Dimension 1/T=M) (Dimension L2=M-2) Momenta of final state forms phase space Transition rate x Number of final states Cross section = Initial flux For a single particle the number of final states in volume V with momenta in element 3 d p is Vd 3 p (2 )3 n Vd 3 p i 1 (2 )3 Fundamental experimental objects (a1 b1b2 ...bn ) Decay width = 1/lifetime (a1a2 b1b2 ...bn ) Cross section n Vd 3 p i 1 (2 )3 Transition rate x Number of final states Cross section = Initial flux a1 v a1 va1 V ( incident 1) V1 T fi d 4 x *f ( x)V ( x )i ( x ) ... The transition rate i , f f p e e.g. A C B D W fi Transition rate per unit volume ip. x 1 2 p0V T fi N V e ip . x 2 TV f ,i e T fi N A NVB N2 C N D (2 ) 4 4 ( pC pD p A pB )M fi 4 1 1 1 1 4 ( pC pD p A pB ) M 2 W fi (2 ) V4 2 E 2 E 2 E 2 E A B C D ip. x The cross section Transition rate x Number of final states Cross section = Initial flux V2 1 d M 4 v A 2 E A 2 EB V 2 d 3 pC d 3 pD 2 (2 ) 4 4 ( pC pD p A pB ) V 6 (2 ) 2 EC 2 ED 2 M d dQ F dQ (2 )4 4 ( pC pD p A pB ) F v A 2 EA 2 EB 4(( pA . pB )2 mA2 mB2 )1/ 2 3 3 d pC d pD (2 )3 2 EC (2 )3 2 ED Lorentz Invariant Phase space The decay rate 1 2 d M dQ 2EA dQ (2 ) 4 4 ( p A pB1 ... pBn ) d 3 pB1 (2 ) 2 EB1 3 ... d 3 pBn (2 )3 2 EBn Compton scattering of a π meson k ', ' k, p pk Feynman rules k ', ' k, p p' pk Klein Gordon k ', ' k, p' p' p ( m2 ) V V ie( A A ) e2 A2 k, k ', ' k, ie( p p ' ) ie 2 p p' p' p p i p 2 m2 External photon Compton scattering of a π meson k ', ' k, p pk iM k ', ' k, p p' pk' k ', ' k, p' p p' i '.(2 p ' k ') fi 2 2 ( p k) m i .(2 p ' k ) '.(2 p k ') 2i . '] 2 2 ( p k ') m (ie)2 [ .(2 p k ) V2 1 d M 4 v A 2 E A 2 EB V 2 d 3 pC d 3 pD 2 (2 ) 4 4 ( pC pD p A pB ) V 6 (2 ) 2 EC 2 ED Compton scattering of a π meson k ', ' k, p pk M fi k ', ' k, p p' pk p p' p' i '.(2 p ' k ') 2 2 ( p k) m i .(2 p ' k ') '.(2 p ' k ') 2i . ' 2 2 ( p k) m .(2 p k ) ( . ') d 2 2 d lab m 1 mk (1 cos ) 2 total k ', ' k, d 8 2 |k 0 .d d 3m2 2 8.10 GeV 2 2 2 3.10 mb ( . p '. p 0 gauge) total |k / m1 2 2 mk Construction of a relativistic field theory Lagrangian L T V (Nonrelativistic mechanics) t2 Action S L dt t1 Classical path … minimises action Quantum mechanics … sum over all paths with amplitude Lagrangian invariant under all the symmetries of nature -makes it easy to construct viable theories eiS / Lagrangian formulation of the Klein Gordon equation L L d 3 x, L Klein Gordon field lagrangian density ( x) L = ( x) ( x) m2 ( x)† ( x) † } } T V L L 0 ( ) ( m2 ) 0 Manifestly Lorentz invariant Euler Lagrange equation Klein Gordon equation The Lagrangian and Feynman rules Associate with the various terms in the Lagrangian a set of propagators and vertex factors The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. The remaining terms in the Lagrangian are associated with interaction vertices. The Feynman vertex factor is just given by the coefficient of the corresponding term in iL e.g. ( ieA ) ( x) ( ieeA ) ( x) ieA ( * ( * ) ) † k, iL ie( pi p f ) A * } p p' New symmetries L = ( x) ( x) m2 ( x)† ( x) † Is invariant under ( x) ei ( x) …an Abelian (U(1)) gauge symmetry A symmetry implies a conserved current and charge. e.g. Translation Rotation Momentum conservation Angular momentum conservation What conservation law does the U(1) invariance imply? Noether current L = ( x) ( x) m2 ( x)† ( x) † Is invariant under ( x) ei ( x) i …an Abelian (U(1)) gauge symmetry i L L 0 L = ( ) ( † ) ( ) 0 (Euler lagrange eqs.) L L L i i ( † ) ( ) ( ) j 0, ie L L † j † 2 ( ) ( ) Noether current The Klein Gordon current L = ( x) ( x) m2 ( x)† ( x) † Is invariant under ( x) ei ( x) j 0, …an Abelian (U(1)) gauge symmetry ie L L † j † 2 ( ) ( ) j ie KG * * This is of the form of the electromagnetic current we used for the KG field Q d 3 x j 0 is the associated conserved charge Suppose we have two fields with different charges : 1,2 ( x) ei Q 1,2 ( x) 1,2 L = 1 ( x) 1 ( x) m21 ( x)† 1 ( x) † 2 ( x) 2 ( x) m22 ( x)† 2 ( x) † ..no cross terms possible (corresponding to charge conservation) U(1) local gauge invariance and QED ( x) ei ( x )Q ( x) L = ( x) ( x) m2 ( x)† ( x) (Q 1) † not invariant due to derivatives ei ( x ) ei ( x) iei ( x) To obtain invariant Lagrangian look for modified derivative transforming covariantly D ei ( x ) D D ieA A A 1e ( x) ei ( x ) ( x) D ei ( x ) D A A 1e L = D ( x) D ( x) m2 ( x)† ( x) † Note : D ieA is invariant under local U(1) is equivalent to p p eA universal coupling of electromagnetism follows from local gauge invariance L = ( x) ( x) m2 ( x)† ( x) j A O(e2 ) † The electromagnetic Lagrangian 0 E1 E2 E3 F A A F F , A A 1e LEM 14 F F j A M 2 A A E1 0 B3 E2 B3 0 B2 B1 E3 B2 B1 0 Forbidden by gauge invariance The Euler-Lagrange equations give Maxwell equations ! L L 0 A ( A ) F ` j .E , .B 0, B 0 t E B j t E The photon propagator 1 A F A ( A ) j Gauge ambiguity A A 1e A A 1e 2 i.e. with suitable “gauge” choice of α (“ξ” gauge) want to solve 1 1 A (1 ) ( A ) ( g 2 (1 ) ) A j In momentum space 1 p p 2 1 i g p (1 ) p p g (1 ) p2 p2 (‘t Hooft Feynman gauge ξ=1) The inclusion of fermions Weyl spinors ( 12 , 0) (0, 12 ) L R 2-component spinors of SU(2) Rotations and Boosts L( R) SL( R) L( R) SL( R) e i σ2 .ω SL( R) e σ2 . ν : Rotations : Boosts Dirac spinor Can combine L, R Lorentz transformations where 0 0 I ei , I 0 , i 0 σi (Dirac gamma matrices, Note : to form a 4-component “Dirac” spinor 2i , σi I 0 3 0 1 2 , 5 i o 0 I Weyl basis …new 4-vector) L ( R ) 12 (1 5 ) , L = R , 2g The Dirac equation † 0 New 4-vector Fermions described by 4-cpt Dirac spinors Lorentz invariant The Lagrangian L = i m From Euler Lagrange equation obtain the Dirac equation p (i m) 0 U(1) symmetry ei i ( p m) j e ie The Standard Model SU (3) SU (2) U (1) e.g. SU (2) local gauge invariance i ( x ). Q e 0 1 0 i 1 0 1 2 3 1 0 i 0 0 1 2 Q L = i QD Q mQ Q D ig i 2 Wi u Q d W ,W , Z where W ,i 1 W ,i i ijk jW ,k g i j k , i ijk 2 2 2 The strong interactions QCD Quantum Chromodynamics Strong coupling, α3 q q q Symmetry : Local conservation of 3 strong colour charges a e iα ( x ).λ a b a Gr Gr g13 r f rst s Gt q q ( ig3r Gr ) Ga=1..8 Gauge boson (J=1) “Gluons” QCD : a non-Abelian (SU(3)) local gauge field theory Weak Interactions Fermi theory of decay n pe e u n (1 5 )u p u e (1 5 )ue 2 GF } velocity Left-handed m=0 Right-handed ue - u L c e+ R Weak Interactions Fermi theory of decay L n pe e u n (1 5 )u p u e (1 5 )ue 2 GF } p p n V-A P, C n 2 GF g2 2 8 M 2 W W+ e e e e g2 Weak Interactions Symmetry : SU(2) local gauge theory u d L e L uR , d R , eR e a e iα ( x ). a b a Wr Wr g12 r f rst sWt Weak coupling, α2 u Local conservation of 2 weak isospin charges ( ig2rWr ) d Wa=1..3 Gauge boson (J=1) e W , Z Neutral currents A non-Abelian (SU(2)) local gauge field theory Massive vector propagator (W, Z bosons) 2 2 g ( M ) B j 2 i ( g p p / M ) 2 2 1 ( g ( p M ) p p ) p2 M 2 B eip.x Free particle solution ( 1) (0,1, i, 0) / 2 ( 0) ( p , 0, 0, E ) / M g ( )* ( ) p p M2 Helicity polarisation vectors Propagation of unstable scalar particle iJ 2 F ( p ) F ( p) iJ iJ . . iJ Particle decays into final state n iJ F ( p) D(n) …. F ( p) No decay iD (n) Optical theorem – conservation of probability, time evolution is unitary S fi fi iT fi S † S SS † 1 } Im(Tkk ) 12 Tnk mtot 2 n J Im( F ( p)) 2 1 2 iJ 2 F ( p) D J F ( p) n F ( p) 1 p 2 m 2 imtot 2 2 1 2 2 D(n) dQ ( F ( x) e m tot t ) e Z e q , l , q , l , 12 ( Z ee)( Z hadrons) (e e hadrons) ( E 2 M Z2 ) 2 M Z2 2Z e e Z q , l , q , l , 2 (e e hadrons) n 3 f 1 Q f (e e ) f (k ) decay e ( q1 ) e (q2 ) Fermi theory (‘40s) GF M u (k ) (1 5 )u ( p) u (q ) (1 5 )v( p) 2 tot 1 5 2 m G F 192 3 The hard part! exp t 1 2.19703(4) 106 sec GF 1.16637(1) 105 GeV 2 (k ) decay (k ) e ( q1 ) W (Q) e ( q1 ) e (q2 ) e (q2 ) Q Q M W2 M igW u (k ) (1 5 )u ( p) In μ decay g MW2 2 W gW2 M W2 0 gW u (q ) (1 5 )v( p) Q2 O(m2 ) MW2 gW2 Q 2 M W2 M W2 Q Q Q M i 2 m me gW2 GF M W2 2 Fundamental principles of particle physics Introduction - Fundamental particles and interactions Symmetries I - Relativity Quantum field theory Cross sections and decay rates - theory confronts experiment Symmetries II– Gauge symmetries and QED Mainly fermions and the weak interactions