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Introduction lecture 1. 2. 3. 4. 5. 6. 7. Introduction Angular momentum and spin Anti-matter Width - Lifetime Experimental spectra Cross section Luminosity 8. Interactions : introduction 9. Electromagnetism 10. Strong interaction 11. Weak interaction Few concepts, first glance in the particles’ world Overview of the 3 interactions (detailed in the next lectures) 12. Symmetry : what is it ? 13. Relationship between conservation and symmetries 14. Continuous symmetries 15. Discrete symmetries : P, C, T 16. Conservation law : summary MH Schune, Kiev, Feb 2008 Symmetries in particle physics 1 Warning … This lecture : • is going to be too fast • will probably repeat many things you already know • Will hopefully say with my words some concepts which will prove to be useful in the following lectures MH Schune, Kiev, Feb 2008 2 Few concepts, first glance in the particles’ world MH Schune, Kiev, Feb 2008 3 1. Introduction A bit of history • • • • • 1897 (Thompson) electron discovery 1912 (Rutherford) proton discovery ~1930 (Pauli/Fermi) neutrino νe hypothesis 1958 (Reines-Cowan) : experimental evidence 1932 (Chadwick) neutron discovery 1932 (Anderson) positron discovery :e+ 1st antiparticle In the ’30s one knows : e- p n and νe the electromagnetic force and the γ Framework to try to explain the forces between p, then between p and n⇒ Strong interaction Observation of unstable particles in cosmic rays + β decays When the muon was discovered the physicist I. Rabi said : Many particles : Increasing mass •1937 μ→eνeνμ • ⇒ Weak interaction Discovered in 1962 •1947 π→ μνμ •1947 strange particles : K Λ It remains in fact a very good question….. 1955 (Chamberlain, Segre, Wiegand, Ypsilantis) antiproton discovery MH Schune, Kiev, Feb 2008 4 1950 Larger statistics Accelerators era : More and more hadrons … •1948 first π produced Control on the particles •1950 π0 discovery •1954 K+ K0 , Σ production … •1964 Ω discovery Greek : 4 elements : earth, air, fire, water 100 China : 5 elements : earth, wood, metal, fire and water India (3000a.C) : 5 elements :space, air, fire, water and Philosophers + Alchemists add : ether, mercury, sulphur, salt Chemistry Partic les “Z o o” Number of particles 85 elements 4 1661 Boyle defines chemistry 1789 Lavoisier makes the list of 33 elements 1868 Mendeleïev classify the elements 1914 85 elements are known e,p 0 1500 1800 1900 MH Schune, Kiev, Feb 2008 years 5 1964 Zweig-GellMann-Neeman : theoretical introduction of quarks ’70 SLAC Deep inelastic scattering experiments Experimental evidence of quarks p≡(uud) n ≡(udd) Strange particles (Λ,K…): quark s ’90 HERA …. New particles.. •1974 (Richter/Ting) J/ψ discovery (cc̅) : c quark •1975 (Perl) discovery of the τ lepton •1977 (Ledermann) Υ discovery (bb̅ bound state) : b quark •1995 (CDF/DØ coll.) t quark •2000 (Donut) : ντ MH Schune, Kiev, Feb 2008 6 100 3 quarks 4 leptons Particl es “Zo o Number of particles ” 85 elements chemistry 4 12 : 6 quarks 6 leptons e,p 0 1500 years 1800 1900 12 matter particles to explain all known particles ! Hadrons : any particle which undergoes the strong interaction q q q (Nucleon : neutron and proton) q q Baryons half integer spin (ex: p =(uud) ) Mesons integer spin (ex: π = (ud) ) Leptons : Any particle which does not undergo the strong interaction (e,μ,τ) (νe,νυ,ντ) MH Schune, Kiev, Feb 2008 7 Interaction particles • Classical mechanics : interactions = force field • Modern physics : interactions = interaction particles which are field quanta 1973 Observation at CERN of the “weak neutral currents” Interactions between neutrinos Æ “Z0” ? 1976 Standard Model. Electroweak unification They are vectors of the weak interaction, their masss ⇒ γ, Z0 W+are predicted 1983 Observation at CERN of the Z0 and W+- bosons 1989 « Mass production» of Z0 at LEP at CERN 1996 Production of W+W- pairs at LEP at CERN MH Schune, Kiev, Feb 2008 8 The elementary particles of the Standard Model matter : 3 families interactions : W±, Z0 (weak) NB : the gravitational force is extremely weak in the particles world ⇒ not discussed here g : gluon (strong) γ (electromagnetic) All tests of the SM have been successful up to now ! But it does not answer to many fundamental questions. For example : • Why 3 families ? + antiparticles • Why several interactions with very different intensities ? •Origin of the masses of the particles (ad-hoc Higgs boson) which has not been seen yet … •Mass hierarchy •The neutrinos masses : mν ≲ qq eV and mν ≠ 0 Charged leptons MH Schune, Kiev, Feb 2008 quarks 9 The mass Defined by : m2c4=E2-p2c2 Invariant length of the Energy-momentum 4-vector With c=1 E, p and m are expressed using the same unity (GeV/MeV ….) – When p =0 ⇒ E = mc2 – When v increases ⇒ E2 et p2c2 increase but their difference remains constant – m is a Lorentz invariant New particles production: Mass/energy It is not “divisibility” ! Since c is large small mass = Large energy mass energy energy mass A particle is a lump of energy MH Schune, Kiev, Feb 2008 10 Quantum mechanics Relativity Quantum field theory Few words on quantum field theory It is our main working tool for particles physics in the Standard Model and beyond It comes from the marriage between quantum mechanics and relativity E = mc2 (High energy physics) mass/energy New particles production p= h/λ (physics on extremely small scales) Particle/Wave Study the matter h = 6.62606876(52) 10−34 Js = = h / 2π MH Schune, Kiev, Feb 2008 11 2. Angular momentum and spin Angular momentum Classical mechanics : G G G L=r∧ p 3 components : • can be measured with infinite precision • can have all values Quantum Mechanics : G same definition, with the operators R and P (notation L or L ) •The algebra of the components of L : [Li,Lj] = i εijkLk ; εijk= 0, +1, -1 according to ijk. One also has : L²=Li²+Lj²+Lk² ; [L²,Li]=0 •2 independent operators (usually : L2 et Lz ) 2= (2 useful quantum numbers) = •quantification : •L2 : ℓ(ℓ+1) ħ² ; ℓ is an integer •Lz: mħ with m = -ℓ, -ℓ+1, …, -1, 0, 1,…, ℓ-1, ℓ −= −2= • Addition of 2 angular momenta : j1, j 2 ; m1, m2 = j1 + j 2 J ∑ ∑ J = j1 − j 2 M =− J J,M J,M j1, j 2 ; m1, m2 MH Schune, Kiev, Feb 2008 ClebschGordan(CG) coefficients 12 Spin • The spin is the intrinsic kinetic momentum of a particle. • it can be half-integer • It determines the behavior of a given particle. • Few examples of experimental evidences for the spin : • Fine structure of the atoms spectral lines : each line is made of several components very close in frequency • “Abnormal » Zeeman effect : Each spectral line is divided in a given number of equidistant lines when the atom is in an uniform magnetic field. «Anomaly» : the atoms of Z odd (ex. Hydrogen) divide into an even number of sub-level. In fact the number of levels is 2A+1 Æ proof of half integer kinetic momentum ! • The spin has no classical equivalent. Trying to explain it saying that the particle rotates on its own axis does not work. e,p,n have very different characteristics (charge/ mass/interaction) but they have the same spin : ½ MH Schune, Kiev, Feb 2008 13 • The spin obeys the same laws as the other kinetic momenta : – Algebra similar as the L one – S2 can have the values s ( s + 1) = 2 (s can be half integer) – And Sz : m= with m = − s, − s + 1,... − 1, 0,1,...., s − 1, s – One can add a spin with • An other spin (S = S1 ⊕ S2) • With an total angular momentum (J = L ⊕ S) A particle can have any angular momentum L but its spin S is fixed integer spin (Bosons) Half integer spin (Fermions) spin 0 spin 1 spin 1/2 spin 3/2 Elementary - Vectors of the interactions quarks, leptons - Composite pseudo-scalar mesons (p,K..) Vector mesons (ρ,K*) some baryons (octet) MH Schune, Kiev, Feb 2008 some baryons (decuplet) 14 spin/statistics theorem (Pauli 1940) Pauli’s exclusion principle : two particles of half integer spin (fermions) cannot be simultaneously in the same quantum state Pauli’s principle anti-symmetry of the wave function by the exchange of 2 particles (for the fermions) Bohr and Pauli For 2 particles one in the state ψα, the other one in the state ψβ, one can write : ψ (1,2) = 1 ψ α (1)ψ β ( 2) + ψ β (1)ψ α ( 2) ) ( 2 Symmetric (bosons) ψ (1,2) = 1 ψ α (1)ψ β ( 2) − ψ β (1)ψ α ( 2) ) ( 2 anti-symmetric (fermions) If 2 fermions are in the same state (α= β) their wave function is 0 ! This problem does not exist for bosons which can occupy the same state (ex. supra- conductors). This can be generalized for a larger system of particles. MH Schune, Kiev, Feb 2008 15 Helicity JG • Particle of spin S • • G Axis orientation in the momentum direction n JG Helicity : JG JG JG p JG JG JG G JG G JG G Λ = n ⋅ S with n= JG Λ = n ⋅ J because p ⋅ L = p ⋅ r ∧ p = 0 p ( • Eigenvalues −s ≤ λ ≤ s • if mass=0 ) 2s + 1 values only 2 eigenvalues : ±s s Right-handed particle s Left-handed particle The helicity is invariant under rotation (scalar product of 2 vectors). MH Schune, Kiev, Feb 2008 16 3. Anti-matter • • In a magnetic field the number of particles which turn right = number of particles which turn left Hypothesis : electron and proton wrong ! Anderson 1932 MH Schune, Kiev, Feb 2008 17 • • • • • The radius of curvature is smaller above the plate. The particle is slow down in the lead Æ the particle in incoming from the bottom The magnetic field direction is known Æ positive charge From the density of the drops one can measure the ionizing power of the particle Æ minimum ionizing particle Similar ionizing power before and after the plate Æ same particle on the 2 sides Curvature measurement after the lead : particle of ~23MeV Æ it is not a non-relativistic proton because he would have lost all its energy after ~5mm (a track of ~5 cm is observed) Lead plate Momentum direction 1 cm Particle of positive electric charge and with a mass much smaller than the proton mass (< 20 me) : the positron MH Schune, Kiev, Feb 2008 18 Matter and anti-matter particles are produced in the interaction of particles with matter + π and π − particles having the same mass, spin… but Opposite electric charge opposite curvature in a magnetic field MH Schune, Kiev, Feb 2008 19 4. Width and lifetime Lifetime : the exponential law Instable particles and nuclei : number of decays per unit of time (ΔN/ΔT) proportional to the number of particles/nuclei (N) ΔN= cte × N × Δt ⇒ exponential law N0 τ N(t) = N0e-t/τ N0/e Mean lifetime (defined in the particle rest frame) The majority of the particles are instable τ from 10-23 sec (resonances) to ~10+3 sec (neutron) t The probability for a radioactive nucleus to decay during a time interval t, does not depend on the fact that the nucleus has just been produced or exists since a time T : ⎡Survival probability ⎤ ⎡ Survival probability ⎢after the time T + t ⎥ = ⎢ after the time T ⎣ ⎦ ⎣ ⎤ ⎡Survival probability ⎤ ⎥ × ⎢ after the time t ⎥ ⎦ ⎣ ⎦ MH Schune, Kiev, Feb 2008 ea+b = ea × eb 20 Few important examples of different lifetimes • Stable particles : γ,e,p,ν Æ the only ones ! proton stability τ(p)> ~1032 ans • particles with long lifetimes : n → p + e- + ν̅e μ− → e- + ν̅e + νμ π+→μ+ νμ (mainly) K+ • particle with short lifetimes : D+ B+ Δ++→N π τ =6.13 10+2 sec, β decay τ = 2.2 10-6 sec, cosmic rays τ =2.6 10-8 sec τ =1.2 10-8 sec τ =1.04 10-12 sec τ =1.6 10-12 sec τ ~ 10-23 sec MH Schune, Kiev, Feb 2008 21 particles which can be directly detected • • The lifetimes are given in the particle rest frame What we see is the lifetime in the laboratory rest frame « Event display » of the BELLE experiment (e+e- Æ BB̅, ECM=10.58 GeV) Æ one should take into account the relativistic time dilation Æ In real life one measures lengths in the detector L= βγ × Boost × cτ lifetime K+ π- • Some particles are seen as stable in the detectors. • Example a pion (cτ = 7.8m) : if Eπ = 20 GeV Æ γ= 20/mπ = 142.9 ; β = 0.999975 γ Β0 → Κ*0 γ K+ π - Æ L = 1114.3m particles which can be directly detected in the detector : n, γ ,e, p, μ, π±, K± MH Schune, Kiev, Feb 2008 22 Width • The uncertainty principle from Heisenberg for an unstable particle is : Heisenberg : ΔE Δt ~ = Δmc2 = Γc2 τ By definition : Γc ≡ 2 Uncertainty on the mass (width Γ) due to τ = The faster the decay, the larger the uncertainty of on m τ Stable particle ↔ well defined mass state 197 × 10 −15 −22 =c = 197 MeV × 1fm ; = = = 6.582 10 MeV.s 8 3.10 Measuring widths, one is able to have information on very small lifetimes. This is the way one can have information on a phenomenon extremely fast (the fastest in Nature?...) : a particle with a lifetime of 10-23 sec) Decay mc2 τ Γc2 Κ∗0→ K- π+ 892 MeV 1.3 10-23 s 51 MeV π0→ γ γ 135 MeV 8.4 10-17 s 8 eV Ds → φπ+ 1969 MeV 0.5 10-12 s 10-3 eV Measurable width Measurable lifetimes MH Schune, Kiev, Feb 2008 23 Breit-Wigner • Schrödinger equation (free particle with energy E0): ∂ψ i= = Hψ = E0ψ ∂t ⇒ ψ = ae (approximate computations) i − E0 t = −i c2 m0 t = 2 (particle rest frame E 0 =m0 c ) ⇒ ψ = ae – stable particle : ψ (t ) = ψ (0) = a0 2 2 2 c2 ⎛ Γ⎞ – unstable particle : − i ⎜ m0 − i ⎟t 2⎠ =⎝ ⇒ ψ (t ) = a0e ⇒ a = a0e − t 2τ ⇒| ψ (t ) |2 = |ψ (0) |2 e −t / τ We want the probability to find a state of energy E A(E ) = 1 2π +∞ i Et = ∫ ψ (t )e dt ∝ 0 1 (E − m c ) 2 0 Γc 2 +i 2 1 2 ⇒ A ∝ Γ ≡ half maximum width 0.5 Probability = |A|2 (E − m c ) 2 0 2 + Γ 2c 4 / 4 MH Schune, Kiev, Feb 2008 E-m0=-Γ/2 m0 E-m0=+Γ/2 E 24 Several possible final states (decay modes/channels) : ⇒ branching ratios (BRi) : probability to obtain a final state i (Σi BRi=1) partial width Γi (definition) : BRi=Γi/Γ Example: Relation between lifetime, partial widths and branching ratios : Λ→pπ in 64 % of the cases Λ→nπ0 in 36 % of the cases τ= = 1 = BRi = 2 2 c Γ c Γi Relation for 2 particles with : Part. a ; decay mode 1 Part. b ; decay mode 2 BR1 Γ1 τ a = BR2 Γ 2 τ b Example : Z0 partial widths (PDG2002) MH Schune, Kiev, Feb 2008 25 5. Experimental spectra experimental spectrum K-π+ : • Search for a K- and a π+ in the detector and computation of the invariant mass Fitted by a Breit-Wigner Γ =51MeV This is a K* ! BaBar 892 MeV « combinatorial » background MH Schune, Kiev, Feb 2008 26 π0 experimental spectrum : 2 γ reconstruction and computation of the invariant mass. PDG Æ τ = 8.4 x10-17 s Fit by a gaussian Γ= 8 eV σ ~ 7MeV ? Detector resolution effect « combinatorial » background 135 MeV MH Schune, Kiev, Feb 2008 27 Ds experimental spectrum : (Ds → φπ+ and φ → π+π-) Γ ~ 10-3 eV PDG Æ τ = 500 x10-15 s But one sees >> 10-3 eV Fit by a gaussian σ ~ 10MeV Detector resolution effect D Ds ⇒ One measures directly «long » lifetimes not through widths 1969 MeV « combinatorial » background MH Schune, Kiev, Feb 2008 28 π τ(Ds) : L Measurement of the Ds :lifetime L ⋅ m t = p Ds φ t : proper time Experiment CLEO : τ(Ds) = 486.3±15.0±5.0 fs MH Schune, Kiev, Feb 2008 29 6. Cross section dNint = n1v1n2σ dtdV • σ is thus defined by : – The number of interactions per unit of volume and time the physics processes are « hidden » in this term and depend upon the beam energy) – The beam energy (usually v1~c !) – The number of particles per unit of volume in the beam (n1) – The number of particles per unit of volume in the target (n2) • For a i → f transition : 1 dσ = F • • Phase space σ : [L]2 1 barn = ∑ int d 3 pk f | T | i (2π ) δ (Q f − Qi )∏ 3 ( 2 π ) 2 Ek k =1 2 4 4 n Flux factor 10-24 cm2 MH Schune, Kiev, Feb 2008 30 7. Luminosity Instantaneous luminosity dN = L ⋅σ dt Cross section Number of interactions /s dNint = n1v1n2σ dtdV dN1,2/dV=n1,2 luminosity cm-2 sec-1 k bunches kfN+ N− L= 4π sx sy f (=c/circumference) frequency N+ : number of electrons in a bunch N- : number of positrons in a bunch y e+ x2 ez − 2 1 2s x ρ ± ( x, y ) = e e 2π sx sy − y2 2s 2y x MH Schune, Kiev, Feb 2008 31 charge ⎡C ⎤ I (e) = ⎢ ⎥ ⎣s⎦ An example : PEP-2 time I (e ) = N (e ) × qe × N Circumference 2200 m I(e-) 0.75 A I(e+) 2.16 A Npaquets 2 x 1658 N(e-)/bunch 2.1 1010 N(e+)/bunch 6.0 1010 Beams size sx=150 μm, sy=5 μm L= e bunches c × Lcirc kfN+ N− 4π sx sy ⇒ L=3 1033 cm-2 s-1 Macroscopic quantity → relates the microscopic world (σ) to a number of events dN = L ⋅σ dt MH Schune, Kiev, Feb 2008 32 Integrated luminosity • Product of the luminosity of a characteristic time (1 year .. , experiment lifetime …) Lint = ∫ L dt cm −2 ( ) or barn−1 b−1 •PEP-2 example L=3 1033 cm-2 s-1 1 year (~ 107 seconds) • Lint = 3 1040 cm-2 = 30 fb-1 1 b = 10-24 cm2 • N = σ Lint 1fb = 10-15 b • production cross section of the l’Υ(4s) : ~1.1 nb 1fb-1= 1039 cm-2 ⇒ 33 106 Υ(4s) produced by year by the PEP-2 machine Lint takes into account the machine operation : convenient ! MH Schune, Kiev, Feb 2008 33 Summary Elementary particles : 3 families of fermions + vector bosons of the interactions (+ anti-matter) leptons, hadrons, mesons, baryons Kinetic momentum : angular (integer) or intrinsic (spin: integer or half-integer) eigenvalues of J2 and Jz: kinetic momentum is quantified addition of angular momenta identical particles; Pauli’s exclusion principle Few properties : helicity: Λ = n.S Width: Γ= h̷/τ ↔ lifetime Breit-Wigner, branching ratios stable particles, can be directly detected cross section Luminosity: instantaneous and integrated MH Schune, Kiev, Feb 2008 34 Overview of the 3 interactions (detailed in the next lectures) MH Schune, Kiev, Feb 2008 35 8. Interactions : introduction Classical physics : «modern» physics: The particle P1 creates around it a force field. If one introduces the particle P2 in this field it undergoes the force. P1 and P2 exchange a field quantum; the interaction boson P1 P2 Electrostatic example : P1 JG JG F E P2 q1 r The heavier the ball, the more difficult it will be to throw it far away q2 JG JG kq1 JJG F = q 2 E (r ) = q 2 2 ur r Interaction vector Range of the interaction ∝1/mass of the vector MH Schune, Kiev, Feb 2008 36 Range of an interaction • Creation and exchange of an interaction particle ⇒ violation of the energy conservation principle during a limited time Δt ≈ • = = = ΔE mc 2 Heisenberg principle During Δt the particle can travel R =c Δt =c R= mc 2 Range → « reduced » wave length (Compton) with ħc ≅ 197.3 MeV fm Example : an interaction particle with m = 200 MeV ⇔ R = 1 fm MH Schune, Kiev, Feb 2008 37 Shape of the interaction potential Klein-Gordon equation for a spin 0 particle : E 2 = p 2c 2 + m 2c 4 ∂ 2ψ 2 ( i = ) 2 = ( i = ) c 2∇2ψ + m2c 4ψ ∂t 2 4 ∂ 2ψ m c − 2 = −c 2∇ 2ψ + 2 ψ ∂t = 2 2 2 ∂ ψ m c 1 ⇒ ∇ 2ψ − 2 ψ − 2 2 = 0 = c ∂t 2 ∂ ∂t p = − i =∇ E = i= operators (one only deals with stationary states) 2 2 m c ∇ 2ψ − 2 ψ = 0 = 1 d ⎛ 2 dU (r ) ⎞ m 2c 2 = r U (r ) In spherical symmetry : ψ=U(r) and ΔU(r)=∇2U(r)= 2 ⎜ ⎟ 2 r dr ⎝ dr ⎠ = if m≠0 : g 2 −r / R U (r ) = − e r = R= mc if m=0 : r>0 Yukawa potential g coupling constant ΔU (r ) = 0 U (r ) = − 1 q1q2 4πε 0 r r>0 qi = charge In this case the Yukawa potential is equivalent MH Schune, Kiev, Feb 2008 to the Coulomb one 38 Force Vector Lifetime (order of magnitude) Strong Relative intensity (order of magnitude) 1 Gluons 10-24 s electromagnetic 10-2 Photon 10-19 - 10-20 s Weak 10-5 W and Z0 10-16 - 10+3 s Gravitation 10-40 Graviton ??? For the strong, electromagnetic and gravitational interactions these orders of magnitudes can be obtained comparing the binding energy of 2 protons separated by ~1fm The intensity of the interactions dictates the particles lifetimes and their interaction cross sections. MH Schune, Kiev, Feb 2008 39 9 electromagnetism (QED) • • • Between charged particles Vector of the interaction : the photon (γ) One Feynman graph for QED: e- e- R. Feynman √α γ (virtual) 1/q2 √α e- et electrons exchanging a photon or An e- which emits a γ and moves back. The γ is absorbed by an other e- whose direction is modified MH Schune, Kiev, Feb 2008 40 Feynman graph • • • A powerful « graphical » method to display the interaction in perturbations theory (each diagram is a term in the perturbation series) Each graph is equivalent to « a number » → computation of the matrix elements and of the transition probabilities Vector boson of the interaction particle • • • Horizontal axis : the time Lines are particles which propagate in space-time The represent the vertices «location» of the interaction (where there is quantum number conservation) e-(k’) e-(k) Feynman rules : External lines: fields (spinors, vectors, …) Vertex: √α factor in the matrix element « interaction intensity » γ (q) Propagator: μ- (p) t μ- (p’) factor igυν /(q2-m2) (depends also on spin …) MH Schune, Kiev, Feb 2008 41 α= e2 4πε 0 =c = Virtual particles 1 137 Example QED : e+e- symmetric collision in the rest frame e- a Re s e l c rti a lp √α √α γ a Re e+ In the rest frame : Ee + + Ee − = Eγ JJG JJG JJG p+ + p− = pγ s e l tic r a lp mγ2 = 2me2 + 2Ee + Ee − − 2 p+ p− cosθ It can be interpreted as : JJG JJG JJG G p+ + p− = pγ = 0 θ = π ⇒ mγ2 = 2me2 + 2Ee + Ee − + 2 p+ p− incompatible with mγ = 0 The γ is « off-shell » e- √α γ Or Creation of a massive virtual photon during a « short » time the γ can only exist virtually thanks to ΔE.Δt ≈ ħ 2 γ production going in opposite directions Virtual e- e+ Violation of the energymomentum conservation law √α → energy-momentum conservation γ MH Schune, Kiev, Feb 2008 42 e+e- Æ e+e- interaction √α e- e- e- √α √α e- ~α + e+ e+ √α e+ γ exchange between an e+ and an e- e- √α √α e- e- √α e- e+ √α √α √α √α √α e+ e+ e- √α + e+ e+ e+e- pair annihilation in γ and γ conversion in an e+ and an e- √α √α +… ~α2 e+ exchange of 2 γ between an e+ and an e- √α √α e- ~α3 +… √α e+ α small (1/137) : one can develop in perturbation series MH Schune, Kiev, Feb 2008 43 The way we see the electron and the photon is modified electron : photon : e- ee- The electron emits and absorbs all the time virtual γ, it can be seen as : γ eγ e- e- e- e- ee- eγ e- e- γ e+ e- γ ee- … e- => Theoretical (α « running ») and experimental (g-2) consequences MH Schune, Kiev, Feb 2008 44 Theoretical consequences of the way we see the electron α (μ2 ) α (Q 2 ) = The electron is surrounded by e+e- pairs with the e+ preferentially oriented towards the e- 1− → Screening of the electron charge α (μ2 ) 3π ⎛ Q2 ⎞ log ⎜ 2 ⎟ ⎝μ ⎠ - - - - -+ - + + + + ++ + e- - + + - - + + + + + - - - - - Vacuum polarization + Test charge Measured charge of the e- Measurement of the electron charge with a test charge : the closer we are, the larger the charge we see high energy low energy 1/137 distance MH Schune, Kiev, Feb 2008 45 (g-2) : Experimental evidence of the vacuum polarisation Gyro-magnetic ratio g • The magnetic moment associated associated to the angular momentum of the electron JG n r • e n : unity vector v : electron speed S : surface I : intensity = charge / time v JG JG JG JG e e 2 μ =I S n= πr n = ( mvr ) n Angular 2π r 2m momentum v μ = μB A with G μ = μB L G e= μB = 2m =A Bohr magneton Intrinsic magnetic momentum : JG JG μ = g μB S Dirac : for spin ½ point-like particles : g=2 spin gyro-magnetic spin ratio MH Schune, Kiev, Feb 2008 46 The value of g is modified by : +… « e-» One defines a = « e-» g −2 g α 1 = −1= + ... ≈ 2 2 2π 800 a=0.00115965241 ± 0.00000000020 experiment (10-11 precision ) a=0.00115965238 ±0.00000000026 theory (α3) MH Schune, Kiev, Feb 2008 47 10 Strong interaction Yukawa : Yukawa’s pontential g 2 −r / R U (r ) = − e r = R= mc σ forte g2 ⇔ r>0 , g coupling constant . Analogy with QED : ⎛ =c ⎞ ~ 30mb ~ π ⎜ 2 ⎟ m c ⎝ π ⎠ 2 e2 4πε 0 Geometrical size R π nucleon-nucleon scattering ⇒mπ~200 MeV Yukawa (1934) : range of the interaction ~1fm (short distance interaction between the nucleons) due to the existence of field quantum with a mass ~200 MeV p π discovery in 1947 √αstrong p n π MH Schune, Kiev, Feb 2008 √αstrong n α strong g2 = =c 48 strong interaction : quarks and gluons (QCD « Quantum ChromoDynamics ») • • Colour = charge of the strong interaction Only quarks feel the strong interaction (leptons are not colored) But: As for QED : q q g QCD QED 3 colours One charge Gluons :coloured Æ Coupling between 3 and 4 gluons Photon : neutral Æ no coupling between photons g q q g γ g γ γ Asymptotic freedom : αs decreases when the energy increases (or the distance decreases) MH Schune, Kiev, Feb 2008 49 11. Weak interaction • The particles lifetimes are very different : – – – – – – Γ∝ Δ++→p π Σ0→Λγ π0→γγ Σ→nπ π→μν n→pνe 1 τ ~ 10-23 sec ~610-20 sec ~10-16 sec ~10-10 sec ~10-8 sec ~15 minutes strong Electromagnetic weak 2 ∝ M ∝ ~ coupling constant τ ( Δ → nπ ) 10 sec ⎛ αW ⎞ = −10 =⎜ ⎟ τ ( Σ → nπ ) 10 sec ⎝ α s ⎠ −23 2 2 ⇒ αw~10-6 ~same phase space MH Schune, Kiev, Feb 2008 50 • All fermions (leptons+quarks) carry a « weak charge» : Neutral Vertex Charged Vertex l+ νl Z0 W- → short range l- l- MW~MZ~100 GeV leptons Charged Vertex q’(-2/3) Neutral Vertex W- Z0 q q q(-1/3) quarks MH Schune, Kiev, Feb 2008 51 Interactions : summary • • The interactions are mediated by vector bosons interaction range ∝ 1/mass Feynman graph = display of a matrix element of the transition in the perturbations series framework Virtual particles (off-shell particles during a short time) • • QED: electric charge, γ, vacuum polarisation, α ↗ with energy QCD: colour, gluons (self-interaction), αs ↘ with energy (asymptotic freedom) • Weak: concerns all fermions, W±,Z0 • MH Schune, Kiev, Feb 2008 52 Symmetries in particle physics MH Schune, Kiev, Feb 2008 53 12. What is a symmetry ? • A physics law is symmetric wrt a transformation if the related equation does not change under this transformation G JG d r F =m 2 dt 2 example: symmetric wrt time reversal t→-t z An object is not symmetric “by chance”, the existence of a symmetry gives information about the object Various types of symmetries: • Geometrical symmetries : rotations, translations, t→-t • Internal symmetries (related to Quantum mechanics): isospin transformation, Charge MH Schune, Kiev, Feb 2008 54 13 Relationship between conservation laws and symmetries In classical mechanics : symmetry principle ↔ non observable quantity ↔ invariance Momentum conservation law Absolute position non Invariance under ⇔ ⇔ observable translation Absolute direction Angular momentum Invariance under ⇔ ⇔ observable conservation rotation E. Noether => Noether’s theorem : For any continuous symmetry for a given system corresponds a conservation law for this system. In quantum mechanics : - Noether’s theorem also works : invariance/symmetry ⇔ conservation law - symmetry operator T: unitary (T+T = 1, Q’ = T Q T+) - a given operator T (which can be an observable) is a symmetry operator for H (= does not change H) if it commutes with H ([H,T]=0) => The associated quantum numbers are conserved ( selection rules ) MH Schune, Kiev, Feb 2008 55 14. continuous symmetries Continuous symmetry : additive quantum number (conserved) - space-time symmetry (translation, rotation) For a unitary transformation Tα one can write Tα = exp(- iαQ) Q is called the transformation’s generator # of generators = # of parameters in the transformation (eg : 3 generators for the rotation) The momentum operators are the generators of the translation - internal symmetry (gauge symmetry : EM) : if global : quantum number conservation (eg baryonic one) ; if local : « appearance » of a vector field (the photon) see later… MH Schune, Kiev, Feb 2008 56 15. discrete symmetries : multiplicative quantum numbers G G G P ψ ( r ) = ψ ( −r ) = ε ψ ( r ) G G G G 2 2 x’ P ψ (r ) = ε ψ (r ) = P ψ ( −r ) = ψ (r ) G G ⇒ ε = ±1 ψ + (r ) et ψ − (r ) parity P: z x P y y’ z’ Examples: Intrinsic parity of a particle : JP : Spinparity (the spin is invariant under P) G G Pr = − r G JG G JG P(r ⋅ p ) = r ⋅ p G JG G JG P(r ∧ p ) = r ∧ p G JG G G JG G P[(r ∧ p ) ⋅ r '] = −(r ∧ p ) ⋅ r ' spin 0 spin 1 vector (parity -1) scalar (parity +1) pseudovector (parity +1) pseudoscalar (parity -1) 0+ parity +1 scalar 0- parity -1 Pseudo-scalar 1+ parity +1 Pseudo-vector 1- parity -1 vector Parity for a two-particles system: Intrinsic parity of the 2 components + parity of the angular momentum L (the part related to the spin is invariant under P) Ptot=P1 P2 (-1)L MH Schune, Kiev, Feb 2008 57 Charge conjugation (C) • particle ↔ antiparticle • + − + C π → π ≠ ± π Not defined in some cases : In particular, the fermions are not C-eigenstates • C can be defined for a neutral boson or for a particle-antiparticle system C π (γ, Z0, π0, ρ, η, …): • 0 = ηC π η2 =1 0 C π C 0 = ± π 0 Example: π0 →2 γ decay Cγ=-1 ⇒ Cπ0 =1 Experimentally : ( BR (π ) < 3.1⋅ 10 → γγ ) BR π 0 → γγγ Time reversal (T) t → −t G G ⇒r =r JG JG p = −p G G L = −L 0 t=t1 −8 90 % CL C is conserved by the elm interaction t=-t1 t=0 t=0 t=t1 t=-t1 t complicated in Quantum Mechanics -t MH Schune, Kiev, Feb 2008 58 16. Conservation laws : summary • • Preserved by all interactions : – energy and momentum – Total angular momentum – electric charge – Baryonic number – Leptonic numbers Practically the difference between N(quarks) and N(anti-quarks) Apart from the weak interaction, all other interactions preserve : – The number of quarks of each type (u,d,s,c,b,t) – Charge conjugation C – Parity P The flavour : – Time reversal T strangeness Ns − Ns MH Schune, Kiev, Feb 2008 charm Nc − Nc beauty Nb − Nb 59 We now have with us all what we need to travel in the particles’ world MH Schune, Kiev, Feb 2008 60