Particle Physics WS 2012/13 (13.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101
Content of Today Up to now: first order non-resonant e+e- cross-section dσ e.g. (e + e μ + μ ) = 4πα dω 3s Intermediate resonances e f e + (ρ, ω, φ, J/Ψ, ϒ, ) f higher order corrections and running coupling constants 2
Hadronic Resonances Cross sections computed up to now are valid only for q 2 values outside the vector resonances (ρ, ω, φ, J/Ψ, ϒ, ) In the neighbourhood of the resonances these particles contribute as Intermediate states to the total cross section: e M = + e + f f steps indicate new quark flavour e e + (ρ, ω, φ, J/Ψ, ϒ, ) f f σ = 4πα 3s + 3π s Γ e Γ f s M 2 + 1 4 Γ2 + interference Γ e, Γ f = Γ (V e + e, ff ) Γ = i Γ i Breit-Wigner resonance (for Vector mesons J=1 particles) 3
Two Jet Event e + e - q q quarks from jets 4
November Revolution 1974 Quark Model at that time: several hundred hadronic states explained as bound states of 3 different quarks: u, d, s Two independent groups found almost simultaneously a new particle J/ѱ = cc >: BNL (Berkly National Laboratory): proton beam on fix target (Berilium) group leader Samuel Ting p (28 GeV) + Be J + X e + e - m(j) = 3.1 GeV Γ: consistent with detector resolution (20 MeV) reconstructed invariant mass of final state particles 5
November Revolution 1974 SLAC-Mark-I experiment (see tau discovery) group leader Burton Richter e + + e - Ψ multi hadrons e + e - μ + μ - + K + K - + π + π - m(ѱ) = 3.105 ± 0.003 GeV, Γ(ѱ) < 1.3 MeV e e + c c ѱ performed scan of E CMS (e + + e - ) 6
Name of Ψ Particle e + + e - J/ѱ(3700) J/ѱ (3100) + π + + π - e + + e - 7
Nobel Prize 1976: Discovery of J/Ψ Particle 8
OZI Rule width of J/ѱ: Γ = 93 ± 2 kev [PDG], τ ~ 7.2 10-21 s m(ρ) = 775 MeV ρ = ½ uu d d > Γ = 149 MeV m(ω) = 782 MeV ω = ½ uu + d d > Γ = 8.4 MeV m(φ) = 782 MeV φ = s s > Γ = 4.3 MeV narrow width of J/ѱ was a surprise OZI (Okubo, Zweig, Iizuka) rule: Decays with disconnected quark lines are suppressed relative to decays where the quark lines are connected s s u π s K+ d + φ φ? u d s π 0 s d u d s K- π u - OZI allowed, but little phase space BR(φ KK) ~ 50% OZI suppressed, but little phase space BR(φ π+π-π0) ~ 15% 9
OZI Rule for J/φ J/ѱ c c c u u c D 0 D 0 J/ѱ c c h h h OZI allowed, but kinematically impossible 2m(D) = 2 x 1865 MeV > m(j/ѱ) J/ѱ c c OZI suppressed h h elm + weak IA suppressed relative to strong IA BR(J/ѱ μμ) ~ 5%, BR(J/ѱ ee) ~ 5%, long decay time and narrow decay width! 10
QCD picture of OZI Rule J/ѱ c c h h colourless gluon exchange Exchange of gluons: 1 g : not possible (colour) 2 g : not possible due to parity violation [P(J/ѱ) = -1 P(g) 2 ] at least 3 gluons are needed, suppression by α s 3 rather narrow decay width/long lifetime (92 kev/ 7.2 x 10-21 s) 11
Hydrogen Atom versus Positronium In both cases 1/r Coulomb potential E n ~ - 1/n 2 (p, e - ) (e +, e - ) Fine structure correction (spin-orbit) ΔE ~ α 2 Hyperfine structure (spin-spin) ΔE ~ α 2 μ p /μ e reduced mass: m m e /2 thus E n E n /2 μ e ~ 650 x μ p fine structure and hyperfine structure corrections of similar size, no clear hierarchy 12
Positronium versus Charmonium Strong interaction potential: V(r)~ 1/r + kr Coulomb potential: V(r) ~ 1/r Kind of same structure but no degeneracy of 2 3 P and 1S states (c, c ) (e +, e - ) (note, historically different nomenclature n 2S+1 L J, Positronium n = N+l+1, Charmonium n=n+1 thus 1P in charmonium correspond to 2P in positronium) 13
Charmonium Spectroscopy Crystall Ball Experiment: photon detection in full solid angle with Sodium Iodid scintillators 14
Charmonium e - e + c c Charmonium produced via elm annihilation must carry quantum numbers of photon Spectra measured by crystal ball experiment photon energy 15
Bottomium: 1977 Y(1S) b b μ + μ - discovery of third quark family 20 years of search for top quark started experiment: p beam on fix target at Fermilab Y(1S) + X wrong sign background m[y(1s)] = 9460 MeV ; Γ[Y(1S)] = 54 kev OZI suppressed decay Y B + B - / B 0 B 0 kinematically allowed for Y(4S) and heavier resonances 16
Radiative corrections Computed up to now lower level tree level Born approximation dσ QED 0 dω e f f: fermion f : anti-fermion e + f Bremsstrahlungs corrections: δ QED soft + δ QED hard soft + hard photon emission (separation kind of arbitrary depend on detector resolution) Initial state radiation (ISR) final state radiation (FSR) 17
Radiative corrections virtual corrections: δ QED virtual box correction propagator correction electroweak corrections: δ EW vertex corrections Z similar for other 4 legs dσ exp dω = dσ 0 QED (1 + δ dω QED + δ EW ) δ QED = δ QED soft + δ QED hard + δ QED virtual 18
Bremsstrahlung with hard γ hard γ : E(γ) > ΔE detection treshold final state: f + f + γ γ is detectable Initial state radiation (ISR): reduced effectively the CMS energy s = zs Z resonance real intermediate Z z max 4m 2 f /s G z σ ff0 zs dz z max = (1- ΔE s )2 radiator function largest effect if σ ff 0 has larger s-dependence In general: distributions are shifted to higher energies and are washed out Example: e + e - μ + μ at LEP dσ QED 0 + δ dω QED y for s ~ m(z) Born: ~1/s EW diagrams dominate 19
Bremsstrahlung with hard γ Final state radiation (FSR): σ ff (γ) = σ ff0 (1+ 3α 4π + ) 1.0017 σ ff 0 (for Z mass measurement at LEP and ξ<20 ) Cross section depend on experimental cuts: which ξ values are allowed to consider e + e - μ + μ in the analysis 20
photon propagator vertex factor virtual electron propagator vertex factor electron propagator photon propagator Propagator Corrections tree level diagramm propagator correction μ ν μ ρ ξ ν igμν q 2 p 1 p 2 =q-p 1 integral over all possible virtual momenta 0 d 4 p 2π 4 ( igμρ q 2 ) (-ieγ ρ ) i- (γπ p 1π +m) p 1 2 m 2 (-ieγ ξ ) i- (γλ p 2λ +m) p 2 2 m 2 ( igνξ q 2 ) 21
Propagator Corrections igμν q 2 ig μν + q 2 0 d 4 p 2π 4 ( igμρ q 2 ) (-ieγ ρ ) i- (γπ p π +m) p 2 m 2 (-ieγ ξ ) i- γλ (p q) λ +m) (p q) 2 m 2 ( igνξ q 2 ) for p >> m ~ p 3 d p ~ 1 p ~ 1 p p d p quadratic divergent! due to conspiratorial cancelation of other terms only logarithmically divergent: igμν q 2 ig μν [ 1 + I(q 2 ) + O(q 4 ) ] q 2 higher order propagator corrections I(q 2 ) = α M2 log 3π q 2 M: upper cut of M for correct result Matrix element M fi ~ e2 q 2 M fi ~ e2 q 2 [1 + I q 2 + O q 4 ] 22
Redefinition of el. Charge Matrix element M fi ~ e2 q 2 M fi ~ e2 q 2 [1 + I q 2 + O q 4 ] Instead of redefining the propagator, redefine charge! e 2 e 2 [1 + I(q 2 ) + O(q 4 )] e = or e = or e =? Higher order contributions can either be assigned to the charge (coupling) and we keep a simple 1 q 2 propagator or equivalently we keep a simple e 2 term for the coupling and modify the propagator 23
Redefinition of el. Charge How is measured charge related to charge entering vertex in Feynman computation? Measured (physcial) charge is NOT a factor contributing to all vertices, but: e e = - ± O(q 4 ) e0 q 2 =μ 2 (contributions from each diagram enter with factor (-1) n, n: nb of loops) μ: scale of experiment Instead of redifining the charge e 2 introduce bare charge associated to vertices e 2 = 2 [1 - I(q 2 =μ 2 ) + O(q 4 )] 2 = e 2 [1 + I(q 2 =μ 2 ) + O(q 4 )] Definition of electric charge depend on the μ 2 of the experiment 24
2 = e 2 [1 + I(q 2 =μ 2 ) + O(q 4 )] Renormalization e 2 = 2 [1 - I(q 2 =μ 2 ) + O(q 4 )] M fi (e 2 0) ~ - ± O(q 4 ) [equ. 1] e0 q 2 e e e e q 2 + ± O(q 4 ) e e e q 2 = μ 2 - e ± O(q 4 ) e e q 2 [equ. 2] M fi (e 2 ) ~ e 2 q 2 [1 + α 3π log ( M2 μ 2 ) - α 3π M2 log ( ) ± O(q q 4 )] = e2 [1 - α log 2 q 2 3π divergencies cancel! q2 + O(q 4 )] μ 2 25
Renormalization Reordering of the contributions resulted in cancelation of arbitrary term with cut-off mass Still the apparent divergent matrix element as function of (equ[1]) and the reordered matrix element as function of e (equ[2]) are mathematical identical [up to O(q 4 )]: M(e 2 0) = M(e 2 ) The infinite coefficients in the original series, arose because itself is not finite (it is in fact infinitesimal) 26
2 = e 2 [1 - I(q 2 ) + O(q 4 )] e 2 (q 2 ) = 2 1+I(q 2 ) Running Coupling Constants I(q 2 ) = α M2 log 3π q 2 electric charge depends on the q 2 of the experiment! 4πα (q 2 )= e 2 (q 2 ) α(q 2 ) = renormalization (very similar procedure as before) α(q 2 =0) = 1 137 PETRA: α(q 2 = 45 2 GeV 2 ) = 1 139 LEP: α(q 2 = 91 2 GeV 2 ) = 1 128 α 0 1 α 0 log ( q2 3π M 2 ) α(q 2 ) = 1 α(μ2 ) 3π α(μ 2 ) log ( q2) μ 2 Phenomenological explanation: screening at large distances smaller net charge 27
Further Loop Corrections virtual corrections all these corrections are infrared divergent (cross section explodes for ν 0 ) soft Bremsstrahlung (E γ << ΔE) no impact on kinematics of scatter process Redefinition of the charge (renormalization) cannot work here! Contrary to propagator corrections, these diagrams depend on initial and/or final scatter particle, thus we would get different charges e.g. for electrons and muons. Divergencies cancel order by order : Ward Identity, fundamental property auf (gauge) field theory [Sakurai 1967, Bjorken and Drell (1964)] 28
Summary Physical electric charge is not a constant it depends (slightly) on the q 2 value at which we run the experiment, similarly true for α. e e = - ±. e0 q 2 =μ 2 Physical charge is not the coupling constant at each vertex! (Physical) matrix elements must be finite, can mathematically reached by renormalization. Coupling at each vertex is a theory parameter (bare charge), which depends on the used renormalization and the order to which we compute the diagrams. 29
What comes Next 1. Introduction 1.1 Natural units 1.2 Standard model basics 1.3 Relativistic kinematics 2. Interaction of Particles with Matter 3. Detectors for Particle Physics 3.1 General Detector Concepts 3.2 PID Detectors 3.3 Tracking Detectors 4. Scattering process and transition amplitudes 4.1. Fermi s golden rule 4.2. Lorentz invariant phase space and matrix element 4.3 Decay Width and Lifetime 4.4 Two and Three Bod decay rate, Dalitz plot 4.5 Cross section 5. Description of free particles 5.1. Klein-Gordan equation 5.2. Dirac equation 5.3. Plane Wave solutions 6. QED 6.1 Interaction by particle exchange 6.2 Feynman rules 6.3. Electron-Positron annihilation 6.4. Electron Scattering 8. Elastic and inelastic electron Proton scattering and the quark model 9. Strong Interaction (QCD) 9.1. Symmetries and conservation laws 9.2. Local gauge invariance 9.3. Bound states 9.4. Quark Gluon Plasma 10. Weak Interaction 10.1. Parity, Wu Experiment, Goldhaber Experiment 10.2. Left right handed couplings 10.3. CKM Matrix 11. The Standard Model 11.1 Electroweak unification 11.2 Precision tests 11.3 Higgs mechanism 12. Neutral Meson Mixing 13. CP Violation 14. Neutrinos 7. Radiativ correction and renormalization 30