Particle Physics WS 2012/13 (9.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101
QED Feyman Rules Starting from elm potential exploiting Fermi s gold rule derived QED Feyman Rules e p 1 e p 3 incoming electron u(p 1 ) incoming positron v(p 2 ) 1 st vertex p 2 p 4 e + e + -im = i q e v e p 2 outgoing electron u(p 3 ) outgoing positron v(p 4 ) γ μ u e (p 1 ) igμν q 2 iq e u e p 3 γ ν v e (p 4 ) 2 nd vertex fermion current propagator fermion current at 1 st vertex at 2 nd vertex 2
e p 1 p 2 Calculation of cross section e + e - μ + μ - Lecture 3, diff. cross-section in CMS: e + μ + μ p 3 p 4 dσ = 1 p f M dω 64 π 2 s p fi 2 i lowest order diagram M ~ e 2 ~α em (NU: e= 4πα; α ~ 1 137 ) -im = i q e v e p 2 p 1 = p 2 = p i p 3 = p 4 = p f s = (E 1 + E 2 ) 2 γ ξ u e (p 1 ) igξν q 2 iq μ u μ p 3 γ ν v μ (p 4 ) For unpolarized beam need to average over all possible initial helicity combinations. spin averaged matrix element < M 2 > = 1 4 ( M RL XX 2 + M RR XX 2 + M LL XX 2 + M LR XX 2 ) 3
Matrix Element Calculation Computation was done for limit E>>m; note, several helicity combinations vanished! 4 4 4 4 M(RL RL) 2 M(RL LR) 2 M(LR RL) 2 M(LR LR) 2 4
Differential Cross Section The cross section is obtained by averaging over the initial states and summing over all final states. (E>>m μ p f = p ) i = dσ dω = 1 4 = 1 64 π 2 s p f p i M fi 2 1 ( M(RL RL) 2 + M(LR LR) 2 + M(RL LR) 2 + M(LR RL) 2 ) 256 π 2 s 4πα 2 256 π 2 s (2(1+cosθ)2 + 2(1-cosθ) 2 ) dσ = α2 dω 4s (1+cos2 θ) Example: e + e - μ + μ, s = 29 GeV pure QED (α 3 ) calculation QED + Z contribution 5 5
Total Cross Section Total cross section obtained by intregrating over θ and φ using 1 + cos 2 θ dω = 2π (1 + cos 2 θ) dθ = 16π 3 total cross section for (e + e - μ + μ ): σ = 4πα2 3s Lowest order computation provides good description of data Very impressive result: from first principle computation we have reached a 1% precise result! 6
Helicity and Chirality = σ 0 0 σ = σ 1 σ 1 σ 2 σ 2 helicity operator: h = Σp [H,h] = 0 Helicity is a conserved quantity! σ 3 Helicity is not a LI quantity! σ 3 (p is not a LI) chirality operator: γ 5 = i γ 0 γ 1 γ 2 γ 3 = 0 I I 0 in Dirac representation [H,γ 5 ] 0 Chirality is not a conserved quantity! Chirality is however LI! only for E>>m : γ 5 ~ Σp LH chirality eigenstates LH helicity eigenstates RH chirality eigenstates RH helicity eigenstates For massive particles (not ultra relativisitc): helicity states are combination of chirality states! 7
Chirality Eigenstates Only certain combinations of chirality eigenstates take part in elm IA (property of vertex coupling) u L γ μ u L, u R γ μ u R Right handed particles with right handed particles Left handed particles with left handed particles v L γ μ v L, v R γ μ v R Right handed antiparticles with right handed antiparticles Left handed anti particles with left handed antiparticles u L γ μ v R, u R γ μ v L, v R γ μ u L v L γ μ u R Right handed particles with left handed antiparticles Left handed particles with right handed particles Left handed antiparticles right handed particles Right handed antiparticles left handed particles Coupling strength for left handed and right handed particles is the same in elm. IA! 8
Content of Today We do already have e + e + μ + μ + e μ + Calculate cross-section for related process exploiting symmetries and Mandelstam variables e + μ e + μ e e e + μ and e + e + e + e + e e e e + + μ μ e + e e + e + Test of first order QED computations in e + e - collisions at TASSO Luminosity measurements exploiting e + e + e + e + Discovery of τ lepton in e + e - collisions 9
Time and Space Structure of Feynman Diagrams general 4 prong diagram (several first order QED diagrams have this structure) a) Time like s channel process p A p B p c p D choose CMS system p A p c s =q 2 =(p A +p B ) 2 = (E A +E B ) 2 (p A +p B ) 2 (m A +m B ) 2 > 0 real photon: q 2 = E 2 - p 2 = 0 p B t p D q 2 >0 virtual photon b) Space like t channel process p A p c t =q 2 =(p C -p A ) 2 = (E C -E A ) 2 (p C p A ) 2 p B p D t choose A at rest and m A =m C : E C > m C t = E c2-2e c m A +m 2 A - p c2 = m C2 2E C m A + m A2 < 0 q 2 <0 virtual photon 10
b) u channel process p A p B p c p D Mandelstam variables u =q 2 =(p D -p A ) 2 < 0 q 2 <0 virtual photon Mandelstam variables: s = (p A +p B ) 2 = (p C +p D ) 2 t = (p C -p A ) 2 = (p D -p B ) 2 u = (p D -p A ) 2 = (p C -p B ) 2 t Total cross/section or any LI invariant diff. cross-section can only depend on scalar products of 4-momenta p A, p B, p C, p D. p i p j 10 scalars: p A p A, p A p B, p A p C, p A p D p B p B, p B p C, p B p D, p C p C, p C p D, p D p D p 2 i = E 2 i - p 2 i = m 2 i four constraints energy + momentum conservation: (p A +p B = p C +p D ) four constraints 2 independent scalars describe the process, usually 2 of Mandelstam variables are chosen 11
Lorentz Invariant Representation of X-Section <M fi > 2 = 1 4 M fi 2 e p 1 μ = 1 4 ( M(RL RL) 2 + M(LR LR) 2 + M(RL LR) 2 + M(LR RL) 2 ) p 3 = e4 4 (2(1+cosθ)2 + 2(1-cosθ) 2 ) [Θ in CMS, E >> m] p 2 e + μ + p 4 <M fi > 2 is LI, would be good to express it in form of LI quantities! In CMS: p 1 = (E,0,0,E) p 2 = (E,0,0,-E) p 3 = (E,E sinθ, 0, E cosθ) p 4 = (E, -E sinθ, 0, -E cosθ) (p i ±p j ) 2 = m i 2 + m j 2 ± 2p i p j ~ ±2p i p j E>>m p 3 μ e θ e + p 1 p 4 μ + π-θ x z p 2 p 1 p 2 = 2E 2 ~ ½ (p 1 +p 2 ) 2 = 1 2 s p 1 p 3 = E 2 (1+cosθ) ~ -½ (p 3 -p 1 ) 2 = - 1 2 t p 1 p 4 = E 2 (1-cosθ) ~ -½ (p 4 -p 1 ) 2 = - 1 2 u <M fi > 2 = 2e 4 t 2 +u 2 s 2 12
e - μ - e - μ - e - e + μ - μ + e μ + p 1 p 3 p 1 e - μ - e - μ - e e p 3 e + p 1 p 1 p 2 p μ 4 p 2 μ μ p 2 p 3 p 3 p 2 p 4 p 4 p 4 s = (p 1 +p 2 ) 2 t = (p 3 -p 1 ) 2 u = (p 4 -p 1 ) 2 (p 1 -p 3 ) 2 = t (-p 2 -p 1 ) 2 = s (p 4 -p 1 ) 2 = u <M fi > 2 = 2e 4 t2 +u 2 s 2 <M fi > 2 = 2e 4 s2 +u 2 t 2 13
e - μ - Scattering in CMS In CMS: p 1 = (E,0,0,E); p 2 = (E,0,0,-E); p 3 = (E,E sinθ, 0, E cosθ); p 4 = (E, -E sinθ, 0, -E cosθ) p 1 x s = 4E 2 z t = (p 3 -p 1 ) 2 e π-θ p 3 θ e μ p 2 ~ -2p 3 p 1 = -2E 2 (1-cosθ) u = (p 4 -p 1 ) 2 ~ -2p 4 p 1 = -2E 2 (1+cosθ) <M fi > 2 = 2e 4 s2 +u 2 t 2 = 2e 2 4 4+ 1+cosθ 1 cosθ 2 p 4 μ dσ = 1 p f <M dω 64 π 2 s p fi > 2 i = e4 8π 2 s 1+ 1 4 1+cosθ 2 1 cosθ 2 = e 4 8π 2 s 1+ 1 4 1+cosθ 2 4 sin 4 θ/2 sideremark: almost Rutherford, for Rutherford need non-relativisitc computation 14
Bhabha-Scattering e + e + e + e + M = M 2 M 1 e e + + e e e + e e + e + M 2 = M 1 2 + interference + M 2 2 same as for e - e + μ - μ + same as for e - e + e - e + dσ dω = e4 8π 2 s t 2 + u 2 s 2 + 2u 2 t + s 2 + u 2 2 t 2 15
Overview from Halz from Halzen and Martin 16
dσ dω Scatter Processes in e + e - Bhabha scattering: t+s channel e e e e Z e + e e + e - e + e + e + e + t-channel less sensitiv to contributions from Z exchange, even at larger energies. QED computation have precision of 10-4 Exploit forward region for luminosity measurements e + e γγ s-channel only e + e μ + μ- e μ+ e μ + exploited for luminosity measurements e + μ e + Z μ 17
Luminosity Determination at e + e - Machines Interaction rate: N IA = σ IA x L Number of observed events: N IA = σ IA x L dt = σ IA L dt σ IA : interaction cross-section L: luminosity, depends on machine parameters [cm 2 /s] L int = L dt [barn -1 ] L = 1 n B f n 1 n 2 4π σ x σ y n 1, n 2 : particles per bunch n B : number of buches f : revolution frequency σ x, σ y : beam size at IA point determination of delivered integrated luminosity from machine parameters not precise enough (± 5-10%) Instead a known reference process with known x-section is used to determine the integrated luminosity L int = N ref σ ref 18
Luminosity Determination at e + e - Machines Reference process for e + e - machines: small angle Bhabha scattering, precisely known, very high rate, very little background e + e + e - e - main detector L int = N(e+ e ) θ σ min < θ < θ max ref Luminosity monitors e.g. forward calorimeters Typical ranges for angles: 15 < θ < 35 (other standard candle Z μμ) 19
L3 Detector at LEP LEP: Large Electron Positron Collider, CERN, E CM = 90 GeV. SLUM: Silicon luminosity detector LCAL: Luminosity calorimeter collision center 20
Bhabha Scattering Event in LEP LCAL 21
Experimental Test of QED: Tasso Experiment PETRA: Positron-electron collider (DESY, Hamburg), E CM up to 35 GeV Started data taking in 1976, finished data taking in 1986 Tasso: Two Arm Spectrometer Solenoid [Tasso experiment is famous for discovery of the gluon] 22
How to test for possible deviations in e + e - e + e - ------ theory prediction (dominated by t-channel) data points Perfect agreement! - how to quantify perfect? Possible deviations from QED: Finite extension of lepton modified photon propagator Description and parametrization of deviation by form factor: F(q 2 ) = 1 ± 1 q 2 1 q 2 ± 1 q 2 Λ ± 2 q 2 q 2 2 Λ ± In this choice of form factor parameterization F(q 2 ) describes an add. massive photon which modify the propagator. Parameter Λ ± would correspond to the photon mass of add. photon. 23
How to test for possible deviations in e + e - e + e - dσ dω = e4 8π 2 s t 2 + u 2 s 2 F t 2 + 2u2 t 2 F t F s + s2 + u 2 t 2 F(s) 2 A fit to the Tasso data results in: Λ + > 435 GeV, Λ - > 590 GeV @ 95% CL In the space picture form factor correspond to modified Coulomb potential: 1 r 1 r (1 + e Λr ) extended charge For Λ > 500 GeV Point-like electron: < 0.197 fm/500 = 0.5 10-18 m 24
Test of QED in e + e - μ + μ - Total cross section Very good agreement σ(e+e- μ+μ-) = e4 3s (1 ± s s Λ ± 2) Λ ± > 250 GeV R μμ = σ exp σ th muon substructure < 10-18 m 25
Test of QED in e + e - μ + μ - Total cross-section in very good agreement, however angular distribution deviates from QED predictions effect of electroweak interference e μ + e μ + + e + μ e + Z μ backward region forward region A FB = σ F σ B σ F + σ B Due to interference effect of EW contribution already seen at s = 30 GeV! 26
Discovery of the Tau Lepton 1975 M. L. Perl, leader of Mark-I experiment at SLAC Nobel prize 1995 SPEAR: Standford-positron-electron-accelerator ring E CM up to 8 GeV Mark-I experiment 27
Discovery of the Tau Meson physics process: e + e - τ + τ Signature in experiment: e + e - e ± μ + 2 undetected particles e + v τ ν e μ - v μ ν τ + CC in e+e- collisions we know the initial 4 momenta, thus we can check for missing energy in the final state
Mark-I experiment μ e
Discovery of the Tau Lepton signature: e + e - e ± μ + 2 undetected particles Background process: e + e - e + e and e + e - μ + μ with one leg in the final state mis/identified as electron/muon respectively; cut on θ copl >20 reduced this background significantly e + e - μ + μ - e + e - Very rare decay, cross check: look for same sign e - μ - and e + μ + combinations hadron mis-id background (h: hadron) e + e - μ + h - X, e + e - h + e - X, e + e - h + h - X 4.7 ± 1.5 background candidates from these sources 24 signal candidates clear signal, not compatible with background fluctuations 30
Discovery of the Tau Lepton M m : missing energy M i : invariant mass of (e,μ) No resonance observed, thus at least 2 missing particles in final state threshold behaviour indicates mass of tau particle 2m τ ~ 4 GeV
Measurement of Tau Mass Later measurement at BESS (1994):
Point like nature of τ Particle As for e+e- μ+μ- 1 st order QED prediction of total cross-section for e + e - τ + τ - in wonderful agreement (differential cross-section suffers from interference with Z exchange diagram) Λ± > 200 GeV another point like (elementary) particle 33
Summary Two add. variables (beside all masses and 4 momenta of initial particles) are required to describe scatter process A+B C +D. E.g. use 2 out of 3 (LI) Mandelstam variables Symmetry arguments can be used to relate cross-section of different scatter process (technical: swap of Mandelstam variables in cross-section formular) First order QED correction describe well the total cross section in data (for e + e - e + e -, e + e - μ + μ -, e + e - τ + τ - ) This is mainly related to the strong supression of higher order diagrams α ~1/137 Differential cross-section is affected by interference with Z exchange diagram (can be computed as well) Next time :continue test of QED in e + e - scattering hadronic resonances in e + e - scattering higher order corrections and renormalization (running coupling constants) 34