Lecture 3 Cross Section Measurements. Ingredients to a Cross Section

Transcription

1 Lecture 3 Cross Section Measurements Ingredients to a Cross Section

2 Prerequisites and Reminders... Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Particle Decays Cross Section Matrix Element Phase Space Feynman Diagrams Mandelstam Variables Parton Distributions Bjorken-x... 4-vector scalar product Lorentz invariant All quantities like cross sections etc. should be in terms of scalar products of 4-vectors...

3 Prerequisites and Reminders... Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Particle Decays Cross Section Matrix Element Phase Space Feynman Diagrams Mandelstam Variables Parton Distributions Bjorken-x... Particle momentum as seen in laboratory frame... Particle momentum as viewed from a frame moving with velocity βf... Lorentz Transformation: with

4 Prerequisites and Reminders... pt Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Particle Decays Cross Section Matrix Element Phase Space Feynman Diagrams Mandelstam Variables Parton Distributions Bjorken-x... Proton η=-1 x 1 p Relevant kinematic variables: Transverse momentum: pt Rapidity: y = ½ ln (E-pz)/(E+pz) Pseudorapidity: η = -ln tan ½θ Azimuthal angle: φ s η=0 (=90 ) θ x 2 p Schematic proton-proton scattering Pseudorapidity η=1 (~40 ) Proton η=2 (~15 ) η=3 (~6 )

5 Prerequisites and Reminders... Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Particle Decays Cross Section Matrix Element Phase Space Feynman Diagrams Mandelstam Variables Parton Distributions Bjorken-x... Invariant Mass: Center-of-mass Energy: Particle 2 at rest: Particle Collider:

6 Prerequisites and Reminders Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Cross Section Particle Decays Matrix Element Phase Space Feynman Diagrams Mandelstam Variables Differential Cross Section: Partial Decay Rate: n-body phase space Matrix element with Parton Distributions Bjorken-x...

7 Prerequisites and Reminders... Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Particle Decays Cross Section Matrix Element Phase Space Feynman Diagrams Mandelstam Variables s-channel f Mandelstam variables: f f f f t-channel f f u-channel f Parton Distributions Bjorken-x... f f f f

8 Prerequisites and Reminders... p x1 Natural Units Four-Vector Kinematics Lorentz Transformation Lorentz Boost Lorentz Invariance Rapidity etc. Invariant Mass CMS-Energy Particle Decays Cross Section Matrix Element Phase Space Feynman Diagrams Mandelstam Variables Parton Distributions Bjorken-x... Proton-Proton Cross Section: x2 Parton content: f(x,q 2 ) = q(x,q 2 ) or g(x,q 2 ) x1,2 : Bjorken-x fractional momentum of parton involve in hard process Q 2 f p : scale; spatial resolution invariant parton-parton mass : Parton Distribution function measured e.g. at HERA... Q 2

9 Proton-Proton LHC Hard interaction: qq, gg, qg fusion Initial State Radiation (ISR) Secondary Interaction [ underlying event ]

10 Proton-Proton LHC Proton Hard Process [calculable] Product PDFs Product Proton

11 Some Hard Processes...

12 QCD Matrix Elements

13 Proton-Proton LHC Proton Hard Process [calculable] Product PDFs Product Proton

14 Electron-Proton HERA Electron e q γ e eq eq q Hard Process [calculable] Electron PDFs Quark Proton

15 Electron-Proton HERA Scattered Electron Electron-Proton Scattering Electron Proton Scattered Quark

16 Electron-Proton HERA [H1 NC Event] Electron e p Quark

17 Electron-Proton HERA Ee=27.5 GeV Q 2 : neg. four momentum transfer squared Ep=920 GeV x : fractional momentum of struck quark

18 Electron-Proton HERA Electron (e ± ) k k' Electron (e ± ) Cross Section: dσ ~ dσeq F2 SF γ Q 2 = -q 2 = -(k-k') 2 Quark 4πα 2 /q 4 2 x eq q(x) x Quark Proton SF Structure function describes proton structure probability to find quark with mom. fraction x Proton Remnant

19 Structure Function F2 p = uud x = 1/3 QPM: Structure Functions F2 independent of Q 2

20 Proton Three valence quarks F2(x) 1/3 1 x Proton Three bound valence quarks 1/3 1 x Proton Bound valence quarks + gluon radiation sea valence [see e.g. Halzen/Martin] small x 1/3 1 x

21 ep Scattering at HERA

22 Which region x-q 2 is seen by different experiments?

23 Kinematics of DIS - 1

24 Kinematics of DIS - 2

25 Kinematics of DIS - 3

26 Kinematics of DIS - 4

27 Kinematics of DIS - 5

28 F1 and F2

29 Scaling Behavior [SLAC 1972] Fixed Target Experiment

30 Scaling Violations [SLAC 1972]

31 DGLAP Equations [DGLAP: Dokshitzer, Gribov, Lipatov, Altarelli, Parisi] PDFs [z: momentum fraction of radiated parton]

32

33

34 Proton Parton Densities x Parton Density # Valenzquarks = u v (x) + d v (x) dx = 3 # Gluonen = g(x) dx > 30 u v d v Proton gluon dominated x

35 Proton Parton Densities xf H1 and ZEUS HERA I+II Combined PDF Fit H1 and ZEUS Combined PDF Fit Q 2 = 10 GeV 2 Q 2 = 10 GeV 2 HERA Structure Functions Working Group July x x

36 Proton Parton Densities xf H1 and ZEUS HERA I+II Combined PDF Fit H1 and ZEUS Combined PDF Fit Q 2 = GeV 2 Q 2 = GeV 2 HERA Structure Functions Working Group July x x

37 Parton Q 2 = 10 TeV GeV Gluons

38 Todays Picture of the Proton The most dramatic of these [experimental consequences], that the protons viewed at ever higher resolution would appear more and more as field energy (soft glue), was only clearly verified at HERA... F. Wilczek [Nobel Prize 2004]

39 Proton-Proton LHC Proton Hard Process [calculable] Product PDFs Product Proton

40 Proton-Proton LHC Proton Hard Process [calculable] Hadron-Jets Leptons... PDFs Parton Shower Proton Hadronization [phenomenological] [see later]

41 Particle LHC p M 2 Q 2 [GeV 2 ] LHC parton kinematics p pp XM + remnants XM: particle with mass M e.g. Higgs 100 GeV Higgs M 2 = x1x2 s i.e. to produce a particle with mass M at LHC energies ( s = 14 TeV) x = x1x2 = M/ s [x1 = x2: mid-rapidity] DGLAP LHC needs: Knowledge of parton densities Extrapolation over orders of magnitudes x

42 Gluon Higgs Cross Section Gluon top top top mh Demartin et al., arxiv: v % PDF uncertainty Higgs The left column shows absolute results, the central column results normalized to the MSTW08 result, and the right column results normalized to each group's central result. mh

43 Relative Uncertainty [compared to CTEQ 6.1M] Inclusive Jet Cross LHC [D.Stump et al., JHEP 10 (2003) 046] pt dσ/dpt pt Mid-rapidity: 100 % ET ~ 5 TeV Forw. jets: 100 % ET ~ 2 TeV pt

44 Inclusive Jet-Cross Section Inclusive Jet cross-section [~Tevatron x 100] - Restricted to 17 nb -1 [no pile-up contamination]; - pt > 60 GeV and y < 2.8 Inclusive Jet Cross Section Measured jets corrected to particle level using Monte Carlo - Experimental uncertainties dominated by JES Good data-mc agreement over 5 orders of magnitude! [Important for Searches]

45 W and Z LHC Q 2 for W/Z LHC energies x = pp W + X pp Z + X Considered as luminosity monitor

46 Vector Boson Production Direct γ-production: Singel W/Z production: q W ± q q q Z At LHC energies these processes take place at low values of Bjorken-x Only sea quarks and gluons are involved At EW scales sea is driven by the gluon, i.e. x-sections dominated by gluon uncertainty Constraints on sea and gluon distributions

47 Effect on PDFs of LHC W data Q 2 = 6488 GeV 2 Q 2 = 6488 GeV 2 ZEUS PDF fit before including W data ZEUS PDF fit after including W data e + CTEQ6.1 pseudo-data e + CTEQ6.1 pseudo-data 35% error reduction