Quantum Chromodynamics at LHC

Transcription

1 Quantum Chromodynamics at LHC Zouina Belghobsi LPTh, Université de Jijel EPAM-2011, TAZA 26 Mars 03 Avril

2 Today s high energy colliders

3 past, present and future proton/antiproton colliders Tevatron (1987 ) Fermilab proton-antiproton collisions S = 1.8, 1.96 TeV SppS ( ) CERN proton-antiproton collisions S = 540, 630 GeV LHC (2007 ) CERN proton-proton and heavy ion collisions S = 14 TeV

4 Outline BasicsI: Introduction to QCD - Motivation. Lagrangian, Feynman rules - The running couplings : theory and measurement BasicII: Parton model ideas for DIS - Scaling violatiion & DGLAP - Parton distribution functions Application to hadron colliders hard scattering & basic kinematics the Drell-Yan process in the parton model order α S corrections to DY, singularities, factorisation examples of hard processes and their phenomenology Beyond the fixed order inclusive cross section - Theoretical frontiers

5 Quantum Chromodynamics - a Yang-Mills gauge theory with SU(3) symmetry Rationale evidence that quarks come in 3 colours ++ = (u u u ) requires additional ( 3) internal degrees of freedom to satisfy Fermi-Dirac statistics cross sections and decay rates, e.g. (e + e - hadrons) N c and ( 0 ) N c2, imply N c = 3.0 ±... Thus, put quarks in triplets, iq = (q,q,q), and require invariance under local SU(3) transformations

6 First experimental evidence colour

7 QCD Lagrangian where g s is the QCD coupling constant f abc are the structure constants of SU(3): [T a,t b ] = i f abc T c (a,b,c = 1, 8) A a are the 8 gluon fields T ija are 8 colour matrices, i.e. generators of the SU(3) transformation acting on the fundamental (triplet) representation: Gell-Mann 3 3 matrices, see ESW

8 this corresponds to the normalisation other colour identities include the QCD Lagrangian is invariant under local SU(3) transformations: and from the Lagrangian the Feynman rules can be derived

9 QCD Feynman rules: vertices

10 Feynman Rules for QCD (covariant gauge)

11 Note: gauge fixing to quantise the theory and reduce the number of degrees of freedom of the gauge fields, need to introduce a gauge fixing term: these are covariant gauges, and additional ghost fields are required. b a p c k or these are non-covariant ( axial ) gauges no ghosts required!

12 Asymptotic freedom in QCD - In QED, α gives the the probability of emitting a photon - the charge seen depends on distance from the particle (running) - Due to screening, α is small at long distances, this is the scale at which perturbation theory works When very far from each-other charges behave as free non-interacting particles - α S strengh of strong interaction α S = g S2 /4π, g S is the color charge - In QCD, α S gives the probability of emitting a gluon. The color charge seen depends on distance from the particle (running) - Due to anti-screening, α S is small at short distances, this is the scale at which perturbation theory works When very close together, partons behave as free noninteracting particles

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16 Measurements of the running coupling

17 S measurements and world average

18 therefore a precise measurement of the coupling at a small scale Q can given improved precision on the fundamental parameter S (M Z2 ) however, the small-scale determination may be more contaminated by power corrections or other non-perturbative effects

19 αs from event shapes at NNLO scale variation reduced by a factor 2 scatter between αs from different event-shapes reduced better, central value closer to world average

20 Recap

21 Partons in the initial state HERA/Tevatron/LHC involve protons in the initial state Protons are made of QCD constituents Will focus on aspects related to the initial state effects

22 Deep Inelastic Scattering The reaction equation of DIS is written e + p e + X where X is a system of outgoing hadrons (mostly pions). Observed is only the scattered electron. The unobserved hadronic system is the missing mass. The energy of the incident electron beam is accurately known The proton is the target particle; in the SLAC experiments (and many later experiments at CERN) the target is at rest in the laboratory. This defines the LAB frame.

23 Scaling found a natural explanation in the parton model (Feynman). Partons are constituents of the proton (more generally of hadrons). They are quarks and gluons: quarks are point-like spin-1/2 fermions like the leptons, gluons are masslessspin-1 bosons: they are the carriers of the strong interaction. Unlike leptons, quarks take part in strong as well as electromagnetic and weak interactions. Recall: charged leptons take part only in electromagnetic and weak interactions, neutrinos, i.e. neutral leptons, only in weak interactions.

24 Scaling in DIS ep In high energy ep X we define variables: Structure functions, F 1 and F 2 are include our ignorance of the proton structure in describing the interaction cross section: F 1,F 2 are functions of x,q 2. Bjorken predicted that for Q 2, and x fixed the F i depend only on x Bjorken scaling scatters off (charged) point-like proton constituents.

25 Parton model for DIS

26 DGLAP QCD evolution equation for parton densities: P(y) are splitting functions which can be calculated within the framework of perturbativeqcd (pqcd).

27 The parton distribution functions (PDFs)

28 The data from all DIS experiments taken together in Global Fits using aqcd based theoreticalframework yield distributionsof the individual partons.shown here are the momentumdistributions of the valenceuand dquarks, quarkantiquarksea and gluons.note the scaling of the gluonand sea distributions!the fits of three differentgroups are in good agreement

29 PDF fits including HERA-II high Q^2 data

30 PDF fits including HERA-II high Q^2 data

31 PDF fits including HERA-II high Q^2 data

32 Introduction: What can we calculate? Scattering processes at high energy hadron colliders can be classified as either HARD or SOFT Quantum Chromodynamics (QCD) is the underlying theory for all such processes, but the approach (and the level of understanding) is very different for the two cases For HARD processes, e.g. W, H or high-p T jet production, the rates and event properties can be predicted with some precision using perturbation theory Calculate, Predict & test For SOFT processes, e.g. the total cross section or diffractive processes, the rates and properties are dominated by non-perturbative QCD effects, which are much less well understood. (From Slide_Stirling07) Model, Fit,Explorate & Pray!

33 The QCD factorization theorem and Hard scattering formalism

34 It was subsequently realized that these logarithms were the same as those that arise in DIS structure function calculations, and could therefore be absorbed, via the DGLAP equations, in the definition of the parton distributions, giving rise to logarithmic violations of scaling. The key point was that all logariithms appearing in the Drell-Yan corrections could be factored into renormalized parton distributions in this way, and factorization theorem which showed that this was a general feature of hard scattering processes is derived Taking into account the leading logarithm corrections, (1) simply becomes:

35 μ F : factorization scale ( separates the long-and short distance physics) μ R : renormalization scale for QCD running coupling The cross section calculated to all orders in perturbation theory iis invariant under changes in these parameters. Different choices will yield different (numerical) results. To avoid unnaturally large logarithms reappearing in the perturbation series it is sensible to choose μ R and μ F values of order of the typical momentum scales of the hard scattering process, and μ R = μ F is also often assumed.

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37 Application to hadron colliders The inclusive prompt photon production The inclusive jet production W/Z production??

38 Improved parton model

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40 The implementation of isolation cuts

41 Inclusive photon production

42 Inclusive photon production

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46 Predicions of pqcd for expre

47 Isolated Prompt photon production

48 Conclusions

49 SUMMARY QCD at hadron Colliders means performing precision calculations (LO NLO NNLO) for signals and backgrounds, cross sections and distributions still much work to do! refining event simulation tools (e.g. PS+NLO) extending the calculational frontiers, e.g. to hard + diffractive/forward processes, multiple scattering, particle distributions and correlations etc. particularly important and interesting is p+p p+ X+p chalenge for experiment and theory.

50 Thank you

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52 Local gauge transformations The QCD lagrangian is is invariant under local gauge transformations, i.e. one can refine the quark and gluon fields independently at every point in space and time without changing the physical content of the theory It follows that (covariant = transforms with the field) Therefore e.g. because The QCD lagrangian is gauge invariant