Physics at Hadron Colliders Partons and PDFs Marina Cobal Thanks to D. Bettoni Università di Udine 1
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How to probe the nucleon / quarks? Scatter high-energy lepton off a proton: Deep-Inelastic Scattering (DIS) In DIS experiments point-like leptons + EM interactions which are well understood are used to probe hadronic structure (which isn t). Relevant scales: d probed = p 10 18 m large momentum -> short distance (Uncertainty Principle at work!)
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e-p scattering
DIS (deep inelastic scattering) 6
Inelastic scattering When the scattering is not elastic (new particles are produced) the energy and direction of the scattered electron are independent variables, unlike the elastic scattering situation. W is the mass squared of the produced hadronic system e (k,e) N (P,M) e(k',e') γ (q) θ W From the measurement of the direction θ (solid angle element dω) and the energy E ' of the scattered electron, the four momentum transfer Q 2 =-q 2 can be calculated. The differential cross-section is determined as a function of E ' and Q 2. k ' = k q q 2 = 4 EE ' sin 2 θ 2 W 2 = (P + q) 2 = P 2 + 2P q + q 2 = M 2 + 2Mν Q 2 ν = E E '
Electron - proton inelastic scattering Bloom et al. (SLAC-MIT group) in 1969 performed an experiment with high-energy electron beams (7-18 GeV). Scattering of electrons from a hydrogen target at 6 0 and 10 0. Only electrons are detected in the final state - inclusive approach. The data showed peaks when the mass W of the produced hadronic system corresponded to the mass of the known resonances.
e-p scattering Rutherford x-sec: e: non-relativistic p: does not recoil Mott x-sec: e: relativistic p does not recoil DIS e-p scattering: e relativistic p recoil spin-spin effect Energy increase 9
DIS Kinematics Four-momentum transfer: q 2 = (E E ') 2 ( k k ') ( k k ') = = m 2 e + m 2 e' 2(EE ' k k ' cosθ) = 4EE 'sin 2 θ 2 Q 2 Mott Cross Section (!c=1): L W µν µν : : lepton tensor hadron tensor a virtual photon of fourmomentum q is able to resolve structures of the order!/ q 2
Form Factors
If the proton recoils: Electron-Proton Point-like e-p Scattering scattering
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Measuring Electron-Proton G E (qscattering 2 ) and G M (q 2 )
Electron-Proton High q 2 Scattering results
Summary Electron-Proton of elastic Scattering scattering
Inelastic scattering cross-section Similar to the electron-proton elastic scattering, the differential cross-section of electron-proton inelastic scattering can be written in a general form: dσ dωde ' = α 2 cos 2 θ 2 " W 4E 2 sin 4 θ 2 (ν, q 2 )+W 1 (ν, q 2 )tan 2 θ # 2 2 The cross-section is double differential because θ and E ' are independent variables. The expression contains Mott cross-section as a factor and is analogous to the Rosenbluth formula. It isolates the unknown shape of the nucleon target in two structure functions W 1 and W 2, which are the functions of two independent variables ν and q 2. The structure functions correspond to the two possible polarisation states of the virtual photon: longitudinal and transverse. Longitudinal polarisation exists only because photon is virtual and has a mass. For elastic scattering, (P+q) 2 =M 2 and the two variables ν and Q 2 are related by Q 2 =2Mν. $ %
Scaling To determine W 1 and W 2 separately it is necessary to measure the differential crosssection at two values of θ and E ' that correspond to the same values of ν and Q 2. This is possible by varying the incident energy E. SLAC result: the ratio of σ /σ Mott depends only weakly on Q 2 for high values of W. For small scattering angles σ /σ Mott W 2. Thus, the structure function W 2 does not depend on Q 2. 2004,Torino Aram Kotzinian
Scaling Instead, at high values of W the function νw 2 depends on the single variable ω = 2Mν / Q 2 (at present the variable x=1/ω is widely used) This is the so-called "scaling" behaviour of the cross-section (structure function). It was first proposed by Bjorken in 1967. W 1,2 (ν,q 2 ) W 1,2 (x) when ν,q 2.
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Observe excited resonance states: Nucleons are composite What do we see in the data for W > 2 GeV? 24
First SLAC experiment ( 69): expected from proton form factor: 2 % dσ / de ' dω ( 1 # Q 8 = && 2 2 # (dσ / dω) Mott ' (1 + Q / 0.71) $ First data show big surprise: very weak Q2-dependence form factor -> 1! scattering off point-like objects?. introduce a clever model!
The Quark-Parton Model Assumptions: Proton constituent = Parton Elastic scattering from a quasi-free spin-1/2 quark in the proton Neglect masses and p T s, infinite momentum frame e P e Lets assume: p quark = xp proton parton Since xp 2 M 2 <<Q 2 it follows: 2 2 2 ( xp + q) = p' quark = mquark 0 Check limiting case: 2 2 2 x 1 2 W = M p + 2M pν Q!! M p 2xP q + q 2 2 2 Q Q 0 x = = 2Pq 2Mν ν = (q. p)/m = E e -E e Definition Bjorken scaling variable Therefore: x = 1: elastic scattering and 0 < x < 1
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The quark structure of nucleons Quark quantum numbers: Spin: ½ S p,n = ( ) = ½ Isopin: ½ I p,n = ( ) = ½ Why fractional charges? Extreme baryons: Z = Assign: z up =+ 2/3, z down = - 1/3 Three families: m c,b,t >> m u,d,s : no role in p,n Structure functions: Isospin symmetry: Average nucleon F 2 (x) with q(x) = q v (x) + q s (x) etc. )] ( ) ( ) ( [ )] ( ) ( ) ( [ 9 1 9 4 9 1 2 9 1 9 4 9 1 2 s s n s n s n v n s n s n v n s s p s p s p v p s p s p v p s s u u u d d d x F s s u u u d d d x F + + + + + + + = + + + + + + + = 3 2 3 1-2 3 1 + + q z q z! " # $ % &! " # $ % &! " # $ % & b t s c d u b s d t c u m m m z m m m z ; ; 0.3 GeV) ( 3 1 1.5 GeV) ( 3 2 << << = << << = + p s p s n s n s p v n v p v n v d u d u u d d u = = = = =,, )] ( ) ( [ )) ( ) ( ( ) ( 9 1, 18 5 2 2 2 1 2 x s x s x x q x q x F F F s s d u n p N + + + = + = 2) 1, ( +
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IF, proton was made of 3 quarks each with 1/3 of proton s momentum: no anti-quark! 2 F 2 = x (q(x) + q(x)) e q F 2 or with some smearing 1/3 x The partons are point-like and incoherent then Q 2 shouldn t matter. à Bjorken scaling: F 2 has no Q 2 dependence.
Proton Structure Function F 2 F 2 Movies from R. Yoshida, CTEQ Summer School 2007
Deep Inelastic Scattering experiments Fixed HERA target collider: DIS H1 at and SLAC, ZEUS FNAL, experiments CERN, now 1992 JLab 2007
Modern data PDG 2002 First data (1980): Now.. Scaling violations : weak Q 2 dependence rise at low x what physics??.. QCD
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Quantum Chromodynamics (QCD) Field theory for strong interaction: quarks interact by gluon exchange quarks carry a colour charge exchange bosons (gluons) carry colour self-interactions (cf. QED!) Hadrons are colour neutral: RR, BB, GG or RGB leads to confinement: q q α s g q α s q q, qq or qqq forbidden Effective strength ~ #gluons exch. low Q 2 : more g s: large eff. coupling high Q 2 : few g s: small eff. coupling
QCD brings new possibilities: q q quarks can radiate gluons q q gluons can produce qq pairs gluons can radiate gluons!
Proton e e *(Q 2 ) r ~1.6 fm (McAllister & Hofstadter 56) Virtuality (4-momentum transfer) Q gives the distance scale r at which the proton is probed. r hc/q = 0.2fm/Q[GeV] CERN, FNAL fixed target DIS: HERA ep collider DIS: r min 1/100 proton dia. r min 1/1000 proton dia. HERA: E e =27.5 GeV, E P =920 GeV (Uncertainty Principle again)
F 2 Higher the resolution (i.e. higher the Q 2 ) more low x partons we see. So what do we expect F 2 as a function of x at a fixed Q 2 to look like?
F 2 (x) Three quarks with 1/3 of total proton momentum each. F 2 (x) 1/3 x Three quarks with some momentum smearing. 1/3 x F 2 (x) The three quarks radiate partons at low x. 1/3 x.the answer depends on the Q 2!
Proton Structure Function F 2 How this change with Q 2 happens quantitatively described by the: Dokshitzer-Gribov- Lipatov-Altarelli- Parisi (DGLAP) equations
QCD predictions: scaling violations Originally: F 2 = F 2 (x) but also Q 2 -dependence Why scaling violations? if Q 2 increases: more resolution (~1/ Q 2 ) more sea quarks +gluons Officially known as: Altarelli-Parisi Equations ( DGLAP )
DGLAP equations are easy to understand intuitively.. First we have four splitting functions z z z z 1-z 1-z 1-z 1-z P ab (z) : the probability that parton a will radiate a parton b with the fraction z of the original momentum carried by a. These additional contributions to F 2 (x,q 2 ) can be calculated.
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Running of α
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QCD predictions: the running of a s pqcd valid if a s << 1: Q 2 > 1.0 (GeV/c) 2 pqcd calculation: CERN 2004 PDG 2002 2 12π α s ( Q ) = 2 2 ( 33 2n ) ln( Q / Λ ) with Λ exp = 250 MeV/c: asymptotic freedom f Q 2 α s 0 confinement Q 2 0 α s Running coupling constant is quantitative test of QCD.
Self Interaction
Colour charge strenght