Introduction to Perturbative QCD

Transcription

1 Introduction to Perturbative QCD Lecture 1 Jianwei Qiu Iowa State University/Argonne National Laboratory PHENIX Sinest at RIKEN 007 June 11 - July 7, 007 RIKEN Wako Camus, Wako, Jaan June 5, 007 1

2 The Goal: To understand hadron structure, and strong interaction dynamics in terms o Quantum Chromodynamics (QCD) The Plan: From the Parton Model to QCD and QCD One lecture Purely inrared sae observables in QCD One lecture Cross sections with identiied hadrons in QCD Two lectures June 5, 007

3 Outline or Lecture 1 Nucleons to Quarks Dee Inelastic Scattering (DIS) The Parton Model Extensions o Parton Model beyond DIS Quantum Chromodynamics (QCD) Asymtotic reedom and erturbative QCD Excellent resource CTEQ summer school website htt:// June 5, 007 3

4 Nucleons to Quarks Protons, Neutrons, and Pions m = MeV S = 1 n m = MeV S = 1 I3 =+ 1 I3 = 1 m = MeV π ± 0 S = 0 π I =± 1 3 Historic m = MeV S I 3 = 0 = 0 π as NN bound states ( n ), ( n), ( nn ) + π = π = π = Isosin doublet Isosin trilet N π = n π 0 = π Fermi and Yang, 195; Nambu and Jona-Lasinio, 1960 (dynamics) π + June 5, 007 4

5 Nucleons not oint-like sin ½ Dirac articles June 5, 007 Proton magnetic moment: Neutron magnetic moment: Modern Quarks: π, N 5 g g 0 n common substructure: quarks Quark Model Q= 3e Q u S = 1 d S I =+ 1 I3 3 Gell Mann, Zweig, 1964 = 1 3e = 1 s Q= 1 3e S = 1 = 1 I3 = 0 ( ud ), ( du ), ( uu dd ) + π = π = π = = uud, n = udd, K = us,..., Δ = uuu, ( ) ( ) ( ) ( ) Magnetic moment: μ μ n = g gn = 3 ( good to %) But, need a new quantum number color and the dynamics! Han, Nambu, 1965

6 How to see substructure o a nucleon? Rutherord exeriment: to see the substructure o an atom α Atom α High energy α bounce o something very hard! Discovery o nucleus inside an atom Nucleus SLAC exeriment (1969): Nucleon e Leton-nucleon deely inelastic scattering (DIS) e θ Scattering inormation on the θ-distribution arton Discovery o the oint-like sin-1/ artons June 5, Callan-Gross relation

7 Leton-hadron DIS Process: ek (, λ) + P(, σ) ek ( ', λ') + X Neutral current (NC) Charged current (CC) ν ( k '), λ ' γ,z 0 W - Kinematics: 4-momentum transer: June 5, 007 Q = q 7 Bjorken variable: Q Squared CMS energy: s = ( + k ) = Inelasticity: xb y Q Final-state hadronic mass: W = ( + q) ( 1 xb ) x B Q xb = q y = q k

8 Leton-hadron DIS general analysis Scattering amlitude: Μ ( λλ, '; σ, q) = u ( k' ) ieγ u ( k) Cross section: dσ DIS λ' μ λ i q X ej μμ ( g ' ) em μ ( ) ' 0, σ X dl dk' = Μ s X i Ei E' Letonic tensor: known rom QED i ( λ, λ'; σ, q) ( ) 4 4 π δ 3 3 λλσ, ', = 1 ( π) ( π) DIS σ d μν E ' = L Wμν q dk' s Q ( k, k' ) (, ) e L kk kk kk k kg π μν (, ') = μ ν + ν μ μν μ μ μ μ ' ' ' ( ) X li + k' k i= 1 June 5, 007 8

9 Hadronic tensor (No QCD has been used): iq z Wμν ( q, ) = d z e, σ Jμ ( z) Jν (0), σ 4π σ Structure unctions: Parity invariance (EM current) Time-reversal invariance Current conservation W W μν μν μ qw = W = W μν νμ * μν sysmetric or sin avg. real ν = qw = 0 qq μ ν 1 q q Wμν = gμν F 1 x Q + μ qμ F ν ν x q q q q ( B, ) q ( B, Q ) Reduced to two dimensionless scalar structure unctions or sin-avgeraged DIS Two more structure unctions or sin-deendent DIS Measure cross sections extraction o structure unctions Note: No exlicit QCD was used in above derivation! μν June 5, 007 9

10 The Parton Model Beore the collision: in e - arton cm rame: t e collision ( k ) Ater the collision: ragments 1 Q m t hadron xi xi e ( k ') Feynman, 1969, x i x i = 1 i Lorentz contracted Time dilated Eectively rozen + q ( x ) i + q 0 ie.., q elastic collision June 5, 007 Deely inelastic scattering 10

11 where σ DIS eh Basic Parton Model Relation DIS eh 1 σ, q dx ˆ σ x, q φ x, q ( ) el ˆ e x, σ ( ) φ x ( ) = ( ) el ( ) ( ) q artons 0 DIS cross section or hadron: e h( ) Elastic cross section or arton: Probability or to have x - PDF x ( ) Inelastic hadronic cross section Nontrivial assumtion: = Partonic elastic cross section Probability or = x Quantum mechanical incoherence between the large q scattering and the artonic distribution June 5,

12 Recall: Structure Functions in Parton Model DIS eh 1 1 μν L, 3 μν dσ E ' = dk' s Q ( kk, ') W ( q ) el PM ormula: W ( q, ) = dx Wˆ ( q, x) ( x) 1 1 x μν μν φ artons 0 ˆ 1 1 W q, e Tr q ( ) ( x ) = γ γ ( x + ) γ γ ( x ) ( π ) δ ( x + q ) el μν μ ν 4π qq μ ν 1 xb = gμν e 1 δ q x 1 q q xb + μ qμ q e 1 ν ν x δ q q q x ( ) ( ) ( B, = B φ B = B B, ) F x Q e x x x F x Q 1 Callan-Gross Relation sin ½ arton Q Bjorken scaling - indeendent universal PDFs June 5, 007 1

13 June 5, 007 Fragmentation Functions in PM Cross rom DIS: q X crossing Cross rom Parton Model: x q φ ( x) ' X Parton distribution PM ormula or 1PI: σ 1PI h 1 13 q X Single article inclusive (1PI) q z D ( z ) ' X Fragmentation unction 1PI (, q) dz ˆ σ ( z, q) D ( z) = artons 0

14 Drell-Yan Dileton Production in PM Drell-Yan Process: h + h' ' q + ( ) ( ) + ( ) X with q = Q PM icture: ' PM ormula: dσ x DY hh' q x ' ' (, ' q) dq x ' ' June 5, t collision 1 1, ' = dx dx' φ ( x), ' 0 0 x 1 m thadron Q el d ˆ σ x, x ' ', q ( ) dq ( x, x ' ' q) 1 4πα φ ' ( x' ) el d ˆ σ ', em 1 Q e δ ' δ = 1 dq 3 3Q xx ' s xx ' s ColorFactor δδ ij ji 3 i, j= 1 3 = =

15 Ex/Thy Need to Imrove the PM Total momentum carried by the artons: 1 F dx x φ x q 0 Drell-Yan cross section: DY DY K = σ σ hh' Ex hh' ( ) Thy 0.5 missing momentum articles not directly interact with hoton (or EM charge) the gluon Scaling violation Q-deendence o structure unctions?... Need a better dynamical theory! June 5,

16 Quantum Chromodynamics (QCD) Known Fundamental Interactions: AdS/CFT corresondence Strong QCD Weak Electromagnetic - QED Gravity Electro Weak Standard Model QCD stands as a very solid building block o the SM: Unbroken SU(3) color gauge symmetry Asymtotic reedom at high energy Success o QCD erturbation theory Nonerturbative results rom Lattice calculations Not many surrises so ar June 5,

17 Fields: QCD as a ield theory ( ) ψ i x A μ,a ( x) Lagrangian density: Quark ields, Dirac ermions (like electron) Color trilet: i = 1,,3=N C Flavor: = u,d,s,c,b,t Gluon ields, sin-1 vector ield (like hoton) Color octet: a = 1,,,8 =N C -1 ( ) μ μ, (, ) g ( ) L ψ A = ψ i A t γ m ψ μ QCD i a a ij i 1 A A gc A A 4 μ ν, a v μ, a abc μ, b ν, c + gauge ixing + ghost terms June 5, 007 Color matrix: [ t, t ] = ic t a b abc c 17

18 Gauge invariance: where ( x) ψ ψ ' = U ψ i j ji i 1 i 1 Aμ Aμ ' = U( x) AU μ ( x) + μu( x) U ( x) g A = A t ( x) unitary [ det = 1, SU(3)] μ Gauge ixing: μ, a a, Uij Allow us to deine a roagator: with Feynman gauge June 5,

19 Ghost: ghost ields so that otical theorem (and hence unitarity) may be resected: Im = Σ Sum over all hysical olarizations June 5, 007 Fail without the ghost loo 19

20 Feynman rules Proagators: Quark: Gluon: or covariant gauge Ghost: June 5, 007 0

21 Interactions: June 5, 007 1

22 Scattering amlitude: Renormalization in QCD = = E i PS I E I E i 1 E I + E i UV divergence Sum over states o high mass Uncertainty rincile: high mass states = Local interaction No exeriment has an ininite resolution! June 5, 007

23 Renormalization: UV divergence due to high mass states Exeriments cannot resolve the details o these states = + Low mass state combine the high mass states with LO LO: + = High mass states Renomalized couling NLO: +... No UV divergence! Renormalization = re-arameterization o the exansion arameter in erturbation theory June 5, 007 3

24 Renormalization Grou Physical quantities can t deend on the renormalization scale - μ: d Q μ σ hy g μ μ =, ( ), 0 dμ μ σ The β-unction: α s ( μ) = μ π ( n) hy( Q ) σ ( Q, ) n g( μ) 3 β 1 5 β( g) = μ =+ g + O( g ) μ 16π n α ( μ ) = s g ( μ) 4π 11 4 n β1 = Nc + < 0 or n QCD running couling constant: α s ( μ1) αs ( μ) = 0 as μ or β1 < 0 β 1 μ 1 αs ( μ1) n 4π μ1 Asymtotic reedom June 5, 007 4

25 Λ QCD : QCD running couling constant α ( μ ) = s α ( μ ) 4π Λ s 1 β1 μ μ 1 α s ( μ1 ) n β 1 n 4π μ 1 QCD μ and μ 1 not indeendent June 5, 007 5

26 Eective quark mass Running quark mass: μ dλ m( μ) = m( μ1) ex - [ 1 + γ m( g( λ)) ] 0 as μ λ μ1 Perturbation theory becomes a massless theory when μ m ( μ) Λ or light quarks, u and d, even s, u and d QCD Choice o renormalization scale: μ Q QCD erturbation theory (Q>>Λ QCD ) is eectively a massless theory June 5, 007 6

27 Inrared saety: Inrared Saety ( ) ( ) κ Q m μ Q m μ σ ˆ hy, α ( ),, ( ) s μ σ α s μ + O μ μ μ μ Inrared sae = κ > 0 Asymtotic reedom is useul only or quantities that are inrared sae Asymtotic reedom + Inrared saety = erturbative QCD June 5, 007 7

28 Foundation o erturbative QCD Renormalization QCD is renormalizable Asymtotic reedom weaker interaction at a shorter distance Inrared saety QCD actorization and calculable short distance dynamics June 5, 007 8

29 Summary QCD is a SU(3) color non-abelian gauge theory o quark and gluon ields QCD erturbation theory works at high energy because o the asymtotic reedom QCD erturbation theory is eectively a massless theory renormalization grou equation or the arton mass Perturbative QCD calculations make sense only or inrared sae (IRS) quantities Look or IR sae quantities June 5, 007 9