Introduction to Perturbative QCD

Transcription

1 Introduction to Perturbative QCD Lecture 3 Jianwei Qiu Iowa State University/Argonne National Laboratory PHENIX Spinfest at RIKEN 007 June 11 - July 7, 007 RIKEN Wako Campus, Wako, Japan June 6, 007 1

2 Outline for Lecture 3 Cross section with one identified hadron Lepton-hadron deeply inelastic scattering (DIS) Factorization for IR sensitive cross sections What is the predictive power of pqcd? DGLAP evolution euation Parton distribution functions Excellent resource CTEQ summer school website June 6, 007

3 Deep inelastic scattering Recall: DIS σ 1 1 d μν E ' = 3 W L μν dk' s Q Hadronic tensor: 1 4 i z Wμν (, p, S) = d z e p, S Jμ ( z) Jν (0) p, S 4π μ ν 1 p p Wμν = gμν F 1 x Q + pμ μ p F ν ν x Q p + im ε S g p. ( k, k' ) (, p) ( B, ) ( B, ) ( x Q ) ( p. ) Sσ ( S. ) ( p. ) p, + g x, Q ( ) μνρσ σ σ p ρ 1 B B Structure functions infrared sensitive: ( ) ( ) ( ) ( B,, B,, B,, B, ) F x Q F x Q g x Q g x Q 1 1 June 6, 007 3

4 Perturbative QCD Factorization Cross sections with identified hadrons are infrared sensitive and non-perturbative Typical hadronic scale: 1/R ~ 1 fm -1 ~ Λ QCD Energy exchange in hard collisions: Q >> Λ QCD pqcd works at α s (Q), but not at α s (1/R) PQCD can be useful iff uantum interference between perturbative and nonperturbative scales can be neglected Short-distance σ phy( Q,1/ R) σ( Q) ϕ(1/ R) + O(1/ QR) Power corrections Measured Long-distance Factorization needs a long-lived parton state June 6, 007 4

5 Time evolution: Picture of factorization in DIS Long-lived parton state Time: Past Now Future Unitarity summing over all hard jets: σ DIS tot Im t 1 Q t R Not IR safe Interaction between the past and now are suppressed! June 6, 007 5

6 Factorization in DIS DIS σ tot 1 + O QR Now Past Connection Predictive power of pqcd short-distance and long-distance are separately gauge invariant short-distance part is Infrared-Safe, and calculable long-distance part can be defined to be Universal June 6, 007 6

7 Long-lived parton states Feynman diagram representation: W μν Perturbative pinched poles: June 6, H(, ) T(, ) perturbatively k + iε k iε r 4 dk Q k k Perturbative factorization: k μ μ T μ μ = xp + n + kt dx d k k + k xp n H( Q, k = 0) dk T x k iε k iε Short-distance 7 0 Nonperturbative matrix element T ( k, ) r 0

8 Collinear factorization Collinear approximation, if Q xp n k, k T k O T + Q +UVCT Same as elastic x-section Scheme dependence Parton s transverse momentum is integrated into parton distributions, and provides a scale of power corrections DIS limit: ν, Q, while x B fixed Feynman s parton model and Bjorken scaling Λ QCD F ( xb, Q ) = xb e f ϕ f ( x B ) + O( α s) + O f Q June 6, 007 8

9 Necessary condition for factorization Any uncanceled long-distance divergence of a partonic scattering cross section has to be process-independent On hadron state: On parton state: σ ( Q,1/ R) σ a ( Q) ϕ (1/ R) + O(1/ QR) H a a/ H σ ( Q,1/ R) σ a ( Q) ϕ (1/ R) + O(1/ QR) p a a/ p Process dependent partonic cross section (Feynman diagrams) Process-independent Parton-level pdfs (Feynman diagrams) Example: Eual long-distance physics () 1 W μν All uncanceled divergences are absorbed into PDFs June 6, 007 9

10 Parton distribution functions (PDFs) Predictive power of pqcd relies on the factorization and the universality of PDFs PDFs as matrix elements of two parton fields: + UV CT Gives the μ-dependence June 6,

11 An instructive exercise for high orders Consider a cross section: σ( Q, m ) = σ αsi + O( αs) Leading uantum correction: Analysis of the integral: Result for the cross section: June 6,

12 Scaling violation and factorization NLO partonic diagram to structure functions: Q 0 dk Dominated by k 1 1 k 1 0 t AB Diagram has both long- and short-distance physics Factorization, separation of short- from long-distance: June 6, 007 1

13 Leading power QCD formula QCD corrections: pinch singularities in 4 d k i Logarithmic contributions into parton distributions + + +UVCT xb Q Λ QCD F ( xb, Q ) = Cf,, α s ϕ f ( x, μf ) + O f x μ F Q Factorization scale: μ F To separate collinear from non-collinear contribution Recall: renormalization scale to separate local from non-local contribution June 6,

14 Calculation of perturbative parts Use DIS structure function F as an example: xb Q Λ QCD F h( xb, Q ) = C f,, α s ϕ f / h( x, μf) + O, f x μ F Q Apply the factorized formula to parton states: Feynman diagrams F x Q C x Q x B ( B, ) = f,, α s ϕ f /, F, f x μ F h ( μ ) Express both SFs and PDFs in terms of powers of α s : 0 th order: 1 th order: ( 0) ( 0) ( 0 ) ( B, ) = ( B /, / μf) ϕ/ (, μf) F x Q C x x Q x ( 0) ( 0) C ( x) = F ( x) ( 0 ) ( x) δ δ ( x) ϕ = / 1 Feynman diagrams ( 1) ( 1) ( 0 ) ( B, ) = ( B /, / μf) ϕ/ (, μf) ( 0) () 1 + C ( xb / x, Q / μf) ϕ/ ( x, μf) ( 1) ( 1) ( 0) ( 1) (, / μf ) = (, ) (, ) ϕ/ (, μf) F x Q C x x Q x C x Q F x Q F x Q x June 6,

15 Leading order coefficient function Projection operators for SFs: μ ν 1 p p Wμν = gμν F 1 x Q + pμ μ p F ν ν x Q p (, ) (, ) 1 μν 4x μ ν F1 ( x, Q ) = g + p p (, ) Wμν x Q Q μν 1x μ ν F ( x, Q ) = x g + p p W ( x, Q ) μν Q μν 1 F ( x) = xg Wμν, = xg 4π e 1 = 4π th (0) μν order: ( 0) μν ( xg ) Tr γ pγ μγ ( p ) γν πδ (( p ) ) = ex δ (1 x) ( 0) C ( x) = e xδ (1 x) June 6,

16 NLO coefficient function () 1 ( 1) ( 0) ( 1) ( ) (, / μf ) = (, ) (, ) ϕ/, μf C x Q F x Q F x Q x Projection operators in n-dimension: g g μν μν = n 4 ε μν 4x μ ν 1 F = x g + (3 ε) p p W Q ( ε) Feynman diagrams: μν ( 1) W μν, Real Virtual + + c.c. + + c.c. + UV CT Calculation: μν ( 1) μ ν ( 1) g W and p p W μν, μν, June 6,

17 Contribution from the trace of Wμν Lowest order in n-dimension: ( ) 0 μν μν g W, = e (1 εδ ) (1 x) NLO virtual contribution: () 1 V μν, μν g W = e (1 εδ ) (1 x) ε 4 μf (1 ) (1 ) α s π Γ + ε Γ ε * CF 4 π Q + + Γ(1 ε) ε ε NLO real contribution: ε () 1 R α s 4 πμ F Γ (1 + ε) μν, (1 ε) CF μν g W = e π Q Γ (1 ε) 1 ε x 1 1 ε ε * 1 x ε + 1 x 1 ε + + (1 ε)(1 x) 1 ε June 6,

18 The + distribution: June 6, ε n(1 x) = δ (1 x) + + ε + O ε 1 x ε (1 x) 1 x 1 1 z f( x) f( x) f(1) dx dx + n(1 z) f (1) (1 x) 1 x + z One loop contribution to the trace of Wμν: ( ) μν () 1 α s 1 Q g Wμν, = e (1 ε ) P ( x) + P ( x) n γ E π ε μ F (4 π e ) Splitting function: n(1 x) x + C F ( 1 + x ) n( x) 1 x 1 x + 1 x 9 π + 3 x + δ (1 x) 3 1+ x 3 P ( x) = CF + δ (1 x) (1 x) + +

19 One loop contribution to p μ p ν W μν : μ ν p p W ( 1) V μν, = 0 μ ν p p W () 1 R μν, = ecf One loop contribution to F of a uark: () 1 s γ E ( ε π ) α s Q π 4x α 1 Q π ε μ CO F F ( x, Q ) = e x P ( x) 1 + n(4 e ) + P ( x) n n(1 x) x 9 π + CF (1 + x ) nx ( ) x + δ (1 x) 1 x + 1 x + 1 x 3 as ε 0 One loop contribution to uark PDF of a uark: ϕ () 1 / α 1 1 ( x, μ ) = P ( x) + + UV-CT F s π ε UV ε CO Different UV-CT = different factorization scheme! June 6,

20 Common UV-CT terms: MS scheme: α s 1 UV-CT MS = P ( x) π ε UV MS scheme: α s 1 UV-CT ( ) ( 1 (4 ) MS ) n E = + ε π γ π ε UV DIS scheme: choose a UV-CT, such that ( 1) C ( x, Q / μ F ) DIS = 0 One loop coefficient function: ( 1) ( 1) ( 0) ( 1) ( ) (, / μf ) = (, ) (, ) ϕ/, μf C xq F xq F xq x () 1 αs Q (, / μ ) = P () π μ MS C x Q e x x n n(1 x) x 9 π + CF (1 + x ) n( x) x + δ (1 x) 1 x + 1 x + 1 x 3 June 6, 007 0

21 Dependence on factorization scale Physical cross sections should not depend on the factorization scale d μf F ( x B, Q ) = 0 dμ f F Evolution (differential-integral) euation for PDFs d xb Q xb Q d μf C,, f α s ϕ f ( x, μf) + Cf,, α s μf ϕ f ( x, μf) = 0 dμf x μf f x μf dμf PDFs and coefficient functions share the same logarithms PDFs: Coefficient functions: DGLAP evolution euation: x μf ϕ (, ) i x μf = Pi/ j, αs ϕ j( x', μf) μ j x ' F ( μf μ0 ) ( μf ΛQCD) ( Q μf ) ( Q μ ) log or log log or log June 6, 007 1

22 DGLAP evolution of PDFs DGLAP euations: x μf ϕ (, ) i x μf = Pi/ j, αs ϕ j( x', μf) μ j x ' Splitting functions: F Splitting functions have to be process independent Can be then derived in many different ways from the log part of the C s from the anomalous dimension of the nonlocal operators defining the PDFs Predictive power of pqcd: Once the boundary condition is fixed by the data, the scale dependence of PDFs is a prediction of pqcd June 6, 007

23 Global QCD analysis of PDFs PDFs are extracted by using: x DGLAP μf ϕ (, ) i x μf = Pi/ j, αs ϕ j( x', μf) μ j x ' Factorized hard cross sections, e.g. F Data: F Λ = + to fix the boundary condition of DGLAP xb Q QCD h( xb, Q ) C/ f,, α s ϕ f / h( x, μf) O x μ F Q The order and scheme dependence of PDFs: Leading order (tree-level) C Next-to-Leading order C LO PDF s NLO PDF s Calculation of C at NLO and beyond depends on the UVCT the scheme dependence of C the scheme dependence of PDFs June 6, 007 3

24 PDFs of a spin-averaged proton Modern sets of PDFs with uncertainties: xf(x,q ) NLO Q =10 GeV Q =10 GeV xu xg(x0.05) xd xs(x0.05) x Consistently fit almost all data with Q > GeV June 6, 007 4

25 June 6, 007 5

26 Comparison with DIS data June 6, 007 6

27 Charm uark distributions Large gluon at small-x large charm uark distribution June 6, 007 7

28 Uncertainties of gluon distribution MRST001 CTEQ5M June 6, 007 8

29 PDF uncertainty for observables CTEQ6 PDFs June 6, 007 % uncertainty for strong interaction 9

30 Recover the effect of non-vanishing k T Sources of power corrections: Parton transverse momentum: Target and parton masses: Coherent multiple scattering: k m Q Q k + ( ) Systematics of power corrections: Leading Twist Q 1 Q R F F + Medium length σ = σˆ [1 + α + α +...] T ( x) h i i/ h phys s s σˆ Q i 4 i/ h + [1 + α...] s + α s + T4 ( x) σˆ Q i 6 i/ h + [1 + α...] 4 s + α s + T6 ( x) +... perturbative Power corrections June 6, Factorization may not be true for power corrections! Need to be proved for any given process Qiu and Vitev, PRL 004

31 Improvement from the fixed order Beyond the Born term (lowest order), partonic hard-parts are NOT uniue, due to renormalization of parton distributions Once ϕ(x,μ ) is fixed in one scheme, same ϕ(x,μ ) should be used for all calculations of partonic parts Coefficient has the P Q ( x) n μf Suggests to choose the scale: Coefficient has potentially large logarithms: μ 1 n(1 x) nx ( ),, (1 x) 1 x + + F Q Resummation of the large logarithms June 6,

32 Summary We can actually see and count the uarks and gluons uark and gluon distributions PQCD factorization works for DIS to all orders as well as all powers due to OPE PDFs evolves number of partons is sensitive to the probing scale PQCD global analysis for spin averaged cross sections results into the reasonably well-determined universal PDFs What happen if there are more than one identified hadrons? June 6, 007 3