QCD and Rescattering in Nuclear Targets Lecture 2

Transcription

1 QCD and Rescattering in Nuclear Targets Lecture Jianwei Qiu Iowa State University The 1 st Annual Hampton University Graduate Studies Program (HUGS 006) June 5-3, 006 Jefferson Lab, Newport News, Virginia June 6, 006 1

2 Fundamentals of perturbative QCD Infrared Safety Purely infrared safe cross sections Jets trace of the partons Factorization predictive power of pqcd Factorization for deeply inelastic scattering Evolution of parton distribution functions Factorization for hadronic collisions June 6, 006

3 Running quark mass: Infrared Safety μ dλ m( μ) = m( μ1) exp - [ 1 + γ m( g( λ)) ] 0 as μ λ μ1 Perturbation theory becomes a massless theory when μ Infrared safety: ( ) ( ) κ Q m μ Q m μ σ ˆ phy, α ( ),, ( ) s μ σ α s μ + O μ μ μ μ Infrared safe = κ > 0 Asymptotic freedom is useful for quantities that are infrared safe QCD perturbation theory (Q>>Λ QCD ) is effectively a massless theory June 6, 006 3

4 Infrared and collinear divergence Consider a general diagram: p = 0, k = 0 for a massless theory k 0 p k p = μ ( ) 0 singularity Infrared (IR) divergence k μ p μ k = λp with 0 < λ < 1 ( ) ( λ ) p k 1 p = 0 Collinear (CO) divergence IR and CO divergences are generic problems for massless perturbation theory June 6, 006 4

5 Purely Infrared safe cross sections e+e- hadron total cross section is infrared safe (IRS) Hadrons n Partons m If there is no quantum interference between partons and hadrons, =1 tot σ P P P P P ee + hadrons ee + = n ee + m m n = ee + m m n n n m m n tot σ + P + ee partons ee m m σ tot + + ee hadrons Unitarity tot Finite in perturbation = σee partons Theory KLN theorem June 6, Local of order of 1/Q

6 σ tot for e + e - hadrons in pqcd tot 1 σ = s PS (3) PS () + + UV counter-term 1 = + Re + Re s UV C.T. = σ + σ + σ (0) (1) (1) 3 Born O(α s ) 3-particle phase space June 6, 006 6

7 Lowest order contribution Lowest order Feynman diagram: p 1 k 1 Invariant amplitude square: Tr μ ν M + = ee ee QQ QN c γ pγ γ p1γ s dσ + ee dt QQ ( k1 mq) μ( k mq) Tr γ γ γ γ + ν 4 = e eqn c ( m ) ( ) Q t + mq u + mq s s Lowest order cross section: 1 = 16π s M + ee QQ where s = Q p k s = ( p + p ) 1 t= ( p k) u= ( p k) Threshold constraint 4 em eq N πα = = c 1 + Q 3s (0) σ σ e + e QQ Q m s Q 1 4m s Q June 6, 006 One of the best tests for the number of colors 7

8 σ σ Dimensional regulation for IR and CO n=4-ε dimension: ( 1 ε ) ( 1 3ε ) ε (1) (0) 4 α s 4πμ Γ , ε = σ, ε π Q Γ ε ε 4 (1) (0) s, ε, ε ( 1 ε) ( 1 ε) 1 Γ( ) ε 4 α 4πμ Γ Γ + 3 π = σ π Q 1 ε ε ε α π ( ) (1) (1) (0) s σ 3, ε + σ, ε = σ + O ε tot (0) ( 1) (1) (0 α σ = σ + σ3, ε + σ, ε + O α = σ 1+ + α π Lesson: ( ) ) s ( ) s O s σ tot is independent of the choice of IR and CO regularization σ tot is Infrared safe! June 6, 006 8

9 Jets in e + e - - trace of the partons Jets Inclusive x-section with a limited phase-space Sterman-Weinberg Jet δ E Q: will IR cancellation be completed? Leading partons are moving away from each other Soft gluon interactions should not change the direction of an energetic parton a jet trace of a parton δ E 1 ε s θ Z-axis Jet algorithm June 6, 006 9

10 A clean two-jet event A clean trace of two partons a pair of quark and antiquark June 6,

11 Discovery of a gluon jet Reputed to be the first three-jet event from TASSO June 6,

12 Tagged 3-jet event from LEP June 6, Gluon Jet

13 Basics of jet finding algorithms Recombination jet algorithms: almost universal choice at e+e- colliders June 6,

14 The JADE jet finder June 6,

15 The Durham k T jet finder June 6,

16 The Cone jet finder CDF Collab., Phys. Rev. D45, 1448 (199); OPAL Collab., Z. Phys. C63, 197 (1994) June 6,

17 Inclusive jet cross section at Tevatron Run 1b results CDF Results 0.1< η <0.7 Data and Predictions span 7 orders of magnitude! June 6,

18 Infrared safety for jet cross sections Jet cross section = inclusive cross section with a phase-space constraint For any observable with a phase space constraint, Γ, ( ) 1 dσ dσ ( Γ) dω Γ ( k, k )! 1 dω ( 3) 1 dσ + 3 3( 1,, 3) 3! dω Γ k k k dω ( n) 1 dσ + dω Γ ( k, k,..., k ) +... n! n n 1 n dωn Conditions for IRS of dσ(γ): ( k ) ( ) 1, k,...,(1 λ) k μ n, λk μ n n k1, k,..., k μ n n+ 1 Where Γ n (k 1,k,,k n ) are constraint functions and invariant under Interchange of n-particles Γ =Γ with 0 λ 1 Special case: (,,..., ) Measurement cannot distinguish a state with a zero momentum parton from a state without the parton Γn k1 k kn = 1 for all n σ ( tot) June 6,

19 PQCD Factorization Can pqcd calculate cross sections with identified hadrons? 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 quantum 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 June 6, 006 Factorization forgetting the past 19

20 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, 006 0

21 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, 006 1

22 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 0 Nonperturbative matrix element T ( k, ) r 0

23 Collinear factorization Collinear approximation, if Q xp n k, k T k O T + Q +UVCT DIS limit: Same as elastic x-section ν, Q, while x B Scheme dependence 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, 006 3

24 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, 006 4

25 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, 006 5

26 Dependence on factorization scale Physical cross sections should not depend on the factorization scale f μ F d F x Q dμ ( B, ) = 0 F Evolution (differential-integral) equation 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: log μf μ or log μf Λ Coefficient functions: log Q μf or log Q μ DGLAP evolution equation: x μf ϕ (, ) i x μf = Pi/ j, αs ϕ j( x', μf) μ j x ' F Predictive power of pqcd: ( 0 ) ( QCD) ( ) ( ) Universality of PDFs and their scale dependence Q dependence of physical observables June 6, 006 6

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28 Measurement of F γ When Q M Z, we can neglect the Z 0 -contribution Precision test of QCD: as good as -3% error for such difficult measurement June 6, 006 8

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30 Improvement from the fixed order Beyond the Born term (lowest order), partonic hard-parts are NOT unique, 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 qq 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,

31 Recover the effect of non-vanishing k T Sources of power corrections: Parton transverse momentum: Target and parton masses: Coherent multiple scattering: Systematics of power corrections: k m Leading Twist Q Q k + ( ) 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 Factorization may not be true for power corrections! Need to be proved for any given process June 6,

32 Summary QCD is a SU(3) color non-abelian gauge theory of quark and gluon fields QCD perturbation theory works at high energy because of the asymptotic freedom Perturbative QCD calculations make sense only for infrared safe (IRS) quantities Jets in high energy collisions provide us the trace of energetic quarks and gluons We can actually see and count the quarks and gluons? June 6, 006 3