Introduction to Perturbative QCD

Transcription

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

2 Infrared Safety Outline for Lecture Purely infrared safe cross sections Jets trace of the partons General conditions for purely infrared safe observables Excellent resource CTEQ summer school website June 5, 007

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 ˆ phy, α ( ),, ( ) s μ 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 5, 007 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 5, 007 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 P P P P P ee + hadrons ee + = n ee + m m n = ee + m m n n n m m n tot + P + ee partons ee m m tot + + ee hadrons Unitarity tot Finite in perturbation = ee partons Theory KLN theorem June 5, Local of order of 1/Q

6 tot for e + e - hadrons in pqcd tot 1 = s PS (3) PS () + + UV counter-term 1 = + Re + Re s UV C.T. June 5, 007 = + + (0) (1) (1) 3 Born O(α s ) 6 3-particle phase space

7 Leading order contribution I Lowest order Feynman diagram: k 1 p 1 k p Invariant amplitude square: s= ( k + k ) 1 t= ( k p) u= ( k p) Tr μ ν M + = ee ee QQ QN c γ kγ γ k1γ s ( p1 mq) μ ( p mq) Tr γ γ γ γ + ν 4 = eeqn c ( m ) ( ) Q t + mq u + mqs s Keeps the final state quark mass June 5, 007 7

8 Leading order contribution II Lowest order total cross section: d + ee dt QQ = 1 16π s M + ee QQ where s = Q Threshold constraint (0) 4 em eq N πα = 1 ee + = QQ c + Q Q 3s m s Q 1 4m s Q Normalized total cross section: R One of the best tests for the number of colors = = = + + ee QQ + 4 ee hadrons Q m Q eq Nc e e e e Q s μ μ μ μ One of the best measurements for the N c m s Q June 5, 007 8

9 Next-to-Leading order contribution I Real Feynman diagram: x i = Ei pi. q with i 1,,3 s / = s = pi. q i x i = = s i ( ) = ( θ ) 1 x x x 1 cos, cycl. Contribution to the cross section: d α x x + ee QQg s 1 + = CF 0 dx1dx π 1 x1 1 x ( )( ) + crossing IR as x3 0 CO as θ 13 0 θ 3 0 Divergent as x i 1 Need the virtual contribution and a regulator! June 5, 007 9

10 Next-to-Leading order contribution II Infrared regulator: Gluon mass: m g 0 easier because all integrals at one-loop is finite Dimensional regularization: 4 D = 4 - ε menifestly preserves gauge invariance Gluon mass regulator: (1) (0) 4 α s Q Q π 5 3, m = ln 3ln g 3 π m g m g 6 June 5, 007 (1) (0) 4 α s Q Q π 7, m = ln 3ln g π m g m g 6 4 tot (0) (1) (1) (0 α = + 3, m +, 1 g m + O α g s = + + O π No m g dependence! ) s ( ) ( α s ) 10

11 Next-to-Leading order contribution III Dimensional regulator: ( 1 ε ) ( ε ) ε (1) (0) 4 α s 4πμ Γ 1 3 3, ε =, ε π Q Γ ε ε 4 (1) (0) s, ε, ε ( 1 ε) ( 1 ε) Γ( 1 ε ) ε 4 α 4πμ Γ Γ π = π Q ε ε α π ( ) (1) (1) (0) s 3, ε +, ε = + O ε tot (0) ( 1) (1) (0 α = + 3, ε +, ε + O α = 1+ + α π Lesson: No ε dependence! ( ) ) s ( ) s O s tot is independent of the choice of IR and CO regularization tot is Infrared safe! June 5,

12 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=δ θ Z-axis Jet algorithm June 5, 007 1

13 Two-jet cross section in e + e - Parton-Model = Born term in QCD: 3 = + 8 ( PM) et J 0 ( 1 cos θ ) Two-jet in pqcd: 3 = 1+ cos 1+ 8 ( pqcd) Jet 0 ( θ) with n n ( ) C = C δ Sterman-Weinberg jet: June 5, n= 1 C n α s π ( pqcd) 3 4α s π 5 J et = 0 ( 1+ cos θ) 1 4 ln ( δ) ln ( δ ') + 3ln ( δ ) π 3 total n = Jet as Q

14 A clean two-jet event A clean trace of two partons a pair of quark and antiquark June 5,

15 Discovery of a gluon jet Reputed to be the first three-jet event from TASSO June 5,

16 Tagged 3-jet event from LEP June 5, Gluon Jet

17 Basics of jet finding algorithms Recombination jet algorithms: almost universal choice at e+e- colliders June 5,

18 The JADE jet finder June 5,

19 The Durham k T jet finder June 5,

20 The Cone jet finder CDF Collab., Phys. Rev. D45, 1448 (199); OPAL Collab., Z. Phys. C63, 197 (1994) June 5, 007 0

21 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 5, 007 1

22 Prediction vs CDF Run II data June 5, 007

23 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 d d ( Γ) dω Γ ( k, k )! 1 dω ( 3) 1 d + 3 3( 1,, 3) 3! dω Γ k k k dω ( n) 1 d + dω Γ ( k, k,..., k ) +... n! n n 1 n dωn Conditions for IRS of d(γ): Where Γ n (k 1,k,,k n ) are constraint functions and invariant under Interchange of n-particles ( k ) ( ) 1, k,...,( 1 λ) k μ, k μ λk μ k, k,..., k μ Γ = Γ n+ 1 n n+ 1 n = n 1 n with 0 λ 1 June 5, 007 3

24 Physical meaning for infrared safety Conditions for IRS ( k ) ( ) 1, k,...,( 1 λ) k μ, k μ λk μ k, k,..., k μ Γ = Γ n+ 1 n n+ 1 n = n 1 n with 0 λ 1 Physical meaning: Measurement cannot distinguish a state with an additional zero/collinar momentum parton from a state without the parton June 5, 007 Special case: (,,..., ) Γn k1 k kn = 1 for all n 4 ( tot)

25 Thrust axis: Phase space constraint: Thrust distribution u u ( ) ( μ μ μ) ( μ μ μ p ) 1, p,..., p δ T T p1, p,..., p Γ = n ( μ μ μ ) i 1 1,,..., max = = u i T p p p n n n n n n n pi u p i Contribution from p=0 particles drops out the sum Replace two collinear particles by one particle does not change the thrust = 1 u and June 5, 007 5

26 Another test of quark spin Angle between the thrust axis and the beam axis: u Θ THRUST At LO: 1+ cos Θ THRUST 1 cos Θ THRUST June 5, 007 6

27 Another test of SU(3) color Select 4-jet events: E1 > E > E3 > E4 jet-3 and jet-4 are more likely from radiation Bengtsson-Zerwas angle: June 5, 007 7

28 Question Does perturbative QCD work for cross sections with identified hadrons? Facts: 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) Answer: Perturbative QCD factorization June 5, 007 8

29 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 can we actually see and count the quarks and gluons? June 5, 007 9