Quantum Chromodynamics (QCD)

Transcription

1 Quantum Chromodynamics (QCD) Jianwei Qiu Brookhaven National Laboratory Stony Brook University Weihai High Energy Physics School (WHEPS) Shandong University Weihai, Weihai, Shandong, China, August 1-11, 015

2 Today s big story - discovery of Higgs boson Nobel Prize, 013 François Engler Peter W. Higgs "for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN's Large Hadron Collider" But, the Higgs mechanism generates Too little to be relevant for the mass of our world of visible matter! m q ~ 10 MeV m N ~ 1000 MeV It is QCD that is responsible for the mass of the visible world!

3 The plan for my four lectures q The Goal: To understand the strong interaction dynamics, and hadron structure, in terms of Quantum Chromo-dynamics (QCD) q The Plan (approximately): Fundamentals of QCD, factorization, evolution, and elementary hard processes Two lectures Role of QCD in high energy collider phenomenology One lecture QCD and hadron structure and properties One lecture

4 New particles, new ideas, and new theories q Early proliferation of new particles particle explosion : and many more! Quark Model Proton Neutron Nobel Prize, 1969

5 Deep inelastic scattering (DIS) q Modern Rutherford experiment DIS (SLAC 1968) e(p)+h(p )! e 0 (p 0 )+X ² Localized probe: Q = (p p 0 ) 1fm ² Two variables: 1 Q 1fm Q =4EE 0 sin ( /) x B = Q m N = E E 0 Discovery of spin ½ quarks, and partonic structure! The birth of QCD (1973) Quark Model + Yang-Mill gauge theory Nobel Prize, 1990

6 Quantum Chromo-dynamics (QCD) q Fields: = A quantum field theory of quarks and gluons = Quark fields: spin-½ Dirac fermion (like electron) Color triplet: Flavor: q QCD Lagrangian density: Gluon fields: spin-1 vector field (like photon) Color octet: q QED force to hold atoms together: L QED (,A)= X f q QCD Color confinement: f [(i@µ ea µ ) µ m f ] f 1 4 [@ A µ ] Gluons are dark, No free quarks or gluons ever been detected!

7 q Interaction strength: QCD Asymptotic Freedom μ and μ 1 not independent Discovery of QCD Asymptotic Freedom Collider phenomenology Controllable perturbative QCD calculations Nobel Prize, 004

8 New particles, new ideas, and new theories q Proliferation of new particles November Revolution : November Revolution! Quark Model and EW many more! QCD H 0 Completion of SM?

9 New particles, new ideas, and new theories q Proliferation of new particles November Revolution : November Revolution! Quark Model and EW many more! QCD H 0 X, Y, Completion of SM? Z, Pentaquark, A new particle explosion?

10 The Standard Model The most successful story to tell Found every elementary particles in the table but, not even one extra! All of them behave in the way that they are supposed to behave We found the lonely Higgs too! What is beyond the Higgs and the SM? The true nature of the effective theory? What we have not been able to address within the SM? The magic of glue that binds us all Without gluons, no hadrons, no atomic nuclei, no human,, no visible world!

11 Overarching questions for QCD q What is the role of QCD in the evolution of the universe? q How hadrons are emerged from quarks and gluons? q How does QCD make up the properties of hadrons? Their mass, spin, magnetic moment, q What is the QCD landscape of nucleon and nuclei? Color Confinement Asymptotic freedom 00 MeV (1 fm) GeV (1/10 fm) Q (GeV) Probing momentum q How do the nuclear force arise from QCD? q...

12 Example: The proton q Our understanding of the proton evolves 1970s 1980s/000s Now ² Proton is a strongly interacting, relativistic bound state of quarks and gluons ² Neither quarks nor gluons appear in isolation! ² Understanding such systems completely is still beyond the capability of the best minds in the world q The great intellectual challenges: ² Probe QCD dynamics without seeing quarks and gluons? ² Cross section involving an identified proton (or a hadron) is NOT IR safe and is NOT perturbatively calculable!

13 Theoretical approaches approximations q Perturbative QCD Factorization: Approximation at Feynman diagram level DIS σ tot : e p 1 + O QR Probe Hard-part q Effective field theory (EFT): Approximation at the Lagrangian level Structure Parton-distribution Approximation Power corrections Soft-collinear effective theory (SCET), Non-relativistic QCD (NRQCD), Heavy quark EFT, chiral EFT(s), q Other approximation or model approaches: Light-cone perturbation theory, Dyson-Schwinger Equations (DSE), Constituent quark models, AdS/CFT correspondence, q Lattice QCD: Approximation mainly due to computer power Hadron structure, hadron spectroscopy, nuclear structure,

14 Foundation of perturbative QCD q Renormalization QCD is renormalizable Nobel Prize, 1999 t Hooft, Veltman q Asymptotic freedom weaker interaction at a shorter distance Nobel Prize, 004 Gross, Politzer, Welczek q Infrared safety and factorization calculable short distance dynamics pqcd factorization connect the partons to physical cross sections J. J. Sakurai Prize, 003 Mueller, Sterman Look for infrared safe and factorizable observables!

15 q Running quark mass: Effective quark mass Quark mass depend on the renormalization scale! q QCD running quark mass: q Choice of renormalization scale: q Light quark mass: for small logarithms in the perturbative coefficients QCD perturbation theory (Q>>Λ QCD ) is effectively a massless theory

16 Infrared and collinear divergences q Consider a general diagram: for a massless theory ² Singularity Infrared (IR) divergence ² Collinear (CO) divergence IR and CO divergences are generic problems of a massless perturbation theory

17 Observables without identified hadron(s) q e + e - è hadron total cross section not a specific hadron! Partons m If there is no quantum interference between partons and hadrons, =1 σ P = P P = P P hadrons tot ee ee n ee m m n ee m m n n n m m n σ + P tot ee partons ee + m m σ tot + + ee hadrons q e + e - è parton total cross section: Hadrons n Unitarity tot = σ Finite in perturbation ee partons theory KLN theorem Calculable in pqcd

18 Infrared Safety of e + e - Total Cross Sections q Optical theorem: q Time-like vacuum polarization: IR safety of IR safety of with q IR safety of : If there were pinched poles in Π(Q ), ² real partons moving away from each other ² cannot be back to form the virtual photon again! Rest frame of the virtual photon

19 Lowest order (LO) perturbative calculation q Lowest order Feynman diagram: p 1 k 1 q Invariant amplitude square: M Q 1 1 Tr s 4 µ ν + = ee N ee QQ c γ p γ γ p1γ ( k1 mq) ( k mq) Tr γ γ µ γ γ + ν 4 = eeqn c ( m ) ( ) Q t + mq u + mqs s q Lowest order cross section: dσ + ee QQ 1 = M + ee QQ dt 16π s where s = Q p k s= ( p + p ) 1 t = ( p k ) u = ( p k ) Threshold constraint σ (0) 4 em eq N πα = σ 1 ee + = QQ c + Q Q 3s m s Q 1 4m s Q One of the best tests for the number of colors

20 Next-to-leading order (NLO) contribution q Real Feynman diagram: x i = Ei pi. q with i 1,,3 s / = s = i x i pi. q i = = s ( ) = ( θ ) 1 x x x 1 cos, cycl q Contribution to the cross section: 1 σ d σ 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!

21 How does dimensional regularization work? q Complex n-dimensional space: Im(n) (1) Start from here: UV renormalization a renormalized theory () Calculate IRS quantities here Theory cannot be renormalized! (3) Take εè 0 for IRS quantities only 4 6 Re(n) UV-finite, IR divergent UV-finite, IR-finite

22 Dimensional regularization for both IR and CO q NLO with a dimensional regulator: ( 1 ε ) ( ε ) ε (1) (0) 4 αs 4πµ Γ ² Real: σ3, ε = σ, ε 3 π Q Γ ε ε 4 ² Virtual: ε (1) (0) 4 α ( 1 ) ( 1 s 4πµ Γ ε Γ + ε) 1 3 π σ, ε = σ, ε π Q ( 1 ε Γ ) ε ε ² NLO: ² Total: α π ( ) (1) (1) (0) s σ3, ε + σ, ε = σ + O ε No ε dependence! tot (0) ( 1) (1) (0 α σ = σ + σ3, ε + σ, ε + O α = σ 1+ + α π σ tot is Infrared Safe! ( ) ) s ( ) s O s σ tot is independent of the choice of IR and CO regularization Go beyond the inclusive total cross section?

23 Hadronic cross section in e+e- collision q Normalized hadronic cross section: R e+ e (s) e+ e!hadrons(s) e + e!µ + µ (s) X apple N c = 3 N c e q 1+ apple s(s) + O( s(s)) 1+ s(s) q=u,d,s "! r # X +N c e q 1+ m q 4m q 1 + O( s (s)) s s q=c,

24 Jets trace of partons q Jets total cross-section with a limited phase-space Not any specific hadron! q Q: will IR cancellation be completed? ² Leading partons are moving away from each other δ E ² Soft gluon interactions should not change the direction of an energetic parton a jet trace of a parton δ ε s=δ θ Z-axis q Many Jet algorithms E 1 Sterman-Weinberg Jet

25 An early clean two-jet event A clean trace of two partons a pair of quark and antiquark

26 Reputed to be the first three-jet event from TASSO Discovery of a gluon jet

27 Tagged three-jet event from LEP Gluon Jet

28 Basics of jet finding algorithms q Recombination jet algorithms (almost all e+e- colliders): Recombination metric: ² different algorithm = different choice of : for Durham k T ² Combine the particle pair with the smallest : e.g. Escheme: p k = p i + p j ² iterate until all remaining pairs satisfy: q Cone jet algorithms (CDF,, colliders): ² Cluster all particles into a cone of half angle to form a jet: ² Require a minimum visible jet energy: Recombination metric: d ij =min k p T i,k p with ij =(y i y j ) +( i j ) ² Classical choices: p=1 k T algorithm, p= -1 anti-k T, T j ij R

29 Infrared safety for restricted cross sections q For any observable with a phase space constraint, Γ, ( ) 1 dσ dσ ( Γ) dω Γ ( k, k )! 1 dω ( 3) 1 dσ + dω Γ ( k, k, k ) 3! dω3 ( n) 1 dσ + dω Γ ( k, k,..., k ) +... n! n n 1 n dωn ( k1, 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 q Conditions for IRS of dσ(γ): Γ =Γ with 0 λ 1 Physical meaning: Measurement cannot distinguish a state with a zero/collinear momentum parton from a state without the parton Special case: (,,..., ) Γn k1 k kn = 1 for all n σ ( tot)

30 q Thrust axis: Another example: Thrust distribution! u! u q Phase space constraint: ( ) ( µ µ µ ) ( µ µ µ p ) 1, p,..., p δ T T p1, p,..., p Γ = ( ) = max T n p 1 µ, p µ,..., p n µ n n n n ² Contribution from p=0 particles drops out the sum ² Replace two collinear particles by one particle does not change the thrust! u # % % $ n i=1 n i=1! p i u! & (! p ( i '! u and

31 Another test of quark spin q Angle between the thrust axis and the beam axis:! u Θ THRUST At LO: 1+ cos Θ THRUST 1 cos Θ THRUST

32 q Select 4-jet events: Another test of SU(3) color E1 > E > E3 > E4 jet-3 and jet-4 are more likely from radiation q Bengtsson-Zerwas angle:

33 The harder question q Question: How to test QCD in a reaction with identified hadron(s)? to probe the quark-gluon structure of the hadron q Facts: Hadronic scale ~ 1/fm ~ Λ QCD is non-perturbative Cross section involving identified hadron(s) is not IR safe and is NOT perturbatively calculable! q Solution Factorization: ² Isolate the calculable dynamics of quarks and gluons ² Connect quarks and gluons to hadrons via non-perturbative but universal distribution functions provide information on the partonic structure of the hadron

34 q Scattering amplitude: Μ ( λλ, '; σ, q) = u ( k' ) ieγ u ( k) q Cross section: dσ Inclusive lepton-hadron DIS one hadron DIS * * λ ' µ λ i q µµ ( g ' ) ( ) em XeJµ ' 0 p, σ X dl dk' = Μ s X i Ei E' q Leptonic tensor: known from QED i ( λ, λ'; σ, q) ( ) 4 4 π δ 3 3 λλσ, ', = 1 ( π) ( π) DIS σ 1 1 d µν E ' = 3 W q L µν dk' s Q ( k, k' ) (, p) μ μ μ μ e L k k k k k k k k g π µν (, ') = µ ν + ν µ µν ' ' ' ( ) X li + k' p k i= 1

35 DIS structure functions q Hadronic tensor: 1 4 iq z Wµν ( q, p, S) = d z e p, S Jµ ( z) Jν (0) p, S 4π q Symmetries: ² Parity invariance (EM current) ² Time-reversal invariance ² Current conservation W W = W µν νµ * µν µν qw = W sysmetric for spin avg. real = qw = 0 µ ν µν µν qq µ ν 1 p q p q Wµν = gµν F 1 x Q + pµ qµ p q F ν ν x Q q p q q q + im ε q S g p. q ( B, ) ( B, ) ( x Q ) ( pq. ) Sσ ( S. ) ( pq. ) q p, + g x, Q ( ) µνρσ σ σ p ρ 1 B B q Structure functions infrared sensitive: ( ) ( ) ( ) ( B,, B,, B,, B, ) F x Q F x Q g x Q g x Q 1 1 Q = q x B = Q p q No QCD parton dynamics used in above derivation!

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

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

38 Collinear factorization - approximation q Collinear approximation, if Q xp n k T, k Lowest order: k T + O Q +UVCT Scheme dependence Same as an elastic x-section Parton s transverse momentum is integrated into parton distributions, and provides a scale of power corrections q DIS limit: q Corrections: ν Q x B,, while fixed Feynman s parton model and Bjorken scaling Spin-½ parton!

39 Parton distribution functions (PDFs) q PDFs as matrix elements of two parton fields: combine the amplitude & its complex-conjugate But, it is not gauge invariant! need a gauge link: Z O (µ ) can be a hadron, or a nucleus, or a parton state! corresponding diagram in momentum space: Z d 4 k ( ) 4 (x k + /p + ) Universality process independence predictive power q Quark distribution of a quark LO: Z d 4 k ( ) 4 (x k+ /p + )( ) 4 4 (k p) = (x 1) + UVCT(µ ) p k Z O (µ ) μ-dependence k p

40 QCD high order corrections q NLO partonic diagram to structure functions: Q 0 dk Dominated k 1 1 by k 1 0 t AB Diagram has both long- and short-distance physics q Factorization, separation of short- from long-distance:

41 QCD high order corrections q QCD corrections: pinch singularities in 4 dk i q Logarithmic contributions into parton distributions: + + +UVCT xb Q Λ QCD F ( xb, Q ) = Cf,, α s ϕ f ( x, µ F ) + O f x µ F Q q Factorization scale: µ F To separate the collinear from non-collinear contribution Recall: renormalization scale to separate local from non-local contribution

42 q Use DIS structure function F as an example: xb Q Λ QCD F h( xb, Q ) = Cq f,, α s ϕ f / h( x, µ ) + O q, f x µ Q ² Apply the factorized formula to parton states: Feynman diagrams How to calculate the perturbative parts? xb Q F q( xb, Q ) Cq f,, s ϕ f / q x, µ q, f x µ α = h ( ) ² Express both SFs and PDFs in terms of powers of α s : 0 th order: 1 th order: ( 0) ( 0) ( 0 ) q B = q B µ ϕ q/ q( µ ) F ( x, Q ) C ( x / x, Q / ) x, ( 0) ( 0) C ( x) = F ( x) q q q ( 0 ) ( x) δ δ( x) ϕ = q/ q qq 1 Feynman diagrams ( 1) ( 1) ( 0 ) q ( B, ) = q ( B /, / µ ) ϕ q/ q(, µ ) ( 0) ( 1) + Cq ( xb / x, Q / µ ) ϕ q/ q( x, µ ) ( 1) ( 1) ( 0) ( 1 ) q (, / µ ) = q (, ) q (, ) ϕ q/ q (, µ ) F x Q C x x Q x C x Q F x Q F x Q x

43 PDFs of a parton q Change the state without changing the operator: q Lowest order quark distribution: ² From the operator definition: given by Feynman diagrams q Leading order in α s quark distribution: ² Expand to (g s ) logarithmic divergent: UV and CO divergence

44 Partonic cross sections q Projection operators for SFs: qq µ ν 1 pq pq Wµν = gµν F 1 x Q + pµ qµ p q F ν ν x Q q p q q q (, ) (, ) 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 q 0 th (0) µν order: ( 0) µν 1 F q ( x) = xg Wµν, q = xg 4 π e µν q 1 = xg Tr γ pγ µ γ p q γν πδ ( p q) 4π + + ( ) ( ) ( ) = exδ(1 x) q ( ) 0 q q C ( x) = e xδ (1 x)

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

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

47 q The + distribution: 1+ ε l 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 l + z q One loop contribution to the trace of W μν : + + ( ) µν ( 1) αs 1 Q g Wµν, q = eq(1 ε) P qq ( x) + P qq ( x) l n γ E π ε µ (4 πe ) q Splitting function: l n(1 x) x + CF ( 1 + x ) l n( x) 1 x 1 x + 1 x 9 π + 3 x + δ (1 x) 3 1+ x 3 Pqq( x) = CF + δ (1 x) (1 x) + +

48 q One loop contribution to p μ p ν W μν : µ ν p p W ( 1) V µν, q = 0 µ ν p p W ( ) = e C 1 R µν, q q F q One loop contribution to F of a quark: αs Q π 4x ( 1) αs 1 γ Q E q (, ) = q Pqq( )( 1 + εl (4 π )) + Pq q( x) ln π ε µ CO F x Q e x x n e l n(1 x) x 9 π + CF (1 + x ) l n( x) x + δ (1 x) 1 x + 1 x + 1 x 3 as ε 0 q One loop contribution to quark PDF of a quark: ϕ ( 1) q/ q α 1 1 s ( x, µ ) = Pqq( x) + + UV-CT π ε UV ε CO in the dimensional regularization Different UV-CT = different factorization scheme!

49 q Common UV-CT terms: αs 1 ² MS scheme: UV-CT MS = Pq q( x) π ε UV αs 1 E UV-CT = P ( ) 1 (4 ) MS qq x + εl n πe γ π ε ² MS scheme: ( ) ² DIS scheme: choose a UV-CT, such that q One loop coefficient function: UV ( ) 1 q ( 1) ( 1) ( 0) ( 1 ) q (, / µ ) = q (, ) q (, ) ϕ q/ q (, µ ) C x Q F x Q F x Q x C ( x, Q / µ ) = 0 DIS ( 1) α s Q q (, / µ ) = q Pq q( ) l π µ MS C x Q e x x n l n(1 x) x 9 π + CF (1 + x ) l n( x) x + δ (1 x) 1 x + 1 x + 1 x 3

50 Dependence on factorization scale q Physical cross sections should not depend on the factorization scale d µ F F ( x B, Q ) = 0 dµ F F (x B,Q )= X C f (x B /x, Q /µ F, s ) f (x, µ F ) f Evolution (differential-integral) equation for PDFs ' d µ F dµ C! x B f # F x, Q µ,α $ * ) s &, f () " F % +, ϕ ( x,µ! x ) + C B f F f # x, Q µ,α $ d s & µ F f " F % dµ ϕ x,µ f F F DGLAP evolution equation: x µ F ϕ (, ) i x µ F = Pi/ j, αs ϕ j(', x µ F) µ j x' F ( ) = 0 q PDFs and coefficient functions share the same logarithms PDFs: Coefficient functions: ( µ F µ 0) ( µ F ΛQCD) ( Q µ F ) ( Q µ ) log or log log or log

51 Calculation of evolution kernels q Evolution kernels are process independent ² Parton distribution functions are universal ² Could be derived in many different ways q Extract from calculating parton PDFs scale dependence P k k P P p k k p P + 1 P p p-k k p P + 1 P p k p-k P Collins, Qiu, 1989 Change Gain Loss ² Same is true for gluon evolution, and mixing flavor terms q One can also extract the kernels from the CO divergence of partonic cross sections

52 Scaling and scaling violation Q -dependence is a prediction of pqcd calculation

53 Summary of lecture one q QCD is a SU(3) color non-abelian gauge theory of quark and gluon fields q QCD perturbation theory works at high energy because of the Asymptotic Freedom q Perturbative QCD calculations make sense only for infrared safe (IRS) quantities e + e - total cross section q Jets in high energy collisions provide us the trace of energetic quarks and gluons q Factorization is necessary for pqcd to treat observables (cross sections) with identified hadrons q Predictive power of QCD factorization relies on the universality of PDFs (or TMDs, GPDs, ), the calculations of perturbative coefficient functions hard parts

54 Backup slides

55 Gauge link 1 st order in coupling g q Longitudinal gluon: q Left diagram: Z applez p + dy dx 1 1 e ix 1p + (y 1 y ) n A a (y 1 ) M( igt a p i((x x 1 x B ) p +(Q /x B p + ) n) ) p + (x x 1 x B )Q /x B + i Z applez Z dy1 = g n Aa (y 1 )t a 1 1 dx 1 x 1 + i eix 1p + (y 1 y ) M = ig dy 1 n A(y 1 )M y q Right diagram: Z = g applez p + dy dx 1 1 Z dy1 n Aa (y 1 )t a e ix 1p + y 1 n A a (y 1 ) applez q Total contribution: dx 1 1 x 1 i((x + x 1 x B ) p +(Q /x B p + ) n) (x + x 1 x B )Q (+igt a ) /x B i Z 1 i eix 1p + y 1 M = ig applez 1 ig 0 Z 1 y 0 dy 1 n A(y 1 ) M dy 1 n A(y 1 ) M LO p p + M