Introduction to the Parton Model and Pertrubative QCD

Transcription

1 Introduction to the Parton Model and Pertrubative QCD George Sterman, YITP, Stony Brook CTEQ summer school, July 0, 20 U. of Wisconsin, Madison II. From the Parton Model to QCD A. Color and QCD B. Field Theory Essentials C. Infrared Safety and Jets

2 IIA. From Color to QCD q? q?? q? Enter the Gluon If φ q/h (x) = probability to find q with momentum xp, then, M q = q 0 dx x φ q/h (x) = total fraction of momentum carried by quarks. Experiment gave M q /2 What else? Quanta of force field that holds H together? Gluons but what are they? 2

3 Where color comes from. Quark model problem: s q = /2 fermion antisymmetric wave function, but (uud) state symmetric in spin/isospin combination for nucleons and Expect the lowest-lying ψ( x m, x u, x d) to be symmetric So where is the antisymmetry? Solution: Han Nambu, Greenberg, 968: Color b, g, r, a new quantum number. Here s the antisymmetry: ɛ ijk ψ( x u, x u, x d), (i,j,k)= (b,g,r) 3

4 Quantum Chromodynamics: Dynamics of Color A globe with no north pole r b G gb g Position on hyperglobe phase of wave function (Yang & Mills, 954) We can change the globe s axes at different points in spacetime, and local rotation emission of a gluon. QCD: gluons coupled to the color of quarks (Gross & Wilczek; Weinberg; Fritzsch, Gell-Mann, Leutwyler, 973) 4

5 IIB. Field Theory Essentials Fields and Lagrange Density for QCD q f (x), f = u, d, c, s, t, b: Dirac fermions (like electron) but extra (i, j, k) =(b, g, r) quantum number. A µ a (x) Vector field (like photon) but with extra a (g b...) quantum no. (octet). L specifies quark-gluon, gluon-gluon propagators and interactions. L = q ] f ([i µ ga µa T a γ µ ) m f q f 2 ( µ A νa ν A µa) f 4 g 2 ( µ A νa ν A µa ) C abc A µ b Aν c g2 4 C abca µ b Aν c C adea µd A νe 5

6 From a Lagrange density to observables, the pattern: Fields Lagrangian Symmetries Perturbation Theory Rules Green Functions Renormalization S - Matrix Cross Sections Observables 6

7 UV Divergences (toward renormalization & the renormalization group) Use as an example L φ 4 = µ φ) 2 m 2 λ φ 2 2 4! φ4 The four-point Green function : M(s,t) = d 4 k (k 2 m 2 )((p + p 2 k) 2 m 2 ) d4 k (k 2 ) 2 7

8 Interpretation: The UV divergence is due entirely states of high energy deficit, E in E state S = p 0 + p0 2 i S k 2 i m 2 Made explicit in Time-ordered Perturbation Theory: = E E E E in E out in E out d 4 k (k 2 m 2 )((p + p 2 k) 2 m 2 ) = states + E in E E in E Analogy to uncertainty principle E t 0. 8

9 This suggests: UV divergences are local and can be absorbed into the local Lagrange density. Renormalization. For our full 4-point Green function, two new counterterms : The renormalized 4-point function: M (s,t) = ren 2 counterterm δλ δm + + counterterm + + The combination is supposed to be finite. 9

10 {{ How to choose them? This is the renormalization scheme Renormalization:!m + = 0 (only natural choice) !" = finite But what should we choose for these? A B C D For example: define A+B+C by cutting off d 4 k at k 2 = Λ 2 (regularization). Then A + B + C = a ln Λ2 s + b(s, t, u, m2 ) 0

11 Now choose: so that D = a ln Λ2 µ 2 A + B + C + D = a ln µ2 independent of Λ. s + b(s, t, u, m2 ) Criterion for choosing µ is a renormalization scheme : MOM scheme: µ = s 0, some point in momentum space. MS scheme: same µ for all diagrams, momenta But the value of µ is still arbitrary. µ = renormalization scale. Modern view (Wilson) We hide our ignorance of the true high-e behavior. All current theories are effective theories with the same low-energy behavior as the true theory, whatever it may be.

12 µ-dependence is the price we pay for working with an effective theory: The Renormalization Group As µ changes, mass m and coupling g have to change: m = m(µ) g = g(µ) renormalized but... Physical quantities can t depend on µ: µ d dµ σ s ij µ 2, m2 µ 2, g(µ), µ = 0 The group is just the set of all changes in µ. RG equation (Mass dimension [σ] = d σ ): µ µ + µ g µ g + µ m µ m + d σ σ s ij µ 2, m2 µ 2, g(µ), µ = 0 The beta function : β(g) µ g(µ) µ 2

13 The Running coupling Consider any σ (m = 0, d σ = 0) with kinematic invariants s ij = (p i + p j ) 2 : µ dσ dµ = 0 µ σ µ = β(g) σ g in PT: σ = g 2 (µ)σ () + g 4 (µ) (2) in () σ (2) s ij s kl g 4 τ (2) = 2gσ () β(g) +... β(g) = g3 2 In QCD: τ (2) () + τ (2) ln s (2) µ 2 σ () + O(g5 ) g3 6π 2 β 0 + O(g 5 ) β 0 = 2n f 3 β 0 < 0 g decreases as µ increases. 3

14 Asymptotic Freedom: Solution for the QCD coupling µ g µ = β g3 0 6π 2 dg g 3 = β 0 dµ 6π 2 µ g 2 (µ 2 ) g 2 (µ ) = β 0 6π 2 ln µ 2 µ g 2 (µ 2 ) = g 2 (µ ) + β 0 6π 2 g 2 (µ ) ln µ 2 µ Vanishes for µ 2. Equivalently, α s (µ 2 2 ) g2 (µ 2 2 ) 4π = α s (µ ) + β 0 4π α s(µ ) ln µ 2 µ 4

15 Dimensional transmutation: Λ QCD Two mass scales appear in α s (µ 2 2 ) = α s (µ ) + β 0 4π α s(µ ) ln µ 2 µ but the value of α s (µ 2 ) can t depend on choice of µ. Reduce it to one by defining Λ µ e β 0/α s (µ ), independent of µ. Then α s (µ 2 2 ) = 4π β 0 ln µ Λ 2 Asymptotic freedom strongly suggests a relationship to the parton model, in which partons act as if free at short distances. But how to quantify this observation? 5

16 IIC. Infrared Safety and Jets To use perturbation theory, would like to choose µ as large as possible to make α s (µ) as small as possible. But how small is possible? A typical cross section,, define Q 2 = s 2 and x ij = s ij /Q 2, σ Q 2 µ 2, x ij, m2 i µ 2, α s(µ), µ = n= a n Q 2 µ 2, x ij, m2 i µ 2 αn s (µ) with m 2 i all fixed masses external, quark, gluon (=0!) Generically, the a n depend logarithmically on their arguments, so a choice of large µ results in large logs of m 2 i /µ2. 6

17 But if we could find quantities that depend on m i s only through powers, (m i /µ) p, p > 0, the large-µ limit would exist. σ Q 2 µ 2, x ij, m2 i µ 2, α s(q), µ = σ Q µ, x ij, m2 i µ 2, α s(µ), µ = n= a n Q µ, x ij α n s (µ) + O m 2 i µ 2 p Such quantities are called infrared (IR) safe. Measure σ solve for α s. Allows observation of the running coupling. Most pqcd is the computation of IR safe quantities. 7

18 Consistency of α s (µ) found as above at various momentum scales Each comes from identifying an IR safe quantity, computing it and comparing the result to experiment. (Particle Data Group) α s (µ) ep event shapes Polarized DIS Deep Inelastic Scattering (DIS) Spectroscopy (Lattice) Υ decay Average Hadronic Jets e + e - rates Photo-production Fragmentation Z width τ decays µ GeV α s (M Z ) To find IR safe quantities, need to understand where the lowmass logs come from. 8

19 To analyze diagrams, we generally think of m 0 limit in m/q. Gives IR logs. Generic source of IR (soft and collinear) logarithms: p αp IR logs come from degenerate states: Uncertainty principle E 0 t. For soft emission and collinear splitting it s never too late. But these processes don t change the flow of energy... Problems arise if we ask for particle content. 9

20 For IR safety, sum over degenerate final states in perturbation theory, and don t ask how many particles of each kind we have. This requires us to introduce another regularization, this time for IR behavior. The IR regulated theory is like QCD at short distances, but is better-behaved at long distances. IR-regulated QCD not the same as QCD except for IR safe quantities. 20

21 See how it works for the total e + e annihilation cross section to order α s. Lowest order is 2 2, σ 2 (0) σ LO, σ 3 starts at order α s. Gluon mass regularization: /k 2 /(k 2 m G ) 2 σ (m G) 4α s 3 = σ LO 3 π 2 Q ln2 3 ln Q π2 m g m g σ (m G) 2 = σ LO 4 α s 3 π 2 Q ln2 3 ln Q π2 m g m g which gives σ tot = σ (m G) 2 + σ (m G) 3 = σ LO + α s π Pretty simple! (Cancellation of virtual (σ 2 ) and real (σ 3 ) gluon diagrams.) 2

22 Dimensional regularization: change the area of a sphere of radius R from 4πR 2 to (4π) ( ε) Γ( ε) Γ(2( ε)) R2 2ε with ε = 2 D/2 in D dimensions. σ 3 (ε) 4α s ( ε) 2 4πµ 2 = σ LO 3 π (3 2ε)Γ(2 2ε) Q 2 ε 2 3 2ε π σ 2 (ε) = σ LO [ 4 α s ( ε) 2 4πµ 2 3 π (3 2ε)Γ(2 2ε) Q 2 ε 2 3 2ε π ] which gives again σ tot = σ (m G) 2 + σ (m G) 3 = σ 0 + α s π This illustrates IR Safety: σ 2 and σ 3 depend on regulator, but their sum does not. 22 ε ε

23 Generalized IR safety: sum over all states with the same flow of energy into the final state. Introduce IR safe weight e({p i }) with dσ de = n P S(n) M({p i}) 2 δ (e({p i }) w) e(... p i... p j, αp i, p j+...) = e(... ( + α)p i... p j, p j+...) Neglect long times in the initial state for the moment and see how this works in e + e annihilation: event shapes and jet cross sections. 23

24 Seeing Quarks and Gluons With Jet Cross Sections Simplest example: cone jets in e + e annihilation. fraction ɛ of energy flows into cones of size δ.! All but " Q Intuition: eliminating long-time behavior recognize the impossibility of resolving collinear splitting/recombination of massless particles 24

25 No factors Q/m or ln(q/m) Infrared Safety. In this case, σ 2J (Q, δ, ɛ) = 3 8 σ 0( + cos 2 θ) 4α s π 4 ln δ ln ɛ + 3 ln δ + π Perfect for QCD: asymptotic freedom dα s (Q)/dQ < 0. No unique jet definition. histories. Each event a sum of possible Relation to quarks and gluons always approximate but corrections to the approximation computable. 25

26 The general form of an e + e annihilation jet cross section: σ jet = σ 0 n=0 c n(y i, N, C F )α n s (Q) Dimensionless variables y i include direction and information about the size and shape of the jet: δ, cone size as above To specify the jet direction, may use a Shape variable, e.g. thrust T = s maxˆn i ˆn p i = s maxˆn i E i cos θ i with θ i the angle of particle i to the thust axis, which we can define as a jet axis. T = for back-to-back jets. 26

27 T = s maxˆn i E i cos θ i The thrust is IR safe precisely because it is insensitive to collinear emission (split energy at fixed θ i ) and soft emission (E i = 0). Once jet direction is fixed, we can generalize thrust to any smooth weight function: τ [f] = E i f(θ i ) particles i in jets Using thrust to define a jet axis is useful mostly to describe two, back-to-back, jets (no wide-angle gluon emission the majority, but by no means all events in e + e ). 27

28 The distribution as seen at high energies, compared 4 R.A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e to experiment (Davison & Webber, 0809): + e Annihilation /! d!/dt /! d!/dt /! d!/dt NNLO NNLO+NLL L3 (88.6 GeV) ALEPH (89.0 GeV) DELPHI (89.0 GeV) Q=89 GeV " s= t NNLO NNLO+NLL L3 (200 GeV) ALEPH (200 GeV) DELPHI (200 GeV) Q=200 GeV " s= t NNLO NNLO+NLL ALEPH (206.0 GeV) L3 (206.2 GeV) DELPHI (207.0 GeV) Q=206 GeV " s= t Fig. 3. Fixed-order (NNLO), resummed (NNLO+NLL) and experimental thrust distributions: Q = GeV. n= m= = Lg (α s L)+g 2 (α s L)+α s g 3 (α s L)+..., () L =ln(/y) and D (y, α s ) is a remainder function that α s L a significant improvement on the fixed-order range α s L 2. For thrust, the first two functions can be determined analytically by using the coherent branching formalism [, 2], which uses consecutive branchings from an initial quarkantiquark state to produce multi-parton final states to NLL accuracy. The results of this calculation depend upon the jet mass distribution J Q 2,k 2 the probability of producing a final state jet with invariant mass k 2 from a parent parton produced in a hard process at scale Q 2 and its Laplace transform J ν Q 2. To the required accuracy, the thrust distribution is dσ σ dt = Q2 dνe tνq2 µ J 2πi ν Q 2 2, (2) C where the contour C runs parallel to the imaginary axis on the right of all singularities of the integrand, ln J ν µ Q 2 = and 2 0 du uq 2 dµ 2 e uνq2 u u 2 Q K α s (µ) 2π µ 2 C α s (µ) F π, (3) 67 K = N 8 π N F. (4) This expression demonstrates explicitly that the divergence of α s (µ) at low µ will affect the perturbative thrust distribution such effects are related to the renormalon mentioned earlier. To NLL accuracy, however, we can neglect the low µ region (although we will return to it in Sect. 4) to give the thrust resummation functions [0] g (α s L)=2f (β 0 ᾱ s L), g 2 (α s L)=2f 2 (β 0 ᾱ s L) ln Γ [ 2f (β 0 ᾱ s L) (5) 2β where 0 ᾱ s Lf (β 0 ᾱ s L)], Strongly peaked C (α s near, )=+ C n but not where at, T =, due to radiation. ᾱs n, n= f (x) = C F [( 2x)ln( 2x) n+ β 0 x ln Σ (y, α s )= G nm ᾱs n L m 2 ( x)ln( x)], 28 f 2 (x) = C F K β 2 0 [2 ln ( x) ln ( 2x)] 3CF ln ( x) 2C F γ E [ln ( x)

29 Will be inserted by the editor 4 For possibly multi-jet events, cluster algorithms. y cut Cluster Algorithm: Combine particles i and j into jets until all y ij > y cut, where (e.g., Durham alogrithm for e + e ): y ij = 2min E 2 i, Ej 2 ) ( cos θ ij The number of jets depends on the variable y cut, and the dependence on the number of jets was an early application of Fig. 25. Measured distributions of thrust, T, (left-hand frame) and the C-parameter in comparison with QCD predictions at jet physics. (Reproduced from s =206.2 GeV [From L3 [48]]. Ali & Kramer, 02) Fig. 26. Relative production rates of n-jet events defined in the Durham jet algorithm scheme [54] as a function of the jet resolution parameter y cut. The data are compared to model calculations before and after the hadronization process as indicated on the figure [OPAL[6]]. 29

30 To anticipate: for hadronic collisions, jets are only well-defined away from the beam axis, so (instead of energy, E i ) use kinematic variables defined by the beam directions: transverse momentum, azimuthal angle and rapidity: The beams define the 3-axis. k t φ y = 2 ln E + p 3 E p 3 30

31 Cluster variables for hadronic collisions: d ij = min k ti 2p, 2 k2p ij tj R 2 2 ij = (y i y j ) 2 + (φ i φ j ) 2. R is an adjustable parameter. The classic choices: p = k t algorithm: p = 0 Cambridge/Aachen p = anti-k t 3

32 Summarize: what makes a cross section infrared safe? Independence of long-time interactions: p More specifically: should depend on only the flow of energy into the final state. This implies independence of collinear re-arrangements and soft parton emisssion. But if we prepare one or two particles in the initial state (as in DIS or proton-proton scattering), we will always be sensitive to long time behavior inside these particles. The parton model suggests what to do: factorize. This is the subject of Part III. 32 αp