QCD Phenomenology at High Energy Bryan Webber CERN Academic - PowerPoint PPT Presentation

QCD Phenomenology at High Energy Bryan Webber CERN Academic Training Lectures 2008 Lecture 4: Jet Fragmentation and Hadron-Hadron Processes ● Jet Fragmentation ❖ Fragmentation functions ❖ Small-x fragmentation ❖ Average multiplicity ● Hadronization Models ❖ General ideas ❖ Cluster model ❖ String model ● Hadron-Hadron Processes ❖ Parton-parton luminosities ❖ Lepton pair, jet and heavy quark production ❖ Higgs boson production ● Survey of NLO Calculations for LHC

Jet Fragmentation ● Fragmentation functions F h i ( x, t ) gives distribution of momentum fraction x for hadrons of type h in a jet initiated by a parton of type i , produced in a hard process at scale t . ● Like parton distributions in a hadron, D h i ( x, t ) , these are factorizable quantities, in which infrared divergences of PT can be factorized out and replaced by experimentally measured factor that contains all long-distance effects. ● In e + e − annihilation, for example, the hard process is e + e − → q ¯ q at scale equal to c.m. energy squared s ; distribution of x = 2 p h / √ s is (for s ≪ M 2 Z ) dσ n o Q 2 F h q ( x, s ) + F h X dx = 3 σ 0 q ( x, s ) q ¯ q where σ 0 is e + e − → µ + µ − cross section. ● Fragmentation functions satisfy DGLAP evolution equation Z 1 t ∂ dz α S ∂tF h 2 πP ji ( z, α S ) F h X i ( x, t ) = j ( x/z, t ) . z x j Splitting functions P ji have perturbative expansions of the form ji ( z ) + α S P ji ( z, α S ) = P (0) 2 πP (1) ji ( z ) + · · · 1

Leading terms P (0) ji ( z ) were given earlier. Notice that splitting function is P ji rather than P ij since F h j represents fragmentation of final parton j . ● Solve DGLAP equation by taking moments as explained for DIS. As in that case, scaling violation is clearly seen. 10 3 1/ σ tot (d σ /dx) DELPHI TASSO 14 22 35 44 GeV 10 7 10 7 10 7 10 7 10 7 ALEPH CELLO √ s=91 GeV AMY MARKII 10 6 10 6 10 6 10 2 10 5 10 5 10 5 L E P , S L C 1/ σ tot d σ /dx × c(flavour) 10 4 10 4 10 4 : a x = 0.1 - 0.2 l l L f E l a P v , o S u L r s C 10 3 10 3 10 3 : U p L , 10 E D P o , 300 x = 0.2 - 0.3 S w L n C , S : t 100 C r a h n a g L r e E m P 30 , S x = 0.3 - 0.4 L C 10 : B o t t o m 3 L x = 0.4 - 0.5 E 1 P 1 : ( ❍ 3-jet, ✶ F L,T ) G l u o n 0.3 x = 0.5 - 0.7 0.1 0.03 0.01 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 2 3 4 5 10 10 10 10 x Q 2 (GeV 2 ) 2

Small-x fragmentation ● Evolution of fragmentation functions at small x sensitive to moments near N = 1 . However, anomalous dimensions γ (0) gq , γ (0) gg are not defined at N = 1 : moment integrals for N ≤ 1 are dominated by small z , where P gi ( z ) diverges due to soft gluon emission. ● At small z must take into account coherence effects. Recall evolution variable becomes t ′ < z 2 ˜ ˜ t = E 2 [1 − cos θ ] , with angular ordering condition ˜ t . Thus, redefining t as ˜ t , evolution equation in integrated form is F i ( x, t ) = F i ( x, t 0 ) Z z 2 t Z 1 dt ′ dz α S X 2 πP ji ( z ) F j ( x/z, t ′ ) + t ′ z x t 0 j or in differential form Z 1 t ∂ dz α S 2 πP ji ( z ) F j ( x/z, z 2 t ) . X ∂tF i ( x, t ) = z x j ● Only difference from DGLAP equation is z -dependent scale on the right-hand side — not important for most values of x but crucial at small x . ● For simplicity, consider first α S fixed and neglect sum over j . Taking moments as usual, Z 1 t ∂ F ( N, t ) = α S ˜ dz z N − 1 P ( z ) ˜ F ( N, z 2 t ) . ∂t 2 π x 3

❖ Try solution of form F ( N, t ) ∝ t γ ( N,α S ) . Then anomalous dimension γ ( N, α S ) must satisfy Z 1 γ ( N, α S ) = α S z N − 1+2 γ ( N,α S ) P ( z ) . 2 π 0 ❖ For N − 1 not small, we can neglect 2 γ ( N, α S ) in exponent and obtain usual formula for anomalous dimension. For N ≃ 1 , z → 0 region dominates, where P gg ( z ) ≃ 2 C A /z . Hence γ gg ( N, α S ) = C A α S 1 π N − 1 + 2 γ gg ( N, α S ) "r # = 1 ( N − 1) 2 + 8 C A α S − ( N − 1) 4 π s r C A α S − 1 4( N − 1) + 1 2 π ( N − 1) 2 + · · · = 2 π 32 C A α S ● To take account of running α S , write »Z t γ gg ( N, α S ) dt ′ – ˜ F ( N, t ) ∼ exp , t ′ 4

and note that γ gg ( N, α S ) should be γ gg ( N, α S ( t ′ )) . Use Z t Z α S ( t ) γ gg ( N, α S ) γ gg ( N, α S ( t ′ )) dt ′ t ′ = dα S , β ( α S ) where β ( α S ) = − bα 2 S + · · · , to find s " 1 2 C A 1 ˜ F ( N, t ) ∼ exp − ( N − 1) b πα S 4 bα S s # 1 2 π ( N − 1) 2 + · · · + . C A α 3 48 b S α S = α S ( t ) ● In e + e − annihilation, scale t ∼ s and behaviour of ˜ F ( N, s ) near N = 1 determines form of small- x fragmentation functions. Keeping terms up to ( N − 1) 2 in exponent gives Gaussian function of N which transforms into Gaussian function of ξ ≡ ln(1 /x ) : » − 1 – 2 σ 2 ( ξ − ξ p ) 2 xF ( x, s ) ∝ exp , 5

● Width of distribution ! 1 s 2 1 2 π 3 4 . σ = ∝ (ln s ) C A α 3 24 b S ( s ) 8 e + e − : LEP 206 GeV LEP 189 GeV 7 LEP 133 GeV LEP 91 GeV TOPAZ 58 GeV 6 TASSO 44 GeV TASSO 35 GeV TASSO 22 GeV 5 DIS: 1/ σ d σ /d ξ H1 * 100-8000 GeV 2 4 ZEUS * 80-160 GeV 2 ZEUS * 40-80 GeV 2 H1 * 12-100 GeV 2 3 ZEUS * 10-20 GeV 2 2 1 0 0 1 2 3 4 5 6 ξ =ln(1/x p ) 6

● Peak position 4 bα S ( s ) ∼ 1 1 ξ p = 4 ln s 4.5 e + e − : OPAL 4 L3 DELPHI 3.5 ALEPH AMY CELLO 3 TPC MARK II ξ p TASSO 2.5 BES MLLA QCD fit DIS: 2 without coherence H1 ZEUS 1.5 1 2 3 5 7 10 20 30 50 70 100 200 √ s [ GeV ] ● Energy-dependence of the peak position ξ p tests suppression of hadron production at small x due to soft gluon coherence. Decrease at very small x is expected on kinematical grounds, but this would occur at particle energies proportional to their masses, i.e. at x ∝ m/ √ s , giving ξ p ∼ 1 2 ln s . Thus purely kinematic suppression would give ξ p increasing twice as fast. p → dijets, √ s is replaced by M JJ sin θ where M JJ is dijet mass and θ is jet cone ● In p ¯ angle. 7

θ CDF Preliminary Mjj=82 GeV� Mjj=105 GeV� Mjj=140 GeV� CDF Preliminary � � � � � � 2 CDF M =80-630 GeV/c , cone 0.28 jj � � � Fragmentation without color coherence CDF M =80-630 GeV/c 2 , cone 0.36 jj ) CDF M =80-630 GeV/c 2 , cone 0.47 jj o =ln(1/x - + + e e and e p Data o dN� Peak position x n d x Mjj=183 GeV� o Mjj=229 GeV� Mjj=293 GeV� i t a � � � m i x � � � � � o r N event� p � � � p A g o L g 1� n i d � � a e � � L Mjj=378 GeV� MLLA Fit: (CDF Data only) � Mjj=488 GeV� Mjj=628 GeV� � � � � � � � � + � � � Qeff = 256 13 MeV � � _ � 2 M jj sin q (GeV/c ) 1 _ x =log( ) x 8

Average Multiplicity ● Mean number of hadrons is N = 1 moment of fragmentation function: Z 1 ˜ � n ( s ) � = dx F ( x, s ) = F (1 , s ) 0 „ s s s ∼ exp 1 2 C A 2 C A « πα S ( s ) ∼ exp πb ln Λ 2 b (plus NLL corrections) in good agreement with data. 9

Hadronization Models General ideas ● Local parton-hadron duality ❖ Hadronization is long-distance process, involving small momentum transfers. Hence hadron-level flow of energy-momentum, flavour should follow parton level. ❖ Results on spectra and multiplicities support this. ● Universal low-scale α S ❖ PT works well down to very low scales, Q ∼ 1 GeV. ❖ Assume α S ( Q ) defined (non-perturbatively) for all Q . ❖ Good description of heavy quark spectra, event shapes. 10