Hadronization: Concepts and Models From Perturbative to Non-Perturbative QCD Bryan Webber University of Cambridge Hadronization Workshop ECT*, Trento, 1-5 Sept 2008 Hadronization ECT* 08 1 Bryan Webber What is Hadronization? In practice, two rather distinct meanings: • General concepts/models for soft QCD – Local parton-hadron duality (LPHD) – Universal low-scale effective α S (ULSEA) • Models for formation of individual hadrons – Monte Carlo models: string & cluster – Thermal/statistical models – AdS/QCD Hadronization ECT* 08 2 Bryan Webber General Concepts • Local parton-hadron duality – Momentum & flavour follows parton flow – Predicts asymptotic spectra – Predicts two-particle correlations αS • Universal low-scale effective – Related to “tube” model – Regulates IR renormalons in PT – Predicts power corrections to event shapes – Predicts jet shapes and energy corrections Hadronization ECT* 08 3 Bryan Webber Local parton-hadron duality • Evolution equation for fragmentation function has extra z2 due to soft gluon coherence ∂ 1 dz α t F (x, t) = S P (z)F (x/z, z2t) ∂t z 2π !x • Solution by moments t dt F˜(N, t) exp γ(N, α ) ! F˜(N, t ) ∼ S t 0 !"t0 ! # 1 αS N 1+2γ(N,α ) γ(N, α ) = z − S P (z) S 2π !0 • Anomalous dimension dominates asymptotically Hadronization ECT* 08 4 Bryan Webber CAαS 1 ∼ π N 1 + 2γ(N, α ) − S • This is regular at N=1 1 2 8CAαS γ(N, αS) = (N 1) + (N 1) 4 !" − π − − # CAαS 1 1 1 2π 2 = (NαS1) + N 1+2γ(N,α(N) 1) + γ2(πN, α− )4 = − z32− CAαS S P−(z) · · · ! S 2π ! !0 Hadronization ECT* 08 5 Bryan Webber t αS (t) dt! γ(N, αS) γ(N, α (t!)) = dα S t β(α ) S ! ! ! S where β ( α ) = b α 2 + . Hence S − S · · · ˜ 1 2CA 1 1 2π 2 F (N, t) exp (N 1) + 3 (N 1) + ∼ ! b παS − 4bαS − 48b#CAαS − · · · $ " αS =αS (t) Mean Position Width multiplicity of peak of peak • Gaussian in N Gaussian in ξ ln(1/x) ↔ ≡ • Mean multiplicity 1 n(s) = dx F (x, s) = F˜(1, s) ! " !0 1 2CA 2CA s exp exp ln 2 ∼ b !παS(s) ∼ πb Λ " # $ Hadronization ECT* 08 6 Bryan Webber Average Multiplicity ! Width of distribution ! Mean number of hadrons is N = 1 moment of fragmentation function: 1 2 1 1 2π 3 σ = (ln s)4 . n(s) = dx F (x, s) = F˜(1, s) 24b C α3(s) ∝ ! " s A S ! Z0 LPHD Predictions 1 2CA 2CA s exp exp ln 2 ∼ bsπαS(s) ∼ s πb Λ „ « + # 8 e e : • Good agreement with data LEP 206 GeV (plus NLL corrections) in good agreement with data. LEP 189 GeV 7 LEP 133 GeV LEP 91 GeV TOPAZ 58 GeV 6 TASSO 44 GeV TASSO 35 GeV 5 TASSO 22 GeV ! DIS: * 2 /d H1 100-8000 GeV " ZEUS* 80-160 GeV2 d 4 * 2 " ZEUS 40-80 GeV * 2 1/ H1 12-100 GeV 3 ZEUS* 10-20 GeV2 2 1 0 0 1 2 3 4 5 6 !=ln(1/xp) 6 9 Hadronization ECT* 08 7 Bryan Webber LPHD in pp dijets CDF Preliminary ! Mjj=82 GeV Mjj=105 GeV Mjj=140 GeV CDF Preliminary 678!9::2;<=>?<!"#$%&'@!&-1#!<A'; 678!9::2;<=>?<!"#$%&'@!&-1#!<A?> ( ' - 678!9::2;<=>?<!"#$%& @!&-1#!<ABC #D#=!*1E!#D,!7*0* Mjj=183 GeV Mjj=229 GeV Mjj=293 GeV 231!4%5 d dN - ! ent v e ximation 1 N ,-./0/-1 ! Mjj=378 GeV Mjj=488 GeV Mjj=628 GeV Leading Log Appro )#*+ ragmentation without color coherence F MLLA Fit: (CDF Data only) Qeff = 256 +_ 13 MeV 1 !!"#$%&'( =log( _ ) Mjj sin x Hadronization ECT* 08 8 Bryan Webber where a is a constant that depends on the number of colors and the number of effectively where a is a constant that depends on the number of colors and the number of effectively masmslaesslsessqquuaarrkks.sT. hTe uhneknuownnktneromwOn(1)tiesremxpecOted(1to)biesinedxepeencdetnetdoftτo. Tbhee pirneddicetpedendent of τ. The predicted dependence of the inclusive momentum distribution on jet hardness is shown in Fig. 1. dependence of the inclusive mowmheernetaumis adciosntsrtiabnutttihoant doenpejnedts hoanrtdhne ensusmibsershofowconlorisnanFdigth. e1n.umber of effectively The two-parton momentum correlation function R(ξ1, ξ2) is defined to be the ratio of the massless quarks. The unknown term O(1) is expected to be independent of τ. The predicted Ttwhoe- atnwdoon-pe-paarrttoonnmmomoenmtuemndtiustmribuctioonrrfuenlcattioinosn: function R(ξ1, ξ2) is defined to be the ratio of the dependence of the inclusive momentum distribution on jet hardness is shown in Fig. 1. D(ξ1, ξ2) two- and one-parton momRe(nξ1t,uξ2m) = distribut,ion functions: (2) TDh(eξ1t)wDo(ξ-2p)arton momentum correlation function R(ξ1, ξ2) is defined to be the ratio of the d2N two- and one-parton momentum distribution functions: where D(ξ1, ξ2) = dξ1dξ2 . The momentum distributions Dar(eξn1o,rξm2a)lized as follows: R(ξ1, ξ2) = , (2) D(ξ)dξ = n , where n is the average multiplicity of partons in a jet, and ! " ! " D(ξ1)D(ξ2) D(ξ1, ξ2) R(ξ1, ξ2) = , (2) ! D(ξ1, ξ2)dξ1dξ2 = n(n 1) for all pairs of partons in a jet. The average multiplicity ! − 2" D(ξ1)D(ξ2) d N o!f partons n is a function of the dijet mass Mjj and the size of the opening angle θc. For where D(ξ1, ξ2) = dξ1dξ2 . The momentum d2 istributions are normalized as follows: ! " where D(ξ , ξ ) = d N . The momentum distributions are normalized as follows: θ = 0.5, n varies from 6 to 12 for M in the r1ang2e 80–600dGξ1edVξ2/c2 [4]. D(ξc )dξ ! =" nT,wo-particlew∼ here∼ n jijs th eenergyaverage m correlationsultiplicity of partons in a jet, and The Fong-Webber approximation of EqD. ((ξ2)dfξor t=he twno-p,artwonhemreomentumiscortrheelatiaovnerage multiplicity of partons in a jet, and ! " ! " ! " ! " ) 1.6 ) 2 1.6 2 1.4 CDF Run II 2 CDF Run II ! ! ! function [6] can1 be written as follows: ! ! " " D(ξ1, ξ2)dξ1dξ2 = n(n 1) fDo(rξ1a, ξll2)dpfitξ toa1 CDFdirξ datas2 =of np(anrto1n)s " fionr aalfitl toj pCDFea tdatai.rs oTf hpeartaovnesriangae jmet.ulTtihpeliacvietryage multiplicity , , 1 1.5 uncertainty of the fit 1.51 uncertainty of the fit ! ! − " ! " ! − " Fong/Webber Q =180 MeV " Fong/Webber Q =180 MeV 1.3 eff eff C( 1.4 R.Perez-Ramos Q =230 MeV C( 1.4 R.Perez-Ramos Q =230 MeV ! eff eff ! 0.5 of partons n is a functi2on of the dijet mass Mjj and the size of the opening angle θc. For of partons n isR(a∆ξfu, ∆nξct)i=onr !+2ofr (t∆hξe +d1.3∆ijξeQ=E)t+m# =100*0.5=50r a(∆ssξ M GeV∆ξ )a,nd t1.3he Q=Esiz# e=100*0.5=50(o3f) t hGeVe opening angle θ . For 1 2 0" 1 1 2 jet c 2 1 jj2 jet c c = 1.2 ! " ! 1 − ! " " 1.2 1.2 2 θc = 0.5, n varies from 6 to 12 for Mjj in the range 80–600 GeV/c [4]. 0 1.1 1.1 ! " ∼ 1.1∼ 2 θc =w0he.r5e,∆ξn= ξvaξr0i,easndftrhoemparame6tertsor0, r1,1a2ndfro2rdeMfinjejthiensttrehngethraofntgheeco8r0re–la6ti0on0 GeV/c [4]. − 1 1 ! " ∼ ∼1The Fong-Webber approximation of Eq. (2) for the two-parton momentum correlation " 0.9 0.9 and depend o-0.5n the variable τ =! ln(Q/Q ). Equation (3) is valid only for partons with ξ 1 =- eff " 0.8 0.8 The Fong-Webber appro! xima0.9tion of Eq. (2) for the two-parton momentum correlation 2 function [6] can be written as follows: around the peak of the inclusive parton mom0.7entum distribution, in the ran0.7ge ∆ξ 1. -1 ∼ ± 0.8 0.6 Text 0.6 function [6] can-1 be-0.5wri0tten0.5 as 1follows:-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 The parameters r0, r1, and r2 are calc"u! lated separately for quark and g- luon jets from an + 1 "!1 "!2 "!1 "!2 2 R(∆ξ1, ∆ξ2) = r0 + r1(∆ξ1 + ∆ξ2) + r2(∆ξ1 ∆ξ2) , (3) expansion in powers of 1/√τ using the assumption that the number of effectiveCDFly m,a arXiv:0802.3182ssless − quarks N is 3. Keeping only terms controlled by theory, the parameters are: 2 f R(∆ξ1, ∆ξ2) w=herr0e +∆ξr=1(ξ∆ξ1ξ +, an∆dξt2h)e+para2m(∆eteξr1s r ,∆r ξ,2a)nd, r define the strength(3of)the correlation − 0 −0 1 2 q 0.64 q 1.6 q 2.25 r = 1.75 , anrd =depen,d onr t=he vari,able τ = ln(Q/Qe(ff4)). Equation (3) is valid only for partons with ξ 0 − √τ 1 τ 3/2 2 − τ 2 where ∆ξ = ξ ξ0, and the paarraomunedtethres pre0a,kro1f, tahnedinrcl2usdiveefipnaerttohnemsotmreenntgutmh doisftrtihbueticoon,rrienlathteiornange ∆ξ 1. − CP Fong & BW, ∼ ± g 0.28 Theg pa0r.a7metersg r , 1r.0, and r are cNPalc uB355(1991)54lated separately for quark and gluon jets from an and depend on ther0 =va1r.3i3able τ, =r1ln=(Q/,Qeffr2)=. 0 Eq1 u, ation2 (3) is v(a5l)id only for partons with ξ − √τ τ 3/2 − τ 2 Hadronization ECT* 08 expansion in pow9 ers of 1/√τ using the asBryansum pWtebberion that the number of effectively massless arouwnhderetqhaendpgesaupkersocrfiptshdeeniontectlhuesciovrreelaptiaonrtpoarnammeteorsmfoer npatrutomns indiqsutarkibjeutstaiondn, in the range ∆ξ 1. quarks N is 3. Keeping only terms controlled by theory, the parame∼ter±s are: gluon jets, respectively. f The parameters r0, r1, and r2 are calculated separately for quark and gluon jets from an The theoretical prediction of the shape of the two-parton momqentum correl0a.t6io4n distri-q 1.6 q 2.25 r0 = 1.75 , r1 = , r2 = , (4) expabnutsioinonfunicntiopn oiswsheorwsn oinf F1ig/.