1. Jet Fragmentation Ali Hanks JClub June 21, 2006 Ali Hanks - JClub

2. 06/21/06 Ali Hanks - JClub 1 Motivation Jets provide a connection between pQCD and non-pQCD –Jet fragmentation/structure is driven by soft QCD Fragmentation functions are important for many theory calculations –Indentified particle multiplicities –Particle correlations Jet fragmentation models are a key part of Monte Carlo event generators Modification of fragmentation functions is a signature of medium effects in heavy ion collisions –Jet energy loss –Baryon/Meson suppression

3. 06/21/06 Ali Hanks - JClub 2 Hard Scattering in pp collisions Intial parton distributuions: PDFs –Long range = non-perturbitive Hard scattering of two partons –Short range = perturbative Hadronization of scattered partons –Long range = non-perturbative

4. 06/21/06 Ali Hanks - JClub 3 Factorization Each step can be treated as independent of the others –  ab for any two partons, a and b, calculated from pQCD –PDFs as functions of parton momentum fraction, x –FFs for a parton to fragment to a hadron with momentum fraction z PDFs and FFs are independent of the process used to determine them (universality)

5. 06/21/06 Ali Hanks - JClub 4 Jet Production Two partons collide (perturbative) Scattered parton emits a shower of quarks and gluons –Parton Cascade (perturbative) Hadronization –Partons pick up color matching partner from see of virtual quarks and gluons We can then observe these hadrons or there decays

6. 06/21/06 Ali Hanks - JClub 5 Scale Dependence - FF evolution FFs are independent of the process used to determine them  Scale independence ? No! Evolution is governed by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation P ji = splitting function (more later) This leads to a shift in the x distribution to lower values as the scale increases: –scaling violation

7. 06/21/06 Ali Hanks - JClub 6 Parton Splitting This is the parton showering that occurs prior to hadronization –Calculated perturbatively Dominated by collinear region –z or (1-z)  1  log(Q 2 /  2 ) –Leading log approximation Requires the introduction of a cutoff scale Q cutoff (k T > Q cutoff ) –This usually means k T > 1 GeV Jets are a soft process  most interesting at k T < 1 GeV!

8. 06/21/06 Ali Hanks - JClub 7 Infrared Divergences and Coherence Gluon emission is coherent –Strong interference –Angular ordering of successive radiation Large cutoff is due to infrared divergences in the theory Add angular resolution to soft gluon emission (Msbar subtraction scheme) –Analogous to energy resolution due to soft photon emission in QED Resume and find all IR divergences cancelled! Cutoff scale can be set as low as  QCD ~ 200GeV

9. 06/21/06 Ali Hanks - JClub 8 Hadronization I For inclusive hadron cross-sections there’s a sort of alternative to FFs  LPHD –Local Parton Hadron Duality hypothesis Assumes hadronization occurs locally at the end of parton shower –Hadrons “remember” parton distributions –N hadrons = K LPHD * N partons Naively: as partons move away they drage a color-matching partner from sea of virtual quarks and gluons to become hadrons  each parton becomes a hadron e.g. K LPHD(all hadrons) ~ 1, K LPHD(+/-) ~ 1/2 - 2/3

10. 06/21/06 Ali Hanks - JClub 9 Hadronization II - Fragmentation Functions We obtain our fragmentation functions by solving the DGLAP evolution equation  The normalization N, and parameters , , and  can be expressed as polynomials in a scaling variable  the initial energy scale  0 and  QCD (or  MS ) taken as inputs This is then fit to data to obtain values for these parameters

11. 06/21/06 Ali Hanks - JClub 10 Hadronization II - Fragmentation Functions

12. 06/21/06 Ali Hanks - JClub 11 Hadronization Models Fragmentation in Monte Carlo Three main models (with many variants and hybrids: –Lund String Model –Independent Fragmentation Models –Cluster Fragmentation Models Goal of each is to represent existing data well and provide a framework or predicting future results while remaining internally consistent Partons from parton shower are transformed to colorless hadrons Use the Local parton-hadron duality hypothesis –Hadron level momentum flow and quantum numbers follows the parton level –The flavor of the quark initiating the jet is found in a hadron near the jet axis

13. 06/21/06 Ali Hanks - JClub 12 Cluster Fragmentation Model Preconfinement of color (after parton shower) –partons generated in the branching process tend to be arranged in confined color-singlet clusters The cluster mass is constrained by the infra- red cutoff used in the parton shower After the parton shower these clusters split non-perterbatively into quark anti-quark pairs –enforced due to the small cutoff scale Does not require a fragmentation function to describe the transition or any free parameters Clusters typically decay into two hadrons depending on the mass of the cluster

14. 06/21/06 Ali Hanks - JClub 13 Lund String Model Models are probabilistic and iterative –Process is described in terms of a few simple underlying branchings Color “string” stretched between q and q- bar moving apart –The string is what is fragmenting rather than the partons Confinement with linearly increasing potential (1GeV/fm) String breaks to form 2 color singlet strings –Process continues as long as the invariant mass of the string is greater than the on-shell mass of a hadron

15. 06/21/06 Ali Hanks - JClub 14 Lund String Model (cont’d) When the potential energy in the string gets large enough it breaks, producing a new quark antiquark pair The system splits into two color- singlet systems This will continue if either system has enough mass The simplest model is a color-singlet 2-jet event Energy stored in color dipole field increases linearly –Related to presence of a triple-gluon vertex (self-interaction) Color flux tube formed as partons move apart –Uniform along its length  confinement picture with linear potential

16. 06/21/06 Ali Hanks - JClub 15 Lund String Model (cont’d) Pairs are generated according to the probability of a tunnelling process Leads to a flavor-independent gaussian spectrum for the p T of the pairs The string has now transverse excitations so the p T of the quark and antiquark pair must cancel in the string rest frame This tunnelling picture implies the suppression of heavy-quark production –s quarks are produced with a suppression relative to the lighter quarks but there is still no mechanism for the production of charm and heavier quarks

17. 06/21/06 Ali Hanks - JClub 16 Lund String Model (cont’d) Meson production: choice between the possible multiplets for meson production –Relative composition not given from first principles –Spin counting suggests a 3:1 mixture of vector and pseudoscalar multiplets The mechanism follows naturally from idea that the meson is a short piece of string between two quark antiquark endpoints Baryon production: harder to generalize - two main scenarios are avaiable –Diquark picture: any flavor q could be represented as an antidiquark –Popcorn model: baryons appear from successive production of several qqbar pairs

18. 06/21/06 Ali Hanks - JClub 17 Lund String Model (cont’d) The hadron p T was determined from the p T of the new qqbar pair created Need to determine the energy and longitudinal momentum –Momentum is constrained already In an iteration from the quark end, we then have We can now determine the fragmentation function, i.e. the probability that a given z is picked –Note: result should be same if we start itereation with qbar = left-right symmetry –Two free parameters remain that must be adjusted to fit the data

19. 06/21/06 Ali Hanks - JClub 18 Independent Fragmentation Model Fragmentation of any system of partons is described by an incoherent sum of independent fragmentation procedures for each parton –Carried out in c.m. frame of the jet system Uses an iteretative process: jet  qq 1 + jet remainder where the pair and the remainder jet are collinear The remainder jet is just a scaled version of the original –Momentum sharing is given by a pdf f(z) where z is the momentum fraction of the hadron –f(z) is assumed to be independent of the remaining energy Internal inconsistencies arrise within the details of this model so it is generally used just for special studies