Ali Hanks

1
Jet Fragmentation

• Ali Hanks
• JClub
• June 21, 2006

Ali Hanks - JClub
2
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

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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

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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)

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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

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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

• Pji splitting function (more later)
• This leads to a shift in the x distribution to
  lower values as the scale increases
• scaling violation

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Parton Splitting

• This is the parton showering that occurs prior to
  hadronization
• Calculated perturbatively
• Dominated by collinear region
• z or (1-z) ? 1 ? log(Q2/?2)
• Leading log approximation
• Requires the introduction of a cutoff scale
  Qcutoff (kT gt Qcutoff)
• This usually means kT gt 1 GeV
• Jets are a soft process ? most interesting at kT
  lt 1 GeV!

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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

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Hadronization I

• For inclusive hadron cross-sections theres 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
• Nhadrons KLPHD Npartons
• 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. KLPHD(all hadrons) 1 , KLPHD(/-) 1/2 -
  2/3

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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

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Hadronization II - Fragmentation Functions
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Fragmentation in Monte Carlo
Hadronization Models

• 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

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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

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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

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Lund String Model (contd)

• 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

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Lund String Model (contd)

• Pairs are generated according to the probability
  of a tunnelling process
• Leads to a flavor-independent gaussian spectrum
  for the pT of the pairs
• The string has now transverse excitations so the
  pT 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

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Lund String Model (contd)

• Meson production choice between the possible
  multiplets for meson production
• Relative composition not given from first
  principles
• Spin counting suggests a 31 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

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Lund String Model (contd)

• The hadron pT was determined from the pT 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

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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 ?qq1
  jetremainder 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