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Jet Fragmentation. Ali Hanks JClub June 21, 2006. Ali Hanks - JClub. 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 - PowerPoint PPT Presentation
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Jet Fragmentation
Ali Hanks
JClub
June 21, 2006
Ali Hanks - JClub
06/21/06
Ali Hanks - JClub2
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
06/21/06
Ali Hanks - JClub3
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
06/21/06
Ali Hanks - JClub4
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)
06/21/06
Ali Hanks - JClub5
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
06/21/06
Ali Hanks - JClub6
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
06/21/06
Ali Hanks - JClub7
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 > Qcutoff)
– This usually means kT > 1 GeV
• Jets are a soft process most interesting at kT < 1 GeV!
06/21/06
Ali Hanks - JClub8
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
06/21/06
Ali Hanks - JClub9
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
– 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
06/21/06
Ali Hanks - JClub10
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
06/21/06
Ali Hanks - JClub11
Hadronization II - Fragmentation Functions
06/21/06
Ali Hanks - JClub12
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
06/21/06
Ali Hanks - JClub13
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
06/21/06
Ali Hanks - JClub14
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
06/21/06
Ali Hanks - JClub15
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
06/21/06
Ali Hanks - JClub16
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 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
06/21/06
Ali Hanks - JClub17
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
06/21/06
Ali Hanks - JClub18
Lund String Model (cont’d)
• 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
06/21/06
Ali Hanks - JClub19
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