Jet Fragmentation

Ali Hanks

JClub

June 21, 2006

Ali Hanks - JClub

06/21/06

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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 > Qcutoff)

– This usually means kT > 1 GeV

• Jets are a soft process most interesting at kT < 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 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

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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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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

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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 (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

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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

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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

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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

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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