2 and 3-jet Analysis in Flux-tube 2 and 3-jet Analysis in Flux-tube ModelModel

J.B.Choi, M.Q.Whang, S.K.LeeJ.B.Choi, M.Q.Whang, S.K.Lee(Chonbuk National University, Korea)(Chonbuk National University, Korea)

ⅠⅠ. Generals. Generals

ⅡⅡ. Flux-tubes in Coordinate Space. Flux-tubes in Coordinate Space

ⅢⅢ. Momentum Space Flux-tube Model. Momentum Space Flux-tube Model

ⅣⅣ. 2-jet Analysis. 2-jet Analysis

ⅤⅤ. 3-jet Analysis. 3-jet Analysis

ⅥⅥ. Look forward. Look forward

I. GeneralsI. Generals

1.1. Purpose of LCPurpose of LC

2.2. Production ProcessesProduction Processes

3.3. FactorizationFactorization

4.4. Hadronization into JetsHadronization into Jets

5.5. Jet Overlapping Jet Overlapping

Purpose of LCPurpose of LC

Higgs → 2, 4 jets Higgs → 2, 4 jets

→ → 4, 6 jets4, 6 jets

→ → 8, 10 jets8, 10 jets

6 jets6 jets

SUSYSUSY

Extra-dimExtra-dim

H

ZHHtt

tt

ZWWWWWW ,,

Production ProcessesProduction Processes

ZH

WW

Z

HWW

4 jets4 jets

QQqq ggqq

e

e SUSY

Htt

tt

ZH

Loop correctionsLoop corrections

no. of loops no. of diagrams drawings calculationsno. of loops no. of diagrams drawings calculations

00 ~10 ~1000 H H H H1 1 ~10 ~1011 H H H H22 ~10 ~1022 H H H/CH/C33 ~10 ~1033 H/C H/C H/CH/C44 ~10 ~1044 C C C C55 ~10 ~1055 C C66 ~10 ~1066 C C

(H : Hand)(H : Hand)

(C : Computer)(C : Computer)

FactorizationFactorization

11stst rule rule

; perturbative expansion in ; perturbative expansion in

; non-perturbative models; non-perturbative models

CorrectionsCorrections asymptotic expansionsasymptotic expansions exponentiation + resummationexponentiation + resummation

Uncertainty exists !Uncertainty exists !

)()(~~ F)(~ )( F

s

Hadronization into JetsHadronization into Jets : 2 jets: 2 jets

2 or 4 jets2 or 4 jets : 4 or 6 jets: 4 or 6 jets : 6 jets: 6 jets : 8 or 10 jets: 8 or 10 jets models based on local models based on local parton-hadron dualityparton-hadron duality

cluster → HERWIGcluster → HERWIG

string → JETSETstring → JETSET

… …

qqH

ZZ

WW

ZHttHtt

Jet OverlappingJet Overlapping

4 jets 4 jets Consider the coneConsider the cone

overlap solid angle ;overlap solid angle ;

∴ ∴ probability to overlapprobability to overlap (maybe OK.)(maybe OK.)

5 jets 5 jets for fixed 4 jetsfor fixed 4 jets

; (difficult to check); (difficult to check)

6

2

41d )(~

16

943 2

2

1

64

2743

169

P4

2

1

16

944

169

P5

ProcessesProcesses 6 jets6 jets

8 jets 8 jets BG :BG :

Htttt ,

3

2

64

4545

169

P6

gttZtt ,

64

6347

169

P8

bbW

ttW

b

b

; ; nearly always overlapnearly always overlap need new methodneed new method

II. Flux-tubes in Coordinate II. Flux-tubes in Coordinate SpaceSpace

1.1. Flux-Tube ClassificationFlux-Tube Classification

2.2. Connection AmplitudeConnection Amplitude

3.3. Gluon DensityGluon Density

4.4. MesonsMesons

5.5. BaryonsBaryons

6.6. 4-quark States4-quark States

7.7. Pentaquarks Pentaquarks

Flux-tube ClassificationFlux-tube Classification a : no. of quarks (sources)a : no. of quarks (sources) b : no. of antiquarks (sinks)b : no. of antiquarks (sinks)baF ,

glueballglueball

mesonmeson

baryonbaryon

pentaquarkpentaquark

hexaquarkhexaquark

0F

11F ,

3F

22F ,

14F ,

6F

qqqq

Connection AmplitudeConnection Amplitude

AA : The amplitude for a quark to be connected: The amplitude for a quark to be connected to another one through given flux-tube.to another one through given flux-tube. M(A)M(A) : measure of A: measure of A ▫▫. . assumptionsassumptions

(1) M(A) decreases as A increases.(1) M(A) decreases as A increases. (2) M(A(2) M(A11) + M(A) + M(A22) = M(A) = M(A11AA22))

((when Awhen A11 and A and A22 are independent) are independent) SolutionSolution

AA0 0 : normalization constant : normalization constant k : parameterk : parameter

)( AMk1

0eAA

Form of MForm of M For M ∝ For M ∝ ｜｜ x-yx-y ｜｜ νν , flux-tube shape is determined by , flux-tube shape is determined by ｜｜ x-yx-y ｜｜ ν ν = = ｜｜ x-zx-z ｜｜ νν + + ｜｜ z-yz-y ｜｜ νν

General A becomesGeneral A becomes

For and For and

xx yyzz

shapesphere2

shapeline1

:

:

10

weightdrF

k

1AAA )(exp)(

: Weight : Weight factorfactor

: Integration : Integration limitlimit

)(F

,,)( 21F ArA

r

rr

k

1 2

er

1A ln

Gluon DensityGluon Density

Overlap functionOverlap function probability amplitude to have quark pairsprobability amplitude to have quark pairs For a mesonFor a meson

We can assumeWe can assume probability to have quark pair ∝ gluon densityprobability to have quark pair ∝ gluon density

fiAA

2

222

1

121

230

2

222

1

1212

0f

2

0i

r

rr

r

rr

r

rr

k

1A

r

rr

r

rr

k

1AA

r

rr

k

1AA

lnlnlnexp

lnlnexp

lnexp

z1r 2r

x yr

MesonsMesons

0

50.

BaryonsBaryonsProtonProton

NNeutroeutronn

4-quark States (1)4-quark States (1)squaresquare

4-quark States (2)4-quark States (2)

Pentaquarks-1Pentaquarks-1

Pentaquarks-2Pentaquarks-2

-2 -1 0 1 2-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Pro

babi

lity

y

x=0 x=0.5 x=1.0

a=1.0, z=0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

P : 1.0000e-2 ~ 1.0010e-2 P : 1.0000e-3 ~ 1.0010e-3 P : 1.0000e-4 ~ 1.0010e-4 P : 1.0000e-5 ~ 1.0010e-5 P : 1.0000e-6 ~ 1.0010e-6 P : 1.0000e-7 ~ 1.0010e-7 P : 1.0000e-8 ~ 1.0010e-8

y

x

a=1.0, z=0

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

ⅢⅢ. Momentum Space Flux-tube . Momentum Space Flux-tube ModelModel

1.1. Momentum Space ConnectionMomentum Space Connection

2.2. Definition of JetsDefinition of Jets

3.3. Phase SpacePhase Space

4.4. Angular OrderingAngular Ordering

5.5. Momentum Distributions Momentum Distributions

Momentum Space Momentum Space ConnectionConnection

Final particles are connected in Final particles are connected in momentum!momentum!

→ → momentum space flux-tube modelmomentum space flux-tube model

10 dpG

k

1AA )(exp

Definition of JetsDefinition of Jets

Fragmentation processFragmentation process by quark pair creationsby quark pair creations .. .. .. gluonic flux-tube descriptionsgluonic flux-tube descriptions

(1) (1) Probability amplitude ∝ overlap functionProbability amplitude ∝ overlap function ( in mementum space)( in mementum space) (2) (2) Phase space Phase space ; parton model assumptions; parton model assumptions

A

Phase Space Phase Space

Parton model assumptions about quark fragmentationParton model assumptions about quark fragmentation

(1) Longitudinal momentum components(1) Longitudinal momentum components ∝ ∝ total jet (parton) energytotal jet (parton) energy

(2) Transverse momentum components(2) Transverse momentum components from soft processes (small uncertainty)from soft processes (small uncertainty) → → parametersparameters → → TrapezoidTrapezoid

ee dd PPLL ∝ E (jet) ∝ E (jet)

PPTT : two parameters : two parameters d, ed, e

Angular OrderingAngular Ordering

Prediction of gluon jet direction ?Prediction of gluon jet direction ?

A = AA = A11AA22

(1) for fixed P(1) for fixed P22 (p (p11≡1.0),≡1.0),

vary Pvary P33 and θ and θ

(2) vary P(2) vary P22 and angle and angle

between between PP11 and and PP22

P1

P2

P3

A1

A2

θθ

Angular OrderingAngular Ordering

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

eve

nt

momentum

p2 = 0.1

= /12

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

eve

nt

momentum

p2=0.1

=/3

0.0 0.1 0.2 0.3 0.4 0.52.0

2.2

2.4

2.6

2.8

3.0

3.2

even

t

angle(rad)

p2=0.3

=/6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

even

t

angle(rad)

p2=0.6

=/4

Momentum Momentum DistributionsDistributions

2-jet case2-jet case

P1P2

Pθθ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

= 0angle = 0.001

even

t

momentum

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

2.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

even

t

momentum

= 0angle = 7

ⅣⅣ. 2-jet Analysis. 2-jet Analysis

PP11 PP22

q q

eedd

hh

L1L1 L2L2

p ln

pk1

-expp1

A 2 p

dpp)A(θ,Ph

0

2

Connection Connection amplitudeamplitude

ProbabilityProbability

Phase Phase SpaceSpace

Parameters – k,d,eParameters – k,d,e

FitsFits

ⅤⅤ. 3-jet Analysis. 3-jet Analysis

I.I.

II.II.

III.III.

Phase SpacePhase Space

◎◎. Phase space 2. Phase space 2

h

q

g

q

dd

ee

◦ ◦ gluonic effectgluonic effect ff

df df

efef

AnalysisAnalysis 3 jet3 jet

(A)(A)

(B)(B)

)

lnccc

lnbbb

lnaaa

(k1

-exp)cb(a

)(AC

1

121

1

121

1

121

111

30

1

)

lnccc

lnbbb

lnaaa

(k1

-exp)cb(a

)(AC

2

222

2

222

2

222

222

30

2

dpCCP

2h

0

21

q

q

g1a

2b

1c

2a

1b

2c q

g

q

FitsFits

◎◎. Phase space 3. Phase space 3

aa aa

◎◎

1. A0, a, e, k, d, f1. A0, a, e, k, d, f

2. Data Analysis2. Data Analysis

lnppp

k1

expp

AA

20

ParametParametersers

Parameters - aParameters - a

Parameters – A0Parameters – A0

Parameters - eParameters - e

Parameters - dParameters - d

Parameters - fParameters - f

Parameters - kParameters - k

Fits 1Fits 1

Fits 2Fits 2

Fits 3Fits 3

Fits 4Fits 4

Fits 5Fits 5

Fits 6Fits 6

ⅥⅥ. Look Forward. Look Forward

1.1. 3-dim. Structures3-dim. Structures

2.2. Momentum DistributionsMomentum Distributions

3.3. Jet Parameter CalculationsJet Parameter Calculations

4.4. Discrimination of Overlapped Discrimination of Overlapped

JetsJets

5.5. PossibilitiesPossibilities