Kazutaka Sudoh (KEK) SPIN2007, Vancouver July 30, 2007 Determination of Fragmentation Functions and Their Uncertainties Determination of Fragmentation

Kazutaka Sudoh (KEK)SPIN2007, Vancouver

July 30, 2007

Determination of Fragmentation Functionsand Their Uncertainties

Determination of Fragmentation Functionsand Their Uncertainties

I. Introduction

II. Global Analysis of Fragmentation Functions

III. Uncertainty Estimation

IV. Summary

In collaboration with M. Hirai (TITech), S. Kumano (KEK), and T.-H. Nagai (The Grad. Univ.)

Reference: Phys. Rev. D 75, 094009 (2007) http://research.kek.jp/people/kumanos/ffs.html

Page-2K. Sudoh (KEK)

Fragmentation Function in

• Fragmentation Functions (FFs):

• Fragmentation process occurs from quarks, anti-quarks, and gluons, so that Fh is expressed in terms of their contributions:

2( ,1 (

))

tot

hF zd e hX

zQ

e

d

e e h X

212 22, ,) , ), ( ,(h

zi

i Rhi FF

dF

zQz CQ D

Coefficient Function calculable in pQCD

Fragmentation Function extracted from experiments

energy fraction of hadron and primary quark

2 2,F RQ s : CMS energy

hq

q

2

2 2h h h

q

P q E Ez

Q Q E

: scaling variable

2( , ) probability to find the hadron from a parton

with the energy fraction

hiD z Q h i

z


• Momentum (Energy) Sum Rule– Energy conservation should be hold for each flavor

• DGLAP equation: controls the energy dependence of FFs

• Favored and Disfavored Fragmentation Functions

2(0) (1)( )

( , ) ( ) ( )2s

ij s ij ijP z P z P z

ijP : j→i splitting function

2

12 22

( )( , ) , ,

ln 2hs

j ij s izi

d zD z P D

1 2

0( , ) 1h

ih

zD z Q dz 0 0, , , , , , , , , ,h K K K p p n n

, , , ...u d sD D D

Disfavored FFs:

Favored FFs: ,u dD D

( ), ( ), ( ), ....ud K us p uud

from a quark which exists in a naïve quark model

from a quark which does not exist in a naïve quark model


Purpose for investigating FFs

• Origin of proton spin

• Properties of quark-gluon matter

(RHIC)A A h X

Fragmentation Functions (FFs) are key issue in high energy

hadron production processes.

(e.g. HERMES), (RHIC-Spin)e p e h X p p h X

Quark, anti-quark, gluon contribution to proton spin(gluon polarization, flavor separation)

Nuclear modification (recombination, energy loss,,,)

2

2

, ,

2

( , )

( ,ˆ ( ) )

( , )

a a b ba b c

cD zab cX

f x Q f x Q

Q


Relevant Diagrams

• LO:

• NLO:

• Advantage– No ambiguity of PDF error– Sensitive to low z behavior

• Disadvantage– Insensitive to gluon FFs– Difficult to separate quark flavor (only sum of charged hadron)– Insensitive to high z behavior

No initial gluon function!!


Present Status of FFs

• There are two widely used FFs by Kretzer and KKP.– AKK is an updated version of KKP.

• Independent global analysis of FFs including new data• Estimate their uncertainties (It’s new!!!)

Kretzer S. Kretzer PRD62, 054001 (2000)KKP B.A. Kniehl, G. Kramer, B. Potter, NPB582, 514 (2000)AKK S. Albino, B.A. Kniehl, G. Kramer, NPB725, 181 (2005)

But, these functions are very different.Large differences indicate that the current FFs have much ambiguities.

Just after our analysis, a new parametrization including ep and pp reactions is proposed by D. de Florian, R. Sassot, M. Stratmann (PRD75, 114010 (2007), arXiv:0707.1506)

In this work


Comparison of Setup

• Pion FFs from e+e- annihilation

HKNS (Ours) Kretzer KKP(AKK)

Function form

# of parameters 14 11 15(18)

Mass threshold

mQ2

(mc,b=1.43, 4.3 GeV)

mQ2

(mc,b=1.4, 4.5 GeV)

4mQ2

(2mc,b=2.98, 9.46 GeV)

Initial scale 0

2 [NLO] 1.0 GeV2 0.4 GeV2 2.0 GeV2

AnsatzOne parameter is fixed.

(1 )

2

u u

u ug

D z D

M MM

0

1 2

1 2

.

0

05( , )

( , )

hi i

hi

M zD z Q dz

zD z Q dz

not

(1 )i iiN z z

(1 )i iiN z z

(1 )i iiN z z


Determination of FFs

• Determination of fragmentation functions and their uncertainties in LO and NLO

• Discuss NLO improvement in comparison with LO– Role of higher order corrections in the determination

• Comparison with other parametrizations

• SLD 2004 data (accurate) are included.

New aspects in our analysis


Ansatz (for ±)

• Function form (simplest form)

• Constraint condition– 2nd moment should be finite and less than 1

20,

2, , , 0

2,

2,

20

( , ) (1 )

( , ) (1 )

( , ) (1 )

( , ) (1 )

( , ) (1 )

u u

u u

c c

b b

g g

uu d

u d s s u

c c c c

b bb b

g g

D z N z z

D z N z z

D z m N z z

D z m N z z

D z N z z

1

02, 1, 0 ( ) 12nd h

i i i iM zD z dz

1

0, ( )

( 2, 1)

2nd2ndi

i i ii i

MN M zD z dz

B

2 2 202 2 2

2 2 2

2 2

3,

4,

5,

6,

c

c bf

b t

t

Q m

m Q mn

m Q m

m Q

q qD D

0

2i i

i

D DD


Experimental Data:

# of data

TASSO

TCP

HRS

TOPAZ

SLD

SLD [light quark]

SLD [ c quark]

SLD [ b quark]

ALEPH

OPAL

DELPHI

DELPHI [light quark]

DELPHI [ b quark]

12,14,22,30,34,44

29

29

58

91.2

91.2

91.2

91.2

29

18

2

4

29

29

29

29

22

22

17

17

17

• the number of Data ： 264 2 21 , 0.1( ), 0.05( )GeV Z ZQ z s M z s M

( )GeVs Kinematical coverage

e e X


2 Analysis

• Input parameters and results

–

• Uncertainty estimation: Hessian method

– N=14, 2=15.94: [K(N.s): 2 distribution]

2 20

4

1

0.220 , 0.323

1.43 4.3

fn

QCD

s f

c b

n

m m

GeV

(LO) (NLO)

varies with a change of

GeV, GeV

2 22 2 2

,

ˆ( )ˆ ˆ( ) ( ) ,ij i j ij

i j i j

aa a a H a a H

a a

2 2 1

,

ˆ ˆ( , ) ( , )( ) ij

i j i j

D z a D z aD z H

a a

2 ( ) 453.2(1.81) [ ], 433.5(1.73) [ ]Total /d.o.f LO NLO

2

0( , ) 0.683K N s ds

MRST: EPJC23, 73; PLB531, 216 (2002)


Comparison with pion Data

2 1 ( )( , )

tot

d e e XF z Q

dz

(Dat

a-T

heo

ry)/

Th

eory

Our NLO fit with uncertainties

Our fit is successful to reproduce the pion data.

The DELPHI data deviate from our fit at large z.

Rational deference between data and theory

2 2Data Theory

2Theory

( , ) ( , )

( , )

F z Q F z Q

F z Q


Comparison with pion Data (2)

• (Data-Theory)/Theory


FFs with Uncertainties for pion

• Gluon and light quark FFs have large uncertainties.

• Uncertainties bands become smaller in NLO compared with LO. (The data are sensitive to NLO effects.)

• The NLO improvement is clear especially in gluon and disfavored FFs.

• Heavy quark functions are relatively well determined.


Determined Functions for Kaon

• Gluon and light quark FFs have large uncertainties.

• Uncertainties bands become smaller in NLO compared with LO.

• Heavy quark functions are relatively well determined.

The situation is similar to the pion funcions

Two favored functions for kaon

and K Ku sD D


FFs with Uncertainties (p/p)

2 20 0( , ) 2 ( , )p p

u dD z D z

• Gluon 2nd moment is fixed.

• Gluon uncertainty at the peak position is underestimated.

• NLO improvement is not significant for uncertainties.

( ) / 2

2u d s

g

M M MM

: average of favored and disfavored functions


Comparison with Other Parametrizations for pion

0

. . ( ) / 2u u ui e D D D

( ) / 2 distribution!!i

evolved to Q2=2, 10, 100 GeV2

All functions are different, but consistent within uncertainties bands.

HKNS (Hirai, Kumano, Nagai, Sudoh)KretzerKKP (Kniehl, Kramer, Potter)AKK (Albino, Kniehl, Kramer)DSS (deFlorian, Sassot, Stratmann)

Relevant to RHIC 0 Xsection


Comparison for Kaon and Protonkaon proton

---- DSS


Summary

• Global analysis of FFs was done for independent parametrization– Determine function forms for , K, p in LO, NLO analyses

• Uncertainties of FFs were estimated

– Large uncertainties in gluon and disfavored functions

– Heavy quark functions are well determined.

– Uncertainties could be reduced by performing NLO analysis

• Importance of accurate FFs

– The uncertainties at low Q2 are very important for discussing Nucleon’s spin a

nd/or heavy ion physics. (e.g. hadron production at small pT at RHIC)

– Need for accurate “low-energy” data by Belle & BaBar

• Program code for calculating our FFs is now available at

http://research.kek.jp/people/kumanos/ffs.html

Documents

Kazutaka Sudoh (KEK) SPIN2007, Vancouver July 30, 2007 Determination of Fragmentation Functions and Their Uncertainties Determination of Fragmentation