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The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008

Global QCD Analysis ofFragmentation and

Parton distribution functions

Yoshiyumi Miyachi, Yoshimitsu ImazuYuKi Kobayashi and Toshi-Aki Shibata

Tokyo Tech


Introduction

High Energy : Low Energy :Asymptotic Freedom Color Confinement

Running Coupling :

QCD : Dynamics of quark-gluon (partons) system

High Energy Interaction including Hadronse.g. Deep Inelastic Scattering (DIS)

Energy of a System :

→ 0As → infinite,



Perturbative QCD (pQCD)Parton Distribution Function (PDF)Fragmentation Function (FF)

Non-Perturbative

Factorization

We don’t know exactlyhow a hadron is composedof partons.

Perturbative


Background of the present study

2005

2006

2007

2008

Fragmentation Function, and Helcity distribution frunction analysesDouble spin asymmetry in DIS and SIDIS → Δq(x)Hadron inclusive cross-section

→ Uncertainty on FF and impact on Δq

Fortran based, brute force method (grid in x, z, and Q^2 )stand alone codes for each application

Limited capability:especially due to required computing timecorrelation between FF and PDF

SPIN2006

SPIN2008

Decide to rebuild analysis framework in Mellin spaceC++ base, module structuregeneral framework for FF and PDFgeneral framework for observable

Tokyo Techgroup


Global QCD Analysis Framework

DGLAP Equation

FFPDF

densityhelicity

Observable

DISF1, g1, ....

e+ e-Hadron Inclusive

Cross-section

SIDISMutiplicity, A1

h, ...h-h scattering

Drell-Yanpi productiondirect photon

jet(Xsec, ALL, ...)

DataFit

Inverse Mellin Transformation(numerically)

Parameterization @ initial scale

Mellin Transformation

F N ,M , ...~∑q , q ' ,...C N ,M ,...⋅qN ⋅q ' M ....

∂q∂ t=⋅q

Analytically or Numerically


Global QCD analysis framework: motivation● General framework– FF, PDF, and observables calculated from them

● Evolution, Mellin inversion, ...● Flexibility– Constraint on FF and PDF:

● (g, u, d, s), (g, uv, dv, sea), (g, ΔΣ, a3, a8 ), ....● functional form, favored–disfavored relation in FF, ....

● Uncertainty study– Hessian & Lagrange multiplier methods– Correlation: among parameters, FF and PDF– Error propagation


What you will see...● Contents of this talk– Parton density distribution → Test of the framework– Hadron inclusive cross section → Fragmentation function– Spin structure function → Helicity distribution– Hadron multiplicity in SIDIS– Double spin asymmetry in SIDIS


Parton density: MRST1998

MRST1998, Eur. Phys. J. C4 (1998) 463

line: evolution in the frameworkPDF was analytically transformed to the Mellin space for the evolution.

dotted: based on the grid data.

Functional form and constrain:

xf x ,Q 2=1GeV2

xf x ,Q 2=200GeV2

@Q2=1GeV2


Fragmentation function


Fragmentation functionsKretzer FF: PRD 62, 054001 (2000)

Oscillation in the small z comes from Mellin inversion. ( under control)

z Du z ,Q2=1.0GeV 2

z Dg z ,Q 2=1.0GeV 2

10-2 10-1 1 z

DSS_evo: Phys. Rev. D75, 114010 (2007)Evolution done in the framework

Kretzer_evoEvolution done in the framework

Kretzer_gridfrom the grid data

Kretzer_fitfit data as done in the ref.(next slides)

zD z =N z a1−z b

D z =N za 1−zb11− z

1−zDu=Du=Dd=Ds=DsDu=DdN g=0.5∗N uN u

For g, u, c and b

For g, U, D, ubar, S, C, B


Hadron inclusive cross-section

zD z =N z a 1−z b

N a bg lu o n - F r e e F r e eu F r e e F r e e F r e ec F r e e F r e e F r e eb F r e e F r e e F r e e

1−z D u=D u=D d=D s=D s

D u=D d

N g=0.5∗N uN u

Fit to data under Kretzer's conditionPRD 62, 054001 (2000)

Total 11 free parameters

Data: 0.05 < z < 0.8

χ2 / n.d.f = 78.3 / 58


Difference between data and fit

b-enriched sample

c-enriched sample

2=1


Fragmentation Function Analysis: uncertaintyz D z =N z a 1−z b

(30x30) * 11 * 12 / 2 = 59400 Fits in total

It took about 160 hours.

N a bg lu o n - F r e e F r e eu F r e e F r e e F r e ec F r e e F r e e F r e eb F r e e F r e e F r e e

Fit under Kretzer's condition:

1−z D u=D u=D d=D s=D sD u=D d N g=0.5∗N uN u

Total 11 free parameters

χ2 contour:


Helicity distribution


Helicity distribution function

xf x = x a 1−x b 1c⋅xd⋅x 1/2BB: Nucl.Phys.B636:225-263,2002

uv, dv, sea, gluon

DSSV: Phys.Rev.Lett.101:072001,2008U, D, u-bar, d-bar, s = s-bar, g

AAC, LSS requires MRST for Δq(x)data from the grid are shownNot yet implemented in the framework


Structure function g1 with DSSV

Data: EMC, SMC, COMPASSHERMES, E142, E143, E154, E155

Δq(x):DSSVPhys.Rev.Lett.101:072001,2008

x⋅g1px ,Q2

x⋅g1d x ,Q 2

x⋅g1nx ,Q 2


Observable in SIDIS l N l 'hX

h N ,M ∝∑qC N ,M q N D q

h M

Hadron multiplicity: q(x), D(z)

Double spin asymmetry: q(x), D(z), Δq(x)

q(x): MRST2001 Eur. Phys. J. C23:73-87, 2002

Δq(x):DSSV Phys.Rev.Lett.101:072001,2008

D(z): DSS Phys. Rev. D75, 114010 (2007)


Hadron multiplicity in SIDIS

PDF: MRST2001 FF: DSS

ProtonDeuteron • HERMES Prelimnary


Double spin asymmetry in SIDIS

DATA: HERMES

PDF:q(x) MRST2001Δq(x) DSSV

FF: DSS

eN e '±XProtonDeuteron

Ready to fit the available datain the new framework!!

A1x

A1−x

0.01 0.01 x 1

0.01 0.01 x 1


Summary● Global QCD Analysis is an essential tool for

understanding of proton spin problem– Uncertainty correlation among parameters, FFs, and PDFs

● New analysis frame has been developed– General framework to handle FF, PDF, and observable– Framework was tested with the available parameterizations

● q(x): MRST, D(z): Kretzer, DSS, Δq(x) DSSV– Multiplicity, Double spin asymmetries in SIDIS– Next step:

● Determine FF and Δq(x) by fit to the available data ● Include hard scattering in p-p collision


End


Backup slides


Δq(x) extraction: exercise


Structure function g1 Fit: exercise

xG x= xa1−x 4.0 /Nxx= xa1−xb1c⋅x/Nx a3x=1.269 xa1−x b/Nx a8 x =0.586 xa 1−xb/N


Parton Helicity Distribution xG x= xa1−x 4.0 /Nxx= xa1−xb1c⋅x/Nx a3x=1.269 xa1−x b/Nx a8 x =0.586 xa 1−xb/N

x s x


DSSV in the framework


Uncertainty on extracted FF: exercise


Fragmentation Function: g →π+

Q2=1GeV2

Q2=10GeV2

Q2=100GeV2

Kretzer evolved in the framework

Kretzer grid data

Kretzer re-Fit (with Δχ2=1 band)

DSS evolved in the framework


Fragmentation Function: u →π+

Kretzer evolved in the framework

Kretzer grid data

Kretzer re-Fit (with Δχ2=1 band)

DSS evolved in the framework

Q2=1GeV2

Q2=10GeV2

Q2=100GeV2


Home work● Higher twist contribution– Target mass contribution, CLAS data

● Positivity condition– Treatment at NLO, Implementation in fitting procedure

(Technical issue )● Error estimation and propagation– It is always good if anyone can use “extracted uncertainty”.

● Proton – proton scattering data– Treatment Wilson coefficient in the x and z space.

● Tuning on SIDIS calculation– It is not satisfactory in terms of “computation time”.

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