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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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,
→ 0As → infinite,
→ 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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Fragmentation function
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Difference between data and fit
b-enriched sample
c-enriched sample
2=1
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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:
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Helicity distribution
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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)
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Hadron multiplicity in SIDIS
PDF: MRST2001 FF: DSS
ProtonDeuteron • HERMES Prelimnary
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
End
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Backup slides
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Δq(x) extraction: exercise
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
DSSV in the framework
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
Uncertainty on extracted FF: exercise
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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
The 18th International Symposium on Spin Physics, University of Virginia, October 6-11 2008
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”.