Parton Distribution
Function (PDF)
Jaehyeok Yoo
UCSD
1
What is PDF?
• distribution of momentum fraction x of the
partons in a proton in the relevant kinematic
energy
• function that partitions hadron momentum
to its constituents, partons
• universal, can be applied to any processes
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σAB =
!
What is PDF?
"
2
dxa dxb fa/A xa , Q
#
"
2
fb/B xb , Q
#
σ̂ab→X
X = l+ l− ab = q q̄, q̄q
Drell-Yan process
A hadronic cross section can be
obtained by summing the cross
sections of sub-processes
with weighting factors
fa(b)/A(B)
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x≡
!
"
xa(b) , Q2
p
pmax
PDF
Q : momentum transfer
Jaehyeok Yoo
How do we get PDF?
• Because of the inherent non-perturbative
effect in a QCD binding state, PDFs cannot
be obtained by pQCD
• get the parameters for PDF in certain
regions where are experimentally accessible
• extrapolation to regions where experimental
data not available
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What we learn from data
•
Lepton-Lepton: !s(Q2) and fragmentation
functions of partons into hadrons
• Lepton-Hadron: DIS structure functions
(F2, F3)
• Hadron-Hadron: lepton pair production
cross sections, gluon distribution function
g(x,Q), direct photon and jet final states
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Input Data
PDF is determined by global fits to data from
•
DIS: main contribution
•
•
•
e-p: HERA(H1, ZEUS)
"-Fe: CCFR(Chicago Columbia Fermilab Rochester, E770)
#-p, #-d: BCDMS(Bologna-CERN-Dubna-Munich-Saclay),
NMC(New Muon Collaboration, NA37)
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Input Data
•
•
DY
•
•
p-Cu->#+#-X: E605
p-p -> #+#-: Nusea(Nucleon Sea, E866)
Hadronic jet: CDF, important for
high x gluon distribution
•
W asymmetry : CDF
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Q^2 vs. 1/x
kinematic coverage of current data
Why are they different by experiments ?
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Q^2 vs. x of LHC
LHC coverage
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Evolution to high Q^2 region
2 dGq
!
2
x, Q
dQ2
"
!
2
αs Q
=
2π
"#
$
$
%
$
%%
dy
x 2
x 2
Q
Pqq (y)Gq
, Q + Pqg (y)Gg
,Q
y
y
y
$
(
!
"
!
"
&
'
&
'
#
2
2
dG
x,
Q
α
Q
dy %
x 2
x 2
g
s
2
Q
=
Pgq (y)Gq
, Q + Pgg (y)Gg
,Q
2
dQ
2π
y
y
y
q
• DGLAP-based NLO(and NNLO) pQCD
) eqn:
• DGLAP(
Dokshitzer–Gribov–Lipatov–Altarelli–Parisi
Coupled set of equations whose solutions
show how the PDFs change with variations
in the scale Q
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Global fits
F (x, Q0 ) = A0 xA1 (1 − x)A2 P (x; A3 , ...)
• Q : reference value, 1-2 GeV
• A : small-x behavior
• A : large-x valence counting rules
• P(x; A , ...): smooth function adding more
0
1
2
3
flexibility to PDF parameterization
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Accuracy of Evolution
depends on
•
•
accuracy of original measurement
uncertainty on !s(Q2)
!s=0.1175±0.0050
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2
!s(Q )
uncertainty effect on PDF
!s varies from 0.110 (short-dash) to 0.124 (long-dash) in increments of 0.002.
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LHC uncertainty
parton-parton luminosity uncertainties
g-g
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q-q
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http://durpdg.dur.ac.uk/hepdata/pdf3.html
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Gluon PDF
Large Q2 dependence at small x
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sea up quark
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valence up quark
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Summary
data
DIS, jets, DY,
W-asymmetry
...
Evolution
DGLAPpQCD,
error
18
Global fit
error
References
1) ATL-Physics -99-008
2) arXiv:hep-lat/9609018v1
3) arXiv:hep-ph/0611148v1
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