James Stirling

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QCD Theory a status report and review of
some developments in the past year

• James Stirling
• IPPP, University of Durham

2
more QCD? see also
QCD _at_ HERA Klein QCD _at_ Tevatron Lucchesi QCD
and hadron spectroscopy Close, Shan Jin QCD and
heavy quarks Ali, Shipsey QCD on the Lattice
Hashimoto
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QCD is

• an essential and established part of the toolkit
  for discovering physics beyond the standard
  model, e.g. at Tevatron and LHC
• a Yang-Mills gauge field theory with a very rich
  structure (asymptotic freedom ? confinement),
  much of which is not yet fully understood in a
  quantitative way
• we no longer test QCD!

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QCD in 2004
compare ?tot(pp) and ?tot(ee-?hadrons)
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World Summary of aS(MZ) July 2004 from S.
Bethke, hep-ex/0407021

• New at this conference
• ZEUS DIS jets pdf fit
• HERA jet cross sections and shape variables
• JADE 4-jet rate and jet shape moments
• LEP 1,2 jet shape observables, 4-jet rate
• All NLO and all consistent with world average

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examples of precision phenomenology
and many other examples presented at this
Conference
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status of pQCD calculations
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fixed order d? A aSN 1 C1 aS C2 aS2
. thus LO, NLO, NNLO, etc, or resummed to
all orders using a leading log approximation,
e.g. d? A aSN 1 (c11 L c10 ) aS (c22
L2 c21 L c20 ) aS2 . where L
log(M/qT), log(1/x), log(1-T), gtgt 1 thus LL,
NLL, NNLL, etc.
current frontier

• LO
• automated codes for arbitrary matrix element
  generation (MADGRAPH, COMPHEP, HELAC, )
• jet parton, but easy to interface to
  hadronisation MCs
• large scale dependence aS(?)N therefore not good
  for precision analyses

• NLO
• now known for most processes of interest
• d?V(N) d?R(N1)
• reduced scale dependence (but can still dominate
  aS measurement)
• jet structure begins to emerge
• no automation yet, but many ideas
• now can interface with PS

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interfacing NnLO and parton showers
Benefits of both NnLO correct overall rate,
hard scattering kinematics, reduced scale
dep. PS complete event picture, correct treatment
of collinear logs to all orders
Example MC_at_NLO Frixione, Webber,
Nason, www.hep.phy.cam.ac.uk/theory/webber/MCatNLO
/ processes included so far pp ?
WW,WZ,ZZ,bb,tt,H0,W,Z/?
pT distribution of tt at Tevatron
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not all NLO corrections are known!
the more external coloured particles, the more
difficult the NLO pQCD calculation Example pp
?ttbb X bkgd. to ttH
the leading order O(aS4) cross section has a
large renormalisation scale dependence!
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John Campbell, Collider Physics Workshop, KITP,
January 2004
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NNLO the perturbative frontier
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Example jet cross section at hadron colliders

• The NNLO coefficient C is not yet known, the
  curves show guesses C0 (solid), CB2/A (dashed)
  ? the scale dependence and hence ? sth is
  significantly reduced
• Other advantages of NNLO
• better matching of partons ?hadrons
• reduced power corrections
• better description of final state kinematics
  (e.g. transverse momentum)

(also ee- ? 3 jets)
Tevatron jet inclusive cross section at ET 100
GeV
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anatomy of a NNLO calculation p p ? jet X

• 2 loop, 2 parton final state
• 1 loop 2, 2 parton final state
• 1 loop, 3 parton final states
• or 2 1 final state
• tree, 4 parton final states
• or 3 1 parton final states
• or 2 2 parton final state

the collinear and soft singularities exactly
cancel between the N 1 and N 1-loop
contributions
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• rapid progress in last two years many authors
• many 2?2 scattering processes with up to one
  off-shell leg now calculated at two loops
• to be combined with the tree-level 2?4, the
  one-loop 2?3 and the self-interference of the
  one-loop 2?2 to yield physical NNLO cross
  sections
• the key is to identify and calculate the
  subtraction terms which add and subtract to
  render the loop (analytically) and real emission
  (numerically) contributions finite
• this is still some way away but lots of ideas so
  expect progress soon!

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summary of NNLO calculations (1990 ?)

• DIS pol. and unpol. structure function
  coefficient functions
• Sum Rules (GLS, Bj, )
• DGLAP splitting functions Moch Vermaseren Vogt
  (2004)
• total hadronic cross section, and Z ? hadrons, ?
  ? ? hadrons
• heavy quark pair production near threshold
• CF3 part of ?(3 jet) Gehrmann-De Ridder,
  Gehrmann, Glover(2004)
• inclusive W,Z,? van Neerven et al, Harlander
  and Kilgore corrected (2002)
• inclusive ? polarised Ravindran, Smith, Van
  Neerven (2003)
• W,Z,? differential rapidity disn Anastasiou,
  Dixon, Melnikov, Petriello (2003)
• H0, A0 Harlander and Kilgore Anastasiou and
  Melnikov Ravindran, Smith, Van Neerven (2002-3)
• WH, ZH Brein, Djouadi, Harlander (2003)
• QQ onium and Qq meson decay rates

ep
ee-
pp
HQ
other partial/approximate results (e.g. soft,
collinear) and NNLL improvements
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Note need to know splitting and coefficient
functions to the same perturbative order to
ensure that ??(n)/?log?F O(aS(n1))
The calculation of the complete set of P(2)
splitting functions completes the calculational
tools for a consistent NNLO pQCD treatment of
Tevatron LHC hard-scattering cross sections!
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Full 3-loop (NNLO) DGLAP splitting functions!
Pba
previous estimates based on known moments and
leading behaviours
Moch, Vermaseren and Vogt, hep-ph/0403192,
hep-ph/0404111
Moch
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Moch, Vermaseren and Vogt, hep-ph/0403192,
hep-ph/0404111
7 pages later
then 8 pages of the same quantities expressed in
x-space!
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NNLO phenomenology already under way

• s(W) and s(Z) precision predictions and
  measurements at the Tevatron and LHC
• the pQCD series appears to be under control
• with sufficient theoretical precision, these
  standard candle processes could be used to
  measure the machine luminosity

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resummation
3
Z
Work continues to refine the predictions for
Sudakov processes, e.g. for the Higgs or Z
transverse momentum distribution, where
resummation of large logarithms of the form ?n,m
aSn log(M2/qT2)m is necessary at small qT, to
be matched with fixed-order QCD at large
qT (also event shapes, heavy quark prodn.)

De Florian Marchesini
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resummation contd. - HO corrections to ?(Higgs)

• the HO pQCD corrections to ?(gg?H) are large
  (more diagrams, more colour)
• can improve NNLO precision slightly by resumming
  additional soft/collinear higher-order logarithms
• example s(MH120 GeV) _at_ LHC
• ?spdf ? 3
• ?sptNNL0 ? 10, ?sptNNLL ? 8
• ? ?stheory ? 9

Catani et al, hep-ph/0306211
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dawn of a new calculational era?
4

• (numerical) calculation of QCD tree-level
  scattering amplitudes can be automated but
  method is brute force, and multiparton
  complexity soon saturates computer capability
• no automation in sight for loop amplitudes
• analytic expressions are very lengthy (recall
  P(2))
• a recent paper by Cachazo, Svrcek and Witten may
  be the long-awaited breakthrough

Bern
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slide from Zvi Bern
gg ? ggg
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the Parke-Taylor amplitude mystery

• consider a n-gluon scattering
• amplitude with ? helicity labels
• Parke and Taylor (PRL 56 (1986) 2459)
• this result is an educated guess
• we do not expect such a simple expression for
  the other helicity amplitudes
• we challenge the string theorists to prove more
  rigorously that it is correct
• Witten, December 2003 (hep-th/0312171)
• Perturbative gauge theory as a string theory
  in twistor space

r
s
Maximum Helicity Violating

(colour factors suppressed)
true!
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Cachazo, Svrcek, Witten (CSW)
April 2004, hep-th/0403047

• elevate MHV scattering amplitudes to effective
  vertices in a new scalar graph approach
• and use them with scalar propagators to calculate
  
• tree-level non-MHV amplitudes
• with both quarks and gluons
• and loop diagrams!
• dramatic simplification compact
• output in terms of familiar spinor
• products
• phenomenology? multijet cross sections at LHC etc

Georgiou, Khoze Zhu Wu, Zhu Bena, Bern,
Kosower Georgiou, Khoze, Glover Kosower
Brandhuber, Spence, Travaglini Bern, del Duca,
Dixon, Kosower
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the other frontier
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compare

• p p ? H X
• the rate (?parton, pdfs, aS)
• the kinematic distribtns. (d?/dydpT)
• the environment (jets, underlying event,
  backgrounds, )

with

• p p ? p H p
• a real challenge for theory (pQCD npQCD) and
  experiment (rapidity gaps, forward protons, ..)

b
b
The most sophisticated calculations and input
from many other experiments are needed to
properly address these issues!
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rapidity gap collision events
typical jet event
hard double pomeron
hard color singlet exchange
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anything that couples to gluons
p p ? p ? H ? p at LHC

• For example Khoze, Martin, Ryskin
• (hep-ph/0210094)
• MH 120 GeV, L 30 fb-1 , ?Mmiss 1 GeV
• Nsig 11, Nbkgd 4 ? 3s effect ?!
  
• Need to calculate production amplitude and gap
  Survival Factor mix of pQCD and npQCD ?
  significant uncertainties
• BUT calibration possible via X quarkonia, large
  ET jet pair, ??, etc. at Tevatron

X
selection rules
mass resolution is crucial! Royon et al
S/B
QCD challenge to refine and test calculations
elevate to precision predictions!
mass resolution
Gallinaro Royon
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summary

• QCD theory major advances in the past year,
  with promise of more to come
• pQCD calculations at the NNLO/NNLL frontier (e.g.
  jet cross sections in pp, ee-), but many NLO
  background calculations still missing
• CSW a new approach still in its infancy (4
  months!), but with major potential
• away from hard inclusive, there are many
  calculational challenges (semi-hard, power
  corrections, exclusive, diffractive, ) close
  collaboration with experiment is essential

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extra slides
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• comparison of resummed / fixed-order
  calculations for Higgs (MH 125 GeV) qT
  distribution at LHC
• Balazs et al, hep-ph/0403052
• differences due mainly to different NnLO and
  NnLL contributions included
• Tevatron d?(Z)/dqT provides good test of
  calculations

log scale
linear scale
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technical details
fictitious scalar-gluon vertex
number of diagrams (QGRAF)
etc.

• L log(Q2/?2)
• F A L3 B L2 C L D
• P(2) contained
  in this term

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World Summary of MW from LEPEWWG, Summer 2004
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the interplay of electroweak and QCD precision
physics
2

• role of aS in global electroweak fit
• hadronic contributions to muon g-2
• use of ?(W) and ?(Z) as standard candles to
  measure luminosity at LHC
• inclusion of O(a) QED effects in DGLAP evolution
• effect of hadronic structure on extraction of
  sin2?W from ?N scattering

Hoecker Vainshtein
Ward
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QED effects in pdfs

• QED corrections to DIS include
• ? mass singularity a log(Q2/mq2) when ?q
• these corrections are universal and can be
  absorbed into pdfs, exactly as for QCD
  singularities, leaving finite (as mq ? 0) O(a)
  QED corrections in coefficient functions
• relevant for electroweak correction
• calculations for processes at Tevatron
• LHC, e.g. W, Z, WH,
• (see e.g. U. Baur et al, PRD 59 (2003) 013002)

included in standard radiative correction
packages (HECTOR, HERACLES)
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QED-improved DGLAP equations

• at leading order in a and aS

where

• momentum conservation

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• effect on quark distributions negligible at small
  x where gluon contribution dominates DGLAP
  evolution
• at large x, effect only becomes noticeable (order
  percent) at very large Q2, where it is equivalent
  to a shift in aS of ?aS ? 0.0003
• dynamic generation of photon parton distribution
• isospin violation up(x) ? dn(x)
• first consistent global pdf fit with QED
  corrections included (MRST 2004)

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perturbative generation of s(x) ? s(x)
Pus(x) ? Pus(x) at O(aS3)

• Quantitative study by de Florian et al
• hep-ph/0404240
• ?x(s-s)?pQCD lt? ?0.0005
• cf. from global pdf fit
• (Olness et al, hep-ph/0312322,3)
• ? ?0.001 lt ?x(s-s)?fit lt 0.004

note!
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sin2?W from ?N
3 ? difference
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Conclusion uncertainties in detailed parton
structure are substantial on the scale of the
precision of the NuTeV data consistency with
the Standard Model does not appear to be ruled
out at present
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• For example Higgs at LHC (Khoze, Martin, Ryskin
  hep-ph/0210094)
• MH 120 GeV, L 30 fb-1 , ?Mmiss 1 GeV
• Nsig 11, Nbkgd 4 ? 3s effect ?!
  
• need to calculate production amplitude and gap
  Survival Factor ? big uncertainties
• BUT calibration possible via X quarkonia or
  large ET jet pair, e.g. CDF observation of
• p p ? p ?0c (?J/? ?) p
• ?excl (J/? ?) lt 49 18 (stat) 39 (syst) pb
• cf. ?thy 70 pb (Khoze et al 2004)

mass resolution is crucial! Royon et al
QCD challenge to refine and test such models
elevate to precision predictions!
Gallinaro Royon
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NNLO corrections to Drell-Yan cross sections

• in DY, sizeable HO pQCD corrections since aS
  (M??) not so small
• for s(W), s(Z) at Tevatron and LHC, allows QCD
  prediction to be matched for (finite)
  experimental acceptance in boson rapidity

Anastasiou et al. hep-ph/0306192 hep-ph/0312266
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top quark production
awaits full NNLO pQCD calculation NNLO NnLL
softvirtual approximations exist (Cacciari et
al, Kidonakis et al), probably OK for Tevatron at
?10 level (gt ?spdf )
but such approximations work less well at LHC
energies
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HEPCODE a comprehensive list of publicly
available cross-section codes for high-energy
collider processes, with links to source or
contact person

• Different code types, e.g.
• tree-level generic (e.g. MADEVENT)
• NLO in QCD for specific processes (e.g. MCFM)
• fixed-order/PS hybrids (e.g. MC_at_NLO)
• parton shower (e.g. HERWIG)

www.ippp.dur.ac.uk/HEPCODE/
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pdfs from global fits
Formalism NLO DGLAP MSbar factorisation Q02 functi
onal form _at_ Q02 sea quark (a)symmetry etc.
fi (x,Q2) ? ? fi (x,Q2)
aS(MZ )
Data DIS (SLAC, BCDMS, NMC, E665, CCFR, H1,
ZEUS, ) Drell-Yan (E605, E772, E866, ) High
ET jets (CDF, D0) W rapidity asymmetry (CDF) ?N
dimuon (CCFR, NuTeV) etc.
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(MRST) parton distributions in the proton
Martin, Roberts, S, Thorne
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uncertainty in gluon distribution (CTEQ)
then ?fg ? ?sgg?X etc.
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solid LHC dashed Tevatron Alekhin 2002
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(No Transcript)
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Higgs cross section dependence on pdfs
Djouadi Ferrag, hep-ph/0310209
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Djouadi Ferrag, hep-ph/0310209
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the differences between pdf sets needs to be
better understood!
Djouadi Ferrag, hep-ph/0310209
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why do best fit pdfs and errors differ?

• different data sets in fit
• different subselection of data
• different treatment of exp. sys. errors
• different choice of
• tolerance to define ? ? fi (CTEQ ??2100,
  Alekhin ??21)
• factorisation/renormalisation scheme/scale
• Q02
• parametric form Axa(1-x)b.. etc
• aS
• treatment of heavy flavours
• theoretical assumptions about x?0,1 behaviour
• theoretical assumptions about sea flavour
  symmetry
• evolution and cross section codes (removable
  differences!)

? see ongoing HERA-LHC Workshop PDF Working Group
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at hadron colliders
the QCD factorization theorem for hard-scattering
(short-distance) inclusive processes
where XW, Z, H, high-ET jets, SUSY sparticles,
black hole, , and Q is the hard scale (e.g.
MX), usually ?F ?R Q, and ? is known

DGLAP equations
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F2(x,Q2) ?q eq2 x q(x,Q2) etc
x dependence of fi(x,Q2) determined by global
fit to deep inelastic scattering (H1, ZEUS, NMC,
) and hadron collider data
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Scattering processes at high energy hadron
colliders can be classified as either HARD or
SOFT Quantum Chromodynamics (QCD) is the
underlying theory for all such processes, but the
approach (and the level of understanding) is very
different for the two cases For HARD processes,
e.g. W or high-ET jet production, the rates and
event properties can be predicted with some
precision using perturbation theory For SOFT
processes, e.g. the total cross section or
diffractive processes, the rates and properties
are dominated by non-perturbative QCD effects,
which are much less well understood
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the QCD factorization theorem for hard-scattering
(short-distance) inclusive processes

proton
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momentum fractions x1 and x2 determined by mass
and rapidity of X x dependence of fi(x,Q2)
determined by global fit (MRST, CTEQ, ) to
deep inelastic scattering (H1, ZEUS, ) data, Q2
dependence determined by DGLAP equations
F2(x,Q2) ?q eq2 x q(x,Q2) etc
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what limits the precision of the predictions?

• the order of the perturbative expansion
• the uncertainty in the input parton distribution
  functions
• example s(Z) _at_ LHC
• ?spdf ? 3, ?spt ? 2
• ? ?stheory ? 4
• whereas for gg?H
• ?spdf ltlt ?spt

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pdfs at LHC

• high precision (SM and BSM) cross section
  predictions require precision pdfs ??th ??pdf
  
• standard candle processes (e.g. ?Z) to
• check formalism
• measure machine luminosity?
• learning more about pdfs from LHC measurements
  (e.g. high-ET jets ? gluon, W/W ? sea quarks)

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Full 3-loop (NNLO) non-singlet DGLAP splitting
function!
Moch, Vermaseren and Vogt, hep-ph/0403192
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s(W) and s(Z) precision predictions and
measurements at the LHC
LHC sNLO(W) (nb)
MRST2002 204 4 (expt)
CTEQ6 205 8 (expt)
Alekhin02 215 6 (tot)
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ratio of W and W rapidity distributions
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pdfs from global fits
Formalism LO, NLO, NNLO DGLAP MSbar
factorisation Q02 functional form _at_ Q02 sea quark
(a)symmetry etc.
fi (x,Q2) ? ? fi (x,Q2)
aS(MZ )
Data DIS (SLAC, BCDMS, NMC, E665, CCFR, H1,
ZEUS, ) Drell-Yan (E605, E772, E866, ) High
ET jets (CDF, D0) W rapidity asymmetry (CDF) ?N
dimuon (CCFR, NuTeV) etc.
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summary of DIS data
neutrino FT DIS data
Note must impose cuts on DIS data to ensure
validity of leading-twist DGLAP formalism in the
global analysis, typically Q2 gt 2 - 4 GeV2 W2
(1-x)/x Q2 gt 10 - 15 GeV2
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typical data ingredients of a global pdf fit
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HEPDATA pdf server

• Comprehensive repository of past and present
  polarised and unpolarised pdf codes (with online
  plotting facility) can be found at the HEPDATA
  pdf server web site
• http//durpdg.dur.ac.uk/hepdata/pdf.html
• this is also the home of the LHAPDF project

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(MRST) parton distributions in the proton
Martin, Roberts, S, Thorne