Parton Density Function Fitting Update

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Parton Density Function Fitting Update

• ZEUS Collaboration Meeting
• 19-June-2003
• Chris Collins-Tooth
• Amanda Cooper-Sarkar, Claire Gwenlan

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Introduction

• How we parameterise the PDFs
• Method for fitting data to obtain the PDFs
• Where does the information come from for the
  fits?
• What can we add/What data do we have to study?
• Which parameters are best/worst determined?
• Comparing new ZEUS O fit with old O fit and
  ZEUS-S
• To Do Ideas for the future

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How we parameterise the PDFs

• At some particular value of Q02 (7 GeV2) we
  parameterise the parton momentum distribution
  with the parameters pi
• xf(x) p1.xp2(1-x)p3(1p5x)
• Gives flexibility at low (p2), high (p3) and mid
  (p5) x.
• Distributions we parameterise are
• xuv(x) u-valence p1u,p2u,p3u,p5u
• xdv(x) d-valence p1d,p2d,p3d,p5d
• xS(x) total sea p1S,p2S,p3S,p5S
• xg(x) gluon p1g,p2g,p3g,p5g
• x? x(d-u) p1?,p2?,p3?,p5?

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• 5 distributions ? 4 parameters 20 possible free
  params.
• Luckily, some we can fix (eg ZEUS-S fit 11 free
  params)
• p1u, p1d fixed through number sum rules
• p1g fixed through momentum sum rule
• p2u, p2d 0.5 little information exists for
  low-x valence after data cuts
• p2? 0.5 , p3?(p3S2), p5?0 as per MRST eg EPJ
  C4, 463(1998)EPJ C14 133(2000)
• p5g 0 since this choice constrains high-x gluon
  to be positive (leaves 11 free)
• Additionally, for the ZEUS-O fit (ZEUS data
  only)
• p1? fixed to value determined by ZEUS-S (10 free
  params)
• The values are evolved in Q2 using NLO DGLAP
  equations, convoluted with coefficient functions
  in Roberts-Thorne Variable Flavour Number scheme
• This list of parameters is fed to an evaluation
  function which calculates Chi-squared based on
  the data and theory PDFs

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Chi-squared definition

• ?2 ?i (Fi(p,s)-Fi(meas))2 ?? s?2
• (?2i,stat ?2i,unc)
• Fi(p,s) FiNLOQCD(p) ?? s? ?i?
• Fi(meas) represents a measured data point
• ?2i,stat and ?2i,unc represent stat. and uncorr.
  syst errors.
• ? are systematic error sources. 1 s.d.
  uncertainty on a data point i, due to source ? ,
  is ?i ?
• s? are independent Gaussians, with zero mean and
  unit variance.
• Accounts for systematic errors AND
  normalisations.
• Applied conservatively by OFFSET method
• see J.Phys.G 28(2002) 2717

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Fitting method

• Parameters s? 0 for central values of fit
• Obtain usual Hessian matrix Mjk 1 ?2?2
• 
  2 ?pj?pk
  
• s? allowed to vary for error analysis
• Obtain 2nd Hessian matrix Cj? 1 ?2?2
• 
  2 ?pj?s?
• Systematic covariance matrix Vsy M-1CCTM-1
• Stat. and uncorr. syst. covariance matrix Vst
  M-1
• Uncertainty on any distribution (eg PDF)
  calculated using Vsy,Vst

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• Errors on the PDF parameters are given by the
  error matrices Vij .
• These are propagated to quantities of interest
  like structure functions, parton densities and
  reduced cross sections via lt ? F2gt ?ij ?F Vij
  ?F
• 
  ?pi ?pj
• Clearly, this is easier if V is diagonalised
• Diagonalisation has various other benefits
• It tells you if you have a stable fit - are the
  eigenvalues all positive?
• It tells you if you actually NEED all the
  parameters you are using
• It tells you which parameters are constrained
  best

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Where does the information come from in the fits?

• ZEUS-S, short for ZEUS-Standard is a fit using
  GLOBAL data. Some ZEUS data (96-97) is included
  in this.
• Valence xF3 x(uvdv) from neutrino-Fe heavy
  target data
• F2n/F2p xdv/xuv at high-x
  from muonD/p data (NMC)
• Sea Low-x from HERA F2 e p data
• High-x predominantly from
  fixed target F2 muonp data
• Flavour structure from muonD
  and p
• Gluon Low-x from HERA dF2/dlnQ2 e p data
• High-x from mom-sum rule only
  (unless we add JET DATA)
• ZEUS-O, short for ZEUS-Only is a fit using ZEUS
  data only.
• Still makes some assumptions from ZEUS-S fit, eg
  p1?.
• The old ZEUS-O fit (94-99 ZEUS data) is available
  - see
• http//www-pnp.physics.ox.ac.uk/cooper/zeus2
  002.html for PDF sets.
• Now, using 99/00 CC/NC ep data with correlated
  systematic error sources this fit is being
  improved.

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What can HERA high Q2 data add?

• TODAY HERA-I
• High-x valence info-particularly on xdv from ep
  CC. No deuterium or heavy target corrections.
• TOMORROW HERA-II
• xF3/xG3 measurement valence info from low to
  high-x
• More accurate xdv, xuv flavour separated from CC
  interactions
• More accurate high-x sea distributions too (and
  better jet data for high-x glue)

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What data do we have to study?

• We have about 100pb-1 per experiment of NC and
  CC data (only 16pb-1 of it e-)
• A new round of NLO QCD fits now to ZEUS data
  alone
• Can use ZEUS high-Q2 data to constrain valence
  distributions independent of nuclear corrections
  needed by fixed target data.
• ZEUS-S published fits used fixed target data and
  only 30pb-1 of NC e data.
• Phys Rev D67, 012007 (2003)

http//durpdg.dur.ac.uk/hepdata/zeus2002.html
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Which parameters are best/worst determined?

• For both ZEUS-S and ZEUS-O fits, p1S and p2S are
  the best determined - ie the low-x behaviour of
  the sea.
• Next best is the p2g - the low-x gluon
• After that, p3u (high-x, u valence) is best
  determined from fixed target data, but high Q2
  ZEUS data is nearly as good.
• p3d, p3S (high-x d valence and sea) and p5u
  (mid-x u valence) are moderately well determined
  by fixed target data, but in new ZEUS-O fit, high
  Q2 ZEUS ep CC data now determines p3d (high-x d
  valence) just as well
• p3g (high-x glue) and p5S,p5g,p5d (mid-x
  sea,glue,d valence) are the least well determined
  in both fits, but are better determined from
  fixed target data.

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ZEUS-S
New ZEUS-O 94-00 -amcs

• At one stage, the ZEUS-S fit was used to
  determine as, by allowing it to vary as a fit
  parameter, and this error (blue) is only shown
  for reference.
• Low-x Sea and Gluon in ZEUS-S fits were strongly
  constrained by ZEUS 96/7 ep data so are well
  constrained using ZEUS data alone

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PUBLISHED ZEUS-O Sea and glue distributions from
ZEUS only with 94-99 data 16pb-1 e- and 30pb-1 e
NEW ZEUS-O 94-00 Sea and glue distributions from
ZEUS only with 94-00 data 16pb-1 e- and 93pb-1 e

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ZEUS-S
NEW ZEUS-O 94-00.

• ZEUS-S constrained valence distributions by using
  fixed target data
• Now the new 99/00 ep data allows high-x valence
  distributions to be well constrained from a fit
  to ZEUS data alone

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NEW ZEUS-O 94-00
PUBLISHED ZEUS-O
Valence distributions from ZEUS data alone 16pb-1
e- data and 37pb-1 e data
Valence distributions from ZEUS data alone 16pb-1
e- data and 100pb-1 e data
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ZEUS-S
NEW ZEUS-O 94-00
NEW ZEUS-O Gluon is almost as well determined as
ZEUS-S gluon for all x
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ZEUS-S
NEW ZEUS-O 94-00
Medium to high x sea is not as well determined
for ZEUS-O as for ZEUS-S
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NEW ZEUS-O 94-00
ZEUS-S
Errors on high-x xuv for ZEUS-O not quite as good
as for ZEUS-S but HERA-II will improve
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NEW ZEUS-O 94-00
ZEUS-S
Errors on High-x xdv from ZEUS-O as good as
ZEUS-S and mostly statistical so that HERA-II
will improve more!
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NEW ZEUS-O amcs
2nd Analysis
A second analysis is underway (by me!) . Results
so far look very promising.
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To do...

• Investigate model dependence further - effect of
  fixing/freeing various parameters e.g.
• valence shapes of ZEUS-O and ZEUS-S do differ
  significantly (and hence differ from MRST/CTEQ)
• Look back and try to understand comparison of
  ZEUS-S fit with latest high Q2 data sets (seems
  higher at x0.2, so this diff could be genuine)
• Aside- you will NOT see any difference between
  ZEUS-O and MRST if you parameterise like H1 have
  done in DESY-03-38 (their H1-only fit). In such a
  plot, a comparison ZEUS/H1 or ZEUS/MRST looks
  quite sensible!
• Ideas
• free low-x valence parameters
• investigate effect of freeing/fixing sea
  parameters on valence shapes- the high-x sea also
  differs significantly from ZEUS-S fit and is not
  well determined in ZEUS-O since we have very
  little data for xgt0.4
• vary assumptions on flavour structure of sea
• Other model dependences not yet thought of!

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ZEUS-O vs MRST
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xU xD xU xD xg at Q210 and 1000 Gev2 compared to
MRST2001 (can compare to H1s plots)
But this can hide a multitude of sins!
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For the new ZEUS-O fit, the following data is
used
96/97 ep NC 30 pb-1 2.7 lt Q2 lt 30000 GeV2 242
d.p. 10 corr..err. 2 norms 94/97 ep CC 33 pb-1
280. lt Q2 lt 30000 GeV2 29 d.p. 3 corr. err.
98/99 e-p NC 16 pb-1 200 lt Q2 lt
30000 GeV2 92 d.p. 6 corr err. 1 norm 98/99
e-p CC 16 pb-1 200 lt Q2 lt 30000 GeV2 26 d.p.
3 corr. err. 99/00 ep NC 63 pb-1 200 lt Q2 lt
30000 GeV2 90 d.p. 9 corr. err. 1 norm 99/00
ep CC 61 pb-1 200 lt Q2 lt 30000 GeV2 29 d.p.
3 corr. err.