1. Global QCD analysis of polarized PDFs status & prospects
October 11th, 2007
IWHSS 09
International Workshop on
Hadron Structure and Spectroscopy
March 30 – April 1, 2009
Schloss Waldthausen, Mainz (Germany)
Global QCD analysis of polarized PDFs status & prospects
Marco Stratmann
in collaboration with
Daniel de Florian, Rodolfo Sassot,Werner Vogelsang

2. How to determine PDFs from data?
information on nucleon (spin) structure available from
DIS SIDIS hadron-hadron
task: extract reliable pdfs not just compare some curves to data
each reaction provides insights into different aspects and kinematics
all processes tied together: universality of pdfs & Q2 - evolution
need at least NLO for quantitative analyses; PDFs are not observables!
information on PDFs “hidden” inside complicated (multi-)convolutions
! a “global QCD analysis” is required

3. PDFs data the charge: analyze a large body of data
from many experiments on different processes
with diverse characteristics and errors
within a theoretical model with many parameters
and hard to quantify uncertainties
without knowing the optimum “ansatz” a priori

4. details & results of the DSSV global analysis toolbox
comparison with data
uncertainties: Lagrange multipliers vs. Hessian
emerging picture
next steps
Global analysis of helicity parton densities and their uncertainties,
PRL 101 (2008) (arXiv: [hep-ph])

5. theory “toolbox” QCD scale evolution
due to resolving more and more parton-parton splittings
as the “resolution” scale m increases
the relevant DGLAP evolution kernels are known to NLO accuracy:
Mertig, van Neerven;
Vogelsang
m - dependence of PDFs is a key prediction of pQCD
verifying it is one of the goals of a global analysis

6. allows to separate universal PDFs from
factorization
e.g., pp ! p X
allows to separate universal PDFs from
calculable but process-dependent
hard scatterring cross sections
higher order QCD corrections
essential to estimate/control
theoretical uncertainties
closer to experiment (jets,…)
do not cancel in ALL
scale uncertainty
Jäger,MS,Vogelsang
all relevant observables available at NLO accuracy
except for charm/hadron-pair production at COMPASS, HERMES
recent progress for Q2 ' 0 ! later

7. outline of a global QCD analysis
start: choose fact. scheme (MS,…) & pert. order (NLO, …), select data sets, cuts, …
parametrize quark and gluon PDFs
a la Df(x,m0) ' xa (1-x)b at
some initial scale m0 ' 1 GeV
obtain PDFs at any x, m > m0
relevant for comparing with data
compute DIS, pp, …cross sections
judge goodness of current fit:
optimum set of parameters {ai, bi, …}
recent achievement: also quantify PDF uncertainties and properly
propagate them to any observable of interest

8. global analysis: a computational challenge
one has to deal with O(500) data points from many processes and experiments
need to determine O(20) parameters describing PDFs at m0
NLO expressions often very complicated ! computing time becomes excessive
! develop sophisticated algorithms & techniques, e.g., based on Mellin moments
Kosower; Vogt; Vogelsang, MS
DSSV global analysis uses:
“classic” inclusive DIS data
routinely used in PDF fits
! Dq + Dq
semi-inclusive DIS data
so far only used in DNS fit
! flavor separation
first RHIC pp data (never used before)
! Dg
467 data pts in total (¼10% from RHIC)

9. interlude: fragmentation functions
crucial for pQCD interpretation (factorization!)
of data with detected hadrons, e.g.,
SIDIS (HERMES, COMPASS), pp! pX (PHENIX, …)
no local OPE ! no lattice formulation
bi-local operator:
Collins, Soper ’81, ’83
some properties of Dih(z,m) [very similar to PDFs]:
non-perturbative but universal; pQCD predicts m–dep.
describe the collinear transition of a parton “i” into
a massless hadron “h” carrying fractional momentum z
quark/gluon
hadron
z k k
“leading particle” picture incompatible with inclusive definition of Dih
global analysis & uncertainty estimates are a recent achievement
DSS fit (de Florian, Sassot, MS)
Phys. Rev. D75 (2007)
Phys. Rev. D76 (2007)

10. DSS: good global fit of all e+e-, ep, and pp hadron data
main results:
de Florian, Sassot, MS
results for p§, K§, chg. hadrons
full flavor separation for DiH(z) and DgH
uncertainties (L.M.) well under control
fits all LEP, HERMES, SMC, RHIC, … data
supersede old fits based only on e+e- data

11. How can we use all this in a global PDF fit?
several crucial differences w.r.t. an unpolarized fit:
no sum rule which relates quarks and gluons (unpolarized: momentum sum)
Df(x,m) not restricted to be positive; nodes possible
“positivity bound” |Df(x,m)| · f(x,m) of limited use (valid only at LO !)
much less data:
DIS in limited x,Q2 range ! much less constrained gluon
no nN-DIS data ! flavor separation relies on SIDIS data
possible uncertainties from fragmentation
pp data have to constrain Dg
(more complicated to analyze than DIS scaling violations)

12. details & results of the DSSV global analysis toolbox
comparison with data
uncertainties: Lagrange multipliers vs. Hessian
the emerging picture
next steps

13. setup of DSSV analysis flexible, MRST-like input form
possible nodes
input scale
simplified form for sea quarks and Dg: kj = 0
avoid assumptions on parameters {aj} unless data cannot discriminate
take as from MRST; also use MRST for positivity bounds
NLO fit, MS scheme
need to impose:
let the fit decide about F,D value constraint on 1st moments:
1.269§0.003
fitted (end up close to zero)
0.586§0.031

14. overall quality of the global fit
very good!
no significant tension
among different data sets
c2/d.o.f. ' 0.88
note: for the time being,
stat. and syst. errors
are added in quadrature

15. spin asymmetries in inclusive DIS
[DNS: old analysis by
de Florian, Navarro, Sassot]
we account for kinematical “mismatches” in
no need for any dynamical higher twist (contrary to Leader et al.)

16. spin asymmetries in semi-inclusive DIS
impact of new
FFs noticeable!

17. detour: DSS kaon FF’s DiK(z)
RHIC pp data (BRAHMS,
STAR) explain different Dg
smaller u & larger s-frag.
required by SIDIS
note: some issues with K- data (slope!)
await eagerly final HERMES data

18. gluons are key players at RHIC
many QCD processes with a
dominant gluon contribution
already at the tree-level:
high-pT jet, pion, heavy quark, …
unpolarized “reference data” (p, jets, g) nicely agree with pQCD
Jäger,MS,Vogelsang;
Signer et al.
Jäger,Schäfer,MS,
Vogelsang; de Florian
Bojak,MS;
Riedl,Schäfer,MS
Gordon,Vogelsang;
Contogouris et al.
all available at NLO
decisive data start to emerge from RHIC …

19. RHIC pp data (inclusive p0 or jet)
good agreement
important constraint
on Dg(x) despite
large uncertainties
! later
uncertainty bands estimated
with Lagrange multipliers by
enforcing other values for ALL

20. Dg in lepton-proton scattering
gluons in DIS: a (small) NLO effect [they don’t couple directly to the photon]
! study processes sensitive to photon-gluon-fusion
final states explored (data !!): one/two hadron production, charm
COMPASS, HERMES
Q2 large: “electroproduction” Q2
theory calculations more challenging than in pp: +
if Q2' 0: “photoproduction”
unknown
photon structure
NLO
pQCD
1-hadron: X Jäger,MS,Vogelsang
charm: X Bojak, MS (direct g)
Riedl, Schäfer, MS (resolved g & MC)
hadron pairs: Hendlmeier, Schäfer, MS (direct g)
Jäger,Owens,MS,Vogelsang (resolved g)
nothing
for Q2 ¹ 0
almost
done

21. DSSV gluon agrees well with model-dependent “LO” extractions of Dg/g
not in global fit
[NLO not available]
a future global NLO fit will use measured ALL not derived Dg/g
need to check unpolarized cross section as well

22. details & results of the DSSV global analysis toolbox
comparison with data
uncertainties: Lagrange multipliers vs. Hessian
emerging picture
next steps

23. estimating PDF uncertainties
mainly two methods in use:
[reshaped for PDF analyses by J. Pumplin and CTEQ]
Hessian method: classic tool, explores vicinity of c2-minimum in
quadratic approx.; often unstable for multi-parameter PDF analyses
track c2
Lagrange multiplier: track how the fit deteriorates
when PDFs are forced to give different predictions
for selected observables; explores the full
paramater space indep. of approximations
issue: what value of Dc2 (tolerance) defines a 1-s error?
non Gaussian errors, c2 “landscape” not parabolic
uncertainties with diverse characteristics
theor. errors correlated and poorly known
data sets often marginally consistent for Dc2=1
we present uncertainties bands
for both Dc2 = 1 and
a more pragmatic 2% increase in c2

24. Hessian eigenvector PDF basis sets
cartoon by CTEQ
eigenvectors provide an optimized orthonormal basis near the minimum
construct 2Npar eigenvector basis sets Sk§ by displacing each zk by § 1
the “coordinates” are rescaled such that Dc2 = åk zk2
sets Sk§ can be used to calculate uncertainties of observables Oi
we will make 40 DSSV eigenvector sets available very soon

25. details & results of the DSSV global analysis toolbox
comparison with data
uncertainties: Lagrange multipliers vs. Hessian
emerging picture
next steps

26. DSSV sea polarizations
indications for an SU(2) breaking of light u,d sea
breaking of similar size than in unpol. case
mainly determined by SIDIS data
“bands”: error estimate from Lagr. mult.
similar patterns in many models:
large-NC, chiral quark soliton, meson cloud
Thomas, Signal, Cao; Diakonov, Polyakov, Weiss; …
c2 profiles for truncated moments:

27. DSSV sea polarizations – cont’dx
a strange strangeness polarization
Ds(x) always thought to be negative, but …
mainly determined from SIDIS kaon data
consistent with LO-type analyses by
HERMES and COMPASS
needs further studies – exp. & theory !
striking result, but relies on
kaon fragmentation
more data available soon (BELLE, …)
unpolarized PDFs
unpol. strangeness not well determined

28. similar Ds results in “LO” analyses by HERMES and COMPASS:
s DS dx = 0.037 0.019(stat.)  0.027(syst.)
COMPASS
prel. from SPIN’08

29. of interest not only for nucleon structure enthusiasts:
e.g. elastic scattering of SUSY dark matter
arXiv:

30. DSSV gluon polarization
error estimates more delicate: small-x behavior completely unconstrained x
study uncertainties in 3 x-regions
RHIC
range
0.05· x · 0.2
small-x
0.001· x · 0.05
large-x
x ¸ 0.2
Dg(x) very small at medium x
(even compared to GRSV or DNS)
best fit has a node at x ' 0.1
huge uncertainties at small x
find

31. prospects on Ds
final HERMES data sets for SIDIS & DIS multiplicities crucial;
more from COMPASS; can we distinguish Ds and Ds in the future?
notoriously difficult in pp:
two channels: W+charm (extremely rare probe)
polarized L production
issues to be addressed for L production:
reliable NLO sets of DiL and DDiL
DSV: de Florian, MS, Vogelsang, PRD 57 (1998) 5811
updated global analysis required, Dg too small (STAR data)
AKK: Albino et al., arXiv: v2
DSV: de Florian, MS, Vogelsang, PRD 57 (1998) 5811
sparse data; 3 models considered; update desirable
feed-down from hyperon weak decays; effect on polarization?
compute helicity-transfer subprocesses at NLO (work has started)

32. spin audit: 1st moments and the spin of the proton
“helicity sum rule”
Jaffe, Manohar; Ji; …
A+ = 0 gauge, IMF
partonic interpretation
total u+d+s
quark spin
gluon
spin
angular
momentum
“quotable” properties of the nucleon !
total spin
polarizations
Sq and Sg !
helicity parton densities
momentum fraction
s dx
x-moment 1
nothing is known yet about Lq and Lg
lattice results for “angular momentum” are for a “different” sum rule
(Ji’s version) w/o partonic interpretation – they cannot be mixed!

33. ! case for a high-energy polarized ep-collider
numerical results
Q2 = 10 GeV2
Ds receives a large negative
contribution at small x
Dg: huge uncertainties
below x'0.01 ! 1st moment
still undetermined
very difficult to give reliable estimates for full moments
both quark and gluons may not contribute much to proton spin
but we need to go to smaller x to settle this issue
! case for a high-energy polarized ep-collider

34. details & results of the DSSV global analysis toolbox
comparison with data
uncertainties: Lagrange multipliers vs. Hessian
emerging picture
next steps

35. getting ready to analyze new types of data
from the next long RHIC spin run with O(50pb-1) and 60% polarization
significantly improve existing
inclusive jet + p0 data
(plus p+, p-, …)
first di-jet data from STAR
! more precisely map Dg(x)
X the Mellin technique is
basically in place to analyze
also particle correlations
challenge: much slower MC-type
codes in NLO than for 1-incl.
from 2008 RHIC spin plan

36. planning ahead: the 500GeV RHIC W-boson program just started
flavor separation independent of SIDIS
! important x-check of present knowledge
implementation in global analysis (Mellin technique) still needs to be done
available NLO codes (RHICBos) perhaps too bulky;
new results emerging de Florian, Vogelsang
would be interesting to study impact with some simulated data soon
further improving on uncertainties
Lagrange multipliers more reliable than Hessian with present data
Hessian method perhaps useful for Dc2 = 1 studies, beyond ??
include experimental error correlations if available
work started together with help from the RHIC Spin Collaboration
(aiming at a CTEQ-like collaboration of theory and experiment)

37. conclusions we have just explored the tip of the iceberg
you are here
Dutot, Ddtot
Du, Dd Dg
many avenues for further
important measurements and
theoretical developments Ds
Lq,g
spin sum rule