Transcript Document

October 7th, 2008
1st Global QCD Analysis
of Polarized Parton Densities
Marco Stratmann
work done in collaboration with
Daniel de Florian
Rodolfo Sassot
(Buenos Aires)
(Buenos Aires)
Werner Vogelsang (BNL)
references
 Global analysis of helicity parton densities and their uncertainties,
PRL 101 (2008) 072001 (arXiv:0804.0422 [hep-ph])
 a long, detailled paper focussing on uncertainties is in preparation
DSSV pdfs and further information available from ribf.riken.jp/~marco/DSSV
2
the challenge:
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
3
information on nucleon spin structure available from
 each reaction provides insights into different aspects and x-ranges
 all processes tied together: universality of pdfs & Q2 - evolution
 need to use NLO
task: extract reliable pdfs not just compare some curves to data
4
details & results of
the DSSV global analysis
 toolbox
 comparison with data
 uncertainties from Lagrange multipliers
 comparison with Hessian method
 next steps
5
1. 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
 factorization
e.g., pp ! p X
allows to separate universal PDFs from
calculable but process-dependent
hard scatterring cross sections
Jäger,MS,Vogelsang
 higher order corrections
essential to estimate/control
theoretical uncertainties
closer to experiment (jets,…)
scale uncertainty
all relevant observables available at NLO accuracy
except for hadron-pair production at COMPASS, HERMES
Q2' 0 available very soon: Hendlmeier, MS, Schafer
7
2. “data selection”
initial step: verify that the theoretical framework is adequate !
! use only data where unpolarized results agree with NLO pQCD
DSSV global analysis uses all three sources of data:
“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
! Dg
(never used before)
467 data pts in total (¼10% from RHIC)
8
data with observed hadrons
• SIDIS
• pp ! p X
(HERMES, COMPASS, SMC)
(PHENIX)
strongly rely on fragmentation functions
! new DSS FFs are a crucial input to the DSSV PDF fit
DSS analysis:
(de Florian, Sassot, MS)
 first global fit of FFs including e+e-, ep, and pp data
 describe all RHIC cross sections and HERMES SIDIS multiplicities
(other FFs (KKP, Kretzer) do not reproduce, e.g., HERMES data)
 uncertainties on FFs from robust Lagrange multiplier method
and propagated to DSSV PDF fit !
details:
 Global analysis of fragmentation functions for pions and kaons
and their uncertainties, Phys. Rev. D75 (2007) 114010 (hep-ph/0703242)
 Global analysis of fragmentation functions for protons
and charged hadrons, Phys. Rev. D76 (2007) 074033 (arXiv:0707.1506 [hep-ph])
9
3. setup of DSSV analysis
• flexible, MRST-like input form
possible nodes
input scale
simplified form for sea quarks and Dg: kj = 0
• take as from MRST; also use MRST for positivity bounds
• NLO fit, MS scheme
• avoid assumptions on parameters {aj} unless data cannot discriminate
need to impose:
let the fit decide about F,D value constraint on 1st moments:
1.269§0.003
0.586§0.031
fitted
10
4. fit procedure
change O(20) parameters
{aj} about 5000 times
467 data pts
another 50000+ calls for
studies of uncertainties
bottleneck !
computing time for a global analysis at NLO becomes excessive
problem: NLO expression for pp observables are very complicated
11
! problem can be solved with the
help of 19th century math
idea: take Mellin n-moments
R.H. Mellin
inverse
Finnish mathematician
well-known property: convolutions factorize into simple products
 analytic solution of DGLAP evolution equations for moments
 analytic expressions for DIS and SIDIS coefficient functions
… however, NLO expression for pp processes too complicated
12
here is how it works:
MS, Vogelsang
earlier ideas: Berger, Graudenz, Hampel, Vogt; Kosower
example: pp! p X
express pdfs by their
Mellin inverses
standard
Mellin inverse
fit
completely indep. of pdfs
pre-calculate prior to fit
discretize on 64 £ 64 grid
for fast Gaussian integration
13
applicability & performance
 computing
load
O(10 sec)/data pt.
!
O(1 msec)/data pt.
recall: need thousands of calls to perform a single fit !
 obtaining
the grids
once prior to the fit
64 £ 64 £ 4 £ 10 ' O(105) calls per pp data pt.
n
m
production of grids much improved recently
can be all done within a day with new MC sampling techniques
 method
completely general
tested for pp!gX, pp!pX, pp!jetX
(much progress towards 2-jet production expected from STAR)
14
details & results of
the DSSV global analysis
 toolbox
 comparison with data
 uncertainties from Lagrange multipliers
 comparison with Hessian method
 next steps
15
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
16
inclusive DIS data
data sets used in:
new
the old GRSV analysis
the combined DIS/SIDIS fit of DNS
17
remark on higher twist corrections
 we only account for the “kinematical mismatch” between A1 and g1/F1
in
(relevant mainly for JLab data)
 no need for additional higher twist corrections
(like in Blumlein & Bottcher)
at variance with results of LSS (Leader, Sidorov, Stamenov) – why?
 very restrictive functional form in LSS: Df = N ¢ x a ¢ fMRST
only 6 parameters for pdfs but 10 for HT
 very limited Q2 – range ! cannot really distinguish ln Q2 from 1/Q2
 relevance of CLAS data “inflated” in LSS analysis:
633 data pts. in LSS vs. 20 data pts. in DSSV
in a perfect world this should not matter, but …
18
semi-inclusive DIS data
not in DNS
analysis
impact of new
FFs noticeable!
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
details & results of
the DSSV global analysis
 toolbox
 comparison with data
 uncertainties from Lagrange multipliers
 comparison with Hessian method
 next steps
21
Lagrange multiplier method
see how fit deteriorates when PDFs are forced
to give a different prediction for observable Oi
 finds largest DOi allowed by the global data set
track c2
and theoretical framework for a given Dc2
 explores the full parameter space {aj}
independent of approximations
 Oi can be anything: we have looked at ALL, truncated 1st moments,
and selected fit parameters aj so far
 requires large series of minimizations (not an issue with fast Mellin technique)
22
Dc2
- a question of tolerance
see: CTEQ, MRST, …
What value of Dc2 defines a reasonable error on the PDFs ?
certainly a debatable/controversial issue …
• combining a large number of diverse exp. and theor. inputs
• theor. errors are correlated and by definition poorly known
• in unpol. global fits data sets are marginally compatible at Dc2 = 1
! idealistic Dc2=1 $ 1s approach usually fails
we present uncertainties bands for both Dc2 = 1 and
a more pragmatic 2% increase in c2
also: • Dc2 = 1 defines 1s uncertainty for single parameters
• Dc2 ' Npar is the 1s uncertainty for all Npar parameters
to be simultaneously located in “c2-hypercontour”
used by AAC
23
summary of DSSV distributions:
 robust pattern of flavor-asymmetric
light quark-sea (even within uncertainties)
 small Dg, perhaps with a node
 Du + Du and Dd + Dd very
similar to GRSV/DNS results
 Ds positive at large x
 Du > 0, Dd < 0
predicted in some
models
Diakonov et al.; Goeke
et al.; Gluck, Reya;
Bourrely, Soffer, …
24
a closer look at Du
Dc2
Dc2
x
 small, mainly positive
 negative at large x
 determined by SIDIS data
 mainly charged hadrons
 pions consistent
25
a closer look at Ds
Dc2
striking result!
Dc2
x
 positive at large x
 negative at small x
 determined by SIDIS data
 DIS alone: more negative
 mainly from kaons,
a little bit from pions
26
a closer look at Dg
error estimates more delicate: small-x behavior completely unconstrained
study uncertainties in 3 x-regions
find
 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
x
small-x
RHIC
range
large-x
0.001· x · 0.05
x ¸ 0.2
0.05· x · 0.2
27
1st moments:
Q2 = 10 GeV2
 Ds receives a large negative
contribution at small x
 Dg: huge uncertainties
below x'0.01 ! 1st moment
still undetermined
 eSU(2),SU(3) come out close
to zero
eSU(2)
eSU(3)
28
details & results of
the DSSV global analysis
 toolbox
 comparison with data
 uncertainties from Lagrange multipliers
 comparison with Hessian method
 next steps
29
Hessian method
estimates uncertainties by exploring c2 near minimum:
displacement:
Hessian Hij
taken at minimum
only quadratic
approximation
 easy to use (implemented in MINUIT) but not necessarily very robust
 Hessian matrix difficult to compute with sufficient accuracy
in complex problems like PDF fits where eigenvalues span a huge range
can benefit from a lot of pioneering work by CTEQ
good news:
and use their improved iterative algorithm to compute Hij
J. Pumplin et al., PRD65(2001)014011
30
PDF eigenvector basis sets SK§
cartoon by CTEQ
• eigenvectors provide an optimized orthonormal basis
to parametrize PDFs near the global minimum
• construct 2Npar eigenvector basis sets Sk§ by displacing each zk by § 1
• the “coordinates” are rescaled such that Dc2 = k zk2
• sets Sk§ can be used to calculate uncertainties of observables Oi
31
comparison with uncertainties from Lagrange multipliers
 tend to be a bit larger
for Hessian, in particular
for Dg(x)
 Hessian method goes
crazy if asking for Dc2>1
 uncertainties of truncated
moments for Dc2=1 agree
well except for Dg
32
details & results of
the DSSV global analysis
 toolbox
 comparison with data
 uncertainties from Lagrange multipliers
 comparison with Hessian method
 next steps
33
1. 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)
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
34
planning ahead: at 500GeV the W-boson program starts
 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
 would be interesting to study impact with some simulated data soon
2. 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
35
36
extra slides
37
DSS: good global fit of all e+e- and ep, pp data
main features:
de Florian, Sassot, MS
• handle on gluon fragmentation
• flavor separation
• uncertainties via Lagrange multipl.
• results for p§, K§, chg. hadrons
38
meet the distributions: Dd
Dc2
Dc2
x
 fairly large
 negative throughout
 determined by SIDIS data
 some tension between
charged hadrons and pions
39
c2 profiles of eigenvector directions
#1: largest eigenvector
(steep direction in c2)
…
#19: smallest eigenvector
(shallow direction in c2)
significant deviations
from assumed
quadratic dependence
for a somewhat simplified
DSSV fit with 19 parameters
40
worse for fit parameters: mix with all e.v. (steep & shallow)
look O.K.
Dg
mixed bag
but not
necessarily
parabolic
steep
shallow
41
roughly corresponds to what we get from Lagrange multipliers
the good …
… the bad
… the ugly
42