1 st g lobal qcd analysis of polarized parton densities marco stratmann october 7th, 2008

1st Global QCD Analysis

of Polarized Parton Densities

Marco Stratmann

October 7th, 2008

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work done in collaboration with

Daniel de Florian (Buenos Aires)

Rodolfo Sassot (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

http://ribf.riken.jp/~marco/DSSV

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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

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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

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details & results of

the DSSV global analysis

toolbox comparison with data uncertainties from Lagrange multipliers comparison with Hessian method next steps

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1. theory “toolbox”

QCD scale evolution

due to resolving more and more parton-parton splittings

as the “resolution” scale increases

the relevant DGLAP evolution kernels are known to NLO accuracy:Mertig, van Neerven; Vogelsang

dependence of PDFs is a key prediction of pQCD

verifying it is one of the goals of a global analysis

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factorization

allows to separate universal PDFs from

calculable but process-dependent

hard scatterring cross sections

e.g., pp ! X

higher order corrections

essential to estimate/control

theoretical uncertainties

closer to experiment (jets,…)

scale uncertainty

Jäger,MS,Vogelsang

all relevant observables available at NLO accuracy

except for hadron-pair production at COMPASS, HERMESQ2' 0 available very soon: Hendlmeier, MS, Schafer

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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:

semi-inclusive DIS dataso far only used in DNS fit

! flavor separation

“classic” inclusive DIS dataroutinely used in PDF fits

! q + q

first RHIC pp data (never used before)

! g

467 data pts in total (¼10% from RHIC)

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data with observed hadrons • SIDIS (HERMES, COMPASS, SMC)

• pp ! X (PHENIX)

strongly rely on fragmentation functions

! new DSS FFs are a crucial input to the DSSV PDF fit

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])

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:

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3. setup of DSSV analysis

• flexible, MRST-like input form

input scale

possible nodes

simplified form for sea quarks and g: j = 0

• avoid assumptions on parameters {aj} unless data cannot discriminate

• take s 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

0.586§0.031

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4. fit procedure

467 data pts

change O(20) parameters {aj} about 5000 times

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

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! problem can be solved with the help of 19th century math

R.H. MellinFinnish mathematician

idea: take Mellin n-moments

inverse

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

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standardMellin inverse

fitcompletely indep. of pdfs

pre-calculate prior to fit

example: pp! X

here is how it works:

express pdfs by their Mellin inverses

discretize on 64 £ 64 grid

for fast Gaussian integration

MS, Vogelsangearlier ideas: Berger, Graudenz, Hampel, Vogt; Kosower

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applicability & performance

computing load

O(10 sec)/data pt. ! O(1 msec)/data pt.

recall: need thousands of calls to perform a single fit !

production of grids much improved recentlycan be all done within a day with new MC sampling techniques

obtaining the grids once prior to the fit

64 £ 64 £ 4 £ 10 ' O(105) calls per pp data pt.

n mn,m

complex

# subproc’s

tested for pp!X, pp!X, pp!jetX

(much progress towards 2-jet production expected from STAR)

method completely general

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overall quality of the global fit

2/d.o.f. ' 0.88

note: for the time being, stat. and syst. errorsare added in quadrature

very good!

no significant tensionamong different data sets

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inclusive DIS data

data sets used in: the old GRSV analysis

the combined DIS/SIDIS fit of DNS

new

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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: f = N ¢ x ¢ 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 …

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semi-inclusive DIS data

impact of new FFs noticeable!

not in DNS analysis

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RHIC pp data (inclusive 0 or jet)

good agreement

important constraint

on g(x) despite

large uncertainties! later

uncertainty bands estimatedwith Lagrange multipliers byenforcing other values for ALL

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Lagrange multiplier method

see how fit deteriorates when PDFs are forced

to give a different prediction for observable Oi

Oi can be anything: we have looked at ALL, truncated 1st moments,

and selected fit parameters aj so far

finds largest Oi allowed by the global data set

and theoretical framework for a given 2

explores the full parameter space {aj}

independent of approximations

track 2

requires large series of minimizations (not an issue with fast Mellin technique)

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2 - a question of tolerance

What value of 2 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 2 = 1

! idealistic 2=1 $ 1 approach usually fails

we present uncertainties bands for both 2 = 1 and

a more pragmatic 2% increase in 2

see: CTEQ, MRST, …

also: • 2 = 1 defines 1 uncertainty for single parameters

• 2 ' Npar is the 1 uncertainty for all Npar parameters

to be simultaneously located in “2-hypercontour” used by AAC

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summary of DSSV distributions:

robust pattern of flavor-asymmetric

light quark-sea (even within uncertainties)

small g, perhaps with a node s positive at large x

u + u and d + d very

similar to GRSV/DNS results

u > 0, d < 0 predicted in some models Diakonov et al.; Goeke et al.; Gluck, Reya; Bourrely, Soffer, …

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x

a closer look at u

small, mainly positive negative at large x

2

2

determined by SIDIS data

pions consistent

mainly charged hadrons

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x

a closer look at s

positive at large x negative at small x

striking result!

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2

determined by SIDIS data

mainly from kaons, a little bit from pions

DIS alone: more negative

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a closer look at g

error estimates more delicate: small-x behavior completely unconstrained

x

study uncertainties in 3 x-regions

RHIC range0.05· x · 0.2

small-x0.001· x · 0.05

large-xx ¸ 0.2

g(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

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1st moments: Q2 = 10 GeV2

s receives a large negative contribution at small x g: huge uncertainties below x'0.01 ! 1st moment still undetermined

SU(2) SU(3)

SU(2),SU(3) come out close

to zero

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Hessian methodestimates uncertainties by exploring 2 near minimum:

Hessian Hij

taken at minimum

displacement:

only quadraticapproximation

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

good news: can benefit from a lot of pioneering work by CTEQ

and use their improved iterative algorithm to compute Hij

J. Pumplin et al., PRD65(2001)014011

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PDF eigenvector basis sets SK§

• 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 2 = k zk2

cartoon by CTEQ

• sets Sk§ can be used to calculate uncertainties of observables Oi

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comparison with uncertainties from Lagrange multipliers

tend to be a bit larger

for Hessian, in particular

for g(x)

Hessian method goes

crazy if asking for 2>1

uncertainties of truncated moments for 2=1 agree well except for g

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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 + 0 data (plus +, -, …)

first di-jet data from STAR ! more precisely map g(x)

the Mellin technique isbasically in place to analyzealso particle correlations

challenge: much slower MC-type codes in NLO than for 1-incl.

from 2008 RHIC spin plan

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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 2 = 1 studies, beyond ??

include experimental error correlations if available

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extra slides

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de Florian, Sassot, MS

DSS: good global fit of all e+e- and ep, pp datamain features:

• handle on gluon fragmentation• flavor separation• uncertainties via Lagrange multipl.• results for §, K§, chg. hadrons

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x

meet the distributions: d

fairly large negative throughout

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2

determined by SIDIS data some tension between charged hadrons and pions

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2 profiles of eigenvector directions

for a somewhat simplifiedDSSV fit with 19 parameters

#1: largest eigenvector (steep direction in 2)

…#19: smallest eigenvector (shallow direction in 2)

significant deviations from assumed quadratic dependence

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worse for fit parameters: mix with all e.v. (steep & shallow)

steep shallow

look O.K. but notnecessarily parabolic

gmixed bag

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roughly corresponds to what we get from Lagrange multipliers

the good …

… the bad

… the ugly

1 st g lobal qcd analysis of polarized parton densities marco stratmann october 7th, 2008

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