1. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 1 How to extract  g from measurements of A LL ? Marco Stratmann XIV International Workshop on Deep-Inelastic Scattering Tsukuba, Japan

2. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 2 objective (as far as possible) a model independent determination of Collins, Soper; Manohar gauge link interpretation as diff. of number operators only in A + =0 gauge features: all n · 2 moments, s x n-1... dx, give local operators but there is no gauge-invariant local gluonic operator for n=1 in A + =0 gauge the 1 st moment also collapses into a local operator and has the interpretation as gluon contribution to the spin “sum rule“

3. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 3 requirements at least NLO corrections should be available essential to have an adequate and proven theoretical framework: unbiased analysis procedure w/o unnecessary assumptions comparison with unpol. cross sections prior to any A LL analyses reliable statements about s  g(x,  ) dx require good x coverage scale dependence K-factors reliability of pQCD framework theoretical uncertainties benchmark (pdf‘s ¼ known) map out where pQCD applies relevance of resummations foundation for A LL

4. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 4 pQCD framework d(  )  ' compute as a power series in  s in pQCD extract from e + e - data measure! learn about hadronic/spin structure hard scattering ! assume factorization e.g. pp !  X

5. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 5 more explicitly: ‘‘features‘‘: separation between short- and long-dist. not unique (fact. scheme) theory calculation depends on unphysical fact./renorm. scales factorized ‘‘picture‘‘ good up to power corrections pQCD framework

6. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 6 the harder we work, the less the final result should depend on these artificial scales a powerful gauge of the reliability of a pQCD calculation the scale dependence is inherent to any pQCD calculation: there is no such thing like ‘‘the right scale‘‘ (not even Q in DIS!) if we truncate the series after the first N terms, there will be a residual scale dependence of order (N+1) theor. error pQCD framework a measurable cross section d(  )  has to be independent of  r and  f renormalization group eqs. like DGLAP evolution

7. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 7 complications information on pdfs inside complicated convolutions and always in a sum over many partonic subprocesses no 1:1 correspondence between data and particular values of x no process is only sensitive to  g ! “global analysis“ prospects of learning from data depends on our ability to efficiently evaluate  2 analysis of data requires typically 1000‘s of evalutions of  cross sections beyond LO are numerically very time-consuming tempting to use approximations – but how reliable are they?

8. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 8 K-factor myth often it is assumed that NLO corrections drop out in A LL or that K=const pp !  X (RHIC)  d !  X (COMPASS) Jäger, MS, Vogelsang !! assumptions about K factors can be quite misleading

9. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 9 data $ x-range how to figure out which range in x is probed?difficult! MS, Vogelsang example: x-range spreads out significantly depends on unknown  g complication: possible oscillations obscure h x i

10. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 10 data $ x-range estimates of x § dx very difficult w/o knowing  g a closer look: diff. subprocesses populate different x-ranges gg vs. qg interplay explains all: large pos.  g ! pronounced gg peak small pos.  g ! double peak not too large neg.  g ! oscillations

11. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 11 MC based methods  g extraction through signal/background separation based on MC MC hadronization not compatible with pQCD fact. which defines pdfs e.g., lp ! HX “fractions“ from MC MC crucial to model experiment but cannot replace a full global analysis: in general, expect:  g(MC)   g( pQCD analysis ) MC neither LO nor NLO (parton showers,...) requires kind of “mean-value“ theorem as (also note that )

12. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 12 global analysis outline of pdf analysis: unpolarized pdfs: CTEQ, MRST gluon constrained by scaling-violations 2 nd moment constrained (mom. sum) pp data only for fine-tuning pdfs K-factor approx., etc. often reasonable polarized pdfs: gluon unconstrained by existing DIS data no momentum sum rule; pol. pdfs can have nodes pp data determine  g and other aspects of pdfs completely different situation! full NLO global analysis mandatory; approximations often misleading

13. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 13 Mellin technique earlier ideas: Berger, Graudenz, Hampel, Vogt; Kosower MS, Vogelsang task : handle exact NLO expressions in global  2 analysis ! computing time becomes excessive idea : use Mellin n-moments to get rid of slow multi-convolutions convolutions factorize example : pp !  X express pdfs by their Mellin inverses can be pre-calculated! fit standard Mellin inverse

14. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 14 Mellin technique contains all time-consuming integrations calculated once and forever before the fit stored in large n £ m grids fast numerical Mellin inverse in complex n,m plane exponential fall-off of x -n, x -m along contour optimal integration = summation in n,m Mellin moments of ansatz for pdfs in x-space, e.g., f a (x,  0 ) = N x  (1-x)  parameters determined in standard  2 analysis

15. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 15 Mellin technique applicability: any process; tested for pp !  X, pp !  X, pp ! jetX,... precision: 64 £ 64 grids sufficient for less than 0.5% deviation performance: “before“: typ. NLO code O(1-2 min/pt.) too slow for fitting w/o approximations “after Mellin tune-up“: bullet-train performance 100 evalutions of x-sec takes a few seconds ideal tool for multidim. fitting beyond LO

16. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 16 gg qq Jäger,Schäfer, MS, Vogelsang; de Florian Jäger,MS,Vogelsang; Signer et al. Gordon,Vogelsang; Contogouris et al.; Gordon, Coriano Bojak, MS Weber; Gehrmann; Kamal; Smith et al. avail. NLO results NLO for polarized lepton-proton scattering (COMPASS, HERMES): only photoproduction of single-hadrons and heavy flavors available Jäger,MS,VogelsangBojak, MS RHIC spin:

17. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 17 1 st results Vogelsang, MS (work in progress)  g unconstrained so far (not even the sign known) Asymmetry Analysis Collaboration (Hirai, Kumano, Saito) (set up like unpol. CTEQ fit) Glück, Reya, MS, Vogelsang (errors set by positivity bound) Blümlein & Böttcher (statistical errors only) impact of 1 st A LL incl. pion & jet data from RHIC? important: unpol. cross sections at RHIC agree nicely with pQCD NLO corrections known and relevant; much reduced theor. errors NLO corrections (K factor) depend on unknown  g 2-hadrons from COMPASS not included: no NLO, no unpol. benchmark

18. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 18 1 st results Vogelsang, MS (work in progress) prel. results from a GRSV-type analysis: first moment of  g at input still a rather flat  2 distr. extreme, large (in partic. positive)  g disfavored very similar for STAR jets to-do list: proper error propagation gluons with nodes & non-standard shapes

19. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 19 1 st results Vogelsang, MS (work in progress) A LL for several trial gluons: 1 st moment at input

20. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 20 1 st results Vogelsang, MS (work in progress) some trial gluons:  g/g x Q 2 = 1 GeV 2 max GRSV std -max  g=0 -0.15 -0.3 -0.45 -0.6 -0.75 -0.9 -1.05 0.7 0.6 0.45 0.3 neg.  g: fits slightly prefer soft large-x behavior

21. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 21 1 st results it would be misleading to extract only  g w/o refitting the quarks: considerable variation of quark singlet at input GRSV std Vogelsang, MS (work in progress)

22. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 22 1 st results indications that  g is not large and positive at x around 0.03 ¥ 0.2... but it is still a very long way to the entire 1 st moment small-x range requires 500 GeV RHIC data AND a polarized ep-collider like eRHIC

23. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 23  g “semantics“ we always talk about things like ‘‘large  g‘‘ - but what does it mean? we should probably compare with ½ - the spin of the proton pion data: -0.9.  g(input). 0.5 still pretty large!! ‘‘anomaly inspired‘‘  g‘s are really large:  g(input) & 1.5 beware of pQCD evolution: small  g can turn asymptotically into large  g however, only J g =  g + L g is gauge-invariant Ji (but they are now excluded)

24. 21 Apr 2006XIV Workshop on Deep-Inelastic Scattering (DIS 2006), Tsukuba 24 summary need a solid and tested theoretical framework for A LL (unpol. benchmark cross sections first!) approximations in pol. pdf analyses depend on  g which we don’t know Mellin technique: fast & reliable analysis tool with exact NLO expressions strong indications that  g is not large and positive