Collinear Distributions from Monte Carlo Global QCD Analyses

Transcription

1 Collinear Distributions from Monte Carlo Global QCD Analyses Jacob Ethier On behalf of the JAM Collaboration Light Cone Conference May th, 8

2 Motivation Want to obtain reliable information of nonperturbative dynamics associated with hadron structure and hadroniation Factoriation à separation of short and long distance physics in pqcd epressions of eperimental observables, e.g. Unpolaried deep inelastic scattering (DIS) observable ` +(p, d)! ` + X d (, Q ) ' X f Collinear factoriation à distributions depend on some fraction of longitudinal momentum d f Parton distribution function (PDF),Q dˆf (,Q ) Hard scattering coefficient Nonperturbative distributions are typically determined empirically through global QCD analyses à Objects are parameteried f() N a ( ) b ( + c p + d) à Parameters are optimied with a least-squares fit N X ep NX data (D e i T i ) e i ( e i )

3 Motivation However, many observables can depend on more than one type of distribution Polaried semi-inclusive P DIS observable A h gh f,f f C F h ff Df h P f,f f C ` +(p, d)! ` + h + X ff Df h à Unpolaried (Polaried) PDFs: describe nucleon s momentum (spin) structure à Fragmentation functions (FFs) D f : describe parton-to-hadron fragmentation and there are many issues with performing single chi-squared minimiations à Uncertainties computed by Hessian or Lagrange multiplier method introduce tolerance criteria (uncertainties inflated by arbitrary factor) à Parameters difficult to constrain (flat eigendirections) are typically fied à Highly non-linear chi-squared function means many local minima that a single fit can be trapped in With a consistent theoretical framework and rigorous fitting procedure we can (more effectively):. Test universality. Eplore the limits of collinear factoriation. Study power suppressed corrections

4 JAM Collaboration Efforts Recent efforts by the JAM collaboration: à JAM5: Iterative Monte Carlo Analysis of spin PDFs (DIS only) studies impact of high precision Jefferson Lab data on proton spin structure N. Sato et. al. Phys. Rev. D9 75 (6) à JAM6: First Monte Carlo analysis of FFs (SIA only) preformed to obtain reliable determination of FFs and their uncertainties N. Sato et. al. Phys. Rev. D9 (6) Process Function JAM5 JAM6 JAM7 JAM8 DIS SIA SIDIS DY f f D h f à JAM7: First combined Monte Carlo analysis of polaried DIS, polaried SIDIS, and SIA data studies impact of SIDIS on sea quark helicity distributions à JAM8: Universal etraction of all nonperturbative input (in progress) Other JAM projects: JE, N. Sato, W. MelnitchoukPRL 9 (7) N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) à Monte Carlo etraction of transversity distribution with lattice QCD constraints H.-W. Lin, W. Melnitchouk, A. Prokudin, N. Sato, H. Shows, PRL 55 (8) à Monte Carlo analysis of pion PDFs (see P. Barry s talk Tuesday p.m.) P. C. Barry, N. Sato, W. Melnitchouk, C.-R. Ji, arxiv:8.965 (8)

5 JAM Collaboration Efforts Recent efforts by the JAM collaboration: à JAM5: Iterative Monte Carlo Analysis of spin PDFs (DIS only) studies impact of high precision Jefferson Lab data on proton spin structure N. Sato et. al. Phys. Rev. D9 75 (6) à JAM6: First Monte Carlo analysis of FFs (SIA only) preformed to obtain reliable determination of FFs and their uncertainties N. Sato et. al. Phys. Rev. D9 (6) Process Function JAM5 JAM6 JAM7 JAM8 DIS SIA SIDIS DY f f D h f à JAM7: First combined Monte Carlo analysis of polaried DIS, polaried SIDIS, and SIA data studies impact of SIDIS on sea quark helicity distributions à JAM8: Universal etraction of all nonperturbative input (in progress) Other JAM projects: JE, N. Sato, W. MelnitchoukPRL 9 (7) N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) à Monte Carlo etraction of transversity distribution with lattice QCD constraints H.-W. Lin, W. Melnitchouk, A. Prokudin, N. Sato, H. Shows, PRL 55 (8) à Monte Carlo analysis of pion PDFs (see P. Barry s talk Tuesday p.m.) P. C. Barry, N. Sato, W. Melnitchouk, C.-R. Ji, arxiv:8.965 (8) 5

6 JAM Fitting Methodology Based on Bayesian statistical methods robust determination of observables O (PDFs,FFs,etc.) and their uncertainties E [O] d n ap(~a data)o(~a) V [O] d n ap(~a data) O(~a) E[O] Bayes theorem defines probability P as P(~a data) L(data ~a) (~a) 6

7 JAM Fitting Methodology Based on Bayesian statistical methods robust determination of observables O (PDFs,FFs,etc.) and their uncertainties E [O] d n ap(~a data)o(~a) V [O] d n ap(~a data) O(~a) E[O] Bayes theorem defines probability P as P(~a data) L(data ~a) (~a) L ep Likelihood function (~a) à Gaussian form in data with N X ep e NX data i (D e i T i ) ( e i ) 7

8 JAM Fitting Methodology Based on Bayesian statistical methods robust determination of observables O (PDFs,FFs,etc.) and their uncertainties E [O] d n ap(~a data)o(~a) V [O] d n ap(~a data) O(~a) E[O] Bayes theorem defines probability P as P(~a data) L(data ~a) (~a) Evidence Priors d n al(data ~a) (~a) 8

9 JAM Fitting Methodology Based on Bayesian statistical methods robust determination of observables O (PDFs,FFs,etc.) and their uncertainties E [O] d n ap(~a data)o(~a) V [O] d n ap(~a data) O(~a) E[O] Bayes theorem defines probability P as P(~a data) L(data ~a) (~a) Monte Carlo technique is used to evaluate epectation value and variance integrals à samples parameter space and assigns weights w k to each parameter a k such that E[O(~a)] X k w k O(~a k ) V [O(~a)] X k w k (O (~a k ) E[O]) 9

10 Iterative Monte Carlo (IMC) (JAM7 Analysis) à Samples wide region of parameter space à Data is partitioned for cross-validation training set is fitted via chi-square minimiation à Posteriors used to construct sampler (multi-dimensional Guassian, kernel density estimation, etc) where parameters are chosen for the net iteration à Procedure iterated until converged E[O] n V [O] n nx O(~a k ) k nx (O(~a k ) E[O]) k

11 Nested Sampling (JAM8 Analysis) Statistical mapping of multidimensional integral to -D d n al(data ~a) (~a) dxl(x) where the prior volume dx (~a)d n a L. L... L L. L.. L. L i X L i w i i where w i (X i X i+ ) X X X. L X Fero et al. arxiv:6. [astro-ph] Algorithm: à Initialie X, L and choose N active points X, X,, X N from prior à For each iteration, sample new point and remove lowest L i, replacing with point such that L is monotonically increasing à Repeat until entire parameter space has been eplored

12 Proton Spin Structure from DIS Typically measure longitudinal and transverse spin asymmetries A k "+ "* "+ + "* D (A + A ) A? A (g à Virtual photoproduction asymmetries: Leading contribution to polaried structure function g : g (, Q ) First moment of polaried structure function g : X q e q ") "( ") + "( d (A + A ) g ) A (g + g ) M F F Q ( Cq q + )(, Q )+ ( C g g)(, Q ) + O Q dg p (, Q ) 6 [8 +g A + a 8 ] à DIS requires assumptions about triplet and octet aial charges to etract ΔΣ Assuming eact SU() f and SU() f values from weak baryon decays d u + d + g A.69 d u + + d + s + a s + O( s) + O Q [,.8].

13 Proton Spin Structure from SIDIS Measured via longitudinal double spin asymmetries h A h (,, Q ) gh (,, Q ) F h (,, Q ) p Polaried structure function at NLO defined in terms of -D convolution g h (,, Q ) X q e q q(, Q )Dq h (,Q ) + s(q ) q C qq Dq h + q C gq Dg h + g C qg Dq h To include SIDIS observables in the JAM global analyses, fragmentation functions (FFs) must be known à Choice of FF parameteriations available (HKNS & DSS) differed significantly in kaon sector strongly impacts Δs + etraction JAM7 Analysis: first to fit simultaneously polaried PDFs + FFs and release SU() constraints

14 JAM7 Polaried PDF Distributions u +.. JAM7 JAM5.8 ( ū + d) DSSV d +.8 ( ū d).8 Isoscalar sea distribution consistent with ero Isovector sea slightly prefers positive shape at low à Non-ero asymmetry given by small contributions from SIDIS asymmetries JE, N. Sato, W. MelnitchoukPRL 9 (7) Δu + consistent with previous analysis Q GeV Δd + slightly larger in magnitude à Anti-correlation with s +, which is less negative than JAM5 at ~ JAM7 q COMPASS A p..5

15 JAM7 Resolution of the Strange Polariation + JAM7 + SU() s DSSV9 JAM5.8 JE, N. Sato, W. MelnitchoukPRL 9 (7) Δs + distribution consistent with ero, slightly positive in intermediate range Primarily influenced by HERMES K - data from deuterium target Why does DIS+SU() give large negative s +? Low DIS deuterium data from COMPASS prefers small negative Δs + Negative polariation shifted to intermediate region to satisfy SU() constraint b parameter for s + typically fied to values ~6-, producing a peak at ~..8 JAM7 s + < HERMES A K d..5 5

16 JAM8 Analysis (Preliminary) Q DIS SLAC BCDMS NMC HERA ` +(p, d)! ` + X Q bj 5 SIDIS W > GeV Q > 5 GeV ` +(p, d)! ` + h + X COMPASS bj W > GeV bj Q 5 Drell-Yan E866 (pp) E866 (pd) p +(p, d)! ` ` + X Q SIA e + + e! h + X F Argus Belle BaBar TPC TASSO ALEPH DELPHI SLD OPAL.8 N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) 6

17 JAM8 Data vs Theory (Preliminary) Fp Fp Fd M d /dm d Data/Theory DIS.5 Fd. Q [, 5] DY 8 Q [5, 5] pp! ll.5.5 Q Fp Q Fd/Fp Q NC + red (e p) Q NC + red (e p)..5.5 Q Q Q Q Q [, ] NC + red (e p) NC red (e Q [, 5] Q [5, 5] p). pd! ll Q Q Q Q log( ) K ± Argus.8 d T d.8.8 d T d d T d.8.8 d T d d T d Q [5, 6] d T d SLD.8.8. Q [, ] 5 d T d.8.8 Q [, 6].8 Q [6, ].8. Overall agreement with DIS, DY, and SIA data 7 6 OPAL(c) OPAL(b) SLD(c) OPAL Q [6, 68] d T d ALEPH 6 SLD(b) d T d N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) DELPHI(b) TASSO 5 DELPHI d T d TPC d T d 6 BaBar TPC(b) d T d Q [6, ] TPC(c) Belle d d ± SIA d T d Q [, 6] Q [68, 75] CC red 5. NC + red (e p) Q [68, 75]. Q [6, 68].5 6 Q [5, 6] 6

18 JAM8 Data vs Theory (Preliminary) SIDIS N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) M + + pd! + + X M K pd! K + + X M +.8 pd! + X M K pd! K + X y [.,.5], y [.5, ], 5 y [,.],.5 y [.,.5],.75 y [.,.5], y [.5, ], 5 y [,.],.5 y [.,.5],.75 Difficulty fitting low-q data à only Q > 5 GeV included 8

19 f()..8 JAM8 Unpolaried PDFs (Preliminary) PDFs Q GeV g/ u v d v - CL d/u.5.. Q GeV CJ5 MMHT CT.8 d/ū.6... ū d s JAM8 (DIS, DY) JAM8 (DIS, DY, SIDIS, SIA).8.. Central value and uncertainties from maimum likelihood + data resampling method Distributions mostly consistent with previous analyses à Light sea asymmetry differs at large- SIDIS supports suppression of strange distribution N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) (s + s)/(ū + d)..8 - CL 9

20 f/f f/f JAM8 Impact of SIDIS u Q GeV v JAM8 (DIS, DY) JAM8 (DIS, DY, SIDIS, SIA) d d N. Sato, JE, C. Andrés, W. Melnitchouk, et. al. (8) d v s ūū - CL Decrease in central value and uncertainty of strange PDF with SIDIS Large effect on the gluon distribution à Correlation with strange PDF (momentum sum rule) g f/f f/f D h q () D h q () JAM8 (DIS, DY) JAM8 (DIS, DY, SIDIS, SIA) K + Q GeV u v d d.. + u + Q.7 GeV c + Q m c - CL CL ūū d v.. s d + Q.7 GeV dotted : JAM8 (SIA) solid : JAM8 (DIS, DY, SIDIS, SIA) b + Q m b..8 g.. s + Q.7 GeV.8 g Q.7 GeV.8.8.8

21 Summary and Outlook Monte Carlo statistical methods are important for robust etractions of nonperturbative functions and their uncertainties à Necessary for future global QCD studies that contain large data sets and have many fit parameters (TMDs, GPDs) New approaches being developed: à Likelihood sampling methods (Nested Sampling) à Generaliation of Gaussian likelihood (systematic treatment of incompatible data sets) First universal analysis of unpolaried + polaried measurements underway à Simultaneous etraction of all nonperturbativeinput à Strict test of universality à Can separate individual aligned/anti-aligned helicity distributions Longer term: etracting transverse momentum dependent (TMD) PDFs and FFs