Iterative Monte Carlo analysis of spin-dependent parton distributions

Transcription

1 Iterative Monte Carlo analysis of spin-dependent parton distributions (PRD ) Nobuo Sato In Collaboration with: W. Melnitchouk, S. E. Kuhn, J. J. Ethier, A. Accardi Precision Radiative Corrections for Next Generation Experiments JLab 26 / 2

2 Roadmap α S Towards the Universal fit PDF PDF FF DIS DIS SIA DY SIDIS SIDIS JETS JETS pp πx W ASY pp γx pp πx W ASY JAM5 ( DIS) JAM6 (FF) in preparation JAM6 ( SIDIS, JETS) 2 / 2

3 Iterative Monte Carlo analysis (IMC) Traditional Method Anzats xf(x) = Nx a ( x) b ( + c x + dx) Fix parameters that are difficult to fit (typically the b exponents) Perform a single Chi2 fit (Minuit, levenberg-marquardt) Determination of uncertainty bands (Hessian) Problems with local minima IMC Same anzats xf(x) = Nx a ( x) b ( + c x + dx) Keep all the parameters free. No assumptions on the exponents Avoid over-fitting by Cross-Validation Robust estimation of uncertainties (flat priors, boostrap, iterations) Typical number of fits: iterations 3 / 2

4 Iterative Monte Carlo analysis (IMC) Toy example fitting 2 model parameters α, β I. Flat sampling Initial priors {(α, β)} β α 4 / 2

5 Iterative Monte Carlo analysis (IMC) Toy example fitting 2 model parameters α, β 2. β I. Flat sampling Initial priors {(α, β)} II. First iteration priors {(α, β)} posteriors {(α, β)} α 4 / 2

6 Iterative Monte Carlo analysis (IMC) Toy example fitting 2 model parameters α, β 2. β α I. Flat sampling Initial priors {(α, β)} II. First iteration priors {(α, β)} posteriors {(α, β)} III. Second iteration priors {(α, β)} posteriors {(α, β)}... until convergence 4 / 2

7 Iterative Monte Carlo analysis (IMC) data pseudo data T data V data fit { p (j) } validation a () pseudo data 2 T data 2 V data 2 fit 2 { p (j) } validation a (2) ensemble pseudo data K T data K V data K fit K { p (j) } validation a (K) priors new priors 5 / 2

8 Iterative Monte Carlo analysis (IMC) data pseudo data T data V data fit { p (j) } validation training validation a noncentral () χ 2 dof χ 2 dof 2.4 pseudo data 2 T data 2 V data 2 fit { p (j) } validation a (2) ensemble iterations pseudo data K T data K V data K fit K { p (j) } validation a (K) priors new priors 5 / 2

9 Spin PDFs Spin structure of nucleons spin carried by a quark of type q 2 q() = 2 dx q(x, Q2 ) spin sum rule 2 = 2 Σ() + g () + L How big is L? From existing global analysis Σ () [ 3,].3, g() [.5,.2]. High x SU(6) spin-flavor symmetry: u/u 2/3, d/d /3.8 u + /u + pqcd q/q.6 Higher twists d 2 matrix element d 2 = 2g (3) (Q2 ) + 3g (3) 2 (Q2 ) Color forces experienced by struck quarks F E = 2d 2 + f 2, FB = 4d 2 f d + /d x Q 2 = GeV 2 6 / 2

10 Spin PDFs Data Polarized DIS u +, d + - Polarized SIDIS: d, ū, s - Inclusive Jets/π : g - W production d, ū Theory Target mass corrections Twist-3 and twist-4 contributions in polarized structure functions Nuclear corrections for 3 He and deuteron targets - Threshold resummation (αs m log( x)n ) Tools Numerical codes developed within python framework Development of DGLAP evolution equations in Mellin space Fast calculation of observables Mellin space techniques Q 2 (GeV 2 ) EMC SMC COMPASS HERMES SLAC JLab W 2 = GeV 2 W 2 = 4 GeV x 7 / 2

11 Polarized DIS Asymmetries A = σ σ σ + σ = D(A + ηa 2 ) A = σ σ σ + σ = d(a 2 ξa ) A = (g γ 2 g 2 ) A 2 = γ (g + g 2 ) γ 2 = 4M 2 x 2 F F Q 2 Polarized structure functions g (x, Q 2 ) = g LT+TMC ( u +, d +, g,...) + g T3+TMC (D u, D d ) + g T4 (H p,n ) g 2 (x, Q 2 ) = g LT+TMC 2 ( u +, d +, g,...) + g T3+TMC 2 (D u, D d ) 8 / 2

12 Fitting strategy Parametrization xf(x) = Nx a ( x) b ( + c x + dx) LT quark distributions u +, d +, s +, g T3 quark distributions D u, D d T4 structure functions H p, H n Chi-squared minimization with correlated systematic uncertainties χ 2 = i ( Di T i ( k rk βi k/d i) ) 2 + (r k ) 2 Issues Stability in the moments (e.g. Σ () ) Is the solution given by a single fit unique? False minima Is over-fitting present in our fits? Which parameters should be fixed and at which value? Determination of uncertainty bands. Solution IMC α i k 9 / 2

13 W 2 cut and Q 2 cut Σ.2 G Σ 2 G.2.8 d p 2.4. d n d p 2.4 d n 2.4 h p h n.8.4. h p.4..4 h n W 2 cut (GeV 2 ) W 2 cut (GeV 2 ) Q 2 cut (GeV 2 ) Q 2 cut (GeV 2 ) Wcut 2 (GeV 2 ) # points χ 2 dof Q 2 cut (GeV 2 ) # points χ 2 dof / 2

14 Data vs theory: proton.4 A p.2. JAM5 no HT syst(+) syst( ) E8/E3 Q 2 [3.3, 5.4] x.2 A p E43..5 Q 2 [.2, 4.5].3..3 x A p E55 Q 2 [., 34.7] x A p COMPASS.6 Q 2 [., 62.] x.8 A p.4.2 EMC Q 2 [3.5, 29.5].3.5. x A p.3.2 E8/E3. Q 2 [5.3, 9.9] x A p E43 Q 2 [2.9, 9.5].2. A p x HERMES Q 2 [., 3.].3..3 x A p COMPASS.6 Q 2 [., 29.8] x.8 A p A p SMC 98 Q 2 [.3, 58.].. x E43 Q 2 [.,.4]..2 x.2 A p E55 Q 2 [., 5.9].4.3. x A p HERMES Q 2 [.2, 9.6] x A p COMPASS.6 Q 2 [., 53.] x.6 A p.2.2 A p SMC 99 Q 2 [., 23.]..3 x E43 Q 2 [., 3.]..3 x.4 A p E55 Q 2 [4., 6.]..3 x.8 A p HERMES Q 2 [2.6, 4.3] x A p COMPASS.6 Q 2 [.2, 96.] x / 2

15 Data vs theory: proton.2 A p JAM5 no HT syst(+) syst( ).2 E55 Q 2 [3.7,.6].4..3 x.6 à p E55x Q 2 [5.4, 7.] x.6 A p HERMES 2 Q 2 [.4, 6.8] x.5 A p E43 Q2 [.2, 4.5] x.6 A p HERMES 2 Q 2 [.8,.3].2.2 A p E43 Q2 [2.9, 9.5] x x A p E55 Q2 [7.6, 24.8] à p E55x Q2 [.,.8] x x à p E55x à p E55x Q 2 [., 2.3] Q 2 [6., 2.4] x x.2 A p E55 Q2 [., 3.3]..3. x.8 à p E55x Q 2 [2.3, 6.] x.6 A p HERMES 2 Q 2 [., 2.8] x 2 / 2

16 Data vs theory: proton JLab egb-dvcs A p.35 JAM5 E = 4.8 Q 2 [.,.].25.3 no HT E = 4.8 Q 2 [.,.2] syst(+) syst( ) eg-dvcs x x E = 4.8 Q 2 [.6,.7].4 E = E = x E = 6. Q 2 [.,.].2.4 x E = 6. Q 2 [.9, 2.] x E = 6. Q 2 [3.9, 4.] x Q 2 [.98, 2.4] x E = 6..2 Q 2 [.2,.3] x.4 E = 6. Q 2 [2.3, 2.4] x.8 E = 6. Q 2 [4.6, 4.7] x Q 2 [2.3, 2.4] x E = Q 2 [.3,.6] x.5 E = 6. Q 2 [2.7, 2.9] x E = 4.8 Q 2 [.3,.4] x E = 4.8 Q 2 [2.7, 2.8] x.35 E = 6. Q 2 [.6,.7] x.55 E = 6. Q 2 [3.2, 3.5] x Signal of HT contribution. 3 / 2

17 Data vs theory: proton JLab egb JAM5 no HT syst(+).5 E = Q 2 [.6,.7] x.4 E = 5.7 Q 2 [.,.] x.8 E = 5.7 Q 2 [.9, 2.] x.5 E = 5.7 Q 2 [3.8, 4.].5 A p.4 E = 4.2 Q 2 [.,.] syst( ). E = 4.2 Q 2 [.,.2] egb x.2.25 x.8 E = 4.2 Q 2 [.9, 2.].8 E = 4.2 Q 2 [2.3, 2.4] x x.4 E = 5.7 Q 2 [.,.2] x.8 E = 5.7 Q 2 [2.2, 2.4] x E = 5.7 Q 2 [4.56, 4.63] x x.3.2. E = 5.7 Q 2 [.3,.4] x Q 2 [2.7, 2.9] x E = 4.2 Q 2 [.3,.4].25.3x E = 4.2 Q 2 [2.69, 2.73] x E = 5.7 Q 2 [.6,.7] x E = 5.7 Q 2 [3.2, 3.4] x 4 / 2

18 Data vs theory: 3 He E54 Q 2 [.2, 5.8] JAM5 no HT syst(+) syst( ).3..3 x E6-4 Q 2 [2., 3.4] x E99-7 Q 2 [2.7, 4.8]. E42 Q 2 [., 5.5].2 2 E Q 2 [., 5.5] x x E54 E54 Q 2 [.2, 5.8] Q 2 [4., 5.] x x E6-4 Q 2 [2.6, 5.2] E6-4 Q 2 [2., 3.4] x x E E54 Q 2 [4., 5.]..3.3 x E6-4 Q 2 [2.6, 5.2] x x.2.4 Q 2 [2.7, 4.8].4.5 x 5 / 2

19 Impact of JLab data JAM5 no JLab x u x d Large uncertainties in s+, g removed by JLab data Non vanishing T3 quark distributions T4 distributions consistent with zero..2 x s x g.4 xhp xhn xd d.5..4 Constraints on small x from large x weak baryon decay constraints JLab data. < x <.7 xdu x x 6 / 2

20 Results (a).4 (c). moment u+ d+ s+ Σ G dp2 dn2 hp hn x u x d+ x s+ x g (b) (d) xdu xdd xhp xhn x 2 x JAM5 JAM3 DSSV9 NNPDF4 BB AAC9 LSS x u Negative s+. x d+ 3 2 x g x Non zero T3 quark distributions T4 contribution to g consistent with zero x s full.83 ±..44 ±.. ±..28 ±.4 ± 5.5 ±.2. ±.. ±.. ±.3 Significant constraints on s+ and g truncated.82 ±..42 ±.. ±..3 ±.3.5 ±.4.5 ±.2. ±.. ±.. ± x JAM5 g compatible with recent DSSV fits. 7 / 2

21 d 2 matrix element d (a) JAM5 p JAM5 n lattice (b) E55x RSS E 2 E6 4 JAM Q 2 (GeV 2 ) d 2 (Q 2 ) dxx2 [ 2g T3 (x, Q 2 ) + 3g T3 2 (x, Q 2 ) ] Q 2 (GeV 2 ) d 2 is related to color polarizability or the transverse color force acting on quarks. Existing measurements of d 2 are in the resonance region (contains TMC T4 and beyond.) Agreement with data indicates quark-hadron duality 8 / 2

22 The strange puzzle.5.4 JAM5 x u + JAM3 DSSV9.2.3 NNPDF4..2 BB AAC9.2. LSS x s x d +. x g x x.35 s ū.4.6 JAM.3.2 HKNS DSS u zd K+ (z, Q 2 ) zd K+ (z, Q 2 ) z z z c b z z.3.25 g z o The sign of s + from combined DIS and SIDIS depends on Kaon fragmentation: positive for DSS and negative for HKNS. (Leader, etal) NEW IMC FF analysis (to be published soon) o Only SIA data is used : npts=245, χ 2 = 35.2 (similar to HKNS) o Size of s + similar to HKNS but smaller than DSS o Combined DIS and SIDIS analysis unlikely to change s + 9 / 2

23 Outlook JAM New JAM5 analysis to study impact of all JLab 6 GeV inclusive DIS data at low W and high x New extraction of LT & HT distributions - Upcoming JAM6 analysis to study polarization of sea quarks & gluons. - SIDIS for flavor separation. - polarized pp cross sections (inclusive jet & π production) for g - W boson asymmetries - Threshold resummation impacts on large x - Combined analysis of all inclusive (un)polarized DIS data - Fits to helicity distributions JLab 2 - Measurements at high-x q/q - Wider coverage in Q 2 g - Determination of pure twist-3 d 2 in DIS PDF DIS DY JETS WASY pp γx αs PDF DIS SIDIS JETS pp πx WASY FF SIA SIDIS pp πx 2 / 2

24 Backup 2 / 2

25 Mellin trick Curse of dimensionality Mellin trick (Stratmann, Vogelsang) dy ( ) I(x) = x y f(y) dz x y z g g(ξ) = dnξ N g N yz 2πi = [ dy ( ) ] dng N 2πi x y f(y) dz x N y z yz = dng N M N 2πi = ) wi k jk Im (e iφ g N M kj N Gaussian quadrature kj i,k Time consuming part can be precalculated prior to the fit Extensible to higher dimensional integrals. A single fit that takes about 4 days 2 mins 22 / 2

26 Polarized DIS ξ = 2x +(+4µ 2 x 2 ) /2 µ 2 = M 2 /Q 2 Leading twist structure functions: g LT+TMC (x, Q 2 ) = x g LT (ξ) ξ ( + 4µ 2 x 2 ) 3/2 + 4µ2 x 2 x + ξ ξ( + 4µ 2 x 2 ) 2 g LT+TMC 2 (x, Q 2 ) = x ξ 4µ 2 x 2 2 4µ 2 x 2 2( + 4µ 2 x 2 ) 5/2 In the Bjorken limit (Q 2 ): ξ dz dz ξ z z z glt (z ) g LT (ξ) ( + 4µ 2 x 2 ) 3/2 + x ( 4µ 2 xξ) ξ ( + 4µ 2 x 2 ) 2 ξ + 3 4µ 2 x 2 dz dz 2 ( + 4µ 2 x 2 ) 5/2 ξ z z z glt (z ) g LT+TMC (x, Q 2 ) g LT (x), glt+tmc 2 (x, Q 2 ) g LT (x) + ξ dz z glt (z) dz z glt (z) dz z glt (z) Leading twist quark distributions: g LT (x) = e 2 q [ C qq q(x) + C qg g(x)] 2 q 23 / 2

27 Polarized DIS Twist-3 structure functions: g T3+TMC (x, Q 2 ) =4µ 2 x 2 D(ξ) ( + 4µ 2 x 2 ) 3/2 4µ2 x 2 3 ( + 4µ 2 x 2 ) 2 + 4µ 2 x 2 2 4µ 2 x 2 ( + 4µ 2 x 2 ) 5/2 g T3+TMC 2 (x, Q 2 D(ξ) ) = ( + 4µ 2 x 2 ) 3/2 8µ2 x 2 ( + 4µ 2 x 2 ) 2 2µ 2 x 2 dz ( + 4µ 2 x 2 ) 5/2 ξ z Bjorken limit(q 2 ): ξ dz z dz z ξ z dz z D(z ) z D(z ) dz z D(z) g T3+TMC (x, Q 2 ) g T3+TMC 2 (x, Q 2 dz ) D(x) ξ z D(z) Twist-3 quark distributions: D(x, Q 2 ) = 4 9 D u(x, Q 2 ) + 9 D d(x, Q 2 ) ξ dz z D(z) 24 / 2

28 Polarized DIS Twist-4 structure function (Nucleon d.o.f.): g T4(p,n) (x, Q 2 ) = H (p,n) (x)/q 2 Nuclear corrections: nuclear smearing functions g A i (x, Q 2 ) = N dy y f N ij (y, γ)g N j (x/y, Q 2 ) Addition constraints: weak neutron and hyperon decay constants u +() d +() = F + D =.269(3) u +() + d +() 2 s +() = 3F D =.586(3) 25 / 2

29 Data vs theory: proton.4 A p.2. JAM5 no HT syst(+) syst( ) E8/E3 Q 2 [3.3, 5.4] x.2 A p E43..5 Q 2 [.2, 4.5].3..3 x A p E55 Q 2 [., 34.7] x A p COMPASS.6 Q 2 [., 62.] x.8 A p.4.2 EMC Q 2 [3.5, 29.5].3.5. x A p.3.2 E8/E3. Q 2 [5.3, 9.9] x A p E43 Q 2 [2.9, 9.5].2. A p x HERMES Q 2 [., 3.].3..3 x A p COMPASS.6 Q 2 [., 29.8] x.8 A p A p SMC 98 Q 2 [.3, 58.].. x E43 Q 2 [.,.4]..2 x.2 A p E55 Q 2 [., 5.9].4.3. x A p HERMES Q 2 [.2, 9.6] x A p COMPASS.6 Q 2 [., 53.] x.6 A p.2.2 A p SMC 99 Q 2 [., 23.]..3 x E43 Q 2 [., 3.]..3 x.4 A p E55 Q 2 [4., 6.]..3 x.8 A p HERMES Q 2 [2.6, 4.3] x A p COMPASS.6 Q 2 [.2, 96.] x 26 / 2

30 Data vs theory: proton.2 A p JAM5 no HT syst(+) syst( ).2 E55 Q 2 [3.7,.6].4..3 x.6 à p E55x Q 2 [5.4, 7.] x.6 A p HERMES 2 Q 2 [.4, 6.8] x.5 A p E43 Q2 [.2, 4.5] x.6 A p HERMES 2 Q 2 [.8,.3].2.2 A p E43 Q2 [2.9, 9.5] x x A p E55 Q2 [7.6, 24.8] à p E55x Q2 [.,.8] x x à p E55x à p E55x Q 2 [., 2.3] Q 2 [6., 2.4] x x.2 A p E55 Q2 [., 3.3]..3. x.8 à p E55x Q 2 [2.3, 6.] x.6 A p HERMES 2 Q 2 [., 2.8] x 27 / 2

31 Data vs theory: proton JLab egb-dvcs A p.35 JAM5 E = 4.8 Q 2 [.,.].25.3 no HT E = 4.8 Q 2 [.,.2] syst(+) syst( ) eg-dvcs x x E = 4.8 Q 2 [.6,.7].4 E = E = x E = 6. Q 2 [.,.].2.4 x E = 6. Q 2 [.9, 2.] x E = 6. Q 2 [3.9, 4.] x Q 2 [.98, 2.4] x E = 6..2 Q 2 [.2,.3] x.4 E = 6. Q 2 [2.3, 2.4] x.8 E = 6. Q 2 [4.6, 4.7] x Q 2 [2.3, 2.4] x E = Q 2 [.3,.6] x.5 E = 6. Q 2 [2.7, 2.9] x E = 4.8 Q 2 [.3,.4] x E = 4.8 Q 2 [2.7, 2.8] x.35 E = 6. Q 2 [.6,.7] x.55 E = 6. Q 2 [3.2, 3.5] x Signal of HT contribution. 28 / 2

32 Data vs theory: proton JLab egb JAM5 no HT syst(+).5 E = Q 2 [.6,.7] x.4 E = 5.7 Q 2 [.,.] x.8 E = 5.7 Q 2 [.9, 2.] x.5 E = 5.7 Q 2 [3.8, 4.].5 A p.4 E = 4.2 Q 2 [.,.] syst( ). E = 4.2 Q 2 [.,.2] egb x.2.25 x.8 E = 4.2 Q 2 [.9, 2.].8 E = 4.2 Q 2 [2.3, 2.4] x x.4 E = 5.7 Q 2 [.,.2] x.8 E = 5.7 Q 2 [2.2, 2.4] x E = 5.7 Q 2 [4.56, 4.63] x x.3.2. E = 5.7 Q 2 [.3,.4] x Q 2 [2.7, 2.9] x E = 4.2 Q 2 [.3,.4].25.3x E = 4.2 Q 2 [2.69, 2.73] x E = 5.7 Q 2 [.6,.7] x E = 5.7 Q 2 [3.2, 3.4] x 29 / 2

33 Data vs theory: 3 He E54 Q 2 [.2, 5.8] JAM5 no HT syst(+) syst( ).3..3 x E6-4 Q 2 [2., 3.4] x E99-7 Q 2 [2.7, 4.8]. E42 Q 2 [., 5.5].2 2 E Q 2 [., 5.5] x x E54 E54 Q 2 [.2, 5.8] Q 2 [4., 5.] x x E6-4 Q 2 [2.6, 5.2] E6-4 Q 2 [2., 3.4] x x E E54 Q 2 [4., 5.]..3.3 x E6-4 Q 2 [2.6, 5.2] x x.2.4 Q 2 [2.7, 4.8].4.5 x 3 / 2

34 JAM: moments # fits # fits # fits u +() s +() Σ () # fits # fits d +() g () # fits # fits # fits D u (3) H p (3) d p 2..5 # fits # fits # fits D (3) d H n (3) d n 2 3 / 2