Iterative Monte Carlo analysis of spin-dependent parton distributions

Transcript

1 Iterative Monte Carlo analysis of spin-dependent parton distributions (arxiv:6.7782) Nobuo Sato In Collaboration with: W. Melnitchouk, S. E. Kuhn, J. J. Ethier, A. Accardi Next Generation Nuclear Physics with JLab2 and EIC, Feb 26 / 6

2 Motivations Spin structure of nucleons - Spin sum rule 2 = 2 Σ() + g () + L - The spin contribution from quarks: Σ () = u () + + d () + + s () + - From existing global analysis Σ () [ 3,] 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 2 - Dedicated global QCD anlysis is required High x - SU(6) spin-flavor symmetry: u/u 2/3, d/d /3 - pqcd q/q , g() [.5,.2]. u + /u + d + /d x Q 2 = GeV 2 2 / 6

3 Global Analysis 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 (α m S 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 EMC (npts = ) SMC (npts = 4) SLAC (npts = 835) HERMES (npts = 98) COMPASS (npts = 8) JLab (npts = 45) W 2 cut = 4GeV 2 W 2 cut = GeV x 3 / 6

4 Polarized DIS Asymmetries Theory 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 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 ) 4 / 6

5 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 5 / 6

6 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) 6 / 6

7 Polarized DIS Twist-4 structure function: g T4(p,n) (x, Q 2 ) = H (p,n) (x)/q 2 Nuclear corrections: nuclear smearing functions gi A (x, Q 2 ) = dy y f ij N (y, γ)gj N (x/y, Q 2 ) N 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 7 / 6

8 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 α i k 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 MC approach 8 / 6

9 Iterative Monte Carlo Analysis (IMC) Toy example fitting 2 model parameters α, β I. Flat sampling Initial priors {(α, β)} β α 9 / 6

10 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 9 / 6

11 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 9 / 6

12 Iterative Monte Carlo Analysis (IMC) Each iteration - Generate pseudo data sets via data resampling - Radom data partition Training & Validation - Fit the training set - Validation data pseudo data T data V data fit { p (j) } validation a () training validation noncentral χ 2 dof pseudo data2 T data2 V data2 fit2 { p (j) } validation a (2) ensemble χ 2 dof T datak fitk { p (j) } pseudo 2. datak V datak validation a (K) iterations priors new priors / 6

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 / 6

14 Data vs. Theory.2 A p à p A p JAM5 no HT E55 Q 2 [3.7,.6] syst(+) syst( )..3 x E55x Q 2 [5.4, 7.].2.5 x HERMES Q 2 [.4, 6.8] x.5 A p E43 Q2 [.2, 4.5] x.6 A p E55 Q2 [7.6, 24.8] à p...6 A p x E55x Q 2 [., 2.3].3. x HERMES Q 2 [.8,.3] x.2 A p E43 Q2 [2.9, 9.5]...3 x.3 à p E55x Q2 [.,.8] x.6 à p E55x Q 2 [6., 2.4] x.2 A p E55 Q2 [., 3.3]..8 à p A p x E55x Q 2 [2.3, 6.]..3 x HERMES Q 2 [., 2.8]..3 x 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].4.5 x E42 Q 2 [., 5.5].3..3 x E54 Q 2 [4., 5.]..3 x E6-4 Q 2 [2.6, 5.2] x E99-7 Q 2 [2.7, 4.8].4.5 x E42 Q 2 [., 5.5].3..3 x E54 Q 2 [.2, 5.8].3. x E6-4 Q 2 [2., 3.4] x Q 2 [4., 5.] E54..3 x E6-4 Q 2 [2.6, 5.2] x A p.35 JAM5 E = 4.8 Q 2 [.,.] E = 4.8 Q 2 [.,.2].25.3 no HT.2.25 syst(+).5.2 syst( ) eg-dvcs x.2.25 x.45.5 E = 4.8 Q 2 [.6,.7] E = 4.8 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].3.35 x E = 6. Q 2 [.2,.3].5.2 x E = 6. Q 2 [2.3, 2.4] x E = 6. Q 2 [4.6, 4.7] x Q 2 [2.3, 2.4].35.4 x E = 6. Q 2 [.3,.6] x.5 E = 6. Q 2 [2.7, 2.9] x.35 E = 4.8 Q 2 [.3,.4] x.55 E = 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? - A p marginally visible at large x - Not visible in 3 He data - Significant contribution in eg-dvcs A p 2 / 6

15 Results (a).4 (c). - Significant constraints to large x s+ and g x u x d+ x s+ x g (b) xdu xdd xhp xhn (d) - T4 contribution to g consistent with zero... - Non zero T3 quark distributions x x JAM5 JAM3 DSSV9 NNPDF4 BB AAC9 LSS x u+..2 x s Negative s+ (see Talk by J. Ethier) - JAM5 g compatible with resent DSSV fits x d+ 3 2 x g x x 3 / 6

16 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 Constraints on small x from large x weak baryon decay constraints JLab data. < x <.7 xdu x x 4 / 6

17 d 2 matrix element - d 2 (Q 2 ) dxx2 [ 2g T3 (x, Q2 ) + 3g T3 2 (x, Q2 ) ] - 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 d (a) JAM5 p JAM5 n lattice (b) E55x RSS E 2 E6 4 JAM Q 2 (GeV 2 ) Q 2 (GeV 2 ) 5 / 6

18 JAM: 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 Q 2 EMC (npts = ) SMC (npts = 4) SLAC (npts = 835) JLab 2 HERMES (npts = 98) COMPASS (npts = 8) JLab (npts = 45) - Measurements at high-x q/q - Wider coverage in Q 2 Wcut 2 = 4GeV 2 Wcut g 2 = GeV 2 JLab 2GeV - Determination of pure twist-3 d 2 in DIS x 6 / 6

19 Backup 7 / 6

20 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 8 / 6

21 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 9 / 6

22 Data vs. Theory: proton 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. 2 / 6

23 Data vs. Theory: proton 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 2 / 6

24 Data vs. Theory: deuteron.4 A d A d.2. E43 Q 2 [., 3] JAM5 no HT syst(+) syst( )..3 x..6.3 x A d HERMES Q 2 [2.6, 4.3].2 E55 Q 2 [4, 6]..3 x.6 A d.2 SMC 98 Q 2 [.3, 54.8]..2. x A d E43 Q. [.2, 4.5] x.6 A d E55 Q 2 [, 34.8] x A d COMPASS Q 2 [., 55.3] x.3 A d SMC 99 Q. 2 [., 22.9]..2 A d..5.. x E43 Q 2 [2.9, 9.5]..5.3 x A d HERMES Q 2 [.2, 3.].5..3 x.2 A d E43 Q. 2 [,.4] x A d E55 Q. [, 6] x.3 A d HERMES Q 2 [., 9.6] x 22 / 6

25 Data vs. Theory: deuteron.. JAM5 no HT syst(+) syst( ).3 A d E55 Q2 [3.7,.6]..3 x.4 à d E55x.2.4 Q 2 [5.4, 6.5].2.5 x.2 A d..2 E43 Q 2 [.2, 4.5].3 A d.. E43 Q 2 [2.9, 9.5].3..3 x x A d E55 à d E55x.2 Q 2 Q2 [.,.8] [7.6, 24.9] x x à d E55x Q2 [., 2.3] à d E55x Q 2 [2.5, 7.5] x.8..3 x.2 A d E55 Q2 [., 3.3]..8 à d à d x E55x Q 2 [2.3, 6.]..3 x E55x Q 2 [6., 2.4].2.5 x 23 / 6

26 Data vs. Theory: deuteron DVCS A d.5 JAM5 Q 2 [.,.].. no HT Q 2 [.2,.3] syst(+) syst( ).5.5 eg-dvcs x x Q 2 [.98, 2.5] Q 2 [2.3, 2.4] Q 2 [.4,.5] x Q 2 [2.8, 2.9] Q 2 [.6,.7] x Q 2 [3.3, 3.4] x x x x.2 Q 2 [3.9, 4.] x 24 / 6

27 Data vs. Theory: deuteron EGb A d..4 JAM5 E = 4.2 Q 2 [.,.].3 E = 4.2 Q 2 [.,.2] no HT syst(+) syst( ).5.2. egb x x E = 4.2 Q 2 [.6,.7] E = 4.2 Q 2 [2., 2.3] E = 4.2 Q 2 [2.3, 2.4] x.3 E = 5.7 Q 2 [.,.2] x.4 E = 5.7 Q.3 2 [2.3, 2.4] x x.3 E = 5.7 Q 2 [.36,.44] x.6 E = 5.7 Q 2 [2.7, 2.9] x x.3 E = 5.7 Q 2 [.6,.7] x E = 5.7 Q 2 [3.2, 3.4] x E = 4.2 Q 2 [.36,.43] x.3 E = 5.7 Q 2 [.,.] x.4 E = 5.7 Q.3 2 [.9, 2.] x E = 5.7 Q 2 [3.8, 4.] x 25 / 6

28 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 26 / 6

29 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 - New extraction of Σ () =.3 ±.3 (Only truncated moments are shown x [ 3,.8]) - First extraction of d 2 matrix element in global QCD analysis - JAM6: reduction of gluon uncertainties (jet data) 27 / 6