Moments of Structure Functions in Full QCD

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1 Moments of Structure Functions in Full QCD John W. Negele Lattice 2000 Collaborators MIT Dmitri Dolgov Stefano Capitani Richard Brower Andrew Pochinsky Jefferson Lab Robert Edwards Wuppertal SESAM Klaus Schilling Thomas Lippert Outline Salient aspects of deep inelastic scattering Details of lattice calculation Results Quenched Full QCD at β =5.6 and 5.5 Cooled full QCD Comparison with phenomenology Summary and future work

2 Deep Inelastic Scattering k' e(k)+n(p ) e(k )+X k q q Q 2 def = q 2 =4EE sin 2 θ 2 > 0 P X ν def = P q = M(E E ) x def = Q 2 /2ν Cross section determined by hadronic tensor W µν A 2 4π = α2 W Q 4lµν µν W µν = 1 PS J µ X (2π) 4 δ (4) (P + q P X ) X J ν PS 4π X = d 4 ξe iqξ PS [J µ (ξ),j ν (0)] PS Unpolarized scattering measures symmetric part with 2 structure functions: W {µν} =( g µν + qµ q ν q )F 1(ν, Q 2 ) ν [(P µ ν )(P ν ν )]F q 2qµ q 2qν 2 (ν, Q 2 ) Polarized scattering measures antisymmetric part with 2 structure functions: W [µν] = iɛ µνλρ q λ ( S ρ ν (g 1(ν, Q 2 ) + g 2 (ν, Q 2 )) q SP ρ ν 2 g 2 (ν, Q 2 )) In parton model F 1 = 1 e 2 2 q (q (x)+q (x)) F 2 =2xF 1 g 1 = 1 e 2 2 q (q (x) q (x)) g 2 =0 q q 2

3 Sketch of Operator Product Expansion Im (ω) ω 1 x i 1 1 Re (ω) T µν (ν, q 2 )=i Forward Compton amplitude d 4 ξe iqξ P T (J µ (ξ)j ν (0)) P Dispersion relation, ω = 1/x i W (ω, q 2 )= 1 4π Im T (ω, q2 ) n n T (ω, q 2 )=4 1 dω ω W (ω,q 2 ) ω 2 ω 2 q µ1 q µn C n Q 2n P ψγ {µ1 Dµ2... D µn } ψ P =4 ω n even n 1 dω (ω ) n 1 W (ω,q 2 ) ( ) 2p q n C n Q 2 v n =4 x n 1 dx even n 0 xn 1 W (x, q 2 ) C n v n 1 0 dx xn 1 W (x, q 2 ) 3

4 Moments of Structure Functions 1 dx 0 xn 1 F 1 (x, Q 2 )= 1 2 Cv n(q 2 /µ 2 )v n (µ) v n x n 1 q 1 dx 0 xn 2 F 2 (x, Q 2 )=Cn v (Q2 /µ 2 )v n (µ) 1 dx 0 xn g 1 (x, Q 2 )= 1 4 Ca n (Q2 /µ 2 )a n (µ) 1 2 a n x n q 1 dx 0 xn g 2 (x, Q 2 )= 1 n 4n +1 (Cd n (Q2 /µ 2 )d n (µ) Cn a (Q2 /µ 2 )a n (µ)) 2 v n P µ1...p µn = 1 2 S PS ( i 2)n 1 ψγ {µ1 Dµ 2... D µn } ψ PS a n S {σ P µ1...p µn } = PS ( i 2) n ψγ 5 γ {σ Dµ1... D µn } ψ PS d n S [σ P {µ1 ]...P µ n } = PS ( i 2) n ψγ 5 γ [σ D{µ1 ]... D µn } ψ PS 4

5 Calculation of Connected Diagrams R αα (t i,t o,t f )= = d 3 x f e ipx f d 3 y J α (x f,t f )O(y, t o ) J α (x i,t i ) V d 3 x f e ipx f J α (x f,t f ) J α (x i,t i ) t i t 0 f P 0 P 8 7 Nucleon source: J α = u α au β b (Cγ 5) ββ d β c ɛabc Use upper two components of J <R 2 > 1/2, lattice units Dirichlet boundary conditions for quarks in t-direction Gauge-invariant Wuppertal smearing of sources Number of Iterations Alpha ψ(x, t) (1 + α 3 i=1 [U(x, i)δ x,x+î + U (x î, i)δ x,x î ])N ψ(x, t) 5

6 Sequential Source x x o y o y D/ zx S(x) = δ(z) D/ yx S(x) =S(y)S(y) D/ = γ 5 D/ γ 5 D/ yx S(x) =γ 5 S (y)s (y) e ipy x x o y o y J O J = S (x) γ 5 O(x) S(x) S(x, x 0 ) W (x, x ) S(x,x 0 ) W (x 0,x 0) 6

7 2-point function with equal source and sink J(t)J(0) = J n 2 e E nt n A = J n 2 n Overlap with Ground State A B B = J 0 2 B A = A B B = J 0 2 n J n 2 70% n 0 J n 2 J 0 2 T (AB)/B <r 2 > 1/2, lattice units 7

8 Consistency checks point-point, point-smeared, smeared-point, and smeared-smeared source-sink combinations give consistent results N =0,N = 20, and N = 100 sink-source smearings give consistent results 8

9 Consistency checks Dirichlet boundary conditions vs. periodic boundary conditions Discrepancy between double-precision a 0 and single-precision a 0 with different CG stopping residues 9

10 Increase in error bars with source-sink separation Sink-source separation T = 14 Sink-source separation T = 12 10

11 Summary of Production Parameters Name QCD L 3 x L t β κ sea κ val Sample size SESAM full = κ sea O(200) O(200) O(200) O(200) SCRI full = κ sea O(100) O(100) O(100) DD60Q quenched O(200) O(200) O(200) SESAM50CXK full O(100) (cooled) O(100) 11

12 Autocorrelation Functions C O (τ) = 1 T τ +1 T τ t=0 O(t)O(t + τ) ( 1 T τ +1 T τ t=0 )( 1 O(t) T τ +1 T t=τ ) O(t) SESAM configurations (κ = ) skip 25 trajectories SCRI configurations (κ = ) skip 10 trajectories, alternate front and back 12

13 More convenient notation moments of Parton Distribution Functions (PDFs) x n q x n q x n δq 1 0 dx xn (q (x)+q (x)) 1 0 dx xn (q (x) q (x)) 1 0 dx xn (q (x) q (x)) Relation to a (q) n x n q = v (q) n+1 x n q = 1 2 a(q) n and v (q) n 1 13

14 H(4) mixes p lattice operator xq (a) v no 0 qγ {1 D4} q xq (b) v no 0 qγ 4 D4 q 1 3 ( qγ 1 x 2 q v 8 1 yes 0 qγ {1 D1 D4} q 1 2 (γ {2 D 1 q + qγ 2 D2 q + qγ 3 D3 q) D 2 D4} +γ {3 D3 D4} )q x 3 q v yes 0 qγ {1 D1 D4 D4} q + qγ {2 D2 D3 D3} q (3 4) q v no 0 qγ 5 γ 3 q x q (a) v 6 3 no 0 qγ 5 γ {1 D3} q x q (b) v 6 3 no 0 qγ 5 γ {3 D4} q x 2 q v no 0 qγ 5 γ {1 D3 D4} q δq v no 0 qγ 5 σ 34 q xδq v 8 1 no 0 qγ 5 σ 3{4 D1} q d no 0 qγ 5 γ [3 D4] q d no 0 qγ 5 γ [1 D{3] D4} q 14

15 Renormalization: From lattice to MS Oi MS (Q 2 )= j To convert lattice result to the MS renormalization scheme ( δ ij + g2 0 Nc 2 1( γ MS 16π 2 ij log(q 2 a 2 ) (Bij LAT T 2N c Bij MS ) )) Oj LAT T (a 2 ) γ B LAT T B MS Z(β =6.0) Z(β =5.6) xq (a) 16/ / xq (b) 16/ / x 2 q 25/ / x 3 q 157/ / q x q (a) 16/ / x q (b) 16/ / x 2 q 25/ / δq d 2 7/ / Renormalization Constants 15

16 Plateaus for SESAM (κ =0.1560) p =0 16

17 Plateaus for SESAM (κ =0.1560) p 0 17

18 Unpolarized Quenched PDF: DD60Q (open) and QCDSF (solid) 18

19 Polarized Quenched PDF: DD60Q (open) and QCDSF (solid) 19

20 Unpolarized PDF: Full QCD (open), quenched (solid) 20

21 Polarized PDF: Full QCD (open), quenched (solid) 21

22 Transversity and d n : Full QCD (open), quenched (solid) 22

23 World plot of a N versus β =6/g 2 0 for dynamical Wilson LANL HEMCGC SCRI SESAM (quadratic) SESAM (linear) 0.15 a N (fm) β 23

24 Scaling behavior from SESAM (β =5.6) and SCRI (β =5.5) configurations u d x u x d 0.6 xd xd a N (fm) First two moments of polarized PDF a N (fm) First moment of unpolarized PDF 24

25 Unpolarized PDF: Full QCD (open), cooled full QCD (solid) 25

26 Polarized PDF: Full QCD (open), cooled full QCD (solid) 26

27 Transversity and d n : Full QCD (open), cooled full QCD (solid) 27

28 Comparison with Phenomenology QCDSF QCDSF(a =0) Wuppertal DD60Q SESAM (4 pts) SESAM (3 pts) Phenom. (q val ) xu c 0.452(26) 0.454(29) 0.504(18) 0.459(29) xd c 0.189(12) 0.203(14) 0.213(11) 0.190(17) xu c xd c 0.263(17) 0.251(18) 0.291(14) 0.269(23) x 2 u c 0.104(20) 0.119(61) 0.158(44) 0.176(63) x 2 d c 0.037(10) 0.029(32) (209) (303) x 3 u c 0.022(11) 0.037(36) (247) (392) x 3 d c 0.001(7) 0.009(18) (1179) (1529) u c 0.830(70) 0.889(29) 0.816(20) 0.888(80) 0.719(48) 0.860(69) d c 0.244(22) 0.236(27) 0.237(9) 0.241(58) 0.179(31) 0.171(43) u c d c 1.074(90) 1.14(3) 1.053(27) 1.129(98) 0.898(57) 1.031(81) x u c 0.198(8) 0.215(25) 0.243(17) 0.242(22) x d c 0.048(3) 0.054(16) (86) (129) x 2 u c 0.087(14) 0.027(60) 0.113(34) 0.116(42) x 2 d c 0.025(6) 0.003(25) (1936) (2515) δu c 0.93(3) 0.980(30) 1.01(8) 0.898(46) 0.963(59) δd c 0.20(2) 0.234(17) 0.20(5) 0.213(29) 0.202(36) d u (18) 0.233(86) 0.224(60) 0.228(81) d d (6) 0.040(31) (222) (310) 28

29 Summary Moments of structure functions in full QCD β =5.6, a n =0.085 fm β =5.5, a n =0.115 fm Close agreement between full QCD and quenched connected diagrams Principal Puzzles xu xd (0.18) u d (1.26) General agreement between full QCD and cooled at small m q instanton and zero-mode dominance Future Disconnected diagrams (exploiting zero-mode dominance) Gluon matrix elements Chiral fermions Finite volume corrections 29

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