Distribution Functions
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1 Distribution Functions Also other distribution functions f 1 = g = 1L g 1T = h 1T = f 1T = h 1 = h 1L = h 1T = Full list of PDF at Twist-2 (Mulders et al) Dirk Ryckbosch, Feldberg, Oct.2006 p.1/33
2 Factorization Dirk Ryckbosch, Feldberg, Oct.2006 p.2/33
3 Factorization Dirk Ryckbosch, Feldberg, Oct.2006 p.2/33
4 Factorization Dirk Ryckbosch, Feldberg, Oct.2006 p.2/33
5 Factorization Dirk Ryckbosch, Feldberg, Oct.2006 p.2/33
6 Factorization Dirk Ryckbosch, Feldberg, Oct.2006 p.2/33
7 Helicity structure Distribution function Dirk Ryckbosch, Feldberg, Oct.2006 p.3/33
8 Helicity structure Distribution function Quark-nucleon scattering Dirk Ryckbosch, Feldberg, Oct.2006 p.3/33
9 Helicity structure Distribution function Quark-nucleon scattering Λ λ Λ λ Λ λ Λ λ 1/2 1/2 1/2 1/2-1/2 1/2 1/2 1/2 1/2 1/2 1/2-1/2-1/2 1/2 1/2-1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2 1/2 1/2 1/2-1/2-1/2-1/2 1/2-1/2-1/2 1/2-1/2 1/2 1/2-1/2-1/2 1/2 1/2 1/2-1/2 1/2-1/2-1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2-1/2-1/2-1/2 1/2 1/2-1/2-1/2-1/2-1/2-1/2-1/2-1/2 Dirk Ryckbosch, Feldberg, Oct.2006 p.3/33
10 Helicity structure Helicity conservation: Λ + λ = Λ + λ Distribution function Quark-nucleon scattering Λ λ Λ λ Λ λ Λ λ 1/2 1/2 1/2 1/2-1/2 1/2 1/2 1/2 1/2 1/2 1/2-1/2-1/2 1/2 1/2-1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2 1/2 1/2 1/2-1/2-1/2-1/2 1/2-1/2-1/2 1/2-1/2 1/2 1/2-1/2-1/2 1/2 1/2 1/2-1/2 1/2-1/2-1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2-1/2-1/2-1/2 1/2 1/2-1/2-1/2-1/2-1/2-1/2-1/2-1/2 Dirk Ryckbosch, Feldberg, Oct.2006 p.3/33
11 Helicity structure Parity: Λ Λ, etc Distribution function Quark-nucleon scattering Λ λ Λ λ Λ λ Λ λ 1/2 1/2 1/2 1/2-1/2 1/2 1/2 1/2 1/2 1/2 1/2-1/2-1/2 1/2 1/2-1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2 1/2 1/2 1/2-1/2-1/2-1/2 1/2-1/2-1/2 1/2-1/2 1/2 1/2-1/2-1/2 1/2 1/2 1/2-1/2 1/2-1/2-1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2-1/2-1/2-1/2 1/2 1/2-1/2-1/2-1/2-1/2-1/2-1/2-1/2 Dirk Ryckbosch, Feldberg, Oct.2006 p.3/33
12 Helicity structure exactly 3 independent quark distribution functions Λ λ Λ λ 1/2 1/2 1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2 Dirk Ryckbosch, Feldberg, Oct.2006 p.4/33
13 Helicity structure exactly 3 independent quark distribution functions Λ λ Λ λ 1/2 1/2 1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2 Dirk Ryckbosch, Feldberg, Oct.2006 p.4/33
14 Helicity structure exactly 3 independent quark distribution functions Λ λ Λ λ 1/2 1/2 1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2 + - Dirk Ryckbosch, Feldberg, Oct.2006 p.4/33
15 Helicity structure exactly 3 independent quark distribution functions Λ λ Λ λ 1/2 1/2 1/2 1/2 1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2 + f = 1 - g 1 = -?? Dirk Ryckbosch, Feldberg, Oct.2006 p.4/33
16 Transversity No probabilistic interpretation in helicity basis Dirk Ryckbosch, Feldberg, Oct.2006 p.5/33
17 Transversity No probabilistic interpretation in helicity basis Only in transverse basis - Dirk Ryckbosch, Feldberg, Oct.2006 p.5/33
18 Transversity No probabilistic interpretation in helicity basis Only in transverse basis - + f = 1 - g 1 = - h = - 1 Dirk Ryckbosch, Feldberg, Oct.2006 p.5/33
19 Transversity f = 1 g 1 = - h = - 1 Dirk Ryckbosch, Feldberg, Oct.2006 p.6/33
20 Transversity f = 1 ψγ µ ψ g 1 = - ψγ µ γ 5 ψ h = - 1 ψσ µν γ 5 ψ Dirk Ryckbosch, Feldberg, Oct.2006 p.6/33
21 Transversity f = 1 ψγ µ ψ Quark density g 1 = - ψγ µ γ 5 ψ Helicity h = - 1 ψσ µν γ 5 ψ Transversity Dirk Ryckbosch, Feldberg, Oct.2006 p.6/33
22 Transversity f = 1 ψγ µ ψ Vector charge g 1 = - ψγ µ γ 5 ψ Axial charge h = - 1 ψσ µν γ 5 ψ Tensor charge Dirk Ryckbosch, Feldberg, Oct.2006 p.6/33
23 Transversity Different notations: h q 1 (x) = δq(x) = T q(x) Dirk Ryckbosch, Feldberg, Oct.2006 p.7/33
24 Transversity Different notations: h q 1 (x) = δq(x) = T q(x) Decouples from gluons anomalous QCD-evolution Dirk Ryckbosch, Feldberg, Oct.2006 p.7/33
25 Transversity Different notations: h q 1 (x) = δq(x) = T q(x) Decouples from gluons anomalous QCD-evolution For relativistic quarks, rotation and boost don t commute h 1 (x) g 1 (x) Dirk Ryckbosch, Feldberg, Oct.2006 p.7/33
26 Transversity Involves quark helicity flip chiral odd distribution Not observable in inclusive DIS Dirk Ryckbosch, Feldberg, Oct.2006 p.8/33
27 Transversity Involves quark helicity flip chiral odd distribution Not observable in inclusive DIS But observable in semi-inclusive DIS Dirk Ryckbosch, Feldberg, Oct.2006 p.8/33
28 Semi-inclusive DIS σ ep ehx = q f p q σ eq eq D q h Dirk Ryckbosch, Feldberg, Oct.2006 p.9/33
29 Semi-inclusive DIS σ ep ehx = q f p q σ eq eq D q h Dirk Ryckbosch, Feldberg, Oct.2006 p.9/33
30 Semi-inclusive DIS σ ep ehx = q f p q σ eq eq D q h Chiral-odd function? fragmentation Dirk Ryckbosch, Feldberg, Oct.2006 p.9/33
31 Semi-inclusive DIS σ ep ehx = q f p q σ eq eq D q h Chiral-odd fragmentation function? H 1 (z), H1 (z) Collins function Dirk Ryckbosch, Feldberg, Oct.2006 p.9/33
32 Collins effect Chiral odd fragmentation function Transverse spin of quark transverse motion of hadron Some estimates from LEP: < H1 > < D 1 > = 6.3%, 12.5%, 4%,...??? Can be negative! Correlation between direction of outgoing hadron and transverse spin of quark Manifests itself in Single-Spin Azimuthal asymmetry Dirk Ryckbosch, Feldberg, Oct.2006 p.10/33
33 Collins effect Artru-model for H 1, based on Lund fragmentation Leading pion goes into page H 1 depends on intrinsic k T Dirk Ryckbosch, Feldberg, Oct.2006 p.11/33
34 Single Spin Asymmetries e + p e + π + X study azimuthal distribution of π s: A(Φ) = N + (Φ) N (Φ) N + (Φ) + N (Φ) Φ = φ + φ S Collins angle Transversely polarized target: A sin Φ UT q e2qh q 1 (x)h,q 1 (z) q e2 qf q 1 (x)dq 1 (z) Dirk Ryckbosch, Feldberg, Oct.2006 p.12/33
35 Single Spin Asymmetries Cross section contains convolution integral over intrinsic p T and k T Can be rewritten assuming Gaussian dependence on momenta A h UT = 2 S T sin(φ) B(y) q (e2 qh q 1 1 (x)h ( ) 2 1 (z))/ < p 2 T > + < k2 T > A(x, y) q e2 qf q 1 (x)dq 1 (z) A(x, y), B(y): kinematic factors Dirk Ryckbosch, Feldberg, Oct.2006 p.13/33
36 Collins asymmetries 2 sin(φ+φ S ) π UT π + III HERMES PRELIMINARY virtual photon asymmetry amplitudes not corrected for acceptance and smearing Collins 0 2 sin(φ+φ S ) π UT π - 6.6% scale uncertainty Large negative for π unexpected x z P h [GeV] Dirk Ryckbosch, Feldberg, Oct.2006 p.14/33
37 Interpretation u-quark dominance (e 2 u = 4 e 2 d ) u > 0, d < 0 Expect: A π+ A π0 > 0 A π 0 A π < A π+ Dirk Ryckbosch, Feldberg, Oct.2006 p.15/33
38 Interpretation Unknowns: h u 1 (x), hd 1 (x), H 1 (z) Collins function: Favoured: u π +, d π, ū π, d π + Disfavoured: u π, d π +, ū π +, d π A π+ = k (4hu 1 + h d 1 )H f + (h d 1 + 4hū1 )H d (4u + d)d f + (d + 4ū)D d Dirk Ryckbosch, Feldberg, Oct.2006 p.15/33
39 Interpretation Define: r = d+4ū u+ d/4 η = D d D f δr = hd 1+4hū1 h u 1 +h d 1 /4 η H = H d H f α = Aπ A π+ Use relations like η H = δr 4α (4η + r)/(4 + ηr) α (4η + r)/(4 + ηr)δr 4 Dirk Ryckbosch, Feldberg, Oct.2006 p.15/33
40 Collins: interpretation Solution space for δr h d 1 /hu 1 and η H = H d /H f Dirk Ryckbosch, Feldberg, Oct.2006 p.16/33
41 Collins: interpretation Solution space for δr h d 1 /hu 1 and η H = H d /H f At δr = 0.93 (χqsm) strong indication for H d H f Dirk Ryckbosch, Feldberg, Oct.2006 p.16/33
42 Disfavoured Collins fragmentation Artru-model for H 1, based on Lund fragmentation Assume u-quark hit Leading pion (π + ) goes into page Second pion (probably π ) comes out of page Dirk Ryckbosch, Feldberg, Oct.2006 p.17/33
43 COMPASS measurements COMPASS has data at higher Q 2 on D-target A Coll 0.2 all hadrons leading hadrons leading hadrons leading hadrons A Siv all hadrons leading hadrons leading hadrons leading hadrons x x z h p T [GeV/c] Consistent with 0 Large cancellations between p and n Dirk Ryckbosch, Feldberg, Oct.2006 p.18/33
44 COMPASS vs HERMES data COMPASS at lower x transversity is valence phenomenon Dirk Ryckbosch, Feldberg, Oct.2006 p.19/33
45 COMPASS vs HERMES data COMPASS at lower x transversity is valence phenomenon COMPASS uses deuterium target Dirk Ryckbosch, Feldberg, Oct.2006 p.19/33
46 COMPASS vs HERMES data COMPASS at lower x transversity is valence phenomenon COMPASS uses deuterium target Dirk Ryckbosch, Feldberg, Oct.2006 p.19/33
47 Collins fragmentation function Need independent measurement of H 1 e + e collider experiment BELLE analyses transverse momentum dependent fragmentation Dirk Ryckbosch, Feldberg, Oct.2006 p.20/33
48 Collins fragmentation function Need independent measurement of H 1 e + e collider experiment BELLE analyses transverse momentum dependent fragmentation BELLE, April 2005 H 1 0 Dirk Ryckbosch, Feldberg, Oct.2006 p.20/33
49 Quark Fragmentation functions Dirk Ryckbosch, Feldberg, Oct.2006 p.21/33
50 Quark Fragmentation functions Functions surviving integration over intrinsic transv. momentum Dirk Ryckbosch, Feldberg, Oct.2006 p.21/33
51 Quark Fragmentation functions Functions surviving integration over intrinsic transv. momentum Collins function Dirk Ryckbosch, Feldberg, Oct.2006 p.21/33
52 Quark Distribution functions Dirk Ryckbosch, Feldberg, Oct.2006 p.22/33
53 Quark Distribution functions Distributions surviving integration over intrinsic k T Dirk Ryckbosch, Feldberg, Oct.2006 p.22/33
54 Quark Distribution functions Distributions surviving integration over intrinsic k T Transversity Dirk Ryckbosch, Feldberg, Oct.2006 p.22/33
55 Quark Distribution functions Distributions surviving integration over intrinsic k T Transversity Sivers function Dirk Ryckbosch, Feldberg, Oct.2006 p.22/33
56 Sivers effect Correlation between intrinsic k T and S T Time-reversal odd! Until 2002 believed not to exist in DIS Requires interference in initial state Also Collins FF is T-odd Dirk Ryckbosch, Feldberg, Oct.2006 p.23/33
57 Sivers effect T-odd functions require interference: intuitive for fragmentation function distribution function? Brodsky, Hwang, Schmidt (2002) with Not higher twist, but soft gluon necessary for gauge invariance ( gauge links ) universality of distribution function? Dirk Ryckbosch, Feldberg, Oct.2006 p.24/33
58 Universality, factorization Distribution, fragmentation functions universal? SIDIS e + e Drell-Yan Universality (almost) proven Collins et al T-odd functions change sign from spacelike to timelike Dirk Ryckbosch, Feldberg, Oct.2006 p.25/33
59 T-odd and Single Spin Asymmetries QCD allows T-odd structures??? No, this is naive/artificial time-reversal P p p J J no interchange T p p J J 1 2 A T p p J J no interchange S ( p 1 p 2 ) + under P + under T - under A T Single Spin Asymmetries Dirk Ryckbosch, Feldberg, Oct.2006 p.26/33
60 Sivers effect Burkardt-model for Sivers function: Spatial distortion of q-distribution Attractive QCD-potential Generates SSA Implies L z of quarks Currently no easy way to extract L z from Sivers Dirk Ryckbosch, Feldberg, Oct.2006 p.27/33
61 Collins and Sivers asymmetries Transversely polarized target Sivers < sin(φ φ S ) > moment Collins < sin(φ + φ S ) > moment Dirk Ryckbosch, Feldberg, Oct.2006 p.28/33
62 Collins and Sivers asymmetries Transversely polarized target Sivers < sin(φ φ S ) > moment Collins < sin(φ + φ S ) > moment f 1T (x) D 1(z) h 1 (x) H 1 (z) Dirk Ryckbosch, Feldberg, Oct.2006 p.28/33
63 Collins and Sivers asymmetries Transversely polarized target Sivers < sin(φ φ S ) > moment Collins < sin(φ + φ S ) > moment f 1T (x) D 1(z) h 1 (x) H 1 (z) Dirk Ryckbosch, Feldberg, Oct.2006 p.28/33
64 Sivers asymmetries 2 sin(φ-φ S ) π UT π + III HERMES PRELIMINARY not corrected for acceptance and smearing Sivers 0 2 sin(φ-φ S ) π UT π - 6.6% scale uncertainty As expected for u- quark dominance x z P h [GeV] Dirk Ryckbosch, Feldberg, Oct.2006 p.29/33
65 Longitudinally polarized target HERMES has taken much more data with longitudinally polarized target Dirk Ryckbosch, Feldberg, Oct.2006 p.30/33
66 Longitudinally polarized target HERMES has taken much more data with longitudinally polarized target % transverse component in S SSA observed Dirk Ryckbosch, Feldberg, Oct.2006 p.30/33
67 Longitudinally polarized target HERMES has taken much more data with longitudinally polarized target % transverse component in S SSA observed Dirk Ryckbosch, Feldberg, Oct.2006 p.30/33
68 Longitudinally polarized target HERMES has taken much more data with longitudinally 0.08 e polarized target d e π + X 0.06 e p e π + X 0.04 sinφ A UL e d e π 0 X e p e π 0 X 0.04 sinφ A UL e d e π - X e p e π - X 10-15% transverse component in S e d SSA observed e K + X sinφ A UL sinφ A UL x P t [GeV/c] z Dirk Ryckbosch, Feldberg, Oct.2006 p.30/33
69 Higher twist With transversely polarized target: A sin(φ+φ S) UT P T (h 1 H 1 ) A sin(φ φ S) UT P T (f 1T D 1) With longitudinally polarized target: A sin φ UL S L (g 1 G /z + xh L H 1 + h 1L H/z xf L D 1) h 1L : probably small G, H: twist-3 FF +S T (h 1 H 1 f 1T D 1) h L, f L : twist-3; Wandzura-Wilczek like relation to h 1, f 1 S T : kinematically suppressed; 10 15% Dirk Ryckbosch, Feldberg, Oct.2006 p.31/33
70 Higher twist distributions Classic example: g 2 (x) W µν = g µν F 1 (x, Q 2 ) + pµ p ν ν F 2(x, Q 2 ) +iɛ µνλσ q λν (S σ g 1 (x, Q 2 ) + 1 ν (p qs σ S qp σ )g 2 (x, Q 2 )) g 2 (x) = g 1 (x) + 1 x g 1(x )dx /x + g 2 (x) = g W W 2 (x) + g 2 (x) g 2 (x): twist-3 operator SLAC, JLAB data g 2 n This work P. Anthony et al. X. Zheng et al. g 1.0 (GeV/c) 2 M. Stratmann X. Song H. Weigel et al. M. Wakamatsu x 10 1 Dirk Ryckbosch, Feldberg, Oct.2006 p.32/33
71 Higher twist 2 sinφ Combined analysis of transverse and longitudinal data: 0.05 π sinφ π - q 2 sinφ UL l 2 sinφ UL l UT l UT -2sinθ γ * ( sin(φ+φ S ) + sin(φ-φ S ) ) A UL S T contribution Twist-3 contribution x z Large twist-3 contribution Dirk Ryckbosch, Feldberg, Oct.2006 p.33/33
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