Transverse Force Tomography Fatma Aslan, Matthias Burkardt, Marc - PowerPoint PPT Presentation

Transverse Force Tomography Fatma Aslan, Matthias Burkardt, Marc Schlegel New Mexico State University May 14, 2019

Motivation: Why Twist 3 GPDs 2 disentangling x − ξ dependence of GPDs from QCD evolution dx GP Ds ( x,ξ,t ) � ℜA DV CS − → x ± ξ ℑA DV CS − → GPDs ( ξ, ξ, t ) polynomiality insufficient to uniquely extract GPDs ( x, ξ, t ) additional input needed to disentangle x − ξ dependence → Q 2 dependence described by known evolution kernels! ֒ → use Q 2 dependence of A DV CS to further constrain GPDs ( x, ξ, t ) ֒ twist-3 effects may need to be included in such a program! nucleon structure twist-2 observables tell us what nucleon structure is twist-3 helps us understand what makes nucleon structure transverse force tomography (this talk) twist-3 PDFs: d 2 − →⊥ force twist-3 GPDs: ⊥ position space resolved force

Motivation: Why Twist 3 GPDs 3 weird stuff going on at twist 3 − → Fatma Aslan discontinuities in GPDs δ ( x ) in PDFs yes, measuring twist-3 GPDs will be hard, but... lattice QCD can provide (genuine) twist 3 info much sooner

Outline 4 motivation twist-3 PDFs − → ’the force’ GPDs − → 3D imaging of the nucleon → twist-3 GPDs − → distribution of ’the force’ in ֒ ⊥ plane summary

Twist-3 PDFs − →⊥ Force on Quarks in DIS 5 d 2 ↔ average ⊥ force on quark in DIS from ⊥ pol target polarized DIS: σ LL ∝ g 1 − 2 Mx ν g 2 σ LT ∝ g T ≡ g 1 + g 2 1 → ’clean’ separation between g 2 and Q 2 corrections to g 1 ֒ � 1 dy g 2 = g W W g 2 with g W W + ¯ ( x ) ≡ − g 1 ( x ) + y g 1 ( y ) 2 2 x � 1 � P, S dx x 2 ¯ � q (0) γ + gF + y (0) q (0) � d 2 ≡ 3 � � g 2 ( x ) = P, S � ¯ 2 MP +2 S x color Lorentz Force on ejected quark (MB, PRD 88 (2013) 114502) √ � y � 2 F + y = F 0 y + F zy = − E y + B x = − � v × � E + � for � v = (0 , 0 , − 1) B matrix element defining d 2 ↔ 1 st integration point in QS-integral d 2 ⇒ ⊥ force ↔ QS-integral ⇒ ⊥ impulse magnitude of d 2 sign of d 2 � F y � = − 2 M 2 d 2 = − 10 GeV ⊥ deformation of q ( x, b ⊥ ) fm d 2 → sign of d q ֒ 2 : opposite Sivers |� F y �| ≪ σ ≈ 1 GeV fm ⇒ d 2 = O (0 . 01)

Twist-3 PDFs − →⊥ Force on Quarks in DIS 5 d 2 ↔ average ⊥ force on quark in DIS from ⊥ pol target polarized DIS: σ LL ∝ g 1 − 2 Mx ν g 2 σ LT ∝ g T ≡ g 1 + g 2 1 → ’clean’ separation between g 2 and Q 2 corrections to g 1 ֒ � 1 dy g 2 = g W W g 2 with g W W + ¯ ( x ) ≡ − g 1 ( x ) + y g 1 ( y ) 2 2 x � 1 � P, S dx x 2 ¯ � q (0) γ + gF + y (0) q (0) � d 2 ≡ 3 � � g 2 ( x ) = P, S � ¯ 2 MP +2 S x color Lorentz Force on ejected quark (MB, PRD 88 (2013) 114502) √ � y � 2 F + y = F 0 y + F zy = − E y + B x = − � v × � E + � for � v = (0 , 0 , − 1) B magnitude of d 2 sign of d 2 � F y � = − 2 M 2 d 2 = − 10 GeV ⊥ deformation of q ( x, b ⊥ ) fm d 2 → sign of d q ֒ 2 : opposite Sivers |� F y �| ≪ σ ≈ 1 GeV fm ⇒ d 2 = O (0 . 01) consitent with experiment (JLab,SLAC), model calculations (Weiss), and lattice QCD calculations (G¨ ockeler et al., 2005)

’The Force’ in Lattice QCD (M.Engelhardt) 6 f ⊥ 1 T ( x, k ⊥ ) is k ⊥ -odd term in quark-spin averaged momentum distribution in ⊥ polarized target Force Operator W.Armstrong, F.Aslan, MB, S.Liuti, M.Engelhardt slope at length =0

Twist-3 PDFs − →⊥ Force on Quarks in DIS 7 chirally even spin-dependent twist-3 PDF g 2 ( x ) MB, PRD 88 (2013) 114502 � dx x 2 g 2 ( x ) ⇒⊥ force on unpolarized quark in ⊥ polarized target ֒ → ‘Sivers force’ scalar twist-3 PDF e ( x ) MB, PRD 88 (2013) 114502 � dx x 2 e ( x ) ⇒⊥ force on ⊥ polarized quark in unpolarized target → ‘Boer-Mulders force’ ֒ chirally odd spin-dependent twist-3 PDF h 2 ( x ) M.Abdallah & MB, PRD94 (2016) 094040 dx x 2 h 2 ( x ) = 0 � ֒ → ⊥ force on ⊥ pol. quark in long. pol. target vanishes due to parity dx x 3 h 2 ( x ) ⇒ long. gradient of ⊥ force on ⊥ polarized quark in � long. polarized target → chirally odd ‘wormgear force’ ֒

Physics of GPDs: 3D Imaging MB,PRD 62, 071503 (2000) 8 unpolarized proton � d 2 ∆ ⊥ (2 π ) 2 H ( x, 0 , − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ q ( x, b ⊥ ) = → probabilistic interpretation ֒ F 1 ( − ∆ 2 � dxH ( x, 0 , − ∆ 2 ⊥ ) = ⊥ ) x = momentum fraction of the quark b ⊥ relative to ⊥ center of momentum small x : large ’meson cloud’ larger x : compact ’valence core’ x → 1: active quark = center of momentum → � ֒ b ⊥ → 0 (narrow distribution) for x → 1

Physics of GPDs: 3D Imaging MB, IJMPA 18, 173 (2003) 9 proton polarized in +ˆ x direction � d 2 ∆ ⊥ (2 π ) 2 H q ( x, 0 , − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ q ( x, b ⊥ ) = � d 2 ∆ ⊥ − 1 ∂ (2 π ) 2 E q ( x, 0 , − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ 2 M ∂b y relevant density in DIS is j + ≡ j 0 + j z and left-right asymmetry from j z av. shift model-independently related to anomalous magnetic moments: � b q � � d 2 b ⊥ q ( x, b ⊥ ) b y y � ≡ dx κ q 1 = � dxE q ( x, 0 , 0) = 2 M 2 M

twist-2 GPDs → Impact Parameter Dependent PDFs10 ⊥ localized state � | R ⊥ = 0 , p + , Λ � ≡ N d 2 p ⊥ | p ⊥ , p + , Λ � ⊥ charge distribution (unpolarized quarks) ρ Λ ′ Λ ( b ⊥ ) ≡ � R ⊥ = 0 , p + , Λ ′ | ¯ q ( b ⊥ ) γ + q ( b ⊥ ) | R ⊥ = 0 , p + , Λ � � � = |N| 2 q (0) γ + q (0) | p ⊥ , p + , Λ � e i b ⊥ · ( p ⊥ − p ′ d 2 p ⊥ d 2 p ′ ⊥ , p + , Λ ′ | ¯ ⊥ ) ⊥ � p ′ � � = |N| 2 2 P + d 2 P ⊥ d 2 ∆ ⊥ F Λ ′ Λ ( − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ � d 2 ∆ ⊥ F Λ ′ Λ ( − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ = crucial: � p ′ ⊥ , p + , Λ ′ | ¯ q (0) γ + q (0) | p ⊥ , p + , Λ � depends only on ∆ ⊥ F Λ ′ Λ ( − ∆ 2 ⊥ ) some linear combination of F 1 & F 2 - depending on Λ, Λ ′ similar for various polarized quark densities similar for x -dependent densities − → GPDs

Digression: Why not 3D Fourier Transforms? 11 localized state - an attempt (for simplicity no spin) d 3 p | � √ R = 0 � ≡ N � p ) | � p � 2 ω ( � charge distribution in that state � � r ) γ 0 q ( � r ) | � R = 0 | ¯ q ( � R = 0 � d 3 p ′ d 3 p � p ′ )) F ( t ) ∼ ( ω ( � p ) + ω ( � � � p ′ ) 2 ω ( � 2 ω ( � p ) p ′ )) 2 − � ∆ 2 additional ω s in t = ( ω ( � p ) − ω ( � � d 3 ∆ and � d 3 P → not possible to factorize into ֒ p ′ ) and call it Breit ’frame’ except if you simply assume ω ( � p ) = ω ( � not possible to construct state in which the charge distribution equals ∆ 2 ) e − i� d 3 ∆ F ( − � � ∆ · � r the 3D Fourier transform of the form factor

Transverse Force Tomography F.Aslan,MB,M.Schlegel arxiv:1904.03494 12 ⊥ force distribution (unpolarized quarks) F i Λ ′ Λ ( b ⊥ ) ≡ � R ⊥ = 0 , p + , Λ ′ | ¯ q ( b ⊥ ) γ + gF + i ( b ⊥ ) q ( b ⊥ ) | R ⊥ = 0 , p + , Λ � � � q (0) γ + gF + i (0) q (0) | p ⊥ , p + , Λ � e i b ⊥ · ( p ⊥ − p ′ = |N| 2 d 2 p ⊥ d 2 p ′ ⊥ � p ′ ⊥ , p + , Λ | ¯ ⊥ ) Form factors of qgq correlator (F.Aslan, M.B., M.Schlegel arXiv:1904.03494) � P + M γ + ∆ i M Φ 1 ( t )+ P + � p ′ , λ ′ | ¯ q (0) γ + igF + i (0) q (0) | p, λ � = ¯ u ( p ′ , λ ′ ) M iσ + i Φ 2 ( t ) + P + iσ +∆ Φ 3 ( t ) + P + ∆ + iσ i ∆ M Φ 4 ( t ) + P ⊥ ∆ + iσ +∆ ∆ i � Φ 5 ( t ) u ( p, λ ) . M 3 M M M M M crucial: for p + ′ = p + , � p ′ , λ ′ | ¯ q (0) γ + igF + i (0) q (0) | p, λ � only depends on ∆ ⊥ → similar to ⊥ charge density ... ֒

Transverse Force Tomography F.Aslan,MB,M.Schlegel arxiv:1904.03494 12 ⊥ force distribution (unpolarized quarks) F i Λ ′ Λ ( b ⊥ ) ≡ � R ⊥ = 0 , p + , Λ ′ | ¯ q ( b ⊥ ) γ + gF + i ( b ⊥ ) q ( b ⊥ ) | R ⊥ = 0 , p + , Λ � � � q (0) γ + gF + i (0) q (0) | p ⊥ , p + , Λ � e i b ⊥ · ( p ⊥ − p ′ = |N| 2 d 2 p ⊥ d 2 p ′ ⊥ � p ′ ⊥ , p + , Λ | ¯ ⊥ ) Form factors of qgq correlator (F.Aslan, M.B., M.Schlegel arXiv:1904.03494) � P + M γ + ∆ i M Φ 1 ( t )+ P + � p ′ , λ ′ | ¯ q (0) γ + igF + i (0) q (0) | p, λ � = ¯ u ( p ′ , λ ′ ) M iσ + i Φ 2 ( t ) + P + iσ +∆ Φ 3 ( t ) + P + ∆ + iσ i ∆ M Φ 4 ( t ) + P ⊥ ∆ + iσ +∆ ∆ i � Φ 5 ( t ) u ( p, λ ) . M 3 M M M M M Φ 1 unpolarized target axially symmetric ’radial’ force

Transverse Force Tomography F.Aslan,MB,M.Schlegel arxiv:1904.03494 12 ⊥ force distribution (unpolarized quarks) F i Λ ′ Λ ( b ⊥ ) ≡ � R ⊥ = 0 , p + , Λ ′ | ¯ q ( b ⊥ ) γ + gF + i ( b ⊥ ) q ( b ⊥ ) | R ⊥ = 0 , p + , Λ � � � q (0) γ + gF + i (0) q (0) | p ⊥ , p + , Λ � e i b ⊥ · ( p ⊥ − p ′ = |N| 2 d 2 p ⊥ d 2 p ′ ⊥ � p ′ ⊥ , p + , Λ | ¯ ⊥ ) Form factors of qgq correlator (F.Aslan, M.B., M.Schlegel arXiv:1904.03494) � P + M γ + ∆ i M Φ 1 ( t )+ P + � p ′ , λ ′ | ¯ q (0) γ + igF + i (0) q (0) | p, λ � = ¯ u ( p ′ , λ ′ ) M iσ + i Φ 2 ( t ) + P + iσ +∆ Φ 3 ( t ) + P + ∆ + iσ i ∆ M Φ 4 ( t ) + P ⊥ ∆ + iσ +∆ ∆ i � Φ 5 ( t ) u ( p, λ ) . M 3 M M M M M Φ 2 ⊥ polarized target; force ⊥ to target spin → spatially resolved Sivers force ֒