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■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ❘❡✈❡❛❧✐♥❣ tr❛♥s✈❡rs✐t② ●P❉s t❤r♦✉❣❤ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❛ r❤♦ ♠❡s♦♥ ❛♥❞ ❛ ♣❤♦t♦♥ ❘❡♥❛✉❞ ❇♦✉ss❛r✐❡ ▲❛❜♦r❛t♦✐r❡ ❞❡ P❤②s✐q✉❡ ❚❤é♦r✐q✉❡ ❖rs❛② P❤♦t♦♥ ✷✵✶✺ ❇✉❞❦❡r ■♥st✐t✉t❡✱ ◆♦✈♦s✐❜✐rs❦ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❇✳ P✐r❡ ✭❈P❤❚✱ P❛❧❛✐s❡❛✉✮✱ ▲✳ ❙③②♠❛♥♦✇s❦✐ ✭◆❈❇❏✱ ❲❛rs❛✇✮✱ ❙✳ ❲❛❧❧♦♥ ✭▲P❚ ❖rs❛② ❛♥❞ ❯P▼❈✮

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ❚r❛♥s✈❡rs✐t② ♦❢ t❤❡ ♥✉❝❧❡♦♥ ✉s✐♥❣ ❤❛r❞ ♣r♦❝❡ss❡s ❲❤❛t ✐s tr❛♥s✈❡rs✐t②❄ ❚r❛♥s✈❡rs❡ s♣✐♥ ❝♦♥t❡♥t ♦❢ t❤❡ ♣r♦t♦♥✿ | ↑� ( x ) ∼ | →� + | ←� | ↓� ( x ) ∼ | →� − | ←� s♣✐♥ ❛❧♦♥❣ x ❤❡❧✐❝✐t② st❛t❡s ❖❜s❡r✈❛❜❧❡s ✇❤✐❝❤ ❛r❡ s❡♥s✐t✐✈❡ t♦ ❤❡❧✐❝✐t② ✢✐♣ t❤✉s ❣✐✈❡ ❛❝❝❡ss t♦ tr❛♥s✈❡rs✐t② ∆ T q ( x ) ✳ P♦♦r❧② ❦♥♦✇♥ ❚r❛♥s✈❡rs✐t② ●P❉s ❛r❡ ❝♦♠♣❧❡t❡❧② ✉♥❦♥♦✇♥ ❋♦r ♠❛ss❧❡ss ✭❛♥t✐✮♣❛rt✐❝❧❡s✱ ❝❤✐r❛❧✐t② ❂ ✭✲✮❤❡❧✐❝✐t② ❚r❛♥s✈❡rs✐t② ✐s t❤✉s ❛ ❝❤✐r❛❧✲♦❞❞ q✉❛♥t✐t② ❙✐♥❝❡ ◗❈❉ ❛♥❞ ◗❊❉ ❛r❡ ❝❤✐r❛❧ ❡✈❡♥✱ t❤❡ ❝❤✐r❛❧ ♦❞❞ q✉❛♥t✐t✐❡s ✇❤✐❝❤ ♦♥❡ ✇❛♥t t♦ ♠❡❛s✉r❡ s❤♦✉❧❞ ❛♣♣❡❛r ✐♥ ♣❛✐rs

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ❚r❛♥s✈❡rs✐t② ♦❢ t❤❡ ♥✉❝❧❡♦♥ ✉s✐♥❣ ❤❛r❞ ♣r♦❝❡ss❡s✿ ✉s✐♥❣ ❛ t✇♦ ❜♦❞② ✜♥❛❧ st❛t❡ ♣r♦❝❡ss❄ ❍♦✇ t♦ ❣❡t ❛❝❝❡ss t♦ tr❛♥s✈❡rs✐t② ●P❉s❄ t❤❡ ❞♦♠✐♥❛♥t ❉❆ ♦❢ ρ T ✐s ♦❢ t✇✐st ✷ ❛♥❞ ❝❤✐r❛❧ ♦❞❞ ✭ [ γ µ , γ ν ] ❝♦✉♣❧✐♥❣✮ ✉♥❢♦rt✉♥❛t❡❧② γ ∗ N ↑ → ρ T N ′ = 0 ❚❤✐s ❝❛♥❝❡❧❧❛t✐♦♥ ✐s tr✉❡ ❛t ❛♥② ♦r❞❡r ✿ s✉❝❤ ❛ ♣r♦❝❡ss ✇♦✉❧❞ r❡q✉✐r❡ ❛ ❤❡❧✐❝✐t② tr❛♥s❢❡r ♦❢ ✷ ❢r♦♠ ❛ ♣❤♦t♦♥✳ ❧♦✇❡st ♦r❞❡r ❞✐❛❣r❛♠♠❛t✐❝ ❛r❣✉♠❡♥t✿ γ α [ γ µ , γ ν ] γ α → 0

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ❚r❛♥s✈❡rs✐t② ♦❢ t❤❡ ♥✉❝❧❡♦♥ ✉s✐♥❣ ❤❛r❞ ♣r♦❝❡ss❡s✿ ✉s✐♥❣ ❛ t✇♦ ❜♦❞② ✜♥❛❧ st❛t❡ ♣r♦❝❡ss❄ ❈❛♥ ♦♥❡ ❝✐r❝✉♠✈❡♥t t❤✐s ✈❛♥✐s❤✐♥❣❄ ❚❤✐s ✈❛♥✐s❤✐♥❣ ✐s ♦♥❧② ♦❝❝✉rs ❛t t✇✐st ✷ ❆t t✇✐st ✸ t❤✐s ♣r♦❝❡ss ❞♦❡s ♥♦t ✈❛♥✐s❤ ❬●♦❧♦s❦♦❦♦✈✱ ❑r♦❧❧❪✱ ❬❆❤♠❛❞✱ ●♦❧❞st❡✐♥✱ ▲✐✉t✐❪ ❍♦✇❡✈❡r ♣r♦❝❡ss❡s ✐♥✈♦❧✈✐♥❣ t✇✐st ✸ ❉❆s ♠❛② ❢❛❝❡ ♣r♦❜❧❡♠s ✇✐t❤ ❢❛❝t♦r✐③❛t✐♦♥ ✭❡♥❞✲♣♦✐♥t s✐♥❣✉❧❛r✐t✐❡s✮ ❖♥❡ ❝❛♥ ❛❧s♦ ❝♦♥s✐❞❡r ❛ ✸✲❜♦❞② ✜♥❛❧ st❛t❡ ♣r♦❝❡ss ❬■✈❛♥♦✈✱ P✐r❡✱ ❙③②♠❛♥♦✇s❦✐✱ ❚❡r②❛❡✈❪✱ ❬❊❧ ❇❡✐②❛❞✱ P✐r❡✱ ❙❡❣♦♥❞✱ ❙③②♠❛♥♦✇s❦✐✱ ❲❛❧❧♦♥❪

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ Pr♦❜✐♥❣ t❤❛♥s✈❡rs✐t② ✉s✐♥❣ r❤♦ ♠❡s♦♥ ♣r♦❞✉❝t✐♦♥ Pr♦❝❡ss❡s ✇✐t❤ ✸ ❜♦❞② ✜♥❛❧ st❛t❡s ❝❛♥ ❣✐✈❡ ❛❝❝❡ss t♦ ❛❧❧ ●P❉s ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❝❡ss γ N → γ ρ N ′ ❈♦❧❧✐♥❡❛r ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ t❤❡ ❛♠♣❧✐t✉❞❡ ❢♦r γ + N → γ + ρ + N ′ ❛t ❧❛r❣❡ M 2 γρ t ′ M 2 γρ H ρ GPD t ❚②♣✐❝❛❧ ♥♦♥✲③❡r♦ ❞✐❛❣r❛♠ ❢♦r ❛ tr❛♥s✈❡rs❡ ρ ♠❡s♦♥ ❋❛❝t♦r✐③❡❞ ❛♠♣❧✐t✉❞❡

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ▼❛st❡r ❢♦r♠✉❧❛ ❜❛s❡❞ ♦♥ ❧❡❛❞✐♥❣ t✇✐st ✷ ❢❛❝t♦r✐③❛t✐♦♥ � 1 � 1 1 dz ( T u ( x, z ) − T d ( x, z )) √ A = dx 2 − 1 0 ( H u T ( x, ξ, t ) − H d × T ( x, ξ, t ))Φ ρ ( z ) + · · · ❇♦t❤ t❤❡ ❉❆ ❛♥❞ t❤❡ ●P❉ ❝❛♥ ❜❡ ❡✐t❤❡r ❝❤✐r❛❧ ❡✈❡♥ ♦r ❝❤✐r❛❧ ♦❞❞✳ ❆t t✇✐st ✷ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❤♦ ❉❆ ✐s ❝❤✐r❛❧ ❡✈❡♥ ❛♥❞ t❤❡ tr❛♥s✈❡rs❡ r❤♦ ❉❆ ✐s ❝❤✐r❛❧ ♦❞❞✳ ❍❡♥❝❡ ✇❡ ✇✐❧❧ ♥❡❡❞ ❜♦t❤ ❝❤✐r❛❧ ❡✈❡♥ ❛♥❞ ❝❤✐r❛❧ ♦❞❞ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❛♥❞ ❤❛r❞ ♣❛rts✳

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ◆♦♥ ♣❡rt✉r❜❛t✐✈❡ ❝❤✐r❛❧ ♦❞❞ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❍❡❧✐❝✐t② ✢✐♣ ●P❉ ❛t t✇✐st ✷ ✿ � dz − � − 1 � � 1 � 4 π e ixP + z − � p 2 , λ 2 | ¯ 2 z − iσ + i ψ 2 z − ψ q | p 1 , λ 1 � T ( x, ξ, t ) P + ∆ i − ∆ + P i 1 � T ( x, ξ, t ) iσ + i + ˜ H q H q = 2 P + ¯ u ( p 2 , λ 2 ) M 2 N T ( x, ξ, t ) γ + ∆ i − ∆ + γ i T ( x, ξ, t ) γ + P i − P + γ i � E q + ˜ E q + u ( p 1 , λ 1 ) 2 M N M N ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ s✐♠♣❧❡st ❝❛s❡ ✇❤❡♥ ∆ ⊥ = 0 ✳ ■♥ t❤❛t ❝❛s❡ ❛♥❞ ✐♥ t❤❡ ❢♦r✇❛r❞ ❧✐♠✐t ξ → 0 ♦♥❧② t❤❡ H q T t❡r♠ s✉r✈✐✈❡s✳ tr❛♥s✈❡rs✐t② ❉❆ ❛t t✇✐st ✷ ✿ � 1 i ρ p ν − σ ν du e − iup · x φ ⊥ ( u ) u (0) σ µν u ( x ) | ρ 0 ( p, s ) � = ( σ µ ρ p µ ) f ⊥ � 0 | ¯ √ ρ 2 0

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ◆♦♥ ♣❡rt✉r❜❛t✐✈❡ ❝❤✐r❛❧ ❡✈❡♥ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❍❡❧✐❝✐t② ❝♦♥s❡r✈✐♥❣ ●P❉s ❛t t✇✐st ✷ ✿ � dz − � − 1 � � 1 � 4 π e ixP + z − � p 2 , λ 2 | ¯ 2 z − γ + ψ 2 z − ψ q | p 1 , λ 1 � H q ( x, ξ, t ) γ + + E q ( x, ξ, t ) iσ α + ∆ α 1 � � = 2 P + ¯ u ( p 2 , λ 2 ) 2 m � dz − � − 1 � � 1 � 4 π e ixP + z − � p 2 , λ 2 | ¯ 2 z − γ + γ 5 ψ 2 z − ψ q | p 1 , λ 1 � E q ( x, ξ, t ) γ 5 ∆ + 1 � � H q ( x, ξ, t ) γ + γ 5 + ˜ ˜ = 2 P + ¯ u ( p 2 , λ 2 ) 2 m

■♥tr♦❞✉❝t✐♦♥ ❆❝❝❡ss t♦ ●P❉ t❤r♦✉❣❤ ❛ ✸ ❜♦❞② ✜♥❛❧ st❛t❡ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❤❛r❞ ♣❛rt ❈♦♥❝❧✉s✐♦♥ ◆♦♥ ♣❡rt✉r❜❛t✐✈❡ ❝❤✐r❛❧ ❡✈❡♥ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ❍❡❧✐❝✐t② ❝♦♥s❡r✈✐♥❣ ❉❆s ❛t t✇✐st ✷ ✿ � 1 p µ ǫ.x u (0) γ µ u ( x ) | ρ 0 ( p, s ) � du e − iup · x φ � ( u ) � 0 | ¯ = √ p.xf ρ m ρ 2 0 � 1 1 due − iup · x g ( a ) u (0) γ µ γ 5 u ( x ) | ρ 0 ( p, s ) � = − ǫ µνσδ ǫ ν p σ x δ f ρ m ρ � 0 | ¯ √ ⊥ ( u ) 4 2 0