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❈❤❛r❣❡ ❛s②♠♠❡tr② ♦❢ ❤✐❣❤ ❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❛ ❤❡❛✈② ❛t♦♠ P❡t❡r ❑r❛❝❤❦♦✈ ❇✉❞❦❡r ■♥st✐t✉t❡ ♦❢ ◆✉❝❧❡❛r P❤②s✐❝s ◆♦✈♦s✐❜✐rs❦ ❙t❛t❡ ❯♥✐✈❡rs✐t② ✶✾ ❏✉♥❡✱ ✷✵✶✺ P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶ ✴ ✶✻

P❧❛♥ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✸ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ✹ ❈♦♥❝❧✉s✐♦♥ P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✷ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❚②♣✐❝❛❧ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥❞✐t✐♦♥s ❍✐❣❤ ❡♥❡r❣② E ≫ m ❙♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝ ❛♥❣❧❡ θ ≪ 1 ▲❛r❣❡ ❛t♦♠✐❝ ❝❤❛r❣❡ ♥✉♠❜❡r = ⇒ Z ≫ 1 θ ≪ 1 P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✸ ✴ ✶✻

◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❋✉rr② r❡♣r❡s❡♥t❛t✐♦♥ ❞✐❛❣r❛♠ t❡❝❤♥✐q✉❡ ❋✉rr② r❡♣r❡s❡♥t❛t✐♦♥ ❛❧❧♦✇s t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❡①❛❝t❧② t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ t❤❡ ❡①t❡r♥❛❧ ✜❡❧❞ r 1 r 2 ε 1 G ( r 1 , r 2 | ε ) = � r 2 | → P − m + i 0 | r 1 � ˆ p r u in → p ( r ) G ( r 1 , r 2 | ε ) = ( ˆ P = γ µ P µ ˆ P + m ) D ( r 2 , r 1 | ε ) P µ = ( ε − V ( r ) , i ∇ ∇ ∇ ) 1 D ( r 2 , r 1 | ε ) = � r 2 | | r 1 � P 2 − m 2 + i 0 ˆ P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✹ ✴ ✶✻

◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❤✐❣❤ ❡♥❡r❣② ♣r♦❝❡ss❡s ∆ τ ∼ 1 / m ✖ ❚❤❡ ✈✐rt✉❛❧ ♣❛✐r ❧✐❢❡ t✐♠❡ ✐♥ t❤❡ ❝♦♠♦✈✐♥❣ ❢r❛♠❡ ∆ t = ∆ τγ ∼ E / m 2 ✖ ❚❤❡ ✈✐rt✉❛❧ ♣❛✐r ❧✐❢❡ t✐♠❡ ✐♥ t❤❡ ▲❋❘ ρ ∼ 1 / m ✖ ❚❤❡ ❧♦♦♣ tr❛♥s✈❡rs❡ s✐③❡ ✐♥ t❤❡ ▲❋❘ z = ∆ t ∼ E / m 2 ✖ ❚❤❡ ❧♦♦♣ ❧♦♥❣✐t✉❞✐♥❛❧ s✐③❡ ✐♥ t❤❡ ▲❋❘ l ∼ E ρ ∼ E / m ≫ 1 ✖ ❚❤❡ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✐♥ t❤❡ ▲❋❘ θ ∼ 1 / l ∼ m / E ≪ 1 ✖ ❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❛♥❣❧❡ ✐♥ t❤❡ ▲❋❘ P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✺ ✴ ✶✻

◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❚❤❡ q✉❛s✐❝❧❛ss✐❝❛❧ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ � 1 � 1 D ( r 2 , r 1 | ε )= ie ikr � �� � 1 − r � d 2 q exp iq 2 − ir α 0 dxV ( R x ) 0 dx α α ∇ ∇ ∇ 1 V ( R x ) 4 π 2 r 2 ε � R x = r 1 + x r + q xr / k ⇐ = q✉❛♥t✉♠ ✢✉❝t✉❛t✐♦♥s 2 x ¯ P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✻ ✴ ✶✻

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ▼❛tr✐① ❡❧❡♠❡♥t � d r ¯ k u − q ( r ) e ∗ γ γ u + p ( r ) e − i kr M = γ p q ❈r♦ss s❡❝t✐♦♥ d σ = αω q ε q ( 2 π ) 4 d Ω k d Ω q d ω | M | 2 , d σ ( p , q , k , η ) = d σ s ( p , q , k , η )+ d σ a ( p , q , k , η ) d σ s ( p , q , k , η ) = d σ ( p , q , k , η )+ d σ ( p , q , k , − η ) 2 d σ a ( p , q , k , η ) = d σ ( p , q , k , η ) − d σ ( p , q , k , − η ) 2 P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✼ ✴ ✶✻

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ 1 D ( r 2 , r 1 | ε ) = � r 2 | ∇ V ( r )+ i 0 | r 1 � H + i α α α · ∇ ∇ = � r 2 | 1 H − 1 ∇ V ( r ) 1 H + 1 ∇ V ( r ) 1 ∇ V ( r ) 1 H i α α α · ∇ ∇ H i α α α · ∇ ∇ H i α α α · ∇ ∇ H | r 1 � ∇ 2 + i 0 H = ( ε − V ( r )) 2 − m 2 + ∇ ∇ D ( r 2 , r 1 | ε ) = d 0 ( r 2 , r 1 )+ α α α · d 1 ( r 2 , r 1 )+ Σ Σ Σ · d 2 ( r 2 , r 1 ) d 0 ∼ l c d 1 ∼ l 2 l c ∼ ε / ∆ ≫ 1 c d 2 P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✽ ✴ ✶✻

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ 1 D ( r 2 , r 1 | ε ) = � r 2 | ∇ V ( r )+ i 0 | r 1 � H + i α α α · ∇ ∇ = � r 2 | 1 H − 1 ∇ V ( r ) 1 H + 1 ∇ V ( r ) 1 ∇ V ( r ) 1 H i α α α · ∇ ∇ H i α α α · ∇ ∇ H i α α α · ∇ ∇ H | r 1 � ∇ 2 + i 0 H = ( ε − V ( r )) 2 − m 2 + ∇ ∇ D ( r 2 , r 1 | ε ) = d 0 ( r 2 , r 1 )+ α α α · d 1 ( r 2 , r 1 )+ Σ Σ Σ · d 2 ( r 2 , r 1 ) d 0 ∼ l c d 1 ∼ l 2 l c ∼ ε / ∆ ≫ 1 c d 2 � dQ exp � 1 � iQ 2 − ir � d 0 ( r 2 , r 1 ) = ie i κ r 0 dxV ( R x ) 4 π 2 r � 1 x � 1 + ir 3 × � � dy ( x − y ) ∇ ∇ ∇ ⊥ V ( R x ) · ∇ ∇ ∇ ⊥ V ( R y ) dx 2 κ 0 0 P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✽ ✴ ✶✻

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ▼❛tr✐① ❡❧❡♠❡♥t µ p )[ N 0 ( e ∗ λ , ξ p p ⊥ − ξ q q ⊥ )+ N 1 ( e ∗ M = − δ µ p µ q ( ε p δ λµ p + ε q δ λ ¯ λ , ε p ξ p p ⊥ − ε q ξ q q ⊥ )] − 1 2 m µ p δ µ p ¯ µ q δ λµ p ( ε p − ε q )[ N 0 ( ξ p − ξ q )+ N 1 ( ε p ξ p − ε q ξ q )] √ 2 i � N 0 = dr exp [ − i ∆ ∆ ∆ · r − i χ ( ρ )] ∆ ∆ ∆ ⊥ · ∇ ∇ ∇ ⊥ V ( r ) ω m 2 ∆ 2 ⊥ ∞ 1 � � N 1 = dr exp [ − i ∆ ∆ ∆ · r − i χ ( ρ )] dxx ∇ ∇ ⊥ V ( r − x ν ∇ ν ν ) · ∇ ∇ ∇ ⊥ V ( r ) ω m 2 ε p ε q 0 � ∞ m 2 m 2 ρ χ ( ρ ) = − ∞ V ( z , ρ ρ ) dz ξ p = ξ q = ∆ ∆ ∆ = q + k − p m 2 + p 2 m 2 + q 2 ⊥ ⊥ P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✾ ✴ ✶✻

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ❙q✉❛r❡ ♦❢ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥t | M | 2 = S 0 + S 1 + S 2 ∑ λ µ q � ∆ 2 S 0 = m 2 | N 0 | 2 � m 2 ( ε 2 p + ε 2 q ) ξ p ξ q − 2 ε p ε q ( ξ p − ξ q ) 2 2 � S 1 = m 2 ❘❡ ( N 0 N ∗ 1 ) ∆ 2 m 2 ( ε 2 p + ε 2 q )( ε p + ε q ) ξ p ξ q 2 � � � ( ε 2 p + ε 2 + q )( ε p − ε q ) − 4 ε p ε q ( ε p ξ p − ε q ξ q ) ( ξ p − ξ q ) S 2 = − µ p ■♠ ( N 0 N ∗ 1 ) ω 2 ( ε p + ε q ) ξ p ξ q [ p ⊥ × q ⊥ ] · ν ν ν P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✵ ✴ ✶✻