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SLIDE 1

❈❤❛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 ✶ ✴ ✶✻

SLIDE 2

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 ✷ ✴ ✶✻

SLIDE 3

■♥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 ✸ ✴ ✶✻

SLIDE 4

◗✉❛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♥❛❧ ✜❡❧❞

r1 r2 ε

→ G(r1,r2|ε) = r2|

1 ˆ P−m+i0|r1

r p

→ uin

p (r)

G(r1,r2|ε) = ( ˆ P +m)D(r2, r1|ε) ˆ P = γµPµ Pµ = (ε −V(r),i∇ ∇ ∇) D(r2,r1|ε) = r2| 1 ˆ P2 −m2 +i0 |r1

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✹ ✴ ✶✻

SLIDE 5

◗✉❛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/m2 ✖ ❚❤❡ ✈✐rt✉❛❧ ♣❛✐r ❧✐❢❡ t✐♠❡ ✐♥ t❤❡ ▲❋❘ ρ ∼ 1/m ✖ ❚❤❡ ❧♦♦♣ tr❛♥s✈❡rs❡ s✐③❡ ✐♥ t❤❡ ▲❋❘ z = ∆t ∼ E/m2 ✖ ❚❤❡ ❧♦♦♣ ❧♦♥❣✐t✉❞✐♥❛❧ s✐③❡ ✐♥ t❤❡ ▲❋❘ l ∼ Eρ ∼ 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 ✺ ✴ ✶✻

SLIDE 6

◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡ q✉❛s✐❝❧❛ss✐❝❛❧ ●r❡❡♥✬s ❢✉♥❝t✐♦♥

D(r2,r1|ε)= ieikr 4π2r

d2qexp
iq2−ir

1

0 dxV(Rx)

1− r

2ε

1

0 dxα

α α∇ ∇ ∇1V(Rx)

Rx = r1 +xr+q
2x¯

xr/k ⇐ = q✉❛♥t✉♠ ✢✉❝t✉❛t✐♦♥s

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✻ ✴ ✶✻

SLIDE 7

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

▼❛tr✐① ❡❧❡♠❡♥t

p q k

M =

dr¯

u−

q (r)e∗γ

γ γu+

p (r)e−ikr

❈r♦ss s❡❝t✐♦♥

dσ = αωqεq (2π)4 dΩk dΩq dω|M|2 , dσ(p,q,k,η) = dσs(p,q,k,η)+dσa(p,q,k,η) dσs(p,q,k,η) = dσ(p,q,k,η)+dσ(p,q,k,−η) 2 dσa(p,q,k,η) = dσ(p,q,k,η)−dσ(p,q,k,−η) 2

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✼ ✴ ✶✻

SLIDE 8

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

r❡❡♥✬s ❢✉♥❝t✐♦♥

D(r2, r1|ε) = r2| 1 H +iα α α ·∇ ∇ ∇V(r)+i0|r1 = r2| 1 H − 1 H iα α α ·∇ ∇ ∇V(r) 1 H + 1 H iα α α ·∇ ∇ ∇V(r) 1 H iα α α ·∇ ∇ ∇V(r) 1 H |r1 H = (ε −V(r))2 −m2 +∇ ∇ ∇2 +i0 D(r2, r1|ε) = d0(r2,r1)+α α α ·d1(r2,r1)+Σ Σ Σ·d2(r2,r1) d0 ∼ lcd1 ∼ l2

cd2

lc ∼ ε/∆ ≫ 1

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✽ ✴ ✶✻

SLIDE 9

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

r❡❡♥✬s ❢✉♥❝t✐♦♥

D(r2, r1|ε) = r2| 1 H +iα α α ·∇ ∇ ∇V(r)+i0|r1 = r2| 1 H − 1 H iα α α ·∇ ∇ ∇V(r) 1 H + 1 H iα α α ·∇ ∇ ∇V(r) 1 H iα α α ·∇ ∇ ∇V(r) 1 H |r1 H = (ε −V(r))2 −m2 +∇ ∇ ∇2 +i0 D(r2, r1|ε) = d0(r2,r1)+α α α ·d1(r2,r1)+Σ Σ Σ·d2(r2,r1) d0 ∼ lcd1 ∼ l2

cd2

lc ∼ ε/∆ ≫ 1 d0(r2,r1) = ieiκr

4π2r

dQexp

iQ2 −ir

1

0 dxV(Rx)

×
1+ ir3

2κ 1

dx

x

dy(x−y)∇

∇ ∇⊥V(Rx)·∇ ∇ ∇⊥V(Ry)

P❡t❡r ❑r❛❝❤❦♦✈

✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✽ ✴ ✶✻

SLIDE 10

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

▼❛tr✐① ❡❧❡♠❡♥t

M =−δµpµq(εpδλµp+εqδλ ¯

µp)[N0(e∗ λ,ξpp⊥−ξqq⊥)+N1(e∗ λ,εpξpp⊥−εqξqq⊥)]

− 1

√ 2mµpδµp ¯ µqδλµp(εp −εq)[N0(ξp −ξq)+N1(εpξp −εqξq)]

N0 = 2i ωm2∆2

⊥

drexp[−i∆

∆ ∆·r −iχ(ρ)]∆ ∆ ∆⊥ ·∇ ∇ ∇⊥V(r) N1 = 1 ωm2εpεq

drexp[−i∆

∆ ∆·r −iχ(ρ)]

∞

dxx∇

∇ ∇⊥V(r −xν ν ν)·∇ ∇ ∇⊥V(r) χ(ρ) =

∞

−∞ V(z,ρ

ρ ρ)dz ξp = m2 m2 +p2

⊥

ξq = m2 m2 +q2

⊥

∆ ∆ ∆ = q+k −p

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✾ ✴ ✶✻

SLIDE 11

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

❙q✉❛r❡ ♦❢ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥t

∑

λ µq

|M|2 = S0 +S1 +S2 S0 = m2|N0|2 2 ∆2 m2 (ε2

p +ε2 q)ξpξq −2εpεq(ξp −ξq)2

S1 = m2❘❡(N0N∗

1)

2

∆2

m2 (ε2 p +ε2 q)(εp +εq)ξpξq

+

(ε2

p +ε2 q)(εp −εq)−4εpεq(εpξp −εqξq)

(ξp −ξq)
S2 = −µp■♠(N0N∗

1)ω2(εp +εq)ξpξq [p⊥ ×q⊥]·ν

ν ν

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✵ ✴ ✶✻

SLIDE 12

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧

rscr ≫ r ≫ Rnucl V(r) = Zα r N0 = 8πη(L∆)2iη ωm2∆2 Γ(1−iη) Γ(1+iη) = N0B Γ(1−iη) Γ(1+iη)(L∆)2iη N1 = 2π2η2(L∆)2iη ωm2εpεq∆ Γ(1/2−iη) Γ(1/2+iη) = N1B Γ(1/2−iη) Γ(1/2+iη)(L∆)2iη |N0|2 = |N0B|2 = 8πη ωm2∆2 2 g(η) = η Γ(1−iη)Γ(1/2+iη) Γ(1+iη)Γ(1/2−iη) ❘❡N0N∗

1 = π❘❡g(η)∆

4εpεq |N0|2 ■♠N0N∗

1 = π■♠g(η)∆

4εpεq |N0|2

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✶ ✴ ✶✻

SLIDE 13

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8

Η gΗ

❉❡♣❡♥❞❡♥❝❡ ♦❢ ❘❡g(η) ✭s♦❧✐❞ ❝✉r✈❡✮ ❛♥❞ ✲■♠g(η) ✭❞❛s❤❡❞ ❝✉r✈❡✮ ♦♥ η✳

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✷ ✴ ✶✻

SLIDE 14

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❜r❡♠sstr❛❤❧✉♥❣ ❢r♦♠ ♠✉♦♥s

❆❝❝♦✉♥t✐♥❣ ♦❢ t❤❡ ✜♥✐t❡ s✐③❡ ♦❢ t❤❡ ♥✉❝❧❡✉s

R♥✉❝ ∼ 7fm λµ = 1.87fm VF(∆2) = −4πηF(∆2) ∆2

∆2(|N0|2 −|N0B|2)d∆

∆ ∆⊥ = ∓128π3η2f(η) ω2m4 f(η) = ❘❡ψ(1+iη)−ψ(1)

1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8

Β G0

F(∆2) = Λ2 ∆2 +Λ2 β = ∆ Λ G0 = |N0|2/|N0B|2 −1

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✸ ✴ ✶✻

SLIDE 15

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

❋♦r t❤❡ ♣♦t❡♥t✐❛❧ VF(∆2) = −4πηF(∆2)

∆2

✱ F(∆2) =

Λ2 ∆2+Λ2

1 2 3 4 5 6 1.5 1.0 0.5 0.0 0.5 1.0 1.5

Β G1

❉❡♣❡♥❞❡♥❝❡ ♦❢ G1 = Σ−1

R ❘❡N0N∗ 1/|N0|2 ♦♥

β = ∆/Λ ❢♦r ❛ ❢❡✇ ✈❛❧✉❡s ♦❢ η✱ η = 0.34 ✭❆❣✱ s♦❧✐❞ ❝✉r✈❡✮✱ η = 0.6 ✭P❜✱ ❞❛s❤❡❞ ❝✉r✈❡✮✱ è η = 0.67 ✭❯✱ ❞♦tt❡❞ ❝✉r✈❡✮✳ ΣR = π❘❡g(η)∆

4εpεq

1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Β G2

❉❡♣❡♥❞❡♥❝❡ ♦❢ G2 = Σ−1

I ■♠N0N∗ 1/|N0|2 ♦♥

β = ∆/Λ ❢♦r ❛ ❢❡✇ ✈❛❧✉❡s ♦❢ η✱ η = 0.34 ✭❆❣✱ s♦❧✐❞ ❝✉r✈❡✮✱ η = 0.6 ✭P❜✱ ❞❛s❤❡❞ ❝✉r✈❡✮✱ è η = 0.67 ✭❯✱ ❞♦tt❡❞ ❝✉r✈❡✮✳ ΣI = π■♠g(η)∆

4εpεq

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✹ ✴ ✶✻

SLIDE 16

❈♦♥❝❧✉s✐♦♥

❚❤❡ q✉❛s✐❝❧❛ss✐❝❛❧ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛r❜✐tr❛❧② ❧♦❝❛❧❛✐③❡❞ ♣♦t❡♥t✐❛❧✱ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t✱ ✜rst q✉❛s✐❝❧❛ss✐❝❛❧ ❝♦rr❡❝t✐♦♥ ❛r❡ ♦❜t❛✐♥❡❞✳ ❚❤❡ ❝❤❛r❣❡ ❛s✐♠♠❡tr② ✐♥ t❤❡ ♣r♦❝❡ss ♦❢ ❤✐❣❤ ❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❛ ❤❡❛✈② ❛t♦♠ ✐s ✐♥✈❡st✐❣❛t❡❞✳ ❊✛❡❝t ♦❢ s❝r❡❡♥✐♥❣ ❛♥❞ ✜♥✐t❡ s✐③❡ ♥✉❝❧❡✉s ❡✛❡❝t ❛r❡ ✐♥✈❡st✐❣❛t❡❞✳ ■t ✐s s❤♦✇♥ t❤❛t t❤❡ ❈♦✉❧♦♠❜ ❝♦rr❡❝t✐♦♥s ❛r❡ ✐♠♣♦rt❛♥t✳

P❡t❡r ❑r❛❝❤❦♦✈ ✭❇■◆P✮ ❈❤❛r❣❡ ❛s②♠♠❡tr② ✳✳✳ ✶✻✴✵✻✴✷✵✶✺✱ ❇■◆P ✶✺ ✴ ✶✻

SLIDE 17

❈❤❛r❣❡ ❛s②♠♠❡tr② ♦❢ ❤✐❣❤ ❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣ ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❛ ❤❡❛✈② ❛t♦♠

P❡t❡r ❑r❛❝❤❦♦✈

✶✾ ❏✉♥❡✱ ✷✵✶✺

P❧❛♥

■♥tr♦❞✉❝t✐♦♥

❍✐❣❤ ❡♥❡r❣② E ≫ m ❙♠❛❧❧ ❝❤❛r❛❝t❡r✐st✐❝ ❛♥❣❧❡ θ ≪ 1 ▲❛r❣❡ ❛t♦♠✐❝ ❝❤❛r❣❡ ♥✉♠❜❡r = ⇒ Z ≫ 1

θ ≪ 1

❋✉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♥❛❧ ✜❡❧❞

→ G(r1,r2|ε) = r2|

→ uin

G(r1,r2|ε) = ( ˆ P +m)D(r2, r1|ε) ˆ P = γµPµ Pµ = (ε −V(r),i∇ ∇ ∇) D(r2,r1|ε) = r2| 1 ˆ P2 −m2 +i0 |r1

◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❤✐❣❤ ❡♥❡r❣② ♣r♦❝❡ss❡s

◗✉❛s✐❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡ q✉❛s✐❝❧❛ss✐❝❛❧ ●r❡❡♥✬s ❢✉♥❝t✐♦♥

D(r2,r1|ε)= ieikr 4π2r

2ε

α α∇ ∇ ∇1V(Rx)

xr/k ⇐ = q✉❛♥t✉♠ ✢✉❝t✉❛t✐♦♥s

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

▼❛tr✐① ❡❧❡♠❡♥t

M =

u−

γ γu+

❈r♦ss s❡❝t✐♦♥

dσ = αωqεq (2π)4 dΩk dΩq dω|M|2 , dσ(p,q,k,η) = dσs(p,q,k,η)+dσa(p,q,k,η) dσs(p,q,k,η) = dσ(p,q,k,η)+dσ(p,q,k,−η) 2 dσa(p,q,k,η) = dσ(p,q,k,η)−dσ(p,q,k,−η) 2

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

lc ∼ ε/∆ ≫ 1

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

lc ∼ ε/∆ ≫ 1 d0(r2,r1) = ieiκr

∇ ∇⊥V(Rx)·∇ ∇ ∇⊥V(Ry)

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

▼❛tr✐① ❡❧❡♠❡♥t

M =−δµpµq(εpδλµp+εqδλ ¯

− 1

N0 = 2i ωm2∆2

∆ ∆·r −iχ(ρ)]∆ ∆ ∆⊥ ·∇ ∇ ∇⊥V(r) N1 = 1 ωm2εpεq

∆ ∆·r −iχ(ρ)]

∇ ∇⊥V(r −xν ν ν)·∇ ∇ ∇⊥V(r) χ(ρ) =

ρ ρ)dz ξp = m2 m2 +p2

ξq = m2 m2 +q2

∆ ∆ ∆ = q+k −p

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

❙q✉❛r❡ ♦❢ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥t

∑

|M|2 = S0 +S1 +S2 S0 = m2|N0|2 2 ∆2 m2 (ε2

+

ν ν

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❤✐❣❤✲❡♥❡r❣② ❜r❡♠sstr❛❤❧✉♥❣

❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧

4εpεq |N0|2 ■♠N0N∗

4εpεq |N0|2

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8

Η gΗ

❉❡♣❡♥❞❡♥❝❡ ♦❢ ❘❡g(η) ✭s♦❧✐❞ ❝✉r✈❡✮ ❛♥❞ ✲■♠g(η) ✭❞❛s❤❡❞ ❝✉r✈❡✮ ♦♥ η✳

❈❤❛r❣❡ ❛s②♠♠❡tr② ✐♥ ❜r❡♠sstr❛❤❧✉♥❣ ❢r♦♠ ♠✉♦♥s

❆❝❝♦✉♥t✐♥❣ ♦❢ t❤❡ ✜♥✐t❡ s✐③❡ ♦❢ t❤❡ ♥✉❝❧❡✉s

R♥✉❝ ∼ 7fm λµ = 1.87fm VF(∆2) = −4πηF(∆2) ∆2

∆ ∆⊥ = ∓128π3η2f(η) ω2m4 f(η) = ❘❡ψ(1+iη)−ψ(1)

F(∆2) = Λ2 ∆2 +Λ2 β = ∆ Λ G0 = |N0|2/|N0B|2 −1

❋♦r t❤❡ ♣♦t❡♥t✐❛❧ VF(∆2) = −4πηF(∆2)

✱ F(∆2) =

❉❡♣❡♥❞❡♥❝❡ ♦❢ G1 = Σ−1

β = ∆/Λ ❢♦r ❛ ❢❡✇ ✈❛❧✉❡s ♦❢ η✱ η = 0.34 ✭❆❣✱ s♦❧✐❞ ❝✉r✈❡✮✱ η = 0.6 ✭P❜✱ ❞❛s❤❡❞ ❝✉r✈❡✮✱ è η = 0.67 ✭❯✱ ❞♦tt❡❞ ❝✉r✈❡✮✳ ΣR = π❘❡g(η)∆

❉❡♣❡♥❞❡♥❝❡ ♦❢ G2 = Σ−1

β = ∆/Λ ❢♦r ❛ ❢❡✇ ✈❛❧✉❡s ♦❢ η✱ η = 0.34 ✭❆❣✱ s♦❧✐❞ ❝✉r✈❡✮✱ η = 0.6 ✭P❜✱ ❞❛s❤❡❞ ❝✉r✈❡✮✱ è η = 0.67 ✭❯✱ ❞♦tt❡❞ ❝✉r✈❡✮✳ ΣI = π■♠g(η)∆

❈♦♥❝❧✉s✐♦♥

❚❤❛♥❦ ②♦✉ ❢♦r ❛tt❡♥t✐♦♥✦