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SLIDE 1

❉P❙ ❅ ▼P■ ✷✵✶✻ ❉P❉ s✉♠ r✉❧❡s ✐♥ ◗❈❉

❉❡❝❡♠❜❡r ✶✱ ✷✵✶✻ P✳ P❧öÿ❧ 1 ❆✳ ❙❝❤ä❢❡r 1 ▼✳ ❉✐❡❤❧ 2

1■♥st✐t✉t ❢ür t❤❡♦r❡t✐s❝❤❡ P❤②s✐❦

❯♥✐✈❡rs✐tät ❘❡❣❡♥s❜✉r❣✱ ✾✸✵✺✸ ❘❡❣❡♥s❜✉r❣

2❉❡✉ts❝❤❡s ❊❧❡❦tr♦♥❡♥✲❙②♥❝❤r♦tr♦♥ ❉❊❙❨

✷✷✻✵✸ ❍❛♠❜✉r❣

SLIDE 2

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❉❡✜♥✐t✐♦♥s O(αs) ❡①❛♠♣❧❡ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ❊①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r♦♦❢ t♦ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❞❉●▲❆P ❊q✉❛t✐♦♥ ❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❙✉♠♠❛r②

✷ ✴ ✶✹

SLIDE 3

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

✸ ✴ ✶✹

SLIDE 4

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ♠♦t✐✈❛t❡❞ ❜② ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♣❛rt♦♥ ♠♦❞❡❧

❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♥str✉❝t ❝♦♥s❡r✈❡❞ q✉❛♥t✐t✐❡s

✸ ✴ ✶✹

SLIDE 5

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ♠♦t✐✈❛t❡❞ ❜② ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♣❛rt♦♥ ♠♦❞❡❧ ◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♥str✉❝t ❝♦♥s❡r✈❡❞ q✉❛♥t✐t✐❡s ✸ ✴ ✶✹

SLIDE 6

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ♠♦t✐✈❛t❡❞ ❜② ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♣❛rt♦♥ ♠♦❞❡❧ ◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♥str✉❝t ❝♦♥s❡r✈❡❞ q✉❛♥t✐t✐❡s

j1,j2

1

dx1

1−x1

dx2

x1x2 M − x1 F j1j2(x1, x2) = M = 1

✸ ✴ ✶✹

SLIDE 7

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ♠♦t✐✈❛t❡❞ ❜② ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♣❛rt♦♥ ♠♦❞❡❧ ◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♥str✉❝t ❝♦♥s❡r✈❡❞ q✉❛♥t✐t✐❡s

1

dx1

1−x1

dx2
F j1j1,v (x1, x2)

Nj1,v − 1 − F j1j1,v (x1, x2) Nj1,v + 1

= Nj1,v

✸ ✴ ✶✹

SLIDE 8

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦✿ ♣❡r❢♦r♠✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ✉s✐♥❣ ❡✐t❤❡r t❤❡ ❉P❉

♥✉♠❜❡r✭♠♦♠❡♥t✉♠✮ s✉♠ r✉❧❡ ❛♥❞ t❤❡ P❉❋ ♠♦♠❡♥t✉♠✭♥✉♠❜❡r✮ s✉♠ r✉❧❡ s❤♦✉❧❞ ②✐❡❧❞ t❤❡ s❛♠❡ r❡s✉❧t

j2

1

dx1

1−x1

dx2 x2 F j1,vj2 (x1, x2)

♣✉t ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❉P❉s ❛♥❞ ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ ✉s❡❞ t♦ r❡✜♥❡ ❉P❉✲♠♦❞❡❧s ♣r♦✈❡ t❤❛t t❤❡s❡ s✉♠ r✉❧❡s ❛r❡ ❢✉❧✜❧❧❡❞ ✐♥ ◗❈❉

✸ ✴ ✶✹

SLIDE 9

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦✿ ♣❡r❢♦r♠✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ✉s✐♥❣ ❡✐t❤❡r t❤❡ ❉P❉

♥✉♠❜❡r✭♠♦♠❡♥t✉♠✮ s✉♠ r✉❧❡ ❛♥❞ t❤❡ P❉❋ ♠♦♠❡♥t✉♠✭♥✉♠❜❡r✮ s✉♠ r✉❧❡ s❤♦✉❧❞ ②✐❡❧❞ t❤❡ s❛♠❡ r❡s✉❧t

j2

1

dx1

1−x1

dx2 x2 F j1,vj2 (x1, x2) = Nj1,v − xj1,v

♣✉t ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❉P❉s ❛♥❞ ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ ✉s❡❞ t♦ r❡✜♥❡ ❉P❉✲♠♦❞❡❧s ♣r♦✈❡ t❤❛t t❤❡s❡ s✉♠ r✉❧❡s ❛r❡ ❢✉❧✜❧❧❡❞ ✐♥ ◗❈❉

✸ ✴ ✶✹

SLIDE 10

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦✿ ♣❡r❢♦r♠✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ✉s✐♥❣ ❡✐t❤❡r t❤❡ ❉P❉

♥✉♠❜❡r✭♠♦♠❡♥t✉♠✮ s✉♠ r✉❧❡ ❛♥❞ t❤❡ P❉❋ ♠♦♠❡♥t✉♠✭♥✉♠❜❡r✮ s✉♠ r✉❧❡ s❤♦✉❧❞ ②✐❡❧❞ t❤❡ s❛♠❡ r❡s✉❧t

j2

1

dx1

1−x1

dx2 x2 F j1,vj2 (x1, x2) = Nj1,v − xj1,v

◮ ♣✉t ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❉P❉s ❛♥❞ ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ ✉s❡❞ t♦ r❡✜♥❡ ❉P❉✲♠♦❞❡❧s

♣r♦✈❡ t❤❛t t❤❡s❡ s✉♠ r✉❧❡s ❛r❡ ❢✉❧✜❧❧❡❞ ✐♥ ◗❈❉

✸ ✴ ✶✹

SLIDE 11

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❉P❉ ❙✉♠ ❘✉❧❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

1−x1

dx2 F j1j2,v(x1, x2) =
Nj2,v + δj1,j2 − δj1,j2
fj1(x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

j2

1−x1

dx2 x2 F j1j2(x1, x2) = (M − x1)fj1(x1)

❙t✐r❧✐♥❣✱ ●❛✉♥t ✷✵✶✵

◮ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦✿ ♣❡r❢♦r♠✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ ✉s✐♥❣ ❡✐t❤❡r t❤❡ ❉P❉

♥✉♠❜❡r✭♠♦♠❡♥t✉♠✮ s✉♠ r✉❧❡ ❛♥❞ t❤❡ P❉❋ ♠♦♠❡♥t✉♠✭♥✉♠❜❡r✮ s✉♠ r✉❧❡ s❤♦✉❧❞ ②✐❡❧❞ t❤❡ s❛♠❡ r❡s✉❧t

j2

1

dx1

1−x1

dx2 x2 F j1,vj2 (x1, x2) = Nj1,v − xj1,v

◮ ♣✉t ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❉P❉s ❛♥❞ ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ ✉s❡❞ t♦ r❡✜♥❡ ❉P❉✲♠♦❞❡❧s ◮ ♣r♦✈❡ t❤❛t t❤❡s❡ s✉♠ r✉❧❡s ❛r❡ ❢✉❧✜❧❧❡❞ ✐♥ ◗❈❉ ✸ ✴ ✶✹

SLIDE 12

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❉❡✜♥✐t✐♦♥s

fj1 (x1, k1) = dz−

1

2π eix1z−

1 p+ d2z−

1

(2π)2 eiz1k1p|qj1(− z1 2 )Γa qj1( z1 2 )|p F j1j2 (x1, x2, k1, k2, ∆) = 2

i=1

dz−

i

2π eixiz−

i p+ d2z−

i

(2π)2 eiziki

2p+

dy−

1

2π d2y1 (2π)2 eiy1∆

× p|qj2(− z2

2 )Γa qj2( z2 2 )qj1(y1 − z1 2 )Γa qj1(y1 + z2 2 )|p ❉✐❡❤❧✱ ❖st❡r♠❡✐❡r✱ ❙❝❤ä❢❡r ✷✵✶✶ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✐♥ t❡r♠s ♦❢ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s✱ ❡✳❣✳

✹ ✴ ✶✹

SLIDE 13

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❉❡✜♥✐t✐♦♥s

fj1 (x1, k1) = dz−

1

2π eix1z−

1 p+ d2z−

1

(2π)2 eiz1k1p|qj1(− z1 2 )Γa qj1( z1 2 )|p F j1j2 (x1, x2, k1, k2, ∆) = 2

i=1

dz−

i

2π eixiz−

i p+ d2z−

i

(2π)2 eiziki

2p+

dy−

1

2π d2y1 (2π)2 eiy1∆

× p|qj2(− z2

2 )Γa qj2( z2 2 )qj1(y1 − z1 2 )Γa qj1(y1 + z2 2 )|p ❉✐❡❤❧✱ ❖st❡r♠❡✐❡r✱ ❙❝❤ä❢❡r ✷✵✶✶

◮ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✐♥ t❡r♠s ♦❢ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s✱ ❡✳❣✳

fji (x1, k1) =

dz−

1

(2π)4

✹ ✴ ✶✹

SLIDE 14

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

✺ ✴ ✶✹

SLIDE 15

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s

✺ ✴ ✶✹

SLIDE 16

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s

u|d d|u

fg

u|d d|u

✺ ✴ ✶✹

SLIDE 17

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s

u|d d|u

fg

u|d d|u u ¯ d ¯ d u

F gu

u ¯ d ¯ d u

✺ ✴ ✶✹

SLIDE 18

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❈♦♥tr✐❜✉t✐♥❣ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s

u|d d|u

fg

u|d d|u u ¯ d ¯ d u

F gu

u ¯ d ¯ d u ¯ d u u ¯ d

F g ¯

d

¯ d u u ¯ d

✺ ✴ ✶✹

SLIDE 19

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ ❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉

✺ ✴ ✶✹

SLIDE 20

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ ❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉

✺ ✴ ✶✹

SLIDE 21

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳ ❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉

✺ ✴ ✶✹

SLIDE 22

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳

. . .

❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉

✺ ✴ ✶✹

SLIDE 23

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

O(αs) ❡①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ t♦②✲♠♦❞❡❧ ♦❢ ❛ ♠❡s♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❛♥ u✲q✉❛r❦ ❛♥❞ ¯ d✲❛♥t✐q✉❛r❦✱ s♣❧✐tt✐♥❣ ✐♥t♦ ✐ts ❝♦♥st✐t✉❡♥ts ✈✐❛ ❛ ♣♦✐♥t❧✐❦❡ ❝♦✉♣❧✐♥❣✳ ❋♦r j1 = g ♦♥❧② t❤❡ ❢♦❧❧♦✇✐♥❣ P❉❋s ✉♥❞ ❉P❉s ❝❛♥ ❜❡ r❡❛❧✐③❡❞ t♦ O(αs)✿ fg, F gu, F g ¯

d

❉P❉ ❣r❛♣❤s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ P❉❋ ❣r❛♣❤s ❜② ✧❝✉tt✐♥❣✧ ♦♥❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❧✐♥❡s✱ ✐✳❡✳

. . .

❯s❡ ❧✐❣❤t✲❢r♦♥t ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② t♦ s❤♦✇ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ P❉❋ ❛♥❞ ❉P❉

✺ ✴ ✶✹

SLIDE 24

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ ✲✈❛❧✉❡ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 25

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱

✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉ ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ ✲✈❛❧✉❡ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 26

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱

✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

+

t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 27

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱

✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞

✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 28

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✱

✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞

✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 29

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s P❉❋ ❛♥❞ ❉P❉ ❞❡✜♥✐t✐♦♥s

fj1

B (x1) =

t
c
x1 p+n1 p+
dD−2k1

(2π)D−1 N(t)

i=2

dxidD−2ki (2π)D−1 p+ × Φj1

P DFt,c,o ({x}, {k}) δ

1 −

M(c)

i=1

xi

1−x1
dx2 F j1j2

B

(x1, x2) =

t
c
l

δf(l),j2

x1p+n12p+

dD−2k1 (2π)D−1 N(t)

i=2

dxidD−2ki (2π)D−1 p+ ×

xl p+nl Φj1j2

DP Dt,c,o ({x}, {k}) δ

1 −

M(c)

i=1

xi

✻ ✴ ✶✹

SLIDE 30

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡

❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞

✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ ✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 31

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡

❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞

✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ 2

xl p+nl Φj1, j2

DP Dt,c,o ?

= Φj1

P DFt,c,o

✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ ✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ ✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 32

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡

❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞

✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ 2

xl p+nl Φj1, j2

DP Dt,c,o ?

= Φj1

P DFt,c,o

✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ j1j2✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ j1✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s ❛♥❞ s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞ t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 33

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙t❡♣s t♦✇❛r❞s ❛ ♣r♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◮ ❛❢t❡r ❛ s✉♠ ♦✈❡r ❝✉ts✱ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ♦❢ ❛ ❣✐✈❡♥ P❉❋ ❣r❛♣❤ ❤❛✈❡ t♦ ❜❡

❝♦♥s✐❞❡r❡❞✱ ✇❤❡r❡ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s

◮ ♣❡r❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ t❤❡ t✇♦ ❛❝t✐✈❡ ♣❛rt♦♥s ✐♥ ❛ ❉P❉

✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡♠ t♦ t❤❡ s❛♠❡ x+✲✈❛❧✉❡

◮ t❤✉s ❢♦r ❉P❉s ❛❧s♦ ♦♥❧② s✉❝❤ ▲❈ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ ✧st❛t❡✧ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞

✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❝❢✳ ❉✐❡❤❧✱ ●❛✉♥t✱ ❖st❡r♠❡✐❡r✱ P❧öÿ❧✱ ❙❝❤ä❢❡r ✷✵✶✻ ▼❛✐♥ ✐♥❣r❡❞✐❡♥t ❢♦r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✐s t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ 2

xl p+nl Φj1, j2

DP Dt,c,o = Φj1 P DFt,c,o

✭♦❜t❛✐♥❡❞ ❢r♦♠ ✐♥t❡❣r❛t✐♥❣ ❛ j1j2✲❉P❉ ♦✈❡r t❤❡ ♠♦♠❡♥t✉♠ ❢r❛❝t✐♦♥ ♦❢ ♣❛rt♦♥ ✷ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧t t♦ ❛ j1✲P❉❋✮ ❈❛r❡❢✉❧ ❛♥❛♥❧②s✐s ♦❢ t❤❡ ▲❈P❚ ❡①♣r❡ss✐♦♥s Φj1, j2

DP Dt,c,o ❛♥❞ Φj1 P DFt,c,o s❤♦✇s t❤❛t t❤✐s ✐s ✐♥❞❡❡❞

t❤❡ ❝❛s❡

✻ ✴ ✶✹

SLIDE 34

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿

✼ ✴ ✶✹

SLIDE 35

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡

✐s t❤❡ ♥✉♠❜❡r ♦❢ ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿

✼ ✴ ✶✹

SLIDE 36

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡

✐s t❤❡ ♥✉♠❜❡r ♦❢ ✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ ❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿

✼ ✴ ✶✹

SLIDE 37

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡ N
j2
t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j2✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φj1

P DFt,c,o

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ ✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ ✐♥ t❡r♠s ♦❢ ✿

✼ ✴ ✶✹

SLIDE 38

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡ N
j2
t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j2✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φj1

P DFt,c,o

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ Nj2,v j2✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j2j2✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N

j2
t,c,o − N
j2
t,c,o ✐♥ t❡r♠s ♦❢ j1✿

✼ ✴ ✶✹

SLIDE 39

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡ N
j2
t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j2✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φj1

P DFt,c,o

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ Nj2,v j2✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j2j2✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N

j2
t,c,o − N
j2
t,c,o ✐♥ t❡r♠s ♦❢ j1✿

j1 = j2, j2

Nj2,v + x
− x = Nj2,v

✼ ✴ ✶✹

SLIDE 40

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡ N
j2
t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j2✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φj1

P DFt,c,o

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ Nj2,v j2✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j2j2✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N

j2
t,c,o − N
j2
t,c,o ✐♥ t❡r♠s ♦❢ j1✿

j1 = j2, j2

Nj2,v + x
− x = Nj2,v

j1 = j2

Nj2,v + x
− (x − 1) = Nj2,v + 1

✼ ✴ ✶✹

SLIDE 41

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡ N
j2
t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j2✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φj1

P DFt,c,o

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ Nj2,v j2✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j2j2✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N

j2
t,c,o − N
j2
t,c,o ✐♥ t❡r♠s ♦❢ j1✿

j1 = j2, j2

Nj2,v + x
− x = Nj2,v

j1 = j2

Nj2,v + x
− (x − 1) = Nj2,v + 1

j1 = j2

Nj2,v + x − 1
− x = Nj2,v − 1

✼ ✴ ✶✹

SLIDE 42

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❯s✐♥❣ t❤❡ r❡❧❛t✐♦♥ st❛t❡❞ ❛❜♦✈❡✱ s❤♦✇✐♥❣ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s r❡❞✉❝❡s t♦ s❤♦✇✐♥❣ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

l
δf(l), j2 − δf(l), j2
=
Nj2,v + δj1, j2 − δj1, j2
N
j2
t,c,o − N
j2
t,c,o
=
Nj2,v + δj1, j2 − δj1, j2
✇❤❡r❡ N
j2
t,c,o ✐s t❤❡ ♥✉♠❜❡r ♦❢ j2✲q✉❛r❦s r✉♥♥✐♥❣ ❛❝r♦ss t❤❡ ✜♥❛❧ st❛t❡ ❝✉t ✐♥ Φj1

P DFt,c,o

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ Nj2,v j2✲✈❛❧❡♥❝❡ q✉❛r❦s ✐♥s✐❞❡ t❤❡ ❤❛❞r♦♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ♣❧✉s ❛♥ ❛r❜✐tr❛② ♥✉♠❜❡r ♦❢ j2j2✲♣❛✐rs ♦♥❡ ❝❛♥ ❞❡t❡r♠✐♥❡ N

j2
t,c,o − N
j2
t,c,o ✐♥ t❡r♠s ♦❢ j1✿

j1 = j2, j2

Nj2,v + x
− x = Nj2,v

j1 = j2

Nj2,v + x
− (x − 1) = Nj2,v + 1

j1 = j2

Nj2,v + x − 1
− x = Nj2,v − 1
=Nj2,v +δj1, j2 −δj1, j2

✼ ✴ ✶✹

SLIDE 43

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿ ✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ ✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣

✽ ✴ ✶✹

SLIDE 44

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿

l
DN(t)

2

[xi] DN(t)

1

[ki] xl Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   = (1 − x1)

DN(t)

2

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✇❤❡r❡

Db

a [xi] = b

i=a

1 dxi p+

Db

a [ki] = b

i=a
dD−2ki

(2π)D−1 , ✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ ✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣

✽ ✴ ✶✹

SLIDE 45

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿

l
DN(t)

2

[xi] DN(t)

1

[ki] xl Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   = (1 − x1)

DN(t)

2

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ x2✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣

✽ ✴ ✶✹

SLIDE 46

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❤❛s t♦ s❤♦✇ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ✐s ❢✉❧✜❧❧❡❞✿

l
DN(t)

2

[xi] DN(t)

1

[ki] xl Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   = (1 − x1)

DN(t)

2

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✇✳❧✳♦✳❣✳ ♣❡r❢♦r♠✐♥❣ t❤❡ x2✲✐♥t❡❣r❛t✐♦♥ ♦♥ ❜♦t❤ s✐❞❡s✱ ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧✇✐♥❣

DN(t)

3

[xi] DN(t)

1

[ki]  1 − x1 −

M(c)

i=3

xi +

M(c)

j=3

xj  

(1−x1)

Φj1

P DFt,c,o ({x}, {k})

x2=x2, 0

= (1 − x1)

DN(t)

3

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o ({x}, {k})

x2=x2, 0

✽ ✴ ✶✹

SLIDE 47

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋

fj1 (x1) =

i1

1

x1

dz1 z1 Zi1→j1 x1 z1

fi1

B (z1)

✇✐t❤ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs Zi1→j1✱ ✇❤✐❝❤ ✐♥ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣❛♥s✐♦♥ ✐♥ αs Zi1→j1 (x1) = δ (1 − x1) δi1,j1 + αs Zi1→j1;11 ε + α2

s

Zi1→j1;22 ε2 + Zi1→j1;21 ε

+ . . . ,

✾ ✴ ✶✹

SLIDE 48

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋

fj1 (x1) = Zi1→j1 ⊗ fi1

B

✾ ✴ ✶✹

SLIDE 49

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋

fj1 (x1) = Zi1→j1 ⊗ fi1

B

❉P❉

F j1j2 (x1, x2) =

i1,i2

1−x2

x1

dz1 z1

1−z1

x2

dz2 z2 Zi1→j1 x1 z1

Zi2→j2

x2 z2

F i1i2

B

(z1, z2) +

i1

1

x1+x2

dz1 z2

1

Zi1→j1j2 x1 z1 , x2 z2

fi1

B (z1)

✇✐t❤ t❤❡ ♥❡✇ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs Zi1→j1j2✱ ✇❤✐❝❤ ❛r❡ ✐♥ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥ ❣✐✈❡♥ ❜② Zi1→j1j2 = αs Zi1→j1j2;11 ε + α2

s

Zi1→j1j2;22 ε2 + Zi1→j1j2;21 ε2

+ . . . ,

✾ ✴ ✶✹

SLIDE 50

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋

fj1 (x1) = Zi1→j1 ⊗ fi1

B

❉P❉

F j1j2 (x1, x2) = Zi1→j1 ⊗ Zi2→j2 ⊗ F i1i2

B

+ Zi1→j1j2 ⊗ fi1

B

✾ ✴ ✶✹

SLIDE 51

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋

fj1 (x1) = Zi1→j1 ⊗ fi1

B

❉P❉

F j1j2 (x1, x2) = Zi1→j1 ⊗ Zi2→j2 ⊗ F i1i2

B

+ Zi1→j1j2 ⊗ fi1

B

❋✐♥❛❧❧② ✇❡ ❞❡✜♥❡ ❛ ✐♥✈❡rs❡ P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦r Z−1

i′

1→i1✱ ♦❜❡②✐♥❣

✐♥✈❡rs❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦r

i1

1

x1

du1 u1 Z−1

i′

1,i1

x1 u1

Zi1,j1 (x1) = δi′

1,j1δ (1 − x1) . ✾ ✴ ✶✹

SLIDE 52

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❘❡♥♦r♠❛❧✐s❡❞ P❉❋s ❛♥❞ ❉P❉s P❉❋

fj1 (x1) = Zi1→j1 ⊗ fi1

B

❉P❉

F j1j2 (x1, x2) = Zi1→j1 ⊗ Zi2→j2 ⊗ F i1i2

B

+ Zi1→j1j2 ⊗ fi1

B

✐♥✈❡rs❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦r

Z−1

i′

1,i1 ⊗ Zi1,j1 = δi′ 1,j1δ (1 − x1)

fi1

B = Z−1 i′

1,i1 ⊗ fi′ 1 ✾ ✴ ✶✹

SLIDE 53

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1)

✇❤❡r❡ ✐s ❣✐✈❡♥ ❜② ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

✶✵ ✴ ✶✹

SLIDE 54

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1)

✇❤❡r❡ R′(x1, u1) ✐s ❣✐✈❡♥ ❜② R′ (x1, u1) =

i1

u1

x1

dz1 z1 Z−1

i′

1→i1

x1 u1



  

Zi1→j1

x1 z1

− δ
1 − x1

z1

δi1,j1
δi1,j2 −δi1,j2 −δj1,j2 +δj1,j2
+

1− x1

z1

du2
Zi1→j1j2

x1 z1 , u2

− Zi1→j1j2

x1 z1 , u2



   . ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

✶✵ ✴ ✶✹

SLIDE 55

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1) ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s

t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ✐♥ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ ✐♥ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ✐✳❡✳ ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

✶✵ ✴ ✶✹

SLIDE 56

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1) ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R′ ❤❛✈❡ t♦ ❝❛♥❝❡❧

❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ ✐♥ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t ❛r❡ ✜♥✐t❡ ❢♦r ✐✳❡✳ ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

✶✵ ✴ ✶✹

SLIDE 57

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1) ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ◮ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ Zi1→j1 ✐♥ R′ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t

❛r❡ ✜♥✐t❡ ❢♦r ε = 0 ✐✳❡✳ ✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

✶✵ ✴ ✶✹

SLIDE 58

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1) ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ◮ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ Zi1→j1 ✐♥ R′ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t

❛r❡ ✜♥✐t❡ ❢♦r ε = 0

◮ ✐✳❡✳ R′ = 0✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥

❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

✶✵ ✴ ✶✹

SLIDE 59

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❙✉❜tr❛❝t✐♥❣ t❤❡ r❤s ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❢r♦♠ t❤❡ ❧❤s ❛♥❞ ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜❡❢♦r❡✱ ✇❡ ✜♥❞

1−x1

dx2F j1j2,v(x1, x2) −
Nj2v + δj1,j2 − δj1,j2
fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R′ (x1, u1) ◮ ❧❤s ♦❢ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ✜♥✐t❡ ❢♦r ε = 0 ❛s ✐t✬s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ◮ t❤✉s t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r t❤❡ r❤s✱ ✐✳❡✳ ❛❧❧ ♣♦❧❡s ✐♥ ε ✐♥ R′ ❤❛✈❡ t♦ ❝❛♥❝❡❧ ◮ ❛s ✇❡ s✉❜tr❛❝t❡❞ t❤❡ tr❡❡❧❡✈❡❧ t❡r♠ ❢r♦♠ Zi1→j1 ✐♥ R′ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② t❡r♠s t❤❛t

❛r❡ ✜♥✐t❡ ❢♦r ε = 0

◮ ✐✳❡✳ R′ = 0✱ s✉❝❤ t❤❛t t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r ▼❙✲r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✭❝❛♥

❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ▼❙✲r❡♥♦r♠❛❧✐s❛t✐♦♥✮ ❆s ✇❡ ♥♦✇ ❦♥♦✇ t❤❛t R′ = 0 ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ❢♦r t❤❡ ✐♥❤♦♠♦❣❡♥❡♦✉s t❡r♠ ❛♥❞ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝♦trs

1−x1

dx2
Zi1→j1j2(

x1, x2)−Zi1→j1j2( x1, x2)

=
δi1,j2 −δi1,j2 +δj1,j2 −δj1,j2
Zi1→j1(

x1)

✶✵ ✴ ✶✹

SLIDE 60

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ❢♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ✜♥❞s

j2

1−x1

dx2 x2F j1j2 (x1, x2) − (1 − x1) fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R (x1, u1)

✇❤❡r❡ ✐s ❣✐✈❡♥ ❜②

✶✶ ✴ ✶✹

SLIDE 61

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ❢♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ✜♥❞s

j2

1−x1

dx2 x2F j1j2 (x1, x2) − (1 − x1) fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R (x1, u1)

✇❤❡r❡ R(x1, u1) ✐s ❣✐✈❡♥ ❜② R (x1, u1) =

i1

u1

x1

dz1 z1 Z−1

i′

1→i1

x1 u1



  

Zi1→j1

x1 z1

−δ
1− x1

z1

δi1,j1
(

x1−z1) +z1

j2

1− x1

z1

du2 u2 Zi1→j1j2

x1 z1 , u2



  

✶✶ ✴ ✶✹

SLIDE 62

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ❢♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ✜♥❞s

j2

1−x1

dx2 x2F j1j2 (x1, x2) − (1 − x1) fj1 (x1) =
i′

1

x1

du1 u1 fi′

1 (u1) R (x1, u1)

❯s✐♥❣ t❤❡ s❛♠❡ r❡❛s♦♥✐♥❣ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ♦♥❡ ❝❛♥ t❤✉s ❝♦♥❝❧✉❞❡✱ t❤❛t ❛❧s♦ t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ❤♦❧❞s ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s✳ ❚❤❡ ❝♦♥str❛✐♥t✱ t❤❛t R = 0 ②✐❡❧❞s t❤❡ ❢♦❧❧✇✐♥❣ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ Zi1→j1j2 ❛♥❞ Zi1→j1

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

j2

1−x1

dx2 x2 Zi1→j1j2(

x1, x2)=( 1−x1)Zi1→j1( x1)

✶✶ ✴ ✶✹

SLIDE 63

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥

d d log (µ2) fj1 (x1) =

i1

1

x1

dz1 z1 Pi1→j1 x1 z1

fi1 (z1)

✇❤❡r❡ Pi1→j1 ❛r❡ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ❉●▲❆P s♣❧✐tt✐♥❣ ❦❡r♥❡❧s✳

✶✷ ✴ ✶✹

SLIDE 64

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥

d d log (µ2) fj1 = Pi1→j1 ⊗ fi1

✶✷ ✴ ✶✹

SLIDE 65

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥

d d log (µ2) fj1 = Pi1→j1 ⊗ fi1

♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥

d d log (µ2) F j1j2 (x1, x2) =

i1

1−x2

x1

dz1 z1 Pi1→j1 x1 z1

F i1j2 (z1, x2)

+

i2

1−x1

x2

dz2 z2 Pi2→j2 x2 z2

F j1i2 (x1, z2) +
i1

1

x1+x2

dz1 z2

1

Pi1→j1j2 x1 z1 , x2 z1

fi1 (z1)

✇❤❡r❡ t❤❡ Pi1→j1j2 ❛r❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛❜♦✉t ✇❤✐❝❤ ♥♦t ♠✉❝❤ ✐s ❦♥♦✇♥ ❛ ♣r✐♦r✐

✶✷ ✴ ✶✹

SLIDE 66

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥

d d log (µ2) fj1 = Pi1→j1 ⊗ fi1

♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥

d d log (µ2) F j1j2 = Pi1→j1 ⊗ F i1j2 + Pi2→j2 ⊗ F j1i2 + Pi1→j1j2 ⊗ fi1

✶✷ ✴ ✶✹

SLIDE 67

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥

d d log (µ2) fj1 = Pi1→j1 ⊗ fi1

♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥

d d log (µ2) F j1j2 = Pi1→j1 ⊗ F i1j2 + Pi2→j2 ⊗ F j1i2 + Pi1→j1j2 ⊗ fi1

◮ t❤❡ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▲❖ ❛♥❞ ◆▲❖ r❡s✉❧ts

❑✐rs❝❤♥❡r ✶✾✼✾ ❈❡❝❝♦♣✐❡r✐ ✷✵✶✶✱✷✵✶✹ ❜② ❝♦♠♣❛r✐♥❣ ♦✉r ♣r♦♣♦s❡❞ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ t♦ t❤❡ ❡①♣❧✐❝✐t ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ ❛♥❞ ✉s✐♥❣ t❤❡ r❡❧❛t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ s✉♠ r✉❧❡s ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✇❡ ✇❡r❡ ❛❜❧❡ t♦ ❞❡r✐✈❡ ❛♥❛❧♦❣♦✉s s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

✶✷ ✴ ✶✹

SLIDE 68

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

◗❈❉ ❡✈♦❧✉t✐♦♥ ♦❢ P❉❋s ❛♥❞ ❉P❉s ❉●▲❆P ❊q✉❛t✐♦♥

d d log (µ2) fj1 = Pi1→j1 ⊗ fi1

♣r♦♣♦s❡❞ ❞❉●▲❆P ❡q✉❛t✐♦♥

d d log (µ2) F j1j2 = Pi1→j1 ⊗ F i1j2 + Pi2→j2 ⊗ F j1i2 + Pi1→j1j2 ⊗ fi1

◮ t❤❡ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▲❖ ❛♥❞ ◆▲❖ r❡s✉❧ts

❑✐rs❝❤♥❡r ✶✾✼✾ ❈❡❝❝♦♣✐❡r✐ ✷✵✶✶✱✷✵✶✹

◮ ❜② ❝♦♠♣❛r✐♥❣ ♦✉r ♣r♦♣♦s❡❞ ❢♦r♠ ♦❢ t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥ t♦ t❤❡ ❡①♣❧✐❝✐t µ✲❞❡♣❡♥❞❡♥❝❡ ♦❢

t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ ❛♥❞ ✉s✐♥❣ t❤❡ r❡❧❛t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ s✉♠ r✉❧❡s ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✇❡ ✇❡r❡ ❛❜❧❡ t♦ ❞❡r✐✈❡ ❛♥❛❧♦❣♦✉s s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

✶✷ ✴ ✶✹

SLIDE 69

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s

◮ ❝♦♠♣❛r✐♥❣ t❤❡ µ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ t♦ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ ♦♥❡

✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥

❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

d d log (µ2) Zi′

1→j1j2 (x1, x2) =

i1

1−x2

x1

dz1 z1 Pi1→j1 x1 z1

Zi′

1→i1j2 (z1, x2)

+

i2

1−x1

x2

dz2 z2 Pi2→j2 x2 z2

Zi′

1→j1i2(

x1, z2) +

i1

1

x1+x2

dz1 z2

1

Pi1→j1j2 x1 z1 , x2 z1

Zi′

1→i1(

z1) ❡①❛❝t❧② t❤❡ s❛♠❡ str✉❝t✉r❡ ❛s t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥✱ ❥✉st ❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ r❡❣✉❧❛r P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

✶✸ ✴ ✶✹

SLIDE 70

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s

◮ ❝♦♠♣❛r✐♥❣ t❤❡ µ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r❡♥♦r♠❛❧✐s❡❞ ❉P❉ t♦ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ ♦♥❡

✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥

❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

d d log (µ2) Zi′

1→j1j2 (x1, x2) =

i1

1−x2

x1

dz1 z1 Pi1→j1 x1 z1

Zi′

1→i1j2 (z1, x2)

+

i2

1−x1

x2

dz2 z2 Pi2→j2 x2 z2

Zi′

1→j1i2(

x1, z2) +

i1

1

x1+x2

dz1 z2

1

Pi1→j1j2 x1 z1 , x2 z1

Zi′

1→i1(

z1)

◮ ❡①❛❝t❧② t❤❡ s❛♠❡ str✉❝t✉r❡ ❛s t❤❡ ❞❉●▲❆P ❡q✉❛t✐♦♥✱ ❥✉st ❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ r❡❣✉❧❛r

P❉❋ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

✶✸ ✴ ✶✹

SLIDE 71

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

d d log (µ2) Zi′

1→j1j2 = Pi1→j1 ⊗ Zi′ 1→i1j2 + Pi2→j2 ⊗ Zi′ 1→j1i2 + Pi1→j1j2 ⊗ Zi′ 1→i1 ✶✸ ✴ ✶✹

SLIDE 72

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

d d log (µ2) Zi′

1→j1j2 = Pi1→j1 ⊗ Zi′ 1→i1j2 + Pi2→j2 ⊗ Zi′ 1→j1i2 + Pi1→j1j2 ⊗ Zi′ 1→i1

■♥ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs✱ t❤✐s ❛❧❧♦✇s t♦ ♦❜t❛✐♥ ❛♥❛❧♦❣♦✉s ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ ♥❡✇ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

1−x1

dx2
Zi1→j1j2(

x1, x2)−Zi1→j1j2( x1, x2)

=
δi1,j2 −δi1,j2 +δj1,j2 −δj1,j2
Zi1→j1(

x1) b

j2

1−x1

dx2 x2 Zi1→j1j2(

x1, x2)=( 1−x1)Zi1→j1( x1)

✶✸ ✴ ✶✹

SLIDE 73

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ❞❉●▲❆P ❡q✉❛t✐♦♥ ❢♦r r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs

d d log (µ2) Zi′

1→j1j2 = Pi1→j1 ⊗ Zi′ 1→i1j2 + Pi2→j2 ⊗ Zi′ 1→j1i2 + Pi1→j1j2 ⊗ Zi′ 1→i1

◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

1−x1

dx2
Pi1→j1j2(

x1, x2)−Pi1→j1j2( x1, x2)

=
δi1,j2 −δi1,j2 +δj1,j2 −δj1,j2
Pi1→j1(

x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

j2

1−x1

dx2 x2 Pi1→j1j2(

x1, x2)=( 1−x1)Pi1→j1( x1)

✶✸ ✴ ✶✹

SLIDE 74

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

1−x1

dx2
Pi1→j1j2(

x1, x2)−Pi1→j1j2( x1, x2)

=
δi1,j2 −δi1,j2 +δj1,j2 −δj1,j2
Pi1→j1(

x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

j2

1−x1

dx2 x2 Pi1→j1j2(

x1, x2)=( 1−x1)Pi1→j1( x1)

◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ s❤♦✇ st❛❜✐❧✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ✉♥❞❡r ◗❈❉ ❡✈♦❧✉t✐♦♥

❛s ✐t s❤♦✉❧❞ ❛❧r❡❛❞② ❜❡ ❝❧❡❛r ❛❢t❡r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s✱ t❤❛t t❤❡② ❛r❡ ❛❧s♦ st❛❜❧❡ ✉♥❞❡r ❡✈♦❧✉t✐♦♥✱ t❤✐s ❛❝ts ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ❢♦r ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥

✶✸ ✴ ✶✹

SLIDE 75

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❈♦♥s✐st❡♥❝② ❈❤❡❝❦s ◆✉♠❜❡r ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

1−x1

dx2
Pi1→j1j2(

x1, x2)−Pi1→j1j2( x1, x2)

=
δi1,j2 −δi1,j2 +δj1,j2 −δj1,j2
Pi1→j1(

x1)

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡ ❢♦r 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

j2

1−x1

dx2 x2 Pi1→j1j2(

x1, x2)=( 1−x1)Pi1→j1( x1)

◮ ❝❛♥ ❜❡ ✉s❡❞ t♦ s❤♦✇ st❛❜✐❧✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ✉♥❞❡r ◗❈❉ ❡✈♦❧✉t✐♦♥ ◮ ❛s ✐t s❤♦✉❧❞ ❛❧r❡❛❞② ❜❡ ❝❧❡❛r ❛❢t❡r t❤❡ ♣r♦♦❢ t❤❛t t❤❡ s✉♠ r✉❧❡s ❤♦❧❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞

q✉❛♥t✐t✐❡s✱ t❤❛t t❤❡② ❛r❡ ❛❧s♦ st❛❜❧❡ ✉♥❞❡r ❡✈♦❧✉t✐♦♥✱ t❤✐s ❛❝ts ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ❢♦r ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥

✶✸ ✴ ✶✹

SLIDE 76

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙✉♠♠❛r②

◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝

❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥

✶✹ ✴ ✶✹

SLIDE 77

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙✉♠♠❛r②

◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝

❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚

◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r

r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❢❛❝t♦rs ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥

✶✹ ✴ ✶✹

SLIDE 78

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙✉♠♠❛r②

◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝

❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚

◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r

r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s

◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥

❢❛❝t♦rs ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥

✶✹ ✴ ✶✹

SLIDE 79

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙✉♠♠❛r②

◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝

❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚

◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r

r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s

◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥

❢❛❝t♦rs

◮ ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r

♦r❞❡rs t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥

✶✹ ✴ ✶✹

SLIDE 80

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙✉♠♠❛r②

◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝

❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚

◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r

r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s

◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥

❢❛❝t♦rs

◮ ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r

♦r❞❡rs

◮ t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s

❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠ r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥

✶✹ ✴ ✶✹

SLIDE 81

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ Pr❡❧✐♠✐♥❛r✐❡s ❘❡♥♦r♠❛❧✐s❛t✐♦♥ ◗❈❉ ❊✈♦❧✉t✐♦♥ ❙✉♠♠❛r②

❙✉♠♠❛r②

◮ ✇❡ s❤♦✇❡❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❉P❉ s✉♠ r✉❧❡s ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s ✉s✐♥❣ ❛ ❞✐❛❣r❛♠❛t✐❝

❛♣♣r♦❛❝❤ ❛♥❞ ▲❈P❚

◮ ✇❡ t❤❡♥ ❞✐s❝✉ss❡❞ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ s❤♦✇❡❞ t❤❛t t❤❡ s✉♠ r✉❧❡s ❛r❡ ❛❧s♦ ✈❛❧✐❞ ❢♦r

r❡♥♦r♠❛❧✐s❡❞ q✉❛♥t✐t✐❡s

◮ ✐♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡❞ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 r❡♥♦r♠❛❧✐s❛t✐♦♥

❢❛❝t♦rs

◮ ✜♥❛❧❧② ✇❡ ❝♦♥s✐❞❡r❡❞ ◗❈❉ ❡✈♦❧✉t✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ t❤❡ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t♦ ❤✐❣❤❡r

♦r❞❡rs

◮ t❤✐s ❛❧❧♦✇❡❞ ✉s t♦ ❞❡r✐✈❡ ♥✉♠❜❡r ❛♥❞ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡s ❢♦r t❤❡ 1 → 2 s♣❧✐tt✐♥❣ ❦❡r♥❡❧s ◮ ❛s ❛ ❝♦♥s✐st❡♥❝② ❝❤❡❝❦ ✇❡ s❤♦✇❡❞ t❤❛t ✇✐t❤ ♦✉r ♣r♦♣♦s❡❞ ❞❉●▲❆P✲❡q✉❛t✐♦♥ t❤❡ s✉♠

r✉❧❡s ❛r❡ ♣r❡s❡r✈❡❞ ✉♥❞❡r ❡✈♦❧✉t✐♦♥

✶✹ ✴ ✶✹

SLIDE 82

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥

❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ3 t❤❡♦r②✿

p k p − k

✶ ✴ ✾

SLIDE 83

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥

❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ3 t❤❡♦r②✿

p k p − k

■♥ ❝♦✈❛r✐❛♥t P❚ t❤❡ ❧♦♦♣ ✐s ❣✐✈❡♥ ❜②

dDk

(2π)D 1 p2 − m2 + iǫ 1 (p − k)2 − m2 + iǫ

✶ ✴ ✾

SLIDE 84

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥

❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ3 t❤❡♦r②✿

p k p − k

■♥ ❝♦✈❛r✐❛♥t P❚ t❤❡ ❧♦♦♣ ✐s ❣✐✈❡♥ ❜②

dDk

(2π)D 1 p2 − m2 + iǫ 1 (p − k)2 − m2 + iǫ P❡r❢♦r♠✐♥❣ t❤❡ k− ✐♥t❡❣r❛t✐♦♥ ✉s✐♥❣ ❈❛✉❝❤②✬s t❤❡♦r❡♠ ♦♥❡ ✜♥❞s

p+

dk+

2π

dD−2k

(2π)D−2 1 (2k+)(2(p+ − k+)) 1 p− − k2+m2

2k+

− (p−k)2+m2

2(p+−k+) + iǫ

✶ ✴ ✾

SLIDE 85

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■✿ ▼♦t✐✈❛t✐♦♥

❆s ❛♥ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r ❛ q✉❛r❦ ❧♦♦♣ ✐♥ φ3 t❤❡♦r②✿

p k p − k

❡♥❡r❛❧❧② t❤❡ ❞❡♥♦♠✐♥❛t♦r ❢♦r ❛ st❛t❡ ζi ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s xi ❛♥❞ xi+1 ✐s ❣✐✈❡♥ ❜②✿

1 P −

i

−

l∈i k− l, on−shell + iǫ

✇❤❡r❡ Pi ✐s t❤❡ s✉♠ ♦❢ ❛❧❧ ❡①t❡r♥❛❧ ♠♦♠❡♥t❛ ❡♥t❡r✐♥❣ t❤❡ ❣r❛♣❤ ❜❡❢♦r❡ ✈❡rt❡① i ❛♥❞ t❤❡ s✉♠ ✐s ♦✈❡r t❤❡ ♦♥✲s❤❡❧❧ ♠✐♥✉s ♠♦♠❡♥t❛ ♦❢ ❛❧❧ ❧✐♥❡s ✐♥ t❤❡ st❛t❡

✶ ✴ ✾

SLIDE 86

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■■✿ ❘✉❧❡s

◮ ❙t❛rt✐♥❣ ❢r♦♠ ❛ ❣✐✈❡♥ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠ ♦♥❡ ❤❛s t♦ ❝♦♥s✐❞❡r ❛❧❧ ♣♦ss✐❜❧❡ x+✲♦r❞❡r✐♥❣s ♦❢

t❤❡ ✈❡rt✐❝❡s✳ ■♥ ♦r❞❡r t♦ ✈✐s✉❛❧✐s❡ t❤❡s❡ ♦r❞❡r✐♥❣s ♦♥❡ ✉s❡s t❤❛t x+ ✐♥❝r❡❛s❡s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ♦♥ t❤❡ ❧❤s ♦❢ t❤❡ ❝✉t ✇❤✐❧❡ ✐t ✐♥❝r❡❛s❡s ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ♦♥ t❤❡ r❤s ♦❢ t❤❡ ❝✉t✳

◮ ❈♦✉♣❧✐♥❣ ❝♦♥st❛♥ts ❛♥❞ ✈❡rt❡① ❢❛❝t♦rs ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ❝♦✈❛r✐❛♥t P❚✳ ◮ P❧✉s ❛♥❞ tr❛♥s✈❡rs❛❧ ♠♦♠❡♥t❛✱ k+

l

✉♥❞ kl✱ ♦❢ ❛ ❧✐♥❡ l ❛r❡ ❝♦♥s❡r✈❡❞ ❛t t❤❡ ✈❡rt✐❝❡s

◮ ❊❛❝❤ ❧✐♥❡ l ✐♥ ❛ ❣r❛♣❤ ❝♦♠❡s ✇✐t❤ ❛ ❢❛❝t♦r

1 2k+

l

❛♥❞ ❛ ❍❡❛✈✐s✐❞❡ ❢✉♥❝t✐♦♥ Θ(k+

l )✱

❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♣r♦♣❛❣❛t✐♦♥ ❢r♦♠ ❧♦✇❡r t♦ ❤✐❣❤❡r x+

◮ ❋♦r ❡❛❝❤ ❧♦♦♣ t❤❡r❡s ❛♥ ✐♥t❡❣r❛❧ ♦✈❡r ♣❧✉s ❛♥❞ tr❛♥s✈❡rs❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❧♦♦♣

♠♦♠❡♥t✉♠ ℓ✿ dℓ+dd−2ℓ (2π)d−1

◮ ❋♦r ❡❛❝❤ st❛t❡ ζi ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s x+

i ✉♥❞ x+ i+1 ♦♥❡ ❣❡ts t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❢❛❝t♦r

1 P −

i

−

l∈i k− l, on−shell + iǫ

✷ ✴ ✾

SLIDE 87

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■■■✿ P❉❋ ❛♥❞ ❉P❉ ❉❡✜♥✐t✐♦♥s ✐♥ ▲❈P❚ P❉❋

fj1

B (x1) =

t
c
k+

1

n1 dk−

1 dD−2k1

(2π)D M(c)

i=2

dk+

i dD−2ki

(2π)D−1

N(t)
i=M(c)+1

dk+

i dD−2ki

(2π)D−1

× Φj1

P DFt,c,o

{k+}, {k}
2πδ
p− − k− −

N

i=2

k−

i ,on−shell

δ
p+ −

N

i=1

k+

i

✇❤❡r❡ n1 = 1 ✐❢ ♣❛rt♦♥ 1 ✐s ❛ ❣❧✉♦♥ ♦r ❛ s❝❛❧❛r q✉❛r❦✱ ✇❤✐❧❡ ❢♦r ❉✐r❛❝ q✉❛r❦s ♦♥❡ ❤❛s n1 = 0✳

❉P❉

✸ ✴ ✾

SLIDE 88

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■■■✿ P❉❋ ❛♥❞ ❉P❉ ❉❡✜♥✐t✐♦♥s ✐♥ ▲❈P❚ P❉❋

fj1

B (x1) =

t
c
k+

1

n1 dk−

1 dD−2k1

(2π)D M(c)

i=2

dk+

i dD−2ki

(2π)D−1

N(t)
i=M(c)+1

dk+

i dD−2ki

(2π)D−1

× Φj1

P DFt,c,o

{k+}, {k}
2πδ
p− − k− −

N

i=2

k−

i ,on−shell

δ
p+ −

N

i=1

k+

i

❉P❉

F j1j2

B

(x1, x2) =

t
c
l

δf(l),j2

k+

1

n1 k+

2

n2 2p+(2π)D−1 × dk−

1 dk− l d∆−dD−2k1dD−2kl

(2π)3D M(c)

i=2, i=l

dk+

i dD−2ki

(2π)D−1

N(t)
i=M(c)+1

dk+

i dD−2ki

(2π)D−1

× Φj1j2

DP Dt,c,o

{k+}, {k}
2πδ
p− − k−

1 − k− l − M(c)

i=2, i=l

k−

i ,on−shell

δ
p+ −

M(c)

i=1

k+

i

✸ ✴ ✾

SLIDE 89

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ▲❈P❚ P❉❋ ❣r❛♣❤

F (FA) H H′ FA I I′ k k

❚❤✐s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ✇❤❡r❡

st❛t❡s st❛t❡s st❛t❡s st❛t❡s

✹ ✴ ✾

SLIDE 90

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ▲❈P❚ P❉❋ ❣r❛♣❤

F (FA) H H′ FA I I′ k k

❚❤✐s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦP DF =I F (FA) I′ ✇❤❡r❡

st❛t❡s st❛t❡s st❛t❡s st❛t❡s

✹ ✴ ✾

SLIDE 91

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ▲❈P❚ P❉❋ ❣r❛♣❤

F (FA) H H′ FA I I′ k k

❚❤✐s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦP DF =I F (FA) I′ ✇❤❡r❡ I =

st❛t❡s ζ

ζ<H

1 p− −

l∈ζ k− l, o.s. + iǫ

I′ =

st❛t❡s ζ

ζ<H′

1 p− −

l∈ζ k− l, o.s. − iǫ

F(FA) =

st❛t❡s ζ

H<ζ<FA

1 p− − k− −

l∈ζ k− l, o.s. + iǫ

st❛t❡s ζ

H′<ζ<FA

1 p− − k− −

l∈ζ k− l, o.s. − iǫ

× 2πδ  p− − k− −

l∈FA

k−

l, o.s.

 

✹ ✴ ✾

SLIDE 92

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t FA✳ ❙✉♠♠✐♥❣ F(FA) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣

✺ ✴ ✾

SLIDE 93

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t FA✳ ❙✉♠♠✐♥❣ F(FA) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣

FA

F (FA) =

N

c=1

 

c−1

f=1

1 p−− k−− Df + iǫ 2πδ

p−− k−− Dc
N
f=c+1

1 p−− k−− Df − iǫ   ✇❤❡r❡ Df =

l∈f

k−

l, on−shell ,

✺ ✴ ✾

SLIDE 94

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t FA✳ ❙✉♠♠✐♥❣ F(FA) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣

FA

F (FA) =

N

c=1

 

c−1

f=1

1 p−− k−− Df + iǫ 2πδ

p−− k−− Dc
N
f=c+1

1 p−− k−− Df − iǫ   r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s 2π δ(x) = i

1

x + iǫ − 1 x − iǫ

✺ ✴ ✾

SLIDE 95

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t FA✳ ❙✉♠♠✐♥❣ F(FA) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣

FA

F (FA) =

N

c=1

 

c−1

f=1

1 p−− k−− Df + iǫ 2πδ

p−− k−− Dc
N
f=c+1

1 p−− k−− Df − iǫ   r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s 2π δ(x) = i

1

x + iǫ − 1 x − iǫ

t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s
FA

F (FA) =i  

N

f=1

1 p− − k− − Df + iǫ −

N

f=1

1 p− − k− − Df − iǫ  

✺ ✴ ✾

SLIDE 96

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t FA✳ ❙✉♠♠✐♥❣ F(FA) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣

FA

F (FA) =

N

c=1

 

c−1

f=1

1 p−− k−− Df + iǫ 2πδ

p−− k−− Dc
N
f=c+1

1 p−− k−− Df − iǫ   r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s 2π δ(x) = i

1

x + iǫ − 1 x − iǫ

t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s
FA

F (FA) =i  

N

f=1

1 p− − k− − Df + iǫ −

N

f=1

1 p− − k− − Df − iǫ   ❋♦r N ≥ 2 t❤✐s ❡①♣r❡ss✐♦♥ ✈❛♥✐s❤❡s ❛❢t❡r ✐♥t❡❣r❛t✐♦♥ ♦✈❡r k− ✇❤✐❧❡ ❢♦r N = 1 t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ✐s r❡♣r♦❞✉❝❡❞✳

✺ ✴ ✾

SLIDE 97

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ■❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r P❉❋s

❆ss✉♠✐♥❣ t❤❛t t❤❡r❡ ❛r❡ N ❞✐st✐♥❝t st❛t❡s ❜❡t✇❡❡♥ H ❛♥❞ H′ t❤❡r❡ ❛r❡ t❤✉s ❛❧s♦ N ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r t❤❡ ✜♥❛❧ st❛t❡ ❝✉t FA✳ ❙✉♠♠✐♥❣ F(FA) ♦✈❡r ❛❧❧ ❝✉ts ♦♥❡ ✜♥❞s t❤❡ ❢♦❧❧♦✇✐♥❣

FA

F (FA) =

N

c=1

 

c−1

f=1

1 p−− k−− Df + iǫ 2πδ

p−− k−− Dc
N
f=c+1

1 p−− k−− Df − iǫ   r❡✇r✐t✐♥❣ t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ❛s 2π δ(x) = i

1

x + iǫ − 1 x − iǫ

t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s
FA

F (FA) =i  

N

f=1

1 p− − k− − Df + iǫ −

N

f=1

1 p− − k− − Df − iǫ   ❋♦r N ≥ 2 t❤✐s ❡①♣r❡ss✐♦♥ ✈❛♥✐s❤❡s ❛❢t❡r ✐♥t❡❣r❛t✐♦♥ ♦✈❡r k− ✇❤✐❧❡ ❢♦r N = 1 t❤❡ ♦♥✲s❤❡❧❧ δ ❢✉♥❝t✐♦♥ ✐s r❡♣r♦❞✉❝❡❞✳ ❖♥❡ ❝❛♥ t❤✉s ❝♦♥❝❧✉❞❡✱ t❤❛t ♦♥❧② s✉❝❤ x+ ♦r❞❡r✐♥❣s ✇✐t❤ ♦♥❧② ♦♥❡ st❛t❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❤❛r❞ ✈❡rt✐❝❡s ❤❛✈❡ t♦ ❜❡ ❝♦♥s✐❞❡r❡❞✳

✺ ✴ ✾

SLIDE 98

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r ❉P❉s

❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦDP D =I1 I2 F (FA) I′

2 I′ 1

✻ ✴ ✾

SLIDE 99

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r ❉P❉s

❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦDP D =I1 I2 F (FA) I′

2 I′ 1

t♦ ❜❡ ❛❜❧❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ❜❡❢♦r❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ x+ ♦r❞❡r✐♥❣s

F (FA) H1 H′

1

FA I1 I′

1

K − k′ K − k′′ H2 H′

2

I2 I′

2

k′ k′′ F (FA) H1 H′

1

FA I1 I′

1

K − k′ K − k′′ H2 H′

2

˜ I2 I′

2

k′ k′′ ✻ ✴ ✾

SLIDE 100

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r ❉P❉s

❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦDP D =I1 I2 F (FA) I′

2 I′ 1

t♦ ❜❡ ❛❜❧❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ❜❡❢♦r❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ x+ ♦r❞❡r✐♥❣s

F (FA) H1 H′

1

FA I1 I′

1

K − k′ K − k′′ H2 H′

2

I2 I′

2

k′ k′′ F (FA) H1 H′

1

FA I1 I′

1

K − k′ K − k′′ H2 H′

2

˜ I2 I′

2

k′ k′′

❈♦♥s✐❞❡r ♥♦✇ t❤❡ st❛t❡s ❜❡t✇❡❡♥ H1 ❛♥❞ H2✱ I2 ❛♥❞ ˜ I2 I2 = 1 p− − (K− − k′−) − DI2 + iǫ ˜ I2 = 1 p− − k′− − D˜

I2 + iǫ .

✻ ✴ ✾

SLIDE 101

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r ❉P❉s

❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦDP D =I1 I2 F (FA) I′

2 I′ 1

t♦ ❜❡ ❛❜❧❡ t♦ ✉s❡ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ❜❡❢♦r❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ x+ ♦r❞❡r✐♥❣s

F (FA) H1 H′

1

FA I1 I′

1

K − k′ K − k′′ H2 H′

2

I2 I′

2

k′ k′′ F (FA) H1 H′

1

FA I1 I′

1

K − k′ K − k′′ H2 H′

2

˜ I2 I′

2

k′ k′′

❆s k′− ♦♥❧② ♦❝❝✉rs ✐♥ t❤❡s❡ ❡♥❡r❣② ❞❡♥♦♠✐♥❛t♦rs ✇❡ ❝❛♥ s✉♠ t❤❡s❡ t✇♦ x+ ♦r❞❡r✐♥❣s ❛♥❞ ✐♥t❡❣r❛t❡ ♦✈❡r k′− dk− 2π

I2 + ˜

I2

=

dk′− 2π   2p− − K− − D˜

I2 − DI2

p−− (K−− k′−) − DI2 + iǫ

p−− k′−− D˜

I2 + iǫ



 = −i ❘❡♣❡❛t✐♥❣ t❤❡ s❛♠❡ ♦♥ t❤❡ r❤s ②✐❡❧❞s ❛ ❢❛❝t♦r ♦❢ i✳

✻ ✴ ✾

SLIDE 102

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱✿ ❝♦♥tr✐❜✉t✐♥❣ x+ ♦r❞❡r✐♥❣s ❢♦r ❉P❉s

❈♦♥s✐❞❡r ♥♦✇ ❛ ❉P❉✱ ✇❤✐❝❤ ❝❛♥ ❛❣❛✐♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s ΦDP D =I1 I2 F (FA) I′

2 I′ 1

❚❤✉s ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡✱ t❤❛t s✉♠♠✐♥❣ ♦✈❡r t❤❡ ♣♦ss✐❜❧❡ ♦r❞❡r✐♥❣s ♦❢ t❤❡ ❤❛r❞ ✈❡rt✐❝❡s ❛♥❞ ✐♥t❡❣r❛t✐♥❣ ♦✈❡r k′− ❛♥❞ k′′− ✐s t❛♥t❛♠♦✉♥t t♦ s❡tt✐♥❣ t❤❡ ❤❛r❞ ✈❡rt✐❝❡s ♦♥ ❡❛❝❤ s✐❞❡ ♦❢ t❤❡ ✜♥❛❧ st❛t❡ ❝✉t t♦ t❤❡ s❛♠❡ x+ ✈❛❧✉❡

F (FA) FA I1 I′

1

K − k′ K − k′′ H H′ k′ k′′ K K ✻ ✴ ✾

SLIDE 103

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

✻ ✴ ✾

SLIDE 104

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱■✿ ✉♣❞❛t❡❞ P❉❋ ❛♥❞ ❉P❉ ❞❡✜♥✐t✐♦♥s P❉❋ ❛♥❞ ❉P❉ ❞❡✜♥✐t✐♦♥s

fj1

B (x1) =

t
c
x1 p+n1 p+
dD−2k1

(2π)D−1 N(t)

i=2

dxidD−2ki (2π)D−1 p+ × Φj1

P DFt,c,o ({x}, {k}) δ

1 −

M(c)

i=1

xi

1−x1
dx2 F j1j2

B

(x1, x2) =

t
c
l

δf(l),j2

x1p+n12p+

dD−2k1 (2π)D−1 N(t)

i=2

dxidD−2ki (2π)D−1 p+ ×

xl p+nl Φj1j2

DP Dt,c,o ({x}, {k}) δ

1 −

M(c)

i=1

xi

❈♦♠♣❛r✐♥❣ t❤❡s❡ ❡①♣r❡ss✐♦♥s✱ ♦♥❡ ✜♥❞s t❤❛t t❤❡ r❤s ✐s ❜❛s✐❝❛❧❧② t❤❡ s❛♠❡ ✭♥❡❣❧❡❝t✐♥❣ t❤❡ s✉♠

♦✈❡r ✮ ✐❢ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t

✼ ✴ ✾

SLIDE 105

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▲❈P❚ ❱■✿ ✉♣❞❛t❡❞ P❉❋ ❛♥❞ ❉P❉ ❞❡✜♥✐t✐♦♥s P❉❋ ❛♥❞ ❉P❉ ❞❡✜♥✐t✐♦♥s

fj1

B (x1) =

t
c
x1 p+n1 p+
dD−2k1

(2π)D−1 N(t)

i=2

dxidD−2ki (2π)D−1 p+ × Φj1

P DFt,c,o ({x}, {k}) δ

1 −

M(c)

i=1

xi

1−x1
dx2 F j1j2

B

(x1, x2) =

t
c
l

δf(l),j2

x1p+n12p+

dD−2k1 (2π)D−1 N(t)

i=2

dxidD−2ki (2π)D−1 p+ ×

xl p+nl Φj1j2

DP Dt,c,o ({x}, {k}) δ

1 −

M(c)

i=1

xi

❈♦♠♣❛r✐♥❣ t❤❡s❡ ❡①♣r❡ss✐♦♥s✱ ♦♥❡ ✜♥❞s t❤❛t t❤❡ r❤s ✐s ❜❛s✐❝❛❧❧② t❤❡ s❛♠❡ ✭♥❡❣❧❡❝t✐♥❣ t❤❡ s✉♠

♦✈❡r l✮ ✐❢ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t 2

xl p+nl Φj1, j2

DP Dt,c,o = Φj1 P DFt,c,o

✼ ✴ ✾

SLIDE 106

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❆ss✉♠✐♥❣ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t 2

xl p+nl Φj1, j2

DP Dt,c,o = Φj1 P DFt,c,o t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❝❛♥

❜❡ r❡✇r✐tt❡♥ ❛s

t
c
l
δf(l), j2 − δf(l), j2

x1 p+n1 p+

dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   =

Nj2v + δj1, j2 − δj1, j2

t

c
x1 p+n1 p+
dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✇❤✐❝❤ r❡❞✉❝❡s t♦

✽ ✴ ✾

SLIDE 107

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

◆✉♠❜❡r ❙✉♠ ❘✉❧❡

❆ss✉♠✐♥❣ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t 2

xl p+nl Φj1, j2

DP Dt,c,o = Φj1 P DFt,c,o t❤❡ ♥✉♠❜❡r s✉♠ r✉❧❡ ❝❛♥

❜❡ r❡✇r✐tt❡♥ ❛s

t
c
l
δf(l), j2 − δf(l), j2

x1 p+n1 p+

dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   =

Nj2v + δj1, j2 − δj1, j2

t

c
x1 p+n1 p+
dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✇❤✐❝❤ r❡❞✉❝❡s t♦

l
δf(l), j2 − δf(l), j2
=
Nj2v + δj1, j2 − δj1, j2
✽ ✴ ✾

SLIDE 108

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

❋♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❛♥❛❧♦❣♦✉s❧② ✜♥❞s

j2
t
c
l

δf(l), j2

x1 p+n1 p+
dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × xl Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   = (1 − x1)

t
c
x1 p+n1 p+
dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✉s✐♥❣ ❛ s❤♦rt❤❛♥❞ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ✐♥t❡❣r❛t✐♦♥ ♠❡❛s✉r❡s

✾ ✴ ✾

SLIDE 109

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

❋♦r t❤❡ ♠♦♠❡♥t✉♠ s✉♠ r✉❧❡ ♦♥❡ ❛♥❛❧♦❣♦✉s❧② ✜♥❞s

j2
t
c
l

δf(l), j2

x1 p+n1 p+
dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × xl Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   = (1 − x1)

t
c
x1 p+n1 p+
dD−2k1

(2π)D−1  

N(t)

i=2

dxidD−2ki (2π)D−1 p+   × Φj1

P DFt,c,o ({x}, {k}) δ

 1 −

M(c)

i=1

xi   ✉s✐♥❣ ❛ s❤♦rt❤❛♥❞ ♥♦t❛t✐♦♥ ❢♦r t❤❡ ✐♥t❡❣r❛t✐♦♥ ♠❡❛s✉r❡s

Db

a [xi] = b

i=a

1 dxi p+

Db

a [ki] = b

i=a
dD−2ki

(2π)D−1 ,

✾ ✴ ✾

SLIDE 110

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

t❤✐s ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s

t
c
l
x1 p+n1p+
DN(t)

2

[xi] DN(t)

1

[ki] xl Φj1

P DFt,c,o({x}, {k}) δ

  1−

M(c)

i=1

xi   = (1−x1)

t
c
x1 p+n1p+
DN(t)

2

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o({x}, {k}) δ

  1−

M(c)

i=1

xi   ✇❤✐❝❤ r❡❞✉❝❡s t♦

✾ ✴ ✾

SLIDE 111

▲❈P❚ Pr♦♦❢ ❢♦r ❜❛r❡ q✉❛♥t✐t✐❡s

▼♦♠❡♥t✉♠ ❙✉♠ ❘✉❧❡

t❤✐s ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s

t
c
l
x1 p+n1p+
DN(t)

2

[xi] DN(t)

1

[ki] xl Φj1

P DFt,c,o({x}, {k}) δ

  1−

M(c)

i=1

xi   = (1−x1)

t
c
x1 p+n1p+
DN(t)

2

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o({x}, {k}) δ

  1−

M(c)

i=1

xi   ✇❤✐❝❤ r❡❞✉❝❡s t♦

l
DN(t)

2

[xi] DN(t)

1

[ki] xl Φj1

P DFt,c,o({x}, {k}) δ

  1−

M(c)

i=1

xi   = (1−x1)

DN(t)

2

[xi] DN(t)

1

[ki] Φj1

P DFt,c,o({x}, {k}) δ

  1−

M(c)

i=1

xi  

✾ ✴ ✾