r tqr s - - PowerPoint PPT Presentation

SLIDE 1

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙❝❛❧❛r ❧❡♣t♦q✉❛r❦ ✐♥ ♥✉❝❧❡♦♥ ❞❡❝❛②s

▼✐t❥❛ ➆❛❞❧ ✇✐t❤ Pr♦❢✳ ❉r✳ ❙✈❥❡t❧❛♥❛ ❋❛❥❢❡r

❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ P❤②s✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ▲❥✉❜❧❥❛♥❛✱ ❙❧♦✈❡♥✐❛ ❏♦➸❡❢ ❙t❡❢❛♥ ■♥st✐t✉t❡✱ ▲❥✉❜❧❥❛♥❛✱ ❙❧♦✈❡♥✐❛

❇r❞❛✱ ✶✵✳ ✶✵✳ ✷✵✶✾

SLIDE 2

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥

✷

▲❡♣t♦q✉❛r❦s

✸

◆❡✉tr♦♥ ❧✐❢❡t✐♠❡

✹

Pr♦t♦♥ ❧✐❢❡t✐♠❡

✺

❈❛❧❝✉❧❛t✐♦♥s

✻

❈♦♥❝❧✉s✐♦♥

SLIDE 3

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥

✷

▲❡♣t♦q✉❛r❦s

✸

◆❡✉tr♦♥ ❧✐❢❡t✐♠❡

✹

Pr♦t♦♥ ❧✐❢❡t✐♠❡

✺

❈❛❧❝✉❧❛t✐♦♥s

✻

❈♦♥❝❧✉s✐♦♥

SLIDE 4

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥

✷

▲❡♣t♦q✉❛r❦s

✸

◆❡✉tr♦♥ ❧✐❢❡t✐♠❡

✹

Pr♦t♦♥ ❧✐❢❡t✐♠❡

✺

❈❛❧❝✉❧❛t✐♦♥s

✻

❈♦♥❝❧✉s✐♦♥

SLIDE 5

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥

✷

▲❡♣t♦q✉❛r❦s

✸

◆❡✉tr♦♥ ❧✐❢❡t✐♠❡

✹

Pr♦t♦♥ ❧✐❢❡t✐♠❡

✺

❈❛❧❝✉❧❛t✐♦♥s

✻

❈♦♥❝❧✉s✐♦♥

SLIDE 6

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥

✷

▲❡♣t♦q✉❛r❦s

✸

◆❡✉tr♦♥ ❧✐❢❡t✐♠❡

✹

Pr♦t♦♥ ❧✐❢❡t✐♠❡

✺

❈❛❧❝✉❧❛t✐♦♥s

✻

❈♦♥❝❧✉s✐♦♥

SLIDE 7

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥

✷

▲❡♣t♦q✉❛r❦s

✸

◆❡✉tr♦♥ ❧✐❢❡t✐♠❡

✹

Pr♦t♦♥ ❧✐❢❡t✐♠❡

✺

❈❛❧❝✉❧❛t✐♦♥s

✻

❈♦♥❝❧✉s✐♦♥

SLIDE 8

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥

t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❡❧❡❝tr✐❝ ❝❤❛r❣❡ ♥♦t q✉❛♥t✐③❡❞ ♠❛♥② ❢r❡❡ ♣❛r❛♠❡t❡rs ✇❤② ✸ ❢❡r♠✐♦♥ ❢❛♠✐❧❧✐❡s❄ ♥❡✉tr✐♥♦ ♠❛ss❡s❄ ↓

• r❛♥❞ ❯♥✐✜❡❞ ❚❤❡♦r✐❡s

♦♥❡ ❢♦r❝❡ ♦♥❡ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t

SLIDE 9

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼♦t✐✈❛t✐♦♥

t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❡❧❡❝tr✐❝ ❝❤❛r❣❡ ♥♦t q✉❛♥t✐③❡❞ ♠❛♥② ❢r❡❡ ♣❛r❛♠❡t❡rs ✇❤② ✸ ❢❡r♠✐♦♥ ❢❛♠✐❧❧✐❡s❄ ♥❡✉tr✐♥♦ ♠❛ss❡s❄ ↓

• r❛♥❞ ❯♥✐✜❡❞ ❚❤❡♦r✐❡s

♦♥❡ ❢♦r❝❡ ♦♥❡ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t

SLIDE 10

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

• r❛♥❞ ❯♥✐✜❡❞ ❚❤❡♦r✐❡s

q✉❛r❦s ❛♥❞ ❧❡♣t♦♥s ✐♥ t❤❡ s❛♠❡ ♠✉❧t✐♣❧❡ts ⇒ ❡❧❡❝tr✐❝ ❝❤❛r❣❡ q✉❛♥t✐③❡❞ ⇒ ♥❡✇ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ q✉❛r❦s ❛♥❞ ❧❡♣t♦♥s ⇒ ❜❛r✐♦♥ ♥✉♠❜❡r ✈✐♦❧❛t✐♦♥ ⇒ ♣r♦t♦♥ ❞❡❝❛② (τp > ✶✵✸✵ ②✮ mX ∼ ✶✵✶✹✕✶✻ ●❡❱

SLIDE 11

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▲❡♣t♦q✉❛r❦s

✭SU(✸)❈, SU(✷)▲, U(✶)❨✮

s♣✐♥ s②♠❜♦❧ ❝♦✉♣❧✐♥❣ ❝♦✉♣❧✐♥❣ ❋ ✭q✉❛r❦ ❛♥❞ ❧❡♣t♦♥✮ ✭♣❛✐r ♦❢ q✉❛r❦s✮ (✸, ✸, ✶/✸) ✵ S✸ ▲▲ ▲▲ −✷ (✸, ✷, ✼/✻) R✷ ❘▲✱ ▲❘ ✵ (✸, ✷, ✶/✻) ˜ R✷ ❘▲✱ ▲❘ ✵ (✸, ✶, ✹/✸) ˜ S✶ ❘❘ ❘❘ −✷ (✸, ✶, ✶/✸) S✶ ▲▲✱ ❘❘✱ ❘❘ ▲▲✱ ❘❘ −✷ (✸, ✶, −✷/✸) S✶ ❘❘ ❘❘ −✷ (✸, ✸, ✷/✸) ✶ U✸ ▲▲ ✵ (✸, ✷, ✺/✻) V✷ ❘▲✱ ▲❘ ▲❘ −✷ (✸, ✷, −✶/✻) ˜ V✷ ❘▲✱ ▲❘ ❘▲ −✷ (✸, ✶, ✺/✸) ˜ U✶ ❘❘ ✵ (✸, ✶, ✷/✸) U✶ ▲▲✱ ❘❘✱ ❘❘ ✵ (✸, ✶, −✶/✸) U✶ ❘❘ ✵

SLIDE 12

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▲❡♣t♦q✉❛r❦s

✭SU(✸)❈, SU(✷)▲, U(✶)❨✮

s♣✐♥ s②♠❜♦❧ ❝♦✉♣❧✐♥❣ ❝♦✉♣❧✐♥❣ ❋ ✭q✉❛r❦ ❛♥❞ ❧❡♣t♦♥✮ ✭♣❛✐r ♦❢ q✉❛r❦s✮ (✸, ✸, ✶/✸) ✵ S✸ ▲▲ ▲▲ −✷ (✸, ✷, ✼/✻) R✷ ❘▲✱ ▲❘ ✵ (✸, ✷, ✶/✻) ˜ R✷ ❘▲✱ ▲❘ ✵ (✸, ✶, ✹/✸) ˜ S✶ ❘❘ ❘❘ −✷ (✸, ✶, ✶/✸) S✶ ▲▲✱ ❘❘✱ ❘❘ ▲▲✱ ❘❘ −✷ (✸, ✶, −✷/✸) S✶ ❘❘ ❘❘ −✷ (✸, ✸, ✷/✸) ✶ U✸ ▲▲ ✵ (✸, ✷, ✺/✻) V✷ ❘▲✱ ▲❘ ▲❘ −✷ (✸, ✷, −✶/✻) ˜ V✷ ❘▲✱ ▲❘ ❘▲ −✷ (✸, ✶, ✺/✸) ˜ U✶ ❘❘ ✵ (✸, ✶, ✷/✸) U✶ ▲▲✱ ❘❘✱ ❘❘ ✵ (✸, ✶, −✶/✸) U✶ ❘❘ ✵

SLIDE 13

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❉✐s❝r❡♣❛♥❝② ❜❡t✇❡❡♥ t❤❡ ♥❡✉tr♦♥ ❧✐❢❡t✐♠❡ ♠❡❛s✉r❡♠❡♥ts

∆τn = ✽.✼ ± ✷.✷ s

SLIDE 14

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❚✇♦ ❞✐✛❡r❡♥t ♠❡t❤♦❞s

❜♦tt❧❡ ♠❡t❤♦❞ Γn =

• τ ❜♦tt❧❡

n

❜❡❛♠ ♠❡t❤♦❞ Γ❙▼

n

= Γ(n → pe−νe) =

• τ ❜❡❛♠

n

Γ❇❙▼

n

= Γ(n → χγ) = Γn−Γ❙▼

n

= ❇r(n → χγ)Γn , ❇r(n → χγ) ≈ ✶%

SLIDE 15

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Pr♦t♦♥ ❧✐❢❡t✐♠❡ ♠❡❛s✉r❡♠❡♥ts

❙✉♣❡r✲❑❛♠✐♦❦❛♥❞❡

SLIDE 16

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈❤❡r❡♥❦♦✈ r❛❞✐❛t✐♦♥ ❞❡t❡❝t✐♦♥

p → e+π✵

SLIDE 17

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙t❛rt✐♥❣ ♣♦✐♥t✿ s❝❛❧❛r ❧❡♣t♦q✉❛r❦ S✶

(SU(✸)❈, SU(✷)▲, U(✶)❨) : (✸, ✶, ✶/✸) LS✶ = +y▲▲

✶ ij Q ❈ i,a ▲

S✶εabLj,b

▲ + y❘❘ ✶ ij u❈ i ❘ S✶ej ❘ + y❘❘ ✶ ij d ❈ i ❘ S✶χj

+ z▲▲

✶ ij Q ❈ i,a ▲

S∗

✶εabQj,b ▲ + z❘❘ ✶ ij u❈ i ❘ S∗ ✶dj ❘ + ❤✳ ❝✳ ,

⇓ LS✶ = y❘❘

✶ ✶✶u❈ ❘S✶e❘ + y❘❘ ✶ ✶✶d ❈ ❘S✶χ + z❘❘ ✶ ✶✶u❈ ❘S∗ ✶d❘ + ❤✳ ❝✳

↑ ❇✳ ❋♦r♥❛❧ ✐♥ ❇✳ ●r✐♥st❡✐♥✱ ❉❛r❦ ▼❛tt❡r ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ◆❡✉tr♦♥ ❉❡❝❛② ❆♥♦♠❛❧②✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✷✵✱ ✶✾✶✽✵✶ ✭✷✵✶✽✮✳

SLIDE 18

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙t❛rt✐♥❣ ♣♦✐♥t✿ s❝❛❧❛r ❧❡♣t♦q✉❛r❦ S✶

(SU(✸)❈, SU(✷)▲, U(✶)❨) : (✸, ✶, ✶/✸) LS✶ = +y▲▲

✶ ij Q ❈ i,a ▲

S✶εabLj,b

▲ + y❘❘ ✶ ij u❈ i ❘ S✶ej ❘ + y❘❘ ✶ ij d ❈ i ❘ S✶χj

+ z▲▲

✶ ij Q ❈ i,a ▲

S∗

✶εabQj,b ▲ + z❘❘ ✶ ij u❈ i ❘ S∗ ✶dj ❘ + ❤✳ ❝✳ ,

⇓ LS✶ = y❘❘

✶ ✶✶u❈ ❘S✶e❘ + y❘❘ ✶ ✶✶d ❈ ❘S✶χ + z❘❘ ✶ ✶✶u❈ ❘S∗ ✶d❘ + ❤✳ ❝✳

↑ ❇✳ ❋♦r♥❛❧ ✐♥ ❇✳ ●r✐♥st❡✐♥✱ ❉❛r❦ ▼❛tt❡r ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ◆❡✉tr♦♥ ❉❡❝❛② ❆♥♦♠❛❧②✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✷✵✱ ✶✾✶✽✵✶ ✭✷✵✶✽✮✳

SLIDE 19

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

n → χγ

LS✶ = y❘❘

✶ ✶✶u❈ ❘S✶e❘ + y❘❘ ✶ ✶✶d ❈ ❘S✶χ + z❘❘ ✶ ✶✶u❈ ❘S∗ ✶d❘ + ❤✳ ❝✳

∆τn = ✽.✻ ± ✷.✶ s

SLIDE 20

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

p → e+π✵

LS✶ = y❘❘

✶ ✶✶u❈ ❘S✶e❘ + y❘❘ ✶ ✶✶d ❈ ❘S✶χ + z❘❘ ✶ ✶✶u❈ ❘S∗ ✶d❘ + ❤✳ ❝✳

τ(p → e+π✵) > ✶.✻ × ✶✵✸✹ ②

u u d d d e+ S1 u d u u u e+ S1

SLIDE 21

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

p → e+γ

LS✶ = y❘❘

✶ ✶✶u❈ ❘S✶e❘ + y❘❘ ✶ ✶✶d ❈ ❘S✶χ + z❘❘ ✶ ✶✶u❈ ❘S∗ ✶d❘ + ❤✳ ❝✳

τ(p → e+γ) > ✻.✼ × ✶✵✸✷ ②

u d u e+ S1 γ p e+ γ

SLIDE 22

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❆t♦♠✐❝ ♣❛r✐t② ✈✐♦❧❛t✐♦♥ ✐♥ ❝❡s✐✉♠

✇❡❛❦ ❝❤❛r❣❡✿ Q❙▼

❲ (Z, N) = −✷(✷Z + N)C✶u − ✷(Z + ✷N)C✶d

♥❡✇ ♣❤②s✐❝s ❝♦♥tr✐❜✉t✐♦♥✿ δC✶q = c▲▲

qq;ee − c▲❘ qq;ee + c❘▲ qq;ee − c❘❘ qq;ee

δC✶u = −c❘❘

✶✶;✶✶ =

v✷ ✹m✷

S✶

• y❘❘

✶ ✶✶

✷ ✇❡❛❦ ❝❤❛r❣❡ ♠❡❛s✉r❡♠❡♥t ✐♥ ✶✸✸❈s ❞✐✛❡rs ❢r♦♠ t❤❡ ❙▼✿ δQ❲ = Q❲ − Q❙▼

❲ = ✵.✻✺(✹✸)

|δC✶u| =

• δQ❲

✸✼✻

• ∼ ✶✵−✸
• y❘❘

✶ ✶✶

• mS✶

∼ ✷.✻ × ✶✵−✹ ●❡❱−✶

SLIDE 23

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❆t♦♠✐❝ ♣❛r✐t② ✈✐♦❧❛t✐♦♥ ✐♥ ❝❡s✐✉♠

✇❡❛❦ ❝❤❛r❣❡✿ Q❙▼

❲ (Z, N) = −✷(✷Z + N)C✶u − ✷(Z + ✷N)C✶d

♥❡✇ ♣❤②s✐❝s ❝♦♥tr✐❜✉t✐♦♥✿ δC✶q = c▲▲

qq;ee − c▲❘ qq;ee + c❘▲ qq;ee − c❘❘ qq;ee

δC✶u = −c❘❘

✶✶;✶✶ =

v✷ ✹m✷

S✶

• y❘❘

✶ ✶✶

✷ ✇❡❛❦ ❝❤❛r❣❡ ♠❡❛s✉r❡♠❡♥t ✐♥ ✶✸✸❈s ❞✐✛❡rs ❢r♦♠ t❤❡ ❙▼✿ δQ❲ = Q❲ − Q❙▼

❲ = ✵.✻✺(✹✸)

|δC✶u| =

• δQ❲

✸✼✻

• ∼ ✶✵−✸
• y❘❘

✶ ✶✶

• mS✶

∼ ✷.✻ × ✶✵−✹ ●❡❱−✶

SLIDE 24

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Γ(p → e+π✵)

Γ(p → e+π✵) = ✶ ✽π|M|✷ |♣❈▼| m✷

p

= ✶ ✸✷π

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ W ❘❘

✵

(✵) ✷

• ✶ −

mπ✵ mp ✷✷ mp

SLIDE 25

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Γ(n → χγ)

L❡✛

NγN = eψ(x)

• F✶(q✷)γµAµ(x) + ✶

✹mF✷(q✷)σµνFµν(x)

• ψ(x)

L❡✛

✶ = L❉✐r❛❝ n′

+ L❉✐r❛❝

χ′

+ L❡✛

n′γn′ + L❡✛ n′↔χ′

= n′(i /

∂ − mn)n′ + χ′(i / ∂ − mχ)χ′ + ean

✹mn n′σµνFµνn′ + ε(n′χ′ + χ′n′) ♠❛ss ♠❛tr✐① ❞✐❛❣♦♥❛❧✐s❛t✐♦♥ (ε ≪ mn − mχ) : −mnn′n′ − mχχ′χ′ + ε(n′χ′ + χ′n′) =

• n′

χ′ −mn ε ε −mχ n′ χ′

SLIDE 26

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Γ(n → χγ)

♠❛ss ❡✐❣❡♥st❛t❡s✿ n = −n′ + ε mn − mχ χ′ , χ = ε mn − mχ n′ + χ′ ❡✛❡❝t✐✈❡ ▲❛❣r❛♥❣✐❛♥ ✐♥ t❤❡ ♠❛ss ❜❛s✐s✿ L❡✛

✶ = L❉✐r❛❝ n

+ L❉✐r❛❝

χ

+ L❡✛

nγn + L❡✛ nγχ

L❡✛

nγχ = − ean

✹mn ε (mn − mχ)χσµνFµνn + ❤✳ ❝✳

SLIDE 27

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Γ(n → χγ)

Γ(n → χγ) = e✷a✷

n

✸✷π mnε✷ (mn − mχ)✷

• ✶ −

mχ mn ✷✸

SLIDE 28

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Γ(n → χγ)

−iM = χ|

• iz❘❘

✶ ✶✶u❈ ❘d❘

−i m✷

S✶

• iy❘❘

✶ ✶✶d ❈ ❘χ

• |n

= i y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

P❘vχ✵|

• u❈P❘d
• d

❈P❘|n

= i y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

β (vnP❘vχ) −iM =

• χ
• iL❡✛

n→χ

• n
• = χ |iεχn| n

= iε(uχP❘un) ♠✐①✐♥❣ ♣❛r❛♠❡t❡r✿ ε = y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

β , β = ✵.✵✶✹✹(✸)(✷✶) ●❡❱✸ Γ(n → χγ) = e✷a✷

n

✸✷π

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ β✷

• ✶ −

mχ mn ✷✸ mn (mn − mχ)✷

SLIDE 29

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

Γ(p → e+γ)

✐♥t❡r❛❝t✐♦♥ ❜❛s✐s✿ L❡✛

✷ = L❉✐r❛❝ p′

+L❉✐r❛❝

e+′

+L❡✛

p′γp′+L❡✛ e+′γe+′+L❡✛ p′↔e+′

L❡✛

p′γp′ = ep′

• γµAµ + ap

✹mp σµνFµν

• p′ ,

L❡✛

e+′γe+′ = e

• e+′γµAµe+′

♠❛ss ❜❛s✐s✿ L❡✛

✷ = L❉✐r❛❝ p

+ L❉✐r❛❝

e+

+ L❡✛

pγp + L❡✛ e+γe+ + L❡✛ pγe+

L❡✛

pγe+ = − eap

✹mp ε (mp − me)e+σµνFµνp + ❤✳ ❝✳ Γ(p → e+γ) = e✷a✷

p

✸✷π

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ β✷

• ✶ −

me mp ✷✸ mp (mp − me)✷

SLIDE 30

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

χ ♠❛ss

❋♦r♥❛❧ ❛♥❞ ●r✐♥st❡✐♥ ❡①❝❧✉❞❡ ♣r♦t♦♥ ❞❡❝❛② ❝❤❛♥♥❡❧ p → n∗e+νe → χe+νe . ■♥ t❤✐s ❝❛s❡ mχ ✐s ❝♦♥str❛✐♥❡❞✿ mn > mχ > mp − me

SLIDE 31

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦✉♣❧✐♥❣ ❝♦♥st❛♥ts

✇❡ ✉s❡ t❤❡ ♣r❡✈✐♦✉s ❝♦♥str❛✐♥t ♦♥ mχ✱ r❡s✉❧ts ❢r♦♠ ❛t♦♠✐❝ ♣❛r✐t② ✈✐♦❧❛t✐♦♥✱ ∆τn = ✽.✻ ± ✷.✶ s ❛♥❞ τ(p → e+π✵) > ✶.✻ × ✶✵✸✹ ② Γ(p → e+π✵) ∝

• y ❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ , Γ(n → χγ) ∝

• y ❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ ⇓

• y ❘❘

✶ ✶✶

• mS✶

> ✼.✵ × ✶✵✶✻ ●❡❱−✶

SLIDE 32

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

τ(p → e+γ)

Γ(p → e+π✵) ∝

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ , Γ(p → e+γ) ∝

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ τ(p → e+π✵) > ✶.✻ × ✶✵✸✹ ② , τ(p → e+γ) > ✻.✼ × ✶✵✸✷ ② ⇓ τ♥❡✇(p → e+γ) > ✹.✵ × ✶✵✸✻ ②

SLIDE 33

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

▼❛ss ♦❢ t❤❡ ❧❡♣t♦q✉❛r❦ S✶

Γ(p → e+π✵) ∝

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ τ(p → e+π✵) > ✶.✻ × ✶✵✸✹ ② ❚❛❦✐♥❣ ❜♦t❤ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥ts ♦❢ ♦r❞❡r ♦♥❡ ⇓ mS✶ ✶✵✶✻ ●❡❱ . ❚❤✐s ✐s ●❯❚ ❡♥❡r❣② s❝❛❧❡✳

SLIDE 34

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❆❧s♦ ❞❡❝❛②s ♦❢ ✈❡r② ❝♦♠♠♦♥ ❤❛❞r♦♥s ❧✐❦❡ ♣r♦t♦♥ ❛♥❞ ♥❡✉tr♦♥ ❝❛♥ ♣✉t ❝♦♥str❛✐♥ts ♦♥ ❤②♣♦t❤❡t✐❝❛❧ ♣❛rt✐❝❧❡s✳

SLIDE 35

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳

SLIDE 36

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣

SLIDE 37

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧

ϕ′ = UG ′ϕ A′ = UG ′AU−✶

G ′

UU(✶)❨ = eiα(x) USU(✷)▲ = eiθa(x) τa

✷

USU(✸)❈ = eiξb(x) λb

✷

G a

µν = ∂µG a ν − ∂νG a µ − gsf abcG b µG c ν

W a

µν = ∂µW a ν − ∂νW a µ − gεabcW b µ W c ν

Bµν = ∂µBν − ∂νBµ Dµ = ∂µ+igsG a

µ

λa ✷ +igW b

µ

τ b ✷ +ig′YBµ Q✵|φ|✵ = ✵ L✸ ⊃ m✷

h

✷ h✷ W ±

µ =

✶ √ ✷

• W ✶

µ ∓ iW ✷ µ

• Zµ

Aµ

• =

❝♦s θ❲ − s✐♥ θ❲ s✐♥ θ❲ ❝♦s θ❲ W ✸

µ

Bµ

• ❝♦s θ❲ =

g

• g✷ + g′✷

s✐♥ θ❲ = g′

• g✷ + g′✷

t❛♥ θ❲ = g′ g

SLIDE 38

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❙▼ ▲❛❣r❛♥❣✐❛♥ ❜❡❢♦r❡ t❤❡ s②♠♠❡tr② ❜r❡❛❦✐♥❣

L = L✶ + L✷ + L✸ + L✹ ❣❛✉❣❡ ❦✐♥❡t✐❝ t❡r♠✿ L✶ = −✶ ✹G a

µνG aµν − ✶

✹W b

µνW bµν − ✶

✹BµνBµν ❢❡r♠✐♦♥ t❡r♠✿ L✷ = L▲ii / DL▲i + e❘ii / De❘i + Q▲ii / DQ▲i + u❘ii / Du❘i + d❘ii / Dd❘i s❝❛❧❛r t❡r♠✿ L✸ = (Dµφ)†Dµφ − V (φ) ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣ t❡r♠✿ L✹ = y(e)

ij L▲iφe❘j + y(u) ij Q▲i ˜

φu❘j + y(d)

ij

Q▲iφd❘j + ❤✳ ❝✳

SLIDE 39

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❍✐❣❣s ♠❡❝❤❛♥✐s♠

❍✐❣❣s ♣♦t❡♥t✐❛❧✿ V (φ) = −µ✷|φ|✷ + λ|φ|✹ |φ|✷ = µ✷ ✷λ = v✷ ✷ ✵|φ|✵ =

• ✵

v/ √ ✷

• φ(x) = ei ✶

v τbξb

• ✵

v+h(x) √ ✷

SLIDE 40

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧

L✸ ⊃ m✷

W W + µ W −µ + m✷ Z

✷ ZµZ µ mW = gv ✷ , mZ = v ✷

• g✷ + g′✷,

m✷

W

m✷

Z

= ❝♦s✷ θ❲ L✹ ⊃ v √ ✷

• y(e)

ij e▲ie❘j + y(u) ij u▲iu❘j + y(d) ij

d▲id❘j + ❤✳ ❝✳

• M(f )

ij

= − v √ ✷ y(f )

ij

f ′

▲i = Sijf▲j,

f ′

❘i = Tijf❘j

f

′ ▲iM(f ) ij f ′ ❘j =

• f

′ ▲kSki

S†

ilM(f ) lm Tmj

T †

jnf ′ ❘n

• = f ▲i
• M(f )

❞

• ij f❘j

S†

ikM(f ) kl Tlj =

• M(f )

❞

• ij ,

S†

iky(f ) kl Tlj =

• y(f )

❞

• ij

SLIDE 41

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ✐♥t❡r❛❝t✐♦♥s

L✷ ⊃ −eQ(e)ei / Aei − eQ(u)ui / Aui − eQ(d)di / Adi e = g s✐♥ θ❲ = g′ ❝♦s θ❲ Aµf

′ ▲✐γµf ′ ▲✐ = Aµf ▲❥S† jiγµSikf▲k = Aµf ▲jδjkγµf▲❦ = Aµf ▲iγµf▲i ,

L✷ ⊃ − g′ ✷ ❝♦s θ❲ f i

• g(f )

❱ /

Z + g(f )

❆ /

Zγ✺

• fi ,

g(f )

❱

= T (f )

✸

− ✷Q(f ) s✐♥✷ θ❲, g(f )

❆

= T (f )

✸

SLIDE 42

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ✐♥t❡r❛❝t✐♦♥s

L✷ ⊃ − g √ ✷

• J❈❈

qµ W +µ + J❈❈ lµ W +µ + ❤✳ ❝✳

• J❈❈

qµ = u′ ▲iγµd′ ▲i = u▲jS(u)† ji

γµS(d)

ik d▲k = u▲jVjkγµd▲k

V = S(u)†S(d) u′

▲i = V † ij u▲j

✐♥ d′

▲i = d▲i

J❈❈

lµ

= ν′

▲iγµe′ ▲i = ν▲jS(ν)† ji

γµS(e)

ik e▲k = ν▲jδjkγµe▲k = ν▲iγµe▲i

S(ν)†

ik

M(ν)

kl T (ν) lj

=

• M(ν)

❞

• ij = ❞✐❛❣(✵, ✵, ✵)

J❈❈†

lµ

= e′

▲iγµν′ ▲i = e▲jS(e)† ji

γµS(ν)

ik ν▲k = e▲jγµUjkν▲k

ν′

▲i = Uijν▲j

✐♥ e′

▲i = e▲i

ν′

▲ =

• νe

νµ ντ T

▲ = U

• ν✶

ν✷ ν✸ T

▲ ,

e′

▲ = e▲ =

• e−

µ− τ −T

▲

L✷ ⊃ −gs ✷ qiλa

ijγµG µ a qj

SLIDE 43

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❝❤❛r❣❡ ❝♦♥❥✉❣❛t✐♦♥ ❛♥❞ ❝❤✐r❛❧✐t②

ψ❈ = Cγ✵ψ∗ = iγ✷ψ∗ ψ▲ = P▲ψ = (✶ − γ✺) ✷ ψ, ψ❘ = P❘ψ = (✶ + γ✺) ✷ ψ ψ▲,❘ ≡ (P▲,❘ψ)†γ✵ = ψ†P▲,❘γ✵ = ψP❘,▲ ψ❈

▲,❘ ≡ (ψ▲,❘)❈ =

• ψ❈

❘,▲

ψ

❈ ▲,❘ ≡ ψ❈ ▲,❘ = (ψ▲,❘)❈ = (ψ❈)❘,▲ = ψ❈P▲,❘

SLIDE 44

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ▲❛❣r❛♥❣✐❛♥ ♦❢ t❤❡ r❛r❡ ♠❡s♦♥ ❞❡❝❛②s

Lqiqjℓℓ′ = − ✹G❋ √ ✷

• c▲▲

ij;ℓℓ′

• qi

▲γµqj ▲

ℓ▲γµℓ′

▲

• + c❘❘

ij;ℓℓ′

• qi

❘γµqj ❘

ℓ❘γµℓ′

❘

• + c▲❘

ij;ℓℓ′

• qi

▲γµqj ▲

ℓ❘γµℓ′

❘

• + c❘▲

ij;ℓℓ′

• qi

❘γµqj ❘

ℓ▲γµℓ′

▲

• + g❘❘

ij;ℓℓ′

• qi

❘qj ▲

ℓ❘ℓ′

▲

• + h❘❘

ij;ℓℓ′

• qi

❘σµνqj ▲

ℓ❘σµνℓ′

▲

• + g▲▲

ij;ℓℓ′

• qi

▲qj ❘

ℓ▲ℓ′

❘

• + h▲▲

ij;ℓℓ′

• qi

▲σµνqj ❘

ℓ▲σµνℓ′

❘

• + g▲❘

ij;ℓℓ′

• qi

▲qj ❘

ℓ❘ℓ′

▲

• + g❘▲

ij;ℓℓ′

• qi

❘qj ▲

ℓ▲ℓ′

❘

+ ❤✳ ❝✳

SLIDE 45

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❲✐❧s♦♥ ❝♦❡✣❝✐❡♥ts ❢♦r t❤❡ ▲❛❣r❛♥❣✐❛♥ ♦❢ t❤❡ r❛r❡ ♠❡s♦♥ ❞❡❝❛②s

c▲▲

ij;ℓℓ′ = − v✷

✹m✷

S✶

• V Ty▲▲

✶

• jℓ′
• V Ty▲▲

✶

∗

iℓ

c❘❘

ij;ℓℓ′ = − v✷

✹m✷

S✶

y❘❘

✶ jℓ′y❘❘∗ ✶ iℓ

, g▲▲

ij;ℓℓ′ = −✹h▲▲ ij;ℓℓ′ =

v✷ ✹m✷

S✶

y❘❘

✶ jℓ′

• V Ty▲▲

✶

∗

iℓ

g❘❘

ij;ℓℓ′ = −✹h❘❘ ij;ℓℓ′ =

v✷ ✹m✷

S✶

• V Ty▲▲

✶

• jℓ′ y❘❘∗

✶ iℓ

SLIDE 46

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❆t♦♠✐❝ ♣❛r✐t② ✈✐♦❧❛t✐♦♥ ✐♥ ❝❡s✐✉♠

Lqiqjℓℓ′ = −✹G❋ √ ✷

• c❘❘

ij;ℓℓ′

• qi

❘γµqj ❘

ℓ❘γµℓ′

❘

• . . .
• + ❤✳ ❝✳

c❘❘

ij;ℓℓ′ = − v✷

✹m✷

S✶

y❘❘

✶ jℓ′y❘❘∗ ✶ iℓ

→ c❘❘

✶✶;✶✶ = − v✷

✹m✷

S✶

• y❘❘

✶ ✶✶

✷ (ue− → ue−) ■♥t❡r❛❝t✐♦♥ ✈✐♦❧❛t✐♥❣ ♣❛r✐t②✿ L❙▼

P❱ = G❋

√ ✷

• q=u,d
• C✶q
• eγµγ✺e
• (qγµq) + C✷q (eγµe)
• qγµγ✺q

SLIDE 47

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❆t♦♠✐❝ ♣❛r✐t② ✈✐♦❧❛t✐♦♥ ✐♥ ❝❡s✐✉♠

Lqiqjℓℓ′ = −✹G❋ √ ✷

• c❘❘

ij;ℓℓ′

• qi

❘γµqj ❘

ℓ❘γµℓ′

❘

• . . .
• + ❤✳ ❝✳

c❘❘

ij;ℓℓ′ = − v✷

✹m✷

S✶

y❘❘

✶ jℓ′y❘❘∗ ✶ iℓ

→ c❘❘

✶✶;✶✶ = − v✷

✹m✷

S✶

• y❘❘

✶ ✶✶

✷ (ue− → ue−) ■♥t❡r❛❝t✐♦♥ ✈✐♦❧❛t✐♥❣ ♣❛r✐t②✿ L❙▼

P❱ = G❋

√ ✷

• q=u,d
• C✶q
• eγµγ✺e
• (qγµq) + C✷q (eγµe)
• qγµγ✺q

SLIDE 48

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❢♦r♠ ❢❛❝t♦r W✵✱ β ❛♥❞ t❤❡ ❞❡❝❛② ✇✐❞t❤ ❞✐✛❡r❡♥t✐❛❧

• P(k′)
• OΓΓ′(q)
• N(k, s)
• =
• W ΓΓ′

✵

(q✷) − i/ q mN W ΓΓ′

✶

(q✷)

• PΓ′uN(k, s)

OΓΓ′ =

• q❈PΓq
• PΓ′q
• ✵
• u❈P❘d
• P❘d
• n
• = βP❘un

❞Γ = ✶ ✸✷π✷ |M|✷ |♣❈▼| m✷ ❞Ω Γ = ✶ ✽π|M|✷ |♣❈▼| m✷

SLIDE 49

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❝♦♠♣❧❡t❡♥❡ss r❡❧❛t✐♦♥ ❛♥❞ tr❛❝❡s

• γ✺, γµ

= ✵,

• γ✺✷ = I

{γµ, γν} = ✷gµνI

• s

ua(p, s)ub(p, s) =

• /

p + mI

• ab
• s

va(p, s)vb(p, s) =

• /

p − mI

• ab
• r

ǫ∗

µ(q, r)ǫν(q, r) = −gµν

❚r [♦❞❞ ♥✉♠❜❡r γµ] = ✵ ❚r

• γ✺

= ✵ ❚r

• γ✺ · ♦❞❞ ♥✉♠❜❡r γµ

= ✵ ❚r

• γ✺γµγν

= ✵ ❚r [γµγν] = ✹gµν

SLIDE 50

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ Γ(p → e+π✵)

−iM =

• e+(k✷)π✵(k✸)
• iz❘❘

✶ ✶✶u❈ ❘d❘

• i

q′✷ − m✷

S✶

• iy❘❘

✶ ✶✶u❈ ❘e❘

• p(k✶)
• = i y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

P❘ve+(k✷, s✷)

• π✵(k✸)
• u❈P❘d
• u❈P❘
• p(k✶)
• = i y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

W ❘❘

✵

(k✷

✷) (vp(k✶, s✶)P❘ve+(k✷, s✷))

|M|✷ = ✶ ✷

• s✶,s✷

|M|✷ =

• y❘❘

✶ ✶✶z❘❘ ✶ ✶✶

m✷

S✶

✷ W ❘❘

✵

(✵) ✷ (k✷ · k✶)

SLIDE 51

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ❛♥❞ Γ(n → χγ)

♣r♦t♦♥✿ F p

✶ (✵) + F p ✷ (✵) = µp = ✶ + ap = ✶ + ✶.✼✾✸ = ✷.✼✾✸

♥❡✉tr♦♥✿ F n

✶ (✵) + F n ✷ (✵) = µn = ✵ + an = ✵ − ✶.✾✶✸ = −✶.✾✶✸

−iM = ean ✷mn ε (mn − mχ)qµǫ∗

ν(q, r)uχ(k✷, s✷)σµνun(k✶, s✶)

|M|✷ = ✶ ✷

• s✶,s✷,r

|M|✷ = ✷

• eanε

mn(mn − mχ) ✷ (k✶ · q)(k✷ · q)

SLIDE 52

■♥tr♦❞✉❝t✐♦♥ ▲❡♣t♦q✉❛r❦s ◆❡✉tr♦♥ ❧✐❢❡t✐♠❡ Pr♦t♦♥ ❧✐❢❡t✐♠❡ ❈❛❧❝✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥

❇❛❝❦✉♣ ✲ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ♣❛rt✐❛❧ ♣r♦t♦♥ ❧✐❢❡t✐♠❡s

♣r♦t♦♥ ❞❡❝❛② ♠♦❞❡ ♣❛rt✐❛❧ ❧✐❢❡t✐♠❡ [✶✵✸✵ ②] p → e+π✵ ✶✻✵✵✵ p → µ+π✵ ✼✼✵✵ p → νπ+ ✸✾✵ p → e+η ✶✵✵✵✵ p → µ+η ✹✼✵✵ p → e+ρ✵ ✼✷✵ p → µ+ρ✵ ✺✼✵ p → νρ+ ✶✻✷ p → e+ω ✶✻✵✵ p → µ+ω ✷✽✵✵ p → e+K ✵ ✶✵✵✵ p → µ+K ✵ ✶✻✵✵ p → νK + ✺✾✵✵ p → e+γ ✻✼✵ p → µ+γ ✹✼✽