Secularly growing loop corrections in strong background fjeld - PowerPoint PPT Presentation

Secularly growing loop corrections in strong background fjeld ❆❦❤♠❡❞♦✈ ❊✳ ❚✳✱ ❇✉r❞❛ P✳✱ P♦♣♦✈ ❋✳✱ ❙❛❞♦❢②❡✈ ❆✳ ❛♥❞ ❙❧❡♣✉❦❤✐♥ ❱✳ ▼■❚P✱ ▼❛✐♥③ ✭❏✉♥❡ ✷✵✶✺✮

Motivation ❙❡❝✉❧❛r ❣r♦✇t❤ ♦❢ ❧♦♦♣ ❝♦rr❡❝t✐♦♥s ✐s ♣r❛❝t✐❝❛❧❧② ✐♥❡✈✐t❛❜❧❡ ✐♥ ♥♦♥✕st❛t✐♦♥❛r② s✐t✉❛t✐♦♥s ✭▲❛♥❞❛✉ ❛♥❞ ▲✐❢s❤✐t③✱ ❳✲t❤ ✈♦❧✉♠❡✮ ❚❤✐s ❣r♦✇t❤ ✐s t❤❡ ■❘ ❡✛❡❝t✳ ◆♦ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ ❯❱ ♣❤②s✐❝s✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❝❧❛ss✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s✱ ✐❢ ♦♥❡ ✇❡✐❣❤ts ❧♦♥❣ ❡♥♦✉❣❤✳ ❞❡ ❙✐tt❡r s♣❛❝❡ ✐♥t❡r❛❝t✐♥❣ ◗❋❚ ✭r❡✈✐❡✇ ❛r❳✐✈✿✶✸✵✾✳✷✺✺✼✮✳ ◗❊❉ ♦♥ str♦♥❣ ❡❧❡❝tr✐❝ ✜❡❧❞ ❜❛❝❦❣r♦✉♥❞ ❜❡②♦♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❛r❳✐✈✿✶✹✵✺✳✺✷✷✺✮✳ ▲♦♦♣ ❝♦rr❡❝t✐♦♥ t♦ ❍❛✇❦✐♥❣ r❛❞✐❛t✐♦♥ ✭❛r❳✐✈✿✶✺✵✾✳ ✳✳✳✮✳ 2 / 21

Adiabatic catastrophe ❙✉♣♣♦s❡ ♦♥❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞✿ ∫︁ t ∫︁ t t 0 dt ′ H ( t ′ ) 𝒫 T e − i ⃒ t 0 dt ′ H ( t ′ ) ⃒ ⟨ ⟩ ⃒ T e i ⟨𝒫⟩ t 0 ( t ) = Ψ ⃒ Ψ , ✭✶✮ ⃒ ⃒ ❡✳❣✳ ⟨ T 𝜈𝜉 ⟩ ♦r ⟨ J 𝜈 ⟩ ✳ ❍❡r❡ H ( t ) = H 0 ( t ) + V ( t ) ✳ T ✖ t✐♠❡✕♦r❞❡r✐♥❣✱ T ✖ ❛♥t✐✕t✐♠❡✕♦r❞❡r✐♥❣✳ t 0 ✖ ✐♥✐t✐❛❧ ♠♦♠❡♥t ♦❢ t✐♠❡✱ | Ψ ⟩ ✖ ✐♥✐t✐❛❧ st❛t❡✱ ⟨ Ψ |𝒫| Ψ ⟩ ( t 0 ) ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❣✐✈❡♥✳ 3 / 21

Adiabatic catastrophe ❚r❛♥s❢❡rr✐♥❣ t♦ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣✐❝t✉r❡✿ ⃒ S + (+ ∞ , t 0 ) T [ 𝒫 0 ( t ) S (+ ∞ , t 0 )] ⃒ Ψ ⃒ ⃒ ⟨︁ ⟩︁ ⟨𝒫⟩ t 0 ( t ) = Ψ . ✭✷✮ ∫︁ t 2 t 1 dt ′ V 0 ( t ′ ) ❀ 𝒫 0 ( t ) ❛♥❞ V 0 ( t ) ❛r❡ ❍❡r❡ S ( t 2 , t 1 ) = T e − i ♦♣❡r❛t♦rs ✐♥ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣✐❝t✉r❡✳ ❙❧✐❣❤t❧② ❝❤❛♥❣✐♥❣ t❤❡ ♣r♦❜❧❡♠✿ ⃒ Ψ ⃒ ⃒ S + ⃒ ⟨︁ ⟩︁ ⟨𝒫⟩ t 0 ( t ) = Ψ t 0 (+ ∞ , −∞ ) T [ 𝒫 0 ( t ) S t 0 (+ ∞ , −∞ )] . ✭✸✮ ❍❡r❡ t 0 ✐s t❤❡ t✐♠❡ ♠♦♠❡♥t ❛❢t❡r ✇❤✐❝❤ t❤❡ ✐♥t❡r❛❝t✐♦♥s✱ V ( t ) ✱ ❛r❡ ❛❞✐❛❜❛t✐❝❛❧❧② t✉r♥❡❞ ♦♥✳ 4 / 21

Discussion ❲❤❡♥ ❞♦❡s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t 0 ❞✐s❛♣♣❡❛r❄ ❖t❤❡r✇✐s❡ ✇❡ ❤❛✈❡ ❛❞✐❛❜❛t✐❝ ❝❛t❛str♦♣❤❡ ❛♥❞ ❜r❡❛❦✐♥❣ ♦❢ ✈❛r✐♦✉s s②♠♠❡tr✐❡s✿ ❊✳❣✳ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s st♦♣ t♦ ❞❡♣❡♥❞ ♦♥❧② ♦♥ | t 1 − t 2 | ✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t 0 ❞✐s❛♣♣❡❛rs ✇❤❡♥ t❤❡ s✐t✉❛t✐♦♥ ✐s ♦r ❜❡❝♦♠❡s st❛t✐♦♥❛r②✳ 5 / 21

Discussion ❚❤❡ s❡♠✐♥❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ st❛t✐♦♥❛r② s✐t✉❛t✐♦♥ ✐s ✇❤❡♥ t❤❡ ❢r❡❡ ❍❛♠✐❧t♦♥✐❛♥ H 0 ✐s t✐♠❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❤❛s ❛ s♣❡❝tr✉♠ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✿ H 0 | vac ⟩ = ✵ ❛♥❞ | 𝜔 ⟩ = | vac ⟩ ✳ ■♥ ❢❛❝t✱ ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ ❜② ❛❞✐❛❜❛t✐❝ t✉r♥✐♥❣ ♦♥ ❛♥❞ t❤❡♥ s✇✐t❝❤✐♥❣ ♦✛ V ( t ) ✇❡ ❞♦ ♥♦t ❞✐st✉r❜ t❤❡ ❣r♦✉♥❞ st❛t❡✿ ⃒ S + (+ ∞ , −∞ ) ⃒ excited state ⟨︁ ⃒ ⃒ ⟩︁ vac = ✵ , ✇❤✐❧❡ ⃒ = ✶ . ⃒ S + (+ ∞ , −∞ ) ⃒ vac ⃒ ⃒⟨︁ ⃒ ⃒ ⟩︁⃒ vac ■t ❞♦❡s ♥♦t ♠❛tt❡r ✇❤❡♥ ♦♥❡ t✉r♥s ♦♥ ✐♥t❡r❛❝t✐♦♥s✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t 0 ❞✐s❛♣♣❡❛r❡❞✦ 6 / 21

Stationary case ❋✉rt❤❡r♠♦r❡✱ ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ ✇❡ ♦❜t❛✐♥✿ ⟨𝒫⟩ ( t ) = ∑︂ ⃒ S + (+ ∞ , −∞ ) ⃒ sta ⟨︁ ⃒ ⃒ ⟩︁ ⟨ sta | T [ 𝒫 0 ( t ) S (+ ∞ , −∞ )] | vac ⟩ = vac sta ⃒ S + (+ ∞ , −∞ ) ⃒ vac ⟨︁ ⃒ ⃒ ⟩︁ = ⟨ vac | T [ 𝒫 0 ( t ) S (+ ∞ , −∞ )] | vac ⟩ = vac = ⟨ vac | T [ 𝒫 0 ( t ) S (+ ∞ , −∞ )] | vac ⟩ . ⟨ vac | S (+ ∞ , −∞ ) | vac ⟩ ❚❤✐s ✇❛② ✇❡ ❛rr✐✈❡ ❛t ❤❛✈✐♥❣ ♦♥❧② t❤❡ T ✕♦r❞❡r❡❞ ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡♥ ❝❛♥ ✉s❡ ❋❡②♥♠❛♥ t❡❝❤♥✐q✉❡✳ ❖t❤❡r s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t 0 ❞✐s❛♣♣❡❛rs ✐❢ t❤❡r❡ ✐s ❛ st❛t✐♦♥❛r② st❛t❡ ✭❡✳❣✳ t❤❡r♠❛❧ ❞❡♥s✐t② ♠❛tr✐① ✐♥ ✢❛t s♣❛❝❡✕t✐♠❡✮✳ 7 / 21

Secularly growing loop corrections in strong background fjeld - PowerPoint PPT Presentation

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Two-Loop Corrections to Top-Quark Pair Production Andrea Ferroglia Johannes Gutenberg

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