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SLIDE 1

❖♣❡♥ q✉❛♥t✉♠ s②st❡♠s ❛♣♣r♦❛❝❤ t♦ t❤❡ st✉❞② ♦❢ q✉❛r❦♦♥✐✉♠ s✉♣♣r❡ss✐♦♥

▼✐❣✉❡❧ ❆✳ ❊s❝♦❜❡❞♦

❯♥✐✈❡rs✐t② ♦❢ ❏②✈äs❦②❧ä

❙❊❲▼ ✷✵✶✽

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶ ✴ ✺✷

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SLIDE 2

P❧❛♥

■♥tr♦❞✉❝t✐♦♥

❖♣❡♥ q✉❛♥t✉♠ s②st❡♠s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❡✛❡❝t✐✈❡ ✜❡❧❞ t❤❡♦r✐❡s✳ ❚❤❡

✶ r ≫ T ❝❛s❡ ✸

▲❛♥❣❡✈✐♥✲❧✐❦❡ ❡q✉❛t✐♦♥s ✐♥ ◗❈❉

❈♦♥❝❧✉s✐♦♥s

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷ ✴ ✺✷

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SLIDE 3

❚❤❡ ♦r✐❣✐♥❛❧ ✐❞❡❛ ♦❢ ▼❛ts✉✐ ❛♥❞ ❙❛t③ ✭✶✾✽✻✮

◗✉❛r❦♦♥✐✉♠ ✐s q✉✐t❡ st❛❜❧❡ ✐♥ t❤❡ ✈❛❝✉✉♠✳ P❤❡♥♦♠❡♥❛ ♦❢ ❝♦❧♦✉r s❝r❡❡♥✐♥❣✱ q✉❛♥t✐t✐❡s ♠❡❛s✉r❛❜❧❡ ✐♥ ▲❛tt✐❝❡ ◗❈❉ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✭st❛t✐❝✮ s✉♣♣♦rt t❤✐s✳ ❋♦r ❡①❛♠♣❧❡ P♦❧②❛❦♦✈ ❧♦♦♣✳ ❉✐ss♦❝✐❛t✐♦♥ ♦❢ ❤❡❛✈② q✉❛r❦♦♥✐✉♠ ✐♥ ❤❡❛✈②✲✐♦♥ ❝♦❧❧✐s✐♦♥s ❞✉❡ t♦ ❝♦❧♦✉r s❝r❡❡♥✐♥❣ s✐❣♥❛❧s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❛ q✉❛r❦✲❣❧✉♦♥ ♣❧❛s♠❛✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸ ✴ ✺✷

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SLIDE 4

❚❤❡ ♦r✐❣✐♥❛❧ ✐❞❡❛ ♦❢ ▼❛ts✉✐ ❛♥❞ ❙❛t③ ✭✶✾✽✻✮

◗✉❛r❦♦♥✐✉♠ ✐s q✉✐t❡ st❛❜❧❡ ✐♥ t❤❡ ✈❛❝✉✉♠✳ P❤❡♥♦♠❡♥❛ ♦❢ ❝♦❧♦✉r s❝r❡❡♥✐♥❣✱ q✉❛♥t✐t✐❡s ♠❡❛s✉r❛❜❧❡ ✐♥ ▲❛tt✐❝❡ ◗❈❉ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✭st❛t✐❝✮ s✉♣♣♦rt t❤✐s✳ ❋♦r ❡①❛♠♣❧❡ P♦❧②❛❦♦✈ ❧♦♦♣✳ ❉✐ss♦❝✐❛t✐♦♥ ♦❢ ❤❡❛✈② q✉❛r❦♦♥✐✉♠ ✐♥ ❤❡❛✈②✲✐♦♥ ❝♦❧❧✐s✐♦♥s ❞✉❡ t♦ ❝♦❧♦✉r s❝r❡❡♥✐♥❣ s✐❣♥❛❧s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❛ q✉❛r❦✲❣❧✉♦♥ ♣❧❛s♠❛✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸ ✴ ✺✷

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SLIDE 5

❚❤❡ ♦r✐❣✐♥❛❧ ✐❞❡❛ ♦❢ ▼❛ts✉✐ ❛♥❞ ❙❛t③ ✭✶✾✽✻✮

◗✉❛r❦♦♥✐✉♠ ✐s q✉✐t❡ st❛❜❧❡ ✐♥ t❤❡ ✈❛❝✉✉♠✳ P❤❡♥♦♠❡♥❛ ♦❢ ❝♦❧♦✉r s❝r❡❡♥✐♥❣✱ q✉❛♥t✐t✐❡s ♠❡❛s✉r❛❜❧❡ ✐♥ ▲❛tt✐❝❡ ◗❈❉ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✭st❛t✐❝✮ s✉♣♣♦rt t❤✐s✳ ❋♦r ❡①❛♠♣❧❡ P♦❧②❛❦♦✈ ❧♦♦♣✳ ❉✐ss♦❝✐❛t✐♦♥ ♦❢ ❤❡❛✈② q✉❛r❦♦♥✐✉♠ ✐♥ ❤❡❛✈②✲✐♦♥ ❝♦❧❧✐s✐♦♥s ❞✉❡ t♦ ❝♦❧♦✉r s❝r❡❡♥✐♥❣ s✐❣♥❛❧s t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❛ q✉❛r❦✲❣❧✉♦♥ ♣❧❛s♠❛✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸ ✴ ✺✷

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SLIDE 6

❈♦❧♦✉r s❝r❡❡♥✐♥❣

V (r) = −αs e−mDr r ❆t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹ ✴ ✺✷

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SLIDE 7

❆♥♦t❤❡r ♠❡❝❤❛♥✐s♠✱ ❝♦❧❧✐s✐♦♥s

❆ s✐♥❣❧❡t ❝❛♥ ❞❡❝❛② ✐♥t♦ ❛♥ ♦❝t❡t✳ ■♥t❡r❛❝t✐♦♥ ✇✐t❤ t❤❡ ♠❡❞✐✉♠ ❝❤❛♥❣❡s t❤❡ ❝♦❧♦r st❛t❡✳ ❉✐ss♦❝✐❛t✐♦♥ ✇✐t❤♦✉t s❝r❡❡♥✐♥❣✳ ❚❤✐s ✐s t❤❡ ♠❡❝❤❛♥✐s♠ ❜❡❤✐♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ✭❋✐rst ❢♦✉♥❞ ❜② ▲❛✐♥❡ ❡t ❛❧✳ ✭✷✵✵✼✮✮✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✺ ✴ ✺✷

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SLIDE 8

❘❡❝♦♠❜✐♥❛t✐♦♥

❚✇♦ ❤❡❛✈② q✉❛r❦s ❝♦♠✐♥❣ ❢r♦♠ ❞✐✛❡r❡♥t ♦r✐❣✐♥ ♠❛② r❡❝♦♠❜✐♥❡ t♦ ❢♦r♠ ❛ ♥❡✇ q✉❛r❦♦♥✐✉♠ st❛t❡✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✻ ✴ ✺✷

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SLIDE 9

❊✈♦❧✉t✐♦♥ ♦❢ q✉❛r❦♦♥✐✉♠ ✐♥ ❛ ✜r❡❜❛❧❧

❲❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❡ s✉r✈✐✈❛❧ ♣r♦❜❛❜✐❧✐t② ♦❢ q✉❛r❦♦♥✐✉♠ ❜♦✉♥❞ st❛t❡s ✐♥s✐❞❡ ♦❢ ❛ ♠❡❞✐✉♠✳ ■t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ✉♥❞❡rst❛♥❞ t❤✐s ♣r♦❜❧❡♠ ❛s ❛♥ ♦♣❡♥ q✉❛♥t✉♠ s②st❡♠✳ ❆❧❧ ❤❡❛✈② q✉❛r❦s❂s②st❡♠✳ ❚❤❡r♠❛❧ ♠❡❞✐✉♠❂❡♥✈✐r♦♥♠❡♥t✳ ❚❤❡ s②st❡♠ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ r❡❞✉❝❡❞ ❞❡♥s✐t② ♠❛tr✐① DQ✳ ❲❡ ❝❛♥ ♦❜t❛✐♥ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t RAA ❜② s♦❧✈✐♥❣ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ t❤❛t ❞❡s❝r✐❜❡s t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ r❡❞✉❝❡❞ ❞❡♥s✐t② ♠❛tr✐①✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✼ ✴ ✺✷

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SLIDE 10

❉✐✛❡r❡♥t str❛t❡❣✐❡s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡

❚❤❡ ✜rst st❡♣ ✐s t♦ ❞❡❞✉❝❡ ❛ ♠❛st❡r ❡q✉❛t✐♦♥ ∂DQ ∂t = F(DQ) t❤❡♥ t❤r❡❡ ❞✐✛❡r❡♥t str❛t❡❣✐❡s ❛r❡ ♣♦ss✐❜❧❡ ✶✮ ❙♦❧✈❡ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ❞✐r❡❝t❧②✳ ❚❤✐s ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✈❡r② ❝♦st❧② ❛♥❞ ❝❛♥ ♦♥❧② ❜❡ ❞♦♥❡ ❢♦r s✐♠♣❧❡ ❝❛s❡s✳

◮ ♣◆❘◗❈❉ ❛♣♣r♦❛❝❤✳ ❉✐s❝✉ss❡❞ ❧❛t❡r✳ ◮ ❆❜❡❧✐❛♥ ❧✐♠✐t st✉❞✐❡❞ ♥✉♠❡r✐❝❛❧❧② ✐♥ ✶❉✳ ✶ ◮ ❆t t❤❡ ♠♦♠❡♥t ✐t ✐s ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ st✉❞② r❡❝♦♠❜✐♥❛t✐♦♥ ❢r♦♠

✐♥✐t✐❛❧❧② ✉♥❝♦rr❡❧❛t❡❞ ❤❡❛✈② q✉❛r❦s✳

✶❉❡ ❇♦♥✐✱ ❏❍❊P ✶✼✵✽ ✭✷✵✶✼✮ ✵✻✹✳ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✽ ✴ ✺✷

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SLIDE 11

❉✐✛❡r❡♥t str❛t❡❣✐❡s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡

❚❤❡ ✜rst st❡♣ ✐s t♦ ❞❡❞✉❝❡ ❛ ♠❛st❡r ❡q✉❛t✐♦♥ ∂DQ ∂t = F(DQ) t❤❡♥ t❤r❡❡ ❞✐✛❡r❡♥t str❛t❡❣✐❡s ❛r❡ ♣♦ss✐❜❧❡ ✷✮ ❯s❡ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥s✳

◮ ❙t♦❝❤❛st✐❝ ♣♦t❡♥t✐❛❧ ✭✐♥ ◗❊❉✮✳ ✷ ◮ ❙❝❤rö❞✐♥❣❡r✲▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥✳ ❙t♦❝❤❛st✐❝ ♣♦t❡♥t✐❛❧ ♣❧✉s ♥♦♥✲❧✐♥❡❛r

t❡r♠ t♦ ✐♠♣♦s❡ t❤❡r♠❛❧✐③❛t✐♦♥✳✸

✷❑❛❥✐♠♦t♦✱ ❆❦❛♠❛ts✉✱ ❆s❛❦❛✇❛ ❛♥❞ ❘♦t❤❦♦♣❢✳ P❘❉ ✾✼ ✭✷✵✶✽✮ ♥♦✳✶ ✵✶✹✵✵✸ ✸●♦ss✐❛✉① ❛♥❞ ❑❛t③✳ ❆♥♥❛❧s P❤②s✳ ✸✻✽ ✭✷✵✶✻✮ ✷✻✼✲✷✾✺ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✾ ✴ ✺✷

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SLIDE 12

❉✐✛❡r❡♥t str❛t❡❣✐❡s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡

❚❤❡ ✜rst st❡♣ ✐s t♦ ❞❡❞✉❝❡ ❛ ♠❛st❡r ❡q✉❛t✐♦♥ ∂DQ ∂t = F(DQ) t❤❡♥ t❤r❡❡ ❞✐✛❡r❡♥t str❛t❡❣✐❡s ❛r❡ ♣♦ss✐❜❧❡ ✸✮ ❙t✉❞② ✐❢ ❛ ❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥ ❜❡ ❥✉st✐✜❡❞ ❛♥❞ s♦❧✈❡ t❤❛t ✐♥st❡❛❞✳

◮ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✐♥ ◗❊❉✳ ◗✉✐t❡ s✉❝❝❡ss❢✉❧ ♣❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧❧②

❝♦♥s✐❞❡r✐♥❣ t❤❡ s✐♠♣❧✐❝✐t②✳ ■♥ t❤✐s ✇❛② ✐t ✐s ♣♦ss✐❜❧❡ t♦ st✉❞② r❡❝♦♠❜✐♥❛t✐♦♥ ❢r♦♠ ✐♥✐t✐❛❧❧② ✉♥❝♦rr❡❧❛t❡❞ ❤❡❛✈② q✉❛r❦s✳ ✹

◮ ❊①t❡♥s✐♦♥ t♦ ◗❈❉ ❞✐s❝✉ss❡❞ ❧❛t❡r✳ ✹❏✲P✳ ❇❧❛✐③♦t✱ ❉✳ ❞❡ ❇♦♥✐✱ P✳ ❋❛❝❝✐♦❧✐ ❛♥❞ ●✳ ●❛r❜❡r♦❣❧✐♦✳ ◆✉❝❧✳P❤②s✳ ❆✾✹✻ ✭✷✵✶✻✮

✹✾✲✽✽✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✵ ✴ ✺✷

slide-13
SLIDE 13

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

■ ❛♠ ❣♦✐♥❣ t♦ ❞✐s❝✉ss t✇♦ ❞✐✛❡r❡♥t str❛t❡❣✐❡s✳ ❚❤❡② ❤❛✈❡ ✐♥ ❝♦♠♠♦♥ ❛ ❢♦❝✉s ♦♥ ❝♦❧♦r ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✳ ❯s✐♥❣ ❊✛❡❝t✐✈❡ ❋✐❡❧❞ ❚❤❡♦r② t❡❝❤♥✐q✉❡s ❛♥❞ s♦❧✈✐♥❣ ❞✐r❡❝t❧② t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥✳ ❙t✉❞②✐♥❣ ✐❢ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❞❡r✐✈❡ ❛ ◗❈❉ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ✐t ✇❛s ❞♦♥❡ ✐♥ ◗❊❉✳✺

✺❋♦r ❝❧❛ss✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐t❤✐♥ t❤❡ ❊❋❚ ❛♣♣r♦❛❝❤ s❡❡ ❨❛♥ ❩❤✉➫s t❛❧❦ ♦♥ ❋r✐❞❛②✳ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✶ ✴ ✺✷

slide-14
SLIDE 14

P❧❛♥

■♥tr♦❞✉❝t✐♦♥

❖♣❡♥ q✉❛♥t✉♠ s②st❡♠s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❡✛❡❝t✐✈❡ ✜❡❧❞ t❤❡♦r✐❡s✳ ❚❤❡

✶ r ≫ T ❝❛s❡ ✸

▲❛♥❣❡✈✐♥✲❧✐❦❡ ❡q✉❛t✐♦♥s ✐♥ ◗❈❉

❈♦♥❝❧✉s✐♦♥s ❙❡❝t✐♦♥ ❜❛s❡❞ ♦♥ ✇♦r❦ ❞♦♥❡ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ◆✳ ❇r❛♠❜✐❧❧❛✱ ❏✳ ❙♦t♦ ❛♥❞ ❆✳ ❱❛✐r♦ ✭P❘❉ ✾✼ ♥♦✳✼ ✵✼✹✵✵✾ ❛♥❞ P❘❉ ✾✻ ♥♦✳✸ ✵✸✹✵✷✶✮

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✷ ✴ ✺✷

slide-15
SLIDE 15

❊✛❡❝t✐✈❡ ❋✐❡❧❞ ❚❤❡♦r✐❡s

QCD NRQCD pNRQCD pNRQCD NRQCD HTL

HTL

m 1/r ∼ mv V ∼ mv2 T mD

❇r❛♠❜✐❧❧❛✱ ●❤✐❣❧✐❡r✐✱ ❱❛✐r♦ ❛♥❞ P❡tr❡❝③❦② ✭P❘❉✼✽ ✭✷✵✵✽✮ ✵✶✹✵✶✼✮ ▼✳ ❆✳ ❊ ❛♥❞ ❙♦t♦ ✭P❘❆✼✽ ✭✷✵✵✽✮ ✵✸✷✺✷✵✮

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✸ ✴ ✺✷

slide-16
SLIDE 16

♣◆❘◗❈❉ ▲❛❣r❛♥❣✐❛♥ ❛t ❚❂✵

✭❇r❛♠❜✐❧❧❛✱ P✐♥❡❞❛✱ ❙♦t♦ ❛♥❞ ❱❛✐r♦✱ ◆P❇✺✻✻ ✭✷✵✵✵✮ ✷✼✺✮✳ LpNRQCD =

  • d✸rTr
  • S† (i∂✵ − hs) S

+O† (iD✵ − ho) O

  • + VA(r)Tr(O†rg❊S + S†rg❊O)

+ VB(r)

Tr(O†rg❊O + O†Org❊) + Lg + Lq ❉❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❛r❡ s✐♥❣❧❡t ❛♥❞ ♦❝t❡ts✳ ❆❧❧♦✇s t♦ ♦❜t❛✐♥ ♠❛♥✐❢❡st❧② ❣❛✉❣❡✲✐♥✈❛r✐❛♥t r❡s✉❧ts✳ ❙✐♠♣❧✐✜❡s t❤❡ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ▲❛tt✐❝❡ ◗❈❉✳ ■❢ ✶/r ≫ T ✇❡ ❝❛♥ ✉s❡ t❤✐s ▲❛❣r❛♥❣✐❛♥ ❛s st❛rt✐♥❣ ♣♦✐♥t✳ ■♥ ♦t❤❡r ❝❛s❡s t❤❡ ♠❛t❝❤✐♥❣ ❜❡t✇❡❡♥ ◆❘◗❈❉ ❛♥❞ ♣◆❘◗❈❉ ✇✐❧❧ ❜❡ ♠♦❞✐✜❡❞✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✹ ✴ ✺✷

slide-17
SLIDE 17

❊✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s✐♥❣❧❡ts

fs(x, y) = Tr(ρS†(x)S(y)) ❲❡ ❝❛♥ ✉s❡ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ❜✉t ❡①♣❛♥❞✐♥❣ ✐♥ r ✐♥st❡❛❞ ♦❢ αs✳ ■♥ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣✐❝t✉r❡ i∂tS = [S, H✵] i∂tρ = [HI, ρ]

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✺ ✴ ✺✷

slide-18
SLIDE 18

❊✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s✐♥❣❧❡ts

∂tfs = −i(Hs,eff fs − fsH†

s,eff ) + F(fo)

Hs,eff = hs + Σs ✇❤❡r❡ Σs ❝♦rr❡s♣♦♥❞s ✇✐t❤ t❤❡ s❡❧❢✲❡♥❡r❣② t❤❛t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✐♥ ♣◆❘◗❈❉ ❜② ❝♦♠♣✉t✐♥❣ t❤❡ t✐♠❡✲♦r❞❡r❡❞ ❝♦rr❡❧❛t♦r✳ F(fo) ✐s ❛ ♥❡✇ t❡r♠ t❤❛t t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣r♦❝❡ss O → g + S✳ ■♥ ❡♥s✉r❡s t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❤❡❛✈② q✉❛r❦s ✐s ❝♦♥s❡r✈❡❞✳ F(fo) ✐s ❛ ❝♦♠♣❧✐❝❛t❡❞ ❢✉♥❝t✐♦♥ ♦❢ Tr(ρO†O) ❛♥❞ E iE j✳ ❚❤❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♠❡❞✐✉♠ ❡♥t❡rs ♦♥❧② ✐♥ t❤❡ ❝❤r♦♠♦❡❧❡❝tr✐❝ ✜❡❧❞ ❝♦rr❡❧❛t♦r✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✻ ✴ ✺✷

slide-19
SLIDE 19

❊✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♦❝t❡t

❱❡r② s✐♠✐❧❛r r❡❛s♦♥✐♥❣✳ f ab

  • (x, y) = Tr(ρO†,a(x)Ob(y))

∂tfo = −i(Ho,eff fo − foH†

  • ,eff ) + F✶(fs) + F✷(fo)

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✼ ✴ ✺✷

slide-20
SLIDE 20

❚❤❡ ✶

r ≫ T, mD ≫ E r❡❣✐♠❡

❇❡❝❛✉s❡ ❛❧❧ t❤❡ t❤❡r♠❛❧ s❝❛❧❡s ❛r❡ s♠❛❧❧❡r t❤❛♥ ✶

r ❜✉t ❜✐❣❣❡r t❤❛♥ E t❤❡

❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✐s ♦❢ t❤❡ ▲✐♥❞❜❧❛❞ ❢♦r♠✳ ∂tρ = −i[H(γ), ρ] +

  • k

(CkρC †

k − ✶

✷{C †

kCk, ρ})

t❤❡r❡ ✐s ❛ tr❛♥s✐t✐♦♥ s✐♥❣❧❡t✲♦❝t❡t C so

i

=

  • κ

N✷

c − ✶ri

  • N✷

c − ✶

  • ❛♥❞ ♦❝t❡t t♦ ♦❝t❡t

C oo

i

=

  • (N✷

c − ✹)κ

✷(N✷

c − ✶) ri

✵ ✵ ✵ ✶

  • ▼✳❆✳ ❊s❝♦❜❡❞♦

✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✽ ✴ ✺✷

slide-21
SLIDE 21

❚❤❡ ♣❛r❛♠❡t❡r κ

κ = g✷ ✻ Nc ❘❡ +∞

−∞

ds ❚ E a,i(s, ✵)E a,i(✵, ✵) q✉❛♥t✐t② ❛❧s♦ ❛♣♣❡❛r✐♥❣ ✐♥ ❤❡❛✈② ♣❛rt✐❝❧❡ ❞✐✛✉s✐♦♥✱ r❡❝❡♥t ❧❛tt✐❝❡ ◗❈❉ ❡✈❛❧✉❛t✐♦♥ ✐♥ ❋r❛♥❝✐s✱ ❑❛❝③♠❛r❡❦✱ ▲❛✐♥❡✱ ◆❡✉❤❛✉s ❛♥❞ ❖❤♥♦ ✭✷✵✶✺✮ ✶.✽ κ T ✸ ✸.✹ P✐❝t✉r❡ t❛❦❡♥ ❢r♦♠ ❖✳ ❑❛❝③♠❛r❡❦ t❛❧❦ ✐♥ ✧✸✵ ②❡❛rs ✐♥ J/Ψ s✉♣♣r❡ss✐♦♥✧✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✶✾ ✴ ✺✷

slide-22
SLIDE 22

❚❤❡ ✶

r ≫ T, mD ≫ E r❡❣✐♠❡

❇❡❝❛✉s❡ ❛❧❧ t❤❡ t❤❡r♠❛❧ s❝❛❧❡s ❛r❡ s♠❛❧❧❡r t❤❛♥ ✶

r ❜✉t ❜✐❣❣❡r t❤❛♥ E t❤❡

❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✐s ♦❢ t❤❡ ▲✐♥❞❜❧❛❞ ❢♦r♠✳ ∂tρ = −i[H(γ), ρ] +

  • k

(CkρC †

k − ✶

✷{C †

kCk, ρ})

H = hs ✵ ✵ ho

  • + r✷

✷ γ(t)

✵ ✵

N✷

c −✷

✷(N✷

c −✶)

  • γ = g✷

✻ Nc ■♠ +∞

−∞

ds ❚ E a,i(s, ✵)E a,i(✵, ✵) ◆♦ ❧❛tt✐❝❡ ◗❈❉ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤✐s ❜✉t ✇❡ ♦❜s❡r✈❡ t❤❛t ✇❡ r❡♣r♦❞✉❝❡ ❞❛t❛ ❜❡tt❡r ✐❢ γ ✐s s♠❛❧❧✳ ■♥ ♣◗❈❉ γ = −✷ζ(✸) CF ✹ ✸Nc + nf

  • α✷

s(µT) T ✸ ≈ −✻.✸ T ✸

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✵ ✴ ✺✷

slide-23
SLIDE 23

■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❤②❞r♦❞②♥❛♠✐❝s

■♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✇❡❧❧ t❤❡ ✉♥❞❡r❧②✐♥❣ ♠❡❝❤❛♥✐s♠ ✇❡ ✇♦r❦❡❞ ✐♥ t❤❡ s✐♠♣❧❡st ♣♦ss✐❜❧❡ ❝♦♥❞✐t✐♦♥s✳ ❍♦✇❡✈❡r ✐t ✐s ♣♦ss✐❜❧❡ ❛♥❞ str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝♦✉♣❧❡ ♦✉r t❤❡♦r② t♦ t❤❡ ❢✉❧❧ ❤②❞r♦ ❡✈♦❧✉t✐♦♥ ❛♥❞ ✇❡ ✇✐❧❧ ❞♦ ✐t ✐♥ t❤❡ ❢✉t✉r❡

■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s

❚♦ ❝r❡❛t❡ ❛ ♣❛✐r ♦❢ ❤❡❛✈② ♣❛rt✐❝❧❡s ✐s ❛ ❤✐❣❤ ❡♥❡r❣② ♣r♦❝❡ss → ♣❛✐r ✐♥✐t✐❛❧❧② ❝r❡❛t❡❞ ✐♥ ❛ ❉✐r❛❝ ❞❡❧t❛ st❛t❡ ✐♥ t❤❡ r❡❧❛t✐✈❡ ❝♦♦r❞✐♥❛t❡✳ ◆❛✐✈❡❧② ✭✇✐t❤♦✉t t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t PT ❞❡♣❡♥❞❡♥❝❡✮ ✐t ✐s αs s✉♣♣r❡ss❡s t♦ ❝r❡❛t❡ ❛ s♣✐♥ ✶ s✐♥❣❧❡t ❝♦♠♣❛r❡❞ t♦ ❛♥ ♦❝t❡t✳

❍②❞r♦❞②♥❛♠✐❝s

❇❥♦r❦❡♥ ❡①♣❛♥s✐♦♥✳ ❖♣t✐❝❛❧ ●❧❛✉❜❡r ♠♦❞❡❧ t♦ ❝♦♠♣✉t❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ ✐♥✐t✐❛❧ t❡♠♣❡r❛t✉r❡ ✇✐t❤ ❝❡♥tr❛❧✐t②✳ ◗✉❛r❦♦♥✐✉♠ ♣r♦♣❛❣❛t❡s ✐♥ t❤❡ ✈❛❝✉✉♠ ❢r♦♠ t = ✵ t♦ t✵ = ✵.✻ ❢♠✱ t❤❡♥ t❤❡ ♣❧❛s♠❛ ✐s ❝r❡❛t❡❞✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✶ ✴ ✺✷

slide-24
SLIDE 24

❘❡s✉❧ts✳ ✸✵ − ✺✵% ❝❡♥tr❛❧✐t②✳ √sNN = ✷.✼✻ ❚❡❱

1.0 1.5 2.0 2.5 3.0 time(fm) 0.0 0.2 0.4 0.6 0.8 1.0 RAA

1S 2S

❊rr♦r ❜❛rs ♦♥❧② t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ✉♥❝❡rt❛✐♥t② ✐♥ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ κ✳ γ ✐s s❡t t♦ ③❡r♦✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✷ ✴ ✺✷

slide-25
SLIDE 25

❘❡s✉❧ts✳ ✺✵ − ✶✵✵% ❝❡♥tr❛❧✐t②✳ √sNN = ✷.✼✻ ❚❡❱

0.6 0.7 0.8 0.9 1.0 1.1 time(fm) 0.0 0.2 0.4 0.6 0.8 1.0 RAA

1S 2S

❊rr♦r ❜❛rs ♦♥❧② t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ✉♥❝❡rt❛✐♥t② ✐♥ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ κ✳ γ ✐s s❡t t♦ ③❡r♦✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✸ ✴ ✺✷

slide-26
SLIDE 26

❘❡s✉❧ts

50 100 150 200 250 < Npart > 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 RAA

❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t❤❡ ❈▼❙ ❞❛t❛ ♦❢ ✷✵✶✼ ✭tr✐❛♥❣❧❡s✮ ❛♥❞ ♦✉r ❝♦♠♣✉t❛t✐♦♥ ✭❝✐r❝❧❡s✮✳ ❯♣♣❡r ✭r❡❞✮ ❡♥tr✐❡s r❡❢❡r t♦ t❤❡ Υ(✶S)✱ ❧♦✇❡r ✭❣r❡❡♥✮ ❡♥tr✐❡s t♦ t❤❡ Υ(✷S)✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✹ ✴ ✺✷

slide-27
SLIDE 27

P❧❛♥

■♥tr♦❞✉❝t✐♦♥

❖♣❡♥ q✉❛♥t✉♠ s②st❡♠s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❡✛❡❝t✐✈❡ ✜❡❧❞ t❤❡♦r✐❡s✳ ❚❤❡

✶ r ≫ T ❝❛s❡ ✸

▲❛♥❣❡✈✐♥✲❧✐❦❡ ❡q✉❛t✐♦♥s ✐♥ ◗❈❉

❈♦♥❝❧✉s✐♦♥s ❙❡❝t✐♦♥ ❜❛s❡❞ ♦♥ ✇♦r❦ ❞♦♥❡ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❏✳ ❇❧❛✐③♦t ✭❏❍❊P ✶✽✵✻ ✭✷✵✶✽✮ ✵✸✹✮✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✺ ✴ ✺✷

slide-28
SLIDE 28

❙❡tt✐♥❣

❲❡ tr❡❛t ❤❡❛✈② q✉❛r❦s ✐♥ ✜rst q✉❛♥t✐③❛t✐♦♥ H = H♣❧ + HQ + H✶ H♣❧ ✐s t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤❡ ♠❡❞✐✉♠✳ HQ ✐s t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ q✉❛r❦♦♥✐✉♠✳ s❂s✐♥❣❧❡t ❛♥❞ ♦❂♦❝t❡t✳ HQ = Hs,o = −∆r M − ∆R ✹M + Vs,o(r) H✶ ✐s t❤❡ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦✳ H✶ = −g

  • r

Aa

✵(r)na(r)

Aa

✵ ✐s t❤❡ t❡♠♣♦r❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❣❧✉♦♥ ✜❡❧❞ ❛♥❞ na ✐s t❤❡ ❝♦❧♦r ❝✉rr❡♥t✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✻ ✴ ✺✷

slide-29
SLIDE 29

❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐①

✹ ❞✐❛❣r❛♠s t❤❛t ❝♦♥♥❡❝t ❛♥② st❛t❡ ❛t t✐♠❡ t ✇✐t❤ ❛ s✐♥❣❧❡t ❛t t✐♠❡ t + dt✳ ❚❤❡s❡ ❞✐❛❣r❛♠s r❡♣r❡s❡♥t t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐①

|ψ(t) |ψ(t + dt) φ(t)| φ(t + dt)|

■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♦❝t❡t✱ ❛❧s♦ ♦❝t❡t t♦ ♦❝t❡t tr❛♥s✐t✐♦♥s ❛r❡ ♣♦ss✐❜❧❡✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✼ ✴ ✺✷

slide-30
SLIDE 30

❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐①

✹ ❞✐❛❣r❛♠s t❤❛t ❝♦♥♥❡❝t ❛♥② st❛t❡ ❛t t✐♠❡ t ✇✐t❤ ❛ s✐♥❣❧❡t ❛t t✐♠❡ t + dt✳ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s ❣❧✉♦♥ ❡①❝❤❛♥❣❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✭✈❛❧✐❞ ✐❢ T ≫ E✮ ♠❡❛♥s t❤❛t Us,o = ✶ ❲❡ ❝❛♥ ❞♦ ❛ ❧✐tt❧❡ ❜✐t ❜❡tt❡r ❜② ✉s✐♥❣ Us,o = ✶ − iHs,o(t − t′) ■♥ ❢❛❝t✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ s❡❡ t❤❛t ✐♥ ◗❊❉ s✉❜st✐t✉t✐♥❣ H ❜② t❤❡ ❦✐♥❡t✐❝ t❡r♠ K ❛❧r❡❛❞② ♣r♦✈✐❞❡s ❛ ❜❡tt❡r ❞❡s❝r✐♣t✐♦♥ Us,o = ✶ − iK(t − t′)

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✽ ✴ ✺✷

slide-31
SLIDE 31

❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐① DQ

❞DQ ❞t ≈ −i[HQ, DQ(t)]− i ✷

  • ①①′ V (① − ①′)[na

①na ①′, DQ]

+ ✶ ✷

  • ①①′ W (① − ①′) ({na

①na ①′, DQ} − ✷na ①DQna ①′)

+ i ✹T

  • ①①′ W (① − ①′)
  • [na

①, ˙

na

①′DQ] + [na ①, DQ ˙

na

①′]

  • ❯♥✐t❛r② ❡✈♦❧✉t✐♦♥✳ ■♥❝❧✉❞❡s s❝r❡❡♥✐♥❣ ❛♥❞ ❝♦♥s❡r✈❡s ❡♥tr♦♣②✳

❚❤❡ ❞✐✛❡r❡♥t tr❛♥s✐t✐♦♥s ♣r❡s❡♥t ✐♥ t❤❡ ❡①❛❝t ✐♥st❛♥t❛♥❡♦✉s ❣❧✉♦♥ ❡①❝❤❛♥❣❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ■♥✢✉❡♥❝❡ ♦❢ t❤❡ ❦✐♥❡t✐❝ ❡♥❡r❣② ✐♥s✐❞❡ Us,o✱ ✇❡ ❤❛✈❡ ✉s❡❞ ✢✉❝t✉❛t✐♦♥✲❞✐ss✐♣❛t✐♦♥ t❤❡♦r❡♠ t♦ ♦❜t❛✐♥ t❤✐s✳ ✻ ■♥ ♦✉r ♥♦t❛t✐♦♥✱ t❤❡ ♣♦t❡♥t✐❛❧ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✐s V (r) + iW (r)✳

✻❆ s✐♠✐❧❛r ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❉✳ ❞❡ ❇♦♥✐✱ ❏❍❊P ✶✼✵✽ ✭✷✵✶✼✮ ✵✻✹ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✷✾ ✴ ✺✷

slide-32
SLIDE 32

❚❤❡ ❝❧❛ss✐❝❛❧ ❧✐♠✐t

❲❡ ❞❡✜♥❡ x ❛♥❞ y s✉❝❤ t❤❛t x + y ✷|DQ|x − y ✷ =

  • d✸p

(✷π)✸ ei♣②DQ(①, ♣) ❛t ❧❛r❣❡ t✐♠❡s ✇❡ ❡①♣❡❝t t❤❛t DQ(①, ♣) → ❡− p✷

✷MT F(①, ♣)

t❤❡r❡❢♦r❡ y ∼

✶ √ MT ✇❤✐❝❤ ✐♥ t❤✐s ❝❛s❡ ✇✐❧❧ ❜❡ ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ x✳ ❚❤✐s

✐♠♣❧✐❡s t❤❛t ❛t ❧❛r❣❡ t✐♠❡s t❤❡ s②st❡♠ ✐s ✐♥ ❛ ❛♣♣r♦①✐♠❛t❡❧② ✇❡❧❧✲❞❡✜♥❡❞ ♣♦s✐t✐♦♥ ❛t ❛ ❣✐✈❡♥ t✐♠❡✳ ❚❤❡r❡ ✐s ❛ s②st❡♠❛t✐❝ ✇❛② t♦ ♦❜t❛✐♥ t❤❡ ❝❧❛ss✐❝❛❧ ❡q✉❛t✐♦♥s ✉s✐♥❣ t❤❡ ❲✐❣♥❡r tr❛♥s❢♦r♠✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✵ ✴ ✺✷

slide-33
SLIDE 33

❚❤❡ ❝❧❛ss✐❝❛❧ ❧✐♠✐t✳ ◗❈❉

■♥ ◗❊❉ t❤❡ str✐❝t y = ✵ ❧✐♠✐t ❣✐✈❡s ∂D ∂t = − P · ∇❘ ✹M + ✷♣ · ∇r M

  • D

■♥ ◗❈❉ ✇❡ ❣❡t ∂Ds ∂t = − P · ∇❘ ✹M + ✷♣ · ∇r M

  • Ds − ✷CFΓ(r)(Ds − Do)

∂Do ∂t = − P · ∇❘ ✹M + ✷♣ · ∇r M

  • Do + ✶

Nc Γ(r)(Ds − Do) ◆❡①t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❝♦♥s✐❞❡r t✇♦ ♦♣t✐♦♥s✳ Γ(r) = W (r) − W (✵)

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✶ ✴ ✺✷

slide-34
SLIDE 34

▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ r❛♥❞♦♠ ❝♦❧♦r ❢♦r❝❡✳ ❖♣t✐♦♥ ✶

❖♥❡ ♦♣t✐♦♥ ✐s t♦ ❞✐❛❣♦♥❛❧✐③❡ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ∂D✵ ∂t = − P · ∇❘ ✹M + ✷♣ · ∇r M

  • D✵

∂D✽ ∂t = − P · ∇❘ ✹M + ✷♣ · ∇r M

  • D✽ − NcΓ(r)D✽

✇❤❡r❡ D✵ = ✶ N✷

c

(Ds + (N✷

c − ✶)Do)

D✽ = ✷ Nc (Ds − Do) D✵ r❡♣r❡s❡♥ts t❤❡ st❛t❡ ♦❢ ♠❛①✐♠✉♠ ❡♥tr♦♣② ✐♥ ❝♦❧♦r ✇❤✐❧❡ D✽ r❡♣r❡s❡♥ts t❤❡ s❤♦rt ❧✐❢❡t✐♠❡ ✢✉❝t✉❛t✐♦♥s ❛r♦✉♥❞ ✐t✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✷ ✴ ✺✷

slide-35
SLIDE 35

▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ r❛♥❞♦♠ ❝♦❧♦r ❢♦r❝❡

◆♦✇ ✇❡ ❞✐❛❣♦♥❛❧✐③❡ t❤❡ s②st❡♠ ❜✉t ✐♥st❡❛❞ ♦❢ ❞♦✐♥❣ ✐t ✐♥ t❤❡ str✐❝t y = ✵ ❧✐♠✐t ✇❡ ❞♦ ✐t ❛t ♦r❞❡r y✷✳ ∂tD′

✵ + ✷♣ · ∇

M D′

✵ − CF

✹ Hij(✵)∆ij

pD′ ✵−✷CFF i(r)F j(r)

N✷

c Γ(r)

∆ij

pD′ ✵

− CF ✷MT Hij(✵)∇i

p(pjD′ ✵) = ✵

❱❧❛s♦✈ ❡q✉❛t✐♦♥ ✇✐t❤ ③❡r♦ ❢♦r❝❡✳ ❙t♦❝❤❛st✐❝ ❢♦r❝❡ ❛♥❞ ❞r❛❣ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ ❛ ❤❡❛✈② q✉❛r❦ ✇✐t❤ t❤❡ ♠❡❞✐✉♠✳ ❙t♦❝❤❛st✐❝ ❢♦r❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❛ttr❛❝t✐✈❡✴r❡♣✉❧s✐✈❡ ❢♦r❝❡ ❜❡t✇❡❡♥ ❤❡❛✈② q✉❛r❦s t❤❛t ✐s ❛✈❡r❛❣❡❞ t♦ ③❡r♦ ❞✉❡ t♦ t❤❡ ❝♦❧♦r st❛t❡✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✸ ✴ ✺✷

slide-36
SLIDE 36

❍❡❛✈② q✉❛r❦♦♥✐✉♠✳ ❍✐st♦❣r❛♠ ♦❢ ❞✐st❛♥❝❡ ❛❢t❡r ∆t = ✺❢♠ ❢♦r ✐♥✐t✐❛❧ ✶❙

t = ✵

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 r (fm) 5 10 15 20 25 30 35 40

t = ✺❢♠

2 4 6 8 10 12 14 16 r (fm) 10 20 30 40 50

P❛r❛♠❡t❡rs✿ Mb = ✹✽✽✶▼❡❱✱ γ = ✵.✷✺T ✷

✷Mb ✱ T = ✸✺✵ ▼❡❱ ✳ αs ❡✈❛❧✉❛t❡❞ ❛t

✷πT ✐♥s✐❞❡ t❤❡ ❉❡❜②❡ ♠❛ss ❛♥❞ ❛t

✶ a✵ = ✶✸✹✽▼❡❱ ✐♥ t❤❡ ❈♦✉❧♦♠❜✐❝ ♣❛rt✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✹ ✴ ✺✷

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SLIDE 37

❉✐st❛♥❝❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ❢♦r ✶✵ ❛r❜✐tr❛r② tr❛❥❡❝t♦r✐❡s

1 2 3 4 5 t (fm) 2 4 6 8 10 12 r (fm)

Pr♦❜❧❡♠ ♥♦t ❢♦✉♥❞ ✐♥ ◗❊❉✳ ❋♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ r t❤❡ r❛♥❞♦♠ ❢♦r❝❡ ❝❛♥ ❣❡t ✉♥♣❤②s✐❝❛❧❧② ❜✐❣✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✺ ✴ ✺✷

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SLIDE 38

❙✐♠✉❧❛t✐♥❣ ♠❛♥② ♣❛rt✐❝❧❡s

x(fm) −20 −10 10 20 30 y ( f m ) −10 0 10 20 −10 10 20

10 12 14 16 18 20 22 24 26 10 20 30 40 50 60 70

❲❡ ❝❛♥ s✐♠✉❧❛t❡ ✺✵ ♣❛✐rs ❛♥❞ ♠❛❦❡ ❛♥ ❤✐st♦❣r❛♠ ♦❢ ❤♦✇ ♠❛♥② ♦❢ t❤❡♠ ✇✐❧❧ ❢♦r♠ ❛ ❜♦✉♥❞ st❛t❡ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❡✈♦❧✉t✐♦♥ ✭r❡❝♦♠❜✐♥❛t✐♦♥✮✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✻ ✴ ✺✷

slide-39
SLIDE 39

❇♦❧t③♠❛♥♥ ❡q✉❛t✐♦♥✳ ❖♣t✐♦♥ ✷

❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛❞✐✉s ♦❢ t❤❡ s♠❛❧❧ y ❡①♣❛♥s✐♦♥ ✐s ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ ✐♥ ◗❊❉✳ ❚❤❡ r❡❛s♦♥ ❝♦✉❧❞ ❜❡ t❤❡ ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡✳ ▲❡t ✉s ❣♦ ❜❛❝❦ t♦ ❧✐♥❡❛r ♦r❞❡r ✐♥ y ❜✉t ✇✐t❤♦✉t ❞✐❛❣♦♥❛❧✐③✐♥❣ ✭◆♦✇ Po = (N✷

c − ✶)Do✮

  • ∂t + ✷♣ · ∇r

M + CF❋(r) · ∇♣

  • Ps = −✷CFΓ(r)
  • Ps −

Po N✷

c − ✶

  • ∂t + ✷♣ · ∇r

M − ✶ ✷Nc ❋(r) · ∇♣

  • Po = − ✶

Nc Γ(r)(Po − (N✷

c − ✶)Ps)

❚❤✐s ✐s ❛ ❇♦❧t③♠❛♥♥ ❡q✉❛t✐♦♥✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✼ ✴ ✺✷

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SLIDE 40

❇♦❧t③♠❛♥♥✲▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥

■❢ t❡r♠s ♦❢ ♦r❞❡r y✷ ❛r❡ ✐♥❝❧✉❞❡❞ t❤❡ ❋♦❦❦❡r✲P❧❛♥❝❦ ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ♠✐①t✉r❡ ❜❡t✇❡❡♥ ❛ ❇♦❧t③♠❛♥♥ ❛♥❞ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥s ❛s ❧♦♥❣ ❛s t❡r♠s t❤❛t ❣♦ ❧✐❦❡ y✷((N✷

c − ✶)Ps − Po) ❛r❡ ♥❡❣❧❡❝t❡❞✳ ❲❡ ♥❡❡❞ t❤❛t t❤❡

♦♥❧② t❡r♠ ✇✐t❤ ♥♦♥✲❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ✐s t❤❡ ❝♦❧❧✐s✐♦♥ t❡r♠✳ ∂tPs + ✷pi∇i

r

M Ps + CFF i(r)∇i

pPs −

CF ✷MT (Hij(✵) + Hij(r))∇i

p(pjPs)

= −✷CFΓ(r)

  • Ps −

Po N✷

c − ✶

  • +CF

✹ (Hij(✵) + Hij(r))∆ij

pPs

❊①❛❝t❧② t❤❡ s❛♠❡ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ❛s ✐♥ ◗❊❉ ♣❧✉s ❝♦❧❧✐s✐♦♥ t❡r♠ t❤❛t ❝❤❛♥❣❡s ❝♦❧♦r st❛t❡

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✽ ✴ ✺✷

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SLIDE 41

❇♦❧t③♠❛♥♥✲▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ♦❝t❡t

∂tPo + ✷pi∇i

r

M Po = − ✶ Nc Γ(r)(Po − (N✷

c − ✶)Ps) +

✶ ✷Nc F i(r)∇i

pPo

+ ✶ ✹

  • CFHij(✵) −

✶ ✷Nc Hij(r)

  • ∆ij

pPo

+ ✶ ✷MT

  • CFHij(✵) −

✶ ✷Nc Hij(r)

  • ∇i

p(pjPo)

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✸✾ ✴ ✺✷

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SLIDE 42

❙t✉❞② ♦❢ J/Ψ ❛t T = ✶✻✵ ▼❡❱

❇❡tt❡r r❡s✉❧t t❤❛t t❤❡ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ r❛♥❞♦♠ ❝♦❧♦r✳ ❆t ❧❛r❣❡ t✐♠❡s ✐t t❡♥❞s t♦ t❤❡ ♠❛①✐♠✉♠ ❝♦❧♦r ❡♥tr♦♣② st❛t❡✱ ❜✉t ✐t ❞♦❡s ♥♦t st❛rt ❢r♦♠ t❤❡r❡✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✵ ✴ ✺✷

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SLIDE 43

Pr♦s ❛♥❞ ❈♦♥s ♦❢ t❤❡ t✇♦ ♠❡t❤♦❞s

❘❛♥❞♦♠ ❝♦❧♦r ❢♦r❝❡

■t ❝❛♥ ❜❡ ✉s❡❞ t♦ st✉❞② ♠❛♥② ❤❡❛✈② ♣❛rt✐❝❧❡s ✇✐t❤ ❛ ❧♦✇ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ❚❤❡r❡ ❛r❡ ✉♥♣❤②s✐❝❛❧ ❧❛r❣❡ ❤✐ts✳

❇♦❧t③♠❛♥♥✲▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥

  • ✐✈❡s ♠♦r❡ r❡❛❧✐st✐❝ r❡s✉❧ts✳

❉✐✣❝✉❧t t♦ ❣❡♥❡r❛❧✐③❡ t♦ ♠❛♥② ♣❛rt✐❝❧❡s✳ ❲❡ ✇♦✉❧❞ ♥❡❡❞ t♦ ❝❧❛ss✐❢② ❛❧❧ ♣♦ss✐❜❧❡ ❝♦❧♦r st❛t❡s ❛♥❞ ❝♦♠♣✉t❡ tr❛♥s✐t✐♦♥ r❛t❡s✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✶ ✴ ✺✷

slide-44
SLIDE 44

P❧❛♥

■♥tr♦❞✉❝t✐♦♥

❖♣❡♥ q✉❛♥t✉♠ s②st❡♠s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❡✛❡❝t✐✈❡ ✜❡❧❞ t❤❡♦r✐❡s✳ ❚❤❡

✶ r ≫ T ❝❛s❡ ✸

▲❛♥❣❡✈✐♥✲❧✐❦❡ ❡q✉❛t✐♦♥s ✐♥ ◗❈❉

❈♦♥❝❧✉s✐♦♥s

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✷ ✴ ✺✷

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SLIDE 45

❲❡ ❤❛✈❡ ❧❡❛r♥❡❞✳✳✳

RAA ❝❛♥ ❜❡ r❡❧❛t❡❞ ✇✐t❤ t❤❡ S†S ♦♣❡r❛t♦r ✐♥ ♣◆❘◗❈❉ ✭S ✐s t❤❡ s✐♥❣❧❡t ✜❡❧❞✮✳ ■♥ t❤❡ t❡♠♣❡r❛t✉r❡ r❡❣✐♠❡ ✶/r ≫ T, mD ≫ E ❛❧❧ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♥❡❡❞❡❞ ❢r♦♠ t❤❡ ♠❡❞✐✉♠ ❝❛♥ ❜❡ ❡♥❝♦❞❡❞ ✐♥ t✇♦ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ ♣❛r❛♠❡t❡rs κ ❛♥❞ γ✳ κ ✐s ❛ ✇❡❧❧✲❦♥♦✇♥ ♣❛r❛♠❡t❡r ✐♥ ♦♣❡♥ ❤❡❛✈② ✢❛✈♦✉r ♣❤②s✐❝s ❛♥❞ ❧❛tt✐❝❡ ❞❡t❡r♠✐♥❛t✐♦♥s ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❉✐r❡❝t❧② s♦❧✈✐♥❣ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ✐♥ t❤✐s r❡❣✐♠❡ ✇❡ ❝❛♥ q✉❛❧✐t❛t✐✈❡❧② r❡♣r♦❞✉❝❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✳ ■♠♣r♦✈❡♠❡♥ts ✐♥ ❤②❞r♦ ❡✈♦❧✉t✐♦♥✱ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ T ∼ E ❝❛s❡ ❝❛♥ ✐♥❝r❡❛s❡ t❤❡ ❛❣r❡❡♠❡♥t ✐♥ t❤❡ ❢✉t✉r❡✳ ❲❡ ❝❛♥ ✜♥❞ ▲❛♥❣❡✈✐♥✲❧✐❦❡ ❡q✉❛t✐♦♥s ✐♥ ◗❈❉ ❜✉t t❤❡✐r r❡❣✐♦♥ ♦❢ ✈❛❧✐❞✐t② ✐s s♠❛❧❧❡r t❤❛♥ ✐♥ ◗❊❉✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s✐♥❣❧❡ts ❛♥❞ ♦❝t❡ts✳ ❆ st✉❞② ♦❢ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ❡♥❡r❣② s❤♦✇s t❤❛t t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❇♦❧t③♠❛♥♥ ❛♥❞ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ❝❛♥ s✉❝❝❡ss❢✉❧❧② ✐♥❝❧✉❞❡ t❤✐s ♣❤②s✐❝s✳✼

✼❙❤♦✇♥ ✐♥ ❛r❳✐✈✿✶✽✵✸✳✵✼✾✾✻✱ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❏✲P✳ ❇❧❛✐③♦t s✉❜♠✐tt❡❞ t♦ P❘❉✳ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✸ ✴ ✺✷

slide-46
SLIDE 46

P❧❛♥

❇❛❝❦✉♣ s❧✐❞❡s

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✹ ✴ ✺✷

slide-47
SLIDE 47

❙t♦❝❤❛st✐❝ ♣♦t❡♥t✐❛❧ ❛♣♣r♦❛❝❤✽

P♦t❡♥t✐❛❧ ♠♦❞❡❧ ✇✐t❤ st♦❝❤❛st✐❝ ♣♦t❡♥t✐❛❧✳ ▼♦♥t❡ ❈❛r❧♦ t❡❝❤♥✐q✉❡ t♦ ❡✣❝✐❡♥t❧② ❝♦♠♣✉t❡ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐①✳ ❱❛❧✐❞ ✐♥ t❤❡ r❡❣✐♠❡ T ≫ ✶

r ≫ mD✳

❇❛s❡❞ ♦♥ ❛ ◗❊❉ ✭❆❜❡❧✐❛♥✮ ❝♦♠♣✉t❛t✐♦♥✳

✽❑❛❥✐♠♦t♦✱ ❆❦❛♠❛ts✉✱ ❆s❛❦❛✇❛ ❛♥❞ ❘♦t❤❦♦♣❢✳ P❘❉ ✾✼ ✭✷✵✶✽✮ ♥♦✳✶ ✵✶✹✵✵✸ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✺ ✴ ✺✷

slide-48
SLIDE 48

❙❝❤rö❞✐♥❣❡r✲▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ❛♣♣r♦❛❝❤ ✾

❙t♦❝❤❛st✐❝ ♣♦t❡♥t✐❛❧ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ ❛ ♥♦♥✲❧✐♥❡❛r t❡r♠ ✐s ❛❞❞❡❞ t♦ ✐♠♣♦s❡ t❤❡r♠❛❧✐③❛t✐♦♥✳ ❆❜❡❧✐❛♥ ❧✐♠✐t✳ ❚❤❡ ♥♦♥✲❧✐♥❡❛r t❡r♠ ✐s ♥♦t ❞❡r✐✈❡❞ ❢r♦♠ ❛ q✉❛♥t✉♠ ✜❡❧❞ t❤❡♦r②✳ ❙✐♠✉❧❛t✐♦♥ ❝♦✉♣❧❡❞ t♦ st❛t❡ ♦❢ t❤❡ ❛rt ❤②❞r♦✳

✾●♦ss✐❛✉① ❛♥❞ ❑❛t③✳ ❆♥♥❛❧s P❤②s✳ ✸✻✽ ✭✷✵✶✻✮ ✷✻✼✲✷✾✺ ▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✻ ✴ ✺✷

slide-49
SLIDE 49

❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐①

❘❡♠✐♥❞❡r ♦❢ t❤❡ ✹ ❞✐❛❣r❛♠s t❤❛t ❝♦♥♥❡❝t ✇❤❛t❡✈❡r st❛t❡ ❛t t✐♠❡ t ✇✐t❤ ❛ s✐♥❣❧❡t ❛t t✐♠❡ t + dt✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✉s❡ t❤❡ ❡①❛❝t ✈❛❧✉❡ ♦❢ Us,o✳ ❚❤❡ q✉❛♥t✉♠ ♠❛st❡r ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ s❝❤❡♠❛t✐❝❛❧❧② ❛s ❞D ❞t + i[HQ, DQ(t)] = −

  • ①①′

t−t✵

❞τ [nA

① , UQ(τ)nA ①′DQ(t − τ)U† Q(τ)] ∆>(τ; ① − ①′))

  • ①①′

t−t✵

❞τ [UQ(τ)DQ(t − τ)nA

①′U† Q(τ), nA ① ] ∆<(τ; ① − ①′),

✇❤❡r❡ ✇❡ ❤❛✈❡ s❡t t − t′ = τ✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✼ ✴ ✺✷

slide-50
SLIDE 50

Us,o ❛♥❞ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥

❚❤❡ r♦❧❡ ♦❢ Us,o ✐s t♦ ✐♥❢♦r♠ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ t❤❡ s✐③❡ ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❡♥❡r❣② ❞✐✛❡r❡♥t ❜❡t✇❡❡♥ t❤❡ st❛t❡ ❛t t′ ❛♥❞ t❤❡ st❛t❡ ❛t t✳ ■❢ ✇❡ ♠❛❦❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ Us,o = ✶ ❛❧❧ tr❛♥s✐t✐♦♥s ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧②✳ ■❢ ✐s ♥♦t ❛♣♣r♦①✐♠❛t❡❞ t❤❡r❡ ❛r❡ s♦♠❡ tr❛♥s✐t✐♦♥s ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ❧✐❜❡r❛t❡ ❡♥❡r❣② ✇❤✐❧❡ ♦t❤❡rs ❛r❡ ❧❡ss ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ♥❡❡❞ t♦ ❛❜s♦r❜ ❡♥❡r❣②✳ ❡ ✐s ✜①❡❞ ❜② ✢✉❝t✉❛t✐♦♥✲❞✐ss✐♣❛t✐♦♥ t❤❡♦r❡♠ ✭❞❡t❛✐❧❡❞ ❜❛❧❛♥❝❡✮✳ ❚❤✐s ✐s ✈❡r② ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ ◗❈❉✳ ❆♥ ♦❝t❡t t♦ s✐♥❣❧❡t tr❛♥s✐t✐♦♥ ✐s ❛❧✇❛②s ❡♥❡r❣❡t✐❝❛❧❧② ❢❛✈♦r❛❜❧❡ ✇❤✐❧❡ t❤❡ ♦♣♣♦s✐t❡ ❤❛♣♣❡♥s ❢♦r s✐♥❣❧❡t t♦ ♦❝t❡t✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✽ ✴ ✺✷

slide-51
SLIDE 51

Us,o ❛♥❞ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥

❚❤❡ r♦❧❡ ♦❢ Us,o ✐s t♦ ✐♥❢♦r♠ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ t❤❡ s✐③❡ ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❡♥❡r❣② ❞✐✛❡r❡♥t ❜❡t✇❡❡♥ t❤❡ st❛t❡ ❛t t′ ❛♥❞ t❤❡ st❛t❡ ❛t t✳ ■❢ ✇❡ ♠❛❦❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ Us,o = ✶ ❛❧❧ tr❛♥s✐t✐♦♥s ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧②✳ ■❢ Us,o ✐s ♥♦t ❛♣♣r♦①✐♠❛t❡❞ t❤❡r❡ ❛r❡ s♦♠❡ tr❛♥s✐t✐♦♥s ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ❧✐❜❡r❛t❡ ❡♥❡r❣② ✇❤✐❧❡ ♦t❤❡rs ❛r❡ ❧❡ss ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ♥❡❡❞ t♦ ❛❜s♦r❜ ❡♥❡r❣②✳ ❡ ✐s ✜①❡❞ ❜② ✢✉❝t✉❛t✐♦♥✲❞✐ss✐♣❛t✐♦♥ t❤❡♦r❡♠ ✭❞❡t❛✐❧❡❞ ❜❛❧❛♥❝❡✮✳ ❚❤✐s ✐s ✈❡r② ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ ◗❈❉✳ ❆♥ ♦❝t❡t t♦ s✐♥❣❧❡t tr❛♥s✐t✐♦♥ ✐s ❛❧✇❛②s ❡♥❡r❣❡t✐❝❛❧❧② ❢❛✈♦r❛❜❧❡ ✇❤✐❧❡ t❤❡ ♦♣♣♦s✐t❡ ❤❛♣♣❡♥s ❢♦r s✐♥❣❧❡t t♦ ♦❝t❡t✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✽ ✴ ✺✷

slide-52
SLIDE 52

Us,o ❛♥❞ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥

❚❤❡ r♦❧❡ ♦❢ Us,o ✐s t♦ ✐♥❢♦r♠ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ t❤❡ s✐③❡ ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❡♥❡r❣② ❞✐✛❡r❡♥t ❜❡t✇❡❡♥ t❤❡ st❛t❡ ❛t t′ ❛♥❞ t❤❡ st❛t❡ ❛t t✳ ■❢ ✇❡ ♠❛❦❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ Us,o = ✶ ❛❧❧ tr❛♥s✐t✐♦♥s ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧②✳ ■❢ Us,o ✐s ♥♦t ❛♣♣r♦①✐♠❛t❡❞ t❤❡r❡ ❛r❡ s♦♠❡ tr❛♥s✐t✐♦♥s ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ❧✐❜❡r❛t❡ ❡♥❡r❣② ✇❤✐❧❡ ♦t❤❡rs ❛r❡ ❧❡ss ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ♥❡❡❞ t♦ ❛❜s♦r❜ ❡♥❡r❣②✳

Γ(A→B) Γ(B→A) = ❡

EA−EB T

✐s ✜①❡❞ ❜② ✢✉❝t✉❛t✐♦♥✲❞✐ss✐♣❛t✐♦♥ t❤❡♦r❡♠ ✭❞❡t❛✐❧❡❞ ❜❛❧❛♥❝❡✮✳ ❚❤✐s ✐s ✈❡r② ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ ◗❈❉✳ ❆♥ ♦❝t❡t t♦ s✐♥❣❧❡t tr❛♥s✐t✐♦♥ ✐s ❛❧✇❛②s ❡♥❡r❣❡t✐❝❛❧❧② ❢❛✈♦r❛❜❧❡ ✇❤✐❧❡ t❤❡ ♦♣♣♦s✐t❡ ❤❛♣♣❡♥s ❢♦r s✐♥❣❧❡t t♦ ♦❝t❡t✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✽ ✴ ✺✷

slide-53
SLIDE 53

Us,o ❛♥❞ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥

❚❤❡ r♦❧❡ ♦❢ Us,o ✐s t♦ ✐♥❢♦r♠ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ t❤❡ s✐③❡ ❛♥❞ t❤❡ s✐❣♥ ♦❢ t❤❡ ❡♥❡r❣② ❞✐✛❡r❡♥t ❜❡t✇❡❡♥ t❤❡ st❛t❡ ❛t t′ ❛♥❞ t❤❡ st❛t❡ ❛t t✳ ■❢ ✇❡ ♠❛❦❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ Us,o = ✶ ❛❧❧ tr❛♥s✐t✐♦♥s ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧②✳ ■❢ Us,o ✐s ♥♦t ❛♣♣r♦①✐♠❛t❡❞ t❤❡r❡ ❛r❡ s♦♠❡ tr❛♥s✐t✐♦♥s ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ❧✐❜❡r❛t❡ ❡♥❡r❣② ✇❤✐❧❡ ♦t❤❡rs ❛r❡ ❧❡ss ❧✐❦❡❧② ❜❡❝❛✉s❡ t❤❡② ♥❡❡❞ t♦ ❛❜s♦r❜ ❡♥❡r❣②✳

Γ(A→B) Γ(B→A) = ❡

EA−EB T

✐s ✜①❡❞ ❜② ✢✉❝t✉❛t✐♦♥✲❞✐ss✐♣❛t✐♦♥ t❤❡♦r❡♠ ✭❞❡t❛✐❧❡❞ ❜❛❧❛♥❝❡✮✳ ❚❤✐s ✐s ✈❡r② ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ ◗❈❉✳ ❆♥ ♦❝t❡t t♦ s✐♥❣❧❡t tr❛♥s✐t✐♦♥ ✐s ❛❧✇❛②s ❡♥❡r❣❡t✐❝❛❧❧② ❢❛✈♦r❛❜❧❡ ✇❤✐❧❡ t❤❡ ♦♣♣♦s✐t❡ ❤❛♣♣❡♥s ❢♦r s✐♥❣❧❡t t♦ ♦❝t❡t✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✽ ✴ ✺✷

slide-54
SLIDE 54

❲❤② ❞♦❡s t❤❡ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✇♦r❦ s♦ ✇❡❧❧ ✐♥ ◗❊❉ ✇❤✐❧❡ t❤❡ ❛♥❛❧♦❣♦✉s ❡q✉❛t✐♦♥ ✐♥ ◗❈❉ ❞♦❡s ♥♦t❄

■♥ ◗❊❉ ❚❤❡ ♣♦s✐t✐♦♥ ✇✐❧❧ ♥♦t ❝❤❛♥❣❡ ❛❢t❡r t❤❡ ❛❜s♦r♣t✐♦♥ ♦❢ t❤❡ ❣❧✉♦♥ → s❛♠❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣②✳ ■❢ t❤❡ ❡❧❡❝tr♦♥s ✐s ❝❧♦s❡❞ t♦ ❜❡ t❤❡r♠❛❧✐③❡❞ t❤❡♥ p ∼ √ MT✳ q ❝❛♥ ❜❡ ❛t ♠♦st ♦❢ ♦r❞❡r T✳ ❚❤❡♥ (♣ + q)✷ − p✷ ✷MT ∼

  • T

M

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✹✾ ✴ ✺✷

slide-55
SLIDE 55

❲❤② ❞♦❡s t❤❡ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✇♦r❦ s♦ ✇❡❧❧ ✐♥ ◗❊❉ ✇❤✐❧❡ t❤❡ ❛♥❛❧♦❣♦✉s ❡q✉❛t✐♦♥ ✐♥ ◗❈❉ ❞♦❡s ♥♦t❄

■♥ ◗❈❉✱ ❡✈❡♥ ✐❢ ♣♦s✐t✐♦♥ ❞♦❡s♥✬t ❝❤❛♥❣❡ t❤❡r❡ ✇✐❧❧ ❜❡ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ∆E T ∼ ∆V T

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✺✵ ✴ ✺✷

slide-56
SLIDE 56

❈♦♠❜✐♥❛t✐♦♥ ♦❢ r❛t❡ ❡q✉❛t✐♦♥ ✰ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥

❚❤❡r❡ ✐s ♥♦ ❣❛♣ ✐♥ ♦❝t❡t t♦ ♦❝t❡t tr❛♥s✐t✐♦♥s✱ t❤❡r❡❢♦r❡ t❤✐s ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ▲❛♥❣❡✈✐♥ ❡q✉❛t✐♦♥ ✭s❛♠❡ ❝❛s❡ ❛s ◗❊❉✮✳ ■♥ t❤❡ ❧❛r❣❡ Nc ❧✐♠✐t t❤❡ ♦❝t❡t ♣♦t❡♥t✐❛❧ ✐s ③❡r♦✳ ❙✐♥❣❧❡t t♦ ♦❝t❡t tr❛♥s✐t✐♦♥s ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❛ r❛t❡ ❡q✉❛t✐♦♥✳ ❆s ❛♥ ✐❧❧✉str❛t✐♦♥ ✇❡ ❝♦♥s✐❞❡r ❛ t♦② ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✐♥❣❧❡t ❡✐❣❡♥✈❛❧✉❡ ✭t❤❡ ✶❙✮✳ ❞ps ❞t = g✷CF

  • p♦

♣ − ps❡−

E♦ ♣ −Es T

q

∆>(ω♦

♣ − E s, q)|s|Sq·ˆ r|♦, ♣|✷ ,

❛♥❞ ∂p♦

∂t − γ∇(♣p♦

♣) − TγM

✷ ∆✷p♦

♣ =

− g✷ ✷Nc ✶ Ω

  • p♦

♣ − ps❡−

E♦ ♣ −Es T

q

∆>(ω♦

♣ − E s, q)|s|Sq·ˆ r|♦, ♣|✷ ,

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✺✶ ✴ ✺✷

slide-57
SLIDE 57

❚♦② ♠♦❞❡❧✳ ❘❡s✉❧ts

Ω = ✶ ❢♠✸ Ω = ✶✵✵ ❢♠✸ ✺ ❢♠/❝ ✶✵✵ ❢♠/❝ ❡q✳ ✺ ❢♠/❝ ✶✵✵ ❢♠/❝ ❡q✳ T = ✷✵✵ ▼❡❱ ✵.✽✻ ✵.✶✸✻ ✵.✵✽✶✹ ✵.✽✺ ✵.✵✹✸✽ ✵.✵✵✵✽✾ T = ✹✵✵ ▼❡❱ ✵.✸✾ ✵.✵✺✶✺ ✵.✵✶✼✺ ✵.✸✻ ✵.✵✵✵✷ ✵.✵✵✵✶✽

❚❛❜❧❡✿ ps ❛s ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ ❡qs✳ ✐♥ ♣r❡✈✐♦✉s s❧✐❞❡

❘❡♠❛r❦s

❚❤❡ ❣❛♣ E o

♣ − E s s✉♣♣r❡ss❡s t❤❡ s✐♥❣❧❡t ❞❡❝❛② ✇✐❞t❤✱ ❤❡❧♣✐♥❣ t♦ ♠❛❦❡

t❤❡ r❡s✉❧ts ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ♦❜s❡r✈❛t✐♦♥s✳ E s ❞❡♣❡♥❞s ♦♥ t❤❡ r❡❛❧ ♣❛rt ♦❢ t❤❡ ♣♦t❡♥t✐❛❧✳ ❙❝r❡❡♥✐♥❣ r❡❞✉❝❡s t❤❡ ❜✐♥❞✐♥❣ ❡♥❡r❣② ❛♥❞ t❤✐s ✐♥❝r❡❛s❡s t❤❡ ❞❡❝❛② ✇✐❞t❤✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♠❡❞✐✉♠ Ω s✉♣♣r❡ss❡s t❤❡ ❞❡❝❛② ♦❢ ♦❝t❡ts ✐♥t♦ ❜♦✉♥❞ s✐♥❣❧❡ts✳

▼✳❆✳ ❊s❝♦❜❡❞♦ ✭❏❨❯✮ ❖◗❙ ❢♦r ❍◗ ❙❊❲▼ ✷✵✶✽ ✺✷ ✴ ✺✷