❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ■♥st✐t✉t❡ ♦❢ ❚❤❡♦r❡t✐❝❛❧ P❤②s✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❲❛rs❛✇ ❘❡♥❝♦♥tr❡s ❞❡ ▼♦r✐♦♥❞✱ ▲❛ ❚❤✉✐❧❡✱ ✶✼t❤ ▼❛r❝❤ ✷✵✶✶ ❜❛s❡❞ ♦♥ t❤❡ ✇♦r❦ ✇✐t❤ ▼❛r❡❦ ❖❧❡❝❤♦✇s❦✐ ❛♥❞ ❙t❡❢❛♥ P♦❦♦rs❦✐ ♣r❡❧✐♠✐♥❛r② r❡s✉❧ts ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
▼✐♥✐♠❛❧ ❙❖✭✶✵✮ ❛♥❞ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ❋♦r ❛ ❣✐✈❡♥ ❣❡♥❡r❛t✐♦♥ ❛❧❧ ♠❛tt❡r ❢❡r♠✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ ν ❘ ✱ s✐t ✐♥ ♦♥❡ ✶✻ ❞✐♠✳ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❙❖✭✶✵✮ ❇♦t❤ ▼❙❙▼ ❍✐❣❣s ❞♦✉❜❧❡ts ❛r❡ ✐♥ t❤❡ ✶✵ ❞✐♠✳ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❙❖✭✶✵✮ ❨✉❦❛✇❛ ✐♥t❡r❛❝t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② ❲ = ❤ ✶✻ ✶✵ ✶✻ ✇❤✐❝❤ ✐♠♣❧② ✉♥✐✜❝❛t✐♦♥ ♦❢ t − ❜ − τ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣ ❛t ▼ ●❯❚ ✳ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
❊①♣❧❛✐♥✐♥❣ ( ❣ − ✷ ) µ ❛♥♦♠❛❧② ❛♥❞ ❨✉❦❛✇❛ ✉♥✐✜❝❛t✐♦♥ ❨✉❦❛✇❛ ✉♥✐✜❝❛t✐♦♥ ♣r❡❢❡rs µ < ✵ ❜✉t ✉s✉❛❧❧② µ > ✵ ✐s ❝♦♥s✐❞❡r❡❞✳ ❲❤②❄ ( ❣ − ✷ ) ❙▼ ♠♦r❡ t❤❛♥ ✸ σ ❜❡❧♦✇ ❡①♣❡r✐♠❡♥t❛❧ ✈❛❧✉❡✳ µ ( ❣ − ✷ ) ❙❯❙❨ ∼ s❣♥ ( µ ▼ ✷ ) ⇒ ( ❣ − ✷ ) µ ♣r❡❢❡rs µ > ✵ ❢♦r ✉♥✐✈❡rs❛❧ µ ❣❛✉❣✐♥♦ ♠❛ss❡s ❜✉t ❯♥✐✈❡rs❛❧ ❣❛✉❣✐♥♦ ♠❛ss❡s ❛r❡ ♥♦t ♦❜❧✐❣❛t♦r② ✐♥ ❙❯❙❨ ●❯❚s ❖✉r ❙tr❛t❡❣② ❈♦♥s✐❞❡r µ < ✵ ❛♥❞ ♥♦♥✲✉♥✐✈❡rs❛❧ ❣❛✉❣✐♥♦ ♠❛ss❡s ✇✐t❤ ▼ ✷ < ✵ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
❚❤❡ ♠♦❞❡❧ µ < ✵ ◆♦♥✲✉♥✐✈❡rs❛❧ s❝❛❧❛r ♠❛ss❡s✿ ♠ ✷ ❍ ❞ = ♠ ✷ ✶✵ + ✷ ❉ ♠ ✷ ❍ ✉ = ♠ ✷ ✶✵ − ✷ ❉ ♠ ✷ ◗ , ❯ , ❊ = ♠ ✷ ✶✻ + ❉ ♠ ✷ ❉ , ▲ = ♠ ✷ ✶✻ − ✸ ❉ ◆♦♥✲✉♥✐✈❡rs❛❧ ❣❛✉❣✐♥♦ ♠❛ss❡s ❣❡♥❡r❛t❡❞ ❜② ❋ ✲t❡r♠ tr❛♥s❢♦r♠✐♥❣ ❛s ✺✹ ❞✐♠✳ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❙❖✭✶✵✮✿ ▼ ✶ = − ✶ ✷ ♠ ✶ / ✷ ▼ ✷ = − ✸ ✷ ♠ ✶ / ✷ ▼ ✸ = ♠ ✶ / ✷ ❯♥✐✈❡rs❛❧ tr✐❧✐♥❡❛r ❝♦✉♣❧✐♥❣s✿ ❆ ❯ = ❆ ❉ = ❆ ❊ = ❆ ✵ ✺ ♣❛r❛♠❡t❡rs ✰ t❛♥ β ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
▼❡t❤♦❞♦❧♦❣② ❚❤❡ s♦❢t ❙❯❙❨ ❜r❡❛❦✐♥❣ ♠❛ss❡s ❛r❡ ✐♠♣♦s❡❞ ❛t t❤❡ ●❯❚ s❝❛❧❡ ❛♥❞ r✉♥ ❞♦✇♥ t♦ t❤❡ ❡❧❡❝tr♦✇❡❛❦ s❝❛❧❡ ✇✐t❤ ❘●❊ ❝♦❞❡ ❙❖❋❚❙❯❙❨ ✳ ❲❡ ✐♠♣♦s❡ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ❘❊❲❙❇✱ ♥❡✉tr❛❧ ▲❙P✱ ♣❛rt✐❝❧❡ ♠❛ss ❜♦✉♥❞s ❛♥❞ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥str❛✐♥ts ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♦❜s❡r✈❛❜❧❡s✿ t❤❡ r❡❧✐❝ ❛❜✉♥❞❛♥❝❡ ♦❢ ❞❛r❦ ♠❛tt❡r Ω ❉▼ ❤ ✷ ❛ µ ≡ ( ❣ − ✷ ) µ / ✷ ❇❘✭ ❜ → s γ ✮ ❇❘✭ ❇ s → µ + µ − ✮ ✇❤✐❝❤ ✇❡ ❝❛❧❝✉❧❛t❡ ✉s✐♥❣ ▼✐❝r❖▼❊●❆s ✳ ❲❡ ❤❛✈❡ ♣❡r❢♦r♠❡❞ r❛♥❞♦♠ s❝❛♥ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ r❛♥❣❡s ♦❢ ♣❛r❛♠❡t❡rs✿ ✵ � ♠ ✶✻ � ✷✵✵✵ ●❡❱ ✵ . ✶ � ♠ ✶✵ / ♠ ✶✻ � ✷ − ✸ � ❆ ✵ / ♠ ✶✻ � ✸ ✵ � ❉ / ♠ ✷ ✵ � ♠ ✶ / ✷ � ✷✵✵✵ ●❡❱ ✶✻ � ✵ . ✸ ✹✵ � t❛♥ β � ✺✺ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
❲✐❞❡ r❛♥❞♦♠ s❝❛♥ ❢♦r ❨✉❦❛✇❛✲✉♥✐✜❡❞ s♦❧✉t✐♦♥s ❘ ≡ ♠❛① ( ❤ t , ❤ ❜ , ❤ τ ) ♠✐♥ ( ❤ t , ❤ ❜ , ❤ τ ) ❜ → s γ ❡①❝❧✉❞❡❞ ❜ → s γ ♦❦ ❨✉❦❛✇❛ ✉♥✐✜❝❛t✐♦♥ ✭ ❘ ≈ ✶✮ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢♦r ❛ ✈❡r② ✇✐❞❡ r❛♥❣❡s ♦❢ ♣❛r❛♠❡t❡rs ❜✉t ♠❛❥♦r✐t② ♦❢ ♣♦✐♥ts ✇✐t❤ ❣♦♦❞ ②✉❦❛✇❛ ✉♥✐✜❝❛t✐♦♥ ❡①❝❧✉❞❡❞ ❜② ❜ → s γ ✳ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
( ❣ − ✷ ) µ ✈s ❜ → s γ ❙tr♦♥❣ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ( ❣ − ✷ ) µ ❛♥❞ ❜ → s γ ❋♦r t❤❡ ♣♦✐♥ts ♥♦t ❡①❝❧✉❞❡❞ ❜② ❜ → s γ ✱ ( ❣ − ✷ ) µ t②♣✐❝❛❧❧② ✈❡r② s♠❛❧❧ ❍♦✇ t♦ ❞✐s❡♥t❛♥❣❧❡ ( ❣ − ✷ ) µ ❢r♦♠ ❜ → s γ ❄ ❲❡ ❤❛✈❡ ✐❞❡♥t✐✜❡❞ ✷ ❝❧❛ss❡s ♦❢ s♦❧✉t✐♦♥s ✇✐t❤ ❣♦♦❞ ②✉❦❛✇❛ ✉♥✐✜❝❛t✐♦♥✱ ❝♦♥s✐st❡♥t ✇✐t❤ ❛❧❧ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥str❛✐♥ts ✭✐♥❝❧✉❞✐♥❣ ❜ → s γ ✮ ✇✐t❤ ❧❛r❣❡ ❙❯❙❨ ❝♦♥tr✐❜✉t✐♦♥ t♦ ( ❣ − ✷ ) µ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
❙♦❧✉t✐♦♥ ✶ ▲❛r❣❡ ❉ ✲t❡r♠s t♦ ♣✉s❤ ✉♣ ♠ ❍ ± ❛♥❞ s✉♣♣r❡ss ❍✐❣❣s ❝♦♥tr✐❜✉t✐♦♥ t♦ ❇❘ ( ❜ → s γ ) ❆ t ≈ ✵ ❛t ▼ ❊❲ ✭t❤✐s r❡q✉✐r❡ ❧❛r❣❡ ♣♦s✐t✐✈❡ ❆ ✲t❡r♠s ❛t ▼ ●❯❚ ✮ t♦ s✉♣♣r❡ss ❝❤❛r❣✐♥♦✲st♦♣ ♠✐①✐♥❣ ❝♦♥tr✐❜✉t✐♦♥ t♦ ❇❘ ( ❜ → s γ ) ✇✐t❤♦✉t s✉♣♣r❡ss✐♥❣ ❝❤❛r❣✐♥♦ ❝♦♥tr✐❜✉t✐♦♥ t♦ ( ❣ − ✷ ) µ ❜✐♥♦✲❧✐❦❡ ▲❙P ✇✐t❤ t❤❡ r❡❧✐❝ ❛❜✉♥❞❛♥❝❡ s❛t✐s❢②✐♥❣ ❲▼❆P ❜♦✉♥❞ ❞✉❡ t♦ st❛✉ ❝♦❛♥♥✐❤✐❧❛t✐♦♥s ( ❣ − ✷ ) µ ✇✐t❤✐♥ ✷ σ ❢r♦♠ ❡①♣✳ ✈❛❧✉❡ ✭✶ σ r❡❧❛①✐♥❣ ❍✐❣❣s ♠❛ss ❜♦✉♥❞✮ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
❙♦❧✉t✐♦♥ ✷ ♠ ✶ / ✷ ≪ ♠ ✶✻ ⇒ ❧✐❣❤t ❣❧✉✐♥♦s ✭✺✵✵✲✼✵✵ ●❡❱✮ ▲❛r❣❡ ❝❤❛r❣✐♥♦ ♠✐①✐♥❣ ✭ ▼ ✷ ≈ µ ❛t ▼ ❊❲ ✮ ✇❤✐❝❤ r❡s✉❧ts ✐♥ ♥❡❣❛t✐✈❡ ❝❤❛r❣✐♥♦ ❝♦♥tr✐❜✉t✐♦♥ t♦ ❇❘ ( ❜ → s γ ) ❜✐♥♦✲❧✐❦❡ ▲❙P ✇✐t❤ ♥♦♥✲♥❡❣❧✐❣✐❜❧❡ ❤✐❣❣s✐♥♦ ❝♦♠♣♦♥❡♥t ✇❤✐❝❤ ❛❧❧♦✇s ❢♦r r❡s♦♥❛♥t ❛♥♥✐❤✐❧❛t✐♦♥s t❤r♦✉❣❤ ❤ ✵ ♦r ❩ ❜♦s♦♥s ( ❣ − ✷ ) µ ✇✐t❤✐♥ ✷ σ ❢r♦♠ ❡①♣✳ ✈❛❧✉❡ ✭✶ σ r❡❧❛①✐♥❣ ❍✐❣❣s ♠❛ss ❜♦✉♥❞✮ ▼❛r❝✐♥ ❇❛❞③✐❛❦ ❨✉❦❛✇❛ ❯♥✐✜❝❛t✐♦♥ ✐♥ ❙❯❙❨ ❙❖✭✶✵✮ ●❯❚
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