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slide-1
SLIDE 1

❚❤❡ ❘♦❧❡ ♦❢ ❋❧✉❝t✉❛t✐♦♥s ✐♥ t❤❡ P❤❛s❡ ❉✐❛❣r❛♠♠ ♦❢ ◗❈✷❉ ❋✉♥❝t✐♦♥❛❧ ▼❡t❤♦❞s

◆❛s❡❡♠✉❞❞✐♥ ❑❤❛♥✱ ❏❛♥ ▼✳ P❛✇❧♦✇s❦✐✱ ▼✐❝❤❛❡❧ ❙❝❤❡r❡r✱ ❋❛❜✐❛♥ ❘❡♥♥❡❝❦❡

❯♥✐✈❡rs✐t② ♦❢ ❍❡✐❞❡❧❜❡r❣

◗❈❉ ♠❡❡ts ❈♦❧❞ ❆t♦♠s ✲ ❊♣✐s♦❞❡ ■■■

✷✻t❤ ❆✉❣✉st ✷✵✶✷

slide-2
SLIDE 2

▼♦t✐✈❛t✐♦♥

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m

π

µB / m

π

Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid

[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]

◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠

✷ ✴ ✷✾

slide-3
SLIDE 3

▼♦t✐✈❛t✐♦♥

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m

π

µB / m

π

Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid

[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]

◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠

✷ ✴ ✷✾

slide-4
SLIDE 4

▼♦t✐✈❛t✐♦♥

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m

π

µB / m

π

Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid

[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]

◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠

✷ ✴ ✷✾

slide-5
SLIDE 5

▼♦t✐✈❛t✐♦♥

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m

π

µB / m

π

Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid

[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]

◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s → ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠

✷ ✴ ✷✾

slide-6
SLIDE 6

▼♦t✐✈❛t✐♦♥

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m

π

µB / m

π

Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid

[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]

◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s → ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠

✷ ✴ ✷✾

slide-7
SLIDE 7

▼♦t✐✈❛t✐♦♥

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m

π

µB / m

π

Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid

[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]

◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s → ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠

✷ ✴ ✷✾

slide-8
SLIDE 8

▼♦t✐✈❛t✐♦♥

❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠

✸ ✴ ✷✾

slide-9
SLIDE 9

▼♦t✐✈❛t✐♦♥

❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠

✹ ✴ ✷✾

slide-10
SLIDE 10

▼♦t✐✈❛t✐♦♥

❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠

✺ ✴ ✷✾

slide-11
SLIDE 11

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯ ✷ ❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t❛ t❚

t✷t❛t✷ ❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉ t✷❈

✺❉ ✺❈t✷

❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯ ◆❢

❙❯ ◆❢

❯ ✶ ❇ ❙❯ ✷◆❢

◗❈✷❉ ▲✐

▲ ❘✐

✐ ❉

▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-12
SLIDE 12

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉ t✷❈

✺❉ ✺❈t✷

❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯ ◆❢

❙❯ ◆❢

❯ ✶ ❇ ❙❯ ✷◆❢

◗❈✷❉ ▲✐

▲ ❘✐

✐ ❉

▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-13
SLIDE 13

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯ ◆❢

❙❯ ◆❢

❯ ✶ ❇ ❙❯ ✷◆❢

◗❈✷❉ ▲✐

▲ ❘✐

✐ ❉

▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-14
SLIDE 14

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ )

◗❈✷❉ ▲✐

▲ ❘✐

✐ ❉

▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-15
SLIDE 15

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ )

◗❈✷❉ ▲✐

▲ ❘✐

✐ ❉

▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-16
SLIDE 16

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†

▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘

✐ ❉

▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-17
SLIDE 17

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†

▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘ = Ψ†✐σµ❉µΨ ▲ ❘

✉▲ ❞▲ ✉❘ ❞❘

❘ ✷t✷ ❘

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-18
SLIDE 18

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†

▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘ = Ψ†✐σµ❉µΨ

Ψ = ψ▲ ˜ ψ❘

  • =

    ✉▲ ❞▲ ˜ ✉❘ ˜ ❞❘     , ˜ ψ❘ = σ✷t✷ψ∗

❛❧❧♦✇s ✉s t♦ r♦t❛t❡

▲ ❘✱ s✐♠✐❧❛r❧②

✻ ✴ ✷✾

slide-19
SLIDE 19

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗

❛ = t❚ ❛ = −t✷t❛t✷

❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†

▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘ = Ψ†✐σµ❉µΨ

Ψ = ψ▲ ˜ ψ❘

  • =

    ✉▲ ❞▲ ˜ ✉❘ ˜ ❞❘     , ˜ ψ❘ = σ✷t✷ψ∗

❛❧❧♦✇s ✉s t♦ r♦t❛t❡ ψ▲ → ˜ ψ❘✱ s✐♠✐❧❛r❧② ¯ ψψ → ψψ

✻ ✴ ✷✾

slide-20
SLIDE 20

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙❯(✹) ❙❯ ✷ ▲ ❙❯ ✷ ❘ ❯ ✶ ❇ ✵ ❙❯ ✷ ▲ ❙❯ ✷ ❘ ♠ ✵ ❙♣ ✹ ♥♦ ❙❙❇ ❢♦r ✵ ❙❯ ✷ ❱ ❯ ✶ ❇ ✵ ❙❯ ✷ ❱

✼ ✴ ✷✾

slide-21
SLIDE 21

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙❯(✹)

µ ❙❯(✷)▲ × ❙❯(✷)❘ × ❯(✶)❇

ψψ = ✵ ❙❯(✷)▲ × ❙❯(✷)❘

♠ψ → ¯ ψψ = ✵ ❙♣(✹)

✲ ♥♦ ❙❙❇ ❢♦r µ = ✵ ❄

µ ❙❯(✷)❱ × ❯(✶)❇

✲ ❄

ψψ = ✵ ❙❯(✷)❱

✼ ✴ ✷✾

slide-22
SLIDE 22

❋❡❛t✉r❡s ♦❢ ◗❈✷❉

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙❯(✹)

♠ψ → ¯ ψψ = ✵ ❙♣(✹)

µ ❙②♠♠❡tr② ❜❡t✇❡❡♥ ¯ ψψ ❛♥❞ ψψ ✐s ❜r♦❦❡♥ ❙❯(✷)❱ × ❯(✶)❇

❙❙❇ ψψ = ✵ ❙❯(✷)❱

✽ ✴ ✷✾

slide-23
SLIDE 23

❋✉♥❝t✐♦♥❛❧ ❘●

❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥

∂❦Γ❦[Φ] = ✶ ✷ ❙❚r ✶ Γ(✷)

❦ [Φ] + ❘❦

∂❦❘❦

❬❲❡tt❡r✐❝❤ ✬✾✶❪ ❦ ✐♥t❡r♣♦❧❛t❡s ❜❡t✇❡❡♥ ♠✐r❝♦s❝♦♣✐❝ ❛❝t✐♦♥ ❙ ❛♥❞ ❢✉❧❧ q✉❛♥t✉♠

❡✛❡❝t✐✈❡ ❛❝t✐♦♥

❙❜❛r❡

❦ ✵

✾ ✴ ✷✾

slide-24
SLIDE 24

❋✉♥❝t✐♦♥❛❧ ❘●

❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥

∂❦Γ❦[Φ] = ✶ ✷ ❙❚r ✶ Γ(✷)

❦ [Φ] + ❘❦

∂❦❘❦

❬❲❡tt❡r✐❝❤ ✬✾✶❪

Γ❦ ✐♥t❡r♣♦❧❛t❡s ❜❡t✇❡❡♥ ♠✐r❝♦s❝♦♣✐❝ ❛❝t✐♦♥ ❙ ❛♥❞ ❢✉❧❧ q✉❛♥t✉♠ ❡✛❡❝t✐✈❡ ❛❝t✐♦♥ Γ

Γ❦=Λ = ❙❜❛r❡ Γ❦=✵ ≡ Γ

[Gies ’06] ✾ ✴ ✷✾

slide-25
SLIDE 25

❋✉♥❝t✐♦♥❛❧ ❘●

❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥

✐♥t❡❣r❛t❡ ♦✉t ✢✉❝t✉❛t✐♦♥s✱ Φ =

  • ϕ, ψ, ¯

ψ, ❆, ❝, ¯ ❝

  • ∂kΓk[Φ] = 1

2 + 1 2 − + 1 2

✶✵ ✴ ✷✾

slide-26
SLIDE 26

❋✉♥❝t✐♦♥❛❧ ❘●

❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥

✐♥t❡❣r❛t❡ ♦✉t ✢✉❝t✉❛t✐♦♥s✱ Φ =

  • ϕ, ψ, ¯

ψ

  • ∂kΓk[Φ] = 1

2 + 1 2 − + 1 2

T=μ=0

✶ ●❡❱ [Diehl ❡t al ’10] ✶✶ ✴ ✷✾

slide-27
SLIDE 27

❋✉♥❝t✐♦♥❛❧ ❘●

❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧

❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷

✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷

  • ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s

✱ ✿ ◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷

❝ ♠✷

✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷

✵ ❋❧♦✇

✶✷ ✴ ✷✾

slide-28
SLIDE 28

❋✉♥❝t✐♦♥❛❧ ❘●

❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧

❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷

✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷

  • ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s σ✱ |∆|✿

◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷ − µ✷ > ✵ , σ =

❝ ♠✷ ,

|∆| = ✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷ − µ✷ < ✵ , σ =

❝ µ✷ ,

|∆| = ✵ ❋❧♦✇

✶✷ ✴ ✷✾

slide-29
SLIDE 29

❋✉♥❝t✐♦♥❛❧ ❘●

❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧

❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷

✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷

  • ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s σ✱ |∆|✿

◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷ − µ✷ > ✵ , σ =

❝ ♠✷ ,

|∆| = ✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷ − µ✷ < ✵ , σ =

❝ µ✷ ,

|∆| = ✵ ❋❧♦✇

✶✷ ✴ ✷✾

slide-30
SLIDE 30

❋✉♥❝t✐♦♥❛❧ ❘●

❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧

❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷

✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷

  • ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s σ✱ |∆|✿

◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷ − µ✷ > ✵ , σ =

❝ ♠✷ ,

|∆| = ✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷ − µ✷ < ✵ , σ =

❝ µ✷ ,

|∆| = ✵ ❋❧♦✇

∂tUk = 1 2 −

✶✷ ✴ ✷✾

slide-31
SLIDE 31

❋✉♥❝t✐♦♥❛❧ ❘●

■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥

❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

∂t =

−1

+ ∂t =

−1

+

❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ ❤❦ ❤❦

✶✸ ✴ ✷✾

slide-32
SLIDE 32

❋✉♥❝t✐♦♥❛❧ ❘●

■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥

❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

∂t =

−1

+ ∂t =

−1

+

❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ = ❤❦ ¯ ψψ = ❤❦σ

✶✸ ✴ ✷✾

slide-33
SLIDE 33

❋✉♥❝t✐♦♥❛❧ ❘●

■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥

❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

∂t =

−1

+ ∂t =

−1

+

❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ = ❤❦ ¯ ψψ = ❤❦σ

✶✸ ✴ ✷✾

slide-34
SLIDE 34

❋✉♥❝t✐♦♥❛❧ ❘●

■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥

❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

∂t =

−1

+ ∂t =

−1

+

❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ = ❤❦ ¯ ψψ = ❤❦σ

✶✸ ✴ ✷✾

slide-35
SLIDE 35

❘❡s✉❧ts

❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠

✶✹ ✴ ✷✾

slide-36
SLIDE 36

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

µ ❉❡♣❡♥❞❡♥❝❡

U

  • k 0

U

  • k , Z
  • k 0

U

  • k , h
  • k 0

U

  • k , Z
  • k, h
  • k 0

Σ T 0

50 100 150 200 250 300ΜMeV 20 40 60 80 100 MeV

✶✺ ✴ ✷✾

slide-37
SLIDE 37

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

❚❡♠♣❡r❛t✉r❡ ❉❡♣❡♥❞❡♥❝❡

U

  • k 0

U

  • k , Z
  • k 0

U

  • k , h
  • k 0

U

  • k , Z
  • k, h
  • k 0

Μ 0

50 100 150 200 250 300TMeV 10 20 30 40 50 60 70 ΣMeV

U

  • k 0

U

  • k , Z
  • k 0

U

  • k , h
  • k 0

U

  • k , Z
  • k, h
  • k 0

Μ 200 MeV

50 100 150 200TMeV 20 40 60 80 MeV

✶✻ ✴ ✷✾

slide-38
SLIDE 38

❘❡s✉❧ts

Pr❡❝♦♥❞❡♥s❛t✐♦♥

40 80 120 160 200 400 600 800 1000 [MeV] k [MeV] m - µ <∆> <∆> T = 220 MeV T = 205 MeV T = 190 MeV T = 175 MeV 20 40 60 1 10 100 1000

← µ = ✷✸✵ ▼❡❱

12 10 8 6 4 2 0.05 0.10 0.15

t ρ0,k T < Tc Tc < T < Tpc

❬❇♦❡tt❝❤❡r ❡t ❛❧ ✷✵✶✷❪ ✶✼ ✴ ✷✾

slide-39
SLIDE 39

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠

❬❘❛♥❞❡r✐❛✱ ◆❛t✉r❡ P❤②s✐❝s ✻ ✭✷✵✶✵✮❪ ✶✽ ✴ ✷✾

slide-40
SLIDE 40

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠

✶✾ ✴ ✷✾

slide-41
SLIDE 41

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠

✶✾ ✴ ✷✾

slide-42
SLIDE 42

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠

✶✾ ✴ ✷✾

slide-43
SLIDE 43

❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮

❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠

Ψ Ψ 0 Ψ Ψ 0 U

  • k 0

U

  • k , Z
  • k 0

U

  • k , h
  • k 0

U

  • k , Z
  • k, h
  • k 0

50 100 150 200 250 300ΜMeV 50 100 150 200 250 TMeV

ΨΨ 0 BEC BCS ΨΨ 0 U

  • k 0

U

  • k , Z
  • k 0

U

  • k , h
  • k 0

U

  • k , Z
  • k, h
  • k 0

50 100 150 200 250 300 ΜMeV 50 100 150 200 250 TMeV

✷✵ ✴ ✷✾

slide-44
SLIDE 44

❘❡s✉❧ts

❈❤✐r❛❧ ▲✐♠✐t ❝ = ✵✱ ♠π = ✵

✷✶ ✴ ✷✾

slide-45
SLIDE 45

❘❡s✉❧ts

❲❛✈❡ ❋✉♥❝t✐♦♥ ❘❡♥♦r♠❛❧✐③❛t✐♦♥

❩∆ ❩∆,❱❛❝ ❩φ ❩φ,❱❛❝

✷✷ ✴ ✷✾

slide-46
SLIDE 46

❘❡s✉❧ts

▼❛ss ❙♣❡❝tr✉♠

100 200 300 400 500 50 100 150 200 250 300 [MeV] T [MeV] mσ mψ mπ , m∆

← µ = ✵

100 200 300 400 500 600 700 50 100 150 200 250 300 350 400 [MeV] µ [MeV] mσ m∆* m∆ mπ mσ

~

m∆

~*

m∆

~

❞❡t●❇♦s♦♥(♣✵, ♣ = ✵) = ✵ ❚ = ✵ →

✷✸ ✴ ✷✾

slide-47
SLIDE 47

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩

❦✱ ❩ ❦✱ ❩ ❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞

❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-48
SLIDE 48

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩

❦✱ ❩ ❦✱ ❩ ❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞

❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-49
SLIDE 49

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩

❦✱ ❩ ❦✱ ❩ ❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞

❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-50
SLIDE 50

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞

❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-51
SLIDE 51

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞

❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-52
SLIDE 52

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞ φ✹ ❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-53
SLIDE 53

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞ φ✹ ❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-54
SLIDE 54

❙✉♠♠❛r②✴❖✉t❧♦♦❦

❙✉♠♠❛r②

♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦

❖✉t❧♦♦❦

▲P❆ ❜❡②♦♥❞ φ✹ ❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉

✷✹ ✴ ✷✾

slide-55
SLIDE 55

❇❛❝❦✉♣ ❙❧✐❞❡s

✷✺ ✴ ✷✾

slide-56
SLIDE 56

❇❛❝❦✉♣ ❙❧✐❞❡s

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙②♠♠❡tr② ❣r♦✉♣

  • ❡♥❡r❛t♦rs

Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲

❙♣ ✹ ✶✵ ✺ ● ✭ ✱ ✱

✷✮

❙❯ ✷ ▲ ❙❯ ✷ ❘ ✻ ✹ P● ✭ ✱ ✮✱ ✶ ● ✭

✷✮

♠ ❙❯ ✷ ❱ ✸ ✸ PP● ✭ ✮✱ ✶ P● ✭ ✮✱ ✶● ✭

✷✮

✷✻ ✴ ✷✾

slide-57
SLIDE 57

❇❛❝❦✉♣ ❙❧✐❞❡s

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙②♠♠❡tr② ❣r♦✉♣

  • ❡♥❡r❛t♦rs

Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲

ψψ = ∆✶ ↓

❙♣(✹) ✶✵ ✺ ● ✭ π✱ σ✱ ∆✷✮ ❙❯ ✷ ▲ ❙❯ ✷ ❘ ✻ ✹ P● ✭ ✱ ✮✱ ✶ ● ✭

✷✮

♠ ❙❯ ✷ ❱ ✸ ✸ PP● ✭ ✮✱ ✶ P● ✭ ✮✱ ✶● ✭

✷✮

✷✻ ✴ ✷✾

slide-58
SLIDE 58

❇❛❝❦✉♣ ❙❧✐❞❡s

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙②♠♠❡tr② ❣r♦✉♣

  • ❡♥❡r❛t♦rs

Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲

ψψ = ∆✶ ↓

❙♣(✹) ✶✵ ✺ ● ✭ π✱ σ✱ ∆✷✮ µ ↓ ❙❯(✷)▲ × ❙❯(✷)❘ ✻ ✹ P● ✭ π✱ σ✮✱ ✶ ● ✭∆✷✮ ♠ ❙❯ ✷ ❱ ✸ ✸ PP● ✭ ✮✱ ✶ P● ✭ ✮✱ ✶● ✭

✷✮

✷✻ ✴ ✷✾

slide-59
SLIDE 59

❇❛❝❦✉♣ ❙❧✐❞❡s

❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪

❙②♠♠❡tr② ❣r♦✉♣

  • ❡♥❡r❛t♦rs

Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲

ψψ = ∆✶ ↓

❙♣(✹) ✶✵ ✺ ● ✭ π✱ σ✱ ∆✷✮ µ ↓ ❙❯(✷)▲ × ❙❯(✷)❘ ✻ ✹ P● ✭ π✱ σ✮✱ ✶ ● ✭∆✷✮ ♠ψ ↓ ❙❯(✷)❱ ✸ ✸ PP● ✭ π✮✱ ✶ P● ✭σ✮✱ ✶● ✭∆✷✮

✷✻ ✴ ✷✾

slide-60
SLIDE 60

❇❛❝❦✉♣ ❙❧✐❞❡s

❍❛❞r♦♥✐③❛t✐♦♥

◆❏▲ ♠♦❞❡❧

λψ g g g g

= ⇒ λψ

  • ( ¯

ψψ)✷ − ( ¯ ψγ✺ τψ)✷ −

  • ψ❚ǫψ

❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥

❤ ✶ ✷♠✷

✇✐t❤ ❤✷ ✷♠✷ ❛♥❞ ❊♦▼

✷✼ ✴ ✷✾

slide-61
SLIDE 61

❇❛❝❦✉♣ ❙❧✐❞❡s

❍❛❞r♦♥✐③❛t✐♦♥

◆❏▲ ♠♦❞❡❧

λψ g g g g

= ⇒ λψ

  • ( ¯

ψψ)✷ − ( ¯ ψγ✺ τψ)✷ −

  • ψ❚ǫψ

❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥ λψ( ¯ ψψ)✷ = ❤σ ¯ ψψ + ✶ ✷♠✷σ✷ ✇✐t❤ λψ = − ❤✷ ✷♠✷ ❛♥❞ ❊♦▼(σ) → σ = ¯ ψψ

✷✼ ✴ ✷✾

slide-62
SLIDE 62

❇❛❝❦✉♣ ❙❧✐❞❡s

❍❛❞r♦♥✐③❛t✐♦♥

◆❏▲ ♠♦❞❡❧

λψ g g g g

= ⇒ λψ

  • ( ¯

ψψ)✷ − ( ¯ ψγ✺ τψ)✷ −

  • ψ❚ǫψ

❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥ λψ( ¯ ψψ)✷ = ❤σ ¯ ψψ + ✶ ✷♠✷σ✷ ✇✐t❤ λψ = − ❤✷ ✷♠✷ ❛♥❞ ❊♦▼(σ) → σ = ¯ ψψ = ✵ → ♠❛sst❡r♠ ♠q = ❤σ

✷✽ ✴ ✷✾

slide-63
SLIDE 63

❇❛❝❦✉♣ ❙❧✐❞❡s

❍❛❞r♦♥✐③❛t✐♦♥

▲❛r❣❡ ❢♦✉r✲❢❡r♠✐♦♥ ❝♦✉♣❧✐♥❣ ❧✐♠✐t

λψ→∞

= ⇒ ...

▲❛r❣❡ ❤❛❞r♦♥ ♠❛ss ❧✐♠✐t ❉②♥❛♠✐❝❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ◗✉❛r❦s ✱ ●❧✉♦♥s ❆ ✱ ♠❡s♦♥s ✱ ❜❛r②♦♥s ✱ ❆

✷✾ ✴ ✷✾

slide-64
SLIDE 64

❇❛❝❦✉♣ ❙❧✐❞❡s

❍❛❞r♦♥✐③❛t✐♦♥

▲❛r❣❡ ❢♦✉r✲❢❡r♠✐♦♥ ❝♦✉♣❧✐♥❣ ❧✐♠✐t

λψ→∞

= ⇒ ...

▲❛r❣❡ ❤❛❞r♦♥ ♠❛ss ❧✐♠✐t

m→∞

= ⇒

❉②♥❛♠✐❝❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ◗✉❛r❦s ✱ ●❧✉♦♥s ❆ ✱ ♠❡s♦♥s ✱ ❜❛r②♦♥s ✱ ❆

✷✾ ✴ ✷✾

slide-65
SLIDE 65

❇❛❝❦✉♣ ❙❧✐❞❡s

❍❛❞r♦♥✐③❛t✐♦♥

▲❛r❣❡ ❢♦✉r✲❢❡r♠✐♦♥ ❝♦✉♣❧✐♥❣ ❧✐♠✐t

λψ→∞

= ⇒ ...

▲❛r❣❡ ❤❛❞r♦♥ ♠❛ss ❧✐♠✐t

m→∞

= ⇒

❉②♥❛♠✐❝❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ◗✉❛r❦s ψ✱ ●❧✉♦♥s ❆ = ⇒ ψ✱ ♠❡s♦♥s φ ∼ ¯ ψψ✱ ❜❛r②♦♥s ∆ ∼ ψψ✱ ❆

✷✾ ✴ ✷✾