ssstr t - - PowerPoint PPT Presentation
ssstr t r r qt tt r r
❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ▼✉♥✐❝❤
P✉❜❧✐❝❛t✐♦♥s✿ P❘❈ ✽✾✱ ✵✻✹✵✵✾ ✭✷✵✶✹✮❀ P❘❈ ✾✷✱ ✵✶✺✽✵✶ ✭✷✵✶✺✮❀ P❘❈ ✾✸✱ ✵✺✺✽✵✷ ✭✷✵✶✻✮
❲♦r❦ s✉♣♣♦rt❡❞ ✐♥ ♣❛rt ❜② ❉❋● ❛♥❞ ◆❙❋❈ ✭❈❘❈ ✶✶✵✮
❝❤❛♥❞r❛✳❤❛r✈❛r❞✳❡❞✉
✇✇✇✷✳❛str♦✳♣✉❝✳❝❧ ❘♦ss✇♦❣✿ P❤✐❧✳ ❚r❛♥s✳ ❘✳ ❙♦❝✳ ❆ ✸✼✶ ✭✷✵✶✸✮
→ ❊♦❙ ♥✉♠❡r✐❝❛❧ ✐♥♣✉t ❢♦r s✐♠✉❧❛t✐♦♥s ♦❢ s✉♣❡r♥♦✈❛❡ ❛♥❞ ♥❡✉tr♦♥✲st❛r ♠❡r❣❡rs ◆♦✈❡❧ ❞❡✈❡❧♦♣♠❡♥ts ✐♥ t❤❡♦r❡t✐❝❛❧ ♥✉❝❧❡❛r ♣❤②s✐❝s✿ ❝❤✐r❛❧ ❊❋❚✱ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❣r♦✉♣
❝❤✐r❛❧ ❊❋❚✿ ❣❡♥❡r❛❧ ❧♦✇✲❡♥❡r❣② ❡✛❡❝t✐✈❡ ✜❡❧❞ t❤❡♦r② ❝♦♥s✐st❡♥t ✇✐t❤ s②♠♠❡tr✐❡s ♦❢ ◗❈❉✱ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✿ ♥✉❝❧❡♦♥s ✫ ♣✐♦♥s s②st❡♠❛t✐❝ ❤✐❡r❛r❝❤② ♦❢ ♥✉❝❧❡❛r ✐♥t❡r❛❝t✐♦♥s ❝♦♥tr♦❧❧❡❞ ❜② ❡①♣❛♥s✐♦♥ ♣❛r❛♠❡t❡r ◗/Λχ = s♦❢t s❝❛❧❡/❤❛r❞ s❝❛❧❡✱ ✇❤❡r❡ Λχ ∼ ✶ ●❡❱
NN Force 3N Force LO (Q/Λχ)0 NLO (Q/Λχ)2 NNLO (Q/Λχ)3 + . . . + . . . + . . . N3LO (Q/Λχ)4 4N Force + . . . not considered here cE not considered here cD c1, c3, c4
r❡str✐❝t r❡s♦❧✉t✐♦♥ ✈✐❛ ❯❱ ❝✉t♦✛ Λ < Λχ ✐♥ ♠♦♠❡♥t✉♠ s♣❛❝❡ ▲❊❈s ❝✐(Λ) ✜①❡❞ ❜② ❤✐❣❤✲♣r❡❝✐s✐♦♥ ✜ts t♦ ❢❡✇✲♥✉❝❧❡♦♥ ♦❜s❡r✈❛❜❧❡s → ◆◆ ❛♥❞ ✸◆ ♥✉❝❧❡❛r ♣♦t❡♥t✐❛❧s ❢♦r ♠❛♥②✲❜♦❞② ❝❛❧❝✉❧❛t✐♦♥s ◆✉❝❧❡❛r ♣♦t❡♥t✐❛❧s ❛r❡ ♥♦t ✉♥✐q✉❡✦ → ✉♥❝❡rt❛✐♥t② ❡st✐♠❛t✐♦♥s ✭❜✉t✿ ❛rt✐❢❛❝ts ♣♦ss✐❜❧❡✮ ▲♦✇✲♠♦♠❡♥t✉♠ ♣♦t❡♥t✐❛❧s Λ ✹✺✵ ▼❡❱✿ ▼❇P❚ ❜❡❝♦♠❡s ✈❛❧✐❞ ❛♣♣r♦❛❝❤✦
❧✐♥❦❡❞✲❝❧✉st❡r ❡①♣❛♥s✐♦♥ ✭✏●♦❧❞st♦♥❡ ❡①♣❛♥s✐♦♥✑✮ ❢♦r ❣r♦✉♥❞ st❛t❡ ❡♥❡r❣② ✭❚ = ✵✮ t❡①t❜♦♦❦ ❛♣♣r♦❛❝❤ ❛t ✜♥✐t❡ ❚✿ ❡①♣❛♥s✐♦♥ ♦❢ ❣r❛♥❞✲❝❛♥♦♥✐❝❛❧ ♣♦t❡♥t✐❛❧ Ω(❚, µ) = Ω✵(❚, µ) + Ω✶(❚, µ) + Ω✷(❚, µ) + K❛♥♦♠(❚, µ) + . . . ❇✉t✿ ♥♦t ❝♦♥s✐st❡♥t ✇✐t❤ ●♦❧❞st♦♥❡ ❡①♣❛♥s✐♦♥✱ ❝❛♥♥♦t ❞❡s❝r✐❜❡ s♣✐♥♦❞❛❧ ✐♥st❛❜✐❧✐t② Pr♦♣❡r ✜♥✐t❡✲t❡♠♣❡r❛t✉r❡ ▼❇P❚✿ ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ❡①♣❛♥s✐♦♥ ❢♦r ❢r❡❡ ❡♥❡r❣② ✏♥❛✐✈❡✑ ❛♣♣r♦❛❝❤✿ ❧✐♥❦❡❞✲❝❧✉st❡r ❡①♣❛♥s✐♦♥ ♦❢ ❢r❡❡ ❡♥❡r❣②❀ ❞♦❡s ♥♦t ✇♦r❦ ❜❡❝❛✉s❡ ❝❛♥♦♥✐❝❛❧✲❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡s ❛r❡ ❝♦♥str❛✐♥❡❞ ✭✜①❡❞ ◆✮ ✐♥st❡❛❞✿ ❡✈❛❧✉❛t❡ ❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡s ✈✐❛ ▲❡❣❡♥❞r❡ tr❛♥s❢♦r♠ ♦❢ ❝✉♠✉❧❛♥ts❀ ❣✐✈❡s ❋(❚, ˜ µ) = ❋✵(❚, ˜ µ) + ❆✶(❚, ˜ µ) + ❆✷(❚, ˜ µ) + K❛♥♦♠(❚, ˜ µ) + K❝♦rr(❚, ˜ µ) + . . . ˜ µ ✜①❡❞ ❜② ρ(❚, ˜ µ) = ∂❋✵/∂ ˜ µ → ❝♦♥s✐st❡♥t ✇✐t❤ ●♦❧❞st♦♥❡ ❡①♣❛♥s✐♦♥✦ ❛❞❞✐t✐♦♥❛❧ ❝♦♥tr✐❜✉t✐♦♥s ❢r♦♠ ✉♥❧✐♥❦❡❞ ❞✐❛❣r❛♠s K❝♦rr✱ r❡♥♦r♠❛❧✐③❡ ˜ µ ❝❛♥♦♥✐❝❛❧ s❡r✐❡s ❝❛♥ ❜❡ ❛❧s♦ ❞❡r✐✈❡❞ ✈✐❛ r❡♦r❣❛♥✐③❛t✐♦♥ ♦❢ ❣r❛♥❞✲❝❛♥♦♥✐❝❛❧ s❡r✐❡s ✭❑♦❤♥✲▲✉tt✐♥❣❡r ♠❡t❤♦❞✮✱ ❜✉t ❡q✉✐✈❛❧❡♥❝❡ ✐s ♦♥❧② ❢♦r♠❛❧ ✭❛s②♠♣t♦t✐❝ s❡r✐❡s✦✮ ▼❡❛♥✲✜❡❧❞ ❜❡♥❝❤♠❛r❦ ❢✉❧❧② r❡♥♦r♠❛❧✐③❡❞ ▼❇P❚✿ ❙❈❍❋ ❧❛r❣❡ K❛♥♦♠ ✭ε r❡♥♦r♠❛❧✐③❛t✐♦♥✮✱ ❜✉t K❛♥♦♠ + K❝♦rr ✭ε ❛♥❞ ˜ µ r❡♥♦r♠❛❧✐③❛t✐♦♥✮ ✐s s♠❛❧❧ s♣✐♥♦❞❛❧ r❡❣✐♦♥ ♦♥❧② ❢r♦♠ ❝❛♥♦♥✐❝❛❧ ❛♥❞ ❢✉❧❧② r❡♥♦r♠❛❧✐③❡❞ ▼❇P❚
5 0.05 0.1 0.15 0.2 0.25 F [MeV] ρ [fm-3]
T=15 MeV
free HF (G.C.) HF+K (G.C.) HF HF+K SCHF
0.05 0.1 0.15 0.2 0.25
Kanom. Kanom.+Kcorr.
■s♦s♣✐♥✲s②♠♠❡tr✐❝ ♥✉❝❧❡❛r ♠❛tt❡r✿ δ := (ρ♥ − ρ♣)/ρ = ✵✱ ❨ := ρ♣/ρ = ✶/✷
0.05 0.1 0.15 0.2 0.25 0.3 F _ [MeV] ρ [fm-3]
T=0 MeV T=25 MeV
n3lo414 n3lo450 n3lo500 VLK21 VLK23
(a) NN first order, no 3N
0.05 0.1 0.15 0.2 0.25 0.3 F _ [MeV] ρ [fm-3]
T=0 MeV T=25 MeV
n3lo414 n3lo450 n3lo500 VLK21 VLK23
(b) NN second order, no 3N
0.05 0.1 0.15 0.2 0.25 0.3 F _ [MeV] ρ [fm-3]
T=0 MeV T=25 MeV
n3lo414 n3lo450 n3lo500 VLK21 VLK23
(c) NN second order, 3N first order
0.05 0.1 0.15 0.2 0.25 0.3 F _ [MeV] ρ [fm-3]
T=0 MeV T=25 MeV
n3lo414 n3lo450 n3lo500 VLK21 VLK23
(d) NN second order, 3N second order
♥✸❧♦✹✶✹ ✫ ♥✸❧♦✹✺✵✿ ❣♦♦❞ ♣❡rt✉r❜❛t✐✈❡ ❜❡❤❛✈✐♦r ❋✶ > ❋✷ ≫ ❋✸ ✭t❤✐r❞ ♦r❞❡r✿ ❍♦❧t ✫ ❑❛✐s❡r✿ ✶✻✶✷✳✵✹✸✵✾ ✭✷✵✶✻✮✮ ♥✸❧♦✺✵✵✿ ❧❡ss ♣❡rt✉r❜❛t✐✈❡ ✭❋✶,◆◆ ✫ ❋✷,◆◆ s✐♠✐❧❛r ♠❛❣♥✐t✉❞❡✮ ❱▲❑✷✶ ✫ ❱▲❑✷✸✿ ◆◆ ♣❡rt✉r❜❛t✐✈❡✱ ❜✉t ❧❛r❣❡ ❝♦♥tr✐❜✉t✐♦♥s ❢r♦♠ ✸◆ ♣♦t❡♥t✐❛❧
■s♦s♣✐♥✲s②♠♠❡tr✐❝ ♥✉❝❧❡❛r ♠❛tt❡r✿ δ := (ρ♥ − ρ♣)/ρ = ✵✱ ❨ := ρ♣/ρ = ✶/✷
0.05 0.1 0.15 0.2 0.25 0.3 F _ [MeV] ρ [fm-3]
T=0 MeV T=25 MeV
n3lo414 n3lo450 n3lo500 VLK21 VLK23
1 2 3 4 5 6 7 0.05 0.1 0.15 0.2 0.25 0.3 P [MeV fm-3] ρ [fm-3]
T=0 MeV T=25 MeV
n3lo414 n3lo450 n3lo500 VLK21 VLK23
❡♠♣✐r✐❝❛❧ s❛t✉r❛t✐♦♥ ♣♦✐♥t✿ ♥✸❧♦✹✶✹✱ ♥✸❧♦✹✺✵✱ ♥✸❧♦✺✵✵✱ ❱▲❑✷✶✱ ❱▲❑✷✸ ❱▲❑✷✶ ✫ ❱▲❑✷✸ r✉❧❡❞ ♦✉t ❜② t❤❡r♠♦❞②♥❛♠✐❝s ✭♣r❡ss✉r❡ ✐s♦t❤❡r♠ ❝r♦ss✐♥❣✮
P✉r❡ ♥❡✉tr♦♥ ♠❛tt❡r ✭δ = ✶✱ ❨ = ✵✮ ❙②♠♠❡tr② ❢r❡❡ ❡♥❡r❣② ¯ ❋s②♠
20 0.05 0.1 0.15 0.2 F _ (δ=1) [MeV] ρ [fm-3]
free energy
n3lo414 n3lo450 VEoS T=0 MeV T=10 MeV T=15 MeV T=20 MeV T=25 MeV E _
sym=F
_
sym [MeV]
ρ [fm-3]
T=0 MeV
n3lo414 n3lo450 Drischler et al. Akmal et al. IAS+NS 10 20 30 0.05 0.1 0.15 0.2
■❆❙✫◆❙✿ ❉❛♥✐❡❧❡✇✐❝③ ✫ ▲❡❡✱ ◆P❆ ✾✷✷ ✭✷✵✶✹✮
❋s②♠ := ¯ ❋(δ = ✶) − ¯ ❋(δ = ✵)
P❤❛s❡ s♣❛❝❡ ❝♦✈❡r❡❞ ✐♥ s✉♣❡r♥♦✈❛❡✿ ❚ ∼ ✵ − ✺✵ ▼❡❱✱ ρ ∼ ✵ − ✻ ρs❛t✱ δ ∼ ✵ − ✶ ❈❤✐r❛❧ ✈s✳ ♣❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧ ❊♦❙ ✭❚ = ✵✱ δ = ✶✮
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10 10
−1
10 10
1
10
2
Baryon density, nB [fm−3] Temperature, T [MeV] Ye 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 6 7 8 9 10 11 12 13 14 15
Baryon density, log10(ρ [g/cm3])
❋✐s❝❤❡r ❡t ❛❧✳✿ ❆str♦♣❤②s✳ ❏✳ ❙✉♣♣❧✳ ✶✾✹ ✭✷✵✶✶✮
0.05 0.1 0.15 n [fm-3] 5 10 15 20 E/N [MeV]
this work LS 180 LS 220 LS 375 FSU2.1 NL3 TM1 DD2 SFHo SFHx ❑r✉❡❣❡r✱ ❚❡✇s ❡t ❛❧✳✿ P❘❈ ✽✽ ✭✷✵✶✸✮
❙t❡✐♥❡r✿ P❘❈ ✼✹ ✭✷✵✵✻✮
♥=✵ ❆✷♥(❚, ρ) δ✷♥,
✶ (✷♥)! ∂✷♥ ¯ ❋(❚,ρ,δ) ∂δ✷♥
10 20 30 40 0.05 0.1 0.15 0.2 0.25 0.3 A _
2 (MeV)
ρ (fm-3)
n3lo414 n3lo450
2 0.05 0.1 0.15 0.2 0.25 0.3 A _
4 (MeV)
ρ (fm-3) 10-1 100 101 102 103 104 0.05 0.1 0.15 0.2 0.25 0.3 A _
6 (MeV)
ρ (fm-3)
T=0 MeV T=2 MeV T=3 MeV T=4 MeV T=5 MeV T=7 MeV T=15 MeV T=25 MeV
0.05 0.1 0.15 0.2 0.25 0.3
2 4 F _
sym-A
_
2 (MeV)
ρ (fm-3)
noninteracting
10 20 0.25 0.5 0.75 1 F _ (MeV) δ
T=3 MeV ρ=0.20 fm-3
F _ (T,ρ,δ) F _
[2](T,ρ,δ)
F _
[4](T,ρ,δ)
F _
[6](T,ρ,δ) 0.02 0.04 0.06 0.08 0.05 0.1 0.15 0.2
|∆F _ | (keV) δ
T (MeV) ρ (fm-3) 2 4 6 0.05 0.1 0.15 0.2 0.25 0.3
divergent convergent
✽
✷❇
✷❇ ♥✶♥✷¯ ♥✸¯ ♥✹−¯ ♥✶¯ ♥✷♥✸♥✹ ε✸+ε✹−ε✶−ε✷
❚→✵
❋(❚ = ✵, ρ, δ) = ❆✵(✵, ρ) + ❆✷(✵, ρ) δ✷ +
∞
❆✷♥,r❡❣(ρ) δ✷♥+
∞
❆✷♥,❧♦❣(ρ) δ✷♥ ❧♥ |δ|
10 20 30 0.25 0.5 0.75 1 F _ (MeV) δ
T=0 MeV ρ=0.25 fm-3
F _ (T,ρ,δ) F _
[2](T,ρ,δ)
F _
[4,nonlog](T,ρ,δ)
F _
[4,log](T,ρ,δ) 0.1 0.2 0.3 0.4 0.25 0.5 0.75 1
|∆F _ | (MeV) δ
27.5 28 28.5 0.98 0.99 1
F _ (MeV)
∂ρ✐ ∂ρ❥
5 10 15 20 0.2 0.4 0.6 0.8 1 T [MeV] δ
TκT(δ) Tc(δ)
n3lo414 n3lo450 5 10 15 20 0.2 0.4 0.6 0.8 1 T [MeV] δ
self-bound liquid TFP(δ) TSP(δ)
n3lo414 n3lo450
❡①tr❛♣♦❧❛t✐♦♥ ♥♦t ✇❡❧❧✲❜❡❤❛✈❡❞ ❢♦r ♠❛♥② ♣❛r❛♠❡tr✐③❛t✐♦♥s ✭❡✳❣✳✱ ✏❙♦♠♠❡r❢❡❧❞✑✲♣♦❧②♥♦♠✐❛❧ ♥ α♥❚✷♥✮ ❙◆▼✿ ❝♦♠♣✉t❡❞ ❞❛t❛ ❤❛s t❡♥❞❡♥❝② t♦✇❛r❞s ♣r❡ss✉r❡ ✐s♦t❤❡r♠ ❝r♦ss✐♥❣ P◆▼✿ ❛♣♣r♦①✐♠❛t❡❧② ❚✲✐♥❞❡♣❡♥❞❡♥t ❢♦r ❚ ✷✺ ▼❡❱ ❛♥❞ ρ ✷ ρs❛t
❆✷♥(❚, ρ) ¯ ❆✷♥(❚, ρ) ≃ ¯ ❆◆,∆δ
✷♥
(❚, ρ) = ✶ (✷♥)! (∆δ)✷♥
◆
ω◆,❦
✷♥
¯ ❋(❚, ρ, ❦∆δ).
❋♦r♥❜❡r❣✿ ▼❛t❤✳❈♦♠♣ ✺✶ ✭✶✾✽✽✮ 0.5 1 0.05 0.1 0.15 AN,∆δ
6 (MeV)
∆δ T=5 MeV, ρ=0.15 fm-3
1,NN 1,3N 1,DDN 2,NN 2,NN (iterat) 2,NN+DDNN
67 68 69 70
AN,∆δ
6 (MeV)
ln(∆δ) T=4 MeV, ρ=0.30 fm-3
0.05 0.1 0.15 AN,∆δ
6 (MeV)
∆δ T=15 MeV, ρ=0.15 fm-3
AN,∆δ
8 (GeV)
ln(∆δ) T=4 MeV, ρ=0.30 fm-3
st❡♣s✐③❡ ✭∆δ✮ ❛♥❞ ❣r✐❞ ❧❡♥❣t❤ ✭◆✮ ✈❛r✐❛t✐♦♥s ❛s ❛❝❝✉r❛❝② ❝❤❡❝❦s s②st❡♠❛t✐❝❛❧❧② ✐♥❝r❡❛s❡ ♣r❡❝✐s✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ r♦✉t✐♥❡
✜♥✐t❡ ❞✐✛❡r❡♥❝❡s ♦❢ ③❡r♦✲t❡♠♣❡r❛t✉r❡ ❧♦❣❛r✐t❤♠✐❝ s❡r✐❡s ✭∼ δ✷♥≥✹ ❧♥ |δ|✮✿ ¯ ❆◆,∆δ
✹
=¯ ❆✹,r❡❣ + ❈ ✹
✶ (◆)¯
❆✹,❧♦❣ + ¯ ❆✹,❧♦❣ ❧♥(∆δ) + ❈ ✹
✷ (◆)¯
❆✻,❧♦❣∆δ✷ + O(∆δ✹), ✭✵✳✶✮ ¯ ❆◆,∆δ
✻
=¯ ❆✻,r❡❣ + ❈ ✻
✶ (◆)¯
❆✹,❧♦❣∆δ−✷ + ¯ ❆✻,❧♦❣ ❧♥(∆δ) + ❈ ✻
✷ (◆)¯
❆✻,❧♦❣ + O(∆δ✷). ✭✵✳✷✮ ❡①tr❛❝t ❧❡❛❞✐♥❣ ❧♦❣❛r✐t❤♠✐❝ t❡r♠ ✈✐❛✿ Ξ✹(◆✶, ◆✷) := ¯ ❆◆✶,∆δ
✹
− ¯ ❆◆✷,∆δ
✹
❈ ✶
✹ (◆✶) − ❈ ✶ ✹ (◆✷) ≃ ¯
❆✹,❧♦❣, ✭✵✳✸✮ Ξ✻(◆✶, ◆✷) := ¯ ❆◆✶,∆δ
✻
− ¯ ❆◆✷,∆δ
✻
❈ ✶
✻ (◆✶) − ❈ ✶ ✻ (◆✷) ∆δ✷ ≃ ¯
❆✹,❧♦❣, ✭✵✳✹✮ ❜❡♥❝❤♠❛r❦ ❛❣❛✐♥st ❛♥❛❧②t✐❝❛❧ r❡s✉❧ts ❢♦r ❙✲✇❛✈❡ ❝♦♥t❛❝t ✐♥t❡r❛❝t✐♦♥
0.1 0.15 0.2 0.25 AN,∆δ
4 (MeV)
∆δ
N=3 (Γnp) N=4 (Γnp) N=3 (n3lo414) N=4 (n3lo414)
0.2 0.1 0.15 0.2 0.25
np (regularized)
2 4 6 8 0.1 0.15 0.2 0.25 AN,∆δ
6 (MeV)
∆δ
N=4 (Γnp) N=5 (Γnp) N=4 (n3lo414) N=5 (n3lo414)
0.5 0.1 0.15 0.2 0.25
np (regularized)
❲❤❛t ❛❜♦✉t ♣❛✐r✐♥❣❄ → ♣❡rt✉r❜❛t✐♦♥ s❡r✐❡s ❛❜♦✉t ❇❈❙ ❣r♦✉♥❞ st❛t❡ ✭∼ ❇♦❣♦❧✐✉❜♦✈✮ ❇❈❙ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✿ ♥❇❈❙
❦
= ✶ ✷
❦ + ξ✷ ❦
−✶/✷ , ¯ ♥❇❈❙
❦
= ✶ ✷
❦ + ξ✷ ❦
−✶/✷ → ❝♦♠♣❛r❡ ✇✐t❤ ✜♥✐t❡✲t❡♠♣❡r❛t✉r❡ ❋❡r♠✐✲❉✐r❛❝ ❞✐str✐❜✉t✐♦♥s
0.25 0.5 0.75 1 0.25 0.5 0.75 1 1.25 1.5 nk k (fm-1) T=0 MeV T=3 MeV T=5 MeV T=10 MeV ∆k=1 MeV ∆k=3 MeV ∆k=5 MeV
→ ❡①♣❛♥s✐♦♥ ❞✐✈❡r❣❡♥t ❛❧s♦ ❢♦r ❇❈❙ ♣❡rt✉r❜❛t✐♦♥ s❡r✐❡s✱ s✐♠✐❧❛r t♦ ❧♦✇✲❚ r❡s✉❧ts ❖✈❡r❛❧❧✿ ❧♦❣❛r✐t❤♠✐❝ t❡r♠s ❢♦r ❚ → ✵ ∧ {◆, Ω} → ∞ ∧ ∆❦ → ✵ ❞✐✈❡r❣❡♥t ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ ✐♥ t❤❡ r❡❣✐♦♥ ✏❝❧♦s❡ ❡♥♦✉❣❤✑ t♦ t❤❡s❡ ❧✐♠✐ts
❑❛✐s❡r✿ ❊P❏❆ ✹✽ ✭✷✵✶✹✮
¯ ❊✵,r❡s✉♠(❦♥
❋ , ❦♣ ❋ ) = −
✷✹ π▼
❋ )✸ + (❦♣ ❋ )✸
r❡s✉♠(❛s) + Γ ♣♣ r❡s✉♠(❛s) + Γ ♥♣ r❡s✉♠(❛s) + ✸Γ ♥♣ r❡s✉♠(❛t)
Γ ♥♥✴♣♣
r❡s✉♠ (❛s) = ✶
❞s s✷ √
✶−s✷
❞κ κ (❦♥✴♣
❋
)✺ ❛r❝t❛♥ ■(s, κ) (❛s❦♥✴♣
❋
)−✶ + π−✶❘(s, κ) Γ ♥♣
r❡s✉♠(❛s/t) = (❦♥ ❋ +❦♣ ❋ )/✷
❞P P✷
q♠❛①
❞q q ❛r❝t❛♥ Φ(P, q, ❦♥
❋ , ❦♣ ❋ )
(❛s/t)−✶ + (✷π)−✶ ❦♥
❋ ❘( P ❦♥ ❋
,
q ❦♥ ❋
) + ❦♣
❋ ❘( P ❦♣ ❋
,
q ❦♣ ❋
)
❋ , ❦♣ ❋ ) ❛r❡ ❣✐✈❡♥ ❜②
■(s, κ) =κ Θ(✶ − s − κ) + ✶ − s✷ − κ✷ ✷s Θ(s + κ − ✶), ❘(s, κ) =✷ + ✶ − (s − κ)✷ ✷s ❧♥ ✶ + s + κ |✶ − s − κ| + ✶ − s✷ − κ✷ ✷s ❧♥ ✶ + s − κ ✶ − s + κ Φ(P, q, ❦♥
❋ , ❦♣ ❋ ) =
q ❢♦r P + q < ❦♣
❋ (❦♣ ❋ )✷−(P−q)✷ ✹P
❢♦r ❦♣
❋ < P + q < ❦♥ ❋ ∧ |P − q| < ❦♣ ❋ (❦♥ ❋ )✷+(❦♣ ❋ )✷−✷(P✷−q✷) ✹P
❢♦r ❦♥
❋ < P + q ∧ P✷ + q✷ < (❦♥ ❋ )✷+(❦♣ ❋ )✷ ✷
▲❛❞❞❡r r❡s✉♠♠❛t✐♦♥✿ ✐s♦s♣✐♥✲❛s②♠♠❡tr② ❞❡♣❡♥❞❡♥❝❡
5 10 15 20 25 0.05 0.1 0.15 0.2 0.25 A _N,∆δ
4 (MeV)
∆δ
N=3 N=4 N=3 (A _N,∆δ
2 )
N=4 (A _N,∆δ
2 ) 2 4 6 0.05 0.1 0.15 0.2 0.25
np (regularized)
0.1 0.2
nn+pp
0.25 0.5 0.75 1 E _
0,resum
(MeV) δ
exact parabolic quadratic quartic+log
0.5 1 0.99 1 1 2 3 4 5 0.1 0.2 0.3
∆E _
0,resum
(keV) δ
❋✐♥✐t❡✲❞✐✛❡r❡♥❝❡ r❡s✉❧ts✿ q✉❛❞r❛t✐❝ ❝♦❡✣❝✐❡♥t ¯ ❆✷ r❡❣✉❧❛r q✉❛rt✐❝ ❝♦❡✣❝✐❡♥t ¯ ❆✹✿ ❧♦❣❛r✐t❤♠✐❝ ❢♦r ♥♣✲❝❤❛♥♥❡❧✱ ❜✉t r❡❣✉❧❛r ✐♥ ♥♥✰♣♣ ■♥t❡r❛❝t✐♦♥ ❝♦♥tr✐❜✉t✐♦♥ t♦ ❣r♦✉♥❞✲st❛t❡ ❡♥❡r❣② ♣❡r ♣❛rt✐❝❧❡ ✭♥♣✲❝❤❛♥♥❡❧✮✿ ❜r❡❛❦❞♦✇♥ ♦❢ ✏♣❛r❛❜♦❧✐❝ ❧❛✇✑✱ ¯ ❋s②♠ − ¯ ❆✷ ≃ ✶✺.✻ − ✽.✸ ❧❛r❣❡ q✉❛rt✐❝✰❧♦❣ ❛♣♣r♦①✐♠❛t✐♦♥ ✈❡r② ❛❝❝✉r❛t❡ ❢♦r δ ✵.✺✱ ❜✉t ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ✐♥ ✈❡r② ♥❡✉tr♦♥✲r✐❝❤ r❡❣✐♦♥ ❡①❛❝t r❡s✉❧ts✿ ♠❛①✐♠✉♠ ❛t δ ✵.✾✾ ✭✈❛♥✐s❤❡s ❢♦r ❧❛r❣❡r ❛✮✱ ❛♥❞ ❡✈❡♥ ❛ ❦✐♥❦❄ → ♥♦♥❛♥❛❧②t✐❝ t❡r♠s s❤♦✉❧❞ ❛r✐s❡ ❛❧s♦ ✐♥ s❡❧❢✲❝♦♥s✐st❡♥t s❝❤❡♠❡s✱ ❡✳❣✳✱ ❇❍❋✱ ❙❈●❋
❍❛♠✐❧t♦♥✐❛♥ H = T + V✱ ✇✐t❤ H |Ψ♣ = ❊♣ |Ψ♣ ❛♥❞ T |Φ♣ = E♣ |Φ♣✱ ✇❤❡r❡ T =
✐❥ φ✐|❚|φ❥ ❛† ✐ ❛❥ = ✐ ε✐❛† ✐ ❛✐
V = ✶
✷!
✐ ❛† ❥ ❛❧❛❦
❣r❛♥❞✲❝❛♥♦♥✐❝❛❧ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥✿ ❨ =
♣
Ψ♣
♣
U (β) = ❡−βT ❡βH = ∞
♥=✵ (−✶)♥ ♥! β
❞β♥ · · · ❞β✶ P
∆❆ = − ✶
β ❧♥
∞
♥=✶ (−✶)♥ ♥! β
❞β♥ · · · ❞β✶
❝♦♥tr❛❝t✐♦♥ r✉❧❡s ✭✇❤❡r❡ ❢ −
✐
= ✶/
✐
= ✶ − ❢ −
✐
✮✿ ❛†
✐ ❛❦ = δ✐❦❢ − ✐
✭❤♦❧❡✮ ❛❦❛†
✐ = δ❥❦❢ + ✐
(♣❛rt✐❝❧❡) ❧✐♥❦❡❞✲❝❧✉st❡r t❤❡♦r❡♠✿ ∆❆ = ✶
β
∞
♥=✵(−✶)♥
❞β♥ · · · ❞β✵ V■(β♥) · · · V■(β✵)❧✐♥❦❡❞ ◆♦t❡✿ ❛❧❧ t❤✐s ✇♦r❦s ❛❧s♦ ❢♦r ❝❛♥♦♥✐❝❛❧ ❡♥s❡♠❜❧❡✱ ❜✉t t❤❡ ❝♦♥str❛✐♥t Φ♣|N|Φ♣ = ◆ ✐♠♣❧✐❡s t❤❛t s✐♥❣❧❡✲♣❛rt✐❝❧❡ st❛t❡s ❝❛♥♥♦t ❜❡ s✉♠♠❡❞ ♦✈❡r ✐♥❞❡♣❡♥❞❡♥t❧② ⇒ ✉s❡❧❡ss✦
(a) self-energy (b) one-loop (c) one-loop (d) self-energy (e) one-loop (f) one-loop (a) two-loop (b) two-loop (c) two-loop (d) three-loop (e) three-loop
♥♦♥✲s❦❡❧❡t♦♥s ❛r❡ ✐♥s❡rt✐♦♥s✿ ❝✉t ❛rt✐❝✉❧❛t✐♦♥ ❧✐♥❡s ❝♦❧❧❡❝t✐♦♥ ♦❢ ✉♥❧✐♥❦❡❞ ❝❧✉st❡rs ✏s❡❧❢✲❡♥❡r❣②✑ ❛♥❞ ❛♥♦♠❛❧♦✉s ♦♥❡✲❧♦♦♣ ❞✐❛❣r❛♠s r❡❧❛t❡❞ ❜② ❝②❝❧✐❝ ✈❡rt❡① ♣❡r♠✉t❛t✐♦♥s s♣✉r✐♦✉s t❡r♠s ✐♥ ♣❡rt✉r❜❛t✐♦♥ s❡r✐❡s
G❦✶···❦♠
✐✶···✐♥
= ❛†
✐✶❛✐✶ · · · ❛† ✐♥❛✐♥❛❦✶❛† ❦✶ · · · ❛❦♠❛† ❦♠ ❤❛s ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ Y ✿
G❦✶···❦♠
✐✶···✐♥
= ✶
Y ∂ ∂(−βε✐✶ ) · · · ∂ ∂(−βε✐♥ )
∂ ∂(−βε❦✶ )
∂ ∂(−βε❦♠ )
❡✈❛❧✉❛t❡ ✐♥ t❡r♠s ♦❢ ❝✉♠✉❧❛♥ts K✐✶...✐♥ =
∂♥ ❧♥ Y ∂(−βε✐✶ )···∂(−βε✐♥ ) ✿
G❦✶···❦♠
✐✶···✐♥
=
(−✶)|P|G✐✶···✐♥
G✐✶···✐♥ =
♦❢ {✶,...,♥}
K
ν∈■ ✐ν
s❦❡❧❡t♦♥s ✉♥❝❤❛♥❣❡❞✿ K✐✶···✐♥ = G✐✶···✐♥ ❢♦r ✐✶ = ✐✷ = . . . = ✐♥ s❡❧❢✲❡♥❡r❣② ❞✐❛❣r❛♠s✿
G❦✶···❦♠
✐✶···✐♥❛❛ ∼ K❛K❛ + K❛❛ = ❢ − ❛ ❢ − ❛
+ ❢ −
❛ ❢ + ❛ = ❢ − ❛
G❦✶···❦♠
✐✶···✐♥❛❛❛ ∼ K❛K❛K❛ + ✸K❛❛K❛ + K❛❛❛ = ❢ − ❛ ❢ − ❛ ❢ − ❛
+ ✸❢ −
❛ ❢ + ❛ + ❢ − ❛ ❢ + ❛ (❢ + ❛ − ❢ − ❛ ) = ❢ − ❛
✳ ✳ ✳
♥♦ ❝♦♥tr✐❜✉t✐♦♥s ❢r♦♠ ❛♥♦♠❛❧♦✉s ❞✐❛❣r❛♠s✿
G❛···❛
✐✶···✐♥❛···❛ =
(−✶)|P|G✐✶···✐♥❛···❛
ν∈P ❛ν =
(−✶)|P|G✐✶···✐♥❛ = ✵
❡①♣❛♥s✐♦♥ ♦❢ ❧♦❣❛r✐t❤♠ ②✐❡❧❞s
∆❆ = ∞
β❜✶+...+❜❦ −✶❜✶ + . . . + ❜❦ ❜✶, . . . , ❜❦ (❆❛✶ )❜✶ · · · (❆❛❦ )❜❦ ❜✶ + . . . + ❜❦
❡❛❝❤ t❡r♠ ❆❛✐ ❤❛s ❧✐♥❦❡❞ ❛♥❞ ✉♥❧✐♥❦❡❞ ❝♦♥tr✐❜✉t✐♦♥s ❆♥✱✉♥❧✐♥❦❡❞ =
β ✶ α✶!···αν! (Γ♥✶)α✶ · · · (Γ♥ν )αν
✶
+...+♥αν
ν
=♥
❜② ❢❛❝t♦r✐③❛t✐♦♥ t❤❡♦r❡♠ t❤❡ ♦♥❧② t❡r♠s t❤❛t s✉r✈✐✈❡ ❛r❡ s✐♠♣❧②✲❝♦♥♥❡❝t❡❞ ✐♥ t❡r♠s ♦❢ ❤✐❣❤❡r ❝✉♠✉❧❛♥ts✱ ❡✳❣✳✱
❦❧
✐❥
✐❥;❛❜
K❧ ¯ K❦ − ✷✷ × δ❦❛K❦❦K✐ K❥ K❜ ¯ K❧ (a) (Γ2,normalΓ1)[Gkl
ij;ab]s.-c.
(b) (Γ2,normalΓ1)[Gkl
ij;ab]s.-c.
(c) (Γ2,normalΓ1)[Gkl
ij;ab]s.-c.
r❡s✉♠♠❛t✐♦♥ ♦❢ ✐♥s❡rt✐♦♥s r❡♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s ❢ ±
❛ [S] = ∞ ♠=✵ ✶ ♠!
♠ ∂♠
∂ε♠
❛ ❢ ±
❛
=
✶ ✶+❡①♣
❈♦rr❡❧❛t✐♦♥✲❇♦♥❞ ❋♦r♠❛❧✐s♠ st❛rt ✇✐t❤ st❛♥❞❛r❞ ♣❡rt✉r❜❛t✐♦♥ s❡r✐❡s ❢♦r ∆❋ = ❋ − F✿
∆❋ = ∞
β❜✶+...+❜❦ −✶❜✶ + . . . + ❜❦ ❜✶, . . . , ❜❦ (❋❛✶ )❜✶ · · · (❋❛❦ )❜❦ ❜✶ + . . . + ❜❦
t❤❡ ❝✉♠✉❧❛♥ts ❛r❡ ♥♦✇ ❣✐✈❡♥ ❜② K✐✶...✐♥ =
∂♥ ❧♥ Z ∂[−βε✐✶ ]···∂[−βε✐♥ ]❀ ❡✈❛❧✉❛t❡ ✉s✐♥❣
❧♥ Z(❚, ˜ µ, Ω) = ❧♥ Y(❚, ˜ µ, Ω) − ˜ µ ∂ ❧♥ Y(❚,˜
µ,Ω) ∂ ˜ µ
✇❤❡r❡ ˜ µ ✐s ❣✐✈❡♥ ❜② ◆(❚, ˜ µ, Ω) =
✐ ˜
❢ −
✐
❀ s✐♥❝❡ ◆ ✐s r❡❣❛r❞❡❞ ✜①❡❞✱ ˜ µ ✐s ❛ ❢✉♥❝t✐♦♥❛❧ ♦❢ t❤❡ s♣❡❝tr✉♠ {εα}✱ ✐✳❡✳✱ ❛s ✐♠♣❧✐❝✐t ❡q✉❛t✐♦♥✿ J (˜ µ, {εα}) =
α ˜
❢ −
α − ◆ = ✵
❘✳ ❇r♦✉t ✫ ❋✳ ❊♥❣❧❡rt✱ P❘ ✶✷✵ ✭✶✾✻✵✮
❚❤✐s ♠❡t❤♦❞ ❡✛❡❝t✐✈❡❧② ✏s❤✐❢ts✑ t❤❡ ❝♦♥str❛✐♥t Φ♣ | N | Φ♣ = ◆ t♦ t❤❡ ❧❡✈❡❧ ♦❢ ❞✐❛❣r❛♠s✱ r❡s✉❧t✐♥❣ ✐♥ ♥❡✇ s✐♠♣❧②✲❝♦♥♥❡❝t❡❞ ❝♦♥tr✐❜✉t✐♦♥s ✭✏❝♦rr❡❧❛t✐♦♥ ❜♦♥❞s✑✮✱ ❡✳❣✳✱ K✐❛ ∼ K✐❛ + δ✐❛K✐✐✱ ✇✐t❤ K✐✶✐✷ =
∂[−βε✐✷]
= ∂K✐✶ ∂[β˜ µ]
µ] ∂[−βε✐✷]
= − ˜ ❢ −
✐✶ ˜
❢ +
✐✶ ˜
❢ −
✐✷ ˜
❢ +
✐✷
❢ −
α ˜
❢ +
α
❘❡s✉♠♠❛t✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ ❜♦♥❞s r❡♥♦r♠❛❧✐③❡s ❛✉①✐❧✐❛r② ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧ ˜ µ