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Symmetries of the similarity renormalizationgroup for nuclear forces
E. Ruiz Arriola (with Sergio Szpigel and Varese S. Timoteo)
Departmento de Fısica Atomica, Molecular y NuclearUniversidad de Granada (Spain)
JLAB, August-9th 2012
Chiral Dynamics 2012
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References
V. S. Timoteo, S. Szpigel and E. Ruiz ArriolaarXiv:1108.1162 [nucl-th]E. Ruiz Arriola, nucl-th/1009.4161
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Introduction
How much do we need to know light nuclei to predict heavynuclei ?Nucleon size a ∼ 1fm
Nuclear Force ∼ 1/mπ = 1.4fm
Nuclear matter (interparticle distance)
ρnm = 0.17fm−3 =1
(1.8fm)3
Fermi Momentum
kF = 270MeV λF = π/kF = 2.3fm� 1/√
mπMN = 0.5fm
Can we ignore explicit core and explicit pions ?
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Nuclear many body Hamiltonian H
H =∑
i
Ti +∑i<j
V2,ij +∑
i<j<k
V3,ijk +∑
i<j<k<l
V4,ijkl + . . .
NN: V2,ij (deuteron+NN scattering data)3N: Triton+ N-deuteron scattering4N: α−particle, dd ,tp etc, scatteringTypical Range of multinucleon forces e−mπd ∼ 0.2
VNN ∼ e−mπd VNNN ∼ e−2mπd VNNNN ∼ e−3mπd
Typical NN wavelengths ≥ 1/√
mπMN ∼ 0.5fm
→ Few wavelengths within a range(Coarse grained Effective interactions) SYMMETRIES
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WIGNER SYMMETRY
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Long wavelength limit (Short range interactions)
1S0 and 3S1 channels (Square well potential)
α1S0 = −23.74fm r1S0 = 2.75fm α3S1 = 5.4fm r3s1 = 1.75fm
0 1 2 3 4 5
-40
-20
0
20
40
r HfmL
VHrLHMeVL
0.0 0.5 1.0 1.5 2.0 2.5-2.5
-2.0
-1.5
-1.0
-0.5
0.0
p Hfm-1L
VHp,pLHfmL
0.0 0.5 1.0 1.5 2.0 2.5-2.5
-2.0
-1.5
-1.0
-0.5
0.0
p Hfm-1L
VHp,0LHfmL
0.0 0.5 1.0 1.5 2.0
0
50
100
150
p Hfm-1L
∆HpLHdegreesL
V JSl ′,l (p′,p) = MN
∫ ∞0
jl ′(p′r)jl(pr)V JSl ′l (r)r2
Wigner symmetry: V1S0(p,p′) ∼ V3S1(p,p′)Exact if
∫d3xV1S0(~x) =
∫d3xV3S1(~x) but NOT if
α1S0, α3S1 →∞Enrique Ruiz Arriola SRG Symmetries
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Long wavelength limit (EFT-approach)
Effective NN-Lagrangian
L = −12CS(N†N)2 − 1
2CT (N†~σN)2
Potential in momentum space
V0(p′,p) = C0θ(Λ− p)θ(Λ− p′)
Lippmann-Schwinger equation
→ C0(Λ) =4π2α0
MN
11 + Λα0/π
α0 = α1S0, α3S1
Wigner symmetry exact if α0 = −∞ [Mehen,Steward,Wise,1999]
BUT there is Regularization dependence
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SU(4) Wigner symmetry
Generators
T a =12
∑A
τaA , ISOSPIN
Si =12
∑A
σiA , SPIN
Gia =12
∑A
σiAτ
aA , GAMOW− TELLER
Casimir operator (two body)
CSU(4) = T aTa + SiSi + GiaGia ,
Irreducible representations (λ, µ, ν)
CSU(4) = µ(µ+ 4) + ν(ν + 2) + λ2
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Selection rules in Gamow-Teller weak decays betweensupermultiplets
〈λµν|Gia|λ′µ′ν ′〉 = 0
SU(4) mass formula [Franzini+Radicatti 63]
E = c1A(A + 1) + c2
[µ(µ+ 4) + ν(ν + 2) + λ2 − 15
4A]
Anomalously large double binding energy for even-evenN = Z nuclei [Van Isaacker,Warnerr,Brenner,1995].SU(4) inequalities for nuclei on the lattice [Chen, Lee ,Schaffer
2004]
SU(4) is crucial for Nuclear Lattice importance sampling[Lee, Epelbaum, 2008]
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One nucleon state
4 = (p ↑,p ↓,n ↑,n ↓) = (S = 1/2,T = 1/2) Quartet
Two nucleon states
CSTSU(4) =
12
(σ + τ + στ) +152,
τ = τ1 · τ2 = 2T (T + 1)− 3 ,σ = σ1 · σ2 = 2S(S + 1)− 3Sextet and decuplet (−1)S+L+T = −1
6A = (1,0)⊕ (1,0) L = 0,2, . . . (1S0,3 S1), (1D2,
3 D1,2,3)
10S = (0,0)⊕ (1,1) L = 1,3, . . . (1P1,3 P0,1,2)
At large Nc Wigner symmetry holds ONLY for even-Lchannels [Kaplan,Savage,Manohar,97]
This is confirmed by the data !![Calle Cordon,Ruiz Arriola,08]
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EFFECTIVE INTERACTIONS
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Standard ways to coarse grain
Nuclear shell model (energy)
~ω = 41A−13 MeV b = 1A
16 fm
Vlow,k (momentum) [Kuo,Brown,Holt,Bogner,2000]
ΛCM = 450MeV π/Λ = 1.5fm
Nuclear lattice (space) [D. Lee]
a = 2fm k ≤ π/a = 300MeV
Coarse grained delta-shells [Navarro,Amaro,Ruiz Arriola,2011]
∆r ∼ 1/√
MNmπ n∆r ∼ 1/mπ
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Similarity Renormalization Group
[Wilson,Glazek,Wegner]
Unitary transformation with generator ηs Hs = eηsHe−ηs
Running Hamiltonian (SRG trajectory)
dHs
ds= [ηs,Hs] , (1)
Initial condition Hs=0 = HWegner generator
ηs = [D(Hs),Hs]
Glazek-Wilsonηs = [T ,Hs]
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SRG long distance symmetries
Operator evolution
dOs
ds= [[T ,Vs],Os] . (2)
Symmetry group generator X
dXs
ds= [[T ,Vs],Xs] . (3)
Jacobi’s identity [[A,B],C] + [[C,A],B] + [[B,C],A] = 0
dXs
ds= − [[Xs,T ],Vs]− [[Vs,Xs],T ] = 0 . (4)
Long distance symmetry is a fixed-point of the SRGevolution
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Momentum space representation
dVs(~p′, ~p)
ds= −(Ep − E ′p)2Vs(~p′, ~p)
+
∫d3q
(2π)3
(Ep + Ep′ − 2Eq
)Vs(~p′, ~q)Vs(~q, ~p) . (5)
At large momenta (off-diagonal suppression)
Vs(~p′, ~p) ≈ Vs=0(~p′, ~p)e−(Ep′−Ep)2s + . . . (6)
Zero momentum states
dVs(~0, ~0)
ds= −2
∫d3q
(2π)3 Eq〈~q|VsV †s |~q〉 ≤ 0 , (7)
Momentum grid pmax = 5000MeV and N = 250 points
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General Structure of the potential
Mopentum space
V (~p′, ~p) = VC + ~τ1 · ~τ2 WC + (VS + ~τ1 · ~τ2 WS)~σ1 · ~σ2
+ (VLS + ~τ1 · ~τ2 WLS) i(~σ1 + ~σ2) · (~p′ × ~p)
+ (VT + ~τ1 · ~τ2 WT ) S12(~p′ − ~p)
+ (VQ + ~τ1 · ~τ2 WQ) S12(~p′ × ~p)
+ (VP + ~τ1 · ~τ2 WP) S12(~p′ + ~p) ,
where the tensor operator is defined as
S12(~q) =
[~σ1 · ~q ~σ2 · ~q −
13
q2~σ1 · ~σ2
], (8)
Pauli principle
V1′,2′;1,2(~p′, ~p) = −V2′,1′;1,2(−~p′, ~p)
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Partial wave decomposition
〈~p′|V Sλ |~p〉 = N
∑JMLL′
YJMLS (p′)V JS
LL′(p′,p)YJM
L′S†(p) (9)
, Coupled-channel equations,
dVs(p,p′)ds
= −(p2 − p′2)2 Vs(p,p′)
+2π
∫ ∞0
dq q2 (p2 + p′2 − 2q2) Vs(p,q) Vs(q,p′), (10)
where Vs(p,p′) is used as a brief notation for the projected NNpotential matrix elements,
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Integrating out vs Similarity Renormalization Group
Λ0
Λ1
Λ2
k’
k
λ0
λ1
λ2
k’
k
Vlowk → Scattering reproduced until the cut-off.
δlowk(k ,Λ) = δ(k)θ(Λ− k)
VSRG Scattering reproduced at ALL eneries.
δSRG(k , λ) = δ(k)
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Infrared fixed points of SRG
Fixed points for discretized equations
dVik
ds= −(Ei − Ek )2Vik +
∑k
(Ei + Ek − 2El)VilVlk = 0 (11)
Diagonal potential in momentum space basis
Vαβ = vαδαβ
R−matrix
R(p′,p; k) = V (p′,p) +2π
P∫ ∞
0q2dq
V (p′,q)R(q,p; k)
k2 − q2
Excluding the mean value
R(p,p; p) = − tan δ(p)
p= lim
λ→0Vλ(p,p)
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Sum rules
V3PC=
19(V3P0
+ 3V3P1+ 5V3P2
), (12)
V3PT= − 5
72(2V3P0
− 3V3P1+ V3P2
), (13)
V3PLS= − 1
12(2V3P0
+ 3V3P1− 5V3P2
), (14)
for triplet D-waves
V3DC=
115(3V3D1
+ 5V3D2+ 7V3D3
), (15)
V3DT= − 7
120(3V3D1
− 5V3D2+ 2V3D3
), (16)
V3DLS= − 1
60(9V3D1
+ 5V3D2− 14V3D3
), (17)
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for triplet F-waves
V3FC=
121(5V3F2
+ 7V3F3+ 9V3F4
), (18)
V3FT= − 5
112(4V3F2
− 7V3F3+ 3V3F4
), (19)
V3FLS= − 1
168(20V3F2
+ 7V3F3− 27V3F4
), (20)
and for triplet G-waves
V3GC=
127(7V3G3
+ 9V3G4+ 11V3G5
), (21)
V3GT= − 77
2160(5V3G3
− 9V3G4+ 4V3G5
), (22)
V3GLS=
1360
(− 35V3G3
− 9V3G4+ 44V3G5
). (23)
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0 0.5 1 1.5 2 2.5p [fm
-1]
-5
-4
-3
-2
-1
0
1
V(p
,p)
[fm
]
1S
03S
1
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-5
-4
-3
-2
-1
0
1
V(p
,p)
[fm
]
1S
03S
1
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-5
-4
-3
-2
-1
0
1
V(p
,p)
[fm
]
1S
03S
1
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
V(p
,p)
[fm
]
1P
13P
C3P
T3P
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
V(p
,p)
[fm
]
1P
13P
C3P
T3P
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
V(p
,p)
[fm
]
1P
13P
C3P
T3P
LS
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.2
-0.15
-0.1
-0.05
0
0.05
V(p
,p)
[fm
]
1D
23D
C3D
T3D
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.2
-0.15
-0.1
-0.05
0
0.05
V(p
,p)
[fm
]
1D
23D
C3D
T3D
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.2
-0.15
-0.1
-0.05
0
0.05
V(p
,p)
[fm
]
1D
23D
C3D
T3D
LS
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.01
0
0.01
0.02
0.03
0.04
0.05
V(p
,p)
[fm
]
1F
33F
C3F
T3F
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.01
0
0.01
0.02
0.03
0.04
0.05
V(p
,p)
[fm
]
1F
33F
C3F
T3F
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.01
0
0.01
0.02
0.03
0.04
0.05
V(p
,p)
[fm
]
1F
33F
C3F
T3F
LS
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.03
-0.02
-0.01
0
0.01
V(p
,p)
[fm
]
1G
43G
C3G
T3G
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.03
-0.02
-0.01
0
0.01
V(p
,p)
[fm
]
1G
43G
C3G
T3G
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.03
-0.02
-0.01
0
0.01
V(p
,p)
[fm
]
1G
43G
C3G
T3G
LS
λ = 1 fm-1
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0 0.5 1 1.5 2 2.5p [fm
-1]
-5
-4
-3
-2
-1
0
1
V(p
,0)
[fm
]
1S
03S
1
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-5
-4
-3
-2
-1
0
1
V(p
,0)
[fm
]
1S
03S
1
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-5
-4
-3
-2
-1
0
1
V(p
,0)
[fm
]
1S
03S
1
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.0003
0
0.0003
0.0006
0.0009
V(p
,0)
[fm
]
1P
13P
C3P
T3P
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.0003
0
0.0003
0.0006
0.0009
V(p
,0)
[fm
]
1P
13P
C3P
T3P
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-0.0003
0
0.0003
0.0006
0.0009
V(p
,0)
[fm
]
1P
13P
C3P
T3P
LS
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-3e-07
-2.5e-07
-2e-07
-1.5e-07
-1e-07
-5e-08
0
5e-08
V(p
,0)
[fm
]
1D
23D
C3D
T3D
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-3e-07
-2.5e-07
-2e-07
-1.5e-07
-1e-07
-5e-08
0
5e-08
V(p
,0)
[fm
]
1D
23D
C3D
T3D
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-3e-07
-2.5e-07
-2e-07
-1.5e-07
-1e-07
-5e-08
0
5e-08
V(p
,0)
[fm
]
1D
23D
C3D
T3D
LS
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-5e-11
0
5e-11
1e-10
1.5e-10
2e-10
2.5e-10
3e-10
V(p
,0)
[fm
]
1F
33F
C3F
T3F
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-5e-11
0
5e-11
1e-10
1.5e-10
2e-10
2.5e-10
3e-10
V(p
,0)
[fm
]
1F
33F
C3F
T3F
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-5e-11
0
5e-11
1e-10
1.5e-10
2e-10
2.5e-10
3e-10
V(p
,0)
[fm
]
1F
33F
C3F
T3F
LS
λ = 1 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-1e-13
-5e-14
0
5e-14
1e-13
V(p
,0)
[fm
]
1G
43G
C3G
T3G
LS
Initial
0 0.5 1 1.5 2 2.5p [fm
-1]
-1e-13
-5e-14
0
5e-14
1e-13
V(p
,0)
[fm
]
1G
43G
C3G
T3G
LS
λ = 2 fm-1
0 0.5 1 1.5 2 2.5p [fm
-1]
-1e-13
-5e-14
0
5e-14
1e-13
V(p
,0)
[fm
]
1G
43G
C3G
T3G
LS
λ = 1 fm-1
Enrique Ruiz Arriola SRG Symmetries
![Page 24: Symmetries of the similarity renormalization group for ... · PDF fileSymmetries of the similarity renormalization group for nuclear forces E. Ruiz Arriola (with Sergio Szpigel and](https://reader031.vdocuments.site/reader031/viewer/2022022505/5abba35e7f8b9a441d8d0aa8/html5/thumbnails/24.jpg)
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
Enrique Ruiz Arriola SRG Symmetries
![Page 25: Symmetries of the similarity renormalization group for ... · PDF fileSymmetries of the similarity renormalization group for nuclear forces E. Ruiz Arriola (with Sergio Szpigel and](https://reader031.vdocuments.site/reader031/viewer/2022022505/5abba35e7f8b9a441d8d0aa8/html5/thumbnails/25.jpg)
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0V
(p,p
) [f
m]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
V(p
,p)
[fm
]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0V
(p,p
) [f
m]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
V(p
,p)
[fm
]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0
V(p
,p)
[fm
]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
V(p
,p)
[fm
]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0
V(p
,p)
[fm
]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
V(p
,p)
[fm
]
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.08
-0.06
-0.04
-0.02
0
0 1 2
-0.08
-0.06
-0.04
-0.02
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
V(p
,0)
[fm
]
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
0 0.5 1 1.5 2 2.5p [fm-1]
-2e-07
-1.5e-07
-1e-07
-5e-08
0
0 1 2-2e-07
-1.5e-07
-1e-07
-5e-08
0
1D23DC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
Enrique Ruiz Arriola SRG Symmetries
![Page 26: Symmetries of the similarity renormalization group for ... · PDF fileSymmetries of the similarity renormalization group for nuclear forces E. Ruiz Arriola (with Sergio Szpigel and](https://reader031.vdocuments.site/reader031/viewer/2022022505/5abba35e7f8b9a441d8d0aa8/html5/thumbnails/26.jpg)
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0V
(p,p
) [f
m]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
V(p
,p)
[fm
]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0V
(p,p
) [f
m]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
V(p
,p)
[fm
]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0
V(p
,p)
[fm
]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
V(p
,p)
[fm
]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0
V(p
,p)
[fm
]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
V(p
,p)
[fm
]
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-0.02
-0.015
-0.01
-0.005
0
0 1 2-0.02
-0.015
-0.01
-0.005
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1V
(p,p
) [f
m]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]0 1 2
-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,p)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
V(p
,0)
[fm
]
0 1 2-3
-2
-1
0
1
1S03S1
0 0.5 1 1.5 2 2.5p [fm-1]
-3
-2
-1
0
1
0 1 2-3
-2
-1
0
1
1S03S1
! = 5 fm-1! = 4 fm-1
! = 3 fm-1 ! = 2 fm-1
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
V(p
,0)
[fm
]
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
0 0.5 1 1.5 2 2.5p [fm-1]
-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
0 1 2-6e-14
-5e-14
-4e-14
-3e-14
-2e-14
-1e-14
0
1G43GC
! = 5 fm-1! = 4 fm-1
! = 3 fm-1! = 2 fm-1
Enrique Ruiz Arriola SRG Symmetries
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Wigner and Serber Symmetries
Wigner symmetry
V1S0= V3SC
, V1D2= V3DC
, V1G4= V3GC
. (24)
Serber symmetry
0 = V3PC= V3FC
= V3HC= . . . (25)
The Wigner SRG scale:
λWigner = 3fm−1
Symmetry pattern consistent with large Nc
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BINDING ENERGIES
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Binding Energies
Mean field Slater Determinant
ψ(~p1, . . . , ~pA) = A[φn1,l1,s,ms1,t ,mt1
(~p1) . . . φnA,lA,s,msA,t ,mtA(~pA)
].(26)
Single particle states (Harmonic oscillator)
Pnl(p) = Nnle−12 b2p2
(bp)l Ll+ 1
2n−1
(b2p2
)(27)
Two body interaction (Talmi-Moshinsky)
〈V2〉A =∑nlJS
gnlJS〈nl |V JST |nl〉 , (28)
Nuclei: Shell model (mean field)
d : (1s)2 t : (1s)3 4He : (1s)4 ,16O : (1s)4(1p)12 40Ca : (1s)4(1p)12(2s)4(1d)20
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Binding Energies
0.5 1 1.5 2 2.5 3b (fm)
-10
-5
0
5
10
15
20
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
3H
1 1.5 2 2.5 3 3.5 4rrms (fm)
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
Exp
40Ca
Binding Energies - AV18
1 1.5 2 2.5 3 3.5 4rrms (fm)
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
ExpCCBHF
16O
0 0.5 1 1.5 2 2.5 3 3.5kF (fm-1)
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
AFDMC
nuclear matter
0.5 1 1.5 2 2.5 3b (fm)
-2
-1
0
1
2
3
4
5
6
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
d
0.5 1 1.5 2 2.5 3rrms (fm)
-40
-30
-20
-10
0
10
20
30
40
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
ExpGFMCUCOM
4He
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Binding Energies
Binding Energies - NijII
0.5 1 1.5 2 2.5 3b (fm)
-2
-1
0
1
2
3
4
5
6
7
8
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
d
0.5 1 1.5 2 2.5 3b (fm)
-10
-5
0
5
10
15
20
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
3H
0.5 1 1.5 2 2.5 3rrms (fm)
-40
-30
-20
-10
0
10
20
30
40
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
ExpGFMCUCOM
4He
1 1.5 2 2.5 3 3.5 4rrms (fm)
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
ExpCCBHF
16O
1 1.5 2 2.5 3 3.5 4rrms (fm)
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
Exp
40Ca
0 0.5 1 1.5 2 2.5 3 3.5kF (fm-1)
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
AFDMC
nuclear matter
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Binding Energies
Binding Energies - N3LO
0.5 1 1.5 2 2.5 3b (fm)
-2
-1
0
1
2
3
4
5
6
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
d
0.5 1 1.5 2 2.5 3b (fm)
-10
-5
0
5
10
15
20
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
3H
0.5 1 1.5 2 2.5 3rrms (fm)
-40
-30
-20
-10
0
10
20
30
40
B (M
eV)
λ = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
ExpGFMCUCOM
4He
1 1.5 2 2.5 3 3.5 4rrms (fm)
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
ExpCCBHF
16O
1 1.5 2 2.5 3 3.5 4rrms (fm)
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
Exp
40Ca
0 0.5 1 1.5 2 2.5 3 3.5kF (fm-1)
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
B /
A (M
eV)
λ = infinityλ = 2 fm-1
λ = 1 fm-1
AFDMC
nuclear matter
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From a variational point of view: GFM=SRG+Shell modelB4He = −24MeV ;
minΨ〈Ψ|Hλ=∞Av18 |Ψ〉 = minb〈(1s)4|Hλ∼fm−1
Av18 |(1s)4〉= minb〈(1s)4|Hλ∼fm−1
Nijm |(1s)4〉
= minb〈(1s)4|Hλ∼fm−1
N3LO |(1s)4〉
Core + TPE are marginal for 4He. Other nuclei ?SRG Wigner scale λWigner = 3fm−1 implies large 3-bodyand 4-body forcesAre the N-body forces SU(4)-invariant ?
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CONCLUSIONS
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Wigner symmetry looks very different when finite rangeforces are considered.In Vlowk Wigner symmetry is saturated, i.e. it holds above acut-offIn SRG Wigner symmetry happens at one SRG cut-off
Enrique Ruiz Arriola SRG Symmetries