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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
Enrique Ruiz Arriola SRG Symmetries
References
V. S. Timoteo, S. Szpigel and E. Ruiz ArriolaarXiv:1108.1162 [nucl-th]E. Ruiz Arriola, nucl-th/1009.4161
Enrique Ruiz Arriola SRG Symmetries
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 ?
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
WIGNER SYMMETRY
Enrique Ruiz Arriola SRG Symmetries
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
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
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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]
Enrique Ruiz Arriola SRG Symmetries
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]
Enrique Ruiz Arriola SRG Symmetries
EFFECTIVE INTERACTIONS
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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]
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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)
Enrique Ruiz Arriola SRG Symmetries
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,
Enrique Ruiz Arriola SRG Symmetries
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)
Enrique Ruiz Arriola SRG Symmetries
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)
Enrique Ruiz Arriola SRG Symmetries
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)
Enrique Ruiz Arriola SRG Symmetries
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)
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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
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
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
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
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
Enrique Ruiz Arriola SRG Symmetries
BINDING ENERGIES
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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
Enrique Ruiz Arriola SRG Symmetries
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 ?
Enrique Ruiz Arriola SRG Symmetries
CONCLUSIONS
Enrique Ruiz Arriola SRG Symmetries
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