fixed points of the similarity …efb22.if.uj.edu.pl/talks/ruizarriola.pdf · implicit vs explicit...
TRANSCRIPT
![Page 1: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/1.jpg)
FIXED POINTS OF THE SIMILARITYRENORMALIZATION GROUP AND THE
NUCLEAR MANY BODY PROBLEM
E. Ruiz Arriola (with Sergio Szpigel and Varese S. Timoteo)
Departmento de Fısica Atomica, Molecular y NuclearUniversidad de Granada (Spain)
22nd European Conference onFew Body Problems in Physics
Krakow, Poland9 - 13 September 2013
Enrique Ruiz Arriola SRG Fixed
![Page 2: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/2.jpg)
References
Implicit vs Explicit Renormalization and EffectiveInteractionse-Print: arXiv:1307.1231Long distance symmetries for nuclear forces and thesimilarity renormalization groupAIP Conf.Proc. 1520 (2013) 346-348Nuclear Symmetries of the similarity renormalization groupfor nuclear forcesPoS CD12 (2013) 106Symmetries of the Similarity Renormalization Group forNuclear ForcesPhys.Rev. C86 (2012) 034002
Enrique Ruiz Arriola SRG Fixed
![Page 3: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/3.jpg)
Introduction
How much do we need to know light nuclei to predict heavy nuclei ?
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
1 Can we ignore explicit core and explicit (and/or chiral) pions ? → R. NavarroPerez
2 What are the errors in the interaction→J. E. Amaro
Enrique Ruiz Arriola SRG Fixed
![Page 4: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/4.jpg)
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 scattering
4N: α−particle, dd ,tp etc, scattering
Chiral hierarchy of few body multipionic forces (Weinberg)
Typical 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)
Enrique Ruiz Arriola SRG Fixed
![Page 5: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/5.jpg)
The off-shell problem
Two-body NN Interactions are not uniquely determined by perfect scatteringdata, or spectrum.
How large is the ambiguity ?
Polyzou-Glockle (Few Body System 1990)1 “Different off-shell extensions of two-body forces can be
equivalently realized as three-body interactions”2 “There are no experiments measuring only three-body
binding energies and phase shifts that can determine ifthere are no three-body forces in a three-body system.”
3 “There may be some systems for which it is possible to finda representation in which three-body forces are notneeded.”
Linear correlation (Tjon line) between triton and α particle binding energykeeping two body scattering fixed
Bα = aBt + b
Enrique Ruiz Arriola SRG Fixed
![Page 6: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/6.jpg)
Isospectral flow in SRG
Wilson-Glazek generator is unitary
dVs
ds= [[T ,Hs],Hs] = [[T ,Vs],T + Vs]→ TrHn
s = TrHn0
Convergence in Frobenius norm and metric (potentials can be compared)
||V ||2 ≡ TrV 2 d(A,B) ≡ ||A− B||
Monotonous decrease
dds
TrV 2s = 2Tr[T ,Vs]2 = −2Tr[T ,Vs]†[T ,Vs] ≤ 0
s0 < s 0 < TrV 2s ≤ TrV 2
s0
Limiting Potential is the smallest possible with the same spectrum
lims→∞
TrV 2s = min
VTrV 2
∣∣∣T +V =UH0U†
High energy states are enhanced by Frobenius norm
1 =2π
∫ ∞0
p2dp|p〉〈p| → TrV 2 =
(2π
)2 ∫ ∞0
p2dp∫ ∞
0k2dk |V (p, k)|2
Enrique Ruiz Arriola SRG Fixed
![Page 7: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/7.jpg)
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 Fixed
![Page 8: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/8.jpg)
Operator space
In NN system most states are continuum states (except deuteron)
Equations need discretization and cut-off in momentum space
pn (n = 1, . . . ,N)→ ∆pn ≡ wn → pmax = Λ
Closure relation
1 =2π
N∑n=1
wnp2n |pn〉〈pn|
Standard matrix multiplication
Anm =2π
pn√
wnAnmpm√
wm → 〈A,B〉 =N∑
n,m=1
A∗nk Bkn
SRG equations
dVnm
ds= −(en − em)2Vnm +
∑k
(en + em − 2ek )Vnk Vkm
Enrique Ruiz Arriola SRG Fixed
![Page 9: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/9.jpg)
Fixed points and stability analsis
Fixed points (Wilson)
dds
∑nm|Vnm|2 = −
∑nm|Vnm|2(εn − εm)2 = 0→ Vnk = Vnδnk
Energy eigenvaluesHψn = Enψn ≡ (εn + Vn)ψn
Perturbation around the equilibrium point
Vnk = Vnδnk + ∆nk → ∆V ′nk = −∆Vnk (εn − εm)(En − Em)
Only ordered as free ones are asymptotically stable (crossing forbbiden)
Hnm(s) = Enδn,m + Cnme−(εn−εm)(En−Em)s + . . .
Enrique Ruiz Arriola SRG Fixed
![Page 10: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/10.jpg)
LS equation on the grid
Rij = Vij +∑k 6=i
2π
wk p2k
Rik Vkl
p2i − p2
k
Phase shifts
Rnn = −tan δLS
n
pn≡ Vn
Limiting potential has no off-shellness
limλ→0
Vnm(λ) = −tan δLS
n
pnδnm
However, the LS phase shifts are not independent of λ in a finite grid
δ(pn, λ) 6= δ(pn, λ′)
Enrique Ruiz Arriola SRG Fixed
![Page 11: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/11.jpg)
Wegner generator
Evolution equation
dHds
= [[HD ,H],H] HD = diagH
dds
Tr(H − HD)2 = 2Tr[HD ,H]2 = −2Tr[HD ,H]†[HD ,H] ≤ 0
so that ||H − HD || → 0
lims→∞
H = HD = minH=UH0U† ||H − HD ||
Wilson generator and Wegner generators provide the same final fixed points uptp permutations
Wegner generator (all points are stable, crossing allowed)
Hnm(s) = Enδn,m + Cnme−(En−Em)2s + . . .
Enrique Ruiz Arriola SRG Fixed
![Page 12: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/12.jpg)
Toy model for S-waves
Separable interaction
Vα(p, p′) = Cαe−(p2+p′2)/L2α α =1 S0,
3 S1 (1)
0.0 0.5 1.0 1.5 2.0 2.5 3.00
50
100
150
p Hfm-1L
∆HpLHdegreesL
Parameter α0 r0 C LUnits (fm) (fm) (fm) (fm−1)1S0 -23.74 2.77 -1.9158 0.69133S1 5.42 1.75 -2.3006 0.4151
Enrique Ruiz Arriola SRG Fixed
![Page 13: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/13.jpg)
SRG evolution (Wilson generator)
Enrique Ruiz Arriola SRG Fixed
![Page 14: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/14.jpg)
SRG evolution (Wegner generator)
Enrique Ruiz Arriola SRG Fixed
![Page 15: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/15.jpg)
Diagonal Matrix Elements Evolution
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5
- 1 0
- 5
0
5V ii
(fm)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )
1 S 0 - W i l s o n - 1 0 p t s
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5
- 1 0
- 5
0
5
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )
1 S 0 - W e g n e r - 1 0 p t s
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5
- 1 0
- 5
0
5
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )
1 S 0 - W i l s o n - 2 0 p t s
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5
- 1 0
- 5
0
5
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )
1 S 0 - W e g n e r - 2 0 p t s
Enrique Ruiz Arriola SRG Fixed
![Page 16: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/16.jpg)
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 5 0
- 4 0
- 3 0
- 2 0
- 1 0
0
1 0
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )
3 S 1 - W i l s o n - 1 0 p t s
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5
- 1 0
- 5
0
5
1 0
1 5
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )
3 S 1 - W e g n e r - 1 0 p t s
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5 0
- 1 0 0
- 5 0
0
5 0
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )
3 S 1 - W i l s o n - 2 0 p t s
0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 3 0
- 2 0
- 1 0
0
1 0
2 0
V ii (fm
)
λ ( f m - 1 )
( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )
3 S 1 - W e g n e r - 2 0 p t s
Enrique Ruiz Arriola SRG Fixed
![Page 17: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/17.jpg)
Eigenvalues ordering
1 2 3 4 5- 5
0
5
1 0
1 5
2 0
2 5
3 0
3 5
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 1 0 p t s
1 2 3 4 5 6 7 8 9- 5
0
5
1 0
1 5
2 0
2 5
3 0
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 2 0 p t s
1 2 3 4 5 6 7 8 9 1 0 1 1- 5
0
5
1 0
1 5
2 0
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 3 0 p t s
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5- 5
0
5
1 0
1 5
2 0
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 4 0 p t s
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7- 2
0
2
4
6
8
1 0
1 2
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 5 0 p t s
1 5 1 0 1 5 2 0 2 5 3 0- 2
0
2
4
6
8
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 1 0 0 p t s
Enrique Ruiz Arriola SRG Fixed
![Page 18: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/18.jpg)
Eigenvalues ordering
1 2 3 4 5- 1 0
0
1 0
2 0
3 0
4 0
5 0
6 0
f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .
E i (MeV
)
i
3 S 1 - 1 0 p t s
1 2 3 4 5 6 7 8 9- 5
0
5
1 0
1 5
2 0
2 5
3 0
3 5
f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .
E i (MeV
)
i
3 S 1 - 2 0 p t s
1 2 3 4 5 6 7 8 9 1 0 1 1- 5
0
5
1 0
1 5
2 0
f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .
E i (MeV
)
i
3 S 1 - 3 0 p t s
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5- 5
0
5
1 0
1 5
2 0
f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .
E i (MeV
)
i
3 S 1 - 4 0 p t s
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7- 5
0
5
1 0
1 5
f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .
E i (MeV
)
i
3 S 1 - 5 0 p t s
1 5 1 0 1 5 2 0 2 5 3 0- 4
- 2
0
2
4
6
8
1 0
f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .
E i (MeV
)
i
3 S 1 - 1 0 0 p t s
Enrique Ruiz Arriola SRG Fixed
![Page 19: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/19.jpg)
Binding Energies
Mean field Slater Determinant
ψ(~p1, . . . , ~pA) = A[φn1,l1,s,ms1,t,mt1
(~p1) . . . φnA,lA,s,msA,t,mtA(~pA)
]. (2)
Single particle states (Harmonic oscillator)
Pnl (p) = Nnl e− 1
2 b2p2(bp)l L
l+ 12
n−1
(b2p2
)(3)
Two body interaction (Talmi-Moshinsky)
〈V2〉A =∑nlJS
gnlJS〈nl|V JST |nl〉 , (4)
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 Fixed
![Page 20: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/20.jpg)
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 Fixed
![Page 21: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/21.jpg)
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 Fixed
![Page 22: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/22.jpg)
SRG Correlations
The Wilson and Wegner binding energy results for SRG evolved forces
{−Bt ,−Bα} = minb
[(A− 1)〈
p2
2M〉+
A(A− 1)
212〈V1S0,λ + V3S1,λ〉
] ∣∣∣A=3,4
Ê
ÊÊÊ
Ê
ÊÊÊ
‡
‡‡‡‡‡
‡‡
Ï
ÏÏÏÏÏ
ÏÏ
Ú
ÚÚÚÚÚÚ
Ú
Ù
ÙÙÙÙÙÙ
Ù
-8 -6 -4 -2 0-30-25-20-15-10-50
BtHMeVL
BhHMeVL
Ê
ÊÊ
Ê
ÊÊÊÊ
‡
‡‡‡
‡‡‡‡
Ï
ÏÏÏ
Ï
ÏÏÏ
Ú
ÚÚÚÚ
ÚÚÚ
Ù
ÙÙÙÙ
ÙÙÙ
-8 -6 -4 -2 0-30-25-20-15-10-50
BtHMeVLBhHMeVL
Linear correlations in two regimes
∆Bα/∆Bt ∼ 2(λ→ 0) ∆Bα/∆Bt ∼ 4(λ ∼ 1)
Enrique Ruiz Arriola SRG Fixed
![Page 23: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/23.jpg)
The on-shell limit
Wilson and Wegner generator results (N=50)
ÊÊÊÊÊÊ
Ê
‡‡‡‡‡‡
‡
Ï
Ï
Ï
ÏÏÏ
Ï
0.0 0.1 0.2 0.3 0.4 0.5 0.6-15
-10
-5
0
l Hfm-1L
BHMeVL
Ê Deuteron‡ TritonÏ Helium
ÊÊÊÊ
ÊÊÊ
‡‡‡‡
‡‡‡
Ï
Ï
Ï
Ï
ÏÏÏ
0.0 0.1 0.2 0.3 0.4 0.5 0.6-15
-10
-5
0
l Hfm-1L
BHMeVL
Ê Deuteron‡ TritonÏ Helium
On-shell results
limλ→0
Et (λ) = −32
Bd limλ→0
Eα(λ) = −3Bd
Enrique Ruiz Arriola SRG Fixed
![Page 24: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/24.jpg)
SRG view of off-shellness and three-body force
Isospectral transformations
dVij
ds=
[[Tij ,Vij
],Tij + Vij
], (5)
dV123
ds= [[T12,V12] ,V13 + V23 + V123]
+ [[T13,V13] ,V12 + V23 + V123]
+ [[T23,V23] ,V12 + V13 + V123]
+ [[Trel,V123] ,Hs] . (6)
What is the initial condition ?
Final condition is unique
[T12,V12] = 0 [Trel,V123] = 0 (7)
Diagonal potential in momentum space (no off-shellness)
Enrique Ruiz Arriola SRG Fixed
![Page 25: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/25.jpg)
Correlations with on-shell 3-body forces
The on-shell triton (3 doublets) and α ( 6 doublets) binding
−Bt = −32
Bd︸ ︷︷ ︸3.3MeV
+ 〈t |V3|t〉︸ ︷︷ ︸off−shellness
−Bα = − 3Bd︸︷︷︸6.6MeV
+ 〈α|V3|α〉︸ ︷︷ ︸off−shellness
Taking 〈α|V3|α〉 = 4〈t |V3|t〉 ( 4 triplets )
Bα = 4Bt − 3Bd
= 4× 8.482− 3× 2.225 = 27.53 (exp.28.296) MeV
BΑ= 4Bt -3BdBΑ= 4Bt -3Bd
++Exp.Exp.
CD-BonnCD-Bonn
AV18AV18
NijmINijmINijmIINijmII
Vlowk HAV18LVlowk HAV18L
SRG HN3LOLSRG HN3LOL
6 7 8 9 1020
22
24
26
28
30
BtHMeVL
BΑ
HMeV
L
Enrique Ruiz Arriola SRG Fixed
![Page 26: FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance](https://reader031.vdocuments.mx/reader031/viewer/2022030511/5abba35e7f8b9a441d8d0aa9/html5/thumbnails/26.jpg)
Conclusions
1 SRG methods allow to reduce off-shell ambiguitycompletely
2 Only measurable two-body information is needed3 Simple explanation of the observed linear correlations
(Tjon line)4 On-shell 3-body forces are large and 4-body forces are
moderate5 Extension to other nuclei, neutron and nuclear matter is
possible
Enrique Ruiz Arriola SRG Fixed