Green's Function Monte Carlo Calculations of Light Nuclei...
Transcript of Green's Function Monte Carlo Calculations of Light Nuclei...
![Page 1: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/1.jpg)
Green’s Function Monte Carlo Calculations of Light Nuclei withTwo- and Three-Nucleon Interactions from Chiral Effective Field
Theory
Joel Lynn
Theoretical Division, Los Alamos National Laboratory
withJ. Carlson, E. Epelbaum, S. Gandolfi, A. Gezerlis, A. Schwenk, I. Tews
International Collaborations in Nuclear Theory: Theory for open-shellnuclei near the limits of stability
May 14, 2015
![Page 2: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/2.jpg)
Outline
1 Motivation and BackgroundBig QuestionsAb-Initio Calculations of Nuclei - GFMCNuclear Interactions
2 NN Results - A ≤ 4Theoretical UncertaintiesChiral ConvergenceShort-Distance Behavior
3 NNN Interaction and FitsThe NNN InteractionThe FitsResults for Light Nuclei
4 ConclusionSummaryFuture WorkAcknowledgments
![Page 3: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/3.jpg)
MotivationBig Questions
Low-energy nuclear theoryconnects several researchareas in physics.
QMC+χEFT is acompelling piece of thepuzzle.
Low-EnergyNuclear Theory
FundamentalSymmetries
NuclearAstrophysics
NuclearStructure
![Page 4: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/4.jpg)
MotivationBig Questions
Low-energy nuclear theoryconnects several researchareas in physics.QMC+χEFT is acompelling piece of thepuzzle.
QMC+χEFT
FundamentalSymmetries
NuclearAstrophysics
NuclearStructure
![Page 5: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/5.jpg)
MotivationBig Questions - Nuclear Structure
�� ��Nuclear Structure
What are the limits ofnuclear existence?How far can we pushab initio calculations?How can we build a coherentframework for describingnuclei, nuclear matter, andnuclear reactions?This talk: GFMC+χEFT
![Page 6: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/6.jpg)
MotivationBig Questions - Nuclear Astrophysics
�� ��Nuclear Astrophysics
How did the elementscome into existence?What is the structure ofneutron stars and how dotheir properties depend onthe underlyingHamiltonian?What role does pairing playin properties of neutronstars?
![Page 7: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/7.jpg)
MotivationBig Questions - Fundamental Symmetries
�� ��Fundamental Symmetries
What explains thedominance of matterover antimatter in theuniverse?What is the nature ofneutrinos and how do theyinteract with nuclei?Electric Dipole Moments oflight nuclei?
![Page 8: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/8.jpg)
BackgroundThe Framework
Quantum Chromodynamics (QCD) isultimately responsible for stronginteractions: nucleons are the relevantdegrees of freedom for low-energynuclear physics→Nucleon-Nucleonpotentials.
![Page 9: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/9.jpg)
BackgroundA Hard Problem!
Nuclei are strongly-interacting many-body systems.What method do we use to solve the many-body Schrodinger equation?
H |Ψ〉 = E |Ψ〉
2A(AZ)
coupled differential equations in 3A− 3 variables in the chargebasis. (Can be lowered to 2A 2T+1
A/2+T+1( A
A/2+T)
using an isospin-conservingbasis).
4He→ 96 equations in 9 variables.6Li→ 1 280 equations in 15 variables.
8Be→ 17 920 equations in 21 variables.10B→ 258 048 equations in 27 variables.
12C→ 3 784 704 equations in 33 variables.14N→ 56 229 888 equations in 39 variables.
What is the Hamiltonian?
![Page 10: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/10.jpg)
BackgroundAb-Initio Calculations of Nuclei - Monte Carlo
Green’s function Monte Carlo (GFMC): the most accurate method for solvingthe many-body Schrodinger equation for light nuclei 4 < A ≤ 12.First: Variational Monte Carlo.
Guess a trial wave function ΨT . Generate a random position:R = r1, r2, . . . , rA.Metropolis algorithm: generate new positions R′ based on the probabilityP = |ΨT (R′)|2
|ΨT (R)|2 .Yields a set of “walkers” distributed according to the trial wave function.A walker:
∑β cβ |Rβ〉 . 3A positions and 2A(A
Z)
spin/isospin states in thecharge basis.Variational principle: ET = 〈ΨT |H |ΨT 〉
〈ΨT |ΨT 〉 ≥ E0.
![Page 11: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/11.jpg)
BackgroundAb-Initio Calculations of Nuclei - Monte Carlo
Second: Green’s Function Monte Carlo.The wave function is imperfect: |ΨT 〉 =
∑∞i=0 αi |Ψi〉 .
Propagate in imaginary time to project out the ground state |Ψ0〉 :
|Ψ(t)〉 = e−(H−E0)t |ΨT〉 = e−(E0−E0)t[α0 |Ψ0〉 +
∑i 6=0 αie−(Ei−E0)t |Ψi〉
]⇒ lim
t→∞|Ψ(t)〉 ∝ |Ψ0〉 .
![Page 12: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/12.jpg)
BackgroundAb-Initio Calculations of Nuclei - Monte Carlo
Second: Green’s Function Monte Carlo.The wave function is imperfect: |ΨT 〉 =
∑∞i=0 αi |Ψi〉 .
Propagate in imaginary time to project out the ground state |Ψ0〉 :
|Ψ(t)〉 = e−(H−E0)t |ΨT〉 = e−(E0−E0)t[α0 |Ψ0〉 +
∑i 6=0 αie−(Ei−E0)t |Ψi〉
]⇒ lim
t→∞|Ψ(t)〉 ∝ |Ψ0〉 .
![Page 13: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/13.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
H = p2x
2m + V (x), V ={
0, 0 < x < L∞, otherwise
~ = m = L = 1�� ��Solution
ψn(x) =√
2 sin(nπx), En = n2π2
2 .
“Trial wave function.”
ψT(x) = −√
30[(
x − 12
)2− 1
4
].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1
ψ(x)
x
ψ1
ψT
![Page 14: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/14.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
H = p2x
2m + V (x), V ={
0, 0 < x < L∞, otherwise
~ = m = L = 1�� ��Solution
ψn(x) =√
2 sin(nπx), En = n2π2
2 .
“Trial wave function.”
ψT(x) = −√
30[(
x − 12
)2− 1
4
].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1
ψ(x)
x
ψ1
ψT
![Page 15: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/15.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.0(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 16: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/16.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.1(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 17: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/17.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.2(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 18: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/18.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.3(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 19: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/19.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.4(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 20: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/20.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.5(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 21: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/21.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.6(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 22: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/22.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.7(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 23: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/23.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.8(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 24: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/24.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 0.9(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 25: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/25.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 1.0(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 26: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/26.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 1.1(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 27: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/27.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 1.2(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 28: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/28.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 1.3(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 29: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/29.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
�� ��A cartoon
E(t) = 〈ψT |He−(H−E1)t |ψT〉〈ψT |e−(H−E1)t |ψT〉
, ψT(x) =∞∑
n=1cnψn(x),
t = 1.4(1/Esep.)
4.93
4.94
4.95
4.96
4.97
4.98
4.99
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E(t)
t [1/Esep.]
π2/2
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0 0.2 0.4 0.6 0.8 1
cn ψ
n(x)
x
n=3n=5n=7
![Page 30: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/30.jpg)
BackgroundGFMC - An Example
For 4He, 1/Esep. = 1/|Eα − Et | ≈ 0.05 MeV−1.
0 1 2 3 4 5 6 7 8
t (1/Esep.)
-29
-28.5
-28
-27.5
-27
-26.5
-26
-25.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
E(t)
(MeV
)
t (MeV-1
)
Eave.Exp.
GFMC
![Page 31: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/31.jpg)
BackgroundSome Limitations of GFMC
�� ��Limitations
A walker consists of the 3A positions and 2A(AZ)
spin-isospin states (in thecharge basis).
12C:→Remember 3 784 704 spin-isospin states!
However: See some exciting work by S. Gandolfi, A. Lovato, J. Carlson, andKevin E. Schmidt in auxiliary-field diffusion Monte Carlo (AFDMC)1.
We can calculate “mixed estimates”: 〈Ψ(t)|O|ΨT 〉〈Ψ(t)|ΨT 〉
〈O(t)〉 = 〈Ψ(t)|O|Ψ(t)〉〈Ψ(t)|Ψ(t)〉 ≈ 〈O(t)〉Mixed + [〈O(t)〉Mixed − 〈O〉T ].
1arXiv:1406.3388 [nucl-th]
![Page 32: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/32.jpg)
BackgroundSome Limitations of GFMC
�� ��However...
For ground-state energies, O = H , and [H ,G] = 0:
〈H 〉Mixed = 〈ΨT |e−(H−ET )t/2He−(H−ET )t/2|ΨT 〉〈ΨT |e−(H−ET )t/2e−(H−ET )t/2|ΨT〉
, limt→∞〈H 〉Mixed = E0.
![Page 33: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/33.jpg)
BackgroundNuclear Interactions - The Hamiltonian
H =A∑
i=1
p2i
2mi+
A∑i<j
vij +A∑
i<j<kVijk + · · ·
Until recently, there were two broad choices for vij .Local, real-space, phenomenological: Argonne v18
2 - informed by theory,phenomenology, and experiment (well tested and very successful).Non-local, momentum-space, effective field theory (EFT): N3LO3 -informed by chiral EFT and experiment (well liked and often used inbasis-set methods, such as the no-core shell model).
2R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995).3e.g. D. R. Entem and R. Machleidt, PRC 68, 041001 (2003)
![Page 34: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/34.jpg)
BackgroundNuclear Interactions - GFMC + Argonne v18
Many excellent results using GFMC and phenomenological potentials 4.Ab-initio few-nucleon calculation
-100
-90
-80
-70
-60
-50
-40
-30
-20
Ener
gy (M
eV)
AV18AV18+IL7 Expt.
0+
4He0+2+
6He 1+3+2+1+
6Li3/2−1/2−7/2−5/2−5/2−7/2−
7Li
0+2+
8He 2+2+
2+1+
0+
3+1+
4+
8Li
1+
0+2+
4+2+1+3+4+
0+
8Be
3/2−1/2−5/2−
9Li
3/2−1/2+5/2−1/2−5/2+3/2+
7/2−3/2−
7/2−5/2+7/2+
9Be
1+
0+2+2+0+3,2+
10Be 3+1+
2+
4+
1+
3+2+
3+
10B
3+
1+
2+
4+
1+
3+2+
0+
0+
12C
Argonne v18with Illinois-7
GFMC Calculations24 November 2012
This is great! But... Until recently the nucleon-nucleon potentials used havebeen restricted to the phenomenological Argonne-Urbana/Illinois family ofinteractions.
4Figure courtesy of Steven C. Pieper and R. B. Wiringa
![Page 35: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/35.jpg)
BackgroundNuclear Interactions - Chiral EFT
NN NNN
LO O(
QΛb
)0 —
NLO O(
QΛb
)2 —
N2LO O(
QΛb
)3+ · · ·
N3LO O(
QΛb
)4+ · · ·
+ · · ·
Chiral EFT is an expansion inpowers of Q/Λb.Q ∼ mπ ∼ 100 MeV;Λb ∼ 800 MeV.Long-range physics: givenexplicitly (no parameters tofit) by pion-exchanges.Short-range physics:parametrized through contactinteractions with low-energyconstants (LECs) fit tolow-energy data.Many-body forces,CP-violating operators, andcurrents enter systematicallyand are related via the sameLECs.
![Page 36: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/36.jpg)
BackgroundNuclear Interactions - Chiral EFT
NN NNN
LO O(
QΛb
)0 —
NLO O(
QΛb
)2 —
N2LO O(
QΛb
)3+ · · ·
N3LO O(
QΛb
)4+ · · ·
+ · · ·
Chiral EFT is an expansion inpowers of Q/Λb.Q ∼ mπ ∼ 100 MeV;Λb ∼ 800 MeV.Long-range physics: givenexplicitly (no parameters tofit) by pion-exchanges.Short-range physics:parametrized through contactinteractions with low-energyconstants (LECs) fit tolow-energy data.Many-body forces,CP-violating operators, andcurrents enter systematicallyand are related via the sameLECs.
![Page 37: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/37.jpg)
BackgroundNuclear Interactions - Chiral EFT
NN NNN
LO O(
QΛb
)0 —
NLO O(
QΛb
)2 —
N2LO O(
QΛb
)3+ · · ·
N3LO O(
QΛb
)4+ · · ·
+ · · ·
Chiral EFT is an expansion inpowers of Q/Λb.Q ∼ mπ ∼ 100 MeV;Λb ∼ 800 MeV.Long-range physics: givenexplicitly (no parameters tofit) by pion-exchanges.Short-range physics:parametrized through contactinteractions with low-energyconstants (LECs) fit tolow-energy data.Many-body forces,CP-violating operators, andcurrents enter systematicallyand are related via the sameLECs.
![Page 38: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/38.jpg)
BackgroundNuclear Interactions - Chiral EFT
NN NNN
LO O(
QΛb
)0 —
NLO O(
QΛb
)2 —
N2LO O(
QΛb
)3+ · · ·
N3LO O(
QΛb
)4+ · · ·
+ · · ·
Chiral EFT is an expansion inpowers of Q/Λb.Q ∼ mπ ∼ 100 MeV;Λb ∼ 800 MeV.Long-range physics: givenexplicitly (no parameters tofit) by pion-exchanges.Short-range physics:parametrized through contactinteractions with low-energyconstants (LECs) fit tolow-energy data.Many-body forces,CP-violating operators, andcurrents enter systematicallyand are related via the sameLECs.
![Page 39: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/39.jpg)
BackgroundNuclear Interactions - Chiral EFT
NN NNN
LO O(
QΛb
)0 —
NLO O(
QΛb
)2 —
N2LO O(
QΛb
)3+ · · ·
N3LO O(
QΛb
)4+ · · ·
+ · · ·
Chiral EFT is an expansion inpowers of Q/Λb.Q ∼ mπ ∼ 100 MeV;Λb ∼ 800 MeV.Long-range physics: givenexplicitly (no parameters tofit) by pion-exchanges.Short-range physics:parametrized through contactinteractions with low-energyconstants (LECs) fit tolow-energy data.Many-body forces,CP-violating operators, andcurrents enter systematicallyand are related via the sameLECs.
![Page 40: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/40.jpg)
BackgroundChiral EFT: Local Constructon Possible
Local construction possible5 up to N2LO.Definitions.
q = p− p′, k = p + p′
Regulator:
f (p, p′) = e−(p/Λ)ne−(p′/Λ)n
→ flong(r) = 1− e−(r/R0)4: R0 = 1.0, 1.1, 1.2 fm.
Contacts:
∝ q and k→ Choose contacts ∝ q (As much as possible!)
5A. Gezerlis, I. Tews, E. Epelbaum, M. Freunek,S. Gandolfi, K. Hebeler, A. Nogga, A. Schwenk, PRC 90 054323 (2014)
![Page 41: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/41.jpg)
BackgroundChiral EFT: Local Constructon Possible
Local construction possible5 up to N2LO.Definitions.
q = p− p′, k = p + p′
Regulator:
f (p, p′) = e−(p/Λ)ne−(p′/Λ)n
→ flong(r) = 1− e−(r/R0)4: R0 = 1.0, 1.1, 1.2 fm.
Contacts:
∝ q and k→ Choose contacts ∝ q (As much as possible!)
5A. Gezerlis, I. Tews, E. Epelbaum, M. Freunek,S. Gandolfi, K. Hebeler, A. Nogga, A. Schwenk, PRC 90 054323 (2014)
![Page 42: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/42.jpg)
BackgroundChiral EFT: Local Constructon Possible
NN
LO O(
QΛb
)0
NLO O(
QΛb
)2
N2LO O(
QΛb
)3+ · · ·
V (0)cont. = α1 + α2(σ1 · σ2) + α3(τ 1 · τ 2)
+ α4(σ1 · σ2)(τ 1 · τ 2)
(Pauli exclusion principle) → Only twoindependent contacts!
V (0)cont. = CS + CT(σ1 · σ2)
![Page 43: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/43.jpg)
BackgroundChiral EFT: Local Constructon Possible
NN
LO O(
QΛb
)0
NLO O(
QΛb
)2
N2LO O(
QΛb
)3+ · · ·
V (2)cont. = γ1q2 + γ2q2(σ1 · σ2)
+ γ3q2(τ 1 · τ 2) + γ4q2(σ1 · σ2)(τ 1 · τ 2)+ γ5k2 + γ6k2(σ1 · σ2) + γ7k2(τ 1 · τ 2)+ γ8k2(σ1 · σ2)(τ 1 · τ 2)+ (σ1 + σ2)(q × k)(γ9 + γ10(τ 1 · τ 2))+ (σ1 · q)(σ2 · q)(γ11 + γ12(τ 1 · τ 2))+ (σ1 · k)(σ2 · k)(γ13 + γ14(τ 1 · τ 2))
![Page 44: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/44.jpg)
BackgroundChiral EFT: Local Constructon Possible
NN
LO O(
QΛb
)0
NLO O(
QΛb
)2
N2LO O(
QΛb
)3+ · · ·
V (2)cont. = γ1q2 + γ2q2(σ1 · σ2)
+ γ3q2(τ 1 · τ 2) + γ4q2(σ1 · σ2)(τ 1 · τ 2)+ γ5k2 + γ6k2(σ1 · σ2) + γ7k2(τ 1 · τ 2)+ γ8k2(σ1 · σ2)(τ 1 · τ 2)+ (σ1 + σ2)(q × k)(γ9 + γ10(τ 1 · τ 2))+ (σ1 · q)(σ2 · q)(γ11 + γ12(τ 1 · τ 2))+ (σ1 · k)(σ2 · k)(γ13 + γ14(τ 1 · τ 2))
![Page 45: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/45.jpg)
BackgroundChiral EFT: Operator Structure
Local chiral EFT potential ∼ a v7 potential
vij =7∑
p=1vp(rij)Op
ij +18∑
p=15vp(rij)Op
ij .
Charge-independent operatorsOp=1,14
ij =[1,σi · σj ,Sij ,L · S,L2,L2(σi · σj), (L · S)2]⊗ [1, τ i · τ j ] .
Charge-independence-breaking operatorsOp=15,18
ij = [1,σi · σj ,Sij ]⊗ Tij , and (τzi + τzj).
Tensor operatorsSij = 3(σi · rij)(σj · rij)− σi · σj , Tij = 3τziτzj − τ i · τ j
![Page 46: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/46.jpg)
BackgroundChiral EFT: Phase Shifts
Phase shifts for the np potential6.
0 50 100 150 200 250
Lab. Energy [MeV]
0
10
20
30
40
50
60
70
Ph
ase
Sh
ift
[deg
]
LONLO
N2LO
0 50 100 150 200 250
Lab. Energy [MeV]
0
50
100
150
200
0 50 100 150 200 250
Lab. Energy [MeV]
-25
-20
-15
-10
-5
0
0 50 100 150 200 250
Lab. Energy [MeV]
0
1
2
3
4
1S03S1
3D1 e1
0 50 100 150 200 250
Lab. Energy [MeV]
-30
-25
-20
-15
-10
-5
0
Ph
ase
Sh
ift
[deg
]
0 50 100 150 200 250
Lab. Energy [MeV]
-10
-5
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250
Lab. Energy [MeV]
-30
-25
-20
-15
-10
-5
0
1P1
3P03P1
6A. Gezerlis et al. PRC 90 054323 (2014)
![Page 47: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/47.jpg)
NN Results7
Investigate
Theoretical uncertainties – Ground-state energies and radii at LO, NLO,and N2LO
Chiral convergenceI Ground-state energies and radii at LO, NLO, and N2LOI 1st-order perturbation theory calculations for 4He: Vpert. = VN2LO − VNLOI 1st-, 2nd-, and 3rd-order perturbation theory calculations for the deuteron
Short-distance behavior – Contrast the short-distance (high-energy)features of one- and two-body distributions
7JEL, J. Carlson, E. Epelbaum, S. Gandolfi, A. Gezerlis, A. Schwenk,PRL 113 192501 (2014)
![Page 48: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/48.jpg)
NN Results4He Ground-State Energies - 〈H〉
Theoretical uncertainties & Chiral convergence4He ground-state energies at LO, NLO, and N2LO: each with cutoff
R0 = 1.0, 1.1, and 1.2 fm, compared with the value for AV8′.
-48
-46
-44
-42
LO NLO N2LO v8
′
E (
MeV
)
Chiral Order
-30
-28
-26
-24
-22
-20
-18
-16
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
![Page 49: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/49.jpg)
NN ResultsA = 3 Ground-State Energies - 〈H〉
Theoretical uncertainties & Chiral convergence3H and 3He ground-state energies at LO, NLO, and N2LO: each with cutoff
R0 = 1.0, 1.1, and 1.2 fm.
3H
-12
-11
-10
-9
-8
-7
LO NLO N2LO
E (
MeV
)
Chiral Order
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
3He
-11
-10
-9
-8
-7
-6
LO NLO N2LO
E (
MeV
)
Chiral Order
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
![Page 50: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/50.jpg)
NN Results4He Radii - 〈r2
pt.〉1/2
Theoretical uncertainties & Chiral convergence4He point proton radii – r2
pt. = r2ch. − r2
p − NZ r2
n (IA) at LO, NLO, and N2LO:each with cutoff R0 = 1.0, 1.1, and 1.2 fm.
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
LO NLO N2LO
r pt. (
fm)
Chiral Order
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
![Page 51: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/51.jpg)
NN ResultsA = 3 Radii - 〈r2
pt.〉1/2
Theoretical uncertainties & Chiral convergence3H and 3He point proton radii – r2
pt. = r2ch. − r2
p − NZ r2
n (IA) at LO, NLO, andN2LO: each with cutoff R0 = 1.0, 1.1, and 1.2 fm.
3H
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
LO NLO N2LO
r pt. (
fm)
Chiral Order
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
3He
1.3
1.4
1.5
1.6
1.7
1.8
1.9
LO NLO N2LO
r pt. (
fm)
Chiral Order
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
![Page 52: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/52.jpg)
NN Results4He Perturbation - 〈ΨNLO|HN2LO|ΨNLO〉
Chiral convergence4He ground-state energy in first-order perturbation theory.ENLO + 〈ΨNLO|Vpert.|ΨNLO〉, with Vpert. = VN2LO −VNLO.
-48
-46
-44
-42
LO NLO N2LO NLO+pert.
E (
MeV
)
Chiral Order
-30
-28
-26
-24
-22
-20
-18
-16
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
![Page 53: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/53.jpg)
NN Results2H Perturbation - 〈ΨNLO|HN2LO|ΨNLO〉
Chiral convergenceWrite H → 〈k ′JMJL′S |H |kJMJLS〉.Diagonalize→ {ψ(i)
D (r)}.Second- and third-order perturbation theory calculations for 2H possible.
CalculationE (MeV)
R0 =1.0 fm R0 =1.1 fm R0 =1.2 fm
ENLO -2.15 -2.16 -2.16ENLO + V (1)
pert. -1.44 -1.80 -1.90ENLO + V (2)
pert. -2.11 -2.17 -2.18ENLO + V (3)
pert. -2.13 -2.18 -2.19EN2LO -2.21 -2.21 -2.20
![Page 54: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/54.jpg)
NN ResultsProton Distribution - 4He
Short-distance behaviorProton distribution: ρ1,p(r) = 1
4πr2 〈Ψ|∑
i1+τz(i)
2 δ(r − |ri −Rc.m.|)|Ψ〉.
0.01
0.1
0 0.5 1 1.5 2
ρ1
, p (
fm-3
)
r (fm)
v8′
LO
NLO
N2LO
![Page 55: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/55.jpg)
NN ResultsTwo-Body T = 1 Distribution - 4He
Short-distance behaviorTwo-body T = 1 distribution: ρ(T=1)
2 (r) = 14πr2 〈Ψ|
∑i<j
3+τ i ·τ j4 δ(r − |rij |)|Ψ〉.
0.0001
0.001
0.01
0.1
ρ2(T
=1) (
fm-3
)R0=1.2 fm
v8′
LONLO
N2LO
0
0.01
0 0.5 1.0 1.5 2.0
r (fm)
N2LO
R0=1.2 fm
R0=1.1 fm
R0=1.0 fm
v8′
![Page 56: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/56.jpg)
NNN Interaction8
Investigate
The form of the force – Fourier transforming, regulating, some ambiguity.
Fitting cD and cE – 4He binding energy and n-α scattering phase shifts.
Results for Light Nuclei – A ≤ 4 radii and binding energies.
8Current work
![Page 57: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/57.jpg)
NNN InteractionThe Form of the Force – Momentum Space
π π
c1,c3,c4
π
cD cE
VC = 12
(gA2fπ
)2 ∑π(ijk)
(σi · qi)(σj · qj)(q2
i + m2π)(q2
j + m2π)Fαβ
ijk ταi τ
βj ,
Fαβijk = δαβ
[−4c1m2
π
f 2π
+ 2c3
f 2π
(qi · qj)]
+ c4
f 2π
εαβγτγk [σk · (qi × qj)],
VD = − gA8f 2π
cDf 2πΛχ
∑π(ijk)
σj · qjq2
j + m2π
(τ i · τ j)(σi · qj),
VE = cE2f 4πΛχ
∑i 6=j
τ i · τ j .
![Page 58: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/58.jpg)
NNN InteractionThe Form of the Force – Momentum Space
π π
c1,c3,c4
π
cD cE
VC = 12
(gA2fπ
)2 ∑π(ijk)
(σi · qi)(σj · qj)(q2
i + m2π)(q2
j + m2π)Fαβ
ijk ταi τ
βj ,
Fαβijk = δαβ
[−4c1m2
π
f 2π
+ 2c3
f 2π
(qi · qj)]
+ c4
f 2π
εαβγτγk [σk · (qi × qj)],
VD = − gA8f 2π
cDf 2πΛχ
∑π(ijk)
σj · qjq2
j + m2π
(τ i · τ j)(σi · qj),
VE = cE2f 4πΛχ
∑i 6=j
τ i · τ j .
![Page 59: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/59.jpg)
NNN InteractionThe Form of the Force – Momentum Space
π π
c1,c3,c4
π
cD cE
VC = 12
(gA2fπ
)2 ∑π(ijk)
(σi · qi)(σj · qj)(q2
i + m2π)(q2
j + m2π)Fαβ
ijk ταi τ
βj ,
Fαβijk = δαβ
[−4c1m2
π
f 2π
+ 2c3
f 2π
(qi · qj)]
+ c4
f 2π
εαβγτγk [σk · (qi × qj)],
VD = − gA8f 2π
cDf 2πΛχ
∑π(ijk)
σj · qjq2
j + m2π
(τ i · τ j)(σi · qj),
VE = cE2f 4πΛχ
∑i 6=j
τ i · τ j .
![Page 60: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/60.jpg)
NNN InteractionThe Form of the Force – Momentum Space
π π
c1,c3,c4
π
cD cE
VC = 12
(gA2fπ
)2 ∑π(ijk)
(σi · qi)(σj · qj)(q2
i + m2π)(q2
j + m2π)Fαβ
ijk ταi τ
βj ,
Fαβijk = δαβ
[−4c1m2
π
f 2π
+ 2c3
f 2π
(qi · qj)]
+ c4
f 2π
εαβγτγk [σk · (qi × qj)],
VD = − gA8f 2π
cDf 2πΛχ
∑π(ijk)
σj · qjq2
j + m2π
(τ i · τ j)(σi · qj),
VE = cE2f 4πΛχ
∑i 6=j
τ i · τ j .
![Page 61: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/61.jpg)
NNN InteractionThe Form of the Force – Momentum Space
π π
c1,c3,c4
π
cD cE
VC = 12
(gA2fπ
)2 ∑π(ijk)
(σi · qi)(σj · qj)(q2
i + m2π)(q2
j + m2π)Fαβ
ijk ταi τ
βj ,
Fαβijk = δαβ
[−4c1m2
π
f 2π
+ 2c3
f 2π
(qi · qj)]
+ c4
f 2π
εαβγτγk [σk · (qi × qj)],
VD = − gA8f 2π
cDf 2πΛχ
∑π(ijk)
σj · qjq2
j + m2π
(τ i · τ j)(σi · qj),
VE = cE2f 4πΛχ
∑i 6=j
τ i · τ j .
![Page 62: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/62.jpg)
NNN InteractionThe Form of the Force – Definitions
F
{π
i j}→ Xij(rij)
where
Xij(rij) = Sij(rij)T (rij) + σi · σjY (rij),
T (r) =(
1 + 3mπr + 3
(mπr)2
)Y (r),
Y (r) = exp (−mπr)mπr
(1− e−(r/R0)4
),
δ(3)(rij)→ ∆ij ≡1
πΓ(3/4)R30
exp(−(rij/R0)4).
![Page 63: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/63.jpg)
NNN InteractionThe Form of the Force – Real Space: 1π-Exchange + Contact
F{
cD
}→ 1π-Exchange + Contact
Some ambiguity here: which nucleons are participating in the pion exchange?Two possible short-range structures.
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik −
22π
m3π
(σi · σk)∆ik
](∆ij + ∆kj)
or
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik(rkj)∆ij + Xik(rij)∆kj −
23π
m3π
(σi · σk)∆ij∆kj
]
A1π+C = gAcDm3π
253πΛχf 4π
![Page 64: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/64.jpg)
NNN InteractionThe Form of the Force – Real Space: 1π-Exchange + Contact
limmσ→∞
F{
σcD
}→ 1π-Exchange + Contact
Some ambiguity here: which nucleons are participating in the pion exchange?Two possible short-range structures.
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik −
22π
m3π
(σi · σk)∆ik
](∆ij + ∆kj)
or
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik(rkj)∆ij + Xik(rij)∆kj −
23π
m3π
(σi · σk)∆ij∆kj
]
A1π+C = gAcDm3π
253πΛχf 4π
![Page 65: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/65.jpg)
NNN InteractionThe Form of the Force – Real Space: 1π-Exchange + Contact
F{
cD
}→ 1π-Exchange + Contact
Some ambiguity here: which nucleons are participating in the pion exchange?Two possible short-range structures.
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik −
22π
m3π
(σi · σk)∆ik
](∆ij + ∆kj)
or
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik(rkj)∆ij + Xik(rij)∆kj −
23π
m3π
(σi · σk)∆ij∆kj
]
A1π+C = gAcDm3π
253πΛχf 4π
![Page 66: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/66.jpg)
NNN InteractionThe Form of the Force – Real Space: 1π-Exchange + Contact
F{
cD
}→ 1π-Exchange + Contact
Some ambiguity here: which nucleons are participating in the pion exchange?Two possible short-range structures.
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik −
22π
m3π
(σi · σk)∆ik
](∆ij + ∆kj)
or
A1π+C∑
i<j<k
∑cyc.
(τ i · τ k)[Xik(rkj)∆ij + Xik(rij)∆kj −
23π
m3π
(σi · σk)∆ij∆kj
]
A1π+C = gAcDm3π
253πΛχf 4π
![Page 67: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/67.jpg)
NNN InteractionThe Form of the Force – Real Space: Contact
F{
cE
}→Contact
The same ambiguity here:
AC∑
i<j<k
∑cyc.
(τ i · τ k)∆ij∆kj
AC
2∑
i<j<k
∑cyc.
(τ i · τ k)∆ij(∆kj + ∆ik)
AC = cEΛχf 4
π
![Page 68: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/68.jpg)
NNN InteractionThe Form of the Force – Real Space: Contact
F{
cE
}→Contact
The same ambiguity here:
AC∑
i<j<k
∑cyc.
(τ i · τ k)∆ij∆kj
AC
2∑
i<j<k
∑cyc.
(τ i · τ k)∆ij(∆kj + ∆ik)
AC = cEΛχf 4
π
![Page 69: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/69.jpg)
NNN InteractionThe Fits
What to fit cD and cE to?Uncorrelated observables.Probe properties of light nuclei: 4He EB.Probe T = 3/2 physics: n-α scattering phase shifts.
![Page 70: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/70.jpg)
NNN InteractionCurrent Work - 4He EB Fit
Curves of cE vs. cD, fitting to 4He EB.
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
-1 0 1 2 3
cE
cD
Fitting the 4He binding energy
Preliminary
R0 = 1.0 fm
R0 = 1.2 fm
![Page 71: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/71.jpg)
NNN Interaction5He Fit
For low-energy scattering and one open channel of total angular momentum J ,
Ψ ∝ {Φc1Φc2YL}J [cos δJLjL(krc)− sin δJLnL(krc)] ,
Impose9
n ·∇rc Ψ = γΨ at rc = R0
⇒ tan δJL = γjL(kR0)− kj ′L(kR0)γnL(kR0)− kn′L(kR0)
9Kenneth M. Nollet, Steven C. Pieper, R. B. Wiringa, J. Carlson, G. M. Hale,PRL 99, 022502 (2007)
![Page 72: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/72.jpg)
NNN Interaction5He Fit
Reject samples with rc > R0, but
Ψn+1(R′) =∫|rc|<R0
dRc1dRc2drcG(R′,R; ∆t)Ψn(R)
+∫|re|>R0
dRc1dRc2drcG(R′,R; ∆t)
maps to
Ψn+1(R′) =∫|r|<R0
dRc1dRc2drcG(R′,R; ∆τ)Ψn(R)
×
[Ψn(R) + G(R′,Re; ∆τ)
G(R′,R; ∆τ)
(rerc
)3Ψn(Re)
]
The wave function at the (n + 1)th ∆t step gets a contribution from theprevious point R and an “image” point at Re.
![Page 73: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/73.jpg)
NNN InteractionCurrent Work - 5He Fit
Results showed the need for greaterspin-orbit splitting than was providedby the largely Fujita-Miyazawa-likeUIX NNN interaction.
![Page 74: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/74.jpg)
NNN InteractionCurrent Work - 5He Fit
5He fit to P3/2− wave. R-matrix analysis provided by G. M. Hale.
0
20
40
60
80
100
120
140
160
180
0 1 2 3 4 5
δ (
deg.)
Ecm (MeV)
n-α elastic P-wave phase shifts
3/2–
1/2–
Preliminary
R0 = 1.0 fm
R0 = 1.2 fm
R Matrix
![Page 75: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/75.jpg)
NNN InteractionCurrent Work - Results for Light Nuclei
-29
-28
-27
3H
3He
4He
Eb (
MeV
)
Preliminary-9
-8
-7
Preliminary
3H
3He
4He
1
1.25
1.5
1.75
2
rpt. (fm
)Preliminary
R0 = 1.0 fmR0 = 1.2 fm
Exp.
![Page 76: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/76.jpg)
ConclusionSummary
Low-energy nuclear theory: QMC+χEFT may help answer manyinteresting questions in physics.Phenomenological potentials have been very successful but there arecompelling reasons to investigate and compare them to chiral EFTinteractions.GFMC calculations of light nuclei are now possible with chiralEFT interactions.The softest of the potentials with R0 = 1.2 fm is more perturbative in thedifference between N2LO and NLO.The high-momentum (short-range) behavior of chiral EFT interactions isdistinct from the phenomenological interactions.The consistent 3N interaction at N2LO appears to be able to fitsimultaneously properties of light nuclei and n-α scatteringphase shifts.
![Page 77: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/77.jpg)
ConclusionFuture Work
Include 2-nucleon force at N3LO (which will be partly non-local).Extend to larger nuclei with 4 < A ≤ 12: compare with Argonne+Illinoisresults.Derive and implement electroweak currents in a consistent chiral EFTframework.Include ∆-full N3LO NN chiral EFT interaction of M. Piarulli et al. inGFMC. Derive and implement consistent 3N interaction.Extend n-α scattering framework to other (multichannel?) reactions, e. g.n-α→ d + t.Investigate the role of pairing correlations in AFDMC calculations ofneutron matter with small proton fractions and what effect this has onneutron-star properties.Parity- and CP-violating observables: Electric dipole moments of lightnuclei: 6Li?
![Page 78: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/78.jpg)
ConclusionAcknowledgments
Los Alamos National Laboratory:S. Gandolfi, J. CarlsonUniversity of Guelph:
A. GezerlisTechnische Universitat Darmstadt:
A. Schwenk, I. TewsUniversitat Bochum:
E. Epelbaum
![Page 79: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/79.jpg)
ConclusionAcknowledgments
Thank you!
![Page 80: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/80.jpg)
MotivationAb-Initio Calculations of Nuclei - GFMC
The trial wave function is a symmetrized product of correlation operatorsacting on a Jastrow wave function.
Trial Wave Function
|ΨT〉 =[S∏i<j
(1 + Uij)]|ΨJ 〉 ,
Uij =m∑
p=2up(rij)Op
ij , |ΨJ〉 =∏i<j
fc(rij) |ΦA〉 ,
|Φ4〉 = A |p↑ p↓ n↑ n↓〉 ,
|ΨT〉 =[S∏i<j
(1 + uσ(rij)σi · σj + utτ (rij)Sijτ i · τ j)]∏
i<jfc(rij) |Φ4〉
![Page 81: Green's Function Monte Carlo Calculations of Light Nuclei ...nuclearphysicsworkshops.github.io/.../doc/.../lynn.pdf · Big Questions Low-energy nuclear theory connects several research](https://reader034.fdocuments.in/reader034/viewer/2022042206/5ea940a2d9b9e134ca050e7d/html5/thumbnails/81.jpg)
NN Results4He Binding Energies - 〈H〉
-48
-46
-44
-42
LO NLO N2LO Av18
Eb (
Me
V)
Chiral Order
-30
-28
-26
-24
-22
-20
-18
-16
Exp.R0 = 1.0 fmR0 = 1.1 fmR0 = 1.2 fm
Figure: 4He binding energy at different chiral orders and cutoff values. SFR dependence weak.