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

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

MotivationBig Questions

Low-energy nuclear theoryconnects several researchareas in physics.

QMC+χEFT is acompelling piece of thepuzzle.

Low-EnergyNuclear Theory

FundamentalSymmetries

NuclearAstrophysics

NuclearStructure

MotivationBig Questions

Low-energy nuclear theoryconnects several researchareas in physics.QMC+χEFT is acompelling piece of thepuzzle.

QMC+χEFT

FundamentalSymmetries

NuclearAstrophysics

NuclearStructure

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

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?

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?

BackgroundThe Framework

Quantum Chromodynamics (QCD) isultimately responsible for stronginteractions: nucleons are the relevantdegrees of freedom for low-energynuclear physics→Nucleon-Nucleonpotentials.

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?

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.

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〉 .

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〉 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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]

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.

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)

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

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.

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.

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.

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.

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.

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)

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)

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)

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))

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))

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

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)

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)

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

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

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

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

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

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

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

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′

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

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 .

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 .

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 .

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 .

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 .

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).

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

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

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

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

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

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

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.

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

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)

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.

NNN InteractionCurrent Work - 5He Fit

Results showed the need for greaterspin-orbit splitting than was providedby the largely Fujita-Miyazawa-likeUIX NNN interaction.

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

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.

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.

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?

ConclusionAcknowledgments

Los Alamos National Laboratory:S. Gandolfi, J. CarlsonUniversity of Guelph:

A. GezerlisTechnische Universitat Darmstadt:

A. Schwenk, I. TewsUniversitat Bochum:

E. Epelbaum

ConclusionAcknowledgments

Thank you!

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〉

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.