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arXiv:0912.4195v2 [nucl-th] 22 Mar 2010

Lattice effective field theory calculations for A = 3,4,6,12 nuclei

Evgeny Epelbauma,b, Hermann Krebsb,a, Dean Leec,b, Ulf-G. Meißnerb,a,d

aInstitut f¨ ur Kernphysik (IKP-3) and J¨ ulich Center for Hadron Physics,

Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany

bHelmholtz-Institut f¨ ur Strahlen- und Kernphysik (Theorie) and

Bethe Center for Theoretical Physics,

Universit¨ at Bonn, D-53115 Bonn, Germany

cDepartment of Physics, North Carolina State University,

Raleigh, NC 27695, USA

dInstitute for Advanced Simulation (IAS),

Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany

We present lattice results for the ground state energies of tritium, helium-3, helium-4, lithium-6,

and carbon-12 nuclei. Our analysis includes isospin-breaking, Coulomb effects, and interactions up

to next-to-next-to-leading order in chiral effective field theory.

PACS numbers: 21.10.Dr, 21.30.-x, 21.45-v, 21.60.De

Several ab initio approaches have been used to calcu-

late the properties of various few- and many-nucleon sys-

tems. Some recent work includes the no-core shell model

[1–5], constrained-path [6–9] and fixed-node [10, 11]

Green’s function Monte Carlo, auxiliary-field diffusion

Monte Carlo [12–14], and coupled cluster methods [15–

17]. The diversity of methods is useful since each tech-

nique has its own computational scaling, systematic er-

rors, and range of accessible problems.

quantities not directly measured in experiments can be

benchmarked with calculations using other methods.

Furthermore,

Another ab initio approach in the recent literature is

lattice effective field theory. This method combines the

theoretical framework of effective field theory (EFT) with

numerical lattice methods. When compared with other

methods it is unusual in that all systematic errors are

introduced up front when defining the truncated low-

energy effective theory.This eliminates approximation

errors tied with a specific calculational tool, physical sys-

tem, or observable.By including higher-order interac-

tions in the low-energy effective theory, one can reason-

ably expect systematic and systemic improvement for all

low-energy observables. The approach has been used to

simulate nuclear matter [18] and neutron matter [19–24].

The method has also been applied to nuclei with A ≤ 4

in pionless EFT [25] and chiral EFT [26, 27]. A review

of lattice effective field theory calculations can be found

in Ref. [28].

In this letter we present the first lattice calculations for

lithium-6 and carbon-12 using chiral effective field the-

ory. We address a fundamental question in the nuclear

theory community: Can effective field theory be applied

to nuclei beyond the very lightest? While there are sev-

eral calculations that probe this question using interac-

tions derived from chiral effective field theory, we present

the first calculations posed and computed entirely within

the framework of effective field theory. Our results show

that lattice-regularized effective field theory can be ap-

plied to the ground state of carbon-12.

there is a clear path towards larger nuclei and nuclear

matter. We also describe the first lattice calculations to

include isospin-breaking and Coulomb interactions, and

compute the energy splitting between helium-3 and the

triton. Our discussion focuses on new features of the

calculation and new results.

the calculational method is contained in a separate paper

[29].

The low-energy expansion in effective field theory is

organized in powers of Q/Λ, where Q is the low momen-

tum scale associated with external nucleon momenta or

the pion mass, and Λ is the high momentum scale at

which the effective theory breaks down.

of chiral effective field theory can be found in Ref. [30–

33]. At leading order (LO) in the Weinberg power count-

ing scheme the nucleon-nucleon effective potential con-

tains two independent contact interactions and instanta-

neous one-pion exchange.

ies we make use of an “improved” leading-order action.

This improved leading-order action is treated completely

non-perturbatively, while higher-order interactions are

included as a perturbative expansion in powers of Q/Λ.

We use the improved LO3 lattice action intro-

duced in Ref. [23] with spatial lattice spacing a =

(100 MeV)−1= 1.97 fm and temporal lattice spacing

at= (150 MeV)−1= 1.32 fm. The interactions provide

a good description of the neutron-proton S-wave and P-

wave phase shifts at low energies as well as the S-D mix-

ing angle. Plots of the scattering data for the LO3lat-

tice action can be found in Ref. [23]. The corrections at

next-to-leading order (NLO) and next-to-next-to-leading

order (NNLO) are calculated using perturbation theory.

A description of these interactions on the lattice is doc-

umented in Ref. [27].

At NLO there are corrections to the two leading-order

Furthermore

A complete description of

Some reviews

As in previous lattice stud-

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2

coefficients and seven additional unknown coefficients for

operators with two powers of momentum. These nine co-

efficients are determined by fitting to the neutron-proton

S-wave and P-wave phase shifts and S-D mixing an-

gle at low energies. At NNLO there are two addi-

tional cutoff-dependent coefficients associated with three-

nucleon interactions. These are parameterized by two

dimensionless coefficients cDand cE, corresponding with

the three-nucleon one-pion exchange diagram and three-

nucleon contact interaction respectively.

cE by requiring that the triton energy equals the phys-

ical value of −8.48 MeV. However the parameter cD is

relatively unconstrained by low-energy phenomena such

as the deuteron-neutron spin-doublet phase shifts. Cur-

rently we are investigating other methods for constrain-

ing cD, including one recent suggestion to determine cD

from the triton beta decay rate [34].

we simply use the estimate cD ∼ O(1) and check the

dependence of observables upon changes in cD.

In addition to isospin-symmetric interactions, we also

include isospin-breaking (IB) and electromagnetic (EM)

interactions.Isospin violation in effective field theory

has been addressed extensively in the literature [35–40].

In the counting scheme proposed in Ref. [40], the isospin-

breaking one-pion exchange interaction and Coulomb po-

tential are numerically the same size as O(Q2/Λ2) cor-

rections at NLO. On the lattice we treat the Coulomb

potential in position space with the usual αEM/r depen-

dence. However this definition is singular for two pro-

tons on the same lattice site and requires short-distance

renormalization via a proton-proton contact interaction.

In this study we include all possible contact interactions,

namely interactions for neutron-neutron, proton-proton,

spin-singlet neutron-proton, and spin-triplet neutron-

proton. The two neutron-proton contact interactions are

already included at NLO and determined from neutron-

proton scattering. The other two coefficients are de-

termined from fitting to S-wave phase shifts for proton-

proton scattering and the neutron-neutron scattering

length. Details of this calculation will be presented in a

separate paper [29].

The first results we present are for helium-3 and the tri-

ton. The three-nucleon system is sufficiently small that

we can use iterative sparse-matrix eigenvector methods

to compute helium-3 and the triton on cubic periodic lat-

tices. We consider cubes with side lengths L up to 16 fm

and extract the infinite volume limit using the asymp-

totic parameterization [41], E(L) ≈ E(∞) − ce−L/L0/L.

While the triton energy at infinite volume is used to set

the unknown coefficient cE, the energy splitting between

helium-3 and the triton is a prediction that can be com-

pared with experiment. The energy difference between

helium-3 and the triton is plotted in Fig. 1 as a function

of cube length.We show several different asymptotic

fits using different subsets of data points.

der at which we are working there is no dependence of

We constrain

In this analysis

To the or-

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25

E3He − Etriton (MeV)

L (fm)

lattice

physical (infinite volume)

FIG. 1: Plot of the energy difference between helium-3 and

the triton as a function of periodic cube length.

the energy splitting upon the value of cD.

lations at next-to-next-to-leading order give a value of

0.780 MeV with an infinite-volume extrapolation error of

±0.003 MeV. To estimate other errors we take into ac-

count an uncertainty of ±1 fm in the neutron scattering

length and a 5% relative uncertainty in our lattice fit of

the splitting between neutron-proton and proton-proton

phase shifts at low energies. Our final result for the en-

ergy splitting with error bars is 0.78(5) MeV. This agrees

well with the experimental value of 0.76 MeV.

For systems with more than three nucleons, we use pro-

jection Monte Carlo with auxiliary fields and extract the

properties of the ground state using Euclidean-time pro-

jection. The transfer matrix, M, is the normal-ordered

exponential of the Hamiltonian over one temporal lattice

spacing. As in previous lattice Monte Carlo simulations

we first define a transfer matrix MSU(4)?πwhich is invari-

ant under Wigner’s SU(4) symmetry rotating all spin and

isospin components of nucleons.

acts as an approximate low-energy filter that happens to

be computationally inexpensive. Starting from a Slater

determinant of free-particle standing waves,

construct the trial state |Ψ(t′)? by successive multiplica-

tion,

Our calcu-

This transfer matrix

??Ψfree?, we

|Ψ(t′)? =?MSU(4)?π

?Lto??Ψfree?, (1)

where t′= Ltoatand Ltois the number of “outer” time

steps.The trial function |Ψ(t′)? is then used as the

starting point for the calculation.

is defined as

The amplitude Z(t)

Z(t) = ?Ψ(t′)|(MLO)Lti|Ψ(t′)?,(2)

where t = Ltiat and Ltiis the number of “inner” time

steps. The transient energy E(t) is proportional to the

logarithmic derivative of Z(t), and the ground state en-

ergy is given by the limit of E(t) as t → ∞.

of the transfer matrices are functions of the auxiliary

fields and pion fields, and the Monte Carlo integration

over field configurations is performed using hybrid Monte

Each

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-32

-30

-28

-26

-24

-22

-20

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

< E4He > (MeV)

LO

+ ∆NLO

+ ∆IB + ∆EM

+ ∆NNLO

+ 4N Contact

-2

0

2

4

6

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

∆NLO

∆IB + ∆EM

∆NNLO

4N Contact

FIG. 2: Ground state energy for helium-4 as a function of

Euclidean time projection.

Carlo.

tions, isospin breaking (IB), and electromagnetic interac-

tions (EM) are incorporated using perturbation theory.

In Fig. 2 we show lattice results for the ground state

of helium-4 in a periodic cube of length 9.9 fm. For the

numerical extrapolation in Euclidean time we use the de-

caying exponential functions described in Ref. [27]. The

plot on the left shows the contributions from leading-

order and higher-order contributions added cumulatively.

The plot on the right shows the higher-order corrections

separately. For each case we show the best fit as well

as the one standard-deviation bound. We estimate this

bound by generating an ensemble of fits determined with

added random Gaussian noise proportional to the er-

ror bars of each data point and varying the number of

fitted data points.These results are similar to those

found in Ref. [27] using the LO2 action.

we get −30.5(4) MeV at LO, −30.6(4) MeV at NLO,

−29.2(4) MeV at NLO with IB and EM corrections, and

−30.1(5) MeV at NNLO. The helium-4 energy decreases

0.4(1) MeV for each unit increase in cD.

The size of the corrections at NNLO gives an estimate

of the remaining error from higher-order terms in the

effective field theory expansion.

mentum scale of Λ = π/a = 314 MeV, an error of 1

to 2 MeV is consistent with the expected size of higher-

order contributions. Interactions at higher order than

NNLO are beyond the scope of the current calculation.

However if it happens that the higher-order effects are

most important when all four nucleons are in close prox-

imity, then we should see universal behavior which can be

reproduced by an effective four-nucleon contact interac-

tion. We test this universality hypothesis by introducing

an effective four-nucleon contact interaction tuned to re-

produce the physical helium-4 energy of −28.3 MeV. The

contribution of this interaction in helium-4 is shown in

Fig. 2.

In Fig. 3 we show lattice results for the ground state of

lithium-6 in a periodic cube of length 9.9 fm. For cD= 1

Contributions due to NLO and NNLO interac-

For cD = 1

Given our cutoff mo-

-36

-34

-32

-30

-28

-26

-24

-22

-20

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

< E6Li > (MeV)

LO

+ ∆NLO

+ ∆IB + ∆EM

+ ∆NNLO

+ 4N Contact

-4

-2

0

2

4

6

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

∆NLO

∆IB + ∆EM

∆NNLO

4N Contact

FIG. 3: Ground state energy for lithium-6 as a function of

Euclidean time projection.

we get −32.6(9) MeV at LO, −34.6(9) MeV at NLO,

−32.4(9) MeV at NLO with IB and EM corrections, and

−34.5(9) MeV at NNLO.

the effective four-nucleon interaction to the NNLO re-

sult gives −32.9(9) MeV. This lies within error bars of

the physical value −32.0 MeV. However we expect some

overbinding due to the finite periodic volume, and the

deviation of 0.9 MeV is consistent with the expected size

of the finite volume correction.

varying volumes will be needed to determine this volume

dependence. Without the effective four-nucleon interac-

tion, the lithium-6 energy decreases 0.7(1) MeV for each

unit increase in cD. With the effective four-nucleon in-

teraction the lithium-6 energy decreases 0.35(5) MeV per

unit increase in cD.

In Fig. 4 we show lattice results for the ground state

of carbon-12 in a periodic cube of length 13.8 fm.

cD = 1 we get −109(2) MeV at LO, −115(2) MeV at

NLO, −108(2) MeV at NLO with IB and EM corrections,

and −106(2) MeV at NNLO. Adding the contribution of

the effective four-nucleon interaction to the NNLO result

gives −99(2) MeV. This is an overbinding of 7% from the

physical value, −92.2 MeV. We note that an overbinding

of 7% is actually a reasonable estimate of the finite vol-

ume correction for carbon-12 in a periodic box of length

13.8 fm. This suggests that at infinite volume the error

is significantly smaller than 7%. Further calculations at

varying volumes will be needed to measure the volume

dependence. Without the effective four-nucleon interac-

tion, the carbon-12 energy decreases 1.7(3) MeV for each

unit increase in cD. With the effective four-nucleon in-

teraction the carbon-12 energy decreases 0.3(1) MeV per

unit increase in cD.

The results for lithium-6 and carbon-12 appear to con-

firm the universality hypothesis regarding higher-order

interactions. The much reduced dependence upon on cD

is also consistent with the universality hypothesis. The

effective four-nucleon contact interaction can be viewed

as absorbing the dependence on cD.

Adding the contribution of

Further calculations at

For

We note a re-

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-120

-110

-100

-90

-80

-70

-60

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

< E12C > (MeV)

LO

+ ∆NLO

+ ∆IB + ∆EM

+ ∆NNLO

+ 4N Contact

-10

-5

0

5

10

15

20

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1)

∆NLO

∆IB + ∆EM

∆NNLO

4N Contact

FIG. 4: Ground state energy for carbon-12 as a function of

Euclidean time projection.

cent related paper on the renormalization group evo-

lution of higher-nucleon interactions [42].

racy of these lattice calculations are competitive with

recent calculations obtained using other ab initio meth-

ods. Constrained-path Green’s function Monte Carlo

calculations and no-core shell model calculations have

an accuracy of 1% − 2% in energy for nuclei A ≤ 12.

Coupled cluster calculations without three-nucleon in-

teractions are accurate to within 1 MeV per nucleon for

medium mass nuclei. Future lattice studies should look

at probing large volumes, including higher-order effects,

and decreasing the lattice spacing.

Lattice effective field theory combines the generality of

effective field theory with the flexibility of lattice meth-

ods. The computational scaling of the calculations pre-

sented here indicates that larger systems with more nu-

cleons should be possible. By applying different lattice

boundary conditions in the spatial and temporal direc-

tions, it is possible to probe nuclear systems of many dif-

ferent varieties: few-body and many-body systems; zero

temperature and nonzero temperature; nuclear matter,

neutron matter, and asymmetric nuclear matter.

Partial financial support provided by the Deutsche

Forschungsgemeinschaft, Helmholtz Association, U.S.

Department of Energy, and EU HadronPhysics2 Project.

Computational resources provided by the J¨ ulich Super-

computing Centre.

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