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

Abstract

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. Comment: 4 pages, 4 figures, published version to appear in Phys. Rev. Lett.

Full text

Page 1

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-

Page 2

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

Page 3

3
-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-

Page 4

4
-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.
The accu-
[1] C. Forssen, P. Navratil, W. E. Ormand, and E. Caurier,
Phys. Rev. C71, 044312 (2005).
[2] A. Nogga, P. Navratil, B. R. Barrett, and J. P. Vary,
Phys. Rev. C73, 064002 (2006).
[3] I. Stetcu, B. R. Barrett, and U. van Kolck, Phys. Lett.
B653, 358 (2007).
[4] P. Navratil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand,
and A. Nogga, Phys. Rev. Lett. 99, 042501 (2007).
[5] P. Maris, J. P. Vary, and A. M. Shirokov, Phys. Rev.
C79, 014308 (2009).
[6] S. C. Pieper, K. Varga, and R. B. Wiringa, Phys. Rev.
C66, 044310 (2002).
[7] S. C. Pieper, Nucl. Phys. A751, 516 (2005).
[8] L. E. Marcucci, M. Pervin, S. C. Pieper, R. Schiavilla,
and R. B. Wiringa (2008), arXiv:0810.0547.
[9] S. C. Pieper, B. Am. Phys. Soc. 54, 70 (2009).
[10] S. Y. Chang et al., Nucl. Phys. A746, 215 (2004).
[11] A. Gezerlis and J. Carlson, Phys. Rev. C77, 032801
(2008); arXiv:0911.3907.
[12] S. Gandolfi, F. Pederiva, S. Fantoni, and K. E. Schmidt,
Phys. Rev. Lett. 99, 022507 (2007).
[13] S. Gandolfi, A. Y. Illarionov, K. E. Schmidt, F. Pederiva,
and S. Fantoni (2009), arXiv:0903.2610.
[14] S. Gandolfi et al. (2009), arXiv:0909.3487.
[15] M. Wloch et al., Phys. Rev. Lett. 94, 212501 (2005).
[16] G. Hagen, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock,
and A. Schwenk, Phys. Rev. C76, 044305 (2007).
[17] G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-
Jensen, Phys. Rev. Lett. 101, 092502 (2008).
[18] H. M. M¨ uller, S. E. Koonin, R. Seki, and U. van Kolck,
Phys. Rev. C61, 044320 (2000).
[19] D. Lee and T. Sch¨ afer, Phys. Rev. C72, 024006 (2005).
[20] D. Lee, B. Borasoy, and T. Sch¨ afer, Phys. Rev. C70,
014007 (2004).
[21] T. Abe and R. Seki, Phys. Rev. C79, 054002 (2009).
[22] B. Borasoy, E. Epelbaum, H. Krebs, D. Lee, and U.-G.
Meißner, Eur. Phys. J. A35, 357 (2008).
[23] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Eur.
Phys. J. A40, 199 (2009).
[24] G. Wlazlowski and P. Magierski (2009), arXiv:0912.0373.
[25] B. Borasoy, H. Krebs, D. Lee, and U.-G. Meißner, Nucl.
Phys. A768, 179 (2006).
[26] B. Borasoy, E. Epelbaum, H. Krebs, D. Lee, and U.-G.
Meißner, Eur. Phys. J. A31, 105 (2007).
[27] E. Epelbaum, H. Krebs, D. Lee, and U. G. Meißner, Eur.
Phys. J. A41, 125 (2009).
[28] D. Lee, Prog. Part. Nucl. Phys. 63, 117 (2009).
[29] E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner
(2010), in preparation.
[30] U. van Kolck, Prog. Part. Nucl. Phys. 43, 337 (1999).
[31] P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part.
Sci. 52, 339 (2002).
[32] E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006).
[33] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev.
Mod. Phys. 81, 1773 (2010).
[34] D. Gazit, S. Quaglioni, and P. Navratil, Phys. Rev. Lett.
103, 102502 (2009).
[35] U. van Kolck, J. L. Friar, and J. T. Goldman, Phys. Lett.
B371, 169 (1996).
[36] U. van Kolck, M. C. M. Rentmeester, J. L. Friar, J. T.
Goldman, and J. J. de Swart, Phys. Rev. Lett. 80, 4386
(1998).
[37] E. Epelbaum and U.-G. Meißner, Phys. Lett. B461, 287
(1999).
[38] M. Walzl, U. G. Meißner, and E. Epelbaum, Nucl. Phys.
A693, 663 (2001).
[39] J. L. Friar, U. van Kolck, G. L. Payne, and S. A. Coon,
Phys. Rev. C68, 024003 (2003).
[40] E. Epelbaum and U.-G. Meißner, Phys. Rev. C72,
044001 (2005).
[41] M. L¨ uscher, Commun. Math. Phys. 104, 177 (1986).
[42] E. D. Jurgenson, P. Navratil, and R. J. Furnstahl, Phys.
Rev. Lett. 103, 082501 (2009).