Pion-less effective field theory for atomic nuclei and lattice nuclei

I. INTRODUCTION

• 16O was the heaviest nucleus previously computed in pionless EFT and was unstable against breakup into four 4He nuclei at LO.The introduction notes that applications beyond A ≥4 were few, while chiral EFT calculations had reached the A = 100 region [28].Source: §I · p. 1
• The work targets 16O and 40Ca with pionless EFT, fits low-energy coefficients to light-nucleus experimental data or lattice-QCD results, and uses the finite oscillator space itself as regulator.The implementation uses a DVR and no external regulator functions.Source: §I · p. 1
• Because lattice-QCD nuclear calculations disagree on whether binding rises or falls with pion mass, this work constrains its EFT coefficients using the lattice results of Ref..The competing conclusions come from plateau-based calculations and potential-based calculations.Source: §I · p. 1

A. Pion-less EFT

• Pionless EFT uses neutrons and protons below a breakdown scale set by the pion mass, while large S-wave scattering lengths motivate KSW power counting and demote other NLO-naive terms to N2LO.The S-wave LO coefficients scale as 1/ℵ rather than 1/mπ, with ℵ≈40 MeV; the scattering length appears at LO and effective range at NLO.Source: A. Pion-less EFT · p. 2
• The NLO potential is solved non-perturbatively because 6Li and 16O are unbound against alpha emission at LO, with cutoffs limited to about 700 MeV.The authors note that finite-order perturbation theory therefore cannot yield bound-state wave functions.Source: A. Pion-less EFT · p. 2

II. PIONLESS EFFECTIVE FIELD THEORY IN THE OSCILLATOR BASIS

• The harmonic-oscillator basis is chosen to enable computations of 16O and 40Ca, addressing the lack of established oscillator-basis infrastructure for pionless EFT.The authors identify a possible infrastructure mismatch with many-body solvers that start from oscillator-basis interactions.Source: §I · p. 1, A. Pion-less EFT · p. 2

B. Discrete Variable Representations

• A finite oscillator basis imposes infrared and ultraviolet cutoffs corresponding to hard walls in position and momentum space, and the position-space wall alters bound-state tails and observables.These cutoffs depend on the maximum oscillator quanta and oscillator length.Source: B. Discrete Variable Representations · pp. 2–3
• DVR momentum eigenfunctions localize around discrete eigenvalues, and the representation makes EFT matrix elements convenient while reproducing exact overlaps for functions within the finite oscillator space.For momenta below the basis IR cutoff, the DVR interaction does not vanish, motivating an improvement to its IR behavior.Source: B. Discrete Variable Representations · pp. 3–4

C. IR improvement of the NN interaction

• Changing the highest-momentum DVR function’s weight enforces the contact’s value at k=0, improving low-momentum accuracy at the possible expense of high-momentum accuracy.The unmodified DVR contact differs from the original interaction at the lowest momenta because the finite oscillator basis imposes an IR cutoff.Source: C. IR improvement of the NN interaction · pp. 4–5
• The IR-improved contact equals v(k)=1 at k=0 and the discrete DVR momenta, with substantially reduced oscillations and zero-momentum curvature decreasing as N^-1.This suppresses effective-range corrections, while the interaction remains smooth and rapidly vanishes near the finite-basis UV cutoff.Source: C. IR improvement of the NN interaction · p. 5, Figure 2 · p. 5
• The NLO DVR function has incorrect value and curvature at k=0; adjusting two coefficients gives the correct IR behavior while retaining agreement with w(k) at N-2 DVR points.The resulting IR improvement is shown for the NLO interaction and its momentum-space matrix elements.Source: C. IR improvement of the NN interaction · p. 6, Figure 4 · p. 6, Figure 5 · p. 6

D. IR improvement of the NNN contact

• The treatment considers a three-body contact with LEC C_NNN and two non-local regulators in the oscillator basis.Momenta k and k′ describe particles 1 and 2, while p and p′ describe particle 3 relative to their center of mass.Source: D. IR improvement of the NNN contact · p. 6
• For a contact interaction, the corresponding orbital angular momenta are zero, so the angular-momentum label is omitted.Source: D. IR improvement of the NNN contact · p. 6
• The matrix element (31) is not fully antisymmetrized, which the discussion says is not relevant here.Source: D. IR improvement of the NNN contact · p. 6

1. Cutoff in Jacobi momenta

• The cutoff can be applied separately to incoming Jacobi momenta k and p, using regulator factors f(p)f(k) multiplied with the interaction.Source: §1 · p. 6
• The DVR interaction in this scheme is a product of vDVR factors for the incoming and outgoing Jacobi momenta.Source: §1 · p. 6
• The NNN contact's infrared improvement is identical to the NN contact's; Eq. (32) uses vDVR(k) replaced by Eq. (23).Source: §1 · pp. 6–7
• Figure 6 plots vDVR(k)vDVR(p) for S waves in both Jacobi momenta with N = 8 and ℏω = 22 MeV.Source: Figure 6 · p. 7, §1 · p. 7

2. Hyperspherical cutoff

• The hyperspherical NNN contact depends only on hyperradial momentum ρ, and its relevant wave function has vanishing hyperspherical angular momentum.The contact is isotropic in hyperspherical coordinates; the associated hyperradial state is a six-dimensional harmonic-oscillator eigenstate.Source: §2 · p. 7
• The hypermomentum DVR uses discrete momenta ρµ at zeros of a Laguerre polynomial and represents the NNN contact in that basis.The DVR construction uses hyperradial momentum eigenfunctions to form the contact interaction.Source: §2 · p. 7
• Below the IR cutoff, the unmodified DVR contact deviates strongly from a constant, while IR improvement restores a near-constant low-momentum behavior.The integration measure dρρ5 suppresses low-momentum deficiencies in the usual scalar product.Source: §2 · pp. 7–8, Figure 7 · p. 7
• The DVR simplifies transformation of the interaction to Jacobi coordinates by providing Gauss-Laguerre integration exact for polynomials of degree N in k and p.This exact integration applies to the basis functions used in the transformation.Source: §2 · p. 8
• The hyperspherical cutoff becomes a triangular cutoff in the Jacobi oscillator basis because overlaps vanish when i + j > N.Accordingly, the NNN contact coupling carries the subscript “tr”; the paper uses the hyperspherical formulation unless otherwise specified.Source: §2 · p. 8

E. Discussion

• The harmonic-oscillator DVR potential matches the momentum-space potential only at discrete momenta, and its low-momentum behavior can be corrected to agree at zero momentum for NN and NNN forces.The paper presents this IR improvement as necessary for EFTs valid at the lowest momenta, while noting that its impact on well-bound nuclear observables is uncertain.Source: E. Discussion · p. 8
• Appendix studies cover DVR implementation and scaling, controlled IR improvement, the Wigner bound, regulator differences, NNN-force truncations, and IR extrapolations.The paper states that regulator differences at fixed UV cutoff are higher-order effects and that IR extrapolations work well in the DVR approach.Source: E. Discussion · p. 8

A. Atomic nuclei

• The oscillator-basis calculation uses N = 8 interaction matrix elements and enlarges the kinetic-energy space to achieve IR convergence, while oscillator spacing probes cutoff dependence.Reported binding energies for A = 2, 3, and 4 are described as virtually converged.Source: A. Atomic nuclei · p. 9
• NLO LECs are fitted using the deuteron binding energy and matter radius, low-energy neutron-proton phase shifts, and the triton binding energy.The NNN contact is adjusted after fixing the NN interaction; the radius input is 1.976 fm.Source: A. Atomic nuclei · p. 9
• NLO extends phase-shift agreement from momenta up to a_s,t^-1 at LO to r_s,t^-1, while IR improvement greatly reduces oscillations.The comparison is to CD-Bonn phase shifts; the plots use IR-improved potentials in a model space with N = 8 and ℏω = 22 MeV.Source: A. Atomic nuclei · p. 9, Figure 9 · p. 10
• At LO, including the NNN force makes cutoff dependence of light-nucleus binding energies and point-proton radii much weaker, but 4He remains underbound.The results are for 3H and 3,4He; the reported table covers A ≤ 4 nuclei with NN and NN + NNN interactions.Source: A. Atomic nuclei · pp. 9–10, Table III · p. 10
• At NLO, NN-only binding energies are E(3H) = 8.48 MeV, E(3He) = 7.5 MeV, and E(4He) = 28.5 MeV, with weak cutoff dependence.Adding the NNN contact further reduces cutoff dependence and brings 4He close to its physical point.Source: Table IV · p. 10, A. Atomic nuclei · p. 10
• The estimated 4He binding-energy uncertainty falls from about 8 MeV at LO to about 3 MeV at NLO, and the 28.3 MeV experimental value agrees with NLO within uncertainty.The estimates correspond to about 30% at LO and 10% at NLO; model-space convergence corrections are very small.Source: A. Atomic nuclei · pp. 10–11

B. Lattice nuclei

• LO contacts fit deuteron, di-neutron, and triton binding energies; NLO also fits singlet and triplet S-channel effective-range expansions alongside binding energies.The lattice-QCD inputs include the deuteron and di-neutron binding energies, effective-range data, and triton binding energy.Source: B. Lattice nuclei · p. 11
• At LO, the NN interaction’s 4He binding energy varies by a factor of two over the cutoff range, and adding the NNN contact reduces this dependence.Table VIII presents LO binding energies and point-proton radii for A = 3, 4 lattice nuclei.Source: Table VIII · p. 11, B. Lattice nuclei · p. 11
• At NLO, the EFT agrees with lattice-QCD effective-range input over a considerable range of momenta.Figure 10 compares the LO and NLO phase shifts with the lattice-QCD effective-range expansion and its uncertainty band.Source: B. Lattice nuclei · p. 11, Figure 10 · p. 12
• The inferred breakdown scale is Λb ≈ 1150 MeV, pF ≈ 370 MeV, and Q ≈ 0.4, yielding an NLO 4He binding-energy uncertainty of ≈ 15 MeV; lattice-input errors are the major uncertainty.The fit could not simultaneously reproduce binding energies and effective-range expansions at ℏω = 70 MeV; heavier-nucleus uncertainty discussion uses LECs fit to central lattice-QCD values.Source: B. Lattice nuclei · p. 12
• At NLO, cutoff dependence in A = 3, 4 binding energies and point-proton radii is strong with NN forces alone and much reduced when NNN forces are included.Table IX reports the binding energies and radii for both interaction choices.Source: B. Lattice nuclei · pp. 11–12, Table IX · p. 12

IV. RESULTS FOR 16O AND 40CA

• The calculations use coupled-cluster singles and doubles with a normal-ordered two-body approximation for the NNN potential, and the intrinsic Hamiltonian supports approximate separation of center-of-mass motion.The CCSD reference product state has the usual harmonic-oscillator shell closures for 4He, 16O, and 40Ca; the Hamiltonian eigenstates factor approximately into intrinsic and Gaussian center-of-mass wave functions.Source: §IV · p. 12

A. Atomic nuclei

• For larger cutoffs, the NLO binding energies of 16O and 40Ca are within 20% of experimental values of about 128 and 342 MeV, respectively.The coupled-cluster correlation energy is about 10% of the binding energy; three-body-space convergence is excellent at ℏω = 22 MeV and slower at other oscillator spacings.Source: A. Atomic nuclei · p. 13, Table X · p. 13
• At ℏω = 5 MeV, 40Ca has E/A ≈14 MeV and kF ≈210 MeV, marginally below the Λ ≈232 MeV cutoff, so the authors consider results at ℏω ≥10 MeV safest to discuss.The estimated NLO interaction uncertainty is about 10%, corresponding to ∆ENLO(16O) ∼15 MeV and ∆ENLO(40Ca) ∼40 MeV.Source: A. Atomic nuclei · p. 13

A. Atomic nuclei; B. Lattice nuclei

• Pionless EFT at NLO binds atomic 16O and 40Ca at about 9 and 10 MeV per nucleon, respectively, and lattice nuclei at approximately 30 MeV per nucleon for ℏω = 22 MeV.The atomic results are shown against UV cutoff in Fig. 11; the lattice result is reported for ℏω = 22 MeV.Source: A. Atomic nuclei · p. 13, Figure 11 · p. 13, B. Lattice nuclei · p. 14

B. Lattice nuclei

• For lattice nuclei, calculations are considered probably meaningful only at ℏω = 22 and 40 MeV, where the Fermi momentum is below the pion mass and EFT cutoff.The NLO binding-energy error is of order 15%, giving ∆ENLO(16O) ≈75 MeV and ∆ENLO(40Ca) ≈200 MeV; ℏω = 40 MeV carries an additional estimated 10% uncertainty from the NNN normal-ordering approximation.Source: B. Lattice nuclei · p. 14
• Using lattice-nucleus results at ℏω = 22 MeV, the authors find Bethe-Weizsäcker volume and surface terms aV ≈35 to 40 MeV and aS ≈14 to 22 MeV.The terms are obtained from the binding energy E(A) of an A-nucleon system.Source: B. Lattice nuclei · p. 14

V. SUMMARY

• The harmonic-oscillator DVR implementation tailors the UV cutoff and regulator to the basis, simplifies matrix-element computation, and enables direct interaction optimization in that basis.The authors report that the DVR yields an EFT with the correct low-momentum behavior; their calibration uses A = 2 and 3 systems, and NLO remedies the lack of alpha-decay stability at LO for 16O and 40Ca.Source: §V · p. 14
• Further progress connecting medium-mass nuclei to lattice QCD requires resolving disagreements among lattice approaches to light nuclei, improving input precision, and moving toward the physical pion mass.Source: §V · p. 14

Appendix A: Overview of Appendices

• Appendix C finds parametrically small, inverse-state-count effective-range corrections from LO-contact IR improvement; Appendix D derives frequency- and mass-scaling laws and recovers the Thomas effect and Tjon correlations.The scaling laws relate potential matrix elements across UV cutoffs and nucleon masses.Source: Appendix A · p. 15
• Appendix E confirms the Wigner bound, while Appendix F studies regulator dependence by comparing (N, ℏω) combinations with similar UV cutoffs.Source: Appendix A · p. 15
• Appendix G studies oscillator-truncation effects on NNN contacts with Jacobi-momentum cutoffs; Appendix H shows Lüscher-like formulas account for finite-size corrections in finite oscillator spaces.Source: Appendix A · p. 15

Appendix B: DVR with a zero-momentum point

• Varying R produces a continuous family of DVRs; the R choice shown with black triangles includes k = 0, unlike the R = 0 points used in the main text.Figure 12 shows the intersections that define these two sets of DVR points.Source: Appendix B · p. 15, Figure 12 · p. 16
• For the zero-momentum DVR, the points are κ = 0 plus the N_l roots of a Laguerre polynomial, with discrete momenta differing by partial wave.The appendix identifies the roots as those of L_l+3/2_N_l(κ^2b^2).Source: Appendix B · p. 15
• At large N_0, the main-text l = 0 DVR approaches a Dirichlet-like condition near r = L, whereas the zero-momentum DVR approaches a Neumann condition and other R values give mixed conditions.The respective momentum points approach μ π/L and (2μ + 1)π/(2L).Source: Appendix B · p. 16
• The zero-momentum DVR reproduces the original momentum-space interactions better than the non-IR-improved DVR, but its NLO phase shifts are slightly oscillatory because of incorrect curvature at k = 0.The appendix presents it as a simple alternative to IR improvement.Source: Appendix B · p. 16

• At low momenta and long wavelengths, the harmonic-oscillator basis behaves like a spherical cavity of radius L = π/k_0,0, enabling finite-size analysis in the cavity.Source: Appendix C · p. 16
• Adding a basis function at k_N+1 with half the usual amplitude gives v(k_μ) = 1 at the DVR momenta and v(0) = 1, with reduced oscillations.The improved function is even in k; the construction is motivated by the oscillatory, nonconstant contact function.Source: Appendix C · p. 18
• The IR-improved contact’s effective-range correction is proportional to 1/N and vanishes as N → ∞, identifying it as a finite-size effect.The quadratic correction’s scaling is also confirmed numerically.Source: Appendix C · p. 18
• Reducing the largest-momentum eigenfunction’s weight by about 0.5 motivates a simple non-local regulator that works well, particularly for chiral interactions.Source: Appendix C · p. 18

• At fixed shell number N, changing oscillator length or nucleon mass varies the UV cutoff and rescales contact-interaction and kinetic-energy matrix elements.The oscillator length changes through the oscillator frequency ℏω.Source: Appendix D · p. 18
• Rescaling b gives ˜ψn,l(k) ∝ b^3/2, cµ,l ∝ b^-3/2, and kµ,l ∝ b^-1, while the generalized Laguerre roots remain unchanged.Source: Appendix D · p. 18
• For infinite scattering length or a zero-energy bound state, CLO ∝ (ℏω)^-1/2, the oscillator-EFT counterpart of CLO ∝ (mΛ)^-1 in momentum space.The approximation is generally good when E2/(ℏω) ≪ 1.Source: Appendix D · p. 19
• The scaled A-body Hamiltonian is dimensionless and independent of oscillator length b, providing the scaling relation used to explain the binding-energy trends.Source: Appendix D · p. 19
• Increasing the cutoff (or ℏω) increases A = 3 binding (Thomas effect), while A = 3 and A = 4 binding energies correlate (Tjon line), as illustrated for triton and 4He.The figure shows LO results with pionless-EFT NN interactions at different UV cutoffs.Source: Appendix D · p. 19, Figure 15 · p. 19

Appendix E: Large UV cutoffs and the Wigner bound

• Beyond Λ ≈650 MeV, the NN effective range cannot be reproduced; the EFT obeys the Wigner bound, approaches zero effective range at very large cutoffs, and does not realize negative effective ranges.The UV cutoff is increased by raising the oscillator spacing in a finite harmonic-oscillator basis with N = 8.Source: Appendix E · p. 19, Figure 16 · p. 20

Appendix F: Regulator effects

• At fixed Λ ≈487 MeV, model spaces N = 6, 8, and 10 yield similar IR-improved NN potentials and regulator functions, particularly at low momenta.The corresponding oscillator frequencies are 26.63, 22, and 18.74 MeV; IR improvement slightly lowers the effective UV cutoff as N decreases.Source: Appendix F · p. 20, Figure 17 · p. 20
• NN LECs show only mild model-space dependence at both LO and NLO for the fixed-cutoff interactions.At NLO, the effective ranges are added as constraints in determining the LECs.Source: Appendix F · p. 20, Table XII · p. 20, Table XIII · p. 21
• The NNN contact interactions are similar across the three model spaces, especially at low momenta, including when regulated in each Jacobi momentum.The reported similarity also extends to NNN contacts regulated in each Jacobi momentum.Source: Appendix F · p. 20, Figure 18 · p. 21
• With the NNN coupling fitted to the triton binding energy, the study compares A = 3, 4 ground-state energies and matter radii across model spaces and NNN regulators.The coupling is fitted to Bt = 8.482 MeV; the comparison of hyperspherical and Jacobi-momentum regulation finds small NNN-contact regulator differences, as expected in an EFT.Source: Appendix F · pp. 20–21, Table XIV · p. 21

Appendix G: Effects of oscillator basis truncation on NNN contact

• The hyperspherical NNN contact is unaffected when N3 ≥ N, whereas the Jacobi-momentum-cutoff contact is affected when N3 < 2N = 16.Lowering N3 below 16 significantly modifies the Jacobi-regulated contact after truncation; this motivates the paper’s choice of hyperspherical regulator.Source: Appendix G · p. 21, Figure 19 · p. 21
• For N3 < 2N = 16, the 4He ground-state energy depends strongly on N3; with the alternative truncation, its binding energy increases with N.The alternative sets the NNN contact to vanish for n1, n2 > N/2 while retaining NNN excitations up to Nℏω, using N and ℏω combinations with similar UV cutoffs.Source: Appendix G · pp. 21–22

Appendix H: IR extrapolations

• IR extrapolations address finite-volume effects using the harmonic-oscillator radius L, while the model-space-tailored UV cutoff makes UV extrapolations unnecessary.The extrapolation formulas generalize Lüscher’s approach to the harmonic oscillator.Source: Appendix H · p. 22
• For the deuteron, the energy difference converges exponentially with L as a exp(−2k∞L), with a ≈15 MeV and k∞ = 0.2316 fm−1.The passage attributes this decay to the separation momentum and notes that its equality with k∞ is more accurate than reported in Ref., presumably because UV convergence is achieved.Source: Appendix H · p. 22, Figure 20 · p. 22
• For 3H, extrapolations from Nmax = 8, 10, 12 are much more accurate than finite-volume results from Nmax = 14 onward.The extrapolated k∞ and theoretical ksep become close but not identical as the model space grows; their agreement is better at LO with NN forces only.Source: Appendix H · p. 22
• For 4He, extrapolated k∞ is somewhat smaller than ksep, and the results are not yet converged as the model space increases.The example is an NLO calculation with a potential defined in N = 8 and ℏω = 40 MeV; the lowest-energy breakup channel is α → t + p.Source: Appendix H · pp. 22–23
• IR extrapolations for the EFT implemented as a harmonic-oscillator DVR work well overall and agree with expectations.The approach addresses finite-volume effects; the harmonic-oscillator basis provides a model-space-tailored UV cutoff, making UV extrapolations unnecessary.Source: Appendix H · pp. 22–23