The two-nucleon system at next-to-next-to-next-to-leading order

E. Epelbaum; Ulf-G. Mei{\ss}ner; W. Gl\"ockle

REVIEW

The two-nucleon system at next-to-next-to-next-to-leading order

E. Epelbaum, W. Glöckle, Ulf-G. Mei{ss}ner

Nuclear Theory · Submitted 18 May 2004 · 121 references

Pith's verdict: A landmark N3LO two-nucleon potential, solid on central values, with an error band that is honestly rough rather than demonstrated.

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CitationE. Epelbaum, Ulf-G. Mei{\ss}ner, W. Gl\"ockle. "The two-nucleon system at next-to-next-to-next-to-leading order." arXiv:nucl-th/0405048 (2004). https://pith.science/paper/WUJEBQQV

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Editor's note

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This is the Epelbaum–Glöckle–Meißner N3LO paper, and it is one of the two papers that set the state of the art for chiral two-nucleon forces. If you work anywhere near nuclear EFT you already know it; the question is what to trust in it now, twenty years on. Short answer: trust the potential and the central values, treat the uncertainty bands as a rough guide, and do not quote the quadrupole moment or the matter radius as N3LO predictions without adding the two-body current the authors themselves say is missing in §6.4. What is genuinely new here, relative to Entem–Machleidt's earlier N3LO potential, is the spectral function regularization of multi-pion exchange, the explicit demonstration in §2.3 and Figs. 1–2 that three-pion exchange collapses to a few percent of the N3LO 2PE once the short-range part of the spectrum is cut off, and a complete 26-parameter fit with all LECs tabulated for four cutoff pairs (Tables 2–4). The deuteron binding energy in Table 10 is a real prediction, not a fit, and it lands within 0.4% — that is the strongest single number in the paper. The soft spot is the error estimate. The stress-test note is right that the cutoff band narrows in S/P/D waves and is excused from narrowing in F/G waves because those channels carry no contact terms until N5LO/N7LO. Both explanations are physically sensible, but together they mean the band cannot fail as a diagnostic. The authors call the estimate 'fairly rough' in §6.1, so this is acknowledged, not concealed; what is missing is any check that the band actually brackets the order-to-order shift. One point the note raises that the authors do not flag: §5 extends the fit range to Elab = 200 MeV at N3LO while NLO and NNLO used a narrower interval, so some of the high-energy improvement is a fit-range artifact. That is worth knowing when you compare Figs. 3–6 across orders. The multiple-minima caveat in §6.1.1 is real but not load-bearing, since NLO and NNLO were fit the same way. For a nuclear forces person this is required reading and a standard citation. It deserves referees, and it got them.

Referee report

Significance

If the result holds, this is the first systematic N3LO treatment of the two-nucleon system with a quantified error estimate and with the pion-nucleon sector kept consistent, and it establishes that the chiral expansion for NN is converging order by order in a controlled way. Several features deserve explicit credit: the peripheral F- and G-waves are parameter-free predictions and are reproduced; the deuteron binding energy in Table 10 is a genuine prediction, since the LECs were fixed on scattering and Ed was not used in the fit; the neglect of 3PE is argued with an explicit, falsifiable DR-versus-SFR comparison rather than asserted; and the prediction r_nn = 2.76...2.80 fm in Eq. (6.7) is a clean, checkable consequence of using a_nn as input. The LECs c1, c2, c3, c4 and the d-bar combinations are imported from pi-N analyses rather than tuned, which is a real consistency gain over purely phenomenological fits.

Major objections (4)

[Section 5 and Section 6.1] The central claim of abstract:s7, that the description is improved relative to NLO and NNLO, is supported only qualitatively. Nowhere in the paper is a chi-squared per degree of freedom, or any other quantitative fit-quality measure, tabulated for any channel at any order; Section 6.1.1 and 6.1.2 speak of results being 'visibly improved' and 'excellent'. For a paper whose headline is a quantitative order-by-order comparison, a table of chi-squared per partial wave (1S0, 3S1-3D1, the four P-waves, the D-waves) at NLO, NNLO and N3LO, evaluated over a common energy interval, is the natural and inexpensive evidence. Without it the reader cannot judge whether the visual improvement in Figs. 3-6 is larger than the residual scatter of the fit.
[Section 5, paragraph on fit ranges; Figs. 3-6] Section 5 states that at N3LO the fit range had to be extended to Elab = 200 MeV, whereas the NLO and NNLO analyses of Ref. [21] used a narrower interval. The order-to-order comparison in Figs. 3-6 is therefore not like-for-like above roughly Elab = 100 MeV, which is precisely where the N3LO curves look closest to the Nijmegen PWA. Part of the apparent improvement at higher energy may be a fit-range artifact rather than evidence of convergence. Either a refit of the lower orders over the same 200 MeV interval, or an explicit statement in Section 6.1 and in the abstract that the high-energy comparison is not apples-to-apples, would resolve this.
[Section 6.1, Eq. (5.1), and Section 6.1.3] The identification of the cutoff-variation spread with the theoretical uncertainty is the load-bearing assumption behind the 'controlled error' half of the claim (abstract:s8). The band is read as evidence of improvement when it narrows in the S-, P- and D-waves, and its failure to narrow in the F- and G-waves is explained in Section 6.1.3 as expected because those channels carry no contact terms until N5LO/N7LO. Both readings are physically motivated, but taken together they leave the band unable to fail as a diagnostic. The authors already call the estimate 'fairly rough' in Section 6.1, so the roughness is acknowledged; what is missing is a quantitative statement of whether the band brackets the actual order-to-order shift. A short table comparing, for a few representative channels and energies, the N3LO band width against the NNLO-to-N3LO central-value shift would make the proxy testable rather than interpretive.
[Sections 6.1.1 and 6.1.2, Eq. (6.3)] The paper reports multiple chi-squared minima for the contact LECs, states that for two of the four cutoff combinations several local minima were found in the 1S0 channel, and states in Section 6.1.1 that for the 3S1-3D1 channel 'we cannot definitely claim that the found values for the LECs correspond to a true global minimum'. Since the LECs are refit independently at each cutoff pair, part of the quoted cutoff band may reflect jumps between solution branches rather than genuine cutoff dependence. The authors should state explicitly whether the same branch was followed across all four pairs in Eq. (5.1), and should quote the chi-squared values of the discarded minima in Eq. (6.3) and in the 1S0 case so the reader can see how degenerate the alternatives are.

Minor objections (7)

[Fig. 13 caption] The caption specifies Lambda = 550 MeV and Lambda-tilde = 600 MeV, which is not one of the four pairs listed in Eq. (5.1). Either use a listed pair or state why this intermediate choice was made.
[Table 2 and Section 6.1.1] The LECs C-tilde_3S1 and C_3S1 change sign between the {450, 500} and {600, 600} cutoff pairs. Section 6.1.1 states that the cutoff uncertainty in the 3S1 channel is smaller than in 1S0; a brief remark on whether the sign change is consistent with that statement would help.
[Section 2.3 and Fig. 2] The text states that 3PE reaches at most 2%-8% of the N3LO 2PE for r > 0.5 fm. Please state explicitly whether this residual is folded into the uncertainty budget of Section 6.1 or is argued to be negligible at the energies considered.
[Section 5, Eq. (5.8)] The relation between the calculated pp 1S0 phase shift and the Nijmegen value uses the quantities Delta-tilde_0, rho_0 and tau_0 taken from Ref. [50], which were computed with the Nijmegen N78 potential. The residual model dependence of this correction, and hence of a_pp and r_pp in Eq. (6.4), should be stated.
[Abstract versus Table 10 and Section 6.4] The abstract's unqualified statement that deuteron observables are improved (abstract:s7) is not uniformly true: Qd and the rms matter radius in Table 10 are further from experiment at N3LO than at NNLO. The text acknowledges this in Section 6.4 and in summary item viii, so only the abstract wording needs qualifying.
[Sections 2.3, 2.4.1, 6.1.2 and Section 4] Several typographical errors should be corrected: 'per constraction' in Section 2.3, 'not well determined at presence' in Section 2.4.1, 'table 3 four these four pairs' in Section 6.1.2, and 'proton and neutron ms-radii' in Section 4.
[References] Reference [53] and Eq. (2.56) give 'E.A. Ueling' for Uehling; Ref. [49] gives 'G.J.M. Ausin' for Austen; Ref. [102] gives 'C. Alderlisten' for Alderliesten; Ref. [86] gives 'M.K. Nefkens' for B.M.K. Nefkens. The name Gonzalez Trotter is spelled inconsistently between Refs. [75] and [87].

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The paper

The paper claims

The authors push the chiral description of the two-nucleon force to fourth order, fit 26 short-range couplings to Nijmegen phase shifts, and argue that scattering and deuteron observables improve order by order with a cutoff-derived error.

The reading

This paper asks whether the chiral expansion for the force between two nucleons is actually converging in a controlled way. The authors construct the two-nucleon potential at next-to-next-to-next-to-leading order (N$^3$LO, fourth order in the low-momentum expansion): one-, two- and three-pion exchange plus contact interactions with zero, two and four derivatives, together with isospin-breaking and relativistic corrections. They regulate the multi-pion loops by cutting off the spectral functions rather than by dimensional regularization, which keeps the long-distance part of the force fixed by chiral symmetry while shunting the short-distance part into contact terms. Twenty-six four-nucleon low-energy constants are then fixed by a combined fit to selected np and pp phase shifts from the Nijmegen analysis plus the nn scattering length, and the deuteron is left as a prediction rather than an input. The payoff, if the argument holds, is that nuclear forces can be improved systematically order by order with a stated error bar, which is what any claim of precision nuclear theory from quantum chromodynamics has to deliver.

What carries the argument

The load-bearing device is spectral function regularization. Instead of regulating pion loop integrals with dimensional regularization, the authors write the two-pion exchange potential as an integral over a spectral function of the exchanged mass $mu$, and cut that integral off at $tilde{Lambda}$ between 500 and 700 MeV. Everything above the cutoff is short-distance and is absorbed into the contact interactions, so the long-distance asymptotics that chiral symmetry constrains is untouched while the short-range piece becomes a fit parameter. This is what lets them take the pion-nucleon low-energy constants $c_1, c_2, c_3, c_4$ and the $bar{d}_i$ from single-nucleon chiral perturbation theory analyses without re-extracting them. The second device is the scattering equation: a Lippmann-Schwinger equation with relativistic kinetic energy $2sqrt{p^2+m^2}-2m$, regulated by multiplying the potential with $exp(-p^6/Lambda^6)$ at $Lambda$ between 450 and 600 MeV. The pair of cutoffs $(Lambda, tilde{Lambda})$ is varied over four combinations, and the spread of the resulting predictions is presented as the theoretical uncertainty.

Extended reading notes

Core claim

The central claim is that the N$^3$LO two-nucleon potential, built with spectral function regularization and 26 fitted contact couplings, describes np and pp phase shifts and the deuteron better than the same scheme at NLO and NNLO. Concretely: the S-waves, P-waves and D-waves are visibly closer to the Nijmegen partial-wave analysis than at the previous orders, the parameter-free F- and G-waves are reproduced with bands of the expected width, the np and pp S-wave scattering lengths and effective ranges land near the Nijmegen values, and the deuteron binding energy comes out within about 0.4% of experiment without being fitted. A second, more structural claim is that three-pion exchange, which first appears at this order, is negligible once the spectral cutoff is imposed, so it can be dropped. The authors also flag where the calculation is admittedly incomplete: the deuteron quadrupole moment and matter radius are computed without the short-range two-nucleon current, which they say is why those two quantities do not improve.

Load-bearing premise

The quoted theoretical uncertainty is taken to be the spread of predictions under variation of the two cutoffs, $Lambda$ in 450-600 MeV and $tilde{Lambda}$ in 500-700 MeV, which assumes that cutoff sensitivity faithfully tracks the size of the omitted fifth-order terms.

Editorial extensions

If this is right

Nuclear forces can be improved order by order in a single systematic scheme, with a quantified error band attached to each prediction rather than a single number with no stated accuracy.
Three-pion exchange can be dropped from practical two-nucleon calculations in this regularization scheme, simplifying the potential without loss at the quoted accuracy.
The same potential, because it is derived from one effective Lagrangian and iterated in a relativistic equation, extends to three- and four-nucleon systems and to processes with external electroweak probes.
The deuteron binding energy, asymptotic S-wave normalization and asymptotic D/S ratio come out close to experiment without being fitted, so they count as genuine predictions of the scheme at this order.
Isospin-breaking effects from the pion mass difference in one-pion exchange and from the Coulomb interaction are of comparable size and often of opposite sign, as the authors' power-counting scheme predicts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The uncertainty bands are the weakest link in the 'controlled accuracy' story: they come from varying two cutoffs, not from computing the omitted fifth-order terms, so an independent check would be to run the same fit at N$^4$LO in one or two channels and see whether the true shift falls inside the quoted band.
Because the contact couplings are fitted to phase shifts rather than to scattering data directly, the scheme inherits whatever model dependence lives in the Nijmegen partial-wave analysis, including its treatment of electromagnetic corrections and its coordinate-space cutoff of the long-range potential at 1.4 fm.
The authors report multiple local minima in the coupling space for several cutoff choices and say they cannot confirm a global minimum in the eight-parameter triplet channel; a reader should treat the tabulated couplings as one viable solution rather than as uniquely determined constants.
The natural next test of the framework is whether the same couplings, transported unchanged, reproduce three-nucleon binding and scattering, since a scheme that only works in the two-body sector has not yet shown that its power counting survives iteration in larger systems.

Original abstract

We consider the two-nucleon system at next-to-next-to-next-to-leading order (N^3LO) in chiral effective field theory. The two-nucleon potential at N^3LO consists of one-, two- and three-pion exchanges and a set of contact interactions with zero, two and four derivatives. In addition, one has to take into account various isospin-breaking and relativistic corrections. We employ spectral function regularization for the multi-pion exchanges. Within this framework, it is shown that the three-pion exchange contribution is negligibly small. The low-energy constants (LECs) related to pion-nucleon vertices are taken consistently from studies of pion-nucleon scattering in chiral perturbation theory. The total of 26 four-nucleon LECs has been determined by a combined fit to some np and pp phase shifts from the Nijmegen analysis together with the nn scattering length. The description of nucleon-nucleon scattering and the deuteron observables at N^3LO is improved compared to the one at NLO and NNLO. The theoretical uncertainties in observables are estimated based on the variation of the cut-offs in the spectral function representation of the potential and in the regulator utilized in the Lippmann-Schwinger equation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,931 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.30 · randomized assignment 0.01 · states quantitative result 0.84 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We consider the two-nucleon system at next-to-next-to-next-to-leading order (N^3LO) in chiral effective field theory. The two-nucleon potential at N^3LO consists of one-, two- and three-pion exchanges and a set of contact interactions with zero, two and four derivatives. In addition, one has to take into account various isospin-breaking and relativistic corrections. We employ spectral function regularization for the multi-pion exchanges. Within this framework, it is shown that the three-pion exchange contribution is negligibly small. The low-energy constants (LECs) related to pion-nucleon vertices are taken consistently from studies of pion-nucleon scattering in chiral perturbation theory. The total of 26 four-nucleon LECs has been determined by a combined fit to some np and pp phase shifts from the Nijmegen analysis together with the nn scattering length. The description of nucleon-nucleon scattering and the deuteron observables at N^3LO is improved compared to the one at NLO and NNLO. The theoretical uncertainties in observables are estimated based on the variation of the cut-offs in the spectral function representation of the potential and in the regulator utilized in the Lippmann-Schwinger equation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the two-nucleon system at next-to-next-to-next-to-leading order (N^3LO) in chiral effective field theory.	contribution 0.74	0.65	0.04	asserted 0.80
s1	The two-nucleon potential at N^3LO consists of one-, two- and three-pion exchanges and a set of contact interactions with zero, two and four derivatives.	method 0.35	0.24	0.05	asserted 1.00
s2	In addition, one has to take into account various isospin-breaking and relativistic corrections.	method 0.45	0.12	0.02	asserted 0.99
s3	We employ spectral function regularization for the multi-pion exchanges.	method 1.00	0.79	0.03	asserted 0.98
s4	Within this framework, it is shown that the three-pion exchange contribution is negligibly small.	result 1.00	0.89	0.15	asserted 1.00
s5	The low-energy constants (LECs) related to pion-nucleon vertices are taken consistently from studies of pion-nucleon scattering in chiral perturbation theory.	method 1.00	0.30	0.03	asserted 1.00
s6	The total of 26 four-nucleon LECs has been determined by a combined fit to some np and pp phase shifts from the Nijmegen analysis together with the nn scattering length.	method 0.93	0.82	0.07	asserted 1.00
s7	The description of nucleon-nucleon scattering and the deuteron observables at N^3LO is improved compared to the one at NLO and NNLO.	result 0.98	0.88	0.74	asserted 1.00
s8	The theoretical uncertainties in observables are estimated based on the variation of the cut-offs in the spectral function representation of the potential and in the regulator utilized in the Lippmann-Schwinger equation.	method 0.97	0.86	0.05	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from simulation or computation 0.59.

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sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Isoscalar spin–spin 3PE potential using dimensio

**Figure 2.** The ratio of the isoscalar spin–spin 3PE and 2PE N

**Figure 3.** S–wave np phase shifts versus the nucleon laboratory energy. The grid, light shaded and dark shaded bands show the NLO, NNLO [21] and N3LO results, respectively. The cut–offs Λ and Λ at N ˜ 3LO are varied as specified in eq. (5.1). The filled circles depict the Nijmegen PWA results [38] and the open triangles are the results from the Virginia Tech PWA [106]. 52

**Figure 4.** P–wave np phase shifts and mixing angle ǫ1 versus the nucleon laboratory energy. For notation see

**Figure 5.** 3P1–wave np phase shift versus the nucleon laboratory energy. The solid and dashed lines correspond to the LECs C3P1 and D3P1 from the first and second lines in eq. (6.3), respectively. For remaining notations see

**Figure 6.** D–wave np phase shifts and mixing angle ǫ2 versus the nucleon laboratory energy. For notation see

**Figure 7.** F–wave np phase shifts and mixing angle ǫ3 versus the nucleon laboratory energy. For notation see

**Figure 8.** G–wave np phase shifts and mixing angle ǫ4 versus the nucleon laboratory energy. For notation see

**Figure 9.** np differential cross section and vector analyzing power at Elab = 25 MeV. The Nijmegen PWA result is taken from [107]. Data for the cross section are taken from [108] and for the analyzing power from [109, 110]. The cut–offs Λ and Λ are ˜ varied as specified in eq. (5.1). 58

**Figure 10.** np differential cross section and vector analyzing power at Elab = 50 MeV. Data for the cross section are taken from [108, 111] and for the analyzing power from [112, 113, 114, 115, 110]. For remaining notations see

**Figure 11.** np differential cross section and vector analyzing power at Elab = 96 MeV. Data for the cross section are taken from [118, 119, 116]. Data for the analyzing power are at Elab = 95 MeV and taken from [117]. For remaining notations see

**Figure 12.** np differential cross section and vector analyzing power at Elab = 143 MeV. Data for the cross section are at Elab = 142.8 MeV and taken from [120] and for the analyzing power from [121]. For remaining notations see

**Figure 13.** Coordinate space representation of the S– (upper

Figures reproduced from arXiv:nucl-th/0405048 by the authors.

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How later papers used it

107 later papers cite it

Evidence: substantial. 84 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 210 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 3 used its method or tool and 1 used its data or values. 2 more papers by the same authors are listed: 2 used its method or tool and 1 used its data or values. 1 of them appears under more than one label.

Uses its method or tool 5 of 14 papers with this label · 2 by the same authors

We evaluate the Wolfenstein amplitudes using the $χ^EFT$ $t$-matrix and optical potential, employing both the Gaussian regulator and EGM's SFR scheme with $(Λ,Λ̃) = (450,500)$, (550,600), and (600,600) MeV, for pp and pn scattering at 100 and 200 MeV [ref] .

Nucleon Nucleon Potential Using N$3$LO Chiral Effective Field Theory (2026) · Results and discussions · 0.97
In this paper, to treat isospin-breaking effects, we closely follow the analyzes proposed in Ref. [ref] .

Charge dependent nucleon-nucleon potentials in covariant chiral effective field theory (2025) · Isospin-breaking potentials · 1.00
Used this paper's standard spectral-function regularization with a cutoff of 700 MeV to regularize the 2-pi exchanges.

Delta isobars and nuclear saturation (2017) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

Employed the two-nucleon interaction derived by this paper using effective field theory.

Saturation of nuclear matter in effective field theory (2008) (same authors) · 0.85 · paraphrased: the licence does not allow quoting
Used the N2LO wave functions corresponding to the chiral NN forces introduced in this paper.

Neutron-neutron scattering length from the reaction gamma d --> pi^+ nn employing chiral perturbation theory (2007) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

Uses its data or values 2 of 9 papers with this label · 1 by the same authors

Adopted the coupling constants (c3,c4) = (-3.4,3.4) in GeV^-1 from this paper for its results with TBC.

The quenching of the axial-vector coupling constant $g_A$ in $\beta$-decay: joint effects from chiral two-body currents and many-body correlations (2026) · 0.86 · paraphrased: the licence does not allow quoting

By the same authors

Adopted the values employed in this paper that enter the long-range part of the two- and three-nucleon force.

Saturation of nuclear matter in effective field theory (2008) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

73 arXiv papers cite this in 134 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 82 sentences without a confident label

Uses its method or tool 11 papers

We employ the two-nucleon interaction derived by Epelbaum, Glöckle, and Meißner using effective field theory (EFT) [ref] .

Saturation of nuclear matter in effective field theory · The method · 0.85
To derive the scalar currents consistent with the nuclear potentials in Refs. [ref] and electroweak currents in Refs. [ref] , we first switch from the effective pion-nucleon Lagrangian to the corresponding Hamiltonian $H[s]$ using the canonical formalism and then apply the unitary transformations $U_Okubo$, $U_η$ and $U[s]$.

Subleading contributions to the nuclear scalar isoscalar currents · Two-nucleon scalar operators · 0.87
In our work, we use the N$^2$LO wave functions corresponding to the chiral NN forces introduced in Ref. [ref] and based on the spectral function regularization (SFR) scheme [ref] .

Neutron-neutron scattering length from the reaction gamma d --> pi^+ nn employing chiral perturbation theory · ChPT calculation for $γd→π^+nn$ · 0.98

Uses its data or values 6 papers

In the present calculations, we adopt the values (in units of GeV$^-1$) [equation] employed in Ref. [ref] which enter the long-range part of the two- and three-nucleon force.

Saturation of nuclear matter in effective field theory · The method · 1.00
The choice of these parameters consistent with the two-nucleon potentials of Ref. [ref] corresponds to $β̅_8 = 1/4$ and $β̅_9=0$.

Two-nucleon electromagnetic current in chiral effective field theory: one-pion exchange and short-range contributions · Comparison with the work by Pastore et al. · 0.82
It was found that for the chiral interactions of [ref] , the 4NF contribution is of the order of 300 keV.

Signatures of three-nucleon interactions in few-nucleon systems · Are four-nucleon forces important? · 0.94

Compares with it 4 papers

We compare our findings with the neutron-matter calculations of Ref. [ref] using many-body perturbation theory () and nonlocal interactions from Refs. [ref] for reference.

Direct Astrophysical Tests of Chiral Effective Field Theory at Supranuclear Densities · Introduction · 1.00
The essential new feature of these interactions as compared to the older potentials of Refs. [ref] and the new potentials developed by Entem et al. [ref] is the usage of a local regulator for pion-exchange contributions, which allowed us to substantially reduce finite-cutoff artifacts.

Light nuclei with semilocal momentum-space regularized chiral interactions up to third order · Introduction · 0.85
As our fitting code is written in momentum space, we first solve for the momentum-space wavefunctions and then obtain the coordinate-space representations as, [equation] In coordinate space, our models do not contain any oscillations in the wave function at intermediate or large $r$, unlike some chiral interactions developed in the past [ref] .

Maximally local two-nucleon interactions at N$^3$LO in $\Delta$-less chiral effective field theory · Deuteron · 0.80

Cites it as support 1 paper

Recent progress in the accurate description of the two- [ref] and more-nucleon systems [ref] within the chiral EFT, see also [ref] and references therein, gives a strong motivation to apply this framework to the abovementioned processes.

Two-pion exchange electromagnetic current in chiral effective field theory using the method of unitary transformation · Introduction · 0.96

Cites it as background 25 papers

The potential is then inserted into the Schrödinger or Lippmann-Schwinger equation in order to obtain theoretical predictions [ref] .

Perturbative renormalizability of chiral two pion exchange in nucleon-nucleon scattering · Introduction · 0.91
Precise two-nucleon potentials have been developed at next-to-next-to-next-to-leading order (N$^3$LO) in the chiral expansion, see Refs. [ref] . 3NFs first appear at N$^2$LO in the Weinberg counting scheme [ref] and have been analyzed and scrutinized in [ref] .

Subleading contributions to the chiral three-nucleon force I: long-range terms · Introduction · 0.89
This program has since been pursued by a number of groups [ref] , with very impressive fits to phase shift data at $N^3LO$.

New approach to NN with perturbative pions · Weinberg and KSW · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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