Sigma exchange in the nuclear force and effective field theory

John F. Donoghue

REVIEW

Sigma exchange in the nuclear force and effective field theory

John F. Donoghue

Nuclear Theory · Submitted 27 Feb 2006 · 19 references

Pith's verdict: A clean, honest construction showing the nuclear sigma is two-pion exchange with Omnes rescattering; the shape claim holds, the magnitude claim needs one more sensitivity test.

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arXiv IDnucl-th/0602074 (version 3)

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CitationJohn F. Donoghue. "Sigma exchange in the nuclear force and effective field theory." arXiv:nucl-th/0602074 (2006). https://pith.science/paper/GKT3F3AY

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Editor's note

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Short version: this paper earns its keep. It shows that the scalar-isoscalar attraction in the nuclear force, traditionally blamed on a sigma meson, is reproduced by chiral two-pion exchange plus the Omnes function from pi-pi rescattering. The construction is explicit, the inputs are external, and no step fits the nuclear sigma strength or shape. That is a real step forward relative to the older dispersive work, which had the same spirit but not this clean a realization in EFT language.

The best part is the shape comparison in Fig. 3: the coordinate-space potential from Eq. (5) is nearly identical to a narrow 600 MeV Yukawa with normalization chosen to match. The author is honest that the magnitude comparison is not a sharp test, because c3 carries large errors and the definition of the scalar contact term depends on how you treat iterated one-pion exchange. The discussion of the contact interaction in EFT terms is useful and appropriately hedged.

Where I push back, and I think the stress-test note has the right target: Eq. (5) is the load-bearing object, and the only robustness probe shown is replacing c3 with a form factor. That does not vary the other major input, the I=0 S-wave pi-pi phase shift. The CGL analysis ends at 850 MeV; the Omnes principal-value integral needs the phase shift above that, and the text says a smooth extension has little effect but gives no quantitative evidence. Because Omega(mu) is an integral over all s, the continuation can shift Re Omega and |Omega|^2 at mu values around 0.5-0.8 GeV, which is exactly the interval controlling rV(r) at nuclear distances. If that shift is large, the "magnitude and shape" match is not controlled, even though the qualitative conclusion that there is no genuine sigma resonance is probably safe. The factorization into Re Omega for diagrams a,b and |Omega|^2 for diagram c is a model assumption; left-hand cuts and inelasticity above 1 GeV are not treated, and these could matter for the quantitative claim.

This is a minor-to-moderate soft spot, not a fatal one. The paper is clear, honest, and the central conceptual claim holds up. The audience is people working in nuclear EFT and hadronic physics who want to understand the status of the sigma in the nuclear force. It deserves a serious referee; I would send it out, and I would ask the referee to do or demand a sensitivity study of the Omnes continuation above 850 MeV. On the evidence here, I would accept after that check.

Referee report

Significance

If the central claim holds, the paper provides a clean resolution of the sigma puzzle in nuclear physics: the intermediate-range attraction traditionally attributed to sigma exchange is an emergent property of correlated two-pion exchange, and modern EFT treatments need not postulate a scalar resonance. The result is significant because it quantitatively links the phenomenological meson-exchange description to chiral perturbation theory and yields a first-principles estimate of the scalar contact interaction. The paper has notable strengths: the inputs (c1, c2, c3 from pion-nucleon data; the pi-pi phase shift from the CGL analysis) are independent of nuclear binding; the computation is compact enough to be checked; the shape comparison in Fig. 3 is striking; and the author is explicit about the limitations, stating that the magnitude is not a precise test given the uncertainty in c3 and the scheme dependence associated with box and crossed-box diagrams. The qualitative conclusion that no sigma resonance is required is robust; the remaining concern is whether the quantitative shape claim is controlled against the unconstrained high-energy continuation of the phase shift.

Major objections (2)

[Eqs. (4)-(5), Fig. 3] The central numerical claim rests on the Omnes function built from the I=0 S-wave pi-pi phase shift, but no quantitative sensitivity test is provided for the extension of delta_0^0 above the 850 MeV endpoint of the CGL analysis. The text asserts that 'as long as this extension is smooth it has little effect on this calculation,' yet because the Omnes exponent in Eq. (4) is a principal-value integral over all s, the region above 0.85 GeV contributes to ReOmega and |Omega|^2 precisely in the interval mu ~ 0.5-0.8 GeV that dominates rV(r) for r between roughly 1 and 6 GeV^-1. I regard this as the main gap in an otherwise sound argument: please add a targeted study that varies the continuation (for example, saturation of delta at different values, different functional forms, and different cutoffs) and replots Figs. 3 and 5; this would settle whether the near-identity of the sigma shape is a controlled prediction. The qualitative conclusion that no actual sigma resonance is required would likely survive any of these variations, but the quantitative 'magnitude and shape' claim is not yet secured against the dominant unconstrained input.
[Eq. (5)] The factorization in Eq. (5), rho_S = rho_{a,b} ReOmega + rho_c |Omega|^2, is presented as the Omnes solution matched to the leading-order chiral result, but the matching is not spelled out. This is not the standard Omnes representation of a single form factor, and the treatment of the left-hand singularities of the chiral NN amplitudes and of inelastic contributions above roughly 1 GeV (the K-Kbar threshold) is not discussed. The gamma-gamma to pi-pi precedent in Ref. [14] is encouraging, but the amplitude here has additional structure. Please state the validity assumptions behind Eq. (5) explicitly and, if feasible, estimate the sensitivity of Figs. 3 and 5 to the matching polynomial and to inelasticity.

Minor objections (5)

[Fig. 2 caption] The caption of Fig. 2 spells 'Omnes' as 'Onmes'; please correct the spelling.
[Before Eq. (6)] In the sentence before Eq. (6), 'let as modify' should read 'let us modify', and the units of m in Eq. (6) (stated as m = 800 in the text) should be given as MeV.
[Figs. 4-5 and text] The text says 'The result for GS as a function of c3 is shown in Fig. 4', but the corresponding figure is numbered Fig. 5; in that figure the y-axis label 'GS' lacks units, and its sign convention should be clarified because the text quotes positive values of 300-450 MeV^-2 while the figure shows negative values.
[Title page] The title-page date 'November 4, 2018' is inconsistent with the arXiv header date '7 Mar 2006'; please reconcile or remove the compilation date.
[References] In the reference list, 'arXive:nucl-th/0503025' should be 'arXiv:nucl-th/0503025', and 'B. A Nilolaus' in Ref. [8] should be 'B. A. Nikolaus'; several references also list only partial titles or proceedings series.

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The paper

The paper claims

The nuclear force's sigma attraction is reproduced by two-pion exchange plus pion rescattering, with no sigma particle in the calculation.

The reading

This paper argues that the scalar-isoscalar '$\sigma$' exchange traditionally placed at the center of nuclear binding is not evidence for a real particle; the same force emerges from two-pion exchange once pion rescattering is included. The claim is that combining leading-order chiral perturbation theory for the two-pion spectral function with the Omnes function for elastic $\pi\pi$ rescattering reproduces both the shape and the magnitude of the coordinate-space potential attributed to a 600 MeV $\sigma$ meson. Because all ingredients, namely chiral amplitudes, measured $\pi\pi$ phase shifts, and measured low-energy constants, are independently controlled, the result would tie the nuclear force more directly to QCD and lift the ambiguity surrounding the $\sigma$. It also gives a parameter-controlled estimate of the scalar contact interaction used in modern effective field theory descriptions of nuclei.

What carries the argument

The load-bearing object is the unsubtracted dispersion integral for the scalar-isoscalar potential, $V_S(q^2)=\frac{2}{\pi}\int_{2m_\pi}^\infty d\mu\,\mu\,\rho_S(\mu)/(\mu^2+q^2)$, together with a spectral function formed from the leading-order chiral $\pi\pi$ amplitudes of diagrams 1a,b,c multiplied by the Omnes function $\Omega(\mu)=\exp\left[\frac{\mu^2}{\pi}\int ds\,\frac{\delta(s)}{s(s-\mu^2)}\right]$. The Omnes function encodes the elastic $\pi\pi$ rescattering via the measured $I=0$, $J=0$ phase shift and supplies the energy dependence that turns monotonically growing chiral amplitudes into a broad, resonance-like peak. The work done by the machinery is to replace the phenomenological $\sigma$ Yukawa term with a parameter-free resummation; diagrams 1a,b use $\mathrm{Re}\,\Omega$ and diagram 1c uses $|\Omega|^2$, while the box and crossed-box diagrams are excluded from $G_S$ because in effective field theory their iterated one-pion-exchange content belongs to the dynamical part.

Extended reading notes

Core claim

The central discovery is that the $I=0$, $J=0$ two-pion exchange contribution to the nucleon-nucleon potential, when the pion-pion final-state interaction is resummed through the Omnes function, is nearly indistinguishable from the potential produced by exchanging a narrow 600 MeV scalar meson. There is no resonance in the input: the $\pi\pi$ phase shift does not pass through $90^\circ$ in the 300–600 MeV region, and the description contains only pions as explicit degrees of freedom. Nevertheless the resulting spectral function $\rho_S(\mu)$, given by $\rho_S(\mu)=\rho_{a,b}^S\,\mathrm{Re}\,\Omega(\mu)+\rho_c^S\,|\Omega(\mu)|^2$, peaks where a $\sigma$ would sit, and the coordinate-space potential $rV(r)$ matches the narrow-$\sigma$ curve to the point where the two cannot be distinguished. The same construction yields a value of the scalar contact strength $G_S$ that falls in the phenomenological range $G_S=300$–$450\,\mathrm{MeV}^{-2}$ for the allowed values of $c_3$.

Load-bearing premise

The calculation assumes that the Omnes-resummed spectral function of Eq. (5), built from leading-order chiral amplitudes and the measured $\pi\pi$ phase shift, stays accurate up to roughly 1 GeV; this is the region where the peak and the shape comparison live, and the paper tests it with only one ad hoc form factor while the phase shifts have to be extended smoothly above 850 MeV.

Editorial extensions

If this is right

If the claim is right, the nuclear force's central scalar attraction is explained by pions plus pion rescattering, and no elementary $\sigma$ or $f_0(600)$ state is required in the description of nuclei.
The scalar contact interaction $G_S$ of effective field theory becomes a controlled integral of known spectral functions, with its main uncertainty tied to $c_3$ rather than to an unknown sigma coupling.
Coordinate-space potentials from chiral perturbation theory plus the Omnes function should remain close to phenomenological sigma-exchange potentials over the relevant radii, making the two descriptions interchangeable in fits.
Quark-mass dependence of nuclear binding can be addressed from the known quark-mass dependence of chiral amplitudes and $\pi\pi$ scattering, since the sigma-exchange piece no longer requires an independent hadronic mass parameter.

Reading between the lines

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

A natural extension is to apply the same Omnes-resummed spectral construction to other two-pion channels; for example, the $\rho$-meson-like vector two-pion exchange may acquire its effective mass more from correlated pions than from a genuine narrow state.
If the sigma is an artifact of correlated pions, then the 'mass' extracted from a Yukawa fit to a nuclear potential should depend on the momentum range being fitted rather than being a universal hadron mass.
The paper's method could be tested in lattice QCD by computing the two-pion spectral function with physical pion masses and verifying that the $I=0$ scalar channel contains no narrow state near 600 MeV even though the reconstructed inter-nucleon potential is attractive.

Original abstract

In the phenomenological description of the nuclear interaction a crucial role is traditionally played by the exchange of a scalar I=0 meson, the sigma, of mass 500-600 MeV, which however is not seen clearly in the particle spectrum and which has a very ambiguous status in QCD. I show that a remarkably simple and reasonably controlled combination of ingredients can reproduce the features of this part of the nuclear force. The use of chiral perturbation theory calculations for two pion exchange supplemented by the Omnes function for pion rescattering suffices to reproduce the magnitude and shape of the exchange of a supposed $\sigma$ particle, even though no such particle is present in this calculation. I also show how these ingredients can describe the contact interaction that enters more modern descriptions of the internucleon interaction.

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, 3,739 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.33 · compares with prior results 0.59 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

In the phenomenological description of the nuclear interaction a crucial role is traditionally played by the exchange of a scalar I=0 meson, the sigma, of mass 500-600 MeV, which however is not seen clearly in the particle spectrum and which has a very ambiguous status in QCD. I show that a remarkably simple and reasonably controlled combination of ingredients can reproduce the features of this part of the nuclear force. The use of chiral perturbation theory calculations for two pion exchange supplemented by the Omnes function for pion rescattering suffices to reproduce the magnitude and shape of the exchange of a supposed $\sigma$ particle, even though no such particle is present in this calculation. I also show how these ingredients can describe the contact interaction that enters more modern descriptions of the internucleon interaction.

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	In the phenomenological description of the nuclear interaction a crucial role is traditionally played by the exchange of a scalar I=0 meson, the sigma, of mass 500-600 MeV, which however is not seen clearly in the particle spectrum and which has a very ambiguous status in QCD.	background 0.99	0.05	0.04	asserted 1.00
s1	I show that a remarkably simple and reasonably controlled combination of ingredients can reproduce the features of this part of the nuclear force.	contribution 0.72	0.93	0.34	asserted 1.00
s2	The use of chiral perturbation theory calculations for two pion exchange supplemented by the Omnes function for pion rescattering suffices to reproduce the magnitude and shape of the exchange of a supposed $\sigma$ particle, even though no such particle is present in this calculation.	result 0.87	0.89	0.80	asserted 1.00
s3	I also show how these ingredients can describe the contact interaction that enters more modern descriptions of the internucleon interaction.	result 0.76	0.91	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.54.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

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

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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.** Two pion exchange diagrams which arise in chiral pe

**Figure 2.** The left graph gives the pion phaseshifts that are t

**Figure 3.** The left figure shows the our results for the spectra

**Figure 4.** The spectral function ρ(µ)/µ in the presence of higher order energy dependent modifications as described in the text have the value g 2 σ /m2 σ . There is some uncertainty in the appropriate value of GS, because the preferred fit magnitudes depend somewhat on the calculational scheme used. However, phenomenological studies of the nuclei tend to require GS = 300 − 450 MeV−2 . The contact interaction is given by the strength of the momentum space potential of Eq. 1 evaluated at q 2 = 0. This is just integral under the integrand shown in

**Figure 5.** The strength of the scalar interaction as a functio

Figures reproduced from arXiv:nucl-th/0602074 by the authors.

References

Works cited (19)

[14]

J. F. Donoghue and B. R. Holstein, “Photon-photon scatt ering, pion polarizability and chiral symmetry,” Phys. Rev. D 48, 137 (1993) [arXiv:hep-ph/9302203]. D. Morgan and M. R. Pennington,“Is low-energy gamma gamma → pi0 pi0 predictable?,” Phys. Lett. B 272, 134 (1991)

View on Pith Pith review arXiv 1993
[1]

G. E. Brown and A. D. Jackson, The Nucleon-Nucleon Interac- tion,(North Holland, 1977)

View on Pith 1977
[2]

T. E. O. Ericson and W. Weise, Pions And Nuclei ,(Clarendon, Oxford, 1988)

View on Pith 1988
[3]

U. G. Meissner, “Chiral Dynamics: Where Are The Scalars? ,” Com- ments Nucl. Part. Phys. 20, 119 (1991)

View on Pith 1991
[4]

I. Caprini, G. Colangelo and H. Leutwyler, “Mass and widt h of the lowest resonance in QCD,” arXiv:hep-ph/0512364

View on Pith Pith review arXiv
[5]

W. Cottingham and R. Vinh Mau, “Theoretical nucleon-nuc leon po- tential”, Phys. Rev. 130, 735 (1963). W. N. Cottingham, M. Lacombe, B. Loiseau, J. M. Richard and R. Vinh Mau, “Nucleon Nucleon Interaction From Pion Nucleon Phase S hift Analysis,” Phys. Rev. D 8, 800 (1973). M. Chemtob, J. W. Durso and D. O. Riska, “Two-Pion-Exchange Nucleon-Nucleon Potent...

View on Pith 1963
[6]

W. Grein and P. Kroll, “Two Pion And Three Pion Cut Contrib utions To Nucleon-Nucleon Scattering,” Nucl. Phys. A 338, 332 (1980). F. Wang, G. h. Wu, L. j. Teng and T. Goldman, “Quark delocaliza tion, 8 color screening, and nuclear intermediate range attractio n,” Phys. Rev. Lett. 69, 2901 (1992) [arXiv:nucl-th/9210002]. E. Oset, H. Toki, M. Mizobe and T. ...

View on Pith Pith review arXiv 1980
[7]

E. Epelbaum, W. Glockle, A. Kruger and U. G. Meissner, “Eﬀe ctive theory for the two-nucleon system,” Nucl. Phys. A 645, 413 (1999) [arXiv:nucl-th/9809084]. M. J. Savage, “Eﬀective ﬁeld theory for nuclear physics,” arXiv:nucl-th/0301058. P. F. Bedaque and U. van Kolck, “Eﬀective ﬁeld theory for few- nucleon systems,” Ann. Rev. Nucl. Part. Sci. 52, 339 (20...

View on Pith Pith review arXiv 1999
[8]

B. A Nilolaus, T. Hoch and D. G. Madland, “Nuclear ground s tate properties in a relativistic point coupling model”, Phys. R ev. C46, 1757 (1992) B. D. Serot and J. D. Walecka, “Eﬀective ﬁeld theory in nuclear many- body physics,” arXiv:nucl-th/0010031. B. D. Serot and J. D. Walecka, “Recent progress in quantum had rody- namics,” Int. J. Mod. Phys. E 6, 5...

View on Pith Pith review arXiv 1992
[9]

N. Kaiser, R. Brockmann and W. Weise, “Peripheral nucleo n nucleon phase shifts and chiral symmetry,” Nucl. Phys. A 625, 758 (1997) [arXiv:nucl-th/9706045]. E. Epelbaum, W. Glockle and U. G. Meissner, “Improving the co nver- gence of the chiral expansion for nuclear forces. I: Periphe ral phases,” arXiv:nucl-th/0304037

View on Pith Pith review arXiv 1997
[10]

N. Kaiser, “Chiral 2pi exchange N N potentials: Two-loo p contribu- tions,” Phys. Rev. C 64, 057001 (2001) [arXiv:nucl-th/0107064]. 9 D. R. Entem and R. Machleidt, “Chiral 2 π exchange at order four and peripheral N N scattering,” Phys. Rev. C 66, 014002 (2002) [arXiv:nucl-th/0202039]

View on Pith Pith review arXiv 2001
[11]

U. G. Meissner, “Quark mass dependence of baryon proper ties,” PoS LA T2005, 009 (2005) [arXiv:hep-lat/0509029]

View on Pith Pith review arXiv 2005
[12]

R. Omnes, “On the solution of certain singular integral equations of quantum ﬁeld theory,” Nuovo Cim. 8, 316 (1958)

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[13]

J. F. Donoghue, J. Gasser and H. Leutwyler, “The decay of a light Higgs boson,” Nucl. Phys. B 343, 341 (1990)

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[15]

S. Bellucci, J. Gasser and M. E. Sainio, “Low-energy pho ton-photon collisions to two loop order,” Nucl. Phys. B 423, 80 (1994) [Erratum- ibid. B 431, 413 (1994)] [arXiv:hep-ph/9401206]

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[16]

G. Colangelo, J. Gasser and H. Leutwyler, “ ππ scattering,” Nucl. Phys. B 603, 125 (2001) [arXiv:hep-ph/0103088]

View on Pith Pith review arXiv 2001
[17]

J. Sucher, “The concept of potential in quantum ﬁeld the ory,” arXiv:hep-ph/9412388

View on Pith Pith review arXiv
[18]

R. Machleidt and D. R. Entem, “Towards a consistent ap- proach to nuclear structure: EFT of two- and many-body force s,” arXiv:nucl-th/0503025

View on Pith Pith review arXiv
[19]

J. F. Donoghue (in preparation) 10

View on Pith

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Weak nuclear decays deep-underground as a probe of axion dark matter
hep-ph 2024-12 unverdicted novelty 6.0 of 10

A framework is developed to predict axion-induced time modulations in weak nuclear decays, used to derive constraints on the axion decay constant from reanalyzed Gran Sasso data on 40K and 137Cs and to propose future ...
Equivalence Principle Violations and Couplings of a Light Dilaton
gr-qc 2010-07 conditional novelty 6.0 of 10

A dilaton coupling to the average light quark mass produces equivalence-principle violation through nuclear binding that scales as A^{-1/3}, comparable to or larger than the Coulomb-driven term for light elements.
Gravitons and Pions
nucl-th 2019-08 accept

An invited essay arguing that effective field theory makes quantum gravity and chiral nuclear forces predictable at low energies, with no new results.

How the literature uses it

13 arXiv papers cite this in 20 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 · 7 sentences without a confident label

Builds on it 1 paper

One of the authors has recently extended this work to include the constraints of unitarity [ref] .

Constraints on the variability of quark masses from nuclear binding · Anthropic constraints from the existence of nuclei using effective field theory · 0.90

Uses its method or tool 7 papers

The framework used below follows [ref] in employing dispersion relations, which can be used to express the desired couplings as integrals over reactions involving physical intermediate states.

Constraints on the variability of quark masses from nuclear binding · Anthropic constraints from the existence of nuclei using effective field theory · 0.99
Similar to Ubaldi [ref] , we employ the parameterization of Refs. [ref] .

$\theta$-dependence of light nuclei and nucleosynthesis · $θ$-dependence of hadron properties · 0.99
Such a modification can be accomplished [ref] by introducing the Omnes function, Ω^(I=0) (μ) = exp[ μ^2 π ∫ds s δ^(I=0) (s) s - μ^2] ,

Unbinding the Deuteron · The Two-pion Isoscalar (`$σ (600)$') Component of the Two-nucleon Potential · 0.89

Uses its data or values 1 paper

The central values in Table [ref] are obtained by setting $√(s_0)$ to 0.8 GeV as in Ref. [ref] and utilizing the central values of the NLO LECs provided in Ref. [ref] .

$\sigma$ exchange in the one-boson exchange model involving the ground state octet baryons · Matching $s$-channel amplitudes · 0.96

Cites it as support 1 paper

This accounts for much of the strength typically ascribed to the sigma [ref] .

Equivalence Principle Violations and Couplings of a Light Dilaton · The central nuclear force terms: $a_v$ and $a_s$ · 0.97

Cites it as background 1 paper

A key role for the binding energy of heavy nuclei is played by the $σ$ and $ω$ channels, via the contact interactions [ref] H_strong ⊃G_S (N̅ N) (N̅ N) + G_V (N̅ γ_μN) (N̅ γ^μN) , which control, respectively, the scalar (attractive) and vector (repulsive) part of the nucleon-nucleon interaction [ref] .

Weak nuclear decays deep-underground as a probe of axion dark matter · θ-dependence of mesons, nucleons and nuclei · 0.82

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.

[1] [14]

J. F. Donoghue and B. R. Holstein, “Photon-photon scatt ering, pion polarizability and chiral symmetry,” Phys. Rev. D 48, 137 (1993) [arXiv:hep-ph/9302203]. D. Morgan and M. R. Pennington,“Is low-energy gamma gamma → pi0 pi0 predictable?,” Phys. Lett. B 272, 134 (1991)

View on Pith Pith review arXiv 1993

[2] [1]

G. E. Brown and A. D. Jackson, The Nucleon-Nucleon Interac- tion,(North Holland, 1977)

View on Pith 1977

[3] [2]

T. E. O. Ericson and W. Weise, Pions And Nuclei ,(Clarendon, Oxford, 1988)

View on Pith 1988

[4] [3]

U. G. Meissner, “Chiral Dynamics: Where Are The Scalars? ,” Com- ments Nucl. Part. Phys. 20, 119 (1991)

View on Pith 1991

[5] [4]

I. Caprini, G. Colangelo and H. Leutwyler, “Mass and widt h of the lowest resonance in QCD,” arXiv:hep-ph/0512364

View on Pith Pith review arXiv

[6] [5]

W. Cottingham and R. Vinh Mau, “Theoretical nucleon-nuc leon po- tential”, Phys. Rev. 130, 735 (1963). W. N. Cottingham, M. Lacombe, B. Loiseau, J. M. Richard and R. Vinh Mau, “Nucleon Nucleon Interaction From Pion Nucleon Phase S hift Analysis,” Phys. Rev. D 8, 800 (1973). M. Chemtob, J. W. Durso and D. O. Riska, “Two-Pion-Exchange Nucleon-Nucleon Potent...

View on Pith 1963

[7] [6]

W. Grein and P. Kroll, “Two Pion And Three Pion Cut Contrib utions To Nucleon-Nucleon Scattering,” Nucl. Phys. A 338, 332 (1980). F. Wang, G. h. Wu, L. j. Teng and T. Goldman, “Quark delocaliza tion, 8 color screening, and nuclear intermediate range attractio n,” Phys. Rev. Lett. 69, 2901 (1992) [arXiv:nucl-th/9210002]. E. Oset, H. Toki, M. Mizobe and T. ...

View on Pith Pith review arXiv 1980

[8] [7]

E. Epelbaum, W. Glockle, A. Kruger and U. G. Meissner, “Eﬀe ctive theory for the two-nucleon system,” Nucl. Phys. A 645, 413 (1999) [arXiv:nucl-th/9809084]. M. J. Savage, “Eﬀective ﬁeld theory for nuclear physics,” arXiv:nucl-th/0301058. P. F. Bedaque and U. van Kolck, “Eﬀective ﬁeld theory for few- nucleon systems,” Ann. Rev. Nucl. Part. Sci. 52, 339 (20...

View on Pith Pith review arXiv 1999

[9] [8]

B. A Nilolaus, T. Hoch and D. G. Madland, “Nuclear ground s tate properties in a relativistic point coupling model”, Phys. R ev. C46, 1757 (1992) B. D. Serot and J. D. Walecka, “Eﬀective ﬁeld theory in nuclear many- body physics,” arXiv:nucl-th/0010031. B. D. Serot and J. D. Walecka, “Recent progress in quantum had rody- namics,” Int. J. Mod. Phys. E 6, 5...

View on Pith Pith review arXiv 1992

[10] [9]

N. Kaiser, R. Brockmann and W. Weise, “Peripheral nucleo n nucleon phase shifts and chiral symmetry,” Nucl. Phys. A 625, 758 (1997) [arXiv:nucl-th/9706045]. E. Epelbaum, W. Glockle and U. G. Meissner, “Improving the co nver- gence of the chiral expansion for nuclear forces. I: Periphe ral phases,” arXiv:nucl-th/0304037

View on Pith Pith review arXiv 1997

[11] [10]

N. Kaiser, “Chiral 2pi exchange N N potentials: Two-loo p contribu- tions,” Phys. Rev. C 64, 057001 (2001) [arXiv:nucl-th/0107064]. 9 D. R. Entem and R. Machleidt, “Chiral 2 π exchange at order four and peripheral N N scattering,” Phys. Rev. C 66, 014002 (2002) [arXiv:nucl-th/0202039]

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