The Two-Nucleon Potential from Chiral Lagrangians

Contribution

• The central contribution is developing the NN potential within the framework of the general effective chiral Lagrangian [14, 15].The formulation includes the pion and nucleon and transforms under chiral symmetry as the QCD Lagrangian.Source: L. Ra · p. 3

Parameter determination

• The analysis relates effective-Lagrangian parameters to measured low-energy NN scattering data and deuteron properties rather than attempting to solve QCD.The coupling constants encode underlying QCD dynamics, while the paper focuses on determining the low-energy parameters phenomenologically.Source: L. Ra · p. 3

Method

• The method avoids adopting a massive-meson-exchange picture or a particular quark model, remaining compatible with QCD and known low-energy phenomenology.The stated compatibility includes NN, πN, and N scattering and meson-exchange currents.Source: L. Ra · p. 3

Expansion scheme

• Assuming natural low-energy parameters consistent with naive dimensional analysis permits a perturbative nuclear-potential expansion in powers of momentum divided by a typical QCD mass scale.At each specified order, the effective chiral Lagrangian determines precisely which terms appear.Source: L. Ra · pp. 3–4

• A successful overall data description would indicate that the perturbation expansion reaches the precision required, whereas missing phenomenology could signal an unexpectedly important operator, diagram, or dynamical mechanism.Such mechanisms may provide information about QCD.Source: L. Ra · p. 4
• The aim is to bridge QCD and nuclear physics with a chiral nucleon–nucleon potential whose off-shell structure can support calculations of other nuclear processes.The general chiral Lagrangian parametrizes both ignorance of QCD and knowledge of nuclear physics.Source: L. Ra · p. 4
• The paper derives the two-nucleon potential to a specified chiral order in momentum and coordinate space, using a momentum-space Gaussian cutoff before treating scattering and bound-state properties.The coordinate-space potential and data fits are presented in later sections of the paper.Source: L. Ra · p. 4
• Interactions are organized by powers of soft momentum Q divided by a QCD mass scale, using locality, QCD symmetries, naturalness, and a nonrelativistic treatment of nucleons and Δs.Anomalously large or small couplings may require special low-energy treatment or indicate dynamical or symmetry effects in QCD.Source: L. Ra · pp. 4–5
• The low-energy treatment includes nucleons as principal constituents of nuclei and retains the pseudo-Goldstone boson as a fundamental ingredient, while not explicitly keeping certain heavier meson states.Source: L. Ra · p. 5
• The two-nucleon potential sums irreducible diagrams, while the full scattering amplitude is obtained by iterating it in the Lippmann–Schwinger or Schrödinger equation.Irreducible intermediate states contain at least one pion or isobar; reducible diagrams are automatically included through equation-of-motion iteration.Source: L. Ra · p. 7
• The construction uses two light quark flavors, neglects isospin breaking, and evaluates diagrams only through one loop, omitting higher-field operators that do not contribute at this order.The treatment also notes that the estimate of the relativistic scale is crude, affecting the expected size of relativistic corrections.Source: L. Ra · pp. 7–8
• The effective chiral Lagrangian includes all terms built from the relevant fields and covariant derivatives that respect QCD’s parity and time-reversal symmetries.Explicit chiral-symmetry breaking from quark masses is incorporated through operators transforming appropriately under the broken symmetry, with coefficients proportional to quark-mass powers.Source: L. Ra · p. 8
• Chiral symmetry supplies a natural low-energy ordering because pion operators require derivatives or quark-mass powers, while nucleon bilinears require derivatives.The interaction index organizes the effective Lagrangian into successive orders, with higher orders generated by loops, interactions, and fewer connected pieces.Source: L. Ra · p. 8

Parameter determination

• The potential’s parameters are fitted to low-energy nucleon-nucleon data because one-loop determinations from one-nucleon processes are not yet complete.The authors note that the number of parameters could decrease when sufficient one-nucleon information becomes available.Source: L. Ra · p. 10

Many-nucleon implications

• The chiral framework is consistent with the empirical observation that three- and higher-body forces are smaller than two-body forces.The remainder of the work restricts the analysis to the two-nucleon system.Source: L. Ra · p. 10

Momentum-space potential

• The leading potential is the well-known static one-pion-exchange interaction supplemented by contact terms.One-pion exchange supplies the longest-range NN force, but the lowest-order result does not include spin-orbit, strong short-range repulsion, or intermediate-range attraction.Source: L. Ra · p. 11
• No corrections occur at the first subleading order Q/M because parity conservation forbids the required odd number of momenta.The absence of suitable vertices and the elimination of time derivatives leave no parity-conserving terms at this order.Source: L. Ra · p. 11
• Second-order corrections include tree-level contributions from higher-order interactions and one-loop diagrams with nucleon and isobar intermediate states.The tree terms include recoil effects, while loop intermediate states may contain two nucleons, one nucleon and one isobar, or two isobars.Source: L. Ra · pp. 11–12

• The passage fragment says the diagrams in Fig. c,d,e also do not …, without specifying what they do not show or establish.The supplied text is truncated, so no further claim can be grounded.Source: L. Ra · p. 13

Momentum-space potential

• Higher-order corrections include more difficult two-loop graphs and additional tree graphs with new undetermined coefficients, which the authors do not include.The traditional s-wave correlated TPE is likewise higher order in the formalism discussed here.Source: N N couplings in L · pp. 14–15
• Chiral symmetry fixes the coefficients of the Eq. (20) terms and introduces new nonlinear pion–nucleon contributions that are not usually included in meson-exchange potentials.These contributions represent the only form of “correlated” pion exchange in this treatment, while the traditional s-wave correlated TPE appears at higher order.Source: N N couplings in L · p. 14

Regularization and contact terms

• Loop integrals diverge and iterating even the lowest-order potential generates further infinities, so regularization and counterterms are required.After renormalization, contact terms absorb regulator dependence and include all parity- and time-reversal-compatible spin and momentum structures through second order.Source: N N couplings in L · p. 15

The Two-Nucleon Potential in Coordinate-Space

• For the coordinate-space potential, the authors use a momentum cutoff with a Gaussian regulator because iterating the dimensionally regularized potential to all orders in arbitrary dimension remains unresolved.Cutoff variations are partly compensated by redefining the theory’s free parameters.Source: N N couplings in L · p. 15
• The coordinate-space NN potential is expressed through operator structures multiplied by sixty radial functions, some vanishing, whose energy dependence is linear and whose regularization makes them smooth at the origin.Each radial function is determined by chiral-Lagrangian parameters and involves at most one one-dimensional integral.Source: N N couplings in L · p. 16

Solution of the Schrödinger Equation

• Derivative terms are handled by redefining the coupled-channel radial function so the auxiliary equation contains no first derivatives before numerical integration.The Schrödinger equation uses partial-wave radial equations, with coupled triplet channels requiring two-component radial functions and Runge–Kutta integration for the auxiliary function.Source: N N couplings in L · pp. 16–17

Fitting Results for Phase Shifts and Deuteron

• The best fits use a cutoff of 1.0 fm, are in quantitative agreement with the Nijmegen phase shifts at 100 MeV, and show significantly better agreement at 25 and 50 MeV.Source: Prop erties · pp. 18–19
• The model predicts singlet and triplet L = 0 scattering lengths of -23.6 and 5.40 fm, compared with measured values of -23.7(4) and 5.419(7) fm.Source: Prop erties · p. 19
• Predicted phase shifts and mixing angles agree qualitatively with Nijmegen solutions over 100–300 MeV, but the model was not fitted above 100 MeV because of its low-momentum expansion.The comparison covers most partial-wave parameters, with an explicitly noted exception in the supplied passage.Source: Prop erties · p. 19

• The chiral-symmetry terms have minor effects on phase shifts and mixing angles in some cases but significantly affect deuteron properties, while additional terms contribute substantially to both.The short-range, high-momentum wave-function component is not quantitatively reliable, although small-radius d-state depletion contributes to the low d-state probability.Source: Prop erties · pp. 19–20
• The potential is derived from an effective chiral Lagrangian using a controlled expansion in powers of Q/M, expressed in momentum and coordinate space, and fitted by varying Lagrangian parameters.The corresponding Schrödinger equation is solved in coordinate space using the transformed potential.Source: Prop erties · p. 21
• Unlike dispersion-relation approaches, the effective chiral Lagrangian explicitly incorporates QCD symmetries and provides a natural perturbative expansion, while differing from Refs. [7, 10] by being controlled.The chiral Lagrangians used generally agree with pion-phenomenology data at the 20% level.Source: Prop erties · p. 21
• Chiral symmetry generates new potential terms beyond previously considered pion-exchange contributions, and energy dependence arises naturally with implications for few-body forces [2].The short-range parts retain the necessary spin and isospin structure.Source: Prop erties · p. 21
• Up to 100 MeV incident nucleon kinetic energy, the potential gives reasonable qualitative deuteron fits and quantitative fits to most scattering phase shifts.These results indicate that the approach captures the principal features of the nucleon–nucleon potential.Source: Prop erties · p. 21
• The chiral potential cannot compete with phenomenological approaches for quantitative NN descriptions over a wide energy range.Source: Prop erties · p. 21
• Higher energies and improved data agreement require extending the model to higher orders in chiral perturbation theory.Source: Prop erties · p. 21
• Covariant derivatives for the pion, nucleon, and isobar fields transform correctly under the chiral symmetry group.Source: App endix · pp. 23–24
• Isoscalars built from the fields and their covariant derivatives are automatically invariant under the full chiral group.Source: App endix · p. 24

• The appendix gives relations between the A's appearing in Eq. (17).Source: B App endix · p. 24

• Chiral Lagrangians allow systematic inclusion of nucleon recoil corrections, including energy-dependent terms.Source: C App endix · p. 24
• Recoil corrections have been associated with cancellations between reducible and irreducible graphs in the three-nucleon problem [2, 31].Source: C App endix · p. 25
• The Lippmann–Schwinger equation uses outgoing and incoming wave boundary conditions and labels A, B, and C for free many-nucleon quantum numbers.Source: C App endix · p. 25
• The perturbative expansion is obtained by iterating the Lippmann–Schwinger equation, while the potential in Eq. (51) is not energy dependent.Source: C App endix · p. 25
• The effective potential is constructed from irreducible diagrams containing at least one pion or isobar in intermediate states.Source: C App endix · p. 25

• For heavy nucleons, the relativistic expression reduces to the nonrelativistic kinetic energy up to small, systematically treatable corrections.Source: Notice that V · pp. 25–26
• Energy dependence vanishes in the infinite-nucleon-mass static limit and appears only in corrections to the lowest-order term.Source: Notice that V · p. 26

• Coordinate-space potentials are obtained by Fourier transforms using a Gaussian cutoff with parameter β.Source: D App endix · p. 26
• The calculation defines auxiliary functions involving the complementary error function and associated integrals.Source: D App endix · pp. 26–27
• The appendix introduces a function f of q and a mass-related parameter for the transform formulas.Source: D App endix · p. 27

E App endix

• The appendix defines combinations of radial functions and their derivatives used to express the coordinate-space potential.Here f=f(r) and g=g(r) denote functions defined in Appendix D, while a prime denotes differentiation with respect to r.Source: E App endix · p. 27
• The effective chiral Lagrangian potential parameters are fitted to the Nijmegen phase shifts for a cutoff of 3.0 fm.Source: E App endix · p. 32
• The appendix tabulates experimental and fitted values for deuteron binding energy, magnetic moment, and electric quadrupole moment.The table compares experiment with the deuteron fit and SP fits.Source: E App endix · p. 33

Figure Captions

• The potential excludes two-loop graphs of the illustrated type, and some graph orderings require at least one pion or isobar intermediate state.Source: Figure Captions · p. 37
• Figures show best fits to the I=0 np and I=1 pp phase shifts and mixing angles, with Nijmegen solutions used for comparison.The captions specify cutoff-dependent curves and cite Nijmegen phase-shift solutions [21] and reported phase-shift errors [22].Source: Figure Captions · p. 37
• Predictions using the 3.0 fm cutoff and Table I parameters are compared with the Nijmegen I=0 np and I=1 pp phase-shift solutions up to 300 MeV.Source: Figure Captions · p. 37
• The captions include deuteron s-state and d-state radial wave functions calculated from the present NN potential.The calculation uses the cutoff and parameters in Table I.Source: Figure Captions · p. 37
• Radial potentials are plotted for a partial-wave state at 50 MeV, with distinct line styles assigned to the potentials W defined in Eqs. (35) and (36).Source: Figure Captions · p. 37