# Chiral Symmetry and the Nucleon-Nucleon Interaction

## Abstract

**:**

## 1. Historical Perspective

## 2. Effective Field Theory for Low-Energy QC

If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition, and the assumed symmetry principles.

- Identify the soft and hard scales, and the degrees of freedom appropriate for (low-energy) nuclear physics.
- Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken.
- Construct the most general Lagrangian consistent with those symmetries and symmetry breakings.
- Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion.
- Guided by the expansion, calculate Feynman diagrams for the process under consideration to the desired accuracy.

#### 2.1. Symmetries of Low-Energy QCD

#### 2.1.1. Chiral Symmetry

#### 2.1.2. Explicit Symmetry Breaking

#### 2.1.3. Spontaneous Symmetry Breaking

#### 2.2. Chiral Effective Lagrangians

**π**and the heavy baryon nucleon field by N ($\overline{N}={N}^{\u2020}$). Furthermore, ${g}_{A}$, ${f}_{\pi}$, ${m}_{\pi}$, and ${M}_{N}$ are the axial-vector coupling constant, pion decay constant, pion mass, and nucleon mass, respectively. Numerical values for these quantities will be given later. The ${c}_{i}$ are low-energy constants (LECs) from the dimension two $\pi N$ Lagrangian and α is a parameter that appears in the expansion of the pion fields, see Ref. [15] for more details. Results are independent of α.

## 3. Nuclear Forces from EFT: Overview

#### 3.1. Chiral Perturbation Theory and Power Counting

#### 3.2. The Hierarchy of Nuclear Forces

## 4. Pion-Exchange Contributions to the $NN$ Interaction

#### 4.1. Leading Order (LO)

#### 4.2. Next-to-Leading Order (NLO)

#### 4.3. Next-to-Next-to-Leading Order (NNLO)

#### 4.4. Next-to-Next-to-Next-to-Leading Order (N${}^{3}$LO)

#### 4.4.1. Football diagram at N${}^{3}$LO

#### 4.4.2. Leading Two-Loop Contributions

#### 4.4.3. Leading Relativistic Corrections

#### 4.4.4. Leading Three-Pion Exchange Contributions

#### 4.5. Next-to-Next-to-Next-to-Next-to-Leading Order (N${}^{4}$LO)

#### 4.5.1. Two-Pion Exchange Contributions at N${}^{4}$LO

#### 4.5.2. Three-Pion Exchange Contributions at N${}^{4}$LO

#### 4.6. Next-to-Next-to-Next-to-Next-to-Next-to-Leading Order (N${}^{5}$LO)

#### 4.6.1. Two-Pion Exchange Contributions at N${}^{5}$LO

#### 4.6.2. Three-Pion Exchange Contributions at N${}^{5}$LO

#### 4.6.3. Four-Pion Exchange at N${}^{5}$LO

## 5. Perturbative $NN$ Scattering in Peripheral Partial Waves

- N${}^{4}$LO.
- The previous curve plus the $1/{M}_{N}^{2}$-corrections (denoted by “1/M2”) [44].

## 6. Constructing Chiral $NN$ Potentials

#### 6.1. $NN$ Contact Terms

#### 6.1.1. Zeroth Order (LO)

#### 6.1.2. Second Order (NLO)

#### 6.1.3. Fourth Order (N${}^{3}$LO)

#### 6.1.4. Sixth Order (N${}^{5}$LO)

#### 6.2. Definition of $NN$ Potential

#### 6.3. Regularization and Non-Perturbative Renormalization

#### 6.3.1. Renormalization Beyond Leading Order

#### 6.3.2. Back to the Beginnings

- Incorporate the correct long-range behavior: The long-range behavior of the underlying theory must be known, and it must be built into the effective theory. In the case of nuclear forces, the long-range theory is, of course, well known and given by one- and multi-pion exchanges.
- Introduce an ultraviolet cutoff to exclude high-momentum states, or, equivalent, to soften the short-distance behavior: The cutoff has two effects: First it excludes high-momentum states, which are sensitive to the unknown short-distance dynamics; only states that we understand are retained. Second it makes all interactions regular at $r=0$, thereby avoiding the infinities that beset the naive approach.
- Add local correction terms (also known as contact or counter terms) to the effective Hamiltonian. These mimic the effects of the high-momentum states excluded by the cutoff introduced in the previous step. In the meson-exchange picture, the short-range nuclear force is described by heavy meson exchange, like the $\rho (770)$ and $\omega (782)$. However, at low energy, such structures are not resolved. Since we must include contact terms anyhow, it is most efficient to use them to account for any heavy-meson exchange as well. The correction terms systematically remove dependence on the cutoff.

#### 6.4. $NN$ Potentials Order by Order

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 1.**Hierarchy of nuclear forces in chiral perturbation theory (ChPT). Solid lines represent nucleons and dashed lines pions. Small dots, large solid dots, solid squares, triangles, diamonds, and stars denote vertices of index $\Delta =\phantom{\rule{0.166667em}{0ex}}$ 0, 1, 2, 3, 4, and 6, respectively. Further explanations are given in the text.

**Figure 2.**Next-to-Next-to-Next-to-Leading Order (N${}^{3}$LO) two-pion exchange contributions with (

**a**) the N${}^{3}$LO football diagram; (

**b**) the leading 2PE two-loop contributions; and (

**c**) the relativistic corrections of NLO diagrams. Notation as in Figure 1. Shaded ovals represent complete $\pi N$-scattering amplitudes with their order specified by the number in the oval. Open circles denote relativistic $1/{M}_{N}$ corrections.

**Figure 4.**N${}^{4}$LO two-pion-exchange contributions. (

**a**) The leading one-loop $\pi N$ amplitude is folded with the chiral $\pi \pi NN$ vertices proportional to ${c}_{i}$; (

**b**) The one-loop $\pi N$ amplitude proportional to ${c}_{i}$ is folded with the leading order chiral $\pi N$ amplitude; (

**c**) Relativistic corrections of next-to-next-to-leading order (NNLO) diagrams. Notation as in Figure 1 and Figure 2.

**Figure 6.**N${}^{5}$LO two-pion-exchange contributions. (

**a**) The subleading one-loop $\pi N$-amplitude is folded with the chiral $\pi \pi NN$-vertices proportional to ${c}_{i}$; (

**b**) The leading one-loop $\pi N$-amplitude is folded with itself; (

**c**) The leading two-loop $\pi N$-amplitude is folded with the tree-level $\pi N$-amplitude. Notation as in Figure 1, Figure 2, and Figure 4.

**Figure 8.**Effect of individual N${}^{4}$LO (fifth-order) contributions on the neutron-proton phase shifts of some selected peripheral partial waves. The individual contributions are added up successively in the order given in parenthesis next to each curve. Curve (1) is N${}^{3}$LO and curve (5) is the complete N${}^{4}$LO. The KH low-energy constants (LECs) are used and $\tilde{\Lambda}$ = 1.5 GeV. The filled and open circles represent the results from the Nijmegan multi-energy $np$ phase-shift analysis [50] and the VPI/GWU single-energy $np$ analysis SM99 [51], respectively.

**Figure 9.**Phase-shifts of neutron-proton scattering at various orders up to N${}^{4}$LO. The colored bands show the variation of the predictions when the spectral-function renormalization (SFR) cutoff $\tilde{\Lambda}$ is changed over the range 0.7 to 1.5 GeV. The KH LECs are applied. Empirical phase shifts as in Figure 8.

**Figure 11.**Effect of individual N${}^{5}$LO (sixth-order) contributions on the neutron-proton phase shifts of two G-waves. The individual contributions are added up successively in the order given in parentheses next to each curve. Curve (1) is N${}^{4}$LO and curve (6) contains all N${}^{5}$LO contributions calculated in Ref. [35]. A SFR cutoff $\tilde{\Lambda}=800$ MeV is applied and the GW LECs are used. The filled and open circles represent the results from the Nijmegen multi-energy $np$ phase-shift analysis [50] and the GWU $np$-analysis SP07 [52], respectively.

**Figure 12.**Phase-shifts of neutron-proton scattering in G and H-waves at N${}^{3}$LO, N${}^{4}$LO, and N${}^{5}$LO. The colored bands show the variations of the predictions when the SFR cutoff $\tilde{\Lambda}$ is changed over the range 700 to 900 MeV. The GW LECs are applied. Empirical phase shifts are as in Figure 11.

**Figure 13.**Phase-shifts of neutron-proton scattering in G- and H-waves at all orders from LO to N${}^{5}$LO. A SFR cutoff $\tilde{\Lambda}=800$ MeV is used and the GW LECs are applied. Empirical phase shifts are as in Figure 11.

**Figure 14.**${\chi}^{2}$/datum for the reproduction of the $np$ data in the energy range 35–125 MeV (

**upper**frame) and 125–183 MeV (

**lower**frame) as a function of the cutoff parameter Λ of the regulator function Equation (95). The (black) dashed curves show the ${\chi}^{2}$/datum achieved with $np$ potentials constructed at order NLO and the (red) solid curves are for NNLO.

**Figure 15.**Phase shifts of $np$ scattering as calculated from $NN$ potentials at different orders of ChPT. The black dotted line is LO(500), the blue dashed is NLO(550/700) [41], the green dash-dotted NNLO(600/700) [41], and the red solid N${}^{3}$LO(500) [32], where the numbers in parentheses denote the cutoffs in MeV. Phase parameters with total angular momentum $J\le 2$ are displayed. Empirical phase shifts (solid dots and open circles) as in Figure 8.

**Table 1.**Low-energy constants as determined in Ref. [26]. The sets “GW” and “KH” are based upon the $\pi N$ partial wave analyses of Refs. [47,48], respectively. The ${c}_{i}$ appear in Equation (18) and are in units of GeV${}^{-1}$. The ${\overline{d}}_{i}$ and ${\overline{e}}_{i}$ belong to ${\widehat{\mathcal{L}}}_{\pi N}^{(3)}$ and ${\widehat{\mathcal{L}}}_{\pi N}^{(4)}$ (cf. Equations (19) and (20)) and are in units of GeV${}^{-2}$ and GeV${}^{-3}$, respectively.

LEC | GW | KH |
---|---|---|

${c}_{1}$ | –1.13 | –0.75 |

${c}_{2}$ | 3.69 | 3.49 |

${c}_{3}$ | –5.51 | –4.77 |

${c}_{4}$ | 3.71 | 3.34 |

${\overline{d}}_{1}+{\overline{d}}_{2}$ | 5.57 | 6.21 |

${\overline{d}}_{3}$ | –5.35 | –6.83 |

${\overline{d}}_{5}$ | 0.02 | 0.78 |

${\overline{d}}_{14}-{\overline{d}}_{15}$ | –10.26 | –12.02 |

${\overline{e}}_{14}$ | 1.75 | 1.52 |

${\overline{e}}_{15}$ | –5.80 | –10.41 |

${\overline{e}}_{16}$ | 1.76 | 6.08 |

${\overline{e}}_{17}$ | –0.58 | –0.37 |

${\overline{e}}_{18}$ | 0.96 | 3.26 |

**Table 2.**Columns three and four show the ${\chi}^{2}$/datum for the reproduction of the 1999 $np$ database (defined in Ref. [9]) by families of $np$ potentials at NLO and NNLO constructed by the Bochum group [41]. The ${\chi}^{2}$/datum is stated in terms of ranges which result from a variation of the cutoff parameters used in the regulator functions. The values of these cutoff parameters in units of MeV are given in parentheses. ${T}_{\mathrm{lab}}$ denotes the kinetic energy of the incident neutron in the laboratory system.

${\mathit{T}}_{\mathbf{lab}}$ (MeV Energy Bin) | # of np Data | Bochum np Potentials | |
---|---|---|---|

NLO (550/700–400/500) | NNLO (600/700–450/500) | ||

0–100 | 1058 | 4–5 | 1.4–1.9 |

100–190 | 501 | 77–121 | 12–32 |

190–290 | 843 | 140–220 | 25–69 |

0–290 | 2402 | 67–105 | 12–27 |

**Table 3.**Number of parameters needed for fitting the $np$ data in the Nijmegen phase-shift analysis and by the high-precision CD-Bonn potential versus the total number of $NN$ contact terms of EFT based potentials to different orders.

State | Nijmegen PWA93 | CD-Bonn Pot. | EFT Contact Potentials [36] | |||
---|---|---|---|---|---|---|

Ref. [50] | Ref. [9] | ${\mathit{Q}}^{\mathbf{0}}$ | ${\mathit{Q}}^{\mathbf{2}}$ | ${\mathit{Q}}^{\mathbf{4}}$ | ${\mathit{Q}}^{\mathbf{6}}$ | |

${}^{1}{S}_{0}$ | 3 | 4 | 1 | 2 | 4 | 6 |

${}^{3}{S}_{1}$ | 3 | 4 | 1 | 2 | 4 | 6 |

${}^{3}{S}_{1}$-${}^{3}{D}_{1}$ | 2 | 2 | 0 | 1 | 3 | 6 |

${}^{1}{P}_{1}$ | 3 | 3 | 0 | 1 | 2 | 4 |

${}^{3}{P}_{0}$ | 3 | 2 | 0 | 1 | 2 | 4 |

${}^{3}{P}_{1}$ | 2 | 2 | 0 | 1 | 2 | 4 |

${}^{3}{P}_{2}$ | 3 | 3 | 0 | 1 | 2 | 4 |

${}^{3}{P}_{2}$-${}^{3}{F}_{2}$ | 2 | 1 | 0 | 0 | 1 | 3 |

${}^{1}{D}_{2}$ | 2 | 3 | 0 | 0 | 1 | 2 |

${}^{3}{D}_{1}$ | 2 | 1 | 0 | 0 | 1 | 2 |

${}^{3}{D}_{2}$ | 2 | 2 | 0 | 0 | 1 | 2 |

${}^{3}{D}_{3}$ | 1 | 2 | 0 | 0 | 1 | 2 |

${}^{3}{D}_{3}$-${}^{3}{G}_{3}$ | 1 | 0 | 0 | 0 | 0 | 1 |

${}^{1}{F}_{3}$ | 1 | 1 | 0 | 0 | 0 | 1 |

${}^{3}{F}_{2}$ | 1 | 2 | 0 | 0 | 0 | 1 |

${}^{3}{F}_{3}$ | 1 | 2 | 0 | 0 | 0 | 1 |

${}^{3}{F}_{4}$ | 2 | 1 | 0 | 0 | 0 | 1 |

${}^{3}{F}_{4}$-${}^{3}{H}_{4}$ | 0 | 0 | 0 | 0 | 0 | 0 |

${}^{1}{G}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0 |

${}^{3}{G}_{3}$ | 0 | 1 | 0 | 0 | 0 | 0 |

${}^{3}{G}_{4}$ | 0 | 1 | 0 | 0 | 0 | 0 |

${}^{3}{G}_{5}$ | 0 | 1 | 0 | 0 | 0 | 0 |

Total | 35 | 38 | 2 | 9 | 24 | 50 |

**Table 4.**Columns three to five display the ${\chi}^{2}$/datum for the reproduction of the 1999 $np$ database (defined in Ref. [9]) by various $np$ potentials. For the chiral potentials, the ${\chi}^{2}$/datum is stated in terms of ranges which result from a variation of the cutoff parameters used in the regulator functions. The values of these cutoff parameters in units of MeV are given in parentheses. ${T}_{\mathrm{lab}}$ denotes the kinetic energy of the incident nucleon in the laboratory system.

${\mathit{T}}_{\mathbf{lab}}$ (MeV) | # of np Data | Idaho N${}^{3}$LO | Bochum N${}^{3}$LO | Argonne ${\mathit{V}}_{\mathbf{18}}$ |
---|---|---|---|---|

Energy Bin | (500–600) [32] | (600/700–450/500) [85] | Ref. [84] | |

0–100 | 1058 | 1.0–1.1 | 1.0–1.1 | 0.95 |

100–190 | 501 | 1.1–1.2 | 1.3–1.8 | 1.10 |

190–290 | 843 | 1.2–1.4 | 2.8–20.0 | 1.11 |

0–290 | 2402 | 1.1–1.3 | 1.7–7.9 | 1.04 |

${\mathit{T}}_{\mathbf{lab}}$ (MeV) | # of pp Data | Idaho N${}^{3}$LO | Bochum N${}^{3}$LO | Argonne ${\mathit{V}}_{\mathbf{18}}$ |
---|---|---|---|---|

Energy Bin | (500–600) [32] | (600/700–450/500) [85] | Ref. [84] | |

0–100 | 795 | 1.0–1.7 | 1.0–3.8 | 1.0 |

100–190 | 411 | 1.5–1.9 | 3.5–11.6 | 1.3 |

190–290 | 851 | 1.9–2.7 | 4.3–44.4 | 1.8 |

0–290 | 2057 | 1.5–2.1 | 2.9–22.3 | 1.4 |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Machleidt, R.
Chiral Symmetry and the Nucleon-Nucleon Interaction. *Symmetry* **2016**, *8*, 26.
https://doi.org/10.3390/sym8040026

**AMA Style**

Machleidt R.
Chiral Symmetry and the Nucleon-Nucleon Interaction. *Symmetry*. 2016; 8(4):26.
https://doi.org/10.3390/sym8040026

**Chicago/Turabian Style**

Machleidt, Ruprecht.
2016. "Chiral Symmetry and the Nucleon-Nucleon Interaction" *Symmetry* 8, no. 4: 26.
https://doi.org/10.3390/sym8040026