# Low Scale Saturation of Effective NN Interactions and Their Symmetries

## Abstract

**:**

## 1. Introduction

## 2. The ${\mathit{V}}_{\mathbf{lowk}}$ Approach

## 3. ${\mathit{V}}_{\mathbf{highr}}$ vs. ${\mathit{V}}_{\mathbf{lowk}}$ Potentials

## 4. Skyrme Forces from Renormalization

#### 4.1. Partial Waves Decomposition

#### 4.2. Analysis of Counterterms

#### 4.3. Numerical Results

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix: Derivation of Analytical Results

#### General Considerations

#### Explicit Analytical $\mathcal{O}\left({p}^{2}\right)$ Results

- Uncoupled ${}^{1}{S}_{0}$ wave$$\begin{array}{ccc}\hfill -\frac{1}{{\alpha}_{{}^{1}{S}_{0}}\Lambda}& =& \frac{2\left(90{\pi}^{2}\left({\tilde{c}}_{{}^{1}{S}_{0}}+2{\pi}^{2}\right)-4{c}_{{}^{1}{S}_{0}}^{2}+60{\pi}^{2}{c}_{{}^{1}{S}_{0}}\right)}{9\pi \left(-10{\pi}^{2}{\tilde{c}}_{{}^{1}{S}_{0}}+{c}_{{}^{1}{S}_{0}}^{2}\right)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{1}{\left({c}_{{}^{1}{S}_{0}}+6{\pi}^{2}\right){}^{2}}& =& \frac{\left(-3{\Lambda}^{2}{\alpha}_{{}^{1}{S}_{0}}^{2}\left(\pi \Lambda {r}_{{}^{1}{S}_{0}}-16\right)-36\pi \Lambda {\alpha}_{{}^{1}{S}_{0}}+9{\pi}^{2}\right)}{324{\pi}^{4}\left(\pi -2\Lambda {\alpha}_{{}^{1}{S}_{0}}\right){}^{2}}.\phantom{\rule{0.166667em}{0ex}}\hfill \end{array}$$The second equation has a solution provided the numerator is positive definite$$\begin{array}{c}\hfill \left(-3{\Lambda}^{2}{\alpha}_{{}^{1}{S}_{0}}^{2}\left(\pi \Lambda {r}_{{}^{1}{S}_{0}}-16\right)-36\pi \Lambda {\alpha}_{{}^{1}{S}_{0}}+9{\pi}^{2}\right)>0.\phantom{\rule{0.166667em}{0ex}}\end{array}$$
- Waves ${}^{1}{P}_{1}$, ${}^{3}{P}_{0}$, ${}^{3}{P}_{1}$ and ${}^{3}{P}_{2}$.$$\begin{array}{ccc}\hfill \phantom{\rule{-28.45274pt}{0ex}}-\frac{1}{{\alpha}_{P}{\Lambda}^{3}}& =& \frac{8\left(350{\pi}^{3}\left({c}_{P}+24{\pi}^{3}\right)-{d}_{P}^{2}+420{\pi}^{3}{d}_{P}\right)}{75\pi \left({d}_{P}^{2}-56{\pi}^{3}{c}_{P}\right)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \phantom{\rule{-28.45274pt}{0ex}}{r}_{P}/\Lambda & =& \frac{16\left(98000{\pi}^{6}\left({d}_{P}^{2}-{c}_{P}^{2}\right)+140{\pi}^{3}{d}_{P}^{2}(25{c}_{P}+14{d}_{P})-19{d}_{P}^{4}+1568000{\pi}^{9}{d}_{P}\right)}{125\pi {\left({d}_{P}^{2}-56{\pi}^{3}{c}_{P}\right)}^{2}}.\hfill \end{array}$$
- Coupled channels ${}^{3}{S}_{1}{-}^{3}{D}_{1},{E}_{1}$.$$\begin{array}{ccc}\hfill -\frac{1}{{\alpha}_{{}^{3}{S}_{1}}\Lambda}& =& \frac{2\left(90{\pi}^{2}\left({\tilde{c}}_{{}^{3}{S}_{1}}+2{\pi}^{2}\right)-9{c}_{{E}_{1}}^{2}-4{c}_{{}^{3}{S}_{1}}^{2}+60{\pi}^{2}{c}_{{}^{3}{S}_{1}}\right)}{9\pi \left(-10{\pi}^{2}{\tilde{c}}_{{}^{3}{S}_{1}}+{c}_{{E}_{1}}^{2}+{c}_{{}^{3}{S}_{1}}^{2}\right)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {c}_{{E}_{1}}& =& \frac{2{\Lambda}^{3}\left({c}_{{}^{3}{S}_{1}}+6{\pi}^{2}\right){\alpha}_{{E}_{1}}}{3\left(\pi -2\Lambda {\alpha}_{{}^{3}{S}_{1}}\right)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{1}{\left({c}_{{}^{3}{S}_{1}}+6{\pi}^{2}\right){}^{2}}& =& \frac{\left(-4{\Lambda}^{6}{\alpha}_{{E}_{1}}^{2}-3{\Lambda}^{2}{\alpha}_{{}^{3}{S}_{1}}^{2}\left(\pi \Lambda {r}_{{}^{3}{S}_{1}}-16\right)-36\pi \Lambda {\alpha}_{{}^{3}{S}_{1}}+9{\pi}^{2}\right)}{324{\pi}^{4}\left(\pi -2\Lambda {\alpha}_{{}^{3}{S}_{1}}\right){}^{2}}.\hfill \end{array}$$The third equation has a solution provided the numerator is positive definite$$\begin{array}{c}\hfill \left(-4{\Lambda}^{6}{\alpha}_{{E}_{1}}^{2}-3{\Lambda}^{2}{\alpha}_{{}^{3}{S}_{1}}^{2}\left(\pi \Lambda {r}_{{}^{3}{S}_{1}}-16\right)-36\pi \Lambda {\alpha}_{{}^{3}{S}_{1}}+9{\pi}^{2}\right)>0.\end{array}$$

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**Figure 1.**(Color on-line) The separation method illustrated for the ${}^{1}{S}_{0}$ channel. (

**Left panel**): Full (

**red solid line**) and Truncated (

**blue dotted line**) Potentials. (

**Right panel**): Zero energy wave function. We show the free and regular solution ${u}_{0}\left(r\right)=ar$ (

**black thin solid line**), the full regular solution (

**red solid line**) and the equivalent solution (

**blue dotted line**) corresponding to the truncated potnential. The matching point $d=1.05\phantom{\rule{3.33333pt}{0ex}}\mathrm{fm}$ corresponds to the joining of the inner free wave function with the integrated-in wave function from the full potential. We take the Granada Gauss-OPE potential [48].

**Figure 2.**(Color on-line) The separation method illustrated for the ${}^{1}{S}_{0}$ channel for two equivalent methods. (

**Left panel**): The separation distance as a function of the CM momentum p. Below this distance the interaction vanishes. (

**Right panel**): The short distance potential below the zero energy separation distance $d=1.05\phantom{\rule{3.33333pt}{0ex}}\mathrm{fm}$ as a function of the CM momentum. We take the Granada Gauss–OPE potential [48].

**Figure 3.**(Color on-line) Counterterms for the S- (in $\mathrm{MeV}{\mathrm{fm}}^{3}$, upper panel) and P-waves ($\mathrm{MeV}{\mathrm{fm}}^{5}$, lower panel) as a function of the momentum scale Λ (in fm${}^{-1}$). Cs from Equation (27) solving Equation (8) including the Ds using just the low energy threshold parameters from Ref. [41] (

**thick solid**). Cs extracted from the diagonal ${V}_{\mathrm{lowk}}(p,p)$ potentials [21] at fixed $\Lambda =420\mathrm{MeV}$ for the Argonne-V18 [43] (

**dashed**). Cs for P-waves including D-terms without mixings (

**thick dotted**).

**Table 1.**Counterterms obtained from the low energy threshold parameters compiled in Ref. [33] and compared with the corresponding potential integrals in different partial waves compiled in Ref. [52] for six different potentials which fit 6713 np and pp scattering data up to ${T}_{\mathrm{LAB}}\le 350\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$. Errors quoted for each potential are statistical; errors in the last column are systematic and correspond to the sample standard deviation of the six previous columns. See main text for details on the calculation of systematic errors. Units are: $\tilde{C}$’s are in ${10}^{4}$${\mathrm{GeV}}^{-2}$, C’s are in ${10}^{4}$${\mathrm{GeV}}^{-4}$. Λ is the renormalization scale, ${\Lambda}_{\mathrm{lowk}}$ the Vlowk cut-off and ${\Lambda}_{\mathrm{Fit}}$ the maximum CM fitting momentum used to determine the interaction (all in fm${}^{-1}$).

Λ | Λ | Λ | Λ | Λ | ${\Lambda}_{\mathrm{Vlowk}}=2.1$ | ${\Lambda}_{\mathrm{Vlowk}}=2.1$ | ${\Lambda}_{\mathrm{Fit}}<2$ | ${\Lambda}_{\mathrm{Fit}}<1.25$ | |
---|---|---|---|---|---|---|---|---|---|

$0.25$ | $0.5$ | $0.75$ | 1 | $1.25$ | N3LO | AV18 | 6 Gr | χ TPE | |

${\tilde{C}}_{{}^{1}{S}_{0}}$ | –0.3825 | –0.241 | –0.198 | –0.178 | –0.174 | –0.168 | –0.164 | –0.13 (1) | –0.15 (1) |

${C}_{{}^{1}{S}_{0}}$ | –28.06 | 1.08 | 2.538 | 2.436 | 2.333 | 4.105 | 3.997 | 4.15 (6) | 4.20 (8) |

${\tilde{C}}_{{}^{3}{S}_{1}}$ | 0.795 | –0.381 | –0.297 | –0.243 | –0.214 | –0.168 | –0.164 | –0.045 (19) | –0.006 (19) |

${C}_{{}^{3}{S}_{1}}$ | –450 | –15.6 | 0.588 | 1.767 | 1.843 | 3.689 | 3.851 | 3.7 (2) | 3.34 (4) |

${C}_{{E}_{1}}$ | 8.530 | –3.905 | –2.297 | –1.714 | –1.468 | –7.912 | –7.716 | –8.42 (7) | –8.72 (6) |

${C}_{{}^{1}{P}_{1}}$ | 6.233 | 6.690 | 8.353 | 16.18 | –29.63 | 6.092 | 5.939 | 6.47 (6) | 6.45 (3) |

${C}_{{}^{3}{P}_{0}}$ | –5.248 | –4.962 | –4.323 | –3.457 | –2.599 | –4.296 | –4.456 | –4.89 (5) | –4.94 (1) |

${C}_{{}^{3}{P}_{1}}$ | 3.324 | 3.449 | 3.844 | 4.946 | 9.376 | 3.452 | 3.411 | 3.68 (6) | 3.72 (3) |

${C}_{{}^{3}{P}_{2}}$ | –0.617 | –0.612 | –0.602 | –0.581 | –0.551 | –0.559 | –0.556 | –0.43 (1) | –0.486 (8) |

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Ruiz Arriola, E.
Low Scale Saturation of Effective NN Interactions and Their Symmetries. *Symmetry* **2016**, *8*, 42.
https://doi.org/10.3390/sym8060042

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Ruiz Arriola E.
Low Scale Saturation of Effective NN Interactions and Their Symmetries. *Symmetry*. 2016; 8(6):42.
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Ruiz Arriola, Enrique.
2016. "Low Scale Saturation of Effective NN Interactions and Their Symmetries" *Symmetry* 8, no. 6: 42.
https://doi.org/10.3390/sym8060042