#
Weinberg’s Compositeness^{ †}

^{1}

^{2}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. EFT of Short-Range Forces

- For ${a}_{0}<0$, either two resonance poles in the lower half of the complex k plane, which are symmetric with respect to the imaginary axis (when $|{a}_{0}|<2|{r}_{0}|$), or two (separate or coincident) poles on the negative imaginary axis (when $|{a}_{0}|\ge 2|{r}_{0}|$).
- For ${a}_{0}>0$, one pole on the negative and the other on the positive imaginary axis.

## 3. Poles

## 4. Diparticle

## 5. Resonances

## 6. Conclusions

- For $\mathcal{Z}=\mathcal{O}(\aleph R)$, physics is dominated by a single pole, which represents a bound or virtual state. The kinetic term of the diparticle field is an NLO effect. If the diparticle is integrated out, the only LO interaction is a non-derivative contact interaction.
- For $\mathcal{Z}\sim 1/2$, the two poles are important. The kinetic term of the diparticle field is LO. If the diparticle is integrated out, the LO interactions consist of non-derivative and two-derivative contact interactions.
- For $1-\mathcal{Z}=\mathcal{O}(\aleph R)$, the two poles are nearly symmetrical with respect to the origin of the complex momentum plane, either on the imaginary axis or very close to the real axis. The poles arise from a weakly coupled diparticle or, if it is integrated out, from an infinite number of correlated contact interactions.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Full diparticle propagator at LO. Solid lines represent particles and a double line, a bare diparticle.

**Table 1.**Schematic representation of interaction orderings in short-range EFT with different numbers of shallow S-wave poles. At each of the three lowest orders—leading (LO), next-to-leading (NLO), and next-to-next-to-leading (N${}^{2}$LO)—the relevant low-energy constants (LECs) ${C}_{2n}$ are indicated. For an LO LEC, all powers are understood.

# S-Wave Poles | 0 | 1 | 2 |
---|---|---|---|

LO | 0 | ${C}_{0}$ | ${C}_{0,2}$ |

NLO | ${C}_{0}$ | ${C}_{2}$ | ${C}_{4}$ |

N${}^{2}$LO | ${C}_{0}^{2}$ | ${C}_{2}^{2}$ | ${C}_{4}^{2}$, ${C}_{6}$ |

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van Kolck, U.
Weinberg’s Compositeness. *Symmetry* **2022**, *14*, 1884.
https://doi.org/10.3390/sym14091884

**AMA Style**

van Kolck U.
Weinberg’s Compositeness. *Symmetry*. 2022; 14(9):1884.
https://doi.org/10.3390/sym14091884

**Chicago/Turabian Style**

van Kolck, Ubirajara.
2022. "Weinberg’s Compositeness" *Symmetry* 14, no. 9: 1884.
https://doi.org/10.3390/sym14091884