# Nucleon Polarizabilities and Compton Scattering as Playground for Chiral Perturbation Theory

## Abstract

**:**

## 1. Introduction

## 2. Baryon Chiral Perturbation Theory

#### 2.1. B$\chi $PT with Pions and Nucleons

#### 2.2. Inclusion of the $\mathsf{\Delta}\left(1232\right)$ and Power Counting

- low-energy region: $p\sim {m}_{\pi}$;
- resonance region: $p\sim \mathsf{\Delta}$.

#### 2.3. Low-Energy Constants and Predictive Orders

#### 2.4. Heavy-Baryon Expansion

## 3. Compton Scattering Formalism

- Real Compton scattering (RCS): ${q}^{2}={q}^{\prime \phantom{\rule{0.166667em}{0ex}}2}=0$;
- Virtual Compton scattering (VCS): ${q}^{2}=-{Q}^{2}<0$ and ${q}^{\prime \phantom{\rule{0.166667em}{0ex}}2}=0$;
- Forward doubly-virtual Compton scattering (VVCS): $q={q}^{\prime}$ (thus $p={p}^{\prime}$) and ${q}^{2}=-{Q}^{2}<0$.

## 4. Nucleon Polarizabilities

## 5. Conclusions and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

B$\chi $PT | Baryon chiral perturbation theory |

$\chi $PT | Chiral perturbation theory |

CS | Compton scattering |

EFT | Effective-field theory |

HB$\chi $PT | Heavy-baryon chiral perturbation theory |

LEC | Low-energy constant |

PWA | Partial-wave analysis |

RCS | Real Compton scattering |

VCS | Virtual Compton scattering |

VVCS | Forward doubly-virtual Compton scattering |

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**Figure 1.**Compton scattering (CS) off the nucleon in general kinematics: ${\gamma}^{*}\left(q\right)N\left(p\right)\to {\gamma}^{*}\left({q}^{\prime}\right)N\left({p}^{\prime}\right)$.

**Figure 2.**Summary for the electric dipole polarizability of the proton ${\alpha}_{E1p}$ (upper panel) and neutron ${\alpha}_{E1n}$ (lower panel). Theoretical predictions from chiral effective-field theories (EFT) and lattice QCD are compared with extractions based on CS data. Note that the lattice QCD calculations are done at unphysical pion masses. For the proton one observes a small tension between the dispersive approaches to CS and the B$\chi $PT results.

**Figure 3.**Summary for the magnetic dipole polarizability of the proton ${\beta}_{M1p}$ (upper panel) and neutron ${\beta}_{M1n}$ (lower panel). Theoretical predictions from chiral EFT and lattice QCD are compared with extractions based on CS data. Note that the lattice QCD results are extrapolated to the physical pion mass. For the proton one observes a small tension between the dispersive approaches to CS and the B$\chi $PT results.

**Figure 4.**Summary for the longitudinal-transverse polarizability of the proton ${\delta}_{LTp}$ (upper panel) and neutron ${\delta}_{LTn}$ (lower panel). Theoretical predictions from chiral EFT are compared to the MAID unitary isobar model.

**Figure 5.**Longitudinal-transverse spin polarizability, Equation (38), for the proton (left) and neutron (right) as function of ${Q}^{2}$. The black dotted line is the MAID model [57,58]; note that for the proton we use the updated estimate from the work in [28] obtained using the $\pi ,\eta ,\pi \pi $ channels. The red line shows the leading-order B$\chi $PT result. The blue band is the $\mathcal{O}({p}^{4}/\mathsf{\Delta})$ B$\chi $PT result from the work in [45]. The gray band is the $\mathcal{O}({\u03f5}^{3}+{p}^{4})$ B$\chi $PT result from the work in [59]. The orange dot-dashed and purple short-dashed lines are the $\mathcal{O}\left({p}^{3}\right)$ and $\mathcal{O}\left({p}^{4}\right)$ HB results from the work in [60]. The experimental points for the neutron are from the work in [61] (blue diamonds).

**Figure 6.**Summary for the fifth-order forward spin polarizability of the proton ${\overline{\gamma}}_{0p}$ (upper panel) and neutron ${\overline{\gamma}}_{0n}$ (lower panel). Theoretical predictions from chiral EFT are compared to empirical evaluations of the fifth-order forward spin polarizability sum rule (36) at the real-photon point and the MAID unitary isobar model.

**Figure 7.**Summary for the quadrupole polarizabilities ${\alpha}_{E2p}$ and ${\beta}_{M2p}$ of the proton. Theoretical predictions from chiral EFT are compared with extractions based on CS data.

**Figure 8.**Summary for the dispersive polarizabilities of the proton, ${\alpha}_{E1\nu p}$ and ${\beta}_{M1\nu p}$. Theoretical predictions from chiral EFT are compared with extractions based on CS data. Note that Pasquini et al. (2017) [83] presented the first extraction of the dispersive polarizabilities from proton real Compton scattering (RCS) data below pion-production threshold.

**Figure 9.**Summary for the lowest-order spin polarizabilities ${\gamma}_{E1E1p}$, ${\gamma}_{M1M1p}$, ${\gamma}_{E1M2p}$, and ${\gamma}_{M1E2p}$ of the proton. Theoretical predictions from chiral EFT are compared with extractions based on CS data. The experimental results are combinations of different beam asymmetry and double-polarization observable measurements at MAMI and LEGS: ${\mathsf{\Sigma}}_{2x}$ [86,87], ${\mathsf{\Sigma}}_{2z}$ [88], and ${\mathsf{\Sigma}}_{3}$ [89,90]. Krupina et al. [91] performed a partial-wave analysis (PWA) of proton RCS data below pion-production threshold.

**Figure 10.**Summary for the forward spin polarizability of the proton ${\gamma}_{0p}$ (upper panel) and neutron ${\gamma}_{0n}$ (lower panel). Theoretical predictions from chiral EFT are compared with empirical evaluations of the forward spin polarizability sum rule (35) at the real-photon point.

**Figure 11.**Summary for the fourth-order Baldin sum rule of the proton ${M}_{1p}^{\left(4\right)}$ (upper panel) and neutron ${M}_{1n}^{\left(4\right)}$ (lower panel). Theoretical predictions from chiral EFT are compared with empirical evaluations of the fourth-order Baldin sum rule (31) at the real-photon point.

**Figure 12.**Upper panel: Generalized forward spin polarizability, Equation (35), for the proton (left) and neutron (right) as function of ${Q}^{2}$. The black dotted line is the MAID model prediction [57,58,99], which is taken from the works in [28] (proton) and [61] (neutron). The red line shows the leading-order B$\chi $PT result. The blue band is the $\mathcal{O}({p}^{4}/\mathsf{\Delta})$ B$\chi $PT result from the work in [45]. The gray band is the $\mathcal{O}({\u03f5}^{3}+{p}^{4})$ B$\chi $PT result from the work in [59]. The purple short-dashed lines is the $\mathcal{O}\left({p}^{4}\right)$ HB results from in [60]; note that the corresponding proton curve is outside of the plotted range. The experimental points for the proton are from the works in [100] (blue dots), [101] (purple square), and [102] (orange triangle; uncertainties added in quadrature). The experimental points for the neutron are from the works in [61] (blue diamonds) and [103] (green dots; statistical and systematic uncertainties added in quadrature). Lower Panel: Fifth-order generalized forward spin polarizability, Equation (36), for the proton (left) and neutron (right) as function of ${Q}^{2}$. The black dotted line is the MAID model prediction [104]. The experimental points for the proton are from the works in [101] (purple square) and [105] (orange dot).

**Figure 13.**Summary for the longitudinal polarizability of the proton ${\alpha}_{Lp}$ (upper panel) and neutron ${\alpha}_{Ln}$ (lower panel). Theoretical predictions from chiral EFT are compared with the MAID unitary isobar model.

**Table 1.**Low-energy constants (LECs) and other parameters and the orders at which they appear in the chiral expansion when employing the low-energy $\delta $-expansion counting scheme.

Order in Chiral Expansion | $\mathit{\chi}$PT Parameters | Values | Sources | |
---|---|---|---|---|

$\mathcal{O}\left({p}^{2}\right)$ | fine-structure constant | $\alpha =MM{e}^{2}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}4\pi $ | $\simeq 1/137.04$ | |

nucleon mass | ${M}_{N}$ | $938.27$ MeV | ||

$\mathcal{O}\left({p}^{3}\right)$ | nucleon axial charge | ${g}_{A}$ | $1.27$ | neutron decay $n\to p\phantom{\rule{0.166667em}{0ex}}{e}^{-}\phantom{\rule{0.166667em}{0ex}}{\overline{\nu}}_{e}$ [47] |

pion decay constant | ${f}_{\pi}$ | $92.21$ MeV | pion decay ${\pi}^{+}\to {\mu}^{+}{\nu}_{\mu}$ [47] | |

pion mass | ${m}_{\pi}$ | $139.57$ MeV | ||

$\mathcal{O}({p}^{4}/\Delta )$ | $\mathcal{N}$-to-$\Delta $ axial coupling | ${h}_{A}$ | $2.85$ | ${P}_{33}$ partial wave in $\pi N$ scattering and $\Delta \left(1232\right)$ decay width [30,48,49] |

$\Delta \left(1232\right)$ mass | ${M}_{\Delta}$ | 1232 MeV | ||

magnetic (M1) coupling | ${g}_{M}$ | $2.97$ | pion electroproduction ${e}^{-}N\to {e}^{-}N\pi $ [50] | |

electric (E2) coupling | ${g}_{E}$ | $-1.0$ | ||

Coulomb (C2) coupling | ${g}_{\mathrm{C}}$ | $-2.6$ |

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Hagelstein, F.
Nucleon Polarizabilities and Compton Scattering as Playground for Chiral Perturbation Theory. *Symmetry* **2020**, *12*, 1407.
https://doi.org/10.3390/sym12091407

**AMA Style**

Hagelstein F.
Nucleon Polarizabilities and Compton Scattering as Playground for Chiral Perturbation Theory. *Symmetry*. 2020; 12(9):1407.
https://doi.org/10.3390/sym12091407

**Chicago/Turabian Style**

Hagelstein, Franziska.
2020. "Nucleon Polarizabilities and Compton Scattering as Playground for Chiral Perturbation Theory" *Symmetry* 12, no. 9: 1407.
https://doi.org/10.3390/sym12091407