Nucleon polarizabilities in covariant baryon chiral perturbation theory with explicit $\mathrm{\Delta }$ degrees of freedom

We compute various nucleon polarizabilities in chiral perturbation theory implementing the $\mathrm{\Delta }$-full ($\mathrm{\Delta }$-less) approach up to order ${\epsilon }^3+q^4$ ($q^4$) in the small-scale (chiral) expansion. The calculation is carried out using the covariant formulation of $\chi \mathrm{PT}$ by utilizing the extended on-mass shell renormalization scheme. Except for the spin-independent dipole polarizabilities used to fix the values of certain low-energy constants, our results for the nucleon polarizabilities are pure predictions. We compare our calculations with available experimental data and other theoretical results. The importance of the explicit treatment of the $\mathrm{\Delta }$ degree of freedom in the effective field theory description of the nucleon polarizabilities is analyzed. We also study the convergence of the $1/m$ expansion and analyze the efficiency of the heavy-baryon approach for the nucleon polarizabilities.

Figure 1

Tree-level diagrams for nucleon Compton scattering which are taken into account in our analysis. Vertices of order $\mathcal{O}\left(q\right)$, $\mathcal{O}\left(q^2\right)$, $\mathcal{O}\left(q^3\right),$ and $\mathcal{O}\left(q^4\right)$ are denoted by dots, circles, squares and diamonds, respectively. Solid, wavy and double lines refer to nucleons, photons and $\mathrm{\Delta }$-isobars, respectively. Time-reversed and crossed diagrams as well as the diagrams with insertions of the nucleon self-energy contact terms are not shown.

Figure 2

$\mathcal{O}\left(q^3\right)$ loop diagrams for nucleon Compton scattering. Dashed lines refer to pions. All vertices are from the leading order Lagrangians ${\mathcal{L}}_{\pi \pi }^{\left(2\right)}$ and ${\mathcal{L}}_{\pi N}^{\left(1\right)}$. Time-reversed and crossed diagrams are not shown.

Figure 3

$\mathcal{O}\left({\epsilon }^3\right)$ loop diagrams for nucleon Compton scattering. All vertices are from the leading order Lagrangians ${\mathcal{L}}_{\pi \pi }^{\left(2\right)}$, ${\mathcal{L}}_{\pi N}^{\left(1\right)}$, ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(1\right)}$, and ${\mathcal{L}}_{\pi \mathrm{\Delta }\mathrm{\Delta }}^{\left(1\right)}$. Double lines denote the $\mathrm{\Delta }$. Time-reversed and crossed diagrams are not shown.

Figure 4

$\mathcal{O}\left(q^4\right)$ loop diagrams for nucleon Compton scattering. Dots denote the leading order vertices and circles denote the vertices from ${\mathcal{L}}_{\pi N}^{\left(2\right)}$. Time-reversed and crossed diagrams are not shown.

Figure 5

$Q^2$-dependence of the scalar and spin polarizabilities for the proton (dotted and dash-dotted cyan lines) and the neutron (dashed and solid purple lines). The dotted and dashed lines correspond to the $\mathrm{\Delta }$-less $\mathcal{O}\left(q^4\right)$ results, whereas the dash-dotted and solid lines correspond to the $\mathrm{\Delta }$-full $\mathcal{O}({\epsilon }^3+q^4)$ results. The bands indicate the theoretical truncation errors.

Figure 6

$Q^2$-dependencies of the forward polarizabilities ${\gamma }_0$ and ${\delta }_{\text{LT}}$, the backward polarizability ${\gamma }_{\pi }$ and the combined higher-order polarizability ${\overline{\gamma }}_0$ for the proton (left) and the neutron (right). The thick solid blue lines indicate our $\mathrm{\Delta }$-full $\mathcal{O}(q^4+{\epsilon }^3)$ calculations with a $1\sigma$ truncation error band corresponding to $68\%$ degree-of-belief intervals and the thick dashed blue lines show our $\mathrm{\Delta }$-less $\mathcal{O}\left(q^4\right)$ calculations. The red loosely dashed lines represent the NLO $\mathrm{B}\chi \mathrm{PT}$ calculation from Ref. [39] with the red error bands. The black dash-dotted line presents the MAID model predictions from Ref. [44] (proton) and Ref. [8] (neutron). The green double-dash-dotted line is the $\mathcal{O}\left(p^4\right)$ calculation from Ref. [83]. Empirical data are: for ${\gamma }_0^{\left(p\right)}$ from [10] (triangle) and [7] (squares); for ${\gamma }_0^{\left(n\right)}$ from Ref. [11] (preliminary, triangles), Ref. [8] (square), and Ref. [9] (diamonds); for ${\delta }_{\text{LT}}^{\left(n\right)}$ from Ref. [8].

Figure 7

The $\omega$-dependence of the real parts of the dipole polarizabilities and the spinless quadrupole polarizabilities ${\alpha }_{\text{E2}}$ and ${\beta }_{\text{M2}}$ for the proton. The solid blue lines represent our $\mathrm{\Delta }$-full $\mathcal{O}(q^4+{\epsilon }^3)$ result. The inner (outer) blue bands stand for the $1\sigma$ ($2\sigma$) truncation error. The red dashed lines are the $\mathrm{B}\chi \mathrm{PT}$ calculation [40] with the red error bands. The black dash-dotted lines correspond to the fixed-$t$ dispersion-relations calculation [45] and the green double-dash-dotted lines correspond to the results of Ref. [84].

Figure 8

The $\omega$-dependence of the real parts of the dipole polarizabilities and the spinless quadrupole polarizabilities ${\alpha }_{\text{E2}}$ and ${\beta }_{\text{M2}}$ for the neutron. The solid blue lines represent our $\mathrm{\Delta }$-full $\mathcal{O}(q^4+{\epsilon }^3)$ result. The inner (outer) blue bands stand for the $1\sigma$ ($2\sigma$) truncation error. The red dashed lines are the $\mathrm{B}\chi \mathrm{PT}$ calculation [40] with the red error bands. The black dash-dotted lines correspond to the fixed-$t$ dispersion-relations calculation [45].

Figure 9

Logarithmic difference of the absolute value between the HB-expanded contributions and the final, nonexpanded value for the spin-independent and spin-dependent dipole polarizabilities of the proton. $i$ stands for the $i$-th order in the HB expansion. The black squares represent the $q^3$ contribution, the red triangles represent the $q^4$ contribution, the blue circles represent the ${\epsilon }^3$-loop contribution and the green diamonds represent the ${\epsilon }^3$-tree contributions. The dashed lines stand for the corresponding linear regression.

Figure 10

Logarithmic difference of the absolute value between the HB-expanded contributions and the final, nonexpanded value for the spin-independent and spin-dependent dipole polarizabilities of the neutron. $i$ stands for the $i\mathrm{th}$ order in the HB expansion. The black squares represent the $q^3$ contribution, the red triangles represent the $q^4$ contribution, the blue circles represent the ${\epsilon }^3$-loop contribution and the green diamonds represent the ${\epsilon }^3$-tree contributions. The dashed lines stand for the corresponding linear regression.