Pion photoproduction off nucleons in covariant chiral perturbation theory

Pion photoproduction off the nucleon close to threshold is studied in covariant baryon chiral perturbation theory at $\mathrm{O}(p^3)$ in the extended-on-mass-shell scheme, with the explicit inclusion of the $\mathrm{\Delta }(1232)$ resonance using the $\delta$ counting. The theory is compared to the available data of cross sections and polarization observables for all the charge channels. Most of the necessary low-energy constants are well known from the analysis of other processes and the comparison with data constrains some of the still unknown ones. The $\mathrm{\Delta }(1232)$ contribution is significant in improving the agreement with data, even at the low energies considered.

Figure 1

Kinematics of the pion photoproduction process. $k$ and $p$ are the incoming photon and nucleon momenta, while $q$ and $p^{\prime }$ correspond to the outgoing pion and nucleon momenta, respectively.

Figure 2

Topologies of tree-level Feynman diagrams for the reaction $\gamma N\to \pi N^{\prime }$. Diagrams (a)–(d) correspond to the $\mathcal{O}(p^k)$ amplitudes ${\mathcal{T}}_{(a)}^{(k)}$ - ${\mathcal{T}}_{(d)}^{(k)}$ given in Appendix pp1.

Figure 3

Direct (a) and crossed (b) Feynman diagrams for the reaction $\gamma N\to \pi N^{\prime }$ including the $\mathrm{\Delta }$ resonance. The labels in the circles specify the chiral order for each vertex.

Figure 4

One-loop topologies for the reaction $\gamma N\to \pi N^{\prime }$ from which Feynman diagrams are generated. Solid lines indicate nucleons, dashed lines stand for pions and the crossed-circle vertices denote the position where incoming photons can be inserted. The topologies that lead to corrections in the external pion and nucleon legs are not shown because they are taken into account by the wave function renormalization.

Figure 5

${\chi }^2$ per d.o.f. as a function of the maximum photon energy of the data included in the fit. Solid diamonds: full model at order $O(p^3)$ with $\mathrm{\Delta }$ resonance, empty circles: model without the $\mathrm{\Delta }$ inclusion. Lines to guide the eye.

Figure 6

Angular cross section for the $\gamma p\to {\pi }^{0}p$ channel at various energies. Solid line: theoretical model (fit I). Dashed line: Fit III. Data from Ref. [17] marked as red points and from [68] as violet squares. The inner band represents the statistical errors obtained by varying the LECs within their uncertainties (as shown in Table 1) in the fit. The outer band stands for the total errors where the theoretical uncertainties (due to the chiral truncation) are added to the statistical ones in quadrature.

Figure 7

Beam asymmetry for the $\gamma p\to {\pi }^{0}p$ channel at various energies. Data from Ref. [17] marked as red points and from [68] as a violet square. Description same as Fig. 6.

Figure 8

Cross section for the $\gamma p\to {\pi }^{0}p$ channel. Solid line: theoretical model, circles: data from Ref. [67], triangles: data from Ref. [83], not included in the fit. Description same as Fig. 6.

Figure 9

Angular cross section for the $\gamma p\to {\pi }^{+}n$ channel at various energies. Going from low to high energy, the data from Ref. [84] are marked as red diamonds, [77] as black squares, [76] as blue triangles, [68] as violet squares. Finally, in the lowest right panel, data for different energies from Ref. [75] are marked as magenta circles and from [65] as dark-green squares. In this latter panel the theory has been calculated at precisely the energies and angles of the data points, and the lines and bands have been interpolated. Description same as Fig. 8.

Figure 10

Cross section for the $\gamma p\to {\pi }^{+}n$ channel at various energies. Data from Ref. [85] presented as red circles. In the same way as in Fig. 9, data from [76] as blue triangles and [77] as black squares. Description same as Fig. 8.

Figure 11

Beam asymmetry for the $\gamma p\to {\pi }^{+}n$ channel at $E_{\gamma }=212.9\mathrm{MeV}$. Data from Ref. [68]. Description same as Fig. 8.

Figure 12

Cross section for the $\gamma n\to {\pi }^{-}p$ channel at various energies. Data from Ref. [86] presented as red square and data from [87] as blue filled circles. Description same as Fig. 8.

Figure 13

Angular cross section for the $\gamma n\to {\pi }^{-}p$ channel at various energies. Data from Ref. [88] are presented as red filled squares, [71] marked as black triangles, [69] as blue squares, [70] as violet filled triangles and data from [72] as magenta filled diamonds. Description same as Fig. 8.