Radiative pion photoproduction in covariant chiral perturbation theory

We present a calculation of radiative pion photoproduction in the framework of covariant chiral perturbation theory with explicit $\mathrm{\Delta }\left(1232\right)$ degrees of freedom. The analysis is performed employing the small-scale expansion scheme adjusted for the $\mathrm{\Delta }$ region. Depending on the channel, we include contributions up to next-to-next-to-leading order. We fit the available experimental data for the reaction $\gamma p\to \gamma p{\pi }^0$ and extract the value of the ${\mathrm{\Delta }}^{+}$ magnetic moment. Errors from the truncation of the small-scale expansion are estimated using the Bayesian approach. We compare our results both with the previous studies within the $\delta$-expansion scheme and with the $\mathrm{\Delta }$-less theory. We also give predictions for radiative charged-pion photoproduction.

Figure 1

An example of a loop diagram included in Ref. [34] and neglected in the present work. All vertices are from the leading-order Lagrangians ${\mathcal{L}}_{\pi \pi }^{\left(2\right)}$, ${\mathcal{L}}_{\pi N}^{\left(1\right)}$, and ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(1\right)}$, except the open circle that denotes the subleading vertex from ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(2\right)}$.

Figure 2

$\mathrm{\Delta }$-less fits to the real parts of the $s$- and $p$-wave photoproduction multipoles. The solid, dashed, and dotted lines denote the $q^3$, $q^2$, and $q^1$ calculations, respectively. The bands indicate the estimated truncation errors at order $q^3$. The filled circles show the results of the MAID partial wave analysis from Ref. [87], while the squares (diamonds) are the results of the energy-dependent (energy-independent) SAID partial wave analysis from Refs. [88, 89] (Ref. [90]).

Figure 3

$\mathrm{\Delta }$-full fits to the real parts of the $s$- and $p$-wave photoproduction multipoles. The solid, dashed, and dotted lines denote the ${\epsilon }^3$, ${\epsilon }^2$, and ${\epsilon }^1$ calculations, respectively. The bands indicate the estimated truncation errors at order ${\epsilon }^3$. The filled circles show the results of the MAID partial wave analysis from Ref. [87], while the squares (diamonds) are the results of the energy-dependent (energy-independent) SAID analysis from Refs. [88, 89] (Ref. [90]).

Figure 4

$\mathrm{\Delta }$-full results at order ${\epsilon }^3$ for the imaginary parts of the $E_{1+}^{3/2}$ and $M_{1+}^{3/2}$ multipoles. The notation is as in Fig. 3.

Figure 5

Differential cross sections and the ratio $R$ (defined in text) for the reaction $\gamma p\to \gamma p{\pi }^0$. The dashed-double-dotted lines correspond to the $\mathrm{\Delta }$-less order-$q^3$ calculation. The solid lines denote the NLO $\mathrm{\Delta }$-full fit with the bands indicating the truncation errors. The double-dashed lines stand for the results of Ref. [34]. The data are from Refs. [26] (filled circles) and [22] (diamonds).

Figure 6

Convergence of the small-scale expansion for the reaction $\gamma p\to \gamma p{\pi }^0$ and sensitivity of observables to the value of ${\mu }_{{\mathrm{\Delta }}^{+}}$. The dashed (solid) lines correspond to the leading (next-to-leading) order result. The dash-dotted lines denote the result of the calculation with ${\mu }_{{\mathrm{\Delta }}^{+}}=0$. The data are from Refs. [26] (filled circles) and [22] (diamonds).

Figure 7

Double differential cross section and polarization asymmetries for the reaction $\gamma p\to \gamma p{\pi }^0$ as functions of the outgoing photon energy at $\sqrt{s}=1277\text{MeV}$. The dashed and solid lines correspond to the predictions at order ${\epsilon }_{\text{eff}}^2$ (LO) and ${\epsilon }_{\text{eff}}^3$ (NLO), respectively. The bands denote the truncation errors. The double-dashed lines denote the results of Ref. [34], while the dash-dotted lines denote the result of our calculation with ${\mu }_{{\mathrm{\Delta }}^{+}}=0$. The data points correspond to the linear photon polarization asymmetry in the reaction $\gamma p\to {\pi }^{0}p$ [109, 110], as provided in Ref. [34].

Figure 8

Differential cross sections for the reaction $\gamma p\to \gamma n{\pi }^{+}$. The dash-double-dotted lines correspond to the $\mathrm{\Delta }$-less $q^3$ calculation. The dotted, dashed, and solid lines denote the LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}\mathrm{\Delta }$-full results, respectively, with the bands indicating the truncation errors. The dash-dotted lines denote the result of the calculation with ${\mu }_{{\mathrm{\Delta }}^{+}}={\mu }_{{\mathrm{\Delta }}^0}=0$.

Figure 9

Double differential cross section and polarization asymmetries for the reaction $\gamma p\to \gamma n{\pi }^{+}$ as functions of the outgoing photon energy at $\sqrt{s}=1277\text{MeV}$. The dotted, dashed, and solid lines denote the LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}\mathrm{\Delta }$-full results, respectively, with the bands indicating the truncation errors. The dash-dotted lines denote the results of the calculation with ${\mu }_{{\mathrm{\Delta }}^{+}}={\mu }_{{\mathrm{\Delta }}^0}=0$. The double-dashed lines denote the results of Ref. [34]. The data points correspond to the linear photon polarization asymmetry in the reaction $\gamma p\to {\pi }^{+}n$ [109], as provided in Ref. [34].

Figure 10

Leading-order $O\left(q^1\right)$ tree-level radiative-pion-photoproduction diagrams. All vertices are of the lowest order, i.e., from ${\mathcal{L}}_{\pi N}^{\left(1\right)}$ and ${\mathcal{L}}_{\pi \pi }^{\left(2\right)}$.

Figure 11

Tree-level radiative-pion-photoproduction diagrams of order $q^2$. The open circles represent second-order vertices from ${\mathcal{L}}_{\pi N}^{\left(2\right)}$. Diagrams with crossed photon lines and the second-order corrections to the nucleon self-energy are not shown.

Figure 12

Third-order tree-level radiative-pion-photoproduction topologies. The open circles, the filled circles, and the squares represent vertices from ${\mathcal{L}}_{\pi N}^{\left(2\right)}$, ${\mathcal{L}}_{\pi N}^{\left(3\right)}$, and ${\mathcal{L}}_{\pi \pi }^{\left(4\right)}$, respectively. Diagrams with crossed photon lines are not shown.

Figure 13

Further tree-level radiative-pion-photoproduction topologies of order $q^3$. The filled circles are vertices from ${\mathcal{L}}_{\pi N}^{\left(3\right)}$; the square represents the vertex from ${\mathcal{L}}_{\mathrm{WZW}}^{\left(4\right)}$.

Figure 14

First set of radiative-pion-photoproduction loop diagrams. Diagrams that can be obtained by permutations of the external boson (photon and/or pion) lines are not shown.

Figure 15

Second set of radiative-pion-photoproduction loop diagrams. Crossed diagrams are not shown.

Figure 16

Third set of radiative-pion-photoproduction loop diagrams. Crossed diagrams are not shown.

Figure 17

Fourth set of radiative-pion-photoproduction loop diagrams. Diagrams with crossed photon lines are not shown.

Figure 18

Fifth set of radiative-pion-photoproduction loop diagrams. Crossed diagrams are not shown.

Figure 19

Sixth set of radiative-pion-photoproduction loop diagrams. Diagrams with crossed photon lines are not shown.

Figure 20

Seventh set of radiative-pion-photoproduction loop diagrams. Diagrams with crossed photon lines are not shown.

Figure 21

Tree-level radiative-pion-photoproduction diagrams with $\mathrm{\Delta }$ lines of order ${\epsilon }^2$. The open circle denotes the second-order vertex from ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(2\right)}$. Diagrams with crossed photon lines are not shown.

Figure 22

Tree-level radiative-pion-photoproduction diagrams with $\mathrm{\Delta }$ lines of order ${\epsilon }^3$. The open circles denote second-order vertices from ${\mathcal{L}}_{\pi N}^{\left(2\right)}$, ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(2\right)}$, and ${\mathcal{L}}_{\pi \mathrm{\Delta }}^{\left(2\right)}$. The filled circle is the vertex from ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(3\right)}$; the square represents the vertex from ${\mathcal{L}}_{\mathrm{WZW}}^{\left(4\right)}$. Diagrams with crossed photon lines are not shown.

Figure 23

First set of loop corrections to the $\mathrm{\Delta }$-pole radiative-pion-photoproduction graphs of order ${\epsilon }^3$. Diagrams with crossed photon lines are not shown.

Figure 24

Second set of loop corrections to the $\mathrm{\Delta }$-pole radiative-pion-photoproduction graphs of order ${\epsilon }^3$. Diagrams with crossed photon lines are not shown.

Figure 25

Third set of loop corrections to the $\mathrm{\Delta }$-pole radiative-pion-photoproduction graphs of order ${\epsilon }^3$. Diagrams with crossed photon lines are not shown.

Figure 26

Leading-order, i.e., order-$q^1$, tree-level pion-photoproduction diagrams. All vertices are of the lowest order, i.e., from ${\mathcal{L}}_{\pi N}^{\left(1\right)}$ and ${\mathcal{L}}_{\pi \pi }^{\left(2\right)}$.

Figure 27

Tree-level pion-photoproduction diagrams of order $q^2$. The open circles represent second-order vertices from ${\mathcal{L}}_{\pi N}^{\left(2\right)}$. Diagrams with the second-order corrections to the nucleon self-energy are not shown.

Figure 28

Third-order tree-level pion-photoproduction diagrams. The filled circles and the squares represent vertices from ${\mathcal{L}}_{\pi N}^{\left(3\right)}$ and ${\mathcal{L}}_{\pi \pi }^{\left(4\right)}$, respectively.

Figure 29

First set of pion-photoproduction loop diagrams.

Figure 30

Second set of pion-photoproduction loop diagrams.

Figure 31

Third set of pion-photoproduction loop diagrams.

Figure 32

Fourth set of pion-photoproduction loop diagrams.

Figure 33

Tree-level pion-photoproduction diagrams involving $\mathrm{\Delta }$ lines of order ${\epsilon }^2$. The open circle denotes the second-order vertex from ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(2\right)}$.

Figure 34

Tree-level pion-photoproduction diagrams involving $\mathrm{\Delta }$ lines of order ${\epsilon }^3$. The filled circle denotes the vertex from ${\mathcal{L}}_{\pi N\mathrm{\Delta }}^{\left(3\right)}$.

Figure 35

Loop corrections to the $\mathrm{\Delta }$-pole pion-photoproduction graphs of order ${\epsilon }^3$.