Electromagnetic form factors of the nucleon in effective field theory

We calculate the electromagnetic form factors of the nucleon to third chiral order in manifestly Lorentz-invariant effective field theory. The $\rho$ and $\omega$ mesons as well as the $\Delta \left(1232\right)$ resonance are included as explicit dynamical degrees of freedom. To obtain a self-consistent theory with respect to constraints we consider the proper relations among the couplings of the effective Lagrangian. For the purpose of generating a systematic power counting, the extended on-mass-shell renormalization scheme is applied in combination with the small-scale expansion. The results for the electric and magnetic Sachs form factors are analyzed in terms of experimental data and compared to previous findings in the framework of chiral perturbation theory. The pion-mass dependence of the form factors is briefly discussed.

Figure 1

Tree-level Feynman diagrams contributing to the electromagnetic form factors of the nucleon up to and including $O\left(q^3\right)$. Wiggly, solid, and double-wiggly lines refer to photons, nucleons, and vector mesons, respectively. The numbers in the interaction blobs denote the chiral order of the respective vertex.

Figure 2

One-loop Feynman diagrams with nucleons, pions, and vector mesons up to and including $O\left(q^3\right)$. Wiggly, solid, dashed, and double-wiggly lines refer to photons, nucleons, pions, and vector mesons, respectively. The numbers in the interaction blobs denote the chiral order of the respective vertex.

Figure 3

One-loop Feynman diagrams with nucleons, pions, and vector mesons up to and including $O\left(q^3\right)$. For notation, see Fig. 2.

Figure 4

One-loop Feynman diagrams with nucleons, pions, vector mesons, and the $\Delta$ resonance up to and including $O\left(q^3\right)$. Wiggly, solid, dashed, double-wiggly, and double-solid lines refer to photons, nucleons, pions, vector mesons, and the $\Delta$ resonance, respectively. The numbers in the interaction blobs denote the chiral order of the respective vertex.

Figure 5

Sachs form factors of the nucleon at $O\left(q^3\right)$ including $\rho$, $\omega$, and $\Delta$, fitted to different ranges of momentum transfer $Q^2$. Solid lines correspond to a fit up to $Q_{\max }^2=0.2$ GeV${}^2$, dashed lines up to $Q_{\max }^2=0.3$ GeV${}^2$, and dotted lines up to $Q_{\max }^2=0.4$ GeV${}^2$, respectively. The gray bands represent empirical fits of the form factors to the measured cross sections.

Figure 6

Decomposition of the Sachs form factors of the nucleon at $O\left(q^3\right)$ including vector mesons and the $\Delta$ resonance. Solid lines: total results; long-dashed lines: tree contribution; short-dashed lines: contribution of set 1 of Fig. 2; dash-dotted lines: contribution of set 2 of Fig. 3; dotted lines: contribution of set 3 of Fig. 4.

Figure 7

Sachs form factors of the nucleon at $O\left(q^3\right)$ including $\rho$ and $\omega$ fitted to different ranges of momentum transfer $Q^2$. Solid lines correspond to a fit up to $Q_{\max }^2=0.2$ GeV${}^2$, dashed lines up to $Q_{\max }^2=0.3$ GeV${}^2$, and dotted lines up to $Q_{\max }^2=0.4$ GeV${}^2$, respectively. The gray bands represent empirical fits of the form factors to the measured cross sections.

Figure 8

Sachs form factors including vector mesons. The solid lines refer to the $O\left(q^3\right)$ results of this work including $\rho$ and $\omega$ mesons and the dashed lines to the $O\left(q^4\right)$ results of Ref. [26] including $\rho$, $\omega$, and $\varphi$. The gray bands represent empirical fits of the form factors to the measured cross sections.

Figure 9

Sachs form factors including vector mesons and the $\Delta$ resonance. The solid lines refer to the $O\left(q^3\right)$ results, whereas the dashed lines are supplemented by additional higher-order terms according to Eq. (31). The gray bands represent empirical fits of the form factors to the measured cross sections.

Figure 10

Isoscalar and isovector Sachs form factors including vector mesons and the $\Delta$ resonance. The solid lines refer to the $O\left(q^3\right)$ results, whereas the dashed lines are supplemented by additional higher-order terms according to Eq. (31).

Figure 11

Electric and magnetic Sachs form factors in the isoscalar and isovector channels including $\rho$ and $\omega$ for different values of the pion mass $M_{\pi }$. The solid lines refer to $M_{\pi }=140$ MeV, the dashed lines to $M_{\pi }=200$ MeV, the dash-dotted lines to $M_{\pi }=300$ MeV, and the dotted lines to $M_{\pi }=400$ MeV, respectively.

Figure 12

Pion-mass dependence of the anomalous magnetic moment in the isoscalar and isovector channels and of the mean-square isovector Dirac and Pauli radii; the vertical dashed line indicates the physical pion mass.