Nucleon-to-$\mathrm{\Delta }$ transition form factors in chiral effective field theory using the complex-mass scheme

We calculate the form factors of the electromagnetic nucleon-to-$\mathrm{\Delta }$-resonance transition to third chiral order in manifestly Lorentz-invariant chiral effective field theory. For the purpose of generating a systematic power counting, the complex-mass scheme is applied in combination with the small-scale expansion. We fit the results to available empirical data. To improve our results, we include a contribution of the $\rho$ meson at tree level in a semiphenomenological approach.

Figure 1

At $s\approx m_{\mathrm{\Delta }}^2$, the process is dominated by the $s$-channel pole diagram due to the propagator of the $\mathrm{\Delta }$(1232). The nucleon is represented by a single line, the $\mathrm{\Delta }$(1232) by a double line, the photon by a wiggly line, and the pion by a dashed line. The circles represent dressed vertices.

Figure 2

$\gamma N\to \mathrm{\Delta }$ vertex: The momentum of the incoming nucleon is called $p_i.q$ and $p_f$ denote the momenta of the photon and of the outgoing $\mathrm{\Delta }$(1232), respectively.

Figure 3

Contributions to the $\gamma N\to \mathrm{\Delta }$ transition form factors up to and including $O\left(q^3\right)$. The vertices of the diagrams (c)–(j) are always of the lowest possible order. Diagram (a) represents the contact contributions and diagram (b) shows the counter term.

Figure 4

Contribution of the $\rho$ meson (wiggly double line) to the transition form factors in a semiphenomenological approach.

Figure 5

Magnetic, electric, and charge transition form factors $G_M^{*},G_E^{*}$, and $G_C^{*}$ at the Breit-Wigner position $W=m_{\mathrm{\Delta }}=1232$ MeV. The dotted, dashed, and solid lines show fits I, II, and III, respectively. The data are fitted up to $Q^2=0.3{\mathrm{GeV}}^2$. For the magnetic form factor, the fits are also compared with experiment for a much larger range of $Q^2$. The data points for $G_M^{*}$ are from Refs. [6] (black circle), [5] (blue circles), and [3] (green triangles); for $G_E^{*}$ (from $R_{EM}$) from [6] (black circle), [13] (black diamonds), [15] (blue open circles), [8] (black triangle), and [14] (red diamonds); and for $G_C^{*}$ (from $R_{SM}$) from [15] (blue open circles), [13] (black diamonds), [7] (black circle), [11] (black triangle), and [14] (red diamonds).

Figure 6

Results for the ratios $R_{EM}$ and $R_{SM}$. For further details, see caption of Fig. 5.

Figure 7

Results of fits II (dashed black lines) and III (solid black lines) for the ratios $R_{EM}$ and $R_{SM}$ for low $Q^2$ and extrapolations into the timelike region down to the Siegert limit. The dash-dotted and long-dashed green lines show the predictions of the Siegert theorem for solutions II and III, respectively (see text). The experimental data are as in Fig. 6.

Figure 8

Form factors $G_E^{*}$ (blue solid line) and $\kappa G_C^{*}$ (red dashed line) with $\kappa =(m_{\mathrm{\Delta }}-m_N)/2m_{\mathrm{\Delta }}$ from timelike to spacelike regions. The blue circles and the red triangles show the experimental data for the electric form factor and the re-scaled charge form factor, respectively. The data are as in Fig. 5.

Figure 9

Real (left) and imaginary (right) parts of the magnetic, electric, and charge transition form factors $G_M,G_E$, and $G_C$ at the pole position $W_p=(1210-i50)$ MeV. The solid lines show the results including the $\rho$ meson (fit III) and the dotted and dashed lines show fits I and II without the $\rho$. The data points are taken as the averaged MAID and SAID results from Ref. [37].

Figure 10

Magnetic, electric, and charge transition form factors $G_M,G_E$, and $G_C$ (fit III) at the pole position split in tree diagrams with only photon couplings, tree diagrams with only $\rho$ couplings, loop diagrams, and total result. The (blue) solid lines and the blue data points show the real parts and the (red) dashed lines and the red data points show the imaginary parts of the form factors. The data points are taken as the averaged MAID and SAID results from Ref. [37].

Figure 11

Comparison of the transition form factors at the pole position with heavy-baryon chiral perturbation theory (HBChPT) [31]. The real (left) and imaginary (right) parts of the HBChPT and our calculation are shown as dashed and solid lines, respectively. The data points are taken as the averaged MAID and SAID results from Ref. [37].