Electromagnetic Nucleon-to-Delta Transition in Chiral Effective-Field Theory

We perform a relativistic chiral effective-field theory calculation of pion electroproduction off the nucleon ($e^{-}N\to e^{-}N\pi$) in the $\Delta (1232)$-resonance region. After fixing the three low-energy constants, corresponding to the magnetic ($M1$), electric ($E2$), and Coulomb ($C2$) $\gamma N\Delta$ couplings, our calculation provides a prediction for the momentum transfer and pion-mass dependence of the $\gamma N\Delta$ form factors. The prediction for the pion-mass dependence resolves the discrepancy between the recent lattice QCD results and the experimental value for the “$C2/M1$ ratio” at low $Q^2$.

Figure 1

Diagrams for the $eN\to e\pi N$ reaction at NLO in the $\delta$ expansion, considered in this work. Double lines represent the $\Delta$ propagators. The crossed nucleon-exchange graph is not shown in (b) but is included in the calculation.

Figure 2

$\chi \mathrm{EFT}$ NLO results for the ${\Theta }_{\pi }$ dependence of the ${\gamma }^{*}p\to {\pi }^{0}p$ cross sections at $\sqrt{s}=1.232\mathrm{GeV}$ and $Q^2=0.127{\mathrm{GeV}}^2$. The theoretical error bands are described in the text. Data points are from BATES experiments [3, 24].

Figure 3

$Q^2$ dependence of the NLO results for $R_{EM}$ (upper panel) and $R_{SM}$ (lower panel). The blue circles are data points from MAMI for $R_{EM}$ [1] and $R_{SM}$ [25, 26]. The green squares are data points from BATES [3].

Figure 4

$m_{\pi }$ dependence of the NLO results at $Q^2=0.1{\mathrm{GeV}}^2$ for $R_{EM}$ (upper panel) and $R_{SM}$ (lower panel). The blue circle is a data point from MAMI [25]; the green squares are data points from BATES [3]. The solid black diamonds are lattice calculations [11], whereas the dashed lines and open diamonds represent their extrapolation assuming linear dependence in $m_{\pi }^2$.