Chiral effective-field theory in the $\Delta (1232)$ region: Pion electroproduction on the nucleon

We develop an extension of chiral perturbation theory to the $\Delta (1232)$-resonance energy region and apply it to investigate the pion electroproduction off the nucleon ($e^{-}N\to e^{-}N\pi$). We present a complete calculation of this process, in the $\Delta$-resonance region, up to next-to-leading order in the $\delta$ expansion. At this order, the only free parameters are the three low-energy constants corresponding to the magnetic ($M1$), electric ($E2$), and Coulomb ($C2$) $\gamma N\to \Delta$ transition strength. After fitting these parameters to a few well-known data, our calculation provides a prediction for observables and multipole amplitudes of pion electroproduction. These results compare favorably with the phenomenological multipole solutions and recent experimental results from MIT-Bates and MAMI. Our prediction for the pion-mass dependence of the $\gamma N\Delta$ form factors offers an explanation for the discrepancy between the recent lattice-QCD results and the experimental value for the “$C2/M1$ ratio” at low $Q^2$.

Figure 1

Nucleon self-energy to one loop.

Figure 2

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.

Figure 3

${\gamma }^{*}N\Delta$ vertex. The 4-momenta of the nucleon ($\Delta$) and of the photon are given by $p$ ($p^{\prime }$) and $q\equiv p^{\prime }-p$ respectively. The 4-vector index of the spin-$3/2$ field is given by $\alpha$, and $\mu$ is the 4-vector index of the photon field.

Figure 4

The higher-order $\gamma N\Delta$ vertex correction required by the electromagnetic current conservation.

Figure 5

Absorptive part of the $\gamma p{\Delta }^{+}$ vertex. (a) ${\pi }^{+}n$ loop where the photon couples to the ${\pi }^{+}$; (b) ${\pi }^{0}p$ loop where the photon couples to the charge of the proton. The vertical dotted lines indicate that the $\pi N$ intermediate state is taken on shell.

Figure 6

Examples of ${\mathrm{N}}^2\mathrm{LO}$ contributions to the $eN\to e\pi N$ reaction neglected in this work. Open circles denote the e.m. coupling to the anomalous magnetic moment of the nucleon.

Figure 7

Multipole amplitudes $M_{1+}^{(3/2)}$ (left panel) and $E_{1+}^{(3/2)}$ (right panel) for pion photoproduction as a function of the invariant mass $W$ of the $\pi N$ system. Green dashed curves: $\Delta$ contribution without the $\gamma N\Delta$-vertex corrections, [i.e., Fig. 2a without Figs. 2e, 2f]. Blue dotted curves: adding the Born contributions, Fig. 2b, to the dashed curves. Black solid curves: complete NLO calculation, includes all graphs from Fig. 2. In all curves the low-energy parameters are chosen as $g_M=2.9$, $g_E=-1.0$. The data points are from the SAID analysis (FA04K) [37] (red circles), and from the MAID 2003 analysis [11] (blue squares).

Figure 8

Nonresonant multipole amplitudes (in units ${10}^{-3}/m_{\pi }$) for pion photoproduction as a function of the invariant mass $W$ of the $\pi N$ system. The solid curves result from our NLO calculation. The data points are from the SAID analysis (FA04K) [37] (red circles), and from the MAID 2003 analysis [11] (blue squares).

Figure 9

The NLO results for the ${\Theta }_{\pi }$ dependence of the ${\gamma }^{*}p\to {\pi }^{0}p$ cross sections at $W=1.232\mathrm{GeV}$ and $Q^2=0.127{\mathrm{GeV}}^2$. The theoretical error bands are described in the text. Data points are from BATES experiments [3, 42].

Figure 10

Nonresonant multipole amplitudes (in units ${10}^{-3}/m_{\pi }$) for pion electroproduction at the resonance position ($W=1.232\mathrm{GeV}$) as a function of $Q^2$. The curves are the results of our NLO calculation. The data points are from the SAID analysis (FA04K) [37] (red circles), and from the MAID 2003 analysis [11] (blue squares).

Figure 11

$Q^2$ dependence of the NLO results (solid curves) for $R_{EM}$ (upper panel) and $R_{SM}$ (lower panel). The blue dashed curves represent a phenomenological estimate of ${\mathrm{N}}^2\mathrm{LO}$ effects by including $Q^2$ dependence in $g_E$ according to Eq. (60). The blue circles are data points from MAMI for $R_{EM}$ [1], and $R_{SM}$ [43, 44]. The green squares are data points from BATES [3].

Figure 12

Pion-mass dependence of the nucleon and $\Delta (1232)$ masses. The curves are two-parameter expressions for the $\pi N$ loop contributions to $M_N$ and $M_{\Delta }$ according to Eqs. (19, 25) respectively, using $M_N^{(0)}=0.883\mathrm{GeV}$, $c_{1N}=-0.87{\mathrm{GeV}}^{-1}$, and $M_{\Delta }^{(0)}=1.20\mathrm{GeV}$, $c_{1\Delta }=-0.40{\mathrm{GeV}}^{-1}$ respectively. The red squares are lattice results from the MILC Collaboration [45]. The stars represent the physical mass values.

Figure 13

The $m_{\pi }$ dependence of the real part of the Jones-Scadron $\gamma N\Delta$ form factor $G_M^{*}$ for $Q^2=0$ and $Q^2=0.127{\mathrm{GeV}}^2$. The solid (dashed) curves are the NLO results for $Q^2=0.127{\mathrm{GeV}}^2$ ($Q^2=0$), respectively, including the $m_{\pi }$ dependence of $M_N$ and $M_{\Delta }$. The green dotted curve is the corresponding result for $Q^2=0.127{\mathrm{GeV}}^2$ where the $m_{\pi }$ dependence of $M_N$ and $M_{\Delta }$ is not included. The blue circle for $Q^2=0$ is a data point from MAMI [1] , and the green square for $Q^2=0.127{\mathrm{GeV}}^2$ is a data point from BATES [3]. The three filled black diamonds at larger $m_{\pi }$ are lattice calculations [25] for $Q^2$ values of 0.125, 0.137, and $0.144{\mathrm{GeV}}^2$ respectively, whereas the open diamond near $m_{\pi }\simeq 0$ represents their extrapolation assuming linear dependence in $m_{\pi }^2$.

Figure 14

$m_{\pi }$ 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 [43], the green squares are data points from BATES [3]. The three filled black diamonds at larger $m_{\pi }$ are lattice calculations [25], whereas the open diamond near $m_{\pi }\simeq 0$ represents their extrapolation assuming linear dependence in $m_{\pi }^2$. Red solid curves: NLO result when accounting for the $m_{\pi }$ dependence in $M_N$ and $M_{\Delta }$; green dashed curves: NLO result of Ref. 23, where the $m_{\pi }$ dependence of $M_N$ and $M_{\Delta }$ was not accounted for.