Local chiral interactions and magnetic structure of few-nucleon systems

The magnetic form factors of ${}^2\mathrm{H},{}^3\mathrm{H}$, and ${}^3\mathrm{He}$, deuteron photodisintegration cross sections at low energies, and deuteron threshold electrodisintegration cross sections at backward angles in a wide range of momentum transfers are calculated with the chiral two-nucleon (and three-nucleon) interactions including $\mathrm{\Delta }$ intermediate states that have recently been constructed in configuration space. The $A=3$ wave functions are obtained from hyperspherical-harmonics solutions of the Schrödinger equation. The electromagnetic current includes one- and two-body terms, the latter induced by one- and two-pion exchange (OPE and TPE, respectively) mechanisms and contact interactions. The contributions associated with $\mathrm{\Delta }$ intermediate states are only retained at the OPE level and are neglected in TPE loop (tree-level) corrections to two-body (three-body) current operators. Expressions for these currents are derived and regularized in configuration space for consistency with the interactions. The low-energy constants that enter the contact currents are determined by reproducing the magnetic moments of these few-nucleon systems. The predicted form factors and deuteron electrodisintegration cross section are in excellent agreement with experiment for momentum transfers up to 2–$3{\mathrm{fm}}^{-1}$. However, the experimental values for the deuteron photodisintegration cross section are consistently underestimated by theory, unless use is made of the Siegert form of the electric dipole transition operator. A complete analysis of the results is provided, including the clarification of the origin of the aforementioned discrepancy.

Figure 1

Diagrams illustrating the contributions to the electromagnetic current up to N2LO (with power scaling up to $Q^0)$. Nucleons, $\mathrm{\Delta }$-isobars, pions, and external fields are denoted by solid, thick-solid, dashed, and wavy lines, respectively. The square in panel (d) represents relativistic corrections to the LO current. Only a single time ordering is shown in panels (b), (c), and (e).

Figure 2

Diagrams illustrating the contributions to the electromagnetic current at N3LO (with power scaling $Q)$ without the inclusion of $\mathrm{\Delta }$ intermediate states. Nucleons, pions, and external fields are denoted by solid, dashed, and wavy lines, respectively. The solid circle in panel (f) is associated with $\gamma \pi N$ interaction vertices generated by ${\mathcal{L}}_{\pi N}^{\left(3\right)}$ [17].

Figure 3

The “densities” relative to ${\mu }_d$ and corresponding to the terms proportional to $I_1^{\pi }$ and $I_2^{\pi }$ in the N3LO(OPE) current of Eq. (2.12) (curves labeled $I_1$ and $I_2)$ as well to the full current (curve labeled “sum”), shown as functions of the internucleon separation $r$ for models ${\mathrm{Ia}}^{*}$ and ${\mathrm{Ib}}^{*}$.

Figure 4

Regularized correlation functions associated with the NLO and N3LO(NM) magnetic moment operators.

Figure 5

Left panel: Predictions for the deuteron magnetic form factor, obtained with currents at LO and up to N3LO for models ${\mathrm{Ia}}^{*}/{\mathrm{b}}^{*}$ and ${\mathrm{IIa}}^{*}/{\mathrm{b}}^{*}$, are compared to the experimental data. Right panel: Cumulative contributions to the deuteron magnetic form factor, obtained at LO, N2LO, and N3LO for models ${\mathrm{Ia}}^{*}$, are compared to experimental data; note that the contributions at NLO, being isovector, vanish for this observable.

Figure 6

Magnitudes of individual contributions to the deuteron magnetic form factor (normalized to ${\mu }_d$ at $Q=0)$ obtained with the N3LO(OPE) and N3LO(NM) currents for models ${\mathrm{Ia}}^{*}$ and ${\mathrm{Ib}}^{*}$. Note that the N3LO(MIN) contribution is not shown, since it is much smaller than the N3LO(NM) one (see text); also, to reduce clutter, since the LO and N2LO(RC) contributions for models ${\mathrm{Ia}}^{*}$ and ${\mathrm{Ib}}^{*}$ are very close to each other, they are only shown for model ${\mathrm{Ia}}^{*}$. The signs are as follows: positive for LO, N3LO(OPE)-${\mathrm{Ia}}^{*}$, N3LO(NM) and negative for N2LO and N3LO(OPE)-${\mathrm{Ib}}^{*}$ (for $Q\lesssim 6{\mathrm{fm}}^{-1})$.

Figure 7

Top panels: Predictions for the ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$ magnetic form factors obtained with models ${\mathrm{Ia}}^{*}/{\mathrm{b}}^{*}$ and ${\mathrm{IIa}}^{*}/{\mathrm{b}}^{*}$ by including contributions up to N3LO in the current operator, are compared to data; also shown are the results for model ${\mathrm{Ia}}^{*}$ with the LO current operator. Bottom panels: Predictions for the isoscalar and isovector combinations of trinucleon magnetic form factors obtained with models ${\mathrm{Ia}}^{*}/{\mathrm{b}}^{*}$ and ${\mathrm{IIa}}^{*}/{\mathrm{b}}^{*}$ by including contributions up to N3LO in the current operator, are compared to data.

Figure 8

Top panels: Predictions for the ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$ magnetic form factors obtained with models ${\mathrm{Ia}}^{*}/{\mathrm{b}}^{*}$ and Ia/b by including contributions up to N3LO in the current operator are compared to data; also shown are the results for models ${\mathrm{Ia}}^{*}$ and Ia with the LO current operator. Bottom panels: Cumulative contributions to the isoscalar and isovector combinations of trinucleon magnetic form factors, obtained at LO, NLO, N2LO, and N3LO for models ${\mathrm{Ia}}^{*}$, are compared to the experimental data.

Figure 9

Magnitudes of individual contributions to the isoscalar (top panels) and isovector (lower panels) form factors obtained for models ${\mathrm{Ia}}^{*}$ and ${\mathrm{Ib}}^{*}$. The signs for $F_M^S\left(Q\right)$ are as follows: positive for LO, N3LO(MIN), N3LO(OPE)-${\mathrm{Ia}}^{*}$, and N3LO(NM) and negative for N2LO and N3LO(OPE)-${\mathrm{Ib}}^{*}$; the signs for $F_M^V\left(Q\right)$ are as follows: negative for LO, NLO, $\text{N}2\text{LO}\left(\mathrm{\Delta }\right)$ (for $Q\lesssim 1.5{\mathrm{fm}}^{-1})$, N3LO(NM), and N3LO(LOOP) and positive for N2LO(RC) and N3LO(MIN) for both models.

Figure 10

Left panel: The low-energy deuteron photodisintegration data (black, blue, and red filled circles) are compared to predictions obtained with models ${\mathrm{Ia}}^{*},{\mathrm{Ib}}^{*},{\mathrm{IIa}}^{*}$, and ${\mathrm{IIb}}^{*}$, including terms up to N3LO in the current operator. Right panel: Cumulative contributions to the low-energy deuteron photodisintegration cross section, obtained at LO, NLO, N2LO, and N3LO for models ${\mathrm{Ia}}^{*}$, are compared to experimental data; also shown are the results obtained with the Siegert form of the $E1$ transition operator.

Figure 11

The low-energy deuteron photodisintegration cross section, obtained with the LO interaction of Ref. [3] and currents at LO and up to NLO; also shown are the results obtained with the Siegert form of the $E1$ transition operator: The curves labeled “Siegert” and “NLO current” overlap. The data are represented by the black, blue, and red filled circles.

Figure 12

Left panel: The deuteron threshold electrodisintegration data at backward angles are compared to predictions obtained with interaction models ${\mathrm{Ia}}^{*},{\mathrm{Ib}}^{*},{\mathrm{IIa}}^{*}$, and ${\mathrm{IIb}}^{*}$, including terms up to N3LO in the current operator. Right panel: Cumulative contributions obtained with currents at LO, NLO, N2LO, and N3LO for model ${\mathrm{Ia}}^{*}$; also shown are the results at N3LO but excluding the $\text{N}2\text{LO}\left(\mathrm{\Delta }\right)$ current.

Figure 13

Integration contour.