Charge and magnetic properties of three-nucleon systems in pionless effective field theory

A method to calculate the form factor for an external current with nonderivative coupling for the three-body system in an effective field theory (EFT) of short-range interactions is shown. Using this method, the point charge radius of ${}^3\mathrm{He}$ is calculated to next-to-next-to-leading order ($\mathrm{NNLO}$) in pionless EFT (EFT($\pi /$)), and the magnetic moment and magnetic radius of ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ are calculated to next-to-leading order (NLO). For the ${}^3\mathrm{He}$ charge and magnetic form factors Coulomb interactions are ignored. The ${}^3\mathrm{He}$ point charge radius is given by 1.74(4) fm at $\mathrm{NNLO}$. This agrees well with the experimental ${}^3\mathrm{He}$ point charge radius of 1.7753(54) fm [Angeli and Marinova, At. Data Nucl. Data Tables 99, 69 (2013)]. The ${}^3\mathrm{H}$ (${}^3\mathrm{He}$) magnetic moment in units of nuclear magnetons is found to be 2.92(35) $(-2.08(25\left)\right)$ at NLO in agreement with the experimental value of 2.979 ($-2.127$). For ${}^3\mathrm{H}$ (${}^3\mathrm{He}$) the NLO magnetic radius is 1.78(11) fm (1.85(11) fm), which agrees with the experimental value of 1.840(182) fm (1.965(154) fm) [Sick, Prog. Part. Nucl. Phys. 47, 245 (2001)]. The fitting of the low-energy constant $L_1$ of the isovector two-body magnetic current and the consequences of Wigner-SU(4) symmetry for the three-nucleon magnetic moments are also discussed.

Figure 1

The top equation shows the LO dressed spin-triplet dibaryon propagator, which can be solved analytically via a geometric series. Nucleons are single lines, solid lines are the bare spin-triplet dibaryon propagator $i/{\mathrm{\Delta }}_t$, and the double line is the dressed spin-triplet dibaryon. The cross represents a NLO order effective range insertion from $c_{0t}^{\left(0\right)}$ and the star is a $\mathrm{NNLO}$ correction from $c_{0t}^{\left(1\right)}$.

Figure 2

The coupled-channel integral equations for the LO three-nucleon vertex function, where the triple line is the three-nucleon system and the filled circle is the LO three-nucleon vertex function.

Figure 3

The coupled-channel integral equations for the NLO correction to the three-nucleon vertex function.

Figure 4

The coupled-channel integral equations for the $\mathrm{NNLO}$ correction to the three-nucleon vertex function.

Figure 5

Diagrams for the LO three-nucleon charge and magnetic form factor. All wavy green lines represent either a magnetic or ${\widehat{A}}_0$ photon and the black dot their respective coupling to the nucleons.

Figure 6

Diagrams for the NLO correction to the three-nucleon charge and magnetic form factors. The boxed diagram is subtracted to avoid double counting. For the charge form factor, diagram (d) comes from gauging the dibaryon kinetic term and for the magnetic form factor from the $L_1$ and $L_2$ contact terms given in Eq. (6). Diagrams related by time-reversal symmetry are not shown.