Charge and matter form factors of two-neutron halo nuclei in halo effective field theory at next-to-leading order

By using halo effective field theory (EFT), an expansion in $R_{\text{core}}/R_{\text{halo}}$, where $R_{\text{core}}$ is the radius of the core and $R_{\text{halo}}$ the radius of the halo nucleus, the charge and neutron form factors of the two-neutron halo nuclei ${}^{11}\mathrm{Li}, {}^{14}\mathrm{Be}$, and ${}^{22}\mathrm{C}$ are calculated to next-to-leading order (NLO) by treating them as an effective three-body system. From the form factors, the point-charge and point-matter radii, inter-neutron distances, and neutron opening angles are extracted. Agreement is found with existing experimental extractions. Results are given for the point-charge and point-matter radii for arbitrary neutron core scattering effective range ${\rho }_{cn}$, which can be used for predictions once ${\rho }_{cn}$ is measured. Estimates for ${\rho }_{cn}$ are also used to make NLO predictions. Finally, the point-charge radii of this work are compared with other halo-EFT predictions, and setting the core mass equal to the neutron mass the point-charge radius is found to agree with an analytical prediction in the unitary limit.

Figure 1

Infinite sum of bubble diagrams that give the LO $cn$- and $nn$-dimer propagators. Solid lines are neutrons, dashed lines are cores, light rectangles are bare $cn$-dimer propagators $i/{\mathrm{\Delta }}_1$, and dark rectangles are bare $nn$-dimer propagators $i/{\mathrm{\Delta }}_0$. The NLO dimer propagators receive range corrections, represented by a cross, to the LO dressed dimer propagators.

Figure 2

Coupled integral equations for the LO trimer vertex function. The trimer field is given by the triple line and the LO trimer vertex function by the red circle.

Figure 3

Coupled integral equations for the NLO correction to the trimer vertex function. The box with a “1” inside represents the NLO correction to the trimer vertex function.

Figure 4

Diagrams representing the LO ${\mathrm{\Sigma }}_0\left(E\right)$ and NLO ${\mathrm{\Sigma }}_1\left(E\right)$.

Figure 5

Diagrams for the LO charge form factor of two-neutron halo nuclei. The wavy blue lines represent minimally coupled ${\widehat{A}}_0$ photons.

Figure 6

Diagrams for the LO neutron form factor of two-neutron halo nuclei. The green zigzags represent an external current that only couples to neutrons. Note there are two (c)-type diagrams, one with an intermediate $cn$ dimer and the other with an intermediate $nn$ dimer.

Figure 7

Diagrams for the NLO correction to the charge form factor of two-neutron halo nuclei. The boxed diagram (e) is subtracted to avoid double counting from diagram (a) and its time-reversed version, and diagram (d) comes from gauging the $cn$-dimer kinetic term. Diagrams related by time-reversal symmetry are not shown.

Figure 8

Diagrams for the NLO correction to the neutron form factor of two-neutron halo nuclei. The boxed diagram (e) is subtracted to avoid double counting. Note that there are two type-(c) and type-(d) diagrams. Diagrams related by time-reversal symmetry are not shown.

Figure 9

Geometric representation of the two-neutron halo nucleus. The value $r_c$ is the point-charge radius and $r_n$ the neutron radius, which both extend from from the c.m. of the two-neutron halo nucleus to the core (large circle) and neutron (small circle), respectively.