Properties of Lithium-11 and Carbon-22 at leading order in halo effective field theory

We study the $^{11}\mathrm{Li}$ and $^{22}\mathrm{C}$ nuclei at leading order (LO) in halo effective field theory (Halo EFT). Using the value of the $^{22}\mathrm{C}$ rms matter radius deduced in Ref. [1] as an input in a LO calculation, we simultaneously constrained the values of the two-neutron (2$n$) separation energy of $^{22}\mathrm{C}$ and the virtual-state energy of the $^{20}\mathrm{C}-$neutron system (hereafter denoted $^{21}$C). The 1$-\sigma$ uncertainty of the input rms matter radius datum, along with the theory error estimated from the anticipated size of the higher-order terms in the Halo EFT expansion, gave an upper bound of about 100 keV for the 2$n$ separation energy. We also study the electric dipole excitation of 2$n$ halo nuclei to a continuum state of two neutrons and the core at LO in Halo EFT. We first compare our results with the $^{11}\mathrm{Li}$ data from a Coulomb dissociation experiment and obtain good agreement within the theoretical uncertainty of a LO calculation. We then obtain the low-energy spectrum of $B(E1)$ of this transition at several different values of the 2$n$ separation energy of $^{22}\mathrm{C}$ and the virtual-state energy of $^{21}\mathrm{C}$. Our predictions can be compared to the outcome of an ongoing experiment on the Coulomb dissociation of $^{22}\mathrm{C}$ to obtain tighter constraints on the two- and three-body energies in the $^{22}\mathrm{C}$ system.


I. INTRODUCTION
The separation of scales between the size of the core and its distance from the halo nucleons allows the low-energy properites of halo nuclei to be studied using Halo EFT [2,3], which is written in terms of the core and halo nucleons as degrees of freedom. Halo EFT yields relations for the low-energy observables as systematic expansions in the ratio of the short-distance scale set by the core size and excitation energies to the long-distance scale associated with the properties of the halo nucleons. At LO, the three-body wavefunction of the 2n halo nucleus is constructed with zero-range two-body interactions, which can be completely characterized by the neutron-neutron (nn) and the neutron-core (nc) scattering lengths [4]. However, a three-body coupling also enters at LO [5], necessitating the use of one piece of three-body data as input to render the theory predictive. It is convenient to fix the three-body force by requiring the three-body bound state to lie at −E B , where E B is the 2n separation energy. The only inputs to the equations that describe a 2n halo are, therefore, E B together with the energies of the nc virtual/real bound state, E nc , and the nn virtual bound state, E nn . The effects of interactions that are higher order in the Halo EFT power counting are estimated from the size of the ignored higher-order terms and then included as theory error bands.
In Ref. [1], Tanaka et al. measured the reaction cross-section of 22 C on a hydrogen target and, using Glauber calculations, deduced a 22 C rms matter radius of 5.4 ± 0.9 fm, implying that 22 C is an S-wave two-neturon halo nucleus. This conclusion is also supported by data on high-energy two-neutron removal from 22 C [6]. We used Halo EFT in Ref. [7], to calculate the rms matter radius of 22 C as a model-independent function of E B and E nc . Since the virtual-state energy of the unbound [8] 21 C is not well known [9], we used Halo EFT to find constraints in the (E B , E nc ) plane using Tanaka et al.'s value of the rms matter radius.
We have also derived universal relations for the electric dipole excitation of two-neutron halo nuclei into the three-body continuum consisting of the core and the two neutrons in Halo EFT. Our LO calculation of the B(E1) of this transition includes all possible rescatterings with S-wave nn and nc interactions, in both the initial and the final state. We compare our results with the 11 Li data from Ref. [10] and obtain a good agreement within the theoretical uncertainty. We predict the B(E1) spectrum of 22 C for selected values of E B and E nc . These findings will be published in Ref. [11].

II. MATTER RADIUS CONSTRAINTS ON BINDING ENERGY
In Fig. 1, we plot the sets of (E B , E nc ) values that give a 22 C rms matter radius, Ref. [12], which set an upper bound of 120 keV on E B . Similarly, Ref. [13] used a correlation between the binding energy and the matter radius derived from a potential model to exclude E B > 220 keV. Our constraint is stricter than the ones set by these studies. Although our conclusion is consistent with the experimental value of −140 (460) keV from a direct mass measurement [14], more studies are needed to further reduce the large uncertainty in the 2n separation energy. In this spirit, we study the E1 excitation of 2n halo nuclei to the three-body continuum.

III. THE B(E 1) SPECTRUM
We first present the result of our LO Halo EFT calculation of the B(E1) for the break up of 11 Li into 9 Li and two neutrons at energy E in their center of mass frame. Only S-wave 9 Li − n interactions are included. After folding with the detector resolution, we obtain the curve shown in Fig. 2 for E B = 369.15(65) keV [15] and E nc = 26 keV [16]. The sensitivity to changes in E nc is much smaller than the EFT error, represented by the purple band.
Within the uncertainty of a LO calculation, a good agreement with the RIKEN data [10] is seen, despite the fact that 10 Li has a low-lying P-wave resonance which is not included in this calculation.   Fig. 1. These results agree qualitatively with those of a potential model calculation by Ref. [17]. A comparison of Fig. 3 with the forthcoming data [18] can provide further constraints on the (E B , E nc ) plane. However, the individual values of these energies thus extracted will have large error bars because different sets of (E B , E nc ) values can give similar curves. This ambiguity can be removed by looking at the neutron-momentum distribution of the Coulomb dissociation cross section [11].

IV. CONCLUSION
The matter radius and the E1 response of S-wave 2n halo nuclei were studied. We put constraints on the (E B , E nc ) parameter space using the value of the 22 C matter radius. The calculated B(E1) spectrum of 11 Li agrees with the experimental result within our theoretical uncertainty. Our 22 C result can be tested once the experimental data is available. Further improvements can be made by rigorously calculating the higher-order terms in the EFT expansion and by including higher partial waves.
We thank our collaborators Chen Ji, Hans-Werner Hammer and Philipp Hagen. This work was supported by the US Department of Energy under grant DE-FG02-93ER40756.
BA is grateful to the organizers of the conference for the opportunity to present this work and to UT for sponsoring his attendance. 5