Signatures of few-body resonances in finite volume

We study systems of bosons and fermions in finite periodic boxes and show how the existence and properties of few-body resonances can be extracted from studying the volume dependence of the calculated energy spectra. We use and briefly review a plane-wave-based discrete variable representation, which allows a convenient implementation of periodic boundary conditions. With these calculations we establish that avoided level crossings occur in the spectra of up to four particles and can be linked to the existence of multibody resonances. To benchmark our method we use two-body calculations, where resonance properties can be determined with other methods, as well as a three-boson model interaction known to generate a three-boson resonance state. Finding good agreement for these cases, we then predict three-body and four-body resonances for models using a shifted Gaussian potential. Our results establish few-body finite-volume calculations as a new tool to study few-body resonances. In particular, the approach can be used to study few-neutron systems, where such states have been conjectured to exist.

Figure 1

$S$-wave phase shift of two particles interacting via the potential given in Eq. (33) as a function of the (dimensionless) relative kinetic energy $E$ for $V_0=6.0$ (blue solid curve) and $V_0=2.0$ (red dashed curve).

Figure 2

Energy spectrum of two particles interacting via the potential given in Eq. (33) in finite volume for different box sizes $L$. The left panel shows results for $V_0=6.0$ in the $A_1^{+}$ representation, whereas for the right panel a weaker barrier $V_0=2.0$ was used. All crossings are avoided because the spectrum is fully projected on states with the same quantum numbers. The crosses mark the inflection points used to extract the resonance energy (see text).

Figure 3

Energy spectrum of three bosons in finite volume for different box sizes $L$ interacting via the potential given in Eq. (36). States corresponding to the irreducible representation $A_1$ of the cubic symmetry group are shown as solid lines, whereas $E^{+}$ and $T_2^{+}$ states are indicated as dashed and dotted lines, respectively. The shaded area indicates the resonance position and width as calculated in Ref. [42], whereas the cross marks the inflection point used here to extract the resonance energy (see text).

Figure 4

Energy spectrum of three bosons in finite volume for different box sizes $L$. The solid lines shows the spectrum for three bosons interacting purely via the shifted Gaussian potential given in Eq. (33) with $V_0=6.0$ while the dashed and dotted lines show results with an additional attractive three-body force as in Eq. (37). With increasing three-body force, the avoided level crossing is shifted to lower energy, while the rest of the spectrum remains unaffected. For each choice of the three-body force, all crossings are avoided because the spectrum is fully projected on states with the same quantum numbers. The crosses mark the inflection points used to extract the resonance energy (see text).

Figure 5

Energy spectrum of three bosons in finite volume for different box sizes $L$. The solid line shows the spectrum for three bosons interacting purely via the shifted Gaussian potential given in Eq. (33) with $V_0=2.0$ while the dashed and dotted lines show results with an additional attractive three-body force as in Eq. (37). With increasing three-body force the avoided level crossing is shifted to lower energy, while the rest of the spectrum remains unaffected. The dashed rectangle in the left panel indicates the zoomed region shown in the right panel. For each choice of the three-body force, all crossings are avoided because the spectrum is fully projected on states with the same quantum numbers. The crosses mark the inflection points used to extract the resonance energy (see text).

Figure 6

Energy spectrum of four bosons in finite volume for different box sizes $L$ interacting via the shifted Gaussian potential given in Eq. (33) with $V_0=2.0$. The dashed rectangle in the left panel indicates the zoomed region shown in the right panel. All crossings are avoided because the spectrum is fully projected on states with the same quantum numbers. The crosses mark the inflection points used to extract the resonance energy (see text).

Figure 7

Negative-parity energy spectrum of three fermions in finite volume for different box sizes $L$ interacting via the shifted Gaussian potential given in Eq. (33) with $V_0=2.0$. All levels shown in the plot were found to belong to the $T_1^{-}$ cubic representation by performing fully projected calculations at selected volumes. Results are shown in the spin $S=1/2$ and $S=3/2$ channels. The crosses mark the inflection points used to extract the resonance energy (see text).