Numerical study of the relativistic three-body quantization condition in the isotropic approximation

We present numerical results showing how our recently proposed relativistic three-particle quantization condition can be used in practice. Using the isotropic (generalized s-wave) approximation, and keeping only the leading terms in the effective range expansion, we show how the quantization condition can be solved numerically in a straightforward manner. In addition, we show how the integral equations that relate the intermediate three-particle infinite-volume scattering quantity, ${\mathcal{K}}_{\mathrm{df},3}$, to the physical scattering amplitude can be solved at and below threshold. We test our methods by reproducing known analytic results for the $1/L$ expansion of the threshold state, the volume dependence of three-particle bound-state energies, and the Bethe-Salpeter wave functions for these bound states. We also find that certain values of ${\mathcal{K}}_{\mathrm{df},3}$ lead to unphysical finite-volume energies, and give a preliminary analysis of these artifacts.

Figure 1

Examples of solving the quantization conditions in the two-particle (left) and three-particle (right) sectors for $\overrightarrow{P}=0$ and $mL=6$. The two-particle condition in the left panel can be written as ${\tilde{F}}_2^s=-1/(2\omega {\mathcal{K}}_2^s)$, where $\omega$ is the energy of the spectating third particle. This is satisfied when the two curves intersect, as indicated by the gray circles. (Here we take the spectator to have $\overrightarrow{k}=0$ and thus $\omega =m$.) This is closely analogous to the isotropic three-particle quantization condition given by Eq. (6), again satisfied at the indicated intersection points in the right panel. For this example, we take ${\mathcal{K}}_2^s$ from the leading order effective-range expansion with $ma=-10$, corresponding to an attractive two-particle interaction that pulls the lowest level below $E_2^{*}/m=2$. We take $1/{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$ to be a simple polynomial in $E/m$.

Figure 2

The four lowest-energy solutions to the quantization condition for ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=0$ and $ma=-10$ (thick, orange) together with the lowest eleven noninteracting levels, i.e., solutions for ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=a=0$, (thin, black). We only show results in the energy range for which our formalism is valid, namely $E_n<5m$. Noninteracting levels are clustered according to momenta that are degenerate in the nonrelativistic theory, as discussed in the text.

Figure 3

$E_n(L)/m$ vs $mL$ for ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=0$ and various values of the scattering length, $a$. Notation as in Fig. 2, although a larger range of $mL$ is displayed here, as well as additional noninteracting levels. The dashed black curve shows the threshold expansion, Eq. (36) through $\mathcal{O}(1/L^5)$.

Figure 4

Finite-volume energy levels for $ma=-10$ and various negative values of $m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$. The left plot shows results from two nonzero values of ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$, as well as reproducing the ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=0$ results and the noninteracting levels from Fig. 2. Note that the extent to which ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$ shifts the energy depends significantly on the level being considered. The right panel magnifies the region shown by the dashed rectangle in the left panel, displaying results for the lowest energy state from a larger number of nonzero values of ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$.

Figure 5

Finite-volume energies for $ma=-10$ and a “Breit-Wigner” ansatz for ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$, Eq. (34). The constant $c$ characterizes the strength of coupling between the resonance of mass $M_R=3.5m$ and the scattering states. For nonzero $c$, the crossing of $E(L)=M_R$ with the scattering states is replaced with an avoided level crossing. The left panel gives an overview of the position of the crossing in the overall spectrum, while the right panel zooms in on the crossing itself.

Figure 6

Finite-volume energy dependence for the bound state that arises for $m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=2500$ and $ma=-{10}^4$. In all three figures the solutions to the quantization condition are marked in orange, as points in (a) and (b) and as the curved solid line in (c). The curving (turquoise) line in panel (a) is a fit of Eq. (35) (neglecting the higher-order corrections) to the data in this panel. The same fit line is shown in panel (b) for lower values of $mL$, along with a horizontal, solid (red) line showing the infinite-volume energy of the bound state $E_B(\infty )$. The horizontal dashed (black) line shows the threshold energy $E=3m$. Panel (c) displays $E_B(L)$ for smaller $mL$, along with the same two horizontal lines as in (b) and the asymptotic prediction.

Figure 7

Log-log plot of $E(L)/m-3$ vs $mL$ (top curve) determined from the quantization condition for $ma=0.41315$ and $m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=10$, together with various subtracted curves as labeled. The results are indistinguishable for the two definitions of ${\tilde{F}}_s$, with the exception of the lowest (maximally subtracted) curve, where the $H$-function regulator is shown in orange and that based on Ref. [30] in blue. The oscillations in the former are discussed in the text.

Figure 8

Plot of $R_6(L)$ [defined in Eq. (43)] versus $1/(mL)$ for $ma=0.41315$ and $m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}=10$. The oscillating (blue) points use ${\tilde{F}}_{\mathrm{HS}}^s$, while the smooth (red) points use ${\tilde{F}}_{\mathrm{KSS}}^s$. The solid curves show quadratic and cubic fits in $1/(mL)$ to the ${\tilde{F}}_{\mathrm{KSS}}^s$ data up to $1/(mL)=0.05$. We take the average of these curves at $1/(mL)=0$ as the central value for the infinite-volume limit, and half the difference as the uncertainty.

Figure 9

Extrapolation in $1/(mL)$ of the left-hand side of Eq. (44) evaluated at $1/(m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}})=0.1$ and $ma=0.41315$. Linear and quadratic fits are done to the region of points indicated by the curves. We stress that this data was generated using ${\tilde{F}}_{\mathrm{HS}}^s$, but in this case there are only weak oscillations, unlike in Fig. 8.

Figure 10

Dependence of $-(m^4/48)\partial {\mathcal{M}}_{3,\mathrm{thr}}/\partial (1/{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}})$ vs $1/(m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}})$ for $ma=0.41315$. The points are obtained from the threshold energy, using extrapolations such as that in Fig. 9, while the solid curve is obtained from solving the integral equation relating ${\mathcal{M}}_{3,\mathrm{thr}}$ and ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$.

Figure 11

$F_3^{\mathrm{iso}}/m^2$ vs $ma$ for $E=2.99m$ and $mL=40–65$, together with an extrapolation to $L\to \infty$ using $mL=50–65$. The inset shows the small $ma$ region, in which $F_3^{\mathrm{iso}}$ changes sign.

Figure 12

$\mathcal{L}(\overrightarrow{k}=0)$ vs $ma$ for $E=2.99m$ and $mL=40–65$, together with an extrapolation to $L\to \infty$ using $mL=50–65$. Here $\mathcal{L}(\overrightarrow{k})$ for finite $L$ is given by Eq. (a15). The inset shows the small $ma$ region, within which $\mathcal{L}(0)$ changes sign. Note that $\mathcal{L}(0)=1/3$ when $ma=0$.

Figure 13

$\mathcal{L}(k)$ versus $k/m$ for choices of $ma$ shown in the legend. Results using either choice of finite-volume quantity, Eq. (a14) or (a15), and using any choice of $mL\ge 50$, lie on a common curve. Here we show the results using Eq. (a15) and $mL=70$. Note that, if $a=0$, $\mathcal{L}(k)=1/3$ independent of $k$. For sufficiently large $k$, $\mathcal{L}(k)=1/3$ for all $a$, since the cutoff functions vanish and remove the correction term.

Figure 14

Momentum dependence of the magnitude squared of the bound-state residue function. The points are predictions following from Eqs. (48) and (49), as described in the text. Different values of $L$ lead to consistent results, indicating that we have reached the infinite-volume limit. The curve shows the prediction of Eq. (50), with the value $|A{|}^2=0.948$ found in Sec. 3c.

Figure 15

$F_3^{\infty }/m^2$ vs $1/(mL)$ for $ma=0.41315$ at threshold. The line indicates a fit of the points shown in blue to a constant, which we use as our estimate of the $L\to \infty$ value.

Figure 16

$\mathcal{L}(0)$ vs $1/(mL)$ for $ma=0.41315$ at threshold. Quadratic and cubic fits to the entire set give very consistent results at $L=\infty$. (Both curves are plotted but are indistinguishable aside from a thickening of the line.) The average and half the difference of these results is used to determine the central value and uncertainty respectively.

Figure 17

${\tilde{I}}_1$ for $ma=0.41315$ plotted versus $1/(mL)$, together with linear and quadratic fits to the range indicated by the curves. (Both curves are plotted but are indistinguishable aside from a thickening of the line.) The average of the two $L\to \infty$ extrapolations is used as the central value and half the difference as the uncertainty.

Figure 18

${\tilde{I}}_2$ for $ma=0.41315$ plotted versus $1/(mL)$, together with quadratic and cubic fits to the range indicated by the curves. The average of the two $L\to \infty$ extrapolations is used as the central value and half the difference as the uncertainty.

Figure 19

$S_I$ for $ma=0.41315$ plotted versus $1/(mL)$, together with quadratic and cubic fits to the range indicated by the curves. The average of the two $L\to \infty$ extrapolations is used as the central value and half the difference as the uncertainty.

Figure 20

Plot showing an example of the unphysical solutions that arise for certain choices of parameters, here for $ma=-10$ and large, negative values of ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$. The left panel shows the lowest finite-volume level, with $-{10}^{-4}{m}^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$ ranging from 1 to 19 in unit steps. This level also appears in Fig. 4, but here we extend the results to more negative values of ${\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}$. The unphysical behavior is the doubling back of the spectrum, so that there are three, rather than one, levels for a range of $mL$ around 5.4 (the value shown by the vertical dashed line). To understand how this doubling back arises, we show in the right two panels plots of $F_3^{\mathrm{iso}}/m^2$ vs $E/m$ for $mL=5.4$, together with the values of $-1/(m^2{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}})$ whose intersections give the solutions. The middle panel looks reasonable, but the enlargement shown in the right panel reveals that $F_3^{\mathrm{iso}}$ is not decreasing monotonically, leading to the triplet of solutions for the largest three values of $|{\mathcal{K}}_{\mathrm{df},3}^{\mathrm{iso}}|$. As explained in the text, the middle solution corresponds to a pole with an unphysical residue.

Figure 21

$F_3^{\mathrm{iso}}/m^2$ vs $ma$ for $E=4m$ and $mL=10$. There are eight poles in total (shown by vertical [red] dashed lines), one per momentum shell.