More on the Universal Equation for Efimov States

Abstract

Efimov states are a sequence of shallow three-body bound states that arise when the two-body scattering length is much larger than the range of the interaction. The binding energies of these states are described as a function of the scattering length and one three-body parameter by a transcendental equation involving a universal function of one angular variable. We provide an accurate and convenient parametrization of this function. Moreover, we discuss the effective treatment of range corrections in the universal equation and compare with a strictly perturbative scheme.

Appendices

Appendix A: Details of the Fitting Methodology

Since plots of the new data set $\mathbb {N}$ together with the values at the thresholds $\mathbb {T}$  questioned the consistency of the resulting data set $\mathbb {N}\cup \mathbb {T}$, we modified the first step of the fitting procedure to address this question. The single Lawson fit is replaced by a series of Lawson fits in the following way: First a set of problematic data points from $\mathbb {N}$ is defined (typically some points around the thresholds from $\mathbb {T}$). Then a Lawson fit without these problematic points is carried out and the deviations of these data points from the resulting function are calculated. Those data points whose deviation is smaller than a given threshold get included and a new fit is done. This procedure is repeated until all points are included or no more points meet the condition.

We used as threshold $1.1 d_{\mathrm {max},0}$, where ${d_{\mathrm {max},0}}$ is the maximum absolute deviation of the initial fit. This algorithm had to be employed for all fits with a different number of coefficients. The result of this procedure is that all fits are based on the complete data set $\mathbb {N}\cup \mathbb {T}$ consisting of 5031 data points.

Fig. 5

Left panel: maximum absolute deviations $d_\mathrm{max}$ of the f3d2 fits as function of the number of coefficients. Right Panel: $d_\mathrm{max}$ and the deviation of the derivative at unitarity $|d^\prime \left( -\pi /2\right) |$  after optimized rounding in the f3d2 scheme

Appendix B: Optimized Rounding Procedure

The optimized rounding improves the quality of the parametrization over standard rounding of the fit coefficients, especially when the coefficients are rounded to a low number of decimal digits. Our procedure was as follows: all coefficients $c_i$, which are rounded to $n_i$ digits, were varied independently within a certain range with a step size of $10^{-n_i}$ in order to minimize $d_\mathrm{max}$, which usually increases by rounding. This procedure was carried out for fits from $6$ up to $11$ coefficients, which were rounded to two and three digits except for the zeroth coefficient. It was rounded to higher number of digits (three or four), since $c_0=\varDelta \left( -\pi /4\right)$ holds in case of our parametrization. As a consequence we ended up with four different rounding schemes: f3d2, f4d2, f3d3 and f4d3. Here the notation f$x$d$y$ is used with $x$ as the number of decimal digits of the zeroth coefficient $c_0$ and $y$ as number of decimal digits of the other coefficients $c_{i>0}$. Thus in total $2 \cdot 2 \cdot 6 = 24$ optimized rounding procedures had to be carried out. The interval in which the coefficients were varied in each optimized rounding process was chosen so that in each process effectively about $10^{12}$ variations were tested. This corresponds to 100 variations per coefficient in a fit with 6 coefficients and 13 variations per coefficient in a fit with 11 coefficients.

In Fig. 5, we show the deviations of the different fits. These plots clearly show that the optimized rounding leads to a significant reduction of the deviations.⁵ We find that rounding the $c_{i>0}$ to two decimal places is enough to have an maximum absolute deviation smaller than $10^{-2}$. In this case at least seven coefficients are necessary. In comparison with the fit with seven coefficients the fit with eight ones has a lower $d_\mathrm{max}$ and a deviation at $-\pi /4$, which is smaller by more than one order of magnitude. Thus the fit with eight coefficients was chosen. Another advantage of this fit is a much lower deviation of its derivative from $\varDelta ^\prime \left( -\pi /2\right)$, which is approximately given to $2.125850069373$ in [27]. These observations hold for the rounding schemes f3d2 and f4d2. We selected f3d2, as the absolute deviation at $-\pi /4$ of f4d2 is also greater than $10^{-4}$ and this implies giving $c_0$ with four decimal digits is not justifiable. It should be a good compromise between accuracy and usability. It is comparable to the complexity of the old parametrization (8).

Appendix C: Comparison of Different Parametrizations

In Table 5 the new parametrization is compared to the parametrization from [15] (respectively [1]) and the one given in [14].

Table 5 Comparison of fits. One should note that we do not know which data set was used for the creation of the parametrization from [14]. $d_\mathrm{max}$ is computed using $\mathbb {N}\cup \mathbb {T}$, $d \left( \xi \right)$ denotes the deviation from $\mathbb {N}\cup \mathbb {T}$ at $\xi$. $d^\prime \left( \xi \right)$ denotes the deviation of the derivative at $\xi$