Studies of quasiclassical approach applicability to true three-body decays

TL;DR

This paper evaluates the quasiclassical method's effectiveness in modeling true three-body decays within the hyperspherical harmonics framework, highlighting its accuracy and limitations through specific nuclear decay examples.

Contribution

It demonstrates the applicability and limitations of the quasiclassical approach in three-body decay calculations, especially when reducing to single-channel formalism.

Findings

Quasiclassical approach is accurate for typical three-body potential profiles.

Reduction to single-channel overestimates two-proton decay widths.

Application to $^{17}$Ne decay questions the approximation's general validity.

Abstract

Within the hyperspherical harmonics approach the three-body problem is reduced to a motion of one effective particle in a "strongly deformed" field, which is described in coupled-channel formalism. This method is especially suited to studies of phenomena characterized by genuine three-body dynamics, e.g. Borromean haloes and true three-body decays. The reduction of the hyperspherical equations set to a single-channel Schr\"odinger equation provides the basis for the use of the standard quasiclassical expression for calculations of widths for true three-body decays. We demonstrate that the quasiclassical approach by itself is quite precise in application to typical profiles of the three-body effective potentials. However, the reduction to single-channel formalism leads to significant overestimation of the two-proton width $\Gamma_{2p}$. This is demonstrated by the example of the $^{17}$Ne first excited $3/2^-$ state decay, questioning, however, the applicability of such an approximation in general.

Tables

Table 1. Table 1: The 17 Ne 3 / 2 − 3 superscript 2 3/2^{-} state widths (in MeV) obtained by Eqs. ( 6 ) and ( 3 ) for hyperspherical adiabatic effective potentials V eff subscript 𝑉 eff V_{\mbox{\scriptsize eff}} from [ 3 , 6 ] . Column 1 corresponds to the “non-converged” result of [ 3 ] (dotted black curve in Fig.  5 ), column 2 to the “accurate converged” result from [ 6 ] (solid black curve in Fig.  5 ), and column 3 shows the result from [ 6 ] which aims to imitate the calculations of Ref.  [ 12 ] (dashed black curve in Fig.  5 ).

Case:	$K_{\max }=70,l_x,l_y\le 2$	$K_{\max }=70,l_x,l_y\le 9$	$K_{\max }=20,l_x,l_y\le 10$
Ref. [3, 6]	$3.6\times {10}^{-12}$	$1.7\times {10}^{-14}$	$5.4\times {10}^{-16}$
IF, Eq. (6)	$5.1\times {10}^{-6}$	$5.7\times {10}^{-12}$	$7.7\times {10}^{-14}$
QC, Eq. (3)	$5.6\times {10}^{-6}$	$6.3\times {10}^{-12}$	$7.2\times {10}^{-14}$

Table 2. Table 2: Widths of the 17 Ne 3 / 2 − 3 superscript 2 3/2^{-} state in 10 − 15 superscript 10 15 10^{-15} MeV units as a function of the hyperspherical basis size K max subscript 𝐾 K_{\max} . IF and QC calculations with diagonalized HH potentials from [ 2 ] , three-body calculations of Ref.  [ 2 ] (see Fig. 15), and their ratio. Column “Asympt.” contains values obtained by exponential extrapolation to the infinite basis.

$K_{\max }$	$12$	$16$	$20$	$24$	$32$	$40$	$48$	Asympt.
IF, Eq. (6)	14.1	27.0	49.9	81.7	171	285	420	1100
QC, Eq. (3)	9.83	21.8	42.2	71.0	151	251	369	971
3-body, Ref. [2]	0.91	1.32	1.70	2.15	3.04	3.86	4.55	6.90
Ratio QC/3-body	10.8	16.5	24.9	33.0	49.6	65.0	81.0	141

Equations

$$\Gamma \sim \exp \left[-2{\int }_{r_2}^{r_3}p(r)dr\right],$$

$${\sigma }_l(E)\sim (\pi /k^2){\sin }^2{\delta }_l(E),$$

$${\delta }_l(E)=\arctan \left[\frac{\Gamma }{2(E_r-E)}\right]\to {\frac{d{\delta }_l(E)}{dE}|}_{E_r}=\frac{2}{\Gamma }.$$

$$\Gamma =\nu \exp \left[2i{\int }_{r_2}^{r_3}p(r)dr\right],\nu ={\left[2M{\int }_{r_1}^{r_2}\frac{dr}{p(r)}\right]}^{-1},p(r)=\sqrt{2M[E_T-V(r)]}.$$

$$\Gamma =\frac{4M}{k{\cos }^2({\delta }_l)}{\left|{\int }_0^{r_{\text{mbox}in}}{\phi }_l^{\ast }(kr)(V-\bar{V}){\tilde{\psi }}_l(kr)dr\right|}^2,$$

$${\tilde{\psi }}_l(k{r}_{\text{mbox}in})\sim G_l(k{r}_{\text{mbox}in}),{\int }_0^{r_{\text{mbox}in}}|{\tilde{\psi }}_l(kr){|}^{2}dr\equiv 1.$$

$$\Gamma =\frac{4M}{k}{\left|{\int }_0^{r_{\text{mbox}in}}{F}_l(kr)(V-V_{\text{mbox}coul}){\tilde{\psi }}_l(kr)dr\right|}^2,$$

$$\left[\frac{d^2}{d{\rho }^2}-\frac{\mathcal{L}(\mathcal{L}+1)}{{\rho }^2}+2M\{E-V_{K\gamma ,K\gamma }(\rho )-V_{\text{mbox}3b}\}\right]{\chi }_{K\gamma }(\rho )=\sum _{K^{\prime }\gamma {}^{\prime }}2M{V}_{K\gamma ,K^{\prime }\gamma {}^{\prime }}(\rho ){\chi }_{K^{\prime }\gamma {}^{\prime }}(\rho ),$$

$$\mathcal{L}=K+3/2,$$

$$V_{K\gamma ,K^{\prime }\gamma {}^{\prime }}(\rho )=\int d{\Omega }_{\rho }{\mathcal{J}}_{K\gamma }^{†}({\Omega }_{\rho })\left[{\sum }_{i>j}\hat{V}({\text{r}}_{ij})\right]{\mathcal{J}}_{K^{\prime }\gamma {}^{\prime }}({\Omega }_{\rho }),$$

$$\tilde{V}=O^{T}VO,{\tilde{V}}_{ij}={\delta }_{ij}{V}_{ii},$$

$$\left[-\frac{d^2}{d{\rho }^2}+2M\left[V_{\text{mbox}3b}(\rho )-E_T\right]+\frac{1}{{\rho }^2}\left\{{\lambda }_n(\rho )+\frac{15}{4}\right\}\right]f_n(\rho )+{\sum }_{n^{\prime }}\left(-2P_{n{n}^{\prime }}\frac{d}{d\rho }-Q_{n{n}^{\prime }}\right)f_{n^{\prime }}(\rho )=0.$$

$$V_{\text{mbox}eff}(\rho )=\frac{{\lambda }_n(\rho )+15/4}{2M{\rho }^2}+V_{\text{mbox}3b}(\rho ).$$

$$V_{\text{mbox}eff}(\rho )=\frac{\mathcal{L}(\mathcal{L}+1)}{2M{\rho }^2}+\frac{Z_{\text{mbox}eff}\alpha }{\rho }+V_{\text{mbox}3b}(\rho ).$$

Full text

1 Introduction

Conventional methods of width determination for resonant states, such as elastic phase shift energy dependence, or via S-matrix pole position in the complex energy plane could be technically complicated for very small widths $\Gamma\ll E$. Therefore, studies of radioactive decays require specific methods for the decay width determination. Among them are “natural” width definition via wave function (WF) with pure outgoing asymptotics [1], “Kadmensky-type” integral formulas (IF) [2], and quasiclassical (QC) approach of Gamow type [3].

The use of a quasi-classical approach of Gamow type for the decay width evaluation

[TABLE]

requires the reduction of the few-body problem to a single-channel formalism of some form, where Gamow integral over the sub-barrier trajectory $\{r_{2},r_{3}\}$ can be defined. Here both the validity of the few-body problem reduction and the applicability of the quasiclassical approximation for barriers of specific for few-body physics shapes can be questioned.

The formalism of the Gamow type has been repeatedly used in recent years for the determination of three-body decay widths [3, 4, 5, 6, 7, 8, 9, 10]. In this work, we examine the validity of the Gamow-type approximation by the example of the width of the first excited $3/2^{-}$ state of 17Ne [3, 6]. This state is known to decay via the so-called “true” two-proton decay mechanism [11, 1]. There are several topics of interest about this state discussed below in Section 1.1. There is also a certain story of theoretical controversy concerning width calculations for this state [12, 3, 2, 6], see Section 1.2.

1.1 Motivation for 17Ne $(3/2^{-})$ decay studies

The 17Ne nucleus is a kind of “test bench” case for several interesting concepts of nuclear structure and dynamics:

(i) The 17Ne $1/2^{-}$ ground state (g.s.) is not strongly bound. The lowest-energy threshold is $2p$ one with $S_{2p}=0.933$ MeV. This is a Borromean system since its core+$p$ subsystem, the 16F isotope, is particle-unbound with $S_{p}=-0.535$ MeV. Because of its low binding, the 17Ne has been considered as a candidate to possess $2p$ halo [13, 12, 14]. The question remains open, see e.g. the discussions in Refs. [15, 16].

(ii) The $3/2^{-}$ first excited state of 17Ne is only slightly unbound relative to $2p$ threshold with $Q_{2p}=0.355$ MeV. Because none of the 16F states is accessible for sequential proton emission, the decay mode of this nucleus is “true” $2p$ decay [11, 1]. Because of the small $Q_{2p}$ the radioactivity lifetime scale is expected for the $2p$ decay branch. This possible decay mode is quite rare in the light nuclei and also the opportunity to study such emission from an excited state is unique so far. Theoretical calculations of this width have produced considerable controversy [12, 3, 2, 6] which we are going to further discuss in this work.

(iii) There is a topic of interest from the nuclear astrophysics side, as 15O is the rp-process “waiting point”. The reaction 15O+$p$+$p\rightarrow^{17}$Ne+$\gamma$ provides a “bypass” of the 15O waiting point together with the more “conventional” 15O+$\alpha\rightarrow^{19}$Ne+$\gamma$ reaction [17]. In astrophysical conditions, the 15O+$p$+$p\rightarrow^{17}$Ne+$\gamma$ reaction has two major reaction mechanisms resonant and nonresonant.

The nonresonant contribution at temperatures of astrophysical interest is mainly defined by the low-energy behavior of the E1 electromagnetic strength function. The latter has the character of “soft dipole mode”, closely connected with the halo characteristics of 17Ne g.s. WF [18, 19, 15]. The resonant contribution to the $2p$ radiative capture on 15O is practically entirely defined by the $2p$ width of the $3/2^{-}$ first excited state of 17Ne [20, 21].

1.2 History of the question

For the first time, the particle decay modes of 17Ne were experimentally studied in paper [22] with the first (as it was understood later, erroneous) ideas about a possible observation of $2p$ emission form the $3/2^{-}$ state. These ideas were based on the $3/2^{-}$ state lifetime estimates obtained in the diproton model: $\Gamma_{2p}=1.8\times 10^{-12}$ MeV.

In 2000 the first quantum-mechanical theory of $2p$ radioactivity was developed [11] in the framework of the three-body core+$p$+$p$ model treated by the hyperspherical harmonics (HH) formalism. It was understood that the diproton model is giving the upper limit estimates for $2p$ widths, rather then realistic results. The three-body model provided much smaller $\Gamma_{2p}=1.4\times 10^{-15}$ MeV for 17Ne $3/2^{-}$ state. It became clear that there was no chance to get evidence for $2p$ emission from $3/2^{-}$ in paper [22].

The improved experiment of [23] established the limit for the ratio of $2p$ and $\gamma$ widths for the 17Ne $3/2^{-}$ state $\Gamma_{2p}/\Gamma_{\gamma}<7.7\times 10^{-3}$. This ratio derivation was based on the gamma width value $\Gamma_{\gamma}=5.5\times 10^{-9}$ MeV deduced in [22].

A more advanced than in [11] structure model of 17Ne was developed in [12]. This was taking into account core spin and provided accurate treatment of the excitation spectra in 16F subsystem of 17Ne. In the improved model, the width estimate for the $3/2^{-}$ state decay was reduced to $\Gamma_{2p}=4.1\times 10^{-16}$ MeV. The results [11, 12] were obtained with relatively limited basis sizes computationally feasible at that time.

Shortly later the work [3] criticized the results of [12] and also provided very different $\Gamma_{2p}=3.6\times 10^{-12}$ MeV. This was unexpectedly large width value, larger than the one provided by the diproton model (as we mentioned, the latter is known to give the strict upper limit for the $2p$ decay width).

The strong disagreement between $\Gamma_{2p}$ in Refs. [12] and [3] inspired us to perform considerable theoretical developments, including large basis three-body calculations and construction of “exact” semianalytical models which were free of basis convergence issues [2, 24]. The revised $2p$ width calculations allowed to confine the possible 17Ne $3/2^{-}$ state width in the limits $\Gamma_{2p}=(5-8)\times 10^{-15}$ MeV. There was no chance to close the gap with the results of [3]. Consequently, the results of Ref. [3] were revised in Ref. [6] claiming the important effect of basis convergence. The revised width value $\Gamma_{2p}=1.7\times 10^{-14}$ MeV was concluded to be consistent with the results of [2]. So, it may seem that the controversy [12, 3, 2, 6] about the 17Ne $3/2^{-}$ state width was resolved.

In recent experimental work [21], the significantly improved limit $\Gamma_{2p}/\Gamma_{\gamma}<1.6\times 10^{-4}$ was established. Using the 17Ne $3/2^{-}$ state gamma width from [22] the limit $\Gamma_{2p}<8.8\times 10^{-13}$ MeV can be found. This experimental advance inspired us to revisit the issue of the 17Ne $3/2^{-}$ state width. We found important inconsistencies in the quasiclassical three-body widths treatment in [3, 6]. Moreover, in the course of this activity, we arrived at a conclusion about the poor applicability of the quasiclassical approach to the 17Ne $3/2^{-}$ state decay. This sheds doubts on the applicability of such an approach to three-body decays in general.

2 Width definitions

The most widespread width definitions are connected with the determination of elastic scattering phase shifts $\delta_{l}(E)$. Then the resonance width can be defined either as FWHM for the resonance peak in the elastic cross section

[TABLE]

or, based on the R-matrix expression for phase shift on proximity of resonance

[TABLE]

Both methods are very inconvenient for small decay widths $\Gamma\ll E_{r}$, where a search for the resonance position and “energy scan” in its proximity becomes computationally a bit not straightforward.

In this case, the quasiclassical Gamow formula may be applied

[TABLE]

Here $r_{1}$, $r_{2}$ are inner, $r_{3}$ is outer classical turning points of the potential $V(r)$ and $M$ is the reduced mass for the channel. The preexponent $\nu$ with a dimension of energy is typically evaluated as an “assault frequency” of classical motion in the potential well $\{r_{1},r_{2}\}$. For the $2p$ width calculations, both the validity of the three-body problem reduction to a single channel formalism and the applicability of the QC approximation for barriers of specific shape can be questioned.

There exists a somewhat more complicated approach of integral (sometimes called Kadmensky-type) formulas for the decay width determination [25, 26, 2], which allows solving the Schrödinger equation only for one selected (resonance) energy. In that sense, the IF method is an analogue of QC approach and can be used to cross-check the QC results. The width value in IF method is defined as

[TABLE]

where quasistationary WF $\tilde{\psi}_{l}(kr)$ is normalized in the “internal region” $r_{\mbox{\scriptsize in}}$ and obtained by solving the Schrödinger equation with potential $V$ with the quasistationary boundary condition

[TABLE]

In general case, the $G_{l}$ is irregular at the origin Coulomb WF. The auxiliary scattering WF $\varphi_{l}(kr)$ is obtained with potential $\bar{V}$ and normalized by diagonalizing S-matrix and providing phase shifts $\delta_{l}$. This formula has an especially simple form in the case $\bar{V}\equiv V_{\mbox{\scriptsize coul}}=Z_{1}Z_{2}\alpha/r$

[TABLE]

where $F_{l}$ is regular at the origin Coulomb WF. It is clear that the radial convergence of the integral in Eq. (6) is provided when the value of $r_{\mbox{\scriptsize in}}$ is selected outside the nuclear interaction region, where $V-V_{\mbox{\scriptsize coul}}\equiv 0$. In this case, some uncertainty remains in the Eq. (6), which is connected with an uncertainty of normalization of $\tilde{\psi}_{l}$ in the Eq. (5). This uncertainty is quite sizable for the case of small barriers. It is possible to show, see [2], that in such a case $r_{\mbox{\scriptsize in}}\sim r_{3}$ should be selected for the best match between Eq. (1) and Eq. (6) results.

2.1 Reliability of the Gamow formula

Let us consider first applicability of the QC formula (3) in different conditions.

Fig. 1 shows the application of Eq. (3) to the system “dineutron”+“dineutron” (no Coulomb interaction, the reduced mass is just equal to the nucleon mass). Integral formula results are provided for cross-checking with standard potential formalism. It was also checked that for relatively large widths (e.g. corresponding to the decay energies $E>10$ eV) the IF provides exactly the same results as standard potential scattering calculations (1).

The results of two tests with Coulomb interaction are shown in Figs. 2 and 3. Fig. 2 shows the calculations with different angular momenta. Fig. 3 shows the calculations with fixed angular momentum, but for different diffusenesses. Here we test also the cases of diffusenesses much larger than those typical for two-body nuclear potentials, e.g. $a\lesssim 1$ fm.

We can see that the QC formula is quite precise by itself (from a few percents to few tens of percent). It has a trend to become more precise for larger barriers and lower (tending to zero) decay energies. There is also no problem to operate it for potentials with large diffuseness (in three-body case there is no well defined “nuclear radius” for effective potential and large diffesenesses may take place).

3 Three-body problem reduction

3.1 Hyperspherical harmonics method

The application of the hyperspherical harmonics method to three-body system provides a set of coupled differential equations in $\rho$ variable, see e.g. [27] for more details

[TABLE]

where $E_{T}$ is the energy relative to the three-body breakup threshold, $M$ is a “scaling” average nucleon mass, and $V_{\mbox{\scriptsize 3b}}$ is short-range phenomenological three-body potential used to fine-tune the decay energy. Effective angular momentum $\mathcal{L}$ in HH equations is expressed via the principal hyperspherical quantum number $K$ as

[TABLE]

so, in contrast with a two-body case, the centrifugal barrier in the three-body case is never equal to zero. The hyperspherical potentials are matrix elements of the pairwise intercluster potentials over hyperspherical harmonics $\mathcal{J}_{K\gamma}$

[TABLE]

where $\Omega_{\rho}$ is the 5-dimensional “hyperangle”, which together with hyperradius $\rho$ provides the complete description of the internal degrees of freedom for three-body systems.

The easiest way for a transition to single channel representation suitable for quasiclassical treatment is a potential diagonalization

[TABLE]

provided by the orthogonal matrix $O$. Some lowest terms of the diagonalized potential matrix for HH potentials from [12, 2] are shown in Fig. 4. The diagonal terms of the potential matrix are intersecting and the effective potential is taken be assuming that quasiclassical motion is taking place all the time along the lowest-energy branch of the potential matrix.

3.2 Hyperspherical adiabatic expansion method

Within the hyperspherical adiabatic expansion (HAE) method the following equations are solved (e.g., [3, 6])

[TABLE]

These equations contain the effective adiabatic potentials, which take the form

[TABLE]

The adiabatic terms $\lambda_{n}(\rho)$ in this approach have complicated radial behavior: they may intersect. For width calculations the lowest-energy branch of the adiabatic potential is taken.

Although this procedure was used in numerous works [3, 4, 5, 6, 7, 8, 9, 10], it has never been justified theoretically and was just accepted as a reasonable approach.

4 Three-body width of the 17Ne $3/2^{-}$ state

The paper [6] is dedicated to finding “necessary conditions for accurate computations of three-body partial decay widths”. The Fig. 5 shows the effective potential curves from papers [3, 6] (see Fig. 4 of the latter). The black dotted curve corresponds to the result of [3] which was claimed to be “non-converged” in [6] (basis size $K_{\max}=70,\,l_{x},l_{y}\leq 2$). The solid black curve corresponds to the “accurate converged” result from [6] (basis size $K_{\max}=70,\,l_{x},l_{y}\leq 9$). The dashed black curve was to imitate calculation conditions of [12] with $K_{\max}=20,\,l_{x},l_{y}\leq 10$. We repeated the QC calculations with $V_{\mbox{\scriptsize eff}}$ of the Fig. 5, see Table 1. For the decay of the 17Ne $3/2^{-}$ state we have found that we cannot reconcile the width values quoted in Ref. [6] with the potential curves provided in this work.

In more details the procedure was like follows. At first, we performed the calculations with the integral formula for width, see Eq. (6), appropriately modified for the three-body case. Effective potentials from Ref. [6] (see Fig. 4 therein) were scanned and interpolated. Since the $V_{\mbox{\scriptsize eff}}$ behavior inside the potential well ($E_{T}<0$) cannot be found in Ref. [6], we used the short-range potential $V_{\mbox{\scriptsize 3b}}$ with the Woods-Saxon formfactor to reproduce the $Q_{2p}=0.34$ MeV in the integral formula formalism. The potentials $V_{\mbox{\scriptsize 3b}}$ were also selected in such a way, that for $E_{T}>0$ they do not affect the $V_{\mbox{\scriptsize eff}}$ behavior in the barrier region. The long-range behavior of the $V_{\mbox{\scriptsize eff}}$ was fitted to the expected asymptotic form of the three-body potential in the systems with Coulomb interaction

[TABLE]

The $\mathcal{L}=7/2$ is taken as we know that only the penetration through the three-body centrifugal barrier defined by the lowest possible hyperspherical excitation $K=2$ is important on long-range asymptotics. The values $\mathcal{L}=7/2$ and $Z_{\mbox{\scriptsize eff}}$ are used to define the Coulomb WFs used in the IF calculations by Eqs. (5) and (6). For consistency, the QC calculations were performed with the same $V_{\mbox{\scriptsize 3b}}$ which were fitted to the correct $Q_{2p}$ value in IF calculations. The results of QC calculations are typically within $10\%$ around the IF values. So, QC approximation is quite precise and cannot be a source of the problems here.

Can it be the effective potential curves are provided in Ref. [6] somehow in a wrong way or we interpret them incorrectly? We have performed our own HH calculations with three-body potentials from Ref. [12, 2] using the diagonalization procedure Eq. (10), see gray and orange curves in Fig. 5. The same procedure was used for the determination of $Z_{\mbox{\scriptsize eff}}$, as described above in Eq. (13). Despite the fact that our three-body potentials are based on somewhat different two-body interactions, there are large overlap regions for $V_{\mbox{\scriptsize eff}}$ produced by HH and HAE. There is a very good overlap in the region of the Coulomb interaction dominance, where different methods should be providing close results. In the particular case of Fig. 5, exact overlap can be seen for $50<\rho<80$ fm. Evidently, we correctly interpret the effective potentials provided in Ref. [6].

The potentials obtained by a trivial single-channel reduction of the hyperspherical potentials of Ref. [2] (just diagonalization) have an analogous basis convergence trend with potential derived in [6]. It can be seen that $K_{\max}=20$ calculations in HAE (black dashed curve in Fig. 5) and HH (gray dotted curve) do not correspond well to each other in contrast with expectations of [6]. However, with the basis increase, the HH results are beginning to follow the HAE results in a larger and larger interval of $\rho$ values. It is reasonable to assume that they finally converge to the same profile, which is quite close to the converged HAE result (black solid curve in Fig. 5).

So, the single-channel effective potentials obtained in HAE and HH are consistent and should provide consistent QC and IF results. However, the QC results for $\Gamma_{2p}$ obtained with effective HH potentials are not consistent with the results of dynamical three-body calculations, which were accurately validated in Ref. [2]. This comparison for the different basis sizes is provided in Table 2. The disagreement is modest for small $K_{max}$ values, but for asymptotically large $K_{max}$ the difference exceeds two orders of the magnitude. So, we find that the reduction of three-body problem to one-channel approximation leads to the significant width overestimation, compared to fully dynamical three-body calculations.

We can draw several conclusions of different nature from our calculations here:

(i) The effective potential provided in Ref. [3] (see the black dotted curve in Fig. 5) was a pure mistake. For example, the long-range asymptotic behavior of this potential cannot be correct. Thus there is no way to obtain it as a reduction of any three-body potential whatever is the convergence. In addition, the QC width calculation by itself for this potential was also erroneous ($\sim 6$ orders of the magnitude away) in [3].

(ii) The effective potential provided in Ref. [6] (see the black solid curve in Fig. 5) looks reasonable. However, the QC width calculations for this potential were also erroneous ($\sim 2$ orders of the magnitude away).

(iii) The results for the decay widths analogous to the results of [3, 6] obtained in HAE can be obtained in HH method just without any dynamical calculation by trivial diagonalization of the potential matrix.

(iv) The QC and IF widths for single-channel reduction of the three-body problem produce much larger (up to more than two orders of the magnitude) width values than the dynamical HH calculations [2]. This is true in a broad range of basis selections.

(v) The width value $\Gamma_{2p}=6.3\times 10^{-12}$ MeV is recalculated by us from the best-converged effective potential from [6]. This value exceeds the recent experimental limit $\Gamma_{2p}<8.8\times 10^{-13}$ MeV from [21] by an order of the magnitude. This also adds confidence in the erroneous character of this result.

5 Conclusion

The standard quasiclassical approximation is easy to formally generalize for the true three-body decays, the decays in which two protons are emitted simultaneously. The motion of such a system can be considered in a certain approximation as a single-channel motion in the hyperradius $\rho$ value. In this work, we have explored the application of quasiclassical approximation to to the true three-body decays.

As a first step, we have systematically compared the “ordinary” two-body width calculations in quasiclassical approximation with potential model calculations by means of an integral formula. These approaches are found to be highly consistent (within $10\%$) for different combinations of angular and Coulomb barriers and nuclear potential diffuseness. So, for the effective hyperspherical barrier with large effective angular momentum, charge, and diffuseness the quasiclassical approximation by itself is not expected to be an obstacle.

Specifically for the decay of the 17Ne $3/2^{-}$ state we have found problems of two kinds, specific for Refs. [3, 6] and generic for quasiclassical approximation:

(i) We can not reconcile the width values quoted in Ref. [6] with the potential curves provided in this work. In general the paper [6] is dedicated to finding “necessary conditions for accurate computations of three-body partial decay widths”, so its results are expected to be specifically accurate. The potential provided as a final result of these studies for the 17Ne $3/2^{-}$ state decay gives the width value which (according to our calculations) exceeds the recent experimental limit for about an order of the magnitude.

(ii) The potentials obtained by a single-channel reduction of the hyperspherical potentials of Ref. [2] look reasonably consistent with potential derived in [6], despite the method is quite different. However, in the case of the HH method the cross-check with complete three-body calculations is available. The quasiclassical results obtained with HH method effective potentials are found to be more than an order of the magnitude larger than the results of the complete three-body calculations [2] for each considered basis size.

Finally, we conclude that quasiclassical approximation for single channel is quite precise, but the reduction to one-channel approximation leads to significant overestimation of the three-body width. We think that in the view of our results all the three-body width calculations in the papers [3, 4, 5, 6, 7, 8, 9, 10] should be questioned and applicability of the quasiclassical formalism in this case reexamined in general.

Acknowledgements. O.M.S. and L.V.G. were partly supported by the Russian Science Foundation grant No. 17-12-01367.