# Tractable $\displaystyle T$-matix model for reaction processes in muon
catalyzed fusion $\displaystyle(dt\mu)_{J=v=0}\to\;\alpha+n+\mu+17.6\,{\rm
MeV}\;\mbox{or}\;(\alpha\mu)_{nl}+n+17.6\,{\rm MeV}$

Qian Wu<EMAIL_ADDRESS>School of Physics, Nanjing University, Nanjing 21000,
China  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou
730000, China  Masayasu Kamimura<EMAIL_ADDRESS>Meson Science
Laboratory, RIKEN Nishina Center, RIKEN, Wako 351-0198, Japan

###### Abstract

Reaction processes in muon catalyzed fusion ($\displaystyle\mu$CF),
$\displaystyle(dt\mu)_{J=v=0}\to\alpha+n+\mu+17.6\,{\rm MeV}\>$ or
$\displaystyle\;(\alpha\mu)_{nl}+n+17.6\,{\rm MeV}$ in the D-T mixture was
comprehensively studied by Kamimura, Kino and Yamashita [Phys. Rev. C 107,
034607 (2023)] by solving the $\displaystyle dt\mu$-$\displaystyle\alpha n\mu$
coupled channel (CC) Schrödinger equation under a boundary condition where the
muonic molecule $\displaystyle(dt\mu)_{J=v=0}$ was set as the initial state
and the outgoing wave was in the $\displaystyle\alpha n\mu$ channel. We
approximate this CC framework and propose a considerably more tractable model
using the $\displaystyle T$-matrix method based on the Lippmann-Schwinger
equation. Nuclear interactions adopted in the $\displaystyle T$-matrix model
are determined by reproducing the cross section of the reaction $\displaystyle
d+t\to\alpha+n+17.6\,{\rm MeV}$ at low energies. The cross section of the
strong-coupling rearrangement reaction is presented in a simple closed form
based on our new model. This $\displaystyle T$-matrix model have reproduced
most of the calculated results on the above $\displaystyle\mu$CF reaction
reported by Kamimura et al. (2023) and is applicable to other
$\displaystyle\mu$CF systems such as $\displaystyle(dd\mu)$,
$\displaystyle(tt\mu)$, $\displaystyle(dt\mu)^{*}$,
$\displaystyle(dd\mu)^{*}$.

##  I INTRODUCTION

To ensure that nuclear fusion occurs within nuclear distance (a few fms) the
Coulomb barrier between two nuclei must be overcome, which typically requires
very high temperatures. At low temperatures, negative muons injected into a
mixture of deuterium (D) and tritium (T) can catalyze the fusion reaction

$\qquad\;\;d+t\rightarrow\alpha+n+17.6\;{\rm MeV},$ (1)

which is energetically the most effective nuclear fusion reaction. Following
the catalyzed reaction, free muons can facilitate another fusion reaction
taking the well-known cycle illustrated in Fig. 1. This cyclic reaction is
called muon catalyzed fusion ($\displaystyle\mu$CF). The fusion of the muonic
molecule $\displaystyle dt\mu$ has attracted more attention compared to that
of molecules such as $\displaystyle dd\mu$, $\displaystyle tt\mu$ from the
perspective of utilization as a future energy source. The history of
$\displaystyle\mu$CF since 1947 has been reviewed in Refs. Breunlich:1989vg ;
Ponomarev:1990pn ; Kamimura1998anp ; Froelich1992 . The present status of the
study of $\displaystyle dt\mu$ fusion is briefly summarized in Ref.
Kamimura:2021msf .

Recently, the $\displaystyle\mu$CF has regained considerable attention owing
to experimental and theoretical developments i) in the production of energy by
$\displaystyle\mu$CF using the high-temperature gas target of a D/T mixture
with high thermal efficiency and ii) in an ultra-slow negative muon beam by
utilizing $\displaystyle\mu$CF for various applications including scanning
negative muon microscope and an injection source for the muon collider. This
is explained in detail in the Introduction of Ref. Kamimura:2021msf .

Figure 1:  Schematic diagram of the $\displaystyle\mu$CF cycle by a muon
injected into the D/T mixture. (a) Formation of $\displaystyle dt\mu$
molecule, (b) fusion reaction, (c) $\displaystyle\alpha$-$\displaystyle\mu$
initial sticking, and (d) muon reactivation. This illustration is taken from
Ref. Kamimura:2021msf .

The study of Ref. Kamimura:2021msf  performed a comprehensive examination of
the following $\displaystyle dt\mu$ fusion reaction:

$\displaystyle\displaystyle(dt\mu)_{J=v=0}$
$\displaystyle\displaystyle\to\;\alpha\,+n\,+\mu+17.6\,\mbox{MeV}$ (2a)
$\displaystyle\displaystyle\searrow\;(\alpha\mu)_{nl}+n+17.6\,\mbox{MeV}.$
(2b)

For the first time, this study solved a coupled-channel (CC) Schrödinger
equation for the reaction (2) using the appropriate boundary conditions where
$\displaystyle(dt\mu)_{J=v=0}$ was set as the initial state and the outgoing
wave was expressed in the $\displaystyle\alpha n\mu$ channel. All interactions
were selected such that the low-energy cross sections of the reaction (1) were
reproduced using the CC calculations for the reaction. They calculated the
fusion rate of the $\displaystyle(dt\mu)_{J=v=0}$ molecule, energy (momentum)
spectra of the muon emitted by $\displaystyle\mu$CF, and
$\displaystyle\alpha$-$\displaystyle\mu$ sticking probability Kamimura:2021msf
.

As investigated in Ref. Yamashita:2022rtu , the $\displaystyle dd\mu$ and
$\displaystyle tt\mu$ fusions play important roles for the new kinematics of
the $\displaystyle\mu$CF cycle in case of high temperature D-T mixtures.
Although the $\displaystyle dd\mu$ fusion is known to be considerably weaker
than $\displaystyle dt\mu$ fusion, the former should be studied more precisely
because the $\displaystyle dd\mu$ experiment is an important preliminary
experiment of $\displaystyle dt\mu$, which has a difficulty in the tritium
treatment. Although a study on reaction processes in the $\displaystyle dd\mu$
and $\displaystyle tt\mu$ fusion is underway Kamimura2023x  using the same CC
method of Ref. Kamimura:2021msf , solving the CC equations for the reactions
expressed as Eq. (2) is difficult. Therefore, creating a starting point from
the CC method Kamimura:2021msf , such as an approximation method that
simulates their results more easily would be beneficial.

The purpose of the present paper is to propose considerably more tractable
$\displaystyle T$-matrix model than the aforementioned CC model of Ref.
Kamimura:2021msf . Instead of directly solving the CC Schrödinger equations
for the reactions (1) and (2), we approximate the $\displaystyle T$-matrix
based on the Lippmann-Schwinger equation Lippmann  that is equivalent to the
Schrödinger equation. Nuclear interactions are selected to reproduce the cross
section of reaction (1) at the center-of-mass (c.m.) energy
$\displaystyle\sim\\!1\\!-\\!300$ keV. The total angular momentum
$\displaystyle I$ and parity of the scattering states is known to be
$\displaystyle I^{\pi}=3/2^{+}$. In the following, we explain the scenario of
our model in Steps i) – v).

Step i) First, we reproduce the cross section of low-energy reaction (1) by
using the $\displaystyle d$-$\displaystyle t$ optical-potential model adopted
in Refs. KAMIMURA1989-BENCHMARK . Absorption cross section is regarded as the
cross section of reaction (1) because no other open channels exist at these
energies than the $\displaystyle d$-$\displaystyle t$ and
$\displaystyle\alpha$-$\displaystyle n$ channels. The $\displaystyle
d$-$\displaystyle t$ scattering wave function is denoted as
$\displaystyle\Phi_{dt,\frac{3}{2}M}^{\rm(opt)}$ (Sec. II A).

Step ii) We diagonalize the $\displaystyle dt\mu$ Hamiltonian composed of the
Coulomb potentials and $\displaystyle d$-$\displaystyle t$ optical potential
to obtain the complex eigenenergy ($\displaystyle E_{\rm R}+iE_{\rm I}$) and
wave function $\displaystyle\Phi_{J=v=0}(dt\mu)$ of the ground state of
$\displaystyle(dt\mu)$ molecule. The fusion rate (decay rate) of the molecule
is expressed as $\displaystyle-2E_{\rm I}/\hbar$ (Sec. II B). Here, we
introduce $\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ as the product
of $\displaystyle\Phi_{J=v=0}(dt\mu)$ and the $\displaystyle d$-$\displaystyle
t$ spin function $\displaystyle\chi_{\frac{3}{2}M}(dt)$.

Step iii) The cross section of reaction (1) can also be expressed by the exact
$\displaystyle T$-matrix introduced in Eq. (4.1) of Ref. Kamimura:2021msf ,
where $\displaystyle\Psi^{(+)}_{\alpha}$ denotes the exact solution of the CC
wave function for the reaction (1) and $\displaystyle V_{\beta}$ stands for a
coupling potential $\displaystyle V_{dt,\alpha n}$ between the $\displaystyle
d$-$\displaystyle t$ and $\displaystyle\alpha$-$\displaystyle n$ channels. In
our model, we replace the exact $\displaystyle\Psi^{(+)}_{\alpha}$ with the
$\displaystyle d$-$\displaystyle t$ wave function
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}$ obtained in Step i).
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}$ is considered to include
the effects of the outgoing $\displaystyle\alpha$-$\displaystyle n$ channel
using the imaginary part of the $\displaystyle d$-$\displaystyle t$ potential.
The coupling potential $\displaystyle V_{dt,\alpha n}$ is determined by
reproducing the cross section of reaction (1). Notably, the cross section of a
strong coupling rearrangement reaction as shown in (1.1) is expressed, based
on our model, in a simple closed form that can successfully reproduce the
observed data at low energies (Sec. III). This coupling potential
$\displaystyle V_{dt,\alpha n}$ is used in Steps iv) and v).

Step iv) In the work of Ref. [5], the $\displaystyle T$-matrix (4.1) was used
to study the three-body fusion reaction (1.2) in the manner of (5.2)–(5.7)
with outgoing waves in the $\displaystyle(\alpha\mu)\\!-\\!n$ channel. There,
the exact $\displaystyle\Psi^{(+)}_{\alpha}$ in the $\displaystyle T$-matrix
was replaced with the three-body CC wave function
$\displaystyle\Psi^{(+)}_{\frac{3}{2}M}$ of Eq.(3.3) [5]. In our model, the
exact $\displaystyle\Psi^{(+)}_{\alpha}$ is replaced with the wave function
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ of the $\displaystyle
dt\mu$ molecule obtained in Step ii). Further, the
$\displaystyle\alpha$-$\displaystyle\mu$ sticking probability is derived using
the fusion rates to the $\displaystyle\alpha$-$\displaystyle\mu$ continuum
states and those to the $\displaystyle\alpha$-$\displaystyle\mu$ bound states.

Step v) We make another calculation of the fusion rate of reaction (2) using
$\displaystyle T$-matrix (4.1) in Ref. Kamimura:2021msf  with outgoing waves
in the $\displaystyle(\alpha n)\\!-\\!\mu$ channel. The exact
$\displaystyle\Psi^{(+)}_{\alpha}$ of Eq. (3.3) in Ref. [5] is again replaced
with $\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ as in iv). We further
calculate the momentum and energy spectrum of the muons emitted by
$\displaystyle\mu$CF.

We shall examine 20 sets of the parameters for the nuclear interactions and
show that the results do not significantly depend on the choice of the
parameter sets as long as the reaction (1) is explained by using them. We
shall report that most of the results obtained in Ref. Kamimura:2021msf  are
well reproduced by the present model.

This paper is organized as follows: In Sec. II, using the optical-potential
model, we calculate the cross section of reaction (1.1) and fusion rate of the
$\displaystyle(dt\mu)_{J=v=0}$ molecule. In Sec. III, the coupling potential
between $\displaystyle d$-$\displaystyle t$ and
$\displaystyle\alpha$-$\displaystyle n$ channels is determined using the
$\displaystyle T$-matrix method. In Sec. IV, we calculate the
$\displaystyle\alpha$-$\displaystyle\mu$ sticking probability and the fusion
rate of $\displaystyle(dt\mu)_{J=v=0}$ with the method described in Step iv).
In Sec. V, the spectra of the muons emitted by $\displaystyle\mu$CF and the
fusion rate of the $\displaystyle(dt\mu)_{J=v=0}$ are calculated using the
method in Step v). The conclusions are presented in Sec. VI.

##  II Optical-potential model for fusion processes

###  II.1 Fusion cross section

Following Step i), we first investigate the fusion rection (1) by using the
optical-potential model of Ref. [9]. The potential parameters of the nuclear
$\displaystyle d$-$\displaystyle t$ potential are determined by reproducing
the cross section of the reaction. The total angular-momentum and parity
$\displaystyle J^{\pi}$ of the reaction (1) at low energies is $\displaystyle
I^{\pi}=3/2^{+}$ with $\displaystyle S$-wave and spin 3/2 in the
$\displaystyle d$-$\displaystyle t$ channel while $\displaystyle D$-wave and
spin 1/2 in the $\displaystyle\alpha$-$\displaystyle n$ channel.

We present the $\displaystyle d$-$\displaystyle t$ scattering wave function
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}(E,{\bf r})$ at the c.m.
energy $\displaystyle E$ as

$\displaystyle\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}(E,{\bf
r})=\phi_{dt,00}^{\rm(opt)}(E,{\bf r})\,\chi_{\frac{3}{2}M}(dt),$ (3)

where $\displaystyle\phi_{dt,00}^{\rm(opt)}(E,{\bf r})$ is the spatial part of
the $\displaystyle S$-wave function and $\displaystyle\chi_{\frac{3}{2}M}(dt)$
is the $\displaystyle d$-$\displaystyle t$ spin 3/2 function. Schrödinger
equation for $\displaystyle\phi_{dt,00}^{\rm(opt)}(E,{\bf r})$ is presented as

$\displaystyle\displaystyle(H_{dt}-E)\,\phi_{dt,00}^{\rm(opt)}(E,{\bf r})=0,$
(4) $\displaystyle\displaystyle H_{dt}=T_{{\bf
r}}+V_{dt}^{{\rm(N})}(r)+iW_{dt}^{{\rm(N})}(r)+V_{dt}^{({\rm Coul})}(r),$ (5)

where the spin-independent $\displaystyle d$-$\displaystyle t$ optical
potential is given by

$\displaystyle\displaystyle\\!V_{dt}^{{\rm(N})}(r)=V_{0}/\\{1+e^{(r-R_{0})/a}\\},$
(7)
$\displaystyle\displaystyle\\!\\!W_{dt}^{{\rm(N})}(r)=W_{0}/\\{1+e^{(r-R_{\rm
I})/a_{\rm I}}\\},$ $\displaystyle\displaystyle\\!\\!\\!\\!\\!V_{dt}^{({\rm
Coul})}(r)=\begin{cases}(e^{2}/(2R_{\rm c}))(3-r^{2}/R_{\rm
c}^{2})\,,&\text{$\displaystyle\;r<R_{\rm c}$},\\\
e^{2}/r\,,&\text{$\displaystyle\;r\geq R_{\rm c}$}\end{cases}$ (8)

with taking the charge radius $\displaystyle R_{\rm c}=R_{0}$.

In the energy regions shown in Fig. 2, only the
$\displaystyle\alpha$-$\displaystyle n$ channel is open, except for the
incoming channel. Therefore, the absorption cross section becomes the cross
section of the reaction (1) as

$\displaystyle\displaystyle\sigma_{dt\to\alpha
n}(E)=\frac{2I+1}{(2I_{d}+1)(2I_{t}+1)}\frac{\pi}{k^{2}}(1-|S_{J}(E)|^{2}),\;\;$
(9)

with $\displaystyle S$-matrix $\displaystyle S_{J}(E)$. In Eq. (2.7),
$\displaystyle I_{d}=1$, $\displaystyle I_{t}=1/2$, and $\displaystyle
k=\sqrt{2\mu_{dt}E}/\hbar$. The astrophysical $\displaystyle S$-factor
$\displaystyle S(E)$ is derived from the cross section as

$\displaystyle\displaystyle\sigma_{dt\to\alpha
n}(E)=S(E)\,e^{-2\pi\eta(E)}/E,$ (10)

where $\displaystyle\eta(E)$ denotes the Sommerfeld parameter.

Owing to the lack of $\displaystyle d$-$\displaystyle t$ elastic scattering
information for $\displaystyle E\lesssim$ 300 keV demonstrating the nuclear-
interaction effect, it is impossible to determine a unique $\displaystyle
d$-$\displaystyle t$ optical potential based on the observed $\displaystyle
S$-factor in Fig. 2 (black solid line). Therefore, in Ref.
KAMIMURA1989-BENCHMARK , five sets of $\displaystyle d$-$\displaystyle t$
optical potentials, denoted as A-E, were selected to reproduce the observed
data. In the present study, the parameter $\displaystyle W_{0}$ is changed
slightly to improve the agreement with the experimental $\displaystyle
S$-factor $\displaystyle S(E)$ for $\displaystyle E\leq$ 10 keV. The
calculated $\displaystyle S(E)$ values with optical potential A-E are shown in
Fig. 2; within the experimental error range [10], the observed $\displaystyle
S(E)$ is reproduced well. The potential parameters are listed in Table. 1.

Figure 2: Calculated $\displaystyle S$-factor $\displaystyle S(E)$ of the reaction $\displaystyle d+t\to\alpha+n+17.6\,$ MeV using five different $\displaystyle d$-$\displaystyle t$ optical potentials A to E listed in Table 1. The black solid line (Exp.) is taken from a review paper Serpico:2004gx ; it fits the literature data using the function $\displaystyle S(E)=(26-0.361E+248E^{2})/{1+[(E-0.0479)/0.0392]^{2}}$ MeV b ($\displaystyle E$ in MeV).  Table 1: Five sets (A to E) of the $\displaystyle d$-$\displaystyle t$ optical-potential parameters. |  $\displaystyle V_{0}$ |  $\displaystyle R_{0}$ |  $\displaystyle a$ |  $\displaystyle W_{0}$ |  $\displaystyle R_{I}$ |  $\displaystyle a_{I}$
---|---|---|---|---|---|---
|  (MeV) |  (fm) |  (fm) |  (MeV) |  (fm) |  (fm)
A |  58.52 |  2.5 |  0.3 |  $\displaystyle-0.30$ |  2.5 |  0.3
B |  38.01 |  3.0 |  0.5 |  $\displaystyle-0.30$ |  3.0 |  0.5
C |  28.27 |  3.0 |  1.0 |  $\displaystyle-0.66$ |  2.0 |  1.0
D |  16.04 |  5.0 |  0.3 |  $\displaystyle-0.22$ |  2.5 |  0.3
E |  13.19 |  5.0 |  1.0 |  $\displaystyle-0.33$ |  3.0 |  1.0

###  II.2 Fusion rate of muonic molecule

Figure 3: Three-body Jacobi coordinates used in this study.

Following Step ii), we calculate the fusion rate of the reaction (1.2) by
diagonalizing the $\displaystyle dt\mu$ three body Hamiltonian including the
Coulomb force and the $\displaystyle d$-$\displaystyle t$ nuclear complex
potentials determined in the previous subsection. We perform a non-adiabatic
three-body calculation of the ground-state wave function of the $\displaystyle
dt\mu$ molecule, $\displaystyle\Phi_{J=v=0}(dt\mu)$, using the Gaussian
Expansion Method for few-body systems (GEM) Kamimura:1988zz ; Kameyama89 ;
Hiyama03 :

$\displaystyle\displaystyle(H_{dt\mu}-E_{00})\,\Phi_{J=v=0}(dt\mu)=0,$ (11)
$\displaystyle\displaystyle H_{dt\mu}=T_{{\bf r}_{c}}+T_{{\bf R}_{c}}+V^{({\rm
C})}(r_{1})+V^{({\rm C})}(r_{2})$ $\displaystyle\displaystyle\qquad+V^{({\rm
N})}_{dt}(r_{3})+iW^{({\rm N})}_{dt}(r_{3})+V^{({\rm C})}_{dt}(r_{3})\,.$ (12)

$\displaystyle\Phi_{J=v=0}(dt\mu)$ is constructed as the sum of amplitudes of
the three rearrangement channels c=1, 2, and 3 as shown in Fig. 3:

$\\!\\!{\Phi}_{J=v=0}(dt\mu)=\Phi_{0}^{(1)}\\!\left(\mathbf{r}_{1},\mathbf{R}_{1}\right)+\Phi_{0}^{(2)}\\!\left(\mathbf{r}_{2},\mathbf{R}_{2}\right)+\Phi_{0}^{(3)}\\!\left(\mathbf{r}_{3},\mathbf{R}_{3}\right).$
(13)

The amplitude of each channel $\displaystyle c$ is expanded in terms of
Gaussian basis functions of the Jacobian coordinates $\displaystyle{\rm
r}_{c}$ and $\displaystyle{\rm R}_{c}$:

$\Phi_{0}^{(c)}\left(\mathbf{r}_{c},\mathbf{R}_{c}\right)=\\!\\!\sum_{n_{s}l_{c},N_{c}L_{c}}A_{n_{c}l_{c},N_{c}L_{s}}^{(c)}\left[\phi_{n_{c}l_{c}}\left(\mathbf{r}_{c}\right)\psi_{N_{c}L_{c}}\left(\mathbf{R}_{c}\right)\right]_{00},$
(14)

where $\displaystyle c=1-3$ and

$\displaystyle\phi_{nlm}(\mathbf{r})=\phi_{nl}(r)Y_{lm}(\hat{\mathbf{r}}),$
(15)
$\displaystyle\phi_{nl}(r)=N_{nl}r^{l}e^{-v_{n}r^{2}},\quad\left(n=1-n_{\max}\right),$
$\displaystyle\psi_{NLM}(\mathbf{R})=\psi_{NL}(R)Y_{LM}(\widehat{\mathbf{R}}),$
$\displaystyle\psi_{NL}(R)=N_{NL}R^{L}e^{-\lambda_{N}R^{2}},\quad\left(N=1-N_{\max}\right),$

with normalization constants $\displaystyle N_{nl}$ and $\displaystyle
N_{NL}$. The Gaussian range parameters $\displaystyle\nu_{n}$ and
$\displaystyle\lambda_{n}$ are chosen in geometric progression.

$\displaystyle v_{n}=1/r_{n}^{2},\quad
r_{n}=r_{1}a^{n-1},\left(n=1-n_{\max}\right),$ (16)
$\displaystyle\lambda_{N}=1/R_{N}^{2},\quad
R_{N}=R_{1}A^{N-1},\left(N=1-N_{\max}\right).$

Subsequently, the eigenenergy and wave function are obtained using the
Rayleigh-Ritz variational method. The advantages of using the GEM basis
functions are explained in detail in Sec. III A in Ref. arxiv2112.08399v1 .

Table 2: Nonlinear variational parameters of the Gaussian basis functions in Eqs. (15)–(16). $\displaystyle r_{1}$($\displaystyle R_{1}$) and $\displaystyle r_{\rm max}$($\displaystyle R_{\rm max}$) are in units of $\displaystyle a_{\mu}$ where $\displaystyle a_{\mu}=\hbar^{2}/m_{\mu}e^{2}=255.9$ fm, $\displaystyle m_{\mu}$ being the muon mass. $\displaystyle\mathrm{c}$ |  $\displaystyle l_{c}$ |  $\displaystyle n_{\max}$ |  $\displaystyle\begin{array}[]{c}r_{1}\\\ {[a_{\mu}]}\end{array}$ |  $\displaystyle\begin{array}[]{c}r_{n_{\max}}\\\ {[a_{\mu}]}\end{array}$ |  $\displaystyle L_{c}$ |  $\displaystyle N_{\max}$ |  $\displaystyle\begin{array}[]{c}R_{1}\\\ {[a_{\mu}]}\end{array}$ |  $\displaystyle\begin{array}[]{c}R_{N_{\max}}\\\ {[a_{\mu}]}\end{array}$
---|---|---|---|---|---|---|---|---
1 |  0 |  25 |  0.1 |  10 |  0 |  15 |  0.05 |  10
2 |  0 |  25 |  0.1 |  10 |  0 |  15 |  0.05 |  10
3 |  0 |  25 |  0.1 |  10 |  0 |  15 |  0.05 |  10
1 |  1 |  15 |  0.2 |  10 |  1 |  15 |  0.3 |  15
2 |  1 |  15 |  0.2 |  10 |  1 |  15 |  0.3 |  15
3 |  1 |  15 |  0.2 |  10 |  1 |  15 |  0.3 |  15
3 |  0 |  25 |  0.001 |  0.05 |  0 |  15 |  0.05 |  10
Table 3: Fusion rate of the reaction (2) using the five sets (A to E) of the $\displaystyle d$-$\displaystyle t$ optical-potential parameters in Table 1. |  A |  B |  C |  D |  E
---|---|---|---|---|---
$\displaystyle\lambda^{(0)}_{f}(10^{12}{\rm s}^{-1})$ |  1.14 |  1.15 |  1.12 |  1.07 |  1.07

As the eigenenergy $\displaystyle E_{00}$ is a complex number, we write
$\displaystyle E_{00}=E_{00}^{(\rm real)}+iE_{00}^{(\rm imag)}$. We introduce
$\displaystyle\varepsilon_{00}=E_{00}^{(\rm real)}-E_{\rm th}$, with
$\displaystyle E_{\rm th}(=-2711.24$ eV) being the
$\displaystyle(t\mu)_{1s}+d$ threshold energy. The diagonalization in the
cases of $\displaystyle l_{\rm max}=4$ and $\displaystyle l_{\rm max}=1$
yield, respectively, $\displaystyle\varepsilon_{00}=-319.14$ eV and
$\displaystyle-319.12$ eV. According to Ref. Kamimura:2021msf , the digits
below 1 eV did not affect the reaction calculation. Thus, we employ
$\displaystyle l_{\rm max}=1$. The input Gaussian basis is shown in Table 2.
We took 7 lines of Gaussian basis parameters where the final line is effective
to the $\displaystyle d$-$\displaystyle t$ nuclear interaction.

The fusion rate $\displaystyle\lambda^{(0)}_{\rm f}$ of reaction (2) can be
derived by $\displaystyle\lambda^{(0)}_{\rm f}=-2E_{00}^{({\rm imag})}/\hbar$
and is given as $\displaystyle\lambda^{(0)}_{\rm f}=(1.11\pm 0.04)\times
10^{12}{\rm s}^{-1}$ using five sets of the $\displaystyle d$-$\displaystyle
t$ potentials as presented in Table 3. It is consistent with the results
$\displaystyle 1.15\times 10^{12}{\rm s}^{-1}$ obtained in Ref.
Kamimura:2021msf .

We multiply $\displaystyle\Phi_{J=v=0}(dt\mu)$ by the $\displaystyle
d$-$\displaystyle t$ spin function $\displaystyle\chi_{\frac{3}{2}M}(dt)$:

$\displaystyle\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)=\Phi_{J=v=0}(dt\mu)\,\chi_{\frac{3}{2}M}(dt),$
(17)

which will be used in Secs. IV and V as the ground-state wave function the
$\displaystyle(dt\mu)$ molecule.

##  III  $\displaystyle T$-matix model for $\displaystyle d+t\to\alpha+n+17.6$
MeV

In this section, following Step iii), we propose tractable $\displaystyle
T$-matrix model for the reaction (1.1) to approximate the CC model described
in Ref. [5] (cf. Eqs. (2.1)-(2.10)) using the results of the optical-potential
model in the previous section.

At low energies, the $\displaystyle d$-$\displaystyle t$ wave function has the
total angular momentum $\displaystyle I=3/2$ with $\displaystyle S$-wave in
the $\displaystyle d$-$\displaystyle t$ channel and $\displaystyle D$-wave in
the $\displaystyle\alpha$-$\displaystyle n$ channel. The authors of Ref.
Kamimura:2021msf  solved the following $\displaystyle dt$-$\displaystyle\alpha
n$ CC Schrödinger equation ($\displaystyle Q=17.6$ MeV) using the coordinates
$\displaystyle{\bf r}_{3}$ and $\displaystyle{\bf r}_{4}$ for the
$\displaystyle d$-$\displaystyle t$ and $\displaystyle\alpha$-$\displaystyle
n$ motion, respectively (cf. Fig.3):

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!(H_{dt}-E)\,\Phi_{dt,\,\frac{3}{2}M}({\bf
r}_{3})=-V^{({\rm T})}_{dt,\alpha n}({\bf r}_{3},{\bf
r}_{4})\>\Phi^{(+)}_{\alpha n,\,\frac{3}{2}M}({\bf r}_{4}),\\!\\!$
$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(H_{\alpha
n}-(E+Q)\right)\Phi^{(+)}_{\alpha n,\,\frac{3}{2}M}({\bf r}_{4})=-V^{({\rm
T})}_{\alpha n,dt}({\bf r}_{4},{\bf r}_{3})\>\Phi_{dt,\,\frac{3}{2}M}({\bf
r}_{3})$ (18)

with trivial notations. The channel-coupling potential $\displaystyle V^{({\rm
T})}_{\alpha n,dt}$ ($\displaystyle V^{({\rm T})}_{dt,\alpha n}$) is of the
tensor type. The reaction cross section is presented as

$\displaystyle\displaystyle\sigma_{dt\to\alpha
n}(E)=\frac{2I+1}{(2I_{d}+1)(2I_{t}+1)}\frac{\pi}{k^{2}}\,|S_{2}^{(dt,\alpha
n)}|^{2},$ (19)

where the $\displaystyle S$-matrix $\displaystyle S_{2}^{(dt,\alpha n)}$
appears in the asymptotic form of the outgoing wave,
$\displaystyle\Phi^{(+)}_{\alpha n,\,\frac{3}{2}M}$ [5].

According to the Lippmann-Schwinger theory, the cross section is alternatively
expressed exactly as following using the solution
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}$ and $\displaystyle\Phi^{(+)}_{\alpha
n,\,\frac{3}{2}M}$ of Eq. (3.1) (cf. Sec. IV A of Ref. Kamimura:2021msf ):

$\displaystyle\displaystyle\\!\\!\\!\\!\sigma_{dt\to\alpha
n}(E)=\frac{v_{4}}{v_{3}}\left(\frac{{\mu_{r_{4}}}}{2\pi\hbar^{2}}\right)^{2}\sum_{m_{s}}\int|\,T_{m_{s}}^{(1)}+T_{m_{s}}^{(2)}\,|^{2}\,{\rm
d}{\bf\widehat{K}},\quad$ (20)
$\displaystyle\displaystyle\\!\\!\\!\\!\\!T_{m_{s}}^{(1)}=\langle\,e^{i{\bf
K}\cdot{\bf r}_{4}}\,\chi_{\frac{1}{2}m_{s}}(n)\,|\,V^{({\rm T})}_{\alpha
n,dt}\,|\,\Phi_{dt,\,\frac{3}{2}M}\,\rangle,$ (21)
$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!T_{m_{s}}^{(2)}=\langle\,e^{i{\bf
K}\cdot{\bf r}_{4}}\,\chi_{\frac{1}{2}m_{s}}(n)\,|\,V_{\alpha
n}\,|\,\Phi^{(+)}_{\alpha n,\,\frac{3}{2}M}\,\rangle,$ (22)

where $\displaystyle\chi_{\frac{1}{2}m_{s}}(n)$ is the neutron spin function,
$\displaystyle V_{\alpha n}$ is the $\displaystyle\alpha$-$\displaystyle n$
potential, $\displaystyle v_{3}$ ($\displaystyle v_{4}$) is the velocity of
the $\displaystyle d$-$\displaystyle t$ ($\displaystyle\alpha$-$\displaystyle
n$) relative motion along $\displaystyle{\bf r}_{3}$ ($\displaystyle{\bf
r}_{4}$), $\displaystyle\mu_{r_{4}}$ is the reduced mass associated with
$\displaystyle{\bf r}_{4}$, and $\displaystyle{\bf K}$ is wave number vector.

The coupling potential $\displaystyle V^{({\rm T})}_{\alpha n,dt}$ between the
$\displaystyle\alpha n$-$\displaystyle dt$ channels are taken in the following
form Kamimura:2021msf :

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!V_{\alpha
n,dt}^{({\rm T})}({\bf r}_{4},{\bf r}_{3})$ $\displaystyle\displaystyle\\!\\!\
=\\!\\!\\!$ $\displaystyle\displaystyle v_{0}^{({\rm
T})}r_{34}^{2}e^{-\mu\,r_{34}^{2}-\mu^{\prime}R_{34}^{2}}\,\big{[}Y_{2}({\widehat{\bf
r}}_{34})\,{\cal S}_{2}(dt,\alpha n)\big{]}_{00},$ (23)

where $\displaystyle{\bf r}_{34}={\bf r}_{3}-{\bf r_{4}}$ and
$\displaystyle{\bf R}_{34}={\bf r}_{3}+{\bf r_{4}}$. In Eq. (23),
$\displaystyle{\cal S}_{2}(dt,\alpha n)$ is a spin-tensor operator comprising
the spins of $\displaystyle dt$\- and $\displaystyle\alpha n$-pairs. However,
the explicit form of $\displaystyle{\cal S}_{2}(dt,\alpha n)$ need not to be
determined as explained below (cf. Eq. (3.4)).

In our model, we perform another $\displaystyle T$-matrix calculation for
$\displaystyle\sigma_{dt\to\alpha n}(E)$ by using Eq. (20) considering the
following approximation. We replace $\displaystyle\Phi_{dt,\,\frac{3}{2}M}$ in
$\displaystyle T_{m_{s}}^{(1)}$ with
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}$ in Eq. (2.1) and neglect
$\displaystyle T_{m_{s}}^{(2)}$ because
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}$ does not include
$\displaystyle\alpha$-$\displaystyle n$ component explicitly. However,
$\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}$ is considered to reflect
the effect of the $\displaystyle\alpha$-$\displaystyle n$ channel using the
imaginary potential $\displaystyle iW_{dt}^{({\rm N})}(r_{3})$ in Eq. (2.2).
Thus, Eqs. (20)–(22) are approximated as

$\displaystyle\displaystyle\sigma^{\rm(our)}_{dt\to\alpha
n}(E)=\frac{v_{4}}{v_{3}}\left(\frac{\mu_{r_{4}}}{2\pi\hbar^{2}}\right)^{2}\sum_{m_{s}}\int|\,T_{m_{s}}^{(1^{\prime})}|^{2}\,{\rm
d}{\bf\widehat{K}},\quad$ (24) $\displaystyle\displaystyle
T_{m_{s}}^{(1^{\prime})}=\langle\,e^{i{\bf K}\cdot{\bf
r}_{4}}\,\chi_{\frac{1}{2}m_{s}}(n)\,|\,V^{({\rm T})}_{\alpha n,dt}({\bf
r}_{4},{\bf r}_{3})\,|\,\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}(E,{\bf
r}_{3})\,\rangle.$ (25)

Instead of Eq.(23), we employ the following separable nonlocal form:

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!V_{\alpha n,dt}^{({\rm
T})}({\bf r}_{4},{\bf r}_{3})=v_{0}^{({\rm
T})}r_{4}^{2}\,e^{-\mu_{4}\,r_{4}^{2}-\mu_{3}r_{3}^{2}}\,\big{[}Y_{2}({\widehat{\bf
r}}_{4})\,{\cal S}_{2}(dt,\alpha n)\big{]}_{00},$ (26)

which is easier to handle than Eq. (23). Consequently, the cross section (24)
can be explicitly expressed as

$\displaystyle\displaystyle\sigma^{\rm(our)}_{dt\to\alpha
n}(E)=\frac{v_{4}}{v_{3}}\left(\frac{\mu_{r_{4}}}{2\pi\hbar^{2}}\right)^{2}\left|\,v_{0}^{(T)}S_{0}^{(T)}F_{0}\,J_{2}\,\right|^{2}$
(27)

with

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!F_{0}=\\!\int\phi_{dt,00}^{\rm(opt)}(E,{\bf
r}_{3})\,e^{-\mu_{3}r_{3}^{2}}\,{\rm d}{\bf r}_{3},$ (28)
$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!J_{2}=4\pi\\!\\!\int\\!\\!j_{2}(Kr_{4})\,r_{4}^{2}\,e^{-\mu_{4}r_{4}^{2}}\,r_{4}^{2}\,{\rm
d}r_{4}=\frac{1}{4}\left(\frac{\pi}{\mu_{4}}\right)^{\frac{3}{2}}\\!\left(\frac{K}{\mu_{4}}\right)^{2}\\!e^{-\frac{\mu_{4}K^{2}}{4}}\\!,$
(29)

where $\displaystyle\phi_{dt,00}^{\rm(opt)}(E,{\bf r}_{3})$ is normalized
asymptotically as

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\phi_{dt,00}^{\rm(opt)}(E,{\bf
r}_{3})\stackrel{{\scriptstyle
r_{3}\to\infty}}{{\longrightarrow}}e^{i\sigma_{0}}\frac{F_{0}(k,r_{3})}{kr_{3}}\,+\,\mbox{(outgoing
wave)}\quad\;\;$ (30)

with the S-wave Coulomb function $\displaystyle F_{0}(k,r)$ and phase shift
$\displaystyle\sigma_{0}$. $\displaystyle j_{2}(kr_{4})$ is the spherical
Bessel function of order 2. In Eq. (27), the constant $\displaystyle
S_{0}^{({\rm T)}}$ is presented, independently of $\displaystyle m_{s}$, as
(cf. Eq. (2.12) of Ref. Kamimura:2021msf )

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!S^{{\rm(T)}}_{0}=\frac{1}{\sqrt{10}}\langle\,\chi_{\frac{1}{2}m_{s}}(\alpha
n)\,|\,\big{[}{\cal S}_{2}(dt,\alpha
n)\,\chi_{\frac{3}{2}}(dt)\big{]}_{\frac{1}{2}m_{s}}\rangle\>.$ (31)

Therefore, the explicit form of $\displaystyle S_{2}(dt,\alpha n)$ need not to
be known, and $\displaystyle v_{0}^{{\rm(T)}}S_{0}^{{\rm(T)}}$ can be
considered as an adjustable parameter for the $\displaystyle T$-matrix
calculations.

Table 4: Parameters of the $\displaystyle dt$-$\displaystyle\alpha n$ coupling potential $\displaystyle V_{\alpha n,dt}^{({\rm T})}({\bf r}_{4},{\bf r}_{3})$ in Eq. (26). Sets A1-A4 were determined using the optical potential Set A in Table I; similarly for the others.  potential set | $\displaystyle v_{0}^{({\rm T})}S_{0}^{({\rm T})}$ |  $\displaystyle\mu^{-1/2}_{3}$ |  $\displaystyle\mu^{-1/2}_{4}$
---|---|---|---
|  (MeV fm-5) |  (fm) |  (fm)
Set A1 | 2.307 | 1.6 | 1.6
Set A2 | 0.138 | 2.0 | 5.2
Set A3 | 0.245 | 3.2 | 1.6
Set A4 | 0.402 | 2.0 | 2.8
Set B1 | 1.001 | 2.8 | 1.6
Set B2 | 0.016 | 4.0 | 2.4
Set B3 | 0.065 | 3.6 | 2.0
Set B4 | 0.003 | 4.0 | 5.6
Set C1 | 0.122 | 2.4 | 2.4
Set C2 | 0.052 | 1.6 | 6.4
Set C3 | 0.029 | 3.2 | 4.8
Set C4 | 0.191 | 4.4 | 1.6
Set D1 | 1.291 | 3.6 | 1.6
Set D2 | 0.151 | 3.2 | 2.0
Set D3 | 0.405 | 4.4 | 1.6
Set D4 | 0.032 | 4.0 | 5.6
Set E1 | 0.054 | 5.2 | 2.0
Set E2 | 0.166 | 2.0 | 2.0
Set E3 | 0.054 | 4.4 | 2.8
Set E4 | 0.019 | 4.4 | 5.2
Figure 4: Calculated $\displaystyle S$-factor $\displaystyle S(E)$ of the
reaction $\displaystyle d+t\to\alpha+n+17.6\,$ MeV. Five lines A1-E1 denote
the cases using the $\displaystyle dt$-$\displaystyle\alpha n$ coupling
potentials with the parameter Sets A1-E1, respectively, in Table IV. The black
solid line indicating the experimental data is from a review paper
Serpico:2004gx .

We determined the parameter sets for the $\displaystyle
dt$-$\displaystyle\alpha n$ coupling potential $\displaystyle V_{\alpha
n,dt}^{({\rm T})}({\bf r}_{4},{\bf r}_{3})$ in Eq. (3.9) so as to reproduce
the observed $\displaystyle S$-factor $\displaystyle S(E)$. We employed, in
Eq. (3.8), $\displaystyle\Phi_{dt,\,\frac{3}{2}M}^{\rm(opt)}(E,{\bf r}_{3})$
obtained in Sec. II A using the optical potentials A-E in Table I. The
resulting parameters are listed in Table IV. Sets A1-A4 were obtained using
the optical potential of Set A, and similarly for others.

In Fig. 4, the calculated $\displaystyle S(E)$ factors using Sets A1-E1 are
illustrated. The experimental data are well reproduced with the same quality
of fitting as in Fig. 4 by the CC calculation in of Ref. [5]. The use of the
other sets yielded a similar agreement. Subsequently, we employ all
$\displaystyle dt$-$\displaystyle\alpha n$ tensor coupling potentials in Secs.
IV and V. The cross section of the strong coupling rearrangement reaction
(1.1) is expressed in a simple closed form (3.10)–(3.12) that can reproduce
observed data by tuning the parameters of $\displaystyle V_{\alpha
n,dt}^{({\rm T})}({\bf r}_{4},{\bf r}_{3})$ —- This is a key finding of this
study.

Here, we highlight the consistency of our model in terms of its potentials. In
the process of tuning the potential parameter sets, we considered the plane
wave ($\displaystyle D$-wave) of the $\displaystyle\alpha$-$\displaystyle n$
relative motion in the $\displaystyle T$-matrix (25) where the
$\displaystyle\alpha$-$\displaystyle n$ potential $\displaystyle V_{\alpha
n}(r_{4})$ did not appear explicitly. We consider that the effect of
$\displaystyle\alpha$-$\displaystyle n$ potential is effectively renormalized
into the coupling potential $\displaystyle V^{({\rm T})}_{\alpha n,dt}$ which
is tuned to reproduce the observed $\displaystyle S(E)$ without $\displaystyle
V_{\alpha n}(r_{4})$. In Secs. IV and V, we consistently use the same
potentials without $\displaystyle V_{\alpha n}(r_{4})$ in the $\displaystyle
T$-matrix calculations of the three-body $\displaystyle
dt\mu$-$\displaystyle\alpha n\mu$ system. As shown later, most of the results
of Ref. Kamimura:2021msf  are well reproduced by our model.

##  IV  $\displaystyle\alpha$-$\displaystyle\mu$ sticking probability

After fusion occurs in the $\displaystyle dt\mu$ molecule, the emitted muon
can be captured by the $\displaystyle\alpha$ particle or freely emitted, as
shown in Eq. (2). In this section, according to Step iv) of the introduction,
we calculate the fusion rate of the reaction (2) and the
$\displaystyle\alpha$-$\displaystyle\mu$ sticking probability by referring to
the $\displaystyle T$-matrix calculation of those quantities in Sec. IV of
Ref. Kamimura:2021msf  with the outgoing waves in channel $\displaystyle c=5$
(Fig. 3).

The authors of Ref. Kamimura:2021msf  solved the CC Schrödinger equation (3.7)
for the reaction (2) and generated the wave function
$\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(E)$ expressed as Eq. (3.3). They
constructed the $\displaystyle T$-matrix elements (5.2)–(5.3) for use in
another fusion calculation by substituting their
$\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(E)$ into the exact wave function
$\displaystyle\Psi_{\alpha}^{(+)}(E_{\alpha})$ in the definition of
$\displaystyle T$-matrix (4.1) in Ref. Kamimura:2021msf .

Because the functions $\displaystyle\phi_{\beta}(\xi_{\beta})$ in the outgoing
channel $\displaystyle\beta$ in the $\displaystyle T$-matrix are ortho-
normalized functions of discrete states, they discretized and ortho-normalized
the $\displaystyle\alpha$-$\displaystyle\mu$ wave function of the
$\displaystyle k$-continuum state, for example, $\displaystyle\phi_{lm}(k,{\bf
r}_{5})$, to generate $\displaystyle{\widetilde{\phi}}_{ilm}({\bf r}_{5})$ and
energy $\displaystyle{\widetilde{\varepsilon}}_{i}$. This discretization was
performed by employing the procedure (Fig. 5):

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!{\widetilde{\phi}}_{ilm}({\bf
r}_{5})=\frac{1}{\sqrt{\mathit{\Delta}k_{i}}}\int_{k_{i-1}}^{k_{i}}\\!\\!\phi_{lm}(k,{\bf
r}_{5})\,dk,\;\;\;\;i=1-N,$ (32)
$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!{\widetilde{\varepsilon}}_{i}=\frac{\hbar^{2}}{2\mu_{r_{5}}}{\widetilde{k}}_{i}^{\,2},\qquad{\widetilde{k}_{i}}^{\,2}=\Big{(}\frac{k_{i}+k_{i-1}}{2}\Big{)}^{2}+\frac{\mathit{\Delta}k_{i}^{2}}{12}.\;$
(33)

This has often been used in the continuum-discretized couple-channels (CDCC)
method for studying the projectile-breakup reactions (for example, see review
papers Kamimura86 ; Austern ; Yahiro12 ). On the other hand,
$\displaystyle\phi_{nlm}({\bf r}_{5})$ stands for the bound states
$\displaystyle(\alpha\mu)_{nlm}$. Then, it can be said that the Ref.
Kamimura:2021msf  investigated the reaction

$\displaystyle\displaystyle(dt\mu)_{J=v=0}$ $\displaystyle\displaystyle\to$
$\displaystyle\displaystyle(\alpha\mu)_{il}+n+17.6\,\mbox{MeV}$ (34)
$\displaystyle\displaystyle\searrow$
$\displaystyle\displaystyle(\alpha\mu)_{nl}+n+17.6\,\mbox{MeV},$ (35)

considering a precise discretization of the
$\displaystyle\alpha$-$\displaystyle\mu$ continuum.

Figure 5: Schematic illustration of Eq. (4.1) to construct the continuum-
discretized wave function $\displaystyle{\widetilde{\phi}}_{ilm}({\bf r})$ by
averaging the continuum wave functions $\displaystyle\phi_{lm}(k,{\bf r})$ in
each momentum bin $\displaystyle\Delta k_{i}=k_{i}-k_{i-1}$.

In our model, we replace $\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(E)$ with
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ in the $\displaystyle
T$-matrix elements (5.2)–(5.3) in Ref. [5] —- This is a key point of the
proposed model. $\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ is the
ground-state wave function of the $\displaystyle(dt\mu)$ molecule with the
total angular momentum spin 3/2 and was obtained in Eqs. (2.9) and (2.15) with
the $\displaystyle d$-$\displaystyle t$ optical potential. As
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ does not explicitly
include the $\displaystyle\alpha n\mu$ amplitude, the third lines of Eq.
(5.2)–(5.3) in Ref. [5] were excluded.
$\displaystyle\mathring{\Psi}_{\frac{3}{2}M}^{({\rm
C})}(dt\mu)+\Psi_{\frac{3}{2}M}^{({\rm N})}(dt\mu)$ that is the dominant
component of $\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(E)$ is replaced with
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$. Consequently, we obtain
our $\displaystyle T$ matrix elements with the outgoing wave functions in
channel $\displaystyle c=5$:

$\displaystyle\displaystyle\\!{\widetilde{T}}^{(5)}_{il,mm_{s}}\\!\\!=\\!\langle\,e^{i{\widetilde{\bf
K}}_{i}\cdot{\bf R}_{5}}\,{\widetilde{\phi}}_{ilm}({\bf
r}_{5})\chi_{\frac{1}{2}m_{s}}\\!(n)\,|\,V^{({\rm T})}_{\alpha
n,dt}\,|\,\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)\,\rangle\quad$ (36)

for the transition to discretized states $\displaystyle(\alpha\mu)_{ilm}$, and

$\displaystyle\displaystyle\\!T^{(5)}_{nl,mm_{s}}\\!\\!=\\!\langle\,e^{i{\bf
K}_{n}\cdot{\bf R}_{5}}\,\phi_{nlm}({\bf
r}_{5})\chi_{\frac{1}{2}m_{s}}\\!(n)\,|\,V^{({\rm T})}_{\alpha
n,dt}\,|\,\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)\,\rangle\quad$ (37)

for the transition to bound states $\displaystyle(\alpha\mu)_{nlm}$. Here, we
take $\displaystyle{\bf R}_{4}={\bf R}_{3}$ [5] (cf. Fig. 3). The
$\displaystyle(\alpha\mu)$-$\displaystyle n$ plane waves along
$\displaystyle{\bf R}_{5}$ are denoted by $\displaystyle e^{i{\widetilde{\bf
K}}_{i}\cdot{\bf R}_{5}}$ and $\displaystyle e^{i{\bf K}_{n}\cdot{\bf R}_{5}}$
in which the momentum $\displaystyle{\widetilde{K}}_{i}$ and $\displaystyle
K_{n}$ are derived from energy conservation as follows (cf. Fig. 2 and Sec. V
of Ref. Kamimura:2021msf ):

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!{\widetilde{E}}_{i}+{\widetilde{\varepsilon}}_{i}=E_{00}+Q,\;\;\;\quad{\widetilde{E}}_{i}=\frac{\hbar^{2}}{2\mu_{R_{5}}}{\widetilde{K}}_{i}^{2},$
(38)
$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!E_{n}+{\varepsilon}_{n}=E_{00}+Q,\;\;\;\quad
E_{n}=\frac{\hbar^{2}}{2\mu_{R_{5}}}K_{n}^{2}\,,$ (39)

where $\displaystyle{\widetilde{\varepsilon}}_{i}$
($\displaystyle\varepsilon_{n}$) is the energy of
$\displaystyle{\widetilde{\phi}}_{ilm}({\bf r}_{5})$
($\displaystyle\phi_{nlm}({\bf r}_{5})$) and $\displaystyle E_{00}\simeq$
3.030 keV which is negligible in the present scattering problem compared with
$\displaystyle Q=17.6$ MeV.

For the above discretization of the $\displaystyle\alpha$-$\displaystyle\mu$
continuum states $\displaystyle\phi_{lm}(k,{\bf r}_{5})$ , we considered
$\displaystyle N=200$ for $\displaystyle l=0$ to 25, and maximum momentum
$\displaystyle\hbar k_{N}=10{\rm MeV}/c\,({\widetilde{\varepsilon}}_{N}=487$
keV) with the constant $\displaystyle\Delta k_{i}$, which is the same as those
used in Ref. Kamimura:2021msf .

Reaction rates of the reactions (34) and (35) are expressed as follows:

$\displaystyle\displaystyle{r}_{nl}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{v}_{nl}\left(\frac{\mu_{R_{5}}}{2\pi\hbar^{2}}\right)^{2}\times\\!\sum_{m,m_{s}}\int\big{|}{T}^{(5)}_{nl,mm_{s}}\big{|}^{2}\,{\rm
d}{\widehat{\bf K}_{n}},$ (40)
$\displaystyle\displaystyle{\widetilde{r}}_{il}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{v}_{il}\left(\frac{\mu_{R_{5}}}{2\pi\hbar^{2}}\right)^{2}\times\\!\sum_{m,m_{s}}\int\big{|}{\widetilde{T}}^{(5)}_{il,mm_{s}}\big{|}^{2}\,{\rm
d}{\widehat{\widetilde{\bf K}}_{i}},$ (41)

which is derived respectively by approximating Eqs.(5.6) and (5.7) in Ref.
Kamimura:2021msf  according to our proposed model. Here, $\displaystyle
v_{il}=\hbar{\widetilde{K}}_{i}/\mu_{R_{5}}$ is the velocity of the
$\displaystyle(\alpha\mu)_{il}$-$\displaystyle n$ relative motion associated
with $\displaystyle{\bf R}_{5}$, and similarly for
$\displaystyle{v}_{nl}=\hbar K_{n}/\mu_{R_{5}}$. The sum of the quantum number
$\displaystyle n$ for $\displaystyle r_{nl}$ and $\displaystyle i$ for
$\displaystyle{\widetilde{r}}_{il}$ yields the total reaction rate
$\displaystyle r^{\rm bound}_{l}$ for the bound states and $\displaystyle
r^{\rm cont}_{l}$ for the continuum states:

$\displaystyle\displaystyle r_{l}^{\rm bound}=\sum_{n}r_{nl},\qquad r_{l}^{\rm
cont}=\sum_{i=1}^{N}{\widetilde{r}}_{il}.$ (42)

In Fig. 6, we show the calculated $\displaystyle r^{\rm bound}_{l}$ and
$\displaystyle r^{\rm cont}_{l}$ with $\displaystyle l$ up to 20 under the
potential Set B1 in Table 4. For comparison, we presented the $\displaystyle
r^{\rm bound}_{l}$ and $\displaystyle r^{\rm cont}_{l}$ obtained in Ref.
Kamimura:2021msf . Both results have similar tendency with respect to
$\displaystyle l$.

Figure 6: Calculated reaction rate $\displaystyle r^{\rm bound}_{l}$ (black closed circle) for the bound state and $\displaystyle r^{\rm cont}_{l}$ (red closed circle) for the continuum states with respect to the angular momentum $\displaystyle l$. The potential Set B1 is used. The black and red open circles denote the $\displaystyle r^{\rm bound}_{l}$ and the $\displaystyle r^{\rm cont}_{l}$ from Ref. Kamimura:2021msf , respectively. Table 5: Fusion rates $\displaystyle\lambda_{\rm f}^{(5)}$ calculated on the channel $\displaystyle c=5$ with Eq. (44). $\displaystyle\omega_{S}^{0}$ is the $\displaystyle\alpha$-$\displaystyle\mu$ sticking probability calculated with Eq. (45). See Table 4 for the potential sets. Another type of the fusion rate $\displaystyle\lambda_{\rm f}^{(4)}$ is discussed in Sec. V on the channel $\displaystyle c=4$. $\displaystyle\lambda_{\rm f}^{(5)}=(1.15\pm 0.05)\times 10^{12}{\rm s}^{-1}$ and $\displaystyle\lambda_{\rm f}^{(4)}=(1.15\pm 0.04)\times 10^{12}{\rm s}^{-1}$, on average.  potential set | $\displaystyle\;\;\lambda^{(5)}\;\;$ | $\displaystyle\;\omega_{S}^{0}\;$ | $\displaystyle\;\;\quad\quad\lambda^{(4)}\;\;$
---|---|---|---
| $\displaystyle\quad(10^{12}\,{\rm s}^{-1})\quad$ | $\displaystyle\quad(\%)\quad$ | $\displaystyle\quad\quad(10^{12}\,{\rm s}^{-1})\quad$
Set A1 | 1.12 | 0.942 |  1.11
Set A2 | 1.15 | 0.943 |  1.15
Set A3 | 1.11 | 0.942 |  1.11
Set A4 | 1.13 | 0.944 |  1.12
Set B1 | 1.14 | 0.942 |  1.13
Set B2 | 1.14 | 0.940 |  1.12
Set B3 | 1.13 | 0.940 |  1.12
Set B4 | 1.14 | 0.939 |  1.15
Set C1 | 1.14 | 0.928 |  1.14
Set C2 | 1.18 | 0.931 |  1.19
Set C3 | 1.18 | 0.945 |  1.18
Set C4 | 1.20 | 0.936 |  1.19
Set D1 | 1.13 | 0.943 |  1.13
Set D2 | 1.13 | 0.941 |  1.14
Set D3 | 1.15 | 0.937 |  1.13
Set D4 | 1.17 | 0.930 |  1.18
Set E1 | 1.16 |  0.935 |  1.15
Set E2 | 1.15 | 0.933 |  1.14
Set E3 | 1.16 | 0.938 |  1.15
Set E4 | 1.17 | 0.937 |  1.17

The total reaction rates to the $\displaystyle\alpha$-$\displaystyle\mu$ bound
states and continuum states are defined as follows:

$\displaystyle\displaystyle\lambda_{\rm f}^{\rm
bound}=\sum_{l=0}^{5}\>r_{l}^{\rm bound},\qquad\lambda_{\rm f}^{\rm
cont.}=\sum_{l=0}^{20}\>r_{l}^{\rm cont.}.$ (43)

Consequently, their sum

$\displaystyle\displaystyle\lambda^{(5)}_{\rm f}=\lambda_{\rm f}^{\rm
bound}+\lambda_{\rm f}^{\rm cont.}.$ (44)

is the fusion rate of the $\displaystyle(dt\mu)_{J=v=0}$ molecule calculated
on channel $\displaystyle c=5$.

Table 5 lists the fusion rates $\displaystyle\lambda^{(5)}_{\rm f}$ for all 20
different potential Sets in Table 4. All the fusion rates
$\displaystyle\lambda^{(5)}_{\rm f}$ are within a small range of
$\displaystyle 1.1\\!-\\!1.2\times 10^{12}{\rm s}^{-1}$. This proves that the
choice of the potentials does not influence the fusion rate significantly.
This fusion rate is consistent with $\displaystyle\lambda_{\rm f}^{(0)}$,
which was derived in Sec. II B from the imaginary part of the complex
eigenenergy of the $\displaystyle(dt\mu)_{J=v=0}$ molecule. Notably, in the
proposed model, the fusion rate $\displaystyle\lambda^{(5)}_{\rm f}$ have been
calculated based on the amplitude of the outgoing $\displaystyle\alpha n\mu$
wave.

The initial $\displaystyle\alpha$-$\displaystyle\mu$ sticking probability
$\displaystyle\omega_{S}^{0}$ is probability of the muon being captured by an
$\displaystyle\alpha$ particle after the fusion happens. This is expressed by
the following formula in the same manner as in Eq. (5.3) in Ref. [5]:

$\displaystyle\displaystyle\omega_{S}^{0}=\frac{\lambda_{\rm f}^{\rm
bound}}{\lambda_{\rm f}^{\rm bound}+\lambda_{\rm f}^{\rm cont.}}.$ (45)

The second column of Table 5 presents the initial sticking probabilities
$\displaystyle\omega^{0}_{s}$ for different potential sets. The
$\displaystyle\omega^{0}_{s}(=0.938\pm 0.07\%)$ are consistent and are close
to the results ($\displaystyle 0.91\%\sim 0.93\%$) obtained by using optical-
potential models KAMIMURA1989-BENCHMARK ; Bogdanova:1981mj  and by the
R-matrix methods Struensee88a ; Szalewicz90 ; Hale93 ; Hu1994 ; Cohen1996 ;
Jeziorski91  considering the $\displaystyle d$-$\displaystyle t$ nuclear
interaction; note that our calculation is based on the absolute values of
$\displaystyle\lambda_{\rm f}^{\rm bound}$ and $\displaystyle\lambda_{\rm
f}^{\rm cont.}$.

However, the value of $\displaystyle\omega^{0}_{s}$ in Table V are
$\displaystyle\sim\\!9\%$ larger than
$\displaystyle\omega^{0}_{s}\,(=\\!0.857\%)$ given by Ref. Kamimura:2021msf
in which the coupling to the $\displaystyle\alpha n\mu$ channel is included
explicitly. This may be attributed to the following reason. In the CC work
[5], strong coupling between the $\displaystyle\alpha n\mu$ outgoing amplitude
$\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(\alpha n\mu)$ and the nuclear
$\displaystyle dt\mu$ amplitude $\displaystyle\Psi_{\frac{3}{2}M}^{({\rm
N})}(dt\mu)$ is expected to enhance the contribution of the transition to the
$\displaystyle\alpha$-$\displaystyle\mu$ continuum states in Eqs. (5.2)-(5.3)
of Ref. [5] more than that to the $\displaystyle\alpha$-$\displaystyle\mu$
bound states, which enhances the $\displaystyle\lambda_{\rm f}^{\rm cont.}$,
and then reduces $\displaystyle\omega_{S}^{0}$. However, the present model
does not exhibit such a CC effect.

##  V Momentum and energy spectra of muon emitted by $\displaystyle\mu$CF

In this section, according to Step v), we calculate the momentum and energy
spectra of the muons emitted by reaction (1.2) and derive another type of
fusion rate of the $\displaystyle(dt\mu)_{J=v=0}$ molecule. We perform a
$\displaystyle T$-matrix calculation by referring to Sec. VI of Ref.
Kamimura:2021msf  with $\displaystyle(\alpha n)$-$\displaystyle\mu$ outgoing
waves in channel $\displaystyle c=4$ (Fig. 3). We discretize and ortho-
normalize the wave functions of the $\displaystyle\alpha$-$\displaystyle n$
continuum states; for example, $\displaystyle\phi_{lm}(k,{\bf r}_{4})$ with
$\displaystyle l=2$, generating $\displaystyle{\widetilde{\phi}}_{ilm}({\bf
r}_{4})$ in the same manner as described in Sec. IV.

We begin with the $\displaystyle T$-matrix expressed as Eq. (6.2) in Ref.
Kamimura:2021msf  where $\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(E)$ is the
total wave function obtained using the CC Schödinger equation (3.7) in Ref.
Kamimura:2021msf . In the proposed model, we replace
$\displaystyle\Psi_{\frac{3}{2}M}^{(+)}(E)$ with
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ which is the ground-state
wave function of the $\displaystyle(dt\mu)$ molecule obtained from Eqs. (2.9)
and (2.15) including the $\displaystyle d$-$\displaystyle t$ optical potential
—- This is the same key point of our model as mentioned in Sec. IV. Thus, in
the three $\displaystyle T$-matrix elements in Eq. (6.2) of Ref.
Kamimura:2021msf , we exclude the third line because the wave function
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$ does not have an
$\displaystyle\alpha n\mu$ scattering amplitude, and replace
$\displaystyle\mathring{\Psi}_{\frac{3}{2}M}^{({\rm
C})}(dt\mu)+\Psi_{\frac{3}{2}M}^{({\rm N})}(dt\mu)$ with
$\displaystyle\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)$.

We obtain $\displaystyle T$-matrix elements as follows (cf. Eq. (4.5)):

$\displaystyle\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!T^{(4)}_{il,mm_{s}}\\!\\!=\\!\langle\,e^{i{\widetilde{\bf
K}}_{i}\cdot{\bf R}_{4}}\,{\widetilde{\phi}}_{ilm}({\bf
r}_{4})\,\chi_{\frac{1}{2}m_{s}}\\!(n)\,|\,V^{({\rm T})}_{\alpha
n,dt}\,|\,\Phi^{(J=v=0)}_{\frac{3}{2}M}(dt\mu)\,\rangle.\qquad\;$ (46)

The outgoing wave is located in channel $\displaystyle c=4$ and composed of
the plane wave $\displaystyle e^{i{\widetilde{\bf K}}_{i}\cdot{\bf R}_{4}}$
and discretized ortho-normalized $\displaystyle\alpha$-$\displaystyle n$
continuum states $\displaystyle{\widetilde{\phi}}_{ilm}({\bf r}_{4})$, which
is constructed in the same manner as that in Eq. (4.1) in Sec. IV.

$\displaystyle\displaystyle{\widetilde{\phi}}_{ilm}({\bf
r}_{4})=\frac{1}{\sqrt{\mathit{\Delta}k_{i}}}\int_{k_{i-1}}^{k_{i}}\\!\\!\phi_{lm}(k,{\bf
r}_{4})\,dk\,,\;(i=1-N).$ (47)

The average energy $\displaystyle\widetilde{\varepsilon}_{i}$ and momentum
$\displaystyle{\widetilde{k}}_{i}$ of
$\displaystyle{\widetilde{\phi}}_{ilm}({\bf r_{4}})$ are given, similarly to
Eq. (4.2). Therefore, the momentum $\displaystyle{\widetilde{K}}_{i}$ of the
plane wave $\displaystyle e^{i{\widetilde{\bf K}}_{i}\cdot{\bf R}_{4}}$ is
derived from the following energy conservation ($\displaystyle Q=17.6$ MeV):

$\displaystyle\displaystyle{\widetilde{E}}_{i}+{\widetilde{\varepsilon}}_{i}=E_{00}+Q,\qquad{\widetilde{E}}_{i}=\frac{\hbar^{2}}{2\mu_{R_{4}}}{\widetilde{K}}_{i}^{2}.$
(48)

Figure 7: Schematic illustration for discretization of the momentum space
$\displaystyle[K_{0},K_{N}]$ of the ($\displaystyle\alpha
n$)-$\displaystyle\mu$ relative motion along $\displaystyle{\bf R}_{4}$ (left
half) and that of the $\displaystyle\alpha$-$\displaystyle n$ relative motion
$\displaystyle[k_{N},k_{0}]$ along $\displaystyle{\bf r}_{4}$ (right half)
while maintaining $\displaystyle E_{i}+\varepsilon_{i}=E_{00}+Q$. The
resulting discretized $\displaystyle\alpha$-$\displaystyle n$ continuum states
$\displaystyle{\widetilde{\phi}}_{ilm}({\bf r}_{4})(i=1-N)$ are indicated by
the red lines. The associated muon plane waves $\displaystyle
e^{i{\widetilde{\bf K}}_{i}\cdot{\bf R}_{4}}$ are indicated by the blue lines
in the left half. This illustration is taken from Ref. Kamimura:2021msf .

A new problem in Sec. V is that we aim to generate the set
$\displaystyle\\{{\widetilde{K}}_{i};i=1-N\\}$ with equal intervals and obtain
the momentum spectrum as a smooth function of $\displaystyle K$ to determine
its peak easily. Figure 7 illustrates the manner in which $\displaystyle
k$-space is discretized. Following Ref. [5], we first assume the maximum value
$\displaystyle K_{N}$ of $\displaystyle K$-space $\displaystyle[K_{0},K_{N}]$
with $\displaystyle K_{0}=0$ at the left side of the figure. We then divide
the $\displaystyle K$-space into $\displaystyle N$ bins ($\displaystyle
K_{i},i=0-N$) with equal intervals $\displaystyle\mathit{\Delta}K$.
Correspondingly, we divide the $\displaystyle k$-space
$\displaystyle[k_{N},k_{0}]$ on the right side of the figure to $\displaystyle
N$ bins ($\displaystyle k_{i},i=0-N$) with the energy conservation kept as

$\displaystyle\displaystyle\\!\\!\\!\\!\\!E_{i}+\varepsilon_{i}=E_{00}+Q,\quad
E_{i}=\frac{\hbar^{2}}{2\mu_{R_{4}}}K_{i}^{2},\quad\varepsilon_{i}=\frac{\hbar^{2}}{2\mu_{r_{4}}}k_{i}^{2},$
(49)

where $\displaystyle K_{i}$ increases ($\displaystyle k_{i}$ decreases) with
increase in $\displaystyle i$. The bin width
$\displaystyle\mathit{\Delta}k_{i}=|k_{i}-k_{i-1}|$ depends on $\displaystyle
i$. Subsequently, $\displaystyle{\widetilde{\phi}}_{ilm}({\bf r}_{4})$ is
generated by using Eq. (5.2) with energy
$\displaystyle{\widetilde{\varepsilon}}_{i}$. Finally,
$\displaystyle{\widetilde{E}}_{i}$ is expressed as Eq. (5.3) as shown in Fig.
7.

Similar to Ref. Kamimura:2021msf , we consider $\displaystyle N=200$ setting
$\displaystyle\hbar K_{N}=6.0$ MeV/$\displaystyle c\,(E_{N}=175$ keV) in Fig.
7. This is sufficient for deriving the muon momentum spectrum with a smooth
function.

###  V.1 Fusion rate of the $\displaystyle dt\mu$ molecule

In our model, instead of Eq. (6.4) in Ref. [5], we can write the reaction rate
to a continuum discretized state $\displaystyle(\alpha n)_{il}$,

$(dt\mu)_{J=v=0}\to(\alpha n)_{il}+\mu+17.6\,\mbox{MeV},$ (50)

as follows:

$\displaystyle\displaystyle{r}_{il}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{v}_{il}\left(\frac{\mu_{R_{4}}}{2\pi\hbar^{2}}\right)^{2}\sum_{m,m_{s}}\int\big{|}{T}^{(4)}_{il,mm_{s}}\big{|}^{2}\,{\rm
d}{\widehat{\widetilde{\bf K}}}_{i}\,,$ (51)

where $\displaystyle{v}_{il}=\hbar{\widetilde{K}}_{i}/\mu_{R_{4}}$ is the
$\displaystyle(\alpha n)_{il}$-$\displaystyle\mu$ relative velocity.

The sum of the transition rates

$\displaystyle\displaystyle\lambda_{\rm f}^{(4)}=\sum_{il}\;{r}_{il}$ (52)

is the fusion rate of the $\displaystyle(dt\mu)_{J=v=0}$ molecule using the
$\displaystyle T$-matrix based on channel $\displaystyle c=4$ (cf. Eq. (6.5)
in Ref. Kamimura:2021msf ). The Contribution to $\displaystyle\lambda_{\rm
f}^{(4)}$ from the final states $\displaystyle{\widetilde{\phi}}_{ilm}({\bf
r}_{4})$ with $\displaystyle l\neq 2$ is negligible under the present
$\displaystyle dt$-$\displaystyle\alpha n$ tensor coupling interaction.

The calculated fusion rates $\displaystyle\lambda_{\rm f}^{(4)}$ are presented
in the final column of Table 5 for 20 sets of potentials listed in Table 4.
$\displaystyle\lambda_{\rm f}^{(4)}=1.1\\!-\\!1.2\times 10^{12}{\rm s}^{-1}$
exhibits minimal dependence on the potential sets and agrees with the result
($\displaystyle 1.15\times 10^{12}{\rm s}^{-1}$) presented in Ref.
Kamimura:2021msf . Moreover, the fusion rates $\displaystyle\lambda_{\rm
f}^{(4)}$ and $\displaystyle\lambda_{\rm f}^{(5)}$ under the same potential
set yield almost the same values.

###  V.2 Momentum and energy spectrum of the ejected muon

This subsection presents the muon momentum and energy spectra derived as
continuous functions of $\displaystyle K$ and the kinetic energy
$\displaystyle E$, respectively, following Sec. VI B of Ref. Kamimura:2021msf
. The momentum spectra, $\displaystyle r(K)$, is obtained by smoothing
$\displaystyle{r}_{il}$ of Eq. (5.6) as

$\displaystyle\displaystyle\\!\\!\\!\\!\lambda_{\rm
f}^{(4)}=\sum_{il}\Big{(}\frac{{r}_{il}}{\mathit{\Delta}\\!K}\Big{)}\,\mathit{\Delta}\\!K\stackrel{{\scriptstyle\mathit{\Delta}\\!K\to
0}}{{\longrightarrow}}\int_{0}^{K_{N}}\\!r(K)\,{\rm d}K,$ (53)

where the present case $\displaystyle\mathit{\Delta}K=0.03$ MeV is
sufficiently small. The energy distribution, $\displaystyle{\bar{r}}(E)$, is
derived as follows:

$\displaystyle\displaystyle{\bar{r}}(E)\,{\rm d}E=r(K)\,{\rm d}K,\qquad
E=\hbar^{2}K^{2}/2\mu_{{\rm R}_{4}}.$ (54)

Figure 8: Momentum spectrum $\displaystyle r(K)$ of the muon emitted by the
$\displaystyle\mu$CF reaction (1.2), which are calculated with Eq. (53) using
the potential Sets A1, B1, C1, D1, and E1 listed in Table 4.  Figure 9: Energy
spectrum $\displaystyle{\bar{r}}(E)$ of the muon emitted by the
$\displaystyle\mu$CF reaction (1.2), which are calculated with Eq. (54) using
the potential Sets A1, B1, C1, D1, and E1 listed in Table 4.

Figures 8 and 9 illustrate the muon momentum spectrum $\displaystyle r(K)$ and
the energy spectrum $\displaystyle{\bar{r}}(E)$ calculated using the potential
Sets A1, B1, C1, D1, and E1 (Table 4). The five lines are close to each other
in both the figures. This demonstrates that our results are potentially
independent of each other. Further, we verified that the figures obtained
using the potentials A3-E3 are similar to Figs. 8 and 9.

In Figs. 10 and 11, we compare our results for $\displaystyle r(K)$ and
$\displaystyle{\bar{r}}(E)$ (red lines using Set B1) with the results in Ref.
[5] (black lines) taken from Figs. 14 and 15 using Eqs. (6.9) and (6.11),

Figure 10: Momentum spectrum $\displaystyle r(K)$ of the ejected muon
calculated with the potential Set B1 in Fig. 8, which is compared with the
momentum spectrum $\displaystyle r(K)$ presented in Fig. 14 of Ref.
Kamimura:2021msf .  Figure 11: Energy spectrum $\displaystyle{\bar{r}}(E)$ of
the ejected muon calculated with the potential Set B1 in Fig. 9, which is
compared with the energy spectrum $\displaystyle{\bar{r}}(E)$ presented in
Fig. 15 of Ref. Kamimura:2021msf .

respectively. Our results agree well with those reported in Ref. [5].The use
of any other potential set also reproduces the result of Ref. [5] as is
understood from Figs. 8 and 9.

In Table. 6, the peak and average energies of the muon energy spectrum
$\displaystyle{\bar{r}}(E)$ and those reported in Ref. Kamimura:2021msf  are
compared. The peak energy is located at $\displaystyle E\sim 1.1\\!$ keV both
in our $\displaystyle T$-matrix model and in Ref. Kamimura:2021msf . The
average energy of 8.9 keV (Set B1), is also consistent with the result (9.5
keV) from Ref. Kamimura:2021msf . This large average energy is caused by the
long high-energy tail of the energy spectrum as evident in Fig. 11. Therefore,
it can be concluded that muons with peak energy of $\displaystyle\\!\sim\\!1$
keV and the average energy of $\displaystyle\\!\sim\\!10$ keV are emitted by
$\displaystyle dt\mu$ fusion, which is the same as that in Ref.
Kamimura:2021msf . These results will be helpful to the ongoing experimental
project for generate an ultra-slow negative muon beam using
$\displaystyle\mu$CF for various applications.

Table 6: Property of the muon energy spectrum $\displaystyle{\bar{r}}(E)$ by the present result with the use of potential Set B1 (Fig. 11). The case of $\displaystyle{\bar{r}}(E)$ given by Ref. Kamimura:2021msf  is also listed for comparison.  Muon energy spectrum |  | Peak |  | Average |  | Peak
---|---|---|---|---|---|---
|  | energy |  | energy |  | strength
|  | (keV) |  | (keV) |  | $\displaystyle({\rm s}\cdot{\rm keV})^{-1}$
Present, $\displaystyle{\bar{r}}(E)$ with B1 |  | 1.1 |  | 8.9 |  | $\displaystyle 1.54\times 10^{11}$
$\displaystyle{\bar{r}}(E)$ (Ref. Kamimura:2021msf ) |  | 1.1 |  | 9.5 |  | $\displaystyle 1.60\times 10^{11}$

##  VI Conclusion

Recently, a comprehensive study of the $\displaystyle\mu$CF reaction (1.2) was
performed in Ref. Kamimura:2021msf  by solving two coupled channel (CC)
Schrödinger equations for the reactions (1.1) and (1.2). In the present study,
we have proposed a considerably more tractable $\displaystyle T$-matrix model
to simulate the CC model Kamimura:2021msf  by considering scenarios i) – v):

i) We reproduced the low-energy cross sections of reaction (1.1) by using an
optical-potential (OP) model (Fig. 2 and Table I). The effect of the
$\displaystyle\alpha$-$\displaystyle n$ channel is considered to be included
using the imaginary part of the optical potential.

ii) The exact $\displaystyle T$-matrix for the reaction (1.1) was approximated
by replacing the exact CC wave function with the OP-model wave function
obtained in i). The cross section of the reaction (1.1) was expressed by the
simple closed form (3.10) based on our model and reproduced the observed cross
section by properly selecting the $\displaystyle dt$-$\displaystyle\alpha n$
coupling potential (Fig. 4 and Table IV).

iii) We calculated the $\displaystyle(dt\mu)_{J=v=0}$ molecular wave function
including the OP-model $\displaystyle d$-$\displaystyle t$ potential
determined in i).

iv) We approximated the full $\displaystyle T$-matrix in Ref. [5] for the
reaction (1.2) as follows: We replaced the CC wave function for this reaction
with the muonic $\displaystyle(dt\mu)_{J=v=0}$ wave function obtained in iii).

v) Using the approximated $\displaystyle T$-matrix in iv), we calculated the
reaction rates to the $\displaystyle\alpha$-$\displaystyle\mu$ continuum and
the bound states, fusion rates of reaction (1.2),
$\displaystyle\alpha$-$\displaystyle\mu$ sticking probability, and momentum
and energy spectra of muons emitted by reaction (1.2). Most of the results
obtained in Ref. [5] were well reproduced.

All the calculated results were insensitive to 20 sets of $\displaystyle
d$-$\displaystyle t$ potential and $\displaystyle dt$-$\displaystyle\alpha n$
coupling potential. In practical calculations, the use of only a few sets is
sufficient.

Thus, the proposed tractable $\displaystyle T$-matrix model was constructed,
such that it reproduced most of the results obtained in Ref. Kamimura:2021msf
. This model is applicable to other $\displaystyle\mu$CF systems such as
$\displaystyle(dd\mu)$, $\displaystyle(tt\mu)$, $\displaystyle(dt\mu)^{*}$,
and $\displaystyle(dd\mu)^{*}$.

###### Acknowledgements.

The authors would like to thank Prof. Y. Kino and Dr. T. Yamashita for their
valuable discussions in the study. This work is supported by the Grant-in-Aid
for Scientific Research on Innovative Areas, “Toward new frontiers: Encounter
and synergy of state-of-the-art astronomical detectors and exotic quantum
beams”, JSPS KAKENHI Grant Number JP18H05461. This work is also supported by
Natural Science Foundation of Jiangsu Province (Grant no. BK20220122), and
National Natural Science Foundation of China (Grant no. 12233002). The main
computing was conducted at the Nanjing Nengmao Space Cloud Computing
Technology Co., Ltd.

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