Reaction processes of muon-catalyzed fusion in the muonic molecule $ddμ$ studied with the tractable $T$-matrix model

Abstract

Muon-catalyzed fusion has recently regained significant attention due to experimental and theoretical developments being performed. The present authors [Phys. Rev. C {\bf 109} 054625 (2024)] proposed the tractable $T$-matrix model based on the Lippmann-Schwinger equation to approximate the elaborate two- and three-body coupled-channel (CC) calculations [Kamimura, Kino, and Yamashita, Phys. Rev. C {\bf 107}, 034607 (2023)] for the nuclear reaction processes in the muonic molecule $dtμ$, $(dtμ)_{J=0} \to\!^4{\rm He} + n + μ+ 17.6 \, {\rm MeV}$. % or $(^4{\rm He}μ)_{nl} + n + 17.6 \,{\rm MeV}$. The $T$-matrix model well reproduced almost all of the results generated by the CC work. In the present paper, we apply this model to the nuclear reaction processes in the $ddμ$ molecule, $(ddμ)_{J=1} \to\!^3{\rm He} + n + μ+3.27 \,$ MeV or $t + p + μ+ 4.03 \,$ MeV, in which the fusion takes place via the $p$-wave $d$-$d$ relative motion. Recently, significantly different $p$-wave astrophysical $S(E)$ factors of the reaction $d + d \to\!^3{\rm He} + n$ or $t + p$ at $E \! \simeq \! 1$ keV to 1 MeV have been reported experimentally and theoretically by five groups. Employing many sets of nuclear interactions that can reproduce those five cases of $p$-wave $S(E)$ factors, we calculate the fusion rate of the $(ddμ)_{J=1}$ molecule using three kinds of methods where results are consistent with each other. We also derive the $^3{\rm He}$-$μ$ sticking probability and the absolute values of the energy and momentum spectra of the emitted muon. The violation of charge symmetry in the $p$-wave $d$-$d$ reaction and the $ddμ$ fusion reaction is discussed. Information on the emitted 2.45-MeV neutrons and \mbox{1 keV-dominant} muons should be useful for the application of $ddμ$ fusion.

Figures (11)

• Figure 1: $p$-wave astrophysical $S(E)$ factors of reactions (1.1) and (1.2), reported by Angulo and Decouvemont Angulo1998 (Angulo+), Nebia et al.Nebia2002 (Nebia+), Arai et al.Arai2011 (Arai+), Tumino et al.Tumino2014 (Tumino+), and Solovyev Solovyev2024 (Solovyev). Tumino+ and Angulo+ have been multiplied by 0.1 to avoid crowds of lines. Nebia+ is up to 100 keV. No result reports error bar.
• Figure 2: Nine Jacobi coordinates used in this work for the $dd\mu$, $^3{\rm He}n\mu$, and $tp\mu$ systems, referred to as channel $c=1$ to $c=9$, respectively.
• Figure 3: $p$-wave $S$-factor $S(E)$, with black lines given as sum of the two $S(E)$ of reactions (1.1) and (1.2) for each reference in Fig. \ref{['fig:8line-sfactor']}. Each black line is well reproduced by the five sets $d$-$d$ optical-potentials listed in Table \ref{['tab:vdd']}.
• Figure 4: $p$-wave $S(E)$ factor of reaction (1.1), $S_{dd\to^3{\rm He}n}(E)$. Five black lines are those reported by Angulo and Decouvemont Angulo1998, Nebia et al.Nebia2002, Arai et al.Arai2011, Tumino et al.Tumino2014, and Solovyev Solovyev2024. Lines A1-E1 closely reproducing each black line are derived by the present $T$-matrix calculation using the $dd-^3$He$n$ coupling potentials A1-E1 listed in Table \ref{['tab:vdtan']}; use of the other coupling potential Ai-Ei ($i=2-4$) give similar results.
• Figure 5: $p$-wave $S(E)$ factor of reaction (1.2), $S_{dd\to tp}(E)$. Same meaning for lines as in Fig. \ref{['fig:newT-S-hn']}.