Reaction processes of muon-catalyzed fusion in the muonic molecule $dd\mu$ 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 $dt\mu$, $(dt\mu {)}_{J=0}\to {}^4\mathrm{He}+n+\mu +17.6\mathrm{MeV}$. % or ${(}^4\mathrm{He}\mu {)}_{nl}+n+17.6\mathrm{MeV}$. 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 $dd\mu$ molecule, $(dd\mu {)}_{J=1}\to {}^3\mathrm{He}+n+\mu +3.27$ MeV or $t+p+\mu +4.03$ 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 $d+d\to {}^3\mathrm{He}+n$ 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 $(dd\mu {)}_{J=1}$ molecule using three kinds of methods where results are consistent with each other. We also derive the ${}^3\mathrm{He}$-$\mu$ 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 $dd\mu$ 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 $dd\mu$ fusion.

Comments: 16 pages, 11 figures