Muon-catalyzed fusion has recently regained significant attention due to experimental and theoretical developments. The present authors proposed Q. Wu and M. Kamimura, [Phys. Rev. C 109, 054625 (2024)] the tractable $T$-matrix model based on the Lippmann-Schwinger equation to approximate the elaborate two- and three-body coupled-channel (CC) calculations M. Kamimura, Y. Kino, and T. Yamashita, [Phys. Rev. C 107, 034607 (2023)] for the nuclear reaction processes in the muonic molecule $dt\mu , {\left(dt\mu \right)}_{J=0}{\to }^4\mathrm{He}+n+\mu +17.6\mathrm{MeV}$. 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, ${\left(dd\mu \right)}_{J=1}{\to }^3\mathrm{He}+n+\mu +3.27$ MeV or $t+p+\mu +4.03$ MeV, in which the fusion takes place via the $p$-wave $d\text{−}d$ relative motion. Recently, significantly different $p$-wave astrophysical $S\left(E\right)$ 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 $S\left(E\right)$ factors, we calculate the fusion rate of the ${\left(dd\mu \right)}_{J=1}$ molecule using three kinds of methods where results are consistent with each other. We also derive the ${}^3\mathrm{He}\text{−}\mu$ 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 $d\text{−}d$ reaction and the $dd\mu$ fusion reaction is discussed. Information on the emitted 2.45 MeV neutrons and 1 keV dominant muons should be useful for the application of $dd\mu$ fusion.

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• Lepton induced nuclear reactions
• Nuclear fusion
• Nuclear reactions
• Muonic atoms & molecules