Abstract Resonant formation of the muonic molecular ion ( α μ t ) 01 2 + in the collision of the muonic tritium atom with the helium hydride ion, ( t μ ) 1 s + 4 HeH + → [ ( α μ t ) 01 2 + , p , e e ] + , is proposed. The ( α μ t ) 01 2 + ion is formed in the rotational-vibrational state (0,1) as one of the two nuclei of the final complex. Back-decay process [ ( α μ t ) 01 2 + , p , e e ] K n + → ( t μ ) 1 s + 4 HeH K ′ n ′ + , where K and n ( K ′ and n ′ ) is the rotational and vibrational quantum number, respectively, is considered in detail and the corresponding decay widths for all possible rotational-vibrational ( K , n ) → ( K ′ , n ′ ) transitions are calculated. Numerous inelastic decay channels of ( α μ t ) 01 2 + are also presented and discussed. The possible fast resonant formation of the ( α μ t ) 01 2 + ion may provide an interesting tool for experimental studies of t ( α , γ ) 7 Li nuclear fusion, which is one of the most important reactions in Big-Bang nucleosynthesis models. Studying this reaction may prove important for solving the lithium problem.