Antikaon-induced $K_1(1270)^+$ meson production on nuclei near threshold

We study the inclusive strange axial-vector meson $K_1(1270)^+$ production in ${K^-}A$ reactions at near-threshold laboratory incident antikaon momenta within a nuclear spectral function approach, which describes incoherent direct $K_1(1270)^+$ meson production in $K^-$ meson--proton ${K^-}p \to {K_1(1270)^+}\Xi^-$ production processes and accounts for three different options for its in-medium mass shift (or for its effective scalar potential) at central density $\rho_0$. We calculate the absolute differential and total cross sections for the production of $K_1(1270)^+$ mesons on $^{12}$C and $^{184}$W target nuclei at laboratory angles of 0$^{\circ}$--45$^{\circ}$ by $K^-$ mesons with momenta of 2.5, 2.8 and 3.5 GeV/c, which are close to the threshold momentum ($\approx$ 2.95 GeV/c) for $K_1(1270)^+$ meson production off the free target proton at rest. The intrinsic properties of carbon and tungsten target nuclei have been described in terms of their spectral functions, which take into account the momenta of target protons and the energies of their separation from the considered nuclei. We show that the differential and total (absolute and relative) $K_1(1270)^+$ antikaon-induced production cross sections at initial momenta not far from threshold -- at momenta $\sim$ 2.8--3.5 GeV/c, at which there is yet no strong drop in their strength, reveal a distinct sensitivity to changes in the in-medium shift of the $K_1(1270)^+$ mass, studied in the paper, both in the $K_1(1270)^+$ meson low-momentum (0.1--1.0 GeV/c) and in its full-momentum ranges. This would permit evaluating this shift. Experimental data necessary for this aim can be obtained in a dedicated experiment at the J-PARC Hadron Experimental Facility.