  We calculate inclusive cross section for $f_2(1270)$ tensor meson production
via color singlet gluon-gluon fusion in the $k_t$-factorization approach with
unintegrated gluon distribution functions (UGDFs). The process may be
potentially interesting in the context of searches for saturation effects. The
energy-momentum tensor, equivalent to helicity-2 coupling, and helicity-0
coupling are used for the $g^* g^* \to f_2(1270)$ vertex. Two somewhat
different parametrizations of helicity-2 and helicity-0 tensorial structure
from the literature are used in our calculations. Some parameters are extracted
from $\gamma \gamma \to f_2(1270) \to \pi \pi$ reactions. Different modern
UGDFs from the literature are used. The results strongly depend on the
parametrization of the $g^* g^* \to f_2(1270)$ form factor. Our results for
transverse momentum distributions of $f_2$ are compared to preliminary ALICE
data. We can obtain agreement with the data only at larger $f_2(1270)$
transverse momenta only for some parametrizations of the $g^* g^* \to
f_2(1270)$ form factor. No obvious sign of the onset of saturation is possible.
At low transverse momenta one needs to include also the $\pi\pi$ final-state
rescattering. The agreement with the ALICE data can be obtained by adjusting
probability of formation and survival of $f_2(1270)$ in a harsh quark-gluon and
multipion environment. The pomeron-pomeron fusion mechanism is discussed in
addition and results are quantified.
