Total $\pi^+ p$ cross section extracted from the leading neutron spectra at the LHC
We use the very forward neutron energy spectra measured by the Large Hadron Collider forward (LHCf) experiment at 7 TeV to extract the $\pi^+ p$ total cross section at centre-of-mass energies in the range 2.3$-$3.5 TeV. To do this we have to first isolate the $\pi$-exchange pole in forward neutron production in $pp$ collisions, by evaluating other possible contributions. Namely, those from $\rho$ and $a_2$ exchange, from both eikonal and enhanced screening effects, from migration, from neutron production by $\Delta$-isobar decay and from diffractive nucleon excitations. We discuss the possible theoretical uncertainties due to the fact that the data do not exactly reach the $\pi$ pole. We choose the kinematical domain where the pion contribution dominates and demonstrate the role of the different corrections which could affect the final result.