  We report on a detailed investigation of four-body $p p \to p p \pi^+ \pi^-$
and $p \bar{p} \to p \bar{p} \pi^+ \pi^-$) reactions which constitute an
irreducible background to three-body processes $p p \to p p M$, where $M$ is a
broad resonance in the $\pi^+ \pi^-$ channel, e.g. $M=\sigma, \rho^{0},
f_{0}(980), f_{2}(1275), f_{0}(1500)$. We include double-diffractive
contribution (both pomeron and reggeon exchanges) as well as the pion-pion
rescattering contributions. The first process dominates at higher energies and
small pion-pion invariant masses while the second becomes important at lower
energies and higher pion-pion invariant masses. We compare our results with the
experimental data. We make predictions for future experiments at PANDA, RHIC,
Tevatron and LHC energies. The two-dimensional distribution in rapidity space
of pions $(y_{\pi^+}, y_{\pi^-})$ is particularly interesting. The higher the
incident energy, the higher preference for the same-hemisphere emission of
pions. The processes considered constitute a sizeable contribution to the total
nucleon-nucleon cross section as well as to pion inclusive cross section.
