General limit on the relation between abundances of D and $^7$Li in big
  bang nucleosynthesis with nucleon injections
  The injections of energetic hadrons could have occurred in the early universe
by decays of hypothetical long-lived exotic particles. The injections induce
the showers of nonthermal hadrons via nuclear scattering. Neutrons generated at
these events can react with $^7$Be nuclei and reduce $^7$Be abundance solving a
problem of the primordial $^7$Li abundance. We suggest that thermal neutron
injection is a way to derive a model independent conservative limit on the
relation between abundances of D and $^7$Li in a hadronic energy injection
model. We emphasize that an uncertainty in cross sections of inelastic $n+p$
scattering affects the total number of induced neutrons, which determines final
abundances of D and $^7$Li. In addition, the annihilations of antinucleons with
$^4$He result in higher D abundance and trigger nonthermal $^6$Li production.
It is concluded that a reduction of $^7$Li abundance from a value in the
standard big bang nucleosynthesis (BBN) model down to an observational two
$\sigma$ upper limit is necessarily accompanied by an undesirable increase of D
abundance up to at least an observational 12 $\sigma$ upper limit from
observations of quasi-stellar object absorption line systems. The effects of
antinucleons and secondary particles produced in the hadronic showers always
lead to a severer constraint. The BBN models involving any injections of extra
neutrons are thus unlikely to reproduce a small $^7$Li abundance consistent
with observations.
