The sum rules relating the axial-vector coupling constant to ππ and πN total cross sections are derived using the commutation relations between the chiral currents χ+(t) and χ3(t). The sum rules obtained are the same as those obtained by other authors using the commutation relation between χ+(t) and χ-(t). It is also shown that the assumption of the existence of the σ meson as a scalar unitary singlet helps in saturating the ππ and KK̄ sum rules. A definite conclusion cannot be arrived at about the existence of σ meson from the ππ sum rule, since the sum rule can equally well be saturated by postulating a large low-energy ππ scattering in I=0 state, but the KK̄ sum rule does seem to require the presence of a particle with properties similar to those of the σ meson. SU(3) symmetry is assumed in the latter case.