  We study the perturbative expansion of the ground state energy in the
presence of an external field coupled to a conserved charge in the integrable
two-dimensional O$(4)$ nonlinear sigma model. By solving Volin's algebraic
equations for the perturbative coefficients we study the large order asymptotic
behaviour of the perturbative series analytically. We confirm the previously
numerically found leading behaviour and study the nearest singularities of the
Borel transformed series and the associated alien derivatives. We find a
'resurgence' behaviour: the leading alien derivatives can be expressed in terms
of the original perturbative series. A simplified 'toy' model is also
considered: here the perturbative series can be found in a closed form and the
resurgence properties are very similar to that found in the real problem.
