Under the assumption of confinement and chiral symmetry breaking the low energy description of QCD is in terms of an effective theory of pions whose properties (couplings) should be derived from the high energy Lagrangian (QCD). In this talk I will describe recent work in collaboration with Yifei He where we propose a bootstrap method to compute pion scattering phase shifts and low energy effective action coefficients. The method looks for the most general S-matrix that matches at low energy the tree level amplitudes of the non-linear sigma model and at high energy, QCD sum rules and form factors. This is a theoretical/numerical calculation that uses as only data the pion mass, pion decay constant fpi and the QCD parameters Nc=3, Nf=2, mq and alpha_s. The results are in reasonable agreement with experiment. In particular, we find the rho(770), f2(1270) and rho(1450) resonances. The interplay between the UV gauge theory and low energy pion physics is an example of a general situation where we know the microscopic theory as well as the effective theory of long wavelength fluctuations but we want to solve the strongly coupled dynamics at intermediate energies. The bootstrap builds a bridge between the low and high energy by determining the consistent S-matrix that matches both and provides, in this case, a new direction to understand the strongly coupled physics of gauge theories.