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We exactly solve Dyson-Schwinger equations for a massless quartic scalarfield theory. n-point functions are computed till n=4 and the exact propagatorcomputed from the two-point function. The spectrum is so obtained, being thesame of a harmonic oscillator. Callan-Symanzik equation for the two-pointfunction gives the beta function. This gives the result that this theory hasonly trivial fixed points. In the low-energy limit the coupling goes to zeromaking the theory trivial and, at high energies, it reaches infinity. No Landaupole appears, rather this should be seen as a precursor, in a weak perturbationexpansion, of the coupling reaching the trivial fixed point at infinity. Usinga mapping theorem, recently proved, between massless quartic scalar fieldtheory and gauge theories, we derive some properties of the latter.

Exact solution of Dyson-Schwinger equations for a scalar field theory