Recent observations of anomalous angular correlations of electron-positron pairs, stemming from the products of several nuclear reactions, have been interpreted by assuming the existence of a neutral boson of rest mass of about 17 MeV/c^2, called the X17 particle. Similarly, the results of another set of experiments on photon pair spectra around the invariant mass 38 MeV/c^2, indicates the existence of the so-called E38 particle, which may mediate energy and momentum from the excited products. In the present talk we rely on quantum electrodynamics, in order to show a possible derivation of the above-mentioned masses. We use the exact solutions of the Dirac equation of the system of a charged particle and quantized electromagnetic plane waves. By applying these non-perturbative solutions to the proton, we derive a formula, yielding the invariant mass of 17.0087 MeV/c^2 for the dressed radiation. An analytic expression can also be derived with the udd quarks of the neutron, which delivers 37.9937 MeV/c^2 for the invariant mass of the radiation. Besides the Sommerfeld fine structure constant and the nucleon masses, the derived formulas contain merely some first-principle statistical factors.