Radiative corrections to the Dalitz plot of K l3 ± decays: Contribution of the four-body region

We calculate the radiative corrections to the Dalitz plot of K_{l3}^\pm decays to order (alpha/pi)(q/M_1), where q is the momentum transfer and M_1 is the mass of the kaon. We restrict the analysis to the so-called four-body region, which arises when no discrimination of real photons is made either kinematically or experimentally. We present our results in two ways: the first one with the triple integration over the photon kinematical variables ready to be performed numerically and the second one in a fully analytical form. Our results can be useful in experimental analyses of the Dalitz plot, by evaluating the model-independent coefficients of the quadratic products of the form factors; we provide some numbers as a case example. We find a small, albeit non-negligible, contribution from the four-body region to the radiative correction to the total decay rate of K_{l3}^\pm decays.