Scalar one-loop 4-point integral with one massless vertex in loop regularization

The scalar one-loop 4-point function with one massless vertex is evaluated analytically by employing the loop regularization method. According to the method a characteristic scale $\mu_{s}$ is introduced to regularize the divergent integrals. The infrared divergent parts, which take the form of $\ln^{2}(\lambda^{2}/\mu^{2}_{s})$ and $\ln(\lambda^{2}/\mu^{2}_{s})$ as $\mu_{s}\rightarrow 0$ where $\lambda$ is a constant and expressed in terms of masses and Mandelstam variables, and the infrared stable parts are well separated. The result is shown explicitly via $44$ dilogarithms in the kinematic sector in which our evaluation is valid.