Scalar one-loop four-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 method a characteristic scale $\mu_{s}$ introduced regularize divergent integrals. infrared parts, which take form of $\ln^{2}(\lambda^{2}/\mu^{2}_{s})$ and $\ln(\lambda^{2}/\mu^{2}_{s})$ as $\mu_{s}\rightarrow 0$ where $\lambda$ constant expressed in terms masses Mandelstam variables, stable parts are well separated. result shown explicitly via $44$ dilogarithms kinematic sector our evaluation valid.