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

The scalar one-loop four-point function with one massless vertex is evaluated analytically by employing the loop regularization method. According to the method, a characteristic scale μ_s is introduced to regularize the divergent integrals. The infrared divergent parts, which take the form of ${\mathrm{ln}}^{2}({\lambda }^{2}/{\mu }_{s}^{2})$ and $\mathrm{ln}({\lambda }^{2}/{\mu }_{s}^{2})$ as μ_s → 0 where λ 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.