Evaluation of two-loop self-energy basis integrals using differential equations

I study the Feynman integrals needed to compute two-loop self-energy functions for general masses and external momenta. A convenient basis for these functions consists of the four integrals obtained at the end of Tarasov's recurrence relation algorithm. The basis functions are modified here to include one-loop and two-loop counterterms to render them finite; this simplifies the presentation of results in practical applications. I find the derivatives of these basis functions with respect to all squared-mass arguments, the renormalization scale, and the external momentum invariant, and express the results algebraically in terms of the basis. This allows all necessary two-loop self-energy integrals to be efficiently computed numerically using the differential equation in the external momentum invariant. I also use the differential equations method to derive analytic forms for various special cases, including a four-propagator integral with three distinct non-zero masses.