Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations

The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method, whose features are discussed in details, offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.Comment: 1+21 pages, Latex, 5 ps-figure

The Threshold Expansion of the 2-loop Sunrise Selfmass Master Amplitudes

The threshold behavior of the master amplitudes for two loop sunrise self-mass graph is studied by solving the system of differential equations, which they satisfy. The expansion at the threshold of the master amplitudes is obtained analytically for arbitrary masses.Comment: 1+18 pages, Latex, no figures, as in Journal reference with more changes in Eq.(31),(42),(45

Numerical evaluation of the general massive 2-loop 4-denominator self-mass master integral from differential equations

The differential equation in the external invariant p^2 satisfied by the master integral of the general massive 2-loop 4-denominator self-mass diagram is exploited and the expansion of the master integral at p^2=0 is obtained analytically. The system composed by this differential equation with those of the master integrals related to the general massive 2-loop sunrise diagram is numerically solved by the Runge-Kutta method in the complex p^2 plane. A numerical method to obtain results for values of p^2 at and close to thresholds and pseudo-thresholds is discussed in details.Comment: Latex, 20 pages, 7 figure

Simulation of the process e+e−↦e+e−γe^+ e^- \mapsto e^+ e^- \gamma within electroweak theory with longitudinally polarized initial electrons

We present simple analytic expressions for the distributions of the Bhabha scattering process with emission of one hard photon, including weak boson exchanges, and with longitudinal polarization of the initial electron. The results from the Monte Carlo generator BHAGEN-1PH, based on these expressions, are presented and compared, for the unpolarized case, with those existing in literature.Comment: 9 pages, plain Tex, no figures, small change in Table

The Pseudothreshold Expansion of the 2-loop Sunrise Selfmass Master Amplitudes

The values at pseudothreshold of two loop sunrise master amplitudes with arbitrary masses are obtained by solving a system of differential equations. The expansion at pseudothreshold of the amplitudes is constructed and some lowest terms are explicitly presented.Comment: 1+22 pages, Latex, no figures, changes in Eq.(41),(44),(47

Using differential equations to compute two-loop box integrals

The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop calculations can be reduced to a small number of master integrals. An efficient method to compute these master integrals is to derive and solve differential equations in the external invariants for them. As an application of the differential equation method, we compute the ${\cal O}(\epsilon)$-term of a particular combination of on-shell massless planar double box integrals, which appears in the tensor reduction of $2 \to 2$ scattering amplitudes at two loops.Comment: 5 pages, LaTeX, uses espcrc2.sty; presented at Loops and Legs in Quantum Field Theory, April 2000, Bastei, German

The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass

We consider the two-loop self-mass sunrise amplitude with two equal masses $M$ and the external invariant equal to the square of the third mass $m$ in the usual $d$-continuous dimensional regularization. We write a second order differential equation for the amplitude in $x=m/M$ and show as solve it in close analytic form. As a result, all the coefficients of the Laurent expansion in $(d-4)$ of the amplitude are expressed in terms of harmonic polylogarithms of argument $x$ and increasing weight. As a by product, we give the explicit analytic expressions of the value of the amplitude at $x=1$, corresponding to the on-mass-shell sunrise amplitude in the equal mass case, up to the $(d-4)^5$ term included.Comment: 11 pages, 2 figures. Added Eq. (5.20) and reference [4