Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case

We present a double precision routine in Fortran for the precise and fast numerical evaluation of the two Master Integrals (MIs) of the equal mass two-loop sunrise graph for arbitrary momentum transfer in d=2 and d=4 dimensions. The routine implements the accelerated power series expansions obtained by solving the corresponding differential equations for the MIs at their singular points. With a maximum of 22 terms for the worst case expansion a relative precision of better than a part in 10^{15} is achieved for arbitrary real values of the momentum transfer.Comment: 11 pages, LaTeX. The complete paper is also available via the www at http://www-ttp.physik.uni-karlsruhe.de/Preprints/ and the program can be downloaded from http://www-ttp.physik.uni-karlsruhe.de/Progdata

Progress on two-loop non-propagator integrals

At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one loop, as further progress was hampered up to very recently by the greater computational problems encountered in the study of multi-leg amplitudes beyond one loop. We discuss the progress made lately in the evaluation of two-loop multi-leg integrals, with particular emphasis on two-loop four-point functions.Comment: 9 pages, LaTeX, Invited talk at 5th International Symposium on Radiative Corrections (RADCOR-2000), Carmel CA, USA, 11--15 September, 200

Two-Loop Form Factors in QED

We evaluate the on shell form factors of the electron for arbitrary momentum transfer and finite electron mass, at two loops in QED, by integrating the corresponding dispersion relations, which involve the imaginary parts known since a long time. The infrared divergences are parameterized in terms of a fictitious small photon mass. The result is expressed in terms of Harmonic Polylogarithms of maximum weight 4. The expansions for small and large momentum transfer are also givenComment: 13 pages, 1 figur

Analytic treatment of the two loop equal mass sunrise graph

The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.Comment: 33 pages, LaTeX, v2: +1 figure; v3: changes in the conclusions; simplifications in the recurrences (6.3) and (6.9

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

Analytic evaluation of Feynman graph integrals

We review the main steps of the differential equation approach to the analytic evaluation of Feynman graphs, showing at the same time its application to the 3-loop sunrise graph in a particular kinematical configuration.Comment: 5 pages, 1 figure, uses npb.sty. Presented at RADCOR 2002 and Loops and Legs in Quantum Field Theory, 8-13 September 2002, Kloster Banz, Germany. Revised version: minor typos corrected, one reference adde

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

The analytic value of a 3-loop sunrise graph in a particular kinematical configuration

We consider the scalar integral associated to the 3-loop sunrise graph with a massless line, two massive lines of equal mass $M$, a fourth line of mass equal to $Mx$, and the external invariant timelike and equal to the square of the fourth mass. We write the differential equation in $x$ satisfied by the integral, expand it in the continuous dimension $d$ around $d=4$ and solve the system of the resulting chained differential equations in closed analytic form, expressing the solutions in terms of Harmonic Polylogarithms. As a byproduct, we give the limiting values of the coefficients of the $(d-4)$ expansion at $x=1$ and $x=0$.Comment: 9 pages, 3 figure