Research Activities

 
 

Master differential equations for Feynman graph amplitudes

It is by now well established that, by means of the integration by part identities of Tkachov and Chetyrkin, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. A new method for the calculation of master integrals is proposed, where the integration by part identities can be further used for obtaining a linear system of first order differential equations for the master integrals themselves. These master equations can then be used for the numerical evaluation of the amplitudes as well as for investigating their analytic properties, such as the asymptotic and threshold behaviours and the corresponding expansions (and for analytic integration purposes, when possible). The new method is at first pedagogically applied to the case of the one loop self-mass amplitude, by explicitly working out expansions and quadrature formulas, both in arbitrary continuous dimension n and in the n going to 4 limit. Then the master equations, differential in the external square momentum p², in n-continuous dimensions and for arbitrary values of the internal masses, are derived for the master integrals of the two-loop sunrise graph. The equations are then used for working out the values at p²= 0 and the expansions, in (n-4) at n going to 4 limit, in p² at p²=0 and in 1/p² for large values of p². Also the values for p² at pseudothresholds and thresholds, for 3 different internal masses, are obtained. For the general case (arbitrary n, masses and p²) is in progress the implementation of a program, for obtaining numerical solutions from a system of linear differential equations with the Runge-Kutta method. The proposed method is expected to work in the more general case of multi-point, multi-loop amplitudes and a constructive developement is planned. A computerized method for the generation of the algebraic solution of the system of equations provided by the integration by part identities is implemented. Several particular cases of graphs, namely 2-loop box graphs, with only massless lines (2 and 3 jets phenomenology in QCD) or with also one massive line (Bhabha scattering in QED) are under examination. With this method also the contribution to the electron anomalous magnetic moment at 3-loop is expected to be recovered in a more authomatic way. A new family of functions, called Harmonic Polylogarithms, is introduced, suitable for the analytic calculation of Feynman graph amplitudes.