Two-loop Self-energy Diagrams Research Articles

Zero-threshold expansion of two-loop self-energy diagrams

An algorithm is constructed to derive a small-momentum expansion for two-loop two-point diagrams in all cases where, due to the presence of physical thresholds, there are singularities at zero external momentum. The coefficients of this “zero-threshold” expansion are calculated analytically for arbitrary masses. Numerical examples, using diagrams occurring in the Standard Model, illustrate the convergence of the expansion below the first non-zero threshold.

Analytical and numerical methods for massive two-loop self-energy diagrams

Motivated by the precision results in the electroweak theory studies of two-loop Feynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions. This is done in three respects: firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals. Finally one-dimensional integral representations with elementary functions are derived. They are very well suited for the numerical evaluations.

Evaluation of two-loop self-energy diagram with three propagators.

A small momentum expansion of the ``sunset'' diagram with three different masses is obtained. Coefficients at powers of ${\mathit{p}}^{2}$ are evaluated explicitly in terms of dilogarithms and elementary functions. Also some power expansions of the ``sunset'' diagram in terms of different sets of variables are given.

Calculation of two-loop self-energies in the electroweak Standard Model

Motivated by the results of the electroweak precision experiments, studies of two-loop self-energy Feynman diagrams are performed. An algebraic method for the reduction of all two-loop self-energies to a set of standard scalar integrals is presented. The gauge dependence of the self-energies is discussed and an extension of the pinch technique to the two-loop level is worked out. It is shown to yield a special case of the background-field method which provides a general framework for deriving Green functions with desirable theoretical properties. The massive scalar integrals of self-energy type are expressed in terms of generalized multivariable hypergeometric functions. The imaginary parts of these integrals yield complete elliptic integrals. Finally, one-dimensional integral representations with elementary integrands are derived which are well suited for numerical evaluation.

On the numerical evaluation of scalar two-loop self-energy diagrams

In this paper a method is presented for the numerical evaluation of scalar two-loop self-energy diagrams. It is an extension of Kreimer's two-dimensional integral representation for convergent scalar integrals to the general case where divergencies occur. Where possible, the method is extensively compared with alternative techniques. From this it can be concluded, that the described two-dimensional numerical integration leads to a practical numerical answer for scalar two-loop self-energy diagrams with arbitrary masses.

Large momentum expansion of two-loop self-energy diagrams with arbitrary masses

For two-loop two-point diagrams with arbitrary masses, an algorithm to derive the asymptotic expansion at large external momentum squared is constructed. By using a general theorem on asymptotic expansions of Feynman diagrams, the coefficients of the expansion are calculated analytically. For some two-loop diagrams occurring in the Standard Model, comparison with results of numerical integration shows that our expansion works well in the region above the highest physical threshold.

Two-loop self-energy diagrams with different masses and the momentum expansion

An algorithm for obtaining the power series in the external momentum of two-loop self-energy diagrams with arbitrary masses of the internal particles is examined. The coefficients of the expansion are represented in terms of vacuum two-loop integrals which are calculated for general values of the masses. By comparison with a numerical calculation of some two-loop diagrams occuring in the Standard Model, it is shown that for values of the external momentum below the threshold the first few terms of the expansion provide good approximations to the complete diagrams.

Massive Yang-Mills fields

Two problems are studied in the paper: (i) the relation between Lagrangian and Feynman rules if the Lagrangian contains derivative couplings and/or vector meson fields and (ii) the behaviour of certain two closed loop diagrams in the perturbation theory of Yang-Mills fields. With respect to (i), using an extremely simple regularisation procedure, the canonical formalism is shown to give directly manifestly covariant Feynman rules. With respect to (ii), a certain set of Feynman rules involving ghost particles is known to be valid for diagrams of the Yang-Mills theory with no or one close loop. It is shown however that with the same set of rules the imaginary part of the two-loop self-energy diagrams receives a non-zero contribution of the unphysical three-ghost intermediate state.