Analytical evaluation of certain on-shell two-loop three-point diagrams

An analytical approach is applied to the calculation of some dimensionally-regulated two-loop vertex diagrams with essential on-shell singularities. Such diagrams are important for the evaluation of QED corrections to the muon decay, QCD corrections to top quark decays t->W^{+}b, t->H^{+}b, etc.Comment: 2 pages, LaTeX, contribution to proceedings of ACAT2002 (Moscow, June 2002

Equivalence of Recurrence Relations for Feynman Integrals with the Same Total Number of External and Loop Momenta

We show that the problem of solving recurrence relations for L-loop (R+1)-point Feynman integrals within the method of integration by parts is equivalent to the corresponding problem for (L+R)-loop vacuum or (L+R-1)-loop propagator-type integrals. Using this property we solve recurrence relations for two-loop massless vertex diagrams, with arbitrary numerators and integer powers of propagators in the case when two legs are on the light cone, by reducing the problem to the well-known solution of the corresponding recurrence relations for massless three-loop propagator diagrams with specific boundary conditions.Comment: 8 pp., LaTeX with axodraw.st

Two-Loop Gluon-Condensate Contributions To Heavy-Quark Current Correlators: Exact Results And Approximations

The coefficient functions of the gluon condensate $$, in the correlators of heavy-quark vector, axial, scalar and pseudoscalar currents, are obtained analytically, to two loops, for all values of $z=q^2/4m^2$. In the limiting cases $z\to0$, $z\to1$, and $z\to-\infty$, comparisons are made with previous partial results. Approximation methods, based on these limiting cases, are critically assessed, with a view to three-loop work. High accuracy is achieved using a few moments as input. A {\em single} moment, combined with only the {\em leading} threshold and asymptotic behaviours, gives the two-loop corrections to better than 1% in the next 10 moments. A two-loop fit to vector data yields $\approx0.021$ GeV$^4$.Comment: 9 page

Small-threshold behaviour of two-loop self-energy diagrams: some special cases

An algorithm to construct analytic approximations to two-loop diagrams describing their behaviour at small non-zero thresholds is discussed. For some special cases (involving two different-scale mass parameters), several terms of the expansion are obtained.Comment: 7 pages, plain latex; talk given at DESY-Zeuthen Workshop "QCD and QED in Higher Order", Rheinsberg, April 1996, to appear in Proceeding

Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators

We present an algorithm for the analytical evaluation of dimensionally regularized massless on-shell double box Feynman diagrams with arbitrary polynomials in numerators and general integer powers of propagators. Recurrence relations following from integration by parts are solved explicitly and any given double box diagram is expressed as a linear combination of two master double boxes and a family of simpler diagrams. The first master double box corresponds to all powers of the propagators equal to one and no numerators, and the second master double box differs from the first one by the second power of the middle propagator. By use of differential relations, the second master double box is expressed through the first one up to a similar linear combination of simpler double boxes so that the analytical evaluation of the first master double box provides explicit analytical results, in terms of polylogarithms \Li{a}{-t/s}, up to $a=4$, and generalized polylogarithms $S_{a,b}(-t/s)$, with $a=1,2$ and $b=2$, dependent on the Mandelstam variables $s$ and $t$, for an arbitrary diagram under consideration.Comment: LaTeX, 16 pages; misprints in ff. (8), (24), (30) corrected; some explanations adde

Asymptotic expansion of Feynman integrals near threshold

We present general prescriptions for the asymptotic expansion of massive multi-loop Feynman integrals near threshold. As in the case of previously known prescriptions for various limits of momenta and masses, the terms of the threshold expansion are associated with subgraphs of a given graph and are explicitly written through Taylor expansions of the corresponding integrands in certain sets of parameters. They are manifestly homogeneous in the threshold expansion parameter, so that the calculation of the given Feynman integral near the threshold reduces to the calculation of integrals of a much simpler type. The general method is illustrated by two-loop two-point and three-point diagrams. We discuss the use of the threshold expansion for problems of physical interest, such as the next-to-next-to-leading order heavy quark production cross sections close to threshold and matching calculations and power counting in non-relativistic effective theories.Comment: 24 pages, LaTeX, 4 figures included via epsf.st

Threshold expansion of the sunset diagram

By use of the threshold expansion we develop an algorithm for analytical evaluation, within dimensional regularization, of arbitrary terms in the expansion of the (two-loop) sunset diagram with general masses m_1, m_2 and m_3 near its threshold, i.e. in any given order in the difference between the external momentum squared and its threshold value, (m_1+m_2+m_3)^2. In particular, this algorithm includes an explicit recurrence procedure to analytically calculate sunset diagrams with arbitrary integer powers of propagators at the threshold.Comment: 26 pages (23 pages in LaTeX and 3 PS figures), a typo in eq.(23) corrected; final version to appear in Nucl.Phys.

Analytical Result for Dimensionally Regularized Massive On-Shell Planar Double Box

The dimensionally regularized master planar double box Feynman diagram with four massive and three massless lines, powers of propagators equal to one, all four legs on the mass shell, i.e. with p_i^2=m^2, i=1,2,3,4, is analytically evaluated for general values of m^2 and the Mandelstam variables s and t. An explicit result is expressed in terms of polylogarithms, up to the third order, depending on special combinations of m^2,s and t.Comment: 10 pages, LaTeX with axodraw.st

Heavy mass expansion, light-by-light scattering and the anomalous magnetic moment of the muon

Contributions from light-by-light scattering to (g_\mu-2)/2, the anomalous magnetic moment of the muon, are mediated by the exchange of charged fermions or scalar bosons. Assuming large masses M for the virtual particles and employing the technique of large mass expansion, analytical results are obtained for virtual fermions and scalars in the form of a series in (m_\mu /M)^2. This series is well convergent even for the case M=m_\mu. For virtual fermions, the expansion confirms published analytical formulae. For virtual scalars, the result can be used to evaluate the contribution from charged pions. In this case our result confirms already available numerical evaluations, however, it is significantly more precise.Comment: revtex4, eps figure

Finite calculation of divergent selfenergy diagrams

Using dispersive techniques, it is possible to avoid ultraviolet divergences in the calculation of Feynman diagrams, making subsequent regularization of divergent diagrams unnecessary. We give a simple introduction to the most important features of such dispersive techniques in the framework of the so-called finite causal perturbation theory. The method is also applied to the 'divergent' general massive two-loop sunrise selfenergy diagram, where it leads directly to an analytic expression for the imaginary part of the diagram in accordance with the literature, whereas the real part can be obtained by a single integral dispersion relation. It is pointed out that dispersive methods have been known for decades and have been applied to several nontrivial Feynman diagram calculations.Comment: 15 pages, Latex, one figure, added reference