[en] 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 m₁, m₂ and m₃ near its threshold, i.e. in any given order in the difference between the external momentum squared and its threshold value, (m₁ + m₂ + m₃)². In particular, this algorithm includes an explicit recurrence procedure to analytically calculate sunset diagrams with arbitrary integer powers of propagators at the threshold

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[en] We discuss the origin of the Wilson polygon-MHV amplitude duality at a perturbative level. It is shown that the duality for the MHV amplitudes at the one-loop level can be proven upon a particular change of variables in Feynman parametrization and with the use of the relation between Feynman integrals at different space-time dimensions. Some generalization of the duality which implies the insertion of a particular vertex operator at the Wilson triangle is found for the 3-point function. We discuss the analytical structure of Wilson loop diagrams and present the corresponding Landau equations. The geometrical interpretation of the loop diagram in terms of the hyperbolic geometry is discussed.

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[en] The connection between Feynman integrals and GKZ A-hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new possibilities; in this paper we continue to explore this connection. To each such hypergeometric system there is an associated toric ideal, we prove that the latter has the Cohen-Macaulay property for two large families of Feynman integrals. This implies, for example, that both the number of independent solutions and dynamical singularities are independent of space-time dimension and generalized propagator powers. Furthermore, in particular, it means that the process of finding a series representation of these integrals is fully algorithmic.

[en] We show how to interpret the scalar Feynman integrals which appear when reducing tensor integrals as scalar Feynman integrals coming from certain nice matroids.

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[en] We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the nth and (n - 1)th minors, whose solution is a representation of the nth minor as an n x n determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order n with respect to the kernel. Our formula is a linear combination of the nth and the (n ± 1)th minors

[en] Feynman integrals are easily solved if their system of differential equations is in ε-form. In this letter we show by the explicit example of the kite integral family that an ε-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The ε-form is obtained by a (non-algebraic) change of basis for the master integrals.