  Loop-Tree Duality (LTD) is a framework in which the energy components of all
loop momenta of a Feynman integral are integrated out using residue theorem,
resulting in a sum over tree-like structures. Originally, the LTD expression
exhibits cancellations of non-causal thresholds between summands, also known as
dual cancellations. As a result, the expression exhibits numerical
instabilities in the vicinity of non-causal thresholds and for large loop
momenta. In this work we derive a novel, generically applicable, Manifestly
Causal LTD (cLTD) representation whose only thresholds are causal thresholds,
i.e. it manifestly realizes dual cancellations. Consequently, this result also
serves as a general proof for dual cancellations. We show that LTD, cLTD, and
the expression stemming from Time Ordered Perturbation Theory (TOPT) are
locally equivalent. TOPT and cLTD both feature only causal threshold
singularities, however LTD features better scaling with the number of
propagators. On top of the new theoretical perspectives offered by our
representation, it has the useful property that the ultraviolet (UV) behaviour
of the original 4D integrand is maintained for every summand. We show that the
resulting LTD integrand expression is completely stable in the UV region which
is key for practical applications of LTD to the computation of amplitudes and
cross sections. We present explicit examples of the LTD expression for a
variety of up to four-loop integrals and show that its increased computational
complexity can be efficiently mitigated by optimising its numerical
implementation. Finally, we provide computer code that automatically generates
the LTD expression for an arbitrary topology.
