  Nuclear cross sections are basic inputs to any nuclear computation. Campaigns
of experiments are fitted with the parametric R-matrix model of quantum nuclear
interactions, and the resulting cross sections are documented - both point-wise
and as resonance parameters (with uncertainties) - in standard evaluated
nuclear data libraries (ENDF, JEFF, BROND, JENDL, CENDL, TENDL): these
constitute our common knowledge of fundamental nuclear physics. In the past
decade, a collaborative effort has been deployed to establish a new nuclear
cross section library format - the Windowed Multipole Library - with the goal
of considerably reducing the cost of cross section calculations in nuclear
transport simulations. This article lays the theoretical foundations
underpinning these efforts. From general R-matrix scattering theory, we derive
the windowed multipole representation of nuclear cross sections. Though
physically and mathematically equivalent, the windowed multipole representation
is particularly well suited for subsequent temperature treatment of
angle-integrated cross sections: we show that accurate Doppler broadening can
be performed analytically up to the first reaction threshold; and we derive
cross sections temperature derivatives to any order. Furthermore, we here
establish a way of converting the R-matrix resonance parameters uncertainty
(covariance matrices) into windowed multipole parameters uncertainty. We show
that generating stochastic nuclear cross sections by sampling from the
resulting windowed multipole covariance matrix can reproduce the cross section
uncertainty in the original nuclear data file. Through this foundational
article, we hope to make the Windowed Multipole Representation accessible,
reproducible, and usable for the nuclear physics community, as well as provide
the theoretical basis for future research on expanding its capabilities.
