This paper is focused on the loop-level understanding of the
Bern-Carrasco-Johansson double copy procedure that relates the integrands of
gauge theory and gravity scattering amplitudes. At four points, the first
non-trivial example of that construction is one-loop amplitudes in N=2
super-Yang-Mills theory and the symmetric realization of N=4 matter-coupled
supergravity. Our approach is to use both field and string theory in parallel
to analyze these amplitudes. The closed string provides a natural framework to
analyze the BCJ construction, in which the left- and right-moving sectors
separately create the color and kinematics at the integrand level. At tree
level, in a five-point example, we show that the Mafra-Schlotterer-Stieberger
procedure gives a new direct proof of the color-kinematics double copy. We
outline the extension of that argument to n points. At loop level, the
field-theoretic BCJ construction of N=2 SYM amplitudes introduces new terms,
unexpected from the string theory perspective. We discuss to what extent we can
relate them to the terms coming from the interactions between left- and
right-movers in the string-theoretic gravity construction.Comment: 46 pages, 8 figures, 2 tables; v3 significantly revised published
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