  We use on-shell methods to study the non-supersymmetric and supersymmetric
low-energy S-matrix on a probe D3-brane, including both the 1-loop
contributions of massless states as well as the effects of higher-derivative
operators. Our results include: (1) A derivation of the duality invariance of
Born-Infeld electrodynamics as the dimensional oxidation of the group of
spatial rotations transverse to a probe M2-brane; this is done using a novel
implementation of subtracted on-shell recursion. (2) The first explicit
loop-level BCJ double-copy in a non-gravitational model, namely the calculation
of the 4-point self-dual amplitude of non-supersymmetric Born-Infeld. (3) From
previous results for $n$-point self-dual 1-loop BI amplitudes and the
conjectured dimension-shifting relations in Yang-Mills, we obtain an explicit
all-multiplicity, at all orders in $\epsilon$, expression for the 1-loop
integrand of the MHV sector of $\mathcal{N}=4$ DBI. (4) For all $n>4$, the
explicitly integrated duality-violating 1-loop amplitudes (self-dual and
next-to-self-dual in pure BI as well as MHV in $\mathcal{N}=4$ DBI) are shown
to be removable at $\mathcal{O}(\epsilon^0)$ by adding finite local
counterterms; we propose that this may be true more generally at 1-loop order.
(5) We find that in non-supersymmetric Born-Infeld, not all finite local
counterterms needed to restore electromagnetic duality can be constructed using
the double-copy with higher-derivative corrections, suggesting a fundamental
tension between electromagnetic duality and color-kinematics duality at
loop-level. Finally we comment on oxidation of duality symmetries in
supergravity and the parallels it has to the M2-brane to D3-brane oxidation
demonstrated in this paper.
