  A search for forward proton scattering in association with light-by-light
scattering mediated by an axion-like particle is presented, using the ATLAS
Forward Proton spectrometer to detect scattered protons and the central ATLAS
detector to detect pairs of outgoing photons. Proton-proton collision data
recorded in 2017 at a centre-of-mass energy of $\sqrt{s} = 13$ TeV were
analysed, corresponding to an integrated luminosity of 14.6 fb$^{-1}$. A total
of 441 candidate signal events were selected. A search was made for a narrow
resonance in the diphoton mass distribution, corresponding to an axion-like
particle (ALP) with mass in the range 150-1600 GeV. No excess is observed above
a smooth background. Upper limits on the production cross section of a narrow
resonance are set as a function of the mass, and are interpreted as upper
limits on the ALP production coupling constant, assuming 100% decay branching
ratio into a photon pair. The inferred upper limit on the coupling constant is
in the range 0.04-0.09 TeV$^{-1}$ at 95%confidence level.
