Search for Higgs boson pair production in the two bottom quarks plus two photons final state in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi><mml:mi>p</mml:mi></mml:math> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>13</mml:mn><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:mi>TeV</mml:mi></mml:math> with the ATLAS detector

Searches are performed for nonresonant and resonant di-Higgs boson production in the $b\overline{b}\ensuremath{\gamma}\ensuremath{\gamma}$ final state. The dataset used corresponds to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton--proton collisions at a center-of-mass energy of 13 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. No excess above the expected background is found and upper limits on the di-Higgs boson production cross sections are set. A 95% confidence-level upper limit of 4.2 times the cross section predicted by the Standard Model is set on $pp\ensuremath{\rightarrow}HH$ nonresonant production, where the expected limit is 5.7 times the Standard Model predicted value. The expected constraints are obtained for a background hypothesis excluding $pp\ensuremath{\rightarrow}HH$ production. The observed (expected) constraints on the Higgs boson trilinear coupling modifier ${\ensuremath{\kappa}}_{\ensuremath{\lambda}}$ are determined to be $[\ensuremath{-}1.5,6.7]$ $([\ensuremath{-}2.4,7.7])$ at 95% confidence level, where the expected constraints on ${\ensuremath{\kappa}}_{\ensuremath{\lambda}}$ are obtained excluding $pp\ensuremath{\rightarrow}HH$ production from the background hypothesis. For resonant production of a new hypothetical scalar particle $X$ ($X\ensuremath{\rightarrow}HH\ensuremath{\rightarrow}b\overline{b}\ensuremath{\gamma}\ensuremath{\gamma}$), limits on the cross section for $pp\ensuremath{\rightarrow}X\ensuremath{\rightarrow}HH$ are presented in the narrow-width approximation as a function of ${m}_{X}$ in the range $251\text{ }\text{ }\mathrm{GeV}\ensuremath{\le}{m}_{X}\ensuremath{\le}1000\text{ }\text{ }\mathrm{GeV}$. The observed (expected) limits on the cross section for $pp\ensuremath{\rightarrow}X\ensuremath{\rightarrow}HH$ range from 640 fb to 44 fb (391 fb to 46 fb) over the considered mass range.

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