A search has been conducted for the $b\bar b$ decay of the Standard Model Higgs boson produced through vector boson fusion in association with a photon and two jets. The search in this $b\bar b \gamma jj$ signature benefits from a large reduction of QCD jet background relative to the inclusive $b\bar b j j$ signature and from the presence of a high-tranverse-momentum photon for triggering. Results are reported from the analysis of 12.6 fb$^{-1}$ of LHC proton-proton collision data at $\sqrt{s} = 13$ TeV collected with the ATLAS detector. The observed 95\% confidence level upper limit on the production cross section times branching ratio for a Higgs mass of 125 GeV is $4.0$ times the Standard Model expectation, and the expected upper limit is $6.0^{+2.3}_{-1.7}$. The measured signal strength is $\mu=-3.9^{+2.8}_{-2.7}$ times the Standard Model value. The analysis methods are also used to search for $Z+\gamma$ vector boson fusion production in the same $b\bar b \gamma j j$ signature. The observed upper limit on the $Z+\gamma$ cross section times branching ratio is $2.0$ times the Standard Model expectation, and the expected limit is $1.8^{+0.7}_{-0.5}$. The measured $Z+\gamma$ signal strength is $\mu=0.3 \pm 0.8$ times the Standard Model value.

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