HL-LHC projections for signal and background yield measurements of the $H\to\gamma\gamma$ when the Higgs boson is produced in association with $t$ quarks, $W$ or $Z$ bosons

In this note the expected yields and the signal significances of standard model Higgs bosons produced in association with top quarks or $Z$ or $W$ bosons are estimated. The signature of the two photons consistent with the Higgs mass is used to discriminate the signal from the background. A data set corresponding to 3000 fb$^{-1}$ recorded by an upgraded ATLAS detector running at $\sqrt{s}$=14 TeV with the high-luminosity LHC is assumed. Evidence for the $WH$ and $ZH$ is expected as well as an observation with a significance of 8.3 $\sigma$ for the $t\overline{t} H$ production mode. A signal strength precisions of the order of 17% to 35% depending on the production mode is expected.