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A search for flavour-changing neutral interactions involving the top quark,the Higgs boson and an up-type quark $q$ ($q = c, u$) is presented. Theproton-proton collision data set used, with an integrated luminosity of 139fb$^{-1}$, was collected at $\sqrt{s} = 13$\~TeV by the ATLAS experiment at theLarge Hadron Collider. Both the decay process $t \to qH$ and the productionprocess $pp \to tH$, with the Higgs boson decaying into two photons, areinvestigated. No significant excess is observed and upper limits are set on the$t\rightarrow cH$ and the $t\rightarrow uH$ branching ratios of $4.3\times10^{-4}$ and $3.8\times 10^{-4}$, respectively, at the 95% confidence level,while the expected limits in the absence of signal are $4.7\times 10^{-4}$ and$3.9\times 10^{-4}$. Combining this search with ATLAS searches in the $H \to\tau^+\tau^-$ and $H \to b\bar{b}$ final states yields observed (expected)upper limits on the $t\to cH$ branching ratio of $5.8\times 10^{-4}\ (3.0\times10^{-4})$ at the 95% confidence level. The corresponding observed (expected)upper limit on the $t\rightarrow uH$ branching ratio is $4.0 \times 10^{-4}\(2.4 \times 10^{-4})$

Search for flavor-changing neutral $tqH$ interactions with $H\rightarrow  \gamma\gamma$ in $pp$ collisions at $\sqrt{s}$ = 13 TeV using the ATLAS  detector