The first observation of the $D^0 \to \pi^+\pi^\mu^+\mu^$ and $D^0 \to K^+K^\mu^+\mu^$ decays is reported using a sample of protonproton collisions collected by LHCb at a centerofmass energy of 8$\,$TeV, and corresponding to 2$\,$fb$^{1}$ of integrated luminosity. The corresponding branching fractions are measured using as normalization the decay $D^0 \to K^ \pi^+[\mu^+\mu^]_{\rho^0/\omega}$, where the two muons are consistent with coming from the decay of a $\rho^0$ or $\omega$ meson. The results are $\mathcal{B}(D^0 \to \pi^+\pi^\mu^+\mu^)=(9.64\pm0.48\pm0.51\pm0.97)\times10^{7}$ and $\mathcal{B}(D^0 \to K^+K^\mu^+\mu^)=( 1.54\pm0.27\pm0.09\pm0.16)\times10^{7}$, where the uncertainties are statistical, systematic, and due to the limited knowledge of the normalization branching fraction. The dependence of the branching fraction on the dimuon mass is also investigated.
Example diagrams describing the (left) short and (right) longdistance contributions to $ D ^0 \rightarrow h^+h^\mu^+\mu^$ decays, where $q=d,s$ and $h=\pi,K$. 
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Distributions of $ m( D ^0 )$ for the $ D ^0 \rightarrow \pi ^+ \pi ^ \mu^+\mu^$ candidates in the low $ m(\mu ^+ \mu ^ )$ , $\eta$, $\rho ^0 /\omega $, $\phi$ and high $ m(\mu ^+ \mu ^ )$ regions. Fit projections are overlaid. 
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Distributions of $ m( D ^0 )$ for the $ D ^0 \rightarrow K ^+ K ^ \mu^+\mu^$ candidates in the low $ m(\mu ^+ \mu ^ )$ , $\eta$ and $\rho ^0 /\omega $ regions. Fit projections are overlaid. No fit is performed in the $\eta$ region, where only two candidates are observed. 
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Animated gif made out of all figures. 
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Yields of (top) $ D ^0 \rightarrow \pi ^+ \pi ^ \mu^+\mu^$ and (bottom) $ D ^0 \rightarrow K ^+ K ^ \mu^+\mu^$ signal decays, their significance with respect to the backgroundonly hypothesis, and ratio of efficiencies between signal and normalization decays ($R_\epsilon^i$) for each dimuonmass region. The yield and the significance ($\mathcal{S}$) are not reported for the $\eta$ region of $ D ^0 \rightarrow K ^+ K ^ \mu^+\mu^$ , where only two candidates are observed. 
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Branching fractions of (top) $ D ^0 \rightarrow \pi ^+ \pi ^ \mu^+\mu^$ and (bottom) $ D ^0 \rightarrow K ^+ K ^ \mu^+\mu^$ decays in different ranges of dimuon mass, where the uncertainties are statistical, systematic and due to the limited knowledge of the normalization branching fraction. The reported upper limits correspond to 90% (95%) CL. The correlations between the various dimuonmass ranges are reported in the supplemental material [23]. 
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Correlation coefficients between the $ D ^0 \rightarrow \pi ^+ \pi ^ \mu^+\mu^$ branching fractions in the dimuonmass ranges. 
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Correlation coefficients between the $ D ^0 \rightarrow K ^+ K ^ \mu^+\mu^$ branching fractions in the dimuonmass ranges. 
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This ZIP file contains supplemetary material for the publication LHCbPAPER2017019. The files are: SupplementaryLHCbPAPER2017019.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in variuous formats 
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Created on 20 April 2019.Citation count from INSPIRE on 25 April 2019.