Searches are performed for both promptlike and longlived dark photons, $A'$, produced in protonproton collisions at a centerofmass energy of 13 TeV. These searches look for $A' \to \mu^+\mu^$ decays using a data sample corresponding to an integrated luminosity of 5.5/fb collected with the LHCb detector. Neither search finds evidence for a signal, and 90 confidencelevel exclusion limits are placed on the $\gamma$$A'$ kineticmixing strength. The promptlike $A'$ search explores the mass region from near the dimuon threshold up to 70 GeV, and places the most stringent constraints to date on dark photons with $214 < m(A') \lesssim 740$ MeV and $10.6 < m(A') \lesssim 30$ GeV. The search for longlived $A' \to \mu^+\mu^$ decays places worldleading constraints on lowmass dark photons with lifetimes $\mathcal{O}(1)$ ps.
Promptlike mass spectrum, where the categorization of the data as prompt $\mu^+\mu^$, $\mu_Q\mu_Q$, and $hh+h\mu_Q$ is determined using the $ {\rm min}[\chi^2_{\mathrm{IP}}(\mu^{\pm})]$ fits described in the text (examples of these fits are provided in the Supplemental Material). The anti$k_{\rm T}$based isolation requirement is applied for $ m( A^{\prime} ) > 1.1\mathrm{ Ge V} $. 
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Regions of the $ [ m( A^{\prime} ) ,\varepsilon^2]$ parameter space excluded at 90% CL by the promptlike $ A^{\prime}$ search compared to the best published [35,38,83] and preliminary [104] limits. 
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Expected reconstructed and selected longlived $ A^{\prime} \to \mu^+\mu^$ yield. 
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Ratio of the observed upper limit on $ n_{\rm ob}^{ A^{\prime} }[ m( A^{\prime} ) ,\varepsilon^2]$ at 90% CL to the expected darkphoton yield, $ n_{\rm ex}^{ A^{\prime} }[ m( A^{\prime} ) ,\varepsilon^2]$ , where regions less than unity are excluded. The only constraints in this region are from (hashed) the previous LHCb search [83]. 
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Validation of the datadriven approach used to predict the longlived $ A^{\prime}$ properties from prompt $\gamma^* \to \mu^+\mu^$ candidates in data is shown using two simulated $ A^{\prime} \to \mu^+\mu^$ samples, each generated with the same production kinematics, for (black points) $\tau( A^{\prime} ) =2 ps $ reconstructed and satisfying the longlived selection criteria except the photonconversion criterion, and (red line) the prediction for $\tau( A^{\prime} ) =2 ps $ based only on the prompt $ A^{\prime} \to \mu^+\mu^$ sample. { N.b.}, there is no free normalization factor, the red yield is predicted entirely from the prompt sample. 
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Expected reconstructed and selected promptlike $ A^{\prime} \to \mu^+\mu^$ yield divided by $\varepsilon^2$, where the displayed uncertainties include the systematic contributions. The gray boxes cover the regions with large SM resonance contributions, where no search for dark photons is performed. The anti$k_{\rm T}$based isolation requirement is applied for $ m( A^{\prime} ) > 1.1\mathrm{ Ge V} $. 
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Example $ {\rm min}[\chi^2_{\mathrm{IP}}(\mu^{\pm})] ^{1/2}$ distributions with fit results overlaid for promptlike candidates near (left) $ m( A^{\prime} ) =0.5$, (middle) 5, and (right) 50 $\mathrm{ Ge V}$ . The square root of $ {\rm min}[\chi^2_{\mathrm{IP}}(\mu^{\pm})]$ is used in the fits to increase the bin occupancies at large $ {\rm min}[\chi^2_{\mathrm{IP}}(\mu^{\pm})]$ values. The background categories, as described in the Letter, are as follows: ($hh$) two prompt hadrons misidentified as muons; ($h\mu_Q$) a misidentified prompt hadron combined with a muon produced in the decay of a heavyflavor quark, $Q$, that is misidentified as prompt; and ($\mu_Q\mu_Q$) two muons produced in $Q$hadron decays that are both misidentified as prompt. 
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Signed local significances at each scan mass from (black points) all fits and (red) the expected distribution; if the bestfit signalyield estimator is negative, the signed significance is negative and { vice versa}. The error bars account for the correlation between nearestneighbor fit results, which often produces outliers in pairs due to the $\sigma[ m(\mu^+\mu^) ] /2$ step size. 
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Threedimensional distribution of $\chi^2_{\rm DF}$ versus $t$ versus $ m(\mu^+\mu^)$ , which is fit to determine the longlived signal yields. The data are consistent with being predominantly due to $b$hadron decays at small $t$, and due to $ K^0_{\mathrm{ \scriptscriptstyle S}}$ decays for large $t$ and $ m(\mu^+\mu^) \gtrsim 280\mathrm{ Me V} $. The remaining few candidates at large $t$ and small $ m(\mu^+\mu^)$ are likely photon conversions. 
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Longlived $ A^{\prime} \to \mu^+\mu^$ candidates (black points) showing $t$ versus $ m(\mu^+\mu^)$ in bins of $\chi^2_{\rm DF}$, compared to the pulls from binned fits performed without a signal contribution (color axis). Positive pulls denote an excess of data candidates. 
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Comparison of the results presented in this Letter to existing constraints from previous experiments in the fewloop $\varepsilon$ region (see Ref. [102] for details about previous experiments). 
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Comparison of the results presented in this Letter to existing constraints from previous experiments in the fewloop $\varepsilon$ region (see Ref. [102] for details about previous experiments), restricted to the mass region motivated by selfinteracting dark matter [4]. 
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Animated gif made out of all figures. 
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Created on 06 March 2021.