Prompt-like 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} $.

Regions of the $ [ m( A^{\prime} ) ,\varepsilon^2]$ parameter space excluded at 90% CL by the prompt-like $ A^{\prime}$ search compared to the best published [35,38,83] and preliminary [104] limits.

Expected reconstructed and selected long-lived $ A^{\prime} \to \mu^+\mu^-$ yield.

Ratio of the observed upper limit on $ n_{\rm ob}^{ A^{\prime} }[ m( A^{\prime} ) ,\varepsilon^2]$ at 90% CL to the expected dark-photon 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].

Validation of the data-driven approach used to predict the long-lived $ 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 long-lived selection criteria except the photon-conversion 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.

Expected reconstructed and selected prompt-like $ 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} $.

Example $ {\rm min}[\chi^2_{\mathrm{IP}}(\mu^{\pm})] ^{1/2}$ distributions with fit results overlaid for prompt-like 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 heavy-flavor 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.

Signed local significances at each scan mass from (black points) all fits and (red) the expected distribution; if the best-fit signal-yield estimator is negative, the signed significance is negative and { vice versa}. The error bars account for the correlation between nearest-neighbor fit results, which often produces outliers in pairs due to the $\sigma[ m(\mu^+\mu^-) ] /2$ step size.

Three-dimensional distribution of $\chi^2_{\rm DF}$ versus $t$ versus $ m(\mu^+\mu^-)$ , which is fit to determine the long-lived 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.

Long-lived $ 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.

Comparison of the results presented in this Letter to existing constraints from previous experiments in the few-loop $\varepsilon$ region (see Ref. [102] for details about previous experiments).

Comparison of the results presented in this Letter to existing constraints from previous experiments in the few-loop $\varepsilon$ region (see Ref. [102] for details about previous experiments), restricted to the mass region motivated by self-interacting dark matter [4].

Animated gif made out of all figures.