Expected and observed 95% CL for the 1SFH mu Majorana model.

Expected and observed 95% CL for the 2QDH NH Dirac model.

Expected and observed 95% CL for the 2QDH NH Majorana model.

Expected and observed 95% CL for the 2QDH IH Dirac model.

Expected and observed 95% CL for the 2QDH IH Majorana model.

Cutflow for six simulated signal channels showing the weighted number of expected events based on the single-flavour mixing model in the Majorana limit. Each column uses the generated signal sample with the mass hypothesis $m_N = 10$ GeV and proper decay length $c\tau_N = 10$ mm.

Cutflow for the six channels in data showing the number of events passing each successive signal selection for Majorana HNLs.

The event selection efficiency for each mass-lifetime point in all six studied channels. Shown is the fraction of the produced MC simulation events that pass all signal region selections. An entry of 0 indicates no events were selected.

The dominant signal uncertainty is due to differences in reconstruction of displaced vertices and tracks between data and MC. This is evaluated by comparing $K^{0}_{S} \rightarrow \pi^+\pi^-$ event yields in the VR and in MC produced with Pythia8.186 in bins of $p_\mathrm{T}$ and $r_\mathrm{DV}$. The data/MC ratio is normalized to the bin nearest the IP where the tracking and vertexing reconstruction algorithms are expected to be most robust. The symmetrized difference from 1.0 is applied to each signal vertex as a per-event systematic variation.

Expected and observed yields in the different analysis regions (prefit) for the 1SFH e Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 1SFH e Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 1SFH e Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 1SFH e Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 1SFH u Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 1SFH u Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 1SFH u Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 1SFH u Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 2QDH (NH) Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 2QDH (NH) Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 2QDH (NH) Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 2QDH (NH) Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 2QDH (IH) Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 2QDH (IH) Dirac model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (prefit) for the 2QDH (IH) Majorana model (10 GeV, 10mm).

Expected and observed yields in the different analysis regions (postfit) for the 2QDH (IH) Majorana model (10 GeV, 10mm).

The total displaced vertexing efficiency as a function of $r_{DV}$ for the custom configuration used in this analysis. The definition of the secondary vertex efficiency can be found in defined in \cite{ATL-PHYS-PUB-2019-013}. The efficiency is shown for $\mu-\mu\mu$, $\mu-\mu e$ and $\mu-ee$ signals with $m_N=10$~GeV and $c\tau_N=10$~mm.

Differential excess $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $m_{\rm ee}$ for $p_{\rm T,ee} < 0.1$ GeV/$c$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$.

Differential $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $p_{\rm T,ee}$ for 0.4 $< m_{\rm ee} < 0.7$ GeV/$c^{\rm 2}$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$.

Differential $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $p_{\rm T,ee}$ for 0.7 $< m_{\rm ee} < 1.1$ GeV/$c^{\rm 2}$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$. The quoted upper limits correspond to a 90% confidence level.

Differential $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $p_{\rm T,ee}$ for 1.1 $< m_{\rm ee} < 2.7$ GeV/$c^{\rm 2}$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$. The quoted upper limits correspond to a 90% confidence level.

Differential excess $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $p^{2}_{\rm T,ee}$ for 0.4 $< m_{\rm ee} < 0.7$ GeV/$c^{\rm 2}$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$. The quoted upper limits correspond to a 90% confidence level.

Differential excess $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $p^{2}_{\rm T,ee}$ for 0.7 $< m_{\rm ee} < 1.1$ GeV/$c^{\rm 2}$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$. The quoted upper limits correspond to a 90% confidence level.

Differential excess $e^+e^-$ yield in 70--90\% Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV as a function of $p^{2}_{\rm T,ee}$ for 1.1 $< m_{\rm ee} < 2.7$ GeV/$c^{\rm 2}$. Electrons are measured within $|\eta_{\rm e}| < 0.8$ and $p_{\rm T,e} > 0.2$ GeV/$c$.

Fig. 6b: Number density $N_{\rm ch}$ (left) and $\Sigma p_{\rm T}$ (right) distributions as a function of $p_{\rm T}^{\rm trig}$ in Away and Toward regions after the subtraction of Number density $N_{\rm ch}$ and $\Sigma p_{\rm T}$ distributions in the transverse region for p-Pb collisions for $p_{\rm T} >$ 0.5 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Fig. 7a: $<p_{\rm T}>$ as a function of $p_{\rm T}^{\rm trig}$ in Toward (left) and Away (right) regions after the subtraction of transverse region for pp collisions for $p_{\rm T} >$ 0.5 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Fig. 7b: $<p_{\rm T}>$ as a function of $p_{\rm T}^{\rm trig}$ in Toward (left) and Away (right) regions after the subtraction of transverse region for p-Pb collisions for $p_{\rm T} >$ 0.5 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Fig. A1: Number density $N_{\rm ch}$ (left) and $\Sigma p_{\rm T}$ (right) distributions as a function of $p_{\rm T}^{\rm trig}$ in Transverse, Away, and Toward regions for $p_{\rm T} >$ 0.15 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Fig. A2: Number density $N_{\rm ch}$ (left) and $\Sigma p_{\rm T}$ (right) distributions as a function of $p_{\rm T}^{\rm trig}$ in Transverse, Away, and Toward regions for $p_{\rm T} >$ 1.0 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Fig. A3: Number density $N_{\rm ch}$ (left) and $\Sigma p_{\rm T}$ (right) distributions as a function of $p_{\rm T}^{\rm trig}$ in Transverse, Away, and Toward regions for $p_{\rm T} >$ 0.15 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Fig. A4: Number density $N_{\rm ch}$ (left) and $\Sigma p_{\rm T}$ (right) distributions as a function of $p_{\rm T}^{\rm trig}$ in Transverse, Away, and Toward regions for $p_{\rm T} >$ 1.0 GeV/$c$. The shaded areas and the error bars around the data points represent the systematic and statistical uncertainties, respectively.

Dependence of $\overline{p}$-$\overline{d}$ correlation on pseudorapidity acceptance of $\overline{p}$ and $\overline{d}$ selection in Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV. Results are for 20.0--30.0$\%$ collision centrality.

Dependence of $\overline{p}$-$\overline{d}$ correlation on pseudorapidity acceptance of $\overline{p}$ and $\overline{d}$ selection in Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV. Results are for 50.0--60.0$\%$ collision centrality.

Example of combinatorial background subtracted invariant mass distributions and the extracted yields as a function of $\cos \theta^*$ for $\phi$ and $K^{*0}$ mesons. \textbf{a)} example of $\phi \rightarrow K^+ + K^-$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{b)} example of $K^{*0} (\overline{K^{*0}}) \rightarrow K^{-} \pi^{+} (K^{+} \pi^{-})$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{c)} extracted yields of $\phi$ as a function of $\cos \theta^*$; \textbf{d)} extracted yields of $K^{*0}$ as a function of $\cos \theta^*$.

Example of combinatorial background subtracted invariant mass distributions and the extracted yields as a function of $\cos \theta^*$ for $\phi$ and $K^{*0}$ mesons. \textbf{a)} example of $\phi \rightarrow K^+ + K^-$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{b)} example of $K^{*0} (\overline{K^{*0}}) \rightarrow K^{-} \pi^{+} (K^{+} \pi^{-})$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{c)} extracted yields of $\phi$ as a function of $\cos \theta^*$; \textbf{d)} extracted yields of $K^{*0}$ as a function of $\cos \theta^*$.

Example of combinatorial background subtracted invariant mass distributions and the extracted yields as a function of $\cos \theta^*$ for $\phi$ and $K^{*0}$ mesons. \textbf{a)} example of $\phi \rightarrow K^+ + K^-$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{b)} example of $K^{*0} (\overline{K^{*0}}) \rightarrow K^{-} \pi^{+} (K^{+} \pi^{-})$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{c)} extracted yields of $\phi$ as a function of $\cos \theta^*$; \textbf{d)} extracted yields of $K^{*0}$ as a function of $\cos \theta^*$.

Efficiency corrected $\phi$-meson yields as a function of cos$\theta$* and corresponding fits with Eq.1 in the method section.

Efficiency and acceptance corrected $K^{*0}$-meson yields as a function of cos$\theta$* and corresponding fits with Eq.4 in the method section.

$\phi$-meson $\rho_{00}$ obtained from 1st- and 2nd-order event planes. The red stars (gray squares) show the $\phi$-meson $\rho_{00}$ as a function of beam energy, obtained with the 2nd-order (1st-order) EP.

$\phi$-meson $\rho_{00}$ with respect to different quantization axes. $\phi$-meson $\rho_{00}$ as a function of beam energy, for the out-of-plane direction (stars) and the in-plane direction (diamonds). Curves are fits based on theoretical calculations with a $\phi$-meson field. The corresponding $G_s^{(y)}$ values obtained from the fits are shown in the legend.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

Global spin alignment measurement of $\phi$ and $K^{*0}$ vector mesons in Au+Au collisions at 0-20\% centrality. The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for $K^{*0}$-meson, obtained with the 2nd-order EP.

Global spin alignment measurement of $\phi$ and $K^{*0}$ vector mesons in Au+Au collisions at 0-20\% centrality. The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for $K^{*0}$-meson, obtained with the 2nd-order EP.

Invariant yield of direct photons, every 20% centrality

Invariant yield of nonprompt photons, every 20% centrality

$T_{eff}$ vs charged particle multiplicity

Integrated direct photon yield of 1-5GeV vs charged particle multiplicity

Integrated nonprompt direct photon yield of different $p_T$ integration window vs charged particle multiplicity

Scaling factor $\alpha$ vs charged particle multiplicity

Direct photon spectra for minimum bias (0-86%) Au+Au collision at $\sqrt{s_{NN}} = 62.4$ GeV (a) and $39$ GeV (b). For $62.4$ GeV also centrality bins of 0-20% (c) and 20-40% (d) are shown. Data points are shown with statistical and systematic uncertainties, unless the central value is negative (arrows) or is consistent with zero within the statistical uncertainties (arrows with data point). In these cases upper limit with CL = 95$%$ are given.

Inverse slopes, $T_{eff}$ , obtained from fitting the combined data from central collisions shown in Fig. 11 is compared to the fit results of the individual data sets at $\sqrt{s_{NN}} = 62.4$ GeV, $200$ GeV, and $2760$ GeV. Also included is the value for $\sqrt{s_{NN}} = 39$ GeV obtained from fitting the min. bias data set in the lower $p_T$ range.

Integrated invariant direct-photon yields vs. charged particle multiplicity for $p_{T}$ integrated from (a) 0.4 GeV/c, (b) 1.0 GeV/c, (c) 1.5 GeV/c, and (d) 2.0 to 5.0 GeV/c for all available A+A datasets.

Integrated invariant direct-photon yields vs. charged particle multiplicity for $p_{T}$ integrated from (a) 0.4 GeV/c, (b) 1.0 GeV/c, (c) 1.5 GeV/c, and (d) 2.0 to 5.0 GeV/c for all available A+A datasets.

Integrated invariant direct-photon yields vs. charged particle multiplicity for $p_{T}$ integrated from (a) 0.4 GeV/c, (b) 1.0 GeV/c, (c) 1.5 GeV/c, and (d) 2.0 to 5.0 GeV/c for all available A+A datasets.

Integrated invariant direct-photon yields vs. charged particle multiplicity for $p_{T}$ integrated from (a) 0.4 GeV/c, (b) 1.0 GeV/c, (c) 1.5 GeV/c, and (d) 2.0 to 5.0 GeV/c for all available A+A datasets.

Integrated direct-photon yields from A+A collisions for $p_{T,min}$ of $5$ GeV/c (a) and $8$ GeV/c. The representation is the same as in Fig. 13. Also shown are the results from pQCD calculations scaled by $N_{coll}$

Integrated direct-photon yields from A+A collisions for $p_{T,min}$ of $5$ GeV/c (a) and $8$ GeV/c. The representation is the same as in Fig. 13. Also shown are the results from pQCD calculations scaled by $N_{coll}$

The $\alpha$ values extracted using fits to integrated direct photon yields. The dashed line gives the average $\alpha$ value for the 4 lower $p_{T,min}$ points. Also shown is a model calculation for $\alpha$ discussed in the text.

Azimuthal anisotropy $v_2\{BF\}$ as a function of transverse momentum $p_T$ in $d$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2\{BB\}$ as a function of transverse momentum $p_T$ in $^3$He+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2\{BF\}$ as a function of transverse momentum $p_T$ in $^3$He+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of transverse momentum $p_T$ in $p$+$p$ collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of centrality in $p$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of centrality in $d$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of centrality in $^3$He+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $p$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $d$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $^3$He+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy $v_2$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $p$+$p$ collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy ratio $R=v_2\{BF\}/v_2\{BB\}$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $p$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy ratio $R=v_2\{BF\}/v_2\{BB\}$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $d$+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy ratio $R=v_2\{BF\}/v_2\{BB\}$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $^3$He+Au collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Azimuthal anisotropy ratio $R=v_2\{BF\}/v_2\{BB\}$ as a function of charged particle multiplicity $dN_{ch}/d\eta$ in $p$+$p$ collisions at $\sqrt{s_{NN}} =$ 200 GeV.

Summary of the total efficiencies. Quoted uncertainties correspond to statistical uncertainties of the simulated samples used. For the $B_c^+\to J/\psi \pi^+$ channel, the efficiency $\epsilon_{B_c^+\to J/\psi \pi^+}$ entering the equations for $R_{D_s^{(*)+}/\pi^+}$ is shown, while the efficiency for the full dataset is not defined.