The invariant yields of $\phi$ and $\omega+\rho$ mesons as a function of $\langle N_{\rm part}\rangle$ in Au+Au collisions. The systematic uncertainties of type-A (uncorrelated) are combined with statistical uncertainties in quadrature and are labeled as stat. Type-B (correlated) systematic uncertainties are listed as sys.

$R_{AA}$ of $\phi$ meson as a function of $p_T$ for in Au+Au collisions at $\sqrt{s_{_{NN}}}=200$~GeV, compared with Cu+Au collisions at $1.2<|y|<2.2$. The systematic uncertainties of type-A (uncorrelated) are combined with statistical uncertainties in quadrature and are labeled as stat. Type-B (correlated) systematic uncertainties are listed as sys.

$R_{AA}$ of $\phi$ meson as a function of $\langle N_{\rm part}\rangle$ for $1.2<|y|<2.2$ in Au+Au collisions at $\sqrt{s_{_{NN}}}=200$~GeV. The systematic uncertainties of type-A (uncorrelated) are combined with statistical uncertainties in quadrature and are labeled as stat. Type-B (correlated) systematic uncertainties are listed as sys.

$\xi$ distributions for different jet $p_T$ bins.

$\xi$ distributions for different jet $p_T$ bins.

The $j_T/p_{T}^{\rm jet}$ distributions for different jet $p_T$ bins.

The $j_T/p_{T}^{\rm jet}$ distributions for different jet $p_T$ bins.

$r$ distributions for different jet $p_T$ bins.

$M_{\tau}$ distribution in signal region, (OS$\mu$, Belle II)

$M_{\tau}$ distribution in signal region, (SS$e$, Belle)

$M_{\tau}$ distribution in signal region, (SS$e$, Belle II)

$M_{\tau}$ distribution in signal region, (SS$\mu$, Belle)

$M_{\tau}$ distribution in signal region, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$e$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$e$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$e$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$e$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$e$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$e$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$\mu$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$\mu$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$e$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$e$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$\mu$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$\mu$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$e$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$e$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$\mu$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$\mu$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$e$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$e$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$\mu$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$\mu$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (OS$e$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (OS$e$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (OS$\mu$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (OS$\mu$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (SS$e$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (SS$e$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (SS$\mu$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (SS$\mu$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$e$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$e$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$e$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$e$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle II)

Signal-selection efficiency as a function of the di-tau invariant mass squared $q^2$ for simulated events in the SR. The error bars indicate the statistical uncertainty.

$p_{T}$ dependence of $v_{2}$ for $\pi^{+}$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $\pi^{+}$ in Au+Au collisions at 4.5 GeV

$p_{T}$ dependence of $v_{2}$ for $\pi^{-}$ in Au+Au collisions at 3 GeV

$p_{T}$ dependence of $v_{2}$ for $\pi^{-}$ in Au+Au collisions at 3.2 GeV

$p_{T}$ dependence of $v_{2}$ for $\pi^{-}$ in Au+Au collisions at 3.5 GeV

$p_{T}$ dependence of $v_{2}$ for $\pi^{-}$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $\pi^{-}$ in Au+Au collisions at 4.5 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{+}$ in Au+Au collisions at 3 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{+}$ in Au+Au collisions at 3.2 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{+}$ in Au+Au collisions at 3.5 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{+}$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{+}$ in Au+Au collisions at 4.5 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{-}$ in Au+Au collisions at 3 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{-}$ in Au+Au collisions at 3.2 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{-}$ in Au+Au collisions at 3.5 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{-}$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{-}$ in Au+Au collisions at 4.5 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{0}_{S}$ in Au+Au collisions at 3 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{0}_{S}$ in Au+Au collisions at 3.2 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{0}_{S}$ in Au+Au collisions at 3.5 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{0}_{S}$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $K^{0}_{S}$ in Au+Au collisions at 4.5 GeV

$p_{T}$ dependence of $v_{2}$ for $p$ in Au+Au collisions at 3 GeV

$p_{T}$ dependence of $v_{2}$ for $p$ in Au+Au collisions at 3.2 GeV

$p_{T}$ dependence of $v_{2}$ for $p$ in Au+Au collisions at 3.5 GeV

$p_{T}$ dependence of $v_{2}$ for $p$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $p$ in Au+Au collisions at 4.5 GeV

$p_{T}$ dependence of $v_{2}$ for $\Lambda$ in Au+Au collisions at 3 GeV

$p_{T}$ dependence of $v_{2}$ for $\Lambda$ in Au+Au collisions at 3.2 GeV

$p_{T}$ dependence of $v_{2}$ for $\Lambda$ in Au+Au collisions at 3.5 GeV

$p_{T}$ dependence of $v_{2}$ for $\Lambda$ in Au+Au collisions at 3.9 GeV

$p_{T}$ dependence of $v_{2}$ for $\Lambda$ in Au+Au collisions at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{+}$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{+}$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{+}$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{+}$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{+}$ at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{-}$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{-}$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{-}$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{-}$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\pi^{-}$ at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{+}$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{+}$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{+}$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{+}$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{+}$ at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{-}$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{-}$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{-}$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{-}$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{-}$ at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{0}_{S}$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{0}_{S}$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{0}_{S}$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{0}_{S}$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $K^{0}_{S}$ at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $p$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $p$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $p$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $p$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $p$ at 4.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\Lambda$ at 3 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\Lambda$ at 3.2 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\Lambda$ at 3.5 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\Lambda$ at 3.9 GeV

The number of constituent quarks $n_q$ scaled $v_2$ as a function of $n_q$ scaled transverse kinetic energy for $\Lambda$ at 4.5 GeV

The energy dependence of $p_T$ integrated $v_2$ for $\pi^{\pm}$, $K^{\pm}$, $K^{0}_{S}$, $p$, and $\Lambda$ in 10-40\% centrality from Au+Au collisions at $\sqrt{s_{\text{NN}}}$ = 3.0, 3.2, 3.5, 3.9, and 4.5 GeV.

The energy dependence of $n_{q}$ scaled $v_{2}$ ratios of $v_{2}^{q}(\pi^{+}) / v_{2}^{q}(K^{+})$ and $v_{2}^{q}(p) / v_{2}^{q}(K^{+})$ at ($m_T - m_0$)/$n_q$ = 0.4 GeV/$c^2$

UrQMD model calculations for net-proton cumulant ratios and proton factorial cumulant ratios in Au+Au collisions from $\sqrt{s_{NN}}$ = 7.7 - 200 GeV for 0-5$\%$ centrality. Same choices of centrality definition are used as used for experimental data.

Significance of deviations of central 0-5$\%$ data from various choices of baselines or references.

Cumulant ratios C2/C1 and C4/C2 of net-proton distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 19.6 GeV as a function of multiplicity (RefMult3X).

Cumulant ratios C2/C1 and C4/C2 of net-proton distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 19.6 GeV as a function of average multiplicity (RefMult3X) for 8 centrality bins from 0-10$\%$ to 70-80$\%$. Data with and without CBWC are given.

Cumulant ratios C2/C1 and C4/C2 of net-proton distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 19.6 GeV as a function of average multiplicity (RefMult3X) for 16 centrality bins from 0-5$\%$ to 75-80$\%$. Data with and without CBWC are given.

Reference multiplicity distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 11.5 GeV from BES-II. Three choices of multiplicity distributions (RefMult3 scaled by efficiency factor of $\epsilon$ = 0.83 or 83$\%$ to mimic RefMult3 of BES-I, RefMult3, and RefMult3X) are presented.

Cumulant ratios C2/C1 and C4/C2 of net-proton distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 11.5 GeV from BES-II as a function of centrality obtained with various centrality definitions.

Reference multiplicity distributions (RefMult3, RefMult3X, and RefMult3A) in Au+Au collisions at $\sqrt{s_{NN}}$ = 19.6 GeV from UrQMD model. The RefMult3 and RefMult3X distributions have been scaled by an efficiency factor of $\epsilon$ = 77.9$\%$ to mimic the distribution in experimenal data.

Cumulant ratios C2/C1 and C4/C2 of net-proton distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 19.6 GeV from UrQMD model as a function of centrality obtained with various centrality definitions (RefMult3, RefMult3X, and RefMult3A). The RefMult3 and RefMult3X distributions have been scaled by an efficiency factor of $\epsilon$ = 77.9$\%$ to mimic the distribution in experimenal data.

Collision energy dependence of net-proton C4/C2 in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7 - 27 GeV from BES-II obtained with RefMult3 centrality definition.

Difference between net-proton C4/C2 (0-5$\%$) data from various choices of baselines or references as a function of collision energy from $\sqrt{s_{NN}}$ = 7.7 - 200 GeV.

The distributions of the variables used in the simultaneous fit for the SR. The black points with error bars represent the data and their statistical uncertainties, whereas the shaded band represents the predicted uncertainties. The bottom panel in each figure shows the ratio of the number of events observed in data to that of the total SM prediction. The last bin of each plot has been extended to include the overflow contribution.

Expected and observed 95% upper limits on the product of the cross section and branching fraction$\sigma(\mathrm{pp} \ ightarrow W\mathrm{a})\mathcal{B}(W ightarrow \ell^{+} v_{\ell})\mathcal{B}(\mathrm{a} \rightarrow Z\gamma)\mathcal{B}(Z \rightarrow \ell^{+} \ell^{-})$ s a function of the ALP mass from 110 to 400 GeV

Expected and observed 95% upper limits on the photophobic ALP model parameter $1/f_{\mathrm{a}}$ as a function of ALP mass reinterpreted from $1/f_{\mathrm{a}}=2~\mathrm{TeV}^{-1}$

Pre-fit MC statistical uncertainties per bin. These values are used to quantify the MC statistical uncertainty contributions to the total uncertainty in $m_t$ based on elements from the covariance matrix obtained in the profile likelihood fit. Each MC statistical contribution is estimated by dividing the relevant covariance matrix entry (see Table 2) by the corresponding pre-fit MC statistical uncertainty for that bin.

The contribution of each source of systematic uncertainty to the total uncertainty in $m_t$ is provided. Each source's contribution is taken directly from the relevant element of the covariance matrix for the profile likelihood fit to $\overline{m_J}$, $m_{jj}$, and $m_{tj}$ using data. The sign of the effect of each uncertainty is included.

The fitted $m_t$ is provided for each replica generated using bootstrapping. Each replica is a 3D histogram of $m_{J}$, $m_{jj}$, and $m_{tj}$, where $m_{J}$ has 50 bins between 145.0 GeV and 205.0 GeV. This method uses the BootstrapGenerator class in ROOT to initialise a pseudo-random number generator for each event, which is seeded by the value of the eventNumber and runNumber variables for a given event in the input dataset. The pseudo-random numbers are drawn from a Poisson distribution with rate parameter equal to 1. These are used to fluctuate the amount by which each replica histogram is filled. There are 1000 replicas, numbered from 0 to 999.

Corrected distributions of the normalized EEC within jets, differential in $ \left\langle p_{\rm T,jet} \right\rangle R_{L} $ at $R_{\rm jet} =$ 0.6 for one $p_{\rm T, jet}$ selection. Each distribution is normalized to integrate to one in $R_{L}$ prior to shifting.

Corrected distributions of the charge-selected EEC compared with the inclusive case for $R_{\rm jet}$ = 0.6 with a jet transverse momentum selection of 20 $ < p_{\rm T, jet} <$ 30 GeV/c.

Corrected distributions of the EEC charged ratio for $R_{\rm jet}$ = 0.6 with a jet transverse momentum selection of 20 $ < p_{\rm T, jet} <$ 30 GeV/c.

Corrected distributions of the normalized EEC differential in $R_{L}$ for $R_{\rm jet}=$ 0.4, with jet transverse momentum selections 15 $< p_{\rm T, jet} <$ 20 GeV/c and 30 $< p_{\rm T, jet} <$ 50 GeV/c

Corrected distributions of the normalized EEC within jets, differential in $ \left\langle p_{\rm T,jet} \right\rangle R_{L} $ at $R_{\rm jet} =$ 0.4 for one $p_{\rm T, jet}$ selection. Each distribution is normalized to integrate to one in $R_{L}$ prior to shifting.

Corrected distributions of the normalized EEC within jets, differential in $ \left\langle p_{\rm T,jet} \right\rangle R_{L} $ at $R_{\rm jet} =$ 0.4 for one $p_{\rm T, jet}$ selection. Each distribution is normalized to integrate to one in $R_{L}$ prior to shifting.

Corrected distributions of the normalized EEC within jets, differential in $ \left\langle p_{\rm T,jet} \right\rangle R_{L} $ at $R_{\rm jet} =$ 0.4 for one $p_{\rm T, jet}$ selection. Each distribution is normalized to integrate to one in $R_{L}$ prior to shifting.

Corrected distributions of the charge-selected EEC compared with the inclusive case for $R_{\rm jet}$ = 0.4 with a jet transverse momentum selection of 20 $ < p_{\rm T, jet} <$ 30 GeV/c.

Corrected distributions of the EEC charged ratio for $R_{\rm jet}$ = 0.4 with a jet transverse momentum selection of 20 $ < p_{\rm T, jet} <$ 30 GeV/c.

Suppl. Fig. 7. The reconstructed shower angle for all single photon events. The individual signal and background event type categories added together form the unconstrained prediction.

Suppl. Fig. 7. The constrained covariance matrix for the reconstructed shower angle for all single-photon events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Suppl. Fig. 7. The unconstrained covariance matrix for reconstructed shower angle for all single-photon events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 3. The reconstructed number of protons, with a proton KE energy threshold of 35 MeV, for all single-photon events. The individual signal and background event type categories added together form the unconstrained prediction.

Fig. 4, Suppl. Fig. 8. The reconstructed shower energy for zero proton events binned from 0 to 600 MeV. The individual signal and background event type categories added together form the unconstrained prediction.

Fig. 4. The constrained covariance matrix for the reconstructed shower energy for zero proton events binned from 0 to 600 MeV. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 4. The unconstrained covariance matrix for reconstructed shower energy for zero proton events binned from 0 to 600 MeV. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 6. The reconstructed shower energy for zero proton events binned from 150 to 1250 MeV, with LEE prediction. The individual signal and background event type categories added together form the unconstrained prediction. The "LEE" category is the unfolded MiniBooNE LEE model prediction scaled under the neutrino-target nucleon interaction assumption.

Fig. 6. The constrained covariance matrix for the reconstructed shower energy for zero proton events binned from 150 to 1250 MeV. Both with and without the LEE model added. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 6. The unconstrained covariance matrix for reconstructed shower energy for zero proton events binned from 150 to 1250 MeV. Both with and without the LEE model added. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Suppl. Fig. 10. The reconstructed shower backwards projected distance for zero proton events. The individual signal and background event type categories added together form the unconstrained prediction.

Suppl. Fig. 9. The reconstructed shower angle for zero proton events. Includes LEE model prediction. The individual signal and background event type categories added together form the unconstrained prediction. The "LEE" category is the unfolded MiniBooNE LEE model prediction scaled under the neutrino-target nucleon interaction assumption.

Suppl. Fig. 9. The constrained covariance matrix for the reconstructed shower angle for zero proton events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Suppl. Fig. 9. The unconstrained covariance matrix for reconstructed shower angle for zero proton events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

1$\gamma$Xp shower energy and angle selection efficiencies for true 1$\gamma$Xp events. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. Energy and Angle values represent bin lower edges.

1$\gamma$0p shower energy and angle selection efficiencies for true single-photon events with no other true particles of any energy. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. Energy and Angle values represent bin lower edges.