




\section{Supplemental Material}

\begin{table*}[h]
\centering
\begin{tabular}{@{}lllrllll@{}}
    \toprule
    Model                & $m_\chi$ [\si{GeV/c^2}]                       & $\sigma$ [\si{{cm}^2}]                    & \multicolumn{2}{c}{S2 ROI [PE]} & Observed & Expected signal  & Expected bg. \\ \colrule
    SI NR &  4 & \SI{6.4e-43}{} & {[}165,       & 271{]}       & 16    & $24.6 \pm 4.0$ & $3.7 \pm 0.6$ \\
    SI NR &  6 & \SI{4.8e-44}{} & {[}165,       & 276{]}       & 16    & $23.3 \pm 3.2$ & $4.0 \pm 0.6$ \\
    SI NR & 17 & \SI{1.8e-45}{} & {[}201,       & 578{]}       & 10    & $12.4 \pm 1.4$ & $5.9 \pm 0.8$ \\
    SI NR & 18 & \SI{5.2e-46}{} & {[}306,       & 1190{]}       & 3    & $4.4 \pm 0.5$ & $3.7 \pm 0.6$ \\
    SI-LM, $m_\phi = \SI{1}{MeV/c^2}$ & 4 & \SI{2.9e-38}{} & {[}165,       & 271{]}       & 16    & $24.8 \pm 4.2$ & $3.7 \pm 0.6$ \\
    SD (neutron-only) & 4 & \SI{9.8e-38}{} & {[}165,       & 271{]}       & 16    & $24.6 \pm 4.0$ & $3.7 \pm 0.6$ \\
    DM-electron scattering & 1 & $\SI{2.5e-40}{}$ & {[}165,       & 271{]}       & 16    & $24.7 \pm 4.1$ & $3.7 \pm 0.6$ \\ \colrule
    ALP & \SI{0.5e-6}{} & $g_\mathrm{ae} = \SI{7.7e-14}{}$ & {[}653,       & 897{]}       & 0    & $2.7 \pm 0.3$ & $0.5 \pm 0.1$ \\
    Dark photon & \SI{0.5e-6}{} & $\kappa = \SI{2.2e-16}{}$ & {[}653,       & 897{]}       & 0    & $2.7 \pm 0.3$ & $0.5 \pm 0.1$ \\
    \botrule 
    \\
\end{tabular}
\caption{Results for specific DM models excluded at exactly 90\% confidence level. The columns show successively: DM model (abbreviated as in the text, LM denotes light mediator and ALP axion-like particle); DM mass $m_\chi$; interaction strength, specified by the cross section for elastic scattering of DM with a free nucleon (or an electron, for DM-electron scattering) at zero momentum transfer for most models, by the axio-electric coupling constant $g_\mathrm{ae}$ for ALPs, and the kinetic mixing parameter $\kappa$ for dark photons; the left and right bounds of the S2 ROI in photoelectrons (PE); the number of observed events in the ROI; the expected number of events from the excluded signal model; the expected number of background events from the modeled backgrounds. The final two columns also show aggregated systematic uncertainties.}
\label{tbl:intervals}
\end{table*}

\twocolumngrid

\clearpage
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\newpage
$\!$
\pagebreak
$\!$

\subsection{Details on the energy scale}

Figure \ref{fig:s2_resolution} shows the assumed response in the S2 region of this analysis to ER and NR signals at different energies and including all efficiencies. As stated in the text, for each dark matter model and mass, a specific S2 ROI is used, whose lower bound is never below 150 PE.

The average energy of events observed at a particular S2 depends on the underlying energy spectrum. For a flat energy spectrum, the majority of low-S2 events are underfluctuations, i.e.~events with an S2 below the median S2 response for their energy. Thus, the average energy of events from a flat spectrum observed at a given S2 is higher than a naive conversion based on the median S2 response of different energies would suggest. This is an instance of `Eddington bias', a systematic effect caused by statistical fluctuations. Similar effects are discussed in \cite{lux_dd}.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.49\textwidth]{s2_energy_resolution.pdf}
    \caption{S2 response after all efficiencies for ER (blue) and NR (red). Solid lines show the median S2 response to signals of a specific energy, considering S2s in [90, 3000] PE only. The shaded regions indicate the $\pm 1 \sigma$ spread of the response. No NR (ER) response is assumed below 0.7 (0.186) \si{keV}; the hypothetical S2 response to those regions is shown lightly shaded for illustration. Dotted lines show the mean energy of events observed at a particular S2, in a flat (in linear energy) spectrum with no $Q_y$ cutoff.}
    \label{fig:s2_resolution}
\end{figure}

\vfill\eject


\subsection{Details on the S2 width model}

The S2 waveform width is defined as 
\begin{equation}
w = \big( \Delta t_{50\%} + \Delta t_r /2 \big) /2 ,
\end{equation}
where $\Delta t_{50\%}$ is the time between the first and third quartile of the S2 waveform, and $\Delta t_r$ the time between the start and median of the S2 waveform. This definition showed a slightly better correlation with depth than using $\Delta t_{50\%}$ alone.
We estimate the distribution of S2 widths using XENON1T's waveform simulator \cite{ap1}. Figure \ref{fig:width_validation} compares the S2 widths from simulation with those observed in neutron generator calibration data, showing that the median width agrees to within $\sim\!\SI{50}{ns}$. S2 widths of \ce{^{220}Rn}\xspace calibration data are in similarly good agreement. S1-tagged cathode events are broader than the model (by $\sim\! \SI{120}{ns}$), as expected from the higher diffusion constant and lower drift velocity in the low-field region at and immediately above the cathode.

The decrease of the mean S2 width with S2 at a fixed depth, clearly visible in Figures~\ref{fig:monster} and \ref{fig:cathode} for cathode events, is not discernible in Figure~\ref{fig:width_validation} due to a compensatory selection effect. At low S2, events with S1s are more common deep in the detector, where the charge loss to impurities and the photon detection efficiency are largest, but the average S2 width is also larger.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.49\textwidth]{wvsim_validation.pdf}
    \caption{First quartile (blue), median (green) and third quartile (red) of S2 widths in simulation (bands) and neutron generator calibration data (dots), as a function of depth $z$ (left) and S2 size (right), integrated over the other dimension (S2 on the left, $z$ on the right). The lower panels show the same, for the median only, with the nominal model subtracted. This figure uses events with S1s (so their $z$ is known) that pass event selections besides those on $z$ and S2 width; the multiple-S2 and SE-before-S2 cuts were relaxed to obtain additional statistics. The width of the bands denotes the uncertainty due to limited calibration data statistics (100 events/bin).}
    \label{fig:width_validation}
\end{figure}

\clearpage

\subsection{Details on the cathode background estimate}
Here we provide more details on our estimate of the background from decays on the cathode wires. As explained in the main text, most cathode events have detectable S1s even at very low S2. These cathode events with S1s are trivial to select, as they have a reconstructed depth $z$ within $\sim\! \SI{1}{cm}$ of the cathode, which sits at a drift length of $\SI{-96.9}{cm}$, with a spread that results from imperfectly corrected effects of the drift field inhomogeneity. In contrast, these events are homogeneously distributed in $(x,y)$, and their rate shows no significant time-dependence during SR1. We also observe \ce{^{214}Po} and \ce{^{210}Po} alpha decays at the cathode, and thus hypothesize that $\beta$ decays from \ce{^{214}Pb} and especially \ce{^{210}Pb} (due to its low $Q = \SI{64}{keV}$) contribute to the observed low-energy cathode events.

Figure~\ref{fig:cathode} illustrates the procedure for estimating the background from cathode events without S1s, after the width cut and inside the \SI{4}{GeV/c^2} NR analysis' S2 ROI, shown in \textbf{black} in figure \ref{fig:cathode}. First, we count cathode events with $S1 < \SI{20}{PE}$ that would appear in the S2 ROI if not for the width and z cuts. This region is shown in \textcolor{red}{red} in Figure~\ref{fig:cathode}. Next, we multiply this by the efficiency $\epsilon_w = (10.5 \pm 1.9) \%$ of the width cut on cathode events in the ROI, determined on cathode events with larger S1s ($20-\SI{200}{PE}$, not shown). Finally, we multiply by the ratio of cathode events without S1s and those with S1s $< \SI{20}{PE}$ in a control region at high S2 and high S2 width -- the count in the \textcolor{green}{green} divided by that in the \textcolor{orange}{orange} boxes in Figure~\ref{fig:cathode}. The count in the \textcolor{green}{green} box is corrected for (an insignificant) contamination by ordinary ER events. This is estimated by first counting events with non-cathode S1s in the control region -- the \textcolor{blue}{blue} box in Figure~\ref{fig:cathode} -- then scaling by one minus the S1 detection efficiency ($\epsilon_\mathrm{S1} > 90\%$) on ordinary ER events in the control region (determined as in \cite{ap1}). The cathode background estimate in the S2 ROI is thus:
\begin{equation} 
\textcolor{red}{r} \cdot \epsilon_w \cdot [\textcolor{green}{g} - \textcolor{blue}{b} \cdot (1 - \epsilon_\mathrm{S1}) ] / \textcolor{orange}{o} ,
\end{equation}
where $\textcolor{red}{r}$, $\textcolor{green}{g}$, $\textcolor{blue}{b}$ and $\textcolor{orange}{o}$ are the event counts in the red, green, blue and orange boxes, respectively. 

This procedure assumes that the ratio of cathode events with and without S1s is the same in the ROI as in the control region. However, events in the ROI have a lower S2, thus on average a lower energy, and are hence more likely to lack an S1 than events in the control region. Our estimate is therefore only a lower bound, and using it in the inference produces conservative limits.

We find $\textcolor{red}{r} = 46$, $\textcolor{orange}{o} = 336$, $\textcolor{green}{g} = 123$, $\textcolor{blue}{b} = 10$, leading to a cathode background estimate of $(1.62 \pm 0.46)$ events. The uncertainty includes statistical uncertainties on all the counts and the uncertainties on $\epsilon_w$ and $\epsilon_\mathrm{S1}$. The cathode background uncertainty is included in the aggregate background uncertainty described in the main text. We verified the estimate is stable well within its uncertainty under variations of the control region's S2 and S2 width bounds or the \SI{20}{PE} S1 threshold.

\onecolumngrid

\begin{figure}[b]
    \centering
    \includegraphics[width=\textwidth]{cathode_bg_estimate_summary.pdf}
    \caption{Illustration of the cathode background estimate. The scatterplots show (A) cathode events with $S1 < \SI{20}{PE}$, (B) events without S1s, and (C) events with $S1 < \SI{200}{PE}$ not from the cathode. Dot colors denote the S1 size in PE according to the colorbar in panel C. The colored boxes are used as described in the text to estimate the background in the DM ROIs after the width cut; the black box shows this region for \SI{4}{GeV/c^2} spin-independent NR DM. All events shown are from the search data, without the S2 width cut, and with the radial cut restricted to $R^2 < \SI{700}{cm}$ to avoid including misreconstructed events from the TPC walls in the blue and green boxes.}
    \label{fig:cathode}
\end{figure}
\twocolumngrid

