


\subsection*{Neutrino emission from AT2019dsg.} With this strong evidence for three distinct emission zones derived purely from multi-wavelength observations, we consider whether this picture is consistent with AT2019dsg being the source of the neutrino IC191001A. In particular, neutrino production requires protons to be accelerated to sufficiently high energies, and to collide with a suitably abundant target. The detection of a single high-energy neutrino implies a mean expectation in the range $0.05 < N_{\nu, \textup{tot}} < 4.74$ at 90\% confidence, where $N_{\nu, \textup{tot}}$ is the cumulative neutrino expectation for all TDEs that ZTF has observed, while for an individual object the expectation will be significantly lower\cite{2019A&A...622L...9S}. AT2019dsg emits $f_{\rm bol} \sim 0.16$ of the population bolometric energy flux, and if we take this as a proxy for neutrino emission, we would expect $0.008 \lesssim N_{\nu} \lesssim 0.76$ for this source.


Radio observations confirm that particle acceleration is indeed occurring, and that this continues without decline through to the detection of the neutrino at $\sim$180 days post-discovery. Given that neutrinos typically take a fraction $\eta_{p\nu} \sim 0.05$ of the parent proton energy, our accelerator must be capable of accelerating protons to at least 4 PeV. We evaluate the Hillas criterion\cite{1984ARA&A..22..425H} that the proton Larmor radius be less than the system size, to determine whether this is possible. We use  our estimates for conditions in the synchrotron zone at the time of neutrino detection, with $B \sim$ 0.07 G and $R \sim 7 \times 10^{16}$ cm for the near-contemporaneous radio epoch. Taking this as a baseline, we find a maximum proton energy of $\sim$160 PeV, far in excess of our requirements. The Hillas criterion can also be satisfied within the engine that powers the radio-emitting outflow because the product $BR$ is not expected to decrease at smaller radii (e.g. $B \propto R^{-1}$ for a toroidal configuration). 

The target for neutrino production can be either photons (p$\gamma$ interactions) or protons (pp interactions). For a photon target, neutrino production occurs above an energy determined by the mass of the $\Delta$ resonance, $m_{\Delta}$. For a thermal spectrum, of temperature $T$, we then find $\epsilon_{\nu} \sim \eta_{p\nu}[(m_{\Delta}^{2} - m_{p}^{2})/ 4 \epsilon_\gamma ] \approx 0.3 \times \left(T/10^{5} \rm K \right)^{-1} \rm PeV$. For the UV photosphere of the TDE, we find $\epsilon_\nu \sim$ 0.8 PeV, while for the compact X-ray source, we find $\epsilon_\nu \sim$ 0.05 PeV. Both of these values are compatible with the observed neutrino, for which there is a typical uncertainty of one energy decade\cite{2018Sci...361.1378I}, so either photon field could serve as a target. For the UV photosphere, we find that the mean free path for the parent proton of a PeV neutrino ($\sim2 \times 10^{13}$ cm, see Supplementary Information; SI) is much smaller than the photosphere radius, so the UV photosphere is indeed optically thick. At smaller radii, the X-rays would overtake the UV photons as dominant scattering targets. 

In the multi-zone model, shown in Figure \ref{fig:diagram}, the thermal photons provide a guaranteed target for pion production. However hadrons could in principle also serve as a target, leading us to consider a single-zone scenario in which the protons are accelerated at the same location as the synchrotron-emitting electrons, with the neutrino spectrum following the same intrinsic energy power law as the protons and electrons. For pp neutrino production, high target densities of $n_{\rm p}\sim 1/(\sigma_{\rm pp} R)\sim 10^8$~cm$^{-3}$ would be required for efficient production of neutrinos, where $\sigma_{\rm pp}$ is the proton-proton cross section and $R \sim 10^{17}$~cm is the characteristic size of the radio region at the time of neutrino production. This high density could be provided by the unbound stellar debris, although this component moves with a typical maximum velocity\cite{2016ApJ...827..127K} of $0.05\,c$ , and therefore the majority of this debris would have to be swept up with the outflow. Alternatively, the density could be provided by pre-existing gas, although since this gas orbits in the sphere of influence of the black hole, it would be challenging to satisfy the upper limits on pre-disruption accretion. 

To obtain the expected neutrino flux from this source we have to estimate the energy carried by protons ($E_{\textup{p}}$) that are accelerated above the energy threshold needed to produce high-energy neutrinos. 
The outflow energy of $2 \times 10^{50}$~erg that we derived from the radio observations (Figure~\ref{fig:radio_spectrum}) represent a lower bound to the energy that is available for particle acceleration in a central engine.  Indeed, the total energy budget for a TDE is set by the mass of the disrupted star, with $E_{\textup{TDE}} \sim (1/2)\, 0.1 \, \mbox{$M_{\odot}$} \, c^{2} \sim 10^{53}$ erg for a solar-mass star. We will assume 1\% of this total energy budget is carried by relativistic protons, $E_{\textup{p}} \sim 10^{51}$ erg. The total energy in muon neutrinos would then be $E_{\nu, \rm tot} = (1/8) E_{\rm p} \sim 10^{50}$ erg for efficient optically-thick pion production, after accounting for the pion decay yield and subsequent neutrino flavour oscillations. Convolving this implied energy $E_{\nu, \rm tot}$ with the effective area, $A_{\rm eff}$, of IceCube's high-energy neutrino alert selection\cite{2019ICRC...36.1021B}, we estimate the expected number of neutrino alerts. Approximating the sharply-peaked p$\gamma$ neutrino spectrum as a monoenergetic flux anywhere between  0.2 PeV $\lesssim \epsilon_\nu \lesssim 1$ PeV, we find $N_\nu = (E_{\nu, \rm tot}$ /$\epsilon_\nu)(A_{\rm eff}$ / 4$\pi D_{\rm L}^{2})  \sim 0.03$. Thus any optically-thick p$\gamma$ scenario would be sufficient to produce the neutrino under these assumptions. 

In contrast to a peaked p$\gamma$ neutrino spectrum, for pp production the neutrinos would instead follow a power law. Many of these neutrinos would then fall below the threshold of IceCube's alert selection. The associated gamma rays would however fall within the sensitive range of gamma-ray telescopes, so this scenario could be securely identified through a joint neutrino-gamma ray signal. While no gamma-ray emission was measured using the \textit{Fermi}-LAT telescope for AT2019dsg (see Methods), gamma-ray Cherenkov telescopes may be sensitive to the expected gamma-ray signal, and the corresponding low-energy (TeV) neutrino emission could confirm a hadronic origin. Conversely, the high optical depth of the UV photosphere would absorb any gamma rays accompanying p$\gamma$ neutrino emission\cite{2016PhRvD..93h3005W}. Some contribution from gamma-dark sources is required to explain the large astrophysical neutrino flux\cite{2016PhRvL.116g1101M}.

Given the different neutrino spectrum expectations, a search for accompanying lower-energy neutrinos could be used to probe the conditions at the site of proton interaction. IceCube has already searched for correlations between a sample of TDEs and a neutrino dataset dominated by lower-energy events, and reported that thermal TDEs account for less than 39\% of the diffuse astrophysical flux under the assumption of standard candles following a power-law spectrum\cite{2019ICRC...36.1016S}. The detection of a single TDE-neutrino association with our program would imply that at least $3\%$ of the astrophysical neutrino alerts arise from the TDE population, fully compatible with these pre-existing limits (see SI). 

As TDE discovery rates have increased substantially since the previous IceCube analysis\cite{2020arXiv200101409V, 2019ICRC...36.1016S}, future searches will be able to study neutrino emission from TDEs with much greater sensitivity. A measurement of O($\sim$1-10) TeV neutrinos without accompanying gamma rays would indicate that neutrino production is occurring in the X-ray photosphere, rather than in the UV photosphere. Indeed, such a detection would confirm the presence of a hidden X-ray source in the first place, while our electromagnetic observations cannot. Conversely, a lack of complementary low-energy neutrinos and gamma rays implies that only UV photons serve as a target. Neutrinos can uniquely serve as probes of the inner region of TDEs, using this novel method of extragalactic neutrino tomography. Now that a persistent central engine has been revealed in coincidence with a high-energy neutrino, we can begin to shed light on the role of TDEs as astrophysical accelerators.






\newpage

{\bf References}\\[-5pt]


\section{Discovery and Classification History of AT2019dsg}
AT2019dsg was discovered by ZTF on 2019 April 9, initially named ZTF19aapreis, and reported on 2019 April 22 as a likely extragalactic transient\cite{2019TNSTR.615....1N}. AT2019dsg was subsequently classified as a TDE on 2019 May 13 by ePESSTO+\cite{2019ATel12752....1N}. Radio emission was tentatively reported on 2019 May 23 by AMI-LA\cite{2019ATel12798....1S}, and confirmed on 2019 July 26 by e-MERLIN\cite{2019ATel12960....1P}. In addition to observations as part of a systematic ZTF search for TDEs\cite{2020arXiv200101409V}, the association with IC191001A prompted additional follow-up. 

\section{Neutrino Production in AT2019dsg}
\label{sec:max_energy}
The Hillas criterion\cite{1984ARA&A..22..425H} for a system of magnetic field strength $B$ and physical radius $R$ can be expressed as\cite{1984ARA&A..22..425H}:

\begin{equation}
\frac{E_{\rm max}}{\rm PeV} \approx
1600 \times \frac{B}{\rm Gauss} \times \frac{R}{10^{16} \, \rm cm} \times
\beta Z
\end{equation}
where $Z$ is the particle charge, $\beta \sim 0.2$ is the outflow velocity in units of c and $E$ is the maximum charged-particle energy. In order for particle acceleration to occur, the timescale required for particle acceleration must be shorter than the associated particle cooling timescale. Previous work has found this condition can be satisfied in TDEs for relevant energies\cite{2017ApJ...838....3S, 2017PhRvD..95l3001L}, although a detailed calculation is beyond the scope of this work.

These accelerated protons then need sufficient target density. For a photon target, with p$\gamma$ pion production via the $\Delta$ resonance, we expect that neutrino production will occur above a threshold:

\begin{equation}
E_{\gamma}E_{p} \sim \Gamma ^{2} 0.16 \, \rm{GeV}^{2}
\end{equation} With this constraint, we can derive the necessary photon energies required for a target to produce IC191001A. Taking the reconstructed neutrino energy of $\sim$0.2 PeV directly, we find a threshold photon target of $E_{\gamma} > $40 eV. However, these reconstructed neutrino energies typically have upper bounds an order of magnitude or more above the central estimate\cite{2018Sci...361.1378I}, so the true neutrino energy could be substantially higher. For example, with a true neutrino energy of $\sim$1 PeV, we would instead require photons $E_{\gamma} > $8 eV for pion production.

During pion production roughly half of the energy will be lost through the neutrino-less $\pi^{0}$ channel\cite{2010ApJ...721..630H}, while for the charged pion channel energy is shared roughly equally among the decay products $\pi^{\pm} \rightarrow e^{\pm} + \overset{\brobor}{\nu_{e}} + \overset{-}{\nu_{\mu}} + \nu_{\mu}$\cite{Waxman:1998yy}. Thus $\sim$3/8 of the pion energy is transferred to neutrinos, with a 1:2:0 flavour composition at source. However, across the cosmological baseline travelled, neutrino oscillations will lead to a mixed 1:1:1 composition on Earth. The IceCube realtime event selection is dominated by muon neutrinos, a channel which will carry no more than $\sim$1/8 of the pionic energy. Thus we find:

\begin{equation}
E_{\nu} \approx f_{\pi} \frac{E_{p}}{8}
\end{equation} where $f_{\pi} \leq1$ is the conversion efficiency of proton energy to pion energy. We can derive the mean free path, $\lambda$, for a proton:

\begin{equation}
\lambda = \frac{1}{\sigma_{p\gamma} n_{\gamma}}
\end{equation} with cross section $\sigma_{p\gamma} \sim 5 \times 10^{-28}$ cm$^{2}$ and photon number density $n_{\gamma}$. For a blackbody of temperature $T_{BB} \sim 10^{4.6}$ K, the mean free path for the parent proton of a 1 PeV neutrino is $\lambda \sim 2 \times 10^{13}$ cm. Accounting for the fact that each proton interaction will lead to a typical energy reduction of 20\%, we then find:

\begin{equation}
f_{\pi}(x) = 1 - e^{\left( \frac{-0.2x}{\lambda} \right)}
\end{equation} for path $x$. Equating $x$ with the radius of the UV photosphere $x \approx 10^{14.6}$cm, we then find that each proton or neutron will typically undergo $\sim$ 10 interactions, which would represent a high efficiency $f_{\pi} \sim 0.9$. We caution that this estimate is only approximate, and that detailed numerical simulations are required to accurately calculate the pion production efficiency\cite{2010ApJ...721..630H}.

We then calculate the effective area for a single high-energy neutrino, under the assumption of a mono-energetic neutrino spectrum which approximates the expectation for p$\gamma$ production. The effective area for IceCube varies from 50-200 m$^{2}$ for a 0.2-10 PeV neutrino energy. Below 1 PeV, this corresponds to a roughly-constant threshold of $6 \times 10^{-4}$ erg cm$^{-2}$ for an expectation of one neutrino alert. Given the redshift of AT2019dsg, we find a required total energy in neutrinos $E_{\nu} \approx 4 \times 10^{51}$ erg to produce a single neutrino alert. 
We can thus express the expected number of detected neutrinos as:

\begin{equation}
N_\nu \approx  0.03 \times \frac{E_{\nu}}{10^{50} \rm erg}
\end{equation} 

This expectation would also be valid for any power-law distribution in the same energy range.

\section{Compatibility with existing constraints}
\label{sec:diffuse}

We can estimate the contribution of TDEs to the diffuse neutrino flux that would be required to produce an observation of one association with our ZTF follow-up program. As outlined in Table \ref{tab:nu_alerts}, a total of eight neutrino alerts were observed. For all but one of these, IceCube reported an estimate of the \emph{signalness}, i.e., the probability for each to be astrophysical. We note that this quantity is not an absolute value, but is rather derived under specific assumptions about the underlying neutrino source population. Nonetheless, if we take these estimates at face value, and assume that the additional event had the reported signalness mean of 0.5, we would expect a total of $\sim$4.3 astrophysical neutrinos in our sample. Taking the implied ZTF population expectation of $0.05 < N_{\nu, \textup{tot}} < 4.74$, we would then require that a fraction $0.01 < f < 1.00$ of the astrophysical neutrino flux was produced by ZTF-detected TDEs at 90\% confidence.

We can further consider the contribution of those TDEs that ZTF has not detected, to  estimate the cumulative contribution of all TDEs to the diffuse neutrino flux. The IceCube collaboration has already constrained the contribution of such TDEs to be less than 39\% of the total, under the assumption of an unbroken E$^{-2.5}$ power law and a negative source evolution \cite{2019ICRC...36.1016S, Sun:2015bda}. We follow the same convention, with any power-law contribution of a transient source population to the diffuse neutrino flux is given by:

\begin{equation}
\frac{dN(E, z_{max})}{dEdAdt} = \int_{0}^{z_{max}} \left[ \ (1+z)^{2 - \gamma} \times \frac{\rho(z)\phi_{0}}{4 \pi D_{L}^{2}} \times \left( \frac{E}{E_{0}}\right) ^{-\gamma}  \right] \frac{dV_{C}}{dz} dz
\label{eq:nu_flux_tot}
\end{equation}

where $\rho(z)$ is the source rate density, $\phi_{0}$ is the time-integrated particle flux normalisation at reference energy $E_{0}$. We can use this to calculate the cumulative distribution of neutrino flux as a function of redshift\cite{flarestack}. This CDF is illustrated in Figure \ref{fig:cdf} for an E$^{-2.5}$ power law, though the distribution only depends weakly on the assumed neutrino spectrum.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figure_s1_completeness.pdf}
    \caption{\textbf{Cumulative distribution function (CDF) for TDE neutrino emission as a function of redshift.} These values are derived under the assumption of an E$^{-2.5}$ power law and a negative source evolution\cite{Sun:2015bda}. Roughly half of all neutrinos will come from sources with $z<0.2$.
    }
    \label{fig:cdf}
\end{figure}

It is clear that, for this negative source evolution, the vast majority of TDE neutrinos are expected to arrive from sources in the local universe. This statement is independent of both the overall level of TDE neutrino production and the absolute TDE rate. ZTF has, thus far, reported the detection of TDEs up to a maximum redshift 0.212\cite{2020arXiv200101409V}. If we simply assume that ZTF can routinely detect TDEs up to a redshift of 0.15, fully 40\% of the total population flux should come from ZTF-detected sources. We would thus require that at least 2.8\% of the astrophysical neutrino alerts are produced by TDEs, which is fully compatible with the previous IceCube limit. TDE-neutrino associations can thus be detected even if the vast majority of the astrophysical neutrino flux is produced by other source classes. The large relative contribution of detectable TDEs to the population neutrino flux is in marked contrast to supernova-like populations which are dominated by distant sources\cite{ps_iceube_19}, so follow-up searches for TDEs are significantly more sensitive than for these other potential sources.

In any case, IceCube limits are derived under the assumptions of unbroken power laws which extend across a broad energy range (100 GeV - 10 PeV), where many additional neutrinos would be expected at lower energies.  However, for the case of neutrino spectra dominated by high-energy components (as expected for p$\gamma$ neutrino production), no such low-energy neutrinos would be expected, and these existing constraints would then be substantially weakened.

{\bf Supplementary References}\\[-5pt]
