text:
\section{Introduction}

Dark matter-electron scattering is a promising search strategy for sub-GeV dark matter (DM)~\cite{Essig:2011nj,Graham:2012su,Essig:2015cda,Lee:2015qva,Hochberg:2015pha,Hochberg:2015fth,Alexander:2016aln,Derenzo:2016fse,Hochberg:2016ntt,Kavanagh:2016pyr,Emken:2017erx,Emken:2017qmp,Battaglieri:2017aum,Essig:2017kqs,Cavoto:2017otc,Hochberg:2017wce,Essig:2018tss,Emken:2018run,Ema:2018bih,Geilhufe:2018gry,Baxter:2019pnz,Essig:2019xkx,Emken:2019tni,Hochberg:2019cyy,Trickle:2019nya,Griffin:2019mvc,Coskuner:2019odd,Geilhufe:2019ndy,Catena:2019gfa,Blanco:2019lrf,Kurinsky:2019pgb,Kurinsky:2020dpb,Griffin:2020lgd,Radick:2020qip,Gelmini:2020xir,Trickle:2020oki,Du:2020ldo,Hochberg:2021pkt,Knapen:2021run,Essig:2012yx,Tiffenberg:2017aac,Romani:2017iwi,Crisler:2018gci,Agnese:2018col,Agnes:2018oej,Settimo:2018qcm,Akerib:2018hck,Abramoff:2019dfb,Aguilar-Arevalo:2019wdi,Aprile:2019xxb,Barak:2020fql,Arnaud:2020svb,Amaral:2020ryn}. In molecular or solid-state systems, atoms are close enough that electronic orbitals overlap significantly, lowering the electronic excitation energies to the eV scale and thus allowing detection of DM particles with $\sim {\rm MeV}$-scale mass which carry eV-scale kinetic energy. Moreover, solid-state systems can exhibit anisotropic electronic wavefunctions (see for example \cite{Hochberg:2016ntt,Hochberg:2017wce,Coskuner:2019odd,Geilhufe:2018gry,Geilhufe:2019ndy,Griffin:2020lgd}), enabling directional detection schemes which leverage the characteristic signature of the daily modulation of the direction of the DM wind in the lab frame (first noted in the context of multiple scattering from terrestrial overburden in \cite{Collar:1992qc,Collar:1993ss,Hasenbalg:1997hs}, followed by the connection to directional detection in \cite{Avignone:2008cw}).

In this paper, we focus on, and advocate for, a particular class of detector materials for DM-electron scattering: aromatic organic crystals. These compounds have numerous advantages, both practical and theoretical, over existing detectors. Their molecular structures consist primarily of hexagonal carbon rings with alternating single and double bonds (see Fig.~\ref{fig:struct}), and the excited molecular electronic levels at $\mathcal{O}(5 \ {\rm eV})$ above the ground state can de-excite by emitting a scintillation photon with $\mathcal{O}(1)$ bulk quantum efficiency. As a subset of the authors showed in a previous paper \cite{Blanco:2019lrf}, simple organic liquids like benzene and its analogs have superior reach per unit target mass compared to single-electron threshold semiconductor detectors for DM heavier than about 10 MeV. Furthermore, as noted in \cite{Blanco:2019lrf} and as discussed at length in the present work, the planar structure of the molecules leads to a marked anisotropy in the electronic wavefunctions which is absent in silicon and germanium detectors and which allows for directional detection: even though the scintillation \emph{emission} is isotropic, the \emph{excitation rate} to the scintillation level depends strongly on the direction of the incoming DM.

Building on our previous work, in this paper we focus on larger organic molecules which are solids at room temperature with known bulk quantum efficiencies at cryogenic temperatures. Furthermore, we focus on molecules with reduced in-plane symmetry. This additional asymmetry means that the lowest-energy transitions are not suppressed, distinguishing them from simpler molecules like benzene. The crystal lattice effects in these organic crystals are small enough that the electronic structure closely resembles that of the isolated molecules. Single-crystal scintillators can have Avogadro's number of unit cells containing the same relative orientations of the molecules, allowing the anisotropic response to persist. From a practical standpoint, single-crystal samples of trans-stilbene (t-stilbene), which we focus on in this paper, can be manufactured at kilogram scale with order-1 scintillation efficiency, such that even a single scintillation photon produced from a DM scattering event has a high probability of being detected from a large-mass sample~\cite{Zaitseva2015}.

These practicalities suggest that running a t-stilbene experiment with a large exposure is feasible in the near future, so we consider possible interpretations of plausible near-future experimental data.
As most sources of background noise (such as radioactive impurities in the target material) are constant in time,
daily modulation provides a way to detect dark matter even without reducing the background rate to zero.
Cosmic rays, the primary time-varying external background, have a daily modulation that has been constrained to be below the level of 0.1\% in underground facilities~\cite{COSINE:2020jml}, so a modulating signal with a significantly larger amplitude would provide a clear detection of dark matter.

For example,
assuming an average observed rate of $1/60 {\rm \, Hz \, kg^{-1}}$, and without incorporating expectations for daily modulation, the assumed (constant) background rate would limit the reach to approximately an order of magnitude above the interesting parameter space, with no prospects for improvement over time or with a larger experiment. However, a 1$\,$kg$\cdot$yr exposure reaches DM relic density targets if the data are interpreted with the expected modulation information. The sensitivity of a modulation analysis continues to scale with $({\rm exposure})^{1/2}$ even without mitigating backgrounds, so a larger 100$\,$kg$\cdot$yr exposure improves the reach by an order of magnitude. This probes the relic density target for DM masses $1.3 \lesssim m_\chi \lesssim 14\MeV$ if the DM interacts through a heavy mediator, or the range $1.5 \lesssim m_\chi \lesssim 1000\MeV$ if the DM achieves its relic abundance through freeze-in via a light kinetically-mixed dark photon. Solid-state organic scintillator detectors would therefore greatly reduce the necessity for a low-threshold zero-background experiment in order to conclusively discover or exclude DM.

As a consequence of our analysis of organic crystals, we point out a generic feature of DM-electron scattering in condensed matter systems: daily modulation is governed by the relationship between the DM velocity and an effective electron velocity $v^* \equiv \Delta E/q^*$, where $\Delta E$ is the energy of an electronic transition and $q^*$ is the typical momentum scale for the electron wavefunctions governing the transition. A necessary condition for daily modulation is anisotropy of the molecular form factor for electronic transitions, but for the anisotropy to be kinematically accessible, the transition needs to be either near a kinematic threshold (for small DM masses) or have $v^*$ close to, but smaller than, the maximum DM velocity in the lab frame $v_{\rm max}$ (for large DM masses). Intriguingly, this suggests that daily modulation, like annual modulation, is driven by the high-velocity tail of the DM velocity distribution. The centrality of these kinematic relations has recently been noted by \cite{Hochberg:2021pkt}, which discusses $v^*$ in the context of maximizing the total rate, and by \cite{Coskuner:2021qxo}, which uses a related $v^*$ to study modulation in the context of single-phonon production.

This paper is organized as follows. In Sec.~\ref{sec:Model}, we review the quantum chemistry relevant for describing the molecular orbitals in t-stilbene. In Sec.~\ref{sec:crystal} we describe the crystal structure of t-stilbene and justify our use of the isolated-molecule orbitals based on experimental measurements of the absorption and emission spectra. In Sec.~\ref{sec:Rate} we set up our calculation of the DM scattering rate, including the relevant molecular form factors and the daily modulation of the velocity distribution. In Sec.~\ref{sec:Results} we determine the daily modulation signal from DM in the Standard Halo Model (SHM) and derive a convenient test statistic for daily modulation in the presence of a non-modulating background. In Sec.~\ref{sec:Modulation} we study the kinematic features of t-stilbene which lead to a large modulation amplitude, and explore how a system with a different $v^*$ could lead to large daily modulation even for DM up to the GeV scale. We conclude in Sec.~\ref{sec:Conclusions}.

\section{Molecular Orbital Model}
\label{sec:Model}

\begin{table*}
\centering
\begin{minipage}{\textwidth}
\begin{tabular}{c | c c c c l l l l l}
$s$	& Platt Symbol &  Symmetry	& $ \Delta E \, [\!\eV]$ & Configuration amplitudes   \\ \hline
$s_1$ & $^1B$  & $B_u$ 		& 4.240 	& $d_{7,8} = 0.94$, & $d_{4,11} = -0.24$ && &&	\Tstrut\\
$s_2$ & $^1G^-$ & $B_u$ 	& 4.788 	& $d_{7,10} = 0.53$,& $d_{5,8} = 0.53$, &&$d_{6,11} = 0.37$, &&$d_{4,9} = -0.37$	 \\
$s_3$ & $^1G^-$ & $A_g$ 	& 4.800 	& $d_{7,9} = 0.53$, & $d_{6,8} = 0.53$, &&$d_{5,11} = 0.37$, &&$d_{4,10} = -0.37$	\\
$s_4$ & $^1(C,H)^+$ & $A_g$ 	& 5.137 	& $d_{7,11} = 0.41$, &$d_{5,9} = -0.41$, &&$d_{6,10} = -0.41$, && $d_{4,8}= -0.59$	\\
$s_5$ & $^1H^+$ & $B_u$ 	& 5.791 	& $d_{5,10} = 0.54$, & $d_{6,9} = 0.54$, &&$d_{7,12} = 0.33$, &&$d_{3,8} = 0.33$	\\
$s_6$ & $^1G^+$ & $A_g$ 	& 6.264 	& $d_{7,9} = 0.68$, & $d_{6,8} = -0.68$ && && 	\\
$s_7$ & $^1C^-$ & $A_g$ 	& 6.013 	& $d_{7,11} = 0.66$, & $d_{4,8} = 0.54$, && &&	\\
$s_8$ & $^1G^+$ & $B_u$ 	& 6.439 	& $d_{7,10} = 0.65$, & $d_{5,8} = -0.65$ && &&	\\
\end{tabular}
\caption{The first eight excited states $s_{n = 1 \ldots 8}$, with their energy eigenvalues $ \Delta E(s_n)$ with respect to the ground state and coefficients $d^{(n)}_{ij}$ as calculated by Ting and McClure~\cite{ting1971}. \label{table:ting}}
\end{minipage}
\end{table*}

\begin{figure}
    \centering
    \includegraphics[width=65mm]{tStilbeneLM.pdf}
    \includegraphics[width=65mm]{struct001.pdf}
    \caption{\textbf{Top:} The chemical  structure of trans-stilbene. Following the convention common in organic chemistry, vertices are taken to be carbon atoms, single lines are carbon--carbon single bonds, and double lines are carbon--carbon double bonds. Our numbering convention for the atoms is shown at each vertex, along with the $\hat{{L}}$ and $\hat{{M}}$ unit vectors used in our coordinate system.
    \textbf{Bottom:} A diagram of the unit cell of the trans-stilbene crystal, adopted from Ref.~\cite{Hoekstra1975}. Here $b$ is the axis of symmetry for the crystal: the positions of the molecules in the middle row are related to those of the upper or lower rows  by a translation of $\frac{1}{2} b$ and a rotation of $180^\circ$ about the $b$ axis. The long axis of each molecule ($\hat{{L}}_i$) is not perfectly perpendicular to the $b$ crystal axis. Molecules $(1)$ and $(-1)$ are shown with thin lines, $(2)$ and $(-2)$ with thick lines.
    \label{fig:struct} }
\end{figure}

Here we determine the many-electron wavefunctions for the electronic transitions which are responsible for the electronic-optical properties of trans-stilbene, following methods appropriate for all aromatic molecules. As shown in Fig.~\ref{fig:struct} (top), t-stilbene is an alternant hydrocarbon hosting two phenyl groups joined by an ethene bridge in the trans configuration (trans-1,2-diphenylethylene). This 14-carbon molecule is planar in the solid state and presents $C_{2h}$ molecular symmetry~\cite{Beveridge1965,Molina1997}, which is a $Z_{2}\times Z_{2}$ symmetry group composed of a two-fold symmetry axis, a center of inversion, a horizontal mirror plane, and the identity. Since this is the symmetry of the Hamiltonian, the electronic states of t-stilbene should transform as irreducible representations of $C_{2h}$. In order to construct electronic states which accurately describe the energy eigenstates of t-stilbene, we first construct the H$\text{\"u}$ckel molecular orbitals (HMOs) using a simple Linear Combination of Atomic Orbitals (LCAO) model taking into account only direct bonding interactions. We then take into account the configurational interactions and construct fully antisymmetric many-body states following the method of Pople, Pariser, and Parr (PPP)~\cite{pople1955electronic,Pariser1953,Pariser1953a}.

The HMOs, $\Psi_i$, of t-stilbene are constructed as linear combinations of Slater-type atomic orbitals (SAOs)
\begin{equation}
    \Psi_i = \sum_{j=1}^{14} c_{i}^{j} \phi_{2p_z}(\mathbf{r}-\mathbf{R}_j),
    \label{eq:LCAO}
\end{equation}
where $c_i^j$ are the coefficients to be determined, $\phi_{2p_z}$ are the atomic orbitals, and $\mathbf{R}_i$ are the equilibrium locations of the carbon nuclei using the numbering conventions in Fig.~\ref{fig:struct}. The $2p_z$ Slater atomic orbital is parameterized as
\begin{equation}
\label{eq:phidef}
    \phi_{2 p_{z}}(\mathbf{r})=\sqrt{\frac{Z^{5}_\text{eff}}{2^5   \pi a_{0}^{3}}}  \frac{ r \cos \theta}{a_{0}} \exp \left(-\frac{Z_{\text {eff }} r}{2 a_{0}}\right),
\end{equation}
where $a_0=(\alpha m_e)^{-1}$ is the Bohr radius and $Z_\text{eff} = 3.15$ is the effective nuclear charge of the carbon $2p_z$ orbital \cite{Pariser1953}. The HMOs diagonalize the $14 \times 14$ core Hamiltonian matrix, $H_{l m} = \langle \phi_l |\mathcal{H}_{core}| \phi_m \rangle$, where $\langle \vec{r} | \phi_m \rangle = \linebreak \phi_{2p_z} (\vec{r} - \vec{R}_m)$ and only bonding atoms interact. Diagonalization is done by solving the following system,
\begin{equation}
    \sum_{j=1}^{14}\left[\left(H_{l j}-E_m \delta_{l j}\right) c_{m}^{j}\right]=0, \text { for } m=1,2, \ldots, 14.
\end{equation}
The core Hamiltonian contains two types of matrix elements; diagonal on-site energies and off-diagonal interaction energies, with values given in Appendix~\ref{sec:OrbitalDetails}. The onsite energy is an empirical quantity which is determined by the atomic species, while the off-diagonal energy is a measure of the bonded nuclear interactions determined by the effective charge and bond length.
The ambiguity in the construction of degenerate states is resolved by requiring that all electronic states transform as irreducible representations of the $C_{2h}$ point group. Since the HMOs are constructed from 14 SAOs, diagonalization of the core Hamiltonian results in 14 HMOs $\Psi_1, \dots, \Psi_{14}$, numbered in order of increasing energy (up to degeneracies). See \cite{Blanco:2019lrf} for an example of this procedure performed on the simpler benzene molecule containing only six carbon atoms.

\begin{table*}
\centering
\begin{minipage}{\textwidth}
\begin{tabular}{c|rl c rl}
Molecule    & & $(L_x, L_y, L_z)$ &&& $(M_x, M_y, M_z)$ \\ \hline
    $(1)$ & $\hat{L}=$  & $ ( 0.153, 0.988, - 0.022 )$ & {\color{white}.}\hspace{1cm}{\color{white}.}& $\hat{M}=$ & $  (0.467, - 0.068, 0.882 )$ \Tstrut\\
    $(2)$ & $\hat{L}=$ & $ ( -0.809, 0.565, 0.162)$ && $\hat{M}=$ & $  (- 0.130, - 0.458, 0.879 )$ \Tstrut\\
    $(-1)$ & $\hat{L}=$ & $(  -0.153, - 0.988, - 0.022 ) $&& $\hat{M}=$ & $  (-0.467 , 0.068, 0.882 )$ \Tstrut\\
    $(-2)$ & $\hat{L}=$ & $ (0.809, - 0.565 , 0.162 )$ && $\hat{M}=$ & $  (0.130 , 0.458, 0.879 )$ \Tstrut\\
         \hline
\end{tabular}
    \caption{The vectors $\hat{L}$ and $\hat{M}$ describe the orientations of each of the four molecular constituents of the unit cell, shown here in the right-handed crystal basis $\hat{x}= \hat{c}$, $\hat{y}= \hat{a}'$, $\hat{z}= \hat{b}$.}
    \label{table:geometry}
\end{minipage}
\end{table*}

In order to form antisymmetric many-electron wave functions, we take Slater determinants of the filled HMOs. The ground state $| g \rangle$ is approximately given by the following combination of orbitals,
\begin{equation}
   \psi_G = |\Psi_{1} \overline{\Psi}_{1}\Psi_{2} \overline{\Psi}_{2}\Psi_{3} \overline{\Psi}_{3}\Psi_{4} \overline{\Psi}_{4}\Psi_{5} \overline{\Psi}_{5}\Psi_{6} \overline{\Psi}_{6}\Psi_{7} \overline{\Psi}_{7}|,
\end{equation}
where $|\cdots|$ denotes the antisymmetrized product of the HMOs and $\overline{\Psi}_i$ is the opposite spin state as $\Psi_i$. Following standard conventions in quantum chemistry, we label some $Z_2$ symmetries of the many-electron wavefunction by $A(B)$ and $g(u)$, corresponding to (anti)symmetry with respect to transformation under $180^\circ$ rotation about the $z$-axis normal to the molecular plane and inversion through the center of mass, respectively. Notice that the ground state represents an electronic configuration in which the lowest 7 HMO's are filled by pairs of electrons in the spin-singlet configuration. Therefore, the ground state transforms as the $A_{g}$ representation, being totally symmetric under the transformations in the $C_{2h}$ group. This is a generic feature of the ground state of alternant hydrocarbons. Reduction of the 14-dimensional t-stilbene representation of $C_{2h}$ predicts 7 $A_{g}$ and 7 $B_{u}$ states~\cite{Beveridge1965}.  These states correspond to the multi-electron configurations of the HMOs which each have either $A_{u}$ or $B_{g}$ symmetry~\cite{Molina1997}.

We construct the  multi-electron states starting with the $A_{g}$ ground state and proceeding upwards in energy through the one-electron singlet excitation configurations, $\psi_i^j$ as follows,
\begin{equation}
    \psi_i ^j = \frac{1}{\sqrt{2}} (|\Psi_{1} \overline{\Psi}_{1}...\Psi_{i} \overline{\Psi}_{j}...\Psi_{7} \overline{\Psi}_{7}| - |\Psi_{1} \overline{\Psi}_{1}...\Psi_{j} \overline{\Psi}_{i}...\Psi_{7} \overline{\Psi}_{7}|).
\end{equation}
The electronic repulsion is taken into account by appending the electronic two-body interaction to the core H$\text{\"u}$ckel Hamiltonian:
\begin{equation}
    \mathcal{H}_{\rm PPP} = \mathcal{H}_{core} + \sum_{\langle ij \rangle}{\frac{4\pi \alpha}{r_{ij}}},
\end{equation}
where $\alpha$ is the fine-structure constant, $r_{ij} = |\vec{r}_i - \vec{r}_j|$, and the sum runs over all pairs of electrons $\langle ij \rangle$ with $i \neq j$. This PPP Hamiltonian perturbs the energy levels of the $\psi_i^j$ configurations and mixes degenerate states of like symmetries to produce PPP HMOs of either $A_{g}$ or $B_{u}$ symmetry~\cite{Beveridge1965,Molina1997,ting1971}. The PPP energy eigenstates are expressed as a linear combination of $\psi_i^j$,
\begin{align}
| s_n \rangle = \sum_{i,j>i} d^{(n)}_{ij} | \psi_{i}^j \rangle ,
&&
\sum_{ij} | d^{(n)}_{ij} |^2 = 1.
\end{align}
The leading coefficients $d^{(n)}_{ij}$ (known as configuration amplitudes) for the first $n=1 \dots 8$ excited states of t-stilbene are tabulated in Tab.~\ref{table:ting}~\cite{ting1971}. To better match our analysis to experimental data, we take the experimentally-determined energy eigenvalues, which are also listed in Tab.~\ref{table:ting}~\cite{ting1971}.

The spin-singlet configurations we have focused on are responsible for the radiative de-excitations known as fluorescence that could be seen with single-photon detectors in a DM experiment. The triplet states are classically forbidden from decaying down to the ground state and hence are responsible for the delayed fluorescence component of photoluminescence which generically has a significantly lower quantum yield~\cite{1964a}.

\section{Crystal Structure}
\label{sec:crystal}
In the solid state, t-stilbene is a monoclinic crystal belonging to the space group $C5_{2h}(P2_{1}/c)$, with unit cell parameters $a=12.29$ \AA, $b=5.66$ \AA, $c=15.48$ \AA, and $\gamma=112^{\circ}$~\cite{Hoekstra1975}. In this convention, the crystal coordinate basis is defined by a unit vector $\hat{b} = \vec{b}/b$ that is orthogonal to both $\hat{a}$ and $\hat{c}$, and $\gamma$ is the opening angle between $\hat{a}$ and $\hat{c}$. The coordinate system is right-handed, such that $\vec{c}\times \vec{a} \parallel \vec{b}$. To form an orthonormal basis we define an $\hat{a}'$ unit vector, $\hat{a}' = \hat{b} \times \hat{c}$. The crystal is symmetric with respect to translations of $\vec{a}$ and $\vec{c}$, and to the twofold screw action composed of the translation $\vec{b}/2$ with $180^\circ$ rotation about $\hat{b}$.

Four distinct molecules of t-stilbene inhabit each unit cell of the crystal (see Fig.~\ref{fig:struct}, bottom): an $M_1$ and $M_2$ with different orientations, and an $M_{-1}$ and $M_{-2}$, which are the images of molecules $M_{1,2}$ (respectively) under the twofold screw action along $\vec{b}$~\cite{Robertson1937}. In Tab.~\ref{table:geometry} we provide the orientations of each of the four molecules in terms of their unit vectors $\hat{L}$ and $\hat{M}$ identified in Fig.~\ref{fig:struct}, in a crystal coordinate system where the $\hat{z}$ direction is assigned to the $\vec{b}$ symmetry axis. The position of each molecule within the crystal is listed in Refs.~\cite{Hoekstra1975,Robertson1937}, but the DM--stilbene scattering rate depends only on their rotational orientation because the kinematics of the scattering process do not permit coherent scattering over an entire unit cell (see Secs.~\ref{sec:Rate} and~\ref{sec:Results} below).

The molecular orbital model derived in the preceding section is empirically valid for macroscopic single-crystal samples of t-stilbene. Although lattice effects are known to perturb the energy bands of molecular crystals~\cite{davydov1953theory}, the Davydov splitting of the molecular bands, which is due to the dipole-dipole (and to a lesser extent quadrupole-quadrupole) interaction of neighboring molecules in a molecular crystal, is known to be very small in single-crystal t-stilbene~\cite{Chaudhuri1969}. Furthermore, the lowest two UV-absorption bands (labeled A and B in the literature) of t-stilbene in the liquid state remain relatively untouched when observed in the solid state~\cite{Suzuki_1960}. The A band is thought to arise from the $g \to s_1$ through $g \to s_4$ transitions while the B band is thought to arise from the $g \to s_5$ transition~\cite{Beveridge1965}. These bands are a direct measurement of transitions between the many-electron configurations described in the previous section which represent the non-interacting single-molecule electronic state, usually taken to be most like the molecular environment of the low-temperature liquid  or gas state. Since these bands remain the same in the solid states, we conclude that the PPP model accurately describes the molecules of t-stilbene in the solid state where lattice effects are only very weak.

A mole of t-stilbene has a mass $180.24$ g and occupies a volume of $185.69 \cm^{3}$. Thus, a kilogram of detector material can be fabricated from a cube of t-stilbene of 10.1 cm per side, or a 1 cm thin sheet of approximately one square foot. These sizes will be convenient to instrument with conventional photodetectors, such as photomultiplier tubes or charge-coupled devices (CCDs).

\section{Rate Calculation}
\label{sec:Rate}

The interaction of DM with the electronic system of a molecule produces a detectable signal through a process analogous to photoluminescence. This process can be separated into two stages: electronic excitation from DM-electron scattering followed by radiative deexcitation. Specifically, DM-electron scattering will produce detectable scintillation photons from t-stilbene at a rate given by \cite{Blanco:2019lrf}
\begin{widetext}
\begin{equation}
    R = \frac{\Phi_{FB} N_\text{A} m_T}{m_{\rm mol}^\text{(t-stil)}} \frac{\rho_\chi}{m_\chi} \frac{\bar\sigma_e}{\mu_{\chi e}^2} \sum_{i = 1} \int\! \frac{d^3 \vec{q}}{4\pi} \int\! d^3 \vec{u} \, f_\chi(\vec{u})  \delta\left( \Delta E(s_i) + \frac{q^2}{2 m_\chi} - \vec{q} \cdot \vec{u} \right)F_\text{DM}^2(q) |f_{g \rightarrow s_i}(\vec{q})|^2.
\label{eq:ratedef}
\end{equation}
\end{widetext}
Here, $N_{\rm A} \simeq 6.022 \times 10^{23}$ is Avogadro's number; $m_T$ is the total detector mass; the molar mass of t-stilbene is $m_{\rm mol}^\text{(t-stil)}= 180.25$ g; $\rho_\chi$ and $m_\chi$ are the DM mass density and mass, respectively; $\mu_{\chi e}$ is the DM--electron reduced mass; $\bar{\sigma}_e \equiv \frac{\mu_{\chi e}^2}{16 \pi m_\chi^2 m_e^2}\langle |\mathcal{M}(q_0)|^2 \rangle$ is a fiducial DM--electron cross section proportional to the free-particle spin-averaged squared matrix element $\langle |\mathcal{M}|^2 \rangle$ for $\chi-e$ scattering, evaluated at $q_0 = \alpha m_e$; $f_\chi(\vec{u})$ is the DM velocity distribution in the lab frame; $F_{\rm DM}(q)$ is a form factor parameterizing the fundamental DM-electron interactions, which has the limits $F_{\rm DM}(q) = 1$ for a contact interaction and $F_{\rm DM}(q) =(\alpha m_e/q)^2$ for a long-range interaction; and $\Delta E(s_i)$ is the excitation energy for each $s_i$ above the ground state as given in Tab.~\ref{table:ting}.

The detector-dependent quantities are the molecular form factor $|f_{g \rightarrow s_i}(\vec{q})|^2$, representing the transition amplitude from the ground state to a singlet excited state $s_i$, and the bulk fluorescence quantum efficiency $\Phi_{FB}$, representing the probability that a molecular excitation will produce a photon through radiative deexcitation that will exit the detector without being absorbed. As is the case with photoluminescence, the emission lines are broadened by vibrational energy sublevels, thermal motion, and lattice effects. The emission spectrum of the single crystal is a continuum of peaks which closely resembles the molecular and micro-crystalline emission spectra but is modified by self-absorption and lattice effects~\cite{Birks1954}. Here we focus on computing the form factor and the quantum efficiency, which determine the signal rate, leaving a detailed investigation of the emission spectrum (which determines the signal photon wavelength and hence the detection mechanism) for future work. We compute the molecular form factors using the first 8 singlet transitions since these are the transitions responsible for the first three lowest-lying absorption bands of trans-stilbene~\cite{Beveridge1965}. Since the probability of interaction is suppressed for higher energy thresholds (see Fig.~\ref{fig:rateByTransition}), it is expected that the rate will be driven primarily by the strongest low-lying excitations.  Using a simple, spherically-symmetric DM velocity distribution, the SHM, we also calculate daily modulation effects due to the rotation of the Earth.

\subsection{Molecular Form Factors}

\begin{figure*}
    \centering

    \includegraphics[width=0.98\textwidth]{draft-formfactors-th90.jpg}
    \caption{Molecular form factors for the $g\to s_1$ transition for momentum transfers $\vec{q} = (q_x, q_y, 0)$ in the plane of the molecule. \textbf{Left:} contour plot of $|f_{g \rightarrow s_1}|^2$, showing the hexagonal symmetry of the benzene ring with secondary inner peaks arising from the extended structure of the joined benzene rings. \textbf{Right:} the product of $|f_{g \rightarrow s_1}|^2$ with $F_\text{DM}^2$ for $F_\text{DM} \propto 1/q^2$, proportional to the rate integrand \eqref{eq:RTotg0} for a light mediator (logarithmic color scale). }
    \label{fig:formFactors}
\end{figure*}

The molecular form factor as calculated for PPP configurational states $s_n$ is given by the following,
\begin{align}
f_{g \to s_n}(\vec{q}) &=  \left\langle \psi_{s_n} (\vec{r}_1 \ldots \vec{r}_{14}) \left|  \sum_{m=1}^{14} e^{i \vec{q} \cdot \vec{r}_m } \right| \psi_G(\vec{r}_1 \ldots \vec{r}_{14}) \right\rangle \nonumber \\
&=\sum_{ij} d^{(n)}_{ij}
 \left\langle \psi_i^j \left|   e^{i \vec{q} \cdot \vec{r} } \right| \psi_G \right\rangle  \nonumber \\
&=\sqrt{2} \sum_{ij}  d^{(n)}_{ij} \langle \Psi_j(\vec{r}) |e^{i\vec{q} \cdot \vec{r}}|\Psi_i (\vec{r})\rangle.
 \label{eq:fgs}
\end{align}
where $\vec{r}_m$ is the position of electron $m$. In the second line we have isolated the contribution from the singlet states which only contain a single-electron excitation above the ground state, and in the third line we have transformed to the basis of HMOs, where the factor of $\sqrt{2}$ is effectively a spin degeneracy factor. The matrix element of HMOs may be readily computed from Eq.~(\ref{eq:LCAO}) in momentum space, where the wavefunctions are simply the momentum-space $2p_z$ orbitals times a product of phase factors determined by the positions of the carbon atoms \cite{Blanco:2019lrf}. An example of the single-molecule form factor for the $g \to s_1$ transition is shown in Fig.~\ref{fig:formFactors} (see also App.~\ref{sec:OrbitalDetails}), showing strong damping of the form factor beyond a characteristic momentum scale given by $q^* \simeq Z_{\rm eff}/(2 a_0) \simeq 6 \ {\rm keV}$. There are also secondary ``inner'' peaks at $q \simeq 2\pi/\ell \simeq 1.2 \ {\rm keV}$ where $\ell \approx 0.83 \; \text{nm}$ is the length scale of the long axis of t-stilbene.

Measurements of the spectrum of trans-stilbene observe three absorption bands, $A$, $B$ and $C$, which have been identified primarily with the $s_1$, $s_5$ and $s_8$ molecular transitions, respectively~\cite{Beveridge1965}.
Compared with the $B$ and $C$ bands, the $A$ band is larger in magnitude and broader in frequency space, overlapping with the $s_{2 \ldots 4}$ transitions. Our analysis for dark matter scattering reproduces these features: for example, the $s_1$ transition dominates the scattering rate, comprising 50--70$\%$ of the rate both near-threshold and at large $m_\chi$. If $m_\chi$ is large enough that the higher-energy excitations are kinematically accessible, the scattering rate receives secondary contributions from $s_3$ and $s_4$, and typically smaller contributions from $s_2$, $s_5$ and $s_8$. The $s_6$ and $s_7$ transition rates remain negligibly small at all values of $m_\chi$. We provide additional detail regarding the separate molecular transitions that contribute to the scattering rate in Appendix~\ref{sec:ffdetails}.

In a t-stilbene crystal, the unit cell contains four molecules of different orientations, as described in Section~\ref{sec:crystal}. As discussed in more detail below, the momentum transfers required to deposit energy above $\Delta E(s_1)$ are sufficient to localize the interaction to a single molecule within a unit cell, so to compute the rate we treat the scattering as incoherent between different molecules, and we sum over four different squared form factors $|f^{(i)}_{g \to s}(\vec{q})|^2$ rotated to give the appropriate orientations of each molecule with respect to $\vec{q}$. The interaction rate of DM with a crystal of t-stilbene then scales like the product of the rate calculated as described with the total number of unit cells $N_{\rm uc} = N_{\rm mol}/4$ in the entire crystal, which is proportional to the total crystal mass $m_T$.

\subsection{Quantum Efficiency}
 Given the molecular fluorescence quantum efficiency ($\Phi_F$), defined as the ratio of emitted photons to absorbed, the probability $\Phi_{FB}$ of a photon exiting the bulk target after an excitation is then given by
 \begin{equation}
   \Phi_{FB} = (1-a_{xx})\Phi_F,
 \end{equation}
where $a_{xx}$ is the probability of self absorption. At liquid nitrogen temperatures, $\Phi_F \simeq 97\%$~\cite{Charlton1977,Birch1976,Sharafy1971}, approaching unity at cryogenic temperatures. Furthermore, using the photoluminescent spectra of t-stilbene in the liquid, microcrystalline, and macroscopic single-crystal state as measured via reflection and transmission, Birks et al.~conclude that $a_{xx} \approx 0.35$ as the continuous reabsorption and emission of the photon gradually red-shifts the radiation into wavelengths to which the bulk crystal is transparent~\cite{Birks1954}. Thus, the bulk fluorescence quantum efficiency of t-stilbene is at least $\Phi_{FB} = 0.63$~\cite{Katoh2009,Zaitseva2015}, likely approaching $0.65$ at cryogenic temperatures, though we use the lower value to be conservative. We propose such a detector to be run around 100K in order to maximize bulk quantum efficiency while maintaining a high enough temperature to run CCD based photo-detectors~\cite{Tiffenberg:2017aac}.

\subsection{DM Velocity Distribution}
\label{sec:Velocity}
We denote the velocity of a particle in the Milky Way reference frame by $\vec v$. We adopt the SHM ansatz for the bulk of the Milky Way DM distribution. The DM velocity is then distributed according to $f_0(\vec v) = \exp(- |\vec v|^2/2\sigma_0^2) \Theta(|\vec v|^2 - v_{\rm esc}^2)/N_0,$ where the dispersion $\sigma_0$ is related to the velocity of the Local Standard of Rest by $\sigma_0 = v_0/\sqrt2$ and the normalization is
\begin{equation} \label{vel-N0}
    N_0 \! = \! \pi^{3/2} v_0^3 \left[ \erf \left( \! \frac{v_{\rm esc}}{v_0}\! \right) - \frac2{\sqrt\pi}\frac{v_{\rm esc}}{v_0} \exp \! \left( \! -\frac{v_{\rm esc}^2}{v_0^2} \! \right)\! \right].
\end{equation}
Numerically, the Local Standard of Rest has value $v_0 \simeq 220$ km/s and the escape velocity is near $v_{\rm esc} \simeq 544$ km/s \cite{Evans:2018bqy}; the uncertainties on these values are nonzero, but $\lesssim 10\%$. We use this velocity distribution in order to facilitate comparison with previous studies, but future exploration of the implications of more realistic velocity distributions will be important for interpreting any future experimental results.

\begin{figure*}
    \centering
    \includegraphics[width=0.98\textwidth]{draft-modplots.pdf}

    \caption{Normalized modulation signals for a variety of DM masses, $m_\chi = \text{2--1000}\,\MeV$, for a crystal in $\beta=0^\circ$ and $\beta=90^\circ$ orientations. Above $10\,\MeV$, the rate relaxes into a function of time that is nearly independent of the DM mass and with modulation amplitude only mildly dependent on the crystal orientation. The peak-to-trough modulation amplitudes are as large as 60\% at low masses and 10\% at high masses for $F_\text{DM}=1$, increasing to  70\% at low masses and 25\% at high masses for $F_\text{DM} = (\alpha m_e/q)^2$.
    }
    \label{fig:rateNorm}
\end{figure*}

To calculate the rate in \eqref{eq:ratedef}, we integrate over the velocity measured in the laboratory $\vec u$, which is related to the velocity in the Milky Way frame by $\vec v = \vec u + \vec v_\oplus$, where $\vec v_\oplus(t)$ is the velocity of the Earth as measured in the Milky Way frame. Following the conventions of \cite{Coskuner:2019odd}, we use the energy conservation $\delta$-function to resolve the integral over the velocity distribution:
\begin{align}
    g_0(\vec{q}, t) &\equiv \int d^3 \vec{u}\, f(\vec v_\oplus(t) + \vec u) \, \delta\left(\Delta E - \vec q \cdot \vec u + \frac{q^2}{2m_\chi} \right) \nonumber
    \\ & = \frac{\pi v_0^2}{q N_0} \left( e^{-v_-^2/v_0^2} - e^{-v_{\rm esc}^2/v_0^2} \right),  \label{vel-g}
\end{align}
where
\begin{equation}
    v_-(\vec{q}, t) = \min \! \left( \! v_{\rm esc}, \frac{\Delta E}{q} + \frac{q}{2m_\chi} + \vec v_\oplus(t) \cdot \vec{\hat q} \right).
    \label{eq:vMinus}
\end{equation}
The momentum transfer $\vec q = q \vec{\hat q}$ has been related to the velocity by the requirement that $\Delta E = \vec q \cdot \vec u - q^2/(2m_\chi)$, and all dependence on the local velocity $\vec{v}_\oplus$ is contained in $v_-$.

The rotation of the Earth over a 24-hour period enters the rate by casting the direction of Earth's velocity (more precisely, the velocity vector of a fixed laboratory location on the Earth's surface) as a function of time \cite{Coskuner:2019odd}:
\begin{equation}
\vec{\hat v}_\oplus (t) \! = \! \!
\left( \begin{array}{c c c} \cos\beta & -\sin\beta & 0 \\  \sin\beta & \cos \beta & 0 \\ 0 & 0 & 1 \end{array} \right) \!
\left( \! \begin{array}{c} \sin\theta_e \sin \vartheta \\ \sin\theta_e \cos\theta_e ( \cos \vartheta - 1) \\ \cos^2 \theta_e + \sin^2 \theta_e \cos\vartheta \end{array} \! \right),
\label{eq:vE}
\end{equation}
where $\vartheta(t) = 2\pi \times \left( \tfrac{t}{24\, \text{h} } \right)$,
$\theta_e \approx \, 42^\circ$, and we have chosen to align the $(x,y)$ plane of the crystal to be perpendicular to the direction of the DM wind at time $t=0$ with the initial orientation of the crystal with respect to rotations about the $\hat z$ axis given by $\beta$. As is clear from \eqref{eq:vMinus}, the only aspects of the Earth velocity vector that we need to know when calculating rates in the context of the SHM are the Earth's speed, for which we adopt $|\vec v_\oplus| \equiv v_\oplus = 234$ km/s, and the angle between the Earth's north pole and its velocity in the Milky Way frame, $\theta_e$. Our formalism is easily extended to other velocity distributions by making the substitution in \eqref{eq:vMinus} of $\vec{v}_\oplus \to \vec{v}_\oplus - \langle \vec v \rangle$ where $\langle \vec v \rangle$ is the mean velocity of the DM distribution as measured in the Milky Way frame. The vector $\langle \vec v \rangle$ is zero by definition for the SHM, but would be nonzero for substructure in the form of a stream.

In terms of $g_0$, the total (time-dependent) rate per unit mass is
\begin{equation}
    \frac{R(t)}{m_T} = \frac{\Phi_{FB} N_\text{A}}{m_{\rm mol}^\text{(t-stil)}} \frac{\rho_\chi}{m_\chi} \frac{\bar\sigma_e}{\mu_{\chi e}^2} \sum_{i = 1} \int\! \frac{d^3 \vec{q}}{4\pi} g_0(\vec{q}) F_\text{DM}^2(q) |f_{s_i}(\vec{q})|^2,
    \label{eq:RTotg0}
\end{equation}
where we emphasize that $g_0(\vec{q})$ is implicitly also a function of $\vec{v}_{\oplus}(t)$.
Eqs.~(\ref{vel-g})--(\ref{eq:RTotg0}) indicate that, from the perspective of kinematics alone, the largest modulation will occur when $v_-(\vec{q})$ modulates around $v_{\rm esc}$. However, as we will see in Secs.~\ref{sec:Results} and~\ref{sec:Modulation} below, the morphology of the molecular form factors will also play a large role in determining the modulation.

\section{Daily modulation reach}
\label{sec:Results}

\begin{figure*}
    \centering
    \includegraphics[width=0.98\textwidth]{draft-exclusions-min.pdf}
    \caption{The capability of a $1\, \text{kg}\cdot \text{year}$ t-stilbene experiment to detect or exclude DM models with $F_\text{DM}=1$ (left) or $F_\text{DM} = (\alpha m_e/q)^2$ couplings to electrons, shown with existing limits from SENSEI~\cite{Barak:2020fql}, XENON~10~\cite{Essig:2017kqs}, and XENON~1T~\cite{Aprile:2019xxb}. The dotted and dashed lines show the 90\% CL exclusions that can be set from the total number of events, without considering modulation effects, for $R=1/60\,\text{Hz}\, \text{kg}^{-1}$ ($N_\text{events} \approx 5.26 \times 10^5$) and for $N_\text{events}=0$, respectively. The orange shaded regions indicate parameter space that leads to a sufficiently large modulation signal that a $1~\text{kg}\cdot\text{year}$ experiment could observe a $3 \sigma$ detection, given a total observed rate of $R=1/60\,\text{Hz}\, \text{kg}^{-1}$.  The solid black ``$\Delta N=0$'' lines show the improved limit that can be set from a null result exhibiting no daily modulation but the same total observed rate. Each plot also shows (in blue) a benchmark model from Ref.~\cite{Battaglieri:2017aum} as a target for the experimental sensitivity. In the $F_\text{DM}=1$ example the scalar DM abundance is set by freeze-out mediated by a dark photon of mass $m_{A'} = 3 m_\chi$,  while for $F_\text{DM}\propto 1/q^2$ we show freeze-in via light mediator, $m_{A'} \ll 3\,\keV$.
    }
    \label{fig:exclusionsSHM}
\end{figure*}

\begin{figure*}
\centering
    \includegraphics[width=0.98\textwidth]{draft-discovery-nevents.pdf}
    \caption{ As a demonstration of the utility of daily modulation, we show the $3\sigma$ discovery and 90\% CL exclusion potential for a trans-stilbene experiment with a background rate of $1/60\,\text{Hz}\,\text{kg}^{-1}$, for exposures of 0.01\,$\text{kg}\cdot\text{year}$, 1\,$\text{kg}\cdot\text{year}$, and 100\,$\text{kg}\cdot\text{year}$. The dashed lines, labelled ``90\% CL 1/60\,Hz\,$\text{kg}^{-1}$,'' show the 90\% CL exclusion from an analysis that does not consider the daily modulation effects. The inclusion of daily modulation in the statistical analysis allows even the $0.01\, \text{kg}\cdot\text{year}$ exposure to set a significantly stronger limit on $\bar\sigma_e$. For the $F_\text{DM}=1$ and $F_\text{DM} = (\alpha m_e/q)^2$ form factors we show the benchmark freeze-out and freeze-in models, respectively, from Ref.~\cite{Battaglieri:2017aum}.
    }
    \label{fig:discovery}
\end{figure*}

With the molecular form factors in hand, we can compute the total DM-induced excitation rate by summing over the eight lowest transitions $s_i$ for a given choice of velocity distribution and DM form factor. As we show in Appendix~\ref{sec:OrbitalDetails}, the lowest-energy transition $g \to s_1$ dominates both the daily modulation effect and the total average rate. Near the mass threshold, $m_\chi \lesssim 10 \, \MeV$, the $g \to s_3$ transition contributes at the $20\%$ level, with all other transitions contributing less than $10\%$ of the total rate. Above 10\,\MeV\ the $s_1$ transition remains dominant, accounting for about 50\% of the total rate for both DM form factors. With $F_\text{DM}=1$ and $m_\chi \gtrsim 100\,\MeV$ the $s_3$ and $s_4$ transitions contribute equally, at the 15\% level. For the same masses and $F_\text{DM}\propto 1/q^2$, the $s_3$ transition provides a larger 20\% correction to the total rate, compared to less than 15\% from $s_4$ and less than 10\% from each of the other excited states. This behavior is distinct from the case of benzene, where the lowest-energy transition is dipole-forbidden \cite{Blanco:2019lrf}.

Fig.~\ref{fig:rateNorm} shows the modulating rate $R(t)$ over a 24-hour period (one sidereal day) for two different alignment angles of the detector crystal, $\beta = 0^\circ$ and $\beta = 90^\circ$,
normalized by the average scattering rate,
\begin{align}
\langle R \rangle = (24\,\text{h})^{-1} \int_0^{24 \,\text{h} } \! dt\, R(t) .
\label{eq:rangle}
\end{align}
We see that the peak-to-trough modulation amplitude is as large as 60\% (10\%) for a low-mass (high-mass) DM particle interacting via a heavy mediator, climbing to 70\% (25\%) for a low-mass (high-mass) DM particle interacting via a light mediator. This is on the same scale, or larger than, the \emph{annual} modulation amplitude for WIMP-nuclear scattering well above threshold \cite{Drukier:1986tm, Freese:1987wu, Freese:2012xd, DelNobile:2015nua, Witte:2016ydc}, as well as for DM-electron scattering in semiconductors at high masses \cite{Lee:2015qva,Radick:2020qip}.

Assessing evidence in favor of a signal will be an important step in making a DM discovery, and the daily modulation is an important handle for improving our statistical power. As we discuss in more detail in Appendix \ref{sec:Stats}, the statistical significance that we formally assign to a modulating signal is
\begin{equation}
    \Delta L = - 2 \sum_k n_k \ln \left[\nu_k^m(\bm{\theta}^m)/\nu_k^0(\bm{\theta}^0) \right],
\label{eq:LMod}
\end{equation}
where $k$ labels the data bins, $\nu_k^m$ is the number of expected events in bin $k$ assuming a modulating signal, $\nu_k^0$ is the expected number of events in bin $k$ assuming a constant rate, and $\bm{\theta}^{m,0}$ are parameters describing the expected rate in the modulating and non-modulating scenario, respectively. The values of $\Delta L$ are distributed as a $\chi^2$ distribution of the number of additional degrees of freedom needed to characterize the modulating (as opposed to the non-modulating) signal; in the case of two bins, this would be a $\chi^2$ with two degrees of freedom. Although we focus on the two-bin case in the remainder of this analysis, we emphasize that \eqref{eq:LMod} is appropriate for any binning of data, including an unbinned analysis. We provide more general explorations of this test statistic in Appendix \ref{sec:Stats}.

A particularly simple limit of \eqref{eq:LMod} is one for which we take two bins per day and describe the modulation simply by a single parameter, the integrated modulation fraction $f_2$, defined as the fractional difference in integrated rate between the two bins, averaged over a day:
\begin{equation} \label{eq:frac-def}
    f_2 = \frac{1}{(24\, \text{h}) \langle R \rangle} \left( \int_{t_0}^{t_0+12{\rm h}} \! dt\, R(t) -  \int_{t_0+12{\rm h}}^{t_0+24{\rm h}} \! dt \, R(t) \right) .
\end{equation}
For a perfectly sinusoidal signal, $f_2$ equals the peak-to-trough amplitude divided by $\pi$. Our choice in \eqref{eq:vE} to align the crystalline symmetry axis, $\hat{b}$, with the lab frame DM wind at $t=0$ ensures that the dominant part of the modulation signal has a 24-hour period, with only small contributions from higher harmonics. In this orientation, the integrated modulation amplitude \eqref{eq:frac-def} is maximized by $t_0 \approx 18\,\text{hours}$, based on the results shown in Fig.~\ref{fig:rateNorm}. This observable is particularly well suited for describing the daily modulation, because it is unaffected by the non-modulating background rate and thus does not require any knowledge of the background.

As explored in detail in Appendix~\ref{sec:Stats}, this simple binning is amenable to analytic results in the large-$N$ limit of the Skellam distribution or in the small-modulation limit of the Poisson distribution. In each case, we find that the statistical significance we may assign to either the modulating or non-modulating hypothesis based on an experiment in which $N_{\rm tot}$ events are observed is
\begin{equation}
    N_\sigma= \frac{f_2\,  T_\text{exp} \langle R \rangle }{\sqrt{ N_\text{tot}}} ,
    \label{eq:NSigma2Bin}
\end{equation}
where $\langle R \rangle$ is the time average of the signal event rate $R(t)$ defined in \eqref{eq:rangle}, and where $T_\text{exp}$ is the total exposure time for the experiment. Since the number of signal and background events both grow linearly with exposure, the significance of a modulating signal improves with exposure as long as the integrated modulation fraction $f_2$ is nonzero. Our \eqref{eq:NSigma2Bin} matches the $\chi^2_{sb}$ statistic suggested by Ref.~\cite{Geilhufe:2019ndy}.

\begin{figure*}
    \centering

    \includegraphics[width=0.98\textwidth]{draft-beanmod-phys-2MeV.jpg}
    \caption{Molecular form factors and modulating rates for DM masses near threshold, $m_\chi = 2 \ {\rm MeV}$.  In the contour plots, the gridded shaded regions indicate the kinematically accessible momentum transfers $\vec{q}$ for the four molecules $M_{1, 2, -1, -2}$ that comprise the unit cell of the crystal, shown at $t=0$ and $t=10\,\text{h}$. Here, $\vec{q}$ is given in the molecular basis, $q_x = \vec{q} \cdot \hat{\vec{L}}$, $q_y = \vec{q} \cdot \hat{\vec{M}}$, and the kinematically accessible region is defined by $v_-(\vec{q}) < v_\text{esc}$, following \eqref{eq:vMinus}. \textbf{Top left:} Contour plot of the molecular form factor $|f(\vec{q})|^2$ for the $g \rightarrow s_1$ transition in the $(q_x, q_y)$ plane, with $q_z =0$. \textbf{Bottom left:} Contour plot for fixed $\phi\equiv \arctan (q_y/q_x) = 55^\circ$, showing the strong anisotropy in $q_z$ with maxima at $q_z = 0$. \textbf{Top right:} The scattering rate (summed over all $g \to s_i$ transitions) as a function of time, $R(t)$, normalized by the average daily rate $R_\text{avg}$. The modulation is dominated by the $s_1$ transition (dashed). \textbf{Bottom right:} A closer look at the form factor near the peak at $\phi \simeq 55^\circ$, plotting $|f_{s_1}|^2$ as a function of $|\vec{q}|$ for fixed $\theta=90^\circ$ and various $\phi$.
    }
    \label{fig:beanplots}
\end{figure*}

In Fig.~\ref{fig:exclusionsSHM} we show the expected results of a $1\, \text{kg}\cdot \text{year}$ t-stilbene experiment operated under a number of different assumptions. As a benchmark to facilitate comparison with other experiments, we demonstrate the reach with an entirely background-free experiment using no modulation information. The potential for parameter space exclusion in this scenario is $\bar \sigma_e \simeq 10^{-41} \cm^2 \ ({\rm few} \times 10^{-41} \cm^2)$ for DM interacting with a form factor $F_{\rm DM} = 1 \ (F_{\rm DM} \propto 1/q^2)$ and with a mass in the range $5 \MeV \lesssim m_\chi \lesssim 10 \MeV$. This is within a factor of 2 or 3 from the expected reach of a silicon CCD experiment like SENSEI or Oscura for an equivalent target mass \cite{Essig:2015cda,OSCURA}. Taking the more realistic scenario that the observed rate for a $1\,\text{kg}$ detector is $R=1\min^{-1} = 1/60\, {\rm Hz}$ (including both signal and background components), the future reach depends on analysis strategy. Without leveraging modulation information, the limit we obtain is slightly stronger than the current exclusion from SENSEI~\cite{Barak:2020fql} at low masses below $\sim 5 \MeV$, and comparable at higher masses. We also comment in passing on the prospects for the detectability of the (non-modulating) absorption of DM: since $\rho_T \simeq 1\, {\rm g/cm^3}$, we anticipate a rate of $\sim \mathcal O (1)/{\rm kg/min}$ for a dark photon kinetic mixing parameter of $\epsilon \simeq 10^{-13}$, assuming a dielectric loss of order $\sim \mathcal O(10^{-2})$, similar to that in benzene \cite{ronne2000benzenedielectric} and comparable to those in semiconductors \cite{Hochberg:2016sqx}. This setup would set leading limits on dark photons in the mass range $4.2 \eV < m_{A'} \lesssim 10 \eV$.

For DM scattering, our sensitivity to exclusion and discovery can be dramatically extended by utilizing the information in the rate via the simple two-bin analysis. Using the significance from \eqref{eq:NSigma2Bin}, a $1\, \text{kg}$, $1\,\text{year}$ t-stilbene experiment that observes a constant $R=1\min^{-1} = 1/60\, {\rm Hz}$ event rate can exclude at 90\% CL a DM particle with a scattering cross section as small as $\bar \sigma_e \simeq 10^{-37} \cm^2$. This cross section lies below the well-motivated line from freeze-out production of scalar DM for $2\MeV \lesssim m_\chi \lesssim 7\MeV$ with a heavy dark photon mediator mediator, and also probes a wide range of masses $2\MeV \lesssim m_\chi \lesssim 200\MeV$ for freeze-in production through a light mediator~\cite{Battaglieri:2017aum}. The $3\sigma$ discovery reach for a modulating signal for a total $R= 1/60\, {\rm Hz}/\text{kg}$ background event rate is nearly as strong, reaching just below (above) the cross section $\bar \sigma_e \simeq 10^{-37} \cm^2$ for $F_{\rm DM} = 1 \  (F_{\rm DM} \propto 1/q^2)$.

Very meaningfully, as shown in \eqref{eq:NSigma2Bin}, the discovery or exclusion significance grows with cumulative exposure, even without background mitigation: this improvement in significance is absent in a non-modulating signal. We demonstrate this explicitly in Fig.~\ref{fig:discovery}, displaying the $90\%$ CL exclusion and the $3\sigma$ discovery reach for a t-stilbene experiment with a constant observed rate $R=1\min^{-1}{\rm kg}^{-1}$ and increasing exposures of $0.01, 1, 100$ kg$\cdot$yr. For the lowest exposure proposed here, the background rate is very nearly equal to the $2e^-$ rate observed by the SENSEI experiment with a $\sim$2g detector \cite{Barak:2020fql}. The sensitivity improves with $\sqrt{N_{\rm tot}}$, so given the assumption of constant total rate in counts per unit time per unit mass, the sensitivity improves with $\sqrt{\rm exposure}$. This conservative expectation for scaling of the background rate essentially assumes that bulk events will dominate the background. There will be an irreducible background from the low-energy tail of $^{14}{\rm C}$ decays which would yield only a single scintillation photon, but assuming scintillators can be manufactured with the $10^{-18}$ g/g $^{14}{\rm C}$ levels achieved by Borexino \cite{Arpesella:2001iz,Benziger:2007aq}, the \emph{total} $^{14}{\rm C}$ decay rate would be 0.01 events/min/kg, well below the background rates we have assumed here. These background rates also include radio contamination from heavy metals (e.g. Th and U) whose beta decay spectrum is not compatible with the one-photon signal that such an experiment will look for. Finally, the cosmic ray background is expected to be the bulk of the exogenous rate and the only major background that might vary over the time scale of a day. This rate can be minimized by running under sufficient overburden, and the daily modulation of this background is constrained to be $\lesssim$ 0.1\% in a deep underground facility~\cite{COSINE:2020jml}. If the background rate were dominated by the dark rate in the photodetector, which would likely scale with area, a large-volume experiment and/or light-focusing scheme would improve the significance even further. We plan to return to these issues in future work.

\section{Kinematics and Target Selection For Daily Modulation}
\label{sec:Modulation}

\begin{figure*}
    \centering

    \includegraphics[width=0.98\textwidth]{draft-beanmod-phys-100MeV.jpg}
    \caption{Same as Fig.~\ref{fig:beanplots} for large DM masses, $m_\chi = 100 \ {\rm MeV}$. Only the nearly-spherical region near $q \sim 0$ with inner boundary $q_{\rm min} \simeq 1.6 \ {\rm keV}$ is kinematically forbidden. As a result, the daily modulation amplitude is smaller, driven by the anisotropy of the inner secondary peaks and the tails of the primary peaks.
    }
    \label{fig:beanplots100MeV}
\end{figure*}

\begin{figure*}
    \centering

    \includegraphics[width=0.98\textwidth]{draft-beanmod-fq2.jpg}
    \caption{ Same as Figs.~\ref{fig:beanplots} and~\ref{fig:beanplots100MeV} (top) for a light mediator DM form factor $F_\text{DM} =(\alpha m_e/q)^2$. Here, the contour plots show $F_\text{DM}^2 |f(s_1)|^2$ which appears in the rate integrand \eqref{eq:RTotg0}; the scattering is dominated by the smallest kinematically-allowed $q$. \textbf{Top:} Molecular form factors with $q_z = 0$ and rate modulations for $m_\chi = 2 \ {\rm MeV}$.  \textbf{Bottom:} Molecular form factors with $q_z = 0$ and rate modulations for $m_\chi = 100 \ {\rm MeV}$.
    }
    \label{fig:beanplotsFq2}
\end{figure*}

\begin{figure*}[t]
    \centering

    \includegraphics[width=0.98\textwidth]{draft-beanmod-8eV-100MeV.jpg}
    \caption{Same as Fig.~\ref{fig:beanplotsFq2} (bottom), but with a counterfactual transition energy of $\Delta E = 8 \ {\rm eV}$. This larger transition energy leads to a larger daily modulation for the $g \to s_1$ transition in the large $m_\chi$ limit. To facilitate a comparison with Fig.~\ref{fig:beanplots100MeV}, we also provide the modulation signal of the $s_1$ transition with $\Delta E = 4.2\, \eV$ as the dotted gray line with the smaller amplitude.
    }
    \label{fig:counterfactuals}
\end{figure*}

Given the large daily modulation amplitude present in t-stilbene, and the associated improvement in discovery and exclusion significance, it is worth examining which characteristics of our target molecule govern the size of the modulation, and whether other choices of organic molecules could improve the modulation amplitude even further. Indeed, in other systems sensitive to sub-MeV DM (Dirac materials, for example), the modulation amplitude can be even larger, $\mathcal{O}(1)$ even for DM masses well above threshold \cite{Geilhufe:2019ndy,Coskuner:2019odd}.

\subsection{Daily modulation in t-stilbene}
\label{sec:StilbeneKinematics}
The peaks of the t-stilbene molecular form factors define a preferred momentum scale $q^* \simeq 6 \ {\rm keV}$ where the rate is largest, so for the purposes of understanding the daily modulation, we may approximate all DM interactions as imparting momentum $q^*$. The $s_1$ transition has $\Delta E = 4.2 \ {\rm eV}$, which defines an effective velocity scale
\begin{equation}
    v^* \equiv \frac{\Delta E}{q^*} \simeq 200 \ {\rm km/s},
\end{equation}
on the same order as $v_\oplus \simeq 230 \ {\rm km/s}$. For sufficiently small $m_\chi$ such that $q^*/(2m_\chi) \simeq v_{\rm esc}$, Eq.~(\ref{eq:vMinus}) shows that $v_-(\vec{q})$ will be driven to $v_{\rm esc}$ unless $\vec{v}_\oplus$ is antiparallel to $\vec{q}$, and hence the rate will be nonzero only for a very narrow range of directions of $\vec{q}$.

Fig.~\ref{fig:beanplots} illustrates this phenomenon, with the gridded ``bean-shaped'' shaded regions representing the kinematically-accessible region $v_-(\vec{q}) < v_\text{esc}$ overlaid on contour plots of the $s_1$ molecular form factor at $\beta = 90^\circ$ for DM mass of 2 MeV and a heavy-mediator form factor $F_{\rm DM} = 1$. There are four such regions for the four different t-stilbene orientations within a unit cell. The daily modulation arises from the movement of the kinematically-allowed region over the course of a day, in particular as these regions rotate out of the plane of the molecule and the peaks at $q_z = 0$ become inaccessible.

On the other hand, for sufficiently large $m_\chi$, $q/(2m_\chi) \to 0$ and $v_-(\vec{q^*}) < v_{\rm esc}$ for any direction of $\hat{\vec{q}}$. Thus, the kinematically-allowed region in $\vec{q}$-space always includes $q^*$ but has inner boundary
\begin{equation}
q_{\rm min} = \frac{\Delta E}{v_{\rm esc} + v_\oplus} \simeq 1.6 \ {\rm keV}.
\end{equation}
Fig.~\ref{fig:beanplots100MeV} provides the same information as Fig.~\ref{fig:beanplots} except now for a heavier DM particle, with mass $m_\chi = 100\MeV$. The form factor remains the same, but the ``beans'' have now expanded to fill in across the plane, leaving only circular ``holes'' with inner boundary $q_{\rm min}$. Because the kinematically-accessible region now includes the full peaks of the form factor, the rate modulation of the course of the day arises only due to the mismatch of the circular inner boundary with the hexagonal symmetry of the form factor and the presence of the inner secondary peaks, compounded by the vector addition of $\hat q$ and $\hat v_\oplus$. This leads to a smaller $\sim 10$\% peak-to-trough modulation amplitude for all $m_\chi \gtrsim 10 \ {\rm MeV}$.

For DM scattering through a light mediator, $F_{\rm DM}=(\alpha m_e/q)^2$, the rate integrand \eqref{eq:RTotg0} is weighted toward small $q$. In Fig.~\ref{fig:beanplotsFq2} we show the molecular form factors multiplied by $F_{\rm DM}^2$; the rescaled form factors are peaked more strongly towards low momenta, as expected. Because the inner peaks are kinematically forbidden for DM of all masses, but the tails of these peaks are also probed by all DM masses, this increases the magnitude of the peak-to-trough modulation amplitude to $ \simeq 70\%$ for $m_\chi = 2 \MeV$ and remains as large as $\simeq 30\%$ for $m_\chi = 100\MeV$.

\subsection{Target selection for daily modulation}

The analysis of Sec.~\ref{sec:StilbeneKinematics} suggests a strategy for designing target materials to obtain a large anisotropic response to electron scattering, and correspondingly large rate modulation, even in the limit of heavy DM. In this limit, the time-independent part of the argument of $v_-$ (\eqref{eq:vMinus}) is simply $\Delta E/q$. Now consider a material with a form factor peaked at a momentum $q^*$ and with a lowest-lying excitation energy $\Delta E$. To maximize the modulation, we look for a material for which $q^*$ and $\Delta E$ are related by $q^* \simeq \frac{\Delta E}{v_{\rm max}}$, where $v_{\rm max} =  v_{\rm esc} + v_\oplus$ is the maximum DM velocity attainable in the lab frame. Equivalently, the ``effective velocity'' characterizing the lowest-lying molecular transition, defined as $v^* = \Delta E/q^*$, should be $v^* \lesssim v_{\rm max} $. In t-stilbene, the primary outer peaks have $v^* \simeq 200 \ {\rm km/s}$, which is a factor of a few too small to lead to the maximal rate, whereas the secondary inner peaks have $v^* \simeq 1200 \ {\rm km/s}$, and these peaks are always kinematically forbidden.

An ideal target for daily modulation would have either larger $\Delta E$ or a larger spatial extent (smaller $q^*$), so as to match $v^* \lesssim v_{\rm max}$ for the primary peaks. To illustrate this, Fig.~\ref{fig:counterfactuals} shows the molecular form factor for the $g \to s_1$ transition in t-stilbene but with the kinematically-allowed region defined by a transition energy $\Delta E = 8 \ {\rm eV}$, rather than the 4.2 eV in t-stilbene. Here, we have chosen a form factor $F_{\rm DM} \propto 1/q^2$, which weights the kinematically-forbidden inner peaks more, but the modulation is still driven by the forbidden region in $q$ which has comparable radius to the outer peaks. As the forbidden region moves in $\vec{q}$-space, the peak-to-trough modulation amplitude can be as large as 20\% for all $m_\chi \gtrsim 20 \ {\rm MeV}$ for the $g \to s_1$ transition alone in this hypothetical material, almost a factor of 2 larger than the modulation amplitude for the equivalent transition in t-stilbene.

Taking a broader perspective, the anisotropic response of a condensed matter target to DM-electron scattering arises from an interplay of preferred scales $q^*$ set by the molecular size and a coincidence between the effective transition velocity $v^*$ and the maximum DM velocity in the lab frame. In the case of organic molecular solids, the conjugated $\pi$-electron system provides two length (momentum) scales given by the extent of the molecule along the molecular plane ($q^* \simeq 1.2 \ {\rm keV}$ for t-stilbene), and the extent of a single 2p orbital $(q^* \simeq 6 \ {\rm keV})$, which sets both the carbon-carbon bond length and the extent of the out-of-plane $\pi$ orbitals. The large hierarchy between these scales in large organic molecules means that the excitation dynamics in the plane are largely separated from excitations along the normal to this plane. Transitions along the normal direction will typically require larger imparted momenta than in the extended directions, and thus the form factor for the lowest transition will be peaked at $q_z = 0$, with peaks in the $x-y$ plane corresponding to the characteristic scales of the molecular (sub)structure. The kinematically-allowed regions which dominate the rate integral rotate in $\vec{q}$-space over the course of the day, where the planar anisotropy (and to a lesser extent, the hexagonal structure of a benzene ring which breaks rotational symmetry to a discrete subgroup) gives the modulation for small $m_\chi$, and the anisotropy of the two displaced benzene rings contributes significantly to the residual modulation for large $m_\chi$. Having electronic transitions with $v^*$ slightly smaller than $v_{\rm max}$ (as in our counterfactual example with $\Delta E = 8 \ {\rm eV}$ in Fig.~\ref{fig:counterfactuals}) will maximize the anisotropy for masses above threshold. That said, there is an inevitable tradeoff between the modulation amplitude and the total rate (consistent with the analysis of Ref.~\cite{Coskuner:2021qxo} for single-phonon production) because as $v^*$ approaches $v_{\rm max}$, the kinematically-allowed transitions rely more and more on the high-velocity tail of the DM velocity distribution.

From this perspective, we can understand why daily modulation amplitudes are typically small or nonexistent for electron scattering in conventional semiconductor and noble liquid detectors. In noble liquids, the filled electron shells are spherically symmetric (ignoring small effects due to van der Waals attraction and dimerization between noble atoms), and thus the form factor will be isotropic and no daily modulation will occur. On the other hand, solid-state lattices have only discrete translational symmetries, which may be expected to lead to anisotropies like those due to the hexagonal structure of the benzene rings. However, the dominant low-energy electronic transitions in conventional semiconductors with eV-scale gaps are due to \emph{delocalized valence electrons}, which lead to a continuous energy spectrum and smooth form factors without a preferred momentum scale, at least for $q$ smaller than the inverse lattice spacing $\sim 3 \ {\rm keV}$.\footnote{In fact, the form factors in silicon and germanium have peaks at $q = 0$ and $\Delta E = 18 \ {\rm eV}$ from the plasmon, as well as at $v^* = v_F \simeq 10^{-2}$ from the approximately free Fermi gas behavior of the valence electrons, but neither of these peaks are kinematically-accessible for halo DM \cite{Hochberg:2021pkt}.} For larger $q$, scattering will probe core electron shells of single atoms at individual lattice sites, but these filled shells will be spherically symmetric and give isotropic form factors.  That said, more exotic solid-state systems like Dirac materials, where a combination of a narrow gap (which permits small $q$) and an anisotropic linear dispersion $\Delta E \sim \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}$ with the $v_i$ bracketing $v_{\rm max}$, can have order-1 daily modulation \cite{Coskuner:2019odd,Geilhufe:2019ndy} and a fairly large overall rate \cite{Hochberg:2017wce}.

Importantly, $q^*$ is related to the characteristic size of the (sub)structure of the molecule as well as the symmetry of the transition, which determines whether the transition is dipole/quadrupole allowed. Meanwhile, the minimum $\Delta E$ is set by the HOMO/LUMO gap which is sensitive to the topology of the conjugated electron system, as well as the presence of functional groups which could donate or accept electronic density. For example, 1,2-diphenylacetylene is the acetylene-bridged analog of t-stilbene and presents a HOMO/LUMO gap of $\mathcal{O}(10\%)$ larger than that of t-stilbene~\cite{Takabe1974}. This implies that the two quantities are somewhat decoupled and can be independently tuned, at least for $\Delta E$ in the range $1-10$ eV where the efficiency for scintillation photon detection is high. Furthermore, theoretical computations of the DM form factor as detailed in this paper can be verified by the complementary experimental probe known as electron energy-loss spectroscopy (EELS), which can be used to extract the generalized oscillator strength (\textit{i.e.}~dielectric function) of the molecular excitations in a given target; such a measurement automatically includes many-body effects~\cite{Hochberg:2021pkt} such as the ones parameterized by the PPP Hamiltonian as well as multi-electron excitations. Since t-stilbene and related chromophores have been identified as good scintillators for decades, their single-crystal synthesis is mature and can be scaled up to $\mathcal{O}(10\; \text{cm})$ crystals~\cite{Carman_2013}. Therefore, it is entirely within the reach of existing methods and technology to implement $\mathcal{O}(10\; \text{kg})$ of organic crystal scintillation target since $\mathcal{O}(10\; \text{cm})$ single crystals of anthracene, t-stilbene and p-terphenyl are commercially available already, though ensuring the radiopurity of samples will be paramount to reduce backgrounds. In principle, this is no more challenging than obtaining radiopure liquid scintillator since crystals are readily grown from liquid stock.

\subsection{Daily modulation from dark matter kinematic substructure}

Another mechanism for exploring different scattering kinematics is supplied in the form of a cold, co-rotating stream of DM particles. The exemplary such sub-distribution of DM particles is the putative Nyx stream \cite{Necib:2019zbk}. This stream has velocity $\vec v_{\rm Nyx} \simeq (150, 0, 140)$ km/s \cite{Adhikari:2020gxw} for components $(v_r, v_\theta, v_\phi)$. In reality Nyx appears slightly anisotropic \cite{Necib:2019zbk, Adhikari:2020gxw}, but it has a relatively low spatially average velocity dispersion, $\bar \sigma_{\rm Nyx} \simeq 60$ km/s. Given its inferred size, we can attribute to it a low escape velocity $w_{\rm esc} = 150$ km/s. To calculate the rate for the Nyx stream, $\vec v_{\rm Nyx}$ is subtracted from $\vec v_\oplus \simeq (40, 10, 230)$ km/s \cite{OHare:2019qxc, Adhikari:2020gxw} in \eqref{eq:vMinus} and $\sigma_{\rm Nyx}$ and $w_{\rm esc}$ replace $v_0/\sqrt2$ and $v_{\rm esc}$ in \eqref{vel-g}, respectively.

Because of the smaller escape velocity $w_\text{esc}$, activating the $4.2\,\eV$ transition requires larger momentum transfers,  $q \gtrsim \mathcal O(10)\,\keV$. As a result the inner peaks of $|f_{s_i}|^2$ at $q \simeq 1.2 \ \keV$ are kinematically inaccessible, and the peaks at $q^* \simeq 6 \ {\rm keV}$ are only accessible at the high velocity tail of the distribution $w \approx w_\text{esc}$, even for $m_\chi \gtrsim 100\,\MeV$. Keeping the peaks of $|f_{s_i}|^2$ at the edge of the kinematically-accessible region can induce a large modulation amplitude for a wide range of $m_\chi$, but at the price of significantly lowering the overall scattering rate. Because the Nyx fraction is $\lesssim 10\%$ \cite{OHare:2019qxc} of the local DM density, however, such modulation is unlikely to be a dramatic effect in any experiment, especially for the $F_\text{DM} \propto 1/q^2$ form factor.

\section{Conclusions}
\label{sec:Conclusions}

Among the many target materials proposed for DM-electron scattering, few have demonstrated the necessary anisotropic response to probe the daily modulation of DM, and none (to our knowledge) optimized for DM from the MeV to GeV scale. In this paper we have shown that organic crystals are a promising family of targets with excellent prospects for daily modulation, already at the same level as the expected annual modulation signal for the particular case of t-stilbene, and possibly larger if a compound with a suitable $v^* = \Delta E/q^*$ can be identified. In previous work \cite{Blanco:2019lrf} we have already demonstrated the efficacy of (liquid) organic scintillators in an experimental context, and we expect that many of the same design considerations will hold for solid-state scintillators.

The excellent overall sensitivity of t-stilbene -- within a factor of a few of a comparable mass of silicon -- combined with the additional handle of daily modulation, would make such a detector strongly complementary to the existing experimental program for Oscura \cite{OSCURA} which uses silicon targets. In the event a positive signal is detected, daily modulation will be crucial for confirming a DM origin, and we have also derived a useful test statistic for determining the daily modulation significance in the presence of non-modulating backgrounds. We will explore design considerations for a concrete experimental implementation of a crystal organic scintillator detector in future work.

Beyond the particular case of t-stilbene, we have argued that aromatic organic crystals are a near-optimal compromise between overall rate, daily modulation, and scalability to large target masses, for DM of mass $m_\chi \gtrsim 1 \ {\rm MeV}$. The building blocks of organic scintillators, the sp$^2$-hybridized carbon orbital and its 2p${}_z$ double bonding counterpart, are naturally anisotropic and support delocalized electronic states extended in the molecular plane, while the spatial extent of the 2p orbitals determines a preferred momentum scale. The weak intermolecular forces in organic crystals allow the electronic wavefunctions (and hence the form factors) to retain their molecular character rather than being entirely delocalized as in semiconductors. Furthermore, the discrete transitions at well-defined energies $\Delta E$, combined with the sharply-peaked form factors, give $v^*$ which is close to optimal for t-stilbene, and may give larger modulation in compounds with slightly larger HOMO/LUMO gaps and therefore with a slightly larger $v^*$. The combination of exciting features demonstrated by these results point to the ability to probe extremely well-motivated parameter space with plausible near-future technology, even in the presence of realistic but significant background rates. This indicates great potential for anisotropic organic scintillator detectors.

~\\~\\~\noindent
\textbf{Acknowledgments.} We thank Dan Baxter, Juan Collar, Mike Crisler, Patrick Draper, Adam Kline, Noah Kurinsky, Danielle Norcini, Dario Rodrigues, and Javier Tiffenberg for helpful conversations. YK and CB thank Danna Freedman and Stephen von Kugelgen for numerous organic chemistry discussions. The work of YK and BL was supported in part by DOE grant DE-SC0015655. The work of CB was supported in part by NASA through the NASA Hubble Fellowship Program grant HST-HF2-51451.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555  as well as by the European Research Council under grant 742104. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. This work made use of the Illinois Campus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois at Urbana-Champaign.

redpajama_set_name: RedPajamaArXiv


text:
Q: Can High Energy Light Escape a Black Hole after Low Energy Light Cannot Perhaps there is another way to calculate the escape velocity, or perhaps light always has the same kinetic energy regardless of the energy of the light, but shouldn't the event horizon of a black hole be different distances from the singularity for different colors (and thus energies) of light?
The amount of potential energy due to gravity is calculated via
$U_{grav} = -\frac{Gm_1m_2}{r}$
Escape velocity is the velocity at which the kinetic energy is equal to the gravitational potential energy.
However, with light, the energy depends on the color or frequency.  If that energy is greater than the potential energy due to gravity, then the photon should be able to escape the gravitational force (unless, of course, the energy I'm think about for light is different than the kinetic energy that is necessary to overcome the gravitational potential energy).  Couldn't, then, low energy light be captured by something close to a black hole while high energy light is able to escape?  Could you, then, theoretically, though not practically, have a purple hole?
I also realize that things become more complex than that light can't reach escape velocity to escape a black hole (or else an indestructible human could build an indestructible tower to get out).  So, (1) does my hole argument (pun intended) fall apart with that simply having one form of energy doesn't mean that you can escape gravity, but it has to be kinetic energy or something?  (2) If not, does other aspects of a black hole either forbid high energy light from escaping even when it has more energy than (the absolute value of) gravitational potential energy or allow low energy light to escape even when it has less energy than (the absolute value of) gravitational potential energy?
CLARIFICATION
I know that the escape velocity at the event horizon is greater than the speed of light and that no colors of light can go faster than the speed of light.  However, I'm saying that "escape velocity" is only a special case of "escape energy."  That is, objects don't need some speed in order to escape gravity, they need a certain kinetic energy that is greater than or equal to the absolute value of gravitational potential energy.  The only reason that "escape velocity" works for most objects is because the kinetic energy is calculated as half of its velocity squared times its mass.  Therefore, an object's kinetic energy is determined by its mass and velocity.  Because the gravitational potential energy is also determined by its mass, additional mass of an object does not give it enough energy to escape.  Therefore, for most cases, only velocity contributes to an object's ability to escape gravity.  Nevertheless, if some other property increased the object's kinetic energy, then it would be able to escape gravity at the same speed as something else that didn't have enough kinetic energy to escape.
But light's kinetic energy (as far as I know) is not determined only by its velocity.  For example, different colors of light have different energies while having the same speed (unless that energy doesn't contribute to light's ability to escape gravity).  In order to determine whether light can escape a gravitational field, you would check whether its kinetic energy is greater than its gravitational potential energy rather than whether its speed is greater than this "escape velocity."  It's not the velocity that matters but the energy.  Usually, velocity is the only factor that contributes to the energy (other than mass which is canceled out by the gravitational potential energy's dependence upon it).  But with light, the energy isn't only determined by its speed.  It's also determined by its frequency.  Does that mean that high energy light has enough energy to overcome gravity, or does the energy due to frequency of light not contribute to its ability to overcome gravity (i.e. not kinetic energy)?

A: All wavelengths of light have exactly the same velocity in the vacuum surrounding a black hole. This means that a "purple hole" cannot form in the manner you propose.

A: Although the existing answers are perfectly correct, I think they don't directly address the reason as to why OP thinks it is possible to have different effective horizon radii corresponding to different colors of light--despite the fact that OP seems to (somewhat contradictorily) understand that there would be a true horizon radius beyond which no light can escape. So, I will give it a try.
The proposal by OP seems to suggest that there would be an ordinary true horizon radius for all the massive particles because the only thing that determines their kinetic energy is their speed and thus if their speed needs to be greater than the speed of light then they simply can't escape. And if you go close enough to a black hole, your speed would be required to be greater than the speed of light in order for you to escape and thus, you cannot escape. And this radius beyond which this starts happening is the ordinary horizon radius. Now, OP suggests that the story is a bit different for light because it is not the speed that determines the energy of the light but the frequency. Thus, for different colors of light, there would be different horizon radii beyond which they cannot escape. Now, this is untrue for a multitude of reasons I will try to explain the most basic flaw in this argument.
First of all, this seems to suggest that there would be no true black hole. Given enough frequency, the light will always escape any gravitational pull and thus, there would not be a true black hole which prevents absolutely everything from escaping its pull beyond a certain distance. The proposal by OP seems to be deeply embedded in Newtonian thinking. You see, in a Newtonian black hole, the interior of a black hole is a region in space--so you can think of "starting at a point" inside the black hole and of approaching towards the horizon. If you have enough energy, you will escape just like a rocket with enough energy escapes the earth--if not, too bad--you will eventually reach a zero velocity and then you will be pulled towards the center again. This seems to motivate OP to think that if only a photon had high enough frequency (which it can have--there is no limit on how high the frequency of a photon can be as long as quantum gravity effects don't take over) then it can just escape any pull of gravity. Now, all of this is simply untrue because the black holes we have in nature are not Newtonian black holes. They are general relativistic black holes. The interior of a black hole is not a region in space but rather a chunk of time. As is often said, going towards the center of a black hole inside the event horizon is not a journey towards a point in space but an inevitable flow towards the future that nobody can avoid. Therefore, no matter how high the energy of a photon is, it will simply head towards the center of the black hole if it is inside the event horizon.
The proposal by OP can perhaps be seen to mean something more benign. Let's say we all know and understand why there is only one true horizon radius and no light can escape the black hole if it is once inside this radius. Now, you can think of massive particles with low enough velocities outside the horizon who will inevitably fall into the black hole simply because they don't have the required (and, in principle, achievable) velocity to escape. The proposal by OP can be seen to mean an analog of this phenomenon for light which simply doesn't exist as I will try to elaborate. In particular, OP might suggest that the light can have low enough frequency outside the event horizon so that it will nonetheless fall into the black hole. If so, then we can resurrect the idea of an orange hole suggested by OP which is to say, okay, if you had a high enough frequency, you could escape this potential since you are outside the event horizon, but since you actually have an orange frequency which doesn't have as high an energy as required (as dictated by how close you are to the horizon), you will not really escape. This is not true the moment you start to think about the effect that the pull of the black hole exerts on a light outside the horizon. You see, for massive particles, decreasing energy would mean decreasing velocity and thus, eventually, they will reach a zero velocity (and will return then towards the black hole) if they didn't start with the required amount of energy based on their distance from the horizon. But this is not true of light because decreasing energy simply means decreasing frequency for light. The effect of the pull of gravity will never be to decrease the speed of light as it can't. Thus, the light will keep becoming reddish and reddish but it will inevitably escape the black hole given it started outside the horizon--no matter how low the energy of the light actually was. (This is the famous gravitational redshift one talks about regarding the light observed coming from a black hole.)

Main Points to Keep in Mind (IMHO)

*

*Even if we accept the notion of the speed limit, Newtonian gravity doesn't give us true black holes provided that we also accept that the energy of light depends on its frequency which can have as high a value as one wishes.

*Nature is more relativistic than Newtonian, and thus, luckily, we do have true black holes for which there is a true horizon beyond which nothing can escape including all the variety of light.

*Massive particles outside the horizon will need to have certain velocities depending on how far they are from the horizon to actually escape the black hole, but the light will always escape the black hole once it is outside the horizon--no matter how low its energy is.

Note: I have assumed that all motion is radial, if you have, for example, purely orbital motion, the light will have to be even farther than the horizon to stay outside the horizon forever. Although interesting, that is not much relevant to the question that OP asked here.

A: The speed of light is the speed of light.   But the escape velocity required to overcome the gravitational pull of a black hole is greater than the speed of Light. And as we know that nothing travels faster than light, so light, no matter how energetic or what wave-length cannot escape.

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text:
Adam Boulton
Revealed: Philip's four-letter riposte to Diana funeral plan
Nicholas Watt, chief political correspondent
Fri 19 Sep 2008 19.01 EDT First published on Fri 19 Sep 2008 19.01 EDT
Prince Philip, Duke of Edinburgh. Photograph: Shaun Curry/AFP/Getty
A poised performance by Dame Helen Mirren in the film The Queen has, until now, provided the nation with the only clues about the atmosphere in Buckingham Palace and Balmoral Castle after the death of Diana, Princess of Wales in 1997.
Mirren's portrayal told us that the Queen found Downing Street's involvement irritating at first, though the royal family ultimately accepted Tony Blair's advice to open up. Prince Philip was grumpy, but realised that Blair, who hailed Diana as the "people's princess", had touched a chord.
Now the nation is given a taste of the true atmosphere behind closed doors thanks to an inside account of Tony Blair's Downing Street by the television journalist Adam Boulton, which is serialised in today's Guardian. Boulton reveals that the royals were so upset by No 10's micro management that at one point Prince Philip told No 10 to "fuck off".
The prince, who is known for his risque jokes, lost his temper with Downing Street officials during a discussion about the funeral arrangements for the princess, according to Tony's Ten Years by Boulton, the political editor of Sky News.
In an extract of the book, Boulton writes: "The events of that week in September 1997 were very sad, but as the spinners from Downing Street came to Buckingham Palace and started to kick around what roles Harry and William should play in the funeral, the Queen had relished the moment when Philip had bellowed over the speakerphone from Balmoral: 'Fuck off. We are talking about two boys who have lost their mother'. Once the arrangements had been sorted out Blair read the lesson very melodramatically that day in the abbey."
The book also reveals that:
· Nicolas Sarkozy told Blair that he was "unwilling to stop philandering" even when he was still married to his second wife, Cecilia.
· John Prescott once described a Welsh MP speaking Welsh as "that fucking taff speaking fucking taff".
· Blair described the novelist Robert Harris as a "cheeky fuck" for his novel Ghost, which lampooned the prime minister and his wife.
The revelations by Boulton, the longest serving television political editor, who has held the job since joining Sky at its creation in 1989, have added weight because he is married to Anji Hunter, a senior aide to Blair in 1997.
· Boulton's book is serialised in the Guardian today and in G2 on Monday.
Politics books

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Offshore? I am not sure why I pay taxes in the USA, if I went offshore, if I became a citizen of Canada, taxes would pay for universal healthcare system.
Hmm, I am age 56, I wonder if I can become a citizen of Norway, Denmark, or Sweden. Yes, I would pay 40-50 percent in taxes, but I would know I am safe, I could afford my health care.
Obama used words like "Universal Healthcare System," this is available offshore for free, it should be the big benefit paying Uncle Sam, but instead I get the worlds biggest war machine.
I Filed a Tax Extension.
Why am I paying Uncle Same? I could be offshore, preparing my payment i is truly are ridiculously easy, but the accountant still cost me 350 USD. I went to an H and R Block office, they said they would file an extension for free. I asked them how much to file my 2555 Foreign Exclusion thingy that allows me to earn 90 plus thousand if I live outside ( live offshore ) the USA for over 335 days.
H and R Block wants 350 Dollars.
If I pay my professor accountant, CPA, charges the same, and I could avoid this if I lived offshore there is not tax international.
I do not want to stop paying taxes, but after living offshore for 14 years, the USA can make me angry. Obama tells me one thing, that he is setting up Universal Healthcare System, but he seem to be working for the Doctors, he is going to force me to buy health insurance. the same as I am forced to buy liability insurance to drive a car, that is a good idea, but to force me to buy health insurance is not Universal Healthcare System, it is a joke.
I want to pay the USA taxes, and, also do not want worry about whether I can afford to have cancer.
95-98 percent of the reasons I go to a Doctor are minor, and I went to a Doctor offshore recently, it was free in the DR. I went to a Doctor on Lago Atitlan, GUA, and it only cost 4-5 dollars. Living offshore give me the benefit of free medical and I am not even a citizen in these countries, I just go the Doctor, they do not charge their own people, and they give anyone free medical.
I want to pay the USA, I want the United States to survive, prosper, and be the leader of the free world. But Uncle Same is nuts.
They give you a refund, to make you want to file. I do not get a refund, therefore it only can cost me to file taxes. I am not going to get a refund, but the normal worker pays, and pays again, and suddenly feels great, he or she gets money back, but truly paid and paid all year.
"Hey you, sign me up for taxes."
Now, I would love to be a citizen of Denmark or Sweden, and over 1 million Islamic people agree. If I do not work, they pay me, if I get sick they pay, if I do anything the country pays, I am always safe.
Strangely, Americans are so brainwashed into paying governments, when they go to another country to live offshore, the instantly try to learn about the offshore taxes, or the International taxes of they country..... Too Stupid -- I can live in any country for the rest of my life, complete off the grid, only by trying to become a citizen of that country, or resident, or getting a retirement visa do I volunteer to pay tax international.
It is truly possible to go off the tax grid if you live abroad, or go offshore.
Obama is playing games, the USA government is playing games, they are the reason all the multi-national factories are moving to China. We are all loyal Americans, but when the American dream becomes the American tax nightmare, we see offshore as an easy solution. It is natural to do what is easy, and what makes us the most amount of money for the benefits of living.
Would Americans file if they did not get a refund?
They do not get a total refund, this refund thing is a carrot, fools gold.
Andy Graham, and American Citizen, and trying to like Uncle SAme, but when the USA is giving me almost no benefits, and only because I love my country, and still see this offshore living as another game it feels tricky. I just want to live, pay taxes, and feel safe that if I get cancer, I have a way to go the Doctor.
Tax hell without the services.
If one files U.S. taxes honestly this year, and has more than $10K and/or $50K in total accounts overseas one will have to file FBAR (more like FUBAR) and/or Form 8938. That's $10K in total! So if one has a few thousand bucks in each country to 'hit the ground running" upon arrival in a long-term living location, it is likely one will have this problem.
The penalty of one is 10K USD regardless of the amount! The other form has an up to 400 percent penalty. Both have a criminal penalties as well. The worst thing about at least one of these forms is the off-the-clock statue of limitations on the entire return if one fails to mention even one foreign account with assets.
If one stays in the USA any year more than 30 days...one no longer qualifies for the Foreign Earned Income Exclusion. Of course in the Congressional socialists and fascists' minds overseas Americans are unpatriotic. Expats don't have representation in the Congress, unlike German, French, and other expats have seats in their law-making bodies. Expats overseas sales help keep American factories open, but instead of respect, the IRS swine deride these Americans as tax-evaders and fat cats drinking cocktails in Monte Carlo. The average expat is anything but... probably a teacher or roustabout netting under $50K, paying $350 - $4000 to file an accurate and complete both foreign and domestic returns.
I love it how these Congressional liars and ignorant mainland Americans say the expats don't pay taxes. They pay many taxes that are not creditable on U.S. taxes, and do end up paying the U.S. government income taxes.
For expats there is no Medicare, much less Universal life insurance. Forget American government services... including "rescue" airlifts. If you been in an evac, they make you pay a high commercial rate. And a U.S. Marine rescue only comes if "American business interests" are at stake. And the absolute worst service providers are consular officers at American consulates and embassies. Well-traveled Americans don't need Wiki-leaks to be educated about the lazy goofs at U.S. diplomatic legations.
All have been warned about FBAR and 8938. It's up to the individual to follow the law.
"Universal Life" should have been Congress' "universal health" insurance law.
There is a correlation between the rise of taxation costs to maintain American salespeople overseas, and the exploding trade deficit from 1976 onwards.
Let's face it. The head of the U.S.Treasury, Tim Geithner, was a tax cheat. He is now sicking the IRS attack dogs on Americans living overseas, while he got the velvet glove.
Stereotypical tax evader Marc Rich, just bought himself a presidential pardon from Switzerland. You're probably saying, "It ain't me. I ain't no fortunate one." If you are one of the six million Americans creating value, you ain't the Fortunate Son, so you have been screwed again by the U.S. tax man.
I'll shut up now! I got to get back to work on my 13 forms that must go along with my 1040.