Direct detection of nuclear scattering of sub-Gev dark matter using molecular excitations

We propose a novel direct detection concept to search for dark matter with 100 keV to 100 MeV masses. Such dark matter can scatter off molecules in a gas and transfer an $\mathcal{O}\left(1\right)$ fraction of its kinetic energy to excite a vibrational and rotational state. The excited rovibrational mode relaxes rapidly and produces a spectacular multi-infrared-photon signal, which can be observed with ultrasensitive photodetectors. We discuss in detail a gas target consisting of carbon monoxide molecules, which enable efficient photon emission even at a relatively low temperature and high vapor pressure. The emitted photons have an energy in the range 180 to 265 meV. By mixing together carbon monoxide molecules of different isotopes, including those with an odd number of neutrons, we obtain sensitivity to both spin-independent interactions and spin-dependent interactions with the neutron. We also consider hydrogen fluoride, hydrogen bromide, and scandium hydride molecules, which each provide sensitivity to spin-dependent interactions with the proton. The proposed detection concept can be realized with near-term technology and allows for the exploration of orders of magnitude of new dark matter parameter space.

Figure 1

(a) Schematic showing the proposed experimental setup for detecting dark matter through rovibrational excitations of molecules in a gas. A reservoir of length $L$ with reflective walls contains a gas of cold diatomic heteronuclear molecules (denoted AB) held at low pressure to avoid clustering. The photodetector has surface area $A_{\det }$ and is shown for simplicity to be attached to one of the reservoir walls; in practice, to allow for the photodetector operating at a different temperature than the gas, the photodetector may need to be either insulated from the reservoir or the light must be transported to the photodetector by, for example, fibers. (b) DM scatters off a molecule and excites a rovibrational mode. The excited rovibrational modes are short lived and relax rapidly, producing two types of infrared photons as the signal: (i) cascade photons (c), where the excited molecule cascades to a lower-lying vibrational mode (or to the ground state) emitting a single photon for every vibrational transition, which each have a large mean free path; and (ii) coquench photons (d), where the excited molecule is resonantly quenched by scattering off and exciting neighboring molecules to their first vibrational mode, which each decay to produce a photon. The mean free path of the coquench photons can be enhanced by adding a buffer gas consisting of, e.g., helium, but only those produced close to the photodetector area will be measurable.

Figure 2

The averaged, spin-independent, molecular form factor, $\langle |F_{\mathrm{mol},v^{\prime }{J}^{\prime }}(q,T){|}^2\rangle$, with $f_{P,\mathrm{SI}}^{\left(i\right)}=f_{N,\mathrm{SI}}^{\left(i\right)}=1$, for a gas of CO molecules at 55 K, plotted as a function of vibrational-rotational energy, $\mathrm{\Delta }{E}_{v^{\prime }{J}^{\prime }}$, and momentum transfer, $q$. In every energy-momentum bin, the color corresponds to the sum over all form factor elements within that bin. The red and green curves correspond to the relation expected from energy conservation under the impact approximation, Eq. (12), for interaction with the C or O atom, respectively. For each, the solid curves follow the first term in the equation while the dashed curves correspond to the minimal and maximal values allowed by the approximation. The orange curves correspond to Eq. (8), with a DM velocity of $v_{\chi }=240\mathrm{km}/\sec$, and for three DM masses of $m_{\chi }=5$, 20, and $50\mathrm{MeV}$.

Figure 3

Contours of ${\rho }_{\chi }/m_{\chi }·{\langle \sigma v_{\chi }\rangle }_{v^{\prime }{J}^{\prime }}$ as a function of $(v^{\prime },J^{\prime })$, i.e., the number of events to scatter into a particular $(v^{\prime },J^{\prime })$ state, for a CO molecule in (a) and an HSc molecule in (b). Results take the reference cross section per nucleon to be ${\overline{\sigma }}_n=5\times {10}^{-36}{\mathrm{cm}}^2$, an exposure of 250 gr-yr, and a DM form factor of $F_{\mathrm{DM}}\left(q\right)=1$ (corresponding to a heavy particle mediating DM interactions with molecules). For CO, the dashed curve corresponds to the range of $(v^{\prime },J^{\prime })$ for which the states are unable to mix with the first excited electronic state, which could provide additional nonradiative energy decay processes (see Sec. 3d for further details). For HSc, this curve has been chosen arbitrarily at values of $(v^{\prime },J^{\prime })$ that have energies equal or below that of $E_{v^{\prime }=10,J^{\prime }=0}$, see Sec. 3d for details. Shaded regions in the left panel for CO indicate values of $(v^{\prime },J^{\prime })$ that produce either “cascade” or “coquench” signals. In the right panel for HSc, the photon production is less well-understood and could involve couplings to electronic states (see Sec. 3d). The different colored contours represent different DM masses, and modes below each contour are excited at a rate equal to or larger than the label on that contour.

Figure 4

(a) For various molecular gases, we show the pressure $p_{50}^{v^{\prime },J^{\prime }=0}$ as a function of $v^{\prime }$, for which the branching ratio of the ($v^{\prime },J^{\prime }=0)$ state to photons, $\text{Br}(v^{\prime },J^{\prime }=0)$, is 50%. Below each contour, $\text{BR}(v^{\prime },J^{\prime }=0)$ is larger than 50% for that particular molecule. For halides, $p_{50}^{v^{\prime },J^{\prime }=0}$ decreases monotonically as a function of $v^{\prime }$, while for CO, $p_{50}^{v^{\prime },J^{\prime }=0}$ increases monotonically. For HSc, we show a flat dashed contour (shown in dashed to illustrate that this curve is speculative), valid under the assumption that $V-E$ transfer is extremely efficient. The temperature is chosen to be $T_{\mathrm{BBR}}=55\mathrm{K}$, {44 K, 72 K, 115 K, 83 K, and 60 K} for CO {HSc, HBr, HF, HCl, and HI}. See text for details. (b) The pressure $p_{50}^{v^{\prime },J^{\prime }}$ as a function of $J^{\prime }$ for various $v^{\prime }\to v^{\prime }-1$ transitions in CO. The notation on each curve indicates the transitions with $v^{\prime }\to v^{\prime }-1$. The value of $p_{50}^{v^{\prime },J^{\prime }}$ should be smaller than the pressure above which the CO molecules begin to cluster, which for CO at $55\mathrm{K}$ occurs for pressures above $\sim 5\mathrm{mbar}$ (gray region).

Figure 5

The single-photon branching ratios for a ($v^{\prime }, J^{\prime }$) state in CO, $\mathrm{BR}(v^{\prime },J^{\prime })$ [see Eq. (23)], and the type of expected signal (cascade or coquench photons). We assume a temperature equal to $T_{\mathrm{BBR}}=55\mathrm{K}$ and a pressure $p=5\mathrm{mbar}$. The blue shading corresponds to the value of $\mathrm{BR}(v^{\prime },J^{\prime })$ for each $(v^{\prime },J^{\prime })$ pair. The light blue region corresponds to values of ($v^{\prime },J^{\prime }$) for which a cascade signal is possible. The darker blue region corresponds to values for which only a coquench signal is possible. The red shaded regions are values of ($v^{\prime }, J^{\prime }$) for which there is not enough information currently available to ensure that a photon signal is produced. Two example events are shown: excitation to (1) the state $(v^{\prime },J^{\prime })=(18,7)$ within the cascade region and (2) the state $(v^{\prime },J^{\prime })=(4,1)$ within the coquench region. In event (1), the molecule cascades down several steps of size $\mathrm{\Delta }v=1$ until reaching $v_{\mathrm{b}}$, emitting $v^{\prime }-v_b$ low-energy photons that are transparent to the medium. At $v_b$, the molecule resonantly quenches and the remaining vibrational energy of the state is converted into $v_b$ excited $(1,J)$ states, which each emit a photon with a very small mean free path within the medium. Event (2) immediately gets coquenched to $v^{\prime }$ excited $(1,J)$ states, which each emit a single photon with small mean free path.

Figure 6

Cascade photon spectrum for $m_{\chi }=100\mathrm{MeV}$ and $1\mathrm{GeV}$ and coquench spectrum for $m_{\chi }=10\mathrm{MeV}$. The rate is given for CO at $55\mathrm{K}$ with ${\overline{\sigma }}_n=5\times {10}^{-36}{\mathrm{cm}}^2$, for $p=5\mathrm{mbar}, {\epsilon }_{\mathrm{abs}}={10}^{-5}$, and ${\epsilon }_{\det }\left(n_{\gamma ,\mathrm{col}}\right)=1$. The tank has volume ${\left(2\text{m}\right)}^3$ and a photo-detector surface area of $A_{\det }=100{\mathrm{cm}}^2$. The volume, temperature, and pressure correspond to a total CO mass of 250 gr. (a) corresponds to $F_{\mathrm{DM}}\left(q\right)=1$ and (b) to $F_{\mathrm{DM}}\left(q\right)={(q_0/q)}^2$ with $q_0=\mathrm{\Delta }{p}_e\approx 28\mathrm{keV}$ from Eq. (11). The cascade signal is always much larger than the coquench signal since coquench photons are not transparent to the medium. On the other hand, coquench signals are able to access much lower DM masses since they correspond to a lower energy threshold. The coquench result for $m_{\chi }=10\mathrm{MeV}$ corresponds to a setup with an internal reflective cylinder based on the photo-detector with length twice the mean free path, within the main tank. Solid curves in the figure correspond to values of $(v,J)$ with energies equal to or below that of $(v,J)=(26,0)$, i.e., below electronic energy levels. For illustration, if higher modes are found to be experimentally accessible, the dotted curves correspond to maximal values of $(v,J)$ equal or below the energy of (55,0). In this case, many more photons are observed.

Figure 7

(a) The sensitivity to DM spin-independent nuclear scattering off CO molecules for different target volumes $V$, different photodetector areas $A_{\det }$, and different photon absorption coefficients for the mirrors ${\epsilon }_{\mathrm{abs}}$. The sensitivities to signals of cascade photons are shown with dashed and solid curves for a fixed pressure of $5\mathrm{mbar}$, while the sensitivities to signals of coquench photons are shown in dotted curves and are independent of the partial pressure of CO. (b) The sensitivity to DM spin-independent nuclear scattering off CO molecules for various values for the pressure and for signals of cascade photons only.

Figure 8

The sensitivity to DM spin-independent nuclear scattering off HF and off HBr molecules for various values for the pressure. For these targets, essentially only cascade photons are produced.

Figure 9

(a) The maximum allowed single-photon rate (blue curves) versus temperature for achieving an $n_{\gamma }$-photon coincidence rate of $R_{n\gamma }<0.1/\mathrm{year}$ for CO for $n_{\gamma }=4$ (dot dashed), 3 (dashed), and 2 (solid) coquench photons ($v=1\to 0$ transitions). Red curves show the single-photon rates from blackbody radiation for different detector areas, $A_{\det }=1{\mathrm{cm}}^2$ (solid), $A_{\det }=100{\mathrm{cm}}^2$ (dashed), and $A_{\det }=1{\mathrm{m}}^2$ (dot dashed). The dotted green curve shows a dark count rate of ${10}^{-4}\mathrm{Hz}$, as demonstrated for SNSPDs in [110]. (b) Same as left panel, but for cascade photons. Here the blue curves show the results for $n_{\gamma }=2$, 3, and 4 cascade photons corresponding to the transitions $v^{\prime }=10\to 8, 11\to 8$, and $12\to 8$ in CO, respectively. The shift to lower temperatures of the red curves in the right panel compared to the left panel is a result of the lower energies of the cascade photons, leading to larger single-photon rates from blackbody radiation for a given temperature.

Figure 10

The 95% confidence-level sensitivity for DM spin-independent nuclear scattering off CO molecules at a partial pressure of $5\mathrm{mbar}$ and a temperature of $55\mathrm{K}$, for various tank sizes and photodetector area and for a time exposure of 1 year. The signal consists of two-or-more coincident photons, and we assume zero background events. The panel (a) corresponds to a trivial DM form factor, $|F_{\mathrm{DM}}{\left(q\right)|}^2=1$, while the panel (b) corresponds to $|F_{\mathrm{DM}}{\left(q\right)|}^2={\left(q_0/q\right)}^4$ (with $q_0={({\mu }_{12}/4m_e)}^{1/4}{\alpha }_{\mathrm{EM}}{m}_e$). All results assume $f_{N,\text{SI}}^{\left(i\right)}=f_{P,\text{SI}}^{\left(i\right)}=1$. Colored dashed curves correspond to CO cascade photons while colored solid curves correspond to the sum of cascade and coquench photons. Since coquench photons are produced at a smaller energy threshold, they reach to lower DM masses. The coquench signal assumes the addition of an He buffer gas at a pressure of 4 bar and an internal mirror that helps to focus photons to the detector (see text). The grey dotted curve corresponds to a cascade photon signal from an HSc gas at a pressure of $10\mu \mathrm{bar}$ and at a temperature of $46\mathrm{K}$. The gray shaded regions are current constraints set by other nuclear recoil direct-detection experiments (see text).

Figure 11

The 95% confidence-level sensitivity for DM spin-dependent scattering for various tank sizes and photodetector areas and for a time exposure of 1 year. The signal consists of two-or-more coincident photons, and we assume zero background events. We assume a trivial DM form factor, $|F_{\mathrm{DM}}{\left(q\right)|}^2=1$. The gray shaded regions are current constraints set by other nuclear recoil direct-detection experiments (see text). (a) Sensitivity to spin-dependent neutron interactions ($f_{N,\mathrm{SD}}^{\left(\mathrm{C}\right)}=1, f_{P,\mathrm{SD}}^{\left(\mathrm{C}\right)}=0$) using ${}^{13}\mathrm{C}{}^{16}\mathrm{O}$ molecules at a partial pressure of $5\mathrm{mbar}$ and a temperature of $55\mathrm{K}$. Colored dashed curves correspond to CO cascade photons while colored solid curves correspond to the sum of cascade and coquench photons. Since coquench photons are produced at a smaller energy threshold, they reach to lower DM masses. The coquench signal assumes the addition of an He buffer gas at a pressure of 4 bar and an internal mirror that helps to focus photons to the detector (see text). (b) Sensitivity to spin-dependent proton interactions ($f_{N,\mathrm{SD}}^{\left(\mathrm{H}\right)}=0, f_{P,\mathrm{SD}}^{\left(\mathrm{H}\right)}=1$ and neglecting couplings to F and Sc atoms) using ${}^1\mathrm{H}{}^{19}\mathrm{F}$ molecules at a pressure of $0.1\mu \mathrm{bar}$ and a temperature of $115\mathrm{K}$ (solid curves) and ${}^1\mathrm{H}{}^{45}\mathrm{Sc}$ molecules at a pressure of $10\mu \mathrm{bar}$ and a temperature of $46\mathrm{K}$ (dotted curves). The signal corresponds only to cascade photons with a minimal $v^{\prime }=3$.

Figure 12

Elastic cross section in atomic units for $\mathrm{CO}(v=0)+\mathrm{He}\to \mathrm{CO}(v=0)+\mathrm{He}$ and $\mathrm{CO}(v=0)+\mathrm{CO}(v=0)\to \mathrm{CO}(v=0)+\mathrm{CO}(v=0)$ as a function of the collision energy (in K). The cross sections do not show dependence on the rotational quantum number $J$, since we assume an isotropic interaction and an interaction potential that is spherically averaged.

Figure 13

The function $I(\omega ,b,T)$ from Eq. (C5) in atomic units for $b=7{\mathrm{a}}_0$, which is needed to determine the probability of resonant quenching through the process $\mathrm{CO}\left(0\right)+\mathrm{CO}\left(v\right)\to \mathrm{CO}\left(1\right)+\mathrm{CO}(v-1)$, see Eq. (C3).

Figure 14

Probability distribution for the change in the rotational quantum number, $\mathrm{\Delta }{J}_1$ and $\mathrm{\Delta }{J}_2$, in the resonant quenching process $\mathrm{CO}\left(0\right)+\mathrm{CO}\left(v\right)\to \mathrm{CO}\left(1\right)+\mathrm{CO}(v-1)$, see Eq. (C3). We choose $\sigma =0.5$ in Eq. (C6), and take $\xi \left(0\right)=\xi \left(2\right)$.

Figure 15

Temperature dependence of the vibrational relaxation rate for hydrogen halides. (a) shows the results for HF, where the blue dots are the experimental data taken from Refs. [128, 132, 157, 158]. (b) shows the results for HBr, where the blue dots are the experimental data from Ref. [129]. In both panels, the solid-red curves are fits to the data of exponential functions whose arguments depend on the temperature as $T^{-1/3}$.

Figure 16

The 95% confidence-level sensitivity for DM spin-independent nuclear scattering off CO molecules via a dark photon mediator at a partial pressure of $5\mathrm{mbar}$ and a temperature of $55\mathrm{K}$, for various tank sizes and photodetector area and for a time exposure of 1 year. Limits are converted from ${\overline{\sigma }}_n$ into ${\overline{\sigma }}_e$ according to Eq. (E1) for comparison purposes. The signal consists of two-or-more coincident photons, and we assume zero background events. Colored-dashed curves correspond to CO cascade photons while colored solid curves correspond to the sum of cascade and coquench photons. Since coquench photons are produced at a smaller energy threshold, they reach to lower DM masses. The coquench signal assumes the addition of an He buffer gas at a pressure of 4 bar and an internal mirror that helps to focus photons to the detector. In (a), we show the prospective reach assuming a heavy dark photon with mass $m_{A^{\prime }}=3m_{\chi }$, leading to a trivial DM form factor $|F_{\mathrm{DM}}{\left(q\right)|}^2=1$. Also shown are existing stellar and terrestrial limits as well as the contour where freeze-out sets the correct relic abundance of the DM. In (b), we show the prospective reach assuming an ultra-light dark photon with mass $m_{A^{\prime }}\ll {\alpha }_{\mathrm{EM}}{m}_e$, leading to a DM form factor of $|F_{\mathrm{DM}}{\left(q\right)|}^2={\left({\alpha }_{\mathrm{EM}}{m}_e/q\right)}^4$. Also shown are existing stellar and terrestrial limits as well as the contour where freeze-in sets the correct relic abundance of the DM. At low DM masses, the prospective reach falls sharply due to the small momentum transfer and Thomas-Fermi screening. See text for details.