Light dark matter in superfluid helium: Detection with multi-excitation production

We examine in depth a recent proposal to utilize superfluid helium for direct detection of sub-MeV mass dark matter. For sub-keV recoil energies, nuclear scattering events in liquid helium primarily deposit energy into long-lived phonon and roton quasiparticle excitations. If the energy thresholds of the detector can be reduced to the meV scale, then dark matter as light as $\sim \mathrm{MeV}$ can be reached with ordinary nuclear recoils. If, on the other hand, two or more quasiparticle excitations are directly produced in the dark matter interaction, the kinematics of the scattering allows sensitivity to dark matter as light as $\sim \mathrm{keV}$ at the same energy resolution. We present in detail the theoretical framework for describing excitations in superfluid helium, using it to calculate the rate for the leading dark matter scattering interaction, where an off-shell phonon splits into two or more higher-momentum excitations. We validate our analytic results against the measured and simulated dynamic response of superfluid helium. Finally, we apply this formalism to the case of a kinetically mixed hidden photon in the superfluid, both with and without an external electric field to catalyze the processes.

Figure 1

Leading order contribution to DM scattering via quasiparticle production in superfluid helium. (left) For scattering through a contact interaction, the off-shell intermediate excitation splits into two, nearly back-to-back on-shell excitations. (right) For scattering through an intermediate hidden photon $A^{\prime }$, the hidden photon splits into a real photon, which carries nearly all the energy, and a fluid excitation, which carries nearly all the momentum.

Figure 2

(left) Interpolation of the data for the static structure function $S(\mathbf{q})$ (at $T=1\mathrm{K}$) from neutron scattering experiment [38]. In the small $\mathbf{q}$ limit, $S(\mathbf{q})$ behaves linearly according to Eq. (9). (right) We compare the measured dispersion curve for single excitations in superfluid helium (solid black line) with the Bijl-Feynman relation for excitations, ${\mathbf{q}}^2/(2m_{\mathrm{He}}S(\mathbf{q}))$ (dashed blue line). The measured dispersion curve [39] comprises the phonon modes at low $\mathbf{q}$ and the maxon and roton at high $\mathbf{q}$ [in particular, the modes at around $\mathbf{q}\sim 4\mathrm{keV}$ where $\epsilon (\mathbf{q})$ reaches a local minimum is called the roton], but does not include the broad multi-excitation response centered around the free-particle dispersion at high $\mathbf{q}$. In the Bijl-Feynman theory, which does track the quadratic dispersion at high $\mathbf{q}$ (shown as the dotted black line), these high $\mathbf{q}$ modes are treated as single-particle excitations.

Figure 3

Expansion of the dynamic response function in terms of diagrams, where the dashed lines indicate the excitations as defined in Eq. (19). The two-excitation diagram is the leading contribution to $S(\mathbf{q},\omega )$ for $\mathbf{q}$, $\omega$ away from the dispersion relation.

Figure 4

Self-consistent calculation of the dynamic structure function $S(\mathbf{q},\omega )$, obtained from Ref. [39] (CKL15). For a given $\mathbf{q}$, the onset of the response at a minimum $\omega$ clearly shows the one-excitation component of $S(\mathbf{q},\omega )$. The response at larger $\omega$ corresponds to the multi-excitation component, where the structures at 2 meV and above arise from multi-excitations of rotons/maxons. In the experimental data these structures are less prominent, which is expected once additional interactions are included (see Figs. 21 and 22 and discussion in Ref. [39].)

Figure 5

The numerical results from CKL15 are compared with our leading order calculation of $S(\mathbf{q},\omega )$ at two representative values of $\omega$. The dashed line shows the extrapolation with the $q^4$ power law, which is a good fit at low $q$ and agrees with the scaling we find in the leading order calculations.

Figure 6

We show typical values for the total momentum transfer $q=|\mathbf{q}|$ as a function of dark matter mass $m_X$, considering both a massive mediator (left) and massless mediator (right). We use $S(\mathbf{q},\omega )$ extrapolated as $q^4$ to plot $⟨q⟩$ as well as the variance for $q$ (indicated by the shaded region). The energy deposited is fixed at $\omega =3\mathrm{meV}$, and we consider two values of the initial DM velocity. The range of $q$ covered in the CKL15 results (Ref. [39]) is indicated by the light gray lines; as can be seen, these numerical results start to be insufficient for DM masses below $\sim 50\mathrm{keV}$, and we must rely entirely on the $q^4$ extrapolation of the CKL results for masses below $\sim 30\mathrm{keV}$.

Figure 7

(left) The DM scattering rate via a massive mediator is computed using the $S(\mathbf{q},\omega )$ obtained from CKL15. (right) Here we used the leading order result in Eq. (30), with the Bijl-Feynman dispersion for single excitations. There are significant differences in the structure of the spectrum between the two methods, due to the incorrect energies given by the Bijl-Feynman dispersion. However, we find the total integrated rate is similar to within a factor of 2.

Figure 8

Same as Fig. 7, but for DM scattering via a massless mediator.

Figure 9

Projected reach at 90% C.L. (2.4 events) for DM scattering through multi-excitation production in superfluid helium for a 1 kg-year exposure, for the massive mediator (top) and massless mediator (bottom) cases defined in Eq. (40). The dashed (solid) blue line shows the result using the leading order (CKL15) result for $S(\mathbf{q},\omega )$. We assume a 30% signal efficiency, zero background, and experimental sensitivity down to $\omega \sim \mathrm{meV}$. The reach is derived from the integrated rate with $\omega \in [1.2–8.6]\mathrm{meV}$, where the multi-excitation scattering rate is largest. The reach from ordinary nuclear recoils is also shown, assuming sensitivity to the energy range $\omega \in [3–100]\mathrm{meV}$ (for smaller $\omega$, ordinary nuclear recoils are not possible). The dotted lines show ${\sigma }_p$ for sample mediator masses and couplings, chosen to roughly satisfy self-interaction, neutron scattering, and stellar bounds (see text).

Figure 10

Processes for dark matter scattering via a hidden photon mediator; the diagrams for absorption of hidden photons are identical to these but without the external fields ${\mathbf{p}}_i$ and ${\mathbf{p}}_f$. (left) In the absence of an external $E$-field, the DM scattering creates a photon and quasiparticle excitation (dashed line) in the final state. The coupling of the hidden photon is given in Eq. (53). (right) In the presence of an external electric field $E_0$, the intermediate hidden photon is converted to an off-shell excitation, which subsequently splits into two or more on-shell excitations. See Eq. (54).