Solar neutrino detection in liquid xenon detectors via charged-current scattering to excited states

We compute the CC neutrino-nucleus scattering cross section to an excited final state of energy $E_{x}$ using the equation [23, 24]

where $G_{\mathrm{F}}$ is the Fermi coupling constant, $F(\pm Z,E_{\mathbf{k^{\prime}}})$ is a Fermi function, $\theta_{\mathrm{C}}$ the Cabibbo angle, and $\mathbf{k^{\prime}}$ and $E_{\mathbf{k^{\prime}}}$ are the three-momentum and energy of the final-state lepton, respectively. The ’-’ sign is chosen for neutrino scattering while the ’+’ sign is for antineutrino scattering. In Eq. (1) $\sigma_{\mathrm{CL}}^{J}$ and $\sigma_{\mathrm{T}}^{J}$ refer to the Coulomb-longitudinal and transverse contributions to the cross section. The details of these terms are discussed in [25] and the references therein, and the present calculations follow the same formalism discussed there. When calculating cross sections in this work, we use the experimentally measured final-state energies where available.

The main contributions to the CC scattering cross section come from multipoles with $\Delta J=0,1$. In the limit of zero momentum transfer the relevant operators for these multipoles simplify to Fermi and Gamow-Teller operators, and the total cross section to an individual final state reduces to [26]

where

and

are the Fermi and Gamow-Teller reduced transition probabilities, respectively. Here $g_{\mathrm{V}}$ and $g_{\mathrm{A}}$ are the vector and axial-vector coupling constants, respectively, and $\mathcal{M}(\mathrm{F})$ ($\mathcal{M}(\mathrm{GT}$)) is the Fermi (Gamow-Teller) nuclear matrix element.

The cross sections in this work are computed using the exact Eq. (1), accounting also for non-zero momentum transfer. However, in light of Eq. (2), we fit our computed $B(\mathrm{GT}^{-})$ values to experimentally observed values where available by applying a quenching factor. The $|\mathcal{M}(\mathrm{GT}^{-})|^{2}$ distribution for the ${}^{136}\textrm{Xe}$ to ${}^{136}\textrm{Cs}$ process is measured directly via (³He, t) charge exchange reactions [27, 28]. For the ${}^{131}\textrm{Xe}$ to ${}^{131}\textrm{Cs}$ process we deduce the $B(\mathrm{GT}^{-})$ value for the ground-state to ground-state transition from the electron-capture decay $\log ft$ obtained using the experimentally observed half life from Ref. [20] and phase-space factor from Ref. [29]. It should be noted that Fermi transitions can only occur when $J_{i}=J_{f}$. No experimental data is available for Fermi transitions relevant to this work, and we use the bare value of $g_{\mathrm{V}}=1.0$.

The $|\mathcal{M}(\mathrm{GT}^{-})|^{2}$ values reported in [28] are determined via a charge-exchange reaction which is a strong interaction probe, allowing access to $|\mathcal{M}_{\mathrm{GT^{-}}}|^{2}$ directly [30]. Thus the definition of $B(\mathrm{GT}^{-})$ in Ref. [28] differs from our Eq. (4) by the factor of $g_{\mathrm{A}}^{2}$. To bring the experimental $|\mathcal{M}(\mathrm{GT}^{-})|^{2}$ values into the weak-interaction picture we multiply them with the bare value of $g_{\mathrm{A}}^{2}=1.267^{2}$ to get the experimental weak interaction $B(\mathrm{GT^{-}})$.

An enhanced value of $g_{\textrm{A}}^{\textrm{eff}}=1.40$ is needed to reproduce the experimentally known ground-state to ground-state $B(\mathrm{GT}^{-})$ value of the ${}^{131}\textrm{Xe}$ to ${}^{131}\textrm{Cs}$ process. Due to a lack of experimental data we also use this value to compute cross sections for scattering to the excited states of ${}^{131}\textrm{Cs}$. For the transition to the first $1^{+}$ state in ${}^{136}\textrm{Cs}$ we adopt a strongly quenched value of $g_{\textrm{A}}^{\textrm{eff}}=0.64$ and for the second $1^{+}$ state $g_{\textrm{A}}^{\textrm{eff}}=1.35$. Strong quenching of $g_{\mathrm{A}}$ in similar transitions between the $0^{+}$ ground state of an even-even nucleus and the first $1^{+}$ state in the neighboring odd-odd nucleus is an actively studied phenomenon, see Ref. [31] for a review. The computed cross sections are shown in Fig. 3.

The solar neutrino capture rates are obtained using the computed cross section to each of the Cs product excited states $\sigma_{E_{x}}(E_{\nu})$, folded with the differential spectrum for each solar neutrino species $\mathrm{d}\Phi_{i}/\mathrm{d}E_{\nu}$, and the oscillation-induced survival probability for electron-type neutrinos $P_{ee}(E_{\nu})$:

where $Q$ is the reaction threshold, $i$ is an index representing different solar neutrino production reactions, $M$ is the xenon isotope molar mass, and $N_{A}$ is the Avogadro constant. We use solar neutrino flux values from the high-metallicity GS98-SFII model [32]. For the electron neutrino survival probability, we use the central value of the MSW-LMA solution shown in Ref. [33].

Tables 2 and 3 show the event rates for solar neutrino capture on ${}^{131}\textrm{Xe}$ and ${}^{136}\textrm{Xe}$, respectively, and compare our results with previous calculations in the literature. In the case of scattering on ${}^{131}\textrm{Xe}$ we find some disagreement between our calculations and those in Ref. [34], which can be traced to our inclusion of neutrino oscillations, which lead to a suppression in event rates compared to those in the reference. In the case of scattering on ${}^{136}\textrm{Xe}$, our results are within $\sim 20\%$ of the calculations from Ref. [35], indicating good agreement. The deviation can be traced to differences in the assumed neutrino fluxes and survival probabilities, and the use of the exact Eq. (1) over Eq. (2).

While the level structure of ${}^{131}\textrm{Cs}$ is well measured, existing nuclear data for ${}^{136}\textrm{Cs}$ is sparse. Here we use the shell model to predict the structure and decay of low-lying states in ${}^{136}\textrm{Cs}$ which will influence the neutrino capture event signature. Fig. 4 shows the computed decay scheme of the first $1^{+}$ state of ${}^{136}\textrm{Cs}$ to the ground state, showing branching ratios for each path and lifetimes of the intermediate states. The calculation predicts that in 99.8% of cases the first $1^{+}$ state rapidly decays to a very low-lying $3^{+}$ state at 23 keV in two or three M1-dominated steps. The decay of the 23 keV $3^{+}$ state is then slower, with a predicted lifetime of 624 $\mu$s, due to the low energy of the state and the E2 nature of the decay. Such a long-lived state would produce a delayed coincidence signature in modern experiments, and is discussed further in Sec. III.2. This finding is consistent with the propensity for $\beta\beta$-decay isotopes to exhibit low neutrino capture thresholds and for their product nuclei to have low-lying, longer-lived E1 or E2 transitions, as discussed in [11].

The second $1^{+}$ state of ${}^{136}\textrm{Cs}$ follows a very similar predicted decay path as the first. It has a predicted lifetime of 0.58 ps and decays to the two lowest-lying $2^{+}$ states with a branching ratio of 25% to $2^{+}_{2}$ and 72% to $2^{+}_{1}$. The remaining 3% goes directly to the $3^{+}_{1}$ state via an E2 transition. The total fraction of $1^{+}_{2}$ state decays that go through the delayed $3^{+}_{1}$-to-ground state transition is 99.9%.

The predicted lifetime of the lowest $3^{+}$ state depends strongly on its energy. The mean lifetime of an E2 transition is proportional to $E_{x}^{-5}$. By altering the state’s energy while assuming that the wave function remains the same, we can obtain an estimate of the uncertainty in the predicted lifetime. Lower and upper bounds on the state energy of 10 and 100 keV lead to lifetimes of 40 ms and 0.4 $\mu$s, respectively.

The decay scheme for ${}^{136}\textrm{Cs}$ relies on the ordering of the first $3^{+}$ and $4^{+}$ states. There is an experimentally observed $4^{+}$ state at 105 keV, while the shell model predicts the first $4^{+}$ state at 39 keV. If there is no $3^{+}$ state between the first $4^{+}$ state and the $5^{+}$ ground state, there would likely be a fast decay path of $\Delta J=0,1$ transitions from the first $1^{+}$ to the $5^{+}$ ground state. In our shell-model calculation the transition from the 105 keV $4^{+}$ state to the $5^{+}$ ground state is predicted to be dominantly M1 with a lifetime of $\mathcal{O}$(1 ns). However, this transition has been experimentally characterized as an E2 transition [22]. If we use only the computed E2 transition strength the lifetime increases to 52 $\mu$s (assuming the predicted level energy $E_{x}=39$ keV) or 360 ns (assuming the measured level energy $E_{x}=105$ keV). We therefore predict that the final step in the decay scheme will have a roughly microsecond-scale (or longer) delay regardless of whether the lowest-lying excited state is a $3^{+}$ or $4^{+}$ state.

To check the accuracy of our model we compare the computed E2 transition strengths in ${}^{136}\textrm{Xe}$ to experimental values in Table 4. The measured values are reasonably well reproduced by the shell model calculation. However we note that the odd-odd ${}^{136}\textrm{Cs}$ is a more complex nucleus and the gamma transitions there may not be as accurate. We performed an additional cross-check by computing the known $5^{+}_{1}\rightarrow 4^{+}_{\mathrm{gs}}$ transition of the neighboring ¹³⁴Cs nucleus in the full shell-model valence space. Here the experimental energy difference is 11 keV while in the shell model it is 67 keV. The experimental $B(\mathrm{M1})$ value of the transition is $0.00363\pm 0.00013$ W.u. and the shell-model computed value is 0.029 W.u. For the E2 transition the experimental $B(\mathrm{E2})$ value is $4.6\pm 0.6$ W.u. and our computed value is 6.4 W.u. The transition probabilities are reproduced within an order of magnitude in the shell-model calculation, and we expect similar accuracy from our calculations for ${}^{136}\textrm{Cs}$. We do not expect the uncertainty in transition probabilities to significantly affect the total branching ratio to go through the lowest $3^{+}$ state.

The potential for a delayed coincidence signature in the decay of $1^{+}$ states in ${}^{136}\textrm{Cs}$ likely deserves experimental study for the benefit of future LXe experiments. A suitable set of experimental conditions may readily detect this signature. For example, use of the $(^{3}\textrm{He},t)$ or $(p,n)$ reactions in conjunction with an array of $\gamma$-ray and IC electron detectors could verify the predictions reported here.

The neutrino capture events of interest are comprised of a primary, fast ($\mathcal{O}(100)$ keV typical) electron and $\gamma$-rays/IC electrons from the excited Cs de-excitation. We perform an analysis with the goal of tagging these events by resolving the emitted $\gamma$-rays which may be absorbed or Compton scatter and produce multiple interaction vertices. The summed energy of these vertices must recover the excited state energy. The analysis presented here does not use the temporal information of each event and is therefore meant to be distinct from the delayed coincidence method discussed in Sec. III.2.

Full particle-tracking simulations of neutrino capture events were performed using the GEANT4-based [40] XeSim package [41]. The event generator models neutrino signal events by generating two primary particles: an electron with energy $E_{\nu}-Q-E_{x}$ and a Cs nucleus in an excited state. The incident neutrino energy can be drawn from any component of the solar neutrino spectrum. The events are simulated in a cylinder of LXe that is 1.2 m in both diameter and height, which approximately models few-tonne detectors such as nEXO, LZ, or XENONnT. The de-excitation of ${}^{131}\textrm{Cs}$ is simulated using the nuclear levels, branching ratios, and lifetimes from the appropriate data file in the built-in PhotonEvaporation3.2 library. For ${}^{136}\textrm{Cs}$, we replace the existing data file with one containing the decay scheme shown in Fig. 4.

To mimic the response of an actual detector, simulated neutrino events are processed by the algorithm developed in Ref. [42], which groups individual energy deposits into “clusters” within the assumed position resolution of the detector. The energy for each cluster is also smeared using a Gaussian with width $\sigma_{E}$ defined by

where $E$ is the total energy deposited in the cluster. We use the $a$ and $b$ parameter values from the charge-plus-light and charge-only energy resolution fits in Refs. [43] and [44], respectively. The values are $a=0.317\,\textrm{keV}^{1/2}$ and $b=0.015$ for the combined charge-plus-light energy and $a=0.75\,\textrm{keV}^{1/2}$, $b=0.035$ for charge-only. In a real experiment, charge detection of low-energy clusters may be limited by a threshold, leading to a loss of some clusters in an event. We omit this effect here for simplicity. The detection efficiency of each Cs state is evaluated by checking if, for the $N$ clusters in the event, there is any sub-combination of $N-1$ clusters which fall within $2\sigma$ of the state energy (the $N^{\textrm{th}}$ cluster is assumed to be the signal from the primary fast electron). The efficiency is approximately independent of incoming neutrino energy as the primary fast electron preferentially forms a single-site energy cluster.

The calculated efficiencies for ⁷Be neutrino scattering to each of the low-lying states in ${}^{131}\textrm{Cs}$ and ${}^{136}\textrm{Cs}$ are shown in Table 5 assuming a 5 mm resolution in $z$ and either 10 mm or 30 mm resolution in $x-y$. An efficiency of $\sim$50% is found for the most favorable states. The position-based tagging efficiency is highly dependent on the energy of the state: the higher-energy states are likely to emit more energetic photons, which have longer mean free path, while lower-energy states are more likely to emit lower-energy photons and IC electrons, which deposit energy very close to the primary nucleus. The results presented here cover the range of expected position resolutions in next-generation detectors: dark matter experiments quote projections as good as 3 mm in the $z$ dimension [45] and 15 mm in the $x-y$ dimension [46]. The efficiencies reported here do not change drastically with $x-y$ resolution due to the relatively good resolution assumed for $z$. Worse resolution in $z$ would require better $x-y$ resolution to maintain the same tagging efficiency.

In order for two events to be distinguishable in a coincidence analysis, their primary scintillation signals must be separated in time by more than the typical reconstructed pulse width. This quantity depends on multiple factors including the xenon scintillation decay time, the size of the detector, the chosen photosensor coverage and characteristics, and the detector electronics. Here we take $t_{1}$ to represent the minimum time separation between the first and second scintillation signals for which these two pulses are distinguishable.

The acceptance for a delayed signal to arrive in a coincidence window of length $\Delta T$ which begins at time $t_{1}$ after a primary pulse is

where $\tau$ is the mean lifetime of the excited nuclear state. In the present work we will take $t_{1}=0.5$ $\mu$s and assume that two primary scintillation pulses will be distinguishable if they are separated by at least this amount. As a coincidence window length we take $\Delta T=2000$ $\mu$s, consistent with the expected event acquisition length of a future TPC with drift length $\gtrsim 1.2$ m. For the decay of the 23 keV $3^{+}$ state in ${}^{136}\textrm{Cs}$ with mean lifetime 624 $\mu$s, this coincidence window accepts 96% of decays. If in reality the first $3^{+}$ state lies higher in energy than the shell model predicts and the coincidence results from the experimentally measured E2 transition of the 105 keV $4^{+}$ state with our calculated lifetime of 360 ns, the acceptance is 25%. In this case the loss in efficiency comes from the minimum time separation requirement. Fig. 5 shows more generally how the acceptance changes with the excited state lifetime for a few choices of the window start time $t_{1}$ and our chosen window duration.

To reduce backgrounds further, two additional requirements are imposed to select neutrino capture events. First, the energy associated with the delayed pulse should be consistent with the energy of the final de-exciting nuclear state. For our analysis we will accept secondary pulses in a $3\sigma$ energy window centered on the excited state energy (the efficiency is then 99.7%). Second, the reconstructed position of the delayed event vertex must be sufficiently close to that of the prompt signal, which is comprised of all energy deposits that occur before the long-lived nuclear state is reached. We require that the delayed signal be located within a 5 cm radius of the weighted-average position of the prompt energy deposits. Simulations from Sec. III.1 show that such a cut will accept 93.5% of neutrino capture events. The total efficiencies for selecting events from the 624 $\mu$s and 360 ns states are then 89.8% and 23.4%, respectively.

We now fold the calculated detection efficiencies with the expected interaction rates to predict the number of tagged events in a given experiment. We consider the cases of neutrino capture on ${}^{131}\textrm{Xe}$ and ${}^{136}\textrm{Xe}$ separately. We estimate the event rates that would be observed in DARWIN [4], which proposes to use a 40-tonne target of natural xenon, and nEXO [3], which will deploy a $\sim$4-tonne target of xenon enriched to 90% in ${}^{136}\textrm{Xe}$.

The low cross sections for neutrino capture on ${}^{131}\textrm{Xe}$ result in event rates which are likely too low for next-generation experiments. The most promising signal is topologically-tagged events produced by populating the 373 keV level in ${}^{131}\textrm{Cs}$. Given the efficiencies and interaction rates calculated above and a natural abundance of 21.2%, we calculate an event rate of 0.3 (0.04) events/yr for ⁷Be (CNO) neutrinos in DARWIN. For a ten-year exposure, this would produce a total of $\mathcal{O}(1)$ events, from which it would be difficult to draw any conclusions. An experiment using xenon enriched in ${}^{136}\textrm{Xe}$ will have essentially no ${}^{131}\textrm{Xe}$ in the target volume, so we expect no events from this process in nEXO.

In the case of ${}^{136}\textrm{Xe}$, we find more promise in the possibility of a delayed coincidence analysis, which potentially allows higher detection efficiency for neutrino capture events. In addition, the cross section for exciting the 590 keV state in ${}^{136}\textrm{Cs}$ is substantially higher than that for any of the low-lying states in ${}^{131}\textrm{Cs}$. The expected tagged event rates for ⁷Be and CNO solar neutrino scattering to the 590 keV state in ${}^{136}\textrm{Cs}$ are 5.3 and 0.63 events per tonne of ${}^{136}\textrm{Xe}$ per year. DARWIN and nEXO will contain approximately 3.5 and 3.6 tonnes, respectively, meaning they will each observe $\approx 19$ ⁷Be neutrino capture events per year and $\approx 2.2$ CNO neutrino captures per year. We have neglected to include the effect of an energy threshold when calculating the detection efficiency for the state at 23 keV; while this is expected to be a good approximation for experiments detecting the charge via electroluminescence amplification such as DARWIN, a dedicated study would need to be made for experiments such as nEXO where the ionized charge will be measured directly (and may therefore be dominated by readout noise in the energy regime below $\sim$100 keV).

The primary source of background events for the topological analysis are decays of ²¹⁴Pb. This isotope enters the LXe volume as a daughter of ²²²Rn which emanates from surfaces in the LXe system. Approximately 90% of ²¹⁴Pb $\beta$ decays leave the ²¹⁴Bi daughter in an excited state which decays rapidly ($\tau\lesssim 100$ ps) to the ground state by emitting $\gamma$-rays and IC electrons. The topology of these $\beta+\gamma/e^{-}$ events can mimic that of the neutrino capture signal when the incident neutrino energy falls below the ²¹⁴Pb $\beta$-decay $Q$-value of 1018 keV. For higher energy neutrinos, the dominant multi-scatter background is likely from external $\gamma$-rays which may be suppressed with a detector-specific fiducial cut, and we therefore do not quantify this background here.

The background from ²¹⁴Pb is assessed by simulating its decay and processing the events in the framework described in Sec. III.1. Table 6 shows the efficiency with which ²¹⁴Pb decays pass the selection used to search for each of the Cs excited states as well as the expected background rate in a detector containing 1 $\mu$Bq/kg of ²²²Rn. For comparison, the table also shows the same quantities for the ⁷Be solar neutrino capture signal. We conclude that a future experiment could only hope to detect the capture signal by this method provided the detector be practically radon-free, containing $\lesssim 10^{-5}$ $\mu$Bq/kg ²²²Rn.

The stringent selection requirements for the delayed coincidence signature defined in Sec. III.2 lead to an estimated background rate that is very small. The expected background is comprised of accidental coincidences of low-energy events (within $3\sigma$ of the delayed energy) following any other event with energy consistent with the prompt neutrino capture signature. In this study we consider three sources of low-energy events: $2\nu\beta\beta$ decay of ${}^{136}\textrm{Xe}$, scattering of solar neutrinos on atomic electrons, and $\beta$ decays of ²¹⁴Pb. These are expected to dominate the background rate in the 0–100 keV energy range relevant for tagging the delayed signals in ${}^{136}\textrm{Cs}$ (see for example Ref. [1]). The energy spectra for these backgrounds are shown in Fig. 6. We note that while a delayed timing signature from the $\beta-\alpha$ decay sequence ${}^{214}\textrm{Bi}\rightarrow^{214}\textrm{Po}$ ($\tau=236$ $\mu$s) will also be present, the large (7.7 MeV) energy of the secondary $\alpha$ pulse excludes these events from our selection.

From the rates in Fig. 6, we derive the expected event rate within a 5 cm radius (corresponding to a 1.47 kg mass) and $3\sigma$ energy window centered at 23 keV and 105 keV to be 0.23 evts/yr and 1.1 evts/yr, respectively. This yields probabilities for coincidence with a primary event in the 2000 $\mu$s search window of $<10^{-10}$. We note that this estimate is relatively robust against theoretical uncertainties on the energy of the delayed ${}^{136}\textrm{Cs}$ state, as the event rate changes by less than a factor of $\sim 3$ across the relevant energy range. In a detector using a ${}^{136}\textrm{Xe}$-enriched target, the background rate from $2\nu\beta\beta$ will be approximately an order of magnitude higher, leading to a correspondingly higher coincidence probability.

The total number of events in an experiment that mimic the primary neutrino capture signal then determines the overall background estimate. As an example, the nEXO experiment is expected to observe approximately $10^{8}$ events in the 700–3500 keV range over ten years of operation [47]. Given the coincidence rates calculated above we expect $\ll 1$ background event to pass our selection, making a delayed coincidence analysis essentially background free.

The detection of CNO neutrinos as a means of completing a detailed picture of power generation and dynamics in our Sun has been a long-standing and sought-after experimental goal for modern solar neutrino experiments. After much experimental effort, the first detection of CNO neutrinos was recently reported by the Borexino collaboration [51]. Here we briefly mention the prospects for CNO detection using the capture signature on ${}^{136}\textrm{Xe}$.

Excluding the lines from ⁷Be and $pep$, the flux of solar neutrinos that lies between the $pp$ endpoint at 0.42 MeV and the CNO endpoint at roughly 1.73 MeV is dominated by the CNO contribution. The ratio of CNO flux to the combined ${}^{8}\textrm{B}+hep$ flux in this region is approximately $4\times 10^{3}$. Therefore, any event consistent with a capture signal on ${}^{136}\textrm{Xe}$ with reconstructed energy in this window (less the reaction threshold $Q$) is overwhelmingly likely to be a CNO event, while events around the ⁷Be and $pep$ energies can be easily removed from that sample. As an example, we project using Table 3 that a ten-year exposure of the 90% enriched, 4 tonne fiducial mass of the nEXO detector would expect to see approximately 32 total CNO events with a background of $<1$ event using the delayed coincidence technique. In combination with the uncertainty on the measured $B(\mathrm{GT}^{-})$ values [28], the fractional uncertainty on this count rate is roughly a factor of two lower than that obtained by the current energy spectrum fit in Borexino.

It is interesting to note that this approach contrasts with the case of neutrino-electron scattering in LXe studied in Ref. [52]. There, a CNO detection at $\sim 3\sigma$ significance in a dark matter experiment could only be achieved with significant depletion of the ${}^{136}\textrm{Xe}$ isotope. By contrast, our discussion suggests that this isotope may be well-suited for such a detection with the caveat that one does not accumulate enough events to precisely reconstruct the energy spectrum. However, it has the benefit that such a search can proceed concurrently with the primary science goals of the experiment and the cost of isotope separation can be avoided.

The opportunity for CNO detection in other future experiments was explored in Ref. [53]. Specifically, a CNO measurement in a five-year exposure of the SNO+ [54] detector may be possible provided this run is free of the $2\nu\beta\beta$ decay isotope ¹³⁰Te. In fact, an argument could be made in favor of a search for CNO neutrino capture on the ¹³⁰Te isotope itself during the SNO+ primary physics run. However, the capture rate on ¹³⁰Te calculated by Ejiri in [35] suggests that the roughly 0.8 tonnes of isotope deployed in SNO+ would not be competitive with the analogous rate in a LXe experiment as discussed here. Existing nuclear data for the product ¹³⁰I does suggest a possible delayed coincidence signature [55]; however the energies of the relevant excited states fall below the current SNO+ energy threshold.

In Ref. [56], Bahcall points out that the neutrino energy line shape resulting from the electron-capture decay of ${}^{7}\textrm{Be}$,

in the Sun is distorted compared to that observed by a laboratory source of ${}^{7}\textrm{Be}$. In the stellar core, electrons are captured from unbound continuum states in the thermal bath, and so the resulting neutrino energy spectrum is broadened by thermal effects. The shape of the broadened spectrum and, in particular, its mean energy therefore depend on the Sun’s central temperature distribution. The assumptions used in [56] predict a mean neutrino energy which is 1.27 keV higher than the monoenergetic 861.8 keV line expected from Eq. (8) in the laboratory.

Detection of this shift requires reconstruction of the incident neutrino energy, making neutrino capture an ideal process for this measurement. The predicted capture rate on ${}^{136}\textrm{Xe}$ and the potentially background-free event signature may enable such a detection in future LXe experiments. Capture events on ${}^{136}\textrm{Xe}$ result in a a total energy deposit of $E_{\nu}-Q$, where $Q=90.3$ keV is the reaction threshold. The total event energy expected from a laboratory source of ${}^{7}\textrm{Be}$ neutrinos is therefore 771.5 keV.

To quantify the likelihood of observing the broadened neutrino spectrum of [56] we compute the significance with which an experiment would reject the “laboratory energy” hypothesis in favor of the hypothesis that the incident neutrinos follow the spectrum predicted by Bahcall. As a test statistic TS we use the distance between the mean energy of events $\bar{E}_{\mathrm{obs}}$ and the energy expected from the laboratory neutrino source $E_{\mathrm{lab}}=771.5$ keV normalized by the error on the observed mean energy:

Here, $\sigma_{E_{\mathrm{obs}}}$ is the standard deviation of observed energies and $N_{\mathrm{obs}}$ is the observed number of events in the exposure. For each assumed exposure of ${}^{136}\textrm{Xe}$, a set of Monte Carlo experiments were performed in which a Poisson random number (with mean calculated using the rate in Table 3) of energies are sampled from the broadened neutrino spectrum and smeared by a Gaussian detector energy resolution. The TS value from each experiment is then used to calculate a $p$-value using the distribution of TS’s calculated from experiments which sample the monoenergetic laboratory spectrum.

The median significance and surrounding 68% interval for observing the broadened spectrum is shown as a function of ${}^{136}\textrm{Xe}$ exposure in Fig. 7. In the figure we show curves for three benchmark values of the detector’s multi-site energy resolution at 771.5 keV. We note that a resolution of approximately 1.5% has been demonstrated in the XENON1T detector [43]. Our results rely on the assumption that the absolute energy scale can be known to sufficient accuracy.

Our results indicate that a detection above $3\sigma$ may be attainable, for example, with an 80 tonne natural xenon experiment that runs for ten years (achieving a ${}^{136}\textrm{Xe}$ exposure of 71 tonne-yr). We also note that a more severe shift of the neutrino energy spectrum (e.g. from a re-evaluation of the solar parameters assumed in [56]) would reduce the exposure required for a detection.