Future CEvNS experiments as probes of lepton unitarity and light-sterile neutrinos

We determine the sensitivities of short-baseline coherent elastic neutrino-nucleus scattering (CE$\nu$NS) experiments using a pion decay at rest neutrino source as a probe for non-unitarity in the lepton sector, expected in low-scale type-I seesaw schemes. We also identify the best configuration for probing light-sterile neutrinos at future ton-scale liquid argon CE$\nu$NS experiments, estimating the projected sensitivities on the sterile neutrino parameters. Possible experimental setups at the Spallation Neutron Source, Lujan facility and the European Spallation Source are discussed.


INTRODUCTION
The three-neutrino paradigm has been put in rather solid grounds from the interpretation of solar and atmospheric oscillation data and the complementary results from reactor and accelerator neutrino studies [1]. Underpinning the precise way by which neutrinos get mass is one of the main current challenges in particle physics. One of the leading ideas is that neutrino mass generation proceeds through the mediation of new heavy fermion states, such as in variants of the so-called type-I seesaw mechanism. Since they carry no anomaly, isosinglet "right-handed" mediators can come in an arbitrary number in the Standard Model (SM), so one can envisage low-scale seesaw realizations, where the mediators can lie at the TeV scale with potentially sizeable mixing with the light neutrinos [2][3][4][5]. The admixture of heavy lepton messengers implies that the charged current weak interaction mixing matrix has a rectangular form [6], leading to unitarity violation, as these heavy states are not kinematically accessible. Likewise, one expects universality violation effects. The associated processes could take place below [7], at [8][9][10] or above [11][12][13][14] the Z boson mass scale. In the context of neutrino propagation, the admixture of heavy neutrinos would clearly also imply deviations from unitarity, as the heavy states can not take part in oscillations.
Unitarity violation in neutrino oscillations has been explicitly considered in [15][16][17][18][19][20][21][22]. It has been noticed that the extra CP violation expected in these schemes can fake the one present within the simplest three-neutrino paradigm [23]. As a result, unitarity violation degrades the CP violation sensitivity expected at DUNE [22]. Here we note that the subleading effects of such TeV-scale heavy neutrino mediators can also be probed in future liquid argon coherent elastic neutrino-nucleus scattering (CEνNS) experiments using muon decays as the neutrino source.
On the other hand, controversial anomalies such as those coming from recent reactor data, as well as those hinted by the LSND [24] and MiniBooNE [25] experiments, inspired many phenomenological studies beyond the simplest three-neutrino oscillation picture [26][27][28]. These are based on the existence of a fourth light sterile neutrino state, with eV-scale mass (m 1,2,3 m 4 ). Indeed, under certain circumstances, such as special symmetries [29,30], one may expect such extra light sterile neutrinos to emerge in fermion mediator models of neutrino mass generation.
The importance of neutral currents in oscillation physics has been noticed since the early days, see Refs. [6,31]. The discovery of CEνNS has now brought neutral-current-based experiments to the center of the stage, as a competitive and complementary tool to shed light on fundamental neutrino parameters. Facilities looking for CEνNS have been recognized to be important probes of sterile neutrino oscillations, since about a decade [32][33][34]. In 2017, the COHERENT collaboration reported the first observation of CEνNS on CsI [35] at the Spallation Neutron Source (SNS), a result recently confirmed by the same collaboration on a liquid argon detector [36]. This prompted a new era with a wide range of physics applications concerning open questions within [37][38][39][40][41][42][43] and beyond the SM , including also dedicated sterile neutrino searches [69][70][71][72]. The field is thriving rapidly, with several experiments aiming to measure CEνNS now in preparation worldwide (for a review see Ref. [73]), many of which are planning to employ large liquid argon detectors. Here we quantify the prospects for probing the effects of both light-sterile neutrinos as well as heavy neutrinomass-mediators within future proposals employing large liquid argon scintillation detectors. In particular, we concentrate on the next generation detector subsystem of COHERENT, namely CENNS [74], as well as on the Coherent Captain-Mills (CCM) experiment [75] at the Los Alamos Neutron Science Center -Lujan facility, and on the CEνNS program developed at the European Spallation Source (ESS) [76].
Our work is organized as follows: in Sec. 2 we present the required formalism for simulating CEνNS signals and discuss the experimental sites considered. In Sec. 3 we present our results concerning non-unitarity effects induced by new heavy neutrino admixtures and in Sec. 4 we discuss the sensitivities we have obtained on light sterile neutrinos. Finally, we summarize and conclude in Sec. 5.

SIMULATING COHERENT ELASTIC NEUTRINO NUCLEUS SCATTERING
Our present research on indirect effects of heavy neutrino states or light sterile neutrinos is motivated by future neutral-current CEνNS measurement proposals. Previous work on sterile-neutrino constraints from the CsI COHERENT measurement can be found in [48]. We consider the process ν α + (A, Z) → ν β + (A, Z) where A and Z stand for the mass and atomic number of a nucleus, respectively, while E ν is the neutrino energy and α, β represent the flavor index (α, β = e, µ, τ ). In this section, we summarize the relevant formalism for simulating the expected CEνNS signal and discuss the various experimental configurations considered in our analysis.

Coherent elastic neutrino nucleus scattering
The CEνNS cross section scales as N 2 , where N = A − Z is the number of neutrons and, therefore, leads to an enhanced neutrino interaction cross section [31]. The relevant CEνNS experiments are mainly sensitive to the tiny recoils generated in a scattering event. The differential cross section in terms of the nuclear recoil energy, T A , is [77] dσ Here, G F denotes the Fermi constant, m A is the nuclear mass, and Q V W is the weak charge [78] written in terms of the weak mixing angle sin 2 θ W = 0.2312, taken in the MS scheme. A coherence loss, due to the finite nuclear size, is incorporated through the nuclear form factors for protons and neutrons, F p,n (Q) 2 . Amongst the various available parametrizations in the literature (for a summary see Ref. [56]), here we employ the well-known Helm form factor, given by where the magnitude of the three-momentum transfer is Q = √ 2m A T A , the spherical Bessel function of order one is j 1 (x) = sin(x)/x 2 −cos(x)/x, and R 2 0 = 5 3 (R 2 p,n −3s 2 ). For the relevant liquid argon detectors, the neutron and proton rms radii take the values R n = 3.36 fm and R p = 3.14 fm, while the surface thickness is s = 0.9 fm.
As for the incoming neutrino flux, at spallation source facilities a large number of protons is scattered on a nuclear target (mercury for the SNS and tungsten for CCM and ESS), producing pions. The latter propagate and subsequently decay at rest generating neutrinos (π-DAR neutrinos). A monochromatic neutrino beam is produced from π + → µ + ν µ (prompt flux, with lifetime τ = 26 ns) with a spectrum given by The subsequent muon decay µ + →ν µ e + ν e (delayed flux, τ = 2.2 µs) generates a beam composed of muon antineutrinos and electron neutrinos

Experimental sites
Our present analysis will be focused on three prominent experiments aiming to deploy large liquid argon detectors (see Table I) to measure a CEνNS signal at a π-DAR source. We first consider the next generation CENNS detector of the COHERENT experiment at the SNS [74], which is expected to replace the CENNS-10 detector that provided the first detection of CEνNS on argon [36]. The planned configuration will contain a 750 kg (610 kg fiducial) liquid argon scintillation detector and will operate with a 20 keV threshold and a baseline of 28.4 m. Another interesting experimental site is the proposed CCM experiment, located at Los Alamos National Laboratory, in the Lujan facility. The CCM experiment plans to install a large 7 ton liquid argon detector and is expected to achieve a 1 keV threshold [75]. The detector will be placed 20 m from the source with the goal to search for sterile neutrinos. Another promising facility is the ESS located in Lund, Sweden, that combines the world's most powerful superconducting proton linac with an advanced hydrogen moderator, generating the most intense neutron beam for different purposes. Following the proposal [76], we will assume two different configurations: i) a first phase configuration with a 10 kg liquid argon detector and an ultra-low 0.1 keV threshold and ii) a next generation configuration with a 1 ton liquid argon detector and a 20 keV threshold, both located 20 m from the source.
The main difference among ESS, SNS and Lujan facilities is that the former one is scheduled to reach a power of 5 MW with a goal energy of 2 GeV by 2023, while SNS (Lujan) will have a power of 1.3 MW 1 (80 kW). This will lead to about one order of magnitude increase in the ESS neutrino flux with respect to SNS, resulting into a significantly faster accumulation of CEνNS signal statistics in comparison with the other two facilities. A second difference is the proton beam pulse timing: SNS provides 60 Hz of 1 µs-wide POT spills while ESS can only offer 14 Hz of 2.8 ms spills, reducing the relative capability of separating  the neutrino flavors with timing information. Finally, while the power of the proton beam at Lujan is 1-2 orders of magnitude smaller than in SNS and ESS, it is worth mentioning that, in contrast to SNS, the CCM experiment can deploy very large ton-scale detectors. This feature, together with the fact that Lujan Center can achieve a shorter beam time interval, makes the CCM experiment clearly complementary to the CEνNS searches at SNS and ESS.

Statistical analysis
Our statistical analysis is based on the expected number of events, simulated for each experiment. For the case of CEνNS, the differential number of events is given by where t run is the data taking time (we will assume t run = 1 year), N target is the number of nuclear targets in the detector, and x = (SM, new) denotes the type of interaction.
Here, η denotes a normalization factor given by η = rN POT /4πL 2 , where L is the baseline, N POT is the number of delivered protons on target (POT) and r is the number of produced neutrinos per POT. If not mentioned otherwise, and given the absence of relevant information regarding backgrounds and detection efficiencies of the future experiments considered here, our analysis will be mainly based on a simple statistical analysis following the χ 2 function where N x represents the number of events evaluated by integrating Eq. (7) over the nuclear recoil energy. Here, N SM refers to the number of events expected according to the SM, while N new includes an extra contribution associated to the relevant new physics of interest.

HEAVY SINGLET NEUTRINOS AND NON-UNITARITY
Here we assume that, in addition to the three standard light neutrinos, one has extra singlet neutral heavy leptons that mediate light-neutrino mass generation. It is well-known that such heavy leptons will couple sub-dominantly in the weak charged current, via mixing with the SM isodoublet neutrinos [6]. In the most general case, their presence and mixing with the active neutrinos respects the chiral SM structure. Alternatively, we also consider the possibility of light sterile neutrinos taking part in oscillations. Both lead to new features beyond the minimal three-neutrino oscillation paradigm. Here we note that constraining non-unitarity effects at short-baselines plays a crucial role in mitigating the ambiguities present in testing for leptonic CP violation in long-baseline neutrino oscillation experiments [23]. We propose to do this through the neutral current.
In this section, we consider CEνNS experiments in the presence of unitarity violation effects. To set up notation, we write the relevant generalized charged current weak interaction mixing matrix as where U 3×3 denotes the standard unitary lepton mixing matrix and N NP represents the new physics (NP) matrix which accounts for unitarity violation [21]. The latter is parametrized as with the diagonal (off-diagonal) components α ii (α ij ) being real (complex) numbers. In this context, the oscillation probability for ν α → ν β transitions reads The survival probabilities P ee and P µµ 2 and the transition probablility P µe simplify to [21] P ee =α 4 11 P 3×3 ee , P µµ =α 4 22 Here, P 3×3 ee , P 3×3 µµ and P 3×3 µe denote the standard oscillation probabilities, while the extra terms P I 1 µµ and P I 2 µµ are defined in Ref. [21]. Notice that P I 1 µµ depends on a new CP violation phase, I N P , while P I 2 µµ is phase-independent.
For the short-baseline CEνNS experiments we are interested in here, there is no time for oscillations among active neutrinos to develop. Hence, the baseline-dependence in Eq.(12) is not relevant 3 . Therefore, the effect of the heavy neutrino states at CEνNS experiments will be mainly due to the zero-distance effect, i.e. P αβ (L = 0). The zero-distance probabilities are given as P µe = α 2 11 |α 21 | 2 , P eτ = α 2 11 |α 31 | 2 , P µτ α 2 22 |α 32 | 2 , while the following "triangle inequalities" among the elements of the N NP matrix hold [22? ] Within this context, due to the zero-distance effect, neutrino fluxes at a spallation source are modified as follows: Most generally, the above expression can be written compactly as with P αβ = P (ν α → ν β ). Given appropriate choices for its entries, Eq. (16) holds for any neutrino experiment with an arbitrary type of neutrino source and no charge identification. For the specific spallation case, only the initial ν e , ν µ and ν µ are non-vanishing, so we obtain the expressions in Eq. (15).
FIG. 1: Flavor composition of the continuous π-DAR neutrino spectra in the SM (solid lines) and with non-unitarity effects (dashed lines), taking for these the maximal deviation parameters α ij allowed at 90% C.L. [22].
Note that, since the experiments under study can not distinguish neutrinos from antineutrinos, we combine both contributions in a flavor-dependent signal, as indicated in Eq. (15). There, we have also assumed that neutrino and antineutrino oscillation probabilities are equivalent: P (ν α → ν β ) = P (ν α → ν β ) = P αβ and also that P αβ = P βα . As seen from Eq. (15), an additional monochromatic ν e beam is generated due to ν µ → ν e transition, as well as a continuousν e spectrum due toν µ →ν e conversion. Similarly, a new tau-neutrino flux is also expected due to ν e → ν τ , ν µ → ν τ andν µ →ν τ oscillations. However, one finds that these fluxes are largely suppressed due to the smallness of the appearance probabilities P eτ and P µτ , well constrained by the existing limits on the non-unitarity (NU) parameters α ij . The flavor components of the corresponding continuous fluxes are displayed in Fig. 1. In this figure, we show the modification of the initial neutrino flux due to the zero-distance non-unitarity effect. The modified spectra have been evaluated using the 90% C.L. limits on the α ij parameters reported in Ref. [22].
In what follows, we give a first estimate on the prospects for probing the unitarity violating parameters at future liquid argon detectors. In order to determine the sensitivity limits on unitarity violation, we proceed as explained in Ref. [21]. For definiteness, we will focus on the detection of electron and muon neutrinos, reducing the number of relevant NU parameters to three: α 11 , α 22 and |α 21 | 4 .
Using the χ 2 function defined in Eq. For comparison we also give the corresponding sensitivity obtained from global oscillation data analysis [22].
ing over the other two, while imposing the constraint coming from the triangular inequality of Eq.(14) as well. The "one-at-a-time" sensitivity profiles of future CEνNS experiments for the diagonal parameters α 11 and α 22 are shown in Fig. 2. Comparing these sensitivities with those derived from global neutrino oscillation data [22], one sees that the CEνNS experiments will eventually become competitive to current oscillation searches. Indeed, while the current configuration of ESS with 10 kg detector mass is not expected to be competitive, the next generation of ESS will certainly have the capability of improving current oscillation sensitivities. In Fig. 3 we illustrate the sensitivities on the modulus of the non-diagonal parameter α 21 . Our results are compared with upper limits obtained from global oscillation fits [22] and with the sensitivity of future ICARUS data, as estimated in Ref. [79]. For the prospects on the |α 21 | sensitivity, we can see that most CEνNS experiments can not compete with current bounds. However, the future ESS configuration may offer the chance of improving this situation drastically. For completeness, we now perform a more realistic sensitivity analysis, considering possible backgrounds and systematic uncertainties in our calculation. In what follows, we explore the projected sensitivities on the diagonal and non-diagonal NU parameters, assuming the χ 2 function where the statistical uncertainty is defined as σ stat = N SM + N bg and the number of background events is taken to be N bg = 10%N SM . Here, a denotes a total normalization  [79] as well as global oscillation data [22] is also given.
factor handled as a nuisance parameter accounting for the systematic uncertainty, for which we employ two benchmark values: σ sys = 2% and 5%. A summary of the bounds we extract is given in Table II. As expected, one finds that a better control of the background events and systematic uncertainties will lead to improved sensitivities. For comparison, the current upper bounds derived from oscillation searches [22] are also given in Table II. One can also perform a combined χ 2 analysis through a simultaneous variation of two NU parameters, and marginalizing over the third one. Our results for the CENNS, CCM and ESS experiments (current as well as next generation setups) are presented in Fig. 4. For each CEνNS experiment, the dark-shaded areas in the α 11 − |α 21 | and α 22 − |α 21 | planes located to the right of the lines are allowed at 90% C.L. by the corresponding experiments. The region consistent with the triangle inequality of Eq. (14) is the one below the dashed line in both panels. Therefore, the allowed values in the α ii − |α 12 | plane are eventually determined by the intersection of the gray shaded area with the allowed region determined by each experiment's sensitivity. We find that CENNS and CCM have the potential to probe part of the currently allowed parameter space. As before, the most promising experimental setup is provided by the next phase of ESS with a ton-scale detector.  We also give a comparison with results from the global neutrino oscillation data analysis [22].

LIGHT STERILE NEUTRINOS IN (3+1) SCHEME
Though the theoretical motivation is not specially strong, there could well be singlet neutrinos in nature, light enough to take part in oscillations, usually known as light sterile neutrinos. Although this situation differs from what we have considered above, it can be described within the same formalism developed in [6]. Here we present basically the same reasoning in somewhat more modern form. There is a basic difference compared to most neutrino oscillation experiments, in which neutrinos are produced and detected through the charged current (CC) weak interaction. Here neutrinos are produced conventionally, but detected through the neutral current, as illustrated in Fig. 5. The other important difference is that, since we can not identify neutrino flavors, the process of interest is necessarily inclusive, the observable being simply the recoil of the relevant nucleus.
For definiteness, we take the simplest (3+1) scheme with 3 active neutrinos ν α (α = e, µ, τ ) and one light sterile neutrino. The overall quantum-mechanical amplitude for the process of interest is given as where the initial flavor index α is fixed, while β is summed over the three flavors, and the  14), while the yellow region above corresponds to the unphysical area.
FIG. 5: Feynman diagram representing the charged current production, followed by oscillation and neutral current detection. There is a sum over the subindex β Roman (neutrino) mass index is summed from 1 to 4. One sees that, in the production CC vertex, one has the rectangular lepton mixing matrix K, then one has the evolution factor 5 , and finally the NC detection vertex characterized by the projective matrix P = K † K = P 2 = P . Assuming the charged leptons to be in their diagonal basis we can identify K with the truncation of the 4 × 4 unitary matrix U diagonalizing the neutrinos, so the active flavors are expressed in terms of the four mass eigenstate neutrinos ν i (i = 1, 2, 3, 4) as ν α = 4 i U αi ν i . From this equation, we see that the survival probability to active neutrinos, P α = 3 β P αβ , is given as where Greek indices run up to 3 and Latin ones up to 4. This result corresponds to Eq. (4.13) in Ref. [6]. Taking into account that the propagation factors are too small for the distances under consideration, except when the light sterile neutrino, corresponding to i = 4, is involved, we have Notice that, as explained above, the active survival probability P α includes all the weak neutrino flavor states, ν β . Note also that we can neglect the "appearance" part of this probability (i.e. the sum over the final ν µ and ν τ states for the case of an initial ν e ), in comparison with the "survival" ν e contribution. Indeed, the appearance probabilities will involve products of the form sin 2 θ i4 sin 2 θ j4 , and will be more suppressed than the "survival" part, that goes as sin 2 θ 14 . Hence, the above expression will lead to the usual vacuum survival probability and similarly for muon neutrinos with θ 14 , θ 24 being the mixing angles and ∆m 2 41 ≈ ∆m 2 42 the mass splittings. The presence of the sterile neutrino is taken into account in the CEνNS process through the substitution Q W → Q W P αα (E ν ) in the SM weak charge of Eq.(2).
We will now estimate the sensitivity of future CEνNS experiments to the light sterile neutrino scenario. To do this, we will use the formalism described in previous sections, but replacing the neutrino oscillation probabilities in Eq. (15) by the expressions in Eqs. (21) and (22) above. Note that, unlike the case of non-unitarity, here oscillation probabilities depend on the neutrino energy. Our treatment of this scenario will be also slightly different, and we will consider independently oscillations in the channel ν e → ν s and ν µ → ν s .
As a first step, we explore the optimal baseline for light sterile neutrino searches with CEνNS detectors. For this purpose, we fix the sterile neutrino mixing parameters to benchmark values:  Fig. 6. As discussed before, we estimate independently the sensitivity for the electronic and muonic channel. In all cases, the maximum sensitivity is reached around L = 30 m, very close to the proposed baselines. One sees how the CENNS and CCM experiments have the best sensitivity. One can also remark the larger sensitivity to sterile searches in the ν µ → ν s channel in comparison with ν e → ν s . Note, however, that to distinguish between these two oscillations channels, timing information would be required [70].
We also find it useful to examine the sensitivity of the CEνNS experiments to the mass splittings ∆m 2 41 . In the right panel of Fig. 6 we illustrate the corresponding χ 2 profiles by fixing sin 2 2θ 14 = 0.1 or sin 2 2θ 24 = 0.1 and the baseline to L = 30 m. As previously, CENNS and CCM perform better, while significantly higher sensitivities are reached when muon neutrinos are involved. This is due to the larger flux of muon-like events emitted at spallation sources. One also sees that, for our chosen mixing angle benchmarks, the ∆m 2 41 mass splitting values for which one has better sensitivity are 1.5 eV 2 and 6 eV 2 .
The attainable sensitivities of CENNS and CCM are very similar, despite the large difference with respect to their active detector masses. Indeed, the highly intense neutrino flux available at the SNS can compensate the gain in exposure due to the large detector of CCM (see Table I). On the other hand, the results obtained for the current configuration of ESS with a 10 kg detector mass are promising, yet not competitive to the latter two since the detector size in this case is smaller by 2-3 orders of magnitude. However, ESS offers the most intense neutrino beam, motivating us to perform an alternative analysis regarding its future configuration with a 1 ton detector mass and a 20 keV threshold. As illustrated in the left and right panels of Fig. 7, ESS-based sterile neutrino searches are expected to be very promising in the long run. Indeed, the highly intense neutrino beam available at the ESS can yield a very large number of events, making CEνNS very relevant for short baseline oscillation searches.
We now explore how the sterile neutrino parameter space can be probed via CEνNS mea-surements at future large liquid argon detectors. In our analysis, we vary simultaneously the mixing angle sin 2 2θ i4 and and the mass splitting ∆m 2 41 = ∆m 2 42 , for different baselines. The sensitivity curves at 90% C.L. for the different experimental proposals considered in our study are presented in Fig. 8. The results are rather promising, with the same general conclusions regarding the relative performance of the studied experiments. We stress that the future configuration of the ESS experiment can become competitive to current precision oscillation studies. Indeed, our results illustrate the potential of neutral-current measurements in probing the parameter space constrained by global sterile-neutrino analyses, see e.g. [27,28].

CONCLUSIONS
We have analyzed the potential of future CEνNS experiments in probing new physics phenomena in the presence of heavy isosinglet neutrinos and light sterile neutrinos. The purely neutral character of CEνNS makes it complementary to neutrino-electron scattering experiments. Due to its inclusive nature, there is no need for disentangling the sterile neutrino mixing from that of the active neutrinos. Specifically, we have focused on large liquid argon detectors such as those intended to be installed by the COHERENT collaboration at the SNS, as well as CCM at the Lujan facility, and the future CEνNS program at the ESS. It is well-known that the admixture of heavy neutrino mediators of neutrino mass generation in the weak charged current induces an effective departure from unitarity in the lepton mixing matrix. We have explored how this can affect the initial neutrino fluxes for spallation source experiments, and estimated the projected sensitivities on the unitarity-violating parameters. In contrast to long-baseline oscillation searches, for the case of short-baseline experiments only the zero-distance effect is relevant. Our results indicate that future short-baseline CEνNS experiments provide a new probe of indirect signatures associated to heavy neutrino mediators, with sensitivities competitive with results extracted from global neutrino oscillation data. In long-baseline experiments, the interplay between zero-distance and oscillation effects can make the search for non-unitarity effects more challenging. A combination of both types of experiments can certainly offer very promising results [80]. All in all, provided the systematic and statistical uncertainties remain under control, the attainable sensitivities to fundamental parameters of the lepton sector obtained in CEνNS experiments will be competitive and complementary to conventional charged-current-based oscillation searches.
We have also studied the prospects for probing light sterile neutrinos at short-baseline CEνNS experiments. We first verified that the typical baselines of 20-40 m are promising for searches of sterile neutrinos with mass splittings of the order of 1 eV 2 . Given the large statistics that can be accumulated by the relevant ton-scale liquid argon detectors, we concluded that CEνNS -based sterile neutrino searches are feasible, providing complementary information to the conventional oscillation approaches. All in all, we have seen