Constraints on light singlet fermion interactions from coherent elastic neutrino-nucleus scattering

The exotic singlet fermions $\chi$, with a mass $m_\chi\lesssim 50$ MeV, could be produced at the coherent elastic neutrino-nucleus scattering (CE$\nu$NS) experiments through the $\nu {\mathcal N} \rightarrow \chi {\mathcal N}$ process. Due to the coherent enhancement, it offers a unique way to study how $\chi$ interacts with the Standard Model (SM) sector. Based on the most general dimension-6 effective Lagrangian, we perform a comprehensive study on the relevant interaction between $\chi$ and the SM sector. From the current and future COHERENT and future CONUS experiments, we obtain the upper bounds on the Wilson coefficients for the dipole, scalar, vector, and tensor interactions. For $m_\chi $ below 10 MeV, future CONUS data has the best sensitivity, while for $m_\chi$ between 10 MeV$-50$ MeV, the current and future COHERENT bounds dominate. These limits are complementary to those from neutrino oscillation and collider searches. Moreover, the bounds do not depend on the charge conjugation property of $\chi$, nor whether $\chi$ is dark matter or not.


I. INTRODUCTION
Singlet fermions, collectively denoted as χ in this paper, are gauge singlets under the Standard Model (SM) SU (3) c × SU (2) L × U (1) Y symmetries. χ is widely discussed in models of new physics beyond the SM. To name a few, the singlet fermion(s) could play the role as the sterile neutrino(s) in the neutrino mass generation [1][2][3][4][5][6][7][8]. χ is also a popular candidate to account for the anomalies observed at short baseline neutrino oscillations [9][10][11]. Moreover, if the lightest one of χ is stable or cosmologically long-lived, it could be a viable dark matter (DM) candidate [12,13]. The physical mass of χ is highly model dependent, and allows to vary in this paper. Motivated by the possible sterile neutrino warm DM and many neutrino experiments at low energies, in this work, we only consider the cases that χ is much lighter than the SM electroweak scale.
In the ultra-violet (UV) theory, if χ does not carry any beyond SM quantum number, the renormalizable dim-4 Yukawa coupling termLHχ is allowed by the SM symmetries, where L and H are the SM lepton and Higgs doublet, respectively. Below the SM electroweak spontaneous symmetry breaking (SSB) scale, the SM Higgs acquires a nonzero vacuum expectation value(VEV), v H ∼ 246 GeV, and bestows the Dirac mass connecting the SM neutrino and χ. The Dirac mass term leads to the mixings between χ and the SM neutrinos, and it is crucial for both the Dirac and the seesaw-type neutrino mass generation mechanisms. Also, χ can participate in the SM neutral/charged current (NC/CC) interactions through the mixing with the SM neutrinos. In addition to theLHχ Yukawa term, it is also possible to have new scalar couplings through the Higgs portal, where the SM Higgs mixes with the potentially light exotic scalar field(s). Usually, the scalar couplings of χ to the SM Higgs or the exotic scalar boson(s) are suppressed by the active neutrino mass, thus negligible.
On the other hand, if χ is charged under some symmetry G BSM beyond the SM, the above Yukawa term is forbidden in the UV theory. But the effective couplings with the SM neutrinos can be achieved if the symmetry G BSM is broken and the proper mediator exists.
In this case, the mediator(s) could yield new interactions other than the SM NC/CC interactions as well. Moreover, the additional interactions are not necessarily negligible comparing to the dominate two, χ-ν-Z 0 and χ-l − -W + (l = e, µ, τ ), through the χ-ν mixings.
As an example, the new χ-ν interactions emerge in a 3-portal model recently discussed in [14]. This simple model consists a pair of vector fermion, χ L/R , and a singlet scalar, φ x . They are both charged under a hidden gauged U (1) x . It also employs the sterile neutrinos, N R 's, for the type-1 see-saw neutrino mass generation. The N R 's are assumed to be heavy, around the typical lepton number violating scale M L ∼ O(10 12−14 ) GeV, as in the standard high scale type-1 see-saw models. The U (1) x is SSB when φ x acquires a VEV, v x , and M L v x v H is assumed. In addition to the Weinberg operator (LH) 2 , more effective operators emerge simultaneously after integrating out N R 's. For instance, the new (LH)(χ c L φ * x ) operator couples the singlet fermion to the SM leptons. Moreover, it leads to a dim-4 χ c L LH effective interaction by replacing φ x by its VEV v x . Since v x v H , the scalar coupling in this model is much larger than the traditional one, which stems from the χ-ν mixing alone. In addition to the neutrino portal χ-ν-Z 0 NC interaction, there are more interactions from the Higgs-portal and the U (1) Y -U (1) x kinematical mixing gauge portal as well.
For a general discussion below the electroweak scale, we use θ χi to denote the unknown mixing angle, regardless of its UV origin, between χ and the i-flavor SM neutrino. Depending on its mass m χ and θ χi , the singlet fermion can be probed at the neutrino oscillation experiments, the spectrum endpoint in the beta decays, colliders, or the lepton universality tests, and so on [15][16][17][18][19][20]. Recently the coherent elastic neutrino-nucleus scattering (CEνNS), νN → νN , predicted by the SM [21] has finally been observed and confirmed by the COHERENT collaboration [22]. The CEνNS experiment opens up a new avenue to explore the new physics associated with χ. Given a nonzero mixing between χ and the SM neutrino, the relevant effective low energy 4-fermi operators, (νγ µ L χ)(qγ µ q) + h.c., can be generated by the SM NC interaction. If the energy transfer is much smaller than the nucleus size inverse, the elastic scattering cross section will be coherently enhanced. Then any light enough singlet fermion(s), not limited to the one which serves as the dark matter, could be produced in the final state by the process νN → χN orνN → χ c N within the same experiment setup designed to study the SM CEνNS process 1 . As mentioned earlier, the UV theory could potentially generate more effective operators with Lorentz structure different from the NC one just discussed. To account for this possibility, we consider the most general model-independent set of 4-fermi operators. In this work, we should study 1 The inverse process χN → νN was recently discussed for the novel detection of the fermionic DM [23,24], and could be potentially constrained by the CEνNS experiments. The process χN → χN with χ serving as a dark matter candidate has been studied in the CEνNS experiments [25][26][27].
how the minimal set of 4-fermi operators impacts CEνNS with χ in the final state. The most general model-independent dim-6 effective Lagrangian will be considered in Sec.II, followed by the discussion of tree-level CEνNS cross-section and the nucleus form factors in Sec.III. For completeness, some calculation details are collected in the appendix.
Section IV is devoted to the current and future constraints on the Wilson coefficients derived from the current and future COHERENT and CONUS experiments. In Sec.V, it comes the discussion and summary where a toy UV complete model is considered to illustrate the physics.

II. EFFECTIVE OPERATORS
Since the momentum transfer squared, −q 2 , involved in the neutrino-nucleus νN → χN coherent scattering are relatively small, i.e., −q 2 (GeV) 2 , it makes sense for one to consider the effective theory below the electroweak scale. In this energy scale, all the degrees of freedom heavier than electroweak scale have been integrated out; even the SM Z, W ± bosons and top quark are absent in the effective theory. Therefore, it is natural to set the cutoff at the electroweak scale. The most general SU (3) QCD × U (1) QED invariant dim-6 effective Lagrangian 2 can be parameterized as where v H 246 GeV is the vacuum expectation value of the SM Higgs. Note that we use the above convention such that in the hermitian conjugation all the dimensionless coefficients take their complex conjugations but the signs in the Lagrangian remain unchanged.
Here we have dropped all γ 5 -terms associated with the quark, which do not receive coherent enhancement in the low energy νN → χN elastic scattering. Due to the identity that be replaced by {−a * M , −a * E , C * S , D * P , −C * V , D * A , −C * T } by performing charge conjugation to the Lagrangian. Moreover, for the scattering process with χ produced in the final state, without being detected, its charge conjugation properties does not got involved. So despite whether χ is Majorana or Dirac, our discussion applies to both cases.
In general, the unknown coefficients C's and D's could be quark and neutrino flavor dependent. Here the flavor indices are suppressed, and they will be specified only when needed. Note that the contact interaction description is no longer valid for a light bosonic mediator 3 of mass m X 50 MeV, the typical momentum transfer in ν − N coherent scattering experiment using a stopped pion decay source. In this paper, we are only interested in the cases that m X is much larger than the momentum transfer, and our result can be easily translated to set bounds on the strength of coupling-mass ratio for m X 50 MeV.
where s W is the short hand for the weak mixing, sin θ W , E the energy of incoming neutrino, T the recoil energy of nucleus, and M the mass of the target nucleus. The q 2 -dependent effective neutrino-nucleus couplings are related to the fundamental neutrino-quark couplings as follows [29,30]: where Z (N ) is the number of protons (neutrons) in the nucleus, f p Tq (f n Tq ) the fraction of nucleon mass contributed by a given quark flavor q, δ p q (δ n q ) the tensor charges, and F p (q 2 ) (F n (q 2 )) the nuclear form factor for protons (neutrons). Here we adopt the Helm form factors [31] in our analysis, and assume the neutron form factor is the same as proton.
Note that deviations from this assumption are possible due to uncertainties on the rootmean-square radius of the neutron distribution [30]. For the scalar and tensor parameters, we use the following values [29]: which are taken from Refs. [32,33]. The expression for D P (D A ) [A E ] can be obtained by . Note that each photon propagator from the dipole term gives one 1/T proportionality. Since we are not dealing with the UV model, this IR divergence is expected. However, we are not concerned about this IR divergence because the experiments are not sensitive to such low T regions.
Since the final states are different from the initial states, there is no interference between the νN → χN and the SM νN → νN processes. Because neither SM ν nor singlet χ is detected in the scattering, the total coherent scattering cross-section is the sum of Eq.(2) and the SM one. Moreover, due to the same chirality of final states, the dipole, scalar, and tensor interactions can mix and yield different interference patterns. The last two interference terms in Eq.(2) change signs when the incoming neutrino is replaced by anti-neutrino. The Lorentz structures of the interactions not only affect the matrix elements of the cross section, but also modify the corresponding form factors. Therefore, in principle, each of the dipole, scalar, and tensor coefficient can be disentangled with precisely measured differential cross-sections on various targets from both the neutrino and anti-neutrino sources in the future.
Here we show a plot of the differential cross sections for different types of interactions as a function of the nuclear recoil energy for illustration. We take 133 Cs as the target nucleus, and set m χ = 40 MeV, E ν = m µ /2 ≈ 53 MeV. For the Wilson coefficient of each interaction, we consider four cases: The coefficient unmentioned in each case are assumed to be zero. The differential cross sections of νN → χN as a function of the nuclear recoil energy for the four cases are shown in Fig. 2. The SM differential cross section of νN → νN is also shown as the black solid curve for comparison. From Fig. 2, we see that the shape of the differential cross sections largely varies according to the types of interactions. For the dipole interaction, there is a peak at Here we assume that the target nucleus is 133 Cs, m χ = 40 MeV, and E ν = 53 MeV. The SM differential cross section of νN → νN is also shown as the black solid curve for comparison.
which can be derived from Eq. (2). Also, for the vector interaction, the overall shape of the differential cross section of νN → χN is very similar to the SM case of νN → νN , which can be easily understood from Eq. (2) since the two differential cross sections will only differ by an overall factor when m χ approaches 0.

IV. CONSTRAINTS FROM CEνNS EXPERIMENTS
CEνNS was first observed by the COHERENT experiment in a cesium iodide (CsI) detector in 2017. The neutrinos measured by the COHERENT experiment are produced by the π + and µ + decays at the Spallation Neutron Source (SNS) in the Oak Ridge National Laboratory [22]. The energy distribution of the three neutrino flavors at SNS are well known and given by where N = rtN POT 4πL 2 denotes the normalization factor with r = 0.08 being the number of neutrinos per flavor produced per proton collision, t the number of years of data collection, N POT = 2.1 × 10 23 the total number of protons delivered to the target per year, and L the distance between the source and the detector [22]. Here ν µ is monochromatic with E νµ ≈ 30 MeV, and the energies of ν e andν µ are less than m µ /2 ≈ 53 MeV. The expected number of events with recoil energy in the energy range [T , T + ∆T ] can be calculated by where α = ν µ ,ν µ , ν e , m det is the detector mass, M mol the molar mass of the target nucleus, and N A = 6.022 × 10 23 mol −1 . In the SM, the differential cross section for a given neutrino flavor ν α scattering off a nucleus is given by [21] ( also the SM limit of Eq. (2) ): where M is the mass of the target nucleus, g V p = 1 2 − 2 sin 2 θ W ≈ 0.04 and g V n = − 1 2 are the SM weak couplings.
To compare with the COHERENT data collected by the CsI detector, we convert the nuclear recoil energy to the photoelectrons (PEs) by using the relation n PE = 1.17(T /keV) [22] 5 . We also utilize the acceptance function given in Ref. [36], which is where k 1 = 0.6655, k 2 = 0.4942, x 0 = 10.8507, θ(x) is the Heaviside function and n PE the observed number of PEs.
For simplicity, we assume universal flavor-conserving couplings to quarks and neutrinos, and consider four cases with each of the four Wilson coefficients {a M , C q S , C q V , C q T } to be 5 We do not use the new quenching factor given in Ref. [34] since it is still under investigation by the COHERENT collaboration [35]. nonzero 6 . Since the coefficients are flavor-independent, we do not use the timing information at the COHERENT experiment. To evaluate the statistical significance of a new interaction, we define where N i meas (N i th ) is the number of measured (predicted) events per energy bin, α and β are the nuisance parameters for the signal rate and the beam-on background with their uncertainties σ α = 0.28 and σ β = 0.25 [22]. The statistical uncertainty per energy bin This can be understood from the kinematic constraint from Eq. (A2). After marginalizing over T , the kinematic constraint becomes [28], Since the energy of neutrinos at SNS is smaller than m µ /2, we have There are several phases for future upgrades of the current detectors at the COHERENT experiment [37]. Here we consider an upgrade of the liquid argon (LAr) detector with a fiducial mass m det = 610 kg [38] and it is located at L = 29 m from the source. We assume 4 years of data collection with the same neutrino production rate as the current setup, which corresponds to 8.2×10 23 protons-on-target (POT) in total. To estimate the projected sensitivities at the LAr detector, we simulate the number of event predicted in the SM in each nuclear recoil energy bin, with the bin size being 2 keV in the range of 20 keV < T < 100 keV. For the steady-state background, we assume it is uniform in energy and the total is 1/4 of the SM expectation. We also adopt the normalization uncertainty to be 17.5%, which includes the neutrino flux uncertainty (10%), form factor uncertainty (5%), signal acceptance uncertainty (5%), and a QF uncertainty of 12.5% [39]. The projected limits are shown by the dashed lines in Fig. 3. We see that future COHERENT experiment with an upgraded LAr detector can improve the current bounds by about a factor of 2 − 3.
Since the LAr detector has an low energy threshold of 20 keV, which largely limits its ability to constrain the new interactions at low nuclear recoil energies. Here we also explore the limits from the CONUS experiment [40,41], which utilizes a Ge detector with a very low energy threshold. The CONUS experiment measures reactor antineutrinos from a 3.9 GW nuclear power plant in Brokdorf, Germany, and the distance between the detector and reactor is 17 meters. The current Ge detector contains only 4 kg natural Ge, and do not yield significant limits in our scenarios. Here we consider a future upgraded Ge detector with a mass of 100 kg natural Ge. The contributions of each Ge isotope are weighted by its relative abundance. We also assume the nuclear recoil energy threshold is improved down to 0.1 keV, and take the energy bin from 0.1 keV to 2.0 keV with a bin width of 0.1 keV. We adopt the reactor flux calculated in Refs. [42,43] with a conservative 5% flux uncertainty, and assume the background event rate to be 1 count/(day · keV· kg). After for a given T and m χ . Thus, only the neutrino flux of E 2 MeV can contribute to the coherent scattering. Since the typical nuclear power plant neutrino energy spectrum diminishes exponentially as the energy increases, one can take the benchmark point MeV for a ballpark estimation 7 . The leading terms of the dipole interaction differential cross section, Eq.
(2), behaves as . Thus, the dominant part of the event rate, see Eq. (10), {a M , C q S , C q T } . Note that the CONUS experiment has no sensitivity for such large m χ , and we only consider the current and future COHERENT experiment. Here we assume the Wilson coefficients are real, and allow them to be both positive and negative. The 90% CL allowed regions in the parameter space from the current CsI (future LAr) data are shown as the regions enclosed by the black solid (blue dashed) curves in Fig. 4. We see that the correlations between C q S and C q T (or a M and C q T ) are very weak, but there is a slight degeneracy between C q S and a M . The results can be understood from Eq. (2). The correlation between C q S and C q T (a M and C q T ) is negligible because the interference term is 7 Indeed, about 57% of the effective CONUS neutrino flux ( E > 2 MeV) falls in the energy range of (2 − 3) MeV, and the average neutrino energy for E > 2 MeV is E = 3.1 MeV. 8 The interference term between C S and A M is similar to that between D P and A E . largely suppressed by T /E in the last (antepenultimate) line in Eq. (2). Also, from Eq. (14) and the flux given in Eq. (9), we know that for m χ = 40 MeV, the νN → χN process at COHERENT is dominated by theν µ events. Since the penultimate line in Eq. (2) changes sign for antineutrinos, an anti-correlation between C q S and a M agrees with the result shown in Fig. 4 .

V. DISCUSSION AND CONCLUSION
The limits obtained in previous section do not look impressive at the first sight. However, we want to point out that these bounds from CEνNS are quite interesting. First of all, the process is sensitive to all singlet fermions lighter than ∼ 40 MeV. Secondly, as long as From the scan (see Fig. 3), barring the small correlation effects due to mutual cancela- at 90% CL for m χ 0.5 MeV by using the current COHERENT (future COHERENT) [future CONUS] data. For other mass ranges, they can be easily read from Fig. 3. We stress again that the singlet fermions in our analysis need not to be the dark matter candidate.
Hence, the above bounds are general and apply to any model. In particular, they are independent to those with the assumption that χ is the dark matter. For instance, the cosmic gamma-ray line background can only set limit on the dark matter singlet dipole interaction strength, but it has no say on any unstable or short-lived singlets heavier than COHERENT experiment for m χ = 40 MeV. The gray shaded areas are obtained from the current COHERENT CsI data, and the blue dashed curves enclose the expected allowed regions from future COHERENT experiment with an upgraded LAr detector.
the DM singlet. Also, our constraints cannot be inferred from neutrino oscillation data unless further assumption is made to relate the new physics to the SM sector.
To further illustrate the physics discussed above, let us consider the coherent scattering implication to a UV complete model. Our custom-made toy model is a simple extension of the type-I seesaw model with total n right-handed sterile neutrinos. The model La-grangian is trivial and will not be spelled out here. We denote the mass eigenstates as ν ≡ {ν 1 , ν 2 , ν 3 , ν 4 , · · · , ν 3+n }, where ν 1,2,3 are the sub-eV light active neutrinos. For the flavor basis, the notationÑ ≡ {ν e , ν µ , ν τ , χ 1 , · · · , χ n } is adopted. We assume that three out of the n sterile neutrinos, ν 1+n,2+n,3+n , are the heavy ones, decoupled at the low energies, as in the classic high scale type-I seesaw. Moreover, with some parameter turning, the details are not important here, all the other sterile neutrinos, ν 4,··· ,n , acquire their masses, m 4,··· ,n , in the range of (1 − 40) MeV 9 . The mass and flavor states are related by an unitary transformation,Ñ = Uν, which diagonals the neutral fermion mass matrix.
The ν µ → ν e transition probability in vacuum is given by where E ν is the neutrino source energy, L is the distance neutrinos travel from the source, and m 2 i ≡ m 2 i − m 2 1 . At the near detector, L 0, the matter effect is negligible and the exponential factors can be dropped. By unitarity, the probability becomes From neutrino oscillation data only, the current upper bound on this quantity is 7.0 × 10 −4 at 90% CL [44], and could be pinned down to the O(10 −5 ) level with the planned near detector at Fermilab [45]. Note that all information about the mixings with light sterile neutrinos does not present.
A similar bound can be inferred from the SM Z 0 boson invisible decay width measurement. In this toy model, the SM Z 0 boson can decay into any lightν iνj pair except the heavy three. Thus, The current value of N ν = 2.984 ± 0.008 from LEP [46] can be translated to at two sigma level. Again, it is not sensitive to the properties of the light sterile neutrinos. 9 Here, Dark Matter is not our concern.
On the other hand, due to the mixing, the new ν-χ-Z 0 NC interaction exists, and the SM Z 0 boson is the mediator. Due to the coherent enhancement, we have roughly In the above ballpark estimation, one power of m χ is required to flip the chirality to make the dipole interaction. In addition, m q , the SM quark mass running in the loop, is called for to balance the dimensionality. Therefore,the loop-generating a 1−loop E,M are too small, and the tree-level constraints on other 4-fermi νχqq operator, although weak, still matter.
In summary, we have considered the potential to probe the light singlet fermions and their effective interactions with SM quark sector by the current and planned COHERENT and CONUS experiments. The analysis is based on a model-independent dim-6 effective Lagrangian. We find the current constraints from the COHERENT data, although loose, are profound already and complementary to the neutrino oscillation and collider measurements. Future upgraded COHERENT and CONUS experiments will largely improve the sensitivity to new interactions, which allows us to see more details on the limits. We find that there is a small kink on the CONUS bound on the dipole interaction strength at m χ ∼ 4 MeV which arises due to partial cancelation in the differential cross section. Also, the CONUS bound on the vector interaction strength become weak for m χ 1 MeV. The precise determination of the differential cross-section of coherent scattering is needed to disentangle the contribution from each effective operator, and we will leave the detailed studies to future works. For completeness, here we collect some calculation details of the tree-level ν(p 1 )q(k 1 ) → χ(p 2 )q(k 2 ) elastic scattering cross-section. One should keep in mind that, in reality, nucleus is the target, and the quark contribution should be summed coherently. Moreover, nucleus mass should be used and the couplings at the quark level should be carefully replaced by the relevant form factors as discussed in Sec.III.
The kinematics of this fundamental 2 → 2 process can be easily worked out as follow. In the lab frame, the target quark of mass M q is at rest and k 1 = (M q , 0, 0, 0). The incoming neutrino 4-momentum is denoted as p 1 = (E, E, 0, 0), and p 2 = (M q + T, p cos θ, p cos θ, 0) is for the scattered quark with recoil energy T and scattering angle θ. We use t ≡ p 1 − p 2 to denote the momentum transfer. From the on-shell conditions k 2 1 = k 2 2 = M 2 q and p 2 2 = m 2 χ , one gets p = T (T + 2M q ) and t 2 = −2M q T . Also, the scattering angle can be expressed in terms of E and T as cos θ = T (M q + E) + m 2 χ /2 E T (T + 2M q ) .