Probing neutrino coupling to a light scalar with coherent neutrino scattering

Large neutrino event numbers in future experiments measuring coherent elastic neutrino nucleus scattering allow precision measurements of standard and new physics. We analyze the current and prospective limits of a light scalar particle coupling to neutrinos and quarks, using COHERENT and CONUS as examples. Both lepton number conserving and violating interactions are considered. It is shown that current (future) experiments can probe for scalar masses of a few MeV couplings down to the level of $10^{-4}$ $(10^{-6})$. Scalars with masses around the neutrino energy allow to determine their mass via a characteristic spectrum shape distortion. Our present and future limits are compared with constraints from supernova evolution, Big Bang nucleosynthesis and neutrinoless double beta decay. We also outline UV-complete underlying models that include a light scalar with coupling to quarks for both lepton number violating and conserving coupling to neutrinos.

In the present paper we focus on new neutrino physics caused by a new light scalar. Such a particle can participate in coherent neutrino-nucleus scattering, and interestingly modify the nuclear recoil spectrum in a characteristic manner, both for a light scalar as well as heavy one. Most of our study will focus on the light scalar case. Light new physics is of course motivated by the lack of signals in collider experiments, and its consequences in coherent neutrino-nucleus scattering have been mentioned before [14][15][16][19][20][21][22]. Mostly it was used that light physics does not suffer from limits on neutrino non-standard interactions from high energy scattering experiments such as CHARM-II, as stressed in [23,24]. Discovery limits on light particles in CEνN S have been discussed in Refs. [14,19,22], indicating already before the observation by COHERENT that CEνN S could provide strong constraints on light mediators. In our work, as mentioned before, we focus on a new scalar particle and obtain the current limits from COHERENT on its mass and coupling with SM particles. We also consider explicit future realizations of this experiment and also of CONUS, which will use reactor antineutrinos to probe coherent scattering. The characteristic distortion of the spectrum shape for light scalars with masses around the neutrino energy allows to reconstruct the scalar mass, which we explicitly demonstrate. We also compare our limits with existing ones from a variety of sources in particle and astroparticle physics. UV-complete models that may be behind the existence of such light scalar particles are also outlined. This paper is organized as follows: In Sec. II, we present the framework of a light scalar with couplings to neutrinos and quarks, discuss the coherent scattering cross section and include a discussion of form factors when the coupling to nucleons is considered. In Sec. III, we discuss the constraints on such scalar particles from various particle and astroparticle physics observables. In Secs. IV and V we discuss and interpret the implications of various possible observations. In Sec. VI, we summarize our findings. The calculational details and outlines for UV-complete gauge invariant models are presented in Appendices.

II. LIGHT SCALAR INTERACTIONS IN COHERENT NEUTRINO-NUCLEUS SCATTERING
In this section, we first introduce the possible interactions of neutrinos with quarks (or nuclei) mediated by a light scalar boson and then discuss the corresponding cross section of coherent neutrino-nucleus scattering.

A. New scalar interactions
We consider a scalar field denoted by φ which couples to neutrinos. There are two possibilities for such coupling, namely lepton number violating (LNV) and conserving (LNC) couplings. The latter possibility requires the presence of right-handed neutrinos: L LNC ≡ν (C ν + iD ν γ 5 ) νφ = y ν φν R ν L + H.c., (1) where y ν = C ν − iD ν . For simplicity, throughout this paper we implicitly assume that φ is a real field but most discussion and overall behavior of the bounds and limits remains valid for a complex φ as well. For real φ, the hermicity of the Lagrangian implies that C ν and D ν are real. Note that the couplings have flavor indices, suppressed here for clarity. The lepton number violating form of the interaction can be written in analogy as In both Lagrangians y ν can in general be a complex number.
The same scalar field can couple to quarks. Since in coherent neutrino scattering we are concerned with the effective coupling of φ with the whole nucleus N we write the Lagrangian as: where ψ N is the Dirac spinor of the nucleus, assuming it is a spin-1/2 particle 1 . We can write Again, for real φ hermicity of the Lagrangian demands C N and D N to be real numbers. The conversion from fundamental quark couplings (C q , D q ) to the effective coupling (C N , D N ) will be discussed later. Note that we here consider both scalar and pseudo-scalar interactions. In what follows, the latter contribution is usually very much suppressed, and essentially only the scalar contribution is what matters. In summary, the Lagrangian (in addition to the SM) responsible for coherent neutrino scattering is where L νφ can be either Eq. (1) or Eq. (2). The masses of the scalar and nucleus are respectively denoted by m φ and M . In App. A, we present examples in which couplings to neutrinos and quarks can be embedded in electroweak symmetric models.

B. Cross Section
The first thing to notice is that the Yukawa interaction of both LNC and LNV forms (either Eq. (1) or Eq. (2)) leads to chirality-flipping scattering which will not interfere with chirality-conserving SM weak interactions 2 . Thus, one can separate the cross section into two parts containing the pure SM and the new physics contributions: where T denotes the recoil energy. The SM cross section, assuming full coherence is given by [26] Here C N is the coupling of the scalar with the nucleus, whose connection to the fundamental quark couplings is discussed in the next subsection. The division by the atomic number in the definition of Y makes it almost independent of the type of nucleus-cf. Eq. (19) and Eq. (20). The cross section has little dependence 3 on D N because the pseudoscalar contribution is suppressed by the O T 2 /E 2 ν term. This is in analogy to dark matter direct detection where dark matter nucleon interactions mediated by a pseudo-scalar is well known to be suppressed.
Obviously scalar interactions lead to a spectral shape different from that in the SM case, see e.g. [13], where an effective scalar interaction, corresponding to m 2 φ M T , is considered. For a light scalar under discussion here we see that additional modifications of the spectrum are possible, in particular if the scalar mass is of the same order or smaller than the typical momentum transfer M T ∼ E 2 ν .
C. From the fundamental couplings to the effective couplings The effective couplings C N and D N originate from fundamental couplings of φ with the quarks. The connection between the effective couplings and the fundamental couplings has been well studied in spin-independent dark matter direct detection. Essentially, one needs to know the scalar form factors of quarks in the nuclei. We refer to [27] for the details and summarize the relevant results below.
Since the pseudo-scalar coupling D N has no effect on CEνN S [cf. Eq. (9)], we focus on the scalar coupling, C N . Taking the fundamental scalar interaction of quarks with φ to be of the form the effective coupling C N is related to C q by where the couplings to protons and neutrons are Here m p = 938.3 MeV and m n = 939.6 MeV are masses of proton and neutron; Z and A − Z are proton and neutron numbers in the nucleus; m q are quark masses; f p q and f n q are the scalar form factors in protons and neutrons. According to the updated data for the u and d quarks from [28,29] and the data for the s quark from [30], the form factors are: where f c, b, t are approximately the same for the heavy quarks because their contributions come from heavy quark loops that couple φ to the gluons [31]. Notice that despite the different quark content, the couplings of proton and neutron to the scalar turn out to be almost equal: |(C n − C p )/C n | = O(10%). If C q for the heavy quarks are of the same order of magnitude as for the light quarks, then the contributions of heavy quarks are negligible due to suppression by their masses. However, if C q ∝ m q (as in the case that the φ coupling to the quarks comes from the mixing of a scalar singlet with the SM Higgs), the contribution from all flavors will be comparable. Taking the following quark masses [32]: For Ge and CsI targets, taking the average values of (Z, A) as (32, 72.6) and (54, 130), we can evaluate the explicit dependence of C N on the C q : This means that from the fundamental coupling to the effective coupling an amplification by a factor of O(10 2 ) or O(10 3 ) can be present. The definition of Y in Eq. (9), in terms of the fundamental couplings, can be rewritten as The dependence of Y on the types of targets is weak because for heavy nuclei, A−Z A and Z A are typically close to 1/2. For example, taking the average values of (Z, A) for Ge and CsI targets, we get When comparing the sensitivities of CEνN S experiments using different targets, we will ignore the small difference and assume Y Ge ≈ Y CsI .

III. EXISTING BOUNDS FROM PARTICLE AND ASTROPARTICLE PHYSICS
In this section we review the relevant bounds on C n , y ν or on their product (C n y ν ) from various observations and experiments other than coherent scattering. As we shall see, the bounds on the hadronic couplings are on C n = R[Γ nφ ] (not on D n ) but the bounds on neutrino couplings are on |y ν | 2 = C 2 ν + D 2 ν . The difference originates from the fact that while nuclei in the considered setups are non-relativistic, neutrinos are ultra-relativistic.
• Bounds on C n from neutron nucleus scattering: In the mass range of our interest, the strongest bounds on C n come from low energy neutron scattering off nuclei [33][34][35], in particular using Pb as target. The effect of a new scalar would be to provide a Yukawa-type scattering potential whose effect can be constrained. Notice that like the case of CEνN S, since in these setups the nuclei are non-relativistic, their dominant sensitivity is only to C n (not to D n ).
• Bounds on y ν from meson decay: If the new light boson couples to neutrinos, it can open new decay modes for mesons such as K + → l + νφ or π + → l + νφ. The scalar will eventually decay into a neutrino pair appearing as missing energy. From the absence of a signal for such decay modes, bounds of order 10 −3 on ( α |y ν | 2 eα ) 1/2 and ( α |y ν | 2 µα ) 1/2 have been found from different modes [36]. Notice that as long as the mass of φ is much smaller than the meson mass the bound is independent of m φ . Moreover, the bound similarly applies for the LNV and LNC cases. In the LNC case, sensitivity is to the combination |y ν | 2 = C 2 ν + D 2 ν . • Bounds from double beta decay: A LNV coupling of form φν T e Cν e can cause neutrinoless double beta decay [37] in the form of n + n → p + p + e − + e − + φ, of course provided that φ is lighter than the Q value of the decaying nucleus. From double beta decay of 136 Xe with a Q value of 2.4 MeV the following bound is found [38]: A more recent and a bit weaker bound for m φ < 2.03 MeV comes from 76 Ge double beta decay [39].
• Supernova bounds and limits: Light particles coupled to neutrinos and neutrons can affect the dynamics of a proto-neutron star in several ways. Before discussing the impact of our particular scenario on supernovae, let us very briefly review the overall structure of the core of a proto-neutron star in the first few 10 seconds after explosion when neutrinos are trapped inside the core (i.e., the mean free path of neutrinos is much smaller than the supernova core radius). For more details, the reader can consult the textbook [40]. In the first ∼ 10 sec after collapse, the core has a radius of ∼ few 10 km and a matter density of ρ ∼ 10 14 g cm −3 (comparable to nuclear density). However, because of the high pressure, most of the nucleons are free. As mentioned before, neutrinos are trapped inside the core and are thermalized with a temperature of ∼ 10 MeV. The core can be hypothetically divided into the inner core with a radius of 10-15 km and the outer core. Within the inner core, the chemical potential of the ν e is about 200 MeV which is much larger than their temperature, implying that ν e are degenerate. The chemical potentials of ν µ and ν τ are zero and their temperatures are equal. Thus, inside the inner core nν e n νµ = nν µ = n ντ = nν τ n νe , where n να is the number density of ν α . In the outer core, the chemical potential of ν e decreases and the number densities of ν e andν e are almost equal. Notice that the average energies of ν µ , ν τ and their antiparticles throughout the core are given by their temperature which is of order of few 10 MeV. The energies of ν e andν e in the outer core, where the chemical potential vanishes, are also of the same order but the energy of ν e in the inner core is of order of the chemical potential, 200 MeV. The neutrinos scatter off nucleons with a cross section of σ ∼ G 2 F E 2 ν /(4π). Considering the high density of nucleons, the mean free path for neutrinos will be of order of λ 300 cm, which is much smaller than the core radius R ∼ few 10 km. The diffusion time, given by R(R/λ), is therefore of order of O(1 sec) to O(10 sec). Neutrinos diffusing out of the core carry out the binding energy of the star which is of order of 10 53 erg.
Let us now see how new light particles coupled to neutrinos and matter fields can affect this picture. First, let us discuss the impact on supernova cooling. If the interaction of the new particle is very feeble, it cannot be trapped. Thus, if it is produced inside the core it can exit without hinderance and take energy out of the core leaving no energy for neutrinos to show up as the observed events of SN1987a. This sets an upper bound on the coupling of the new particles. On the other hand, if the coupling is large enough to trap the new particles, the impact on cooling will not be dramatic. Still, if the new particles are stable, they can diffuse out and, along with neutrinos, can contribute to supernova cooling. Considering however the theoretical uncertainty in the evaluation of total binding energy and the observational uncertainty on the energy carried away, such contributions can be tolerated. Thus, supernova cooling consideration, within present uncertainties, can only rule out a range of coupling between an upper bound and a lower bound. New interactions can also affect the mean free path of neutrinos λ and therefore the diffusion time R 2 /λ, which roughly speaking coincides with the observable duration of neutrino emission from a supernova. This sets another limit. Finally, if there is a new process that can remove ν e and/or convert it to any ofν e , ν µ(τ ) orν µ(τ ) , it can have profound effect on the Equation of State (EoS) in the inner core. For example, if ν e (with energy of 200 MeV) are converted to ν µ , the temperature of ν µ will increase dramatically. Conversion of ν e toν e will lead to the production of e + which annihilates with electrons inside the core.
In our case the φ particles can decay into a neutrino pair. The decay length is evaluated to be approximately (10 −5 /|y ν |) 2 (E φ /10 MeV)(5 MeV/m φ ) 2 cm, which is much smaller than the radius of the proto-neutron star (∼ few 10 km). The consequences of new interactions on supernova explosions can be categorized into three effects: (1) change of equation of state in case of LNV interaction; (2) new cooling modes because of righthanded (anti)neutrino emission in case of LNC interaction; (3) prolonging the duration of neutrino emission (R 2 /λ) because of a shorter mean free path λ. In all these three effects, the neutrino scattering plays a key role. Neglecting flavor indices, we find that the neutrino-neutrino scattering cross section is comparable to the ν-nucleon scattering cross section if y ν ∼ C n . The number density of nucleons is larger than that of neutrinos, thus the ν-nucleon scattering should be more important. The cross section of the scattering due to φ both for LNV and LNC cases can be estimated as The scattering cross section for sterile (anti)neutrinos is given by the same formula.
Let us discuss the three effects mentioned above one by one. If ν e has a LNV coupling to φ of the form (y ν ) eα φν T e Cν α , where α ∈ {e, µ, τ }, ν e can be converted intoν α . The produced antineutrino will be trapped. However, if a significant fraction of degenerate ν e in the inner core will convert into antineutrinos, drastically changing the equation of state. That is, for in which E νe ∼ 200 MeV, the equation of state of the supernova core has to be reconsidered. In subsequent plots that summarize the limits on our scenario, we call the associated limit to avoid this feature as "SN core EoS". Since the temperatures of ν µ and ν τ are expected to be the same and their chemical potentials to be zero, conversion of ν µ and ν τ toν µ andν τ due to non-zero (y ν ) µµ , (y ν ) µτ and (y ν ) τ τ will not change the equation of state.
When the interaction is LNC, scattering will convert left-handed neutrinos (right-handed antineutrinos) into sterile right-handed neutrinos (left-handed antineutrinos), which do not participate in weak interactions. If σ(ρ/m n )(10 sec) > ∼ 1, a significant fraction of the active (anti)neutrinos will convert into sterile ones. Avoiding this generates an upper limit on the coupling. If σ(ρ/m n )R > ∼ 100, the produced sterile neutrinos will be trapped. The values of √ y ν C n between these two limits are therefore excluded by supernova cooling considerations. The limits are denoted in Fig. 2 as "SN energy loss" and "SN ν R trapping", respectively.
Drawing these figures we have assumed a nominal temperature of 30 MeV for neutrinos which is the typical energy for all neutrinos in the outer core. In case that the neutrino-nucleon scattering cross section due to φ exchange becomes comparable to the standard weak cross section G 2 F E 2 ν /(4π), the diffusion time R 2 /λ will be significantly affected. This limit is shown in Figs. 2 and 3 as "SN ν diffusion", and holds for both the LNC and LNV cases.
• BBN and CMB bounds: The contribution from the LNC and LNV cases to the additional number of relativistic degrees of freedom (δN eff ) will be quite different so in the following, we address them separately.
i) LNV case: In this case, no ν R exist so we should only check for the φ production. For m φ > ∼ 1 MeV, φ will be produced at high temperatures but it will decay before neutrino decoupling without affecting N eff at the BBN or the CMB era. Thus, within the present uncertainties, there is no bound from BBN for m φ > ∼ 1 MeV. Lighter φ, in turn, can contribute to N eff as one scalar degree of freedom if they enter thermal equilibrium. Taking Notice that for 0.1 MeV < m φ < 1 MeV, although φ decays away before the onset of BBN it still contributes to N eff by warming up the ν andν distributions. At m φ = 1 MeV, combining the above bound on y ν from BBN with that from n-nucleus scattering, yields Y < 5 × 10 −7 . For m φ > 1 MeV, this bound does not apply because φ decays into neutrinos before neutrino decoupling from the plasma. That is why the bound denoted "BBN + n scat." appears as a vertical line in Fig. 3. Our simplified analysis seems to be in excellent agreement with the results of [41] which solves the full Boltzmann equations.
ii) LNC case: In this case, the production of scalars via νν → φ is not possible. Processes like νν → ννφ or νN → νN φ can take place but are suppressed. The t-channel process ν LνL → ν RνR can lead to ν R andν R production. The ν L ν L → ν R ν R process can also take place but because of cancelation between t and u channel diagrams it has a smaller cross section. The cross section of the dominant production mode is where x = 2E 2 ν /m 2 φ , in which E ν is the energy of colliding neutrinos in the center-of-mass frame. Taking n ν σH −1 | T =3 MeV < ∼ 0.3 (3 MeV is the temperature at neutrino decoupling and 0.3 is the bound from CMB on δN eff [42]), we find in which m φ and T are in MeV and x = 2T 2 /m 2 φ . Combining this bound with the one on C n from neutron-Pb scattering gives the limit denoted "BBN + n scat." in Fig. 2.
Let us now discuss the effects of forward scattering off nuclei due to the new interaction on neutrino propagation in matter composed of nuclei, N . The induced effective mass will be of chirality flipping form as follows Notice that m N Am p Am n and C N /m N C p /m p so we can write the effective mass as (C p y ν /m 2 φ )(ρ/m p ). Taking for example the density of the Sun as ρ = 150 g cm −3 , we find a shift in the mass of neutrinos of order of 3 × 10 −12 eV (y ν /10 −5 )(C p /10 −5 )(5 MeV/m φ ) 2 , which is completely negligible compared to ∆m 2 /m ν . One may wonder why forward scattering due to a possible new gauge boson with similar mass and coupling has such a large impact on neutrino propagation in matter, while the present case of a scalar does not. The reason lies in the different Lorentz structure of the induced operators. The vectorial interaction induces a contribution of formν T γ 0 ν = ν † ν which has to be compared with m 2 ν /E ν . In the case of scalar, the matter effects have the operatorial form as the mass themselves and should be compared to the mass splitting, ∆m 2 /m ν . More detailed discussion can be found in [43].

IV. CONSTRAINTS AND FUTURE SENSITIVITIES FROM CEνNS
To collect large statistics at coherent scattering energies, CEνN S experiments require intensive and low-energy ( 50 MeV) neutrino fluxes. Two types of neutrino sources can be invoked to carry out CEνN S experiments: reactor neutrinos (E ν 8 MeV) and pion decay at rest (E ν 50 MeV). Two on-going experiments, CONUS 4 and COHERENT [1], adopt these two sources respectively. In this section, we study the sensitivities of the two experiments on light scalar bosons.

A. CONUS
The CONUS experiment uses a very low threshold Germanium detector setting 17 m away near a nuclear power plant (3.9 GW thermal power) in Brokdorf, Germany. The total antineutrino flux is 2.5 × 10 13 s −1 cm −2 . Data collection started in 2017 and first results are expected soon. To study the sensitivity of CONUS, we compute the event numbers given by Here ∆t is the running time, N Ge is the number of Ge nuclei, (T i , T i + ∆T ) is the range of recoil energy in each bin, Φ(E ν ) is the reactor neutrino flux, and θ (T max (E ν ) − T ) is the Heaviside theta function, equal to 0 for T max (E ν )−T < 0 and 1 for T max (E ν ) − T > 0. It is necessary to insert the θ function in (28) because the cross section dσ dT (T, E ν ) does not automatically vanish when T max (E ν ) < T . We take ∆t = 1 year, ∆T = 0.05 keV and N Ge = 3.32 × 10 25 , corresponding to 4 kg natural Ge (with average atomic number A = 72.6). For Φ(E ν ), we use a recent theoretical calculation of the flux [44], and normalize it to meet the total antineutrino flux (2.5 × 10 13 s −1 cm −2 ) in CONUS.
Using Eq. (28), we compute the event numbers for several examples (m φ = 0.1 MeV, 10 MeV, 30 MeV and 100 GeV) and compare them with the SM value in Fig. 1. The signal strength is quantified by the ratio N/N 0 where N 0 is the SM expectation, and N contains the additional contributions of light scalar bosons. One can see from the figure that the shape of the spectrum when we include the scalar contribution can be dramatically different, in particular for low values of m φ .
To study the sensitivity of CONUS on light scalar bosons, we adopt the following χ 2 -function [13], with The pull parameter a with an uncertainty of σ a = 2% takes care of the uncertainty in the normalization originating from various sources such as the variation of nuclear fuel supply or the uncertainty of the fiducial mass and distance. Other systematic uncertainties that may change the shape of the event spectrum are parameterized by σ f in Eq. (30).
Here we assume they are proportional to the event numbers and take σ f = 1%. We also introduce a background in our calculation by adding N bkg, i to the event number in each bin. The background in CONUS is about 1 count/(day · keV · kg). The threshold of ionization energy detection in CONUS is 0.3 keV, which if divided by the quenching factor (≈ 0.25) corresponds to 1.2 keV recoil energy. The reactor neutrino flux at E ν > 8 MeV has negligible contributions and also large uncertainties, so we set a cut of E ν at 8 MeV, which corresponds to about 1.75 keV recoil energy according to Eq. (7). As a result, in Eq. (29) we only sum over the bins from 1.2 keV to 1.75 keV.
The result is shown in Fig. 2 and Fig. 3 for the LNC and LNV cases respectively. Although the constraints of CEνN S experiments are independent of the LNC/LNV cases, the other constraints depend on the nature of the interaction, as explained in Section III. We therefore present the two cases separately.
We also study future improved sensitivities of CONUS by assuming a 100 kg Germanium detector as the target and an improved threshold down to 0.1 keV. We assume the corresponding systematic uncertainties are also reduced to a matching level, (σ a , σ f ) = (0.5%, 0.1%). This will be possible if the reactor neutrino flux is better understood due to improved theoretical models and measurements. The forecast for CONUS100 with 5 years of data taking is also shown in Figs. 2, 3 with dashed blue lines.
As mentioned above, the shape information for light scalar masses is noteworthy in the spectrum, see Eq. (8). It can in fact be used to determine the value of the mass. In Fig. 4 we show the potential of CONUS100 for determining the mass and coupling of the φ particle assuming two characteristic examples. As long as the mass of the scalar is not much larger than the neutrino energy or the typical momentum exchange, m 2 φ ∼ M T ∼ E 2 ν , reconstruction of the mass is possible.

B. COHERENT
The COHERENT experiment uses a CsI scintillator to detect neutrinos produced by π + and µ + decay at rest. In its recent groundbreaking publication [1] a 6.7σ observation of the SM coherent scattering was announced. There are three types of neutrinos in the neutrino flux, ν µ , ν µ , and ν e . The first is produced in the decay π + → µ + + ν µ while the second and the third are produced in the subsequent decay µ + → e + + ν µ + ν e . Because the first decay is a two-body decay and the pion is at rest, the produced neutrinos will be monochromatic with energy: where m µ = 105.66 MeV and m π = 139.57 MeV are the muon and pion masses, respectively. Remembering that the muon also decays at rest, the neutrino fluxes are given by [45]: where E ν should be in the range (0, m µ /2). The event numbers are computed by which is similar to Eq. (28) except that (i) N Ge is replaced with N Cs/I ; (ii) for the ν µ flux, the delta function in Eq. (31) has been integrated out. For simplicity, we assume that Cs and I have approximately the same proton and neutron numbers, (Z, A) = (54, 130). We also assume that the couplings of neutrinos are flavor universal ((y ν ) αβ = y ν ), so we take equal cross section for all neutrino flavors. Using Eq. (34), we also compute the ratio N/N 0 in COHERENT, shown in the right panel of Fig. 1.
In the COHERENT experiment, the recoil energy of the nucleus is converted to multiple photoelectrons and eventually detected by PMTs. The number of photoelectrons n PE is approximately proportional to the recoil energy [1]: For n PE > 20, the signal acceptance fraction is about 70% (cf. Fig. S9 of [1]). This number drops down quickly for smaller n PE , and becomes approximately zero for n PE < 5. This implies that the threshold for T is about 4 keV in COHERENT. Using Eqs. (34,35) and the signal acceptance fraction data, we can study the constraint of the COHERENT data (from Fig. 3 of [1]) on light scalar bosons. The SM expectation is also provided by [1] which can be used to compute the total normalization factor. We directly use the relevant uncertainties provided by [1]. The result is shown in Figs. 2 and 3 as well. Since reactor neutrinos provide much larger event numbers, CONUS limits will be better, though of course limited to the electron-type couplings, whereas COHERENT will also have muon-type neutrinos.
In the future, the COHERENT experiment will further develop the detection of CEνN S with different targets 5 , including 30 kg liquid argon, 10 kg high purity Ge, and 185 kg NaI crystal. A complete study on the future sensitivities of future COHERENT including all the different targets and different detection technology is beyond the scope of this paper. Considering that the total fiducial mass compared to the current value (14.6 kg CsI) will be increased by a factor of ∼ 20, plus a prolonged running for few years, at best the statistics may be increased by a factor of 100, which corresponds to a reduction of the statistical uncertainties by a factor of 10. It is therefore reasonable to assume that the uncertainties of future measurement will be reduced by a factor between 1 and 10. To show the sensitivity of future COHERENT versions on the light scalar coupling, we plot a black dashed curve in Fig. 2 and Fig. 3 assuming the uncertainties (both systematical and statistical) are reduced by a factor of 10. For the sake of definiteness, we take the liberty to denote this potential situation as COHERENT (stat. × 100). Again, the determination of the mass of the scalar particle is possible if its mass lies below the typical neutrino energy. As seen in Fig. 1, the spectral distortion due to the scalar exchange is less dramatic as for CONUS. This is mostly caused by the larger energy of the neutrinos, the momentum exchange and nuclear recoil. Fig. 4 shows the potential of the assumed future COHERENT version for determining the mass and coupling of the φ particle, assuming two characteristic examples. Due to the larger energy of COHERENT, and also because of the smaller statistics, the reconstruction potential is less promising compared to experiments based on reactor neutrinos.

Figs. 2 and 3
show the constraints and limits on the relevant combination of Y couplings versus the mass of the scalar for lepton number conserving and lepton number violating interactions, respectively. To draw these lines the coupling of φ to neutrinos is taken to be flavor universal. Each limit is however sensitive to a different flavor structure. Let us start by discussing the bounds which apply for both lepton number violating and lepton number conserving interactions. The red-dashed lines show the constraint on Y √ y ν C n from combining the upper bounds on C n and    y ν from the n-Pb scattering and meson decay experiments. As seen from the figures, this bound is relatively weak. The present bound from COHERENT shown by a solid black line is already well below this combined bound. The bounds from meson decay are sensitive to α (y ν ) 2 eα and α (y ν ) 2 µα . Since the beam at the COHERENT experiment is composed of ν µνµ and ν e fluxes, it will be sensitive to similar flavor composition. Our forecast for the future bound by the COHERENT experiment (a factor 10 smaller uncertainties) is shown by dashed black line; the bound on Y can be improved by a factor of 2. The blue solid and dashed lines are the upper bounds that CONUS can set with 1 year × 4 kg and 5 year × 100 kg of data taking, respectively. As seen from these figures, CONUS can improve the bound by one or two orders of magnitudes. Since CONUS is a reactor neutrino experiment, it can only probe C 1/2 n ( α |y ν | 2 eα ) 1/4 . Since the uncertainties of CONUS are mainly limited by statistics, the bound that it can set on the cross section σ φ scales as t −1/2 with data taking time. Since σ φ ∝ Y 4 , the bound on Y will scale as t −1/8 . The violet curve in Fig. 3 up to 2.4 MeV is the combined bound from the n-Pb scattering on C n and from double beta decay on (y ν ) ee . CONUS with only one year of data taking can provide a stronger bound. The dashed green lines in Figs. 2 and 3 denoted "SN ν diffusion" show the limits resulting essentially from a neutrino-nucleon scattering cross section due to φ exchange being equal to that in the SM. As we discussed before, in the vicinity of this line supernova evolution and emitted neutrino flux will be dramatically affected. As seen from the figures, the CONUS experiment with 1 year of data taking can already probe all this range. The green area between solid and dotted-dashed green lines in Fig. 3 is ruled out by supernova cooling and ν R trapping considerations.
In the LNC case, the orange line in Fig. 2 denoted by "BBN + n scat." shows the combined bound from n-Pb scattering and BBN. As seen from the figure, for m φ > 3 MeV, the bound from COHERENT is already stronger. Both for LNC and LNV cases, a φ particle with m φ ∈ (1.5 − 3) MeV and y ν ∼ few × 10 −5 can significantly affect BBN. As seen from the figures this mass range can be probed by CEνNS experiments. For C n |y ν | eα above the dotted-dashed green line in Fig. 3, the equation of state in supernova inner core will drastically change because of ν e + n →ν α + n scattering. As seen from the figure, a significant part of the parameter space above this line can be probed by CONUS. We can therefore deduce that coherent scattering results may have dramatic impact on SN and BBN physics. Fig. 4 displays the prospect of measuring m φ and Y by our future versions of CONUS and COHERENT, assuming true values of (m φ , Y ) are (10 MeV, 10 −5 ) or (60 MeV, 5 × 10 −5 ) for CONUS100, and (10 MeV, 7.5 × 10 −5 ) or (60 MeV, 10 −4 ) for COHERENT (stat. × 100). Here for comparison, we choose the same masses for the two experiments. However, the couplings in COHERENT (stat. × 100) are set to larger values to lead to similar precision as CONUS100 (cf. Figs. 2 and 3). Even with larger couplings, COHERENT still cannot measure (m φ , Y ) as good as CONUS100. As shown in Fig. 4, for true values m φ = 10 MeV and Y = 10 −5 , the mass and coupling can be determined with better than 10% accuracy by CONUS100. In comparison, COHERENT (stat. × 100) loses its capability to determine m φ but it has still reasonable precision in determining Y provided that Y is large enough (close to its present bound). This is understandable because m φ = 10 MeV is larger than the typical energy-momentum transfer in the CONUS100 (m 2 φ > ∼ M T ∼ E 2 ν ) but is smaller than the energy-momentum transfer in COHERENT (m 2 φ < ∼ M T ∼ E 2 ν ). Another reason is the better statistics in CONUS. If the mass is raised to 60 MeV, then both lose their ability to determine the mass and the coupling separately. They however maintain their sensitivity to Y /m φ .
Discovering a positive signal for the effects of φ by CONUS will have drastic consequences for the analysis of supernova evolution. If a value of Y below the green solid line in Fig. 2 is found, the supernova cooling bounds tell us that interaction cannot involve ν R so the interaction should be of lepton number violating form. If φ turns out to have a mass around 2 MeV, it will be more intriguing as it may be discovered at double beta experiments. If, however, double beta decay searches fail to discover φ with expected mass and coupling, we may draw a conclusion that (y ν ) ee (y ν ) eµ , (y ν ) eτ . In any case, in analyzing BBN, effects of such light m φ with sizeable Y has to be taken into account. Comparing Figs. 2 and 3, we conclude that because of the BBN bounds, discovery of m φ < 1 MeV will indicate LNC interaction with light right-handed neutrinos with immediate consequences for supernova evolution.
If CONUS and/or COHERENT finds Y ∼ 10 −5 , the chances of finding a signal for φ in meson decay experiments as well as in n-Pb scattering experiments increase. If CONUS finds a signal for Y > 5 × 10 −5 , this means that signals for K + → e + νφ and for π + → e + νφ will be within reach of next generation [46], and new n-Pb measurements would be very interesting. If CONUS finds Y < 10 −5 and if meson decay experiments find y ν > ∼ 10 −4 , we would conclude C n < 10 −6 , making it difficult to see an effect on n-Pb scattering experiments. Similarly Y < 10 −5 and C n ∼ 10 −4 (close to the present bound) would imply y ν < 10 −6 .
We conclude this section by mentioning some possibilities on inferring the flavor structure or type of interaction that arise due to the complementarity of the various sources and limits. If the ν e couplings are large enough to be within the reach of COHERENT (that is if Y > 5 × 10 −5 ), the CONUS experiment will easily determine α |(y ν ) eα | 2 C 2 N , where α runs over all active flavors for the LNV case (over all light right-handed species for LNC case). The information by COHERENT can then determine α |(y ν ) µα | 2 C 2 N . If COHERENT alone would be able to distinguish the flavor content of the events (e.g. by timing cuts), information on flavor structure of y ν (i.e. on α |(y ν ) µα | 2 / α |(y ν ) eα | 2 ) could be extracted. In the special case that |(y ν ) eα | |(y ν ) µα |, it may be possible that COHERENT will discover new effect but CONUS will report null results for new physics discovery.

VI. SUMMARY AND CONCLUDING REMARKS
Coherent elastic neutrino-nucleus scattering can probe both new light as well as heavy physics. Focussing here on the light case we demonstrated the discovery potential of current and future coherent scattering experiments on the mass and coupling of scalar particles interacting with neutrinos and quarks. The shape of the nuclear recoil spectrum is distorted by the scalar interaction, and allows even to determine the mass of the scalar, if its mass is around the energy of the scattered neutrinos. Even current limits by COHERENT are competitive with a combination of bounds from BBN and from various terrestrial experiments such as meson decay and neutron-scattering experiments. Moreover, these bounds probe areas in parameter space that can have important consequences for BBN and supernova evolution, in particular for lepton number violating interactions. Future versions of the experiment or upcoming reactor experiments such as CONUS will reach not yet explored areas in parameter space. as well as the ones shown in Eqs. (1,2). The coupling of φ to nuclei should of course arise from its coupling to quarks. The latter can originate from the mixing of the singlet scalar φ with an electroweak doublet, Φ. Taking complex couplings of form and a mixing of β between φ and the neutral component of Φ, we find the coupling of φ to the u and d quarks respectively to be given by Γ u = sin β Y * u and Γ d = sin βY * d , or equivalently Notice that taking Y q to be real, the coupling will be parity invariant and therefore D q = 0.
The most economic solution is to identify Φ with the SM Higgs. Remember that the couplings of the SM Higgs to quarks of first generation are O(10 −5 ). Moreover, the mixing of φ with SM Higgs cannot exceed O(10 −2 ), otherwise the rate of invisible decay mode Br(H → φφ) will exceed the experimental limits. Combining these, we conclude that in case φ is taken to be the SM Higgs, |Γ qφ | ∼ 10 −2 m q / H 0 . As we have seen in Sec. II C, the contributions from quarks of different generations to the coupling of a nucleus to φ will be of the same order and too small to lead to discernable effects on current CEνNS experiments.
Taking thus Φ to be a new doublet, its coupling to quarks can in principle be as large as O(1). Taking To avoid a need for fine-tuned cancelation m Φ sin β should be smaller than O(m φ ). For m φ ∼ 5 MeV and m Φ ∼ 1 TeV, this implies sin β ∼ 10 −5 . For Γ u ∼ Γ d ∼ 10 −5 , naturalness (i.e., absence of fine-tuned cancelation) requires the mass of Φ to be within the reach of the LHC and its coupling to u and d quarks to be O(1) which in turn promises a rich phenomenology at the LHC.
By turning to the LNC interactionν R ν L , a mechanism similar to the one described above can provide a lepton number conserving interaction of φν R ν L through mixing of φ with neutral component of the Φ doublet. For a lepton number violating coupling φν T Cν, two scenarios can be realized: • The φν T Cν coupling can be obtained by the mixing of φ with a neutral component of an electroweak triplet ∆ which couples to the left-handed lepton doublet as L T C∆L. The quantum numbers of ∆ should be the same as the triplet scalar whose tiny vacuum expectation value is responsible for neutrino mass in the type II seesaw mechanism. Identifying these two, the flavor structure of the φ coupling to neutrinos will be determined by that of neutrino mass: (C ν ) αβ , (D ν ) αβ ∝ (m ν ) αβ . Naturalness (i.e., m φ ∼ 1 MeV m ∆ ∼ TeV without fine tuned cancelation) again implies that ∆ is not much heavier than TeV and its coupling to leptons are of order of 1 which promises rich phenomenology at the LHC such as production of ∆ ++ and its decay into a same sign pair of charged leptons.
• Another scenario that can provide coupling of form φν T Cν is suggested in [47]. The scenario is very similar to the inverse seesaw mechanism for generating mass for neutrinos and requires a Dirac fermion singlet Ψ with the following Lagrangian When M Ψ m φ , we can integrate out Ψ and arrive at C ν = (y H /M Ψ ) 2 y . As shown in [47], C ν even as large as 10 −3 can be obtained by this mechanism. Moreover, if φ develops a vacuum expectation value, an inverse seesaw mechanism for neutrino mass generation will emerge and the flavor structure of C ν and neutrino mass matrix will be similar. In this scenario, the SM Higgs will have an invisible decay mode H →Ψν L governed by y 2 .
The discovery of a scalar field in coherent scattering experiments will therefore, at least in the models outlined here, hint towards a rich new collider phenomenology.

Appendix B: Calculation of the cross section
In this appendix, we give the analytic calculation of νN and νN cross sections, assuming a lepton number conserving interaction with φ. The cross section for the lepton number violating case is identical. Let us first focus on the νN case. The initial and final momenta are denoted in the way shown in Fig. 5. The scattering amplitude of this diagram is where Γ νφ ≡ C ν + D ν iγ 5 and We have inserted a right-handed projector P R = (1 + γ 5 )/2 in Eq. (B1) because the initial antineutrino produced by the charged current interaction should be right-handed. Because of P R , if the initial antineutrino is left-handed, the amplitude automatically vanishes. We can therefore sum over all the spins and apply the trace technology: = 1 (2M T + m 2 φ ) 2 tr γ · p 1 P R γ · k 1 (C 2 ν + D 2 ν )P L 2M C 2 N (2M + T ) + D 2 N T .
For νN scattering, one needs to change Eq. (B1) to iM φ = u s (p 1 )P L (iΓ νφ ) u s (k 1 ) −i q 2 − m 2 φ u r (k 2 ) (iΓ N φ ) u r (p 2 ). (B5) Consequently, one has to interchange P R ↔ P L in Eq. (B3). From Eq. (B4) we can see that P R ↔ P L does not change the result so the cross sections are equal for νN and νN scattering. From Eq. (B4) we obtain which is the cross section for both neutrino and antineutrino scattering.