Potentialities of a Low-Energy Detector Based on $^ 4$ He Evaporation to Observe Atomic Effects in Coherent Neutrino Scattering and Physics Perspectives

Potentialities of a low-energy detector based on 4He evaporation to observe atomic eﬀects in coherent neutrino scattering and physics perspectives

M. Cadeddu,1, ∗ F. Dordei,2, † C. Giunti,3, ‡ K. A. Kouzakov,4, § E. Picciau,1, ¶ and A. I. Studenikin5, 6, ∗∗ 1Universitàdegli studi di Cagliari and Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy 2Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy 3Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy 4Department of Nuclear Physics and Quantum Theory of Collisions, Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia 5Department of Theoretical Physics, Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia 6Joint Institute for Nuclear Research, Dubna 141980, Moscow Region, Russia We propose an experimental setup to observe coherent elastic neutrino-atom scattering (CEνAS) using electron antineutrinos from tritium decay and a liquid helium target. In this scattering process with the whole atom, that has not beeen observed so far, the electrons tend to screen the weak charge of the nucleus as seen by the electron antineutrino probe. The interference between the nucleus and the electron cloud produces a sharp dip in the recoil spectrum at atomic recoil energies of about 9 meV, reducing sizeably the number of expected events with respect to the coherent elastic neutrino-nucleus scattering case. We estimate that with a 60 g tritium source surrounded by 500 kg of liquid helium in a cylindrical tank, one could observe the existence of CEνAS processes at 3σ in 5 years of data taking. Keeping the same amount of helium and the same data-taking period, we test the sensitivity to the Weinberg angle and a possible neutrino magnetic moment for three different scenarios: 60 g, 160 g, and 500 g of tritium. In the latter scenario, the Standard Model (SM) value 2 SM+0.015 of the Weinberg angle can be measured with a statistical uncertainty of sin ϑW −0.016. This would 2 represent the lowest-energy measurement of sin ϑW , with the advantage of being not affected by the uncertainties on the neutron form factor of the nucleus as the current lowest-energy determination. Finally, we study the sensitivity of this apparatus to a possible electron neutrino magnetic moment −13 and we find that using 60 g of tritium it is possible to set an upper limit of about 7 × 10 µB at 90% C.L., that is more than one order of magnitude smaller than the current experimental limit.

I. INTRODUCTION as taking place on the atom as a whole. In Ref. [21] it has been noted that, in the case of electron neutrino scat- Coherent elastic neutrino-nucleus scattering (CEνNS), tering, the interference between the electron cloud and has been recently observed by the COHERENT experi- the nucleus becomes destructive. As a consequence, the 2 ment [1, 2], after many decades from its prediction [3–5]. scattering amplitude becomes null and changes sign as q This observation triggered a lot of attention from the sci- varies from large to small values, producing a sharp dip in entific community and unlocked a new and powerful tool the differential cross section. In practice, electrons tend to study many and diverse physical phenomena: nuclear to screen the weak charge of the nucleus as seen by an physics [6, 7], neutrino properties [8–10], physics beyond electron neutrino probe. This effect should be visible for 2 2 the Standard Model (SM) [11–17], and electroweak (EW) momentum transfers of the order of q ∼ 100 keV , cor- interactions [18, 19]. The experimental challenge related responding to atomic recoil energies TR ∼ 10 meV. For to the CEνNS observation is due to the fact that in order recoil energies of O(100 meV) this effect starts to become arXiv:1907.03302v2 [hep-ph] 11 Sep 2019 to meet the coherence requirement qR 1 [20], where completely negligible. The small recoils needed are well q = |~q| is the three-momentum transfer and R is the nu- below the thresholds of detectability of currently avail- clear radius, one has to detect very small nuclear recoil able detectors, making it very difficult to observe this effect. energies ER, lower than a few keV. At even lower momentum transfers, such that However, in Ref. [22] a new technology based on the qRatom 1, where Ratom is the radius of the target atom including the electron shells, the reaction can be viewed evaporation of helium atoms from a cold surface and their subsequent detection using field ionization has been proposed for the detection of low-mass dark matter parti- cles. In this configuration, the nuclear recoils induced by ∗ [email protected] dark-matter scattering produce elementary excitations † [email protected] ‡ [email protected] (phonons and rotons) in the target that can result in § [email protected] the evaporation of helium atoms, if the recoil energy is ¶ [email protected] greater than the binding energy of helium to the surface. ∗∗ [email protected] Given that the latter can be below 1 meV, this proposed 2

2 technique represents an ideal experimental setup to ob- where Fe(q ) is the electron form factor and the + sign serve atomic effects in coherent neutrino scattering. applies to νe andν ¯e, while the − sign applies to all the Here, we propose a future experiment that would allow other neutrino species. As explained in Ref. [21], the the observation of coherent elastic neutrino-atom scatter- + sign is responsible for the destructive interference being (CEνAS) processes and we investigate its sensitivity. tween the electron and nuclear contributions. In princi- Since this effect could be visible only for extremely small ple, one should add also the axial contribution from the recoil energies, in order to achieve a sufficient number Atom electron cloud, CA , that however becomes null when of low-energy CEνAS events, electron neutrinos with en- the number of spin up and down electrons is the same, ergies of the order of a few keV need to be exploited. which applies to our scenario. Moreover, at the low mo- Unfortunately, there are not so many available sources of mentum transfers considered in CEνAS processes, one 2 2 such low-energy neutrinos. In this paper, we investigate can safely put FN(q ) = FZ(q ) = 1. This allows to de- the possibility to use a tritium β-decay source, that is rive physics properties from the analysis on CEνAS pro- characterized by a Q-value of 18.58 keV. The PTOLEMY cesses being independent on the knowledge of the neutron project [23, 24], that aims to develop a scalable design to distribution that is largely unknown [6, 28]. detect cosmic neutrino background, is already planning As visible from Eq. (3), in CEνAS processes a key role to use about 100 g of tritium, so we can assume that a is played by the electron form factor, that is defined as the similar or even larger amount could be available in the Fourier transform of the electron density of an atom. In near future. Moreover, we show the potentialities of such contrast to the case of atomic hydrogen, the He electron a detector to perform the lowest-energy measurement of density is not known exactly. In our present study we the weak mixing angle ϑW , also known as the Weinberg employ the following parameterisation of the He electron angle, a fundamental parameter in the theory of Standard form factor, that has proved to be particularly effective Model electroweak interactions, and to reveal a magnetic and accurate (see, for instance, Refs. [29–31]) moment of the electron neutrino below the current limit. 4 ! 2 2 X −bi(q/4π) Fe(q ) = A· ai · e + c . (4) II. ATOMIC EFFECTS IN COHERENT i=1 SCATTERING The parameters ai = {0.8734, 0.6309, 0.3112, 0.178}, bi = In order to derive the cross section for a CEνAS process {9.1037, 3.3568, 22.9276, 0.9821} and c = 0.0064, ex- tracted from Tab. 6.1.1.4 of Ref. [30], are given by a ν` + A → ν` + A, where A is an atom and ` = e, µ, τ, 2 we start from the differential-scattering cross-section of close fit of the numerical calculations for Fe(q ) using a well-established theoretical model of the He electron a CEνNS process as a function of the recoil energy ER wave function. The normalization A is a scaling fac- of the nucleus [21, 25] 2 tor such that Fe(q ) → 1 for q → 0, as in the nuclear dσCEνNS G2 m E form factor definition. This parameterisation assumes = F C2 m 1 − N R , (1) V N 2 that the electron density is spherically symmetric so that dER π 2Eν the value of the Fourier transform only depends on the where GF is the Fermi constant, mN is the nuclear mass, distance from the origin in reciprocal space. The moment E is the neutrino energy and C is the q2-dependent ν V transfer√ q is related to the atomic recoil energy through matrix element of the vector neutral-current charge 1 q = 2mATR, where mA ∼ mN is the atomic mass. 1 For various models of the He electron wave function that C = [(1 − 4 sin2 ϑ )ZF (q2) − NF (q2)] . (2) V 2 W Z N give an accurate value of the electron binding energy, the numerical results for F (q2) are known to be practically Here, Z and N are the number of protons and neutrons e 2 2 identical in a wide range of q values. To illustrate this in the atom and FN(q )(FZ(q )) is the nuclear neutron feature, we compare in Fig. 1 two He electron form fac- (proton) form factor [6, 26, 27]. In principle, one should tors obtained with two different well-known approaches: also consider the axial coupling CA contribution, but for (i) the Roothaan-Hartree-Fock (RHF) method [32] and even-even (spin zero) nuclei it is equal to zero. Since in 4 (ii) the variational method using a strongly correlated this paper we take as a target a He detector, we have ansatz [33]. Despite the fact that the indicated methods CA = 0. treat electron-electron correlations in an opposite man- ner (the first completely neglects them, while the sec- As anticipated in the introduction, when the energy ond takes them into account explicitly), they yield very of the incoming neutrino is low enough, atomic effects CEνAS close electron form-factor values. This illustration shows arise. In the case of CEνAS processes, dσ /dTR can be derived starting from the formula in Eq. (1) with the inclusion of the electron contribution to the vector coupling [21, 25] 1 Here we neglect the binding of the atom in the liquid helium Atom 1 target, since it becomes of relevance only at the energy scale C = C + (±1 + 4 sin2 ϑ )ZF (q2) , (3) T 1 meV. V V 2 W e R . 3 a negligible role of the theoretical uncertainties associ- which also acts as a vessel for the source. Cautions have ated with the He electron form factor in the analysis car- to be taken such that all the materials used in the detec- ried out in the next sections. tor are extremely radiopure, such that the background contaminations are kept under control. Finally, the en- 1.0 ergy deposited by the electrons in the shield would cause a significant heating of the surrounding helium. Thus the shield has to be placed inside a cryocooler to keep 0.8 RHF the source at the desired temperature, surrounded by a vacuum layer to further isolate the source. Thakkar and Smith

) 0.6 R ( T e

F 0.4

0.2

0 0 50 100 150 200 TR [meV]

FIG. 1. The He electron form factor as a function of the recoil energy TR. The solid (black) curve represents the RHF FIG. 2. Schematic representation of the detector proposed model [32] and the dashed (red) curve the strongly correlated to observe CEνAS processes. The recoil of a helium atom model of Thakkar and Smith [33]. after the scattering with an electron antineutrino coming from the tritium source in the center produces phonons and rotons which, upon arrival at the top surface, cause helium atoms to be released by quantum evaporation. A ﬁeld ionization III. EXPERIMENTAL SETUP AND EXPECTED detector array on the top surface, as proposed in Ref. [22], NUMBER OF EVENTS detects the number of helium atoms evaporated.

As mentioned in the introduction, a tritium source The expected CEνAS differential rate dN /(dt dTR) in would provide electron antineutrinos in the needed en- such a cylindrical configuration, that represents the num- ergy range. Indeed, tritium β− decay produces one elec- ber of neutrino-induced events N that would be observed tron, one antineutrino and a 3He atom via the decay in the detector each second and per unit of the atomic 3 3 − recoil energy, is H → He+e +¯νe, with a lifetime τ = 17.74 years. The antineutrino energy spectrum ranges from 0 keV to the dN Z Z Q 1 dN dσCEνAS Q-value, with a maximum at approximately 15 keV [34]. ν = n 2 dV dEν , dt dTR min 4πr dt dEν dTR The number of neutrinos released, Nν , after a time t fol- V Eν lows the simple exponential decay law (6) where n is the number density of helium atoms in the tar- −t/τ Nν (t) = N 1 − e , (5) get, dV is the infinitesimal volume around the position 3H −→ r ≡ (x, y, z) in the detector, dNν /(dt dEν ) is the differ- min where N is the number of tritium atoms in the source ential neutrino rate and E ' pm T /2 is the mini- 3H ν A R at t = 0. Given the rather large tritium lifetime, the mum antineutrino energy necessary to produce an atom antineutrino rate is expected to stay almost constant in recoil of energy TR. Note that, if the atomic effect is ne- CEνNS the first 5 years. glected, the differential number of events dN /dER We consider a detector setup such that the tritium could be straightforwardly obtained with the following CEνAS CEνNS source is surrounded with a cylindrical superfluid-helium set of substitutions: dσ /dTR → dσ /dER, tank, as depicted in Fig. 2. This configuration allows to mA → mN and TR → ER. However, given that maximize the geometrical acceptance, while allowing to mA ' mN , the atomic and nuclear recoil energies are have a top flat surface where helium atoms could evap- practically coincident, TR ' ER. orate after a recoil. This surface should be equipped To illustrate the consequences of the atomic effect on as described in Ref. [22], to detect the small energy de- the expected number of events, we consider a detector posited by the helium evaporation. In order to shield the of height h = 160 cm, and radius d = 90 cm filled with helium detector from the electrons produced by the tri- 500 kg of helium, a tritium source of 60 g, and a data- tium decay, an intermediate thin layer of heavy material taking period of 5 years. The choice of this particular has to be inserted between the source and the detector, configuration will be clarified in Sec. IV. 4

The number of neutrino-induced events as a function of setup described in Sec. III. For this purpose, we build the recoil energy TR is obtained by integrating the differ- the following least-squares function ential rate defined in Eq. (6) for a time period of 5 years. The result is illustrated in Fig. 3, where the CEνNS dif- 2 N CEνAS − N CEνNS ferential rate is shown by the black solid line, while the χ2 = , (8) CEνAS one is shown by the dashed red line. As stated σ in the introduction, when atomic effects are considered, electrons screen the weak charge of the nucleus as seen by where N CEνAS represents the number of neutrino- the antineutrino. The screening is complete for atomic Atom induced events observed considering the atomic effect, recoil energies TR such that CV = 0, or, in accordance while N CEνNS is the number of events expected ignoring with Eqs. (2) and (3), when such an effect, which represents our reference model [36]. These quantities are obtained integrating in time and N − (1 − 4 sin2 ϑ ) Z W recoil energy the differential spectrum in Eq. (6), con- Fe(TR) = 2 , (7) 1 + 4 sin ϑW sidering for the latter the range of 1-184 meV, being the where for 4He atoms N/Z = 1. Using the SM prediction upper limit the maximum recoil energy that a neutrino of the Weinberg angle at near zero momentum transfer with energy Eν can give to an atom with mass mA, which 2 SM is given by sin ϑW = 0.23857(5) [35], calculated in the MS scheme, we obtain from Eq. (7) the following condition: Fe(TR) = 2 0.4883. Thus, the screening is complete for TR ' 9 meV max 2Eν (see Fig. 1). Due to this destructive interference between TR = . (9) mA + 2Eν the nuclear and the electron contributions, the number of events drops rapidly to zero, as shown by the red dashed line in Fig. 3. Assuming that the main uncertainty contribution is due to the available√ statistics, the denominator in Eq. (8) is set to be σ = N CEνAS. CEνNS -12 1 μν= 10 μB add. term In order to find the experimental setup that would allow to reach a sensitivity of at least 3σ, we calculate this CEνAS(μ ν=0) CEνAS(μ = 10-12 μB) 0.500 ν least-squares function in terms of three parameters: the

] amount of helium in the tank, the amount of tritium in - 1 the source, and the time of data-taking (the main con- 0.100 tributor being the amount of tritium used). We ﬁnd that [ meV

R 0.050 a reasonable combination of these parameters that would allow the observation of the atomic eﬀect is 500 kg of he-

d  / dT lium, 60 g of tritium, and 5 years of data-taking. In 0.010 this scenario, the expected number of CEνAS events is 0.005 N CEνAS = 6.7, to be compared with the expected number of CEνNS events N CEνNS = 14.6.

0.001 In order to claim a discovery, i.e. reach a sensitivity 1 5 10 50 100 of 5σ, the amount of tritium needed increases to 160 g, TR [meV] leaving the other parameters unchanged. The expected number of events in this case becomes N CEνAS = 17.7 FIG. 3. Differential number of neutrino-induced events as considering the atomic effect and N CEνNS = 38.9 without a function of the atomic recoil energy TR in a logarithmic the atomic effect. One can note that the atomic screening scale on both axes. The CEνNS differential number is shown reduces the number of events by almost one half with the by the black solid line while the CEνAS one is shown by particular experimental setup proposed in this paper. For the dashed red line. The dashed-dotted blue line represents the additional term appearing in the CEνAS differential completeness, in Fig. 4 one can see the amount of tritium number of events assuming a neutrino magnetic moment of and helium mass needed to observe (3σ) or discover (5σ) −12 µν = 10 µB while the dotted green line represents the total the atomic effect, considering a data taking time of 3 and differential number of CEνAS for the same value of µν (i.e. 5 years. using the differential cross section in Eq. 12) . Clearly, all these estimates are to be refined for a specific experiment by taking into account systematic contributions from backgrounds and detector efficiencies and resolutions. However, neither of the indicated IV. SENSITIVITY TO THE ATOMIC EFFECT contributions appear to be seriously limiting [37] and, hence, the conclusions drawn in this paper should remain We test the feasibility to observe the atomic effect valid. in coherent neutrino scattering using the experimental 5

500 the number of events for a given value of the Weinberg angle.

200 99% 60 g of 3H 100 6 160 g of 3H 500 g of 3H 50 3σ(3 years) 2σ Tritium mass [ g ] 4 5σ(3 years) 2 Δχ 3σ(5 years) 20 90% 5σ(5 years) 2 100 200 500 1000 2000 5000 Helium mass[kg] 1σ

0 FIG. 4. Iso-sigma curves to observe (3σ in dashed red and 0.15 0.20 0.25 0.30 solid black) or discover (5σ in dotted-dashed blue and long 2 sin ϑW dashed green) the atomic effect, as a function of the helium and tritium masses, considering a data taking time of 3 and FIG. 5. With the black solid curve it is shown the 5 years, respectively. 2 2 2 2 ∆χ = χ − χmin, where the χ is defined in Eq. (10), as a 2 function of the Weinberg angle sin ϑW , obtained considering a tritium source of 60 g, while with the dashed red line V. PHYSICS PERSPECTIVES and the dotted-dashed blue line it is shown the ∆χ2 obtained considering a tritium source of 160 g and 500 g, respectively. In the following, we assume that CEνAS has been observed and we discuss the sensitivity of the determination of the Weinberg angle and the neutrino magnetic 2 2 2 Figure 5 shows the ∆χ = χ − χmin profile as a func- moment. Motivated by the studies performed in the pre- 2 tion of sin ϑW for the three different scenarios described vious section, we consider a detector made of 500 kg of above. The black solid, red dashed, and blue dotted- helium, 5 years of data-taking, and three different sce- dashed lines refer to 60 g, 160 g, and 500 g of tritium, narios for the source: 60 g, 160 g, and 500 g of tritium. respectively. The uncertainties achievable in the three The last scenario is considered in order to see the po- +0.04 +0.025 +0.015 scenarios are −0.05, −0.029 and −0.016, respectively. tentialities of such a detector if a large quantity of tri- Figure 6 shows a summary of the weak mixing an- tium will become available. For completeness, the ex- gle measurements at different values of q, together with pected number of events in this optimistic scenario be- the SM prediction. The uncertainty that can be reached CEνAS comes N = 55 taking into account the atomic effect with the method proposed in this paper in the case of a CEνNS and N = 122 without it. 500 g tritium source is shown by the dashed blue line. Despite the fact that this uncertainty is rather large, such measurement would represent a unique opportu- A. DETERMINATION OF THE WEINBERG nity to explore the low-energy sector, since the value ANGLE hqi ' 2×10−5 GeV is several orders of magnitude smaller

Atom than in all the other measurements. Given that the value Since the vector coupling CV in Eq. (3) depends on of the Weinberg angle provides a direct probe of physics 2 sin ϑW , the Weinberg angle can be measured in CEνAS phenomena not included in the SM, such a measurement processes. In order to quantify the sensitivity of a mea- would give complementary information to those at mid surement of the Weinberg angle with the experimental and high-energy. In particular, this measurement would 2 setup described above, we consider a deviation of sin ϑW be highly sensitive to an extra dark boson, Z , whose ex- 2 SM d from the Standard Model value sin ϑW in the least- istence is predicted by grand uniﬁed theories, technicolor squares function models, supersymmetry and string theories [44]. Mea-  2 surements of the Weinberg angle provide constraints on CEνAS CEνAS 2 N − N (sin ϑ ) the properties of this dark boson as its mass, mZd , its χ2(sin2 ϑ ) = SM W . W  q  kinetic coupling to the SM fermions, ε, and its Z-Zd CEνAS NSM mass-mixing coupling, δ [45–47]. As an example, the (10) orange and green regions in Fig. 6 show the low-q de- CEνAS 2 Here, NSM represents the expected number of CEνAS viations of sin ϑW with respect to the SM value pre- 2 2 SM CEνAS 2 events if sin ϑW = sin ϑW , while N (sin ϑW ) is dicted for two particular conﬁgurations of these param- 6

0.255 CEνAS B. EFFECT OF NEUTRINO MAGNETIC MOMENT 0.250 500 g of 3H

0.245 The experiment proposed in this paper would be

_ E158 MS

) highly sensitive to a possible neutrino magnetic moment.

q NuTeV ( 0.240 So far, the most stringent constraints on the electron W ϑ APV Qweak neutrino magnetic moment in laboratory experiments 2 0.235

sin LEP1 have been obtained looking for possible distortions of 0.230 PVDIS SLC the recoil electron energy spectrum in neutrino-electron elastic scattering, exploiting solar neutrinos and reactor 0.225 Tevatron LHC anti-neutrinos. The Borexino collaboration reported the 10-4 0.1 100 best current limit on the effective magnetic moment of 2.8 × 10−11µ at 90% confidence level (C.L.) using the q [GeV] B electron recoil spectrum from 7Be solar neutrinos [48]. 2 The best magnetic moment limit from reactor anti- FIG. 6. Variation of sin ϑW with energy scale q. The SM prediction is shown as the red solid curve, together neutrinos, obtained by the GEMMA experiment, is −11 with experimental determinations in black at the Z-pole [35] very similar and it corresponds to 2.9 × 10 µB (90% (Tevatron, LEP1, SLC, LHC), from atomic parity viola- C.L.) [49]. tion on caesium [18, 38, 39], which has a typical momentum transfer given by hqi ' 2.4 MeV, Møller scattering [40] In the original SM with massless neutrinos the neu- (E158), deep inelastic scattering of polarized electrons on trino magnetic moments are vanishing, but the results 2 deuterons [41] (e H PVDIS) and from neutrino-nucleus scat- of neutrino oscillation experiments have proved that the tering [42] (NuTeV) and the result from the proton’s weak SM must be extended in order to give masses to the neu- charge at q = 0.158 GeV [43] (Qweak). For clarity the Teva- trinos. In the minimal extension of the SM in which neutron and LHC points have been displayed horizontally to the trinos acquire Dirac masses through the introduction of left and to the right, respectively, as indicated by the arrows. In dashed-blue it is shown the result that could be achieved right-handed neutrinos, the neutrino magnetic moment using the experimental setup proposed in this paper, obtained is given by [50–55] exploiting CEνAS effect at very low momentum transfer. The 3 e GF −19 mν orange and green regions indicate the values of the weak mix- µν = √ mν ' 3.2 × 10 µB , (11) ing angle that are obtained for particular masses and cou- 8 2 π2 eV plings of a hypothetical Zd boson, see the text for more de- tails. where µB is the Bohr magneton, mν is the neutrino mass and e is the electric charge. Taking into account the current upper limit on the neutrino mass of the order of 1 eV, this value is about 8 orders of magnitude smaller than the Borexino and GEMMA limits. However, search- eters: the contours of the orange shadowed region corre- ing for values of µ larger than that in Eq. (11) is inter- −3 ν spond to mZd = 30 MeV, δ = 0.015, and ε = 1 × 10 , esting because a positive result would represent a clear while the contours of the green shadowed region have signal of physics beyond the minimally extended SM (see been obtained using mZd = 0.05 MeV, δ = 0.0015, and Ref. [56]). −5 ε = 1 × 10 . These values have been chosen such that The existence of a neutrino magnetic moment could −4 |ε δ| ≤ 8 × 10 [47], which allows to roughly satisfy the have a significant effect on the CEνAS cross-section, that 2 existing upper bounds on these quantities . The shad- acquires an additional term owed regions indicate the values of the weak mixing an- CEνAS CEνAS 2 2 2 gle obtained leaving the mass of the Zd boson invariant, dσ dσ πα Z µν ' + but using smaller values of ε and δ. As it is visible from dT dT m2 µ R µν 6=0 R e B Fig. 6, the impact of the hypothetical Zd boson starts 1 1 2 at values of the transferred momentum equal to its mass · − (1 − Fe(TR)) , (12) and extends to lower values. Thus, a measurement like TR Eν the one proposed in this paper would be useful to better where α is the fine structure constant and me is the elec- constraint even lighter Zd’s. tron mass. The atomic effect is included in the term 2 (1 − Fe(TR)) . In fact, for high energy neutrinos, the electron form factor goes to zero, obtaining the neutrino magnetic moment contribution for CEνNS process. On the contrary, for low energies, where Fe(TR) → 1, neutrinos see the target as a whole neutral object, thus no elec- tromagnetic properties can affect the process. By adding 2 Note that in Ref. [47] the constraint is expressed in terms of δ0, this contribution in Eq. (6), integrating for a time pe- but for our choice of parameters δ0 ' δ. riod of 5 years, and considering a magnetic moment of 7

10 evaluate the physics potentialities of this apparatus. The 60 g of 3H 3σ observation of a coherent scattering of the whole atom re- 160 g of 3H quires the detection of very low atomic recoil energies, of 8 the order of 10 meV. This is achievable thanks to the 500 g of 3H 99% combination of different critical ingredients. First, the − 6 exploitation of the β decay of tritium that is charac- 2 terized by a small Q-value, ensuring a sufficient flux of Δχ low-energy antineutrinos. Second, the usage of a target 4 2σ detector with a new technology based on the evaporation

90% of helium atoms coupled with ﬁeld ionization detector 2 arrays that allows to be sensitive to very small energy

1σ deposits. The usage of liquid helium as a target has the triple advantage of being stable, having a small binding 0 2.× 10 -13 4.× 10 -13 6.× 10 -13 8.× 10 -13 1.× 10 -12 energy to the surface, expected to be below 1 meV, and μν [μB] having a small atomic radius Ratom, which allows to reach the coherence condition for larger values of the momen- FIG. 7. With the black solid curve it is shown the tum transfer. The effect of the interference between the 2 2 2 2 ∆χ = χ − χmin, where the χ is defined in Eq. (10), as a nucleus and electron cloud scattering cross section is to function of the magnetic moment of the neutrino µν , obtained produce a sharp dip in the recoil spectrum at atomic re- considering a tritium source of 60 g, while with the dashed coil energies around TR ' 9 meV, which reduces sizeably 2 red line and the dotted-dashed blue line it is shown the ∆χ the expected number of events with respect to that in obtained considering a tritium source of 160 g and 500 g, re- the case of coherent neutrino-nucleus elastic scattering. spectively. We estimated that in order to observe at 3σ the existence of CEνAS processes it is necessary to use a tritium −12 source of about 60 g, a tank filled with about 500 kg of 10 µB, one obtains the expected differential number of CEνAS events as a function of the atomic recoil energy liquid helium (contained in a cylinder of height 160 cm shown by the blue dotted-dashed line in Fig. 3. Compar- and radius 90 cm) and a data-taking period of about 5 ing it to the red dashed curve obtained considering a null years. magnetic moment, it is clear that there is a large differ- Keeping the same amount of helium and data-taking ence between the two cases and this difference could be period we tested the sensitivity of this experimental setup exploited to put stringent limits on the neutrino magnetic to the measurement of the Weinberg angle and the obser- moment. vation of a possible neutrino magnetic moment for three To estimate the sensitivity, we consider the least- different scenarios corresponding to different amounts of squares function tritium in the source: 60 g, 160 g, and 500 g. In the lat-  2 ter scenario, the SM value of the Weinberg angle can be +0.015 N CEνAS − N CEνAS(µ ) measured with a statistical uncertainty of sin2 ϑSM . χ2(µ ) = SM ν , (13) W −0.016 ν  q  Even if this measurement will be less precise than other N CEνAS SM determinations of the Weinberg angle, it will represent 2 CEνAS the lowest-energy measurement of sin ϑW , at a momen- where NSM represents the number of CEνAS events tum transfer about 2 orders of magnitude smaller than that one would observe if the magnetic moment is zero, CEνAS the current determination using caesium atomic parity while N (µν ) is the number of events for a given 2 violation [38, 39]. Moreover, the measurement proposed value of the magnetic moment. Figure 7 shows the ∆χ in this paper is not affected by the uncertainties on the profile as a function of the neutrino magnetic moment nuclear neutron form factor that contribute to atomic for the three different experimental scenarios described 2 parity violation measurements of sin ϑW [18]. Therefore, above: the black solid, red dashed and blue dotted- the observation of CEνAS would represent an indepen- dashed lines refer to 60 g, 160 g, and 500 g of tri- 2 dent test of the running of sin ϑW predicted by the SM tium, respectively. The respective limits that are achiev- and could be sensitive to low-energy deviations due to able in these three different scenarios at 90% C.L. are −13 −13 −13 physics beyond the SM, as the contribution of low-mass 7.0×10 µB, 5.5×10 µB, and 4.1×10 µB, which dark Z bosons. are almost two orders of magnitude smaller than the current experimental limits. Finally, we studied the sensitivity of the proposed apparatus to a possible electron neutrino magnetic moment and we found that using 60 g of tritium it is possible to VI. CONCLUSIONS −13 set a lower limit of about 7 × 10 µB at 90% C.L., that is more than one order of magnitude smaller than the In this paper we proposed an experimental setup to current experimental limit, showing the great potential- observe coherent elastic neutrino-atom scattering and we ities of this experimental setup. 8

ACKNOWLEDGMENT ulating discussions about the experimental setup of the detector. M.C., F.D. and E.P. would like to thank Michele Saba and Marco Razeti for the useful suggestions and stim-

[1] D. Akimov et al. (COHERENT), Science 357, 1123 ph.IM]. (2017), arXiv:1708.01294 [nucl-ex]. [24] M. G. Betti et al. (PTOLEMY), JCAP 1907, 047 (2019), [2] D. Akimov et al. (COHERENT), (2018), 10.5281/zen- arXiv:1902.05508 [astro-ph.CO]. odo.1228631, arXiv:1804.09459 [nucl-ex]. [25] Y. V. Gaponov and V. Tikhonov, Yadernaya Fizika 26, [3] D. Z. Freedman, Phys. Rev. D 9, 1389 (1974). 594 (1977). [4] D. Z. Freedman, D. N. Schramm, and D. L. Tubbs, Ann. [26] K. Patton, J. Engel, G. C. McLaughlin, and N. Schunck, Rev. Nucl. Part. Sci. 27, 167 (1977). Phys. Rev. C 86, 024612 (2012). [5] A. Drukier and L. Stodolsky, Phys. Rev. D30, 2295 [27] J. Barranco, O. G. Miranda, and T. I. Rashba, JHEP (1984), [,395(1984)]. 12, 021 (2005), arXiv:hep-ph/0508299 [hep-ph]. [6] M. Cadeddu, C. Giunti, Y. F. Li, and Y. Y. Zhang, [28] D. Aristizabal Sierra, J. Liao, and D. Marfatia, JHEP Phys. Rev. Lett. 120, 072501 (2018), arXiv:1710.02730 06, 141 (2019), arXiv:1902.07398 [hep-ph]. [hep-ph]. [29] P. A. Doyle and P. S. Turner, Acta Crystallographica [7] D. K. Papoulias, T. S. Kosmas, R. Sahu, V. K. B. Kota, Section A 24, 390 (1968). and M. Hota, (2019), arXiv:1903.03722 [hep-ph]. [30] P. J. Brown et al., International Tables for Crystallogra- [8] P. Coloma, M. C. Gonzalez-Garcia, M. Maltoni, phy C ch. 6.1, 554 (2006). and T. Schwetz, Phys. Rev. D96, 115007 (2017), [31] A. J. Thakkar and V. H. Smith, Jnr, Acta Crystallo- arXiv:1708.02899 [hep-ph]. graphica Section A 48, 70 (1992). [9] O. G. Miranda, D. K. Papoulias, M. Tórtola, and [32] M. A. Coulthard, Proceedings of the Physical Society 91, J. W. F. Valle, JHEP 07, 103 (2019), arXiv:1905.03750 44 (1967). [hep-ph]. [33] A. J. Thakkar and V. H. Smith, Phys. Rev. A 15, 1 [10] M. Cadeddu, C. Giunti, K. A. Kouzakov, Y. F. Li, A. I. (1977). Studenikin, and Y. Y. Zhang, Phys. Rev. D98, 113010 [34] C. Giunti and C. W. Kim, Fundamentals of Neutrino (2018), arXiv:1810.05606 [hep-ph]. Physics and Astrophysics (Oxford University Press, Ox- [11] B. Dutta, S. Liao, S. Sinha, and L. E. Strigari, Phys. ford, UK, 2007) pp. 1–728. Rev. Lett. 123, 061801 (2019), arXiv:1903.10666 [hep- [35] M. Tanabashi et al. (Particle Data Group), Phys. Rev. ph]. D98, 030001 (2018). [12] O. G. Miranda, G. Sanchez Garcia, and O. Sanders, [36] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, Adv. High Energy Phys. 2019, 3902819 (2019), Eur. Phys. J. C71, 1554 (2011), [Erratum: Eur. Phys. arXiv:1902.09036 [hep-ph]. J.C73,2501(2013)], arXiv:1007.1727 [physics.data-an]. [13] J. Heeck, M. Lindner, W. Rodejohann, and S. Vogl, [37] S. A. Hertel, A. Biekert, J. Lin, V. Velan, and D. N. SciPost Phys. 6, 038 (2019), arXiv:1812.04067 [hep-ph]. McKinsey, (2018), arXiv:1810.06283 [ins-det]. [14] M. Abdullah, J. B. Dent, B. Dutta, G. L. Kane, S. Liao, [38] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, and L. E. Strigari, Phys. Rev. D98, 015005 (2018), J. L. Roberts, C. E. Tanner, and C. E. Wieman, Science arXiv:1803.01224 [hep-ph]. 275, 1759 (1997). [15] Y. Farzan, M. Lindner, W. Rodejohann, and X.-J. Xu, [39] V. A. Dzuba, J. C. Berengut, V. V. Flambaum, and JHEP 05, 066 (2018), arXiv:1802.05171 [hep-ph]. B. Roberts, Phys. Rev. Lett. 109, 203003 (2012), [16] J. Liao and D. Marfatia, Phys. Lett. B775, 54 (2017), arXiv:1207.5864 [hep-ph]. arXiv:1708.04255 [hep-ph]. [40] P. L. Anthony et al. (SLAC E158), Phys. Rev. Lett. 95, [17] D. K. Papoulias and T. S. Kosmas, Phys. Rev. D97, 081601 (2005), arXiv:hep-ex/0504049 [hep-ex]. 033003 (2018), arXiv:1711.09773 [hep-ph]. [41] D. Wang et al. (PVDIS), Nature 506, 67 (2014). [18] M. Cadeddu and F. Dordei, Phys. Rev. D99, 033010 [42] G. P. Zeller et al. (NuTeV), Phys. Rev. Lett. 88, 091802 (2019), arXiv:1808.10202 [hep-ph]. (2002), [Erratum: Phys. Rev. Lett.90,239902(2003)], [19] B. C. Cañas, E. A. Garcés, O. G. Miranda, arXiv:hep-ex/0110059 [hep-ex]. and A. Parada, Phys. Lett. B784, 159 (2018), [43] D. Androić et al. (Qweak), Nature 557, 207 (2018), arXiv:1806.01310 [hep-ph]. arXiv:1905.08283 [nucl-ex]. [20] V. A. Bednyakov and D. V. Naumov, Phys. Rev. D98, [44] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, 053004 (2018), arXiv:1806.08768 [hep-ph]. A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90, [21] L. M. Sehgal and M. Wanninger, Physics Letters B 171, 025008 (2018), arXiv:1710.01833 [physics.atom-ph]. 107 (1986). [45] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Phys. [22] H. J. Maris, G. M. Seidel, and D. Stein, Phys. Rev. Lett. Rev. D85, 115019 (2012), arXiv:1203.2947 [hep-ph]. 119, 181303 (2017), arXiv:1706.00117 [astro-ph.IM]. [46] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Phys. [23] S. Betts et al., in Proceedings, 2013 Community Summer Rev. Lett. 109, 031802 (2012), arXiv:1205.2709 [hep-ph]. Study on the Future of U.S. Particle Physics: Snowmass [47] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Phys. on the Mississippi (CSS2013): Minneapolis, MN, USA, Rev. D92, 055005 (2015), arXiv:1507.00352 [hep-ph]. July 29-August 6, 2013 (2013) arXiv:1307.4738 [astro- 9

[48] M. Agostini et al. (Borexino), Phys. Rev. D96, 091103 [52] B. Kayser, Phys. Rev. D26, 1662 (1982). (2017), arXiv:1707.09355 [hep-ex]. [53] J. F. Nieves, Phys. Rev. D 26, 3152 (1982). [49] A. G. Beda, V. B. Brudanin, V. G. Egorov, D. V. [54] P. B. Pal and L. Wolfenstein, Phys. Rev. D 25, 766 Medvedev, V. S. Pogosov, E. A. Shevchik, M. V. (1982). Shirchenko, A. S. Starostin, and I. V. Zhitnikov, Phys. [55] R. E. Shrock, Nucl. Phys. B206, 359 (1982). Part. Nucl. Lett. 10, 139 (2013). [56] C. Giunti and A. Studenikin, Rev. Mod. Phys. 87, 531 [50] K. Fujikawa and R. E. Shrock, Phys. Rev. Lett. 45, 963 (2015), arXiv:1403.6344 [hep-ph]. (1980). [57] A. G. Beda, V. B. Brudanin, V. G. Egorov, D. V. [51] J. Schechter and J. W. F. Valle, Phys. Rev. D 24, 1883 Medvedev, V. S. Pogosov, M. V. Shirchenko, and A. S. (1981). Starostin, Adv. High Energy Phys. 2012, 350150 (2012).