Probing Neutrino Dipole Portal at COHERENT Experiment

Motivated by the first observation of coherent-elastic neutrino-nucleus scattering at the COHERENT experiment, we confront the neutrino dipole portal giving rise to the transition of the standard model neutrinos to sterile neutrinos with the recently released CENNS 10 data from the liquid argon as well as the CsI data of the COHERENT experiment. Performing statistical analysis of those data, we show how the transition magnetic moment can be constrained for the range of the sterile neutrino mass between 10 keV and 40 MeV.


I. INTRODUCTION
Coherent elastic neutrino-nucleus scattering (CEνNS ) proposed roughly 47 years ago [1] can be a powerful test of the Standard Model (SM), and provides a useful tool to search for new physics (NP) beyond the SM. In 2017, the collaboration COHERENT has announced the discovery of CEνNS with stopped pion neutrinos on a CsI detector showing the reference of its presence against the absence at a 6.7σ confidence level (CL) [2,3]. Since then CEνNS becomes a very active and dynamic field as it opens up new opportunities to probe NP [4][5][6][7][8][9][10][11][12][13][14]. Very recently, the COHERENT has reported the observation of CEνNS for the first time also in argon, using a single-phase 24 kg liquid-argon (LAr) scintillation detector, with two independent analyses that prefer CEνNS to the background-only null hypothesis with a 3σ level [15]. The experimental challenge behind those experiments is the need to observe nuclear recoils with a very small kinetic energy T nr of a few keV in the presence of a larger background. This requirement is necessary for the coherent nucleus recoils which occur for | #» q |R << 1 [16], where | #» q | √ 2m N T nr is the three momentum transfer, R is the nuclear radius of a few fm, and m N is the nucleus mass.
Owing to the fact that neutrinos are massive, the SM of particle physics is incomplete.
Sterile neutrinos are natural extensions to the SM and provide a possible portal to the dark sector. In the last few years, much attention has been paid to the models containing sterile neutrino N that can couple to the SM lepton doublet L and Higgs scalar H. Since there is no constraint on the sterile neutrino mass, a wide range is possible, from the sub-eV scale to the Planck scale. This range can be somewhat narrower if N is indeed taking part in generating masses for the light active neutrinos.
In the SM, neutrinos are charge neutral and have extremely tiny electromagnetic (EM) moments [17][18][19][20] . Those EM properties of active neutrinos in the context of the SM are not detectable due to the small mass in the sub eV range with current experimental sensitivity.

II. CEνNS AT COHERENT AND SIGNAL PREDICTION
The COHRENET experiment uses a high intensity neutrino beam produced at the Spallation Neutron Source (SNS) of the Oak Ridge National Laboratory [2,3,15]. A 1 GeV proton beam incident on a Hg target produces pions. The negative pions are captured almost within the target while the positive pions decay at rest, π + → µ + ν µ , produces neutrinos from which a monoenergetic ν µ beam is generated . The muon neutrinos arrive at the target within a short span of time (< 1.5µs) after passing proton-on-target (POT) and thus call them "prompt" neutrinos. The muons accompanied by ν µ also decay to produce ν e andν µ via µ + → e + ν eνµ , which are known as "delayed" neutrinos and have a lower energy profile as they are produced in a three-body decay. In the end, one can obtain ∼ 0.08 − 0.09 ν e and (−) ν µ neutrinos per POT. As is well known, the neutrino fluxes from the SNS are described by, where r is the number of ν e and (−) ν µ neutrinos produced per proton on target, N POT is the total number of protons on target during the data-taking period and L is the distance between the detector and the neutrino source. For the dataset collected by the COHERENT CsI detector, we use r = (8 ± 0.9) × 10 −2 , N POT = 17.6 × 10 22 and L = 19.3m [2,3].
The SM prediction for the differential cross section of CEνNS with a spin-zero nucleus N with Z protons and N neutrons as a function of the nuclear kinetic recoil energy T nr is given by [21][22][23] where G F is the Fermi constant, l = e, µ, τ denotes the neutrino flavor, E ν is the neutrino energy and Q l, For neutrino-proton coupling, g p V and the neutrino-neutron coupling, g n V , we take the more accurate values which take into account radiative corrections in the minimal subtraction, MS scheme, as follows [24]; In eq. (2), F Z (| #» q | 2 ) and F N (| #» q | 2 ) are, respectively, the form factors of the proton and neutron distributions in the nucleus, respectively. For those, we employ the Helm parameterisation [25], that is known to be practically equivalent to the other two commonly-used symmetrized Fermi [26] and Klein-Nystrand [27] parameterisations. The Helm form factor is given by where R p and R n are the rms radii of the proton and neutron distributions, respectively, 2M N E ν is the absolute value of exchanged three-momentum, s = 0.9 fm is the nuclear skin-width and j 1 (x) is the spherical Bessel function of order one. The nucleon charge radius is given by The size of the neutron distribution radius R n is taken to be 4.7 fm and 4.1 fm for the analyses involving CsI and Ar, respectively [14]. Note that the coherence is lost for | #» q |R p(n) 1 The theoretical prediction of CEνNS event-number N i in each nuclear-recoil energy-bin i is given by where A(T nr ) is the energy-dependent reconstrcution efficiency, N D represents the number of target nuclei in the detector mass and is given by where m det is the detector mass, N A is the Avogadro's number, (m N ) mol is the molar mass of the detector molecule and g mol is the numer of atoms in a single detector molecule. For CsI detector [2,3], m det = 14.6 kg and (M CsI ) mol = 259.8 gram/mol, and for Ar detector [15], m det = 24 kg and (M Ar ) mol = 39.96 gram/mol. The lower integration limit in eq. (8) E min is the minimum neutrino energy required to attain a recoil energy T nr , which is given by In Ref. [3], the recostruction efficiency for CsI detector is given in terms of the detected number of photoelectrons n PE by the function For Ar detector, we take the dector effficiency from the results in Ref. [15].
For CsI detector, we consider the quenching factor [3], describing the number of photoelectrons detected by photomultiplier tubes per keV nuclear recoil energy. It can be used to map n PE to the recoil energy T nr in the analysis. For Ar detector, the electron-equivalent recoil energy T ee [keV ee ], is transformed into the nuclear recoil energy [15] T nr [keV nr ] thanks to the relation where f Q is the quenching factor, which is the ratio between the scintillation light emitted in the nuclear and electron recoils. It is parameterised as f Q (T nr ) = (0.246 ± 0.006 keV nr ) + ((7.8 ± 0.9) × 10 −4 )T nr up to 125 keV nr , and is kept constant for larger values.
In this work, we consider the additional possibility that there is new physics in the neutrino sector, and examine how this may affect the signals observed by the COHERENT experiments.

III. NEUTRINO DIPOLE PORTAL
Given the large interests in the searches of sterile neutrinos N , in this work we willl examine the effect of N coupled to the active neutrinos via the so-called "dipole portal" encoded in the following effective Lagrangian [28,29] where α denotes the flavor index, µ α is magnetic moment, F µν is the electromagnetic field strength tensor and ν L is a SM neutrino. This interaction has been considered in the context of the MiniBooNE events [30][31][32][33][34][35][36] and has also been studied in the context of IceCube data [37] and at the upcoming SHiP experiment [28]. A summary of existing constraints can be found in Refs. [28,37]. Note that this is an effective Lagrangian that needs to be UV completed at energy scales not much larger than Λ ∼ µ −1 . Here, we assume that the Yukawa interaction LN H is so suppressed that the dipole term should play an enhanced role of EM interactions in the production and decay of N , and mixing between the SM neutrinos and the sterile neutrino is so negligible that neutrino oscillation involving sterile neutrino should be suppressed in our study.
The dipole term can admit for a SM neutrino to up-scatter off a nucleus N to the sterile neutrino N given as This up-scattering generates a distinct recoil spectrum. We will particularly investigate the distortion of the event spectrum generated from the up-scattering process in the presence of the dipole interaction given in eq. (15).
Without loss of generality, we assume that µ α are common for all flavors. The differential corss section for the up-scattering process is given by [28,[38][39][40] dσ where Z is the number of proton, M N is the right-handed neutrino mass, α em is the electromagnetic fine structure constant, E ν is the neutrino energy. Inserting Eqs. (2, 17) into Eq. (8), we can calculate the total number of events for neutrino-nucleus scattring.
From the kinematics, we see that the maximum possible recoil energy for a given neutrino energy E ν is reached to In our analysis, instead of applying eq.(18) to eq.(8), we take M N so that T max nr becomes larger than T i nr for a given range of E ν in eq. (8). We found that the most conservative upper limit on M N satisfying T max nr T i nr is about 40 MeV, which is taken as an upper limit for the scan of M N in our analysis.

A. COHERENT data analysis
In the analysis based on the CsI COHERENT dataset, we perform a fit of the data by means of a least-squares function given as [2,3] where N exp and σ βc = 0.25 [2]. σ i is the statistical uncertainty taken from Ref. [2,3]. Following Ref. [41], we employ only the 12 energy bins from i = 4 to i = 15 of the COHERENT data, because they cover the recoil kinetic energy of the more recent Chicago-3 quenching factor measurement [42].
In the analysis based on the Ar COHERENT experiment, we only take the data coming from the analysis A, whose range of interest of the nuclear recoil energy is [0, 120] keV ee with 12 energy bins of size equal to 10 keV ee . Following Ref. [14], we perform a fit of the data by means of the least-squares function given by where B PBRN σ BRNES = 0.058 2 12 = 1.7%, (22) σ CEνNS = 13.4%, In Eq. (21), η CEνNS , η PBRN and η LBRN are nuisance parameters which quantify, respectively, the systematic uncertainty of the signal rate and that of the PBRN and LBRN background rate, with the corresponding standard deviations σ CEνNS , σ PBRN and σ LBRN .

B. Results and Discussion
The parameter regions that we scan are where µ B denotes Bohr magneton. Fig.1 shows the expected number of events for the CsI to the experimental measurements. We found that the best fit of the CENNS 10 data is achieved at µ ν = 0, which is the same as the SM result, and the minimum value of χ 2 for CsI data is 6.72. It turns out that the values of χ 2 for the SM predictions are 7.07 (CsI) and 3.18 (Ar), respectively.
In Fig. 2, we present ∆χ 2 in terms of µ ν in unit of µ B for CsI (Left) and Ar (Right) dector. The plots correpond to M N = 28.9 MeV. The black, blue and red dashed lnes correspond to the 1σ, 90% and 99% C.L, respectively. From the results, we see that the minimum χ 2 is located at µ ν 3.2 × 10 −9 µ B for CsI detector, while it reaches at µ ν = 0 for Ar detector, which means that the SM prediction gives rise to best fit point. The shapes of the plots in Fig. 2 indicate that CsI data is more sensitive to the transition magnetic moment than CENNS 10 data. than that for the SM prediction. We see that for a given M N the result from the data at CsI detector gives more stringent bound on µ ν than that at Ar dector.
Let us compare our results with other experimental constraints shown in [29,40]. In Fig. 4, the colored regions below the blue and black solid lines are allowed at 90% by the data obtained at Ar and CsI detectors, respectively, which are the same as shown in Fig. 3.
The gray, green and blue regions are excluded at 90% C.L. by the data from BOREXINO [43], XENON1T (nuclear recoil) [44,45], and NOMAD [46], respectively. The red line is a part of the contour corresponding to the 90% favored region at ICeCube [47]. Thus, the region above the red line is allowed. The green and black dashed lines are exclusion curves obtained from MiniBooNE [28,48] and CHARM-II [49]. Note that the constraints from the experiments except for the COHERENT correspond to the bounds on the transition magnetic moment between ν µ and N .
As can be seen in Fig.4, the result from BOREXINO shows that the bound on µ ν /µ B at 90% for low M N ( 2MeV) is more stringent than those we obtained in this work [45]. On the other hand, for higher M N ( 2MeV), there is no bound on µ ν from BOREXINO and the results of the nuclear recoil from XENON1T give the lower bound on µ ν /µ B ∼ 10 −8 for M N 10 MeV, whereas our results lead to new or more stringent constraints on µ ν for Once the sterile neutrino is produced via up-scattering in the detector, it will eventually decay. The dominant decay channel is N → ν α + γ. The boosted radiative decay length of the sterile neutrino with energy E s (= E ν − T nr ) is given by where γ is the Lorentz factor and β is the ratio of the velocity to speed of light in a vacuum.  The colored regions below the blue and black solid lines are allowed at 90% by the data obtained at Ar and CsI detectors, respectively, which are the same as shown in Fig. 3. The gray, green and blue regions are excluded at 90% C.L. by the data from BOREXINO [43], XENON1T (nuclear recoil) [44,45], and NOMAD [46], respectively. The red line is a part of the contour corresponding to the 90% favored region at ICeCube [47]. The green and black dashed lines are exclusion curves obtained from MiniBooNE [28,48] and CHARM-II [49].

ACKNOWLEDGMENTS
JEK is supported in part by the NRF grant NRF-2018R1A2A3074631. AD and SKK are supported in part by the National Research Foundation (NRF) grants NRF-2019R1A2C1088953.