Studies of Quantum-Mechanical Coherency Effects in Neutrino-Nucleus Elastic Scattering

V. Sharma, 2 L. Singh, 3 H.T. Wong, ∗ M. Agartioglu, 5 J.-W. Chen, M. Deniz, S. Kerman, † H.B Li, C.-P. Liu, M.K. Singh, 2 and V. Singh 3 (TEXONO Collaboration) 1 Institute of Physics, Academia Sinica, Taipei 11529, Taiwan. 2 Department of Physics, Institute of Science, Banaras Hindu University, Varanasi 221005, India. 3 Department of Physics, School of Physical and Chemical Sciences, Central University of South Bihar, Gaya 824236, India 4 Department of Physics, Dokuz Eylül University, Buca, İzmir 35160, Turkey. 5 Department of Physics, National Dong Hwa University, Shoufeng, Hualien 97401, Taiwan. 6 Department of Physics, CTS and LeCosPA, National Taiwan University, Taipei 10617, Taiwan. (Dated: October 15, 2020)


I. INTRODUCTION
The elastic scattering of a neutrino with a nucleus [1,2] νA el : where A(Z, N ) denotes the atomic nucleus with its respective atomic, charge and neutron numbers, is a fundamental electroweak neutral current process in the Standard Model (SM). Studies of neutrino-nucleus elastic scattering can provide sensitive probes to physics beyond SM (BSM) [3,4] and certain astrophysical processes [1,5]. It offers prospects to study quantum-mechanical coherency effects in electroweak interactions [6], neutron density distributions [7], to detect supernova neutrinos [8] and to provide a compact and transportable neutrino detectors for realtime monitoring of nuclear reactors [9]. The νA el events from solar and atmospheric neutrinos are the irreducible "neutrino floor" background [10] to forthcoming generations of dark matter experiments [11].
There are active experimental programs to observe and measure the processes with neutrinos from reactors [12] or from decay-at-rest pions (DAR-π) [4] with spallation neutron source [13]. Future dark matter experiments may also be sensitive to νA el from solar neutrinos [14]. First positive measurement of νA el was achieved by the COHERENT experiment with CsI(Na) detector [15], followed by measurements with liquid Ar detector [16].
The νA el reaction provides a laboratory to probe the quantum-mechanical coherency effects [6]. Experimental measurements are mostly performed in parameter space where the coherency effects are partial and incomplete. The deviations from perfect coherency would have to be characterized before this interaction can be effectively applied towards other goals like the studies of BSM physics. Our earlier work [6] identified a coherency parameter α(q 2 ) which can consistently characterize the transitions between coherent and decoherent states in νA el with different ν-sources and target nuclei. This article follows and expands on these studies. The relations between α(q 2 ) with the complementary descriptions in terms of nuclear physics with the language of nuclear form factors or with the measurable suppression in cross-sections are discussed in Section II. The dependence of coherency effects with interaction kinematics for various neutrino sources and detector targets are surveyed in Section III. The constraints provided by the COHERENT-CsI [15] and -Ar [16] data are derived in Section IV.

II. FORMULATION AND CHARACTERIZATION
The differential cross-section of νA el scattering at three-momentum transfer q (≡ | q|) and neutrino energy E ν can be expressed as [2,6]: where Γ(q 2 ) is a function describing the contributions due to many-body physics in the target nuclei, since the νA el arXiv:2010.06810v1 [hep-ex] 14 Oct 2020 interactions involve collective contributions of individual nucleons in the nucleus. The relevant kinematics variable is q 2 which characterizes the physics and is universal to all target. The experimental observable is the nuclear recoil energy (T ), expressed in units of keV nr in this article, which depends on the target nuclear mass M and is related to q 2 via q 2 =2M T +T 2 2M T . The minimal observable energy T min for the nuclear recoils is the detector threshold, while kinematics limits the maximum recoil energy to be T max =2E ν 2 /(M +2E ν ) 2E ν 2 /M . These limits can be translated to q 2 min =2M T min and The variations of the νA el differential and integral cross-sections with respect to T are discussed in Appendix A.
Depending on the particular physics aspects to probe, there are complementary formulations on the Γ(q 2 ) function. The usual description is based on nuclear physics, in which where F Z (q 2 )∈[0, 1] and F N (q 2 )∈[0, 1] are, respectively, the proton and neutron nuclear form factors for the nucleus A(Z, N ), while ε≡(1−4 sin 2 θ W )=0.045, indicating the dominant contributions are from the neutrons.
The merit of this description is to connect νA el to nuclear physics so that its studies may benefit from or contribute to the wealth of information and data. Electronnucleus scattering experiments provide important data to the nuclear proton form factor F Z (q 2 ) [17]. The neutron counterpart F N (q 2 ), however, would require weak processes to probe. Studies of νA el have therefore triggered intense activities towards their measurements [7], complementing experiments with parity-violation scattering using polarized electrons [18].
In the kinematics regime relevant to this work − q 2 R 2 π 2 (natural units with =c=1 are used throughout), where R=1.2A 1/3 fm is the typical scale characterizing the radius of nuclei − nucleons can be taken as structureless point-like particles, such that their internal dynamics and QCD effects can be neglected. At q 2 →0, there is a perfect alignment of the scattering amplitude vectors of individual nucleons in the target nucleus [6]. The interactions are completely coherent. As q 2 increases, deviations from this complete coherency condition lead to suppression in the crosssection. The loss of coherency can be described by a parameter α(q 2 )≡cosφ∈[0, 1] where φ(q 2 )∈[0, π/2] is the misalignment phase angle [6]. This leads to a formulation in terms of quantum-mechanical coherency among the various scattering centers, in which: The Γ QM -formulation with α(q 2 ) provides an intuitive physics understanding and quantitative description on the suppression of νA el cross-sections in terms of phaseangle alignment and quantum-mechanical coherency. In particular, it naturally leads to the limiting behavior at the complete coherency (α=1 at q 2 ∼0) and decoherency (α=0 at q 2 [π/R] 2 ) states, corresponding to , respectively. The experimentally measured α(q 2 )-values from different isotope targets can be directly compared to reveal their varying degrees of coherency in the respective processes. An alternative measurement-driven description, denoted by ξ(q 2 ), is the cross-section reduction relative to that of complete coherency condition [6], where The functions Γ N P , Γ QM and Γ DAT A are complementary descriptions of the νA el interactions. The experimentally measurable cross-section suppression (ξ in Γ DAT A ) is related to nuclear form factors and quantum-mechanical I: Summary of the three formulations which characterize the many-body physics in νA el , and the values of the key parameters at the limiting domains where the scattering amplitudes are either completely in phase ("Coherency") or decoupled ("Decoherency").

Conditions
Complete Complete Coherency Decoherency and while the two physics descriptions are connected by: The relations between ξ and Γ N P with α for three representative nuclei are shown in Figures 1a&b, respectively. Contours of maximum-q 2 for different neutrino sources are marked in Figure 1b. The behavior of Γ N P , α and ξ at the limiting domains corresponding to the complete coherency and decoherency conditions are summarized in Table I. In particular, the relation Γ N P =(ε 2 Z + N ) for completely decoherent νA el interactions is a result that emerges by relating Γ N P and Γ QM in Eq. 8, and could not be derived by considerations of nuclear form factor of Eq. 3 alone.

III. PROJECTED EXPERIMENTAL RANGES
The functions Γ N P , Γ QM and Γ DAT A can be directly measured from νA el data without input from the underlying The variation of α and ξ(= F 2 A ) as a function of q 2 of νA el on the three selected nuclei. Different neutrino sources share the same contour for the same target in q 2 -space, but with different ranges. The end-points for reactor, solar and DAR-π neutrinos are marked.
physics. Prior to actual measurements, specific formulations of the nuclear form factors have to be adopted for phenomenological studies and to establish the typical ranges to guide the choices of experimental parameters. To serve these purposes, the frequently adopted approach is to take the nuclear form factors for protons and neutrons are identical: F N (q 2 )=F Z (q 2 )≡F A (q 2 ), and to use the effective "Helm Form Factor" description of Ref. [20]: where j  thickness of the nuclei. In this formulation, the squaredform factor is equivalent to the cross-section suppression fraction: [F A (q 2 )] 2 =ξ(q 2 ). Typical spectra [21] of reactor, solar and atmospheric neutrinos, as well as those due to decay-at-rest π (DAR-π), are used in this study. These are depicted in Figure 2.
The measurable total cross-section is given by convoluting Eq. 2 with the neutrino spectrum Φ ν (E ν ), and integrating over E ν and q 2 ∈[q 2 min , q 2 max ], from which the mean suppression fraction ξ and the mean coherency factor α can be derived [19].
The νA el processes on several nuclei of experimental interest and at different mass ranges are studied − (Ar;Ge;Xe) with Z= (18;32;54). The target that provides the first νA el measurements [15] − CsI, having Z=55 and 53, respectively, can be approximated as Xe in this discussion.
The variations of α and ξ(=F 2 A ) with q 2 of the four neutrino sources, with three selected nuclei (Ar;Ge;Xe) are depicted in Figure 3. The q 2 -dependence is universal for the different neutrino sources, though their q 2 max -values are distinct due to their varying maximum E ν . These spectra end-points for reactor, solar and DAR-π neutrinos are well-defined, and their corresponding ranges in α and q 2 are depicted in Figures 1b&3, respectively. A summary plot on the variations of α with the neutrino sources and target nuclei is illustrated in Figure 4, in which the ranges in E ν are defined by the Full-Width-Half-Maximum (FWHM) of [Φ ν ·σ νA el ]. For completeness, the differential and integral event rates due to the four neutrino sources in measurable nuclear recoil energy T , together with their corresponding α and α values, are discussed and presented in Appendix A.
It can be seen that coherency is mostly complete  [15] and (b) Ar [16] data with DAR-π-ν. The stripe-shaded areas are the 1-σ allowed regions derived from the reduction in cross-section relative to the complete coherency conditions independent of nuclear physics input. The dark-shaded regions are the theoretical expectations adopting the nuclear form factor formulation of Eq. 9 with a ±1σ uncertainty of 10%. The same projection applies to reactor-ν on Ge in (b) with q 2 -range specified by FWHM in [Φν ·σνA el ]. The α(q 2 )-values for different nuclei can be consistently compared.
(α>95%) for νA el with reactor and solar neutrinos, whereas coherency is only partial for DAR-π and weak for atmospheric neutrinos. Accordingly, studies of νA el with different neutrino sources provide complementary information and cover the transitions from completely coherent to decoherent states.

IV. MEASUREMENTS FROM CURRENT DATA
The COHERENT-CsI(Na) [15] and -Ar [16] experiments at the DAR-π beam with the Spallation Neutron Source facility at the Oak Ridge National Laboratory provide positive measurements on νA el . ξ ] and total event rates in kg -1 day -1 for the target nuclei at a threshold of 1 and 10 keVnr and for different ν-sources. Reactor and DAR-π neutrino fluxes are taken to be 10 13 cm -2 s -1 , while DAR-π neutrino flux is 3.4×10 14 cm -2 yr -1 /flavor at 19.3 m from target at beam intensity 2×10 23 POT yr -1 . Rates due to atmospheric neutrinos are from the integration of q 2 -ranges corresponding to α∈[0.01, 1.0].  The p-value significance to probe the specific cases corresponding to the complete coherency (α=1, in red) and decoherency(α=0, in blue) conditions from the COHERENT-CsI data [15].
While the first-generation "discovery" measurements cannot be expected to provide severe constraints on α(q 2 ), it is instructive to go through the data analysis to establish the ranges of the effects and to check consistency. Values and uncertainties of α(q 2 ) are derived from the measured spectra and displayed in Figures 5a&b for CsI and Ar, respectively.
The bin-wise ξ(q 2 ) cross-section suppression relative to the complete coherency condition was provided by measurements. The allowed 1-σ ranges in α(q 2 ) are evaluated according to Eq. 6 and depicted as stripe-shaded regions in Figures 5a&b. The results are data-driven without invoking nuclear physics input. The theoretical expectations adopting the nuclear form factor formulation of Eq. 9 with a ±1σ uncertainty of 10% are superimposed as dark-shaded bands, showing the cases with CsI (equivalently, Xe) and Ar at DAR-π, and with Ge at reactors.
These diverse ranges of α-sensitivity indicate the complementarity of νA el measurements among reactor and DAR-π-neutrinos. Future measurements of solar νA el [14] with multi-ton detectors would probe a similar range of α as reactor neutrinos. Xenon detectors with scale O(100)ton would be required to probe the weaklycoherent region at α<0.2 with atmospheric neutrinos.

V. SUMMARY AND PROSPECTS
Neutrino-nucleus elastic scattering provides a laboratory to study quantum-mechanical coherency effects in electroweak interactions. This interpretation of the process is complementary to the language of nuclear form factors describing the nucleon-nucleus interplay.
We relate the two approaches in this work. Current positive measurements on νA el provide weak constraints to the coherency-parameter α(q 2 ). Data with O(10%) accuracy would allow the studies of coherency transitions over a large range of α.
We note that the interaction νA el of Eq. 1 involves two distinct concepts: elastic kinematics and quantummechanical coherency. The coherency aspect should be characterized by distributions with dependence on A(Z, N ) and q 2 . Descriptions of coherency as a binary state or having both concepts bundled together may have the unintended consequences of missing the complexities of the process and suppressing the potential richness of its physics content. The differential cross-section of Eq. 2 on q 2 can be translated to one on measurable nuclear recoil-energy T by

VI. ACKNOWLEDGEMENT
The measurable differential spectra (dR/dT ) convoluted with the neutrino spectrum Φ ν (E ν ) is given by: (A2) Integration over T ∈[T min , T max ] gives the total event rates.
The universality of Figure 3 no longer applies when q 2 is replaced by T . The variations of α, F A and ξ with T depend on E ν -distributions and therefore neutrino sources. The variations are depicted in Figure 7.
The differential rates derived from the four sources and three targets are displayed in Figure 8. The corresponding total rates are shown in Figure 9, showing their variations with T min and α . The values of α and ξ as well as the total event rates at T min =1(10) keV nr for the various neutrino sources and target nuclei are summarized in Table II. Evaluation of these rates are based on standard solar and atmospheric spectra [21]. Reactor ν e flux is taken to be 10 13 cm -2 s -1 , while DAR-π perflavor neutrino flux is 3.4×10 14 cm -2 yr -1 corresponding to 2×10 23 proton-on-target(POT)/year at 19.3 m from target [15]. There is no high-energy cut-off in E ν for the atmospheric neutrino spectra. The differential and integral spectra of At the detection threshold of 1 keV nr , 90% of the elastic scattering events between Weakly Interacting Massive Particles (WIMPs)-dark matter of mass 1 TeV with (Ar;Ge;Xe)-target have recoil energy up to (99;74;35) keV nr . These kinematics ranges correspond to α as low as (0.49;0.22;0.14) for νA el scattering with atmospheric neutrinos, as indicated in Figure 7d − far from the complete coherency regime. Accordingly, the description of "the neutrino floor originates from coherent neutrino-nucleus scattering" is not applicable for WIMPs at TeV or higher mass scales.