Nuclear Structure Physics in Coherent Elastic Neutrino-Nucleus Scattering
NNuclear Structure Physics in Coherent Elastic Neutrino-Nucleus Scattering
N. Van Dessel, V. Pandey, ∗ H. Ray, and N. Jachowicz † Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium Department of Physics, University of Florida, Gainesville, FL 32611, USA
The prospects of extracting new physics signals in a coherent elastic neutrino-nucleus scattering(CE ν NS) process are limited by the precision with which the underlying nuclear structure physics,embedded in the weak nuclear form factor, is known. We present microscopic nuclear structurephysics calculations of charge and weak nuclear form factors and CE ν NS cross sections on C, O, Ar, Fe and
Pb nuclei. We obtain the proton and neutron densities, and charge and weak formfactors by solving Hartree-Fock equations with a Skyrme (SkE2) nuclear potential. We validate ourapproach by comparing
Pb and Ar charge form factor predictions with elastic electron scatteringdata. In view of the worldwide interest in liquid-argon based neutrino and dark matter experiments,we pay special attention to the Ar nucleus and make predictions for the Ar weak form factor andthe CE ν NS cross sections. Furthermore, we attempt to gauge the level of theoretical uncertaintypertaining to the description of the Ar form factor and CE ν NS cross sections by comparing relativedifferences between recent microscopic nuclear theory and widely-used phenomenological form factorpredictions. Future precision measurements of CE ν NS will potentially help in constraining thesenuclear structure details that will in turn improve prospects of extracting new physics.
I. INTRODUCTION
Coherent elastic neutrino-nucleus scattering (CE ν NS),where the only detectable reaction product is a low mo-mentum recoiling nucleus, was suggested soon after theexperimental discovery of a weak neutral current in neu-trino interactions [1]. Even though for neutrino energiesof some tens of MeV the CE ν NS cross section is a feworders of magnitude larger than competing inelastic pro-cesses, the difficulty in detecting the ∼ keV scale recoil ofa nucleus has hindered experimental detection of this pro-cess for decades. In 2017, the COHERENT collaborationdetected the first CE ν NS signal using a stopped–pionbeam in the Spallation Neutron Source (SNS) at OakRidge National Laboratory with a CsI detectors [2, 3],followed up by another recent measurement in a liquidargon (LAr) detector [4, 5].The detection of CE ν NS has opened up a slew ofopportunities in high-energy physics, astrophysics andin nuclear physics, inspiring new probes into beyond-Standard-Model (BSM) physics and new experimentalmethods. Several extensions of the SM that can be ex-plored at low energy such as non–standard interactions(NSI) [6–9], sterile neutrinos [10, 11], CP–violation [12],as well as exploration of nuclear effects, are being stud-ied [13–15, 66]. Several experimental programs havebeen or are being set up to detect CE ν NS and BSMsignals in the near future using stopped–pion neutrinosources in COHERENT at the SNS [2] and CoherentCAPTAIN-Mills (CCM) at Los Alamos National Lab-oratory (LANL) [17], as well as reactor–produced neu-trinos in CONNIE [18], MINER [19], ν GEN [20], NU-CLEUS [21], RICOCHET [22], and TEXONO [23]. ∗ [email protected]; Present Address: Fermi National AcceleratorLaboratory, Batavia, Illinois 60510, USA † [email protected] The main source of uncertainty in the evaluation ofthe CE ν NS cross section is the accuracy with which theunderlying nuclear structure and nucleon dynamics thatdetermine the distributions of the nucleon density in thenuclear ground state, embedded in the form factor, areknown in the target nucleus. The ground state pro-ton (charge) density distributions are relatively well con-strained through elastic electron scattering experimentspioneered by Hofstadter and collaborators at the Stan-ford Linear Accelerator [24], followed by other measure-ments in the following decades [25–27]. CE ν NS is how-ever primarily sensitive to the neutron density distribu-tions of the nucleus, which are only poorly constrained.Hadronic probes have been used to extract neutron dis-tributions, these measurements are however plagued byill-controlled model–dependent uncertainties associatedwith the strong interaction [28]. More (experimentally)challenging electroweak probes such as parity–violatingelectron scattering (PVES) [28, 29] and CE ν NS providerelatively model-independent ways of determining neu-tron distributions. In recent years, one such PVES ex-periment, PREX at Jefferson lab, has measured the weakcharge of
Pb at a single value of momentum trans-fer [30, 31], while a follow up PREX–II experiment isongoing to improve the precision of that measurement.Another PVES experiment, CREX at Jefferson lab, isunderway to measure the weak form factor of Ca [32].Future ton and multi-ton CE ν NS detectors will enablemore precise measurements and will potentially offer apowerful avenue to constrain neutron density distribu-tions and weak form factors of nuclei at low momentumtransfers where the process remains coherent [13, 14, 33].As long as no precision measurements of neutron den-sity distributions of nuclei are available, the weak nu-clear form factor has to be modeled in order to evaluatethe CE ν NS cross section and event rates. The accuracyof such an assumption is vital to the CE ν NS programsince any experimentally measured deviation from the a r X i v : . [ nu c l - t h ] J u l expected CE ν NS event rate can point to new physics orto unconstrained nuclear physics. It is therefore crucialto treat the underlying nuclear structure physics that isembedded in nuclear form factors with utmost care. Phe-nomenological approaches, such as the Klein–Nystrandform factor [34] adapted by the COHERENT collabo-ration, or the Helm form factor [35] where density dis-tributions are represented by analytical expressions, arewidely used in the CE ν NS community. Empirical val-ues of the proton rms radius, measured in elastic elec-tron scattering, are often used to evaluate the protonform factor and often similar parameterizations are as-sumed for the neutron form factor. Microscopic nuclearphysics approaches which describe a more accurate pic-ture of the nuclear ground state and density distributionssuch as density functional theory [33], coupled–clustertheory from first principles [36], relativistic mean–fieldmodel [37], and Hartree–Fock plus Bardeen–Cooper–Schrieffer model [38] have also been reported in recentyears.In this work we will present a microscopic many–bodynuclear theory model where the nuclear ground stateis described in a Hartree–Fock (HF) approach with aSkyrme (SkE2) nuclear potential. We calculate protonand neutron density distributions, charge and weak formfactors, and CE ν NS cross sections on C, O, Ar, Fe and
Pb, and confront our predictions with theavailable experimental data. In view of the worldwideinterest in liquid–argon–based neutrino and dark mat-ter experiments, we pay special attention to the Arnucleus. We attempt to gauge the level of theoretical un-certainty pertaining to the description of the Ar formfactor and CE ν NS cross section by comparing relativedifferences between recent nuclear theory and widely–used phenomenological form factor predictions.The remainder of this manuscript is organized as fol-lows. In Sec. II, we lay out the general formalism ofcalculating the CE ν NS cross section and form factor. InSec. III, we present results of proton and neutron densi-ties, charge and weak form factors, and CE ν NS cross sec-tions on C, O, Ar, Fe and
Pb obtained withinour HF–SkE2 approach. We focus on Ar in subsec-tion III A, and compare our predictions with experimen-tal data and other theoretical calculations. We presentconclusions of this study in Sec. IV.
II. FORMALISM
In this section, we lay out the general formalism ofcalculating the (differential) cross section and form fac-tor for coherent elastic neutrino scattering off a nucleusgoverned by the weak neutral current where a single Z boson is exchanged between the neutrino and the targetnucleus. FIG. 1. Diagrammatic representation of the CE ν NS processwhere a single Z boson is exchanged between neutrino andtarget nucleus. A. CE ν NS Cross Section
A neutrino with four momentum k i = ( E i , (cid:126)k i ) scattersoff the nucleus, which is initially at rest in the lab framewith p A = ( M A ,(cid:126) Z boson. The neutrinoscatters off carrying away four momentum k f = ( E f , (cid:126)k f )while the nucleus remains in its ground state and receivesa small recoil energy T , so that p (cid:48) A = ( M A + T, (cid:126)p (cid:48) A ) with | (cid:126)p (cid:48) A | = (cid:112) ( M A + T ) − M A and T = q / M A . Here, M A is the rest mass of the nucleus, q = | (cid:126)q | is the absolutevalue of the three–momentum transfer which is of theorder of keV for neutrino energies of tens of MeV, and Q ≈ q = | (cid:126)k f − (cid:126)k i | . The process is schematically shownin Fig. 1.The initial elementary expression for the cross sectionreadsd σ = 1 | (cid:126)v i − (cid:126)v A | m i E i m f E f d (cid:126)k f (2 π ) M A M A + T d (cid:126)p (cid:48) A (2 π ) × (2 π ) (cid:88) fi |M| δ (4) ( k i + p A − k f − p (cid:48) A ) . (1)This expression can be integrated to yield the expressionfor the cross section differential in neutrino scatteringangle θ f :d σ d cos θ f = m i E i m f E f M A M A + T E f π f − rec (cid:88) fi |M| . (2)The recoil factor reads f rec = E i E f M A M A + T . (3)Working out the Feynman amplitude one gets (cid:88) fi |M| = G F L µν W µν , (4)with the nuclear tensor W µν reading W µν = (cid:88) fi ( J µnuc ) † J νnuc . (5)The summation symbols in these expressions denote sum-ming and averaging over initial and final polarizationsrespectively. The nuclear tensor depends on the nuclearcurrent transition amplitudes: J µnuc = (cid:104) Φ | (cid:98) J µ ( (cid:126)q ) | Φ (cid:105) . (6)Under the assumption that the nuclei of interest arespherically symmetric with J π = 0 + and taking the z–axis to be along the direction of (cid:126)q , one only needs to takeinto account the zero and third component of the nuclearcurrent’s vector part, which are furthermore connectedthrough conserved vector current (CVC): q µ (cid:98) J µ ( (cid:126)q ) = 0 . (7)Through performing the necessary algebra, one arrivesat the final expressiond σ d cos θ f = G F π E f E i (cid:20) Q q (1 + cos θ f ) |J V | (cid:21) , (8)where one can safely approximate Q q ≈ θ f as:d σ d cos θ f = G F π E f E i (1 + cos θ f ) Q W F W ( Q ) . (9)Here we have introduced the elastic form factor, F W ( Q ),which we will discuss in the next subsection. In elasticscattering the entire nuclear dynamics is encoded in theform factor. Equivalently one can express the differentialcross section as a function of the nuclear recoil T , whichreads:d σ d T = G F π M A (cid:18) − TE i − M A T E i (cid:19) Q W F W ( Q ) , (10)where G F is the Fermi coupling constant, and Q W is theweak nuclear charge : Q W = [ g Vn N + g Vp Z ] = [ N − (1 − θ W ) Z ] (11)with coupling constants g Vn = − g Vp = (1 − θ W ). N and Z are neutron and proton numbers, re-spectively, and θ W is the weak mixing angle. The valueis such that sin θ W = 0 . ν NSkinematic distribution both in neutrino scattering angle, θ f , and in nuclear recoil energy, T. In most experimentsthe only signal of a CE ν NS event is a nuclear recoil en-ergy deposition. In principle, future experiments withmore advanced detector technologies may be able to de-tect both nuclear recoil and angular distribution simul-taneously. Such capabilities are already being exploredin some dark-matter experiments and will greatly en-hance the physics capabilities of future CE ν NS experi-ments [40].
B. Form Factor
The scattering process’ cross section is proportional tothe squared magnitude of the transition amplitude in-duced by the nuclear current. Since the relevant groundstate to ground state transition for spherically symmetri-cal nuclei is 0 + → + , only the vector part of the currentwill contribute. The amplitude can be expressed as J V = (cid:104) Φ | (cid:98) J V ( (cid:126)q ) | Φ (cid:105) = (cid:90) e i(cid:126)q · (cid:126)r (cid:104) Φ | (cid:98) J V ( (cid:126)r ) | Φ (cid:105) = 12 (cid:2)(cid:0) − θ W (cid:1) f p ( (cid:126)q ) F p ( Q ) − f n ( (cid:126)q ) F n ( Q ) (cid:3) , (12)where we have inserted the impulse approximation (IA)expression for the nuclear current, as a sum of single–body operators: (cid:98) J V ( (cid:126)r ) = (cid:88) i F Z ( Q , i ) δ (3) ( (cid:126)r − (cid:126)r i ) , (13)with F Z ( Q , i ) = (cid:18) − sin θ W (cid:19) ( F p − F n ) τ ( i ) − sin θ W ( F p + F n ) . (14)Furthermore, f p ( (cid:126)q ) and f n ( (cid:126)q ) are the Fourier trans-forms of the proton and neutron densities respectively. F p and F n are proton and neutron form factors. Theoverall structure of the transition amplitude consists ofproducts of the weak charge with two factors: the nu-clear form factor, determined by the spatial distributionof the nucleons in the nucleus, as well as the nucleon formfactor. We arrive at the expression: F W ( Q ) = 1 Q W (cid:2)(cid:0) − θ W (cid:1) f p ( (cid:126)q ) F p ( Q ) − f n ( (cid:126)q ) F n ( Q ) (cid:3) = 2 Q W J V , (15)such that the form factor becomes 1 in the static limit.Note that in writing down the functional dependence wecan make use of the non–relativistic approximation Q ≈| (cid:126)q | , valid in the energy regime considered.We employ a microscopic many–body nuclear theorymodel where the nuclear ground state is described in aHartree–Fock (HF) approach with a Skyrme (SkE2) nu-clear potential, which we will refer to as HF–SkE2. Wesolve the HF equations to obtain single–nucleon wavefunctions for the bound nucleons in the nuclear groundstate. We evaluate proton ( ρ p ( r )) and neutron ( ρ n ( r ))density distributions from those wave functions. The pro-ton and neutron densities are utilized to calculate charge( F ch ( Q )) and weak ( F W ( Q )) nuclear form factors. Thisapproach involves realistic nuclear structure calculations . . . . .
10 2 4 6 8 10 00 . . . . . .
120 2 4 6 8 1010 − − . . − − . . ρ p ( r )( f m − ) r (fm)(a) C O Ar Fe Pb ρ n ( r )( f m − ) r (fm)(b) | F c h ( q ) | q (fm − ) (c) | F W ( q ) | q (fm − ) (d) FIG. 2. (color online) Panels (a) and (b) represent proton and neutron densities of different nuclei obtained using the HF–SkE2approach. Panels (c) and (d) represent charge and weak form factors. of proton and neutron density distributions making itmore reliable compared to the phenomenological ap-proaches that rely on the approximation ρ n ( r ) ≈ ρ p ( r ),utilizing empirical values of ρ p ( r ) extracted from electronscattering experiments. Our approach has been devel-oped and tested against several electron- and neutrino-nucleus scattering scenarios [41–54]. The effects of long-range correlations included through a continuum Ran-dom Phase Approximation (CRPA) approach, describedin aforementioned references, are found to be very smallin evaluating ground state densities of nuclei [55] and aretherefore not included in this work. III. RESULTS AND DISCUSSION
Since the weak charge of the proton is strongly sup-pressed by the weak mixing angle, Eq. (15), the nu- clear weak charge is predominately carried by the neu-trons. The weak form factor F W ( Q ), and hence theCE ν NS cross section, are dominated by the distributionof neutrons within the nucleus. As proton densities arewell constrained by experimental elastic electron scatter-ing data [27] while little reliable neutron density datais available, phenomenological approaches approximate ρ n ( r ) ≈ ρ p ( r ) and thus assume F n ( Q ) ≈ F p ( Q ), mak-ing the nuclear form factor more of a global factor [56].Within the HF–SkE2 approach we treat proton and neu-tron densities and their corresponding form factors sepa-rately and do not have to rely on such assumptions. Thedensities are defined in terms of the radial single particlewave functions as ρ q ( r ) = 14 πr (cid:88) a v a,q (2 j a + 1) | φ a,q ( r ) | , (16) − − − . . | F c h ( q ) | q ( fm − ) PbHF - SkE2Yang et al. - RMFExp. − − − . . | F W ( q ) | q ( fm − ) PbHF - SkE2Yang et al. - RMFPREX
FIG. 3. (color online) Left: the charge form factor of
Pb compared with elastic electron scattering data of Ref. [25]. Right:the weak form factor of
Pb along with the single point measured by the PREX collaboration at the momentum transfer of q = 0.475 fm − [30, 31]. Both form factors are compared with relativistic mean-field predictions of J. Yang et al. [37]. − . − . . . . . . .
060 0 . . F W , s k i n ( q ) = F c h ( q ) − F W ( q ) q ( fm − ) PbHF - SkE2
FIG. 4. (color online) The “weak-skin” form factor of
Pbdepicts the difference between the charge and weak form fac-tors. with v a,q being the occupation probability of orbital a ofnature q (i.e. proton p or neutron n .).In Fig. 2, we present proton (panel (a)) and neutron(panel (b)) density distributions of C, O, Ar, Feand
Pb obtained using our HF–SkE2 approach. Nat-urally the heavier the nucleus, the more broadly the den-sities are distributed. Panel (c) and (d) show the chargeand weak form factors for all the nuclei, respectively. Inboth the charge and weak form factor cases, the heavierthe nuclei the faster the form factor encounters its firstminimum at rising q values. Lighter nuclei have theirminima spread over a larger q range. C has its first − − − − − − − − σ ( c m ) E (MeV) 12C16O40Ar56Fe208Pb FIG. 5. (color online) Total CE ν NS cross sections for a set ofnuclear targets obtained within the HF–SkE2 approach. minimum at q ∼ − while Pb has its first mini-mum around q ∼ − .In the left panel of Fig. 3, we show our predictions forthe charge form factor of Pb. The predictions are com-pared with the experimental charge form factor obtainedfrom a Fourier–Bessel fit to the elastic electron scatteringdata of Ref. [25]. Our predictions describe the experimen-tal data remarkably well. Our predictions almost overlapwith data for q (cid:46) − . We also performed a compar-ison with the relativistic mean-field (RMF) predictions ofYang et al. [37]. There are no visible differences betweenboth models up to q (cid:46) − . The right panel showsour predictions for the weak form factor, again compared − − − − − − − − − − − − − − − − − − − − − − − − σ ( c m ) CCEvNSCCQENCQE O σ ( c m ) E (MeV) Ar E (MeV) Fe FIG. 6. (color online) CE ν NS cross section strength compared to CCQE and NCQE scattering cross sections for several nuclei,above particle emission threshold. with the RMF predictions of [37]. We also show thesingle data point measured at a momentum transfer of q = 0 .
475 fm − by the PREX collaboration [30, 31]. Thisremains the only measurement of the weak form factorobtained with an electroweak probe. The error bars onthe data point are too large to discriminate between the-oretical predictions. The follow–up PREX–II measure-ment at Jefferson lab aims to reduce the error bars by atleast a factor of three.Comparing the left and the right panels of Fig. 3 onecan see that although the charge and weak form factorshave a similar overall structure, the minima and max-ima of both occur at different values of the momentumtransfer. To further illustrate this, in Fig. 4, we showthe “weak-skin” form factor [28] of Pb, defined as thedifference between the charge and weak form factors: F W,skin ( q ) = F ch ( q ) − F w ( q ) , (17)which, near the origin, is proportional to the experimen-tally observable weak skin [28]. The figure shows that the difference between the charge and weak form factorsrises rapidly for q (cid:46) . − and falls off after the firstminimum.The total CE ν NS cross section as a function of neu-trino energy for C, O, Ar, Fe and
Pb is shownin Fig. 5. All nuclei show similar a behavior: there is arapid rise of the cross section up to about ∼
30 MeV,then the steep increase slows down and flattens out onthe log scale thereafter. The cross section increases withthe atomic number, with nearly two to three orders ofmagnitude difference between C and
Pb, reflectingthe ≈ N scaling behavior shown in Eq. 11.To demonstrate the dominance of the CE ν NS strengthover the quasi-elastic one for a neutrino energy of a fewtens of MeV, in Fig. 6 we compare CE ν NS cross sectionsto ν e -nucleus charged-current quasielastic (CCQE) andneutral-current quasielastic (NCQE) cross sections. Inthe quasielastic cross section calculations, the influenceof long-range correlations between the nucleons is intro-duced through the continuum Random Phase Approxi- − − − − . . . | F c h ( q ) | q ( fm − ) Ar HF - SkE2Payne et al. - NNLOsatExp
FIG. 7. (color online) The Ar charge form factor predic-tions compared to elastic electron scattering data taken fromRef. [63]. A comparison is also performed with the coupled–cluster theory predictions of Payne et al. [36]. mation (CRPA) on top of the HF-SkE2 approach. CRPAeffects are vital to describe the quasielastic scatteringprocess where the nucleus can be excited to low-lyingcollective nuclear states. For the energies relevant forpion decay-at-rest neutrinos, E (cid:46)
52 MeV, the CE ν NScross section is roughly two orders of magnitude largerthan inelastic cross sections.
A. Constraining Ar In view of the worldwide interest in liquid argon (LAr)–based detectors in neutrino and dark matter experiments,in this section we will focus on Ar. In the COHERENTcollaboration’s expanding series of detectors at SNS, thecollaboration has recently presented new measurementsfrom a 24-kg, single–phase, LAr CENNS-10 detector [5]while a ton-scale LAr experiment is underway. A 10ton LAr scintillation detector, Coherent CAPTAIN-Mills(CCM), was recently built at LANL to study CE ν NS on Ar and to search for low–mass dark matter that co-herently scatters off Ar nuclei [17]. Several other neu-trino [57, 58] and dark matter experiments [59–62] em-ploy LAr detectors, making it vital to study ground stateproperties of the Ar nucleus.In Fig. 7 we compare our argon charge form factor( F ch ( q )) predictions with the elastic electron scatteringdata of Ref. [63]. Our predictions describe experimentaldata remarkably well for q (cid:46) − , validating our ap-proach. We also compared with the predictions of Payne et al. [36] where form factors are calculated within acoupled–cluster theory from first principles, using a chi-ral NNLO sat interaction. At higher q , q (cid:38) − , bothpredictions diverge from experimental data. Note that − − − − . . . | F W ( q ) | q ( fm − ) Ar HF - SkE2Klein-Nystrand FFKlein-Nystrand FF (ad.)Helm FFYang et al. - RMFPayne et al. - NNLOsat
FIG. 8. (color online) The Ar weak form factor predic-tions compared with calculations of Payne et al. [36], Yang etal. [37], and with the predictions of Klein–Nystrand [34] andHelm [35] form factors. − . . . . .
020 0 . . F W , s k i n ( q ) = F c h ( q ) − F W ( q ) q ( fm − ) ArHF - SkE2
FIG. 9. (color online) The “weak-skin” form factor of Ardepicts the difference between charge and weak form factor. for neutrino energies relevant for pion decay–at–rest theregion above q (cid:38) . − does not contribute to CE ν NScross sections.After validating our approach, we make predictions forthe weak form factor of Ar in Fig. 8. There is no dataavailable for the weak form factor on argon yet. We com-pare our predictions with the nuclear theory predictionof Payne et al. [36] and Yang et al. [37]. We also comparewith two phenomenological form factors which are widelyused in the CE ν NS community: the Klein–Nystrand [34]form factor that is adapted by the COHERENT collab-oration and the Helm form factor [35]. Note that wealso show an adapted version of the Klein–Nystrand form .
05 0 . .
15 0 . .
25 0 .
30 MeV . . . . .
50 MeV . . . . . . .
70 MeV . . . .
100 MeV σ ( q c u t o ff )( − c m ) Ar σ ( q c u t o ff )( − c m ) σ ( q c u t o ff )( − c m ) σ ( q c u t o ff )( − c m ) q cutoff ( fm − ) SkE2Payne et al.Yang et al.Helm FFKlein-NystrandKlein-Nystrand FF (ad.)
FIG. 10. (color online) The Ar cumulative cross sectionas a function of q cutoff compared with calculations done us-ing Payne et al. [36], Yang et al. [37], as well the Klein–Nystrand [34] (standard and adapted) and Helm [35] formfactors. factor that we will describe shortly. Overall, the shapeand structure of the weak form factor is similar to thecharged one, but the position of minima and maxima aresomewhat different. In our HF–SKE2 approach the firstminimum of F c h ( q ) is at q ∼ .
23 fm − while for F W ( q ) itlies at q ∼ .
19 fm − , pointing to the fact that the neu-tron distribution extends further out compared to theproton one. To quantify differences between the chargeand weak form factor, in Fig. 9 we show the “weak-skin”form factor of Ar using Eq. (17).In order to appreciate which values of momentumtransfer q are involved at different neutrino energies, aswell as to see at which q values the differences in thenuclear modeling start causing discrepancies in reactionstrength predictions, we plot cumulative cross sections for Ar at several neutrino energies and for different mod-els in Fig. 10. This is defined as the total cross sectionstrength, integrated up to a cutoff value q cutoff in themomentum transfer: σ ( q cutoff ) = (cid:90) q cutoff d σ ( q )d q . d q (18)The model differences become stronger for increasinglyhigh energies with discrepancies originating from thehigher– q regions of the elastic form factor. The range ofcutoff values also coincides with all kinematically avail-able momentum transfers. At 100 MeV e.g., Ar is onlyprobed up to q ≈ − .In Fig. 11, we show differential cross sections on Aras a function of recoil energy T , and scattering angle θ f , for different incoming neutrino energies according toEq. (9) and (10). For comparison, we have also plottedthe case with no nuclear structure effects i.e. F ( Q ) = 1.The effects of nuclear structure physics are more promi-nent as the neutrino energy increases. Most of the crosssection strength lies in the lower-end of the recoil energyand in the forward scattering as the cross section fallsoff rapidly at higher T (top panels) and higher θ f values(bottom panels). Most CE ν NS detectors are sensitiveonly to the recoil energy deposited in the detector but,in principle, in the future more advanced detector tech-nologies might enable measurement of both nuclear recoiland angular distribution simultaneously. Utilizing suchadditional information can be valuable in disentanglingnew physics signals in CE ν NS experiments [40].In Fig. 8, we come back to the differences between var-ious predictions. Different form factor approaches arebased on different representations of the nuclear den-sities, with no experimental data to constrain neutrondistributions. Identifying the size of the differences be-tween various theoretical predictions is crucial as experi-ments have to assign any deviation from expected eventrates either to new physics or to unconstrained nuclearphysics. We compare five predictions. These includethree nuclear theory approaches: the HF–SkE2 calcu-lation of this work, the predictions of Payne et al. [36]where form factors are calculated within a coupledclustertheory from first principles using a chiral NNLO sat inter- . . . . . . . . − − . . − − . . − − . . d σ d T ( − c m k e V − ) T (keV) E = 10MeV HF - SkE2 F ( Q ) = 1 T (keV) E = 30MeV T (keV) E = 50MeV d σ d c o s θ f ( − c m ) cos θ f cos θ f cos θ f FIG. 11. (color online) Differential cross section on argon as a function of recoil energy and scattering angle. action, and the calculations of Yang et al. [37] whereform factors are predicted within a relativistic mean–field model informed by the properties of finite nucleiand neutron stars. They also contain two phenomenolog-ical approaches: the Helm [35] and Klein–Nystrand [34]form factors where density distributions are representedby analytical expression.In the Helm approach [35] the density distribution isdescribed as a convolution of a uniform nucleonic densitywith a given radius and a Gaussian profile characterizedby the folding width s , accounting for the nuclear skinthickness. The resulting form factor is expressed as: F Helm ( q ) = 3 j ( qR ) qR e − q s / , (19)where j ( x ) = sin( x ) /x − cos( x ) /x is a spherical Besselfunction of the first kind. R is an effective nuclear radiusgiven as: R = (1 . A / − . + π r − s with r = 0.52 fm and s = 0.9 fm, fitted [64, 65] to muon spec-troscopy and electron scattering data compiled in [26].The Klein–Nystrand (KN) form factor, adapted by theCOHERENT Collaboration, is obtained from the convo-lution of a short-range Yukawa potential with range a k = 0.7 fm over a Woods–Saxon distribution approximatedas a hard sphere with radius R A = 1 . A / fm [34]. Theresulting form factor is expressed as: F KN ( q ) = 3 j ( qR A ) qR A (cid:20)
11 + q a k (cid:21) . (20) An adapted version of the KN form factor is often used,where R A is defined as R A = (cid:113) r − a k utilizing mea-sured proton rms radii r of the nucleus [15, 66]. We showboth the standard and the adapted (ad.) KN form fac-tor. For the adapted we use r = 3 .
427 fm, the measuredproton rms radii of Ar [27].We attempt to quantify differences between differentform factors and the CE ν NS cross section due to dif-ferent underlying nuclear structure details. We considerquantities that emphasize the relative differences betweenthe results of different calculations, arbitrarily using HF–SkE2 as a reference calculation, as follows: | ∆ F i W ( q ) | = | F i W ( q ) − F HFW ( q ) || F HFW ( q ) | , (21)and ∆ σ i W ( E ) = | σ i W ( E ) − σ HFW ( E ) | σ HFW ( E ) , (22)where i refers to calculations from different approachesas discussed above.The relative differences are shown in Fig. 12. We showonly the low–momentum part of the weak form factor toa maximum value of q = 0.5 fm − ( ∼
100 MeV) thatcorresponds to a maximum incoming neutrino energy ofE ∼
50 MeV, as shown in Fig. 10. The relative differ-ences are shown on a linear scale. At smaller energies themomentum transfer is low and hence the differences be-tween form factors are also small. For higher energies the0 . . . . . . . .
080 0 . . . . . . . . . .
05 10 20 30 40 50 | ∆ F i W ( q ) | q ( fm − ) ∆ σ i ( E ) E (MeV) Payne et al.Yang et al.Helm FFKlein-Nystrand FFKlein-Nystrand FF (ad.)
FIG. 12. (color online) Relative differences in the weak form factor and CE ν NS cross section predictions of Payne et al. [36],Yang et al. [37], Helm [35], Klein–Nystrand [34] and the adapted Klein–Nystrand [15, 66], all with respect to HF-SkE2. σ W ( − c m ) E (MeV) ArHF - SkE2 F ( Q ) = 1 SkE2 - foldedCOHERENT - ACOHERENT - B
FIG. 13. (color online) The CE ν NS cross section on Ar as afunction of neutrino energy. Recent flux–folded measurementby the COHERENT collaboration [5] is shown along with theflux-folded HF–SkE2 prediction. available momentum transfers increase and therefore thedifferences between the form factors become more preva-lent. The differences in model predictions amount to <
8% over the entire momentum transfer range. The dif-ferences rise rapidly at the higher end of q . This translateinto relative differences in CE ν NS cross sections, ∆ σ ( E ),of <
5% over the whole energy range, where E (cid:46)
55 MeV,relevant for neutrinos from pion decay-at-rest. Note thatmost of the strength in the cross section lies at the lower h σ i ( − c m ) Neutron numberCOHERENT F ( Q ) = 1 HF - SkE2
FIG. 14. (color online) Flux–averaged CE ν NS cross sectionsas a function of neutron number for the C, O, Ar, Feand
Pb nuclei. We also show Ar data measured by CO-HERENT [5]. T end (and therefore at the lower q end), as we have seenin Fig. 11.The CE ν NS cross section on Ar as a function of theneutrino energy is shown in Fig. 13. We also show recentflux–averaged measurements performed by the COHER-ENT collaboration [5]. Measurements from two analy-ses are included, with the horizontal bars indicating theminimum value set by the nuclear recoil threshold en-ergy for each analysis. The flux–averaged measured crosssection is 2.2 ± × − cm (average of both anal-yses), while the HF-SkE2 predicted flux–averaged crosssection is 1.82 × − cm . The total experimental erroris dominated by statistics, amounting to ∼ Ar. In Fig. 14, we also show flux–folded cross sec-tions as a function of neutron number for all five nuclei– C, O, Ar, Fe and
Pb – considered in thispaper. As expected, the deviation of F ( Q ) = 1 fromthe full HF-SkE2 calculation becomes more prominentas the number of neutrons, and hence the influence ofnuclear structure effects, increases. Also included is the Ar data measured by COHERENT [5].
IV. CONCLUSIONS
The experimental observation of coherent elasticneutrino–nucleus scattering processes by the COHER-ENT collaboration has inspired physicists across manyfields. The power of CE ν NS as a probe of BSM physicsand its potential for determining neutron density distri-butions is being realized. The main uncertainty in theevaluation of the CE ν NS cross sections is driven by theweak form factor that encodes the entire nuclear struc-ture contribution to the CE ν NS cross section.We presented microscopic nuclear structure physicscalculations of charge and weak nuclear form factors andthe CE ν NS cross section on C, O, Ar, Fe and
Pb nuclei. We obtain neutron (proton) densities andweak (charge) form factors by solving the Hartree–Fock equations with a Skyrme (SkE2) nuclear potential. Ourpredictions for
Pb and Ar charge form factors de-scribe elastic electron scattering data remarkably well.After validating Ar charge form factor calculations,we make predictions for the Ar weak form factor.Thereby, we calculate differential cross section as a func-tion of recoil energy and neutrino scattering angle. Weattempt to gauge the level of theoretical uncertainty per-taining to the description of Ar form factor and CE ν NScross section by comparing relative differences betweenrecent nuclear theory and widely–used phenomenologicalform factor predictions. We compare our Ar predic-tion with recent measurements of the COHERENT col-laboration. Future precise measurements of CE ν NS withton and multi-ton detectors will aid in constraining neu-tron densities and weak nuclear form factor that will inturn improve prospects of extracting new physics throughCE ν NS.
ACKNOWLEDGMENTS
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