Boosting background suppression in the NEXT experiment through Richardson-Lucy deconvolution
A. Simón, Y. Ifergan, A.B. Redwine, R. Weiss-Babai, L. Arazi, C. Adams, H. Almazán, V. ?lvarez, B. Aparicio, A.I. Aranburu, I.J. Arnquist, C.D.R Azevedo, K. Bailey, F. Ballester, J.M. Benlloch-Rodríguez, F.I.G.M. Borges, N. Byrnes, S. Cárcel, J.V. Carrión, S. Cebrián, E. Church, C.A.N. Conde, T. Contreras, F.P. Cossío, A.A. Denisenko, G. Díaz, J. Díaz, J. Escada, R. Esteve, R. Felkai, L.M.P. Fernandes, P. Ferrario, A.L. Ferreira, F. Foss, E.D.C. Freitas, Z. Freixa, J. Generowicz, A. Goldschmidt, J.J. Gómez-Cadenas, R. González, D. González-Díaz, S. Gosh, R. Guenette, R.M. Gutiérrez, J. Haefner, K. Hafidi, J. Hauptman, C.A.O. Henriques, J.A. Hernando Morata, P. Herrero, V. Herrero, J. Ho, B.J.P. Jones, M. Kekic, L. Labarga, A. Laing, P. Lebrun, N. López-March, M. Losada, R.D.P. Mano, J. Martín-Albo, A. Martínez, M. Martínez-Vara, G. Martínez-Lema, A.D. McDonald, Z.-E. Meziani, F. Monrabal, C.M.B. Monteiro, F.J. Mora, J. Muñoz Vidal, C. Newhouse, P. Novella, D.R. Nygren, E. Oblak, M. Odriozola-Gimeno, B. Palmeiro, A. Para, J. Pérez, M. Querol, J. Renner, L. Ripoll, I. Rivilla, Y. Rodríguez García, J. Rodríguez, C. Rogero, L. Rogers, B. Romeo, C. Romo-Luque, F.P. Santos, J.M.F. dos Santos, M. Sorel, C. Stanford, J.M.R. Teixeira, P. Thapa, J.F. Toledo, J. Torrent, A. Usón, J.F.C.A. Veloso, T.T. Vuong, R. Webb, et al. (3 additional authors not shown)
PPrepared for submission to JHEP
Boosting background suppression in the NEXTexperiment through Richardson-Lucy deconvolution
The NEXT Collaboration
A. Sim´on, ,a Y. Ifergan, , A.B. Redwine, R. Weiss-Babai, ,b L. Arazi, ,c C. Adams, H. Almaz´an, V. ´Alvarez, B. Aparicio, A.I. Aranburu, I.J. Arnquist, C.D.R Azevedo, K. Bailey, F. Ballester, J.M. Benlloch-Rodr´ıguez, , F.I.G.M. Borges, N. Byrnes, S. C´arcel, J.V. Carri´on, S. Cebri´an, E. Church, C.A.N. Conde, T. Contreras, F.P. Coss´ıo, , A.A. Denisenko, G. D´ıaz, J. D´ıaz, M. Diesburg, J. Escada, R. Esteve, R. Felkai, , , L.M.P. Fernandes, P. Ferrario, , A.L. Ferreira, E.D.C. Freitas, Z. Freixa, , J. Generowicz, S. Ghosh, A. Goldschmidt, J.J. G´omez-Cadenas, , ,d R. Gonz´alez, D. Gonz´alez-D´ıaz, R. Guenette, R.M. Guti´errez, J. Haefner, K. Hafidi, J. Hauptman, C.A.O. Henriques, J.A. Hernando Morata, P. Herrero, , V. Herrero, J. Ho, B.J.P. Jones, M. Kekic, , L. Labarga, A. Laing, P. Lebrun, N. L´opez-March, , M. Losada, R.D.P. Mano, J. Mart´ın-Albo, A. Mart´ınez, , M. Mart´ınez-Vara, G. Mart´ınez-Lema, , ,e A.D. McDonald, Z.-E. Meziani, F. Monrabal, , C.M.B. Monteiro, F.J. Mora, J. Mu˜nozVidal, , K. Navarro, P. Novella, D.R. Nygren, ,f E. Oblak, M. Odriozola-Gimeno, B. Palmeiro, , A. Para, J. P´erez, M. Querol, J. Renner, , L. Ripoll, I. Rivilla, , Y. Rodr´ıguez Garc´ıa, J. Rodr´ıguez, C. Rogero, L. Rogers, B. Romeo, , C. Romo-Luque, F.P. Santos, J.M.F.dos Santos, M. Sorel, C. Sofka, ,g C. Stanford, T. Stiegler, J.M.R. Teixeira, P. Thapa, J.F. Toledo, J. Torrent, A. Us´on, J.F.C.A. Veloso, R. Webb, J.T. White, ,h K. Woodruff, N. Yahlali Department of Physics and Astronomy, Iowa State University, 12 Physics Hall, Ames, IA 50011-3160, USA Argonne National Laboratory, Argonne, IL 60439, USA Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA a Corresponding author. b Now in Soreq Nuclear Research Center, Yavneh, Israel. c Corresponding author. d NEXT Co-spokesperson. e Now at the Unit of Nuclear Engineering, Ben-Gurion University, Beer-Sheva, Israel. f NEXT Co-spokesperson. g Now at University of Texas at Austin, USA. h Deceased. a r X i v : . [ phy s i c s . i n s - d e t ] F e b Institute of Nanostructures, Nanomodelling and Nanofabrication (i3N), Universidade de Aveiro,Campus de Santiago, Aveiro, 3810-193, Portugal Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Unit of Nuclear Engineering, Faculty of Engineering Sciences, Ben-Gurion University of theNegev, P.O.B. 653, Beer-Sheva, 8410501, Israel Nuclear Research Center Negev, Beer-Sheva, 84190, Israel Lawrence Berkeley National Laboratory (LBNL), 1 Cyclotron Road, Berkeley, CA 94720, USA Ikerbasque, Basque Foundation for Science, Bilbao, E-48013, Spain Centro de Investigaci´on en Ciencias B´asicas y Aplicadas, Universidad Antonio Nari˜no, SedeCircunvalar, Carretera 3 Este No. 47 A-15, Bogot´a, Colombia Department of Physics, Harvard University, Cambridge, MA 02138, USA Laboratorio Subterr´aneo de Canfranc, Paseo de los Ayerbe s/n, Canfranc Estaci´on, E-22880,Spain LIBPhys, Physics Department, University of Coimbra, Rua Larga, Coimbra, 3004-516, Portugal LIP, Department of Physics, University of Coimbra, Coimbra, 3004-516, Portugal Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242,USA Donostia International Physics Center, BERC Basque Excellence Research Centre, Manuel deLardizabal 4, 20018 San Sebasti´an / Donostia, Spain Escola Polit`ecnica Superior, Universitat de Girona, Av. Montilivi, s/n, Girona, E-17071, Spain Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, Campus de Cantoblanco,Madrid, E-28049, Spain Instituto de F´ısica Corpuscular (IFIC), CSIC & Universitat de Val`encia, Calle Catedr´atico Jos´eBeltr´an, 2, Paterna, E-46980, Spain Pacific Northwest National Laboratory (PNNL), Richland, WA 99352, USA Instituto Gallego de F´ısica de Altas Energ´ıas, Univ. de Santiago de Compostela, Campus sur,R´ua Xos´e Mar´ıa Su´arez N´u˜nez, s/n, Santiago de Compostela, E-15782, Spain Instituto de Instrumentaci´on para Imagen Molecular (I3M), Centro Mixto CSIC - UniversitatPolit`ecnica de Val`encia, Camino de Vera s/n, Valencia, E-46022, Spain Centro de Astropart´ıculas y F´ısica de Altas Energ´ıas (CAPA), Universidad de Zaragoza, CallePedro Cerbuna, 12, Zaragoza, E-50009, Spain University of the Basque Country (UPV/EHU), Faculty of Chemistry, Manuel de Lardizabal 3,20018 San Sebasti´an / Donostia, Spain Materials Physics Center. Manuel de Lardizabal 5, 20018 San Sebasti´an / Donostia, Spain Department of Chemistry and Biochemistry, University of Texas at Arlington, Arlington, TX76019, USA
E-mail: [email protected] , [email protected] bstract: Next-generation neutrinoless double beta decay experiments aim for half-life sensitivities of ∼ years, requiring suppressing backgrounds to a level of < ∼ Tl 1.6 MeV double escape peak (with Compton events asbackground), recorded in the NEXT-White demonstrator at the Laboratorio Subterr´aneode Canfranc, while retaining ∼
72% of the signal. These results were recently improvedthrough the use of a deep convolutional neural network to yield a background rejectionfactor of ∼
10 with 65% signal efficiency. Here, we present a new reconstruction method,based on the Richardson-Lucy deconvolution algorithm, which allows reversing the blurringinduced by electron diffusion and electroluminescence light production in the NEXT timeprojection chamber. The new method yields highly refined 3D images of reconstructedevents, and, as a result, significantly improves the topological background discrimination.When applied to real-data 1.6 MeV electron-positron pairs (at this stage without using deepneural networks), it leads to a background rejection factor of 27 at 57% signal efficiency. ontents − e + events 14 The search for neutrinoless double beta (0 νββ ) decay is the most promising experimentalpath to determine whether the neutrino is a Majorana fermion, with far-reaching impli-cations in particle physics and cosmology [1–5]. Presently, several collaborations pursuedifferent technologies for detecting 0 νββ decay with the leading experiments focusing on Ge [6–8],
Xe [9–15],
Te [10, 16, 17], and
Mo [18, 19]. The long half-life of 0 νββ decay (above 1.8 × yr in Ge [6] and 1.07 × yr in Xe [9]) makes its detectionextremely difficult, with only a few candidate 0 νββ events expected throughout the run-ning life of an experiment, calling for outstanding background suppression capabilities. Thenext generation of 0 νββ decay experiments will aim at half-life sensitivities of ∼ yr,requiring, in turn, a background level below a few counts/tonne/yr.NEXT (Neutrino Experiment with a Xenon TPC) is a staged experimental programaiming at the detection of 0 νββ decay in Xe, using successive generations of high-pressure gaseous xenon electroluminescent time projection chambers (HPXe EL-TPC) [20].– 1 –he choice of gaseous rather than liquid xenon is driven by two considerations: (1) theattainable energy resolution at the Q-value of the decay, Q ββ (which for Xe is 2458 keV),is a factor 3 better in gas than in liquid [11, 21–23]; and (2) whereas in liquid xenon eventsare point-like, the projected length of ionization tracks in high-pressure Xe gas is ∼
10 cm,allowing for background discrimination based on the track topology, as first pioneered bythe Gotthard experiment [24]. The distinguishing topological feature of a two-electrondouble beta decay event in high-pressure gas is the appearance of two blob-like Bragg-peak energy depositions at the opposite ends of the track, in contrast with single-electronbackground events which have only one such feature (figure 1). Thus, by reconstructingthe track in 3D and comparing the energy contained in small regions at its extremities, onecan effectively classify the event as either signal or background.The current stage of the NEXT program is the NEXT-White demonstrator, whoseTPC has an active volume half a meter in diameter and length [25]. NEXT-White (“NEW”)is a radiopure detector, operated underground under low-background conditions at the Lab-oratorio Subterr´aneo de Canfranc (LSC), using xenon enriched to 90% Xe. Its purposeis to validate all aspects of the technology on a large scale, including full characterizationof the background model and the technique’s background rejection power, and to demon-strate its performance on two-neutrino double beta (2 νββ ) decay events. NEXT-White,which has been running continuously since October 2017, will be superseded in 2021 by thetwice-larger NEXT-100 detector, which will deploy 97 kg of enriched xenon and demon-strate sensitivity to 0 νββ decay half-lives on the scale of 10 yr [14]. This will pave theway to a tonne-scale experiment which will be sensitive to half-lives longer than 10 yr[26]. Importantly, the NEXT Collaboration pursues in parallel an extensive R&D programto develop the capability of detecting the Ba daughter resulting from
Xe double betadecays inside a running TPC using single molecule fluorescence imaging [27–31]. If success-ful, these efforts could boost the sensitivity of HPXe EL-TPCs to half-lives on the scale of10 yr.NEXT excellent energy resolution – demonstrated to be ∼
1% full-width half maxi-mum (FWHM) at Q ββ in NEXT-White [22] – allows to reduce background from naturalradioactivity and 2 νββ events in the Q ββ region-of-interest (ROI) by about 5 and 12 or-ders of magnitude, respectively, making the latter completely negligible [32]. NEXT alsoutilizes track multiplicity, i.e., the number of distinct tracks in a given event, to reducebackground from γ -ray interactions or from single beta decays with energy in the Q ββ ROI(from cosmogenically-produced
Xe). This effectively removes multiple Compton scat-ters, as well as fast-electron tracks with accompanying vertices of bremsstrahlung x-rayinteractions, resulting in an additional background rejection factor of ∼
10 while retaining ∼
70% of the signal [26, 32]. Finally, using the topology of the remaining single tracks,background is further suppressed by measuring the energy deposition at the track ends asdescribed above.Tests of the topological background rejection in NEXT-White rely on the use of a
Th calibration source to produce 2615 keV γ -rays by the decay of Tl. Interactions of Named after Prof. James White, our late mentor and friend. – 2 – igure 1 . Simulated GEANT4 signal and background tracks at Q ββ . these γ -rays inside the gas lead to the production of electron-positron pairs with a totalkinetic energy of 1593 keV. These leave a trace with two Bragg-peak energy depositionsat the track endpoints, mimicking double beta decay events. Previous topological analysisof such events in NEXT-White (with Compton electrons serving as background) yielded abackground rejection factor of ∼
5, while retaining 72% of the signal events [33] for optimalperformance. Extended by Monte Carlo to Q ββ , this analysis provided a backgroundrejection factor 7.4 with similar signal efficiency. (Note that these values correspond to theperformance of the topological analysis alone, after the preceding energy and single-trackcuts.) Recently, these results were improved by using a deep convolutional neural networkyielding a background rejection factor of ∼
10 with a signal acceptance of ∼
65% [34].In this work we describe an improved methodology for track reconstruction. The un-derlying idea is to enhance the sharpness of reconstructed tracks – degraded by electrondiffusion and spread of light produced in the electroluminescence (EL) process – throughthe application of an image deblurring procedure, namely the Richardson-Lucy (RL) decon-volution algorithm [35, 36]. We begin by describing signal production and event reconstruc-tion in NEXT-White, with particular emphasis on previous work regarding the topologicalanalysis. This is followed by a discussion of the blurring effects of diffusion and EL lightproduction, and their quantification through a spatially-dependent point spread function(PSF). We then describe the implementation of the RL algorithm, which is outlined in ap-pendix A, within the experiment’s data processing chain, and validate it by demonstratingthe accurate reconstruction of point-like events (individual and pairs) and muon tracks.The procedure is subsequently employed on 1.6 MeV e − e + pairs and gamma-induced back-ground events in both Monte Carlo and data recorded in NEXT-White, yielding a majorimprovement in topological background rejection. We discuss the effect of the key pa-rameters of the deconvolution procedure on its performance, and present the optimizedresults. – 3 – NEXT-White: event reconstruction and prior work on topologicalanalysis
The NEXT-White TPC [25] consists of a cathode and gate grids defining a 53 cm-longdrift region, a transparent anode plate positioned 6 mm behind the gate, and a field cagewith an inner diameter of 45 cm. An array of 12 Hamamatsu R11410-10 3” photomultipliertubes (PMTs), constituting the energy plane , is located 13 cm behind the cathode. At theopposite end of the TPC, 2 mm behind the anode plate, an array of 1792 SensL series-C 1 mm silicon photomultipliers (SiPMs) distributed at a pitch of 10 mm, serves as the tracking plane . The entire tracking plane area is covered by polytetrafluoroethylene (PTFE)to reflect light towards the PMTs, with holes for the SiPMs. The interior wall of the fieldcage comprises a PTFE tube coated with a thin wavelength-shifting layer of tetraphenylbutadiene (TPB) to further optimize light collection. The anode is a 3 mm-thick fusedsilica plate, coated on both faces with transparent resistive and conductive layers, whichare themselves coated by TPB. The detector is presently operated under voltages whichdefine a uniform drift field of 0.42 kV/cm between the cathode and gate, and a nominalelectroluminescence (EL) field of 12.8 kV/cm between the gate and anode (the EL gap ).When an event occurs inside the sensitive volume, the associated primary chargedparticles form Xe excimers and electron-ion pairs along their track. De-excitation of theformer produces prompt vacuum ultra violet (VUV) scintillation light (“S1”) centered at172 nm, lasting a few hundred ns. This light, which is shifted to ∼
430 nm by the TPBcoating the inner surfaces of the TPC and recorded by the PMTs with a sampling timeof 25 ns, provides the start time t of the event. The drift field prevents electron-ionrecombination and drives the electrons at a uniform velocity of 0 .
91 mm/ µ s (for E drift =0 .
42 kV/cm) towards the EL gap. As they cross it (in ∼ . µ s), they produce EL light(“S2”), which is also centered at 172 nm and wavelength-shifted by TPB. For a nominalEL field of 12.8 kV/cm and 6 mm gap, this provides, at 10 bar, a light yield of ∼
450 VUVphotons per electron crossing the gap [37]. Depending on the track length, the duration ofthe S2 signal varies from a few µ s to a few hundred µ s. The S2 light is recorded by boththe PMTs and SiPMs, with the latter integrated in 1 µ s time slices.During operation, NEXT-White is continuously calibrated using m Kr, which is intro-duced into the gas system by an in-line Rb source. m Kr spreads uniformly throughoutthe TPC volume and produces point-like energy depositions of 41.6 keV. These events servefor precise determination of the electron lifetime and xy S2 response of the detector, togenerate 3D correction maps which are then employed for accurate energy measurements ofsignal and background events [38]. Furthermore, as detailed below, m Kr can also be usedto obtain the point-spread function that describes both electron diffusion and the opticalresponse of the tracking plane, a vital ingredient for image deconvolution.When processing events with energy higher than 400 keV, SiPM waveforms are re-binned into 2 µ s, keeping only sensors recording more than 5 photoelectrons (PEs) withineach slice. At this stage a second, optional, higher charge threshold is applied. For eachSiPM passing this threshold in each slice, a 3D “hit” is generated, whose xy coordinates arethose of the associated SiPM, the z coordinate is the product of the electron drift velocity– 4 –nd time difference between the slice and S1, and the magnitude is the number of detectedPEs. The energy measured by the PMTs in the same time slice is divided among thereconstructed hits, proportionally to the charge of their respective SiPMs. Afterwards theenergy assigned to each hit is multiplied by the correction factors derived from the m Kr3D maps. Figure 2 (left) shows an example of a 3D hit map of a reconstructed track withcolor representing the number of PEs detected.The resulting hits are grouped in cubical voxels with an event-dependent fixed size.These are further grouped into connected tracks, using the breadth-first search (BFS) algo-rithm [39, 40]. Voxels are considered to be part of the same track if they have a commonface, side or corner. An event may contain more than one track (for example, in the caseof multiple Compton scatters). After building the tracks, the BFS algorithm further iden-tifies the end-point voxels of each track. These are defined as the pair of voxels with thelongest distance between them, where the distance between any pair of voxels is definedas the shortest path along the track that connects them. The SiPM hits contained in theend-point voxels are used to define the voxel center-of-gravity (COG). Finally, two spheresof a fixed radius are defined around the end-point voxels COGs, and the energy containedin them is summed, designating the sphere containing more energy as blob1 and the onecarrying less energy as blob2 . The final output for analysis consists of a collection of tracksand their “blobs”.To investigate the effectiveness of topological event classification in NEXT-White, theexperimental work described in [33], to which we refer below as the classical analysis ,focused on e − e + pairs produced in Xe by the 2615 keV gamma of Tl, using an externalsource of
Th (parent of the decay chain containing
Tl). The topological analysis ofsuch pairs provides an excellent “training arena” for that of 0 νββ events. Similar to thetrack structure of 0 νββ decay, the pair electron and positron tracks start from a commonvertex, and both end with a Bragg peak of dense ionization. Such pairs have a total kineticenergy of 1593 keV, and their selection is based on the identification of the
Tl doubleescape peak in the energy spectrum. At 10 bar the combined CSDA (continuous slowingdown approximation) range of such pairs is ∼
20 cm, the same as that of 0 νββ events at15 bar (the planned operation pressure of NEXT-100); the typical projected length of thefull event in both cases is (cid:46)
10 cm. For the sake of the analysis, e − e + pairs are thereforeconsidered as signal events, while single-electron events (from Compton scatters) in theregion of the double escape peak, are considered as background.For the classical analysis the SiPM charge cut was set at a high value of 30 PEs, SiPMhits were grouped in cubical voxels with an event-dependent fixed size between 10 and15 mm (figure 2, right), and the blobs were defined by 21 mm-radius spheres. All choicesare revisited in the present work. In the analysis (classical and the new method presentedhere), events are required to be fully contained within the fiducial volume (with all hitsat least 2 cm away from any of the drift volume surfaces) and consist of only one track.Classification to signal or background is done by comparing the energy contained in blob2to a fixed threshold. If blob2 energy exceeds it the event is classified as signal, and if not– as background.To quantify the effectiveness of this topological cut on double escape peak events, one– 5 – igure 2 . An example for a reconstructed event in NEXT-White. Left: SiPM hits, with colorrepresenting detected photons; right: same event voxelized in 15 × ×
15 mm bins (from [33]). defines the signal efficiency (cid:15) as the fraction of signal events ( e − e + pairs) passing the cut,and the background acceptance b as the fraction of background events which survive it. Bothparameters depend on the choice of the energy threshold for blob2. In order to maximize thesensitivity of the experiment, the threshold on blob2 energy is selected such that a figure ofmerit, defined as f.o.m. = (cid:15)/ √ b , is maximal [32]. As shown in [33] for the optimal figure ofmerit, the classical analysis, when applied to experimental data, yields a signal efficiencyof 71.6% with a background acceptance of 20.6% (with small statistical and systematicerrors), i.e., a background rejection factor of 4.9. Similar outcomes were obtained in MonteCarlo (MC) simulations of double-escape peak events, performed with the NEXUS Geant4-based NEXT simulation framework [41]. When evaluating the performance of the classicalanalysis at Q ββ using MC events, the background acceptance was further reduced down to13.6% (background rejection factor of 7.4) while keeping virtually the same signal efficiency[33].As mentioned earlier, a more recent work [34] improved on these results by using adeep convolutional neural network to classify 1.6 MeV double-escape peak events to signaland background, starting from the voxelized tracks, instead of using the BFS algorithm andblob-based analysis. This method yielded a background acceptance level of 10% at 65%signal efficiency. The reconstruction method described in the following sections is closerin nature to the classical analysis, which therefore serves as the reference for comparison.Future works will investigate the benefit of combining image deconvolution and neuralnetwork-based analysis. The classical reconstruction using the SiPM hits and voxels is affected by two blurringmechanisms which degrade the quality of the reconstructed track: electron diffusion andthe spread of EL light on the tracking plane.– 6 – igure 3 . Simulated
Xe 0 νββ event. Left: initial track; Right - same track after drifting40 cm considering the longitudinal and transverse diffusion coefficients to be 0.27 mm / √ cm and1.07 mm / √ cm respectively. Color scale in the right plot represents the number of ionization elec-trons at each xy bin. The initial track structure is a thin trail of ionization, as shown in figure 1. As the ion-ization electrons drift towards the gate, elastic collisions with Xe atoms lead to transverseand longitudinal diffusive spread of the charge cloud around the track “backbone”. Theroot-mean-square (r.m.s.) diffusive spread of each point-like element of the initial track isproportional to the square root of the drift time, and therefore – since the drift velocity isconstant – to the square root of the distance between this element and the gate (i.e., its z coordinate). Under the operating conditions of NEXT-White, this effect can be on the cmscale.The second contribution to the overall blurring occurs as the electrons cross the ELgap, where they emit light isotropically in 4 π . VUV photons emitted towards the anodeare absorbed in TPB and re-emitted (again isotropically) as blue photons, which are subse-quently transmitted through or reflected on the interfaces of the multi-layered anode plate,with additional reflections from the tracking plane PTFE cover. VUV EL photons emittedtowards the gate and absorbed on it may result in secondary photoelectron emission, cre-ating an additional discrete “halo” of diffuse light around the event. All of these processescombine to further optically smear the image of the charge distribution in the EL gap at agiven time slice, with a similar relative contribution as electron diffusion.Both blurring effects can be characterized using m Kr data. In fact, transverse andlongitudinal diffusion in NEXT-White were already studied using m Kr events in [42].The r.m.s. transverse diffusion spread at 10 bar was found to be 1.07 mm × (cid:112) z (cm) andthe longitudinal one – 0.27 mm × (cid:112) z (cm). For the full drift distance in NEXT-White, z = 53 cm, the transverse and longitudinal FWHM spread of an electron cloud starting– 7 –rom a point-like charge distribution are, in this case, 18.3 mm and 4.6 mm, respectively.An example for the effect of diffusion on a simulated 0 νββ event for a drift distance of40 cm is shown in figure 3.Both electron diffusion and the EL light spread can be quantified in terms of pointspread functions. The full diffusion PSF is three dimensional: a point-like initial electroncloud transforms after diffusion to an oblate 3D Gaussian (wider in the transverse planethan along the drift direction), where both the transverse and longitudinal widths areproportional to √ z . This 3D PSF can be projected on the xy plane to yield an effective2D transverse diffusion PSF, F Ddif ( x (cid:48) , y (cid:48) ; z ); (here x (cid:48) and y (cid:48) are the xy coordinates in aframe of reference centered on the PSF axis). Similarly, integrating the total light hittingthe tracking plane for a point-like charge crossing the EL gap produces a 2D EL PSF, F EL ( x (cid:48) , y (cid:48) ). Unlike the diffusion PSF, the EL PSF does not depend on the drift distance z .Detailed analysis of m Kr events show that except for the TPC edges, both the diffusionand EL PSFs do not depend, to leading order, on the absolute xy position with respect tothe TPC axis, and both are axisymmetric.Experimentally, the EL PSF can be determined from m Kr events occurring imme-diately in front of the gate, such that they do not suffer a diffusive spread and can beconsidered point-like. The procedure, similar to that described in [42], involves recordinga large number of m Kr events over a small drift region (drift time < µ s). For eachevent, the SiPM response is integrated over several µ s, to include the full S2 signal. The xy location of the event is determined by calculating the center of gravity (COG) of theSiPM hit map. The coordinates of all SiPMs participating in the event are shifted to areference frame whose origin coincides with the COG of the event, and their charge isbinned in 1 × pixels. The process, repeated over a large number of events, suchthat in each step the SiPM charge is added to the corresponding pixels, converges to thePSF. Figure 4 (top left) shows the profile of the EL PSF constructed from m Kr data.For comparison, we also show the PSF extracted from a Monte Carlo simulation. TheMC PSF is narrower and the wings of the experimental PSF are somewhat larger. Thisindicates that the present simulation does not provide a complete description of the opticalprocesses occurring in the EL gap and multi-layered anode plate (in particular, it does notinclude, at this stage, a contribution of secondary photoelectron emission from the gate,which may lead to single-electron EL signals at some distance from the main event).The effective 2D transverse diffusion PSF is given approximately by a Gaussian whosestandard deviation is: σ t = 1 . √ z , where z is in cm and σ t in mm. This function isshown in the top right panel of figure 4, for several values of z . The two blurring effectsof diffusion and EL light production can be combined in a single z-dependent PSF, which,to a good approximation, is given by their convolution: F dif + EL ( z ) = F Ddif ( z ) ∗ F EL . Thisexpression is only approximate, because in reality longitudinal diffusion introduces “crosstalk” of charge between adjacent slices; however, since longitudinal diffusion is ∼ m Kr events over a range [ z, z + ∆ z ],as was done in [42] to determine the transverse and longitudinal diffusion coefficients. In the– 8 –resent study, events were selected in 25 µ s drift time intervals. This step size was chosento keep variations in transverse diffusion within the interval sufficiently small. Thus, forthe first 100 µ s drift, the relative change within a 25 µ s interval is ∼ ∼ µ s, the change falls below 5%. The bottom panel offigure 4 shows the combined z-dependent PSF for the set of drift distances shown in the topright panel. We show both the experimental PSF and the one obtained by convolving theMonte Carlo EL PSF and the transverse diffusion Gaussian. As indicated by the figure,convolution of the EL PSF with the diffusion PSF washes out most of the differencesbetween the two. Note that for the two different data sources, MC and detector acquireddata, double escape peak events are analyzed below with their respective z-dependent PSFs. Figure 4 . Top left: EL PSF (from m Kr data and Monte Carlo). Top right: The 2D transversediffusion PSF (calculation based on the diffusion measurements in [42]). Bottom left: The com-bined EL+diffusion PSF. Experimental data (solid line) are from m Kr events selected by theirdrift distance, while MC (dashed lines) is a convolution of the EL PSF and the 2D transverse dif-fusion PSF. Bottom right: The 2D PSF for experimental data for the different drifts shown in theaccompanying plots. The intensity value refers to the area-normalized value of the distributions. – 9 –
Track reconstruction in NEXT-White using Richardson-Lucy decon-volution
The Richardson-Lucy (RL) algorithm (also known as the Lucy-Richardson algorithm) wasdeveloped independently by W. H. Richardson [35] and L. B. Lucy [36] in the early 1970’sin the context of observational astronomy. Richardson focused his discussion solely onthe recovery, by deconvolution, of an underlying sharp image from an observed blurredone. Lucy’s work, in contrast, considered a more general case, where one seeks to recoveran underlying frequency distribution from an observed one, with image restoration asa particular application. The method has subsequently become a main tool for imagerestoration in many scientific and engineering fields. The algorithm is iterative, generating asequence of improved approximations for the underlying sharp image based on the observedblurred and noisy one, and the (presumably known) point spread function. It can beapplied on an arbitrary number of spatial dimensions. Appendix A outlines the procedurefor two-dimensional images. Here we describe its implementation within the NEXT eventreconstruction scheme and a series of evaluations performed to cross-check and validatethe methodology.
In NEXT-White, we apply RL deconvolution on individual SiPM time-sliced hit maps(each integrated over a time interval δt , where, in the present analysis δt = 2 µ s). Eachslice is considered to be fully independent from the others and longitudinal spread is nottaken into account. For a slice recorded at time t , we associate a physical slice of width δz = v d δt of the original 3D track at the corresponding drift distance z = v d · ( t − t ) (for v d = 0 . δz = 1 . W ( x, y ), and the corresponding SiPMhit map as a sampled representation of the blurred image ˜ H ( x, y ). The two images areassumed to be related through the combined diffusion+EL PSF F ( x, y ; z ) correspondingto a drift distance z .The implementation of RL deconvolution on each slice is done, for both experimentaldata and MC events, in the following steps:1. Charge cut: SiPMs containing less charge than a predefined threshold q cut are re-moved from the slice. As discussed in appendix A, after exploring several differentchoices for the charge threshold, we adopted a value q cut = 10 PE (compared to 30PE used in the classical analysis).2. Removal of isolated SiPMs: Single SiPM hits which have no adjacent non-zero neigh-bors in the same slice are removed. This is done to avoid filling the region betweenthe main track and isolated SiPMs which fluctuate above the charge threshold q cut by non-physical data in the subsequent interpolation step.3. 2D interpolation: We define a rectangular region surrounding the SiPMs which havesurvived steps (1) and (2) with 10 mm margins. To estimate the full pattern of photon– 10 –it points in this region, we apply bicubic 2D interpolation on the “cleaned up” SiPMhit map over a 1 × grid (the SiPMs cover only 1% of the plane). Note thatno significant differences were observed in the final outcomes of the analysis (signalefficiency and background acceptance for double escape peak events) when replacingbicubic by linear interpolation (see a brief discussion in appendix B).4. RL deconvolution: For each slice, we use the corresponding z-dependent combinedEL+diffusion PSF for the deconvolution process, following equations (A.6)-(A.8) inappendix A, to find successive estimations W ( r ) ( x, y ) for W ( x, y ) in N iter iterations.Data and MC PSFs are used for data and MC events, respectively. The process main-tains the overall charge of each slice constant in all iterations. It was implementedusing the richardson-lucy function from Python’s scikit-image library [43].5. Cleaning cut: Once the iterative process is completed, a cleaning cut with an ad-justable threshold (cid:15) cut is applied to the image intensity given by the iterative process.This is done to remove non-physical backgrounds and reconstruction leftovers, andsharpen the track edges for the topological analysis. For the double escape peakanalysis the cut was set at 0.008 a.u. Details on the optimization can be found inappendix B. No cut was performed when applying the method to Kr events as thereconstruction leftovers where not found to have an impact on the performance.6. Energy allocation: Finally, based on the integrated S2 signal recorded by the PMTsover the entire event duration, and using the m Kr-based lifetime and S2 correctionmaps [22], we find the total energy of each recorded slice and divide it between all ofthe 1 × pixels of the deconvolved image, proportionally to their interpolatedcharge.The first three steps aim to generate a reasonable estimate for the actual photon hitpattern on the tracking plane. They reflect a pragmatic approach to bridge the emptyspaces (and hence lack of information) between the SiPMs, and to avoid distorting theimage by distant effects, such as reflections from the various TPC surfaces, or distant ELlight emission by photoelectrons ejected from the gate mesh. The interpolation step isjustified as the smearing effects of both electron diffusion and EL light spread producegradual changes in light intensity on the tracking plane. Rather than claim for absolutemathematical rigor in this approximation, we provide a series of simple demonstrationsto support its practical value. These include the reconstruction of individual Kr events,adjacent pairs of Kr decays, and straight muon tracks. As a first test, RL deconvolution was applied to individual Kr events from both MC anddetector data. A typical example, from data, is shown in figure 5, with the SiPM sensorresponse for q cut = 10 PE on the first column, bicubic interpolation on the second, anddeconvolved images after 75 RL iterations on the third. When comparing the deconvolvedimages to the MC true information, the r.m.s. error in the reconstructed COG of all Kr-events was ∼ . − . x and y . For 75 RL iterations, the FWHM of the– 11 – igure 5 . Example of a reconstructed m Kr event from NEXT-White data. The event is centeredat (0,0) for convenience. Left: raw sensor response, with a charge cut of 10 PE. Center: bicubicinterpolation. Right: deconvolved image after 75 RL iterations.
Figure 6 . Reconstructed pair of nearby m Kr events from detector data, synthesized by overlayingSiPM response maps for two nearby events. Left column - raw (synthesized) SiPM data; center -bicubic interpolation; right - deconvolved image after 75 iterations. Red dots represent the COGsof individual events. .reconstructed Gaussian-like charge distribution was ∼ z >
100 mm. Although adding iterations wasfound to reduce the FWHM further, the effect was quite marginal (e.g., applying 150iterations reduced the FWHM to ∼ x and y , to bring its COG close to that of the other one. An example is shown in– 12 – igure 7 . Two muon events in NEXT-White data after RL deconvolution, shown in three Cartesianprojections. The top event is of a “clean” muon, while the bottom one also contains a delta electron. figure 6, where the COGs are 19.5 mm apart (for a drift distance of ∼
360 mm). The dotsrepresent the individual Kr COGs. As in figure 5, the left, center and right columns showthe raw (overlaid) sensor response maps, interpolated images and deconvolved ones (with75 iterations).To demonstrate the method over long tracks, we selected, from detector data, a set ofcrossing muon events. A visual inspection was carried over an extensive dataset with noobservable deviations from the expected straight line tracks, with occasional delta electronsbranching out from the main track. Two examples of muon tracks of NEXT-White data(Run V) are shown in figure 7.Figure 8 demonstrates the iterative refinement in the track sharpness obtained by theRL process. The event consists of a 1.6 MeV electron-positron pair, acquired in NEXT-White, which forms a U-shaped track, where the two ends are located roughly in the same xy plane. The first row shows the raw data (sensor response) before deconvolution, binnedin 10 × × . yz plane, is shown on the topleft. The dashed lines mark three particular slices where the effect of applying successivelymore iterations is shown for each one individually. The corresponding slice images areshown in columns 2-4. Rows 2-4 show the same event after 2, 10 and 75 iterations, withthe latter after application of the cleaning cut. The disconnected artifact appearing in slice– 13 – e n s o rr e s p o n s e i t e r a t i o n s i t e r a t i o n s i t e r a t i o n s Figure 8 . Effect of successive RL iterations (2, 10 and 75) in three selected slices. The event is anelectron-positron pair (from data). x = 70 mm, y = 50 mm) does not affect the analysis of themain track. − e + events The detector data used for this analysis were taken during August 2019 under the sameconditions as in [33], but with a much longer electron lifetime of ∼ Th source– 14 –laced in an external calibration port above the center of the drift region, and of internal m Kr events.Similarly to [33], event selection for the topological analysis around the
Tl doubleescape peak was done by applying the following filters: (1) event energy in the range1 . − . × × . voxels with 1.8 mm along z ) were re-binned into larger voxels of adjustable size. Asdiscussed in appendix B, 5-mm voxels provided the optimal results. The BFS algorithmwas then used – as in the classical analysis – to combine the event voxels into one or moretracks. At this stage, events may have contained more than one track either because theyconsisted of multiple nearby physical tracks which were unresolved by the gross single-track cut, or due to artificial breaking of a single track into smaller segments during theRL process. Concretely, it was estimated, based on the true information of simulateddata, that 29% of the events passing the gross single-track cut were actually multi-trackevents. At the same time, for the optimal parameter configuration described below, 27% ofthe deconvolved simulated events exhibited more reconstructed tracks than the number ofsegments in the true information. These additional “satellites” were of low intensity, withessentially no effect on the blob-based analysis.Once the voxels were grouped into tracks, the BFS algorithm was used to find the trackends. Those of the longest track of the event served as the centers for two spherical blobsof a variable radius. As before, we defined blob1 as the more energetic blob and blob2 asthe less energetic one. We set a threshold E ,blob on blob2 energy such that events aboveit were considered signal and below it – background. For the i -th value of blob2 energythreshold E ( i )0 ,blob we defined the signal efficiency (cid:15) i , background acceptance b i and figureof merit f.o.m i as: (cid:15) i = number of signal events with E blob2 > E (i)0 , blob2 total number of signal events with no cut (5.1) b i = number of background events with E blob2 > E (i)0 , blob2 total number of background events with no cut (5.2) f.o.m i = (cid:15) i √ b i (5.3)For MC, the true nature of the event is known: any event containing a positron is signal,and any event without one is background. For each threshold E ( i )0 ,blob , the signal efficiency,background acceptance and figure of merit are found directly from equations (5.1)-(5.3)– 15 – igure 9 . Energy spectrum and event population in the double escape peak region for MC events.In blue, events with a positron, in red – the rest of the interactions (predominantly Comptonelectrons). Left: the spectrum before applying any topological cut. Right: after a blob2 energy cutof 340 keV. using the true information. For experimental detector data, however, the nature of theevent is unknown a priori and one must resort to a different approach, which involvesfitting the ROI data around the double escape peak with an expression that describes bothsignal and background, before and after the application of the cut E blob > E ( i )0 ,blob [33].Figure 9 shows the MC energy spectrum around the double escape peak, highlightingseparately the contribution of e − e + pairs (in blue) and of events with no positron (in red).The figure shows the spectrum before and after the application of the topological cut. Thebackground spectrum in this region can be well described by a decreasing exponential, aswas done in [33]: f bkg ( E ) = A exp( − A E ) (5.4)The signal spectrum consists of a Gaussian centered at 1593 keV, with flat shallowwings for energies below and above the peak, as shown in figure 10. Below the peak,signal events comprise e − e + pairs created by photons of energy below 2615 keV (whichpredominantly result from Compton scatters of 2615 keV gammas prior to pair production),as well as e − e + pairs which lose some energy by bremsstrahlung, where the emitted photondoes not interact in the sensitive volume. Above the peak there is a smaller populationof e − e + events where one of the 511 keV gammas created when the positron annihilatesinteracts close to the main track and is unresolved by the gross single-track cut. The signalspectrum can therefore be approximated as: f sig ( E ) = B (cid:32) √ πσ exp (cid:32) − ( E − µ ) σ (cid:33) + C erfc (cid:18) E − µ √ σ (cid:19) + C (cid:33) (5.5)where a complementary error function (erfc) with the same standard deviation as theGaussian is used to describe events below the peak. This expression was chosen empirically– 16 – igure 10 . Energy spectrum of e − e + pairs around the 1.6 MeV double escape peak, fitted accordingto equation (5.5). based on the MC distribution shown in figure 10 and should be considered only as a proxyto obtain an estimate of the signal population outside the peak with no other physicalmeaning. The parameters C and C were extracted from fitting, through an unbinnedextended maximum likelihood fit, MC data consisting only of e − e + pairs in the region1 . − . C = (4 . ± . · − and C = (1 . ± . · − . Note that theanalysis in [33] disregarded the shallow wings in the e − e + spectrum, and instead assumedthat the signal is completely described by a Gaussian. This assumption is valid whenthe signal population outside the peak is negligible compared to the background, which isthe case before application of the topological cut (figure 9, left). For a modestly effectivetopological analysis, this holds also after applying the cut. However, this consideration losesvalidity as the topological cut becomes more effective, and a realistic fit should consist ofthe sum of equations (5.4) and (5.5): f ( E ) = f bkg ( E ) + f sig ( E ) (5.6)The procedure for estimating the signal efficiency and background acceptance usingthe fit was as follows. The parameters C and C were extracted from the MC fit to the truesignal spectrum and assumed to hold also for detector data, without being affected by theapplication of the topological cut. This approximation, which reflects an assumption thatthe signal efficiency of the the blob cut is energy-independent over the range 1 . − . E ( i )0 ,blob we then fitted the data with f ( E ) (equation (5.6)), with A , A and B as free parameters, before and after applyingthe cut E blob > E ( i )0 ,blob using an unbinned maximum likelihood fit (the parameters µ and σ were extracted once from a fit to the Gaussian alone and kept constant in subsequentfits). The number of background and signal events passing a given value of E ( i )0 ,blob were– 17 –alculated by integrals over ± σ around the peak centroid: N bkg,i = (cid:90) µ +3 σµ − σ A ,i exp( − A ,i E ) dE (5.7) N sig,i = (cid:90) µ +3 σµ − σ B ,i (cid:32) √ πσ exp (cid:32) − ( E − µ ) σ (cid:33) + C erfc (cid:18) E − µ √ σ (cid:19) + C (cid:33) dE (5.8)The estimate was found to be within 1% of the real number of events for both signaland background populations in MC. The signal efficiency and background acceptance forthe cut E blob > E ( i )0 ,blob were calculated as: (cid:15) i = N sig,i /N sig, (5.9) b i = N bkg,i /N bkg, (5.10)where the subscript “0” refers to no cut (i.e., E ,blob = 0). The figure of merit for the i -ththreshold was calculated, as before, by f.o.m i = (cid:15) i / √ b i .Figure 11 shows four examples of detector data events reconstructed using RL decon-volution. The first two rows show events classified as signal, while the third and fourthrows display events classified as background (deconvolution and classification were doneusing the choice of parameters described below). Note that the second background eventcontains a blob-like satellite close to its start point. This event passed the gross single-trackcut, and, in the classical analysis, would have likely been classified as signal. The refinedreconstruction offered by RL deconvolution allows separating this satellite from the maintrack and correctly classify this event as background. The RL process and subsequent analysis involve several parameters, whose values deter-mine the blob identification quality which ultimately reflects on the signal efficiency andbackground acceptance. These parameters can be divided in two groups: those affectingthe quality of the reconstructed image and the end-point identification accuracy, and thosewhich affect the energy calculation inside the blobs. The first group is related to the RLprocess and includes the SiPM charge threshold q cut , number of RL iterations N iter andfinal cleaning threshold (cid:15) cut . The second group is comprised of the BFS characteristicparameters, namely the voxel size l voxel and blob radius R blob . For a given choice of pa-rameters the f.o.m. attains a maximum for a particular value of E ,blob . By carefullyadjusting the values of these parameters one can try to maximize the f.o.m. , and thereforethe experimental sensitivity in the energy region under investigation.The optimization process is highly demanding in computing resources: beyond beinga multivariate problem, the determination of the f.o.m. for every parameter configurationrequires analyzing tens of thousands of events. We therefore adopted here a pragmatic ap-proach, which allowed attaining a high value for the f.o.m. , but without proving that this is– 18 – igure 11 . Experimental detector data events in the 1.6 MeV double escape peak region aftertopological classification. Top two rows: e − e + candidates; bottom two rows: background candi-dates. – 19 – igure 12 . 2D histogram of blob1 and blob2 energies of 1.6 MeV double escape peak events fromdetector data: (A) classical analysis; (B) RL deconvolution with the optimal choice of parameters.Each histogram contains events passing the fiducial cut and gross single-track cut with energiesin the range 1 . − . f.o.m . a global maximum. The analyzed dataset comprised 3 . · MC and 1 . · detector dataevents passing the energy, fiducialization and gross single-track cuts. We performed sepa-rate analyses to MC events using the true information (equations (5.1)-(5.3)), MC eventswhere signal and background estimates were made using the fitting procedure (equations(5.7)-(5.10)), and experimental detector data using the same fitting procedure. The op-timal choice of parameters and resultant f.o.m. vary, to some extent, between MC anddetector data. Appendix B provides a detailed description of the optimization process.Here, we present the final results of the analysis.Our choice of parameters aimed to simultaneously achieve a high f.o.m. for both detec-tor data and MC (using the fitting procedure) while looking for optimal MC (true informa-tion) performance. We refer to the chosen set of parameters as the “optimal configuration,”where: q cut = 10 PE, N iter = 75, (cid:15) cut = 0 . l voxel = 5 mm and R blob = 18 mm.Figure 12 shows 2D histograms of signal and background events from detector data,binned according to their blob1 and blob2 energies, comparing the classical analysis (A)to RL deconvolution with the optimal choice of parameters (B). Each histogram containsboth signal (electron-positron) and background (Compton scatter) events, in the range1 . − . f.o.m . Application of RL deconvolution clearly leads to an improved separation betweenthe two groups.Figure 13A shows the signal efficiency as a function of background rejection (1 − b )– 20 – igure 13 . (A) Signal efficiency vs. background rejection for 1.6 MeV double escape peak events,for detector data, MC (using the fit) and MC (using the true information), for the optimal choice ofparameter configuration (marked rectangles in figure 18 in appendix B). The curve resulting fromthe classical analysis of data (from [33]) is shown for comparison. (B) The figure of merit for theoptimal parameter configuration for detector data, MC (fit and true) and classical analysis vs. thethreshold on blob2 energy. The maximal (optimal) f.o.m. is for a blob2 energy cut at 340 keV. for 1.6 MeV double escape peak events, for the optimal choice of parameters. The figureincludes the curves for detector data, MC (using the fit and true information) and – forcomparison – the classical analysis of detector data events [33]. Increasing backgroundrejection is equivalent to moving the horizontal white lines in figure 12 upward. Figure 13Bshows the f.o.m. as a function of the threshold on blob2 energy for the same datasets andparameters. For RL deconvolution the maximal (optimal) f.o.m. is obtained for E ,blob =340 keV.Table 1 shows the signal efficiency, background acceptance and f.o.m. for detectordata and MC (with the classification based on both the fit and true information), for theoptimal choice of parameters and optimal value of blob2 energy cut. For comparison thetable also includes the results of the classical analysis. The errors represent statisticaluncertainties (standard deviation); systematic effects were found to be dominated by thechoice of reconstruction parameters and are not further considered here. Table 1 . Signal efficiency and background acceptance for the optimal figure of merit.
Dataset/analysis Signal efficiency Background acceptance Figure of Merit
Classical 71 . ± .
5% 20 . ± .
4% 1 . ± . . ± .
2% 3 . ± .
7% 2 . ± . . ± .
6% 4 . ± .
5% 2 . ± . . ± .
0% 3 . ± .
4% 3 . ± .
20– 21 – igure 14 . (A) Reconstructed energy for blob2 for deconvolution and classical analysis. The“true” blob energy is found by integrating over a sphere of 18 mm radius centered on the true trackend-point. (B) Distributions divided by population, either signal or background.
For detector data, the RL-based analysis using the optimal choice of parameters pro-vides a 5.6-fold reduction of background acceptance compared to the classical analysis(overall topological background rejection factor of ∼ igure 15 . End-finding error distributions. Left: overall error distributions obtained by RL-deconvolution and the classical analysis for 1.6 MeV double escape peak events; right: distributionsfor RL-deconvolution divided into signal and background events, with subdivision into blob1 andblob2. Richardson-Lucy deconvolution was shown in this work to be a highly effective tool forenhancing image reconstruction in NEXT, leading to a major improvement in topologicalbackground rejection. The application of the method to detector data in the 1.6 MeVdouble escape peak of
Tl, using a cut on blob2 energy, yielded a background acceptancelevel b = (3 . ± . (cid:15) = (56 . ± . f.o.m. = (cid:15)/ √ b = 2 . ± . f.o.m. = 1 .
58 [33]) and greatly booststhe background rejection realizable by the experiment. The obtained level of backgroundacceptance is similar to the value previously reported by the Gotthard experiment usingvisual inspection of double escape peak events at 1.6 MeV [24]. The new results are alsoconsiderably better than those obtained using a deep convoluted neural network on non-deconvolved tracks to classify double escape peak events, where the background acceptancewas 10% and signal efficiency was 65% ( f.o.m. = 2 .
06) [34].The primary effect of employing RL deconvolution is the attainment of refined 3D trackimages, which allows better identification of the track ends, and therefore improved posi-tioning of the blob centers and better estimates of the energy they contain. Improved endfinding generally results from the enhanced resolving power offered by RL-deconvolution.Several illustrative examples for this are given in appendix C.The focus of this work was on the analysis of experimental detector data recordedat the 1.6 MeV
Tl double escape peak. However, we also probed the possibility ofimplementing the RL-based method to MC signal and background events in the Q ββ ROI,using the NEXT-White detector MC model. Preliminary analysis indicated that similarlevels of background acceptance and signal efficiency are expected in this energy range.– 23 – igure 16 . Deconvolved 2.0 MeV 2 νββ candidate obtained during the current data taking ofNEXT-White.
However, since NEXT-100 will operate under different conditions (15 bar, 113 cm maximaldrift, 15.6 mm SiPM spacing and a different EL and tracking plane geometry), such resultsare only of indicative nature, and a full simulation is deferred to a separate publication.In spite of the improved event classification offered by the new method, the full po-tential Richardson-Lucy deconvolution is yet to be exploited by the collaboration, as theresults presented in this work remain a first-approach evaluation using existing tools (e.g.,the BFS algorithm), which may not be optimal for the fine-grained output of the RL pro-cedure. In particular, although figure 15 shows a clear improvement in track-end finding,the distributions of deconvolved events have significant tails extending to large errors, re-quiring the use of large blob radii for the analysis. Presently, several ideas for potentialimprovement in event classification are under study. These include improved algorithms forend finding, as well as the potential use of Machine Learning approaches for the classifica-tion of high-definition reconstructed events. In addition, RL deconvolution can be furtherdeveloped to a full 3D method instead of following the slice-by-slice approach employed inthis work, which could lead to enhanced image quality along the drift direction. Lastly,image quality may greatly benefit from the use of low-diffusion gas mixtures, such as Xe-He[44–46], or Xe “spiked” with low concentrations of molecular additives [47–49].Prior to further improvement of the method, the collaboration is presently evaluatingthe benefits offered by RL deconvolution in its current form, in particular for the analysisof 2 νββ events in NEXT-White. As an example, figure 16 shows a 2.0 MeV double betacandidate (from NEXT-White detector data) reconstructed by the RL procedure describedabove.
Acknowledgments
The NEXT Collaboration acknowledges support from the following agencies and institu-tions: the European Research Council (ERC) under the Advanced Grant 339787-NEXT;the European Union’s Framework Programme for Research and Innovation Horizon 2020(2014–2020) under the Grant Agreements No. 674896, 690575 and 740055; the Ministe-rio de Econom´ıa y Competitividad and the Ministerio de Ciencia, Innovaci´on y Universi-– 24 –ades of Spain under grants FIS2014-53371-C04, RTI2018-095979, the Severo Ochoa Pro-gram grants SEV-2014-0398 and CEX2018-000867-S, and the Mar´ıa de Maeztu ProgramMDM-2016-0692; the Generalitat Valenciana under grants PROMETEO/2016/120 andSEJI/2017/011; the Portuguese FCT under project PTDC/FIS-NUC/2525/2014 and un-der projects UID/04559/2020 to fund the activities of LIBPhys-UC; the U.S. Departmentof Energy under contracts No. DE-AC02-06CH11357 (Argonne National Laboratory), DE-AC02-07CH11359 (Fermi National Accelerator Laboratory), DE-FG02-13ER42020 (TexasA&M) and DE-SC0019223 / DE-SC0019054 (University of Texas at Arlington); the Univer-sity of Texas at Arlington (USA); and the Pazy Foundation (Israel) under grants 877040and 877041. DGD acknowledges Ramon y Cajal program (Spain) under contract num-ber RYC-2015-18820. JM-A acknowledges support from Fundaci´on Bancaria “la Caixa”(ID 100010434), grant code LCF/BQ/PI19/11690012. AS acknowledges support from theKreitman School of Advanced Graduate Studies at Ben-Gurion University.
A Richardson-Lucy deconvolution
The Richardson-Lucy (RL) algorithm aims to recover, by deconvolution, an underlyingsharp image from an observed blurred and noisy one. The algorithm is iterative, generatinga sequence of improved approximations for the underlying image using the (presumablyknown) point spread function (PSF) of the imaging process. In this appendix we outlinethe mathematical procedure, as employed on 2D images.We denote by W ( x, y ) the underlying sharp image and by F ( x, y ) the PSF. In theabsence of noise, the ideal blurred image H ( x, y ) is obtained as a convolution of W and F : H ( x, y ) = (cid:90) (cid:90) W ( x (cid:48) , y (cid:48) ) F ( x − x (cid:48) , y − y (cid:48) ) dx (cid:48) dy (cid:48) (A.1)In principle, W ( x, y ) could be recovered from H ( x, y ) by solving this integral equation. Thiscould be done by discretization, converting H , W and F into matrices and the integral toa double summation. This generates a system of linear equations with the elements of W as the unknowns: (cid:88) j F i,j W j = H i (A.2)where the indices i and j refer to individual elements in H and W , combining the enumer-ation of both the row and column, and F i,j describes the influence of the j -th element of W on the i -th element of H .In reality, because of the presence of noise, the actual observed image ˜ H ( x, y ) is dif-ferent from the ideal one H ( x, y ). The system of linear equations then becomes: (cid:88) j F i,j W j = ˜ H i (A.3)Attempting to solve equations (A.3) generally yields poor results, with large discontinuitiesin { W j } , as well as non-physical negative values. This occurs because the process tends toamplify short-wavelength errors in ˜ H , which are characteristic of noisy images [36].– 25 –he approach of the RL algorithm is different. It begins by noting that one couldformally write: W ( x, y ) = (cid:90) (cid:90) H ( x (cid:48) , y (cid:48) ) G ( x − x (cid:48) , y − y (cid:48) ) dx (cid:48) dy (cid:48) (A.4)provided that we define the inverse kernel G as: G ( x − x (cid:48) , y − y (cid:48) ) = W ( x, y ) F ( x − x (cid:48) , y − y (cid:48) ) H ( x (cid:48) , y (cid:48) ) (A.5)where (cid:82)(cid:82) F ( x − x (cid:48) , y − y (cid:48) ) dx (cid:48) dy (cid:48) = 1. Since G depends on W , the direct calculation of W from equation (A.4) is impossible. However, the process may work iteratively, if onecould provide successively improved approximations for G , which, in turn, would rely onsuccessive estimates of W .We begin with an initial estimate W (0) for W , where W (0) ( x, y ) is generally taken tobe a flat image. In the r -th iteration we calculate an intermediate blurred image H ( r ) by: H ( r ) ( x, y ) = (cid:90) (cid:90) W ( r ) ( x (cid:48) , y (cid:48) ) F ( x − x (cid:48) , y − y (cid:48) ) dx (cid:48) dy (cid:48) (A.6)This allows finding an estimate for G : G ( r ) ( x − x (cid:48) , y − y (cid:48) ) = W ( r ) ( x, y ) F ( x − x (cid:48) , y − y (cid:48) ) H ( r ) ( x (cid:48) , y (cid:48) ) (A.7)The new estimate for W , W ( r +1) , is then calculated following equation (A.4), with ˜ H replacing H and G ( r ) replacing G : W ( r +1) ( x, y ) = (cid:90) (cid:90) ˜ H ( x (cid:48) , y (cid:48) ) G ( r ) ( x − x (cid:48) , y − y (cid:48) ) dx (cid:48) dy (cid:48) = W ( r ) ( x, y ) (cid:90) (cid:90) ˜ H ( x (cid:48) , y (cid:48) ) H ( r ) ( x (cid:48) , y (cid:48) ) F ( x − x (cid:48) , y − y (cid:48) ) dx (cid:48) dy (cid:48) (A.8)The discussion in [36] shows that if successive changes in W ( r ) are sufficiently small,in the limit r → ∞ the scheme converges to the solution of the corresponding maximumlikelihood problem . It further shows that if the number of elements in H is equal or largerthan those of W (i.e., the system of linear equations (A.3) is over-determined), this solutionis unique. B Parameter optimization
In this appendix we describe in detail the steps taken to optimize the choice of parametersused for the RL process and subsequent analysis.The first step in the analysis was to choose a value for the SiPM charge threshold, q cut . It determines how much of the signal is cleaned out before starting the deconvolution The possibility of track reconstruction using a maximum likelihood approach had been previouslyexplored by the NEXT Collaboration [50], but was later disfavored for the method presented here. – 26 –
10 15 20 25SiPM charge cut (pes)1.82.02.22.42.62.83.03.23.4 f . o . m DataMCTrue
Figure 17 . Variation of the maximal f.o.m. achieved for both experimental detector data and MCfor different cuts on the input signal. The scan on q cut was done with fixed values for the otherparameters, as described in the text. process. Cutting too low may lead to the inclusion of distant signals (reflected light orphotons induced by photoelectrons emitted from the gate), while cutting too high maybias and distort the output. The impact of q cut on the f.o.m. was studied for several cutvalues, over the range 5 −
25 PEs (in 2 µ s). We performed this scan keeping N iter = 90, (cid:15) cut = 0 .
008 (in arbitrary units), l voxel = 5 mm and R blob = 21 mm (these values werechosen as a reasonable starting point after visual inspection of many events). For eachvalue of q cut we calculated the f.o.m. as a function of blob2 energy threshold and found itsmaximal value. Given the results, shown in figure 17, we settled on a 10 PE cut due to amuch better match between detector data and both MC fitted data and true information.Next, we considered the effect of the number of RL iterations and final cleaning cut.Understanding the optimal point to stop applying RL iterations is of prime importance.If not applied enough times, the reconstructed charge distribution remains too blurry,which harms the blob energy estimation. On the other hand, over-iterating can result innoisy artifacts and in breaking up of the track to disconnected segments. These effectsare strongly related to the subsequent application of the cleaning cut. If applied correctly,it can remove artifacts that appear in the iterating process. However, this is a delicateparameter as a too high cut could lead again into track fragmentation.Given the observed relation between the number of iterations and the cleaning cut, wedecided to scan and optimize both parameters simultaneously. The number of iterationswas varied in steps of 15, and the cleaning cut was scanned over the range 0 . − . l voxel = 5 mm and R blob = 21 mm. For each configuration wefound the maximal f.o.m. as a function of E ,blob . The results are shown in the toppart of figure 18 for detector data, MC using the fit and MC using the true information.We chose a configuration which displayed a high f.o.m. for both data and MC, includingthe true information, namely N iter = 75 and (cid:15) cut = 0 . f.o.m. ( data ) = 2 . f.o.m. ( M C f it ) = 2 .
89 and f.o.m. ( M C true ) = 3 .
11 (marked rectangles).– 27 –ata MC True
Figure 18 . Parameter scan for optimizing the figure of merit, equation (5.3), for events in thedouble escape peak ROI for data, MC using the fitting procedure, and MC using the true information(i.e., whether the event contains a positron or not). The result for the optimal configuration ismarked in red. Top row: maximal figure of merit for different combinations of the number of RLdeconvolution iterations and the final threshold for the cleaning cut, keeping a voxel size of 5 mmand blob radius of 21 mm. Bottom row: maximal figure of merit for different combinations of voxelsize and blob radius, for 75 RL iterations and a cleaning cut of 0.008. The SiPM charge cut is 10PE in all cases.
With the above choice of N iter = 75 and (cid:15) cut = 0 .
008 we moved to testing the effect ofvoxel size and blob radius. If voxels are too small, the track may be broken into disconnectedsegments, resulting in identifying internal points as the track ends. On the other hand, ifvoxels are too large, one loses the advantages of track refinement through RL deconvolution:the track is “re-smeared” and its ends are shifted. The blob radius is intimately related tothe choice of voxel size. The blob must be large enough to contain the full energy depositedby the electron as it approaches the Bragg peak (which is shared between several voxels),but not too large, as this can lead to the inclusion of energy outside of the peak. Keeping q cut = 10 PE, N iter = 75 and (cid:15) cut = 0 . l voxel from 3 to 10 mm, and R blob from 12 to 27 mm. As before, for each parameter configuration we scanned the thresholdon blob2 energy and found the maximal f.o.m. The results are shown in the second rowof figure 18, with rectangles marking the final choice of parameters, namely l voxel = 5 mmand R blob = 18 mm. We refer to this choice ( q cut = 10 PE, N iter = 75, (cid:15) cut = 0 . l voxel = 5 mm and R blob = 18 mm) as the “optimal” choice of parameters.As a cross-check and validation of the choice of main RL parameters, namely thecleaning cut and number of iterations, the distance between the true location of the trackends and the reconstructed ones was computed and evaluated (for MC events), generating– 28 – i g n a l All Blob1 Blob2 B a c k g r o und Figure 19 . Blob location error distance for the RL parameter scan for MC events. The first columnshows the overall value while the second and third columns show the blob1 and blob2 distributionrespectively. The result for the optimal configuration is marked in red. Top row: signal eventswithin the double escape peak ROI. Bottom row: background events in the same ROI. error distributions as shown in figure 15. To quantify the analysis, we consider the errorlevel for which the distribution contains 68% of the events. We scanned the main RLparameters over a reasonable range of values, and calculated, for each configuration, the68% error level (to which we refer as the “1 σ location error”).The results of the scan are displayed on figure 19, which shows the 1 σ location errorobtained for different values of the number of iterations and cleaning cut for both blobs(first column), blob1 (second column) and blob2 (third column), where the scan results areshown separately for signal (first row) and background events (second row). Consideringboth blobs together (first column), the optimal choice of N iter and (cid:15) cut (red rectangle) leadsto an overall error close to the minimum, which would have been obtained for N iter = 75and (cid:15) cut = 0 . σ error values of figure19 as a function of the best f.o.m. value obtained for each configuration using the trueinformation (where the best f.o.m. is calculated for the optimal choice of blob2 energythreshold). The data display a clear correlation between high f.o.m. values and smallerrors in end-finding. Again, blob1 location error stays roughly constant while the blob2location error improvement is fully correlated with increasing f.o.m. values. This indicatesthat the overall improvement in the f.o.m. is primarily driven by the error in the locationof blob2. – 29 –ignal Background Figure 20 . Blob location error distance, defined in the text, for all configurations of the RLparameter scan and as a function of the best f.o.m. value achieved with such configuration. Left:location error for signal events within the double escape peak ROI. Right: background events inthe same ROI.
Figure 21 . Comparing different interpolation approaches for NEXT-White data and MC (usingthe fit), for the optimal choice of RL parameters. (A) Signal efficiency vs. background rejection for1.6 MeV double escape peak events. (B) The figure of merit for each dataset as a function of theblob2 cut.
Lastly, a brief study of the impact of the interpolation method was performed. Theevaluation consisted of repeating the topological analysis using the optimal configurationbut with different interpolation approaches, to compare the figure of merit. In additionto the bicubic interpolation, we evaluated a linear interpolation and a nearest-neighborapproach, were the value assigned to each 10 ×
10 mm bin is equal to the number ofphotons detected in the closer SiPM. The results are shown in figure 21. While mostlyall configurations are compatible within error, the maximal performance is achieved withthe bicubic interpolation. This was expected as the parameters were optimized using– 30 –icubic interpolation and it is possible that different results can be achieved with the otherapproaches if optimizing the parameters for those. However, this study is left for futurework. C Examples of successful RL-based classification vs. failure of the clas-sical analysis
In this appendix we discuss three representative examples illustrating the reasons for im-proved classification of signal and background events by the RL-based method. The exam-ples shown in figures 22 and 23 are simulated MC background events in the 1.4-1.8 MeVROI, which are misclassified by the classical analysis as signal, and correctly identified asbackground by the RL process. For each event, the top row shows the three projections ofthe raw sensor response, binned in 10 × × . voxels, with a charge cut of 30 PEfor the classical analysis. The bottom row shows the corresponding projections after RLdeconvolution (charge cut of 10 PE with 75 iterations). Yellow symbols designate the trackends found by the classical analysis and purple symbols - the ones found by the RL process.Squares represent the center of blob2 and circles those of blob1. The deconvolved imagesfurther include the true track overlaid in green. The circles represent the blobs used inboth analysis methods.Event 1 is a photoelectric absorption of a Compton scattered gamma, accompaniedwith a delta electron, where the track is “folded” such that its true start point is close toits main part. Since the classical analysis rebins the SiPM hits in 15 mm voxels, the truestart point (purple square) is merged into the track “body”, and a distant internal point(yellow square) is misidentified as the track extremity. Since the local ionization densitynear this point is high, the energy contained in the classical blob2 centered at it lies abovethe threshold value, and the event is identified as signal. The RL process, on the otherhand, identifies both ends with an error of a few mm, and correctly places the center ofblob2 close to the track starting point.Event 2 is a double Compton scatter of a Tl 2615 keV gamma. The first scatter(starting point of the main track, close to the purple square) gives rise to an energeticCompton electron that creates the main track. The scattered photon interacts ∼ q cut ) in the RL process. The 30 PE cut employed in the classical analysis (after carefuloptimization, as discussed in [33]) eliminates ∼ v e n t : b i nn e dS i P M E v e n t : d ec o n v o l u t i o n E v e n t : b i nn e dS i P M E v e n t : d ec o n v o l u t i o n Figure 22 . Single electron events (background) in the pair production ROI misclassified by theclassical analysis as signal and correctly identified as background by the RL process. For eachevent the top row is the binned SiPM response, and the bottom row – the deconvolved projections.Yellow symbols mark the track ends found by the classical analysis, purple – by the RL process.Squares are blob2 centers, circles – those of blob1. The true track is overlaid on the deconvolvedimages. Misclassification of Event 1 results from a merger between the start point of the track andits main part. Event 2 (double Compton scatter) is misidentified as a single track because of thelimited resolving power of the classical analysis. – 32 – v e n t : b i nn e dS i P M E v e n t : d ec o n v o l u t i o n Figure 23 . An additional background event misidentified as signal by the classical analysis andcorrectly identified as background by the RL process. Symbols have the same meaning as in figure22. Here the mistaken classification results from the high charge cut used in the classical analysiswhich removes ∼ reconstructed in the tail region, with proper placement of blob2 energy at the true startingpoint of the photoelectron. References [1] F. T. Avignone, S. R. Elliott and J. Engel,
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