Analysis methods for the first KATRIN neutrino-mass measurement
M. Aker, K. Altenmüller, A. Beglarian, J. Behrens, A. Berlev, U. Besserer, B. Bieringer, K. Blaum, F. Block, B. Bornschein, L. Bornschein, M. Böttcher, T. Brunst, T. S. Caldwell, L. La Cascio, S. Chilingaryan, W. Choi, D. Díaz Barrero, K. Debowski, M. Deffert, M. Descher, P. J. Doe, O. Dragoun, G. Drexlin, S. Dyba, F. Edzards, K. Eitel, E. Ellinger, R. Engel, S. Enomoto, M. Fedkevych, A. Felden, J. A. Formaggio, F. M. Fränkle, G. B. Franklin, F. Friedel, A. Fulst, K. Gauda, W. Gil, F. Glück, R. Grössle, R. Gumbsheimer, T. Höhn, V. Hannen, N. Hau?mann, K. Helbing, S. Hickford, R. Hiller, D. Hillesheimer, D. Hinz, T. Houdy, A. Huber, A. Jansen, L. Köllenberger, C. Karl, J. Kellerer, L. Kippenbrock, M. Klein, A. Kopmann, M. Korzeczek, A. Kovalík, B. Krasch, H. Krause, T. Lasserre, T. L. Le, O. Lebeda, B. Lehnert, A. Lokhov, J. M. Lopez Poyato, K. Müller, M. Machatschek, E. Malcherek, M. Mark, A. Marsteller, E. L. Martin, C. Melzer, S. Mertens, S. Niemes, P. Oelpmann, A. Osipowicz, D. S. Parno, A. W. P. Poon, F. Priester, M. Röllig, C. Röttele, O. Rest, R. G. H. Robertson, C. Rodenbeck, M. Ryšavý, R. Sack, A. Saenz, A. Schaller, P. Schäfer, L. Schimpf, K. Schlösser, M. Schlösser, L. Schlüter, M. Schrank, B. Schulz, M. ?ef?ík, et al. (29 additional authors not shown)
AAnalysis methods for the first KATRIN neutrino-mass measurement
M. Aker, K. Altenm¨uller,
2, 3
A. Beglarian, J. Behrens,
5, 6
A. Berlev, U. Besserer, B. Bieringer, K. Blaum, F. Block, B. Bornschein, L. Bornschein, M. B¨ottcher, T. Brunst,
2, 10
T. S. Caldwell,
11, 12
L. La Cascio, S. Chilingaryan, W. Choi, D. D´ıaz Barrero, K. Debowski, M. Deffert, M. Descher, P. J. Doe, O. Dragoun, G. Drexlin, S. Dyba, F. Edzards,
2, 10
K. Eitel, E. Ellinger, R. Engel, S. Enomoto, M. Fedkevych, A. Felden, J. A. Formaggio, F. M. Fr¨ankle, G. B. Franklin, F. Friedel, A. Fulst, K. Gauda, W. Gil, F. Gl¨uck, R. Gr¨ossle, R. Gumbsheimer, T. H¨ohn, V. Hannen, N. Haußmann, K. Helbing, S. Hickford, R. Hiller, D. Hillesheimer, D. Hinz, T. Houdy,
2, 10
A. Huber, A. Jansen, L. K¨ollenberger, C. Karl,
2, 10
J. Kellerer, L. Kippenbrock, M. Klein,
6, 5
A. Kopmann, M. Korzeczek, A. Koval´ık, B. Krasch, H. Krause, T. Lasserre, ∗ T. L. Le, O. Lebeda, B. Lehnert, A. Lokhov,
8, 7
J. M. Lopez Poyato, K. M¨uller, M. Machatschek, E. Malcherek, M. Mark, A. Marsteller, E. L. Martin,
11, 12
C. Melzer, S. Mertens,
2, 10
S. Niemes, P. Oelpmann, A. Osipowicz, D. S. Parno, † A. W. P. Poon, F. Priester, M. R¨ollig, C. R¨ottele,
1, 6, 5
O. Rest, R. G. H. Robertson, C. Rodenbeck, M. Ryˇsav´y, R. Sack, A. Saenz, A. Schaller (n´ee Pollithy),
2, 10
P. Sch¨afer, L. Schimpf, K. Schl¨osser, M. Schl¨osser, L. Schl¨uter,
2, 10
M. Schrank, B. Schulz, M. ˇSefˇc´ık, H. Seitz-Moskaliuk, V. Sibille, D. Siegmann,
2, 10
M. Slez´ak,
2, 10
F. Spanier, M. Steidl, M. Sturm, M. Sun, H. H. Telle, T. Th¨ummler, L. A. Thorne, N. Titov, I. Tkachev, N. Trost, D. V´enos, K. Valerius, A. P. Vizcaya Hern´andez, S. W¨ustling, M. Weber, C. Weinheimer, C. Weiss, S. Welte, J. Wendel, J. F. Wilkerson,
11, 12
J. Wolf, W. Xu, Y.-R. Yen, S. Zadoroghny, and G. Zeller (KATRIN Collaboration) Tritium Laboratory Karlsruhe (TLK), Karlsruhe Institute of Technology (KIT),Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany Technische Universit¨at M¨unchen, James-Franck-Str. 1, 85748 Garching, Germany IRFU (DPhP & APC), CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France Institute for Data Processing and Electronics (IPE), Karlsruhe Institute ofTechnology (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany Institute of Experimental Particle Physics (ETP), Karlsruhe Institute ofTechnology (KIT), Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany Institute for Astroparticle Physics (IAP), Karlsruhe Institute of Technology (KIT),Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany Institute for Nuclear Research of Russian Academy of Sciences, 60th October Anniversary Prospect 7a, 117312 Moscow, Russia Institut f¨ur Kernphysik, Westf¨alische Wilhelms-Universit¨at M¨unster, Wilhelm-Klemm-Str. 9, 48149 M¨unster, Germany Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA Departamento de Qu´ımica F´ısica Aplicada, Universidad Autonoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain Department of Physics, Faculty of Mathematics and Natural Sciences,University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany Center for Experimental Nuclear Physics and Astrophysics, andDept. of Physics, University of Washington, Seattle, WA 98195, USA Nuclear Physics Institute of the CAS, v. v. i., CZ-250 68 ˇReˇz, Czech Republic Institute for Technical Physics (ITEP), Karlsruhe Institute of Technology (KIT),Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany Laboratory for Nuclear Science, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Institute for Nuclear and Particle Astrophysics and Nuclear ScienceDivision, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA University of Applied Sciences (HFD) Fulda, Leipziger Str. 123, 36037 Fulda, Germany Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany Project, Process, and Quality Management (PPQ), Karlsruhe Institute of Technology (KIT),Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany (Dated: January 19, 2021)We report on the data set, data handling, and detailed analysis techniques of the first neutrino-mass measurement by the Karlsruhe Tritium Neutrino (KATRIN) experiment, which probes theabsolute neutrino-mass scale via the β -decay kinematics of molecular tritium. The source is highlypure, cryogenic T gas. The β electrons are guided along magnetic field lines toward a high-resolution, integrating spectrometer for energy analysis. A silicon detector counts β electrons above a r X i v : . [ h e p - e x ] J a n the energy threshold of the spectrometer, so that a scan of the thresholds produces a precise measure-ment of the high-energy spectral tail. After detailed theoretical studies, simulations, and commis-sioning measurements, extending from the molecular final-state distribution to inelastic scatteringin the source to subtleties of the electromagnetic fields, our independent, blind analyses allow usto set an upper limit of 1 . CONTENTS
I. Introduction 2II. KATRIN experimental setup 4III. The KNM1 measurement campaign 6A. Tritium source parameters 6B. Column density 6C. Electron starting potential 8D. Analyzing-plane potentials 8E. Electron counting and region of interest 9F. Data pipeline 10G. Acquisition of the integral β decayspectrum 11IV. Tritium-spectrum modeling 11A. Theoretical β -spectrum of moleculartritium 12B. Final-state distribution (FSD) 131. Solutions to the molecular Schr¨odingerequation 132. Energy-resolved FSD 13V. Response function modeling 14A. Response and transmission functions 14B. Inelastic-scattering cross section 15C. Energy-loss function 16VI. Background 17A. Steady-State Background 17B. Background Dependence on Scan-StepDuration 19VII. Assembling spectral data for KNM1 19A. Pixel combination 19B. Scan combination (stacking) 20C. Resulting integral spectrum 20VIII. Systematic uncertainties 21A. Tritium concentration 21B. Column density and expected number ofscatterings 21C. Electron starting potential 21 ∗ [email protected] † [email protected] D. Detector efficiency 22E. Final-state distribution 22F. Response function 23G. Background 23H. Stacking 23I. Neutrino-mass fit range 23IX. Spectral fit 24A. Blinding strategy 24B. Covariance-matrix approach 25C. Monte-Carlo-propagation approach 25D. Fit results 27X. Frequentist bounds on the neutrino mass 27XI. Bayesian bound on the neutrino mass 28XII. Q-value measurement 29XIII. Results and discussion 30XIV. Conclusion 31Acknowledgments 32References 32
I. INTRODUCTION
The absolute mass scale of the neutrino remains akey open question in contemporary physics, with far-reaching implications from cosmology to elementary par-ticle physics. Despite numerous efforts along three com-plementary lines of approach (observational cosmology,the search for neutrinoless double- β decay, and directsearches using the kinematics of weak-interaction pro-cesses such as single β decay or electron capture), onlyupper bounds on the neutrino mass have been found sofar (see, e.g. , [1–3] for reviews on these subjects). Mean-while, neutrino flavor-oscillation experiments ( e.g. , [4, 5])have firmly established the existence of non-zero neutrinomasses, thereby setting a lower mass limit. Depending onthe ordering of the pattern of neutrino-mass eigenstates ν i ( i = 1 , , e.g. , Ref. [6]).With the advent of precision cosmology, correspond-ing bounds on neutrino masses have been dramaticallyimproved, and now form the tightest constraints avail-able. Yet, cosmological bounds on (cid:80) m i (the sum ofthe distinct neutrino-mass eigenvalues m i ) are derivedusing the paradigm of the cosmological standard model(ΛCDM), and the values obtained vary with the selec-tion of data sets included in the analysis. The Planckcollaboration has inferred robust bounds from cosmic-microwave-background power spectra alone: (cid:80) m i < .
26 eV (95 % confidence level, CL), which can be furtherimproved to (cid:80) m i < .
12 eV (95 % CL) by including lens-ing and baryon-acoustic-oscillation data [7]. Meanwhile,laboratory searches for neutrinoless double- β decay aresensitive to the neutrino-mass scale, under the assump-tion that neutrinos are Majorana particles that make thedominant contribution to the decay mechanism. Here,the observable is the coherent sum of weighted neutrinomass values (cid:104) m ββ (cid:105) = | (cid:80) U ei m i | , where U e i denotes theelectron-flavor element coupled to the i th neutrino-massstate in the neutrino mixing matrix. Presently, the mostsensitive limits on (cid:104) m ββ (cid:105) are set by searches in Ge(GERDA, 0.07 - 0.16 eV) [8] and in
Xe (KamLAND-Zen, 0.06 - 0.17 eV) [9]. The ranges of these 90% con-fidence limits arise from uncertainties in nuclear-matrixelements.Direct laboratory-based measurements are an indis-pensable model-independent probe of the neutrino-massscale, resting solely on the determination of kinematic pa-rameters. Two weak processes particularly suitable forthis quest are the electron capture of
Ho [10, 11] andthe β decay of tritium:T → He + + β − + ¯ ν e . (1)The kinematics of these decays provide access to theeffective neutrino-mass square value, an incoherent sumover the weighted squares of the mass values: m ν = (cid:88) i | U e i | m i . (2)Historically, the Mainz and Troitsk experiments usedtritium to set the previous most stringent direct upperlimit at m ν < β -decay spectrumin the vicinity of its kinematic endpoint ( E = 18 .
57 keVfor molecular tritium, T ).The Karlsruhe Tritium Neutrino (KATRIN) experi-ment [14, 15] is further improving this approach to targeta neutrino-mass sensitivity of 0 . m ν observable. To ac-complish this challenging measurement, KATRIN relieson the proven technology of the MAC-E filter (MagneticAdiabatic Collimation with an Electrostatic filter, devel-oped for neutrino-mass measurements by the Mainz andTroitsk groups [16, 17]) and a large β -decay luminosityprovided by a gaseous molecular tritium source (followingpioneering work at the Los Alamos experiment [18]). Af- ter commissioning and characterizing the complex 70 m-long electron beamline, initially with monoenergetic cal-ibration sources [19] and subsequently with first-tritium β electrons [20], the KATRIN collaboration has recentlyreported an improved upper limit on the neutrino massof m ν < . β spectrum was acquired over a“full” energy interval stretching from about 90 eV be-low to about 50 eV above the endpoint E . The ac-tual neutrino-mass analysis was performed in a narrowerinterval, [ E –37 eV, E +49 eV], in which the measure-ment is statistics-dominated. Within this 86 eV analy-sis interval, the data set comprises a total ensemble of2 . × events after data-quality selection cuts. Theensemble was collected over a measurement time of521 . . × β decay electronsbelow E and 0 . × events in a flat background overthe entire analysis interval.We begin this paper with an overview of the exper-imental setup (Sec. II) and the configuration in whichthe KATRIN beamline was operated, including data han-dling and measurement strategy (Sec. III). (For reference,Table I lists abbreviations frequently used in this pa-per.) Two key ingredients of the analysis, the β -spectrummodel and the instrument response function, are pre-sented in Secs. IV and V. Section VI summarizes relevantsources of background and their characteristics. Generalunderlying principles of the analysis, in which data fromthe individual detector pixels and β -spectrum scans arecombined into a single spectrum for fitting, are given inSec. VII.Section VIII presents a detailed assay of individual sys-tematic uncertainties. Section IX documents the strat-egy employed for blind analysis, describes two comple-mentary methods employed to propagate the systematicuncertainties into the neutrino-mass fit, and shows theresulting spectral fit and uncertainty breakdown. Sec-tion X details the construction of the confidence belt andthe derivation of the neutrino-mass upper limit via theFeldman-Cousins [22] and the Lokhov-Tkachov [23] ap-proaches. Our Lokhov-Tkachov result of m ν < . m ν < . TABLE I. Acronyms and abbreviations used in this work.ADC Analog-to-Digital ConverterCL Confidence Levelcps counts per secondDAQ Data-Acquisition Systemd.o.f. degree(s) of freedome-gun electron gunFSD molecular Final-State DistributionFPD Focal-Plane DetectorHV High VoltageKNM1 KATRIN Neutrino Mass run 1LARA LAser RAman spectroscopy systemΛCDM Λ Cold-Dark Matter model(cosmological standard model)MAC-E filter Magnetic Adiabatic Collimation withElectrostatic filterMC Monte Carloppm part per millionp-value Probability of achieving a result as extremeas the one found, through statisticalfluctuationQ-value Kinetic energy released in tritium β decay(for zero neutrino mass)ROI Region of InterestTOF Time of FlightWGTS Windowless Gaseous Tritium Source In Sec. XII, as a consistency check of KATRIN’s ab-solute energy scale, we show that the effective endpointvalue E obtained from the fit to the β spectrum agreeswith independent measurements of the Q-value throughthe He-T mass difference.We conclude by summarizing our findings (Sec. XIII)and discussing them in the wider context of contempo-rary neutrino-mass probes (Sec. XIV).
II. KATRIN EXPERIMENTAL SETUP
Figure 1 gives an overview of the KATRIN apparatus.Briefly, in order to ensure sufficient statistics, a bright tri-tium source produces some 2 . × decays each sec-ond in the KNM1 configuration. In order to perform afine-grained energy analysis near the tritium endpoint,the energies of the resulting β electrons are analyzed bya pair of MAC-E-filter spectrometers [16, 17]. These ba-sic functions require the support of extensive systems forhandling the tritium gas, maintaining vacuum conditions,ensuring adiabatic electron transport, mitigating or elim-inating backgrounds, detecting β electrons, and calibrat-ing and monitoring the apparatus as a whole. The re-sulting 70 m beamline is described in detail in Ref. [24];here, we offer a brief summary.T gas from a temperature- and pressure-controlledbuffer vessel at 313 K is cooled to 30 K and continuouslyinjected via a capillary into the center of the source sys-tem. The resulting Windowless Gaseous Tritium Source(WGTS) freely streams to both ends of the system, where it is continuously pumped away with turbomolecularpumps. This results in a stable pressure distributioninside the source beam tube [25]. Once the T gas ispumped away, it flows over a PdAg membrane filter thatis permeable only to hydrogen isotopes. A constant frac-tion of the circulating gas is also removed at this stagefor later purification, and is replaced with highly pureT directly after the filter. The purified gas is fed backto the temperature- and pressure-controlled buffer vessel,forming a closed loop. The loop system is integrated withthe infrastructure of the Tritium Laboratory Karlsruhe,which provides tritium purification of exhaust gas, tri-tium storage, and fresh tritium supply for KATRIN [26–28].Within the 10 m-long, 90 mm-diameter source beamtube [29], tritium decays produce β electrons that areguided along magnetic field lines [30] through the restof the experimental beamline. At the upstream end, theWGTS terminates in a gold-plated rear wall, which canbe held at a fixed potential and/or illuminated with ul-traviolet light to liberate photoelectrons. At the down-stream end, the windowless nature of the source is es-sential to avoid catastrophic energy loss, but necessitatesother means for the confinement of tritium. The β elec-trons are first guided around magnetic chicanes throughtwo pumping stages, namely a differential pumping sys-tem and a cryogenic pumping system, which collectivelyreduce the partial pressure of tritium by more than 14 or-ders of magnitude [31]. Specially designed electrodeswithin the differential stage [32] prevent the transmissionof tritium ions. β electrons must then pass through a pair of MAC-E-filter spectrometers, operated in tandem. Each MAC-Efilter is characterized by strong magnetic fields at theentrance and exit, with a region of weak magnetic fieldin the center. Since the magnetic moment is conservedin the adiabatic transport of the electrons through thebeamline, the electron momenta rotate to become ap-proximately parallel to the magnetic field lines, produc-ing a broad, roughly collimated beam. A longitudinal re-tarding potential therefore analyzes the total kinetic en-ergy of the electrons at the central “analyzing plane,” atwhich the magnetic field is the weakest. Electrons belowthe resulting energy threshold are reflected upstream, to-ward the source; electrons above the energy threshold aretransmitted downstream, toward the spectrometer exit.The transmission function of the spectrometers was ex-tensively calibrated prior to the measurement (Sec. V).The first MAC-E filter in the tandem pair, the pre-spectrometer [33], has a fixed energy threshold at 10 keVand removes the bulk of the low-energy electrons. Im-mediately downstream, the main spectrometer is thehigh-resolution, adjustable-threshold filter that analyzesthe integral β spectrum. Each data-taking “scan”(Sec. III G) consists of a sequence of main-spectrometerretarding-potential settings, with a new threshold of in-tegration at each setting. The electropolished interiorstainless-steel surface of the main spectrometer is lined Main spectrometer
Rear system
Source system
Differential pumping system
Detector system
Monitor spectrometer
Cryogenic pumpingsystem
P r e - s p e c t r o me t e r
FIG. 1. Overview of the 70 m KATRIN beamline. Moving downstream, from left to right, the major components are: the rearsystem, the source system, the differential pumping system, the cryogenic pumping system, the pre-spectrometer, the mainspectrometer, and the detector system. The monitor spectrometer monitors the retarding potential of the main spectrometer. with two layers of inner, wire electrodes, providing fineshaping of the electric fields and, when operated at anegative potential offset from the main-spectrometer ves-sel, electrostatic rejection of low-energy secondary elec-trons from the main-spectrometer surface [34]. The ves-sel potential is supplied by a commercial system, withadditional regulation and post-regulation designed andbuilt by the collaboration to suppress 50 Hz mains noiseand other sources of interference [35]. Air-cooled mag-netic coils, mounted on a framework surrounding themain spectrometer, compensate for the Earth’s magneticfield, fringe fields of the solenoids, and residual magneti-zation [36]. The ultra-high vacuum in the spectrometeris maintained by non-evaporable getter strips and tur-bomolecular pumps [37]. Liquid-nitrogen-cooled copperbaffles are positioned across the pump ports to suppressbackground electrons due to radon decay in the main vol-ume [38, 39]. To mitigate backgrounds from the Penningtrap between the two MAC-E filters, a conductive elec-tron catcher is inserted into the inter-spectrometer regionat each change in the set voltage of the main spectrom-eter [40]. This device removes trapped electrons thatwould produce secondary ions and electrons.Electrons that pass through the main spectrometer un-dergo additional acceleration via the post-accelerationelectrode, improving rejection of non-spectrometer back-grounds. When they reach the detector system, theyare counted in the focal-plane detector (FPD) [41], amonolithic silicon p-i-n diode segmented into 148 equal-area pixels. The FPD and its readout electronics areelevated to the post-acceleration potential, and pream-plified signals are transmitted to the data-acquisition(DAQ) system via optical fiber. Each FPD pulse isdigitized in a 12-bit analog-to-digital converter (ADC),and its amplitude and timing are reconstructed on-line by the sequential application of two trapezoidal fil-ters [41, 42]. These values are then recorded usingthe Object-oriented Real-time Control and Acquisition (ORCA) framework [43], which can also communicatedirectly with the main-spectrometer high-voltage sys-tem using a web-based database tool [44]. Pulse ampli-tudes are translated into energies in near-time processing(Sec. III F), based on the results of regular calibrationruns with an
Am photon source.Multiple calibration and monitoring systems provideessential information during both neutrino-mass scansand dedicated runs [45]. In the tritium loops feeding thesource, a laser-Raman spectroscopy system (LARA) [46–48] monitors the relative concentrations of hydrogen iso-topologs, particularly T , DT, and HT, within the sourcegas. In the rear system upstream of the source, an elec-tron gun (e-gun), following the design of a similar e-gunused for testing the main spectrometer [49], serves as anangle- and energy-selective calibration source. This e-gun delivers electrons through an aperture in the rearwall at the upstream end of the source. Observed inthe FPD, these electrons test the response function ofthe experiment as a whole. Two radioactive, in-vacuumcalibration sources are also available: gaseous m Kr thatcan be circulated within the source when its temperatureis elevated to about 100 K [50], and a condensed m Krsource that can be inserted into the cryogenic pumpingsystem [51].Upstream of the rear wall, the β -induced x-ray spec-troscopy system continuously monitors the source activ-ity: silicon drift detectors view x-rays produced by β electrons scattering in the rear wall [52]. Further down-stream, within the cryogenic pumping system, a forwardbeam monitor provides complementary activity monitor-ing [53]. This monitor includes two silicon p-i-n diodesfor electron rate and spectrum measurements, a Hall sen-sor, and a temperature gauge. A vacuum manipulatorallows these sensors to be positioned radially within thebeam; normally, the forward beam monitor is positionedat the outer edge of the β electron flux.The main-spectrometer retarding potential, which de-fines the energy analysis, is continuously monitored bothby a voltage divider with demonstrated part-per-million(ppm) precision [54–57] and by the refurbished MAC-Efilter from the historical Mainz experiment [12]. Nowrelocated to KATRIN, this monitor spectrometer refer-ences the main-spectrometer retarding potential to anatomic standard via synchronous scans of a m Kr con-version line [58].Prior to the KNM1 neutrino-mass run, the full KA-TRIN beamline was commissioned with photoelectrons,ions, and m Kr conversion electrons in 2016–2017 [19],and with small amounts of tritium in D carrier gas in2018 [20]. Subsequently, in another campaign with D ,the electron gun was commissioned and gas propertiesof the source were investigated [59]. KNM1 marked thefirst time that the inner surfaces of the injection capil-lary and source system were exposed to large amounts oftritium. Radiochemical reactions between T and thesemetal surfaces produced both CO and tritiated methane,which condensed on the cold metal surface of the cap-illary and partially obstructed tritium flow over time.To improve stability during this burn-in period, KA-TRIN operated at a reduced column density of ρd exp =1 . × molecules / cm . III. THE KNM1 MEASUREMENT CAMPAIGN
In this section we describe the operating conditionsof the KATRIN experiment during its first high-puritytritium campaign (KNM1), which took place from 10 th April to 13 th May 2019. In particular, we characterizethe system performance in terms of the source-gas iso-topic purity (Sec. III A) and column density (Sec. III B),as well as the reproducibility, homogeneity, and sta-bility of the electron starting potential in the source(Sec. III C) and the retarding potential in the analyzingplane (Sec. III D). We also discuss the detection of β elec-trons and the definition of a region of interest (Sec. III E)as well as the processing and analysis pipeline for the data(Sec. III F).The requirements for system stability arise from themethod adopted to measure the tritium β spectrum byrepeatedly scanning the retarding potential in alternatingup and down sweeps (Sec. III G), and from the fact thatKNM1 data from all pixels and all scans are combinedinto a single spectrum for fitting. In the final analysis,then, experimental parameters are essentially averagedover both space (across the detector) and time (acrosslike scan steps throughout the KNM1 data-taking pe-riod). Later on, Sec. VII explores the justification forthis analysis method in the statistics-dominated KNM1data set.For the KNM1 campaign, the sequence of scan steps,each consisting of a retarding-potential set point dis-tributed in the interval [ E −
91 eV, E +49 eV], resultedin a typical scan duration of 2 . A. Tritium source parameters
The average source activity during KNM1 neutrino-mass data-taking was about 2 . × Bq, maintainedby a column density of 1 . × molecules / cm . Thiswas achieved by a cumulative tritium throughput of4 . / d.The gas injected into the source consists mainly ofmolecular T . Due to initial impurities and exchangereactions with the stainless-steel piping and vessel, theother hydrogen isotopologs (H , HD, HT, D , and DT)are also present in minor fractions. A PdAg membrane(permeator) in the tritium loop [60] continuously fil-ters the circulated tritium gas to prevent the recircula-tion of built-up impurities. The relative fractions c x ofthe six hydrogen isotopologs are continuously monitoredby LARA, downstream of the permeator. The relativemolecular isotopolog fractions c x and the atomic tritiumpurity ε T are defined as: c x = N x (cid:80) i N i , and (3) ε T = N T + ( N HT + N DT ) (cid:80) i N i (4)where N x is the number of molecules of isotopolog x inthe source, and the sums are over all six isotopologs.The tritium purity is monitored with better than 10 − statistical precision [48].The time evolution of the relative fractions of thethree tritiated isotopologs injected into the source dur-ing KNM1 is shown in Fig. 2. On average, the concen-trations of the tritiated species throughout the campaignwere c T = 0 . c HT = 0 . c DT = 0 . ε T = 0 . , HD, and D )are only present in trace amounts, as they are stronglysuppressed by shifts of the chemical equilibrium in thepresence of high-surplus T . B. Column density
The column density ρd determines the number of tri-tium atoms N T in the source N T = 2 ε T · ρd · A, (5)where A is the cross-sectional area of the WGTS, and thefactor of 2 is necessary because ρd is defined in terms of T ( % ) (T ) = 95.25 %
11 April 18 April 25 April 02 May 09 May
Date H T ( % ) (HT) = 3.54 % Entries D T ( % ) (DT) = 1.08 %Total uncertainty illustration FIG. 2. Evolution of the relative fractions of the three triti-ated isotopologs injected into the source during KNM1. Thedotted lines show the mean values over KNM1, with the rederror bars illustrating the total uncertainties (statistical andsystematic). The steps and kinks in the trends indicate timesat which a new tritium gas batch was fed into the circulationloop. As the tritium is re-processed in several steps at theTritium Laboratory Karlsruhe [62], its purity varies slightlybetween batches. the number of T molecules. The column density furtherdefines the s -fold scattering probabilities P s of electrons,traveling parallel to magnetic field lines through the en-tire tritium source, with the gas molecules: P s = ( ρdσ ) s s ! e − ρdσ . (6)The product ρdσ , where σ is the cross section for inelasticscattering of electrons from molecular tritium (Sec. V B),gives the expected number of scatterings. It must beknown with high accuracy for the analysis [25].The precise absolute value of ρdσ is obtained from mea-surements with the narrow-angle, quasi-monoenergetic e-gun located in the rear system. This e-gun produces ahigh-intensity beam of electrons via the photoelectric ef-fect according to the principle described in Ref. [49]. Ontheir path towards the detector, the electrons traversethe source, where they can undergo inelastic scatteringand in the process lose energy. Only those electrons withsufficient remaining energy to surpass the spectrometerpotential are counted in the detector. By measuring theelectron rate at different retarding potentials and fittinga model response function (Sec. V) to these data, we maymake a precise determination of ρdσ .E-gun electrons differ from β -electrons with regard totheir starting positions and their energy and angular dis-tributions. For this reason a modified response func-tion, including a precise description of the e-gun beam
11 April 18 April 25 April 02 May 09 May
Date d ( m o l e c u l e s / c m ) ×10 ( d ) = 1.110 × 10 cm Total uncertainty illustration
Entries
FIG. 3. Evolution of the column density during KNM1; theuncertainty is dominated by systematics arising from the rela-tionship of e-gun data to the measured throughput, and fromfluctuations in the latter quantity. The visible decrease of thecolumn density over time is caused by conductance changesof the tritium injection capillary. By increasing the tritiuminjection pressure several times, the column density was sta-bilized. characteristics, is used in the column-density determina-tion. The e-gun electron rate is measured at retardingpotentials where the impact of the column density is thestrongest. The mean energy of the e-gun electrons is setto 18 .
78 keV, allowing a clean separation from β elec-trons that could bias the column-density determination.During KNM1, ρdσ was determined with the e-gun on aweekly basis, achieving relative uncertainties of less than0 . resulted in the production of gas species whichcondensed on the surface of the injection capillary. Thisobstruction caused the tritium injection flow and columndensity to drift over time at constant tritium injectionpressure. By lowering the column density to a value afactor of approximately 5 smaller than the nominal col-umn density ρd nom , and by increasing the tritium injec-tion pressure several times during KNM1, these driftswere kept lower than 3 %. To ensure precise monitor-ing of the column density during the whole measurementperiod, the e-gun measurements were combined with con-tinuous ρd fluctuation data from a mass-flow meter with200 sccm full-scale range [63], applied to the tritium in-jection flow. The reproducibility of the flow meter dur-ing KNM1 is conservatively estimated to be 1 . µ bar · l / s.Based on simulations that show a linear relation between ρdσ and the tritium injection flow for a narrow through-put range [64], a linear calibration function is suitable torelate the measured throughput to ρdσ .With this strategy, we determine the column densitywith high precision for all tritium data-taking. Thetime evolution and distribution of the column-densityvalues are shown in Fig. 3; the average value of ρdσ is 0 .
404 at the molecular tritium endpoint. Usingthe cross-section value from Eq. 17 further below, thisvalue translates to an average column density of ρd =1 . × molecules / cm . C. Electron starting potential
The starting potential of the β electrons is providedby a cold and strongly magnetized plasma in the WGTS.The magnitude of the potential depends on the bound-ary conditions at the rear wall and the grounded beamtube. By optimization of the rear-wall set voltage, a ho-mogeneous, stable plasma potential can be created. Thisis important because both spatial inhomogeneities andtemporal fluctuations of the plasma potential lead to asystematic shift of the measured neutrino mass. For aGaussian variance σ of a continuous variable like thestarting potential of the β electrons, the resulting shiftis given at leading order by [65]:∆ m ν = − σ . (7)The source plasma is generated by the weakly self-ionizing tritium gas. According to simulation, each β electron creates on average 36 secondary electrons, andthus 36 positive ions, through scattering interactions.Throughout the central part of the WGTS, the ions havea mean free path of less than 0 . β electrons ranges from meV to keV.The electric potential inside the plasma depends onthe surface potentials at its boundaries. These are deter-mined in turn by their intrinsic work functions φ , whichare expected to differ by several 100 mV [66], and bythe applied bias voltages. As the beam tube is grounded( U bt = 0 V), only the rear-wall bias voltage U RW remainsto compensate the work-function differences. At an opti-mal U RW , the radial and longitudinal inhomogeneities ofthe plasma potential both vanish, as expected from sim-ulations with the assumption of negligible work-functioninhomogeneities [67].The optimal rear-wall bias voltage was determined bymeasuring the β -rate at various U RW settings. Compar-ing these rates to reference spectra, we extracted the de-pendence of the spectral endpoint E on the FPD ringnumber – which correlates to radius in the source.For U RW = −
150 mV, a flat radial E distribution wasfound. Also, the measurement of the plasma-induced cur-rent on the rear wall showed no drifts and less noise thanat other bias voltages. U RW was therefore set to −
150 mVfor the measurement campaign.The systematic effect of remaining spatial inhomo-geneities and fluctuations of the plasma potential canbe constrained by studying the line widths and posi-tions of quasi-monoenergetic conversion electrons fromgaseous
Kr co-circulating in the T gas [68]. TheL -32 line at 30 472 . E . Second, the37 . ≈ Kr in the absence of tritium gas [19],the KATRIN experiment measured an L -32 line posi-tion of E L -32 = 30472 . ± . stat ± . sys eV anda Lorentzian line-width of Γ L -32 = 1 . ± . stat ± . sys eV [70]. This effective line position includes ashift arising from the absolute work-function differencebetween the source and the main spectrometer.After the KNM1 neutrino-mass campaign ended,plasma studies were performed for two days withco-circulating Kr and T . It should be notedthat the column density during neutrino-mass mea-surements was only 22 % of the nominal value of5 . × molecules / cm , while during the plasma studyit was about 30 % of the nominal value.The krypton admixture did not affect general plasmaproperties, such as charged-particle density or electricpotentials, because the partial pressure and activity ( ≈ ≈
33 GBq). However, the plasmawas affected by the beam-tube temperature of 100 K,elevated from the nominal 30 K. This higher tempera-ture was necessary to prevent the krypton from freez-ing, but also increased the temperature of the dominantlow-energy part of the electron energy distribution [71].The plasma temperature is known to strongly influencethe rate of electron-ion recombination at the meV scale.As the recombination rate is much stronger at 30 K, weexpect plasma effects at elevated source temperature tobe more prominent. We thus use results obtained dur-ing the krypton measurement at 100 K to set an upperlimit of the scale of possible plasma effects. The β -decayelectrons and non-thermalized electrons make only minorcontributions to the number density, but their dominantrole in the energy density of charged particles requires adetailed investigation.The intrinsic Lorentzian line width was measured withgaseous Kr in the absence of tritium, with the ex-perimental conditions as similar as possible to the L -32measurements with co-circulating T / Kr (describedabove). By comparing these two measurements and as-suming an energy-independent background, we find thatthe presence of T results in a Gaussian line broadeningof <
80 mV for rear-wall settings in the range −
350 mV
350 mV. The collaboration is currently investi-gating the impact of a possible radial-dependent back-ground, which could arise due to detector effects.The impact of these findings on the neutrino-massmeasurement is discussed in Sec. VIII C.
D. Analyzing-plane potentials
The threshold energy for electrons to pass through theMAC-E filter is determined by the value of the retard-ing potential U at the analyzing plane. Any unknowninstabilities in the retarding potential directly affect theenergy scale of the tritium spectrum and can introducesystematic effects on m ν . As shown earlier in Eq. 7,to first order, significant continuous inhomogeneities ef-fectively broaden the spectrum and therefore lead to ashift in the fitted m ν . For the target sensitivity of KA-TRIN, the energy scale must be stable to within 60 meVor 3 ppm on a baseline retarding potential of − . σ =34(1) mV. This limitation of thereproducibility is directly related to the digital-to-analogconverter inside the post-regulation setup; for measure-ment phases after KNM1, finer-grained regulation is inplace.The retarding potential is continuously monitored dur-ing the measurements. Therefore, at each scan step, thetime evolution of the retarding potential is known withppm precision. Neglecting this in the analysis introducesan additional broadening of the energy scale, leading to aneutrino-mass shift of ∆ m ν = − × − eV . This shiftis less than half the allotment for the high-voltage-relatedsystematic uncertainty in the KATRIN uncertainty bud-get for full five-year statistics [15], and can be neglectedin the KNM1 analysis. E. Electron counting and region of interest
The FPD records a low-resolution, differential spec-trum of electrons that have passed the high-resolutionenergy threshold set by the main spectrometer. Measur-ing the integrated tritium β spectrum for KNM1, andthereby extracting the neutrino mass, requires an accu-rate count of electrons that arrive at the FPD within anenergy region of interest (ROI) during each scan step.The ROI cut allows rejection of backgrounds and noiseevents generated near or in the FPD.When electrons strike the FPD, its pixels are triggeredindividually, with thresholds set just above the noise FIG. 4. Achieved stability of the retarding potential as afunction of the scan-step duration. Light blue points showthe standard deviation of the measured retarding potentialfor each scan step during KNM1. Dark blue points show themean of this standard deviation for scan steps with the samelength; their error bars show the standard deviation of thisvalue. Outliers arise from a brief period in which the changein HV setpoint was incorrectly synchronized with the DAQ,a problem affecting 0.3% of scan steps. In the inset figure,scans with synchronization errors have been removed to showthe performance of the HV subsystem. The black line givesthe prediction of the statistical model described in the text. floor at around 5 keV. As described in detail in earlierwork [41], the energy and timing for each pulse are recon-structed online using a double trapezoidal filter and thenrecorded; FPD waveforms are not saved during normaloperations. The shaping length of the trapezoidal-filterpair is set to 1 . µ s, optimizing the energy resolution ataround 1 . β scans, rates are too low for significant pileup,but severe pileup during high-rate e-gun measurementscan result in deadtime when multiple coincident eventsdrive the baseline out of the ADC dynamic range. Thiseffect is mitigated by individually adjusting the gain ofeach channel to approximately 5 ADC counts per keV,preserving good energy resolution while defining a dy-namic range (up to 400 keV) sufficient to accommodatepileup. These settings were implemented in the DAQfirmware prior to the KNM1 measurement. Simulationsof the readout chain show that the fraction of time dur-ing which the baseline is shifted out of the ADC inputrange is less than 0 .
05 % for 50 kcps of 28 . β -electron path along the mag-netic flux tube.Electrons that transit the spectrometer (Sec. V) receivean additional 10 keV of kinetic energy from the post-acceleration electrode, and 120 eV from the bias voltageapplied to the FPD. For a retarding potential around18 . FIG. 5. FPD pixel selection for KNM1. All 117 selectedpixels, colored in solid green, are used for the analysis. Thetwo pixels filled with horizontal blue lines are excluded due toshadowing by the forward beam monitor. The six pixels filledwith vertical orange lines are excluded due to intrinsic noisybehaviour. All pixels filled with gray circles are excluded sincethey are partially shadowed by beamline components.
Energy (keV) - - -
10 110 R a t e ( / s / k e V ) qU = 18373 eVqU = 18553 eVqU = 18593 eVROI boundaries FIG. 6. FPD energy spectra recorded during KNM1 at threedifferent scan steps. At qU = E + 20 eV (blue), the retardingpotential eliminates all β electrons. The backgrounds visiblein the spectrum consist of electrons from the main spectrom-eter (energies up to the peak at 28 keV), electrons from thedetector system (distributed across the energy range), anddetector electronics noise (below 7 keV). At qU = E −
20 eV(orange), the spectrum additionally incorporates signal eventsfrom tritium β - decays. Deep in the β spectrum, at qU = E −
200 eV (red), rates are high and pile-up events become visibleabove 32 keV. Events within the ROI, demarcated by the twovertical dotted lines, are included in high-level analysis. electrons and β electrons share this characteristic energyspectrum in the FPD, since the primary background dur-ing KNM1 arises from low-energy electrons that are cre-ated inside the main spectrometer and then acceleratedby the retarding potential (Sec. VI). The FPD energyscale is calibrated with a Am gamma source every two weeks.Our ROI is defined as [14 keV, 32 keV], as measuredby the FPD (Fig. 6). The upper bound of the ROI isdetermined simply from the peak position and the peakwidth; the lower bound is determined for stability androbustness. In contrast to earlier studies that consid-ered backgrounds originating near the detector [41], thechoice of a low-energy KNM1 ROI lower bound does notreduce the signal-to-background ratio, since an energycut cannot differentiate between β electrons and main-spectrometer background. A cut far away from the peak,where the spectrum shape derivative is small, improvesstability against fluctuations of energy scale and resolu-tion. Consequently, corrections for peak-position depen-dence on retarding potential are negligible.The specific lower bound of the ROI, 14 keV, was cho-sen so as to cancel two effects that arise from chargesharing, in which energy from a single incident electronis divided between two neighboring pixels. If a pixel losesmore than half the event charge, its loss from the ROIdecreases the effective rate; if a pixel receives more thanhalf the event charge, its inclusion in the ROI increasesthe effective rate. With the FPD threshold set at half thepeak energy, these two effects exactly compensate eachother. F. Data pipeline
Following each pixel trigger (Sec. III E), the DAQrecords the trigger timestamp from a 20 MHz clock andthe energy information as raw ADC counts integratedover the shaping time of the trapezoidal filter. A scanis divided into scan steps. Each scan step is definedby its HV set point, and its duration is determined ac-cording to the measurement-time distribution of the scan(Sec. III G). Prior to acquisition start at each scan step,handshakes between the DAQ and the HV control systemensure that the HV read-back value has reached the set-point value within a defined accuracy of 50 mV, as mea-sured by a four-point moving average over the last 8 s.The inter-spectrometer electron catcher is inserted andremoved during this change of scan steps, so that it doesnot obstruct the beamline during data-taking. A seriesof pulse-per-second (PPS) pulses from a precision clocksynchronized to the Global Positioning System (GPS) de-fines both the start and stop times of scan steps, provid-ing boundary time accuracy better than 1 ns. The 50 nsdigitization timestamps are also phase-locked to 10 MHzpulses from the same precision clock. The readout sys-tem is capable of handling a pixel rate of 100 kcps and atotal rate of 3 Mcps. Therefore, no deadtime is expectedfor the actual tritium scan, which has a maximum countrate of 7 kcps. A typical two-hour scan produces roughly120 MB of data.Immediately after completion of a scan, data files areprocessed automatically. This processing includes thetransfer to storage computers, time-wise event sorting,1conversion to offline data formats, and indexing into arun database, followed by automated user-side analysisincluding the reduction of data in user-specified data files.Except for the handshakes between the DAQ and HVsystems, slow-control channels are independent from thetritium scans. Each slow-control sensor has a definedrecording interval, typically between 2 and 10 s. This isa heterogeneous system for which timestamps are takenfrom computer timestamps synchronized to the NetworkTime Protocol (NTP). In the offline analysis, special careis taken for synchronization among different slow-controlchannels, as well as between the DAQ and slow controls.An intermediate data layer, consisting of user-sideshared data storage with version management, splits thedata analysis chain. The first half of the chain coversanalysis at the event and time-series levels, and the sec-ond half provides higher-level analysis including modelfitting. For each scan, results of the first-level analy-sis are summarized in digest files that contain analyzedFPD counts with efficiency corrections, individual scansteps, calibrated slow-control values (including LARAisotopolog concentration and column density, and ana-lyzed rates extracted from β -induced x-ray spectroscopyand the forward beam monitor), and data-quality flags.Some experimental parameters, such as beamline align-ment information and magnetic- and electric-field valuesdetermined by measurements and simulations, are sharedacross all scans in a given measurement period; each suchperiod is summarized in a digest file containing the valuesof these parameters.During data-taking, acquisition occasionally began be-fore the HV readback values achieved stability due to mi-nor synchronization errors. The first two seconds of everyscan step were removed from the data to address theseissues. Count-rate, livetime, efficiency, and stability cal-culations are performed after these data-quality cuts. G. Acquisition of the integral β decay spectrum KATRIN measures the integral tritium β decay spec-trum by sequentially applying different retarding energies qU , or equivalently HV settings, to the main spectrom-eter and counting the rate of transmitted β electrons, R ( qU ), with the FPD. Our choice of the scan steps – thatis, the HV set points and the measurement time at eachset point – maximizes the sensitivity for m ν by focus-ing on a narrow region where the impact of the neutrinomass on the spectrum is most pronounced. The locationof this region depends on the experimental conditions; inthe KNM1 campaign, it lies at E −
14 eV [72].Figure 7 shows the measurement-time distributionused during this campaign, developed using a nominalvalue of E = 18 574 eV. The spectrum is scanned re-peatedly over the range [ E −
91 eV, E +49 eV] by se-quentially applying the non-equidistant HV values (eachconstituting one scan step) to the main spectrometer. Acomplete set of measurements at all 39 scan steps is de- Retarding energy (eV) M ea s u r e m e n t ti m e ( hou r s ) Background 37 eV analysis interval - E 91 eV full interval - EEndpoint
FIG. 7. Cumulative measurement-time distribution forKNM1. The 27 scan steps of the [E −
37 eV, E +49 eV] analy-sis interval are shown in blue below E , and in black above E . The most sensitive region to the signal of the neutrinomass is approximately 14 eV below the endpoint, where mostof the measurement time is spent. fined as a scan . Each scan over this energy range takesapproximately 2 . E −
37 eV, E +49 eV],consisting of 27 scan steps. A brief, additional scan stepat E −
201 eV is used for rate-stability monitoring.For each tritium scan, we apply quality cuts to rel-evant slow-control parameters to select a data set withstable run conditions. As Sec. VII describes in detail,data from all active detector pixels are summed, effec-tively converting the detector wafer into a single, uniformpixel for analysis. Furthermore, all 274 scans are com-bined by summing counts from like scan steps, forminga single spectrum for fitting.The 27 scan steps within the analysis interval covera total measurement time of 521 . . × events. Table II summarizes key operationalparameters and figures for events and scans, coveringboth the full interval and the analysis interval. Theevolution of the integrated β -decay luminosity over thecourse of KNM1 is displayed in Fig. 8. IV. TRITIUM-SPECTRUM MODELING
The KNM1 analysis relies on a model of the measuredspectrum, which convolves the theoretical β spectrum(outlined in this section) with the experimental responsefunction (details in Sec. V). We first describe the generaltheory of β -decay in Sec. IV A, along with some straight-forward corrections. To account for the physics of KA-TRIN’s molecular source (T with some HT and DT), wethen address the molecular final-state distribution (FSD)in detail in Sec. IV B.2
14 Apr 21 Apr 28 Apr 05 May 12 May
Date N u m b e r o f β - d eca y s ( ) β -decays providedby the source β -decays providedduring mass scansreduced byscan efficiency FIG. 8. Integrated luminosity over KNM1. Compared tothe accumulated number of β decays delivered by the tritiumsource (blue line), the collection efficiency during neutrino-mass scans is slightly reduced by calibration runs and by thetime it takes for the retarding potential to settle between scansteps (orange line). Dates are in 2019.TABLE II. Summary of data acquisition for the KNM1 mea-surement campaign. (See text for details.) Scan overview
Number of β spectrum scans 274Net (total) time per scan 2 h (2 . E −
91 eVto E +49 eVEnergy range (analysis interval) from E −
37 eVto E +49 eVNumber of scan steps 39in signal region (full interval) 34in signal region (analysis interval) 22in background-only region (both intervals) 5Source activity 2 . × BqEnergy resolution at 18 . . Event ensemble
Accumulated measurement timein full interval (39 scan steps) 541 . . . × in analysis interval (27 scan steps) 2 . × signal region 1 . × background-only region 0 . × A. Theoretical β -spectrum of molecular tritium In KATRIN’s molecular source, the β decay parentin Eq. 1 becomes T , with a molecular decay prod-uct HeT + . To model the resulting differential β spec-trum, we begin with a point-like Fermi interaction, whichcauses the weak decay, and then apply the sudden ap-proximation, in which the Coulomb interaction of the β electron with the remaining molecular system HeT + is neglected. The validity of this approximation was demonstrated in Refs. [73, 74].Choosing the center-of-mass coordinate frame to alignwith the momentum of the neutrino and integrating overthe experimentally unresolved neutrino and electron di-rections and neutrino energy, the decay rate into thenuclear and molecular configuration f of the daughter HeT + at a given electron kinetic energy E reads [73]R f ( E ) = | T f ( E ) | π ( E + m e ) (cid:113) ( E + m e ) − m e · ε f ( E ) (cid:113) ε f ( E ) − m ν Θ( ε f ( E ) − m ν ) , (8)in natural units with c = (cid:126) = 1. m e and m ν are theelectron and neutrino masses, respectively; ε f ( E ) has theform of the neutrino energy after energy conservation hasbeen enforced by the Heaviside function Θ( ε f ( E ) − m ν ). | T f | is the transition matrix element to the nuclear andmolecular state f . Since the derivation of the decay rateis performed in the center-of-mass frame, which almostperfectly coincides with that centered on the decayingmolecule, there is no need to integrate over the recoilmomentum of the molecule; the recoil kinetic energy isnaturally added as a constant energy loss. | T f | may be factorized in the sudden approximationas | T f | = (cid:12)(cid:12) T weak f (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) T lep f (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) T mol f (cid:12)(cid:12) (9)where (cid:12)(cid:12)(cid:12) T weak f (cid:12)(cid:12)(cid:12) is independent of the electron energy forthe superallowed tritium β decay. Similarly, the leptonicpart (cid:12)(cid:12)(cid:12) T lep f (cid:12)(cid:12)(cid:12) is independent of the electron energy in thesudden approximation. As is customary, however, theFermi function F ( E, Z (cid:48) = 2) (as given in Ref. [75]) isincluded in this factor. This allows a partial incorpora-tion of the influence of the Coulomb interaction duringthe decay by accounting for the charge of an isolated He daughter nucleus, leading to an effectively Coulomb-distorted sudden approximation. Meanwhile, (cid:12)(cid:12)(cid:12) T mol f (cid:12)(cid:12)(cid:12) isequal to the probability ζ f that HeT + populates theunresolved set of molecular electronic, vibrational, androtational states with energy V f . Since the motion of thecenter of mass of HeT + must balance the neutrino andelectron momenta, (cid:12)(cid:12)(cid:12) T mol f (cid:12)(cid:12)(cid:12) theoretically depends on theelectron energy after the integrations are performed. TheKNM1 analysis interval is narrow enough to neglect thisdependence.After evaluating | T f | according to Eq. 9, summingover the possible final nuclear states, and explicitly sum-ming over the included range of molecular states, we ob-3tain R β ( E ) = G cos Θ C π | M nucl | F ( E, Z (cid:48) = 2) · ( E + m e ) (cid:113) ( E + m e ) − m e · (cid:88) f ∈ mol ζ f ε f ( E ) (cid:113) ε f ( E ) − m ν Θ( ε f ( E ) − m ν ) . (10)The prefactors include the energy-independent quantities G F (the Fermi constant), Θ C (the Cabibbo angle), and | M nucl | (the nuclear matrix element). Meanwhile, ε f ( E ) = E − V f − E, (11)where the reduced endpoint E represents the total max-imum electron kinetic energy in the case of a masslessneutrino. While E is retrieved from the fit during theneutrino-mass analysis (Sec. IX), the internal molecularexcitation energies V f and the corresponding populationprobabilities ζ f come from computation (see Sec. IV B).The values of all constants are as in Ref. [72].Beyond the molecular effects discussed in detail inSec. IV B, theoretical corrections to the tritium β decayspectrum arise at the particle, nuclear, and atomic levels(see Ref. [76] for details). Of these, we include only theradiative corrections [77] in this work; these have by farthe largest effect on the high-energy tail of the β electronspectrum.Finally, the electron spectrum R β is Doppler-broadened due to the finite motion of tritium moleculesin the source. To account for this effect, we replace eachdiscrete final state with a Gaussian centered at the final-state energy V f , normalized to ζ f and with a standarddeviation of 94 meV according to the Doppler broadeningat 30 K. Effects due to the bulk gas flow are negligible. B. Final-state distribution (FSD)
Within the sudden approximation, the β decay ef-fectively corresponds to a sudden change of the nu-clear charge of one of the tritium nuclei. This induceselectronic and vibrational excitations of the daughtermolecular ion HeT + , possibly including its dissociationand/or ionization. Furthermore, the departing β electronand neutrino induce external (translational) and internal(rotational, vibrational, and – to a smaller extent, ne-glected here – electronic) excitations.Since only the energies of the β electrons are analyzedby KATRIN, the undetected energy associated with theremaining molecular system must be computed ab ini-tio by first solving the Schr¨odinger equation for the ini-tial and final molecular systems, and then computingthe transition probabilities ζ f = (cid:12)(cid:12)(cid:12) T mol f (cid:12)(cid:12)(cid:12) to the molec-ular daughter states f thus found. Earlier calculationseither focused on lower temperatures than KATRIN’s 30 K [78], thus artificially constraining the population ofinitial molecular states, or did not include all the tritium-containing isotopologs [79]. In the following, we provideonly a minimal description of the new computations car-ried out for the initial gas states relevant to KATRIN;a detailed publication is in preparation [80]. The the-oretical prediction of the dissociation probability of thedaughter HeT + ion, following β decay, has recently beenexperimentally verified [81].
1. Solutions to the molecular Schr¨odinger equation
As in previous works, these computations adopttwo fundamental approximations. First, the Coulomb-distorted version of the sudden approximation neglectsthe interaction of the β electron with all but the daugh-ter nucleus He + in the β decay. Second, the Born-Oppenheimer approximation allows a separate treatmentof the electronic and nuclear motions that define the full,internal molecular Schr¨odinger equation.Our solution of the Schr¨odinger equation describ-ing the nuclear motion uses the isotopolog-independentBorn-Oppenheimer electronic potentials generated ac-cording to Ref. [82] and presented explicitly in Ref. [83].Mass-dependent corrections are applied for the electronicground states of specific isotopologs – T , DT, HT, HeT + , HeD + , and HeH + – and the potential curvesare extended up to an internuclear separation of 20 a ,with a the Bohr radius. Because of the rotational sym-metry of the corresponding Schr¨odinger equation, the so-lutions for nuclear motion are expanded as products ofspherical harmonics and radial functions. They are thenaugmented by the rotational barrier for non-zero initialangular momenta J i .The electronic ground state of the daughter moleculesupports about 300 rotational/vibrational bound statesand a large number of predissociative resonances in thedissociation continuum. We have therefore adopted anew approach for solving the nuclear motion in these elec-tronic potentials. Expanding the radial part in B -splinefunctions and adopting vanishing boundary conditions atthe end of the radial grid, the solution of the Schr¨odingerequation is turned into a generalized matrix eigenvalueproblem and requires only the diagonalization of a verysparse matrix. The spectral density and energy range ofthe resulting discretized spectrum may be controlled bythe size of the adopted spherical box and the number of B -splines.
2. Energy-resolved FSD
With the newly obtained nuclear-motion solutions,and the isotopolog-independent Born-Oppenheimer elec-tronic overlaps defined in Refs. [83] (final electronicground state n = 1) and [82] (final electronic states n ∈ (cid:74)
2; 6 (cid:75) ), the transition probabilities between the initial4 . . . . .
020 25 30 35 40DossKNM1
Excitation energy (eV) P r ob a b ilit y P r ob a b ilit y D e n s it y ( / . e V ) FIG. 9. Final-state distribution (FSD) for transitions fromT in its electronic and vibrational ground state but initialrotational angular momentum J i = 1 (likeliest for a source at30 K) to the first 13 (KNM1) or 6 electronic states (Doss [79])of HeT + , as defined by Refs. [82, 84]. The transitions tothe electronic ground state (below 4 eV) and its bound rovi-brational states are given as probabilities (left axis) while thedissociative continuum is naturally plotted as a density (rightaxis); the total transition probability to the electronic groundstate is 57 . n = 2 and n = 3 states. and final states of interest in the KNM1 analysis inter-val can be obtained by integrating the matrix elementsover the internuclear separation vector. The transitionoperator, which can be expanded into spherical Besselfunctions, depends on this vector.Compared to earlier work, our new calculation ex-tends the results of Ref. [79] from the first 6 to the first13 bound electronic states, and employs more accuratemolecular masses than Refs. [78, 79]. These more accu-rate masses are used in the Hamiltonian, in the fractionof the recoil momentum imparted onto the spectator nu-cleus – which selects the population of the states due tothe molecular β decay via the transition operator –, andin the recoil energy of the whole molecular system. Forthe electronic excited final states n ∈ (cid:74)
1; 6 (cid:75) , we have beenable to reproduce the results of Refs. [78, 79] for the pub-lished initial states of T ( J i ∈ (cid:74)
0; 3 (cid:75) ), DT ( J i ∈ (cid:74)
0; 1 (cid:75) ),and HT ( J i = 0), when using the old kinematic inputs.Figure 9 shows a comparison of the current distributionwith Ref. [79] for transitions from the most populated T initial state at T = 30 K. The new distribution of tran-sitions to the electronic ground state is ∼ n >
6, combined with theelectronic continuum, contribute negligibly – at the 10 − level – to the KNM1 analysis interval, with its lowerbound at E – 37 eV. In our new calculation, these havebeen adapted for energy-scale changes from the calcula-tions in Ref. [78]. The n > J i ∈ (cid:74)
0; 3 (cid:75) for all three decaying isotopologs and weight their respec-tive contributions based on the source temperature. TheBoltzmann distributions are calculated at 30 K. How-ever, for the homonuclear T molecule, the resulting J i probability must be multiplied by nuclear-spin proba-bilities characteristic of 700 K. The molecules in thetritium loop dissociate when they arrive at the perme-ator (Sec. II), which is operated at 700 K. After diffu-sion through the permeator, the atoms recombine intomolecules with an ortho-para ratio of 0 .
75, characteris-tic of that temperature. The time for natural conversionto a lower-temperature ortho-para ratio is many ordersof magnitude longer than the O (1 s) passage time of themolecules through the 30 K region of the injection capil-lary and source tube, so the T gas retains an ortho-pararatio of 0 . ,DT and HT, as measured during KNM1, is performed ata subsequent stage of the analysis. V. RESPONSE FUNCTION MODELING
The observed KNM1 tritium integral spectrum R ( qU )is the convolution of the differential β electron spectrum R β (E) from Eq. 10 with the instrumental response func-tion f ( E − qU ), with an added energy-independent back-ground rate R bg : R ( qU ) = A s · N T , eff (cid:90) R β ( E ) · f ( E − qU ) dE + R bg . (12)Here, N T , eff denotes the effective number of tritiumatoms in the source, as adjusted by the detector effi-ciency and by the solid-angle acceptance of the setup∆Ω / π = (1 − cos θ max ) /
2, where θ max ≈ . ° as dis-cussed below. A s is the signal amplitude.As shown in Fig. 10, the response function f ( E − qU ) [72] describes the probability of transmission of anelectron with initial energy E through the beamline asa function of its surplus energy E − qU relative to theretarding potential U . Below, we discuss its calculationin detail. First, Sec. V A defines the response functionand describes the effects of the beamline electromagneticfields on the β electrons. We then treat the inelastic scat-tering cross section for β electrons (Sec. V B) and developa model of energy loss experienced in flight through theKATRIN apparatus (Sec. V C). A. Response and transmission functions
The transmission condition for any electromagneticconfiguration of the KATRIN MAC-E filter determineswhether an electron with starting energy E and startingangle θ is transmitted through a retarding potential U :5 T ( E, θ, U ) = E (cid:18) − sin θ · B min B S · γ +12 (cid:19) − qU >
00 else . (13)Here, θ = ∠ ( (cid:126)p, (cid:126)B ) is defined as the initial pitch angle ofthe electron, the polar angle of its momentum relativeto the magnetic field: p ⊥ = E sin θ · ( γ + 1) · m e . TheLorentz factor γ arises from its relativistic motion andhas a maximum value of about 1.036 at E . Meanwhile, B min = 0 .
63 mT is the magnetic field in the analyzingplane, B max = 4 .
23 T the maximum field of the beamline, and B S = 2 .
52 T the source magnetic field.Only electrons with sufficient surplus energy satisfy thetransmission condition and are included in the measuredintegral spectrum. The KATRIN main spectrometerachieves a magnetic-field ratio B min /B max ≈ / ≈ ∆ E/E , corresponding to a filter width (energy resolu-tion) of ∆ E = 2 . . θ max = arcsin (cid:112) B S /B max ≈ . ° lim-its the range of pitch angles contributing to the integralspectrum. The magnetic fields and the retarding poten-tial are provided by detailed field calculations using the Kassiopeia software [85]. To compute the precise elec-tromagnetic fields across the analyzing plane, we use anas-built geometry of the beamline magnets with a de-tailed three-dimensional model of the main spectrometer.The resulting transmission conditions can be included inthe model individually for each active pixel.The detailed response function of the KATRIN appa-ratus is calculated from Eq. 13, as modified by energylosses (cid:15) between source and analyzing plane [72]: f ( E − qU ) = (cid:90) E − qU(cid:15) =0 (cid:90) θ max θ =0 T ( E − (cid:15), θ, U ) sin θ · (cid:88) s P s ( θ ) f s ( (cid:15) ) dθ d(cid:15) . (14)For an ensemble of electrons, f ( E − qU ) depends onthe acceptance angle θ max and the amount of neutralgas the electrons pass in the WGTS, which is describedby the scattering probability P s ( θ ) and the inelastic-scattering energy-loss function f s ( (cid:15) ) for a given numberof scatters s . As Sec. V C will discuss in detail, we mea-sure f ( E − qU ) using monoenergetic electrons with smallangular spread, and thus deduce f s ( (cid:15) ). Briefly, theseelectrons are produced in the e-gun with surplus ener-gies E − qU spanning a 50 eV range. They follow themagnetic-field lines and pass through the integral col-umn density ρd of the source. This allows us to observesingle ( s = 1) and multiple ( s >
1) electron scatterings inthe source. The scattering probability P s ( θ = 0 ◦ ) (Eq. 6)follows a Poisson distribution with the expected numberof scatterings given by the product of the effective col-umn density ρd and the inelastic-scattering cross section σ (Sec. V B). In an isotropic source like the WGTS, electrons areemitted with an angular distribution ω ( θ ) dθ = sin θdθ ,and we can define an integrated transmission function T ( E, U ): T ( E, U ) = (cid:90) θ max θ =0 T ( E, θ, U ) · sin θ dθ = , (cid:15) < − (cid:113) − E − qUE B S B min γ +1 , ≤ E − qU ≤ ∆ E − (cid:113) − B S B max , E − qU > ∆ E . (15)Although analysis of non-isotropic e-gun data requiresthe full expression in Eq. 14, the neutrino-mass analysisin this work exploits the isotropic nature of the tritium β -source and uses the simplified response function f ( E − qU ) = (cid:90) E − qU(cid:15) =0 (cid:90) θ max θ =0 T ( E − (cid:15), U ) · (cid:88) s P s ( θ ) f s ( (cid:15) ) dθ d(cid:15) . (16)In principle, the response function is slightly modifieddue to the dependence of the path length, and thereforethe effective column density, on the pitch angle of the β -electrons [72]. The resulting effect on the measuredendpoint is small compared to the overall uncertaintiesof the electric potential of the source, and this effect is nottaken into account in the current analysis. Synchrotronenergy losses of β -electrons in the high magnetic fieldin the source and transport systems are included as ananalytical correction to the transmission function [72]. B. Inelastic-scattering cross section
The theoretical total inelastic-scattering cross sectionof electrons with T molecules in the high-energy Bornapproximation can be written as [86–88]: σ inel ( E ) = 4 πa ( E nr /R H ) (cid:20) M · ln (cid:18) c tot · E nr R H (cid:19) − . (cid:21) , (17)where R H = 13 .
606 eV is the Rydberg energy, a =28 . × − cm the Bohr radius squared, and E nr de-notes the non-relativistic kinetic energy of the electron: E nr = 0 . m e β , with β = 1 − m / ( m e + E ) and E the relativistic kinetic energy of the electron. At thespectral endpoint for molecular tritium β decay, we take E = E = 18 .
575 keV and E nr = 17 .
608 keV.The dominant parameter M can be calculated reli-ably and with high accuracy, since it is a special elec-tron expectation value for the ground-state hydrogen-molecule wave function. For the three isotopologs, wehave [89, 90]: M [H ] = 1 . M [D ] = 1 . -1.0 0.0 1.0 2.0 3.0 E qU (eV) R e s pon s e f un c ti on E =2.8 eV
10 20 30 40 50
Scattered eIdeal case, calculation d = 0, calculationActual d , calculation FIG. 10. Response function with infinitesimal filter widthand no scattering (orange dash-dotted line); actual transmis-sion function of the MAC-E filter without source scattering(blue dashed line); and actual transmission function alongwith source scattering (solid red line). Electrons emitted with0 < θ ≤ θ max need additional surplus energy E − qU to over-come the potential barrier. At higher surplus energies, elec-trons that lost energy due to source scattering can pass thefilter as well, and the transmission probability increases. and M [T ] = 1 . c tot is more difficult, and we use the1987 value of Liu [88]: c tot = 1 .
18. With these num-bers, we obtain σ inel [T ]( E ) = 3 . × − cm , withan estimated uncertainty of 0 . . × − cm [91], by 7 % (3 . σ ). However,it is ρdσ , directly measured by the e-gun as described inSec. III B, which is used in the neutrino-mass analysis –not σ as a separate input. C. Energy-loss function
Electrons traversing the WGTS can scatter elasticallyor inelastically from tritium molecules before being an-alyzed in the main spectrometer. (Here, “elastic” scat-tering refers to interactions that do not change the elec-tronic state of the molecule.) While elastic scatteringonly causes a small broadening of the measured responsefunction ( ∼ .
03 eV), inelastic scattering can result in en-ergy losses from ∼
11 eV up to E/
2, where the lowerbound is associated with the lowest electronic excitationsin T .Small inelastic energy losses, in particular, can moveelectrons emitted at energies close to the endpoint (thesensitive region for m ν ) into a region still within theanalysis interval extending 37 eV below the endpoint.Precise knowledge of the energy loss spectrum is, there-fore, a crucial input for the KATRIN response function.During planning, its uncertainty was estimated to be oneof the dominant systematics of the experiment [15]. A de-tailed paper on the energy-loss determination is in prepa-ration [92]. Various electronic excitations, in combination with ro-tational and vibrational states of the T molecule, re-sult in a rich spectrum up to the ionization threshold at15 .
486 eV [93]. Prior to this work, there were no calcu-lations of the energy-loss spectrum with the required ac-curacy. We therefore measured the energy-loss functionwith the e-gun installed in the rear system of the KA-TRIN beamline. In contrast to β electrons originatingwithin the source, these calibration electrons start withan adjustable kinetic energy chosen close to the endpointof the tritium β spectrum and traverse the full lengthof the source. The dependence of the energy-loss func-tion on the kinetic energy of the electrons can be ne-glected within the small fit window around the endpointat ∼ . qU .The e-gun was operated in two different modes: a fastmode with a 100 kHz laser repetition rate to obtain aquasi-continuous electron beam used to record integralspectra as shown in Fig. 11 (top panel) and a slower modewith a 20 kHz repetition rate, in which the electron starttimes were synchronized with the DAQ to record time-of-flight (TOF) spectra as shown in Fig. 11 (center panel).This TOF information allows us to record a differentialenergy spectrum by applying a TOF cut on individualevents [94]. Electrons with energies close to qU take sig-nificantly longer to reach the detector since they are de-celerated to almost zero kinetic energy near the analyzingplane. Selecting electrons with flight times between 35 µ sand 50 µ s, as illustrated in Fig. 11 (bottom panel), effec-tively turns the main spectrometer from a high-pass filterinto a narrow band-pass filter with a width of ∼ .
02 eV.Apart from effects of multiple scattering and finite energyresolution, this method provides direct access to the elec-tron energy-loss spectrum.The energy-loss function is parametrized by a semi-empirical model using three Gaussians to describe thethree groups of lines created by excitations of the(2 pσ Σ + u ), (2 pπ Π u ) and (3 pπ Π u ) molecular statesaround 12 . . I n t e g r a l d a t a ( a r b . un it s ) UnscatteredSingle scatteringDouble scatteringTriple scatteringFitData
E - qU [eV] T O F d a t a ( a r b . un it s ) E qU (eV) T i m e o f f li gh t ( s ) TOF cut FIG. 11. Energy-loss function from e-gun data; e-gun elec-trons must cross the entire 10 m source at a pitch angle θ = 0.Top panel: Integral measurement, showing data, fit func-tion and individual components for single and multiple scat-ters. Center panel: Differential measurements obtained withthe TOF cut, showing data, fit function and the four sub-components of the semi-empirical energy-loss parametriza-tion: three Gaussians and a binary-encounter-dipole (BED)tail. Bottom panel: Time of flight versus surplus energy data,showing the TOF selection region used to obtain the differ-ential spectrum in the center panel. This spectrum of electrons which have not undergone in-elastic scattering naturally includes the effects of elasticscattering and the filter width of the main spectrometer.The resulting curves for single and multiple scatteringare then weighted with the Poisson-distributed scatter-ing probabilities and summed. The expectation valueof this Poisson distribution is a nuisance parameter inthe fit. A combined fit of TOF spectra taken at dif-ferent column densities must also account for differencesin the e-gun laser intensity between the individual mea-surements, leading to changes in the count rate. Ad-ditional normalization factors are therefore included asnuisance parameters in the fit. Finally, additional back-ground components are included in the fit. Backgroundelectrons produced by the impact of positive ions onto thephotocathode of the e-gun, for example, do not exhibita TOF structure and appear in the differential spectrumas a small additional component with the shape of anintegral energy-loss spectrum. The scaling factors of this background are additional nuisance parameters.We performed a combined fit to four TOF datasetsmeasured at different column densities. Each datasetcontains about 12 hours of data, resulting in ∼ × events surviving the TOF cut. The nine model parame-ters of interest are shared between all datasets, whereaseach dataset has its own nuisance parameters as de-scribed above.The resulting best-fit parametrization is shown inFig. 11 top and center for the integral and differentialdata, respectively. The same energy-loss function de-scribes all four datasets well and the fit has a reduced χ close to one.Uncertainties used in this work are of a statistical na-ture only. However, more advanced combined fits thatalso take into account the integral energy-loss measure-ments yield the same parameter values within their sta-tistical uncertainties.Systematic uncertainties in the energy-loss determina-tion are largely canceled by alternating up- and down-ward scans. A study of systematic effects on the param-eter uncertainties has been undertaken using a MonteCarlo (MC) approach and taking into account distur-bances like column-density drifts, background events, de-tector pileup and the binning of the continuous voltageramp. These systematic uncertainties are negligible forthe KNM1 analysis.An improved parametrization of the energy-loss func-tion and its uncertainties is under investigation for future,more sensitive neutrino-mass campaigns. VI. BACKGROUND
The rate of background events during KNM1 was dom-inated by the two steady-state mechanisms described inSec. VI A. In Sec. VI B, we also consider a background de-pendent on the duration of the corresponding scan step.
A. Steady-State Background
The steady-state background originates from excitedor unstable neutral atoms which can propagate freelyin the ultra-high-vacuum environment of the main spec-trometer. It has two primary causes.First, a significant part of the steady-state backgroundarises from hydrogen Rydberg atoms sputtered from theinner spectrometer surfaces by
Pb recoil ions follow-ing α decays of Po. These processes follow the decaychain of the long-lived
Rn progeny
Pb, which wassurface-implanted from ambient air (activity ∼ / m )during the construction phase. A small fraction of theseRydberg atoms is ionized by black-body radiation whenpropagating through the magnetic flux tube. The result-ing sub-eV scale electrons are accelerated to qU by theMAC-E-filter, adding a Poisson component to R bg .8 R bg = 0.293 ± 0.001 cps R bg p e r p i x e l s ( c p s ) FIG. 12. Distribution of pixel-wise, steady-state backgroundrates during the first neutrino mass campaign for the 117included pixels. The background rate increases radially fromthe center by about 50 %. White pixels are excluded from theanalysis (Sec. III E).
The second significant steady-state background mech-anism originates with α decays of single Rn atoms( t / = 3 .
96 s) emanating from the non-evaporable-getterpumps. Each decay releases a large number of electronsup to the keV scale. If the decay occurs in the magneticflux tube, these electrons are stored due to their signif-icant transverse momenta. They subsequently producesecondary electrons by scattering on the residual gas un-til they have cooled to energies of a few eV, when they canescape; both primary and secondary electrons contributeto R bg at qU [97]. Since several background electronsmay originate from each Rn decay in the magneticflux tube, this background source is not purely Poisso-nian. Liquid-nitrogen-cooled copper baffles at the portsto the getter pumps mitigate this effect by preventing
Rn from diffusing into the sensitive volume [39, 98].Due to the formation of a thin layer of H O covering thebaffle surface, the retention of
Rn was hampered suchthat R bg retains an observable non-Poissonian compo-nent during KNM1.In KNM1, the overall steady-state background rate, R bg , is continuously measured through the energy-independent part of the spectrum R( (cid:104) qU (cid:105) ). The wholespectrum is fitted, leading to a value over the 117 selectedpixels of R bg = 0 . E . This value isconsistent with data from independent background runs.Full fit results are given in Sec. IX D.The background is not distributed uniformly acrossthe detector, as shown in Fig. 12. The decrease of R bg towards smaller radii can be explained by radiative de-excitation of the Rydberg atoms as they propagate insidethe main spectrometer. Further from the spectrometer
14 April 21 April 28 April 05 May 12 MayDate0.240.260.280.300.320.34 B a c k g r o un d r a t e ( c p s ) Slope: ( 0.01 ± 0.08) mcps/day
FIG. 13. Evolution of the background rate during KNM1,measured via the five background-region scan steps above theexpected endpoint E . The slope of a linear fit to the datais compatible with zero, indicating the long-term stability ofthe background. O cc u rr e n ce r a t e FIG. 14. Background-event distribution for the fivebackground-region scan steps during KNM1. The fit of aGaussian profile to the measured data yields a width of10 . . . wall, fewer Rydberg atoms are therefore available for ion-ization by the thermal radiation.The steady-state background was monitored for each β -scan with the five dedicated background-region scansteps. Figure 13 shows the time evolution of these back-ground measurements during KNM1. A linear fit wasapplied to the data in order to test the long-term sta-bility of the background. The slope of − . / dis compatible with a background that is stable over longtime scales.The non-Poissonian component of R bg causes a broad-ening of the event distribution of the five background-region scan steps, amounting to 6 . R a t e ( c p s ) Fit: slope (4.4±4.8) mcps/keVData
FIG. 15. Background measurement without tritium sourcein a region near E . The slope of a linear fit to the datais compatible with zero, supporting our assumption that thebackground is independent of qU . Fits to the immediate E region, where scan steps are more evenly spaced, also find nosignificant trend in qU . prediction from pure Poisson statistics (Fig. 14).Our model predicts a background that is independentof qU near E . To test this expectation, we performeda dedicated background-only measurement, without anactive tritium source, in June 2018. As shown in Fig. 15, qU was scanned in 26 steps over an interval of 16 .
975 keVto 18 .
615 keV. We then fit a line with a free slopeparameter to these data. The resulting best-fit slope, − . / keV, is compatible with zero, and we takeits uncertainty as an overall uncertainty on our assump-tion of a qU -independent background (Sec. VIII G). B. Background Dependence on Scan-Step Duration
With both the pre-spectrometer and main spectrom-eter held at negative retarding potentials, a Penningtrap inevitably forms in the strong magnetic field of thegrounded inter-spectrometer region. Electrons trappedin this region slowly lose energy by ionizing residual gasmolecules. The resulting ions may escape into the mainspectrometer, where they can create background elec-trons when their own collisions with the residual gasor the vessel wall release ionization electrons, Rydbergatoms, or photons. The intense WGTS feeds the Penningtrap when β electrons produce positive ions on their wayinto the pre-spectrometer; these ions sputter Rydbergatoms from the pre-spectrometer walls, and the Rydbergatoms in turn produce low-energy ionization electronsthat fill the trap [40]. This mechanism may also play arole in main-spectrometer backgrounds, when β electronsscatter further downstream and the resulting ions strikethe main-spectrometer walls.During each transition to a new scan step, an elec-tron catcher is briefly inserted into the beamline to re- move stored electrons from the Penning trap. At higherpre-spectrometer potential, this has been shown to pro-vide a statistically significant reduction in the baselinebackground [40]. However, since the electron catcher isinserted only at the beginning of a scan step, the Pen-ning trap continues to fill until a new electron-catcheractuation at the beginning of the next scan step. Thecorresponding rise of the background rate is strongly in-fluenced by surface conditions and by the achieved pres-sure between the spectrometers. In principle, however,this mechanism can produce a background that effec-tively increases in rate for longer-duration scan steps (seemeasurement-time distribution in Fig. 7). This effect wasobserved in a subsequent KATRIN scientific run, but forKNM1 – the initial science run, with pristine surfacesand lower column density – no statistically significantdependence on scan-step duration was observed. Sec-tion VIII G will address the impact on the neutrino-massmeasurement. VII. ASSEMBLING SPECTRAL DATA FORKNM1
Data are acquired in a sequence of O (2 h) scans andthe integral spectrum (Eq. 12) is recorded with the FPD.In the final analysis (Sec. IX), the spectral fit uses fourfree parameters: the signal amplitude A s , the effective β -decay endpoint E , the background rate R bg , and thesquared neutrino mass m ν . In this analysis we leave E and A s unconstrained, which is equivalent to a “shape-only” fit. The 4-parameter fit procedure over the aver-aged scan steps (cid:104) qU (cid:105) compares the experimental spec-trum R( (cid:104) qU (cid:105) ) to the model R model ( (cid:104) qU (cid:105) ).Spectra from all of the scans and pixels have to becombined in the final analysis without loss of informa-tion. In the following we describe the strategy appliedto combine all these data prior to the final spectral fit toextract the effective neutrino mass. A. Pixel combination
During KNM1, the electric potential and magneticfield in the analyzing plane of the main spectrometer werenot perfectly homogeneous, but varied radially by about140 mV and 2 µ T, respectively, and to a much smallerextent azimuthally. The pixelation of the detector allowsus to account for these spatial dependencies. Each pixelhas a specific transmission function and records a sta-tistically independent tritium β -electron spectrum. Inthis analysis, we combine these pixel-wise spectra into asingle effective pixel by adding all counts and assumingan average transmission function for the entire detector.The averaging of fields leads to a negligible broadeningof the spectrum which does not affect the filter width,and carries a negligible bias of O (10 − eV) on m ν .Combining all 274 scans that passed data-quality cuts,0 E f i t E f i t ( e V ) FIG. 16. Distribution of the endpoint E fit0 over the detectorpixels. No spatial inhomogeneity beyond statistical fluctua-tions is observed, justifying the merge of the data of all 117pixels for the subsequent analysis. White pixels are excludedfrom the analysis (Sec. III E). single-pixel fits were performed resulting in an endpoint E fit0 for each pixel, as shown in Fig. 16. We find no sys-tematic spatial ( i.e. pixel) dependence of E fit0 . The stan-dard deviation from the mean endpoint is 0 .
16 eV, whichis consistent with statistical fluctuations. This indicatesa good description of the electric potential and magneticfield in the analyzing plane, and the absence of a signif-icantly spatially dependent electron starting potential.We therefore merge the data of all 117 selected pixelsused in the analysis (Fig. 5).
B. Scan combination (stacking)
Combining all pixels in a uniform fit, we can now con-sider the stability of the fit parameters with respect topossible temporal variations. We investigate all four freeparameters in the fit. For single scans of 2 hours, the ac-cumulated statistics are not sufficient to significantly con-strain the neutrino mass. Therefore, the neutrino mass isfixed to zero. The 274 fit values show excellent stabilityover the course of a month (Fig. 17). The standard devi-ation from the mean endpoint is 0 .
25 eV, which is againconsistent with statistical fluctuations.In order to constrain the neutrino mass, the statisticsof all 274 scans must be combined. Based on our sta-bility results, we achieve this by merging the data of all274 scans into a single stacked, integral spectrum. Inthe underlying process, the events at like scan steps aresummed and the corresponding retarding-potential val-ues are averaged over all scans. This procedure yields onehigh-statistics integral spectrum with the same numberof scan steps as a single scan. Since this method doesnot correct for scan-to-scan variations of slow-control pa-
11 April 18 April 25 April 02 May 09 May
Date ° . . . A s ° h A s i Entries ° E ° h E i ( e V ) Scanwise fit results Weighted mean
FIG. 17. Evolution of two fit parameters, the endpoint E fit0 (upper panel) and the signal normalization A s (lower panel),as functions of time during the whole KNM1 data-taking pe-riod. Each plot shows the deviation of the fit parameter,evaluated on a per-scan basis, from its weighted mean duringKNM1. rameters, it relies on good time stability and excellentreproducibility of the individual HV settings from scanto scan. The Gaussian spread of these HV settings is onaverage σ = 34(1) mV (better than 2 ppm) (Sec. III D).The scan stacking results in a minor systematic effect,which is included in the analysis. C. Resulting integral spectrum
The resulting stacked integral spectrum, R( (cid:104) qU (cid:105) ), isdisplayed in Fig. 18. It comprises a total of 2 . × events, with 1 . × β decay electrons below E anda flat background ensemble of 0 . × events in the86 eV analysis interval, [ E −
37 eV, E +49 eV]. Retarding energy (eV) R a t e ( c p s ) KATRIN data with 1 error bars Fit result with 1 uncertaintes (stat. only) Background Tritium signal
Zoom on m ROI KATRIN data with 1 error bars 50
FIG. 18. Display of the measured KNM1 endpoint β spectrumafter scan and pixel combination, superimposed to the best-fitmodel (Sec. IX). TABLE III. Summary of systematic uncertainties used as in-put for the neutrino-mass inference. Details of each entry(including those neglected in this analysis) are given in thetext.Effect Description 1 σ uncertaintyBackground Rate over-dispersion 6 . . / keVRate dependence onscan-step duration neglectedSource effects Expected number ofscatterings ( ρdσ ) 0 .
85 %Energy-loss function O (1 %) β starting potential neglectedScan fluctuations Column density 0 . . . . . . . . VIII. SYSTEMATIC UNCERTAINTIES
Systematic uncertainties generally arise from param-eter uncertainties that enter into the calculation of theintegral spectrum, and from instabilities of experimen-tal parameters. The KNM1 analysis heavily relies on aprecise description of the spectral shape, including all rel-evant systematic effects and a robust treatment of theiruncertainties. Any erroneously neglected effect or un-certainty can lead to a systematic shift of the deducedneutrino mass [99]. The individual systematics are de-scribed in detail below. A summary of these systematicuncertainties is given in Table III, while their ultimateimpacts on the m ν uncertainty budget are collated inTable IV. A. Tritium concentration
The concentration of the tritium isotopologs in thesource affects the model in two different ways.First, the total activity is directly correlated to thetritium purity described in Eq. (4). The absolute num-ber does not impact the neutrino-mass measurement, asthe signal normalization is a free fit parameter. Changesduring a given scan, however, could introduce a slightspectral distortion which would bias the measurement.As described in Sec. III A, the tritium purity was mea-sured continuously by the LARA laser-Raman spectro- scopic system. The precision was determined from theshot noise √ N x of the Raman signal and then prop-agated to ε T and c x ; the resulting precision of betterthan 2 × − for each scan was reported in Ref. [21].Scan-to-scan fluctuations of the tritium purity amountto 0 . × − after accounting for anti-correlations be-tween the isotopologs.Second, each of the three tritium isotopologs also has aslightly different FSD. Systematic uncertainties on theirrelative fractions, mainly determined by the trueness ofthe LARA calibration, thus propagate into the spectralshape. The impact on m ν from this effect is less than2 × − eV and is thus negligible for KNM1. B. Column density and expected number ofscatterings
The determination of the expected number of scatter-ings, ρdσ , is described in Sec. III B. The total uncer-tainty on ρdσ arises from three separate contributions:the limited precision of single column-density measure-ments made with the e-gun; uncertainty on the through-put measurement, arising from fluctuations of the gasthroughput and imperfect reproducibility of the flow me-ter; and the scaling of the inelastic-scattering cross sec-tion to a lower electron energy via Eq. 17. This last op-eration is necessary because the e-gun is operated at anenergy of 18 .
78 keV, well above E , for measurements ofthe column density – but the β electrons, at lower ener-gies, have a slightly different scattering cross section. Wetake 18 .
575 keV as a representative value for our observed β electrons; the variation of the inelastic-scattering crosssection within the analysis interval is negligible.Taking these three contributions into account leads toa total systematic uncertainty on ρdσ of less than 0 .
85 %for all scan steps.
C. Electron starting potential
Spatial inhomogeneities and temporal fluctuations ofthe starting potentials of the β electrons would lead toa shift of the neutrino mass according to Eq. 7. As dis-cussed in Sec. III C, the intrinsic width of the Kr L -32 line is a diagnostic tool to investigate these effects,probing the plasma-potential distribution.In the KNM1 analysis, we treat the fitted Gaussian linebroadening in the presence of a T plasma as a conser-vative upper limit for the inhomogeneity of the plasmapotential, yielding a negative m ν shift with magnitudeless than 0 .
013 eV .Electrons undergo inelastic scattering as described inSec. V. The s -fold scattering probabilities for each β elec-tron depend on the longitudinal position of its creation.As a result, the populations of β electrons with differ-ent scattering multiplicities also have different distribu-tions of starting positions, and therefore different distri-2butions of starting potentials if the plasma potential isinhomogeneous. Analysis of the positions of the kryptonL -32 lines of unscattered and singly scattered electronsshows that a plasma-induced mutual shift of these posi-tions cannot be larger than 70 mV. The correspondingadditional m ν -shift can be neglected for KNM1. We thusconclude that the effective L -32 broadening parametersgiven above serve as a very conservative upper limit ofplasma effects in the neutrino-mass analysis.In addition to the Kr spectroscopy method, radialplasma inhomogeneities can be inferred directly from theneutrino-mass data by radial evaluation of E . The spec-tral fit from twelve separate detector rings (see Fig. 5 fordetector structure) revealed a slope of − / ring,consistent with a slope of zero.A full propagation of the plasma model and its un-certainty was not included in the KNM1 analysis, pri-marily due to the immaturity of the plasma model asapplied to the low KNM1 column density. Adding this O ( − .
01 eV ) uncertainty in quadrature to the total sys-tematic uncertainty does not yield significant leverage onthe total budget.The neutral-gas density strongly affects the chargedensities from secondary electrons and ions, as well asother plasma parameters. For this reason, we are cur-rently investigating the effect of different column densi-ties, gas temperatures, source magnetic-field strengths,and changing boundary conditions on plasma parame-ters. This will inform the consideration of plasma effectsin the data analysis for upcoming campaigns, in whichthe gas throughput will be higher by a factor of up tofour. D. Detector efficiency
Although numerous physical and detector effects canreduce the detector efficiency, any effects which do notdepend on the retarding potential U will not affect theKNM1 fit results due to the overall, free scaling param-eter for each spectrum and the uniform, all-pixel fit.The overall FPD detection efficiency within the ROIhas been estimated by both simulation and commission-ing analysis to be approximately 95 %, with an uncer-tainty of a few percent, and per-pixel variations of aboutthe same size. For KNM1, the ROI is fixed regardlessof U (Sec. III E). However, the shape of the FPD energyspectrum changes with U , primarily due to the β -electronenergy threshold at qU . Additional distortions are due toenergy- or rate-dependent detector effects: energy loss inthe dead layer, charge sharing among pixels, pileup, andback-scattering of electrons and their subsequent reflec-tion back toward the FPD by local electric and magneticfields.We have studied the effects of these spectral shapechanges using a reference spectrum for each pixel, ac-quired at U = −
18 375 V. For each scan step at U i = U + ∆ U , the reference spectrum is shifted by the corre- sponding q ∆ U and a count correction is calculated. As | U | decreases, the corrections become larger, with a max-imum size of about 0 .
05 %. We estimate the error relativeto these correction factors at less than 0 .
05 %, determinedby comparing spectral shapes at nearby U values. Inthe KNM1 analysis, we apply these corrections to FPDcounts while neglecting the corresponding uncertainty.Pileup events also result in event loss, since the en-ergy is erroneously reconstructed above the upper boundof the ROI. We assume that pileup events arise fromrandom coincidences; each coincidence produces a totalenergy deposit that is an integer multiple of 28 . .
02 % at low | U | and, correspond-ingly, high rate. Our conservative estimate of the relativeerror on these correction factors is less than 18 %, basedon the shape of the measured FPD energy spectrum anda simulation of the trapezoidal filter. This error is negli-gible.Our final consideration is electron backscattering fromthe FPD. The majority of backscattered electrons are re-flected back to the FPD, either by magnetic fields in thedetector system, or by the electric potentials of the post-acceleration electrode or main spectrometer. Even withmultiple backscatters, the electron returns to the samepixel each time, always arriving well within the shapingtime of the trapezoidal filter, so that the detector doesnot register the event as separate hits. Our spectral-shape calculations include the resulting reconstructed-energy shifts, due to multiple transits of the detectordead layer and hits distributed within the shaping time.However, an additional correction is in principle neededfor those few backscattered electrons which have enoughenergy to surmount the qU threshold and escape towardsthe source. Simulations show that the resulting event lossis less than 0 .
01 % for the KNM1 analysis window. Thiseffect is therefore neglected in this analysis.
E. Final-state distribution
The uncertainty estimation on the FSD is based ondifferences between the theoretical ab initio calculationsfrom Saenz et al. [78] and Fackler et al [100]. The differ-ence between the calculations for the ground-state vari-ance is found to be small, of O (1 %) [101]. However,the descriptions of the electronic excited states and theelectronic continuum exhibit larger discrepancies.We conservatively estimate the uncertainty on the vari-ance of the ground state (excited states and continuum)to be 1 % (4 %). The uncertainty on the normalization ofthe ground to excited-state populations is taken as 1 %.Our narrow analysis interval, extending 37 eV below E , is dominated by electrons from the ground-state dis-tribution. Consequently, the uncertainty on the FSDonly contributes on the order of O (10 − ) eV to the total3systematics budget on m ν within our analysis interval. F. Response function
Response-function-related systematic uncertainties areconnected with the electromagnetic fields that define thetransmission function (Eq. 16) and with the energy-lossfunction. The electromagnetic fields are computed froma simulation of the beamline magnets and the main-spectrometer vessel. a. Magnetic fields
Systematic uncertainties on themagnetic field at the analyzing plane arise from resid-ual magnetic fields in the spectrometer hall, e.g. due tomagnetized materials, and from model imperfections. Asensor network was used to compare measured fields atthe spectrometer vessel to simulation results. Our as-sessment of the maximum deviation yields a conservativesystematic uncertainty of ∆ B min /B min = 1 %.The maximum magnetic field, located at the exit ofthe main spectrometer, was measured in 2015 at thecenter of the magnet bore [30] and compared to simula-tions. We include a conservative systematic uncertaintyof ∆ B max /B max = 0 . B S /B S = 2 . b. Electric potentials Since any offset of the simu-lated retarding potential at the analyzing plane is com-pensated by the free endpoint parameter, no additionalsystematic uncertainty is assigned for the spectral fit. c. Energy-loss function
The uncertainty of theenergy-loss parametrization is obtained from fits to themeasurements described in Sec. V C. For each of the 9 pa-rameters describing the energy-loss function, an individ-ual fit uncertainty is determined. As stated in Sec. V C,the contribution of systematic effects is about one orderof magnitude lower than the uncertainties related to thecurrent statistics of the e-gun measurements. As a re-sult, only statistical fit uncertainties are considered forthis analysis. Correlations between the energy-loss pa-rameters are taken into account, reducing the overall un-certainty of the energy-loss function with respect to theuncorrelated case.The systematic effect on m ν due to the uncertaintiesof the energy-loss function is determined to be below0 .
01 eV . G. Background
The steady-state background enters the uncertaintybudget in two independent ways: rate and shape.The background rate distribution, as shown in Fig. 14shows an over-dispersion of 6 . E > E −
15 eV.As described in Sec. VI A, we expect the backgroundto be flat with respect to the retarding potential. In thisanalysis we assess the slope uncertainty via a slope pa-rameter, which makes a first-order correction to the con-stant expectation. Based on the dedicated measurementin June 2018 (Fig. 15), the slope parameter is consis-tent with zero, within an uncertainty of 5 mcps / keV. Inthe final spectral fit (Sec. IX), we use a central value of0 mcps / keV.A Penning-induced background (Sec. VI B) may in-crease over the course of each scan step, effectivelyintroducing a higher background for scan steps withlonger duration. Since longer scan steps are concen-trated near E −
14 eV (Sec. III G), the net effect is ashape distortion of the background shape. An analysisof KNM1 scan steps yields a best-fit linear time slope of( − . ± . µ cps / s, which would result in a systematicuncertainty of 0 .
15 eV on the squared neutrino mass.This systematic was not taken into account in the spec-tral fit (Sec. IX), but would not alter the statistics-dominated final uncertainty. H. Stacking
The averaging of the scan steps within the stackingtechniques introduces a small bias on m ν and E . Inorder to quantify these biases, we construct an Asimovdataset [102] by simulating 274 statistically unfluctuated“MC twin” spectra, incorporating the actual variationof slow-control parameters (including measured high-voltage values, isotopic compositions, and column densi-ties) between scans. Later on, the MC spectra are com-bined into a single integral spectrum through the stack-ing procedure, as described in Sec. VII. As a last step,we fit this stacked MC spectrum. Comparing this fit re-sult to the MC truth yielded a 1 σ stacking uncertaintyof 14 × − eV in one analysis approach (Sec. IX B),and 5 × − eV in the other (Sec. IX C), as shown inTable IV further below. The discrepancy between thetwo approaches arises from different treatments of the in-dividual contributions to this subdominant uncertainty;the stacking method and error treatment will be opti-mized in the analysis of future neutrino-mass campaigns,in which scan-to-scan fluctuations are also expected tobe smaller. I. Neutrino-mass fit range
The full spectrum was recorded over a large energyrange down to E −
91 eV. Several systematic uncer-tainties, like those related to inelastic scattering and theFSD, increase further away from the endpoint, while thestatistical uncertainty decreases. The optimization of the4neutrino-mass fit range is performed using MC twin sim-ulations of KNM1 (Sec. VIII H), assuming a zero neutrinomass and using the set of systematics presented earlier inthe section (Table III). The lower bound of the fit inter-val is then varied between E −
91 eV and E −
30 eV,and two fits are performed in turn. The first fit consid-ers statistical uncertainty only, while the second fit usesboth statistical and systematic errors. For each pair offits, the systematic uncertainty is deduced by subtractingthe statistical uncertainty in quadrature from the totalerror. As a result, both statistical and systematic uncer-tainties become equal for the fit range starting at about E −
70 eV, and systematic uncertainties become domi-nant when including data below E −
70 eV. Moreover,the overall sensitivity only marginally improves by in-cluding data at energies below E −
40 eV.This study addresses only the dependence of the mea-surement precision on the fit range. It does not addressthe accuracy of the determination of the neutrino mass,since the same model is used for the fit and for the MCtwins. Indeed, further than about E −
40 eV, the elec-tronic continuum – with less well-validated modeling –dominates the FSD (Sec. IV B). Therefore, before un-blinding the data (Sec. IX A, below), we fixed the analy-sis interval to cover the region of E −
37 eV (22 scansteps) and E + 49 eV (5 scan steps). IX. SPECTRAL FIT
In this section we discuss our blinding method(Sec. IX A) and present two approaches for inferring thevalue of the neutrino mass squared m ν and the endpoint E simultaneously, based on fitting the integrated β spec-trum (Eq. 12) assembled as described in Sec. VII. Inboth approaches, the spectrum is fitted using a shape-only analysis with four free parameters. In addition to m ν and E , these are the signal amplitude A s and thebackground rate R bg .The first approach (Sec. IX B) uses a standard χ estimator and covariance matrices to encode all uncer-tainties. The second approach (Sec. IX C), Monte-Carlopropagation, repeats the final fits many times, for eachfit choosing randomized input values for the systematicnuisance parameters.Three analyses were performed, each with its ownspectrum calculation and analysis software: two us-ing the covariance-matrix approach, and one using theMC-propagation approach. The analyses were per-formed blind and give consistent results, as described inSec. IX D. The resulting breakdown of systematic uncer-tainties is given in Table IV, below. Section X uses thesespectral results to derive frequentist bounds on the neu-trino mass, while Sec. XI uses the same data to deriveBayesian bounds. A. Blinding strategy
For the KNM1 analysis we enforced blind analysis pro-cedures to fix data selection, analysis cuts, and modelcomposition before the model was fitted to the data. Thisstandard technique is designed to avoid observer’s bias.For this first KATRIN m ν limit, we employed modelblinding rather than data blinding. The fit results arehighly dependent on the molecular FSD (Sec. IV B); inparticular, the value of m ν depends on the width of thedistribution of transitions to the electronic ground stateof the daughter molecule HeT + . Using an FSD withtoo large a width pushes m ν towards higher values, whiletoo narrow a width pushes it towards lower values. In-deed, historically, inaccurate FSD models were likely re-sponsible for artificially negative m ν results from the LosAlamos [18] and Livermore [103] experiments, a prob-lem which is resolved by using the more modern theorydescribed in Sec. IV B [101].If we fit the data with a model using an FSD ground-state width that has been picked randomly within a suit-able interval, the true value of m ν cannot be retrieved.That is, the analysis is blind to its parameter of inter-est, while the remaining three parameters are left essen-tially unaffected [64]. The range of possible ground-statewidths was chosen so that the sensitivity of the KATRINblind analysis could not improve upon the results of pre-vious direct m ν measurements [12, 13]. In addition, be-cause the endpoint fit parameter only depends – to a goodapproximation – on the mean of the FSD, leaving thatmean value untouched ensured that the endpoint couldstill be used during a blind analysis, e.g. for comparisonwith other independent measurements (Sec. XII).In practice, the theoretical electronic ground-statemanifold of the FSD was swapped with a Gaussian dis-tribution function, constructed with the true mean and arandomly chosen width. To prevent accidental unblind-ing, the adjusted FSD was provided as an independentsoftware module synchronized with the main fitting soft-ware.The second measure to mitigate biasing is to performthe full analysis, including parameter fitting, using MC-based data sets first, before turning to the experimentaldata. For each experimental scan i we generate an MCtwin (Sec. VIII H) from its averaged slow-control parame-ters to calculate the expected rate R β ( E ) i with the corre-sponding response function f ( E −(cid:104) qU (cid:105) ) i and backgroundrate R bg ,i . Analyzing the MC twins allows us to verifythe accuracy of our parameter inference by recovering thecorrect input MC values for m ν .This MC dataset is used to assess statistical ( σ stat )and systematic ( σ syst ) uncertainties and to compute ourexpected sensitivity. It is also used to benchmark the in-dependent analysis codes. At this stage, all model inputsand systematic uncertainties are frozen.Before the unblinding via incorporation of the unmod-ified FSD, a final benchmark was successfully performedon the data with the blinded FSD to verify that the in-5dependent analysis codes eventually lead to very consis-tent results. After this final test, the “true” FSD wasrevealed to the collaboration for the final neutrino-massanalysis of the data. The first, overnight fits – using theindependent analysis codes – already yielded preliminary,consistent results the very next morning. B. Covariance-matrix approach
Here, we report on our results using the covariance-matrix approach to include and propagate systematic un-certainties in the neutrino-mass fit. The spectrum calcu-lation code and methods used for this analysis are de-scribed in detail in Ref. [104].The free fit parameters in our analysis, θ , are inferredfrom the data points { R i } by minimizing the negativePoisson likelihood function − L ( θ ) = 2 (cid:88) i (cid:20) R model i ( θ , η ) − R i + R i ln (cid:18) R i R model i ( θ , η ) (cid:19)(cid:21) (18)where the summation is over scan steps i .The model points, denoted by R model i , depend on boththe model parameters θ and the systematic nuisance pa-rameters η (including column density and tritium iso-topolog concentrations). In the fit the nuisance terms η are fixed according to our best knowledge of operationalparameters averaged over KNM1.Since the β spectrum measured in this first KATRINscience run comprises a large number of observed eventsin each scan-step bin, the negative Poisson likelihoodfunction (Eq. 18) is replaced by the standard χ esti-mator χ ( θ ) = (cid:0) R − R model ( θ , η ) (cid:1) (cid:124) C − (cid:0) R − R model ( θ , η ) (cid:1) . (19)The covariance matrix C describes the correlated anduncorrelated model uncertainties, including both statis-tical and systematic uncertainties. This fit procedure hasbeen extensively tested by injecting fake neutrino-masssignals in simulated pseudo-experiments. It was verifiedthat the fit results provide an unbiased estimation of theinjected parameters.Systematic uncertainties on the nuisance parameters η are propagated using covariance matrices. For thispurpose the values of η are randomized according totheir associated probability density functions. Correla-tions between parameters are taken into account. Sub-sequently, O (10 ) sample spectra { R sample } are simu-lated [20, 105, 106]. For each sample-spectrum calcu-lation, a different η is drawn from the set { η sample } .The signal normalization A s , being a free fit parameter,is not considered in the uncertainty propagation. There-fore, all fluctuations in { R sample } that translate solelyinto an overall signal normalization uncertainty must be eliminated. The transformation of { R sample } into shape-only sample spectra is achieved by normalizing the statis-tics of each sample spectrum to the statistics of the av-erage sample spectrum.Finally, the shape-only covariance matrix is estimatedfrom { R sample } using the sample covariance as an esti-mator. For any set of uncorrelated systematic effects,the associated covariance matrices can be calculated in-dependently of one another. The sum of all matrices en-codes the total uncertainties on the model points R model and their scan-step-dependent correlations.In the fit, χ ( θ ) is minimized to determine the best-fit parameters ˆ θ , whereas the profile of the χ functionis used to infer the uncertainties on ˆ θ . Once the co-variance matrices are pre-calculated, the spectral fit andmajor diagnoses can be performed within a few hours ona standard personal computer.The data and results of this fit are displayed in Fig. 19.Of the four free parameters, the signal amplitude A s isunconstrained for the shape-only analysis. The effective β -decay endpoint E can be related to the Q-value afterfinal corrections of the energy scale (Sec. XII). The back-ground rate R bg is primarily constrained by the 5 HVscan steps above E . The squared neutrino mass m ν canbe varied freely and therefore can take any positive ornegative value.We find a best-fit value of m ν = ( − . + 0 . − . ) eV with a goodness of fit of χ = 21 . . χ -value at least as large as the one obtained.The total uncertainty budget of m ν is first calculatedon an Asimov data set assuming the null hypothesis.Based on the final fit applied to these simulated data,we derive m ν = 0 . +0 . − . eV . The relative impact ofeach systematic effect is assessed by performing a seriesof fits, each one including solely the selected effect inaddition to statistical uncertainties (stat+1 test). Thestatistical uncertainty is then subtracted in quadrature.The same breakdown is then calculated using the un-blinded data, and is in excellent agreement with our MCexpectations. This data-driven uncertainty breakdown isshown in Table IV. As expected, the total uncertaintyis largely dominated by σ stat (0.94 eV ) as compared to σ syst (0.30 eV ). C. Monte-Carlo-propagation approach
Here we report the fit results using the MC-propagation approach to propagate systematic uncer-tainties. The spectrum-calculation code used is de-scribed in Ref. [107] while the method is adapted fromRefs. [102, 108, 109].In the MC-propagation method, we repeat the fittingprocess ∼ times, each time with newly randomizedinput values for the systematic nuisance parameters η that are held fixed during that fit. Compared to the well-6 a) C oun t r a t e ( c p s ) KATRIN data with 1 error bars 50 Fit resultb) R e s i du a l s () Stat. Stat. and syst.c)
Retarding energy (eV) T i m e ( h ) FIG. 19. a) Spectrum of β electrons R ( (cid:104) qU (cid:105) ) over a 86 eV-wide interval from all 274 tritium scans and best-fit model R model ( (cid:104) qU (cid:105) ) (line). The integral β -decay spectrum extendsup to E on top of a flat background R bg . Experimental dataare stacked at the average value (cid:104) qU (cid:105) of each scan step andare displayed with 1 σ statistical uncertainties enlarged by afactor of 50 for visibility. b) Residuals of R ( (cid:104) qU (cid:105) ) relative tothe 1 σ uncertainty band of the best-fit model. c) Integralmeasurement-time distribution of all 27 scan steps; see alsoFig. 7. Figure reproduced from Ref. [21]. known approach of free nuisance parameters constrainedwith pull terms, this method has two key advantages forthe KATRIN analysis. Foremost, the computationallyexpensive response function does not have to be recom-puted with varying η during the fit. In addition, theminimization is technically simplified due to the reducednumber of free parameters.To retrieve an initial estimate of the best-fit val-ues ˆ θ data of our four fit parameters θ (that is, m ν , E , A s , R bg ) and to assess their statistical uncertain-ties, we fit the original data with the additional parame-ters η fixed to our best knowledge from the experiment.Next, we generate MC spectra assuming the values ˆ θ data for our model and a Poisson distribution of the counts.We then fit each of these statistically randomized MCspectra, retrieving one sample of values ˆ θ statsample for ourfree parameters. The resulting distribution of { ˆ θ statsample } can be used to infer the statistical uncertainty of θ .Our next step is to assess the systematic uncertainties,beginning by varying the values of η according to theiruncertainties. The model is initialized with the values η sample . We then fit the model to the data spectrumwithout statistical randomization to retrieve the sampleof our free parameters ˆ θ systsample . In principle, the resultingdistribution of { ˆ θ systsample } reflects the systematic uncer- TABLE IV. Uncertainty breakdown obtained from the datain the covariance-matrix (Sec. IX B) and MC-propagation(Sec. IX C) approaches. The statistical uncertainty is drawnfrom data in all cases. Covariance-matrix results are aver-aged over positive and negative uncertainties. Section refer-ences are provided for the two systematics that are neglectedin both approaches.Effect m ν uncertainty (1 σ ; 10 − eV )Cov. matrix MC prop.Background rate 22 30Scan fluctuations 14 5Background slope 9 7Final-state distribution 9 2Magnetic fields 7 5Expected number ofscatterings ( ρdσ ) 5 5Detector efficiency 2 NeglectedEnergy loss < < < All included systematics
30 31Statistical 94 97 tainty, taking into account only the external informationon η . However, the data also contain information toconstrain η . To account for this, we weight each sample ˆ θ systsample by the corresponding likelihood L ( ˆ θ systsample ). Theresulting weighted distribution of { ˆ θ systsample } weight is thenused to retrieve the systematic uncertainty on θ .In the final step, we combine the statistics- andsystematics-only steps described above. We first fit themodel, initialized with randomized η sample , to statisti-cally randomized MC spectra to retrieve the values of ˆ θ totsample . This model is then also fit to the unmodified dataspectrum to retrieve the likelihood L ( ˆ θ totsample ). We in-fer the combined statistical and systematical uncertaintyfrom the distribution of { ˆ θ totsample } weight , which is weightedby these likelihood values.Initially we apply this method to the MC twin datadescribed in Sec. VIII H). From the statistics-only fit wederive m ν = 0 . +0 . − . eV . Including the systematic un-certainties described in Sec. VIII, the best-fit value be-comes m ν = 0 . +0 . − . eV . This is only a slight changewith respect to the statistics-only analysis.After freezing the method and inputs on MC spec-tra, we repeat the analysis on the data. Here thestatistics-only fit to the data gives a best-fit value of m ν = − . +0 . − . eV at a goodness-of-fit of − L =23 . . m ν = − . +0 . − . eV . The one-dimensional m ν distribu-tions used to derive these values are shown in Fig. 20.7 m (eV )0.000.050.100.150.200.250.300.350.40 P r o b a b ili t y d e n s i t y ( e V ) FIG. 20. One-dimensional distribution of m ν with statisticaluncertainty only (orange) as well as statistical and systemat-ical uncertainty combined (blue). The dashed lines indicatethe 1 σ confidence interval for each case. Using the MC propagation of uncertainty, it is possi-ble to analyze the impact of individual systematic effectson the parameters of interest. Table IV, further above,shows the uncertainty budget on m ν for KNM1. D. Fit results
The results of the two independent methods ofSecs. IX B and IX C agree to within a few percent ofthe total uncertainty. As a best-fit value for the squaredneutrino mass, we quote m ν = − . +0 . − . eV . This best-fit result corresponds to a 1 σ statistical fluctuation tonegative values of m ν . Assuming the true neutrino massis zero, the probability to retrieve a best-fit value as neg-ative as ours is 16 % and is thus fully compatible withstatistical expectations. The total uncertainty budgetof m ν is largely dominated by σ stat (0 .
97 eV ) as com-pared to σ syst (0 .
32 eV ). The dominant contributionsto σ syst are found to be the non-Poissonian backgroundfrom radon and the uncertainty on the background slope.Uncertainties on the column density, energy-loss func-tion, FSD, and magnetic fields play a minor role in thebudget of σ syst . Likewise, the uncertainties induced byfluctuations of ε T and HV parameters during a scan arenegligibly small compared to σ stat .For the effective β -decay endpoint we find a best fitvalue of 18 573 . m ν and E . The large correlation (0.97) betweenthe two parameters is in line with expectation [3, 99].For completeness, we report here that our best-fitbackground rate is R bg = 293(1) mcps. The signal-normalization parameter A s absorbs the rate effectsof our systematic uncertainties, and does not have astraightforward interpretation. m (eV ) E ( e V ) FIG. 21. Scatter plot of fit values for the squared neutrinomass m ν and the effective β decay endpoint E togetherwith 1 σ (black) and 2 σ (blue) contours around the best-fitpoint (white cross). Results are generated from a large set ofpseudo-experiments emulating our experimental data set andits statistical and systematical uncertainties – each one an in-dividual sample from the MC propagation. Figure reproducedfrom Ref. [21]. X. FREQUENTIST BOUNDS ON THENEUTRINO MASS
The result of a neutrino-mass experiment is commonlypresented in form of a confidence interval for the neutrinomass, or an upper limit if the lower boundary of the confi-dence interval is zero. These values are used by the com-munity for constraining phenomenological models, devel-oping theoretical predictions, and comparing the resultsof different experiments, and as input parameters to bothterrestrial experiments and cosmological observations.There are several methods of constructing the confi-dence intervals with additional information on the esti-mated parameter. To account for the physical bound of m ν ≥
0, despite the fact that m ν is unconstrained inthe fit, we perform full Neyman constructions using themethods of Lokhov and Tkachov and of Feldman andCousins, for completeness. Both methods avoid emptyconfidence intervals for negative best-fit estimates (cid:98) m ν .We briefly compare them below.In the Feldman-Cousins method [22], the likelihood ra-tio L (cid:0) (cid:98) m ν | m ν (cid:1) L ( (cid:98) m ν | max(0 , (cid:98) m ν )) (20)determines the order in which the estimates (cid:98) m ν are addedto the acceptance region for an assumed value of m ν ,thereby constructing the confidence interval. This order-ing principle avoids the problem of empty intervals, butat the same time results in more stringent limits for neg-8ative best-fit estimates that are further from zero, as inFig. 22a. This yields an unreasonably strict upper limitin the case of statistical fluctuations in one direction, orin the presence of an unknown systematic bias as seen inmost neutrino-mass experiments of the early 1990s (seeFig. 26). While our best-fit result is statistically compat-ible with zero, we decided after unblinding to pursue analternative approach to ensure a conservative handling offluctuations.Following the prescription of Lokhov and Tkachov [23],a new estimator (cid:101) m ν can be defined such that (cid:101) m ν = max( (cid:98) m µ , . (21)The estimator is by definition as close as possible to theunknown true non-negative value of the m ν , which isthe fundamental aim of the statistical estimation. Theconfidence interval for the new estimator (cid:101) m ν is then con-structed according to the Neyman procedure, which guar-antees the correct coverage. The non-physical values ofthe best-fit estimate (cid:98) m ν are indistinguishable and givethe same confidence interval from zero to the experimen-tal sensitivity (Fig. 22b). Therefore more negative valuesof m ν , obtained due to a statistical fluctuation or an im-properly treated systematic contribution, do not yieldbetter upper limits. This makes it possible to comparethe upper limits of different measurements directly with-out the need to know the best-fit estimate, as long as m ν is not significantly positive.In order to allow the squared-neutrino-mass estima-tor to become negative in either analysis, the differentialspectrum shape must be extended into the unphysicalregion of m ν <
0. In previous experiments [12, 13] theextension was made by modifying the differential spec-trum shape so that the χ function became symmetricaround m ν = 0. Such a modification depends on theparticular shape of the χ function and consequently onthe experimental setup. In the present analysis we takethe differential spectrum shape in Eq. (8) without anymodification for m ν <
0. This leads to a χ functionwith an asymmetric shape, as shown in Fig. 20. TheLokhov-Tkachov method yields the same upper limit forall m ν <
0. Therefore, by construction, the upper limitdoes not depend on a particular choice of the extension.Using the Lokhov-Tkachov construction we derive anupper limit of m ν < . m ν < . m ν < . m ν < . XI. BAYESIAN BOUND ON THE NEUTRINOMASS
Bayesian analysis methods provide an alternativemeans of handling the unphysical, m ν < A s , E , and R bg ; this choice is moststraightforward for analysis of stacked spectra. An in-formative prior, restricting the result to only physicallyallowed m ν values (equal to or larger than zero), is usedto ultimately obtain an upper credibility limit on the neu-trino mass in a Bayesian interpretation. In the allowedregion, this prior is flat in m ν space. Future work willinvestigate alternate choices of prior, including a priorflat in m ν .First, we extract statistical uncertainties and comparewith other analysis methods using the basic model, in-cluding the four-parameter set θ with flat prior prob-abilities. The global mode (maximum value) of the 4-dimensional posterior for m ν is found at − . . Thetwo-sided 1 σ interval, with equal probability on eitherside, is obtained from the posterior distribution marginal-ized for m ν as [ − . , − .
3] eV .Four of the leading systematic uncertainties are in-cluded in this analysis, and are incorporated into the fitin various ways. A background slope is included as a fifthfree parameter with a Gaussian prior probability centeredaround zero and a width given by its uncertainty. Non-Poissonian background counts are included by wideningthe underlying likelihood distribution in each scan stepaccording to background measurements (Sec. VI). Varia-tions of the response due to uncertainties in the magneticfield or the column density were too computationally ex-pensive at the time of the analysis. Instead of includingthese as free parameters in the model, multiple indepen-dent fits were parallelized on a computing cluster. Eachfit was started with the input systematic fixed at a differ-ent value, following a Gaussian distribution with a widthgiven by the parameter uncertainty. The median valuesof the output posterior distributions were used to ob-tain parameter estimates with systematic uncertainties.The same results are obtained by combining the Markovchains of the individual fits into a single chain, and subse-quently performing the same parameter-estimation pro-cedure. Additional systematics will be analyzed in futurework.The present dataset is strongly dominated by statisti-cal uncertainties, and individual systematic effects arelargely masked below 0 . by numerical uncertain-ties. These uncertainties come from the finite numberof Markov-chain Monte-Carlo samples and are on the or-der of 0 .
006 eV in the 1 σ posterior width. Hence, thesystematic budget was investigated with Asimov data,9 -3 -2 -1 0 1 2 3 Measured m (eV ) T r u e m ( e V )
90% C.L. (stat. and sys.) m m (a) Feldman-Cousins -3 -2 -1 0 1 2 3 Measured m (eV ) T r u e m ( e V )
90% C.L. (stat. and sys.) m m (b) Lokhov-Tkachov FIG. 22. The confidence belts constructed for m ν using the methods of Feldman and Cousins (a) and of Lokhov and Tkachov(b). The blue contour represents the 90 % CL confidence belt, which takes into account statistical and systematic uncertainties.The orange vertical line corresponds to the best-fit estimate (cid:98) m ν , while the horizontal line then defines the upper limit on m ν .In the case of LT, the upper limit coincides with the KNM1 experimental sensitivity to m ν . artificially increasing the amount of data and thus en-hancing each included systematic effect with respect tostatistical uncertainties.Taking these four explicitly included systematic un-certainties into account, the most probable m ν value wasfound at − . and the two-sided, 1 σ , probability-symmetric interval at [ − . , − .
3] eV . Using Table IVto estimate upper bounds on the primary excluded sys-tematics – scan fluctuations and the FSD – we find thatthey affect the total uncertainty on this most probablevalue by about 1%.To determine the limit on the neutrino-mass, we thenperform the same fits with a flat prior in m ν ≥
0. The m ν marginalized posterior distribution is shown in Fig. 23.The best-fit value is found at m ν = 0. The 90 % quantileof the marginalized posterior distribution is at 0 .
78 eV .The Bayesian upper limit is thus m ν < . m ν -space gives equalprobability for statistical fluctuations in the data. In ourcase, the Bayesian 90% credibility limit is numericallycloser than the Feldman-Cousins 90% confidence limitto the sensitivity of the experiment and to the Lokhov-Tkachov limit, as is often observed in the presence oflarger statistical fluctuations.As an additional test, the positive flat prior was slightlymodified by knowledge from oscillation experiments, al-lowing only m ν > m ν >
50 meV (inverted ordering) [6]. The posterior quantilesshow no numerical difference, as is expected with thecurrent data.
XII. Q-VALUE MEASUREMENT
A consistency check of the energy scale of KATRIN canbe performed by extracting the experimental Q-value for m (eV )0.0000.0050.0100.0150.0200.0250.0300.035 P r o b a b ili t y ( m | d a t a ) C u m u l a t i v e p r o b a b ili t y % C.L. m <0.8 eV , m <0.9 eV FIG. 23. Posterior probability distribution using a flat priorfor m ν ≥ (blue curve). Also shown are the cumulativeprobability (orange curve) and the 90 % quantile (solid blackline) used for limit setting (dashed black line). molecular tritium from KATRIN data, and comparing itto Q-values based on Penning-trap measurements of the He-T atomic mass difference. The Q-value representsthe amount of kinetic energy released in β decay for zeroneutrino mass; Fig. 24 shows its relationship to the massdifference and the binding energies of the atomic andmolecular states involved in T β decay. In equationform, we have:∆ M ( He , T) − Q (T ) = E ion (T) − E D ( HeT) + + E D (T ) . (22)Table V summarizes literature values for the relevantenergies. Inserting these values into Eq. 22, we obtain Q (T ) ∆ M = ∆ M ( He , T) − . . FIG. 24. Energy diagram showing the connection betweenthe atomic mass difference ∆M( He,T) and the Q-value ofmolecular tritium Q(T ).TABLE V. Relevant energies for deducing the Q-value of T β decay from measured mass differences. E D denotes a dis-sociation energy and E ion denotes an ionization energy. Theionization energy of tritium is calculated as R H 11+ m e m T , with R H the Rydberg constant.Quantity Value (eV) Reference∆ M ( He , T) 18 592 . E D (T ) 4 . E D ( HeT + ) 1 . E ion ( T ) 13 . R H for the Q-value derived from the measured He-T atomic-mass difference.The KATRIN result for the Q-value in molecular tri-tium β decay is derived from the best-fit value of E withcorrections for the center-of-mass molecular recoil of the HeT + daughter ion, as well as the relative offset of theelectron starting potential in the source to the work func-tion of the inner electrode of the main spectrometer.For the effective endpoint, our two fitting methodsboth obtain a best-fit value of E = 18 573 . HeT + molecule is givenby E rec = E + 2 E m e m HeT + = 1 .
720 eV . (24)E-gun data were used to investigate the work func-tion of the inner electrode system of the main spectrom-eter. First, the work function of this electron source wasmeasured with the Fowler method [114] to be Φ egun =4 . IE = 4 . β -electron starting potential inside the tritium source is defined by the cold and strongly magnetizedplasma within its boundary conditions at the rear walland the grounded beam tube (Sec. III C). By assumingthat the magnitude of the plasma potential is small, asindicated by the Kr measurement campaign, we treatthe electron starting potential as mainly defined by thebias voltage and work function of the gold-plated rearwall, especially at small radii.The work function of the rear wall was measured withthe Fowler method prior to KNM1. Due to the illumina-tion conditions, only the inner two-thirds of its area couldbe used for the measurement. The resulting raw, meanvalue from this measurement is Φ vacRW = 4 .
29 eV. How-ever, this measurement was performed with an evacuatedsource. Previous measurements with deuterium gas indi-cate that the work function changes by about −
100 meVwhen the rear wall is exposed to hydrogen isotopes in thesource, as is the case during tritium operation. This es-timate of the in situ work function of the rear wall has alarge uncertainty, which we estimate at about ±
200 meV.Further, during KNM1 the rear wall was set to a voltageof U RW = −
150 mV, which is numerically equivalent toan increase of the work function by 150 meV. These con-siderations lead us to estimate an actual rear-wall workfunction of Φ RW = 4 . ±
100 mV forthe sum of all involved voltages. The main contributionto this is the uncertainty of the absolute voltage of themain spectrometer, ∆ U abs = ±
94 mV [56]. The dom-inant uncertainty for the Q-value determination is thepossibility of a plasma potential in the source that dif-fers from the rear-wall potential. We assume an uncer-tainty of U plasma = ±
400 mV because we cannot directlyprobe the plasma potential under KNM1 operational con-ditions. Our final result is then: Q (T ) KNM1 = E + E rec − . ± . . . (25) Q (T ) KNM1 and Q (T ) ∆ M agree within uncertainties.Figure 25 shows a comparison of the obtained Q-valuein KATRIN with values derived from Penning-trap mea-surements. The consistency of the Q-values underlinesthe robustness of the energy scale in our scanning mea-surement of the T β spectrum. XIII. RESULTS AND DISCUSSION
In this work we have presented the first neutrino-massmeasurement campaign of the KATRIN experiment. Theacquired high-precision T β decay spectrum, contain-ing a total of 2 million electrons in an energy range of[ E −
37 eV, E +49 eV], was compared against a modelof the theoretical spectrum, incorporating relevant ex-perimental effects such as electromagnetic fields, back-grounds, and scattering. The experiment was operatedat a reduced column density. Taking into account boththe reduced activity and the reduced scattering proba-bilities, the β electrons recorded in the ROI during our1 Van Dyck '93 Nagy '06 Wang '12 Myers '15 KATRIN '19Measurement185721857318574185751857618577 Q - v a l u e o f T ( e V ) Penning trap measurementsKATRIN KNM1
FIG. 25. Comparison of the Q-value of molecular tritiumfound in this work to values derived from Penning-trap mea-surements of the atomic-mass difference. In chronological or-der, the values of the Penning-trap measurements are thosereported in Refs. [115], [116], [117] and [111]. four-week KNM1 campaign correspond to just 9 days ofmeasurement time at the full, design source strength.The full analysis was carried out applying a multi-stageblinding scheme. All analysis inputs were fixed on MCtwin copies of the data; the spectrum model was blindedwith a modified molecular final-state distribution; andfinally the full analysis was performed using two inde-pendent analysis techniques (covariance matrix and MCpropagation) which revealed a high degree of consistency.We find excellent agreement of the calculated spectrumwith the data. The covariance-matrix fit method obtainsa goodness of fit of of χ = 21 . .
56) and the MC-propagationtechnique finds a goodness-of-fit of − L = 23 . . E , which is inferredfrom the spectral fit alongside m ν , can be related to thenuclear Q-value using the molecular recoil and the off-set between the source potential and spectrometer workfunction. Our analysis gives a Q-value of 18 575 . He- H atomic mass difference [111]. While theneutrino-mass result does not depend on the absoluteenergy scale of the spectrum, this consistency check isstill of major importance to our understanding of the ob-tained spectra.The best fit of the squared neutrino mass was foundat m ν = − . +0 . − . eV . The uncertainty is largely domi-nated by the statistical error of σ stat ( m ν ) = 0 .
97 eV .If one were to assume the true neutrino mass to beequal to zero, the probability of obtaining this fit resultgiven our total error budget is 16 %. The best-fit resultsof the covariance-matrix and MC-propagation techniquesagree within 2 %.We have applied three methodologies to derive an up-per limit on the neutrino mass, based on the best-fit re-sult. The Lokhov-Tkachov limit construction was de-veloped in particular for direct neutrino-mass experi-ments [23]. By construction, in the case of a negative best-fit value of m ν it yields the experimental sensi-tivity as an upper limit. Based on this technique wefind m ( ν e ) < . . m ν through aflat, positive prior. The Bayesian result is presented inthis work for the first time, yielding a 90 % credibilityinterval of 0 to 0 . m ν by afactor of two compared to the final results of the Troitskand Mainz experiments [12, 13] (Fig. 26, bottom), whilethe systematic uncertainties are reduced by a factor ofsix (Fig. 26, center).The systematic error budget is expected to improvewith future measurement campaigns. Most notably, newmeans to further suppress the background rate are now inplace. These will increase the signal-to-background ratioand at the same time reduce the dominant systematic un-certainties related to the dependence of the backgroundon time and retarding potential. Furthermore, in thisfirst measurement the activity stability suffered from aburn-in phase, in which the structural material was ex-posed to tritiated gas for the first time. Subsequent tothis first campaign, significant improvements of the ac-tivity stability have been demonstrated at an increasedintensity about four times the KNM1 source strength.Finally, sub-dominant systematic effects, such as uncer-tainties in the final-state distribution, have been conser-vatively estimated for this analysis. Our knowledge ofthese systematics is expected to improve significantly inour future commissioning and measurement phases. XIV. CONCLUSION
The new upper limit m ν < . . -200-150-100-50050 m ( e V ) Los Alamos (1991)Tokyo (1991)Zurich (1992) Mainz (1993)Beijing (1993)Livermore (1995) Troitsk (1995)Mainz (1999)Troitsk (1999) Mainz (2005)Troitsk (2011)KATRIN (2019) m ( e V ) m s y s t e m a ti c un ce r t a i n t y ( e V ) a)b)c) Year m s t a ti s ti ca l un ce r t a i n t y ( e V ) FIG. 26. Overview of the neutrino-mass results obtained fromtritium β decay in the period 1990–2019, plotted against theyear of publication. (a) Squared-neutrino-mass results; theinset shows a more detailed comparison to results from themost recent experiments, Mainz and Troitsk. (b) System-atic uncertainties and (c) statistical uncertainties, both onthe squared neutrino mass. The total uncertainty is reducedby a factor of three. The historical measurements plottedhere are: Los Alamos (1991) [18], Tokyo (1991) [118], Zurich(1992) [119], Mainz (1993) [120], Beijing (1993) [121], Liver-more (1995) [103], Troitsk (1995) [122], Mainz (1999) [123],Troitsk (1999) [124], Mainz (2005) [12], Troitsk (2011) [13]. physics and cosmology. In particle physics, this mea-surement narrows the allowed range of quasi-degenerateneutrino-mass models. In cosmology, it provides laboratory-based input for studies of structure evolutionin ΛCDM and other cosmological models. In the absenceof a definitive observation of dark matter, the neutrino-mass scale is unique as a ΛCDM parameter that is di-rectly observable in the laboratory.Upcoming cosmological probes are expected to achievea determination of the sum of the neutrino masses overthe next 5 to 10 years, making this laboratory measure-ment particularly important for obtaining a consistentpicture of the neutrino as both particle and dark-matterconstituent in the universe. This first KATRIN resultserves as a milestone towards this goal. ACKNOWLEDGMENTS
We acknowledge the support of Helmholtz Association,Ministry for Education and Research BMBF (5A17PDA,05A17PM3, 05A17PX3, 05A17VK2, and 05A17WO3),Helmholtz Alliance for Astroparticle Physics (HAP),Helmholtz Young Investigator Group (VH-NG-1055),Max Planck Research Group (MaxPlanck@TUM), andDeutsche Forschungsgemeinschaft DFG (Research Train-ing Groups GRK 1694 and GRK 2149, Graduate SchoolGSC 1085 - KSETA, and SFB-1258) in Germany;Ministry of Education, Youth and Sport (CANAM-LM2015056, LTT19005) in the Czech Republic; Ministryof Science and Higher Education of the Russian Feder-ation under contract 075-15-2020-778; and the UnitedStates Department of Energy through grants DE-FG02-97ER41020, DE-FG02-94ER40818, DE-SC0004036, DE-FG02-97ER41033, DE-FG02-97ER41041, DE-AC02-05CH11231, DE-SC0011091, and DE-SC0019304, andthe National Energy Research Scientific Computing Cen-ter. 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