\(\upbeta \)Decay spectrum, response function and statistical model for neutrino mass measurements with the KATRIN experiment
Abstract
The objective of the Karlsruhe Tritium Neutrino (KATRIN) experiment is to determine the effective electron neutrino mass \(m(\upnu _\text {e})\) with an unprecedented sensitivity of \(0.2 \hbox {eV}/\hbox {c}^2\) (\(90 \%\,\hbox {C.L.}\)) by precision electron spectroscopy close to the endpoint of the \(\upbeta \)decay of tritium. We present a consistent theoretical description of the \(\upbeta \)electron energy spectrum in the endpoint region, an accurate model of the apparatus response function, and the statistical approaches suited to interpret and analyze tritium \(\upbeta \)decay data observed with KATRIN with the envisaged precision. In addition to providing detailed analytical expressions for all formulae used in the presented model framework with the necessary detail of derivation, we discuss and quantify the impact of theoretical and experimental corrections on the measured \(m(\upnu _\text {e})\). Finally, we outline the statistical methods for parameter inference and the construction of confidence intervals that are appropriate for a neutrino mass measurement with KATRIN. In this context, we briefly discuss the choice of the \(\upbeta \)energy analysis interval and the distribution of measuring time within that range.
1 Introduction
It is the goal of this work to provide a complete and uptodate model of the experiment, such that it can be used as either a prescription or reference for upcoming analyses of tritium \(\upbeta \)decay data observed with KATRIN. For established aspects of this model, we refer to the appropriate publications. For those not yet published at all or not in the required detail, we provide the necessary derivations. The later will mostly be the case for the description of the experimental response function, which has been considerably refined during recent commissioning phases.
In this work we first present a detailed account of the theoretical \(\upbeta \) spectrum of tritium, with an emphasis on molecular effects in \(\mathrm {T_2}\) (Sect. 2). We then outline the experimental configuration of KATRIN (Sect. 3), before we elaborate on the individual characteristics that define the response of our instrument in Sect. 4. The statistical techniques suited to determine the effective neutrino mass from a fit of the modeled \(\upbeta \) spectrum to the measured data are treated in Sect. 5. A summary of this work is given in Sect. 6.
Throughout this article we use natural units (\(c = \hbar = 1\)) for better readability, except for Sects. 4.7 and 4.8 where we use SI units instead.
2 Theoretical description of the differential \(\upbeta \)decay spectrum
In this section we compile a comprehensive analytical description of the differential \(\upbeta \)decay spectrum, with specific focus on gaseous molecular tritium \(\mathrm {T_2}\), the \(\upbeta \) emitter used by KATRIN. We will also evaluate the relevance of various theoretical correction terms on the neutrino mass analysis.
In the following, we use the shorthand notation \( {m_{\upnu }} = m(\upnu _\text {e})\) for better readability. Furthermore, we assume there is no difference between the masses of the neutrinos and the antineutrinos, i.e. \( {m_{\upnu }} = m(\upnu _\text {e}) = m({\bar{\upnu }}_\text {e})\).
2.1 Fermi theory
The full spectrum is an incoherent sum over the three known neutrino mass eigenstates \(m_i\) (\(i=1,2,3\)) with the intensity of each component defined by the squared magnitude of the neutrino mixing matrix elements \(U_{\text {e}i}^2\) [11].
The phasespace factor of the outgoing electron with momentum p is given by the factor \(p\, (E+ {m_\text {e}} )\). The phase space of the emitted neutrino is the product of the neutrino energy \(\epsilon = E_0  E\) and the neutrino momentum \(\sqrt{ \epsilon ^2  m_i^2}\), which determines the shape of the \(\upbeta \)electron spectrum near the tritium endpoint \(E_0\). The Heaviside step function \(\varTheta \) ensures that the kinetic energy cannot become negative.
2.2 Neutrino mass eigenstate splittings
2.3 Molecular tritium \(\mathrm {T_2}\)
2.4 Excited molecular final states
After the decay, the daughter molecular system is left in an excited rotational, vibrational and electronic state. According to theoretical calculations, about 57 % of all \(\mathrm {T_2}\) \(\upbeta \)decays result in the rovibronicallybroadened electronic ground state with an average excitation energy of about 1.7 eV, while the others go to the excited electronic states [17]. Each discrete final state effectively branches into its own \(\upbeta \) spectrum with a distinct endpoint energy.
The accuracy of a neutrino mass measurement critically depends on the knowledge of the distribution of these final states, which have to be taken from theory. Precise calculations of the final state distributions of the hydrogen isotopologues (\(\mathrm {T_2 \rightarrow HeT^+}\), \(\mathrm {DT \rightarrow HeD^+}\) and \(\mathrm {HT \rightarrow HeH^+}\)) have been performed in the endpoint region [15, 16]. The discrete energy states and their transition probabilities have been determined below the dissociation threshold, while continuous distributions are available above the threshold. A comprehensive review of the theory of the tritium finalstate spectrum and current validation efforts can be found in [18].
Figure 3 gives a comparison of the finalstate distributions of \(\mathrm {HeT^+}\) and \(\mathrm {HeD^+}\). The differences in their distributions arise from the mass difference; thus, a precise knowledge of the source gas isotopological composition and its stabilization on the 0.1 %level are necessary. Laser Raman spectroscopy [19] provides two important input parameters for our source model: the tritium purity \(\epsilon _\mathrm {T}\) denoting the fraction of tritium nuclei,^{1} and \(\kappa \) denoting the ratio of \(\mathrm {DT}\) versus \(\mathrm {HT}\).
2.5 Exact relativistic threebody calculation
2.6 Additional correction terms

Radiative corrections: In addition to the Coulomb interaction described by F(Z, E), electromagnetic effects involving contributions from virtual and real photons give rise to a correction factor \(G(E,E_0)\).

Screening: The unscreened F(Z, E), which describes the Coulomb interaction between the daughter nucleus and the departing \(\upbeta \)electron, must be corrected by a factor S(Z, E) that accounts for the screening effect on the Coulomb field by the \(\text {1s}\)orbital electrons left behind by the parent molecule.

Recoil effects: In the relativistic elementary particle treatment of the \(\upbeta \)decay (see for instance [23, 24]), energydependent recoil effects on the order of 1 / M can be calculated, with M being the mass of \(\mathrm {^3He}\). These effects – spectrum shape modification due to a threebody phase space, weak magnetism and \(VA\) interference – are typically combined into a common factor \(R(E, E_0, M)\).

Finite structure of the nucleus: Because the \(\mathrm {^3He^+}\) daughter nucleus is not a pointlike object, the Coulomb field does not scale with an inversesquared relationship within the radius, leading to a correction factor L(Z, E). A proper convolution of the electron and neutrino wave functions with the nucleonic wave function throughout the nuclear volume leads to another factor C(Z, E).

Recoiling Coulomb field: The departing electron does not propagate in the field of a stationary charge, but one which is itself recoiling from the electron emission. This effect introduces another correction factor \(Q(Z, E, E_0, M)\).

Orbitalelectron interactions: A correction factor I(Z, E) is introduced to account for possible quantum mechanical interactions between the departing \(\upbeta \)electron and the \(\text {1s}\)orbital electrons.
In Fig. 5, a graphical overview of these correction factors in the energy interval 30 eV below the tritium endpoint is given. The radiative corrections have the most significant effect with a pronounced energy dependence, as they deplete the spectrum completely towards the endpoint. Most other corrections are negligible in the neutrino mass analysis, as further detailed in Sect. 4.12 and Table 1.
3 The KATRIN experiment
The experimental setup of KATRIN combines a highluminosity windowless gaseous molecular tritium source (WGTS) with an integrating electrostatic spectrometer of MACE filter (magnetic adiabatic collimation with electrostatic filter) type [25, 26, 27], offering a narrow filter width and a wide solidangle acceptance at the same time.
The apparatus depicted in Fig. 6 features several major subsystems. The isotopological composition, temperature, and density fluctuations of the tritium source are monitored by a set of calibration devices housed in the rear section (a). The windowless gaseous tritium source (b) contains a beam tube of length \(L = 10\hbox { m}\) and diameter \(d = 90\hbox { mm}\), residing in a nominal magnetic field of 3.6 T, where repurified molecular tritium (\(\mathrm {T_2}\)) is continuously circulated by injection at the center and pumping at both ends through a closed loop system [28, 29, 30]. To prevent tritiated gas from entering the spectrometer section, the transport section (c) combines differential pumping with cryogenic pumping to reduce the tritium flow by 14 orders of magnitude [31, 32]. The \(\upbeta \)electrons are guided through the entire beamline by a magnetic field [33] into the prespectrometer (d), which acts as a prefilter that blocks the lowenergy electrons of the \(\upbeta \)spectrum [34]. The energy analysis around the endpoint region takes place in the main spectrometer (e), which is operated under ultrahigh vacuum conditions [35] at a retarding voltage of about \(18.6\hbox { kV}\). Both spectrometers are designed as MACE filters, and the main spectrometer achieves a very narrow filter width (\(\lesssim 1\hbox { eV}\)) [9] while providing high luminosity for the \(\upbeta \)electrons. Electrons with sufficient energy pass both the MACE filters and are then counted at a segmented silicon PIN diode detector (f) [36] with 148 individual pixels. An integrated \(\upbeta \)spectrum is recorded by scanning the retarding voltage in the endpoint region.
3.1 MACE filter principle
The electrons emitted isotropically from tritium \(\upbeta \)decay in the gaseous source are guided adiabatically by magnetic fields. In the forward direction the \(\upbeta \)electrons are confined in cyclotron motion along the magnetic field lines towards the MACE filter. Along their path to the analyzing plane (central plane) of the spectrometer, the magnetic field strength decreases by several orders of magnitude.^{3} Due to the conservation of magnetic moment in a slowly varying field, most of the electrons’ transverse momentum is adiabatically transformed into longitudinal momentum. With a high negative potential (\(U \approx 18.6\hbox { kV}\), corresponding to the endpoint energy of tritium) at its center and most of the electron momentum being parallel to the magnetic field lines, the MACE filter acts as an electrostatic highpass energy filter. Only electrons with positive longitudinal energy (the kinetic energy in direction of the magnetic field line) along their entire trajectory are transmitted, while the others are reflected and reaccelerated towards the entrance of the spectrometer.
4 Response function of the KATRIN experiment
In the KATRIN experiment, the energy of the \(\upbeta \)electrons is analyzed using the MACE filter technique as described in Sect. 3. For a specific electrostatic retardation potential U, the count rate of electrons at the detector can be calculated, given the probability of an electron with a starting energy E to traverse the whole apparatus and hit the detector. This probability is described by the socalled transmission function T(E, U). Additional modifications arise from energy loss and scattering in the source, and reflection of signal electrons propagating from their point of origin until detection. These effects are incorporated together with the transmission function into the response function R(E, U), which is vital for the neutrino mass analysis as it describes the propagation of signal electrons that contribute to the integrated \(\upbeta \)spectrum.
In the following, an analytical description of the response function of the KATRIN experiment will be laid out. At first, we derive the transmission function of the MACE filter that is implemented by the main spectrometer (Sect. 4.1). In Sect. 4.2 we consider energy loss in the source and develop a first description of the response function. Inhomogeneities in the MACE filter (Sect. 4.3) and the source (Sect. 4.4) requires extension of the model by a segmentation of the source and spectrometer volume. Further modifications to the response function arise from considering the effective source column density which an individual \(\upbeta \)electron traverses (Sect. 4.5), changes to the electron angular distribution (Sect. 4.6), thermal motion of the source gas (Sect. 4.7), and energy loss by cyclotron radiation (Sect. 4.8). After discussing these contributions, in Sect. 4.9 we arrive at a description of the integrated spectrum that is measured by the KATRIN experiment. We close the discussion with a general note on experimental energy uncertainties (Sect. 4.11) and give a quantitative overview of theoretical corrections and systematic effects (Sect. 4.12) on the neutrino mass analysis.
4.1 Transmission function of the MACE filter
The transmission of \(\upbeta \)electrons through the MACE filter is an important characteristic of the measurement and a significant part of the response function. In the simplest case, one can assume that electrons enter the MACE filter with an isotropic angular distribution and propagate adiabatically towards the detector. In the discussion here we apply the adiabatic approximation (see Eq. (15) below), which is fulfilled in the case of KATRIN.
4.2 Response function and energy loss
In the next step we consider the energy loss when the electron traverses the gaseous source. The dominant energy loss process is the scattering of electrons on gas molecules within the source. Because the pressure decreases rapidly outside the source, scattering processes in the transport section or thereafter are of no concern.
Two ingredients are required to appropriately treat electron scattering in the source. First, the energy loss function \({\tilde{f}}(\epsilon , \delta \vartheta )\) describes the probability for a certain energy loss \(\epsilon \) and scattering angle \(\delta \vartheta \) of the \(\upbeta \)electrons to occur in a scattering process. Because the scattering angles \(\delta \vartheta \) are small,^{5} we will neglect them in the following formulae and describe the scattering energy losses by the function \(f(\epsilon )\). Here we do not consider a dependence of f or \(P_s\) on the incident kinetic energy E of the electrons, since for the KATRIN experiment the energy range of interest amounts to a very narrow interval of a few times 10 eV below the tritium endpoint only, where these functions can be considered as independent of E. The other important ingredients are the scattering probability functions \(P_s(\theta )\) for an electron with pitch angle \(\theta \) to scatter s times before leaving the source. These scattering probabilities depend on \(\theta \), since electrons with a larger pitch angle must traverse a longer path, meaning a larger effective column density, and are thus likely to scatter more often.
The scattering cross section can be divided into an elastic and an inelastic component. The inelastic cross section and the energy loss function for electrons with kinetic energies of \(\approx 18.6\hbox { keV}\) scattering from tritium molecules have both been measured in [40, 41]. In this work, the inelastic scattering cross section was determined to be \(\sigma _\text {inel} = (3.40 \pm 0.07) \times 10^{18}\,\hbox {cm}^2\) and an empirical model was fit to the energy loss spectrum.
This parameterization of the energy loss function is used for the response model presented in this paper. However, the parameters are not precise enough for KATRIN to meet its physics goals. Dedicated electron gun measurements with the full experimental KATRIN setup have been planned for the determination of the inelastic scattering cross section and the energy loss function with higher precision; the analysis of these data will involve a sophisticated deconvolution technique [42].
The elastic energy loss component can be accurately calculated. Due to its narrow width and steep slope, \(\sim \hbox {meV}\) binning is required for incorporating it accurately in the response function, thereby increasing computational cost considerably. We will neglect the elastic scattering component in neutrino mass measurements as the associated systematic error on an \( {m_{\upnu }^2} \) is minute (\(\sim 5.10^{5}\hbox {eV}^2\), see Table 1).
4.3 Radial inhomogeneity of the electromagnetic field
To calculate the transmission and response functions of the KATRIN setup as explained in Sects. 4.1 and 4.2, it is in principle sufficient to only consider the axial position of an electron to identify the initial conditions such as electromagnetic fields or scattering probabilities. In the case of the main spectrometer, radial dependencies must be incorporated in the description of the magnetic field and the electrostatic potential in the analyzing plane. Additional radial dependencies in the source are discussed in Sect. 4.4; these are then incorporated into the model together with the spectrometer effects.
Impact of individual theoretical and experimental model corrections on the measured squared neutrino mass \( {m_{\upnu }^2} \), if neglected or approximated. The analysis energy window is restricted to \([E_0  30\hbox { eV}; E_0 + 5\hbox { eV}]\). For \( {m_{\upnu }} \) a true value of 200 meV is assumed
Source of systematic shift  Systematic shift \(\varDelta _\text {syst}( {m_{\upnu }^2} )\) 

Neglected effect or model component  \(( \times {10}^{5}\,\hbox {eV}^2 )\) 
Relativistic description of \(E_\text {rec}^{\text {a}}\)  0.03 
Neutrino mixing with 3 mass eigenstates (inv. hierarchy)  0.04 
Relativistic Fermi function \(F_\text {rel}(Z,E)^{\text {a}}\)  0.19 
Radiative corrections (G)  214.10 
Screening correction (S)  \(\) 2.82 
Recoil, weak magnetism, \(VA\) interference corr. (R)  \(\) 0.12 
Finite nucl. ext. corr. (LC)  0.01 
Recoiling Coulomb field corr. (Q)  \(\) 0.02 
Orbital electron exch. corr. (I)  \(\) 0.02 
Calculate G, R, Q for each final state\(^{\text {b}}\)  13.50 
Energy loss due to elastic \(\text {e}^{}  \text {T}_2\) scattering  \(\) 5.20 
Transmission function \(T^\star \) (nonisotropic angular distr.)  1027.51 
Energy loss due to cyclotron radiation  \(\) 2939.43 
Radial dependence of analyzing magnetic field in \(R_j(E, U)^{\text {c}}\)  904.20 
Radial dependence of retarding potential in \(R_j(E, U)\)  8470.47 
Doppler effect (thermal and bulk velocity neglected)  \(\) 1554.46 
Doppler effect (only bulk gas velocity neglected)  117.81 
Doppler effect (only approximated by smearing the FSD)  101.41 
Key operational and derived parameters of KATRIN as defined in the technical design report [4]
Parameter  Value 

Column density  \(\mathcal {N}= 5\times 10^{17}\,\hbox {cm}^{2}\) 
Active source crosssection  \(A_\text {S} = 53\,\hbox {cm}^{2}\) 
Magnetic field strength (source)  \(B_\text {S} = 3.6\,\hbox { T}\) 
Magnetic field strength (analyzing plane)  \(B_\text {A} = 3.10^{4}\,\hbox { T}\) 
Magnetic field strength (maximum)  \(B_\text {max} = 6.0\,\hbox { T}\) 
Inelastic scattering cross section  \(\sigma _\text {inel} = 3.45 \times 10^{18}\,\hbox {cm}^{2}\) 
Scattering probabilities  \(P_0 = 41.33 \%\) 
\(P_1= 29.27 \%\)  
\(P_2 = 16.73 \%\)  
\(P_3 = 7.91 \%\)  
\(P_4 = 3.18 \%\)  
Detector efficiency  \(\epsilon _\text {det} = 0.9\) 
Even with these optimizations of the setup, the small radial variations in the electromagnetic fields at the analyzing plane, as shown in Fig. 9, cannot be neglected. The segmentation of the KATRIN main detector into annuli of pixels allows us to incorporate such radial variations in the response function model for each individual detector pixel. Because the tritium source also features radial variations of certain parameters, this segmentation is combined with a full segmentation of the source volume as described in Sect. 4.4. Dependencies of the electromagnetic field are typically averaged over the surface area of a pixel. The specific detector geometry with thinner annuli towards outer radii (each with equal surface area) helps minimize the potential variation within individual annuli, despite the increasing steepness of the potential.
4.4 Source volume segmentation and effects
In addition to radial dependencies of the analyzing plane parameters that govern the energy analysis of the \(\upbeta \)electrons (Sect. 4.3), the tritium source also features radial and axial dependencies of its parameters. In the following, we will briefly outline the most relevant source parameters that are required to accurately model the differential \(\upbeta \) spectrum and the response function. These parameters include the beam tube temperature \(T_\text {bt}\), the magnetic field strength \(B_\text {S}\), plasma potentials \(U_\text {P}\), the particle density \(\rho \) and the bulk velocity u of the gas, all of which may vary slightly in longitudinal, radial and azimuthal directions. The complex gas dynamic simulations, which are needed to calculate these local source parameters, are described in comprehensive detail in [47, 48].
In order to model accurately these effects for each individual detector pixel, the simulation source model is partitioned to match the detector geometry. It is partitioned longitudinally into \(N_L\) slices and segmented radially into \(N_R\) annuli (rings) of \(N_S\) segments each, resulting in a total of \(N_L \cdot N_R \cdot N_S\) segments (see Fig. 10). The geometry of these segments is chosen in such a way, that a longitudinal stack of segments is magnetically projected^{7} onto a corresponding detector pixel. Note that all detector pixels have identical surface area, which leads to broader annuli at the center and thinner annuli towards larger radii. In the following, we index the longitudinal slices by the subscript i and radial/azimuthal segments with their corresponding detector pixel by the subscript j.
4.5 Scattering probabilities
As discussed in Sect. 4.2, inelastic scattering results in an energy loss that directly affects the energy analysis of the signal electrons, and needs to be incorporated accurately into the analytical description. Changes to the angular distribution of the emitted electrons due to scattering processes, which also modify the response function, are discussed in Sect. 4.6.
4.6 Response function for nonscattered electrons
4.7 Doppler effect
With \(\sigma _v \approx 203\hbox { m/s}\) for \(\mathrm {T_2}\) molecules at \(T_\text {bt}=30\hbox { K}\) and the weighted mean bulk velocity at nominal source conditions being \({\bar{u}} \approx 13\hbox { m/s}\), thermal Doppler broadening clearly is a dominating effect. The standard deviation of the broadening function \(g(E_\text {cms}, E_\text {lab})\) at a fixed bulk velocity \(u = 0\) for \(T_\text {bt} = 30\hbox { K}\) and \(E \approx E_0\) evaluates to \(\sigma _E \approx 94\hbox { meV}\) (also see Fig. 13). This value can be interpreted as a significant smearing of the energy scale. Its implication for the neutrino mass measurement is shown in Table 1.
4.8 Cyclotron radiation
As electrons move from the source to the spectrometer section in KATRIN, they lose energy through cyclotron radiation. In contrast to energy loss due to scattering with tritium gas (Sect. 4.5), this energy loss process applies to the entire trajectory of an electron as it traverses the experimental beamline [49].
For complex geometric and magnetic field configurations as in the KATRIN experiment, the overall cyclotron energy loss can be computed using a particle tracking simulation framework such as Kassiopeia [50]. By this means, the cyclotron energy loss from the source to the analyzing point in the main spectrometer can be obtained as a function of the electron’s starting position z and pitch angle \(\theta \). Particles starting in the rear of the source will lose more energy due to their longer path through the whole setup. The total cyclotron energy loss can be up to 85 meV for electrons with the maximum pitch angle \(\theta _\text {max} = 50.8 ^{\circ }\).
4.9 Expected integrated spectrum signal rate
Earlier in this section we have laid out the different contributions to the response function of the experiment, which describes the probability for \(\upbeta \)electrons to arrive at the detector where they contribute to the measured integrated spectrum. The response function describes the energy analysis at the spectrometer (Sects. 4.1 and 4.3), energy loss caused by scattering in the tritium source (Sects. 4.2 and 4.5), and additional corrections (Sect. 4.6 and following).
To first order (due to nearly constant magnetic field and tritium concentration in the source), the integrated signal rate in Eq. (41) depends on \(\mathcal {N}\sigma \) – which can be accurately determined by calibration measurements with a photoelectron source – but is independent of the longitudinal gas density profile \(\rho (z)\) which cannot be measured directly (see [47, 48] for simulation results).
4.10 Scan of the integrated spectrum
A scan of the integrated \(\upbeta \) spectrum comprises a set of detector pixel event counts \(N_j(U_k)\), observed at various retarding potential settings \(U_k\) for the duration of \(\varDelta t_k\) each, with \(k \in \{1 \ldots n_k\}\). In the following, the indices j and k are condensed by writing \(N_{jk} = N_j(U_k)\), with \(N_{jk}\) denoting the event count on a single detector pixel j for a specific retarding potential setting k.
KATRIN will be operated for a duration of 5 calendar years in order to collect 3 live years of spectrum data over multiple runs.
4.11 Energy uncertainties
At the end of this section we will briefly discuss the influence of energy uncertainties on the neutrino mass measurement. In general, any fluctuation with variance \(\sigma ^2\) induces a spectrum shape deformation which – if not considered in the analysis – is indistinguishable to first order from a shift of the measured value of \( {m_{\upnu }^2} \) in the negative direction with \(\varDelta {m_{\upnu }^2} = 2 \sigma ^2\) [11]. This shift of \(\varDelta {m_{\upnu }^2} \) also holds if an accounted fluctuation or distribution of true variance \(\sigma ^2_\mathrm {true}\) is described wrongly in the analysis by the variance \(\sigma ^2_\mathrm {ana} = \sigma ^2_\mathrm {true}  \sigma ^2\).
Different sources of fluctuations and distributions with uncertainties can be distinguished. One group comprises \(\upbeta \)decay and source physics, such as molecular final states, scattering processes and the Doppler effect (all discussed in this work). Others are experimental systematics originating in the energy measurement, which have to be studied during commissioning of the setup and then incorporated into the model. An example is the distortion of the spectrometer transmission function due to retardingvoltage fluctuations [51, 52].
4.12 Impact of theoretical and experimental corrections
In Table 1 we review and quantify the impact of theoretical corrections to the differential \(\upbeta \)spectrum, discussed in Sect. 2, and of experimental corrections which have been introduced above. Many individual model components can be safely neglected, while others need to be considered more accurately, such as the radial dependence of retarding potentials (Sect. 4.3), energy loss due to cyclotron radiation (Sect. 4.8) or the Doppler effect (Sect. 4.7).
5 Measurement of the neutrino mass
Having compiled a complete description of the theoretical \(\upbeta \)decay spectrum and the response function of KATRIN into a parameterizable model, we will now outline the statistical terms and methods required for actual neutrino mass measurements. In the next (Sect. 5.1 and 5.2) we review the process of parameter inference (model fitting) and the construction of confidence intervals in the case of a KATRIN neutrino mass analysis, and we explain the relation between observed data, fit parameters and their uncertainties. After introducing Frequentist methods of inferring \( {m_{\upnu }^2} \) we give an example of a Bayesian approach in Sect. 5.3. We briefly list statistical and systematic uncertainty contributors for KATRIN in Sect. 5.4 and in that context discuss the relevance of the choice of the energy analysis interval in Sect. 5.5 and the distribution of accounted measuring time among that interval in Sect. 5.6. In Sect. 5.7 we give an explanation of negative \( {m_{\upnu }^2} \) estimates and provide a nonphysical extension of the \(\upbeta \)decay spectrum model.
5.1 Parameter inference
The statistical technique for analyzing \(\upbeta \)decay spectrum data is well established. By comparing the observed number of counts \(N^\text {obs}_{jk}\) on each pixel j for each experimental setting k with the prediction from the spectrum and response model \(N_{jk}(U_k, {m_{\upnu }^2} , E_0, \dots )\) (see Eqs. (41) and (43)), \( {m_{\upnu }^2} \) and other unknown model parameters can be inferred. In the case of a KATRINlike neutrino mass measurement, a continuous model that depends on \( {m_{\upnu }^2} \) is fit to unbinned spectral shape data. The method of least squares is most commonly applied.
Our parameter of interest is \( {m_{\upnu }^2} \), which distorts the spectrum shape close to the endpoint. Because the fitted \(\upbeta \)spectrum shape essentially only depends on \( {m_{\upnu }^2} \), with \(\chi ^2\) being approximately parabolic in \( {m_{\upnu }^2} \), it is the preferred fit parameter over \( {m_{\upnu }} \) [54].

The tritium endpoint energy \(E_0\), the maximum electron energy assuming a vanishing neutrino mass, has to be estimated from the data, due to uncertainties in the measured \(\mathrm {T^+}\)/\(\mathrm {^3He^+}\) mass difference [55] and in the experimental energy scale.

The signal amplitude \(A_\text {sig}\), a multiplicative factor close to 1, is applied to the predicted signal rate^{9} \(\dot{N}_j^\text {sig}\) to correct for any energyindependent model uncertainty. \(E_0\) and \(A_\text {sig}\) are estimated from the slope of the spectrum at lower energies of the analysis interval (\(\approx 30  40 \hbox { eV}\) below the endpoint), where the absolute signal rate is highest.

The background rate amplitude \(A_\text {bg}\) is another normalization factor, which is applied to the background model component \(\dot{N}_j^\text {bg}\). It is estimated using the data from retarding potentials above the tritium endpoint, where no signal is expected. Note that we assume a constant background rate without retarding potential dependence in the energy interval near the tritium endpoint. However, such an energy dependence could be incorporated into the model using additional data above the endpoint.
5.2 Confidence intervals
Due to the stochastic nature of the observed data, a single parameter point estimate by itself cannot relate to the unknown true value of a parameter. In parameter inference, a confidence interval defines an interval of parameter values that contain the true value of the parameter to a certain proportion (confidence level), assuming an infinite number of independent experiments. Various methods of constructing such intervals exist.
Using the Neyman construction [56] (a Frequentist method), ensembles of pseudoexperiments are sampled for a range of true values of \( {m_{\upnu }^2} \), leading to the construction of a confidence belt (see Fig. 15). Incorporating an ordering principle proposed by Feldman and Cousins [57], empty confidence intervals for nonphysical estimates of \( {m_{\upnu }^2} \) can be avoided, while ensuring correct Frequentist coverage.
5.3 Bayesian statistics
Fortunately, KATRIN’s \( {m_{\upnu }^2} \) posterior PDF is rather insensitive to the choice of prior on \( {m_{\upnu }^2} \). Assuming, for instance, a true value of \( {m_{\upnu }^2} = 0\hbox { eV}^2\), a Gaussian prior with mean \(\mu _\pi = 0\hbox { eV}^2\) and \(\sigma _\pi = 1\hbox { eV}^2\) (or a value on the order of the Mainz or Troitsk upper limits) will be outweighed by the KATRIN likelihood function. It will thus have no significant effect on the derived Bayesian upper limit compared to a prior that is flat in \( {m_{\upnu }^2} \). This underlines the improved sensitivity of the experiment.
The posterior distributions can be obtained practically with Markovchain Monte Carlo (MCMC) methods [59]. With proper adjustments, this class of algorithms is capable of efficiently traversing highdimensional parameter spaces and sampling from posterior probability distributions of an unknown quantity such as \( {m_{\upnu }^2} \). From these distributions, any choice of credibility interval \([\theta _1, \theta _2]\), with \(P = \int _{\theta _1}^{\theta _2} p(\varvec{\theta }) \,\mathrm {d}\theta \) being the confidence level, can be constructed.
When considering the distribution of only a subspace of all parameters, one speaks of a marginal posterior distribution. To determine the onedimensional posterior distribution of \( {m_{\upnu }^2} \), the fourdimensional posterior distribution of \(( {m_{\upnu }^2} , E_0, A_\text {sig}, A_\text {bg})\) is marginalized over the three nuisance parameters.
\( {m_{\upnu }^2} \)  \(E_0\)  \(A_\text {sig}\)  \(A_\text {bg}\)  

\( {m_{\upnu }^2} \)  1  
\(E_0\)  0.698  1  
\(A_\text {sig}\)  \(\)0.581  \(\)0.953  1  
\(A_\text {bg}\)  0.396  \(\)0.022  0.077  1 
5.4 Statistical and systematic uncertainties
Traditionally, the statistical uncertainty \(\sigma _\text {stat}( {m_{\upnu }^2} )\) is identified with the spread of an \( {m_{\upnu }^2} \) estimate caused by the randomness of the observed data (spectrum count rates \(N^\text {obs}_{k}\)), and usually decreases when data are taken (as \(1/\sqrt{N_k}\) or \(1/\sqrt{\varDelta t_k}\)). A systematic uncertainty \(\sigma _\text {syst}( {m_{\upnu }^2} )\), by contrast, represents an uncertainty in the \( {m_{\upnu }^2} \) estimate due to an uncertainty in the spectrum or response model which does not scale with the amount of data taken in general.
Providing a comprehensive review of all systematics of KATRIN – some of which are not adequately quantifiable until final commissioning and characterization of the experimental apparatus – is beyond the scope of this article. Among the major systematic contributors are the final state distribution (Sect. 2.4), the shape of the energy loss function and the inelastic scattering cross section (Sect. 4.2), the sourcegas column density (Sect. 4.4), and highvoltage fluctuations (Sect. 4.11).
5.5 Choice of the analysis energy interval
5.6 Measuring time distribution
Figure 20 illustrates the relative spectrum rates with a measuring time distribution in the energy interval of \([E_030\hbox { eV}, E_0+5\hbox { eV}]\). In the case of \( \widehat{m_{\upnu }^2} \), sufficient measuring time must be spent on the region slightly below the endpoint, where the spectral distortion due to a nonzero \( {m_{\upnu }} \) is most prominent. This is also the region with a signaltobackground ratio between 2:1 and 1:1. Accordingly, for scenarios of elevated background, this feature of the measuring time distribution must be adapted and shifted to slightly lower energies.
The measuring time distribution can be further optimized to provide even better statistical leverage on the model parameters fit to the spectrum shape (see Sect. 5.1), reducing the statistical uncertainty \(\sigma _\text {stat}^\text {opti}\left( {m_{\upnu }^2} \right) < 0.015\hbox { eV}^2\) for nominal experimental conditions [62]. An example is shown in Fig. 21, which describes a rather sparse measuring time distribution with only four features, covering distinct retarding energy regions qU. The peak at the lower end of the analysis energy interval (\(\approx 30\hbox { eV}\)) is best suited to measure \(E_0\) and \(A_\text {sig}\) due to the higher absolute spectrum rates. At \(qUE_0\approx 14.0\hbox { eV}\) the correlation between \(E_0\) and \(A_\text {sig}\) is broken. \( {m_{\upnu }^2} \) is measured through the \(\upbeta \) spectrum shape distortion around \(qUE_0\approx 4.5\hbox { eV}\), where about one third of the overall measuring time is invested. \(A_\text {bg}\) is measured using data beyond the endpoint energy \(E_0\), where no \(\upbeta \)decay signal is expected. Note that all four of these parameters are correlated, so the measuring time cannot be shifted arbitrarily between these four regions of retarding energy.
This more focused model allows a lower statistical uncertainty of the measured \( {m_{\upnu }^2} \), however, it bears a higher risk of overseeing unexpected spectrum shape distortions in the neglected regions of the \(\upbeta \)decay spectrum. To safeguard against such spectral deviations from the model and against unexpected systematics, a more uniform distribution, such as the one first shown in Fig. 20, seems more appropriate, at least for the initial datataking period.
5.7 Negative \( {m_{\upnu }^2} \) estimates
For the construction of physical \( {m_{\upnu }} \) confidence intervals such a nonphysical continuation of the model is not required. The unified approach ensures correct Frequentist coverage while allowing to respect parameter boundaries in the fit [57]. In a Bayesian framework the physical constraint is typically realized through a prior \(\pi ( {m_{\upnu }^2} ) = 0\) for \( {m_{\upnu }^2} < 0\hbox { eV}^2\).
6 Conclusion
Using \(\upbeta \) spectroscopy, the KATRIN experiment aims to probe the absolute neutrino mass scale with an unprecedented subeV sensitivity. Both the statistical and systematic uncertainties of the model parameter of interest, the squared electron neutrino mass \( {m_{\upnu }^2} \), are required to be on the order of \({\mathcal {O}}(0.01\hbox { eV}^2)\). This demands a solid understanding and consistent implementation of the theoretical \(\upbeta \)decay spectrum model and the experimental response function.
With this work, an effort was made to summarize the \(\upbeta \) spectrum calculation with all known theoretical corrections relevant for spectroscopy in the endpoint region. Furthermore, a response function model of the KATRIN experiment was outlined, including its dependencies on sourcegas dynamics and the spectrometer electromagnetic configuration. Finally, the statistical methods applicable to the intended measurement were investigated and concrete examples of their application to the KATRIN neutrino mass measurement were given.
In Sect. 4.12, an overview of the impact of various model components on the measured squared neutrino mass was given. The purpose is to provide a quantitative measure of their relative importance, indicating components that are negligible in the neutrino mass analysis. Among the most important effects are the radial dependencies of analyzing magnetic field and retarding potential, energy loss of signal electrons due to cyclotron motion and the Doppler broadening of the electron \(\upbeta \)spectrum due to the source gas thermal motion.
The calculations presented here are implemented as part of a common C++ simulation and analysis software framework called Kasper, which is used by the KATRIN collaboration to investigate the effect of model corrections and possible systematics, and to optimize the operational parameters of the setup for the neutrino mass measurement [39, 62, 64, 65, 66].
During the ongoing commissioning measurement campaign of the KATRIN experiment, many aspects of the current response model will be verified with experimental data. The results of recent investigations are described in [38, 49, 67]. This thorough characterization of the complex setup will allow a quantitative evaluation of the systematic effects in the neutrino mass analysis at KATRIN.
Footnotes
 1.
If we denote the fraction of all hydrogen isotopologues X by c(X) with \(\sum _X c(X) = 1\), then the tritium purity is given by \(\epsilon _\mathrm {T}= c(\mathrm {T_2}) + c(\mathrm {DT}) / 2 + c(\mathrm {HT}) / 2\).
 2.
The gas is then injected into the source beam tube and rapidly cooled down to 30 K. Because the gas spends only a short time (\(\lesssim 1.5\hbox { s}\)) at this temperature, the rotational states cannot equilibrate again.
 3.
 4.
At a temperature of 30 K and a magnetic field strength of 3.6 T, the polarization of the tritium nuclei can be neglected.
 5.
As investigated in [39], the direct angular change of \(\upbeta \)electrons due to elastic and inelastic scattering has only negligible effect on the response function shape.
 6.
It is required that \(E_\parallel \) reaches its global minimum in the analyzing plane, which is achieved by optimizing the electromagnetic conditions in the spectrometer. See [37] for details.
 7.
The \(\upbeta \)electrons are guided from source to detector by magnetic field lines, so each detector pixel maps a certain stack of source segments.
 8.
In the nonrelativistic case, the power loss due to cyclotron radiation amounts to \({\dot{E}}_\perp = 0.39 /\hbox {s T}^2 \cdot E_\perp \cdot B^2\).
 9.
Deviations from unity arise mainly from incomplete knowledge of the tritium column density and the detector efficiency (see Eq. (41)).
Notes
Acknowledgements
We acknowledge the support of Helmholtz Association (HGF), Ministry for Education and Research BMBF (05A17PM3, 05A17VK2 and 05A17WO3), Helmholtz Alliance for Astroparticle Physics (HAP), and Helmholtz Young Investigator Group (VHNG1055) in Germany; and the Department of Energy through Grants DEAC0205CH11231 and DESC0011091 in the United States. We thank T. Lasserre, V. Sibille, N. Trost and D. Vénos for contributive discussions, and D. Parno for her careful review and suggestions.
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