Muon-induced background in the KATRIN main spectrometer

The KArlsruhe TRItium Neutrino (KATRIN) experiment aims to make a model-independent determination of the effective electron antineutrino mass with a sensitivity of 0.2 eV/c$^{2}$. It investigates the kinematics of $\beta$-particles from tritium $\beta$-decay close to the endpoint of the energy spectrum. Because the KATRIN main spectrometer (MS) is located above ground, muon-induced backgrounds are of particular concern. Coincidence measurements with the MS and a scintillator-based muon detector system confirmed the model of secondary electron production by cosmic-ray muons inside the MS. Correlation measurements with the same setup showed that about $12\%$ of secondary electrons emitted from the inner surface are induced by cosmic-ray muons, with approximately one secondary electron produced for every 17 muon crossings. However, the magnetic and electrostatic shielding of the MS is able to efficiently suppress these electrons, and we find that muons are responsible for less than $17\%$ ($90\%$ confidence level) of the overall MS background.


Introduction
The discovery of neutrino oscillations [1,2] and the accompanying fact of neutrino mass have made the determination of the absolute neutrino mass scale an important measurement in physics. Investigations of the kinematics of β-decay provide a nearly model-independent method to determine the effective mass of electron antineutrinos. The best upper limit so far is about 2 eV/c 2 (95 % C.L.), measured by the Mainz [3] and the Troitsk [4] experiments. Both experiments used a tritium source and a spectrometer of MAC-E filter 2 type [5][6][7]. The KArlsruhe TRItium Neutrino experiment (KATRIN) is a next-generation experiment based on the same technique, which aims to determine the effective mass of the electron antineutrino with a sensitivity of 0.2 eV/c 2 (90 % C.L.) [8].
To achieve such a high sensitivity, it is essential to have a low background level. As the experiment is built above ground, cosmic-ray muons could be a dominant background source. The average muon flux at sea level is about 189 µ/m 2 /s [9], with typical muon energies on the order of a GeV [10]. Assuming the so-called "cos θ*" angular distribution, which takes the curvature of the atmosphere into account [11], a total flux of 39 500 µ/s passes through the main spectrometer [12]. These muons produce secondary electrons as they make two crossings of the inner surface of the stainless steel vessel. Most of these are "true" secondaries (i.e. have energies below 50 eV [13,14]) which are accelerated by the electric field as they leave the spectrometer. Consequently, true secondaries that reach the detector have similar energies as the signal electrons from tritium β-decay and, therefore, contribute to the background. For the Mainz neutrino experiment, muon coincidence studies indicated that secondary electrons from cosmic-ray muons contributed a significant portion of the observed background rate [3].
In order to investigate the muon-induced background for KATRIN, a muon detector system was installed in the spectrometer hall. With such an apparatus, two complementary approaches to examine the muon background are available. First, one can look for electron events that are coincident with events from the muon detectors. If muons contribute to the background, a surplus of secondary electrons is expected in the time window following a signal from the muon detectors. However, this method fails if muon-induced secondaries are trapped in the spectrometer for a significant time before being detected. A second approach is to use the fact that the muon flux shows variations in time (on the order of hours or days) due to changes in atmospheric pressure and temperature [12]. Thus, one expects the background electron rate to vary in a correlated manner with the muon detector rate, if the background rate is at least partly muon-induced. Both of these methods (coincidence and correlation) were employed to study the muon component of the background electron rate.
Section 2 gives an introduction to the components of the KATRIN experiment, and in Section 3 there is an overview of the background measurements 2 Magnetic Adiabatic Collimation combined with an Electrostatic filter that are relevant to this paper. In Section 4 we describe a validation of the hypothesized mechanism for muon-induced backgrounds. A correlation study of muon and background electron rates is then presented in Section 5, with an estimate of the remaining muon-induced background under normal KATRIN operating conditions in Section 6. Finally, in Section 7 we discuss the relevance of the muon-induced background component for KATRIN.

Measurement apparatus
The KATRIN experiment is located at the Karlsruhe Institute of Technology, Campus North, near the city of Karlsruhe, Germany. The beamline (see Fig. 1) has an overall length of about 70 m. Molecular tritium is injected into the windowless gaseous tritium source (b) where it decays with an activity of 10 11 Bq, thus providing a sufficient number of β-decay electrons close to the endpoint energy E 0 ≈ 18.6 keV. The rear section (a) is responsible for monitoring the source activity and also can produce electrons for transmission studies. The tritium is removed from the beamline in the differential pumping section (c) and the cryogenic pumping section (d) while electrons from the source are magnetically guided towards the spectrometer section. Both pre-spectrometer and main spectrometer (MS) are operated as electrostatic retarding high-pass filters of MAC-E filter type. The pre-spectrometer (e) is operated as a pre-filter that reduces the flux of electrons into the MS (f ), which performs the energy analysis of the β-decay electrons near the endpoint (E 0 ) with an energy resolution of ∆E ≈ 1 eV. The MS is equipped with a dual-layer wire electrode system for electrostatically shielding secondary electrons from the inner vessel surface [15,16]. The transmitted β-decay electrons are counted in the focalplane detector (FPD) system (g) with a segmented silicon detector [17].
The MS and the FPD system are described in Section 2.1. Details of the muon detector system are presented in Section 2.2.

Main spectrometer and focal-plane detector
With a volume of about 1240 m 3 and an internal surface area of about 690 m 2 , the MS is the largest component of the KATRIN experiment. The steel vessel has a length of about 23 m and a central inner diameter of 9.8 m [8]. It works as a MAC-E filter for the energy analysis of signal β-particles. Superconducting magnets at both ends of the MS generate a guiding magnetic field [18]. Signal electrons from the tritium source are guided adiabatically along the field lines towards the detector, always traveling within a flux tube delineated by the local magnetic field. An electrostatic retarding potential U 0 is applied to the MS vessel, such that only electrons with sufficient energy to overcome the resulting potential barrier reach the detector. The potential reaches its largest value in the vertical analyzing plane in the middle of the MS. For the neutrino mass measurements, U 0 will be varied around −18.6 kV in order to scan the β-spectrum close to this endpoint energy.
The thickness of the MS walls varies between 25 mm and 32 mm [8]. The vessel is operated under ultra-high-vacuum conditions [19] in order to minimize the energy loss from scattering of signal electrons off residual gas molecules. An air-coil system, consisting of 14 axial coils and two Earth's magnetic field compensation coils, is installed around the MS for the fine tuning of the magnetic field [20,21]. The polarity of each air coil can be reversed which allows a large variety of magnetic field configurations. Of particular interest for the measurements presented here are the so-called "asymmetric configurations" [21], in which the magnetic field lines connect parts of the inner MS surface to the FPD (see Section 3). In this non-standard running mode, there is no flux tube connecting the entrance and exit of the MS.
During KATRIN operation, one possible background source comes from lowenergy secondary electrons that originate from the inner MS surface. If these electrons enter the flux tube, they can be accelerated toward the detector by the retarding potential U 0 , in a similar way to the signal electrons. This process is shown schematically in Fig. 2. Because the signal electrons have very low energies in most of the MS volume due to the operation of the MAC-E filter, the signal electrons cannot be distinguished from the background electrons originating from MS walls.
In the standard configuration, the magnetic field lines inside the MS are axially symmetric and approximately parallel to the walls. This causes charged particles (e.g. secondary electrons) emitted from the MS walls to be deflected by the Lorentz force back towards the MS surface, or, under favorable circumstances, to follow peripheral field lines outside the flux tube covered by the detector. Hence the magnetic guiding field provides a powerful shield against background electrons emitted from the walls. Additional shielding is provided by an inner wire electrode (IE) system installed in two layers, close to the inner walls of the vessel [22]. The IE system can be held at a negative potential offset ∆U IE of up to a few hundred volts relative to the voltage on the MS vessel, reflecting low-energy, negatively charged secondaries back toward the MS surface.
The FPD system [17] is situated at the downstream end of the MS. The heart of this system is the detector wafer, a silicon PIN diode whose 90-mm-diameter active area is segmented into a dartboard pattern of 148 pixels, each with an area of 44 mm 2 . After the detector signals are amplified, a cascade of two trapezoidal filters [23] is applied in order to extract energy and timing information for recording via the ORCA data-acquisition software [24]. An energy resolution The upper plot shows two particle tracks: a through-going β-particle (solid red line) and a secondary electron produced inside the vessel (dotted blue line). The electrons spiral around the magnetic field lines as they travel, although this motion is too small to be seen in the plot. The middle plot shows the electric potential along the β-particle track, with labels indicating the important voltage contributions. The lower plot shows the energy of the two particles as a function of z-position. Due to the finite energy resolution of the FPD, the two particles cannot be distinguished. of (1.52 ± 0.01) keV (full-width half maximum, FWHM) has been achieved for 18.6-keV electrons with this system; the FWHM timing resolution is 246 ns for a typical 6.4-µs shaping length [17]. Due to an unstable preamplifier module, six detector pixels were excluded from the data analysis described in this paper. While traveling from the MS to the FPD system, electrons must pass through a funnel-shaped post-acceleration electrode (PAE), allowing them to be accelerated by up to 10 keV. By increasing the energy of the electrons, one can then apply an energy cut to separate these electrons from lower-energy background electrons produced in the FPD system. A superconducting magnet with a 3.6 T field focuses electrons onto the detector.

Muon detector system
The muon detector system consists of eight large BICRON BC-412 plastic scintillator modules arranged in three towers around the MS vessel, as well as one smaller module above the vessel (Fig. 3). These modules were repurposed from the muon veto counters used in the KARMEN experiment [25]. Each of the large rectangular modules has a sensitive area of 2.05 m 2 and is equipped with four photomultipliers (PMTs) at both ends. The smaller module has a sensitive area of 0.3 m 2 and only uses two PMTs located on a single side of the module. The PMTs are wrapped in several layers of permalloy foil in order to shield them from magnetic fields near the MS.
A muon passing through the scintillating material will induce about 8500 photons per MeV of deposited energy [26], and these photons are detected by the PMTs. A dedicated ORCA DAQ system processes the signals from the PMTs and is configured to trigger on coincident events that are measured at both ends of a large scintillator module or in both PMTs for the smaller module. Signals are collected in 50-ns time bins. Due to leaks in the permalloy shielding, modules 1 and 2 showed a rate dependence on the magnetic field during the measurements described in Section 3.1; signals from these modules are therefore excluded from the analyses in this paper.
In order to synchronize the FPD and the muon detector systems, both DAQ systems are driven by the same precision clock. The clock provides a 10 MHz reference signal, as well as a pulse-per-second signal. Both signals are routed to the DAQ systems via optical fiber cables of equal length (50 m). The synchronization between the systems is accurate to 50 ns. An independent electronic pulser (about 0.07 Hz) is connected via BNC cables of equal length to both DAQ systems. A comparison of the timestamps of these pulser events allows the detection of possible time offsets between the two systems.

Measurements
To understand the muon-induced background, two separate datasets are of interest: measurements with the muon detectors and FPD running in parallel (Section 3.1), and measurements with only the FPD that focused on investigating the energy spectrum of secondary electrons (Section 3.2).

Muon detector measurements
The muon and FPD systems were simultaneously operated under three different electromagnetic configurations, as shown in Table 1. In setting 1, an asymmetric magnetic field (Fig. 4, left panel) was generated where the field lines connect the surface of the MS to the active area of the FPD, maximizing the detection efficiency of electrons generated on or near the vessel walls. In contrast, settings 2 and 3 utilized a symmetric magnetic field (Fig. 4, right panel) which provides magnetic shielding. These latter two settings are similar to the configuration to be used during KATRIN neutrino mass measurements and thus provided a more realistic background scenario. Settings 2 and 3 differed in terms of the electrostatic shielding applied by the IE system. Due to technical limitations of the available power supply, ∆U IE = 0 was not possible during the measurements; the smallest stable voltage, ∆U IE = −5 V, was used instead.
A run is defined as a fixed length of time during which FPD data were collected. In order to see the variations in the muon flux in each setting, runs were performed in a cyclic manner, iterating through each of the three settings. This sequence was repeated automatically over the course of about 16 days, resulting in 113 completed cycles. A small number of the runs had to be excluded from further analysis due to hardware issues. The total measurement time for each setting is listed in Table 1.
For all three settings, an acceleration voltage U PAE = +4 kV was applied to the PAE, and a bias voltage U bias = +0.12 kV was applied to the FPD wafer. To determine the electron rate for a particular run, an electron region of interest (ROI) was defined using the initial electron energy (assumed to be ∼0 eV for production at the MS surface), the sum of the applied electrostatic potentials (−U 0 − ∆U IE + U PAE + U bias ), and the energy resolution of the FPD.  Table 1: Run settings used for the measurements described in this paper. Settings 2 and 3 have lower electron rates and therefore require additional measurement time to get meaningful statistics. The asymmetric magnetic field during the IE voltage scan used modified magnet and air-coil settings compared to the muon studies.  shows the voltages and ROIs for the measurements described in this paper. For the muon studies, the ROI is 19.7−24.7 keV, and electrons with energies outside of this window were excluded from the data analysis.

Secondary electron spectrum measurements
Several months after the muon data were collected, additional measurements were performed to study secondary electrons emitted from the MS. Utilizing an asymmetric magnetic field configuration similar to setting 1 but with modified magnet and air-coil settings, the FPD rate was measured as a function of the IE offset voltage. Electrons with energies less than −e · ∆U IE would be screened and therefore not detected. By scanning over the offset voltage, the electron energy integral spectrum could be found. A special power supply was available for these measurements which allowed small offset voltages to be reached; ∆U IE spanned a range from −5 V to 0 V for the results reported in this paper.
Prior to the voltage-scan measurements, the MS vessel was vacuum-baked for 8 days at 200 • C [27] to reduce the amount of water and other residual gases on the inner surface of the vessel [19]. Because of the change in the vessel surface properties between the muon and voltage-scan measurements, it is possible that a corresponding effect may be seen in the energies of secondary electrons.

Validation of muon-induced background mechanism
A necessary component for understanding the muon-induced background is the energy spectrum of secondary electrons emitted from the walls of the MS; the derivation of the spectrum is presented in Section 4.1. Using FPD events that were coincident with those from the muon detectors, the time distribution for secondary electrons emitted from the MS surface was determined (Section 4.2). A simulated distribution, produced using the derived energy spectrum, was then compared to the measured data to verify the model of muon-induced events (Section 4.3). Finally, in Section 4.4 we discuss the results of the electron-muon coincidence analysis under nominal magnetic field conditions.

Secondary electron energy spectrum
The method used to determine the energy spectrum of secondary electrons emitted from the walls of the MS follows a similar procedure to that outlined in [12], where the electron energy spectrum for the KATRIN monitor spectrometer was calculated. The electron rate as a function of the IE offset voltage ∆U IE < 0 can be written as where E is the electron energy and F is the energy spectrum of secondaries emitted from the vessel. P is the probability for an electron emitted from the MS surface to reach the FPD. It is given by 39.2 ± 0.6 cps Table 2: Fit parameters obtained from the IE voltage scan data (Fig. 5).
This equation can be derived by considering that a detected electron must overcome both the offset voltage and the magnetic mirror effect [12]. The derivation assumes a cosine angular distribution for the electrons relative to the surface normal, which is a valid approximation for true secondaries [14]. Two forms for F (E) were considered. A phenomenological model for the emission energy of true secondary electrons can be obtained from [14]: where and p are parameters of the model and A is a normalization factor. An alternative, theory-based form for the energy spectrum can be found in [28][29][30]: where B is a normalization factor and Φ is the work function. The rate data from the IE voltage scan measurements, previously described in Section 3.2, were fit to Eq. (1) using F phen (E) and F theory (E) (Fig. 5). It was assumed that the total observed rate consisted of two components, R(∆U IE ) and a constant background rate R 0 . By fitting to data, the parameters in F phen (E) and F theory (E) were fixed ( Table 2). The derived energy spectra are plotted in Fig. 6.
The fit to the theory-based energy spectrum indicates that Φ = (3.475 ± 0.008) eV for the MS surface. This result is in accord with transmission measurements with a photoelectron source in the MS [31], which found that Φ varied between 3.39 eV and 3.65 eV [32]. An older MAC-E experiment found Φ = (4.4 ± 0.2) eV for stainless steel [7]. However, the discrepancy between the measured values for Φ can be attributed to surface effects, as described in [32].

Coincidence analysis
A straightforward method to study the muon-induced background is to perform a coincidence analysis on muon and electron events. If muons passing through the MS vessel are responsible for creating electrons that reach the FPD system, one expects an excess of electron events in the time window following a muon event. (This is only true for the asymmetric magnetic field configuration; for the symmetric configuration, electrons can be trapped in the MS for  Phen. Theory Fig. 6: The secondary-electron energy spectra, generated from the parameters in Table 2. F phen is given by the solid black curve, and F theory is given by the dashed red curve. The spectra have been normalized to unity within the displayed energy range. Residual Fig. 7: The distribution of the time differences between electron and muon events collected with field setting 1 (blue points and fill). On the x-axis, t = 0 µs corresponds to the detection of a three-module muon event. Overlaid are the simulated time distributions (red and black markers) produced with Kassiopeia using the energy spectra in Fig. 6 as input. The simulated distributions are scaled to minimize the χ 2 /ndf (calculated for t > 0 µs) with the offset from zero being fixed by the average electron counts prior to muon detection (−100 µs < t < 0 µs). The error bars are purely statistical. At the bottom of the figure, the residuals (Simulation − Data) are displayed. long durations.) The timing difference between muon and electron events allows the determination of the electron flight time, which can be compared with simulation.
In terms of event selection for a coincidence study with the FPD, it is desirable that all selected muons travel through the walls of the MS in order to have a chance of producing detectable electrons. Out of the available muon modules, modules 6, 7 and 8 are best suited to fulfill this condition (see Fig. 3). The position and orientation of these modules relative to the MS is such that a muon that creates a signal in all three modules is geometrically constrained to have passed through the MS. (The deflection from the Lorentz force is negligible.) Thus, only three-module muon events are used in the coincidence study, where such an event has concurrent signals within a 200-ns window. This time window was chosen to account for the 50-ns timing resolution of the muon modules.
In order to study events originating from the walls, only events from the outer 136 pixels of the detector were included in the analysis, since these pixels directly measure events from a well-defined section of the MS surface. For each electron event, the time difference between the electron event and the most recent muon event was tabulated, and the distribution of these time differences is shown in Fig. 7 for the case of setting 1. An excess above the randomcoincidence level is clearly visible, indicating the presence of muon-coincident electron events. The distribution peaks at time differences of about 15 µs.

Comparison with simulation
To confirm that the time structure of the coincidence peak is consistent with the production of muon-induced electrons, Monte Carlo simulations were performed using Kassiopeia, the particle-tracking simulation package developed for the KATRIN experiment [33]. The simulation geometry included a simplified version of the system apparatus, consisting of the MS vessel and the FPD system, and employed the same electromagnetic field configuration used in setting 1, excluding the IE system. Over 10 5 electrons were produced 10 cm from the MS walls, uniformly spread over axial positions −3.37 m ≤ z ≤ −0.46 m, which is the range corresponding to the magnetic field lines that connect to the FPD (see Fig. 4). The initial distance from the walls was chosen in order to reduce computation time associated with tracking charged particles near the MS surface; test simulations showed that starting electrons closer to the wall had no significant effect on the flight time to the detector. The starting angle of electrons relative to the MS walls was sampled from a cosine angular distribution. For ease of implementation, electrons were started at a single (x, y)-position, which is sufficient due to the axial symmetry of the geometry.
Two simulations of the secondary-electron flight time were performed, each using one of the derived energy spectra as input (see Fig. 6), up to an energy of 50 eV. The flight times for the simulated electrons that reach the FPD are shown in Fig. 7. The simulation results replicate the basic features of the measured distribution of electron events. Both data and simulations show the mode of the time-of-flight spectrum to be about 15 µs, with the distributions showing approximately equal widths for the peak. The theory-based electron energy spectrum provides a better fit at shorter timescales, due to the larger fraction of high-energy electrons.
At longer times (t > 15 µs), simulation replicates the exponential tail of the measurement data but tends to underestimate the number of events. Several factors may explain this disagreement. First, the simulation utilized the energy spectra derived in Section 4.1, which consider all secondary electrons emitted from the MS, not just the muon-induced component. Muon-induced secondaries may have a modified energy spectrum, meaning that the data, which is specific to muon-induced secondaries, may not be directly comparable to the simulation. Second, the baking period that occurred between the muon and IE voltage scan measurements may have changed the observed energy spectrum by altering the surface properties (e.g. work function) of the MS surface.
Third, the simulation excludes any effects from the wire electrodes, which were placed at an offset voltage (∆U IE = −5 V) during the measurement. This voltage is large enough to block a significant fraction of events from vessel walls. However, secondary electrons are also emitted from the wire electrodes and their holding structure (in the same way as from the walls), and these secondaries are not shielded by the wire electrode. The combined effect of the blocked electrons from the walls and the additional events from the wire electrodes should result in a partly distorted energy spectrum for secondary electrons; this implies a modified time distribution of detected events. Overall, however, the general agreement between the measured muon-induced electron time-of-flight distribution and simulation confirms KATRIN's basic model of background production due to muons.

Muon coincidence under nominal conditions
The time distributions of muon-coincident electron events under setting 2 and setting 3 are displayed in Fig. 8. No corresponding increase in the number of electron events following the three-module muon signals is observed.
One can attempt to set an upper limit on the muon-induced background rate by counting the excess number of events for t > 0 compared with t < 0, and then scaling the result appropriately to consider all muon events that pass through the MS, not just those that pass through modules 6, 7, and 8. However, this approach is vulnerable to systematic uncertainties. First, it is challenging to accurately extrapolate the coincidence rate for a particular region of the MS surface to the entire vessel without understanding the efficiency of electron transport as a function of the initial location on the MS surface. This requires significant particle-tracking simulations beyond the scope of the present paper. A second difficulty is the possible time-dependent behavior of the secondaries. Electrons can be magnetically trapped in the symmetric magnetic field of setting 2 and setting 3 for up to several minutes [26]; thus, muon-induced secondaries and any additional electrons produced in the trap can reach the FPD well beyond the 100 µs interval applied in the coincidence study.
To test the statistical sensitivity of using the coincidence data to set an upper limit, a naive extrapolation to the entire MS was performed; the resulting upper limit on the muon-induced rate is comparable to the value derived from the correlation study (see Section 6). Because the uncertainties for the coincidence approach are difficult to calculate, this method was not developed further.

Correlation of cosmic-ray muon rate with detected background
For the correlation analysis of the background electron rate and the muon rate, the following assumptions were made: the background consists of a fluctuating muon-induced component and a constant component of at least one other source; and the muon-induced background component is directly correlated with the muon rate. These assumptions lead to the following formula for the background electron rate R e (t): where R µ (t) is the muon rate measured by the muon modules, R x is the constant background component and K is the coefficient representing the linear relation between the muon rate and the resulting rate of secondary electrons detected by the FPD.
Translating this into normalized rates, Eq. (5) becomes: with R e and R µ being the mean electron and muon rate, respectively. The only unknown parameter is m, which represents the fraction of background that is muon-induced. Plotting the normalized electron rate as a function of the normalized muon rate, m is given by the slope. The correlation analysis was performed for all three field settings; a summary of the results is given in Table 3. The correlation under asymmetric field setting is described in Section 5.1, and in Section 5.2 the measured muon-induced fraction is used to determine the production rate of muon-induced secondary electrons in the MS. The analysis of the symmetric field correlation data is described in Section 6.

Correlation under asymmetric magnetic field conditions
The normalized muon and electron rates as functions of time for setting 1 are displayed in Fig. 9. A large increase in the muon rate is visible near day 5, caused by a low-pressure weather system that passed over the experiment. In Fig. 10, the normalized muon and electron rates are plotted against each other, and the fit to Eq. (6) is also shown. The fraction of muon-induced background is 0.123 ± 0.012. This result indicates that muons make up a sizable fraction but not the majority of secondary electron events originating from the MS surface and IE system. (Because of the electrostatic shielding potential applied during setting 1, a significant portion of the background from low-energy electrons originates from the IE system.) The Pearson correlation coefficient r     was calculated to be 0.70 ± 0.06, which indicates significant linear correlation. The uncertainty was estimated via a case resampling Monte Carlo bootstrap method [34].
The distribution of the time difference ∆t between electron events for setting 1 is shown in Fig. 11, left panel. It can be seen that there are a large number of events with time difference ∆t less than 0.2 ms. At longer time differences, however, the distribution has a constant slope. The distribution in the figure can be explained by two processes with different multiplicity distributions for secondary-electron production. One process with multiplicity M = 1 produces a single electron that is detected at the FPD. These events are called "single" events, and they are Poisson-distributed, contributing to the constant slope at large time differences.
Another process creates clustered electron events (M ≥ 2) which arrive at the FPD within short time intervals (Fig. 11, right panel). The electrons produced from these high-multiplicity events will have a spread of initial energies and pitch angles, resulting in the observed flight time differences of up to 0.2 ms. These events are referred to as "cluster" events. The multiplicity M of a cluster is defined using a sliding time window of duration d = 0.2 ms. All events that fall within d of neighboring events are grouped together, such that the time difference between the first and last event in the group can in principle be greater than d. The event multiplicity is defined as the number of events in the group. Using this criterion, about 45% of the electron events in setting 1 are classified as single events, with the remainder being cluster events. Due to the presence of cluster events, the FPD event rate is non-Poissonian. All FPD rates used in the correlation analyses (and shown in the figures) utilize the RMS error.
The correlation analysis was repeated for different electron event multiplicities (see Table 3). The single (M = 1) electron event rate shows a strong correlation with the muon rate (Fig. 12, left panel). A weaker correlation is observed for the double (M = 2) electron event rate. It should be mentioned, however, that a portion of the cluster event rate comes from "accidental" clusteringsingle events that statistically happen to fall within ∆t of another single event or another cluster. Thus, the measured correlations and slopes for M > 2 are not corrected for the contribution from single events. Nevertheless, no significant correlation is found for cluster events with M ≥ 3 (Fig. 12, right panel). This result strongly indicates that muons dominantly produce events with small multiplicities.

Electron production rate from muons
Knowing the value of m for setting 1, it is possible to make a rough determination of the electron production rate by a single muon crossing the MS surface. This quantity, which we denote as α, can be obtained from the following equation: The numerator gives the number of muon-induced electrons emitted from the inner surface. N FPD is the rate of electrons from the MS surface that reach the FPD for the same magnetic field configuration as setting 1, but without electrostatic shielding (i.e. with ∆U IE = 0). A measurement, described elsewhere [27], found this value to be 790 cps. C is a correction factor that accounts   for the probability of an electron emitted from the surface to be detected by the FPD. From Kassiopeia simulations (see Section 4.3), it was determined that electrons have a 5.1% and 4.8% chance of reaching the FPD when using the phenomenological and theoretical energy spectra, respectively. Taking the average of these two values and considering that the FPD itself has a detection efficiency of about 95% [17], then C ≈ (0.95 · 0.050) −1 = 21.
The denominator, N µ , is the rate of muon hits on the MS surface. A very simple Geant4 simulation [35][36][37] was performed to estimate this value. Muons were started in a 60 m×40 m plane located 7 m above the spectrometer axis, with starting angles given by the cos θ* angular distribution, and the collision points with the spectrometer were calculated. N µ is approximately 10.4 kcps for the portion of the MS surface measured with setting 1. Applying the aforementioned values to Eq. (7), one finds that α ≈ 0.20. This result indicates that one secondary electron is emitted from the MS surface for about every five muon crossings.

Residual muon-induced background with symmetric magnetic field
Turning to settings 2 and 3, no significant correlation was found between the muon and FPD rates (see Table 3). In addition, performing a single/cluster event analysis is not useful in this case since the FPD events for these settings are essentially all single events (> 99%).
The measured muon-induced fraction for setting 2 and setting 3 is consistent with zero. For setting 3, which is closest to the nominal KATRIN operating mode, the assumption of Gaussian errors leads to an upper limit on the true muon-induced fraction of m < 0.27 (95% C.L.). However, it is possible to reduce this upper limit by doing a simultaneous fit of the setting 2 and setting 3 data. Setting 2 has reduced shielding and should therefore have a larger muoninduced background component, if such a background is indeed present. By performing a simultaneous fit of the two datasets, one naively expects to raise the measured upper limit on the muon-induced background compared with an analysis with only setting 3, but the opposite effect is observed since the analysis is statistics-limited. (Although the correlation r should be larger for setting 2 compared with setting 3, this cannot be seen due to the large uncertainties on the correlation coefficients.) Fig. 13 shows the simultaneous fit for setting 2 and setting 3, which results in a value of m = 0.054 ± 0.068 for setting 3. Following the unified approach [38], the upper limit on the muon-induced fraction is m < 0.166 (90% C.L.).
The average background rate under setting 3 is 0.692 cps, after applying a corrective factor (148/142) to account for the six excluded detector pixels. Thus, Eq. (8) indicates that cosmic-ray muons contribute less than 0.115 cps to the total background rate.
A separate study was performed to check the sensitivity of the correlation analysis, using an ensemble of toy measurements generated based on the observed muon and electron rates. For a measurement with the duration reported in this paper, we calculated a 95% probability of detecting m = 0.10 for a simultaneous fit of data from setting 2 and setting 3, which indicates that the correlation analysis is sensitive to m > 0.10. Thus, the measured null result for the muon-induced fraction is consistent with the expected statistical sensitivity of the measurement.

Conclusion
In order to reach KATRIN's design sensitivity, it is necessary to have a good understanding of the background processes inside the MS, including muoninduced backgrounds. Using an electromagnetic configuration in which electrons are directly guided from the surface of the MS to the FPD, it has been possible to use muon-coincident electron events to probe the energy spectrum of the secondary electrons. Rate correlations using the same data found that (12.3 ± 1.2) % of the observed rate from the MS surface was muon-induced. In addition, the fraction of single events that are muon-induced appears to be significantly higher than the fraction of muon-induced cluster events.
A rough calculation indicates that, on average, one secondary electron is produced for every five muons passing through the MS. However, the magnetic shielding of the KATRIN flux tube is highly effective at mitigating this background. In an electromagnetic configuration similar to that planned for neutrino-mass measurements, there is no correlation between the muon rate and the rate of detected electrons. An analysis of all data with magnetic shielding indicates that cosmic-ray muons are responsible for less than 16.6 % of the overall MS background rate, at 90% confidence. This corresponds to an upper limit of 0.115 cps for a total background rate of 0.692 cps. Muon-induced backgrounds are therefore not a significant concern for KATRIN, although they may become more important as other background sources are alleviated.
A significant potential source of background electrons is due to the decay of radon in the MS vessel; however, the installation of liquid-nitrogen cooled baffles between the MS volume and the NEG pumps has effectively mitigated this background process [39,40]. In the current background model for KATRIN, the largest background contribution seems to originate from the ionization of Rydberg atoms, produced from the decay of 210 Pb on the surface of the MS vessel [27]. Additional details regarding this background source can be found in [41].