#### keV-Scale sterile neutrino sensitivity estimation with time-of-flight spectroscopy in KATRIN using self-consistent approximate Monte Carlo

Eur. Phys. J. C
keV-Scale sterile neutrino sensitivity estimation with time-of-flight spectroscopy in KATRIN using self-consistent approximate Monte Carlo
Nicholas M. N. Steinbrink 2
Jan D. Behrens 1 2
Susanne Mertens 0
Philipp C.-O. Ranitzsch 2
Christian Weinheimer 2
0 Physics Department, TU München , James-Franck-Str. 1, 85748 Garching , Germany
1 Institute of Experimental Particle Physics (ETP), Karlsruhe Institute of Technology (KIT) , Wolfgang Gaede-Str. 1, 76131 Karlsruhe , Germany
2 Institut für Kernphysik, WWU Münster , Wilhelm Klemm-Str. 9, 48149 Münster , Germany
We investigate the sensitivity of the Karlsruhe Tritium Neutrino Experiment (KATRIN) to keV-scale sterile neutrinos, which are promising dark matter candidates. Since the active-sterile mixing would lead to a second component in the tritium β-spectrum with a weak relative intensity of order sin2 θ 10−6, additional experimental strategies are required to extract this small signature and to eliminate systematics. A possible strategy is to run the experiment in an alternative time-of-flight (TOF) mode, yielding differential TOF spectra in contrast to the integrating standard mode. In order to estimate the sensitivity from a reduced sample size, a new analysis method, called self-consistent approximate Monte Carlo (SCAMC), has been developed. The simulations show that an ideal TOF mode would be able to achieve a statistical sensitivity of sin2 θ ∼ 5×10−9 at one σ , improving the standard mode by approximately a factor two. This relative benefit grows significantly if additional exemplary systematics are considered. A possible implementation of the TOF mode with existing hardware, called gated filtering, is investigated, which, however, comes at the price of a reduced average signal rate.
1 Introduction
In recent years the interest has grown for sterile neutrinos
with a mass scale of a few keV [
1
]. They are proposed as
dark matter particle candidates in cold dark matter (CDM)
and especially warm dark matter (WDM) scenarios [
2–5
].
WDM has the potential to avoid issues regarding structure
formation on small scales which are not yet solved for WIMP
(weakly interacting massive particle) CDM [
6–12
].
However, the shortcomings of WIMP CDM can possibly be
mitigated via Baryonic feedback [
13
] while any sterile neutrino
dark matter production mechanism needs to be fine-tuned
to yield the correct DM density. Mass-dependent bounds
on the sterile neutrino mixing with active neutrinos have
been established by searches for sterile neutrino decay via
X-ray satellites [
14,15
] and on basis of theoretical
considerations in order to avoid dark matter overproduction [16],
which never exceed sin2 θ 10−7. The mass range has been
constrained by the DM phase-space distribution in dwarf
spheroidal galaxies [
17
] and gamma-ray line emission from
the Galactic center region [
18
] to 1 keV < mh < 50 keV.
In order to produce the existing amount of dark matter, mass
and mixing angle are linked by the production mechanism,
which can be non-resonant [
16,19,20
] or resonant [
21–24
].
Moreover, possible evidence of relic sterile neutrinos with
mass mh = 7 keV has been reported in XMM-Newton data
[
25–27
].
In principle, it can also be searched for keV-scale sterile
neutrinos in ground-based experiments, such as in tritium
βdecay [
28,29
]. A promising example is the Karlsruhe Tritium
Neutrino Experiment (KATRIN) [30], which is the most
sensitive neutrino mass experiment currently under construction.
Sterile neutrinos would be visible by a discontinuity in the
βdecay spectrum if they have a sufficiently large mixing angle
with electron neutrinos. In order to adapt KATRIN, which is
optimized for light neutrinos of ml O((eV)), for keV
sterile neutrinos, different approaches are discussed with the goal
of enhancing statistics and managing systematics. A suitable
idea is to develop a dedicated detector measuring in
differential mode [
31–33
]. As an alternative idea, it is worthwhile to
study the performance of an alternative time-of-flight (TOF)
mode, which has already shown to be promising in theory
for active neutrino mass measurements [
34
].
In this publication the sensitivity of a keV-scale
sterile neutrino search based on TOF spectroscopy with the
KATRIN experiment is discussed both for an ideal
measurement method as for a possible implementation with minimal
hardware modifications.
2 Sterile neutrino search with TOF spectroscopy
2.1 Sterile neutrinos in tritium β-decay and KATRIN
There has been some previous work on sterile neutrinos in
general in tritium β-decay. Most publications focus on
eVscale sterile neutrinos [
35–39
], which are proposed to address
certain anomalies in oscillation experiments [
40–45
].
However, in recent time also dedicated studies, dealing with
keVscale neutrinos have been published, such as [
29,31,32
], as
well as studies involving more exotic models, such as [
46–
48
]. We will quickly summarize the main effect of a
keVscale sterile neutrino on the tritium β-spectrum, while we
refer especially to [31] for deeper insights into systematics
and theoretical corrections.
The tritium β-decay spectrum with a single neutrino with
mass eigenstate mi is given as
cos2(θC)|M |2 F (E , Z ) · p · (E + mec2)
dΓ G2F
d E = N 2π 3 h¯7c5
·
j
Pj · (E0 − V j − E )
· (E0 − V j − E )2 − mi2c4,
[
28,49,50
], where E is the kinetic electron energy, θC the
Cabbibo angle, N the number of tritium atoms, G F the Fermi
constant, M the nuclear matrix element, F (E , Z ) the Fermi
function with the charge of the daughter ion Z , p the electron
momentum, Pj the probability to decay to an excited
electronic and rotational–vibrational state with excitation energy
V j [
51–53
] and E0 the beta endpoint, i.e. the maximum
kinetic energy in case of mi = 0.
The electron neutrino is a superposition of multiple mass
eigenstates. Since the flavor eigenstate is the one which
defines the interaction, but the mass eigenstate the one which
describes the dynamics of the decay, the β-spectrum for the
electron neutrino is an incoherent superposition of the
contributions for each mass eigenstate,
dΓ
d E (mνe ) =
3
i=1
2 dΓ
|Uei | d E (mi ).
In case of an additional keV-scale sterile neutrino, a fourth
mass state m4 is introduced with a significantly lower
mixing with the electron neutrino, |Ue4|2 |Uei |2 (i ∈ 1, 2, 3).
In the following we define the heavy or sterile neutrino
(1)
(2)
mass as mh ≡ m4 and the active-sterile mixing angle as
sin2 θ ≡ |Ue4|2 < 10−7 [
15
]. Since the light mass
eigenstates 1, 2, 3 are not distinguishable by KATRIN [
50
], a light
neutrino mass is defined as ml2 ≡ i3=1 |Uei |2mi2. The
combined β-spectrum with sterile and active neutrino can then
be expressed as
dΓ
d E
(mνe ) = sin2 θ dΓ (mh ) + cos2 θ dΓ (ml ).
d E d E
(3)
An example with exaggerated mixing is shown in Fig. 1.
In probing the absolute neutrino mass scale, the KATRIN
experiment is designed to measure the light neutrino mass
ml with a sensitivity of < 0.2 eV at 90% confidence level
(CL) [
30
]. Therefor it uses a windowless gaseous molecular
tritium source (WGTS) [
54
] with an activity of ∼ 1011 Bq.
The electrons from the β-decay are filtered in the main
spectrometer based on the magnetic adiabatic collimation with
electrostatic filter (MAC-E-Filter) principle [
55
]. The
magnetic field in the center of the main spectrometer, the
analyzing plane, is held small at BA = 3 mT and otherwise
high at BS = 3.6 T in the source and at Bmax = 6 T at the
exit of the main spectrometer just before the counting
detector. Due to adiabatic conservation of the relativistic magnetic
moment, electron momenta are aligned with the field in the
analyzing plane. By additionally applying an electrostatic
retarding potential qU in the analyzing plane, the
MAC-EFilter acts as a high-pass filter with a sharp energy resolution
of ΔE /E = BA/Bmax ≈ 0.9 eV/E0. In the focal plane
detector (FPD) the count rate is then measured. That way,
KATRIN measures the integral β-spectrum as a function of
qU .
0 0
Since the probability of n-fold scattering is a function of
the emission angle (10), the response corrected energy
spectrum (8) itself becomes dependent on the angle. Note that
(mνe ) = sin2 θ d N (mh ) + cos2 θ d N (ml ). (5) the scattering model is simplified, since angular changes in
dτ dτ collisions are neglected and the scattering probabilities are
For each of these two components, the TOF spectrum can averaged over a hypothetical uniform density profile in the
then formally be obtained from the β-spectrum with neu- source. We would like to clarify that in our actual
impletrino mass ml and mh , respectively, using the transformation mentation the n-fold energy loss spectra are not generated
theorem for densities [
56
]: via convolution but via Monte Carlo, which yields, however,
d N ϑmax E0 equivalent results. Furthermore, using Eq. (4), the radial
startdτ = 0 qU dϑ d E g(ϑ ) icnagsepionsKitiAoTn RisINal,wbauytswaessduomneodt etoxpbeecrt s=ign0i,fiwcahnicthc hisanngoetsthine
d N (E , ϑ ) δ (τ − T (E , ϑ )) , (6) the spectral shape for outer radii. For analysis of real
experd E imental data a fully realistic treatment would be necessary,
where g(ϑ ) denotes the angular distribution and d N /d E (E , ϑ ) yet for a principle sensitivity study these approximations are
the response corrected energy spectrum, which itself is a reasonable.
function of the β-spectrum (1) for a given neutrino mass. If The benefits of a TOF measurement can be understood
angular changes from inelastic scattering processes in the tri- from Fig. 2, where the TOF (4) as a function of E for
diftium source are neglected, the angular distribution is approx- ferent angles is shown. It can be seen that energy differences
imately independent from the energy spectrum and given by
isotropic emission
1
g(ϑ ) = 2 sin(ϑ )
within the angular acceptance interval given by the default
KATRIN field settings with ϑmax = √BS/Bmax = 50.77◦.
The response corrected energy spectrum d N /d E (E , ϑ ) in
Eq. (6) is given in good approximation by the β-spectrum
(3), convolved with the inelastic energy loss function in the
tritium source,
d N dΓ
d E (E |ϑ ) = d E ⊗ floss(E , ϑ )
dΓ
= p0(ϑ ) · d E +
∞
n=1
dΓ
pn(ϑ ) · d E ⊗ fn(E ) (8)
where the fn is the energy loss spectrum of scattering order
n which can be approximately defined via recursive
convolution through the single scattering energy loss spectrum f1.
This can be written as
fn = fn−1 ⊗ f1
(n > 1).
The probability pn that an electron is scattered n times
depends on the emission angle ϑ and is given by a Poisson
law
pn(θ ) = λnn(!ϑ ) e−λ(ϑ). (10)
The average number of scattering processes λ is given in
terms of the column density ρd of the tritium source, the
mean free column density ρdfree and the scattering cross
section σscat as
λ(ϑ ) =
1
dx
ρd · x
ρdfree · cos ϑ =
1
dx ρd · x · σscat . (11)
cos ϑ
2.2 Time-of-flight spectroscopy
The idea of using time-of-flight (TOF) spectroscopy for
a measurement of the light neutrino mass is explained in
detail in Ref. [
34
]. In the following, we will recapitulate the
approach briefly and explain the motivations for
investigating this technique for a keV-scale sterile neutrino search as
well.
In contrast to the standard mode of operation, as described
in the last section, TOF spectroscopy allows to measure not
only the count-rate, but a full TOF spectrum at a given
retarding potential qU . The TOF as a function of the energy is
given by integrating the reciprocal velocity over the center
of motion, which we will assume for simplicity to be on the
z-axis,
T (E , ϑ ) =
dz
1
v =
zstop
zstart
dz
E + mec2 − qΔU (z)
p (z) · c2
, (4)
where E and ϑ are the initial kinetic energy and polar angle
of the electron, respectively. zstart and zstop are the positions
on the beam axis between which TOF is measured, ΔU (z) is
the potential difference as a function of position z and p (z)
the parallel momentum. By assuming adiabatic conservation
of the magnetic moment, p (z) can be expressed analytically
as a function of the potential ΔU (z) and magnetic field B(z)
(derivation see Ref. [
34
]). If these are known, the integral in
Eq. (4) can be solved numerically.
Since the TOF is a function of the energy, the β-spectrum
can be transformed into a TOF spectrum d N /dτ , given the
initial angular distribution of the β-decay electrons. A
feature in the β-spectrum such as a sterile neutrino contribution
would then also have a corresponding effect on the TOF
spectrum if the retarding energy qU is sufficiently low. Like the
β-spectrum (2), the TOF spectrum can as well be expressed
as a superposition of a component with a heavy neutrino mass
mh and a light neutrino mass ml :
d N
dτ
(7)
(9)
0 deg polar angle
20 deg polar angle
40 deg polar angle
25000
20000
up to some ∼ 100 eV above the retarding potential translate
into significant TOF differences. Within these regions, TOF
spectroscopy is thus a sensitive differential measurement of
the energy spectrum. Combining multiple TOF spectra
measured at different retarding energies thus allows to measure
a differential equivalent of the β-spectrum throughout the
whole region of interest. As already outlined in Ref. [
31
], a
differential measurement has important benefits for a sterile
neutrino search. On the one hand it enhances the statistical
sensitivity since the sterile neutrino signature can be
measured directly without any intrinsic background from higher
energies as in the classic high pass mode. On the other hand, it
reduces the systematic uncertainty since it improves the
distinction between systematic effects and a real sterile neutrino
signature in the spectrum.
2.3 TOF measurement
As the approach is rather novel, most existing ideas for TOF
measurement are still in an early development phase and
have not been tested. There are ongoing efforts to develop
hardware which is intended to detect passing electrons with
minimal interference with their energy (electron tagger) [
34
].
Approaches are amongst others to measure tiny excitations
induced in an RF cavity or to detect the weak synchrotron
emission of the electrons in the magnetic field via long
antennas (cf. Refs. [
57, 58
]). While promising, there has
unfortunately not been any break-through in the technical
realization for such an electron tagger, yet. Additionally, it seems
unlikely that such an approach is also useful for keV sterile
neutrino searches. For a sufficient sensitivity on sin2 θ the
count-rate needs to be as high as possible. However,
countrates much above 10 kcps would lead to ambiguities in the
combination of a start signal in the electron tagger and the
stop signal in the detector given the overall TOF of order
∼ µs (see Fig. 2).
A method which has already been tested in the predating
Mainz experiment [
59
] is a periodic blocking of the
electron flux, called gated filtering (GF). If electrons are only
transmitted during a short fraction of the time, the arrival
time spectrum would approximate the TOF spectrum. In
KATRIN, this could for instance be achieved by pulsing
the pre-spectrometer potential between one setting with full
transmission and one setting with zero transmission (Fig. 3).
The main downside of the method is that it sacrifices statistics
in order to get time information. However, it would require
minimal hardware modifications since only the capability
to pulse the pre-spectrometer potential by some keV would
have to be added. Since the focal plane detector of KATRIN
is optimized for low rates near the endpoint, the method could
also in principle be utilized for an early keV sterile neutrino
search by using a small duty cycle with sharp pulses and
thereby reducing the count-rate. However, in this scenario
with small hardware modifications, it is unlikely that the
prespectrometer potential can be pulsed by more than some keV.
Due to the capacity of the pre-spectrometer, there is possibly
a non-vanishing ramping time involved, depending on the
ramping interval. If electrons arrive within the ramping time,
they become either accelerated or retarded, giving rise to
non-isochronous background. The problem can be mitigated
partly by using a voltage supply with higher power.
Alternatively, a mechanical high-frequency beam shutter could be
use. However, this would come at the cost of larger
modifications of the set-up and a lower flexibility regarding
finetuning of the timing parameters. We will not discuss this
problem further and just assume an ideally efficient method
of periodically blocking the beam. However, we will restrict
the sensitivity study of the sterile neutrino search with the GF
method to a measurement region spanning only a few keV
below the endpoint.
3 Monte Carlo sensitivity estimation
The TOF spectrum (6) can not be calculated analytically,
since the magnetic field B(z) and electron potential qΔU (z)
are only known numerically. There are two remaining
possibilities of simulating TOF spectra. The first approach is to
evaluate the δ function in the TOF spectrum (6) via
numerical integration. This method has been used in Ref. [
34
] since
it delivers generally precise results and is well scalable. The
bottleneck of this method is, however, the convolution of the
β-spectrum with the n-fold energy loss spectra (8). The
convolution routine is rather performance-intensive especially
for a large spectral surplus E0 − qU (as present in case of
keV scale sterile neutrino search) and requires complicated
optimizations to work successfully. Furthermore, if the
addition of further effects such as angular-changing collisions
might be requested for future studies, the implementation
will become more difficult.
Therefore, we chose to apply the second approach which
is to generate the TOF spectra (6) via Monte Carlo (MC)
simulation. This especially avoids the convolution of the
βspectrum with the energy loss function (8), since the energy
loss can be randomly generated individually without
additional expensive convolutions. While a MC approach is
generally very flexible when it comes to the addition of more
detailed effects and systematics, it is generally not as
scalable in terms of the expected number of events as a purely
numerical approach. KATRIN is designed for measurements
near the β-endpoint with low rates on the order of several
cps. The measurements for the keV-scale sterile neutrino
detection, however, have to be performed over a significantly
broader region of the β-spectrum and thus count rates up to
∼ 1010 cps can be expected. For a data taking period of
three years, one would thus expect up to ∼ 1018 cps. If a
realistic model for a sensitivity analysis shall be simulated
event-by-event, it is obvious that the sample size needs to be
significantly larger than the expected number of events. In
our case, the calculation of flight times of more than 1018
events is simply not possible within a reasonable computing
time.
However, we will show that, if the signal is sufficiently
small compared to the total expected rate, the dominating
“background part” of the model (corresponding to the cos2 θ
term in Eq. (5)) can be approximated. This works due to the
fact that for a pure sensitivity study, as opposed to an analysis
of real data, only the fidelity of the signal is relevant.
3.1 Self-consistent approximate Monte Carlo
In this section, we argue that a modified Monte Carlo strategy,
from here on called self-consistent approximate Monte Carlo
(SCAMC), will be able to reduce the necessary total sample
size in a sufficient amount to address the problems mentioned
above. This works if two requirements are met. These are
1. that the model can be separated into a background part
and a signal part, with the latter sufficiently smaller than
the first, and
2. that model and toy data are self-consistent, i.e. the toy
data are sampled directly from the model.
We will first discuss this approach for a generic case.
Assume, the model distribution Φ can be expressed by a
linear combination
Φ = cSΦS + cBΦB,
consisting of a signal contribution cSΦS, sampled with
maximum precision, and an approximated background
contribution, cBΦB. The distribution of interest is then replaced by a
modified distribution
Φ = cSΦS + cBΦB,
with ΦB ∼ ΦB, where the background component is either
approximated by an analytic expression or simulated by MC
with a reduced sample size. We demand that ΦB is
independent of any parameter of interest, μ (and of any parameter
which is strongly correlated with a parameter of interest):
dΦB
dμ
= 0.
The approximate model (13) can then be used as
replacement for the real model. The sensitivity estimation can then
be continued in the standard frequentist way: toy data are
sampled from Φ for given parameter choices and the
confidence region for the parameter of interest μ can then be
determined via χ 2 fits.
The benefit of this strategy can be understood in the
following way. Since the data have been sampled from the
model, any error in the model will also be passed over to the
data. However, while the total approximated distribution Φ
itself is inaccurate, it still contains all essential information
about the sensitivity, since Φ − cBΦB = cSΦS holds exactly
(Fig. 4). Since only the fidelity of the signal is relevant for the
sensitivity analysis (which we assured with condition (14)),
both the error in the model and in the data approximately
cancel each other in the fit. It can be shown in this case that
the width of the χ 2 minimum stays the same as long as the
background component is at least approximately correct. A
simplified proof can be found in Appendix A.
We shall discuss the method now on the initial case of
the keV scale sterile neutrino search with TOF spectroscopy.
As derived above, the electron TOF spectrum (6) with added
sterile neutrinos can be expressed as a superposition of two
TOF spectra with a light or heavy neutrino mass, ml and mh ,
respectively. We identify the signal with the sterile neutrino
(12)
(13)
(14)
signal + background
Φ = cSΦS + cBΦB
background
cBΦB
signal
cSΦS = Φ − cBΦB
component of the TOF spectrum (5) and the background with
the active neutrino contribution,
d N d N
ΦS = dτ (mh ) ΦB = dτ (ml ).
cS = sin2 Θ cB = cos2 Θ.
ni
λSi
(16)
(17)
(18)
(19)
It is obvious that for a small signal fraction of, e.g.,
sin2 θ 10−6, only a small fraction of the total expected
events needs to be simulated now. However, since signal and
background are always measured together and not separately,
the required sample size is reduced even more. For
demonstration purposes, let us define the signal expectation value
in bin i as
λSi = n · cS · ΦS(x = Xi ),
with n as total number of expected events (see Fig. 4). We
will denote the number of expected events in bin i as ni . To
approximate the necessary sample size, we require that the
numerical uncertainty of λSi needs to be smaller than the
expected measurement uncertainty of the number events in
the corresponding bin, σi :
Assuming a Poissonian measurement uncertainty, σi = √ni
and using ΔλSi /λSi = 1/ NSi , where NSi denotes the signal
sample size in bin i , Eq. (18) gives
⇐⇒
NSi
λ2Si ,
ni
ΔλSi
σi ,
1
NSi
λSi
We now define the total signal sample size as NS = i NSi .
If we assume the signal-background ratio to be roughly
within a constant order of magnitude, we get the required
minimum signal sample size:
NS
i
λn2Sii ≈
ni2 · cS
2
ni
2
= n · cS.
Naively, one would suppose that the signal part still needs to
be sampled with full statistics, i.e. NS n · cS. However,
due to the fact that the signal part is always measured with
background, we have shown that an additional suppression
factor of cS applies. Assuming sin2 θ ∼ 10−6 and a total
event size of n ∼ 1018, we thus get
Note that sin2 θ ∼ 10−6 represents roughly the upper bound
from astrophysical observations. Likewise, n ∼ 1018 is
approximately the maximum number of counts which will
decrease with higher retarding potentials. Thus, for lower
values of either one, the necessary sample size is reduced
even more according to condition (20).
Using a Monte Carlo algorithm, the TOF spectra given by
the transformation (6) can be determined in a
straightforward way. For each MC sample, at first an initial energy and
starting angle is generated. The angular distribution is given
by Eq. (7). For the initial energy, the electronic excited state
is generated from the final state distribution in Eq. (1) and
then the energy is generated from the respective β-spectrum
component in Eq. (2). Given the initial energy and the starting
angle, the number of inelastic scattering process in the source
is generated from Eq. (10) and for each process the energy
loss is generated from Eq. (9) and subtracted from the energy.
In order to further optimize the Monte Carlo method for a
parametrizable heavy neutrino mass, the TOF spectra have
additionally been decomposed into elements
corresponding to different sterile neutrino mass phase space segments,
which is explained in detail in Appendix B. The advantage of
such a scheme is that already simulated Monte Carlo events
can be reused for different sterile neutrino masses.
We found that a sample size of 108 for each sterile
subcomponent is feasible in finite calculation time and
sufficient for an accurate simulation. The active neutrino
component, which contains ∼ 1/ sin2 θ more counts than the total
sterile component, was approximated with a sample size of
109, according to the SCAMC approach. The active neutrino
mass was set to ml = 0 and the endpoint held constant at
τ (ns)
4000
5000
Fig. 5 Electron TOF spectra for a keV-scale sterile neutrino of mh =
1.1 keV and different mixing angles at a fixed retarding potential of
17 keV. The mixing angles have been exaggerated to enhance the
signature and comprise additionally the case of no mixing (sin2 θ = 0)
as well of pure sterile contribution (sin2 θ = 1). Similar to the tritium
β-decay energy spectrum, the signature of a sterile neutrino is a
kinklike discontinuity at a certain point in the TOF spectrum. Figure first
published in [
1
]
mh = 0.8 keV
mh = 0.9 keV
mh = 1.0 keV
mh = 1.1 keV
4000
5000
1000
2000
3000
E0 = 18.575 keV, since there is no correlation to expect
with the sterile neutrino. The bin width was chosen to be
250 ns (compared to the FPD time resolution of about 50 ns)
for reasons of performance and robustness. However, it is
unlikely to expect for any measurement method to achieve a
higher resolution. To all spectra a Gaussian time uncertainty
of Δτ = 50 ns was added to account for the detector time
resolution and a isochronous background of b = 10 mcps.
Figures 5 and 6 show exemplary simulated TOF
spectra for different active-sterile mixings and heavy neutrino
masses, respectively. It can be seen that the spectra show a
dominating peak within the first 2 µs which consists of the
fast electrons more than some 100 eV above the retarding
potential. They are, however, followed by a long tail where
the electron velocity becomes slower and the TOF difference
per given energy difference (see Fig. 2) becomes more
significant. In this region the TOF spectrum is to a good extent
a differential representation of the β spectrum, while the fast
peak region consists only of some bins, thus contributing
to the sensitivity more by its integral. If the sterile neutrino
mass is some 100 eV smaller than the difference of endpoint
and retarding potential, the sterile neutrino signal becomes
similar to that one in the tritium beta spectrum. The sterile
neutrino contribution appears as a discontinuity in shape of
a “kink” at a certain position in the spectrum. Since the
relationship between energy and TOF is non-linear, the position
of the kink allows no direct analytical conclusion about the
sterile neutrino mass. However, given the retarding potential,
the relation in Fig. 2 can be used for an estimation.
4.2 Ideal TOF mode sensitivity
The model described in the last section was utilized to
estimate the sensitivity according to the procedure described in
Sect. 3. The fits have generally been performed by a χ 2
minimizations using MINUIT [
60
] . For statistical sensitivity
estimation, the mixing sin2 θ and overall amplitude S are
free fit parameters, using a range of fixed values for mh . In
those simulations, where the uncertainty on mh is of interest,
also the squared heavy neutrino mass m2h has been included
as fit parameter. Since each fit incorporates a set of
multiple measurements at different retarding potentials, the χ 2
functions of each measurement are added and fitted with
global fit parameters. Instead of a pure ensemble approach,
the parameter uncertainties have been calculated using the
module MINOS from MINUIT [
60
], averaged over
multiple simulations, which gives in case of an approximately
quadratic χ 2 near the minimum an identical result.
Exemplary systematics
In addition to the statistical sensitivity, an exemplary
systematic effect has been studied, which is the inelastic
scattering cross section due to fluctuation in the column density
as described in Eq. (9). This is one of two main systematics
when it comes to keV sterile neutrino search, the other being
the final state distribution [
51–53
]. To incorporate the
systematics, the χ 2 function has been modified by an additional
term:
χ 2 = χ02 +
(ρd − ρd )2
(Δρd)2
0.000
)
.
.U−0.001
A
(
0
Φ
)/−0.002
√
0
Φ
−
)ρd−0.003
(
Φ
(
−0.004
Fig. 7 Difference between TOF spectra with shifted ρd, Φ(ρd) and
default value ρd = 5 × 1014 cm−2, Φ0, weighted proportionally with
the expected Poissonian uncertainty of the data ∝ √Φ0. The imprint
of a shifted column density is present foremost at lower flight times,
due to missing events near the endpoint because of the energy loss.
Fluctuations at higher flight times near the retarding potential are
suppressed by a lower differential count rate. The spectra consist only of
the active neutrino component, sin2 θ = 0, and the retarding potential
is qU = 18 kV
where χ02 is the default binned χ 2 function, ρd the fitted
column density, ρd its expectation value and Δρd the
systematic uncertainty. In order to be able to have ρd as
free fit parameter, the complete model has additionally been
separated by number of inelastic scattering processes and
weighted with the l-fold energy loss probability pl (ρd) as
given by Eq. (10), instead of randomly generating the number
of inelastic scattering events,
(23)
d N
dτ =
l
pl (ρd) ·
d N
dτ
l
.
To determine the influence of the uncertainties Δρd on
the sensitivity, the column density has been shifted by for
the data generation by ρd = ρd + Δρd while still using
the unshifted expectation value ρd in Eq. (22). By this
approach the MINOS error will increase plus a possibly slight
bias in average which is then quadratically added to the
average error bars.
To illustrate the imprint of the systematic uncertainty of
ρd in the TOF spectrum, Fig. 7 shows the difference between
a TOF spectrum with shifted column density, Φ(ρd) =
d N /dτ (ρd + Δρd) and a TOF spectrum with mean
column density, Φ0 = d N /dτ ( ρd ), weighted by √Φ0 which
is proportional to the expected Poissonian uncertainty of the
data. By doing so, the signature becomes visible
proportionally to its impact in the χ 2 function. It can be seen that the
imprint of a shifted column density is present foremost at
lower flight times, which is since the energy loss causes the
count-rate near the endpoint to drop. There are fluctuations
at higher flight times near the retarding potential arising from
10−6
10−7
10−9
the energy loss spectrum (9). However, these are weighted
minimally since the differential rate in the TOF spectrum
drops with higher flight times (see Fig. 5).
Results
Figure 8 shows the sensitivity for an ideal TOF mode. The
results are based on three years measurement time which
was distributed uniformly on the retarding potential within
an interval of [4; 18.5] keV with steps of 0.5 keV. The
setting was chosen in that way that a 7 keV neutrino signal
[
25
] would roughly lie in the center of the potential
distribution. For the exemplary inelastic scattering systematics an
initial uncertainty of Δρd/ρd = 0.002 has been assumed in
accordance with Ref. [
30
]. The statistical sensitivity of the
integral mode in this simulation is in good agreement with
Ref. [
31
]. The statistical sensitivity of the ideal TOF mode
is close to that one of an ideal differential detector in the
aforementioned publication. However, if the uncertainties of
the column density are incorporated, the benefit by the TOF
mode grows even further, since a shifted column density has
a unique imprint on the TOF spectrum (see Fig. 7), which is
not the case in the integral mode.
It should be noted, however, that for low retarding
potentials as used in Fig. 8, adiabaticity of the electron transport is
limited. Yet, that can be maintained by increasing the
magnetic field in the main spectrometer. This lowers the energy
resolution and thus the transformation of transverse into
longitudinal momentum, which would manifest in a stronger
1012
s
t
n
u
o
c1011
1010
Fig. 9 Exemplary fit of a sterile neutrino with mass mh = 2 keV and
low mixing sin2 θ = 10−6 (not visible by eye), assuming an ideal TOF
measurement and using four exemplary retarding potentials of 15, 16,
17 and 18 keV. The fit includes the systematic uncertainty of the column
density ρd, as well as the sterile neutrino mass as free fit parameters.
The overall count rate increases with decreasing retarding potential
angular-dependence of the energy-TOF relation in Fig. 2.
Though, this should have no significant influence on the
sensitivity since the measurement takes place on a keV-scale
where the requirements for magnetic adiabatic collimation
are more relaxed.
An exemplary fit is shown in Fig. 9 for a sterile neutrino
with mass mh = 2 keV and a mixing of sin2 θ = 10−6
assuming an ideal TOF measurement and using four
exemplary retarding potentials of 15, 16, 17 and 18 keV. In this
case the sterile neutrino mass has not been fixed but used as
a free fit parameter to test the ability to fit the sterile neutrino
mass, given a sufficiently high active-sterile mixing. While
it is in principle sufficient to use only one retarding potential
closely below the sterile neutrino kink, in practice a multitude
of retarding potentials is necessary. The reasons are that, on
one hand, the mass of the sterile neutrino is unknown and, on
the other hand, that it is also necessary in order to determine
the other parameters. In contrast to the pure sterile active
mixing sensitivity estimation (Fig. 8) the heavy neutrino mass has
been used as free fit parameter. It shows that the method is
capable of a sensitive mass determination as well, in case the
mixing angle is large enough. However, since most parts of
the sensitive regions of the TOF method are disfavored by
X ray satellite measurements [
15
], it seems unlikely that a
mass fit will be possible.
4.3 Optimization and integrity
SCAMC variance
In order to show that the SCAMC method is really working
as expected, it has been tested using different Monte Carlo
sample sizes. A necessary condition is convergence of the
result towards a constant value with growing sample size. As
described in Appendix B, the signal itself is split into signal
components defined by slices of the signal TOF spectrum in
mh -space (Eq. (B.6)). Figure 10 shows the ideal TOF mode
statistical sensitivity for a 2 keV neutrino as a function of
the sample size used for each such component of the signal.
The components have been sampled with steps of 0.1 keV in
terms of mh . The total signal sample size per TOF spectrum
is thus given by
NS = NC · (E0 −0.1mkhe−V qU ) , (24)
where NC denotes the sample size per component. For
minimum qU and mh it amounts to ∼ 150 · NC. The background
has been simulated with a sample size of 10 · NC.
It can be seen that convergence is met and that with
subsample sizes such as 104 per component the expected result
is approximated with less than 1% uncertainty.
Measurement interval
Figure 11 shows the sensitivity to sin2 θ in a similar way
as Fig. 8, but for a measurement interval of [15; 18.5] keV,
roughly centered around a 2 keV neutrino, as favored in Ref.
[
3
]. It can be seen in comparison that there is no benefit of
restricting the measurement interval to a narrow region in
search for a sterile neutrino with a given energy. This seems
counter-intuitive at first, but is has to be kept in mind that
the sterile neutrino signal is not localized at the kink, but
instead contributes to the whole spectrum below. In contrast
10−6
10−8
4 × 10−8
3 × 10−8
2 × 10−8
integral
TOF
500
1000
1500
step size (eV)
2000
Fig. 12 Statistical sensitivity as a function of the step size between
measurement points of the retarding potential qU for a sterile neutrino
with mh = 2 keV, with constant total sample size. The measurement
interval is [4; 18.5] keV for a total measurement time of 3 years
to dedicated ‘kink-search’ methods [
32
], all spectral parts
contribute to the sensitivity in a χ 2 fit. While the relative
difference made by a sterile neutrino signal might be smaller
at lower retarding potentials, this drawback is however
balanced by a larger count-rate at lower potentials.
Measurement step size
Figure 12 shows the statistical sensitivity as a function of
the spacing between different measurement points of the
retarding potential qU . The total sample size has been kept
constant. The simulations show no preference towards any
particular value. That appears unintuitive, since one would
expect a narrower spacing to have beneficial effects on a
distinct kink search. Yet, as mentioned in the last paragraph, the
sterile neutrino signal is not localized, but manifests itself
in relative count rate differences between the measurement
points with a spectral feature as broad as the mass of the
sterile neutrino mh . Therefore, a larger step size does not weaken
the sensitivity in principle because the measurement time is
distributed over less points. Anyway, it is in general
recommended to use a step size lower then the smallest possible
heavy neutrino mass, since otherwise it is possible that there
are not enough vital measurement points above the kink.
The benefit of a TOF measurement can be explained in
this context as follows: TOF spectra carry extra information
about the differential energy distribution closely above each
measurement point. That equates to knowledge about the
slope of the integral spectrum at these measurement points.
4.4 Gated filter sensitivity
Figure 13 shows exemplary TOF spectra using Gated
Filtering (GF, see Fig. 3). It illustrates how GF works: without the
gate (cyan points), the arrival time spectrum is isochronous.
However, with activated gate, a certain portion is cut away
from the isochronous spectrum. For a given repetition time
tr and duty cycle ξ , the duration in which the gate is open is
given by tr · ξ . The GF arrival time filter thus is smeared with
a step function when compared to the raw TOF spectrum.
Reducing the duty cycle ξ makes the arrival time spectrum
approximate the TOF spectrum of Fig. 9, however with a
loss of overall rate. Electrons with a TOF greater than the
repetition time tr lead to the wrongful attribution of the
corresponding events to a later period, which can be seen in the
first few bins. However, since TOF spectra at several keV
below the endpoint are rather sharp, this effect is small for
repetition times of ∼ 10 µs.
Figure 14 shows the sensitivity for two exemplary gated
filter scenarios with a constant duty cycle of 0.1. The
scenario is based on the assumption that the existing focal plane
detector (FPD) of KATRIN is used, which is optimized for a
measurement near the endpoint of the β spectrum and thus
can not maintain much higher count-rates. The bottleneck
is particularly the per-pixel rate which should not exceed
∼ 103 cps within a window of some µs. This limitation holds
for the current data acquisition and might be improved in the
future. In this simulation, an exemplary overall reduction of
the signal rate by a factor 105 has been chosen which will
ensure a flux compatible with the current hardware. Since the
gated filter periodically blocks the flux of electrons, the rate
can be increased again with respect to the integral mode. The
actual allowed rate with the gated filter depends on the
readout electronics and will effectively be between two extremal
values. In an optimistic case, short-time excess of the rate
is tolerable, while the average rate has to be at the same
level as with the integral mode. In a conservative case, also
short-time excess leads to pile-up, which means that instead
τ (ns)
8000
10000
5 Summary and discussion
Fig. 13 Exemplary Gated Filter arrival time spectra for different duty
cycles. Retarding energy is qU = 17 keV and repetition time tr =
10 µs. The active-sterile mixing has been set to sin2 θ = 0. Activating
the gate and decreasing the duty cycle cuts away portions of the arrival
time spectrum, which is isochronous without gate
20000
15000
)
s
n
/
s
p
(c10000
τ
d
/
N
d
Fig. 14 Sensitivity (1 σ ) of integral mode with the rate reduced by a
factor 1 × 105 (red), compared with a conservative gated filter TOF
scenario (blue) with same peak rate as the integral mode and an optimistic
gated filter TOF scenario (green) with the same total rate as the integral
mode. Both statistical uncertainty (dashed lines) and combined
statistical plus systematic uncertainty including a column density uncertainty
Δρd/ρd = 0.002 (full lines, see Sect. 4.2) are plotted. The duty cycle
is 0.1 for both gated filter scenarios. Measurement interval has been [15;
18.5] keV for three years data taking. The repetition time is tr = 10 µs
for all retarding potentials
the peak rate may not exceed the constant rate of the
integral mode (see Fig. 13). The repetition rate has been fixed at
10 µs, which will ensure coverage the vast part of the TOF
spectrum. The measurement interval has been limited to [15;
18.5] keV since it is not believed to be viable to pulse the
pre-spectrometer more than several keV.
It can be seen that in the gated filter beats the integral
mode in the optimistic case, but not in the conservative case.
This means that the loss of rate in the conservative case is too
high to be compensated by the additional TOF information.
In case the detector readout electronics sufficiently tolerates
short-time excesses, the loss of statistics by the gated filter
can, nevertheless, be compensated and additional TOF
information is gained. Note, however, that in the scenario of an
upgraded future detector which tolerates the full rate from
the tritium source, the integral mode outperforms the gated
filter mode, since there is now way in this case to increase
the rate further.
It has been shown that TOF spectroscopy in a KATRIN
context is in principle able to boost the sensitivity of the sterile
neutrino search significantly. Figure 11 suggests an
improvement of up to a factor two in terms of pure statistical
uncertainty down to at maximum sin2 θ ∼ 5 × 10−9 for a sterile
neutrino of mh = 7 keV at one σ . If the exemplary systematic
uncertainty of the inelastic scattering cross section is
considered, the sensitivity is only mildly weakened in contrast to
the integral mode, which is in that case outperformed by the
TOF mode by up to a factor five. However, the practical
realization of a sensitive TOF measuring method is still work
in progress. Given the current hardware, which requires a
reduction of the signal rate, the gated filter method might be
able to realize a TOF mode with a slight sensitivity increase
compared to the integral mode under the condition that the
detector tolerates short-time excesses of the rate and that it
is possible to ramp the pre-spectrometer potential some keV
within ∼ 0.1 µs. From a long-term point of view, the concept
of an upgraded differential detector [
31
] which is capable of
extreme rates up to 1010 cps is very promising. If there is
sufficient progress in developing a sensitive TOF measurement
method, a beneficial strategy could be a combined
measurement to eliminate systematics and perform cross-checks.
Acknowledgements We would like to thank S. Enomoto for
discussions. This work is partly funded by BMBF under Contract no.
05A11PM2 and DFG GRK 2149.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Unchanged χ 2 properties with SCAMC
In the following it is shown that the properties of the χ 2
function defining the sensitivity, which are position and width
of the minimum with respect to any parameter of interest, are
independent of the choice of the background model ΦB. This
works as well for a Poissonian log-likelihood, but for brevity
we show it on a χ 2 example. First we define the expectation
value for the i -th bin,
λi = λSi + λBi = n (cSΦSi + cBΦBi ),
using the definition of the approximated model (13), and
assume that the background prediction λBi is independent of
the parameter of interest μ,
For the proof we differentiate χ 2 with respect to μ and
demand that the result is approximately independent of the
choice of the background model ΦB:
(ni − λi (μ))2
λi (μ)
λi ddμ (ni − λi )2 − (ni − λi )2 ddμ λi
−2λi (n − λi ) ddμ λSi − (n − λi )2 ddμ λSi
λi2
λi2
i
⎛
(ni2 − λi2) ddμ λSi
λi2
ni2
1 − λ 2
i
d
dμ λSi
ni2
1 − (λBi + λSi )2
⎝ 1 −
λBi λSi
ni + ni
d
dμ λSi
−2⎞
⎠
d
dμ λSi
d
dμ λBi = 0.
χ 2(μ) =
d
dμ
χ 2 =
i
i
i
i
i
i
= −
=
=
=
=
The variable ni is Poisson distributed with mean λi (μ0) =
λSi (μ0) + λ , where μ0 is the null-hypothesis for μ. Due
Bi
λ
Bi is approximately independent from
to self-consistency, ni
the choice of ΦB, as long as the order of magnitude is in
agreement ΦB ∼ ΦB. The latter condition ensures that the
Poissonian uncertainty of ni , which is given by λi (μ0), is
approximately correct.
Note that the proof is only correct in the simplified case
of one parameter of interest and no correlation with nuisance
parameters. However, the simulation results in this paper
show that there is valid reason to expect the method to work
also for more complex problems as long as there is no heavy
parameter correlation.
(A.1)
(A.2)
(A.3)
(A.4)
350
300
)s 250
n
/
s
(cp200
)
j
(m150
τ
d
/
dN100
50
00
Appendix B: Sterile neutrino mass decomposition of TOF
spectra
The simulation of the TOF spectra has further been optimized
with the aim of being able to use the sterile neutrino mass
ml as a free parameter with a minimum of computational
overhead. The idea is to decompose the sterile neutrino
components of the TOF spectra, ΦS, into sub-spectra ΦSk which
can be added subsequently to obtain the signal for a given
sterile neutrino mass mh . That works as follows: at first a
number J of grid points with heavy neutrino masses m j are
chosen. For each grid-point j , the signal spectrum is given
as the sum of all sub-signals from j up to J ,
ΦS(m j ) =
ΦSk .
J
k= j
The sub-signals ΦSk constitute the difference of two TOF
spectra with adjacent sterile neutrino masses. The total TOF
spectrum for the sterile component can then be written as
d N
dτ (m j ) =
d N
dτ (m J ) +
d N d N
dτ (mk ) − dτ (mk+1) .
J −1
k= j
Each sub-component in the sum will be sampled
separately. The difference between two TOF spectra can be
sampled just like any TOF spectrum, as outlined, by replacing
the β-spectrum in (6) also with the difference of two β
spectra corresponding to the neutrino masses mk and mk+1. Via
(B.5)
(B.6)
mj = 0.0 keV
mj = 0.1 keV
mj = 0.2 keV
mj = 0.3 keV
mj = 0.4 keV
mj = 0.5 keV
2000
4000
6000
Fig. 15 Illustration of the calculation of sterile component of the
electron TOF spectrum via subsequent addition of sub-components
according to (B.6). The figure shows the sterile components of the TOF
spectrum (5) for different sterile neutrino masses m j on a grid for a retarding
potential of qU = 18 kV. Each colored area corresponds to a
subcomponent between two adjacent mass values. The component for any
sterile neutrino mass m j is then given by the sum of all areas below the
envelope
(B.6), that gives then the sterile contribution of the TOF
spectrum for each mass value m j on the grid. For sterile neutrino
masses between the grid points, the resulting spectrum is
then calculated by cubic spline interpolation. The strategy is
illustrated in Fig. 15.
In addition to the reuse of already simulated Monte Carlo
events, this strategy has the possible advantage of a smoother
interpolation in bins with small statistics, which are possible
for high flight times 40 µs. By the de-composition and
subsequent addition of the components, monotony between
the interpolation grid points is guaranteed. However, if a
sufficient overall sample size is chosen, this effect should not
matter significantly.
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