Project 8 Collaboration

# Tritium Beta Spectrum and Neutrino Mass Limit from Cyclotron Radiation
Emission Spectroscopy

A. Ashtari Esfahani  Present Address: Department of Physics, Sharif University
of Technology, P.O. Box 11155-9161, Tehran, Iran  Center for Experimental
Nuclear Physics and Astrophysics and Department of Physics, University of
Washington, Seattle, WA 98195, USA  S. Böser  Institute for Physics, Johannes
Gutenberg University Mainz, 55128 Mainz, Germany  N. Buzinsky  Present
Address: Center for Experimental Nuclear Physics and Astrophysics and
Department of Physics, University of Washington, Seattle, WA 98195, USA
Laboratory for Nuclear Science, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  M. C. Carmona-Benitez  Department of Physics,
Pennsylvania State University, University Park, PA 16802, USA  C. Claessens
Center for Experimental Nuclear Physics and Astrophysics and Department of
Physics, University of Washington, Seattle, WA 98195, USA  Institute for
Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany  L. de
Viveiros  Department of Physics, Pennsylvania State University, University
Park, PA 16802, USA  P. J. Doe  Center for Experimental Nuclear Physics and
Astrophysics and Department of Physics, University of Washington, Seattle, WA
98195, USA  M. Fertl  Institute for Physics, Johannes Gutenberg University
Mainz, 55128 Mainz, Germany  J. A. Formaggio  Laboratory for Nuclear Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  J. K. Gaison
Pacific Northwest National Laboratory, Richland, WA 99354, USA  L. Gladstone
Department of Physics, Case Western Reserve University, Cleveland, OH 44106,
USA  M. Grando  Pacific Northwest National Laboratory, Richland, WA 99354, USA
M. Guigue  Laboratoire de Physique Nucléaire et de Hautes Énergies, Sorbonne
Université, Université Paris Cité, CNRS/IN2P3, 75005 Paris, France  J. Hartse
Center for Experimental Nuclear Physics and Astrophysics and Department of
Physics, University of Washington, Seattle, WA 98195, USA  K. M. Heeger
Wright Laboratory and Department of Physics, Yale University, New Haven, CT
06520, USA  X. Huyan  Present Address: LeoLabs, Menlo Park, CA 94025, USA
Pacific Northwest National Laboratory, Richland, WA 99354, USA  J. Johnston
Laboratory for Nuclear Science, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  A. M. Jones  Present Address: Ozen Engineering,
Sunnyvale, CA 94085, USA  Pacific Northwest National Laboratory, Richland, WA
99354, USA  K. Kazkaz  Lawrence Livermore National Laboratory, Livermore, CA
94550, USA  B. H. LaRoque  Pacific Northwest National Laboratory, Richland, WA
99354, USA  M. Li  Laboratory for Nuclear Science, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  A. Lindman  Institute for Physics,
Johannes Gutenberg University Mainz, 55128 Mainz, Germany  E. Machado  Center
for Experimental Nuclear Physics and Astrophysics and Department of Physics,
University of Washington, Seattle, WA 98195, USA  A. Marsteller  Center for
Experimental Nuclear Physics and Astrophysics and Department of Physics,
University of Washington, Seattle, WA 98195, USA  C. Matthé  Institute for
Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany  R.
Mohiuddin  Department of Physics, Case Western Reserve University, Cleveland,
OH 44106, USA  B. Monreal  Department of Physics, Case Western Reserve
University, Cleveland, OH 44106, USA  R. Mueller  Department of Physics,
Pennsylvania State University, University Park, PA 16802, USA  J. A. Nikkel
Wright Laboratory and Department of Physics, Yale University, New Haven, CT
06520, USA  E. Novitski<EMAIL_ADDRESS>Center for Experimental Nuclear Physics
and Astrophysics and Department of Physics, University of Washington, Seattle,
WA 98195, USA  N. S. Oblath  Pacific Northwest National Laboratory, Richland,
WA 99354, USA  J. I. Peña  Laboratory for Nuclear Science, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  W. Pettus
<EMAIL_ADDRESS>Center for Exploration of Energy and Matter and Department
of Physics, Indiana University, Bloomington, IN, 47405, USA  R. Reimann
Institute for Physics, Johannes Gutenberg University Mainz, 55128 Mainz,
Germany  R. G. H. Robertson  Center for Experimental Nuclear Physics and
Astrophysics and Department of Physics, University of Washington, Seattle, WA
98195, USA  D. Rosa De Jesús  Pacific Northwest National Laboratory, Richland,
WA 99354, USA  G. Rybka  Center for Experimental Nuclear Physics and
Astrophysics and Department of Physics, University of Washington, Seattle, WA
98195, USA  L. Saldaña  Wright Laboratory and Department of Physics, Yale
University, New Haven, CT 06520, USA  M. Schram  Present Address: Thomas
Jefferson National Accelerator Facility, Newport News, VA 23606, USA  Pacific
Northwest National Laboratory, Richland, WA 99354, USA  P. L. Slocum  Wright
Laboratory and Department of Physics, Yale University, New Haven, CT 06520,
USA  J. Stachurska  Laboratory for Nuclear Science, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Y.-H. Sun  Department of Physics, Case
Western Reserve University, Cleveland, OH 44106, USA  P. T. Surukuchi  Wright
Laboratory and Department of Physics, Yale University, New Haven, CT 06520,
USA  J. R. Tedeschi  Pacific Northwest National Laboratory, Richland, WA
99354, USA  A. B. Telles  Wright Laboratory and Department of Physics, Yale
University, New Haven, CT 06520, USA  F. Thomas  Institute for Physics,
Johannes Gutenberg University Mainz, 55128 Mainz, Germany  M. Thomas  Present
Address: Booz Allen Hamilton, San Antonio, Texas, 78226, USA  Pacific
Northwest National Laboratory, Richland, WA 99354, USA  L. A. Thorne
Institute for Physics, Johannes Gutenberg University Mainz, 55128 Mainz,
Germany  T. Thümmler  Institute of Astroparticle Physics, Karlsruhe Institute
of Technology, 76021 Karlsruhe, Germany  L. Tvrznikova  Present Address:
Waymo, Mountain View, CA 94043  Lawrence Livermore National Laboratory,
Livermore, CA 94550, USA  W. Van De Pontseele  Laboratory for Nuclear Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  B. A.
VanDevender  Center for Experimental Nuclear Physics and Astrophysics and
Department of Physics, University of Washington, Seattle, WA 98195, USA
Pacific Northwest National Laboratory, Richland, WA 99354, USA  J. Weintroub
Center for Astrophysics $\mid$ Harvard $\&$ Smithsonian, Cambridge, MA 02138,
USA  T. E. Weiss  Wright Laboratory and Department of Physics, Yale
University, New Haven, CT 06520, USA  T. Wendler  Department of Physics,
Pennsylvania State University, University Park, PA 16802, USA  A. Young
Present Address: Department of Astrophysics/IMAPP, Radboud University, PO Box
9010, 6500 GL Nijmegen, The Netherlands  Center for Astrophysics $\mid$
Harvard $\&$ Smithsonian, Cambridge, MA 02138, USA  E. Zayas  Laboratory for
Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  A. Ziegler  Department of Physics, Pennsylvania State University,
University Park, PA 16802, USA

###### Abstract

The absolute scale of the neutrino mass plays a critical role in physics at
every scale, from the particle to the cosmological. Measurements of the
tritium endpoint spectrum have provided the most precise direct limit on the
neutrino mass scale. In this Letter, we present advances by Project 8 to the
Cyclotron Radiation Emission Spectroscopy (CRES) technique culminating in the
first frequency-based neutrino mass limit. With only a cm3-scale physical
detection volume, a limit of $m_{\beta}{<}155$ eV ($152$ eV) is extracted from
the background-free measurement of the continuous tritium beta spectrum in a
Bayesian (frequentist) analysis. Using 83mKr calibration data, an improved
resolution of $1.66{\pm}0.19$ eV (FWHM) is measured, the detector response
model is validated, and the efficiency is characterized over the multi-keV
tritium analysis window. These measurements establish the potential of CRES
for a high-sensitivity next-generation direct neutrino mass experiment
featuring low background and high resolution.

The discovery of neutrino flavor transformation [1, 2] proves that neutrinos
are massive particles, in conflict with the original Standard Model
formulation, and establishes that the weak flavor eigenstates ($\nu_{e}$,
$\nu_{\upmu}$, $\nu_{\uptau}$) are admixtures of the three neutrino mass
eigenstates ($\nu_{1}$, $\nu_{2}$, $\nu_{3}$). The neutrinos, alone among the
fundamental fermions, have masses that remain unmeasured [3].

Neutrino mass is important across nuclear and particle physics, astrophysics,
and cosmology. The origin and magnitude of neutrino mass may hint at new
physics [4] such as the neutrino’s possible Majorana nature [5], with
laboratory searches for neutrinoless double-beta decay [6, 7, 8, 9, 10]
testing this hypothesis. The tightest, though model-dependent, upper limits on
the absolute scale of neutrino mass [11, 12] are derived from measurements of
the large-scale structure analyzed within the $\Lambda$CDM cosmological
framework [13]. Neutrino mass has some degeneracy with other parameters, and
constraints on it are weakened when allowing additional model extensions [14].
Further, emerging cosmological tensions (_e.g.,_ $H_{0}$ determination [11,
15, 16]) might point to new physics [17], highlighting the value of
independent measurements of $\Lambda$CDM parameters.

A direct and model-independent laboratory constraint on the neutrino mass can
be derived from the kinematics of beta decay or electron capture [18, 19]. The
electron-weighted neutrino mass ($m_{\beta}$) observable is

$m_{\beta}=\sqrt{\sum_{i=1}^{3}\left|U_{ei}\right|^{2}m_{i}^{2}},$ (1)

where $m_{i=1,2,3}$ are the neutrino mass eigenvalues and $U_{ei}$ are
elements of the $3\times 3$ unitary leptonic mixing matrix [20, 21]. The
effect of neutrino mass manifests in the decay electron spectrum near the
endpoint as both a shape distortion and a reduction in maximum electron
energy. Neutrino flavor oscillation measurements, sensitive only to the mass-
squared differences, impose an ultimate lower bound of $m_{\beta}{\geq}0.009$
eV/$c^{2}$ ($m_{\beta}{\geq}0.048$ eV/$c^{2}$) for the normal (inverted) mass
ordering [3].

For over 70 years, tritium beta decay experiments have produced the most
sensitive direct $m_{\beta}$ limits [19]. Most recently, KATRIN has set a
limit of $m_{\beta}{<}0.8$ eV/$c^{2}$ (90% C.L.) [22]. Such experiments using
molecular tritium (3H2, or T2) become systematics-limited at
$m_{\beta}{\sim}0.1$ eV/$c^{2}$ due to broadening caused by internal molecular
motion [23, 24]. An alternative method uses electron capture on 163Ho [25,
26], with a current limit of $m_{\beta}{<}150$ eV/$c^{2}$ [27]; the challenges
of complex atomic and solid-state structure, backgrounds, and pileup in this
method are being investigated.

The Project 8 Collaboration has developed a new technique, Cyclotron Radiation
Emission Spectroscopy (CRES), in pursuit of eventual sensitivity to
$m_{\beta}$ down to 0.04 eV/$c^{2}$. CRES uses the cyclotron emission from
electrons or positrons to determine their energies [28, 29, 30]. The cyclotron
frequency ($f_{\mathrm{c}}$) of electrons in a magnetic field ($B$) is a proxy
for their kinetic energy ($E_{\rm{kin}}$):

$f_{\mathrm{c}}=\frac{1}{2\pi}\frac{\lvert e\rvert
B}{m_{e}+E_{\rm{kin}}/c^{2}},$ (2)

where $e$ is the electron charge, $m_{e}$ is the mass of the electron, and $c$
is the speed of light in vacuum. For magnetic field strength of
$\mathcal{O}$(T), an electron’s faint $\mathcal{O}$(fW) cyclotron radiation
can be detected directly at radio frequency (RF). The high precision possible
with a frequency measurement and CRES’s inherent relative immunity to
background make the technique ideally suited to studying electrons emitted in
beta decay [31]. CRES has the statistical advantage of simultaneous
acquisition across the energy range of interest and elimination of systematic
effects associated with integral spectroscopic methods [32, 33]. Project 8
first demonstrated this technique on the conversion electrons emitted by
gaseous 83mKr [29]. Here we present the first continuous-spectrum measurement
using the CRES technique—on molecular tritium beta decay near the endpoint
region—enabling the first neutrino mass limit using CRES. We further
demonstrate eV-scale resolution and modeling of the detector response using
83mKr data. The full details of these analyses are presented in a companion
manuscript [34].

At the core of the CRES apparatus is a cryogenic gas cell (the “CRES cell”),
within which electrons are produced in radioactive decay and magnetically
trapped while they emit cyclotron radiation (Fig. 1).

Figure 1: Cutaway of the cryogenic CRES cell, where electrons are produced in
radioactive decay and magnetically trapped. Cyclotron radiation travels
axially up the waveguide (left in rotated view), toward the amplifiers and
readout electronics.

Radioactive source gas is delivered to this cylindrical cell through an array
of sub-wavelength holes and the gas is confined axially by microwave-
transparent CaF2 vacuum windows. Cooling the cell to 85 K reduces thermal
noise while maintaining sufficient 83mKr in the gas phase. The cell is
positioned in a superconducting magnet, whose 0.959 T axial background
magnetic field induces cyclotron motion and confines electrons radially. A
“field-shifting solenoid” within the magnet bore enables shifting of this
background field by up to $\pm$0.3% for systematic studies. Five coils wound
around the cell provide near-harmonic magnetic trap potentials to confine
electrons axially. Data were taken in composite traps: a “shallow trap” of two
coils with depth ${\sim}0.08$ mT to demonstrate high-resolution CRES and a
“deep trap” of four coils with depth ${\sim}1.4$ mT to increase effective
volume for the tritium endpoint measurement.

Source gases—molecular tritium and 83mKr—are delivered individually from a
custom gas manifold. Tritium is stored in a non-evaporable getter, with its
operating partial pressure stabilized at $10^{-6}$ mbar using a feedback-loop-
controlled heating current to optimally balance the event rate with the rate
of unwanted electron-gas collisions. In a separate calibration mode, 83mKr
emanates from 83Rb adsorbed in zeolite [35], and the rate of electron
collisions is tuned to match that in tritium data by the controlled release of
H2 from a separate getter. The gas composition is monitored using quadrupole
mass analyzers. To remove unwanted 3He from the decay of tritium adsorbed on
gas system walls, in later data sets the gas from the active volume is
continuously pumped through a controllable leak valve and sequestered.

The CRES cell is a circular waveguide segment with an inner diameter of 10.06
mm. The 26 GHz cyclotron radiation couples to the TE11 propagating mode. Only
the upward-propagating radiation is detected; it is transmitted via waveguide
to a series pair of low-noise cryogenic amplifiers held at 30 K. To avoid
unwanted reflections, the downward-propagating radiation is absorbed below the
cell in a conical graphite-epoxy terminator, thermal noise from which is the
dominant contributor to the 132$\pm$7 K system noise temperature. After
amplification, the signal is mixed down in frequency and filtered before being
digitized by a ROACH2 data acquisition (DAQ) system [36] sampling at 3.2 GS/s.
The onboard FPGA performs Fourier transforms and digital downconversion to 200
MS/s in three independently-tunable DAQ frequency windows. The data are
streamed to a compute node that applies trigger logic, writing periods of
time-series data to disk based on high observed power in frequency space.

The Fourier-transformed data form a two-dimensional spectrogram of power as a
function of frequency and time (Fig. 2). A tunable point-clustering algorithm
is used to identify bins with a high signal-to-noise ratio (SNR) that belong
to electron signals and to group these into contiguous “tracks.” Tracks are
always “chirped”—positive-sloped in frequency—due to radiated cyclotron power
(Eq. 2). Inelastic collisions between electrons and gas molecules cause energy
loss and therefore sudden jumps in cyclotron frequency, so tracks in close
time sequence are designated as being a single electron “event.” An electron’s
energy at the time of decay is extracted from the initial frequency of the
earliest track in an event. The on-axis detector geometry introduces a Doppler
shift, shunting power from the main carrier into sidebands and thereby
limiting the SNR and effective volume. All tracks in the spectrogram in Fig. 2
are the main carrier signal; sideband tracks due to modulation are present at
frequencies above and below the main carrier but are too low-power to be
detected in this apparatus [37].

Figure 2:  Spectrogram of the first CRES event detected from a tritium beta
decay electron. Raw time-series data are Fourier-transformed in time bins of
40.96 $\upmu$s, yielding 24.41 kHz frequency bins.

CRES is inherently an extremely low-background technique, with RF noise
fluctuations as the dominant background. By precisely characterizing the RF
background, true events can be sensitively distinguished from noise. In the
event selection, the decision to keep or remove an event is based on the
number of tracks it contains and properties of its first track: the number of
high-power bins it contains (roughly corresponding to its duration) and its
average SNR. The parameters for this cut were chosen before tritium data
acquisition at a level expected to allow less than one RF-noise-induced
background event in the tritium data set with 90% probability.

Analysis is performed on binned cyclotron frequency ($f_{\mathrm{c}}$) data
[34]. The predicted spectral shape ($\mathcal{S}$) as detected in a CRES
apparatus is

$\mathcal{S}=\epsilon\left(\mathcal{Y}*\sum_{j=0}\mathcal{A}_{j}\left(\mathcal{I}*\mathcal{L}^{*j}\right)\right),$
(3)

where all variables can be expressed as functions of $f_{\mathrm{c}}$, and
consequently $E_{\mathrm{kin}}$ and $B$ (Eq. 2). Convolution is denoted by
$*$, with superscript $*j$ representing self-convolution $j$ times. The
detection efficiency ($\epsilon$) uniquely has explicit physical dependence on
both $f_{\mathrm{c}}$ and $E_{\mathrm{kin}}$. The true underlying spectrum
($\mathcal{Y}$) includes all source effects. The summation term characterizes
the broadening elements of the detector response, primarily from scattering
and the inhomogeneity of the magnetic trapping field. Scattering before an
electron’s detection gives rise to a low-energy tail populated by events with
undetected true first tracks. Scatter peak amplitudes ($\mathcal{A}_{j}$) are
the probabilities that an electron is first detected after $j$ scatters. They
are determined by a phenomenological model. The electron’s energy-loss
distribution after $j$ scatters ($\mathcal{L}^{*j}$) depends on the
differential cross sections, the fractions $\gamma_{i}$ of each gas species
$i$, and the loss to cyclotron radiation. The intrinsic instrumental
resolution ($\mathcal{I}$) arises from variation in the average magnetic
fields sampled by electrons with different kinematic parameters [37]. It is
modeled using simulations of monoenergetic electrons in the trap’s magnetic
field profile [38].

Near-monoenergetic conversion electrons from 83mKr are a powerful tool for
characterizing the detector response near the tritium endpoint. The underlying
spectrum of the 83mKr K-line at 17.8 keV ($\mathcal{Y}_{\mathrm{Kr}}$)
consists of a narrow main peak and secondary low-energy peaks from shakeup and
shakeoff [39]. 83mKr calibration data (Fig. 3) are taken in the shallow and
deep trap configurations to verify the validity of the detector response model
across different trap depths and scattering environments. Poisson-likelihood
$\chi^{2}$ fits [40] are performed on these data, with goodness-of-fit tested
using Monte Carlo methods.

Figure 3:  Data and fits of the 17.8 keV 83mKr conversion electron K-line, as
measured in the shallow (high-resolution) and the deep (high-statistics)
electron trapping configurations. The shallow trap exhibits an instrumental
resolution of $1.66{\pm}0.19$ eV (FWHM), while the deep trap provides direct
calibration of the tritium data-taking conditions.

The scattering parameters in $\mathcal{A}_{j}$ and the magnetic field $B$ are
free fit parameters, and no background is observed.

In the shallow trap configuration, resolution is optimized by minimizing the
magnetic field variation experienced by trapped electrons. The observed
primary peak width is 4.0 eV (FWHM); the K-line natural linewidth is 2.774 eV
[41], yielding an instrumental resolution $\mathcal{I}$ of $1.66{\pm}0.19$ eV.
The low-energy satellite peak consists of overlapping contributions from
scattering (61% of satellite peak counts) and shakeup/shakeoff components.
Across the full shallow-trap 83mKr spectrum, 69% of events are detected before
scattering.

The event rate is approximately $40$ times higher in the deep trap
configuration than in the shallow trap, at the expense of a broader peak width
of 54.3 eV (FWHM). This deep-trap configuration is also used for data
acquisition with tritium, making these 83mKr extracted parameters inputs for
the tritium analysis: $B{=}0.9578104(13)$ T is used directly, while
$\mathcal{A}_{j}$ is corrected for small differences in scattering environment
between data sets.

The 83mKr data are also used to measure the frequency variation of efficiency
$\epsilon$ and detector response elements $\mathcal{I}$ and $\mathcal{A}_{j}$.
By varying the background magnetic field in steps of $0.07$ mT over a range of
$\pm$3.2 mT using the field-shifting solenoid, deep trap 83mKr 17.8-keV data
(Fig. 4) are produced at a range of frequencies in the region of interest. The
“notch” in detection efficiency at 25.93 GHz arises from electron interactions
with the TM01 cavity mode due to reflections from waveguide elements.
Electrons in resonance with this mode lose energy to cyclotron radiation
faster than in free space [42, 43], which increases frequency chirp (track
slope) of the CRES signal and reduces the efficiency of the event
reconstruction procedure, which has been optimized for the non-enhanced
slopes. In addition to the frequency-dependent effects described above, the
efficiency $\epsilon$ for tritium data also includes an analytic term that
directly depends on electron energy, in order to account for the radiated
power [37]. Detection efficiencies are determined with uncertainties of 2–6%.
The efficiency $\epsilon$ and the dependence of $\mathcal{A}_{j}$ on
frequency, which are derived from these field-shifted 83mKr data, are passed
as inputs to the tritium data analysis, while variations in $\mathcal{I}$ were
found to have negligible impact.

Figure 4:  The 17.8 keV 83mKr conversion electron line recorded in the deep
trap with varying magnetic background fields (red to blue). The gray curve
shows the efficiency’s response to frequency variation, extrapolated from
single trap data. The green curve is corrected for energy dependence of
emitted cyclotron power and shows the relative efficiency predicted for
tritium data.

The electron spectrum from tritium beta decay extends out to its endpoint of
$E_{0}{=}18574$ eV [44, 24, 22]. All three DAQ frequency windows were used
simultaneously to record around the endpoint, with the combined analysis
window spanning 16.2–19.8 keV (25.81–25.99 GHz). Over the 82-live-day data-
taking period, using the high-statistics deep trap configuration, 3770
distinct tritium events were recorded.

The tritium analysis follows Eq. 3, where the underlying spectrum
$\mathcal{Y}_{\mathrm{tritium}}$ [45] here is an approximated beta spectrum
[46] convolved with the final state distribution of the 3HeT+ decay product
[23]. A flat background component is included as a free fit parameter.
Approximations are made to the instrumental resolution $\mathcal{I}$ and
energy-loss $\mathcal{L}$ to reduce computing time, to account for differences
in scattering environment, and to include an explicit parameter $\sigma$
describing instrumental resolution width. The approximate model produces
correct coverages and no biases for ensembles of Monte Carlo data generated
with an un-approximated model.

The endpoint and neutrino mass limit are determined using both Bayesian and
frequentist analyses validated with Monte Carlo studies. Separate fits were
performed to measure $E_{0}$ (with $m_{\beta}$ constrained near 0 eV) and
constrain $m_{\beta}$ (with $E_{0}$ floating). The frequentist analysis best-
fit interval for $m_{\beta}^{2}$ is constructed using the procedure in [47],
and is converted to the $m_{\beta}$ limit using the Feldman-Cousins method
[48].

Fig. 5 shows the measured tritium spectrum and fits, with results summarized
in Tab. 1.

Figure 5:  Measured tritium endpoint spectrum with Bayesian and frequentist
fits. (Inset) Frequentist neutrino mass and endpoint contours.

The endpoint and mass values are consistent with each other and with
literature values. No counts were detected above the endpoint, setting a
stringent upper limit on backgrounds of ${\leq}3\times 10^{-10}$ counts/eV/s.
The endpoint uncertainty contributions are listed in Tab. 2. Statistical
uncertainty dominates the uncertainty budget, with determination of systematic
effects also statistics-limited.

Table 1: Extracted tritium endpoint values with 1-$\sigma$ uncertainty and neutrino mass 90% credible/confidence intervals. The literature value is $E_{0}{=}18574$ eV [44, 24, 22]. |  Endpoint [eV] |  $m_{\beta}$ limit [eV/$c^{2}$]
---|---|---
Bayesian |  $18553^{+18}_{-19}$ |  $<$155
Frequentist |  $18548^{+19}_{-19}$ |  $<$152
Table 2: Contributions to endpoint uncertainty $\sigma(E_{0})$ in the frequentist analysis. Individual systematic uncertainty contributions were computed via the method of Asimov sets [49]; they therefore do not sum in quadrature to the total systematic uncertainty, which takes into account correlations. Uncertainty |  Parameters |  $\sigma(E_{0})$ [eV]
---|---|---
Magnetic field |  $B$ |  4
Magnetic field broadening |  $\sigma$ |  4
Scattering |  $\gamma_{\mathrm{H}_{2}}$, $\mathcal{A}_{j}$ |  6
Efficiency variation |  $\epsilon$ |  4
Other freq. dependence |  $\sigma(f_{c})$, $\mathcal{A}_{j}(f_{c})$ |  6
Systematics total |  |  9
Statistical |  |  17

These results highlight the capabilities of the frequency-based CRES
technique. Krypton calibration data demonstrate its inherently high
resolution, enabling the full decomposition of the detector response. Energy-
and frequency-dependent effects are measured and controlled to allow analysis
across a multi-keV continuous spectrum. The dominant background is RF noise
fluctuations, consistent with expectation, which is characterized and rejected
to achieve a zero-background measurement. These characteristics combine to
enable the first tritium endpoint measurement and direct neutrino mass limit
with the novel CRES technique.

These measurements demonstrate significant advances for CRES and suggest
avenues for improving its sensitivity to $m_{\beta}$. The analysis is
statistics-limited, motivating pursuit of a large-volume CRES apparatus [33].
The planned cavity-based detection geometry will benefit from increased signal
power due to enhanced spontaneous emission on resonance [42, 43] while also
reducing the Doppler shift, thus simplifying event morphology. Paired with
reduced noise, potentially from the use of quantum amplifiers, the SNR and
thus detection efficiency can be significantly enhanced. More sophisticated
reconstruction techniques, including matched filtering and/or machine learning
[50], have the potential to further increase reconstruction efficiency and
enable the identification of sidebands, providing input for kinematic
corrections to improve resolution [37]. Novel calibration with a tunable
monoenergetic electron source will be required to further improve detector
response characterization, as the CRES resolution has already surpassed the
natural linewidth of 83mKr and atomic shakeup/shakeoff satellites contribute
significantly to the 83mKr lineshape [39].

Project 8 aims to combine these advances with an atomic tritium source to
bypass the molecular final state broadening and uncertainties. This sets the
stage for a next-generation neutrino mass experiment probing the full range of
mβ allowed by the inverted neutrino mass ordering.

Acknowledgments

This material is based upon work supported by the following sources: the U.S.
Department of Energy Office of Science, Office of Nuclear Physics, under Award
No. DE-SC0020433 to Case Western Reserve University (CWRU), under Award No.
DE-SC0011091 to the Massachusetts Institute of Technology (MIT), under Field
Work Proposal Number 73006 at the Pacific Northwest National Laboratory
(PNNL), a multiprogram national laboratory operated by Battelle for the U.S.
Department of Energy under Contract No. DE-AC05-76RL01830, under Early Career
Award No. DE-SC0019088 to Pennsylvania State University, under Award No. DE-
FG02-97ER41020 to the University of Washington, and under Award No. DE-
SC0012654 to Yale University; the National Science Foundation under Award No.
PHY-2209530 to Indiana University, and under Award No. PHY-2110569 to MIT; the
Cluster of Excellence “Precision Physics, Fundamental Interactions, and
Structure of Matter” (PRISMA+ EXC 2118/1) funded by the German Research
Foundation (DFG) within the German Excellence Strategy (Project ID 39083149);
the Karlsruhe Institute of Technology (KIT) Center Elementary Particle and
Astroparticle Physics (KCETA); Laboratory Directed Research and Development
(LDRD) 18-ERD-028 and 20-LW-056 at Lawrence Livermore National Laboratory
(LLNL), prepared by LLNL under Contract DE-AC52-07NA27344, LLNL-JRNL-838683;
the LDRD Program at PNNL; Indiana University; and Yale University. Portions of
the research were performed using the Core Facility for Advanced Research
Computing at CWRU, the Engaging cluster at the MGHPCC facility, Research
Computing at PNNL, and the HPC cluster at the Yale Center for Research
Computing. The 83Rb/83mKr isotope used in this research was supplied by the
United States Department of Energy Office of Science through the Isotope
Program in the Office of Nuclear Physics.

## References

  * Fukuda  _et al._ [1998] Y. Fukuda _et al._ (Super-Kamiokande), Phys. Rev. Lett. 81, 1562 (1998), arXiv:hep-ex/9807003 .
  * Ahmad  _et al._ [2002] Q. R. Ahmad _et al._ (SNO), Phys. Rev. Lett. 89, 011301 (2002), arXiv:nucl-ex/0204008 .
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  * Petcov [2013] S. T. Petcov, Adv. High Energy Phys. 2013, 852987 (2013), arXiv:1303.5819 [hep-ph] .
  * Dolinski  _et al._ [2019] M. J. Dolinski, A. W. P. Poon, and W. Rodejohann, Ann. Rev. Nucl. Part. Sci. 69, 219 (2019), arXiv:1902.04097 [nucl-ex] .
  * Abe  _et al._ [2023] S. Abe _et al._ (KamLAND-Zen), Phys. Rev. Lett. 130, 051801 (2023), arXiv:2203.02139 [hep-ex] .
  * Agostini  _et al._ [2020] M. Agostini _et al._ (GERDA), Phys. Rev. Lett. 125, 252502 (2020), arXiv:2009.06079 [nucl-ex] .
  * Arnquist  _et al._ [2023] I. J. Arnquist _et al._ (Majorana), Phys. Rev. Lett. 130, 062501 (2023), arXiv:2207.07638 [nucl-ex] .
  * Anton  _et al._ [2019] G. Anton _et al._ (EXO-200), Phys. Rev. Lett. 123, 161802 (2019), arXiv:1906.02723 [hep-ex] .
  * Adams  _et al._ [2022] D. Q. Adams _et al._ (CUORE), Nature 604, 53 (2022), arXiv:2104.06906 [nucl-ex] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Alam  _et al._ [2021] S. Alam _et al._ (eBOSS), Phys. Rev. D 103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO] .
  * Lesgourgues and Pastor [2006] J. Lesgourgues and S. Pastor, Phys. Rept. 429, 307 (2006), arXiv:astro-ph/0603494 .
  * Di Valentino  _et al._ [2015] E. Di Valentino, A. Melchiorri, and J. Silk, Phys. Rev. D 92, 121302 (2015), arXiv:1507.06646 [astro-ph.CO] .
  * Riess  _et al._ [2022] A. G. Riess _et al._ , Astrophys. J. Lett. 934, L7 (2022), arXiv:2112.04510 [astro-ph.CO] .
  * Knox and Millea [2020] L. Knox and M. Millea, Phys. Rev. D 101, 043533 (2020), arXiv:1908.03663 [astro-ph.CO] .
  * Abdalla  _et al._ [2022] E. Abdalla _et al._ , JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO] .
  * Drexlin  _et al._ [2013] G. Drexlin, V. Hannen, S. Mertens, and C. Weinheimer, Adv. High Energy Phys. 2013, 293986 (2013), arXiv:1307.0101 [physics.ins-det] .
  * Formaggio  _et al._ [2021] J. A. Formaggio, A. L. C. de Gouvêa, and R. G. H. Robertson, Phys. Rept. 914, 1 (2021), arXiv:2102.00594 [nucl-ex] .
  * Pontecorvo [1957] B. Pontecorvo, Sov. Phys. JETP 6, 429 (1957).
  * Maki  _et al._ [1962] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
  * Aker  _et al._ [2022] M. Aker _et al._ (KATRIN), Nature Phys. 18, 160 (2022), arXiv:2105.08533 [hep-ex] .
  * Saenz  _et al._ [2000] A. Saenz, S. Jonsell, and P. Froelich, Phys. Rev. Lett. 84, 242 (2000).
  * Bodine  _et al._ [2015] L. I. Bodine, D. S. Parno, and R. G. H. Robertson, Phys. Rev. C 91, 035505 (2015), arXiv:1502.03497 [nucl-ex] .
  * Gastaldo  _et al._ [2017] L. Gastaldo _et al._ , Eur. Phys. J. ST 226, 1623 (2017).
  * Alpert  _et al._ [2015] B. Alpert _et al._ , Eur. Phys. J. C 75, 112 (2015), arXiv:1412.5060 [physics.ins-det] .
  * Velte  _et al._ [2019] C. Velte _et al._ , Eur. Phys. J. C 79, 1026 (2019).
  * Monreal and Formaggio [2009] B. Monreal and J. A. Formaggio, Phys. Rev. D 80, 051301 (2009), arXiv:0904.2860 [nucl-ex] .
  * Asner  _et al._ [2015] D. M. Asner _et al._ (Project 8), Phys. Rev. Lett. 114, 162501 (2015), arXiv:1408.5362 [physics.ins-det] .
  * Byron  _et al._ [2022] W. Byron _et al._ , arXiv:2209.02870 [nucl-ex] (2022).
  * Ashtari Esfahani  _et al._ [2017] A. Ashtari Esfahani _et al._ (Project 8), J. Phys. G 44, 054004 (2017), arXiv:1703.02037 [physics.ins-det] .
  * Aker  _et al._ [2021] M. Aker _et al._ (KATRIN), Phys. Rev. D 104, 012005 (2021), arXiv:2101.05253 [hep-ex] .
  * Esfahani  _et al._ [2022] A. A. Esfahani _et al._ (Project 8), in  _2022 Snowmass Summer Study_ (2022) arXiv:2203.07349 [nucl-ex] .
  * Ashtari Esfahani  _et al._ [2022] A. Ashtari Esfahani _et al._ (Project 8) (2022), long-form article in preparation.
  * Vénos  _et al._ [2005] D. Vénos, A. Spalek, O. Lebeda, and M. Fiser, Applied Radiation and Isotopes 63, 323 (2005).
  * Hickish  _et al._ [2016] J. Hickish _et al._ , Journal of Astronomical Instrumentation 5, 1641001-12 (2016), arXiv:1611.01826 [astro-ph.IM] .
  * Ashtari Esfahani  _et al._ [2019a] A. Ashtari Esfahani _et al._ , Phys. Rev. C 99, 055501 (2019a), arXiv:1901.02844 [physics.ins-det] .
  * Ashtari Esfahani  _et al._ [2019b] A. Ashtari Esfahani _et al._ (Project 8), New J. Phys. 21, 113051 (2019b), arXiv:1907.11124 [physics.comp-ph] .
  * Robertson and Venkatapathy [2020] R. G. H. Robertson and V. Venkatapathy, Phys. Rev. 102, 035502 (2020).
  * Baker and Cousins [1984] S. Baker and R. D. Cousins, Nucl. Instrum. Meth. 221, 437 (1984).
  * Altenmüller  _et al._ [2020] K. Altenmüller _et al._ , J. Phys. G 47, 065002 (2020), arXiv:1903.06452 [physics.ins-det] .
  * Purcell [1946] E. M. Purcell, Phys. Rev. 69, 681 (1946).
  * Gabrielse and Dehmelt [1985] G. Gabrielse and H. Dehmelt, Phys. Rev. Lett. 55, 67 (1985).
  * Myers  _et al._ [2015] E. G. Myers, A. Wagner, H. Kracke, and B. A. Wesson, Phys. Rev. Lett. 114, 013003 (2015).
  * Kleesiek  _et al._ [2019] M. Kleesiek _et al._ , Eur. Phys. J. C 79, 204 (2019), arXiv:1806.00369 [physics.data-an] .
  * Ashtari Esfahani  _et al._ [2021] A. Ashtari Esfahani _et al._ , Phys. Rev. C 103, 065501 (2021), arXiv:2012.14341 [physics.data-an] .
  * Kraus  _et al._ [2005] C. Kraus _et al._ , Eur. Phys. J. C 40, 447 (2005), arXiv:hep-ex/0412056 .
  * Feldman and Cousins [1998] G. J. Feldman and R. D. Cousins, Phys. Rev. D 57, 3873 (1998), arXiv:physics/9711021 .
  * Cowan  _et al._ [2011] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, Eur. Phys. J. C 71, 1554 (2011), [Erratum: Eur.Phys.J.C 73, 2501 (2013)], arXiv:1007.1727 [physics.data-an] .
  * Ashtari Esfahani  _et al._ [2020] A. Ashtari Esfahani _et al._ , New J. Phys. 22, 033004 (2020), arXiv:1909.08115 [nucl-ex] .