Direct high-precision measurement of the mass difference of 77 As– 77 Se related to neutrino mass determination

,


I. INTRODUCTION
The discovery of neutrino oscillations has challenged the idea of massless neutrinos, thus requiring extension of the Standard Model to account for neutrino mass [1][2][3].Neutrino oscillations provide the possibility to infer the neutrinomass splittings to a high precision.However, they are not sensitive to the absolute neutrino-mass scale, e.g., the mass of the lightest mass eigenstate.To access the absolute mass scale, three complementary approaches are currently being explored: neutrinoless double-β decay, cosmological observations, and kinematic studies of weak-interaction processes such as single β ± decays and electron capture (EC) [4][5][6][7][8].
Among them, high-precision measurements of single β ± decays or EC are considered to be the most model-independent methods to determine the absolute scale of the (anti)neutrino mass, as they require no prior assumption on the basic nature (Dirac vs. Majorana) of the neutrino.
The neutrino mass can be determined by comparing the measured and the expected endpoint energy (Q β ).The presence of a finite neutrino mass causes a shift in the decay endpoint and alters the shape of the decay spectrum, particularly in the proximity of the endpoint.For β decays, the fraction of decay events that fall into an energy interval just below the endpoint energy is proportional to Q −3 β [9], while for EC the proportionality of the event fraction to Q β can be even steeper.This implies that nuclear transitions with the lowest possible Q-value are desirable [9].Currently, two nuclei 3 H (β − decay) and 163 Ho [10,11] (EC), with low ground-stateto-ground-state (gs-to-gs) decay Q values of 18.59201 (7) keV [12], and 2.8632 (6) keV [10,11,13], are employed for longterm direct neutrino-mass measurements [8] in experiments such as KATRIN (KArlsruhe TRitium Neutrino) [14,15], ECHo (Electron Capture in 163 Ho) [7,[16][17][18], Project 8 [19] and HOLMES [20,21].
Further explorations for nuclear β -decay or EC transitions with low Q-values are desirable for prospective (anti)neutrino  17) keV.The Q * value can be deduced from the high-precision excitation-energy E * evaluation [26], with sub-keV precision, and the gs-to-gs Q value of 683.2 (17) keV from AME2020 [27,28], which provides the main uncertainty of Q * .The gs-to-gs Q value of 77 As in AME2020 is evaluated from reaction and β − -decay data of 80 Se(p,α) 77 As, 76 Ge( 3 He,d) 77 As, and 77 As(β − ) 77 Se, with influence of 32%, 31.8%, and 17.9%, respectively.
In this work, we report on the first direct gs-to-gs β − -decay Q-value measurements of 77 As with the JYFLTRAP double PTMS at the University of Jyväskylä [43][44][45].The Q-value of the transition to the excited state of interest is evaluated and determined with a high precision.Moreover, the half-life of the candidate transition is assessed based on nuclear shellmodel calculations in order to validate the possibility of using 77 As for future long-term antineutrino-mass determination experiments.

II. EXPERIMENTAL DESCRIPTION
The direct measurement of the Q value, based on mass difference measurements of the decay pair ions of 77 As + and 77 Se + , was carried out at the Ion Guide Isotope Separator On-Line facility (IGISOL) using the JYFLTRAP double Penning trap mass spectrometer [43], situated at the University of Jyväskylä [44,45].To produce the ions of interest at IGISOL, a 9-MeV deuteron beam from the K-130 cyclotron was directed onto a thin germanium target with a thickness of approximately 2 mg/cm 2 .Ions of 77 As + and 77 Se + were simultaneously produced through fusion-evaporation reaction.These produced ions were stopped in the gas cell of the IGISOL light-ion ion guide [46] by colliding with high-purity helium gas at a pressure of about 100 mbar.During this process, the highly charged ions undergo recombination, predominantly adopting a singly charged state.The resulting recoils exited the gas cell through a small nozzle into a sextupole ion guide (SPIG) [47], which transports the ions into high vacuum and a subsequent electrode system accelerates them to an energy of 30 keV.A dipole magnet with a mass resolving power of approximately 500 adequately isolated only the isobaric ions with A = 77, where A represents the mass number.Following the separation, the mass-number-selected ions were transported through an electrostatic switchyard housing a fast kicker electrode used to chop the beam for an optimum number of ions.The ions were then injected into a radiofrequency quadrupole (RFQ) cooler-buncher [48], which cooled and bunched the beam.Finally, the resulting bunches were transported to the JYFLTRAP double Penning trap for the actual frequency ratio measurement.
The JYFLTRAP double Penning trap comprises two cylindrical traps located in a 7-T superconducting magnet, as illustrated schematically in Fig. 1(a).The cooled and bunched ions are confined using the combination of a homogeneous magnetic field and a quadrupolar electrostatic potential inside the traps, where they undergo the superposition of three simple harmonic modes, one axial and two radial.The first trap, designated as the preparation trap, serves as a high-resolution mass separator.In contrast, the second trap, referred to as the measurement trap, is utilized for finely detailed mass determination using ion-cyclotron-resonance techniques.
The ion beam contained 77 Ge + as a co-produced impurity.In the first trap, an isobarically purified sample of ions was prepared using the mass-selective buffer gas cooling method [49], providing a typical resolving power M/∆M ≈ 10 5 .This method was sufficient to provide a clean sample of 77 Se + or 77 As + ions.
The determination of the gs-to-gs Q (Q 0 β − ) value is based on the measurement of the cyclotron frequency, ν c = 1 2π q m B, where q/m is the charge-to-mass ratio of the stored ion and B is the magnetic field strength, for both the decay parent and decay daughter ions.In this work, the phase-imaging ion-cyclotron-resonance (PI-ICR) technique [50,51] was employed for measuring the cyclotron frequencies.Specifically, measurement scheme number 2 described in [52] was utilized for directly measuring the cyclotron frequency.
Two timing patterns, denoted as "magnetron" and "cyclotron", were employed (see Refs. [50,51] for detailed information).These patterns were nearly identical, differing only in the initiation moment of the π-pulse that converts the ions' cyclotron motion to magnetron.In the "magnetron" pattern, ions primarily circulated in the trap with magnetron motion for a duration t acc (accumulation time), while in the "cyclotron" pattern, ions revolved with cyclotron motion.These patterns produced magnetron and cyclotron spots or phases on the position-sensitive micro-channel plate (MCP) detector located after the trap, as illustrated in Fig. 1(a).The patterns enable the calculation of the angle between the phases of cyclotron and magnetron motion with respect to the center spot, α c = α + − α − , where α + and α − are the polar angles of cyclotron and magnetron phases, respectively.The acquired "magnetron" and "cyclotron" phase positions of 77 As + ions are depicted in the left and right panels of Fig. 1(b).It is essential to determine the position of the motional center spot, also shown in Fig. 1(b).
Finally, the cyclotron frequency ν c is deduced from: where n c is the number of complete revolutions of the ions during the phase accumulation time t acc .The measurement procedure ensured that α c remained small to reduce systematic shifts caused by image distortion.To achieve this, the duration t acc was selected to closely align with integer multiples of the ν c period.This minimized the angle α c , ensuring minimal impact from the interconversion of magnetron and cyclotron motions [52,53].During the measurements, α c did not exceed a few degrees.Two different accumulation times (330 ms and 321 ms) for both 77 Se + and 77 As + ions during the measurement were used.The chosen duration t acc ensured the separation of the cyclotron spot of the target ion from any potential isobaric, isomeric, or molecular contaminants.The delay of the initiation of the π pulse was repeatedly scanned over one magnetron period and the final extraction delay was varied over one cyclotron period to account for any residual magnetron and cyclotron motion that could shift the different spots.This constituted in a total of 5 × 5 = 25 scan points for both magnetron and cyclotron phase spots.
The Q 0 β − value is directly obtained through the cyclotron frequency ratio, R = ν c,Se /ν c,As , where ν c,As is the cyclotron frequency for 77 As + and ν c,Se for 77 Se + .During this experiment, alternating measurements of the 77 As + and 77 Se + cyclotron frequency measurements were conducted every few minutes to minimize the contribution of magnetic field fluctuations in the measured cyclotron frequency ratio.Still, linear interpolation was employed to determine the magnetic field at the moment of the parent cyclotron frequency measurement.The Q 0 β − value, mass difference of 77 As and 77 Se, can be given as: where M p and M d are the atomic masses of the parent ( 77 As) and daughter ( 77 Se), respectively.m e denotes the mass of an electron, and R represents the cyclotron frequency ratio ( ν c,d ν c,p ) for singly charged ions (q = 1).The electron binding energies of the parent and daughter atoms are denoted as B p (9.78855(25) eV) and B d (9.752390(15) eV) [54].Since both the parent and daughter, as mass doublets, have the same A/q, the mass-dependent error becomes effectively negligible compared to the statistical uncertainty achieved in the measurement [55].Moreover, due to the fact that the mass difference of the parent and daughter is very small (∆M/M < 10 −4 ), the contribution from the reference (daughter) mass uncertainty of 0.06 keV/c 2 is suppressed by that factor and thus can be disregarded.

III. RESULTS AND DISCUSSION
The data collection involved initiating a ν c measurement of 77 Se + for four full scan rounds (one round consisting of 5×5 points for both magnetron and cyclotron phases), followed by the measurement of 77 As + for four full scan rounds.Subsequently, a center spot was recorded with 77 Se + ions.In total, these steps lasted about seven minutes and were repeated.For each repetition, the positions of each spot were determined using the maximum likelihood method and the phase angles were calculated to deduce the cyclotron frequencies [32,50,51]..The consecutive fitted cyclotron frequencies of 77 Se + were linearly interpolated to the time of the measurement of 77 As + .This interpolated frequency was used to deduce the cyclotron resonance frequency ratio R. The contribution of temporal fluctuations of the magnetic field to the final frequency ratio uncertainty was less than 10 −10 since the frequency measurements of the ion pair were tightly interleaved.The incident ion rate was limited to a maximum of 5 detected ions/bunch with the median value being around  77 As to the excited state of the daughter 77 Se.The first and second columns illustrate the experimental spin-parity of the initial ground state and its half-life.The third column gives the measured spin-parity of the excited final state.The fourth column gives the decay type, which in this case is first-forbidden non-unique (1 st FNU).The fifth column gives the gs-to-gs decay Q-value (Q 0 β − ) from AME2020 [27,28] and the sixth column from this work.The seventh and eighth columns give the gs-to-es decay Q value (Q * β − ) from the literature [27] and this work, respectively.The ninth column lists the level of being positive/negative (in σ ).The last column gives the excitation energy E * from [26].All the energy values are in units of keV.

Initial state
Half-life Final state Decay type 2 ions/bunch.Bunches containing more than 5 ions were excluded from the analysis to mitigate a potential cyclotron frequency shift due to ion-ion interactions [55,56].Count-rate class analysis [56] was employed to verify that the frequency indeed did not shift.By maintaining a small angle α c < 10 degrees during the measurements, the frequency shifts due to ion image distortions remained well below the statistical uncertainty.The total measurement period spanned 13.1 hours, which we divided into four time slots: 7.2, 3.7, 0.6, and 1.6 hours as shown in Fig. 2.This division allowed us to maintain the desired small angle of α c .The weighted mean ratio R of the single ratios was computed alongside the inner and outer errors.The maximum of the inner and outer errors of the Birge ratios [57], was adopted as the weights to calculate the final weighted mean cyclotron frequency ratio R. The final frequency ratio R and the resultant Q 0 β − -values are 1.000 009 552 87(98) and 684.463(70) keV, respectively.In Fig. 2, the results of the analysis are contrasted with the literature values.The Q 0 β − value of 77 As from this work is a factor of FIG. 3: (Color online).The transition of 77 As ground-state β − decay to the 680.1035(17) keV 5/2 − state in 77 Se.The horizontal blue line depicts the level with the Q 0 β − taken from AME2020 [27,28] (shaded area shows the 1σ uncertainty) and the red dashed line the Q 0 β − from this work.The data for the level scheme are adopted from [26].
24 more precise and 1.3 (17) keV larger than the value in AME2020 [28].The mass-excess value of 77 As was deduced to be -73915.026(94)keV/c 2 and the precision was improved by a factor of 18.The uncertainty of the mass excess for 77 As has an additional 0.06 keV/c 2 uncertainty in the mass of the daughter 77 Se as reference, which was evaluated based on (n,γ) reaction experiments [58][59][60] in AME2020 [27,28].Combining the new Q 0 β − value together with the nuclear energy level data gives the final Q-values for decays to the potential low Q-value states, as tabulated in Table I and illustrated in Fig. 3.The gs-to-es Q * β − value of 4.360(70) keV for the decay channel 77 As (3/2 − ) → 77 Se (5/2 + , 680.1035(17) keV), is comparable to the presently running direct (anti)neutrino mass experiments using electron-capture decaying 163 Ho with a Q value of 2.8632 (6) keV [13].
The transition of interest is first-forbidden non-unique (1 st FNU) and thus depends on nuclear structure through nuclear matrix elements (NME).To compute these NME and to estimate the half-life of this transition, nuclear shell-model (NSM) calculations utilizing the software KSHELL [61], with the well established effective interactions jj44bpn [62] and jun45pn [63], were performed.Their model spaces consist of adopting 56 Ni as a closed core with the orbitals 0f 5/2 , 1p 3/2 , 1p 1/2 , and 0g 9/2 for both protons and neutrons.We assess the TABLE II: Comparison of the experimental and jj44bpn and jun45pn-computed state energies E exc (in units of MeV), electric quadrupole moments Q e (in units of barn), and magnetic dipole moments µ (in units of nuclear magneton µ N ).The experimental evaluation data are from [26].Effective charges adopted are e p eff = 1.5e and e n eff = 0.5e and the bare g-factors are g l (p) = 1, g l (n) = 0, g s (p) = 5.585, and g s (n) = −3.826.

Experimental Evaluation jj44bpn jun45pn
Isotope reliability of the nuclear Hamiltonians employed by analyzing the magnetic dipole and electric quadrupole moments, alongside the excitation energies, predicted by the models in Table II.The computed electromagnetic properties and the excitation energies are generally in fair agreement with the evaluation data.
The dependence of the partial half-life of the 3/2 − → 5/2 + transition on the Q value is shown in Fig. 4. The computations of the β − transition rate account for screening, radiative, and atomic exchange corrections.The atomic exchange correction was originally derived for allowed β decays by Nitescu et al. [64] and is the most important contribution due to the low Q value of the discussed transition, as can be seen in Fig. 4. The computed half-life is of the order of 10 9 years for this 1 st FNU transition, which rules out this candidate to be a potential source for rare-event experiments searching for the electron antineutrino mass.Due to the short half-life (less than 2 days) of the isotope 77 As, it poses additional challenges for inclusion in a long-term neutrino mass determination experiment.(5/2 + ) for different choices of the axial-vector coupling g A (0.8, 0.9 and 1.0) using the effective Hamiltonians jun45pn (left panel) and jun45pn (right panel).The legend includes half-life values with (W.) and without (No) the atomic exchange correction (cor.).Additionally, green vertical dashed lines shaded in between with green color denote the highest and lowest available energies derived in this work for the transition, which are 4.430 keV and 4.290 keV, respectively.

IV. CONCLUSION AND OUTLOOK
A direct high-precision gs-to-gs 77 As(β − ) 77 Se decay Qvalue measurement was performed using the PI-ICR technique at the JYFLTRAP Penning trap mass spectrometer.A Q value of 684.463(70) keV was obtained from the cyclotron frequency ratio measurements to a relative precision better than 1×10 −9 of the ions of the decay pair.The refined Q-value is in good agreement with the evaluated value in AME2020 but its uncertainty was improved by a factor of 24.The candidate FNU transition 77 As (3/2 − )→ 77 Se (5/2 + , 680.1035 (17) keV) is confirmed to be energetically allowed at a level of more than 60σ .Furthermore, the newly derived gs-to-es Q value allowed us to quantify the order of magnitude of the partial half-life of this transition by using nuclear wave functions obtained with the use of two different established Hamiltonians in nuclear shell-model calculations.The obtained halflife of approximately 10 9 years for this 1 st FNU transition, excludes 77 As as a viable candidate for future long-term neutrino mass determination experiments.

FIG. 1 :
FIG.1:(Color online).(a) Schematic view of ion production and mass measurements using the PI-ICR technique at IGISOL.The 77 As + and 77 Se + ions were produced from the fusion reaction with a 9-MeV deuteron beam from K130 cyclotron, bombarding a germanium target of approximately 2 mg/cm 2 in thickness.Ions having mass number of 77 were selected with a dipole magnet and transported to the JYFLTRAP PTMS for final ion species selection in the preparation trap by means of a buffer-gas cooling technique and cyclotron frequency determination using the phase-imaging technique in the measurement trap.A position-sensitive MCP detector was used to register the images of the motion phases.(b) An illustration of the radial-motion ("magnetron", "cyclotron", and "center") projection of the 77 As + ions onto the positionsensitive MCP detector.The cyclotron phase spot is displayed on the left side and the magnetron phase spot on the right.The angle difference between the two spots relative to the center spot is utilized to deduce the cyclotron frequency of the measured ion.The color bar indicates the number of ions in each pixel.

FIG. 2 :
FIG.2:(Color online).The measured cyclotron frequency ratios R ( ν c ( 77 Se + )/ν c ( 77 As + )) (left axis) and Q value (right axis) in this work compared to evaluated values from AME2020.The red dots with uncertainties are the measured PI-ICR single ratios in four time slots, which are separated with vertical brown dashed lines.The weighted average value in this work R = 1.000 009 552 87(98) is represented by the solid red line and its 1σ uncertainty band is shaded in red.The dashed blue line indicates the value adopted from AME2020 with its 1σ uncertainty area shaded in blue.

8 (
FIG. 4: (Color online).Computed partial half-lives as a function of available energy (Q value) for the transition 77 As (3/2 − ) → 77 Se (5/2 + ) for different choices of the axial-vector coupling g A (0.8, 0.9 and 1.0) using the effective Hamiltonians jun45pn (left panel) and jun45pn (right panel).The legend includes half-life values with (W.) and without (No) the atomic exchange correction (cor.).Additionally, green vertical dashed lines shaded in between with green color denote the highest and lowest available energies derived in this work for the transition, which are 4.430 keV and 4.290 keV, respectively.

TABLE I :
Transition from the ground state of the parent nucleus