Theoretical and Experimental Challenges in the Measurement of Neutrino Mass

Neutrino masses are yet unknown. We discuss the present state of effective electron anti-neutrino mass from $\beta$ decay experiments; effective Majorana neutrino mass from neutrinoless double-beta decay experiments; neutrino mass squared differences from neutrino oscillation: solar, atmospheric, reactor and accelerator based experiments; sum of neutrino masses from cosmological observations. Current experimental challenges in the determination of neutrino masses are briefly discussed. The main focus is devoted to contemporary experiments.


Introduction
Neutrinos are the second most abundant known particles in the Universe.Despite of their abundance in the nature, their hypothetical presence was first announced by Pauli in 1930, when trying to protect the law of conservation of energy in beta radioactivity [1].This particle got its name "Neutrino" by Enrico Fermi in 1934.The neutrinos were introduced as the neutral and massless fermions [2].These neutrinos interact only via weak interaction and their cross-section of interaction is very small [3].
The standard model (SM) of particle physics is based on the gauge group SU 3 C × SU 2 L × U 1 Y [4,5].The electroweak group is represented by SU 2 L × U 1 Y .SM describes the interaction between fundamental matter particles, i.e., quarks and leptons which are fermions, three fields, i.e., electromagnetic, weak and strong field, and their associated gauge bosons along with a scalar Higgs boson.All the charged fermions in the SM are Dirac, leaving neutrinos.Neutrinos are Dirac (ν ≠ ν), or Majorana (ν = ν) is yet to be established [6][7][8].In SM, neutrinos are considered as massless fermion.
The discovery of neutrino oscillations by neutrino experiments came up with the rejection of the idea of massless neutrino.The neutrino oscillation was confirmed by Super-Kamiokande [9] and Sudbury Neutrino Observatory [10]; this remarkable discovery led to the Nobel Prize in Physics in 2015 [11,12].This discovery was the first experimentally confirmed dent in the SM and it opened the door for physics beyond standard model (BSM).

Neutrino Mass
Generally, the neutrino mass can be determined using, (i) cosmological data: sets a most stringent bound on the sum of neutrino masses (Σm ν ); (ii) beta decay: sets a most stringent bound on effective electron antineutrino mass (m ν e ) by observing the kinematics of weak interaction; (iii) neutrinoless double-beta decay: sets a most stringent bound on effective Majorana neutrino mass (m ββ ) by observing the monoenergetic peak (if observed) at the decay Q value.These approaches are discussed below.
2.1.Sum of Neutrino Masses: Cosmological Bounds.Sum of neutrino mass is defined as ∑m ν = m 1 + m 2 + m 3 , where m 1 , m 2 , and m 3 are three neutrino mass eigenstates.Cosmological observations carry imprints of neutrinos, and there-fore, it can be used to extract and constrain the neutrino properties.Cosmology is sensitive to the following neutrino properties: (i) number of active neutrinos, (ii) neutrino density, (iii) sum of neutrino masses.
Generally accepted cosmological model, standard model of cosmology, explains the large-scale structures and their dynamics and answers unresolved puzzles associated with the evolution and fate of the Universe.The Λ-CDM (cold dark matter) model best describes the present parameters, such as density parameter of baryons (Ω b ≃ 0 05) which refers to observable objects in the Universe, density parameter of CDM (Ω c ≃ 0 25) which refers to nonbaryonic and nonrelativistic matter, density parameter of cosmological constant (Ω Λ ≃ 0 70) which refers to vacuum, also called the dark energy, and the Hubble constant (h ≃ 70 km s −1 Mpc −1 ) which refers to the present rate of expansion of the Universe.
The precise estimation of neutrino to photon number density ratio (n ν /n γ ) is important for the determination of the sum of neutrino masses (Σm ν ), and this ratio is fixed in SM including many extensions of SM.The ratio related to Σm ν as Ω ν = ρ 0 ν /ρ 0 crit = Σm ν / 93 14 h 2 eV , where Ω ν is the present total neutrino density in terms of critical density ρ 0 crit .The expression of Ω ν reported in the text assumes the "standard" (n ν /n γ ).This ratio is connected to the physics of neutrino decoupling.
The neutrino to photon energy density ratio (ρ ν /ρ γ ) between the e − e + annihilation time and nonrelativistic transition time of neutrino can be given by the expression ρ ν /ρ γ = 7/8 N eff 4/11 4/3 , where N eff is the effective number of neutrinos estimated as 3.044 from a detailed calculations of the process of neutrino decoupling with at least 10 −4 numerical precision [13][14][15][16].The direct measurement of the invisible width of Z-boson limits the number of active left-handed neutrino states to three, N ν = 2 9963 ± 0 0074, and they are ν e , ν μ , ν τ [17].
The estimated sum of neutrino masses from composite samples (such as Planck, BAO, and RSD) based on ΛCDM + Σm ν model is mentioned in Table 1.There are several challenges in measuring the sum of neutrino masses which needs to be mentioned for imposing more stringent constraints on Σm ν .Detailed overview of the cosmological constraints on the neutrino properties can be found in [18][19][20][21].(v) Subpercent level precision in BAO (baryon acoustic oscillation) measurements of the distance scale (vi) Sum of neutrino mass calculated by different models using composite dataset has to be minimized, because we know, if the neutrino mass variation is in the range 0.025 eV-1 eV, then the error of the order of 5% will be generated on the matter power spectrum in comparison to the current matter power spectrum Next-generation cosmological experiments will address the abovementioned issues and provide better constraints on Σm ν ; few upcoming experiments are DESI [22], Euclid [23], LSST [24], SPHEREx [25], SKA [26], Simon Observatory [27], CMB-S4 [28], and LiteBird [29].

Effective Electron Antineutrino Mass: β Decay Bounds.
Effective electron antineutrino mass is defined as follows: , where U e1 , U e2 , and U e3 are components of neutrino mixing matrix.Determination of m ν e is of urgent importance for cosmology and particle physics.This information will help in understanding the role of neutrinos in the structure formation of the Universe after the Big Bang [34].In addition, the value of effective electron antineutrino mass will help us in identifying the right theories for the prediction of BSM physics [35,36]. 2 Advances in High Energy Physics The β decay experiments are designed in a way to explore effective electron antineutrino mass.The weak interaction process of β decay can be expressed as n ⟶ p + e − + ν e , a careful study of the given reaction can quantitate the effective electron antineutrino mass.β-decay experiment measures a distortion in the spectral shape near the endpoint of β decaying isotope.The phase space of an electron emitted in a β decay process can be expressed ν e , where p is the momentum of outgoing electron possessing energy E, m ν e is the effective electron antineutrino mass, and E 0 is the end point energy of the spectrum.The β decay spectrum of 3 H Q β = 18 6 keV is shown in Figure 1(a), and distortion produced by effective electron antineutrino mass in 3 H energy spectrum is shown in Figure 1(b).
One of the most promising experiments designed to probe the effective electron antineutrino mass by studying the kinematics of β decay is KATRIN (KArlsruhe TRItium Neutrino) experiment by looking at the decay of tritium ( 3 H) as 3 H ⟶ 3 He + e − + ν e .KATRIN put a constrain on m ν e < 0 8 eV at the 90% C.L [39] and have potential to impose constrain on m ν e < 0 2 eV.
KATRIN reduce the statistical uncertainty by a factor of three and systematic uncertainty by a factor of two relative to its earlier campaign.In a first campaign, KATRIN (2019) reached a sensitivity of 1.1 eV at 90% C.L, and in its second campaign, KATRIN (2021) achieved sensitivity of 0.7 eV at 90% C.L. KATRIN would be dominated by systematics, although results of KATRIN first and second campaign are dominated by statistical uncertainties.KATRIN continue reducing its systematic uncertainty to achieve a designed sensitivity of 0.2 eV at 90% C.L on m ν e .The constrain imposed on the upper limit of effective electron antineutrino mass by KATRIN experiment is shown in Table 2. To pin down the effective electron antineutrino mass from β decay experiments, many potential challenges need to be addressed carefully.

Main Challenges in the Measurement of Effective Electron Antineutrino Mass
(i) Spectrometer with good counting rate or high efficiency (ii) Spectrometer with better end-point energy resolution (iii) Intense source of tritium and Holmium-163: high Becquerel activity is recommended (iv) Energy loss of β in the source, [8,40].
Neutrinoless double-beta (0νββ) decay is a hypothetical nuclear transition and is expressed as A, Z ⟶ A, Z + 2 + 2e − .This lepton number violating [47] phenomenon if observed, will assign neutrinos Majorana characteristics of particle.0νββ decay could provide effective Majorana neutrino mass assuming the decay is mediated by light Majorana neutrino.Experiments measuring 0νββ decay measure small peak generated by the sum of energy of energy of two electrons.
Different isotopes used by experiments searching for the signatures of 0νββ decay are, 76  Ge Nd [58].The half-life sensitivity of an experiment is estimated using the expression, T 0ν 1/2 ∝ aε E/ B ΔE (with background) and T 0ν 1/2 ∝ aεE (background free), where a is the isotopic abundance, ε is the efficiency of the detection of 0νββ signal at the region of interest (ROI), B is the background index, ΔE is the energy resolution of the detector, and the exposure (E) is given by the product of the mass of the isotope and run time.
KamLAND-Zen sets the strongest limit on the half-life of any 0νββ decay isotope to date, for 136  Xe as T 0ν 1/2 > 2 3 × 10 26 yr [59].Energy spectrum of 136  Xe measured by KamLAND-Zen experiment (currently running) is shown in Figure 2. Energy spectrum of 76  Ge measured by GERDA experiment (final results) is shown in Figure 3.
By considering neutrinos to be a Majorana particle, its m ββ is estimated for experimentally measured isotopic half-life using relation, T 0ν Here, M 0ν is the nuclear matrix element, G 0ν is the phase space factor, g A is the axial coupling constant, m e is the mass of electron.The tightest bounds imposed on the T 0ν 1/2 of different isotopes by various experiments and estimated m ββ values are mentioned in Table 3.  100Mo) [64], nEXO ( 136 Xe) [65], SNO +( 130Te) [66], SuperNEMO ( 82 Se) [67], LEGEND-1000 ( 76 Ge) [63], KamLAND-Zen 800 ( 136Xe) [68], and NEXT-100 ( 136Xe) [69] (isotopes corresponding to each experiment are shown in bracket).2.4.Neutrino Mass Squared Differences.In the old theory of electroweak interactions, formulated by Glashow, Weinberg, and Salam, lepton flavor was conserved and neutrinos were assumed as massless fermions.This simply means that leptons produced in a particular flavor state will remain in that state forever.As soon the theory of two-component neutrino was developed, Pontecorvo proposed the idea of neutrino oscillation in 1957-1958 [71,72].Later, neutrino oscillation or conversion of the neutrino flavor was observed in the solar [10] and atmospheric [9] neutrino experiments.Therefore, solar and atmospheric neutrino anomaly was resolved by assigning oscillation phenomenon to neutrino.In 2015, the Nobel prize in physics was awarded to Kajita and McDonald for their landmark discovery of neutrino oscillation.These results of neutrino oscillation were subsequently confirmed by reactor experiment, such as KamLAND [73,74] and long baseline experiment, such as NOvA [75].Neutrino oscillation experiment measure the appearance or disappearance channel.
The neutrino oscillations can be analytically expressed using PMNS matrix and two mass-squared differences of active neutrino; this makes minimum six parameters, solar mixing angle θ 12 , atmospheric mixing angle θ 23 , reactor mixing angle θ 13 , solar mass-squared difference Δm 2  21 , atmospheric mass-squared difference Δm 2  31 , and Dirac CP-violating phase δ CP .In a PMNS matrix, δ CP informs about the difference in neutrino and antineutrino oscillations.For baseline (L), neutrino energy (E) and δ CP = 0, in three flavor neutrino oscillation probability equations can be expressed as follows: For small values of L/E, P ν e ⟶ ν μ = sin 2 2θ 13 sin 2 θ 23 sin 2 1 27Δm

Solar Neutrino Experiment.
Sun is an abundant source of neutrino and produces electron neutrino in the process of fusion.The total solar neutrino flux comes from different fusion reactions as shown in Figure 4(a).Among these, the dominant contribution to solar neutrino flux comes from pp reaction (99.6%).
The solar neutrino fluxes predicted by standard solar model (SSM) at one astronomical unit are shown in Figure 4(a), where continuum sources are in units of cm −2 s −1 MeV −1 , and the line fluxes are in units of cm −2 s −1 .
Along with the SSM predictions, Figure 4(b) gives the current picture of the experimentally estimated flux of B, Be with respect to CNO (carbon-nitrogen-oxygen) and Be with respect to B. These estimated solar neutrino fluxes are compared with solar models, SSM B16-GS98 [81] and SSM B16-AGSS09met [81].

Atmospheric Neutrino Experiment.
Cosmic ray particles are mostly protons, these protons after entering the Earth's atmosphere interacts with atmospheric nuclei present at high altitude.These high-energy nuclear interactions produce many pi mesons and less abundantly produced kaons.These mesons are unstable and decay into other particles.The π + meson decays into a μ + and a ν μ .This produced μ + are also unstable particles which further decay into an e + , ν e , and ν μ as shown in Figure 6(a).Similar decay process takes place for unstable π − meson and kaons.The neutrino produced in these processes are known as atmospheric neutrinos.The atmospheric flux consists of both neutrinos and antineutrinos.
Atmospheric neutrino flux as a function of neutrino energy is shown in Figure 6(b).The energy of these atmospheric neutrinos varies from few MeV to few PeV range and their path lengths are suitable to probe many of the prevailing neutrino puzzles.When these neutrinos 21 on sin 2 θ 12 from all solar neutrino data for allowed 1σ, 2σ, 3σ regions where green represents SK+SNO, blue represents KamLAND, and red represents for combined result.Image credit [91,92].7 Advances in High Energy Physics (antineutrinos) pass via Earth, the matter effects [100] influence the oscillation probability as the oscillation parameters sin 2 θ 13 and Δm 2  32 are replaced by their matter equivalents.Matter effects play a significant role in distinguishing neutrino mass hierarchy since atmospheric neutrino flux has L/E dependency.Current atmospheric neutrino experiments are listed in Table 5.

Reactor Neutrino Experiment.
Along with the energy production by nuclear fission, the nuclear reactors also produce flavor pure source of antineutrino (ν e ) flux, which is well understood, and this special feature makes reactors a "free" and copious neutrino source for the study.In reactor neutrino physics, we use inverse beta decay (IBD) where antineutrino will interact with the proton of detector target and produce a positron which annihilates an electron (prompt signal) and a neutron which is captured afterwards (delayed signal).
We can broadly categorize the reactor experiments into (i) short baseline (~1 km) and (ii) long baseline (~100-1000 km) reactor experiments.Three short baseline reactor neutrino experiments which looked for antineutrino disappearance with the main objective to measure last unknown neutrino oscillation angle θ 13 are Double Chooz in France [114], RENO in South Korea [115], and Daya Bay in China [116].All three experiments used detectors which included liquid scintillator target loaded with 0.1% of Gadolinium.The results of the three experiments for sin 2 2θ 13 are Double Chooz: 0 102 ± 0 012 [117]; Daya Bay: 0 0856 ± 0 0029 [118]; RENO: 0 0892 ± 0 0044 stat ± 0 0045 sys [119].These experiments can also add knowledge to the value of the effective combination of mass, which can be expressed as Δm At the same time, we can also extract information regarding the sign (+ for NO, -for IO) of a phase Φ ⊙ which depends on solar parameters.The upcoming reactor experiment JUNO have the potential to determine the neutrino mass ordering at ≥ 3σ to be 31% by 2030 [120].Current reactor neutrino experiments are shown in Table 6.
Main Challenges in the Measurement of Mass-Squared Difference from Reactor Neutrinos.Upcoming experiments resolving the above challenges are SNO+ ( 130 Te) [66] and JUNO (linear alkylbenzene) [87].
Main Challenges in the Measurement of Mass-Squared Difference from Accelerators Neutrinos.
(i) Large fiducial mass of detector to collect high statistics of neutrino event data  Th, etc.) interacting detector volume (viii) Neutrino energy reconstruction due to nuclear effects and nuclear properties, for example, pion produced via neutrino interaction, gives rise to fake neutrino events, [127][128][129][130].

Conclusion
To summarize, determination of neutrino mass is a difficult task.We have given a brief overview of current experimental challenges in a neutrino mass measurement.Current limits on effective electron antineutrino mass from β decay by KATRIN and effective Majorana neutrino mass from 0νββ decay by KamLAND-Zen, GERDA, Majorana Demonstrator, EXO-200, CUORE, CUPID-0, and CUPID-Mo are presented.Current bounds on neutrino mass-squared differences from neutrino oscillation by SNO, Super Kamiokande, KamLAND, IceCube, ANTARES, Daya Bay, RENO, NOvA, T2K, and MINOS+ are discussed.Present bounds on the sum of neutrino masses from cosmological measurements by Planck, combined with BAO, RSD, Pantheon, DES, and Lyman α, are discussed.
Given the effort of many experiments, a measurement of the absolute neutrino mass may be around the corner, especially considering cosmology.And given the interplay of all the observables, the underlying model can be tested.

2. 1 . 1 .
Main Challenges in the Measurement of Sum of Neutrino Masses (i) Measurement of cosmological parameters with utmost accuracy (ii) Dependency on cosmological model (iii) Making scaling to current detectors (iv) Removal of false B-mode signal in the CMB (cosmic microwave background) measurement

Figure 2 :
Figure 2: Energy distribution measured by KamLAND-Zen experiment.(a) Events from short-lived backgrounds.(b) Events from longlived backgrounds.Fit to the 0νββ decay signal Q ββ at 2.458 MeV is shown in cyan.Image credit [59].

Figure 3 :
Figure 3: Energy distributed measured by GERDA experiment, expected 0νββ decay peak Q ββ at 2039 keV is shown in blue.Image credit [49].
(i) Only ν e disappearance channel can be analyzed (ii) Decrement of reactor neutrino flux as a function of distance because antineutrino flux is isotropic (iii) Difficulty in computation of ν e spectrum, because neutrino spectrum of each decay isotope is different (iv) About 75% of ν e produced by reactor remains undetected (v) Suppression of neutron induced by cosmic-ray muons (vi) Elimination of cosmogenic production of radioactive isotopes; 12 B, 8 Li, and 6 He inside the detector volume

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Advances in High Energy Physics (ii) Production of intense beam of neutrinos requires high power proton accelerator (iii) Deep underground location of detector (iv) Reduction of proton beam-related background (v) Large distance between neutrino source and detector (vi) Reduction of uncertainty in mixing angle (θ 23 ) and determination of its octant (vii) Elimination of neutrons (produced from cosmogenic, 238 U/ 238

Table 1 :
Cosmological bounds on sum of neutrino masses based on ΛCDM+Σm ν model, where TT: temperature power spectra; TE: temperature-polarization power spectra; EE: polarization power spectra; low-E: low-l polarization; RSD: redshift-space distortions; DES: dark energy survey; Pantheon: combined sample of supernova Type Ia.

Table 2 :
Current bounds on effective electron antineutrino mass from β decay kinematics.