⁸ALEXANDER 96M bound comes from analyses of $\tau^{-} \rightarrow 3\pi^{-}2\pi^{+}\nu_{\tau}$ and $\tau^{-} \rightarrow h^{-}h^{-}h^{+}\nu_{\tau}$ decays.

⁹BOTTINO 96 assumes three generations of neutrinos with mixing, finds consistency with massless neutrinos with no mixing based on 1995 data for masses, lifetimes, and leptonic partial widths.

¹⁰HANNESTAD 96C limit is on the mass of a Majorana neutrino. This bound assumes $N_{\nu} < 4$ from nucleosynthesis. A wider excluded region occurs with a smaller $N_{\nu}$ upper limit. This paper is the corrected version of HANNESTAD 96; see the erratum: HANNESTAD 96B.

¹¹SOBIE 96 derive their limit from the Standard Model relationship between the tau mass, lifetime, and leptonic branching fraction, and the muon mass and lifetime, by assuming lepton universality and using world average values.

¹²BUSKULIC 95H bound comes from a two-dimensional fit of the visible energy and invariant mass distribution of $\tau \rightarrow 5\pi(\pi^{0})\nu_{\tau}$ decays. Replaced by BARATE 98f.

¹³DOLGOV 95 removes earlier assumptions (DOLGOV 93) about thermal equilibrium below $T_{QCD}$ for wrong-helicity Dirac neutrinos (ENQVIST 93, FULLER 91) to set more stringent limits. DOLGOV 96 argues that a possible window near 20 MeV is excluded.

¹⁴SIGL 95 exclude massive Dirac or Majorana neutrinos with lifetimes between $10^{-3}$ and $10^{8}$ seconds if the decay products are predominantly $\gamma$ or $e^{+}e^{-}$.

¹⁵DODELSON 94 calculate constraints on $\nu_{\tau}$ mass and lifetime from nucleosynthesis for 4 generic decay modes. Limits depend strongly on decay mode. Quoted limit is valid for all decay modes of Majorana neutrinos with lifetime greater than about 300 s. For Dirac neutrinos limits change to < 0.3 or > 33.

¹⁶KAWASAKI 94 excluded region is for Majorana neutrino with lifetime >1000 s. Other limits are given as a function of $\nu_{\tau}$ lifetime for decays of the type $\nu_{\tau} \rightarrow \nu_{\mu}\phi$ where $\phi$ is a Nambu-Goldstone boson.

¹⁷PERES 94 used PDG 92 values for parameters to obtain a value consistent with mixing. Reexamination by BOTTINO 96 which included radiative corrections and 1995 PDG parameters resulted in two allowed regions, $m_{3} < 70$ MeV and 140 MeV $m_{3} < 149$ MeV.

¹⁸CINABRO 93 bound comes from analysis of $\tau^{-} \rightarrow 3\pi^{-}2\pi^{+}\nu_{\tau}$ and $\tau^{-} \rightarrow 2\pi^{-}\pi^{+}2\pi^{0}\nu_{\tau}$ decay modes.

¹⁹DOLGOV 93 assumes neutrino lifetime >100 s. For Majorana neutrinos, the low mass limit is 0.5 MeV. KAWANO 92 points out that these bounds can be overcome for a Dirac neutrino if it possesses a magnetic moment. See also DOLGOV 96.

²⁰ENQVIST 93 bases limit on the fact that thermalized wrong-helicity Dirac neutrinos would speed up expansion of early universe, thus reducing the primordial abundance. FULLER 91 exploits the same mechanism but in the older calculation obtains a larger production rate for these states, and hence a lower limit. Neutrino lifetime assumed to exceed nucleosynthesis time, ~ 1 s.

²¹ALBRECHT 92M reports measurement of a slightly lower $\tau$ mass, which has the effect of reducing the $\nu_{\tau}$ mass reported in ALBRECHT 88B. Bound is from analysis of $\tau^{-} \rightarrow 3\pi^{-}2\pi^{+}\nu_{\tau}$ mode.

²²Assumes neutrino lifetime >1 s. For Dirac neutrinos. See also ENQVIST 93.

²³KOLB 91 exclusion region is for Dirac neutrino with lifetime >1 s; other limits are given.

Revised August 2023 by K.A. Olive (University of Minnesota).

Neutrinos decouple from thermal equilibrium in the early universe at temperatures $\mathcal{O}(1)$ MeV. The limits on low mass ($m_{\nu} \lesssim 1$ MeV) neutrinos apply to $m_{\text{tot}}$ given by

$$m_{\text{tot}} = \sum_{\nu} m_{\nu}.$$

Stable neutrinos in this mass range decouple from the thermal bath while still relativistic and make a contribution to the total energy density of the Universe which is given by

$$\rho_{\nu} = m_{\text{tot}}n_{\nu} \approx m_{\text{tot}}(3/11)(3.045/3)^{3/4}n_{\gamma},$$

where the factor 3/11 is the ratio of (light) neutrinos to photons and the factor $(3.045/3)^{3/4}$ corrects for the fact that the effective number of neutrinos in the standard model is 3.045 when taking into account $e^{+}e^{-}$ annihilation during neutrino decoupling. Writing $\Omega_{\nu} = \rho_{\nu}/\rho_{c}$, where $\rho_{c}$ is the critical energy density of the Universe, and using $n_{\gamma} = 410.7 \text{ cm}^{-3}$, we have

$$\Omega_{\nu}h^2 \approx m_{\text{tot}}/(93 \text{ eV}).$$

While an upper limit to the matter density of $\Omega_{m}h^2 < 0.12$ would constrain $m_{\text{tot}} < 11$ eV, much stronger constraints are obtained from the observations of the CMB, combined with

lensing and baryon acoustic oscillations data. These combine to give an upper limit of around 0.12 eV, and may, in the near future, be able to provide a lower bound on the sum of the neutrino masses. The current lower bound of $m_{\text{tot}} > 0.06$ eV implies a lower limit of $\Omega_{\nu}h^2 > 6 \times 10^{-4}$. See our review on "Neutrinos in Cosmology" for more details.

## SUM OF THE NEUTRINO MASSES, $m_{\text{tot}}$

This is a sum of the neutrino masses, $m_{\text{tot}}$, as defined in the above note, of effectively stable neutrinos, i.e. those with mean lifetimes on cosmological scales. When necessary, we have generalized the results reported so they apply to $m_{\text{tot}}$. For other limits, see SZALAY 76, VYSOTSKY 77, BERNSTEIN 81, FREESE 84, SCHRÄMM 84, and COWSIK 85. For more information see a note on "Neutrinos in Cosmology" in this Review.

<table><thead><tr><th>VALUE (eV)</th><th>CL%</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>&lt; 0.082</td><td>95</td><td>1 BRIEDEN</td><td>22</td><td>COSM BOSS, eBOSS, and CMB</td></tr><tr><td>&lt; 0.116</td><td>95</td><td>2 KUMAR</td><td>22</td><td>COMS BOSS and CMB</td></tr><tr><td>&lt; 0.14</td><td>95</td><td>3 TANSERI</td><td>22</td><td>COSM BOSS and CMB</td></tr><tr><td>&lt; 0.13</td><td>95</td><td>4 ABBOTT</td><td>21A</td><td>COSM DES and Planck</td></tr><tr><td>&lt; 0.12</td><td>95</td><td>5 ALAM</td><td>21</td><td>COSM</td></tr><tr><td>&lt; 0.09</td><td>95</td><td>6 DI-VALENT...</td><td>21</td><td>COSM</td></tr><tr><td>&lt; 0.16</td><td>95</td><td>7 GARNY</td><td>21</td><td>COSM</td></tr><tr><td>&lt; 0.06-0.14</td><td>95</td><td>8 STOCKER</td><td>21</td><td>COSM Normal mass ordering</td></tr><tr><td>&lt; 0.12</td><td>95</td><td>9 AGHANIM</td><td>20</td><td>COSM</td></tr><tr><td>&lt; 0.15</td><td>95</td><td>10 CHOUDHURY</td><td>20</td><td>COSM Normal mass hierarchy</td></tr><tr><td>&lt; 0.16</td><td>95</td><td>11 IVANOV</td><td>20</td><td>COSM Planck and BOSS</td></tr><tr><td>&lt; 0.11</td><td>95</td><td>12 PALANQUE...</td><td>20</td><td>COSM Lyman alpha and CMB</td></tr><tr><td>&lt; 0.26</td><td>95</td><td>13 LOUREIRO</td><td>19</td><td>COSM</td></tr><tr><td>&lt; 0.18</td><td>95</td><td>14 UPADHYE</td><td>19</td><td>COSM BOSS and CMB</td></tr><tr><td>&lt; 0.152</td><td>95</td><td>15 CHOUDHURY</td><td>18</td><td>COSM</td></tr><tr><td>0.064 ±0.061</td><td>95</td><td>16 SIMPSON</td><td>17</td><td>COSM</td></tr><tr><td>-0.005</td><td>95</td><td>17 YECHE</td><td>17</td><td>COSM BOSS and XQ-100</td></tr><tr><td>&lt; 0.14</td><td>95</td><td>18 DIVALENTINO</td><td>16</td><td>COSM</td></tr><tr><td>&lt; 0.0926</td><td>90</td><td>19 HUANG</td><td>16</td><td>COSM Normal mass hierarchy</td></tr><tr><td>&lt; 0.18</td><td>95</td><td>20 ROSSI</td><td>15</td><td>COSM</td></tr><tr><td>&lt; 0.14</td><td>95</td><td>21 ADE</td><td>14</td><td>COSM Planck</td></tr><tr><td>&lt; 0.23</td><td>95</td><td>22 BATTYE</td><td>14</td><td>COSM</td></tr><tr><td>0.320 ±0.081</td><td>95</td><td>23 BEUTLER</td><td>14</td><td>COSM BOSS</td></tr><tr><td>0.35 ±0.10</td><td>95</td><td>24 COSTANZI</td><td>14</td><td>COSM</td></tr><tr><td>0.22 ±0.09</td><td>95</td><td>25 HOU</td><td>14</td><td>COSM</td></tr><tr><td>-0.10</td><td>95</td><td>26 LEISTEDT</td><td>14</td><td>COSM</td></tr><tr><td>0.32 ±0.11</td><td>95</td><td>27 RIEMER-SOR...</td><td>14</td><td>COSM</td></tr><tr><td>&lt; 0.26</td><td>95</td><td>28 MORESCO</td><td>12</td><td>COSM</td></tr><tr><td>&lt; 0.18</td><td>95</td><td>29 XIA</td><td>12</td><td>COSM</td></tr><tr><td>&lt; 0.24</td><td>68</td><td>30 SAITO</td><td>11</td><td>COSM SDSS</td></tr><tr><td>&lt; 0.29</td><td>95</td><td>31 HANNESTAD</td><td>10</td><td>COSM</td></tr><tr><td>&lt; 0.81</td><td>95</td><td>32 SEKIGUCHI</td><td>10</td><td>COSM</td></tr><tr><td>&lt; 0.44</td><td>95</td><td>33 THOMAS</td><td>10</td><td>COSM</td></tr><tr><td>&lt; 0.6</td><td>95</td><td>34 ICHIKI</td><td>09</td><td>COSM</td></tr><tr><td>&lt; 0.28</td><td>95</td><td>35 KOMATSU</td><td>09</td><td>COSM WMAP</td></tr><tr><td>&lt; 1.1</td><td>95</td><td>36 TERENO</td><td>09</td><td>COSM</td></tr><tr><td>&lt; 1.3</td><td>95</td><td>37 VIKHLININ</td><td>09</td><td>COSM</td></tr><tr><td>&lt; 1.2</td><td>95</td><td>38 BERNARDIS</td><td>08</td><td>COSM</td></tr><tr><td>&lt; 0.33</td><td>95</td><td>39 FOGLI</td><td>07</td><td>COSM</td></tr><tr><td>&lt; 0.28</td><td>95</td><td>40 KRISTIANSEN</td><td>07</td><td>COSM</td></tr><tr><td>&lt; 0.17-2.3</td><td>95</td><td>41 ZUNCKEL</td><td>07</td><td>COSM</td></tr><tr><td>&lt; 0.42</td><td>95</td><td>42 CIRELLI</td><td>06</td><td>COSM</td></tr><tr><td>&lt; 0.63-2.2</td><td>95</td><td>43 HANNESTAD</td><td>06</td><td>COSM</td></tr><tr><td>&lt; 0.24</td><td>95</td><td>44 SANCHEZ</td><td>06</td><td>COSM</td></tr><tr><td>&lt; 0.62</td><td>95</td><td>42 SELJAK</td><td>06</td><td>COSM</td></tr><tr><td>&lt; 1.2</td><td>95</td><td>45 ICHIKAWA</td><td>05</td><td>COSM</td></tr><tr><td>&lt; 0.17</td><td>95</td><td>46 BARGER</td><td>04</td><td>COSM</td></tr><tr><td>&lt; 2.0</td><td>95</td><td>47 CROTTY</td><td>04</td><td>COSM</td></tr><tr><td>&lt; 0.75</td><td>95</td><td>48 SPERGEL</td><td>03</td><td>COSM WMAP</td></tr><tr><td>&lt; 1.0</td><td>95</td><td>49 LEWIS</td><td>02</td><td>COSM</td></tr><tr><td>&lt; 0.7</td><td>95</td><td>50 WANG</td><td>02</td><td>COSM CMB</td></tr><tr><td>&lt; 0.9</td><td>95</td><td>51 FUKUGITA</td><td>00</td><td>COSM</td></tr><tr><td>&lt; 4.2</td><td>95</td><td>52 CROFT</td><td>99</td><td>ASTR Ly α power spec</td></tr><tr><td>&lt; 2.7</td><td>95</td><td>SZALAY</td><td>74</td><td>COSM</td></tr><tr><td>&lt; 5.5</td><td>95</td><td>COWSIK</td><td>72</td><td>COSM</td></tr><tr><td>&lt;180</td><td>95</td><td>28 MARX</td><td>72</td><td>COSM</td></tr><tr><td>&lt;132</td><td>95</td><td>28 GERSHTEIN</td><td>66</td><td>COSM</td></tr></tbody></table>

<footnote n="1">¹BRIEDEN 22 combines redshift-space distortions and the shape of the matter power spectrum from BOSS and eBOSS data together with Planck CMB data. Absent the CMB data, the limit is 0.40 eV.</footnote>

<footnote n="2">²KUMAR 22 combine the reconstructed galaxy power spectrum from BOSS data with Planck CMB data.</footnote>

<footnote n="3">³TANSERI 22 combines BOSS galaxy clustering data with measurements of CMB data. Updates VAGNOZZI 17.</footnote>
⁴ABBOTT 21A combines Dark Energy Survey (DES) year 3 results with Planck CMB lensing measurements.

⁵ALAM 21 limit on the sum of neutrino masses by the eBOSS collaboration is based on galaxy, quasar, and Lyman-α 3D clustering data combined with Planck temperature and polarization CMB and supernovae data.

⁶DI-VALENTINO 21 combines CMB temperature and polarization, SN1a luminosity dis- tances and baryon acoustic oscillations data.

⁷GARNY 21 employs a model for the Lyman-α flux power spectrum to set a limit using BOSS data. When combined with Planck CMB temperature and polarization data, a 95% CL range 0.10-0.13 eV is found.

⁸STOCKER 21 use terrestrial and cosmological experiments to set a 95% CL range on the sum of neutrino masses of 0.058-0.139 eV for normal ordering and 0.098-0.174 eV for inverse ordering. They also set an upper limit of 0.037 eV (NO) and 0.042 eV (IO) for the lightest neutrino mass.

⁹AGHANIM 20 limit on the sum of neutrino masses from Planck data combined with lens- ing and baryon acoustic oscillations (BAO). Without BAO, the limit relaxes to <0.24 eV. Several other limits are quoted based on different combinations of data.

¹⁰CHOUDHURY 20 combines 2018 Planck CMB temperature and polarization data plus lensing, together with baryon acoustic oscillation data from BOSS, MGS, and 6dFGS. Assumes ΛCDM model. The upper limit is 0.17 eV for the inverted hierarchy, and 0.12 eV for degenerate neutrinos. Limits are also derived for extended cosmological models.

¹¹IVANOV 20 combines 2018 Planck CMB data with baryon acoustic oscillation data from BOSS. This study is based on a full-shape likelihood for the redshift-space galaxy power spectrum of the BOSS data.

¹²PALANQUE-DELABROUILLE 20 combine Lyman alpha and Planck temperature and polarization data. Limit improves to 0.09 eV when CMB lensing and baryon acoustic oscillation data are included.

¹³LOUREIRO 19 combines data from large scale structure, cosmic microwave background, type la supernovae and big bang nucleosynthesis using physically motivated neutrino mass models.

¹⁴UPADHYE 19 uses the shape of the BOSS redshift-space galaxy power spectrum in combination with the CMB, and supernovae data. Limit weakens to < 0.54 eV if the dark energy equation of state is allowed to vary.

¹⁵CHOUDHURY 18 combines 2015 Planck CMB temperature data, information from the optical depth to reionization from Planck 2016 intermediate results together with baryon acoustic oscillation data from BOSS, MGS, and 6dFGS as well as supernovae Type Ia data from the Pantheon Sample. The limit is strengthened to 0.118 eV when high-/CMB polarization data is also included.

¹⁶SIMPSON 17 uses a combination of laboratory and cosmological measurements to de- termine the light neutrino masses and argue that there is strong evidence for the normal mass ordering.

¹⁷Constrains the total mass of neutrinos using the Lyman-alpha forest power spectrum with BOSS (mid-resolution), XQ-100 (high-resolution) and CMB. Without the CMB data, the limit relaxes to 0.8 eV. Supersedes PALANQUE-DELABROUILLE 15A.

¹⁸Constrains the total mass of neutrinos from Planck CMB data combined with baryon acoustic oscillation and Planck cluster data.

¹⁹Constrains the total mass of neutrinos from BAO data from SDSS-III/BOSS combined with CMB data from Planck. Limit quoted for normal mass hierarchy. The limit for the inverted mass hierarchy is 0.20 eV and for the degenerate mass hierarchy it is 0.15 eV.

²⁰ROSSI 15 sets limits on the sum of neutrino masses using BOSS Lyman alpha forest data combined with Planck CMB data and baryon acoustic oscillations.

²¹Constrains the total mass of neutrinos from Planck CMB data along with WMAP polar- ization, high L, and BAO data.

²²Finite neutrino mass fit to resolve discrepancy between CMB and lensing measurements.

²³Fit to the total mass of neutrinos from BOSS data along with WMAP CMB data and data from other BAO constraints and weak lensing.

²⁴Fit to the total mass of neutrinos from Planck CMB data along with BAO.

²⁵Fit based on the SPT-SZ survey combined with CMB, BAO, and $H_0$ data.

²⁶Constrains the total mass of neutrinos (marginalizing over the effective number of neu- trino species) from CMB, CMB lensing, BAO, and galaxy clustering data.

²⁷Constrains the total mass of neutrinos from Planck CMB data combined with baryon acoustic oscillation data from BOSS, 6dFGS, SDSS, WiggleZ data on the galaxy power spectrum, and HST data on the Hubble parameter. The limit is increased to 0.25 eV if a lower bound to the sum of neutrino masses of 0.04 eV is assumed.

²⁸Constrains the total mass of neutrinos from observational Hubble parameter data with seven-year WMAP data and the most recent estimate of $H_0$.

²⁹Constrains the total mass of neutrinos from the CFHTLS combined with seven-year WMAP data and a prior on the Hubble parameter. Limit is relaxed to 0.41 eV when small scales affected by non-linearities are removed.

³⁰Constrains the total mass of neutrinos from the Sloan Digital Sky Survey and the five-year WMAP data.

³¹Constrains the total mass of neutrinos from the 7-year WMAP data including SDSS and HST data. Limit relaxes to 1.19 eV when CMB data is used alone. Supersedes HANNESTAD 06.

³²Constrains the total mass of neutrinos from a combination of CMB data, a recent mea- surement of $H_0$ (SHOES), and baryon acoustic oscillation data from SDSS.

³³Constrains the total mass of neutrinos from SDSS MegaZ LRG DR7 galaxy clustering data combined with CMB, HST, supernovae and baryon acoustic oscillation data. Limit relaxes to 0.47 eV when the equation of state parameter, $w \neq 1$.

³⁴Constrains the total mass of neutrinos from weak lensing measurements when combined with CMB. Limit improves to 0.54 eV when supernovae and baryon acoustic oscillation observations are included. Assumes ΛCDM model.

³⁵Constrains the total mass of neutrinos from five-year WMAP data. Limit improves to 0.67 eV when supernovae and baryon acoustic oscillation observations are included. Limits quoted assume the ΛCDM model. Supersedes SPERGEL 07.

³⁶Constrains the total mass of neutrinos from weak lensing measurements when combined with CMB. Limit improves to $0.03 < \Sigma_m < 0.54$ eV when supernovae and baryon acoustic oscillation observations are included. The slight preference for massive neutrinos at the two-sigma level disappears when systematic errors are taken into account. Assumes ΛCDM model.

³⁷Constrains the total mass of neutrinos from recent Chandra X-ray observations of galaxy clusters when combined with CMB, supernovae, and baryon acoustic oscillation measure- ments. Assumes flat universe and constant dark-energy equation of state, $w$.

³⁸Constrains the total mass of neutrinos from recent CMB and SOSS LRG power spectrum data along with bias mass relations from SDSS, DEEP2, and Lyman-Break Galaxies. It assumes ΛCDM model. Limit degrades to 0.59 eV in a more general wCDM model.

³⁹Constrains the total mass of neutrinos from neutrino oscillation experiments and cosmo- logical data. The most conservative limit uses only WMAP three-year data, while the most stringent limit includes CMB, large-scale structure, supernova, and Lyman-alpha data.

⁴⁰Constrains the total mass of neutrinos from recent CMB, large scale structure, SN1a, and baryon acoustic oscillation data. The limit relaxes to 1.75 when WMAP data alone is used with no prior. Paper shows results with several combinations of data sets. Supersedes KRISTIANSEN 06.

⁴¹Constrains the total mass of neutrinos from the CMB and the large scale structure data. The most conservative limit is obtained when generic initial conditions are allowed.

⁴²Constrains the total mass of neutrinos from recent CMB, large scale structure, Lyman- alpha forest, and SN1a data.

⁴³Constrains the total mass of neutrinos from recent CMB and large scale structure data. See also GOBAR 06. Superseded by HANNESTAD 10.

⁴⁴Constrains the total mass of neutrinos from the CMB and the final 2dF Galaxy Redshift Survey.

⁴⁵Constrains the total mass of neutrinos from the CMB experiments alone, assuming ΛCDM Universe. FUKUGITA 06 show that this result is unchanged by the 3-year WMAP data.

⁴⁶Constrains the total mass of neutrinos from the power spectrum of fluctuations derived from the Sloan Digital Sky Survey and the 2dF galaxy redshift survey, WMAP and 27 other CMB experiments and measurements by the HST Key project.

⁴⁷Constrains the total mass of neutrinos from the power spectrum of fluctuations derived from the Sloan Digital Sky Survey, the 2dF galaxy redshift survey, WMAP and ACBAR. The limit is strengthened to 0.6 eV when measurements by the HST Key project and supernovae data are included.

⁴⁸Constrains the fractional contribution of neutrinos to the total matter density in the Universe from WMAP data combined with other CMB measurements, the 2dfGRS data, and Lyman α data. The limit does not noticeably change if the Lyman α data are not used.

⁴⁹LEWIS 02 constrains the total mass of neutrinos from the power spectrum of fluctuations derived from the CMB, HST Key project, 2dF galaxy redshift survey, supernovae type Ia, and BBN.

⁵⁰WANG 02 constrains the total mass of neutrinos from the power spectrum of fluctuations derived from the CMB and other cosmological data sets such as galaxy clustering and the Lyman α forest.

⁵¹FUKUGITA 00 is a limit on neutrino masses from structure formation. The constraint is based on the clustering scale $\sigma_8$ and the COBE normalization and leads to a conservative limit of 0.9 eV assuming 3 nearly degenerate neutrinos. The quoted limit is on the sum of the light neutrino masses.

⁵²CROFT 99 result based on the power spectrum of the Ly α forest. If $\Omega_{\text{matter}} < 0.5$, the limit is improved to $m_\nu < 2.4 (\Omega_{\text{matter}}/0.17-1)$ eV.

## Limits on MASSES of Light Stable Right-Handed $\nu$ (with necessarily suppressed interaction strengths)

<table><thead><tr><td>VALUE (eV)</td><td>DOCUMENT ID</td><td>TECN</td><td>COMMENT</td></tr></thead></table>

• • • We do not use the following data for averages, fits, limits, etc. • • •

<table><tr><td>&lt;100-200</td><td>1 OLIVE</td><td>82</td><td>COSM Dirac $\nu$</td></tr><tr><td>&lt;200-2000</td><td>1 OLIVE</td><td>82</td><td>COSM Majorana $\nu$</td></tr></table>

¹Depending on interaction strength $G_R$ where $G_R < G_F$.

## Limits on MASSES of Heavy Stable Right-Handed $\nu$ (with necessarily suppressed interaction strengths)

<table><thead><tr><td>VALUE (GeV)</td><td>DOCUMENT ID</td><td>TECN</td><td>COMMENT</td></tr></thead></table>

• • • We do not use the following data for averages, fits, limits, etc. • • •

<table><tr><td>&gt; 10</td><td>1 OLIVE</td><td>82</td><td>COSM $G_R/G_F < 0.1$</td></tr><tr><td>&gt;100</td><td>1 OLIVE</td><td>82</td><td>COSM $G_R/G_F < 0.01$</td></tr></table>

¹These results apply to heavy Majorana neutrinos and are summarized by the equation: $m_\nu > 1.2$ GeV ($G_F/G_R$). The bound saturates, and if $G_R$ is too small no mass range is allowed.

## $\nu$ CHARGE

<table><thead><tr><th colspan="5">e = electron charge is the unit of values listed below.</th></tr><tr><th>VALUE (e)</th><th>CL%</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>&lt;4 × 10<sup>-35</sup></td><td>95</td><td>1 CAPRINI</td><td>05</td><td>COSM charge neutral universe</td></tr><tr><td>&lt;2.24 × 10<sup>-13</sup></td><td>90</td><td>2 AALBERS</td><td>23A LZ</td><td>Solar $\nu$ spectrum</td></tr><tr><td>&lt;1.5 × 10<sup>-13</sup></td><td>90</td><td>3 ATZORI-COR...23</td><td>FIT</td><td>solar neutrinos</td></tr></tbody></table>
<table><tr><td>&lt;3.3 &times; 10<sup>-12</sup></td><td>90</td></tr><tr><td>&lt;5.4 &times; 10<sup>-12</sup></td><td>90</td></tr><tr><td>1.7-2.3 &times; 10<sup>-12</sup></td><td>68</td></tr><tr><td>&lt;3 &times; 10<sup>-8</sup></td><td>95</td></tr><tr><td>&lt;2.1 &times; 10<sup>-12</sup></td><td>90</td></tr><tr><td>&lt;1.5 &times; 10<sup>-12</sup></td><td>90</td></tr><tr><td>&lt;3.7 &times; 10<sup>-12</sup></td><td>90</td></tr><tr><td>&lt;2 &times; 10<sup>-14</sup></td><td></td></tr><tr><td>&lt;6 &times; 10<sup>-14</sup></td><td></td></tr><tr><td>&lt;4 &times; 10<sup>-4</sup></td><td></td></tr><tr><td>&lt;3 &times; 10<sup>-4</sup></td><td></td></tr><tr><td>&lt;2 &times; 10<sup>-15</sup></td><td></td></tr><tr><td>&lt;1 &times; 10<sup>-13</sup></td><td></td></tr></table>

<table><tr><td>4 BONET</td><td>22A CONU</td><td>nuclear reactor</td></tr><tr><td>5 ABE</td><td>20E XMAS</td><td>solar neutrinos</td></tr><tr><td>6 KHAN</td><td>20</td><td>spectral fit of XENON1T</td></tr><tr><td>7 DELLA-VALLE</td><td>16 LASR</td><td>magnetic dichroism</td></tr><tr><td>8 CHEN</td><td>14A TEXO</td><td>nuclear reactor</td></tr><tr><td>9 STUDENIKIN</td><td>14</td><td>nuclear reactor</td></tr><tr><td>10 GNINENKO</td><td>07 RVUE</td><td>nuclear reactor</td></tr><tr><td>11 RAFFELT</td><td>99 ASTR</td><td>red giant luminosity</td></tr><tr><td>12 RAFFELT</td><td>99 ASTR</td><td>solar cooling</td></tr><tr><td>13 BABU</td><td>94 RVUE</td><td>BEBC beam dump</td></tr><tr><td>14 DAVIDSON</td><td>91 RVUE</td><td>SLAC e<sup>-</sup> beam dump</td></tr><tr><td>15 BARBIELLINI</td><td>87 ASTR</td><td>SN 1987A</td></tr><tr><td>16 BERNSTEIN</td><td>63 ASTR</td><td>solar energy losses</td></tr></table>

1. CAPRINI 05 limit derived from the lack of a charge asymmetry in the universe. Limit assumes that charge asymmetries between particles are not anti-correlated.

2. AALBERS 23A utilize the first 60 days of data collected by the LZ dark matter search to place a limit on the electric charge of solar neutrinos. Low energy electron-recoil events are utilized. This LZ-collaboration analysis supersedes that of the external authors in ATZORI-CORONA 23 because of a more complete treatment of experiment uncertainties.

3. ATZORI-CORONA 23 use LUX-ZEPLIN dark matter search data published by AALBERS 23 to place a limit on neutrino millilcharge.

4. BONET 22A use data collected by four low-threshold Ge detectors, placed 17.1 m from one of the cores of the nuclear reactors at Brokdorf to derive this limit. A spectral analysis is performed on reactor on and off data.

5. ABE 20E obtains this result by assuming that the low-energy excess events in the XMASS detector are produced by neutrino millilcharge which is common for all three neutrino flavors.

6. KHAN 20 performed a constrained spectral fit analysis of the excess observed in the electron recoil energy spectrum by the XENON1T experiment. This range of neutrino millilcharge values is one of the possible interpretations of these excess events. For the individual flavor constraints at 90% C.L. see the original reference.

7. DELLA-VALLE 16 obtain a limit on the charge of neutrinos valid for masses of less than 10 MeV. For heavier neutrinos the limit increases as a power of mass, reaching $10^{-6}$ e for $m = 100$ MeV.

8. CHEN 14A use the Multi-Configuration RRPA method to analyze reactor $\bar{\nu}_e$ scattering on Ge atoms with 300 eV recoil energy threshold to obtain this limit.

9. STUDENIKIN 14 uses the limit on $\mu_\nu$ from BEDA 13 and the 2.8 keV threshold of the electron recoil energy to obtain this limit.

10. GNINENKO 07 use limit on $\bar{\nu}_e$ magnetic moment from LI 03B to derive this result. The limit is considerably weaker than the limits on the charge of $\nu_e$ and $\bar{\nu}_e$ from various astrophysics considerations.

11. This RAFFELT 99 limit applies to all neutrino flavors which are light enough ($<5$ keV) to be emitted from globular-cluster red giants.

12. This RAFFELT 99 limit is derived from the helioseismological limit on a new energy-loss channel of the Sun, and applies to all neutrino flavors which are light enough ($<1$ keV) to be emitted from the Sun.

13. BABU 94 use COOPER-SARKAR 92 limit on $\nu$ magnetic moment to derive quoted result. It applies to $\nu_\tau$.

14. DAVIDSON 91 use data from early SLAC electron beam dump experiment to derive charge limit as a function of neutrino mass. It applies to $\nu_\tau$.

15. Exact BARBIELLINI 87 limit depends on assumptions about the intergalactic or galactic magnetic fields and about the direct distance and time through the field. It applies to $\nu_e$.

16. The limit applies to all flavors.

<table><tr><td>6-26 &times; 10<sup>9</sup></td><td>95</td></tr><tr><td>&gt; 10<sup>5</sup> - 10<sup>10</sup></td><td>95</td></tr><tr><td>&gt; 0.11</td><td>90</td></tr><tr><td>&gt; 0.004</td><td>90</td></tr><tr><td>&gt; 4.4 &times; 10<sup>-5</sup></td><td>90</td></tr><tr><td>&gt; 100</td><td>95</td></tr><tr><td>&gt; 0.067</td><td>90</td></tr><tr><td>&gt; 1.1 &times; 10<sup>-3</sup></td><td>90</td></tr><tr><td>&gt; 8.7 &times; 10<sup>-5</sup></td><td>99</td></tr><tr><td>&ge; 4200</td><td>90</td></tr><tr><td>&gt; 2.8 &times; 10<sup>-5</sup></td><td>99</td></tr></table>

<table><tr><td>7 ESCUDERO</td><td>19 COSM</td><td>Invisible decay $m_\nu \ge 0.05$ eV</td></tr><tr><td>8 CECCCHINI</td><td>11 ASTR</td><td>$\nu_2 \to \nu_1$ radiative decay</td></tr><tr><td>9 MIRIZZI</td><td>07 CMB</td><td>radiative decay</td></tr><tr><td>10 MIRIZZI</td><td>07 CIB</td><td>radiative decay</td></tr><tr><td>11 WONG</td><td>07 CNTR</td><td>Reactor $\bar{\nu}_e$</td></tr><tr><td>12 XIN</td><td>05 CNTR</td><td>Reactor $\nu_e$</td></tr><tr><td>13 XIN</td><td>05 CNTR</td><td>Reactor $\nu_e$</td></tr><tr><td>14 AHARMIM</td><td>04 SNO</td><td>quasidegen, $\nu$ masses</td></tr><tr><td>14 AHARMIM</td><td>04 SNO</td><td>hierarchical $\nu$ masses</td></tr><tr><td>15 CECCCHINI</td><td>04 ASTR</td><td>Radiative decay for $\nu$ mass $> 0.01$ eV</td></tr><tr><td>16 EGUCHI</td><td>04 KLND</td><td>quasidegen, $\nu$ masses</td></tr><tr><td>16 EGUCHI</td><td>04 KLND</td><td>hierarchical $\nu$ masses</td></tr><tr><td>17 BANDYOPA...</td><td>03 FIT</td><td>nonradiative decay</td></tr><tr><td>18 DERBIN</td><td>02B CNTR</td><td>Solar pp and Be $\nu$</td></tr><tr><td>19 JOSHIPURA</td><td>02B FIT</td><td>nonradiative decay</td></tr><tr><td>20 DOLGOV</td><td>99 COSM</td><td></td></tr><tr><td>21 BILLER</td><td>98 ASTR</td><td>$m_\nu = 0.05-1$ eV</td></tr><tr><td>&gt; 2.8 &times; 10<sup>15</sup></td><td>22,23 BLUDMAN</td><td>$m_\nu < 50$ eV</td></tr><tr><td>none 10<sup>-12</sup> - 5 &times; 10<sup>4</sup></td><td>24 DODELSON</td><td>$m_\nu = 1-300$ keV</td></tr><tr><td>&lt; 10<sup>-12</sup> or &gt; 5 &times; 10<sup>4</sup></td><td>24 DODELSON</td><td>$m_\nu = 1-300$ keV</td></tr><tr><td>&gt; 6.4</td><td>90</td><td>25 GRANEK</td></tr><tr><td>&gt; 1.1 &times; 10<sup>15</sup></td><td></td><td>26 KRAKAUER</td></tr><tr><td>&gt; 6.3 &times; 10<sup>15</sup></td><td></td><td>27 WALKER</td></tr><tr><td>&gt; 1.7 &times; 10<sup>15</sup></td><td></td><td>$m_\nu = 0.03 - \sim 2$ MeV</td></tr><tr><td>&gt; 8.3 &times; 10<sup>14</sup></td><td></td><td>23,28 CHUPP</td></tr><tr><td>&gt; 22</td><td>68</td><td>$m_\nu < 20$ eV</td></tr><tr><td>&gt; 38</td><td>68</td><td>$m_\nu < 20$ eV</td></tr><tr><td>&gt; 59</td><td>68</td><td>$m_\nu < 20$ eV</td></tr><tr><td>&gt; 30</td><td>68</td><td>29 RAFFELT</td></tr><tr><td>&gt; 20</td><td>68</td><td>30 RAFFELT</td></tr><tr><td>&gt; 0.11</td><td>90</td><td>31 VONFEILIT...</td></tr><tr><td>&gt; 2 &times; 10<sup>21</sup></td><td></td><td>$\bar{\nu}_e$ at LAMPF</td></tr><tr><td>&gt; 1.0 &times; 10<sup>-2</sup></td><td></td><td>$m_\nu \sim 10-100$ eV</td></tr><tr><td>&gt; 1.7 &times; 10<sup>-2</sup></td><td>90</td><td></td></tr><tr><td>&lt; 3 &times; 10<sup>-11</sup></td><td></td><td>$\bar{\nu}_e$ at LAMPF</td></tr><tr><td>&gt; 2.2 &times; 10<sup>-3</sup></td><td>90</td><td>$m_\nu < 10$ MeV</td></tr><tr><td>&gt; 3. &times; 10<sup>-3</sup></td><td>90</td><td>$\bar{\nu}_e$ at LAMPF</td></tr><tr><td>&gt; 1.3 &times; 10<sup>-2</sup></td><td>90</td><td>$\bar{\nu}_e$ at LAMPF</td></tr></table>

1. KRAKAUER 91 quotes the limit $\tau/m_{\nu_1} > (0.75a^2 + 21.65a + 26.3) \text{ s/eV}$, where a is a parameter describing the asymmetry in the neutrino decay defined as $dN_\gamma/d\cos\theta = (1/2)(1 + a\cos\theta)$ The parameter $a=0$ for a Majorana neutrino, but can vary from $-1$ to $1$ for a Dirac neutrino. The bound given by the authors is the most conservative (which applies for $a = -1$).

2. RAFFELT 85 limit on the radiative decay is from solar $\chi$- and $\gamma$-ray fluxes. Limit depends on $\nu$ flux from pp, now established from GALLEX and SAGE to be $> 0.5$ of expectation.

3. REINES 74 looked for $\nu$ of nonzero mass decaying radiatively to a neutral of lesser mass + $\gamma$. Used liquid scintillator detector near fission reactor. Finds lab lifetime $6 \times 10^7$ s or more. Above value of (mean life)/mass assumes average effective neutrino energy of 0.2 MeV. To obtain the limit $6 \times 10^7$ s REINES 74 assumed that the full $\bar{\nu}_e$ reactor flux could be responsible for yielding decays with photon energies in the interval 0.1 MeV - 0.5 MeV. This represents some overestimate so their lower limit is an over-estimate of the lab lifetime (VOGEL 84). If so, OBERAUER 87 may be comparable or better.

4. IVANEZ-BALLESTEROS 23 reports a limit on the lifetime-to-mass ratio of the mass eigenstates $\nu_1$ and $\nu_2$ for inverted mass ordering. No limit was obtained in the case of normal mass ordering.

5. AHARMIM 19 quotes the limit $\tau/m_{\nu_2}$ for invisible nonradiative decay of $\nu_2$. They obtained this result by analyzing the entire SNO dataset, allowing for the decay of $\nu_2$ which would cause an energy-dependent distortion of the survival probability of electron-type solar neutrinos.

6. AHARMIM 19 quotes the limit $\tau/m_{\nu_2}$ for invisible nonradiative decay of $\nu_2$. They obtained this result by combining the $\tau/m_{\nu_2}$ measurements from SNO and other solar neutrino experiments (Super-Kamiokande, KamLAND, and Borexino $^8$B results; Borexino and KamLAND $^7$Be results; the combined gallium interaction rate from GNO, GALLEX, and SAGE; and the chlorine interaction rate from Homestake). The quoted limit at 99% CL is $> 1.04 \times 10^{-3}$.

7. ESCUDERO 19 sets limits on invisible neutrino decays using Planck 2018 data of $\tau > 1.3-0.3 \times 10^9$ s at 95% C.L. Values in the range $\tau = 2-16 \times 10^9$ s are preferred at 95% C.L. when Planck polarization data is included. Limits scale as $(m_\nu/0.05 \text{ eV})^3$.

8. CECCCHINI 11 search for radiative decays of solar neutrinos into visible photons during the 2006 total solar eclipse. The range of (mean life)/mass values corresponds to a range of $\nu_1$ masses between $10^{-4}$ and 0.1 eV.

9. MIRIZZI 07 determine a limit on the neutrino radiative decay from analysis of the maximum allowed distortion of the CMB spectrum as measured by the COBE/FIRAS. For the decay $\nu_2 \to \nu_1$ the lifetime limit is $\lesssim 4 \times 10^{20}$ s for $m_{\min} \lesssim 0.14$ eV. For transition with the $|\Delta m_{31}|$ mass difference the lifetime limit is $\sim 2 \times 10^{19}$ s for $m_{\min} \lesssim 0.14$ eV and $\sim 5 \times 10^{20}$ s for $m_{\min} \gtrsim 0.14$ eV.

## $\nu$ (MEAN LIFE) / MASS

Measures $[\sum |\nu_{\epsilon j}|^2 \Gamma_j m_j]^{-1}$, where the sum is over mass eigenstates which cannot be resolved experimentally. Some of the limits constrain the radiative decay and are based on the limit of the corresponding photon flux. Other apply to the decay of a heavier neutrino into the lighter one and a Majoron or other invisible particle. Many of these limits apply to any $\nu$ within the indicated mass range.

Limits on the radiative decay are either directly based on the limits of the corresponding photon flux, or are derived from the limits on the neutrino magnetic moments. In the later case the transition rate for $\nu_j \to \nu_j + \gamma$

is constrained by $\Gamma_{ij} = \frac{1}{\tau_{ij}} = \frac{(m_i^2 - m_j^2)^3}{m_j^2} \mu_{ij}^2$, where $\mu_{ij}$ is the neutrino transition moment in the mass eigenstates basis. Typically, the limits on lifetime based on the magnetic moments are nonmany orders of magnitude more restrictive than limits based on the nonobservation of photons.

<table><thead><tr><th>VALUE (s/eV)</th><th>CL%</th></tr></thead><tbody><tr><td>&gt; 15.4</td><td>90</td></tr><tr><td>&gt; 7 &times; 10<sup>9</sup></td><td></td></tr><tr><td>&gt; 300</td><td>90</td></tr></tbody></table>

<table><thead><tr><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>1 KRAKAUER</td><td>91 CNTR</td><td>$\nu_{\mu}, \bar{\nu}_{\mu}$ at LAMPF</td></tr><tr><td>2 RAFFELT</td><td>85 ASTR</td><td></td></tr><tr><td>3 REINES</td><td>74 CNTR</td><td>$\bar{\nu}_e$</td></tr></tbody></table>

• • • We do not use the following data for averages, fits, limits, etc. • • •

<table><tr><td>&gt; 1.2 &times; 10<sup>5</sup></td><td>90</td></tr><tr><td>&gt; 8.08 &times; 10<sup>-5</sup></td><td>90</td></tr><tr><td>&gt; 1.92 &times; 10<sup>-3</sup></td><td>90</td></tr></table>

<table><tr><td>4 IVANEZ-BAL... 23 ASTR</td><td>SN1987A, nonradiative decay</td></tr><tr><td>5 AHARMIM 19 SNO</td><td>$\nu_2$ invisible nonradiative decay</td></tr><tr><td>6 AHARMIM 19 FIT</td><td>$\nu_2$ invisible nonradiative decay</td></tr></table>
¹⁰ MIRIZZI 07 determine a limit on the neutrino radiative decay from analysis of the cosmic infrared background (CIB) using the Spitzer Observatory data. For transition with the $|\Delta m_{31}|$ mass difference they obtain the lifetime limit ~$10^{20}$ s for $m_{min} \sim 0.14$ eV.

¹¹ WONG 07 use their limit on the neutrino magnetic moment together with the assumed experimental value of $\Delta m_{13}^2 \sim 2 \times 10^{-3} \text{ eV}^2$ to obtain $\tau_{13}/m_1^3 > 3.2 \times 10^{27} \text{ s/eV}^3$ for the radiative decay in the case of the inverted mass hierarchy. Similarly to RAFFELT 89 this limit can be violated if electric and magnetic moments are equal to each other. Analogous, but numerically somewhat different limits are obtained for $\tau_{23}$ and $\tau_{21}$.

¹² XIN 05 search for the $\gamma$ from radiative decay of $\nu_e$ produced by the electron capture on $^{51}$Cr. No events were seen and the limit on $\tau/m_\nu$ was derived. This is a weaker limit on the decay of $\nu_e$ than KRAKAUER 91.

¹³ XIN 05 use their limit on the neutrino magnetic moment of $\nu_e$ together with the assumed experimental value of $\Delta m_{13}^2 \sim 2 \times 10^{-3} \text{ eV}^2$ to obtain $\tau_{13}/m_1^3 > 1 \times 10^{23} \text{ s/eV}^3$ for the radiative decay in the case of the inverted mass hierarchy. Similarly to RAFFELT 89 this limit can be violated if electric and magnetic moments are equal to each other. Analogous, but numerically somewhat different limits are obtained for $\tau_{23}$ and $\tau_{21}$. Again, this limit is specific for $\nu_e$.

¹⁴ AHARMIM 04 obtained these results from the solar $\bar{\nu}_e$ flux limit set by the SNO measurement assuming $\nu_2$ decay through nonradiative process $\nu_2 \to \bar{\nu}_1 X$, where X is a Majoron or other invisible particle. Limits are given for the cases of quasidegenerate and hierarchical neutrino masses.

¹⁵ CECCINI 04 obtained this bound through the observations performed on the occasion of the 21 June 2001 total solar eclipse, looking for visible photons from radiative decays of solar neutrinos. Limit is a $\tau/m_{\nu_2}$ in $\nu_2 \to \nu_1 \gamma$. Limit ranges from ~ 100 to $10^7 \text{ s/eV}$ for 0.01 < $m_{\nu_1}$ < 0.1 eV.

¹⁶ EGUCHI 04 obtained these results from the solar $\bar{\nu}_e$ flux limit set by the KamLAND measurement assuming $\nu_2$ decay through nonradiative process $\nu_2 \to \bar{\nu}_1 X$, where X is a Majoron or other invisible particle. Limits are given for the cases of quasidegenerate and hierarchical neutrino masses.

¹⁷ The ratio of the lifetime over the mass derived by BANDYOPADHYAY 03 is for $\nu_2$. They obtained this result using the following solar-neutrino data: total rates measured in CI and Ga experiments, the Super-Kamiokande's zenith-angle spectra, and SNO's day and night spectra. They assumed that $\nu_1$ is the lowest mass, stable or nearly stable neutrino state and $\nu_2$ decays through nonradiative Majoron emission process, $\nu_2 \to \bar{\nu}_1 J + J$, or through nonradiative process with all the final state particles being sterile. The best fit is obtained in the region of the LMA solution.

¹⁸ DERBIN 02B (also BACK 03B) obtained this bound for the radiative decay from the results of background measurements with Counting Test Facility (the prototype of the Borexino detector). The laboratory gamma spectrum is given as $dN_\gamma/d\cos\theta = (1/2)(1 + \alpha\cos\theta)$ with $\alpha=0$ for a Majorana neutrino, and $\alpha$ varying to -1 to 1 for a Dirac neutrino. The listed bound is for the case of $\alpha=0$. The most conservative bound $1.5 \times 10^3 \text{ s eV}^{-1}$ is obtained for the case of $\alpha=-1$.

¹⁹ The ratio of the lifetime over the mass derived by JOSHIPURA 02B is for $\nu_2$. They obtained this result from the total rates measured in all solar neutrino experiments. They assumed that $\nu_1$ is the lowest mass, stable or nearly stable neutrino state and $\nu_2$ decays through nonradiative process like Majoron emission decay, $\nu_2 \to \nu'_1 J + J$ where $\nu'_1$ state is sterile. The exact limit depends on the specific solution of the solar neutrino problem. The quoted limit is for the LMA solution.

²⁰ DOLGOV 99 places limits in the (Majorana) $\gamma$-associated $\nu$ mass-lifetime plane based on nucleosynthesis. Results would be considerably modified if neutrino oscillations exist.

²¹ BILLER 98 use the observed TeV $\gamma$-ray spectra to set limits on the mean life of any radiatively decaying neutrino between 0.05 and 1 eV. Curve shows $\tau_\gamma/B_\gamma > 0.15 \times 10^{21} \text{ s}$ at 0.05 eV, $> 1.2 \times 10^{21} \text{ s}$ at 0.17 eV, $> 3 \times 10^{21} \text{ s}$ at 1 eV, where $B_\gamma$ is the branching ratio to photons.

²² BLUDMAN 92 sets additional limits by this method for higher mass ranges. Cosmological limits are also obtained.

²³ Limit on the radiative decay based on nonobservation of $\gamma$'s in coincidence with $\nu$'s from SN 1987A.

²⁴ DODELSON 92 range is for wrong-helicity keV mass Dirac $\nu$'s from the core of neutron star in SN 1987A decay to $\nu$'s that would be interacted in KAM2 or IMB detectors.

²⁵ GRANEK 91 considers heavy neutrino decays to $\gamma m_L$ and $3m_L$, where $m_{\nu_L} < 100 \text{ keV}$. Lifetime is calculated as a function of heavy neutrino mass, branching ratio into $\gamma m_L$, and $m_{\nu_L}$.

²⁶ KRAKAUER 91 quotes the limit for $\nu_e, \tau/m_\nu > (0.3a^2 + 9.8a + 15.9) \text{ s/eV}$, where a is a parameter describing the asymmetry in the radiative neutrino decay defined as $dN_\gamma/d\cos\theta = (1/2)(1 + a\cos\theta) = a$ = 0 for a Majorana neutrino, but can vary from -1 to 1 for a Dirac neutrino. The bound given by the authors is the most conservative (which applies for $a = -1$).

²⁷ WALKER 90 uses SN 1987A $\gamma$ flux limits after 289 days.

²⁸ CHUPP 89 should be multiplied by a branching ratio (about 1) and a detection efficiency (about 1/4), and pertains to radiative decay of any neutrino to a lighter or sterile neutrino.

²⁹ RAFFELT 89 uses KYULDJIEV 84 to obtain $\tau m^3 > 3 \times 10^{18} \text{ s eV}^3$ (based on $\bar{\nu}_e e^-$ cross sections). The bound for the radiative decay is not valid if electric and magnetic transition moments are equal for Dirac neutrinos.

³⁰ RAFFELT 89B analyze stellar evolution and exclude the region $3 \times 10^{12} < \tau m^3 < 3 \times 10^{21} \text{ s eV}^3$.

³¹ Model-dependent theoretical analysis of SN 1987A neutrinos. Quoted limit is for $\left|\sum_j |\nu_{ej}|^2 \bar{\Gamma}_j m_j\right|^{-1}$, where $\ell=\mu, \tau$. Limit is $3.3 \times 10^{14} \text{ s/eV}$ for $\ell=e$.

³² OBERAUER 87 looks for photons and $e^+e^-$ pairs from radiative decays of reactor neutrinos.

³³ BINETRUY 84 finds $\tau < 10^8 \text{ s}$ for neutrinos in a radiation-dominated universe.

³⁴ These experiments look for $\nu_K \to \nu_j \gamma$ or $\bar{\nu}_K \to \bar{\nu}_j \gamma$.

³⁵ STECKER 80 limit based on UV background; result given is $\tau > 4 \times 10^{22} \text{ s}$ at $m_\nu = 20 \text{ eV}$.

³⁶ FALK 78 finds lifetime constraints based on supernova energetics.

³⁷ COWSIK 77 considers variety of scenarios. For neutrinos produced in the big bang, present limits on optical photon flux require $\tau > 10^{23} \text{ s}$ for $m_\nu \sim 1 \text{ eV}$. See also COWSIK 79 and GOLDMAN 79.

## ν MAGNETIC MOMENT

The coupling of neutrinos to an electromagnetic field is a characterized by a $3 \times 3$ matrix $\lambda$ of the magnetic ($\mu$) and electric ($d$) dipole moments ($\lambda = \mu - id$). For Majorana neutrinos the matrix $\lambda$ is antisymmetric and only transition moments are allowed, while for Dirac neutrinos $\lambda$ is a general $3 \times 3$ matrix. In the standard electroweak theory extended to include neutrino masses (see FUJIKAWA 80) $\mu_\nu = 3eG_F m_\nu / (8\pi^2\sqrt{2}) = 3.2 \times 10^{-19} (m_\mu / eV) \mu_B$, i.e. it is unobservably small given the known small neutrino masses. In more general models there is no longer a proportionality between neutrino mass and its magnetic moment, even though only massive neutrinos have nonvanishing magnetic moments without fine tuning.

Laboratory bounds on $\lambda$ are obtained via elastic $\nu-e$ scattering, where the scattered neutrino is not observed. The combinations of matrix elements of $\lambda$ that are constrained by various experiments depend on the initial neutrino flavor and on its propagation between source and detector (e.g., solar $\nu_e$ and reactor $\bar{\nu}_e$ do not constrain the same combinations). The listings below therefore identify the initial neutrino flavor.

Other limits, e.g. from various stellar cooling processes, apply to all neutrino flavors. Analogous flavor independent, but weaker, limits are obtained from the analysis of $e^+e^- \to \nu\bar{\nu}\gamma$ collider experiments.

<table><thead><tr><th>VALUE (10<sup>-10</sup> μ<sub>B</sub>)</th><th>CL%</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>&lt; 0.064</td><td>90</td><td>1 APRILE</td><td>22B</td><td>Solar ν spectrum</td></tr><tr><td>&lt; 0.29</td><td>90</td><td>2 BEDA</td><td>13</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 6.8</td><td>90</td><td>3 AUERBACH</td><td>01</td><td>LSND ν<sub>e</sub>e, ν<sub>μ</sub>e scattering</td></tr><tr><td>&lt; 3900</td><td>90</td><td>4 SCHWIENHO...01</td><td>DONU</td><td>ν<sub>τ</sub>e<sup>+</sup> e<sup>-</sup> → ν<sub>τ</sub>e<sup>-</sup></td></tr></tbody></table><p>• • • We do not use the following data for averages, fits, limits, etc. • • •</p><table><tbody><tr><td>&lt; 0.136</td><td>90</td><td>5 AALBERS</td><td>23A</td><td>LZ Solar ν spectrum</td></tr><tr><td>&lt; 0.11</td><td>90</td><td>6 ATZORI-COR...23</td><td>FIT</td><td>Solar ν spectrum</td></tr><tr><td>&lt; 0.75</td><td>90</td><td>7 BONET</td><td>22A</td><td>CONU Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 2.8</td><td>90</td><td>8 COLOMA</td><td>22</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 1.8</td><td>90</td><td>9 ABE</td><td>20E</td><td>XMAS Solar ν spectrum</td></tr><tr><td>0.14-0.29</td><td>90</td><td>10 APRILE</td><td>20</td><td>XEIT Solar ν spectrum</td></tr><tr><td>&lt; 0.012</td><td>95</td><td>11 CAPOZZI</td><td>20</td><td>ASTR Tip of the Red-Giant Branch</td></tr><tr><td>0.2-0.4</td><td>68</td><td>12 KHAN</td><td>20</td><td>Spectral fit of XENON1T</td></tr><tr><td>&lt; 0.28</td><td>90</td><td>13 AGOSTINI</td><td>17A</td><td>BORX Solar ν spectrum</td></tr><tr><td>&lt; 0.022</td><td>90</td><td>14 ARCEO-DIAZ</td><td>15</td><td>ASTR Red giants</td></tr><tr><td>&lt; 0.1</td><td>95</td><td>15 CORSICO</td><td>14</td><td>ASTR</td></tr><tr><td>&lt; 0.05</td><td>95</td><td>16 MILLER-BER...14B</td><td>ASTR</td><td>Globular cluster M5</td></tr><tr><td>&lt; 0.045</td><td>95</td><td>17 VIAUX</td><td>13A</td><td>ASTR</td></tr><tr><td>&lt; 0.32</td><td>90</td><td>18 BEDA</td><td>10</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 2.2</td><td>90</td><td>19 DENIZ</td><td>10</td><td>TEXO Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 0.011-0.027</td><td>90</td><td>20 KUZNETSOV</td><td>09</td><td>ASTR ν<sub>L</sub> → ν<sub>R</sub> in SN1987A</td></tr><tr><td>&lt; 0.54</td><td>90</td><td>21 ARPESELLA</td><td>08A</td><td>BORX Solar ν spectrum</td></tr><tr><td>&lt; 0.58</td><td>90</td><td>22 BEDA</td><td>07</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 0.74</td><td>90</td><td>23 WONG</td><td>07</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 0.9</td><td>90</td><td>24 DARAKTCH...</td><td>05</td><td>ASTR ν<sub>e</sub></td></tr><tr><td>&lt; 130</td><td>90</td><td>25 XIN</td><td>05</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 37</td><td>95</td><td>26 GRIFOLS</td><td>04</td><td>FIT Solar 8B ν (SNO NC)</td></tr><tr><td>&lt; 3.6</td><td>90</td><td>27 LIU</td><td>04</td><td>SKAM Solar ν spectrum</td></tr><tr><td>&lt; 1.1</td><td>90</td><td>28 LIU</td><td>04</td><td>SKAM Solar ν spectrum (LMA region)</td></tr><tr><td>&lt; 5.5</td><td>90</td><td>29 BACK</td><td>03B</td><td>CNTR Solar pp and Be ν</td></tr><tr><td>&lt; 1.0</td><td>90</td><td>30 DARAKTCH...</td><td>03</td><td>Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 1.3</td><td>90</td><td>31 LI</td><td>03B</td><td>CNTR Reactor ν<sub>e</sub></td></tr><tr><td>&lt; 2</td><td>90</td><td>32 GRIMUS</td><td>02</td><td>FIT solar + reactor (Majorana ν)</td></tr><tr><td>&lt;80000</td><td>90</td><td>33 TANIMOTO</td><td>00</td><td>RVUE e<sup>+</sup> e<sup>-</sup> → ννγ</td></tr><tr><td>&lt; 0.01-0.04</td><td>90</td><td>34 AYALA</td><td>99</td><td>ASTR ν<sub>L</sub> → ν<sub>R</sub> in SN 1987A</td></tr><tr><td>&lt; 1.5</td><td>90</td><td>35 BEACOM</td><td>99</td><td>SKAM Solar ν spectrum</td></tr><tr><td>&lt; 0.03</td><td>90</td><td>36 RAFFELT</td><td>99</td><td>ASTR Red giant luminosity</td></tr><tr><td>&lt; 4</td><td>90</td><td>37 RAFFELT</td><td>99</td><td>ASTR Solar cooling</td></tr><tr><td>&lt;44000</td><td>90</td><td>ABREU</td><td>97J</td><td>DLPH e<sup>+</sup> e<sup>-</sup> → ννγ at LEP</td></tr><tr><td>&lt;33000</td><td>90</td><td>38 ACCIARRI</td><td>97Q</td><td>L3 e<sup>+</sup> e<sup>-</sup> → ννγ at LEP</td></tr><tr><td>&lt; 0.62</td><td>90</td><td>39 ELMFORS</td><td>97</td><td>COSM Depolarization in early universe plasma</td></tr><tr><td>&lt;27000</td><td>95</td><td>40 ESCRIBANO</td><td>97</td><td>RVUE Γ(Z → νν) at LEP</td></tr><tr><td>&lt; 30</td><td>90</td><td>VILAIN</td><td>95B</td><td>CHM2 ν<sub>μ</sub>e → ν<sub>μ</sub>e</td></tr><tr><td>&lt;55000</td><td>90</td><td>GOULD</td><td>94</td><td>RVUE e<sup>+</sup> e<sup>-</sup> → ννγ at LEP</td></tr><tr><td>&lt; 1.9</td><td>95</td><td>41 DERBIN</td><td>93</td><td>CNTR Reactor ν<sub>e</sub> → ν<sub>e</sub></td></tr></tbody></table>
<table><tr><td>&lt; 5400</td><td>90</td><td>42 COOPER...</td><td>92</td><td>BEBC</td><td>ν<sub>e</sub> e<sup>−</sup> → ν<sub>τ</sub> e<sup>−</sup></td></tr><tr><td>&lt; 2.4</td><td>90</td><td>43 VIDYAKIN</td><td>92</td><td>CNTR</td><td>Reactor ν̄e → ν̄e</td></tr><tr><td>&lt;56000</td><td>90</td><td>44 DORENBOS...</td><td>91</td><td>RVUE</td><td>e<sup>+</sup> e<sup>−</sup> → ννγ</td></tr><tr><td>&lt; 100</td><td>95</td><td>45 KRAKAUER</td><td>91</td><td>CHRM</td><td>ν<sub>μ</sub> e<sup>−</sup> → ν<sub>μ</sub> e</td></tr><tr><td>&lt; 8.5</td><td>90</td><td>46 AHRENS</td><td>90</td><td>CNTR</td><td>ν<sub>μ</sub> e<sup>−</sup> → ν<sub>μ</sub> e</td></tr><tr><td>&lt; 10.8</td><td>90</td><td>47 KRAKAUER</td><td>90</td><td>CNTR</td><td>LAMPF νe → νe</td></tr><tr><td>&lt; 7.4</td><td>90</td><td>48 FUKUGITA</td><td>88</td><td>CNTR</td><td>LAMPF (ν<sub>μ</sub>, ν̄<sub>μ</sub>) e elast.</td></tr><tr><td>&lt; 0.02</td><td></td><td>49 GROTCH</td><td>88</td><td>RVUE</td><td>e<sup>+</sup> e<sup>−</sup> → ννγ</td></tr><tr><td>&lt; 0.1</td><td></td><td>50 NUSSINOV</td><td>89B</td><td>ASTR</td><td>Cooling helium stars</td></tr><tr><td>&lt;40000</td><td>90</td><td>51 SUTHERLAND</td><td>76</td><td>ASTR</td><td>Cosmic EM backgrounds</td></tr><tr><td>≤ .3</td><td></td><td>52 KIM</td><td>74</td><td>RVUE</td><td>ν̄<sub>μ</sub> e → ν̄<sub>μ</sub> e</td></tr><tr><td>&lt; 0.11</td><td></td><td>53 BERNSTEIN</td><td>63</td><td>ASTR</td><td>Solar cooling</td></tr><tr><td>&lt; 0.0006</td><td></td><td>54 COWAN</td><td>57</td><td>CNTR</td><td>Reactor ν̄</td></tr><tr><td>&lt; 0.1-0.2</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>&lt; 0.85</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>&lt; 0.6</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>&lt; 81</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>&lt; 1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>&lt; 14</td><td></td><td></td><td></td><td></td><td></td></tr></table>

1 APRILE 22b use data collected with the XENONnT dark matter detector to place a limit on an enhanced magnetic moment of solar neutrinos. Supersedes APRILE 20.

2 BEDA 13 report ν̄e e scattering results, using the Kalinin Nuclear Power Plant and a shielded Ge detector. The recoil electron spectrum is analyzed between 2.5 and 55 keV. Supersedes BEDA 07. Supersedes BEDA 10. This is the most stringent limit on the magnetic moment of reactor ν̄e.

3 AUERBACH 01 limit is based on the LSND νe and νμ electron scattering measurements. The limit is slightly more stringent than KRAKAUER 90.

4 SCHWIENHORST 01 quote an experimental sensitivity of $4.9 \times 10^{-7}$.

5 AALBERS 23A utilize the first 60 days of data collected by the LZ dark matter search to place a limit on the magnetic moment of solar neutrinos. Low energy recoil events are utilized. This LZ-collaboration analysis supersedes that of the external authors in ATZORI-CORONA 23 because of a more complete treatment of experiment uncertainties.

6 ATZORI-CORONA 23 use LUX-ZEPLIN dark matter search data published by AALBERS 23 to place a limit on an enhanced magnetic moment of solar neutrinos.

7 BONET 22A use data collected by four low-threshold Ge detectors, placed 17.1 m from one of the cores of the nuclear reactors at Borfdorf to derive this limit. A spectral analysis is performed on reactor on and off data.

8 COLOMA 22 present a re-analysis of data taken by the COHERENT and Dresden-II experiments. The combination of both experiments is used to place a limit on the magnetic moment of electron-type neutrinos. The presented value is one-sided limit as recommended by the authors; the two-sided limit is $< 3.2 \times 10^{-10} \mu_B$ at 90% C.L. Results based on FeF and YBE quenching models are reported in the paper. The authors are not part of either collaboration.

9 ABE 20e observed an excess of low-energy events in the XMASS detector, which could be interpreted as a signal produced by a neutrino magnetic moment with this magnitude.

10 APRILE 20 observed an excess of low-energy events in the XENON1T detector, which could be interpreted as a signal produced by a neutrino magnetic moment with this magnitude.

11 CAPOZZI 20 obtains a limit on the neutrino dipole moment from the brightness of the tip of the red-giant branch in ω Centauri. A similar limit of $\mu_ν < 1.5 \times 10^{-12} \mu_B$ is obtained in NGC 4258.

12 KHAN 20 performed a constrained spectral fit analysis of the excess observed in the electron recoil energy spectrum by the XENON1T experiment. This range of the $\mu_B$ values is one of the possible interpretations of these excess events. For the individual flavor constraints at 90% C.L. see the original reference.

13 AGOSTINI 17A obtained this limit using the shape of the recoil electron energy spectrum from the Borexino Phase-II 1291.5 live days of solar neutrino data and the constraints on the sum of the solar neutrino fluxes from the radiochemical gallium experiments SAGE, Gallex, and GNO. Without radiochemical constraints, the 90% C.L. limit of $< 4.0 \times 10^{-11} \mu_B$ is obtained.

14 ARCEO-DIAZ 15 constrains the neutrino magnetic moment from observation of the tip of the red giant branch in the globular cluster ω-Centauri.

15 CORSICO 14 constrains the neutrino magnetic moment from observations of white drarf pulsations.

16 MILLER-BERTOLAMI 14B constrains the neutrino magnetic moment from observations of the white dwarf luminosity function of the Galactic disk.

17 VIAUX 13A constrains the neutrino magnetic moment from observations of the globular cluster M5.

18 BEDA 10 report ν̄e e scattering results, using the Kalinin Nuclear Power Plant and a shielded Ge detector. The recoil electron spectrum is analyzed between 2.9 and 45 keV. Supersedes BEDA 07. Supersedes BEDA 13.

19 DENIZ 10 observe reactor ν̄e e scattering with recoil kinetic energies 3–8 MeV using CsI(Tl) detectors. The observed rate and spectral shape are consistent with the Standard Model prediction, leading to the reported constraint on ν̄e magnetic moment.

20 KUZNETSOV 09 obtain a limit on the flavor averaged magnetic moment of Dirac neutrinos from the time averaged neutrino signal of SN1987A. Improves and supersedes the analysis of BARBIERI 88 and AYALA 99.

21 ARPESELLA 08A obtained this limit using the shape of the recoil electron energy spectrum from the Borexino 192 live days of solar neutrino data.

22 BEDA 07 performed search for electromagnetic ν̄e e scattering at Kalininskaya nuclear reactor. A Ge detector with active and passive shield was used and the electron recoil spectrum between 3.0 and 61.3 keV analyzed. Supersedes by BEDA 10.

23 WONG 07 performed search for non-standard ν̄e e scattering at the Kuo-Sheng nuclear reactor. Ge detector equipped with active anti-Compton shield is used. Most stringent laboratory limit on magnetic moment of reactor ν̄e. Supersedes LI 03B.

24 DARAKTCHIEVA 05 present the final analysis of the search for non-standard ν̄e e scattering component at Bugey nuclear reactor. Full kinematical event reconstruction of both the kinetic energy above 700 keV and scattering angle of the recoil electron, by use of TPC. Most stringent laboratory limit on magnetic moment. Supersedes DARAKTCHIEVA 03.

25 XIN 05 evaluated the νe flux at the Kuo-Sheng nuclear reactor and searched for non-standard νe e scattering. Ge detector equipped with active anti-Compton shield was used. This laboratory limit on magnetic moment is considerably less stringent than the limits for reactor ν̄e, but is specific to νe.

26 GRIFOLS 04 obtained this bound using the SNO data of the solar $^8$B neutrino flux measured with deuteron breakup. This bound applies to $\mu_{eff} = (\mu_{21}^2 + \mu_{22}^2 + \mu_{23}^2)^{1/2}$.

27 LIU 04 obtained this limit using the shape of the recoil electron energy spectrum from the Super-Kamiokande-I 1496 days of solar neutrino data. Neutrinos are assumed to have only diagonal magnetic moments, $\mu_{\nu 1} = \mu_{\nu 2}$. This limit corresponds to the oscillation parameters in the vacuum oscillation region.

28 LIU 04 obtained this limit using the shape of the recoil electron energy spectrum from the Super-Kamiokande-I 1496 live-day solar neutrino data, by limiting the oscillation parameter region in the LMA region allowed by solar neutrino experiments plus KamLAND. $\mu_{\nu 1} = \mu_{\nu 2}$ is assumed. In the LMA region, the same limit would be obtained even if neutrinos have off-diagonal magnetic moments.

29 BACK 03B obtained this bound from the results of background measurements with Counting Test Facility (the prototype of the Borexino detector). Standard Solar Model flux was assumed. This $\mu_ν$ can be different from the reactor $\mu_ν$ in certain oscillation scenarios (see BEACOM 99).

30 DARAKTCHIEVA 03 searched for non-standard ν̄e e scattering component at Bugey nuclear reactor. Full kinematical event reconstruction by use of TPC. Supersedes by DARAKTCHIEVA 05.

31 LI 03B used Ge detector in active shield near nuclear reactor to test for nonstandard ν̄e e scattering.

32 GRIMUS 02 obtain stringent bounds on all Majorana neutrino transition moments from a simultaneous fit of LMA-MSW oscillation parameters and transition moments to global solar neutrino data + reactor data. Using only solar neutrino data, a 90% CL bound of $6.3 \times 10^{-10} \mu_B$ is obtained.

33 TANIMOTO 00 combined $e^+ e^- \to \nu\bar{\nu}$ data from VENUS, TOPAZ, and AMY.

34 AYALA 99 improves the limit of BARBIERI 88.

35 BEACOM 99 obtain the limit using the shape, but not the absolute magnitude which is affected by oscillations, of the solar neutrino spectrum obtained by Superkamiokande (825 days). This $\mu_ν$ can be different from the reactor $\mu_ν$ in certain oscillation scenarios.

36 RAFFELT 99 is an update of RAFFELT 90. This limit applies to all neutrino flavors which are light enough ($< 5$ keV) to be emitted from globular-cluster red giants. This limit pertains equally to electric dipole moments and magnetic transition moments, and it applies to both Dirac and Majorana neutrinos.

37 RAFFELT 99 is essentially an update of BERNSTEIN 63, but is derived from the helioseismological limit on a new energy-loss channel of the Sun. This limit applies to all neutrino flavors which are light enough ($< 1$ keV) to be emitted from the Sun. This limit pertains equally to electric dipole and magnetic transition moments, and it applies to both Dirac and Majorana neutrinos.

38 ACCIARRI 97q result applies to both direct and transition magnetic moments and for $q^2=0$.

39 ELMFORS 97 calculate the rate of depolarization in a plasma for neutrinos with a magnetic moment and use the constraints from a big-bang nucleosynthesis on additional degrees of freedom.

40 Applies to absolute value of magnetic moment.

41 DERBIN 93 determine the cross section for 0.6–2.0 MeV electron energy as ($1.28 \pm 0.63) \times \sigma_{\text{weak}}$. However, the (reactor on – reactor off)/(reactor off) is only $\sim 1/100$.

42 COOPER-SARKAR 92 assume $f_{D_S}/r_π = 2$ and $D_S$, $\bar{D}_S$ production cross section = $2.6 \ \mu b$ to calculate $\nu$ flux.

43 VIDYAKIN 92 limit is from a $e\bar{e}$ elastic scattering experiment. No experimental details are given except for the cross section from which this limit is derived. Signal/noise was $1/10$. The limit uses $\sin^2\theta_W = 0.23$ as input.

44 DORENBOCH 91 corrects an incorrect statement in DORENBOCH 89 that the $\nu$ magnetic moment is $< 1 \times 10^{-9}$ at the 95%CL. DORENBOCH 89 measures both $\nu_\mu e$ and $\bar{\nu}e$ elastic scattering and assume $\mu(\nu) = \mu(\bar{\nu})$.

45 KRAKAUER 90 experiment fully reported in ALLEN 93.

46 RAFFELT 90 limit applies for a diagonal magnetic moment of a Dirac neutrino, or for a transition magnetic moment of a Majorana neutrino. In the latter case, the same analysis gives $< 1.4 \times 10^{-12}$. Limit at 95%CL obtained from $\delta M_C$.

47 Significant dependence on details of stellar models.

48 FUKUGITA 88 find magnetic dipole moments of any two neutrino species are bounded by $\mu < 10^{-16} [10^{-9} G/B_0]$ where $B_0$ is the present-day intergalactic field strength.

49 GROTCH 88 combined data from MAC, ASP, CELLO, and Mark J.

50 For $m_\nu = 8-200$ eV. NUSSINOV 87 examines transition magnetic moments for $\nu_\mu \to \nu_e$ and obtain $< 3 \times 10^{-15}$ for $m_\nu > 16$ eV and $< 6 \times 10^{-14}$ for $m_\nu > 4$ eV.

51 We obtain above limit from SUTHERLAND 76 using their limit $f < 1/3$.

52 KIM 74 is a theoretical analysis of $\bar{\nu}_\mu$ reaction data.
NEUTRINO CHARGE RADIUS SQUARED

We report limits on the so-called neutrino charge radius squared. While the straight-forward definition of a neutrino charge radius has been proven to be gauge-dependent and, hence, unphysical (LEE 77c), there have been recent attempts to define a physically observable neutrino charge radius (BERNABEU 00, BERNABEU 02). The issue is still controversial (FUJIKAWA 03, BERNABEU 03). A more general interpretation of the experimental results is that they are limits on certain nonstandard contributions to neutrino scattering.

<table><thead><tr><th>VALUE (10<sup>-32</sup> cm<sup>2</sup>)</th><th>CL<sub>90</sub></th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>- 2.1 to 3.3</td><td>90</td><td>1 DENIZ</td><td>10</td><td>REactor ν<sub>e</sub> e</td></tr></tbody></table>

¹DENIZ 10 observe reactor $\bar{\nu}_e$ e scattering with recoil kinetic energies 3–8 MeV using CsI(Tl) detectors. The observed rate and spectral shape are consistent with the Standard Model prediction, leading to the reported constraint on $\bar{\nu}_e$ charge radius.

²CADEDDU 18 use the data of the COHERENT experiment, AKIMOV 18. The limit is $\langle r_\nu^2 \rangle$ for $\nu_\mu$ obtained from the time-dependent data. Weaker limits were obtained for charge radii of $\nu_e$ and for transition charge radii. The published value was divided by 2 to conform to the convention of this table.

³Based on analysis of CCFR 98 results. Limit is on $\langle r_\nu^2 \rangle + \langle r_\tau^2 \rangle$ for the CHARM II and E734 at BNL results are reanalyzed, and weaker bounds on the charge radius squared than previously published are obtained. The NuTeV result is discussed; when tentatively interpreted as $\nu_\mu$ charge radius it implies $\langle r_\nu^2 \rangle + \langle r_\tau^2 \rangle = (4.20 \pm 1.64) \times 10^{-33}$ cm<sup>2</sup>.

⁴Results of LEP-2 are interpreted as limits on the axial-vector charge radius squared of a Majorana $\nu_\mu$. Slightly weaker limits for both vector and axial-vector charge radius squared are obtained for the Dirac case, and somewhat weaker limits are obtained from the analysis of lower energy data (LEP-1.5 and TRISTAN).

⁵AUERBACH 01 measure $\nu_e$ e elastic scattering with LSND detector. The cross section agrees with the Standard Model expectation, including the charge and neutral current interference. The 90% CL applies to the range shown.

⁶VIDYAKIN 92 limit is from a $e\bar{\nu}$ elastic scattering experiment. No experimental details are given except for the cross section from which this limit is derived. Signal/noise was $1/10$. The limit uses $\sin^2\theta_W = 0.23$ as input.

⁷Result is obtained from reanalysis given in ALLEN 91, followed by our reduction to obtain 1 σ errors.

⁸GRIFOLS 89B sets a limit of $\langle r_\nu^2 \rangle < 0.2 \times 10^{-32}$ cm<sup>2</sup> for right-handed neutrinos.

REFERENCES FOR Neutrino Properties

<table><tbody><tr><td>AALBERS</td><td>23</td><td>PRL 131 041002</td><td>J. Aalbers et al.</td><td>(LZ Collab.)</td></tr><tr><td>AALBERS</td><td>23A</td><td>PRL D108 072006</td><td>J. Aalbers et al.</td><td>(LZ Collab.)</td></tr><tr><td>ATZOR-LOR...</td><td>23</td><td>PRL D107 053001</td><td>M. Atzori Corona et al.</td><td>(CAGL, CAGLI)</td></tr><tr><td>ESFAHANI</td><td>23</td><td>PRL 131 102502</td><td>A.A. Esfahani et al.</td><td>(Project 8 Collab.)</td></tr><tr><td>IVANEZ-BAL...</td><td>23</td><td>PRL B847 138252</td><td>P. Ivanez-Ballesteros, M.C. Volpe</td><td>(CNRS, UPCIT)</td></tr><tr><td>AKER</td><td>22</td><td>NATP 18 160</td><td>M. Aker et al.</td><td>(KATRIN Collab.)</td></tr><tr><td>APRILE</td><td>22B</td><td>PRL 129 161805</td><td>E. Aprile et al.</td><td>(XENONNT Collab.)</td></tr><tr><td>BRIDEN</td><td>22A</td><td>EPJ C82 813</td><td>H. Bonet et al.</td><td>(CONUS Collab.)</td></tr><tr><td>BRIEDEN</td><td>22</td><td>JCAP 2208 024</td><td>S. Brieden, Hector Gil-Marin, Licia Verde</td><td></td></tr><tr><td>COLOMA</td><td>22</td><td>JHEP 2205 037</td><td>P. Coloma et al.</td><td>(IFT, OSU, STON, ICREA+)</td></tr><tr><td>KUMAR</td><td>22</td><td>JHEP 2209 060</td><td>S. Kumar, R. Nunes, P.Yadav</td><td></td></tr><tr><td>TANSERI</td><td>22</td><td>JHEAP 36 1</td><td>I. Tanseri et al.</td><td></td></tr><tr><td>ABBOTT</td><td>21A</td><td>PRL D105 023520</td><td>T.C.M. Abbott et al.</td><td>(DES Collab.)</td></tr><tr><td>ALAM</td><td>21</td><td>PRL D103 083533</td><td>S. Alam et al.</td><td>(eBOS Collab.)</td></tr><tr><td>DI-VALENT...</td><td>21</td><td>JCAP 104 083504</td><td>E. Di Valentino, S. Gariazzo, O. Mena</td><td>(DURH+)</td></tr><tr><td>GARNY</td><td>21</td><td>JCAP 2103 049</td><td>P. Garny et al.</td><td></td></tr><tr><td>STOCKER</td><td>21</td><td>PRL D103 123508</td><td>P. Stocker et al.</td><td>(GAMBIT Collab.)</td></tr><tr><td>ABE</td><td>20E</td><td>PL B809 135741</td><td>K. Abe et al.</td><td>(XMASS Collab.)</td></tr><tr><td>AGHANIM</td><td>20</td><td>PL B841 A6</td><td>N. Aghanim et al.</td><td>(Planca Collab.)</td></tr><tr><td>APRILE</td><td>20</td><td>PRL D102 072004</td><td>E. Aprile et al.</td><td>(XENON Collab.)</td></tr><tr><td>CAPOZZI</td><td>20</td><td>PRL D102 083007</td><td>F. Capozzi, G. Raffelt</td><td>(MPIM)</td></tr><tr><td>CHOUDHURY</td><td>20</td><td>JCAP 2007 037</td><td>S.R. Choudhury, S. Hannestad</td><td>(IITB, AARH)</td></tr><tr><td>IVANOV</td><td>20</td><td>PRL D101 083504</td><td>M.M. Ivanov, M. Simonovic, M. Zaldarriaga</td><td>(NYU+,)</td></tr><tr><td>KHAN</td><td>20</td><td>PL B809 135782</td><td>A.N. Khan</td><td>(MPK)</td></tr><tr><td>PALANQUE...</td><td>20</td><td>JCAP 2004 0338</td><td>N. Palanque-Delabrouille et al.</td><td>(SACL+)</td></tr><tr><td>AHARMIM</td><td>19</td><td>PRL D99 032013</td><td>B. Aharmim et al.</td><td>(SNO Collab.)</td></tr><tr><td>AKER</td><td>19</td><td>PRL 123 221802</td><td>M. Aker et al.</td><td>(KATRIN Collab.)</td></tr><tr><td>Also</td><td>19</td><td>PRL D104 012005</td><td>M. Aker et al.</td><td>(KATRIN Collab.)</td></tr><tr><td>ESCUDERO</td><td>19</td><td>PRL D100 103531</td><td>M. Escudero, M. Fairbairn</td><td></td></tr><tr><td>LOUREIRO</td><td>19</td><td>PRL 123 083101</td><td>A. Loureiro et al.</td><td></td></tr><tr><td>UPADHYE</td><td>19</td><td>JCAP 1905 041</td><td>A. Upadhye</td><td></td></tr><tr><td>AKIMOV</td><td>18</td><td>arXiv:1804.09459</td><td>D. Akimov et al.</td><td></td></tr><tr><td>CADEDDU</td><td>18</td><td>PRL D98 113010</td><td>M. Cadeddu et al.</td><td>(COHERENT Collab.)</td></tr><tr><td>Also</td><td>18</td><td>PRL D101 059902 (errat.)</td><td>M. Cadeddu et al.</td><td></td></tr><tr><td>CHOUDHURY</td><td>18</td><td>JCAP 1809 017</td><td>S.R. Choudhury, S. Choubey</td><td></td></tr><tr><td>AGOSTINI</td><td>17A</td><td>PRL D96 091103</td><td>M. Agostini et al.</td><td>(Borexino Collab.)</td></tr><tr><td>IMPSON</td><td>17</td><td>JCAP 1706 029</td><td>F. Simpson et al.</td><td></td></tr><tr><td>VAGNOZZI</td><td>17</td><td>JCAP 199 123503</td><td>S. Vagnozzi et al.</td><td></td></tr><tr><td>YECHE</td><td>17</td><td>JCAP 1706 047</td><td>C. Yeche et al.</td><td></td></tr><tr><td>DIVA-VALLE</td><td>16</td><td>JEP 476 24</td><td>E. Della Valle et al.</td><td>(PVLAS Collab.)</td></tr><tr><td>DIVALENTINO</td><td>16</td><td>PRL D93 083527</td><td>E. Di Valentino et al.</td><td></td></tr></tbody></table>

<table><tbody><tr><td>HUANG</td><td>16</td><td>EJ P C76 489</td><td>Q-G. Huang, K. Wang, S. Wang</td></tr><tr><td>ARCEO-DIAZ</td><td>15</td><td>ASP 70 1</td><td>S. Arceo-Diaz et al.</td></tr><tr><td>LOKHOV</td><td>15</td><td>PPN 46 347</td><td>A.V. Lokhov, F.V. Tkachov</td></tr></tbody></table>

<table><tbody><tr><td>TRANSLATED FROM FECAY</td><td>46 3.</td></tr><tr><td>PALANQUE...</td><td>15A</td><td>JCAP 1511 011</td><td>N. Palanque-Delabrouille et al.</td></tr><tr><td>ROSSI</td><td>15</td><td>PR D92 063505</td><td>G. Rossi et al.</td></tr><tr><td>ADE</td><td>14</td><td>AA 571 A16</td><td>P.A.R. Ade et al.</td></tr><tr><td>BATTYE</td><td>14</td><td>PRL 112 051303</td><td>R.A. Battye, A. Moss</td></tr><tr><td>BEUTLER</td><td>14</td><td>MNRAS 444 3501</td><td>F. Beutler et al.</td></tr><tr><td>CHEN</td><td>14</td><td>PRL D90 011301</td><td>J.-W. Chen et al.</td></tr><tr><td>CORSICO</td><td>14</td><td>JCAP 1408 054</td><td>A.H.S. Costanzi et al.</td></tr><tr><td>COSTANZI</td><td>14</td><td>JCAP 1410 081</td><td>Z. Hou et al.</td></tr><tr><td>HOU</td><td>14</td><td>APJ 782 74</td><td>B. Leistedt, H.V. Peiris, L. Verde</td></tr><tr><td>LEISTEDT</td><td>14</td><td>PRL 113 041301</td><td>M.M. Miller Bertolami</td></tr><tr><td>MILLER-BER...</td><td>14B</td><td>AA 562 A123</td><td>(MPIG, LAPL)</td></tr><tr><td>RIEMER-SOR...</td><td>14</td><td>PR D89 103505</td><td>S. Riemer-Sorensen, D. Parkinson, T.M. Davis</td></tr><tr><td>STUDENIKIN</td><td>14</td><td>EPL 107 21001</td><td>A.I. Studenkin</td></tr><tr><td>BEDA</td><td>13A</td><td>PRL 111 13301</td><td>A.G. Beda et al.</td></tr><tr><td>VIAUX</td><td>13A</td><td>JCAP 1207 053</td><td>N. Viaux et al.</td></tr><tr><td>MORESCO</td><td>12</td><td>JCAP 1206 010</td><td>M. Moreco et al.</td></tr><tr><td>XIA</td><td>12</td><td>PR D84 112003</td><td>J.-Q. Xia et al.</td></tr><tr><td>ASEEV</td><td>11</td><td>PR D81 004</td><td>V.N. Aseev et al.</td></tr><tr><td>CECCHINI</td><td>11</td><td>ASP 34 486</td><td>S. Cecchini et al.</td></tr><tr><td>SATTO</td><td>11</td><td>PR D83 043529</td><td>S. Saito, M. Takada, A. Taruya</td></tr><tr><td>BEDA</td><td>10</td><td>PPNL 7 406</td><td>A.G. Beda et al.</td></tr><tr><td>DENIZ</td><td>10</td><td>PR D81 072001</td><td>M. Deniz et al.</td></tr><tr><td>HANNESTAD</td><td>10</td><td>JCAP 1008 001</td><td>S. Hannestad et al.</td></tr><tr><td>PAGLIAROLI</td><td>10</td><td>ASP 33 287</td><td>G. Pagliarol, F. Rossi-Torres, E. Vissani</td></tr><tr><td>SEDIIGUCHI</td><td>10</td><td>JCAP 1003 015</td><td>T. Sekiguchi et al.</td></tr><tr><td>THOMAS</td><td>10</td><td>PRL 105 031301</td><td>S.A. Thomas, F.B. Abdalla, O. Lahav</td></tr><tr><td>ICHIKI</td><td>09</td><td>PR D79 023520</td><td>K. Ichiki, M. Takada, T. Takahashi</td></tr><tr><td>KOMATSU</td><td>09</td><td>APJS 180 330</td><td>E. Komatsu et al.</td></tr><tr><td>KUZNETSOV</td><td>09</td><td>JMP A24 5977</td><td>A. Kuznetsov, N.V. Mikheev, A.A. Okrugin</td></tr><tr><td>TERENO</td><td>09</td><td>AA 500 657</td><td>I. Tereno et al.</td></tr><tr><td>VIKHLININ</td><td>09</td><td>APJ 692 1060</td><td>A. Vikhlinin et al.</td></tr><tr><td>ARPESELLA</td><td>08A</td><td>PRL 101 091302</td><td>C. Arpesella et al.</td></tr><tr><td>BERNARDIS</td><td>08A</td><td>PRL 078 083535</td><td>F. De Bernardis et al.</td></tr><tr><td>BEDA</td><td>07</td><td>PAN 70 1873</td><td>A.G. Beda et al.</td></tr></tbody></table>

<table><tbody><tr><td>TRANSLATED FROM YAF</td><td>70 1925.</td></tr><tr><td>FOLGI</td><td>07</td><td>PR D75 053001</td><td>G.L. Fogli et al.</td></tr><tr><td>GNINENKO</td><td>07</td><td>PR D75 075014</td><td>S.N. Gninenko, N.V. Krasnikov, A. Rubbia</td></tr><tr><td>KRISTIANSEN</td><td>07</td><td>PR D75 083510</td><td>J. Kristiansen, O. Elgaroy, H. Dahle</td></tr><tr><td>MIRIZZI</td><td>07</td><td>PR D76 053007</td><td>A. Mirizzi, D. Montanino, P.D. Serpico</td></tr><tr><td>SPERGEL</td><td>07</td><td>APJS 170 377</td><td>D.N. Spergel et al.</td></tr><tr><td>WONG</td><td>07</td><td>PR D75 012001</td><td>H.T. Wong et al.</td></tr><tr><td>ZUNCKEL</td><td>07</td><td>JCAP 0708 004</td><td>C. Zunckel et al. Ferreira</td></tr><tr><td>CIRELLI</td><td>06</td><td>JCAP 0612 013</td><td>M. Cirelli et al.</td></tr><tr><td>FUKUGITA</td><td>06</td><td>PR D74 027302</td><td>M. Fukugita et al.</td></tr><tr><td>GOOBAR</td><td>06</td><td>JCAP 0606 019</td><td>A. Goobar et al.</td></tr><tr><td>HANNESTAD</td><td>06</td><td>PR D71 0611 016</td><td>S. Hannestad, G. Raffelt</td></tr><tr><td>KRISTIANSEN</td><td>06</td><td>PR D74 123005</td><td>J. Kristiansen, O. Elgaroy, H. Eriksen</td></tr><tr><td>SANCHEZ</td><td>06</td><td>MNRAS 366 189</td><td>A.G. Sanchez et al.</td></tr><tr><td>SELJAK</td><td>06</td><td>JCAP 0610 014</td><td>U. Seljak, A. Slosar, P. McDonald</td></tr><tr><td>CAPRINI</td><td>06</td><td>PR D75 0502 006</td><td>C. Caprini, P.G. Ferreira</td></tr><tr><td>DARAKCTCH...</td><td>05</td><td>PL B615 153</td><td>Z. Darakctchieva et al.</td></tr><tr><td>ICHIKAWA</td><td>05</td><td>PR D71 043001</td><td>K. Ichikawa, M. Fukugita, M. Kawasaki</td></tr><tr><td>KRAUS</td><td>05</td><td>PR EJ P 440 447</td><td>Ch. Kraus et al.</td></tr><tr><td>XIN</td><td>05</td><td>PR D72 012006</td><td>B. Xin et al.</td></tr><tr><td>AHARMIM</td><td>04</td><td>PR D70 093014</td><td>B. Aharmim et al.</td></tr><tr><td>BARGER</td><td>04</td><td>PL B595 55</td><td>V. Barger, D. Marfatia, A. Tregre</td></tr><tr><td>CECCHINI</td><td>04</td><td>ASP 21 183</td><td>S. Cecchini et al.</td></tr><tr><td>CROTTY</td><td>04</td><td>PR D69 123007</td><td>P. Crotty, J. Legourges, S. Pastor</td></tr><tr><td>EUGUCHI</td><td>04</td><td>PL P92 071301</td><td>K. Eguchi et al.</td></tr><tr><td>GRIFOLS</td><td>04</td><td>JCAP 1587 184</td><td>J.A. Grifols, E. Masso, S. Mohanty (BARC, AHMED)</td></tr><tr><td>LIU</td><td>04</td><td>PL P93 021802</td><td>D.W. Liu et al.</td></tr><tr><td>ARNABOLDI</td><td>03A</td><td>PL P91 161802</td><td>C. Arnaboldi et al.</td></tr><tr><td>BACK</td><td>03B</td><td>PL B563 35</td><td>H.O. Back et al.</td></tr><tr><td>BACKYOPA...</td><td>03B</td><td>PL B555 33</td><td>A. Bandyopadhyay, S. Choubey, S. Goswami (SAHA+)</td></tr><tr><td>BERNABEU</td><td>03</td><td>hep-ph/03030220</td><td>J. Bernabeu, J. Papavassiliou, J. Vidal</td></tr><tr><td>DARAKCTCH...</td><td>03</td><td>PL B564 190</td><td>Z. Darakctchieva et al.</td></tr><tr><td>FUJIKAWA</td><td>03</td><td>hep-ph/0303188</td><td>K. Fujikawa, R. Shrock</td></tr><tr><td>HIRSCH</td><td>03</td><td>PR D67 033005</td><td>M. Hirsch, E. Nardi, D. Restrepo</td></tr><tr><td>LI</td><td>03B</td><td>APJS 148 175</td><td>H.B. Li et al.</td></tr><tr><td>SPERGEL</td><td>03</td><td>PR P89 101802</td><td>D.N. Spergel et al.</td></tr><tr><td>BERNABEU</td><td>02</td><td>PR P89 299902 (errat.)</td><td>J. Bernabeu, J. Papavassiliou, J. Vidal</td></tr><tr><td>Also</td><td>02B</td><td>JETPL 76 409</td><td>J. Bernabeu, J. Papavassiliou, J. Vidal</td></tr><tr><td>DERBIN</td><td>02B</td><td>JETPL 76 409</td><td>A.V. Derbin, O.Ju. Smirnov</td></tr></tbody></table>

<table><tbody><tr><td>GRIMUS</td><td>02</td><td>NP B648 376</td><td>W. Grimus et al.</td></tr><tr><td>JOSHIPURA</td><td>02B</td><td>PR D66 113008</td><td>A.S. Joshipura, E. Masso, S. Mohanty</td></tr><tr><td>LEWIS</td><td>02</td><td>PR D66 103511</td><td>A. Lewis, S. Bride</td></tr><tr><td>LOREDO</td><td>02</td><td>PR D65 063002</td><td>T.J. Loredo, D.Q. Lamb</td></tr><tr><td>WANG</td><td>02</td><td>PR D65 123001</td><td>X. Wang, M. Tegmark, M. Zaldarriaga</td></tr><tr><td>AUERBACH</td><td>01</td><td>PR D63 112001</td><td>L.B. Auerbach et al.</td></tr><tr><td>SCHWIENINO...</td><td>01</td><td>PR D61 0213 23</td><td>M. Schwenhorst et al.</td></tr><tr><td>ATHANAS</td><td>00</td><td>PR D61 052002</td><td>M. Athanas et al.</td></tr><tr><td>BERNABEU</td><td>00</td><td>PR D62 113012</td><td>J. Bernabeu et al.</td></tr><tr><td>FUKUGITA</td><td>00</td><td>PR D64 1082</td><td>M. Fukugita, G.C. D. Liu, N. Sugiyama</td></tr><tr><td>TANIMOTO</td><td>00</td><td>PL B478 1</td><td>N. Tanimoto et al.</td></tr><tr><td>ALIA</td><td>99</td><td>A. Ayala, J.C. O’Dilvo, M. Torres</td><td>A. Ayala, J.C. O’Dilvo, M. Torres</td></tr><tr><td>BEACOM</td><td>99</td><td>J.F. Beacom, P. Vogel</td><td>R.A.C. Croft, W. Hü, R. Dave</td></tr><tr><td>CROFT</td><td>99</td><td>PL P83 5222</td><td>A.D. Dolgov et al.</td></tr><tr><td>DOLGOV</td><td>99</td><td>PL P83 1092</td><td>V.M. Lobashev et al.</td></tr><tr><td>LOBASHEV</td><td>99</td><td>PL B460 227</td><td>G.G. Raffelt</td></tr><tr><td>RAFFEL</td><td>99</td><td>PRPL 320 319</td><td>Ch. Weinheimer et al.</td></tr><tr><td>WEINHEIMER</td><td>99</td><td>PL B460 219</td><td>K. Ackerman et al.</td></tr><tr><td>ACKERSTAFF</td><td>98T</td><td>EPJ C5 229</td><td>R. Ammar et al.</td></tr><tr><td>AMMAR</td><td>98</td><td>PL B431 209</td><td>R. Ambar et al.</td></tr><tr><td>BARATE</td><td>98F</td><td>EPJ C2 395</td><td>R. Barate et al.</td></tr><tr><td>BILLER</td><td>98</td><td>PL P80 2992</td><td>S.D. Biller et al.</td></tr><tr><td>FELDMAN</td><td>98</td><td>PL D57 3873</td><td>G.J. Feldman, R.D. Cousins</td></tr><tr><td>LENZ</td><td>98</td><td>PL B416 50</td><td>S. Lenz et al.</td></tr><tr><td>ABREU</td><td>97J</td><td>ZPHY C74 577</td><td>P. Abreu et al.</td></tr><tr><td>ACCIARRI</td><td>97Q</td><td>PL B412 201</td><td>M. Acciarri et al.</td></tr><tr><td>ANASTASSOV</td><td>97</td><td>PR D55 2559</td><td>A. Anastassov et al.</td></tr><tr><td>Also</td><td>97</td><td>PR D58 119903 (errat.)</td><td>A. Anastassov et al.</td></tr><tr><td>ELMFORS</td><td>97</td><td>NP B503 3</td><td>P. Elmfors et al.</td></tr><tr><td>ESCRIBANO</td><td>97</td><td>PL B395 369</td><td>C. Escribano, E. Masso</td></tr><tr><td>FIELDS</td><td>97</td><td>ASP 6 169</td><td>B.D. Fields, K. Kainulainen, K.A. Olive (NDAM+)</td></tr><tr><td>SWAIN</td><td>97</td><td>PR D55 1</td><td>J. Swain, L. Taylor (NEAS)</td></tr><tr><td>ALEXANDER</td><td>96M</td><td>ZPHY C72 231</td><td>G. Alexander et al.</td></tr><tr><td>ASSAMAGAN</td><td>96</td><td>PR D53 6065</td><td>K.A. Assamagan et al.</td></tr><tr><td>BAI</td><td>96</td><td>PR D53 20</td><td>J.Z. Bai et al.</td></tr><tr><td>BOTTINO</td><td>96</td><td>PR D53 6361</td><td>A. Bottino et al.</td></tr><tr><td>DOLGOV</td><td>96</td><td>PL P383 193</td><td>A.D. Dolgov, S. Pastor, J.W.F. Valle (IFIC, VALE)</td></tr><tr><td>HANNESTAD</td><td>96B</td><td>PL P76 2848</td><td>S. Hannestad, J. Madsen (AARH)</td></tr><tr><td>HANNESTAD</td><td>96B</td><td>PL P77 5148 (errat.)</td><td>S. Hannestad, J. Madsen (AARH)</td></tr><tr><td>HANNESTAD</td><td>96C</td><td>PR D54 7894</td><td>R.J. Sobie, R.K. Keeler, I. Lawson (VICT)</td></tr><tr><td>SOBIE</td><td>96</td><td>ZPHY C70 383</td><td>A.J. Bekelev et al.</td></tr><tr><td>BELEV</td><td>95</td><td>PL B350 263</td><td>(INRM, KIAE)</td></tr><tr><td>BUSKULIC</td><td>95H</td><td>PL B349 585</td><td>D. Buskulic et al.</td></tr></tbody></table>

<footnote>(ALEPH Collab.)</footnote>
## Number of Neutrino Types

The neutrinos referred to in this section are those of the Standard SU(2)×U(1) Electroweak Model possibly extended to allow nonzero neutrino masses. Light neutrinos are those with $m < m_Z/2$. The limits are on the number of neutrino mass eigenstates, including $\nu_1$, $\nu_2$, and $\nu_3$.

## THE NUMBER OF LIGHT NEUTRINO TYPES FROM COLLIDER EXPERIMENTS

Revised June 2020 by C.-J. Lin (LBNL). Written by D. Karlen (University of Victoria and TRIUMF).

The most precise measurements of the number of light neutrino types, $N_ν$, come from studies of $Z$ production in $e^+e^-$ collisions. The invisible partial width, $Γ_{inv}$, is determined by subtracting the measured visible partial widths, corresponding to $Z$ decays into quarks and charged leptons, from the total $Z$ width. The invisible width is assumed to be due to $N_ν$ light neutrino species each contributing the neutrino partial width $Γ_ν$ as given by the Standard Model. In order to reduce the model dependence, the Standard Model value for the ratio of the neutrino to charged leptonic partial widths, $(Γ_ν/Γ_ℓ)_{SM} = 1.991±0.001$, is used instead of $(Γ_ν)_{SM}$ to determine the number of light neutrino types:

$$N_ν = \frac{Γ_{inv}}{Γ_ℓ} \left( \frac{Γ_ℓ}{Γ_ν} \right)_{SM} . \quad (1)$$

The combined result from the four LEP experiments is $N_ν = 2.984 ± 0.008$ [1]. Recent analyses applied corrections to the LEP result [1] by including the effect of correlated luminosity systematics and also using an improved Bhabha cross section calculation [2,3] to obtain $N_ν = 2.9963 ± 0.0074$.

In the past, when only small samples of $Z$ decays had been recorded by the LEP experiments and by the Mark II at SLC, the uncertainty in $N_ν$ was reduced by using Standard Model fits to the measured hadronic cross sections at several center-of-mass energies near the $Z$ resonance. Since this method is much more dependent on the Standard Model, the approach described above is favored.

Before SLC and LEP, limits on the number of neutrino generations were placed by experiments at lower-energy $e^+e^-$ colliders by measuring the cross section of the process $e^+e^- → ννγ$. The ASP, CELLO, MAC, MARK J, and VENUS experiments observed a total of 3.9 events above background [4], leading to a 95% CL limit of $N_ν < 4.8$. This process has a much larger cross section at center-of-mass energies near the $Z$ mass and has been measured at LEP by the ALEPH, DELPHI, L3, and OPAL experiments [5]. These experiments have observed several thousand such events, and the combined result is $N_ν = 3.00 ± 0.08$. The same process has also been measured by the LEP experiments at much higher center-of-mass energies, between 130 and 208 GeV, in searches for new physics [6]. Combined with the lower energy data, the result is $N_ν = 2.92 ± 0.05$.

Experiments at $p\bar{p}$ colliders also placed limits on $N_ν$ by determining the total $Z$ width from the observed ratio of

<table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>CHING</td><td>95</td><td>UMP A10 2841</td><td>C.R. Ching et al. (CST, BEIJING, CIAE)</td></tr><tr><td>DOLGOV</td><td>95</td><td>PR D51 4129</td><td>A.D. Dolgov, K. Kainulainen, I.Z. Rothstein (MICH+)</td></tr><tr><td>HIDDEMANN</td><td>95</td><td>JP G21 639</td><td>K.H. Hiddemann, H. Daniel, O. Schwentker (TUM)</td></tr><tr><td>KERNAN</td><td>95</td><td>NP B437 243</td><td>P.J. Kernan, L.M. Krauss (CASE)</td></tr><tr><td>SIGL</td><td>95</td><td>PR D51 1499</td><td>G. Sigl, M.S. Turner (FNAL, EFI)</td></tr><tr><td>STOEFFL</td><td>95</td><td>PRL 75 3237</td><td>W. Stoeffl, D.J. Decman (LLNL)</td></tr><tr><td>VILAIN</td><td>95B</td><td>PL B345 115</td><td>P. Villain et al. (CHARM II Collab.)</td></tr><tr><td>ASSAMAGAN</td><td>94</td><td>PL B335 231</td><td>K.A. Assamanag et al. (PSI, ZURI, VILL+)</td></tr><tr><td>BABU</td><td>94</td><td>PL B321 140</td><td>K.S. Babu, T.M. Gould, I.Z. Rothstein (BART+)</td></tr><tr><td>DODELSON</td><td>94</td><td>PR D49 5068</td><td>S. Dodelson, G. Gyuk, M.S. Turner (FNAL, CHIC+)</td></tr><tr><td>GOULD</td><td>94</td><td>PL B333 545</td><td>T.M. Gould, I.Z. Rothstein (JHU, MICH)</td></tr><tr><td>JECKELMANN</td><td>94</td><td>PL B335 326</td><td>B. Jeckelmann, P.F.A. Goudsmit, H.J. Leisi (WABRN+)</td></tr><tr><td>KAWASAKI</td><td>94</td><td>NP B419 105</td><td>M. Kawasaki et al. (OSU)</td></tr><tr><td>PERES</td><td>94</td><td>PR D50 513</td><td>O.L.G. Peres, V. Pfeitez, R. Zukanovich Funchal</td></tr><tr><td>YASUMI</td><td>94</td><td>PL B334 229</td><td>S. Yasumi et al. (KEK, TSUK, KYOT+)</td></tr><tr><td>ALLEN</td><td>93</td><td>PR D47 11</td><td>R.C. Allen et al. (UCI, LANL, ANL+)</td></tr><tr><td>BALEST</td><td>93</td><td>PR D47 3671</td><td>R. Balest et al. (CLEO Collab.)</td></tr><tr><td>CINABRO</td><td>93</td><td>PL B70 3700</td><td>D. Cinabro et al. (CLEO Collab.)</td></tr><tr><td>DERBIN</td><td>93</td><td>JETPL 57 768</td><td>A.V. Derbin et al. (PNPI)</td></tr><tr><td colspan="4">Translated from ZETF 57 755.</td></tr><tr><td>DOLGOV</td><td>93</td><td>PRL 71 476</td><td>A.D. Dolgov, I.Z. Rothstein (MICH)</td></tr><tr><td>ENQVIST</td><td>93</td><td>PL B301 376</td><td>K. Engist, V. Ulbo (NORD)</td></tr><tr><td>SUN</td><td>93</td><td>CJNP 15 261</td><td>H.C. Sun et al. (CIAE, CST, BEIJING)</td></tr><tr><td>WEINHEIMER</td><td>93</td><td>B300 210</td><td>C. Weinheimer et al. (MAINZ)</td></tr><tr><td>ALBRECHT</td><td>92M</td><td>PL B292 221</td><td>H. Albrecht et al. (ARGUS Collab.)</td></tr><tr><td>BLUDMAN</td><td>92</td><td>PR D45 4720</td><td>S.A. Bludman (CFPA)</td></tr><tr><td>COOPER-....</td><td>92</td><td>PL B280 153</td><td>A.M. Cooper-Sarkar et al. (BEBEC WA66 Collab.)</td></tr><tr><td>DODELSON</td><td>92</td><td>PRL 68 2572</td><td>S. Dodelson, J.A. Friedman, M.S. Turner (FNAL+)</td></tr><tr><td>HOLZSCHUH</td><td>92B</td><td>PL B287 381</td><td>E. Holzschuh, M. Fritschi, W. Kundig (ZURI)</td></tr><tr><td>KAWANO</td><td>92</td><td>PL B275 487</td><td>L.H. Kawano et al. (CIT, UCSD, LLL+)</td></tr><tr><td>MOURAO</td><td>92</td><td>PL B285 364</td><td>A.M. Mourao, J. Pulido, J.P. Ralston (LISB, LISBT+)</td></tr><tr><td>PDG</td><td>92</td><td>PR D45 S1</td><td>K. Hikasa et al. (KEK, LBL, BOST+)</td></tr><tr><td>VIDYAKIN</td><td>92</td><td>JETPL 55 206</td><td>G.S. Vidyakin et al. (KIAE)</td></tr><tr><td colspan="4">Translated from ZETF 55 212.</td></tr><tr><td>ALLEN</td><td>91</td><td>PR D43 1</td><td>R. Allen et al. (UCI, LANL, UMD)</td></tr><tr><td>DAVIDSON</td><td>91</td><td>PR D43 2314</td><td>S. Davidson, B.A. Campbell, D. Bailey (ALBE+)</td></tr><tr><td>DESHPANDE</td><td>91</td><td>PR D43 943</td><td>N.G. Deshpande, K.V.L. Sarma (OREG, TATA)</td></tr><tr><td>DORENBOS...</td><td>91</td><td>ZPHY C51 142 (errat.)</td><td>J. Dorenbosch et al. (CHARM Collab.)</td></tr><tr><td>FULLER</td><td>91</td><td>PR D43 3136</td><td>G.M. Fuller, R.A. Malaney (UCSD)</td></tr><tr><td>GRANEK</td><td>91</td><td>IJMP A6 2387</td><td>H. Granek, B.H.J. McKellar (MELB)</td></tr><tr><td>KAWAKAMI</td><td>91</td><td>PL B256 105</td><td>H. Kawakami et al. (INUS, TOHOK, TINT+)</td></tr><tr><td>KOLB</td><td>91</td><td>PRL 67 533</td><td>E.W. Kolb et al. (FNAL, CHIC)</td></tr><tr><td>KRAKAUER</td><td>91</td><td>PR D44 6</td><td>D.A. Krakauer et al. (LAMPF E225 Collab.)</td></tr><tr><td>LAM</td><td>91</td><td>PR D44 3345</td><td>W.P. Lam, K.W. Ng (AST)</td></tr><tr><td>ROBERTSON</td><td>91</td><td>PRL 67 957</td><td>R.G.H. Robertson et al. (LASL, LLL)</td></tr><tr><td>AHrens</td><td>90</td><td>PL D41 3297</td><td>L.H. Ahrens et al. (BNL, BROW, HIR+)</td></tr><tr><td>AVIGNONE</td><td>90</td><td>PR D41 682</td><td>F.T. Avignone, J.I. Collar (SCUC)</td></tr><tr><td>KRAKAUER</td><td>90</td><td>PL B252 177</td><td>D.A. Krakauer et al. (LAMPF E225 Collab.)</td></tr><tr><td>RAFFELT</td><td>90</td><td>PR D4 2856</td><td>G.G. Raffelt (MPIM)</td></tr><tr><td>WALKER</td><td>90</td><td>PR D41 689</td><td>T.P. Walker (HARV)</td></tr><tr><td>CHIPP</td><td>89</td><td>PL D2 505</td><td>E.L. Chupp, W.T. Vestrand, C. Reppin (UNH, MPIM)</td></tr><tr><td>DORENBOS...</td><td>89</td><td>ZPHY C41 567</td><td>J. Dorenbosch et al. (CHARM Collab.)</td></tr><tr><td>GRIFOLS</td><td>89B</td><td>PR D40 3819</td><td>J.A. Grifols, E. Masso (BARC)</td></tr><tr><td>KOLB</td><td>89B</td><td>PL D2 509</td><td>E.W. Kolb, M.S. Turner (CHIC, FNAL)</td></tr><tr><td>LOREDO</td><td>89</td><td>ANYAS 571 601</td><td>T.J. Loredo, D.Q. Lamb (CHIC)</td></tr><tr><td>RAFFELT</td><td>89B</td><td>PR D39 2066</td><td>G.G. Raffelt (PRIN, UCB)</td></tr><tr><td>RAFFELT</td><td>89B</td><td>APJ 336 61</td><td>G. Raffelt, D. Dearborn, J. Silk (UCB, LLL)</td></tr><tr><td>ALBRECHT</td><td>88B</td><td>PL B202 149</td><td>H. Albrecht et al. (ARGUS Collab.)</td></tr><tr><td>BARBIERI</td><td>88</td><td>PR D61 27</td><td>R. Barbiieri, R.N. Mohapatra (PISA, UMD)</td></tr><tr><td>BORIS</td><td>88</td><td>PR D61 245 (errat.)</td><td>S.D. Boris et al. (ITEP, ASCI)</td></tr><tr><td>FUKUGITA</td><td>88</td><td>PR D60 879</td><td>M. Fukugita et al. (KYOTU, MPIM, ASCI)</td></tr><tr><td>GROTCH</td><td>88</td><td>ZPHY C39 553</td><td>H. Grotch, R.W. Robinett (PSU)</td></tr><tr><td>RAFFELT</td><td>88B</td><td>PR D37 549</td><td>G.G. Raffelt, D.S.P. Dearborn (UCB, LLL)</td></tr><tr><td>SPERGEL</td><td>88</td><td>PL B200 366</td><td>D.N. Spergel, J.N. Bachall (IAS)</td></tr><tr><td>VONFEILIT...</td><td>88</td><td>PL B200 580</td><td>F. von Feilitzsch, L. Oberauer (TUM)</td></tr><tr><td>BARBIELLINI</td><td>88</td><td>NAT 329 21</td><td>G. Barbielini, G. Cocconi (CERN)</td></tr><tr><td>BORIS</td><td>87</td><td>PRL 58 2019</td><td>S.D. Boris et al. (ITEP, ASCI)</td></tr><tr><td>Also</td><td>87</td><td>PR D1 245 (errat.)</td><td>S.D. Boris et al. (ITEP, ASCI)</td></tr><tr><td>BORIS</td><td>87B</td><td>JETPL 45 333</td><td>S.D. Boris et al. (ITEP)</td></tr><tr><td colspan="4">Translated from ZETF 45 267.</td></tr><tr><td>FUKUGITA</td><td>87</td><td>PR D36 3817</td><td>M. Fukugita, S. Yazaki (KYOTU, TOKYO)</td></tr><tr><td>NUSSINOV</td><td>87</td><td>PR D36 2278</td><td>S. Nussinov, Y. Rephaeli (TELA)</td></tr><tr><td>OBERAUER</td><td>87</td><td>PL B198 113</td><td>L.F. Oberauer, F. von Feilitzsch, R.L. Mossbauer (LLNL)</td></tr><tr><td>SPRINGER</td><td>87</td><td>PR A35 679</td><td>S.N. Ketov et al. (KIAE)</td></tr><tr><td>KETOV</td><td>86</td><td>JETPL 44 146</td><td>Translated from ZETF 44 114.</td></tr><tr><td>COWSIK</td><td>85</td><td>PL 151B 62</td><td>R. Cowsik (TATA)</td></tr><tr><td>RAFFELT</td><td>85</td><td>PR D31 3002</td><td>G.G. Raffelt (MPIM)</td></tr><tr><td>BINETRUY</td><td>84</td><td>PL 134B 174</td><td>P. Binetruy, G. Girardi, P. Salati (LAPP)</td></tr><tr><td>FREESE</td><td>84</td><td>NP B233 167</td><td>K. Freese, D.N. Schramm (CHIC, FNAL)</td></tr><tr><td>KYULDJIEV</td><td>84</td><td>NP B243 387</td><td>A.V. Kyuldjiev (SOFI)</td></tr><tr><td>SCHRAMM</td><td>84</td><td>PL 141B 337</td><td>D.N. Schramm, G. Steigman (FNAL, BART)</td></tr><tr><td>VOGEL</td><td>84</td><td>PR D30 1505</td><td>P. Vogel (TATA)</td></tr><tr><td>ANDERHUB</td><td>82</td><td>PL 114B 76</td><td>H.B. Anderhub et al. (ETH, SIN)</td></tr><tr><td>OLIVE</td><td>82</td><td>PR D25 213</td><td>K.A. Olive, M.S. Turner (CHIC, UCBS)</td></tr><tr><td>BERNSTEIN</td><td>81</td><td>PL 101B 39</td><td>J. Bernstein, G. Feinberg (STEV, COLU)</td></tr><tr><td>FRANK</td><td>81</td><td>PR D24 2001</td><td>J.S. Frank et al. (LASL, YALE, MIT+)</td></tr><tr><td>MORGAN</td><td>81</td><td>PL 102B 247</td><td>J.A. Morgan (SUSS)</td></tr><tr><td>FUJIKAWA</td><td>80</td><td>PRL 45 963</td><td>K. Fujikawa, R. Shrock (STON)</td></tr><tr><td>LUBIMOV</td><td>80</td><td>PRL 94B 266</td><td>V.A. Lyubimov et al. (ITEP)</td></tr><tr><td>STECKER</td><td>80</td><td>PL D1 1460</td><td>F.W. Stecker (NASA)</td></tr><tr><td>COWSIK</td><td>79</td><td>PR D19 2219</td><td>R. Cowsik (TATA)</td></tr><tr><td>GOLDMAN</td><td>79</td><td>PR D19 2215</td><td>T. Goldman, G.J. Stephenson (LASL)</td></tr><tr><td>BEG</td><td>78</td><td>PR D17 1395</td><td>M.A.B. Beg, W.J. Marciano, M. Ruderman (ROCK+)</td></tr><tr><td>BLIETSCHAU</td><td>78</td><td>NP B133 205</td><td>J. Blietschau et al. (Gargamelle Collab.)</td></tr><tr><td>FALK</td><td>78</td><td>PL 79B 511</td><td>S.W. Falk, D.N. Schramm (CHIC, CHIC)</td></tr><tr><td>BARNES</td><td>77</td><td>PL 38 1049</td><td>V.E. Barnes et al. (PURD, ANL)</td></tr><tr><td>COWSIK</td><td>77</td><td>PRL 39 784</td><td>R. Cowsik (MPIM, TATA)</td></tr><tr><td>LEE</td><td>77C</td><td>PR D16 1444</td><td>B.W. Lee, R.E. Shrock (STON)</td></tr><tr><td>VYSOTSKY</td><td>77</td><td>JETPL 26 188</td><td>M.I. Vysotsky, A.D. Dolgov, Y.B. Zeldovich (ITEP)</td></tr><tr><td colspan="4">Translated from ZETF 26 200.</td></tr><tr><td>BELLOTTI</td><td>76</td><td>LNC 17 553</td><td>E. Bellotti et al. (MILA)</td></tr><tr><td>SUTHERLAND</td><td>76</td><td>PR D13 2700</td><td>P. Sutherland et al. (PENN, COLU, NYU)</td></tr><tr><td>SZALAY</td><td>76</td><td>AA 49 437</td><td>A.S. Szalay, G. Marx (EOTV)</td></tr><tr><td>CLARK</td><td>74</td><td>PR D9 533</td><td>A.R. Clark et al. (LBL)</td></tr><tr><td>KIM</td><td>74</td><td>PR D9 3050</td><td>J.E. Kim, V.S. Mathur, S. Okubo (ROCH)</td></tr><tr><td>REINES</td><td>74</td><td>PL 32 180</td><td>F. Reines, H.W. Sobel, H.S. Gurr (UCI)</td></tr><tr><td>SZALAY</td><td>74</td><td>APAH 35 8</td><td>A.S. Szalay, G. Marx (EOTV)</td></tr><tr><td>COWSIK</td><td>72</td><td>PRL 29 669</td><td>R. Cowsik, J. McClelland (UCB)</td></tr><tr><td>MARX</td><td>72</td><td>Nu Conf. Budapest</td><td>G. Marx, A.S. Szalay (EOTV)</td></tr><tr><td>GERSHTEIN</td><td>66</td><td>JETPL 4 120</td><td>S.S. Gershtein, Y.B. Zeldovich (KIAM)</td></tr><tr><td colspan="4">Translated from ZETF 4 189.</td></tr><tr><td>BERNSTEIN</td><td>63</td><td>PR 132 1227</td><td>J. Bernstein, M. Ruderman, G. Feinberg (NYU+)</td></tr><tr><td>COWAN</td><td>57</td><td>PR 107 528</td><td>C.L. Cowan, F. Reines (LANL)</td></tr></tbody></table>
$W^{\pm} \to \ell^{\pm}\nu$ to $Z \to \ell^{+}\ell^{-}$ events [7]. This involved a calculation that assumed Standard Model values for the total $W$ width and the ratio of $W$ and $Z$ leptonic partial widths, and used an estimate of the ratio of $Z$ to $W$ production cross sections. Now that the $Z$ width is very precisely known from the LEP experiments, the approach is now one of those used to determine the $W$ width.

## References

1. ALEPH, DELPHI, L3, OPAL, and SLD Collaborations, and LEP Electroweak Working Group, and SLD Electroweak Group, and SLD Heavy Flavour Group, Phys. Reports 427, 257 (2006).

2. P. Janot and S. Jadach, Phys. Lett. B803, 135319 (2020).

3. G. Voutsinas et al., Phys. Lett. B800, 135068 (2020).

4. VENUS: K. Abe et al., Phys. Lett. B232, 431 (1989); ASP: C. Hearty et al., Phys. Rev. D39, 3207 (1989); CELLO: H.J. Behrend et al., Phys. Lett. B215, 186 (1988); MAC: W.T. Ford et al., Phys. Rev. D33, 3472 (1986); MARK J: H. Wu, Ph.D. Thesis, Univ. Hamburg (1986).

5. L3: M. Acciarri et al., Phys. Lett. B431, 199 (1998); DELPHI: P. Abreu et al., Z. Phys. C74, 577 (1997); OPAL: R. Akers et al., Z. Phys. C65, 47 (1995); ALEPH: D. Buskulic et al., Phys. Lett. B313, 520 (1993).

6. DELPHI: J. Abdallah et al., Eur. Phys. J. C38, 395 (2005); L3: P. Achard et al., Phys. Lett. B587, 16 (2004); ALEPH: A. Heister et al., Eur. Phys. J. C28, 1 (2003); OPAL: G. Abbiendi et al., Eur. Phys. J. C18, 253 (2000).

7. UA1: C. Albajar et al., Phys. Lett. B198, 271 (1987); UA2: R. Ansari et al., Phys. Lett. B186, 440 (1987).

## Number from $e^{+}e^{-}$ Colliders

### Number of Light $\nu$ Types

$$2.9963 \pm 0.0074$$

$$1 \text{ JANOT} \quad 20$$

• • • We do not use the following data for averages, fits, limits, etc. • • •

$$2.9918 \pm 0.0081$$

$$2 \text{ VOUTSINAS} \quad 20$$

$$2.9840 \pm 0.0082$$

$$3 \text{ LEP-SLC} \quad 06 \quad \text{RVUE}$$

$$3.00 \pm 0.05$$

$$4 \text{ LEP} \quad 92 \quad \text{RVUE}$$

1 JANOT 20 applies a correction to LEP-SLC 06 using an updated Bhabha cross section calculation. This result also includes a correction to account for correlated luminosity bias as presented in VOUTSINAS 20.

2 VOUTSINAS 20 applies a correction to LEP-SLC 06 to account for correlated luminosity bias.

3 Combined fit from ALEPH, DELPHI, L3 and OPAL Experiments.

4 Simultaneous fits to all measured cross section data from all four LEP experiments.

### Number of Light $\nu$ Types from Direct Measurement of Invisible $Z$ Width

In the following, the invisible $Z$ width is obtained from studies of single-photon events from the reaction $e^{+}e^{-} \to \nu\bar{\nu}\gamma$. All are obtained from LEP runs in the $E_{cm}$ range 88–209 GeV.

$$\text{VALUE} \qquad \text{DOCUMENT ID} \qquad \text{TECN} \qquad \text{COMMENT}$$

#### 2.92±0.05 OUR AVERAGE

Error includes scale factor of 1.2.

$$2.84 \pm 0.10 \pm 0.14$$

$$\text{ABDALLAH} \quad 05\text{B} \quad \text{DLPH} \quad \sqrt{s} = 180-209 \text{ GeV}$$

$$2.98 \pm 0.05 \pm 0.04$$

$$\text{ACHARD} \quad 04\text{E} \quad \text{L3} \quad 1990-2000 \text{ LEP runs} \quad \sqrt{s} = 189-209 \text{ GeV}$$

$$2.86 \pm 0.09$$

$$\text{HEISTER} \quad 03\text{C} \quad \text{ALEP} \quad \sqrt{s} = 189-209 \text{ GeV}$$

$$2.69 \pm 0.13 \pm 0.11$$

$$\text{ABBIENDI,G} \quad 00\text{D} \quad \text{OPAL} \quad 1998 \text{ LEP run}$$

$$2.89 \pm 0.32 \pm 0.19$$

$$\text{ABREU} \quad 97\text{J} \quad \text{DLPH} \quad 1993-1994 \text{ LEP runs}$$

$$3.23 \pm 0.16 \pm 0.10$$

$$\text{AKERS} \quad 95\text{C} \quad \text{OPAL} \quad 1990-1992 \text{ LEP runs}$$

$$2.68 \pm 0.20 \pm 0.20$$

$$\text{BUSKULIC} \quad 93\text{L} \quad \text{ALEP} \quad 1990-1991 \text{ LEP runs}$$

• • • We do not use the following data for averages, fits, limits, etc. • • •

$$2.84 \pm 0.15 \pm 0.14$$

$$\text{ABREU} \quad 00\text{Z} \quad \text{DLPH} \quad 1997-1998 \text{ LEP runs}$$

$$3.01 \pm 0.08$$

$$\text{ACCIARRI} \quad 99\text{R} \quad \text{L3} \quad 1991-1998 \text{ LEP runs}$$

$$3.1 \pm 0.6 \pm 0.1$$

$$\text{ADAM} \quad 96\text{C} \quad \text{DLPH} \quad \sqrt{s} = 130, 136 \text{ GeV}$$

## Limits from Astrophysics and Cosmology

### Effective Number of Light $\nu$ Types

"Light" means here with a mass < about 1 MeV. The quoted values correspond to $N_{\text{eff}}$, where $N_{\text{eff}} = 3.045$ in the Standard Model with $N_{\nu} = 3$. See also reviews on "Big-Bang Nucleosynthesis" and "Neutrinos in Cosmology."

<table><thead><tr><th>VALUE</th><th>CL%</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>3.12±0.38</td><td>95</td><td>1 BRIEDEN</td><td>22</td><td>COSM BOSS, eBOSS, CMB</td></tr><tr><td>2.90±0.15</td><td>68</td><td>2 KUMAR</td><td>22</td><td>COSM BOSS + CMB</td></tr><tr><td>2.89±0.14</td><td>68</td><td>3 YEH</td><td>22</td><td>COSM BBN + CMB</td></tr><tr><td>2.99±0.17</td><td>68</td><td>4 AGHANIM</td><td>20</td><td>COSM</td></tr><tr><td>2.84±0.15</td><td>68</td><td>5 FIELDS</td><td>20</td><td>COSM BBN</td></tr><tr><td>2.88±0.17</td><td>68</td><td>6 IVANOV</td><td>20</td><td>COSM Planck and BOSS</td></tr><tr><td>2.3-3.2</td><td>95</td><td>7 VERDE</td><td>17</td><td>COSM</td></tr><tr><td>2.88±0.16</td><td>68</td><td>8 CYBURT</td><td>16</td><td>COSM BBN</td></tr><tr><td>2.88±0.20</td><td>95</td><td>9 ROSSI</td><td>15</td><td>COSM</td></tr><tr><td>3.3 ± 0.5</td><td>95</td><td>10 ADE</td><td>14</td><td>COSM Planck</td></tr><tr><td>3.78+0.31-0.30</td><td></td><td>11 COSTANZI</td><td>14</td><td>COSM</td></tr><tr><td>3.29±0.31</td><td></td><td>12 HOU</td><td>14</td><td>COSM</td></tr><tr><td>&lt; 3.80</td><td>95</td><td>13 LEISTEDT</td><td>14</td><td>COSM</td></tr><tr><td>&lt; 4.10</td><td>95</td><td>14 MORESCO</td><td>12</td><td>COSM</td></tr><tr><td>&lt; 5.79</td><td>95</td><td>15 XIA</td><td>12</td><td>COSM</td></tr><tr><td>&lt; 4.08</td><td>95</td><td>16 MANGANO</td><td>11</td><td>COSM BBN</td></tr><tr><td>0.9-8.2</td><td></td><td>17 ICHIKAWA</td><td>07</td><td>COSM</td></tr><tr><td>3-7</td><td>95</td><td>17 CIRELLI</td><td>06</td><td>COSM</td></tr><tr><td>2.7-4.6</td><td>95</td><td>18 HANNESTAD</td><td>06</td><td>COSM</td></tr><tr><td>3.6-7.4</td><td>95</td><td>17 SELJAK</td><td>06</td><td>COSM</td></tr><tr><td>&lt; 4.4</td><td></td><td>19 CYBURT</td><td>05</td><td>COSM</td></tr><tr><td>&lt; 3.3</td><td></td><td>20 BARGER</td><td>03C</td><td>COSM</td></tr><tr><td>1.4-6.8</td><td></td><td>21 CROTTY</td><td>03</td><td>COSM</td></tr><tr><td>1.9-6.6</td><td></td><td>21 PIERPAOLI</td><td>03</td><td>COSM</td></tr><tr><td>2-4</td><td></td><td>LISI</td><td>99</td><td>COSM BBN</td></tr><tr><td>&lt; 4.3</td><td></td><td>OLIVE</td><td>99</td><td>COSM BBN</td></tr><tr><td>&lt; 4.9</td><td></td><td>COPI</td><td>97</td><td>Cosmology</td></tr><tr><td>&lt; 3.6</td><td></td><td>HATA</td><td>97B</td><td>High D/H quasar abs.</td></tr><tr><td>&lt; 4.0</td><td></td><td>OLIVE</td><td>97</td><td>BBN; high <sup>4</sup>He and <sup>7</sup>Li</td></tr><tr><td>&lt; 4.7</td><td></td><td>CARDALL</td><td>96B</td><td>COSM High D/H quasar abs.</td></tr><tr><td>&lt; 3.9</td><td></td><td>FIELDS</td><td>96</td><td>COSM BBN; high <sup>4</sup>He and <sup>7</sup>Li</td></tr><tr><td>&lt; 4.5</td><td></td><td>KERNAN</td><td>96</td><td>COSM High D/H quasar abs.</td></tr><tr><td>&lt; 3.6</td><td></td><td>OLIVE</td><td>95</td><td>BBN; ≥ 3 massless $\nu$</td></tr><tr><td>&lt; 3.3</td><td></td><td>WALKER</td><td>91</td><td>Cosmology</td></tr><tr><td>&lt; 3.4</td><td></td><td>OLIVE</td><td>90</td><td>Cosmology</td></tr><tr><td>&lt; 4</td><td></td><td>YANG</td><td>84</td><td>Cosmology</td></tr><tr><td>&lt; 4</td><td></td><td>YANG</td><td>79</td><td>Cosmology</td></tr><tr><td>&lt; 7</td><td></td><td>STEIGMAN</td><td>77</td><td>Cosmology</td></tr><tr><td>&lt;16</td><td></td><td>PEEBLES</td><td>71</td><td>Cosmology</td></tr><tr><td></td><td></td><td>22 SHVARTSMAN69</td><td></td><td>Cosmology</td></tr><tr><td></td><td></td><td>HOYLE</td><td>64</td><td>Cosmology</td></tr></tbody></table>

1 BRIEDEN 22 combines large scale structure data from BOSS and eBOSS including the shape of the matter power spectrum with Planck CMB data.

2 KUMAR 22 combine the reconstructed galaxy power spectrum from BOSS data with Planck CMB data.

3 YEH 22 combines Planck 2018 CMB data with BBN and observations of deuterium and Helium-4. Supersedes FIELDS 20.

4 AGHANIM 20 best fit on number of neutrino types is based on Planck data combined with lensing and baryon acoustic oscillations (BAO). Without BAO, they find $2.89^{+0.18}_{-0.19}$. Several other values are quoted using different combinations of data.

5 FIELDS 20 combines Planck 2018 CMB data with BBN and observations of deuterium and Helium-4.

6 IVANOV 20 combines 2018 Planck CMB data with baryon acoustic oscillation data from BOSS. This study is based on a full-shape likelihood for the redshift-space galaxy power spectrum of the BOSS data.

7 Uses Planck Data combined with an independent standard measure of distance to the sound horizon to set a limit on the total number of neutrinos. Only CMB and early-time information are used.

8 CYBURT 16 combines Planck 2015 CMB data with BBN and observations of deuterium and Helium-4.

9 ROSSI 15 sets limits on the number of neutrino types using BOSS Lyman alpha forest data combined with Planck CMB data and baryon acoustic oscillations.

10 Fit to the number of neutrino degrees of freedom from Planck CMB data along with WMAP polarization, high L, and BAO data.

11 Fit to the number of neutrino degrees of freedom from Planck CMB data along with BAO, shear and cluster data.

12 Fit based on the SPT-SZ survey combined with CMB, BAO, and $H_0$ data.

13 Constrains the number of neutrino degrees of freedom (marginalizing over the total mass) from CMB, CMB lensing, BAO, and galaxy clustering data.

14 Limit on the number of light neutrino types from observational Hubble parameter data with seven-year WMAP data, SPT, and the most recent estimate of $H_0$. Best fit is $3.45 \pm 0.65$.

15 Limit on the number of light neutrino types from the CFHTLS combined with seven-year WMAP data and a prior on the Hubble parameter. Best fit is $4.17^{+1.62}_{-1.26}$. Limit is relaxed to $3.98^{+2.02}_{-1.20}$ when small scales affected by non-linearities are removed.
16 Constrains the number of neutrino types from recent CMB and large scale structure data. No priors on other cosmological parameters are used.

17 Constrains the number of neutrino types from recent CMB, large scale structure, Lyman-alpha forest, and SN1a data. The slight preference for $N_{\nu} > 3$ comes mostly from the Lyman-alpha forest data.

18 Constrains the number of neutrino types from recent CMB and large scale structure data. See also HAMANN 07.

19 Limit on the number of neutrino types based on $^4$He and D/H abundance assuming a baryon density fixed to the WMAP data. Limit relaxes to 4.6 if D/H is not used or to 5.8 if only D/H and the CMB are used. See also CYBURT 01 and CYBURT 03.

20 Limit on the number of neutrino types based on combination of WMAP data and big-bang nucleosynthesis. The limit from WMAP data alone is 8.3. See also KNELLER 01. $N_{\nu} \ge 3$ is assumed to compute the limit.

21 95% confidence level range on the number of neutrino flavors from WMAP data combined with other CMB measurements, the 2dfGRS data, and HST data.

22 SHVARTSMAN 69 limit inferred from his equations.

## Number Coupling with Less Than Full Weak Strength

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th></tr></thead><tbody><tr><td>&lt;20</td><td>1 OLIVE</td><td>81C COSM</td></tr><tr><td>&lt;20</td><td>1 STEIGMAN</td><td>79 COSM</td></tr></tbody></table>

## REFERENCES FOR Limits on Number of Neutrino Types

<table><tbody><tr><td>BRIEDEN</td><td>22</td><td>JCAP 2208 024</td><td>S. Brieden, Hector Gil-Marin, Licia Verde</td></tr><tr><td>KUMAR</td><td>22</td><td>JCAP 2209 060</td><td>S. Kumar, R. Nunes, P.Yadav</td></tr><tr><td>N</td><td>22</td><td>JCAP 2210 046</td><td>T. H. Yeh et al. (ILL, MINN)</td></tr><tr><td>AGHANIM</td><td>20</td><td>AA 641 A6</td><td>N. Aghanim et al. (Planck Collab.)</td></tr><tr><td>FIELDS</td><td>20</td><td>JCAP 2003 010</td><td>B. Fields et al. (ILL, MINN)</td></tr><tr><td>Also</td><td>20</td><td>JCAP 2011 E02 (errat.)</td><td>B. Fields et al. (ILL, MINN)</td></tr><tr><td>IVANOV</td><td>20</td><td>PR D101 083504</td><td>M.M. Ivanov, M. Stimonovic, M. Zaldarriaga (NYU,)</td></tr><tr><td>JANOT</td><td>20</td><td>PL B803 135319</td><td>P. Janot, S. Jadach (CERN, CRAC)</td></tr><tr><td>VOUTSINAS</td><td>20</td><td>PL B800 135068</td><td>G. Voutsinas et al. (CERN, BOHR)</td></tr><tr><td>VERDE</td><td>17</td><td>JCAP 1704 023</td><td>L. Verde et al.</td></tr><tr><td>CYBURT</td><td>16</td><td>RMP 88 015004</td><td>R.H. Cyburt et al. (MSU, ILL, MINN)</td></tr><tr><td>ROSSI</td><td>15</td><td>PR D92 063505</td><td>G. Rossi et al.</td></tr><tr><td>ADE</td><td>14</td><td>AA 571 A16</td><td>P.A.R. Ade et al. (Planck Collab.)</td></tr><tr><td>COSTANZI</td><td>14</td><td>JCAP 1410 081</td><td>M. Costandi et al. (TRST, TRSTI)</td></tr><tr><td>HOU</td><td>14</td><td>APJ 782 74</td><td>Z. Hou et al.</td></tr><tr><td>LEISTEDT</td><td>14</td><td>PRL 113 041301</td><td>B. Leistedt, H.V. Peiris, L. Verde</td></tr><tr><td>MORESCO</td><td>12</td><td>JCAP 1207 053</td><td>M. Moresco et al.</td></tr><tr><td>XIA</td><td>12</td><td>JCAP 1206 010</td><td>J.-Q. Xia et al.</td></tr><tr><td>MANGANO</td><td>11</td><td>PL B701 296</td><td>G. Mangano, P. Serpico</td></tr><tr><td>HAMANN</td><td>07</td><td>JCAP 0708 021</td><td>J. Hamann et al.</td></tr><tr><td>ICHIKAWA</td><td>07</td><td>JCAP 0705 007</td><td>K. Ichikawa, M. Kawasaki, F. Takahashi</td></tr><tr><td>CIRELLI</td><td>06</td><td>JCAP 0612 013</td><td>M. Cirelli et al.</td></tr><tr><td>HANNESTAD</td><td>06</td><td>JCAP 0611 016</td><td>S. Hannestad, G. Raffelt</td></tr><tr><td>LEP-SLC</td><td>06</td><td>PRPL 427 257</td><td>ALEPH, DELPHI, L3, OPAL, SLD and working groups</td></tr><tr><td>SELJAK</td><td>06</td><td>JCAP 0610 014</td><td>U. Soljak, A. Slosar, P. McDonald</td></tr><tr><td>ABDALLAH</td><td>05B</td><td>EPJ C38 395</td><td>J. Abdallah et al. (DELPHI Collab.)</td></tr><tr><td>CYBURT</td><td>05</td><td>ASP 23 313</td><td>R.H. Cyburt et al.</td></tr><tr><td>ACHARD</td><td>04E</td><td>PL B587 16</td><td>P. Achard et al. (L3 Collab.)</td></tr><tr><td>BARGER</td><td>03C</td><td>PL B566 8</td><td>V. Barger et al.</td></tr><tr><td>CROTTY</td><td>03</td><td>PL B67 123005</td><td>P. Crotty, J. Lesgourgues, S. Pastor</td></tr><tr><td>CYBURT</td><td>03</td><td>PL B567 227</td><td>R.H. Cyburt, B.D. Fields, K.A. Olive</td></tr><tr><td>HEISTER</td><td>03C</td><td>EPJ C28 1</td><td>A. Heister et al. (ALEPH Collab.)</td></tr><tr><td>PIERPAOLI</td><td>03C</td><td>MNRS 342 L63</td><td>A. Pierpaoli</td></tr><tr><td>CYBURT</td><td>01</td><td>ASP 17 87</td><td>R.H. Cyburt, B.D. Fields, K.A. Olive</td></tr><tr><td>KNELLER</td><td>01</td><td>PR D64 123506</td><td>J.P. Kneller et al. (OPAL Collab.)</td></tr><tr><td>ABBIENDI, G.</td><td>00D</td><td>EPJ C18 253</td><td>G. Abbiendi et al. (DELPHI Collab.)</td></tr><tr><td>ABREU</td><td>00Z</td><td>EPJ C17 53</td><td>P. Abreu et al. (DELPHI Collab.)</td></tr><tr><td>ACCIARRI</td><td>99R</td><td>PL B470 268</td><td>M. Acciarri et al. (L3 Collab.)</td></tr><tr><td>LISI</td><td>99</td><td>PR D59 123520</td><td>E. Lisi, S. Sarkar, F.L. Villante</td></tr><tr><td>OLIVE</td><td>99</td><td>ASP 11 403</td><td>K.A. Olive, D. Thomas</td></tr><tr><td>ABREU</td><td>97</td><td>ZPHY C74 577</td><td>P. Abreu et al. (DELPHI Collab.)</td></tr><tr><td>COPI</td><td>97</td><td>PR D55 3389</td><td>C.J. Copi, D.N. Schramm, M.S. Turner (CHIC)</td></tr><tr><td>HATA</td><td>97B</td><td>PR D55 540</td><td>N. Hata et al. (OSU, PENN)</td></tr><tr><td>OLIVE</td><td>97</td><td>ASP 7 27</td><td>K.A. Olive, D. Thomas (MINN, FLOR)</td></tr><tr><td>ADAM</td><td>96C</td><td>PL B380 471</td><td>W. Adam et al. (DELPHI Collab.)</td></tr><tr><td>CARDALL</td><td>96B</td><td>APJ 472 435</td><td>C.Y. Cardall, G.M. Fuller (UCSD)</td></tr><tr><td>FIELDS</td><td>96</td><td>New Ast 1 77</td><td>B.D. Fields et al. (NDAM, CERN, MINN+)</td></tr><tr><td>KERNAN</td><td>96</td><td>PR D54 3681</td><td>P.S. Kernan, S. Sarkar (CASE, OXFT)</td></tr><tr><td>AKERS</td><td>95C</td><td>ZPHY C65 47</td><td>R. Akers et al. (OPAL Collab.)</td></tr><tr><td>OLIVE</td><td>95</td><td>PL B354 357</td><td>K.A. Olive, G. Steigman (MINN, OSU)</td></tr><tr><td>BUSKULIC</td><td>93L</td><td>PL B313 520</td><td>D. Bushulic et al. (ALEPH Collab.)</td></tr><tr><td>LEP</td><td>92</td><td>PL B276 247</td><td>LEP Collabs. (LEP, ALEPH, DELPHI, L3, OPAL)</td></tr><tr><td>WALKER</td><td>91</td><td>APJ 376 51</td><td>T.P. Walker et al. (HSCA, OSU, CHIC+)</td></tr><tr><td>OLIVE</td><td>90</td><td>PL B236 454</td><td>K.A. Olive et al. (MINN, CHIC, OSU+)</td></tr><tr><td>YANG</td><td>84</td><td>APJ 281 493</td><td>J. Yang et al. (CHIC, BART)</td></tr><tr><td>OLIVE</td><td>81C</td><td>NP B180 497</td><td>K.A. Olive, D.N. Schramm, G. Steigman (CHIC+)</td></tr><tr><td>STEIGMAN</td><td>79</td><td>PL B43 239</td><td>G. Steigman, K.A. Olive, D.N. Schramm (BARTL+)</td></tr><tr><td>YANG</td><td>79</td><td>APJ 227 697</td><td>J. Yang et al. (CHIC, YALE, UVA)</td></tr><tr><td>STEIGMAN</td><td>77</td><td>PL 66B 202</td><td>G. Steigman, D.N. Schramm, J.E. Gunn (YALE, CHIC+)</td></tr><tr><td>PEEBLES</td><td>71</td><td>Physical Cosmology</td><td>P.Z. Peebles (PRIN)</td></tr><tr><td>Princeton Univ.</td><td></td><td>Press (1971)</td><td></td></tr><tr><td>SHVARTSMAN</td><td>69</td><td>JETPL 9 184</td><td>V.F. Shvartsman (MOSU)</td></tr><tr><td>HOYLE</td><td>64</td><td>Translated from ZETF 9 315.</td><td></td></tr><tr><td></td><td></td><td>NAT 203 1108</td><td>F. Hoyle, R.J. Tayler (CAMB)</td></tr></tbody></table>

## Double-$\beta$ Decay

OMITTED FROM SUMMARY TABLE

## NEUTRINOLESS DOUBLE-$\beta$ DECAY

Revised August 2019 by A. Piepke (University of Alabama) and P. Vogel (Caltech).

Observation of neutrinoless double-beta ($0\nu\beta\beta$) decay would signal violation of total lepton number conservation. The process can be mediated by an exchange of a light Majorana neutrino, or by an exchange of other particles. However, the existence of $0\nu\beta\beta$-decay requires a nonvanishing Majorana neutrino mass, no matter what the actual mechanism is. As long as only a limit on the lifetime is available, limits on the effective Majorana neutrino mass, on the lepton-number violating right-handed current or other possible mechanisms mediating $0\nu\beta\beta$ decay can be obtained, independently of the actual mechanism, by assuming that one of these “new physics” possibilities dominates. These limits are listed in the Double-$\beta$ Decay Listings of the experimental measurements.

In the following we assume that the exchange of light Majorana neutrinos ($m_{\nu_i} \le 10$ MeV) contributes dominantly to the decay rate. Besides a dependence on the phase space ($G^{0\nu}$) and the nuclear matrix element ($M^{0\nu}$), the observable $0\nu\beta\beta$-decay rate is proportional then to the square of the effective Majorana mass $m_{ee}$, $(T_{1/2}^{0\nu})^{-1} = G^{0\nu} \cdot |M^{0\nu}|^2 \cdot m_{ee}^2$, with $m_{ee}^2 = |\sum_i U_{ei}^2 m_{\nu_i}|^2$. The sum contains, in general, complex CP-phases in $U_{ei}^2$, i.e., cancellations may occur. For three neutrino flavors there are two physical phases for Majorana neutrinos ($\eta_1, \eta_2$) and one for Dirac neutrinos ($\delta_{CP}$). The relevant Majorana phases affect only processes to which lepton-number changing amplitudes contribute. Given the general 3 × 3 mixing matrix for Majorana neutrinos, one can construct other analogous lepton number violating quantities, $m_{\ell\ell'} = \sum_i U_{\ell i} U_{\ell' i} m_{\nu_i}$ ($\ell$ or $\ell' \ne e$). However, these are currently much less constrained than $m_{ee}$.

Nuclear structure calculations are needed to deduce $m_{ee}$ from the decay rate. While $G^{0\nu}$ can be calculated accurately, the computation of $M^{0\nu}$ is subject to uncertainty. Comparing different nuclear model evaluations indicates a factor ~2-3 spread in the calculated nuclear matrix elements. Nuclear structure calculation consistently overestimate Gamow-Teller (axial current) matrix elements. This inability of the nuclear models to reproduce Gamow-Teller decay rates is often parametrized in form of a modified coupling constant $g_A$. Many nuclear theorists interpret this shortcoming as evidence that important physics is missing in the modeling of weak nuclear transitions. It is not clear how these observed uncertainties impact $0\nu\beta\beta$-matrix elements. Nevertheless, this constitutes an additional element of uncertainty. Recent work, [1] shows how the discrepancy between experimental and theoretical axial current matrix elements might be resolved. However, application of this approach to the $0\nu\beta\beta$ decay remains to be accomplished. The particle physics quantities to be determined are thus nuclear model-dependent, so the half-life measurements are listed first. Where possible, we reference the nuclear matrix elements used in the subsequent analysis. Since rates for the conventional $2\nu\beta\beta$ decay serve to constrain the nuclear theory models, results for this process are also given.

<footnote n="1">¹ Limit varies with strength of coupling. See also WALKER 91.</footnote>
Oscillation experiments utilizing atmospheric, accelerator, solar, and reactor produced neutrinos and anti-neutrinos show that at least some neutrinos are massive. However, so far the inverted mass ordering (i.e., whether $\Delta m_{31}^2 < 0$) is disfavored only by $2-3 \sigma$ compared to the normal mass ordering (when $\Delta m_{31}^2 > 0$), while the absolute neutrino mass values or the properties of neutrinos under CPT-conjugation (Dirac or Majorana) remain undetermined. All confirmed oscillation experiments can be consistently described using three interacting neutrino species with two mass splittings and three mixing angles. (For values of the mixing angles and mass square differences see the corresponding tables.)

Based on the 3-neutrino analysis:

$m_{ee}^2 = |\cos^2 \theta_{13} \cos^2 \theta_{12} m_1 + e^{2i(\eta_2 - \eta_1)} \cos^2 \theta_{13} \sin^2 \theta_{12} m_2 + e^{-2i(\eta_1 + \delta_{CP})} \sin^2 \theta_{13} m_3|^2$, valid for both mass orderings. Given the present knowledge of the neutrino oscillation parameters one can derive a relation between the effective Majorana mass and the mass of the lightest neutrino, as illustrated in Figure 14.11 in the Neutrino Masses, Mixing and Oscillations review. The three mass orderings allowed by the oscillation data: normal ($m_1 < m_2 \ll m_3$), inverted ($m_3 \ll m_1 < m_2$), and degenerate ($m_1 \approx m_2 \approx m_3$), result in different projections. The width of the colored bands reflects the uncertainty introduced by the unknown Majorana and Dirac phases as well as the experimental errors of the oscillation parameters. The latter causes only minor broadening of the bands. Because of the overlap of the different mass scenarios, a measurement of $m_{ee}$ would not reveal which mass ordering is applicable, provided the value of $m_{ee}$ is in the overlapping range.

Analogous plots depict the relation of $m_{ee}$ with the summed neutrino mass $m_{tot} = m_1 + m_2 + m_3$, constrained by observational cosmology, and $m_{ee}$ as a function of the average mass $m_{\nu_e}^{eff} = [\sum |U_{ei}|^2 m_{\nu_i}^2]^{1/2}$ determined through the analysis of the electron energy distribution in low energy beta decays. (See Fig. 1 of [2].) The oscillation data thus allow to test whether observed values of $m_{ee}$ and $m_{tot}$ or $m_{\nu_e}^{eff}$ are consistent within the 3 neutrino framework. The rather large intrinsic width of the $\beta\beta$-decay constraints essentially does not allow to positively identify the mass ordering, and thus the sign of $\Delta m_{31}^2$, even in combination with these other observables. Naturally, if a value of $0 < m_{ee} \le 0.01$ eV is ever established, then the normal mass ordering becomes the only possible scenario.

It should be noted that systematic uncertainties of the nuclear matrix elements and possible quenching of the axial current matrix elements are sometimes not folded into the mass limits reported by $\beta\beta$-decay experiments. Taking this additional uncertainty into account would further widen the projections. The plots are based on a 3-neutrino analysis. If it turns out that additional, i.e. sterile light neutrinos exist, the allowed regions would be modified substantially.

If neutrinoless double-beta decay is observed, it will be possible to fix a range of absolute values of the masses $m_{\nu_i}$. Unlike the direct neutrino mass measurements, however, a limit

on $m_{ee}$ does not allow one to constrain the individual mass values $m_{\nu_i}$ even when the mass differences $\Delta m_{ij}^2$ are known.

Neutrino oscillation data imply the existence of a lower limit $\sim 0.014$ eV for the Majorana neutrino mass for the inverted mass ordering pattern, while $m_{ee}$ could, by fine tuning, vanish in the case of the normal mass ordering. Several new double-beta searches have been proposed to probe the interesting $m_{ee}$ mass range, with the prospect of full coverage of the inverted mass ordering region within the next decade.

The $0\nu\beta\beta$ decay mechanism discussed so far is not the only way in which the decay can occur. Numerous other possible scenarios have been proposed, however, all of them requiring new physics. It will be a challenging task to decide which mechanism was responsible once $0\nu\beta\beta$ decay is observed. LHC experiments may reveal corresponding signatures for new physics of lepton number violation. If lepton-number violating right-handed weak current interactions exist, its strength can be characterized by the phenomenological coupling constants $\eta$ and $\lambda$ ($\eta$ describes the coupling between the right-handed lepton current and left-handed quark current while $\lambda$ describes the coupling when both currents are right-handed). The $0\nu\beta\beta$ decay rate then depends on $\langle\eta\rangle = \eta \sum_i U_{ei} V_{ei}$ and $\langle\lambda\rangle = \lambda \sum_i U_{ei} V_{ei}$ that vanish for massless or unmixed neutrinos ($V_{\ell j}$ is a matrix analogous to $U_{\ell j}$ but describing the mixing with the hypothetical right-handed neutrinos). The observation of the single electron spectra could, in principle, allow to distinguish this mechanism of $0\nu\beta\beta$ from the light Majorana neutrino exchange driven mode. The limits on $\langle\eta\rangle$ and $\langle\lambda\rangle$ are listed in a separate table. The reader is cautioned that a number of earlier experiments did not distinguish between $\eta$ and $\lambda$. In addition, see the section on Majoron searches for additional limits set by these experiments.

## References

1. P. Gysbers et al., Nature Phys. **15**, 5 (2019); [arXiv:1903.00047].

2. M.J. Dolinski, A.W.P. Poon and W. Rodejohann, Ann. Rev. Nucl. Part. Sci. **49**, 219 (2019); [arXiv:1902.04097].

## Half-life $0\nu$ double-$\beta$ decay

In most cases the transitions $(Z,A) \rightarrow (Z+2,A) + 2e^{-}$ to the $0^{+}$ ground state of the final nucleus are listed. We also list transitions that decrease the nuclear charge $(2e^{+}, e^{+} CC \text{ and double EC})$ and transitions to an excited state of the final nucleus $(0_{i}^{+}, 2^{+}, \text{ and } 2_{i}^{+})$. In the following Listings only the best or comparable limits for the half-lives of each transition are reported and only those with about $T_{1/2} > 10^{23}$ years that are relevant for particle physics.

<table><thead><tr><th>t<sub>1/2</sub>(10<sup>23</sup> yr)</th><th>CL% ISOTOPE</th><th>TRANSITION METHOD</th><th>DOCUMENT ID</th></tr></thead><tbody><tr><td>&gt;2300</td><td>90 <sup>136</sup>Xe</td><td>KamLAND-Zen</td><td><sup>1</sup>ABE</td></tr><tr><td>&gt; 830</td><td>90 <sup>76</sup>Ge</td><td>MAJORANA</td><td><sup>2</sup>ARNQUIST</td></tr><tr><td>&gt; 2.1</td><td>90 <sup>100</sup>Mo</td><td>g.s. → <sup>2+</sup><sub>1</sub> CUPID-Mo</td><td><sup>3</sup>AUGIER</td></tr><tr><td>&gt; 1.2</td><td>90 <sup>100</sup>Mo</td><td>g.s. → <sup>0+</sup><sub>1</sub> CUPID-Mo</td><td><sup>3</sup>AUGIER</td></tr><tr><td>&gt; 13</td><td>90 <sup>136</sup>Xe</td><td>NEXT</td><td><sup>4</sup>NOVELLA</td></tr><tr><td>&gt; 220</td><td>90 <sup>130</sup>Te</td><td>CUORE</td><td><sup>5</sup>ADAMS</td></tr><tr><td>&gt; 36</td><td>90 <sup>128</sup>Te</td><td>CUORE</td><td><sup>6</sup>ADAMS</td></tr></tbody></table>
<table><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>&gt; 12</td><td>90</td><td><sup>136</sup>Xe</td><td>XENON1T</td><td>7</td><td>APRILE 22A</td></tr><tr><td>&gt; 18</td><td>90</td><td><sup>100</sup>Mo</td><td>CUPID-Mo</td><td>8</td><td>AUGIER 22</td></tr><tr><td>&gt; 46</td><td>90</td><td><sup>82</sup>Se</td><td>CUPID-0</td><td>9</td><td>AZZOLINI 22</td></tr><tr><td>&gt; 1.8</td><td>90</td><td><sup>82</sup>Se</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> CUPID-0</td><td>10</td><td>AZZOLINI 22</td></tr><tr><td>&gt; 3.0</td><td>90</td><td><sup>82</sup>Se</td><td>g.s. → 2<sup>+1</sup><sub>1</sub> CUPID-0</td><td>11</td><td>AZZOLINI 22</td></tr><tr><td>&gt; 3.2</td><td>90</td><td><sup>82</sup>Se</td><td>g.s. → 2<sup>+2</sup><sub>2</sub> CUPID-0</td><td>12</td><td>AZZOLINI 22</td></tr><tr><td>&gt; 59</td><td>90</td><td><sup>130</sup>Te</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> CUORE</td><td>13</td><td>ADAMS 21A</td></tr><tr><td>&gt; 15</td><td>90</td><td><sup>100</sup>Mo</td><td>CUPID-Mo</td><td>14</td><td>ARMENGAUD 21</td></tr><tr><td>&gt; 39.9</td><td>90</td><td><sup>76</sup>Ge</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> MAJORANA-Dem</td><td>15</td><td>ARNQUIST 21</td></tr><tr><td>&gt; 21.2</td><td>90</td><td><sup>76</sup>Ge</td><td>g.s. → 2<sup>+1</sup><sub>1</sub> MAJORANA-Dem</td><td>16</td><td>ARNQUIST 21</td></tr><tr><td>&gt; 9.7</td><td>90</td><td><sup>76</sup>Ge</td><td>g.s. → 2<sup>+2</sup><sub>1</sub> MAJORANA-Dem</td><td>17</td><td>ARNQUIST 21</td></tr><tr><td>&gt; 320</td><td>90</td><td><sup>130</sup>Te</td><td>CUORE</td><td>18</td><td>ADAMS 20A</td></tr><tr><td>&gt;1800</td><td>90</td><td><sup>76</sup>Ge</td><td>GERDA</td><td>19</td><td>AGOSTINI 20B</td></tr><tr><td>&gt; 14</td><td>90</td><td><sup>130</sup>Te</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> CUORE-0</td><td>20</td><td>ALDUINO 19</td></tr><tr><td>&gt; 0.95</td><td>90</td><td><sup>100</sup>Mo</td><td>AMoRE</td><td>21</td><td>ALENKOV 19</td></tr><tr><td>&gt; 350</td><td>90</td><td><sup>136</sup>Xe</td><td>EXO-200</td><td>22</td><td>ANTON 19</td></tr><tr><td>&gt; 2.4</td><td>90</td><td><sup>136</sup>Xe</td><td>PANDAX-II</td><td>23</td><td>NI 19</td></tr><tr><td>&gt; 150</td><td>90</td><td><sup>130</sup>Te</td><td>CUORE</td><td>24</td><td>ALDUINO 19</td></tr><tr><td>&gt; 2.5</td><td>90</td><td><sup>82</sup>Se</td><td>NEMO-3</td><td>25</td><td>ARNOLD 18</td></tr><tr><td>&gt; 2.2</td><td>90</td><td><sup>116</sup>Cd</td><td>AURORA</td><td>26</td><td>BARABASH 18</td></tr><tr><td>&gt; 1.1</td><td>90</td><td><sup>134</sup>Xe</td><td>EXO-200</td><td>27</td><td>ALBERT 17C</td></tr><tr><td>&gt; 1</td><td>90</td><td><sup>116</sup>Cd</td><td>NEMO-3</td><td>28</td><td>ARNOLD 17</td></tr><tr><td>&gt; 40</td><td>90</td><td><sup>130</sup>Te</td><td>CUORICINO</td><td>29</td><td>ALDUINO 16</td></tr><tr><td>&gt; 260</td><td>90</td><td><sup>136</sup>Xe</td><td>g.s. → 2<sup>+1</sup><sub>1</sub> KamLAND-Zen</td><td>30</td><td>ASAKURA 16</td></tr><tr><td>&gt; 260</td><td>90</td><td><sup>136</sup>Xe</td><td>g.s. → 2<sup>+2</sup><sub>1</sub> KamLAND-Zen</td><td>31</td><td>ASAKURA 16</td></tr><tr><td>&gt; 240</td><td>90</td><td><sup>136</sup>Xe</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> KamLAND-Zen</td><td>32</td><td>ASAKURA 16</td></tr><tr><td>&gt; 11</td><td>90</td><td><sup>100</sup>Mo</td><td>NEMO-3</td><td>33</td><td>ARNOLD 15</td></tr><tr><td>&gt; 9.4</td><td>90</td><td><sup>130</sup>Te</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> CUORICINO</td><td>34</td><td>ANDREOTTI 12</td></tr><tr><td>&gt; 0.58</td><td>90</td><td><sup>48</sup>Ca</td><td>CaF<sub>2</sub> scint.</td><td>35</td><td>UMEHARA 08</td></tr><tr><td>&gt; 0.89</td><td>90</td><td><sup>100</sup>Mo</td><td>g.s. → 0<sup>+1</sup><sub>1</sub> NEMO-3</td><td>36</td><td>ARNOLD 07</td></tr><tr><td>&gt; 1.6</td><td>90</td><td><sup>100</sup>Mo</td><td>g.s. → 2<sup>+1</sup><sub>1</sub> NEMO-3</td><td>37</td><td>ARNOLD 07</td></tr><tr><td>&gt; 1.1</td><td>90</td><td><sup>128</sup>Te</td><td>Cryog. det.</td><td>38</td><td>ARNABOLDI 03</td></tr><tr><td>&gt; 1.7</td><td>90</td><td><sup>116</sup>Cd</td><td>116CdWO<sub>4</sub> scint.</td><td>39</td><td>DANEVICH 03</td></tr><tr><td>&gt; 157</td><td>90</td><td><sup>76</sup>Ge</td><td>Enriched HPGe</td><td>40</td><td>AALSETH 02B</td></tr></tbody></table>

1 ABE 23 use the combined data set of the KamLAND-Zen 400 and 800 experiments, utilizing 745 kg of isotopically enriched xenon (90.9% $^{136}$Xe), dissolved in liquid scintillator and an exposure of 970 kg·yr of $^{136}$Xe, to derive this limit on $\nu\beta\beta$ decay. A half-life sensitivity of $1.5 \times 10^{26}$ yr is reported.

2 ARNQUIST 23 use the final data set of the MAJORANA DEMONSTRATOR experiment, operating enriched in $^{76}$Ge detectors, to set this limit on the $\nu\beta\beta$ half-life of $^{76}$Ge. The exposure is 64.5 kg·yr. A median sensitivity of $8.1 \times 10^{25}$ yr is reported.

3 AUGIER 23 utilize the complete data, set collected by the CUPID-Mo bolometric calorimeter with particle ID and located at the LSM, to study various double beta decays of $^{100}$Mo to excited states of the daughter nucleus. An exposure of 1.47 kg·yr of $^{100}$Mo is available.

4 NOVELLA 23 use data collected by the NEXT-White experiment to limit the $\nu\beta\beta$ half-life of $^{136}$Xe. The experiment contains 3.5 kg of enriched Xe and is based on a high-pressure gas TPC. Two different limits are reported, based on different data analysis approaches, $> 5.5 \times 10^{23}$ yr and $> 13 \times 10^{23}$ yr.

5 ADAMS 22A use the CUORE TeO<sub>2</sub> experiment with an exposure of 288.8 kg·yr of $^{130}$Te to place a limit on its $\nu\beta\beta$ decay. The median sensitivity is reported as $280 \times 10^{23}$ yr. Supersedes ADAMS 20A.

6 ADAMS 22B use the CUORE bolometric calorimeter to place a limit on the $\nu\beta\beta$ decay half-life of $^{128}$Te.

7 APRILE 22A use 36.16 kg·yr of $^{136}$Xe exposure of the XENON1T not enriched detector to establish the stated limit.

8 AUGIER 22 use the final data set of the CUPID-Mo cryogenic calorimeter, utilizing enriched $^{Li}_2^{100}MoO_4$ and an isotope exposure of 1.47 kg·y, to place a limit on the $\nu\beta\beta$ decay half-life.

9 AZZOLINI 22 use the CUPID-0 scintillating cryogenic bolometer to set a limit on the $\nu\beta\beta$ half-life of $^{82}$Se. The analyzed isotope exposure is 8.82 kg·yr. A median sensitivity of $7 \times 10^{24}$ yr is reported. Supersedes AZZOLINI 19.

10 AZZOLINI 22 use CUPID-0 data with an isotope exposure of 8.82 kg·yr to set a limit on the $\nu\beta\beta$ decay to the first excited $0^+$ state.

11 AZZOLINI 22 use CUPID-0 data with an isotope exposure of 8.82 kg·yr to set a limit on the $\nu\beta\beta$ decay to the first excited $2^+$ state.

12 AZZOLINI 22 use CUPID-0 data with an isotope exposure of 8.82 kg·yr to set a limit on the $\nu\beta\beta$ decay to the second excited $2^+$ state.

13 ADAMS 21A et al. used 101.76 kg yr of $^{130}$Te exposure of the CUORE (LNGS) bolometric detector to place a limit on the decay to the first excited state of $^{130}$Xe, superseding ALDUINO 19 as the most restrictive bound on this particular decay.

14 ARMENGAUD 21 use the CUPID-Mo 4.2 kg array of enriched $^{Li}_2^{100}MoO_4$ scintillating bolometers, with 1.17 kg·yr exposure, to set this limit.

15 ARNQUIST 21 use the MAJORANA demonstrator to set this limit for the $\nu\beta\beta$ decay to the first excited $0^+$ state, with a 41.9 kg yr isotopic exposure. The median sensitivity is $39.9 \times 10^{23}$ yr.

16 ARNQUIST 21 use the MAJORANA demonstrator to set this limit for the $\nu\beta\beta$ decay to the first excited $2^+$ state, with a 41.9 kg yr isotopic exposure. The median sensitivity is $21.2 \times 10^{23}$ yr.

17 ARNQUIST 21 use the MAJORANA demonstrator to set this limit for the $\nu\beta\beta$ decay to the second excited $2^+$ state, with a 41.9 kg yr isotopic exposure. The median sensitivity is $18.6 \times 10^{23}$ yr.

18 ADAMS 20A use the CUORE detector to search for the $\nu\beta\beta$ decay of $^{130}$Te. The exposure was 372.5 kg·yr of TeO<sub>2</sub> corresponding to 103.6 kg·yr of $^{130}$Te. The exclusion sensitivity is $1.7 \times 10^{25}$ yr. Supersedes ALDUINO 18.

19 AGOSTINI 20 present the final data set of the GERDA experiment, searching for $\nu\beta\beta$ decay of $^{76}$Ge with isotopically enriched, high resolution Ge detectors. A final exposure of 127.2 kg·yr is reported. The experiment reports the lowest and longest half life limit ever achieved by any double beta decay experiment. The reported experiment sensitivity equals the limit. Supersedes AGOSTINI 19.

20 ALDUINO 19 use the combined data of the CUORICINO and CUORE-0 experiments to place a lower limit on the half life of the $\nu\beta\beta$ decay of $^{130}$Te to the first excited $0^+$ state of $^{130}$Xe. Supersedes ANDREOTTI 12.

21 ALENKOV 19 report the $\nu\beta\beta$ decay half-life limit based on the 52.1 kg·d exposure of $^{100}$Mo, of a cryogenic dual heat and light detector in the Yangyang underground laboratory. The median sensitivity is $1.1 \times 10^{23}$ years.

22 ANTON 19 uses the complete dataset of the EXO-200 detector to search for the $\nu\beta\beta$ decay. The exposure is 234.1 kg yr. The median sensitivity is $5.0 \times 10^{25}$ yr. Supersedes ALBERT 18 and ALBERT 14b.

23 NI 19 use the PandaX-II dual phase TPC at CJPL to search for the $\nu\beta\beta$ decay of $^{136}$Xe. The half-life limit $2.4 \times 10^{23}$ yr is obtained from 22.2 kg yr exposure with a sensitivity of $1.9 \times 10^{23}$ yr.

24 ALDUINO 18 uses the CUORE detector to search for the $\nu\beta\beta$ decay of $^{130}$Te. The exposure is 86.3 kg·year of natural TeO<sub>2</sub> corresponding to 24.0 kg·year for $^{130}$Te. The median sensitivity is $0.7 \times 10^{25}$ yr. The limit is obtained combining the new data from CUORE with those of CUORE0 (9.8 kg·year of $^{130}$Te) and Cuoricino (19.8 kg·year of $^{130}$Te).

25 ARNOLD 18 use the NEMO-3 tracking calorimeter to place a limit on the $\nu\beta\beta$ decay of $^{82}$Se. This is a slightly weaker limit than in BARABASH 11A, using the same detector. Supersedes ARNOLD 05A.

26 BARABASH 18 use 1.162 kg of $^{116}$CdWO<sub>4</sub> scintillating crystals to obtain this limit. Supersedes DANEVICH 03 with analogous source and is more sensitive than ARNOLD 17.

27 ALBERT 17c uses the EXO-200 detector that contains 19.098 ± 0.014% admixture of $^{134}$Xe to search for the $\nu\beta$ and $2\nu\beta\beta$ decay modes. The exposure is 29.6 kg·year. The median sensitivity is $1.9 \times 10^{21}$ years.

28 ARNOLD 17 use the NEMO-3 tracking calorimeter, containing 410 g of enriched $^{116}$Cd exposed for 5.26 yr, to determine the half-life limit. Supersedes BARABASH 11A.

29 ALDUINO 16 report result obtained with 9.8 kg·y of data collected with the CUORE-0 bolometer, combined with data from the CUORICINO. Supersedes ALFONSO 15.

30 ASAKURA 16 use the KamLAND-Zen liquid scintillator calorimeter ($^{136}$Xe 89.5 kg yr) to place a limit on the $\nu\beta\beta$-decay into the first excited state of the daughter nuclide.

31 ASAKURA 16 use the KamLAND-Zen liquid scintillator calorimeter ($^{136}$Xe 89.5 kg yr) to place a limit on the $\nu\beta\beta$-decay into the second excited state of the daughter nuclide.

32 ASAKURA 16 use the KamLAND-Zen liquid scintillator calorimeter ($^{136}$Xe 89.5 kg yr) to place a limit on the $\nu\beta\beta$-decay into the third excited state of the daughter nuclide.

33 ARNOLD 15 use the NEMO-3 tracking calorimeter with 34.3 kg yr exposure to determine the limit of $\nu\beta\beta$-half life of $^{100}$Mo. Supersedes ARNOLD 2005A and BARABASH 11A.

34 ANDREOTTI 12 use high resolution TeO<sub>2</sub> bolometric calorimeter to search for the $\nu\beta\beta$ decay of $^{130}$Te leading to the excited $0^+$ state at 1793.5 keV.

35 UMEHARA 08 use CaF<sub>2</sub> scintillation calorimeter to search for double beta decay of $^{48}$Ca. Limit is significantly more stringent than quoted sensitivity: $18 \times 10^{21}$ years.

36 Limit on $\nu$-decay to the first excited $0^+$-state of daughter nucleus using NEMO-3 tracking calorimeter. Supersedes DASSIE 95.

37 Limit on $\nu$-decay to the first excited $2^+$-state of daughter nucleus using NEMO-3 tracking calorimeter.

38 Supersedes ALESSANDRELLO 00. Array of TeO<sub>2</sub> crystals in high resolution cryogenic calorimeter. Some enriched in $^{128}$Te. Ground state to ground state decay.

39 Limit on $\nu\beta\beta$ decay of $^{116}$Cd using enriched CdWO<sub>4</sub> scintillators. Supersedes DANEVICH 00.

40 AALSETH 02b limit is based on 117 mol·yr of data using enriched Ge detectors. Background reduction by means of pulse shape analysis is applied to part of the data set. Reported limit is slightly less restrictive than that in KLAPDOR-KLEINGROTHAUS 01b However, it excludes part of the allowed half-life range reported in KLAPDOR-KLEINGROTHAUS 01b for the same nuclide. The analysis has been criticized in KLAPDOR-KLEINGROTHAUS 04b. The criticism was addressed and disputed in AALSETH 04.

## Half-life measurements of the two-neutrino double-$\beta$ decay

The measured half-life values for the transitions $(Z,A) \rightarrow (Z+2,A) + 2e^{-} + 2\pi e$ to the $0^+$ ground state of the final nucleus are listed. We also list the transitions to an excited state of the final nucleus $(0^+, \text{etc.})$. We report only the measurements with the smallest (or comparable) uncertainty for each transition.

<table><thead><tr><th>t<sub>1/2</sub>(10<sup>20</sup> yr)</th><th>ISOTOPE</th><th>TRANSITION METHOD</th><th>DOCUMENT ID</th></tr></thead><tbody><tr><td>20.22 ± 0.18 ± 0.38</td><td>76Ge</td><td>GERDA</td><td>1 AGOSTINI 23</td></tr><tr><td>1.11 ± 0.19 ± 0.17</td><td>150Nd 0<sup>+</sup> → 0<sup>+1</sup><sub>1</sub> NEMO-3</td><td></td><td>2 AGUERRE 23</td></tr></tbody></table>
# Lepton Particle Listings

## Double-$\beta$ Decay

<table><tbody><tr><td>7.5</td><td>± 0.8</td><td>+ 0.4<br/>- 0.3</td><td><sup>100</sup>Mo</td><td>0<sup>+</sup> → 0<sub>1</sub><sup>+</sup> CUPID-Mo</td><td>3 AUGIER</td><td>23</td></tr><tr><td>0.0707</td><td>± 0.0002</td><td>± 0.0011</td><td><sup>100</sup>Mo</td><td>CUPID-Mo</td><td>4 AUGIER</td><td>23A</td></tr><tr><td>0.869</td><td>± 0.005</td><td>+ 0.009<br/>- 0.006</td><td>82<sub>Se</sub></td><td>CUPID-0</td><td>5 AZZOLINI</td><td>23A</td></tr><tr><td>21.6</td><td>+ 6.2<br/>- 4.0</td><td>+ 4.0<br/>- 2.9</td><td>136<sub>Xe</sub></td><td>NEXT</td><td>6 NOVELLA</td><td>23</td></tr><tr><td>21900</td><td>± 700</td><td></td><td>128<sub>Te</sub></td><td>CUORE</td><td>7 ADAMS</td><td>22B</td></tr><tr><td>110</td><td>± 20</td><td>± 10</td><td>124<sub>Xe</sub></td><td>XENON1T</td><td>8 APRILE</td><td>22A</td></tr><tr><td>118</td><td>± 13</td><td>± 14</td><td>124<sub>Xe</sub></td><td>XENONnT</td><td>9 APRILE</td><td>22B</td></tr><tr><td>23.4</td><td>+ 0.8<br/>- 4.6</td><td>+ 3.0<br/>- 1.7</td><td>136<sub>Xe</sub></td><td>NEXT</td><td>10 NOVELLA</td><td>22</td></tr><tr><td>8.76</td><td>+ 0.09<br/>- 0.07</td><td>+ 0.14<br/>- 0.17</td><td>130<sub>Te</sub></td><td>CUORE</td><td>11 ADAMS</td><td>21</td></tr><tr><td>180</td><td>± 50</td><td>± 10</td><td>124<sub>Xe</sub> 2ν/DEC</td><td>XENON1T</td><td>12 APRILE</td><td>19E</td></tr><tr><td>0.0680</td><td>± 0.0001</td><td>+ 0.0038<br/>- 0.0040</td><td><sup>100</sup>Mo</td><td>NEMO-3</td><td>13 ARNOLD</td><td>19</td></tr><tr><td>0.939</td><td>± 0.017</td><td>± 0.058</td><td>82<sub>Se</sub></td><td>NEMO-3</td><td>14 ARNOLD</td><td>18</td></tr><tr><td>0.263</td><td>+ 0.011<br/>- 0.012</td><td></td><td>116<sub>Cd</sub></td><td>AURORA</td><td>15 BARABASH</td><td>18</td></tr><tr><td>> 0.87</td><td></td><td></td><td>134<sub>Xe</sub></td><td>EXO-200</td><td>16 ALBERT</td><td>17C</td></tr><tr><td>8.2</td><td>± 0.2</td><td>± 0.6</td><td>130<sub>Te</sub></td><td>CUORE-0</td><td>17 ALDUINO</td><td>17</td></tr><tr><td>0.274</td><td>± 0.004</td><td>± 0.018</td><td>116<sub>Cd</sub></td><td>NEMO-3</td><td>18 ARNOLD</td><td>17</td></tr><tr><td>0.64</td><td>+ 0.07<br/>- 0.06</td><td>+ 0.12<br/>- 0.09</td><td>48<sub>Ca</sub></td><td>NEMO-3</td><td>19 ARNOLD</td><td>16</td></tr><tr><td>0.0934</td><td>± 0.0022<sup>+</sup></td><td>- 0.0062<br/>- 0.0060</td><td>150<sub>Nd</sub></td><td>NEMO-3</td><td>20 ARNOLD</td><td>16A</td></tr><tr><td>19.26</td><td>± 0.94</td><td></td><td>76<sub>Ge</sub></td><td>GERDA</td><td>21 AGOSTINI</td><td>15A</td></tr><tr><td>0.0693</td><td>± 0.0004</td><td></td><td>100<sup>Mo</sup></td><td>NEMO-3</td><td>22 ARNOLD</td><td>15</td></tr><tr><td>21.65</td><td>± 0.16</td><td>± 0.59</td><td>136<sub>Xe</sub></td><td>EXO-200</td><td>23 ALBERT</td><td>14</td></tr><tr><td>92</td><td>+ 55<br/>- 26</td><td>± 13</td><td>78<sub>Kr</sub></td><td>BAKSAN</td><td>24 GAVRILYAK</td><td>13</td></tr><tr><td>23.8</td><td>± 0.2</td><td>± 1.4</td><td>136<sub>Xe</sub></td><td>KamLAND-Z</td><td>25 GANDO</td><td>12A</td></tr><tr><td>7.0</td><td>± 0.9</td><td>± 1.1</td><td>130<sub>Te</sub></td><td>NEMO-3</td><td>26 ARNOLD</td><td>11</td></tr><tr><td>0.235</td><td>± 0.014</td><td>± 0.016</td><td>96<sub>Zr</sub></td><td>NEMO-3</td><td>27 ARGYRIADES</td><td>10</td></tr><tr><td>6.9</td><td>+ 1.0<br/>- 0.8</td><td>± 0.7</td><td>100<sup>Mo</sup> 0<sup>+</sup> → 0<sub>1</sub><sup>+</sup> Ge coinc.</td><td></td><td>28 BELLI</td><td>10</td></tr><tr><td>5.7</td><td>+ 1.3<br/>- 0.9</td><td>± 0.8</td><td>100<sup>Mo</sup> 0<sup>+</sup> → 0<sub>1</sub><sup>+</sup> NEMO-3</td><td></td><td>29 ARNOLD</td><td>07</td></tr><tr><td>0.96</td><td>± 0.03</td><td>± 0.10</td><td>82<sub>Se</sub></td><td>NEMO-3</td><td>30 ARNOLD</td><td>05A</td></tr><tr><td>0.29</td><td>+ 0.04<br/>- 0.03</td><td></td><td>116<sub>Cd</sub></td><td>CdWO<sub>4</sub> sc.</td><td>31 DANEVICH</td><td>03</td></tr></tbody></table>

1. AGOSTINI 23 report an updated value for the $2\nu\beta\beta$ half-life of $^{76}$Ge; the final result of the GERDA Phase II experiment. A subset of the data, corresponding to an exposure of exposure is 11.8 kg·yr, is utilized. This is one of the most precise measurements of $2\nu\beta\beta$ decay reported in the literature. An effective nuclear matrix element of 0.101 ± 0.001 is derived from this result.

2. AGUERRE 23 report the results of a 5.25 yr search for the $2\nu\beta\beta$ decay to the exited $0^+ \to 0_1^+$ state of the daughter nucleus, using the NEMO-3 tracking calorimeter. 36.6g of $^{150}$Nd isotope were available for the measurement of this decay rate.

3. AUGIER 23 utilize the complete data, set collected by the CUPID-Mo bolometric calorimeter with particle ID and located at the LSM, to measure the $^{100}$Mo $2\nu\beta\beta$ half-life to exited $0_1^+$ state of the daughter nucleus. An exposure of 1.47 kg·yr of $^{100}$Mo is available.

4. AUGIER 23A use full data set collected by the CUPID-Mo experiment to derive an improved $2\nu\beta\beta$ g.s. to g.s. half-life of $^{100}$Mo. An exposure of 1.48 kg·yr of $^{100}$Mo is utilized. Supersedes ARMENGAUD 20.

5. AZZOLINI 23A report an improved measurement of the $2\nu\beta\beta$ decay with an exposure of 8.82 kg·yr of $^{82}$Se, collected with the CUPID-0 detector. Superseded AZZOLINI 19B.

6. NOVELLA 23 used the NEXT-White experiment, with a fiducial mass of 3.5 kg of enriched xenon, to measure the $2\nu\beta\beta$ g.s. to g.s. half-life of $^{136}$Xe. The experiment is based on a high-pressure gas TPC. Supersedes NOVELLA 22.

7. ADAMS 22B derive the $2\nu\beta\beta$ half-life of $^{128}$Te from data of the CUORE bolometric calorimeter and the half-life ratio for $^{130}$Te / $^{128}$Te reported in BERNATOWICZ 92.

8. APRILE 22A report an improved $^{124}$Xe $2\nu$DEC half-life measurement for $^{124}$Xe, using data collected by the XENON1T detector with an isotopically not enriched Xe target. The analyzed $^{124}$Xe exposure is 0.87 kg·yr. The statistical significance of the signal is 7.0 sigma. The stated half-life considers captures from the K shell up to the N5 shell. This result supersedes APRILE 19e, which exclusively considered captures from the K shell.

9. APRILE 22B use data collected by the XENONnT dark matter experiment to derive an improved $^{124}$Xe $2\nu$DEC half-life measurement for $^{124}$Xe. This result supersedes APRILE 22A.

10. NOVELLA 22 report on a high-pressure gas TPC at Canfran underground laboratory, filled with 3.5 kg (fiducial) xenon gas, used to measure the $2\nu\beta\beta$ decay of $^{136}$Xe. Topological track reconstruction is utilized in the data analysis. The measurement is based on comparing runs with isotopically enriched and depleted xenon. Other measurements with smaller error exist.

11. ADAMS 21 use $102.7$ kg·yr of $^{130}$Te exposure, collected by the CUORE bolometric detector at LNGS, to perform a measurement of the $2\nu\beta\beta$ decay of $^{130}$Te. The dataset is more than 10-times that collected by the CUORE-0 experiment. The result has been revised in ADAMS 23A. Supersedes ALDUINO 17.

12. APRILE 19e report first measurement of two-neutrino double electron capture in $^{124}$Xe using the XENON1T detector with a 0.73 t-yr exposure. An excess of 126 ± 29 events is observed at 64.3 ± 0.6 keV decay energy, corresponding to $\sqrt{\Delta\chi^2} = 4.4$ with respect to the background-only hypothesis.

13. ARNOLD 19 use the NEMO-3 tracking calorimeter with 34.3 kg y exposure to determine the $2\nu\beta\beta$ half-life of $^{100}$Mo. Supersedes ARNOLD 15.

14. ARNOLD 18 use the NEMO-3 tracking detector to determine the $2\nu\beta\beta$ half-life of $^{82}$Se. 0.93 kg of $^{82}$Se was observed for 5.25 y. The half-life value was obtained based on the single-state-dominance (SSD) hypothesis, preferred in this case by about 2 σ. Supersedes ARNOLD 05A.

15. BARABASH 18 use 1.162 kg of $^{116}$CdWO$_4$ scintillating crystals to obtain this value. Supersedes DANEVICH 03 with analogous source and agrees with ARNOLD 17 with the NEMO-3 detector.

16. ALBERT 17c uses the EXO-200 detector that contains 19.098 ± 0.014% admixture of $^{134}$Xe to search for the $2\nu\beta\beta$ decay mode. The exposure is 29.6 kg·year. The median sensitivity is $1.2 \times 10^{21}$ years.

17. ALDUINO 17 use the CUORE-0 detector containing 10.8 kg of $^{130}$Te in 52 crystals of TeO$_2$. The exposure was 9.3 kg yr of $^{130}$Te. This is a more accurate rate determination than in ARNOLD 11 and BARABASH 11A.

18. ARNOLD 17 use the NEMO-3 tracking calorimeter, containing 410 grams of enriched $^{116}$Cd exposed for 5.26 years, to determine the half-life value.

19. ARNOLD 16 use the NEMO-3 detector and a source of 6.99 g of $^{48}$Ca. The half-life is based on 36.7 g year exposure. It is consistent, although somewhat longer, than the previous determinations of the half-life. Supersedes BARABASH 11A.

20. ARNOLD 16A use the NEMO-3 tracking calorimeter, containing 36.6 g of $^{150}$Nd exposed for 1918.5 days, to determine the half-life. Supersedes ARGYRIADES 09.

21. AGOSTINI 15A use 17.9 kg yr exposure of the GERDA calorimeter to derive an improved measurement of the $2\nu\beta\beta$ decay half life of $^{76}$Ge.

22. ARNOLD 15 use the NEMO-3 tracking calorimeter with 34.3 kg yr exposure to determine the $2\nu\beta\beta$-half life of $^{100}$Mo. Supersedes ARNOLD 05A and ARNOLD 04.

23. ALBERT 14 use the EXO-200 tracking detector for a re-measurement of the $2\nu\beta\beta$-half life of $^{136}$Xe. A nuclear matrix element of 0.0218 ± 0.0003 MeV$^{-1}$ is derived from this data. Supersedes ACKERMAN 11.

24. GAVRILYAK 13 use a proportional counter filled with Kr gas to search for the $2\nu$2K decay of $^{78}$Kr. Data with the enriched and depleted Kr were used to determine signal and background. A 2.5σ excess of events obtained with the enriched sample is interpreted as an indication for the presence of this decay.

25. GANDO 12A use a modification of the existing KamLAND detector. The $\beta\beta$ decay source/detector is 13 tons of enriched $^{136}$Xe-loaded scintillator contained in an inner balloon. The $2\nu\beta\beta$ decay rate is derived from the fit to the spectrum between 0.5 and 4.8 MeV. This result is in agreement with ACKERMAN 11.

26. ARNOLD 11 use enriched $^{130}$Te in the NEMO-3 detector to measure the $2\nu\beta\beta$ decay rate. This result is in agreement with, but more accurate than ARNABOLDI 03.

27. ARGYRIADES 10 use 9.4 ± 0.2 g of $^{96}$Zr in NEMO-3 detector and identify its $2\nu\beta\beta$ decay. The result is in agreement and supersedes ARNOLD 99.

28. BELLI 10 use enriched $^{100}$Mo with 4 HP Ge detectors to record the 590.8 and 539.5 keV γ rays from the decay of the $0_1^+$ state in $^{100}$Ru both in singles and coincidences. This result confirms the measurement of KIDD 09 and ARNOLD 07 and supersedes them.

29. First exclusive measurement of $2\nu$-decay to the first excited $0_1^+$-state of daughter nucleus. ARNOLD 07 use the NEMO-3 tracking calorimeter to detect all particles emitted in decay. Result agrees with the inclusive $(0\nu + 2\nu)$ measurement of DEBRAECKELEER 01.

30. ARNOLD 05A use the NEMO-3 tracking detector to determine the $2\nu\beta\beta$ half-life of $^{82}$Se with high statistics and low background (389 days of data taking). Supersedes ARNOLD 04.

31. DANEVICH 03 is calorimetric measurement of $2\nu\beta\beta$ ground state decay of $^{116}$Cd using enrichedCdWO$_4$ scintillators. Agrees with EJIRI 95 and ARNOLD 96. Supersedes DANEVICH 00.

## $\langle m_{ee} \rangle$, The Effective Weighted Sum of Majorana Neutrino Masses
Contributing to Neutrinoless Double-$\beta$ Decay

$\langle m_{ee} \rangle = |\sum U_{ei}^2 m_{\nu_i}|$, $i = 1,2,3$. It is assumed that $\nu_i$ are Majorana particles and that the transition is dominated by the known (light) neutrinos. Note that $U_{ei}^2$ and not $|U_{ei}|^2$ occur in the sum, and that consequently cancellations are possible. The experiments obtain the limits on $\langle m_{\nu_i} \rangle$ from the measured ones on $T_{1/2}$ using a range of nuclear matrix elements (NME), which is reflected in the spread of $\langle m_{\nu_i} \rangle$. Different experiments may choose different NME. All assume $g_A = 1.27$. In the following Listings, only the best or comparable limits for each isotope are reported. When not mentioned explicitly the transition is between ground states, but transitions between excited states are also reported.

<table><thead><tr><th>VALUE (eV)</th><th>ISOTOPE</th><th>METHOD</th><th>DOCUMENT ID</th></tr></thead><tbody><tr><td>&lt; 0.036-0.156</td><td>136<sub>Xe</sub> KamLAND-Zen</td><td></td><td>1 ABE</td><td>23</td></tr><tr><td>&lt; 0.113-0.269</td><td>76<sub>Ge</sub> MAJORANA</td><td></td><td>2 ARNUIST</td><td>23</td></tr><tr><td>&lt; 0.48-3.19</td><td>136<sub>Xe</sub> NEXT</td><td></td><td>3 NOVELLA</td><td>23</td></tr><tr><td>&lt; 0.09-0.305</td><td>130<sub>Te</sub> CUORE</td><td></td><td>4 ADAMS</td><td>22A</td></tr><tr><td>&lt; 0.8-2.5</td><td>136<sub>Xe</sub> XENON1T</td><td></td><td>5 APRILE</td><td>22A</td></tr><tr><td>&lt; 0.28-0.49</td><td>100<sub>Mo</sub> CUPID-Mo</td><td></td><td>6 AUGIER</td><td>22</td></tr><tr><td>&lt; 0.263-0.545</td><td>82<sub>Se</sub> CUPID-0</td><td></td><td>7 AZZOLINI</td><td>22</td></tr><tr><td>&lt; 0.31-0.54</td><td>100<sub>Mo</sub> CUPID-Mo</td><td></td><td>8 ARMENGAUD</td><td>21</td></tr><tr><td>&lt; 0.075-0.35</td><td>130<sub>Te</sub> CUORE</td><td></td><td>9 ADAMS</td><td>20A</td></tr><tr><td>&lt; 0.079-0.180</td><td>76<sub>Ge</sub> GERDA</td><td></td><td>10 AGOSTINI</td><td>20B</td></tr><tr><td>&lt; 1.2-2.1</td><td>100<sub>Mo</sub> AMORE</td><td></td><td>11 ALENKOV</td><td>19</td></tr><tr><td>&lt; 0.093-0.286</td><td>136<sub>Xe</sub> EXO-200</td><td></td><td>12 ANTON</td><td>19</td></tr><tr><td>&lt; 1.3-3.5</td><td>136<sub>Xe</sub> PANDAX-II</td><td></td><td>13 NI</td><td>19</td></tr></tbody></table>
<table><tr><td>&lt; 0.11-0.52</td><td><sup>130</sup>Te</td><td>CUORE</td><td>14</td></tr><tr><td>&lt; 1.2-3.0</td><td>82Se</td><td>NEMO-3</td><td>15</td></tr><tr><td>&lt; 1.0-1.7</td><td><sup>116</sup>Cd</td><td>AURORA</td><td>16</td></tr><tr><td>&lt; 1.4-2.5</td><td><sup>116</sup>Cd</td><td>NEMO-3</td><td>17</td></tr><tr><td>&lt; 0.27-0.76</td><td><sup>130</sup>Te</td><td>CUORICINO</td><td>18</td></tr><tr><td>&lt; 1.6-5.3</td><td><sup>150</sup>Nd</td><td>NEMO-3</td><td>19</td></tr><tr><td>&lt; 0.33-0.62</td><td><sup>100</sup>Mo</td><td>NEMO-3</td><td>20</td></tr><tr><td>&lt; 7.2-19.5</td><td>96Zr</td><td>NEMO-3</td><td>21</td></tr><tr><td>&lt; 3.5-22</td><td>48Ca</td><td>CaF<sub>2</sub> scint.</td><td>22</td></tr><tr><td>&lt; 1.5-1.7</td><td><sup>116</sup>Cd</td><td><sup>116</sup>CdWO<sub>4</sub> scint.</td><td>23</td></tr></table>

1 ABE 23 utilize 745 kg of $^{136}$Xe isotope exposure from the combined data set of the KamLAND-Zen 400 and 800 to derive a limit on $\langle m_{\beta\beta} \rangle$. The range reflects the author's assessment of the variability of the theoretically calculated nuclear matrix elements.

2 ARNQUIST 23 use the final data set of the MAJORANA DEMONSTRATOR experiment, with 64.5 kg·yr of isotop exposure, to derive an upper limit for $\langle m_{\beta\beta} \rangle$. The range reflects the author's assessment of the variability of the theoretically calculated nuclear matrix elements.

3 NOVELLA 23 use data collected with the NEXT-White experiment to derive a range of upper limits for $\langle m_{\beta\beta} \rangle$. The range reflects the author's assessment of the variability of the theoretically calculated nuclear matrix elements and both half-life limits stated in NOVELLA 23.

4 ADAMS 22A use 1038.4 kg·yr of TeO$_2$ exposure and collected by the CUORE experiment to determine this range of limits. The range reflects the uncertainty of nuclear matrix element calculations needed for the conversion of half-life to neutrino mass.

5 APRILE 22A use data taken with the XENON1T detector to limit the Majorana neutrino mass. 36.16 kg·yr of $^{136}$Xe exposure were utilized. The reported range of limits is due to uncertainties in the nuclear matrix elements.

6 AUGIER 22 use the final data set of the CUPID-Mo cryogenic calorimeter with an isotop exposure of 1.47 kg·y to derive a range of neutrino mass limits. The range reflects the authors' estimate of the spread of nuclear matrix element calculations.

7 AZZOLINI 22 use 8.82 kg·yr of isotopic exposure of the CPID-0 scintillating cryogenic bolometer to set this range of neutrino mass limits. The range reflects the authors' estimate of the spread of nuclear matrix element calculations.

8 ARMENGAUD 21 use the CUPID-Mo demonstrator, with 1.17 kg·yr exposure of $^{100}$Mo, to set this limit. The range reflects the estimated uncertainty of the calculated nuclear matrix elements.

9 ADAMS 20A use the data of CUORE (372.5 kg·yr exposure of TeO$_2$) to obtain this limit.

10 AGOSTINI 20B use the final data set of the GERDA experiment, representing an exposure of 127.2 kg·yr to derive an upper limit for $\langle m_{\beta\beta} \rangle$. Isotopically enriched Ge detectors were used. The range reflects the variability of the theoretically calculated nuclear matrix elements. Supersedes AGOSTINI 19.

11 ALENKOV 19 report the range of the effective masses $\langle m_{\beta\beta} \rangle$ corresponding to the $0\nu\beta\beta$ decay half-life limit. It is based on the 52.1 kg·d exposure of $^{100}$Mo, in the Yangyang underground laboratory. The median sensitivity is $1.1 \times 10^{23}$ years. The range of $\langle m_{\beta\beta} \rangle$ reflects the uncertainty of nuclear matrix elements.

12 ANTON 19 uses the complete dataset of the EXO-200 experiment to obtain these limits. The spread reflects the uncertainty in the nuclear matrix elements. Supersedes ALBERT 16 and ALBERT 14B.

13 NI 19 use the PandaX-II dual phase TPC at CJPL to search for the $0\nu\beta\beta$ decay of $^{136}$Xe with 22.2 kg·yr exposure. The range in the $m_{\beta\beta}$ limit of 1.3-3.5 eV reflects the range of the calculated nuclear matrix elements. The sensitivity is $1.9 \times 10^{23}$ yr.

14 ALDUINO 18 use the combined data of CUORE, CUORE0, and Cuoricino to obtain this limit.

15 ARNOLD 18 use the NEMO-3 tracking detector to constrain the $0\nu\beta\beta$ decay of $^{82}$Se. The limit on $\langle m_{\beta\beta} \rangle$ is obtained assuming light neutrino exchange; the range reflects different calculations of the nuclear matrix elements. This is a somewhat weaker limit than in BARABASH 11A using the same detector.

16 BARABASH 18 use 1.162 kg of $^{116}$CdWO$_4$ scintillating crystals to obtain these limits. The spread reflects the estimated uncertainty in the nuclear matrix element. Supersedes DANEVICH 03.

17 ARNOLD 17 utilize NEMO-3 data, taken with enriched $^{116}$Cd to limit the effective Majorana neutrino mass. The reported range results from the use of different nuclear matrix elements. Supersedes BARABASH 11A.

18 ALDUINO 16 place a limit on the effective Majorana neutrino mass using the combined data of the CUORE-0 and CUORICINO experiments. The range reflects the authors' evaluation of the variability of the nuclear matrix elements. Supersedes ALFONSO 15.

19 ARNOLD 16A limit is derived from data taken with the NEMO-3 detector and $^{150}$Nd. A range of nuclear matrix elements that include the effect of nuclear deformation have been used. Supersedes ARGYRIADES 09.

20 ARNOLD 15 use the NEMO-3 tracking calorimeter with 34.3 kg·yr exposure to determine the neutrino mass limit based on the $0\nu\beta\beta$-half life of $^{100}$Mo. The spread range reflects different nuclear matrix elements. Supersedes ARNOLD 14 and BARABASH 11A.

21 ARGYRIADES 10 use $^{96}$Zr and the NEMO-3 tracking detector to obtain the reported mass limit. The range reflects the fluctuation of the nuclear matrix elements considered.

22 Limit was obtained using CaF$_2$ scintillation calorimeter to search for double beta decay of $^{48}$Ca. Reported range of limits reflects a spread of QRPA and SM matrix element calculations used. Supersedes OGAWA 04.

23 Limit for $\langle m_{\nu} \rangle$ is based on the nuclear matrix elements of STAUDT 90 and ARNOLD 96. Supersedes DANEVICH 00.

## Limits on Lepton-Number Violating (V+A) Current Admixture

For reasons given in the discussion at the beginning of this section, we list only results from 1989 and later. $\langle \lambda \rangle = \lambda \sum U_{ij} V_{ej}$ and $\langle \eta \rangle = \eta \sum U_{ij} V_{ej}$, where the sum is over the number of neutrino generations. This sum vanishes for massless or unmixed neutrinos. In the following Listings, only best or comparable limits or lifetimes for each isotope are reported.

<table><thead><tr><th>⟨λ⟩ (10<sup>-6</sup>)</th><th>CL%</th><th>⟨η⟩ (10<sup>-8</sup>)</th><th>CL%</th><th>ISOTOPE</th><th>METHOD</th><th>DOCUMENT ID</th></tr></thead><tbody><tr><td>&lt; 2.2-2.6</td><td>90</td><td>&lt; 1.7-2.1</td><td>90</td><td>82Se</td><td>NEMO-3</td><td>1 ARNOLD 18</td></tr><tr><td>&lt; 1.8-22</td><td>90</td><td>&lt; 1.6-21</td><td>90</td><td>116Cd</td><td>AURORA</td><td>2 BARABASH 18</td></tr><tr><td>&lt; 0.9-1.3</td><td>90</td><td>&lt; 0.5-0.8</td><td>90</td><td>100Mo</td><td>NEMO-3</td><td>3 ARNOLD 14</td></tr><tr><td>&lt;120</td><td>90</td><td></td><td></td><td>100Mo</td><td>0<sup>+</sup> → 2<sup>+</sup></td><td>4 ARNOLD 07</td></tr><tr><td>0.692<sup>+0.058</sup><sub>-0.056</sub></td><td>68</td><td>0.305<sup>+0.026</sup><sub>-0.025</sub></td><td>68</td><td>76Ge</td><td>Enriched HPGe</td><td>5 KLPADOR-K... 06A</td></tr><tr><td>&lt; 2.5</td><td>90</td><td></td><td></td><td>100Mo</td><td>0ν, NEMO-3</td><td>6 ARNOLD 05A</td></tr><tr><td>&lt; 3.8</td><td>90</td><td></td><td></td><td>82Se</td><td>0ν, NEMO-3</td><td>7 ARNOLD 05A</td></tr><tr><td>&lt; 1.5-2.0</td><td>90</td><td></td><td></td><td>100Mo</td><td>0ν, NEMO-3</td><td>8 ARNOLD 04</td></tr><tr><td>&lt; 3.2-3.8</td><td>90</td><td></td><td></td><td>82Se</td><td>0ν, NEMO-3</td><td>9 ARNOLD 04</td></tr><tr><td>&lt; 1.6-2.4</td><td>90</td><td>&lt; 0.9-5.3</td><td>90</td><td>130Te</td><td>Cryog. det.</td><td>10 ARNABOLDI 03</td></tr><tr><td>&lt; 2.2</td><td>90</td><td>&lt;2.5</td><td>90</td><td>116Cd</td><td>116CdWO<sub>4</sub> scint.</td><td>11 DANEVICH 03</td></tr><tr><td>&lt; 3.2-4.7</td><td>90</td><td>&lt; 2.4-2.7</td><td>90</td><td>100Mo</td><td>ELEGANT V</td><td>12 EJIRI 01</td></tr><tr><td>&lt; 1.1</td><td>90</td><td>&lt;0.64</td><td>90</td><td>76Ge</td><td>Enriched HPGe</td><td>13 GUENTHER 97</td></tr><tr><td>&lt; 4.4</td><td>90</td><td>&lt;2.3</td><td>90</td><td>136Xe</td><td>TPC</td><td>14 VUILLEUMIER 93</td></tr><tr><td></td><td></td><td>&lt;5.3</td><td></td><td>128Te</td><td>Geochem</td><td>15 BERNATOW... 92</td></tr></tbody></table>

1 ARNOLD 18 use the NEMO03 tracking detector, with 0.93 kg of $^{82}$Se mass and 5.25 y exposure to obtain the limits for the hypothetical right-handed currents. Supersedes ARNOLD 05A.

2 BARABASH 18 use 1.162 kg of $^{116}$CdWO$_4$ scintillating crystals to obtain this limits for the hypothetical right-handed currents in the $0\nu\beta\beta$ decay of $^{116}$Cd.

3 ARNOLD 14 is based on 34.7 kg·yr of exposure of the NEMO-3 tracking calorimeter. The reported range limit on $\langle \lambda \rangle$ and $\langle \eta \rangle$ reflects the nuclear matrix element uncertainty in $^{100}$Mo.

4 ARNOLD 07 use NEMO-3 half life limit for $0\nu$-decay of $^{100}$Mo to the first excited $2^+$-state of daughter nucleus to limit the right-right handed admixture of weak currents $\langle \lambda \rangle$. This limit is not competitive when compared to the decay to the ground state.

5 Re-analysis of data originally published in KLPADOR-KLEINGROTHAUS 04A. Modified pulse shape analysis leads the authors to claim 6σ statistical evidence for observation of $0\nu$-decay. Authors use matrix element of MUTO 89 to determine $\langle \lambda \rangle$ and $\langle \eta \rangle$. Uncertainty of nuclear matrix element is not reflected in stated errors.

6 ARNOLD 05A derive limit for $\langle \lambda \rangle$ based on $^{100}$Mo data collected with NEMO-3 detector. No limit for $\langle \eta \rangle$ is given. Supersedes ARNOLD 04.

7 ARNOLD 05A derive limit for $\langle \lambda \rangle$ based on $^{82}$Se data collected with NEMO-3 detector. No limit for $\langle \eta \rangle$ is given. Supersedes ARNOLD 04.

8 ARNOLD 04 use the matrix elements of SUHONEN 94 to obtain a limit for $\langle \lambda \rangle$, no limit for $\langle \eta \rangle$ is given. This limit is more stringent than the limit in EJIRI 01 for the same nucleus.

9 ARNOLD 04 use the matrix elements of TOMODA 91 and SUHONEN 91 to obtain a limit for $\langle \lambda \rangle$, no limit for $\langle \eta \rangle$ is given.

10 Supersedes ALESSANDRELLO 00. Cryogenic calorimeter search. Reported a range reflecting uncertainty in nuclear matrix element calculations.

11 Limits for $\langle \lambda \rangle$ and $\langle \eta \rangle$ are based on nuclear matrix elements of STAUDT 90. Supersedes DANEVICH 00.

12 The range of the reported $\langle \lambda \rangle$ and $\langle \eta \rangle$ values reflects the spread of the nuclear matrix elements. On axis value assuming $\langle m_{\nu} \rangle=0$ and $\langle \lambda \rangle=\langle \eta \rangle=0$, respectively.

13 GUENTHER 97 limits use the matrix elements of STAUDT 90. Supersedes BALYSH 95 and BALYSH 92.

14 VUILLEUMIER 93 uses the matrix elements of MUTO 89. Based on a half-life limit $2.6 \times 10^{23}$ y at 90%CL.

15 BERNATOWICZ 92 takes the measured geochemical decay width as a limit on the $0\nu$ width, and uses the SUHONEN 91 coefficients to obtain the least restrictive limit on $\eta$. Further details of the experiment are given in BERNATOWICZ 93.

## Double-β Decay REFERENCES

<table><tr><td>ABE</td><td>23</td><td>PRL 130 051801</td><td>S. Abe et al.</td><td>(KamLAND-Zen Collab.)</td></tr><tr><td>ADAMS</td><td>23A</td><td>PRL 131 249902 (errat.)</td><td>D.Q. Adams et al.</td><td>(CUORE Collab.)</td></tr><tr><td>AGOSTINI</td><td>23</td><td>PRL 131 142501</td><td>M. Agostini et al.</td><td>(GERDA Collab.)</td></tr><tr><td>AGUERRE</td><td>23</td><td>EPJ C83 1117</td><td>X. Aguerre et al.</td><td>(NEMO-3 Collab.)</td></tr><tr><td>ARNQUIST</td><td>23</td><td>PRL 130 062501</td><td>I.J. Arquist et al.</td><td>(MAJORANA Collab.)</td></tr><tr><td>AUGIER</td><td>23</td><td>PR C107 025503</td><td>C. Augier et al.</td><td>(CUPID-Mo Collab.)</td></tr><tr><td>AUGIER</td><td>23A</td><td>PR 131 162501</td><td>C. Augier et al.</td><td>(CUPID-Mo Collab.)</td></tr><tr><td>AZZOLINI</td><td>23A</td><td>PR 131 222501</td><td>O. Azzolini et al.</td><td>(CUPID-0 Collab.)</td></tr><tr><td>NOVELLA</td><td>23</td><td>JHEP 2309 190</td><td>P. Novella et al.</td><td>(NEXT Collab.)</td></tr><tr><td>ADAMS</td><td>22A</td><td>NAT 604 53</td><td>D.Q. Adams et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ADAMS</td><td>22B</td><td>PRL 129 222501</td><td>D.Q. Adams et al.</td><td>(XENON1T Collab.)</td></tr><tr><td>APRILE</td><td>22A</td><td>PR C106 024328</td><td>E. Aprile et al.</td><td>(XENON1T Collab.)</td></tr><tr><td>APRILE</td><td>22B</td><td>PRL 129 161805</td><td>E. Aprile et al.</td><td>(XENON1T Collab.)</td></tr><tr><td>AUGIER</td><td>22</td><td>EPJ C82 1033</td><td>C. Augier et al.</td><td>(CUPID-Mo Collab.)</td></tr><tr><td>AZZOLINI</td><td>22</td><td>PRL 129 111801</td><td>O. Azzolini et al.</td><td>(CUPID-0 Collab.)</td></tr><tr><td>NOVELLA</td><td>22</td><td>PR C105 055501</td><td>P. Novella et al.</td><td>(NEXT Collab.)</td></tr><tr><td>ADAMS</td><td>21</td><td>PRL 126 171801</td><td>D.Q. Adams et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ADAMS</td><td>21A</td><td>PRL 126 181802 (errat.)</td><td>D.Q. Adams et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ARMENGAUD</td><td>21</td><td>EPJ C81 567</td><td>D.Q. Adams et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ARNQUIST</td><td>21</td><td>PR C103 015501</td><td>E. Armengaud et al.</td><td>(CUPID-Mo Collab.)</td></tr><tr><td>AGOSTINI</td><td>20A</td><td>PR 124 122501</td><td>I.J. Arquist et al.</td><td>(MAJORANA Collab.)</td></tr><tr><td>AGOSTINI</td><td>20B</td><td>PR 125 252502</td><td>D.Q. Adams et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ARMENGAUD</td><td>20</td><td>EPJ C80 674</td><td>M. Agostini et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ARMENGAUD</td><td>19</td><td>SCI 365 1445</td><td>M. Agostini et al.</td><td>(GERDA Collab.)</td></tr><tr><td>ALDUINO</td><td>19</td><td>EPJ C79 795</td><td>C. Alduino et al.</td><td>(CUORE Collab.)</td></tr><tr><td>ALLENKOV</td><td>19</td><td>EPJ C79 791</td><td>V. Alenkov et al.</td><td>(AMORE Collab.)</td></tr><tr><td>ANTON</td><td>19</td><td>PRL 123 161802</td><td>G. Anton et al.</td><td>(EXO-200 Collab.)</td></tr><tr><td>APRILE</td><td>19E</td><td>NAT 568 532</td><td>E. Aprile et al.</td><td>(XENON1T Collab.)</td></tr><tr><td>ARNOLD</td><td>19</td><td>EPJ C79 440</td><td>R. Arnold et al.</td><td>(NEMO-3 Collab.)</td></tr><tr><td>AZZOLINI</td><td>19</td><td>PRL 123 032501</td><td>O. Azzolini et al.</td><td>(CUPID-0 Collab.)</td></tr><tr><td>AZZOLINI</td><td>19B</td><td>PRL 123 262501</td><td>O. Azzolini et al.</td><td>(CUPID-0 Collab.)</td></tr><tr><td>NI</td><td>19</td><td>CP C43 113001</td><td>K. Ni et al.</td><td>(PandaX-II Collab.)</td></tr></table>
<table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>ALBERT</td><td>18</td><td>PRL 120 072701</td><td>J.B. Albert et al. (EXO-200 Collab.)</td></tr><tr><td>ALDUINO</td><td>18</td><td>PRL 120 132501</td><td>C. Alduino et al. (CUORE Collab.)</td></tr><tr><td>ARNOLD</td><td>18</td><td>EPJ C78 821</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>BARABASH</td><td>18</td><td>PR D98 092007</td><td>A.S. Barabash et al. (AURORA Collab.)</td></tr><tr><td>ALBERT</td><td>17C</td><td>PR D96 092001</td><td>J.B. Albert et al. (EXO-200 Collab.)</td></tr><tr><td>ALDUINO</td><td>17</td><td>EPJ C77 13</td><td>C. Alduino et al. (CUORE Collab.)</td></tr><tr><td>ARNOLD</td><td>17</td><td>PR D95 012007</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>ALDUINO</td><td>16</td><td>PR C93 045503</td><td>C. Alduino et al. (CUORE Collab.)</td></tr><tr><td>ARNOLD</td><td>16</td><td>PR D93 112008</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>ARNOLD</td><td>16A</td><td>PR D94 072003</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>ASAKURA</td><td>16</td><td>NP A946 171</td><td>K. Asakura et al. (KamLAND-Zen Collab.)</td></tr><tr><td>AGOSTINDI</td><td>15A</td><td>EPJ C75 416</td><td>M. Agostini et al. (GERDA Collab.)</td></tr><tr><td>ALFONSO</td><td>15</td><td>PRL 115 102502</td><td>K. Alfonso et al. (CUORE Collab.)</td></tr><tr><td>ARNOLD</td><td>15</td><td>PR D92 072011</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>ALBERT</td><td>14B</td><td>PR C89 015502</td><td>J. Albert et al. (EXO-200 Collab.)</td></tr><tr><td>ALBERT</td><td>14B</td><td>NAT 510 229</td><td>J.B. Albert et al. (EXO-200 Collab.)</td></tr><tr><td>ARNOLD</td><td>14</td><td>PR D89 111101</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>ARNOLD</td><td>13</td><td>PR C87 035501</td><td>Yu.M. Gavriluk et al. (CUORICINO Collab.)</td></tr><tr><td>ANDREOTTI</td><td>12</td><td>PR C85 045503</td><td>E. Andreotti et al. (KamLAND-Zen Collab.)</td></tr><tr><td>GANDO</td><td>12A</td><td>PR C85 045504</td><td>A. Gando et al. (EXO Collab.)</td></tr><tr><td>ACKERMAN</td><td>11</td><td>PRL 107 212501</td><td>N. Ackerman et al. (NEMO-3 Collab.)</td></tr><tr><td>ARNOLD</td><td>11</td><td>PRL 107 062504</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>BARABASH</td><td>11A</td><td>PAN 74 312</td><td>A.S. Barabash et al. (NEMO-3 Collab.)</td></tr><tr><td></td><td></td><td>Translated from YAF 74 330.</td><td></td></tr><tr><td>ARGYRIADES</td><td>10</td><td>NP A847 168</td><td>J. Argyriades et al. (NEMO-3 Collab.)</td></tr><tr><td>BELLI</td><td>10</td><td>NP A846 143</td><td>P. Belli et al. (DAMA-INR Collab.)</td></tr><tr><td>ARGYRIADES</td><td>9</td><td>PR C80 032501</td><td>J. Argyriades et al. (NEMO-3 Collab.)</td></tr><tr><td>KIDD</td><td>09</td><td>NP A821 251</td><td>M. Kidd et al. (EXO-200 Collab.)</td></tr><tr><td>UMEHARA</td><td>08</td><td>PR C78 058501</td><td>S. Umehara et al. (EXO-200 Collab.)</td></tr><tr><td>ARNOLD</td><td>07</td><td>NP A781 209</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>KLAPDOR-K...</td><td>06A</td><td>MPL A21 1547</td><td>H.V. Klapdor-Kleingrothaus, I.V. Krivoshina (NEMO-3 Collab.)</td></tr><tr><td>ARNOLD</td><td>05A</td><td>PRL 95 182302</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td>AALSETH</td><td>04</td><td>PR D70 078302</td><td>C.E. Aalseth et al. (NEMO-3 Collab.)</td></tr><tr><td>ARNOLD</td><td>04</td><td>JETPL 80 377</td><td>R. Arnold et al. (NEMO-3 Collab.)</td></tr><tr><td></td><td></td><td>Translated from ZETF 80 421.</td><td></td></tr><tr><td>KLAPDOR-K...</td><td>04A</td><td>PL B586 198</td><td>H.V. Klapdor-Kleingrothaus et al. (IGEX Collab.)</td></tr><tr><td>KLAPDOR-K...</td><td>04B</td><td>PR D70 078301</td><td>H.V. Klapdor-Kleingrothaus, A. Dietz, I.V. Krivoshina</td></tr><tr><td>OGAWA</td><td>04</td><td>NP A730 215</td><td>I. Ogawa et al. (NEMO Collab.)</td></tr><tr><td>ARNABOLDI</td><td>03</td><td>PL B557 167</td><td>C. Aranaboldi et al. (NEMO Collab.)</td></tr><tr><td>DANEVICH</td><td>03</td><td>PR C68 035501</td><td>F.A. Danevich et al. (NEMO Collab.)</td></tr><tr><td>AALSETH</td><td>02B</td><td>PR D65 092007</td><td>A. Alessandrello et al. (IGEX Collab.)</td></tr><tr><td>DEBRAECKEL...</td><td>01</td><td>PRL 86 3510</td><td>L. De Braeckeleer et al. (NEMO Collab.)</td></tr><tr><td>EJIRI</td><td>01</td><td>PR C63 065501</td><td>H. Ejiri et al. (NEMO Collab.)</td></tr><tr><td>KLAPDOR-K...</td><td>01</td><td>EPJ A12 147</td><td>H.V. Klapdor-Kleingrothaus et al. (NEMO Collab.)</td></tr><tr><td>KLAPDOR-K...</td><td>01B</td><td>MPL A16 2409</td><td>H.V. Klapdor-Kleingrothaus et al. (NEMO Collab.)</td></tr><tr><td>ALESSAND...</td><td>00</td><td>PL B486 13</td><td>A. Alessandrello et al. (NEMO Collab.)</td></tr><tr><td>DANEVICH</td><td>00</td><td>PR C62 045501</td><td>F.A. Danevich et al. (NEMO Collab.)</td></tr><tr><td>ARNOLD</td><td>99</td><td>NP A658 299</td><td>R. Arnold et al. (NEMO Collab.)</td></tr><tr><td>GUENTHER</td><td>97</td><td>PR D55 54</td><td>M. Gunther et al. (Heidelberg-Moscow Collab.)</td></tr><tr><td>ARNOLD</td><td>96</td><td>ZPHY C72 239</td><td>R. Arnold et al. (BCEN, CAEN, JINR+)</td></tr><tr><td>BALSHY</td><td>95</td><td>PL B356 450</td><td>A. Balshy et al. (Heidelberg-Moscow Collab.)</td></tr><tr><td>DASSIE</td><td>95</td><td>PR D51 2090</td><td>S. Dassie et al. (NEMO Collab.)</td></tr><tr><td>EJIRI</td><td>95</td><td>JPSJ 64 339</td><td>H. Ejiri et al. (OSAK, KIEV) (NEMO Collab.)</td></tr><tr><td>SUHONEN</td><td>94</td><td>PR C49 3055</td><td>J. Suhonen, O. Civitarese (NEMO Collab.)</td></tr><tr><td>BERNATOW...</td><td>93</td><td>PR C47 806</td><td>T. Bernatowicz et al. (WUSL, TATA)</td></tr><tr><td>VUILLEUMIER</td><td>93</td><td>PR D48 1009</td><td>J.C. Vailleumier et al. (NEUC, CIT, VILL)</td></tr><tr><td>BALSHY</td><td>92</td><td>PL B283 32</td><td>A. Balshy et al. (MPIK, KIAE, SASSO) (WUSL, TATA)</td></tr><tr><td>BERNATOW...</td><td>92</td><td>PRL 69 2341</td><td>T. Bernatowicz et al. (WUSL, TATA) (NEMO Collab.)</td></tr><tr><td>SUHONEN</td><td>91</td><td>NP A535 509</td><td>J. Suhonen, S.B. Khadkikar, A. Faessler (JYV+)</td></tr><tr><td>STOMADA</td><td>91</td><td>RPP 54 53</td><td>T. Tomoda (NEMO Collab.)</td></tr><tr><td>STAUDT</td><td>90</td><td>RPL 13 31</td><td>A. Staudt, K. Muto, H.V. Klapdor-Kleingrothaus (TINT, MPIK)</td></tr><tr><td>MUTO</td><td>89</td><td>ZPHY A334 187</td><td>K. Muto, E. Bender, H.V. Klapdor (NEMO Collab.)</td></tr></tbody></table>

Neutrino Mixing

With the possible exceptions of "short-baseline anomalies," such as LSND, all neutrino data can be described within the framework of a 3×3 mixing matrix between the mass eigenstates $\nu_1$, $\nu_2$, and $\nu_3$, leading to the flavor eigenstates $\nu_e$, $\nu_\mu$, and $\nu_\tau$, as described in the review "Neutrino masses, mixing and oscillations."

The Listings are divided in the following sections:

**(A) Neutrino fluxes and event ratios:** shows measurements which correspond to various oscillation tests for Accelerator, Reactor, Atmospheric, and Solar neutrino experiments. Typically, ratios involve a measurement in a realm sensitive to oscillations compared to one for which no oscillation effect is expected.

**(B) Neutrino mixing parameters:** shows measurements of $\sin^2(\theta_{12})$, $\sin^2(\theta_{23})$, $\Delta m^2_{21}$, $\Delta m^2_{32}$, and $\delta_{CP}$ as extracted from the measured data in the quoted publications in the frame of the three-neutrino mixing scheme. The quoted averages are not the result of a global fit, as in the review "Neutrino masses, mixing and oscillations," and, as a consequence, might slightly differ from them. In some cases, measurements depend on the mass order (normal when $\Delta m^2_{32} > 0$ or inverted

when $\Delta m^2_{32} < 0$) or octant of $\theta_{23}$ (lower when $\theta_{23} < 45^\circ$ or upper when $\theta_{23} > 45^\circ$).

**(C) Other neutrino mixing results:**

The LSND anomaly [AGUILAR 01], reported a signal which is consistent with $\bar{\nu}_\mu \to \bar{\nu}_e$ oscillations. In a three neutrino framework, this would be a measurement of $\theta_{12}$ and $\Delta m^2_{21}$. This does not appear to be consistent with the interpretation of other neutrino data. It has been interpreted as evidence for a 4th "sterile" neutrino. The following listings include results which might be relevant towards understanding this observation. They include searches for $\nu_\mu \to \nu_e$, $\bar{\nu}_\mu \to \bar{\nu}_e$, sterile neutrino oscillations, and others.

**(A) Neutrino fluxes and event ratios**

**Events (observed/expected) from accelerator $\nu_\mu$ experiments.**

Some neutrino oscillation experiments compare the flux in two or more detectors. This is usually quoted as the ratio of the event rate in the far detector to the expected rate based on an extrapolation from the near detector in the absence of oscillations.

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>1.01 ± 0.10</td><td>1 ABE</td><td>14B</td><td>T2K ν<sub>e</sub> rate in T2K near detect.</td></tr><tr><td>0.71 ± 0.08</td><td>2 AHN</td><td>06A</td><td>K2K to Super-K</td></tr><tr><td>0.64 ± 0.05</td><td>3 MICHAEL</td><td>06</td><td>MINS All charged current events</td></tr><tr><td>0.71 ± 0.09</td><td>4 ALIU</td><td>05</td><td>K2K KEK to Super-K</td></tr><tr><td>0.70 ± 0.10</td><td>5 AHN</td><td>03</td><td>K2K KEK to Super-K</td></tr></tbody></table>

1. The rate of $\nu_e$ from $\mu$ decay was measured to be $0.68 \pm 0.30$ compared to the predicted flux. From $K^\mu$ decay $1.10 \pm 0.14$ compared to the predicted flux.

2. Based on the observation of 112 events when $158.1 \pm 8.6^{+9.2}_{-8.6}$ were expected without oscillations. Including not only the number of events but also the shape of the energy distribution, the evidence for oscillation is at the level of about 4.3 $\sigma$. Supersedes ALIU 05.

3. This ratio is based on the observation of 215 events compared to an expectation of 336 ± 14 without oscillations. See also ADAMSON 08.

4. This ratio is based on the observation of 107 events at the far detector 250 km away from KEK, and an expectation of $151 \pm 12_{-10}^{+10}$.

5. This ratio is based on the observation of 56 events with an expectation of $80.1 \pm 6.2_{-5.4}^{+5.4}$.

**Events (observed/expected) from reactor $\bar{\nu}_e$ experiments.**

The quoted values are the ratios of the measured reactor $\bar{\nu}_e$ event rate at the quoted distances, and the rate expected without oscillations. The expected rate is based on the experimental data for the most significant reactor fuels ($^{235}$U, $^{239}$Pu, $^{241}$Pu) and on calculations for $^{238}$U.

A recent re-evaluation of the spectral conversion of electron to $\bar{\nu}_e$ in MUELLER 11 results in an upward shift of the reactor $\bar{\nu}_e$ spectrum by 3% and, thus, might require revisions to the ratios listed in this table.

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>0.948 ± 0.008 ± 0.033</td><td>1 ALMAZAN</td><td>20</td><td>RHF RHF reactor at ILL</td></tr><tr><td>0.952 ± 0.027</td><td>2 ADEY</td><td>19</td><td>DAYA DayaBay, Ling Ao/Ao II reactors</td></tr><tr><td></td><td>3 AN</td><td>16</td><td>DAYA DayaBay, Ling Ao/Ao II reactors</td></tr><tr><td>1.08 ± 0.21 ± 0.16</td><td>4 DENIZ</td><td>10</td><td>TEXO Kuo-Sheng reactor, 28 m</td></tr><tr><td>0.658 ± 0.044 ± 0.047</td><td>5 ARAKI</td><td>05</td><td>KLND Japanese react. ~ 180 km</td></tr><tr><td>0.611 ± 0.085 ± 0.041</td><td>6 EGUCHI</td><td>03</td><td>KLND Japanese react. ~ 180 km</td></tr><tr><td>1.01 ± 0.024 ± 0.053</td><td>7 BOEHM</td><td>01</td><td>Palo Verde react. 0.75-0.89 km</td></tr><tr><td>1.01 ± 0.028 ± 0.027</td><td>8 APOLLONIO</td><td>99</td><td>CHOZ Chooz reactors 1 km</td></tr><tr><td>0.987 ± 0.006 ± 0.037</td><td>9 GREENWOOD</td><td>96</td><td>Savannah River, 18.2 m</td></tr><tr><td>0.988 ± 0.004 ± 0.05</td><td>ACHKAR</td><td>95</td><td>CNTR Bugey reactor, 15 m</td></tr><tr><td>0.994 ± 0.010 ± 0.05</td><td>ACHKAR</td><td>95</td><td>CNTR Bugey reactor, 40 m</td></tr><tr><td>0.915 ± 0.132 ± 0.05</td><td>ACHKAR</td><td>95</td><td>CNTR Bugey reactor, 95 m</td></tr><tr><td>0.987 ± 0.014 ± 0.027</td><td>10 DECLAIS</td><td>94</td><td>CNTR Bugey reactor, 15 m</td></tr><tr><td>0.985 ± 0.018 ± 0.034</td><td>KUVSHINN...</td><td>91</td><td>CNTR Rovno reactor</td></tr><tr><td>1.05 ± 0.02 ± 0.05</td><td>VUILLEUMIER</td><td>82</td><td>Gösgen reactor</td></tr><tr><td>0.955 ± 0.035 ± 0.110</td><td>11 KWON</td><td>81</td><td>$\bar{\nu}_e p \to e^+ n$</td></tr><tr><td>0.89 ± 0.15</td><td>11 BOEHM</td><td>80</td><td>$\bar{\nu}_e p \to e^+ n$</td></tr></tbody></table>

<footnote n="1">¹ALMAZAN 20 use the RHF research reactor at ILL to compare their measured anti-neutrino event rate to the calculation by HUBER 11. Reported $0.948 \pm 0.008 \pm 0.023 \pm 0.023$ measurement with uncertainties from statistics, systematic, and model. Note that this result is obtained for highly enriched $^{235}$U reactor fuel while most other reactor experiments utilize a low-enrichment mix of fissile nuclides.</footnote>
² ADEY 19 present a re-analysis of 1230 days of Daya Bay near detector data with reduced systematic uncertainties on the neutrino detection efficiency. Note that ADEY 19 report the measured to predicted antineutrino ratio using the reactor model of MUELLER 11 (Huber-Mueller model). The ratio using the older ILL-Vogel model is 1.001 ± 0.015 ± 0.027.

³ AN 16 use 217 days of data (338k events) to determine the neutrino flux ratio relative to the prediction of Mueller-Huber and ILL-Vogel models (see AN 16 for details). The reported flux ratios were corrected for $\theta_{13}$ oscillation effect. The flux measurement is consistent with results from previous short-baseline reactor experiments. The measured inverse beta decay yield is $(1.55 \pm 0.04) \times 10^{-18} \text{ cm}^2/(\text{GW day})$ or $\sigma_f = (5.92 \pm 0.14) \times 10^{-43} \text{ cm}^2/\text{fission}$. About $4\sigma$ excess of events was observed in the 4-6 MeV prompt energy region.

⁴ DENIZ 10 observe reactor $\tau_e e^-$ scattering with recoil kinetic energies 3-8 MeV using CsI(Tl) detectors. The observed rate is consistent with the Standard Model prediction, leading to a constraint on $\sin^2\theta_W = 0.251 \pm 0.031(\text{stat}) \pm 0.024(\text{sys})$.

⁵ Updated result of KamLAND, including the data used in EGUCHI 03. Note that the survival probabilities for different periods are not directly comparable because the effective baseline varies with power output of the reactor sources involved, and there were large variations in the reactor power production in Japan in 2003.

⁶ EGUCHI 03 observe reactor neutrino disappearance at ~ 180 km baseline to various Japanese nuclear power reactors.

⁷ BOEHM 01 search for neutrino oscillations at 0.75 and 0.89 km distance from the Palo Verde reactors.

⁸ APOLLONIO 99, APOLLONIO 98 search for neutrino oscillations at 1.1 km fixed distance from Chooz reactors. They use $\bar{\nu}_e p \rightarrow e^+ n$ in Gd-loaded scintillator target. APOLLONIO 99 supersedes APOLLONIO 98. See also APOLLONIO 03 for detailed description.

⁹ GREENWOOD 96 search for neutrino oscillations at 18 m and 24 m from the reactor at Savannah River.

¹⁰ DECLAIS 94 result based on integral measurement of neutrons only. Result is ratio of measured cross section to that expected in standard V-A theory. Replaced by ACHKAR 95.

¹¹ KWON 81 represents an analysis of a larger set of data from the same experiment as BOEHM 80.

### Atmospheric neutrinos

Neutrinos and antineutrinos produced in the atmosphere induce $\mu$-like and e-like events in underground detectors. The ratio of the numbers of the two kinds of events is defined as $\mu/e$. It has the advantage that systematic effects, such as flux uncertainty, tend to cancel, for both experimental and theoretical values of the ratio. The "ratio of the ratios" of experimental to theoretical $\mu/e$, $R(\mu/e)$, or that of experimental to theoretical $\mu$/total, $R(\mu$/total) with total = $\mu$+e, is reported below. If the actual value is not unity, the value obtained in a given experiment may depend on the experimental conditions. In addition, the measured "up-down asymmetry" for $\mu$ ($N_{\mu\uparrow}(\mu)/N_{\downarrow\uparrow}(\mu)$) or e ($N_{\mu\uparrow}(e)/N_{\downarrow\downarrow}(e)$) is reported. The expected "up-down asymmetry" is nearly unity if there is no neutrino oscillation.

$$R(\mu/e) = (\text{Measured Ratio } \mu/e) / (\text{Expected Ratio } \mu/e)$$

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>0.658 ± 0.016 ± 0.035</td><td>1 ASHIE</td><td>05</td><td>SKAM sub-GeV</td></tr><tr><td>0.702<sup>+0.032</sup><sub>-0.030</sub></td><td>2 ASHIE</td><td>05</td><td>SKAM multi-GeV</td></tr><tr><td>0.69 ± 0.10 ± 0.06</td><td>3 SANCHEZ</td><td>03</td><td>SOU2 Calorimeter raw data</td></tr><tr><td>1.00 ± 0.15 ± 0.08</td><td>4 FUKUDA</td><td>96B</td><td>KAMI Water Cherenkov</td></tr><tr><td>0.60<sup>+0.06</sup><sub>-0.05</sub></td><td>5 DAUM</td><td>95</td><td>FREJ Calorimeter</td></tr><tr><td>0.57<sup>+0.08</sup><sub>-0.07</sub></td><td>6 FUKUDA</td><td>94</td><td>KAMI sub-GeV</td></tr><tr><td></td><td>7 FUKUDA</td><td>94</td><td>KAMI multi-GeV</td></tr><tr><td></td><td>8 BECKER-SZ...</td><td>92B</td><td>IMB Water Cherenkov</td></tr></tbody></table>

¹ ASHIE 05 results are based on an exposure of 92 kton yr during the complete Super-Kamiokande I running period. The analyzed data sample consists of fully-contained single-ring e-like events with 0.1 GeV/c < $p_e$ and $\mu$-like events 0.2 GeV/c < $p_\mu$, both having a visible energy < 1.33 GeV. These criteria match the definition used by FUKUDA 94.

² ASHIE 05 results are based on an exposure of 92 kton yr during the complete Super-Kamiokande I running period. The analyzed data sample consists of fully-contained single-ring events with visible energy > 1.33 GeV and partially-contained events. All partially-contained events are classified as $\mu$-like.

³ SANCHEZ 03 result is based on an exposure of 5.9 kton yr, and updates ALLISON 99 result. The analyzed data sample consists of fully-contained e-flavor and $\mu$-flavor events having lepton momentum > 0.3 GeV/c.

⁴ FUKUDA 96b studied neutron background in the atmospheric neutrino sample observed in the Kamiokande detector. No evidence for the background contamination was found.

⁵ DAUM 95 results are based on an exposure of 2.0 kton yr which includes the data used by BERGER 90b. This ratio is for the contained and semicontained events. The data used also report $R(\mu/e) = 0.99 \pm 0.13 \pm 0.08$ for the total neutrino induced data sample which includes upward going stopping muons and horizontal muons in addition to the contained and semicontained events.

⁶ FUKUDA 94 result is based on an exposure of 7.7 kton yr and updates the HIRATA 92 result. The analyzed data sample consists of fully-contained e-like events with $0.1 < p_e < 1.33 \text{ GeV/c}$ and fully-contained $\mu$-like events with $0.2 < p_\mu < 1.5 \text{ GeV/c}$.

⁷ FUKUDA 94 analyzed the data sample consisting of fully contained events with visible energy > 1.33 GeV and partially contained $\mu$-like events.

⁸ BECKER-SZENDY 92b reports the fraction of nonshowing events (mostly muons from atmospheric neutrinos) as $0.36 \pm 0.02 \pm 0.02$, as compared with expected muons 0.51 ± 0.01 ± 0.05. After cutting the energy range to the Kamiokande limits, BEIER 92 finds $R(\mu/e)$ very close to the Kamiokande value.

$$R(\nu_{\mu}) = (\text{Measured Flux of } \nu_{\mu}) / (\text{Expected Flux of } \nu_{\mu})$$

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>0.84 ± 0.12</td><td>1 ADAMSON</td><td>06</td><td>MINS MINOS atmospheric</td></tr><tr><td>0.72 ± 0.026 ± 0.13</td><td>2 AMBROSIO</td><td>01</td><td>MCRO upward through-going</td></tr><tr><td>0.57 ± 0.05 ± 0.15</td><td>3 AMBROSIO</td><td>00</td><td>MCRO upgoing partially contained</td></tr><tr><td>0.71 ± 0.05 ± 0.19</td><td>4 AMBROSIO</td><td>00</td><td>MCRO downgoing partially contained + upgoing stopping</td></tr><tr><td>0.74 ± 0.036 ± 0.046</td><td>5 AMBROSIO</td><td>98</td><td>MCRO Streamer tubes</td></tr><tr><td>0.95 ± 0.22</td><td>6 CASPER</td><td>91</td><td>IMB Water Cherenkov</td></tr><tr><td>0.62 ± 0.17</td><td>7 AGLIETTA</td><td>89</td><td>NUSX</td></tr><tr><td></td><td>8 BOLIEV</td><td>81</td><td>Baksan</td></tr><tr><td></td><td>CROUCH</td><td>78</td><td>Case Western/UCI</td></tr></tbody></table>

¹ ADAMSON 06 uses a measurement of 107 total neutrinos compared to an expected rate of 127 ± 13 without oscillations.

² AMBROSIO 01 result is based on the upward through-going muon tracks with $E_\mu > 1$ GeV. The data came from three different detector configurations, but the statistics is largely dominated by the full detector run, from May 1994 to December 2000. The total live time, normalized to the full detector configuration, is 6.17 years. The first error is the statistical error, the second is the systematic error, dominated by the theoretical error in the predicted flux.

³ AMBROSIO 00 result is based on the upgoing partially contained event sample. It came from 4.1 live years of data taking with the full detector, from April 1994 to February 1999. The average energy of atmospheric muon neutrinos corresponding to this sample is 4 GeV. The first error is statistical, the second is the systematic error, dominated by the 25% theoretical error in the rate (20% in the flux and 15% in the cross section, added in quadrature). Within statistics, the observed deficit is uniform over the zenith angle.

⁴ AMBROSIO 00 result is based on the combined samples of downgoing partially contained events and upgoing stopping events. These two subsamples could not be distinguished due to the lack of timing information. The result came from 4.1 live years of data taking with the full detector, from April 1994 to February 1999. The average energy of atmospheric muon neutrinos corresponding to this sample is 4 GeV. The first error is statistical, the second is the systematic error, dominated by the 25% theoretical error in the rate (20% in the flux and 15% in the cross section, added in quadrature). Within statistics, the observed deficit is uniform over the zenith angle.

⁵ AMBROSIO 98 result is for all nadir angles and updates AHNLE 95 result. The lower cutoff on the muon energy is 1 GeV. In addition to the statistical and systematic errors, there is a Monte Carlo flux error (theoretical error) of ±0.13. With a neutrino oscillation hypothesis, the fit either to the flux or zenith distribution independently yields $\sin^2 2\theta = 1.0$ and $\Delta(m^2) \sim$ a few times $10^{-3} \text{ eV}^2$. However, the fit to the observed zenith distribution gives a maximum probability for $\chi^2$ of only 5% for the best oscillation hypothesis.

⁶ CASPER 91 correlates showering/nonshowing signature of single-ring events with parent atmospheric-neutrino flavor. They find nonshowing ($\approx \nu_\mu$ induced) fraction is 0.41 ± 0.03 ± 0.02, as compared with expected 0.51 ± 0.05 (syst).

⁷ AGLIETTA 89 finds no evidence for any anomaly in the neutrino flux. They define $\rho = (\text{measured number of } \nu_e \text{s})/(\text{measured number of } \nu_\mu \text{s})$. They report

$$\rho(\text{measured}) = \rho(\text{expected}) = 0.96^{+0.32}_{-0.28}$$

⁸ From this data BOLIEV 81 obtain the limit $\Delta(m^2) \le 6 \times 10^{-3} \text{ eV}^2$ for maximal mixing, $\nu_\mu \leftrightarrow \nu_\mu$ type oscillation.

$$R(\mu/\text{total}) = (\text{Measured Ratio } \mu/\text{total}) / (\text{Expected Ratio } \mu/\text{total})$$

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>1.1<sup>+0.07</sup><sub>-0.12</sub></td><td>1 CLARK</td><td>97</td><td>IMB multi-GeV</td></tr></tbody></table>

¹ CLARK 97 obtained this result by an analysis of fully contained and partially contained events in the IMB water-Cherenkov detector with visible energy > 0.95 GeV.

$$N_{\mu\uparrow}(\mu)/N_{\downarrow\uparrow}(\mu)$$

<table><thead><tr><th>VALUE</th><th>DOCUMENT ID</th><th>TECN</th><th>COMMENT</th></tr></thead><tbody><tr><td>0.71 ± 0.06</td><td>1 ADAMSON</td><td>12B</td><td>MINS contained-vertex muons</td></tr><tr><td>0.551<sup>+0.035</sup><sub>-0.033</sub></td><td>2 ASHIE</td><td>05</td><td>SKAM multi-GeV</td></tr></tbody></table>

¹ ADAMSON 12B reports the atmospheric neutrino results obtained with MINOS far detector in 2,553 live days (an exposure of 37.9 kton-yr). This result is obtained with a sample of high resolution contained-vertex muons. The quoted error is statistical only.

² ASHIE 05 results are based on an exposure of 92 kton yr during the complete Super-Kamiokande I running period. The analyzed data sample consists of fully-contained single-ring $\mu$-like events with visible energy > 1.33 GeV and partially-contained events. All partially-contained events are classified as $\mu$-like. Upward-going events are those with $-1 < \cos(\text{zenith angle}) < -0.2$ and downward-going events are those with $0.2 < \cos(\text{zenith angle}) < 1$. The $\mu$-like up-down ratio for the multi-GeV data deviates from 1 (the expectation for no atmospheric $\nu_\mu$ oscillations) by more than 12 standard deviations.